{
    "componentChunkName": "component---src-templates-markdown-doc-tsx",
    "path": "/2025.1.3/knowledge_base/velocity_misfits",
    "result": {"data":{"mdx":{"id":"e74c4369-5c47-5a39-a50b-07f42ac3e4ae","frontmatter":{"title":"","images":null},"body":"var _excluded = [\"components\"];\nfunction _extends() { _extends = Object.assign || function (target) { for (var i = 1; i < arguments.length; i++) { var source = arguments[i]; for (var key in source) { if (Object.prototype.hasOwnProperty.call(source, key)) { target[key] = source[key]; } } } return target; }; return _extends.apply(this, arguments); }\nfunction _objectWithoutProperties(source, excluded) { if (source == null) return {}; var target = _objectWithoutPropertiesLoose(source, excluded); var key, i; if (Object.getOwnPropertySymbols) { var sourceSymbolKeys = Object.getOwnPropertySymbols(source); for (i = 0; i < sourceSymbolKeys.length; i++) { key = sourceSymbolKeys[i]; if (excluded.indexOf(key) >= 0) continue; if (!Object.prototype.propertyIsEnumerable.call(source, key)) continue; target[key] = source[key]; } } return target; }\nfunction _objectWithoutPropertiesLoose(source, excluded) { if (source == null) return {}; var target = {}; var sourceKeys = Object.keys(source); var key, i; for (i = 0; i < sourceKeys.length; i++) { key = sourceKeys[i]; if (excluded.indexOf(key) >= 0) continue; target[key] = source[key]; } return target; }\n/* @jsxRuntime classic */\n/* @jsx mdx */\n\nvar _frontmatter = {};\nvar layoutProps = {\n  _frontmatter: _frontmatter\n};\nvar MDXLayout = \"wrapper\";\nreturn function MDXContent(_ref) {\n  var components = _ref.components,\n    props = _objectWithoutProperties(_ref, _excluded);\n  return mdx(MDXLayout, _extends({}, layoutProps, props, {\n    components: components,\n    mdxType: \"MDXLayout\"\n  }), mdx(\"h1\", null, \"Misfit functionals involving velocity\"), mdx(\"p\", null, mdx(\"strong\", {\n    parentName: \"p\"\n  }, \"What difference does it make if the misfit functional uses velocity  traces \", mdx(\"span\", {\n    parentName: \"strong\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"v_{\\\\text{obs}}\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.58056em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), \" instead of displacements \", mdx(\"span\", {\n    parentName: \"strong\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"u\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"u_{\\\\text{obs}}\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.58056em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), \"?\")), mdx(\"p\", null, \"The answer might be a bit more complicated than you think.\"), mdx(\"p\", null, \"Usually, we solve the wave equation with displacement as primary state variable. To keep things simple let's consider the scalar wave equation in a homogeneous medium\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"c\"), mdx(\"mrow\", {\n    parentName: \"msup\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"2\"))), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"msubsup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msubsup\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mrow\", {\n    parentName: \"msubsup\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\")), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, \"2\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u0394\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"f\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"c^{-2}\\\\,\\\\partial^2_{tt} u - \\\\Delta u = f,\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.111108em\",\n      \"verticalAlign\": \"-0.247em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"c\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.864108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\"))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.4530000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.247em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u0394\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.8888799999999999em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10764em\"\n    }\n  }, \"f\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"with initial conditions\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mspace\", {\n    parentName: \"mrow\",\n    \"width\": \"1em\"\n  }), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0.\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"u(0) = 0,\\\\quad  \\\\partial_t u(0) = 0.\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"1em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.64444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \".\")))))), mdx(\"p\", null, \"For displacement measurements \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"u\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"u_{\\\\text{obs}}\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.58056em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), \" and the \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"L\"), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, \"2\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"L_2\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.83333em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"L\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), \"-misfit,\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mfrac\", {\n    parentName: \"mrow\"\n  }, mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"1\"), mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"2\")), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2225\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"u\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2225\"), mdx(\"mn\", {\n    parentName: \"msup\"\n  }, \"2\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\frac{1}{2}\\\\| u - u_{\\\\text{obs}} \\\\|^2,\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.00744em\",\n      \"verticalAlign\": \"-0.686em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen nulldelimiter\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mfrac\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.32144em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.314em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"2\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.23em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"frac-line\",\n    \"style\": {\n      \"borderBottomWidth\": \"0.04em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.677em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"1\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.686em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose nulldelimiter\"\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2225\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.1141079999999999em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2225\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"it is well known that the adjoint source is then simply given by the  residuals \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"u\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"u_{\\\\text{obs}} - u\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.73333em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"))))), \".\"), mdx(\"p\", null, \"Instead of displacements \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"u\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"u_{\\\\text{obs}}\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.58056em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), \", we now want to consider the \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"L\"), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, \"2\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"L_2\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.83333em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"L\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), \"-misfit for velocity measurements.\"), mdx(\"h2\", null, \"Strategy 1 - Transforming the wave equation\"), mdx(\"p\", null, \"Under some reasonable regularity assumptions on the source time function, we can just take the time derivative of the wave equation and obtain:\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"c\"), mdx(\"mrow\", {\n    parentName: \"msup\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"2\"))), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"msubsup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msubsup\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mrow\", {\n    parentName: \"msubsup\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\")), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, \"3\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u0394\"), mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"f\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"c^{-2}\\\\,\\\\partial^3_{ttt} u - \\\\Delta \\\\left(\\\\partial_t u\\\\right) = \\\\partial_t f,\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.111108em\",\n      \"verticalAlign\": \"-0.247em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"c\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.864108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\"))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.4530000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"3\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.247em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u0394\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"minner\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \")\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.8888799999999999em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10764em\"\n    }\n  }, \"f\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"with initial conditions\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mspace\", {\n    parentName: \"mrow\",\n    \"width\": \"1em\"\n  }), mdx(\"msubsup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msubsup\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mrow\", {\n    parentName: \"msubsup\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\")), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, \"2\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0.\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\partial_t u(0) = 0,\\\\quad  \\\\partial^2_{tt} u(0) = 0.\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.1141079999999999em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"1em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.4530000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.247em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.64444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \".\")))))), mdx(\"p\", null, \"This results in the same wave equation as the one originally defined but with velocity as state variable and the time-differentiated source time function on the right-hand-side:\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"c\"), mdx(\"mrow\", {\n    parentName: \"msup\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"2\"))), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"msubsup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msubsup\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mrow\", {\n    parentName: \"msubsup\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\")), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, \"2\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"v\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u0394\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"v\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"f\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"c^{-2}\\\\,\\\\partial^2_{tt} v - \\\\Delta v = \\\\partial_t f,\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.111108em\",\n      \"verticalAlign\": \"-0.247em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"c\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.864108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\"))))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.4530000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.247em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u0394\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.8888799999999999em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.10764em\"\n    }\n  }, \"f\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"with initial conditions\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"v\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mspace\", {\n    parentName: \"mrow\",\n    \"width\": \"1em\"\n  }), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"v\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"0.\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"v(0) = 0,\\\\quad  \\\\partial_{t} v(0) = 0.\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"1em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\"))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.64444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"0\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \".\")))))), mdx(\"p\", null, \"For velocity measurements \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"v_{\\\\text{obs}}\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.58056em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), \" the \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"L\"), mdx(\"mn\", {\n    parentName: \"msub\"\n  }, \"2\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"L_2\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.83333em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"L\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.30110799999999993em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"0em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), \"-misfit then reads\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mfrac\", {\n    parentName: \"mrow\"\n  }, mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"1\"), mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"2\")), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2225\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"v\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2225\"), mdx(\"mn\", {\n    parentName: \"msup\"\n  }, \"2\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\frac{1}{2}\\\\| v - v_{\\\\text{obs}} \\\\|^2,\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.00744em\",\n      \"verticalAlign\": \"-0.686em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen nulldelimiter\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mfrac\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.32144em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.314em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"2\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.23em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"frac-line\",\n    \"style\": {\n      \"borderBottomWidth\": \"0.04em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.677em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"1\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.686em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose nulldelimiter\"\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2225\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.1141079999999999em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2225\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"and the adjoint source is given by the velocity residuals \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"v\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"v_{\\\\text{obs}} - v\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.73333em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.43056em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"))))), \".\"), mdx(\"p\", null, \"In summary, we transform the forward equation from displacement to velocity as primary state variable, in which case we can directly deduce the adjoint source in terms of velocity and then also solve the adjoint equation in the velocity formulation to obtain gradients.\"), mdx(\"h2\", null, \"Strategy 2 - Invoking the chain rule\"), mdx(\"p\", null, \"Alternatively, we can keep the wave equation in the displacement formulation and invoke the chain rule to compute the correct adjoint source.\\nIn this case, the velocity misfit is given by\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mfrac\", {\n    parentName: \"mrow\"\n  }, mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"1\"), mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"2\")), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2225\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2225\"), mdx(\"mn\", {\n    parentName: \"msup\"\n  }, \"2\")), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \".\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\frac{1}{2}\\\\| \\\\partial_t u - v_{\\\\text{obs}} \\\\|^2.\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.00744em\",\n      \"verticalAlign\": \"-0.686em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen nulldelimiter\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mfrac\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.32144em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.314em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"2\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.23em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"frac-line\",\n    \"style\": {\n      \"borderBottomWidth\": \"0.04em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.677em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"1\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.686em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose nulldelimiter\"\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2225\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.1141079999999999em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2225\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \".\")))))), mdx(\"p\", null, \"What is the correct adjoint source in this case?\"), mdx(\"ul\", null, mdx(\"li\", {\n    parentName: \"ul\"\n  }, \"is it the residuals \", mdx(\"span\", {\n    parentName: \"li\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\left(v_{\\\\text{obs}}- \\\\partial_t u\\\\right)\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"minner\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \")\")))))), \"?\"), mdx(\"li\", {\n    parentName: \"ul\"\n  }, \"or maybe the time derivative of the residuals \", mdx(\"span\", {\n    parentName: \"li\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\partial_t \\\\left(v_{\\\\text{obs}}- \\\\partial_t u\\\\right)\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"minner\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \")\")))))), \"?\")), mdx(\"p\", null, \"The answer is that both are incorrect!\"), mdx(\"p\", null, \"Let's recall that the adjoint source is defined as the negative derivative of the misfit functional with respect to the state variable. In our case, we have\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C7\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mfrac\", {\n    parentName: \"mrow\"\n  }, mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"1\"), mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"2\")), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2225\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2225\"), mdx(\"mn\", {\n    parentName: \"msup\"\n  }, \"2\")), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mfrac\", {\n    parentName: \"mrow\"\n  }, mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"1\"), mdx(\"mn\", {\n    parentName: \"mfrac\"\n  }, \"2\")), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"msubsup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"msubsup\"\n  }, \"\\u222B\"), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, \"0\"), mdx(\"mi\", {\n    parentName: \"msubsup\"\n  }, \"T\")), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mrow\", {\n    parentName: \"msup\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")), mdx(\"mn\", {\n    parentName: \"msup\"\n  }, \"2\")), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"d\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\chi(u) = \\\\frac{1}{2} \\\\| \\\\partial_t u - v_{\\\\text{obs}}\\\\|^2 = \\\\frac{1}{2}\\\\,\\\\int_0^T \\\\left(\\\\partial_t u(t) - v_{\\\\text{obs}}(t)\\\\right)^2\\\\,dt,\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C7\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.00744em\",\n      \"verticalAlign\": \"-0.686em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen nulldelimiter\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mfrac\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.32144em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.314em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"2\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.23em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"frac-line\",\n    \"style\": {\n      \"borderBottomWidth\": \"0.04em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.677em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"1\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.686em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose nulldelimiter\"\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2225\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.1141079999999999em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2225\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.8641079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.113em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.503181em\",\n      \"verticalAlign\": \"-0.9119499999999999em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen nulldelimiter\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mfrac\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.32144em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.314em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"2\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.23em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"frac-line\",\n    \"style\": {\n      \"borderBottomWidth\": \"0.04em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.677em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"1\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.686em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose nulldelimiter\"\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\",\n    \"style\": {\n      \"marginRight\": \"0.44445em\",\n      \"position\": \"relative\",\n      \"top\": \"-0.0011249999999999316em\"\n    }\n  }, \"\\u222B\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.591231em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.7880500000000001em\",\n      \"marginLeft\": \"-0.44445em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"0\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.8129000000000004em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }, \"T\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.9119499999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"minner\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"minner\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, \")\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.954008em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.2029em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"2\")))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"d\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"and the adjoint source is defined as\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"mfrac\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"mfrac\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mrow\", {\n    parentName: \"mfrac\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"))), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C7\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \".\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"-\\\\frac{\\\\partial}{\\\\partial u}\\\\,\\\\chi(u).\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.05744em\",\n      \"verticalAlign\": \"-0.686em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen nulldelimiter\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mfrac\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.37144em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.314em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.23em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"frac-line\",\n    \"style\": {\n      \"borderBottomWidth\": \"0.04em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.677em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.686em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose nulldelimiter\"\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C7\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \".\")))))), mdx(\"p\", null, \"In variational form, we obtain the directional derivative of \", mdx(\"span\", {\n    parentName: \"p\",\n    \"className\": \"math math-inline\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C7\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\chi\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"0.625em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C7\"))))), \" as\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"mfrac\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"mfrac\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mrow\", {\n    parentName: \"mfrac\"\n  }, mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"))), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C7\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"msubsup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"msubsup\"\n  }, \"\\u222B\"), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, \"0\"), mdx(\"mi\", {\n    parentName: \"msubsup\"\n  }, \"T\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"false\"\n  }, \"(\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"d\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \".\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\left(\\\\frac{\\\\partial}{\\\\partial u}\\\\,\\\\chi(u), \\\\delta u\\\\right) =\\n\\\\int_0^T \\\\big(\\\\partial_t u(t) - v_{\\\\text{obs}}(t)\\\\big)\\\\,\\\\partial_t \\\\delta u(t)\\\\,dt.\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.40003em\",\n      \"verticalAlign\": \"-0.95003em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"minner\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \"(\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen nulldelimiter\"\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mfrac\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.37144em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.314em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.23em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"frac-line\",\n    \"style\": {\n      \"borderBottomWidth\": \"0.04em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.677em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.686em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose nulldelimiter\"\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03C7\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03785em\"\n    }\n  }, \"\\u03B4\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose delimcenter\",\n    \"style\": {\n      \"top\": \"0em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \")\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.503181em\",\n      \"verticalAlign\": \"-0.9119499999999999em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\",\n    \"style\": {\n      \"marginRight\": \"0.44445em\",\n      \"position\": \"relative\",\n      \"top\": \"-0.0011249999999999316em\"\n    }\n  }, \"\\u222B\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.591231em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.7880500000000001em\",\n      \"marginLeft\": \"-0.44445em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"0\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.8129000000000004em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }, \"T\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.9119499999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size1\"\n  }, \"(\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.20001em\",\n      \"verticalAlign\": \"-0.35001em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size1\"\n  }, \")\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03785em\"\n    }\n  }, \"\\u03B4\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"d\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \".\")))))), mdx(\"p\", null, \"To shift the time derivative onto the first term, we apply integration-by-parts which yields:\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msubsup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"msubsup\"\n  }, \"\\u222B\"), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, \"0\"), mdx(\"mi\", {\n    parentName: \"msubsup\"\n  }, \"T\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"false\"\n  }, \"(\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"d\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msubsup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"msubsup\"\n  }, \"\\u222B\"), mdx(\"mn\", {\n    parentName: \"msubsup\"\n  }, \"0\"), mdx(\"mi\", {\n    parentName: \"msubsup\"\n  }, \"T\")), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"false\"\n  }, \"(\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"d\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"t\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\int_0^T \\\\big(\\\\partial_t u(t) - v_{\\\\text{obs}}(t)\\\\big)\\\\,\\\\partial_t \\\\delta u(t)\\\\,dt =\\n- \\\\int_0^T \\\\partial_t \\\\big(\\\\partial_t u(t) - v_{\\\\text{obs}}(t)\\\\big)\\\\,\\\\delta u(t)\\\\,dt\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.503181em\",\n      \"verticalAlign\": \"-0.9119499999999999em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\",\n    \"style\": {\n      \"marginRight\": \"0.44445em\",\n      \"position\": \"relative\",\n      \"top\": \"-0.0011249999999999316em\"\n    }\n  }, \"\\u222B\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.591231em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.7880500000000001em\",\n      \"marginLeft\": \"-0.44445em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"0\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.8129000000000004em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }, \"T\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.9119499999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size1\"\n  }, \"(\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.20001em\",\n      \"verticalAlign\": \"-0.35001em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size1\"\n  }, \")\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03785em\"\n    }\n  }, \"\\u03B4\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"d\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mrel\"\n  }, \"=\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2777777777777778em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"2.503181em\",\n      \"verticalAlign\": \"-0.9119499999999999em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mop op-symbol large-op\",\n    \"style\": {\n      \"marginRight\": \"0.44445em\",\n      \"position\": \"relative\",\n      \"top\": \"-0.0011249999999999316em\"\n    }\n  }, \"\\u222B\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"1.591231em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-1.7880500000000001em\",\n      \"marginLeft\": \"-0.44445em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"0\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.8129000000000004em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }, \"T\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.9119499999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size1\"\n  }, \"(\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.20001em\",\n      \"verticalAlign\": \"-0.35001em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size1\"\n  }, \")\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03785em\"\n    }\n  }, \"\\u03B4\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"d\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"t\")))))), mdx(\"p\", null, \"where we have used the homogeneous initial (resp. final time) conditions of forward and adjoint equation. From the last equation, we deduce that the correct adjoint source in the displacement formulation is thus given by\"), mdx(\"div\", {\n    \"className\": \"math math-display\"\n  }, mdx(\"span\", {\n    parentName: \"div\",\n    \"className\": \"katex-display\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-mathml\"\n  }, mdx(\"math\", {\n    parentName: \"span\",\n    \"xmlns\": \"http://www.w3.org/1998/Math/MathML\",\n    \"display\": \"block\"\n  }, mdx(\"semantics\", {\n    parentName: \"math\"\n  }, mdx(\"mrow\", {\n    parentName: \"semantics\"\n  }, mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"false\"\n  }, \"(\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2202\"), mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"t\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\u2212\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\"\n  }, \"v\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\partial_t\\\\big(\\\\partial_t u - v_{\\\\text{obs}}\\\\big),\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"katex-html\",\n    \"aria-hidden\": \"true\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.20001em\",\n      \"verticalAlign\": \"-0.35001em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size1\"\n  }, \"(\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\",\n    \"style\": {\n      \"marginRight\": \"0.05556em\"\n    }\n  }, \"\\u2202\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.2805559999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.05556em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal mtight\"\n  }, \"t\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"u\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin\"\n  }, \"\\u2212\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.2222222222222222em\"\n    }\n  })), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"base\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"strut\",\n    \"style\": {\n      \"height\": \"1.20001em\",\n      \"verticalAlign\": \"-0.35001em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\",\n    \"style\": {\n      \"marginRight\": \"0.03588em\"\n    }\n  }, \"v\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"msupsub\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-t vlist-t2\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.33610799999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\n      \"marginLeft\": \"-0.03588em\",\n      \"marginRight\": \"0.05em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"2.7em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"sizing reset-size6 size3 mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text mtight\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mtight\"\n  }, \"obs\")))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-s\"\n  }, \"\\u200B\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist-r\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"vlist\",\n    \"style\": {\n      \"height\": \"0.15em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size1\"\n  }, \")\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"i.e., it has the opposite sign of what we might have guessed.\"), mdx(\"p\", null, \"In summary, this strategy solves the adjoint equation in its usual displacement formulation, but correctly accounts for velocities by invoking the chain rule when computing the adjoint source.\"), mdx(\"h2\", null, \"Pros and cons\"), mdx(\"p\", null, \"The gradients will be exactly the same in both cases.\"), mdx(\"p\", null, \"In the second case, the adjoint source needs to account for the time derivative and to apply the chain rule properly. The good news is that \", mdx(\"strong\", {\n    parentName: \"p\"\n  }, \"Salvus does this automatically when the misfit is defined in velocities, so you don't have to worry about this as a user\"), \".\"), mdx(\"p\", null, \"The numerical time integration scheme is more accurate in the primary state variable than in its time derivative. Hence, solving the wave equation for velocity using the time derivative of the source time function would be the preferred way.\"), mdx(\"p\", null, \"The only downside here is that the field labels become ambiguous, because Salvus would still call the primary field \", mdx(\"inlineCode\", {\n    parentName: \"p\"\n  }, \"displacement\"), \".\"), mdx(\"p\", null, \"The whole argument also applies to acceleration.\"));\n}\n;\nMDXContent.isMDXComponent = true;"},"site":{"siteMetadata":{"salvusDocVersions":{"current":"2026.5.0"}}}},"pageContext":{"id":"e74c4369-5c47-5a39-a50b-07f42ac3e4ae"}},
    "staticQueryHashes": ["1756726491","1865182279","3419370438","3597190305","4112489441","519097329"]}