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 }), 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, \"where \", 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    \"mathvariant\": \"bold\"\n  }, \"d\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{d}_{\\\\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\": \"0.84444em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"d\")), 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      \"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  })))))))))), \" denotes an observation of the derived field, and thus has the same dimension as \", 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    \"mathvariant\": \"bold\"\n  }, \"q\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{q}\")))), 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\": \"0.63888em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")))))), \".\\nWe can formally deduce the directional derivative with respect to \", 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    \"mathvariant\": \"bold\"\n  }, \"u\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{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\": \"0.44444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")))))), \" for a direction\\n\", 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  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\delta \\\\mathbf{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\": \"0.69444em\",\n      \"verticalAlign\": \"0em\"\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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")))))), \" 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(\"mtable\", {\n    parentName: \"semantics\",\n    \"rowspacing\": \"0.24999999999999992em\",\n    \"columnalign\": \"right left\",\n    \"columnspacing\": \"0em\"\n  }, mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\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    \"mathvariant\": \"bold\"\n  }, \"u\"))), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C7\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\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    \"mathvariant\": \"bold\"\n  }, \"q\"))), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C7\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"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    \"mathvariant\": \"bold\"\n  }, \"u\"))), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"q\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")), mdx(\"mo\", {\n    parentName: \"mrow\"\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    \"mathvariant\": \"bold\"\n  }, \"q\"))), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C7\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"script\"\n  }, \"D\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")))))), mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"q\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), 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    \"mathvariant\": \"bold\"\n  }, \"d\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"script\"\n  }, \"D\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\"))))), mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\",\n    \"mathvariant\": \"script\"\n  }, \"D\"), mdx(\"mo\", {\n    parentName: \"msup\"\n  }, \"\\u2020\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"q\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), 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    \"mathvariant\": \"bold\"\n  }, \"d\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), 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    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \".\")))))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\begin{aligned}\\n\\\\left(\\\\frac{\\\\partial}{\\\\partial \\\\mathbf{u}} \\\\chi, \\\\delta \\\\mathbf{u} \\\\right)\\n&=\\n\\\\left(\\\\frac{\\\\partial}{\\\\partial \\\\mathbf{q}} \\\\chi, \\\\frac{\\\\partial }{\\\\partial\\\\mathbf{u}}\\\\mathbf{q}(\\\\mathbf{u})\\\\,\\\\delta \\\\mathbf{u} \\\\right)\\n=\\n\\\\left(\\\\frac{\\\\partial}{\\\\partial \\\\mathbf{q}} \\\\chi,\\n\\\\mathcal{D}(\\\\delta \\\\mathbf{u}) \\\\right)\\\\\\\\\\n&=\\n\\\\biggl(\\n(\\\\mathbf{q}(\\\\mathbf{u}) - \\\\mathbf{d}_{\\\\text{obs}}),\\n\\\\mathcal{D}(\\\\delta \\\\mathbf{u})\\n\\\\biggr)\\\\\\\\\\n&=\\n\\\\biggl(\\n\\\\mathcal{D}^\\\\dagger (\\\\mathbf{q}(\\\\mathbf{u}) - \\\\mathbf{d}_{\\\\text{obs}}),\\n\\\\delta \\\\mathbf{u}\\n\\\\biggr).\\n\\\\end{aligned}\")))), 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\": \"8.10009em\",\n      \"verticalAlign\": \"-3.800045000000001em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mtable\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-r\"\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\": \"4.300044999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-6.300045000000001em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.45em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\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\", 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}, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \")\"))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.6000149999999995em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.45em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\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\": \"mopen\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \"(\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"d\")), 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      \"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\": \"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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \")\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-0.8999849999999996em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.45em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\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\": \"mopen\"\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\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")), 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\": \"0.8991079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.1130000000000004em\",\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\": \"mbin mtight\"\n  }, \"\\u2020\")))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"d\")), 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      \"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\": \"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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\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\": \"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\": \"3.800045000000001em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))))), mdx(\"p\", null, \"The adjoint \", 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(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\",\n    \"mathvariant\": \"script\"\n  }, \"D\"), mdx(\"mo\", {\n    parentName: \"msup\"\n  }, \"\\u2020\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathcal{D}^\\\\dagger\")))), 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\": \"0.8491079999999999em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")), 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\": \"0.8491079999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.063em\",\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\": \"mbin mtight\"\n  }, \"\\u2020\")))))))))))), \" is impractical in many situations, in particular, when \", 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    \"mathvariant\": \"script\"\n  }, \"D\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathcal{D}\")))), 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\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")))))), \" contains spatial derivatives. However, the variational form of the spectral-element method enables the use of the second line in the equations above, where \", 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    \"mathvariant\": \"script\"\n  }, \"D\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathcal{D}\")))), 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\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")))))), \" is applied to the test function instead.\\nFor some specific choices 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    \"mathvariant\": \"script\"\n  }, \"D\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathcal{D}\")))), 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\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")))))), \", it will be more convenient to isolate the strain or gradient operator within \", 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    \"mathvariant\": \"script\"\n  }, \"D\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathcal{D}\")))), 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\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")))))), \" and shift only the remaining terms onto the left-hand side of the inner product. This results in moment-tensor or gradient sources for the adjoint equation.\\nBefore giving a few concrete examples, we first recall the definition of a moment-tensor and gradient point sources in variational form.\"), mdx(\"p\", null, \"Consider a vector-valued test function \", 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    \"mathvariant\": \"bold\"\n  }, \"v\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{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\": \"0.44444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\",\n    \"style\": {\n      \"marginRight\": \"0.01597em\"\n    }\n  }, \"v\")))))), \", a 6-component moment tensor \", 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    \"mathvariant\": \"bold\"\n  }, \"M\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{M}\")))), 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\": \"0.68611em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"M\")))))), \" or a gradient source \", 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    \"mathvariant\": \"bold\"\n  }, \"G\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{G}\")))), 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\": \"0.68611em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"G\")))))), \" consisting of 9 individual components. Then the weak form of the source terms 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(\"mtable\", {\n    parentName: \"semantics\",\n    \"rowspacing\": \"0.24999999999999992em\",\n    \"columnalign\": \"right\",\n    \"columnspacing\": \"\"\n  }, mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"M\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B5\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"v\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mspace\", {\n    parentName: \"mrow\",\n    \"width\": \"1em\"\n  }), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"and\"), mdx(\"mspace\", {\n    parentName: \"mrow\",\n    \"width\": \"1em\"\n  }), mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }, mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"G\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2207\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"v\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")))))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\begin{aligned}\\n\\\\left(\\\\mathbf{M}, \\\\varepsilon(\\\\mathbf{v}) \\\\right), \\\\quad\\\\text{and}\\\\quad\\n\\\\left(\\\\mathbf{G}, \\\\nabla\\\\mathbf{v} \\\\right),\\n\\\\end{aligned}\")))), 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\": \"1.5000000000000002em\",\n      \"verticalAlign\": \"-0.5000000000000002em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mtable\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-r\"\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\": \"1em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.16em\"\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\": \"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 mathbf\"\n  }, \"M\")), 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  }, \"\\u03B5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\",\n    \"style\": {\n      \"marginRight\": \"0.01597em\"\n    }\n  }, \"v\")), 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\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), 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 text\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"and\")), 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\": \"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 mathbf\"\n  }, \"G\")), 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\"\n  }, \"\\u2207\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\",\n    \"style\": {\n      \"marginRight\": \"0.01597em\"\n    }\n  }, \"v\")), 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\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))), 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\": \"0.5000000000000002em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))))), mdx(\"p\", null, \"respectively.\"), mdx(\"h2\", null, \"Example 1 - Strains and gradients\"), mdx(\"p\", null, \"Using the relations above, point-wise measurements of the full strain tensor or gradient of the displacement field are straightforward to include in adjoint computations. These cases result in \", 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    \"mathvariant\": \"script\"\n  }, \"D\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B5\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathcal{D}\\\\mathbf{u} = \\\\varepsilon(\\\\mathbf{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\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\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\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03B5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"))))), \" or \", 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    \"mathvariant\": \"script\"\n  }, \"D\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \"\\u2207\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathcal{D}\\\\mathbf{u} = \\\\nabla \\\\mathbf{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\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\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\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"\\u2207\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")))))), \", respectively. In the case of 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, we thus obtain adjoint sources of the form\"), 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    \"mathvariant\": \"bold\"\n  }, \"q\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), 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    \"mathvariant\": \"bold\"\n  }, \"d\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{q}(\\\\mathbf{u}) - \\\\mathbf{d}_{\\\\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\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), 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\": \"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  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"d\")), 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      \"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\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"which either carries 6 or 9 components, respectively, and is injected as moment tensor or gradient point source into the adjoint simulation.\"), mdx(\"h2\", null, \"Example 2 - Axial strain for distributed acoustic sensing\"), mdx(\"p\", null, \"Applications of distributed acoustic sensing consider single-component measurement of the strain tensor in axial direction. Given a direction vector \", 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    \"mathvariant\": \"bold\"\n  }, \"e\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{e}\")))), 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\": \"0.44444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"e\")))))), \", this can be written 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(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"script\"\n  }, \"D\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\",\n    \"mathvariant\": \"bold\"\n  }, \"e\"), mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"T\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B5\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"e\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathcal{D}\\\\mathbf{u} = \\\\mathbf{e}^T \\\\varepsilon(\\\\mathbf{u}) \\\\, \\\\mathbf{e}\")))), 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\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\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\": \"1.1413309999999999em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"e\")), 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\": \"0.8913309999999999em\"\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 mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }, \"T\")))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03B5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"e\"))))))), mdx(\"p\", null, \"In this case, it is more convenient to isolate the strain operator 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(\"mtable\", {\n    parentName: \"semantics\",\n    \"rowspacing\": \"0.24999999999999992em\",\n    \"columnalign\": \"right left\",\n    \"columnspacing\": \"0em\"\n  }, mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\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    \"mathvariant\": \"bold\"\n  }, \"u\"))), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C7\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"q\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), 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    \"mathvariant\": \"bold\"\n  }, \"d\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\",\n    \"mathvariant\": \"bold\"\n  }, \"e\"), mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"T\")), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B5\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"\\u2009\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"e\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\"))))), mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"q\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), 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    \"mathvariant\": \"bold\"\n  }, \"d\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"e\"), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\",\n    \"mathvariant\": \"bold\"\n  }, \"e\"), mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"T\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B5\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"normal\"\n  }, \".\")))))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\begin{aligned}\\n\\\\left(\\\\frac{\\\\partial}{\\\\partial \\\\mathbf{u}} \\\\chi, \\\\delta \\\\mathbf{u} \\\\right)\\n&=\\n\\\\biggl(\\n(\\\\mathbf{q}(\\\\mathbf{u}) - \\\\mathbf{d}_{\\\\text{obs}}),\\n\\\\mathbf{e}^T \\\\varepsilon(\\\\delta \\\\mathbf{u}) \\\\, \\\\mathbf{e}\\n\\\\biggr)\\\\\\\\\\n&=\\n\\\\biggl(\\n(\\\\mathbf{q}(\\\\mathbf{u}) - \\\\mathbf{d}_{\\\\text{obs}}) \\\\mathbf{e} \\\\mathbf{e}^T,\\n\\\\varepsilon(\\\\delta \\\\mathbf{u})\\n\\\\biggr).\\n\\\\end{aligned}\")))), 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\": \"5.40006em\",\n      \"verticalAlign\": \"-2.45003em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mtable\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-r\"\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\": \"2.95003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.95003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.45em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\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\": \"mord mathnormal\"\n  }, \"\\u03C7\"), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\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    \"style\": {\n      \"top\": \"-2.2500000000000004em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.45em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }))), 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\": \"2.45003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-l\"\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\": \"2.95003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.95003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.45em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\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\": \"mopen\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \"(\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"d\")), 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      \"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\": \"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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"e\")), 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\": \"0.8913309999999999em\"\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 mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }, \"T\")))))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03B5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"e\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \")\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.2500000000000004em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.45em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\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\": \"mopen\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \"(\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"d\")), 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      \"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\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"e\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"e\")), 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\": \"0.8913309999999999em\"\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 mathnormal mtight\",\n    \"style\": {\n      \"marginRight\": \"0.13889em\"\n    }\n  }, \"T\")))))))), 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  }, \"\\u03B5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\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\": \"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\": \"2.45003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))))), mdx(\"p\", null, \"Hence, the adjoint source can be interpreted as a moment-tensor point source, where the 6-components are extracted from the upper triangular part 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    \"mathvariant\": \"bold\"\n  }, \"e\"), mdx(\"msup\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msup\",\n    \"mathvariant\": \"bold\"\n  }, \"e\"), mdx(\"mi\", {\n    parentName: \"msup\"\n  }, \"T\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{e} \\\\mathbf{e}^T\")))), 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\": \"0.8413309999999999em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"e\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"e\")), 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\": \"0.8413309999999999em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-3.063em\",\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\")))))))))))), \" and scaled by the scalar residual \", 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    \"mathvariant\": \"bold\"\n  }, \"q\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), 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    \"mathvariant\": \"bold\"\n  }, \"d\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\"))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{q}(\\\\mathbf{u}) - \\\\mathbf{d}_{\\\\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\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), 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\": \"0.84444em\",\n      \"verticalAlign\": \"-0.15em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"d\")), 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      \"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(\"h2\", null, \"Example 3 - Divergence measurement\"), mdx(\"p\", null, \"The divergence of the displacement field is equal to the trace of the strain tensor, i.e.,\"), 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    \"mathvariant\": \"script\"\n  }, \"D\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"tr\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B5\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), 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  }, \"\\\\mathcal{D}\\\\mathbf{u} = \\\\text{tr}(\\\\varepsilon(\\\\mathbf{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\": \"0.68333em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\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\": \"1em\",\n      \"verticalAlign\": \"-0.25em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord text\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"tr\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03B5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \".\")))))), mdx(\"p\", null, \"Similar to the previous example, the scalar measurement leads to a moment tensor adjoint point source 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(\"mtable\", {\n    parentName: \"semantics\",\n    \"rowspacing\": \"0.24999999999999992em\",\n    \"columnalign\": \"right left\",\n    \"columnspacing\": \"0em\"\n  }, mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\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    \"mathvariant\": \"bold\"\n  }, \"u\"))), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C7\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"q\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), 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    \"mathvariant\": \"bold\"\n  }, \"d\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mtext\", {\n    parentName: \"mrow\"\n  }, \"tr\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B5\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\"))))), mdx(\"mtr\", {\n    parentName: \"mtable\"\n  }, mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }))), mdx(\"mtd\", {\n    parentName: \"mtr\"\n  }, mdx(\"mstyle\", {\n    parentName: \"mtd\",\n    \"scriptlevel\": \"0\",\n    \"displaystyle\": \"true\"\n  }, mdx(\"mrow\", {\n    parentName: \"mstyle\"\n  }, mdx(\"mrow\", {\n    parentName: \"mrow\"\n  }), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"=\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \"(\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"q\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), 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    \"mathvariant\": \"bold\"\n  }, \"d\"), mdx(\"mtext\", {\n    parentName: \"msub\"\n  }, \"obs\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"msub\", {\n    parentName: \"mrow\"\n  }, mdx(\"mi\", {\n    parentName: \"msub\",\n    \"mathvariant\": \"script\"\n  }, \"I\"), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"3\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\xD7\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"3\"))), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B5\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")))))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\begin{aligned}\\n\\\\left(\\\\frac{\\\\partial}{\\\\partial \\\\mathbf{u}} \\\\chi, \\\\delta \\\\mathbf{u} \\\\right)\\n&=\\n\\\\biggl(\\n(\\\\mathbf{q}(\\\\mathbf{u}) - \\\\mathbf{d}_{\\\\text{obs}}),\\n\\\\text{tr}(\\\\varepsilon(\\\\delta\\\\mathbf{u}))\\n\\\\biggr)\\\\\\\\\\n&=\\n\\\\biggl(\\n(\\\\mathbf{q}(\\\\mathbf{u}) - \\\\mathbf{d}_{\\\\text{obs}})\\\\mathcal{I}_{3\\\\times 3},\\n\\\\varepsilon(\\\\delta \\\\mathbf{u})\\n\\\\biggr),\\n\\\\end{aligned}\")))), 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\": \"5.40006em\",\n      \"verticalAlign\": \"-2.45003em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mtable\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-r\"\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\": \"2.95003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.95003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.45em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\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\": \"mord mathnormal\"\n  }, \"\\u03C7\"), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\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    \"style\": {\n      \"top\": \"-2.2500000000000004em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.45em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }))), 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\": \"2.45003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  }))))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"col-align-l\"\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\": \"2.95003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-4.95003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.45em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\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\": \"mopen\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \"(\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"d\")), 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      \"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\": \"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 text\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, \"tr\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathnormal\"\n  }, \"\\u03B5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \")\")))), mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.2500000000000004em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"pstrut\",\n    \"style\": {\n      \"height\": \"3.45em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\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\": \"mopen\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \"(\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\n  }, \"(\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"d\")), 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      \"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\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.07382em\"\n    }\n  }, \"I\")), 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\": \"0.301108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\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  }, \"3\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin mtight\"\n  }, \"\\xD7\"), 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\": \"0.208331em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\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  }, \"\\u03B5\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, \")\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mclose\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"delimsizing size3\"\n  }, \")\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))), 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\": \"2.45003em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))))), mdx(\"p\", null, \"where \", 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    \"mathvariant\": \"script\"\n  }, \"I\"), mdx(\"mrow\", {\n    parentName: \"msub\"\n  }, mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"3\"), mdx(\"mo\", {\n    parentName: \"mrow\"\n  }, \"\\xD7\"), mdx(\"mn\", {\n    parentName: \"mrow\"\n  }, \"3\")))), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathcal{I}_{3\\\\times 3}\")))), 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\": \"0.891661em\",\n      \"verticalAlign\": \"-0.208331em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.07382em\"\n    }\n  }, \"I\")), 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\": \"0.301108em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"style\": {\n      \"top\": \"-2.5500000000000003em\",\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  }, \"3\"), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mbin mtight\"\n  }, \"\\xD7\"), 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\": \"0.208331em\"\n    }\n  }, mdx(\"span\", {\n    parentName: \"span\"\n  })))))))))), \" is the identity matrix.\\nHence, the adjoint source results in an explosive moment-tensor point source.\"), mdx(\"h2\", null, \"Extensions to other misfit functionals\"), mdx(\"p\", null, \"Using\"), 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    \"mathvariant\": \"bold\"\n  }, \"u\"))), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C7\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")), mdx(\"mo\", {\n    parentName: \"mrow\"\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    \"mathvariant\": \"bold\"\n  }, \"q\"))), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03C7\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"script\"\n  }, \"D\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \"(\"), mdx(\"mi\", {\n    parentName: \"mrow\"\n  }, \"\\u03B4\"), mdx(\"mi\", {\n    parentName: \"mrow\",\n    \"mathvariant\": \"bold\"\n  }, \"u\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"stretchy\": \"false\"\n  }, \")\"), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"fence\": \"true\"\n  }, \")\")), mdx(\"mo\", {\n    parentName: \"mrow\",\n    \"separator\": \"true\"\n  }, \",\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\left(\\\\frac{\\\\partial}{\\\\partial \\\\mathbf{u}} \\\\chi, \\\\delta \\\\mathbf{u} \\\\right)\\n=\\n\\\\left(\\\\frac{\\\\partial}{\\\\partial \\\\mathbf{q}} \\\\chi,\\n\\\\mathcal{D}(\\\\delta \\\\mathbf{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\": \"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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\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\": \"mord mathnormal\"\n  }, \"\\u03C7\"), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\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: 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  \"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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")))), 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\": \"0.8804400000000001em\"\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 mathnormal\"\n  }, \"\\u03C7\"), 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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathcal\",\n    \"style\": {\n      \"marginRight\": \"0.02778em\"\n    }\n  }, \"D\")), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mopen\"\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\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")), 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\": \"delimsizing size3\"\n  }, \")\"))), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mspace\",\n    \"style\": {\n      \"marginRight\": \"0.16666666666666666em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mpunct\"\n  }, \",\")))))), mdx(\"p\", null, \"the extension to misfit functionals other than 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  })))))))))), \" norm is straightforward. In fact, the equation above readily provides the recipe for general misfit functional acting on \", 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    \"mathvariant\": \"bold\"\n  }, \"q\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{q}\")))), 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\": \"0.63888em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")))))), \", for instance, cross-correlation time-shift, time-frequency phase misfits, or others. All that is required is injecting the 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\"))))), \" with respect to the derived quantity \", 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    \"mathvariant\": \"bold\"\n  }, \"q\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{q}\")))), 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\": \"0.63888em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")))))), \", which is no different than replacing \", 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    \"mathvariant\": \"bold\"\n  }, \"u\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{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\": \"0.44444em\",\n      \"verticalAlign\": \"0em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"u\")))))), \" with \", 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    \"mathvariant\": \"bold\"\n  }, \"q\")), mdx(\"annotation\", {\n    parentName: \"semantics\",\n    \"encoding\": \"application/x-tex\"\n  }, \"\\\\mathbf{q}\")))), 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\": \"0.63888em\",\n      \"verticalAlign\": \"-0.19444em\"\n    }\n  }), mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord\"\n  }, mdx(\"span\", {\n    parentName: \"span\",\n    \"className\": \"mord mathbf\"\n  }, \"q\")))))), \" in a conventional misfit based on displacements.\"));\n}\n;\nMDXContent.isMDXComponent = true;"},"site":{"siteMetadata":{"salvusDocVersions":{"current":"2026.5.0"}}}},"pageContext":{"id":"af025106-a718-50b5-99c3-256f39aee9d9"}},
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