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import os
import numpy as np
import xarray as xr
import salvus.namespace as sn
from salvus.mesh.tools.transforms import interpolate_mesh_to_mesh

# Run simulations on this site.
SALVUS_FLOW_SITE_NAME = os.environ.get("SITE_NAME", "local")

Mesh-to-mesh interpolation

Micro-tutorial

Interpolating between two unstructured meshes is useful in several situations, such as
  • representing two models on the exact same mesh,
  • increasing the frequency band in an ongoing inversion, and
  • re-meshing a model when a decrease in the minimum velocity begins to result in under-resolved waveforms,
among others.

In this tutorial, we demonstrate how to perform mesh-to-mesh interpolation in 2-D (first) and 3-D (second).

# Gaussian
x, y = np.linspace(0.0, 1.0, 100), np.linspace(0.0, 1.0, 100)
xx, yy = np.meshgrid(x, y, indexing="xy")
g = np.exp(-(((xx - 0.5) ** 2 + (yy - 0.5) ** 2) / (2 * 0.2**2)))

# Pars
vp = 2 * g + 1
rho = vp / 2

# Xarray dataset
ds = xr.Dataset(
    coords={"x": x, "y": y},
    data_vars={"vp": (["x", "y"], vp), "rho": (["x", "y"], rho)},
)

# Salvus wrapper.
m = sn.model.volume.cartesian.GenericModel(name="blob", data=ds)

# Plot
m.ds.vp.plot()
<matplotlib.collections.QuadMesh at 0x7377de153910>
p = sn.Project.from_volume_model("proj_2d", m, True)
s = sn.simple_config.source.cartesian.ScalarPoint2D(x=0.25, y=0.25, f=1.0)
r = sn.simple_config.receiver.cartesian.Point2D(
    x=0.75, y=0.75, fields=["phi"], station_code="XX", network_code="XX"
)

p.add_to_project(sn.EventCollection.from_sources(sources=s, receivers=r))
f_max = 10.0

# Common event configuration.
ec = sn.EventConfiguration(
    wavelet=sn.simple_config.stf.Ricker(center_frequency=f_max / 2),
    waveform_simulation_configuration=sn.WaveformSimulationConfiguration(
        end_time_in_seconds=1.0
    ),
)

# Heterogeneous - meshed at f_max.
s0 = sn.SimulationConfiguration(
    name="blob_fmax",
    max_frequency_in_hertz=f_max,
    event_configuration=ec,
    model_configuration=sn.ModelConfiguration(
        background_model=None, volume_models="blob"
    ),
    elements_per_wavelength=2.0,
)

# Homogeneous - meshed at 1.5 * f_max.
s1 = sn.SimulationConfiguration(
    name="homo_fmax_1.5",
    max_frequency_in_hertz=1.5 * f_max,
    event_configuration=ec,
    model_configuration=sn.ModelConfiguration(
        background_model=sn.model.background.homogeneous.IsotropicAcoustic(
            rho=rho.min(), vp=vp.min()
        ),
    ),
    elements_per_wavelength=2.0,
)
for s in [s0, s1]:
    p.add_to_project(s, overwrite=True)
Interpolate the blob mesh onto the homogeneous mesh
# Interpolate.
m2m = interpolate_mesh_to_mesh(
    p.simulations.get_mesh("blob_fmax"),
    p.simulations.get_mesh("homo_fmax_1.5"),
    use_layers=False,
    use_1d_vertical_coordinate=False,
)

# Visualize.
m2m
[2026-06-22 09:14:48,015] INFO: Creating mesh. Hang on.
[2026-06-22 09:14:48,117] INFO: Creating mesh. Hang on.
<unstructured_mesh.unstructured_mesh.UnstructuredMesh object at 0x7377dde64510>
if "blob_fmax_1.5" not in p.simulations.list():
    p.add_to_project(
        sn.UnstructuredMeshSimulationConfiguration(
            name="blob_fmax_1.5", unstructured_mesh=m2m, event_configuration=ec
        )
    )
for name in ["blob_fmax", "blob_fmax_1.5", "homo_fmax_1.5"]:
    p.simulations.launch(
        simulation_configuration=name,
        events=p.events.list(),
        site_name=SALVUS_FLOW_SITE_NAME,
        ranks_per_job=1,
    )
    p.simulations.query(block=True)
[2026-06-22 09:14:49,059] INFO: Submitting job ...
Uploading 1 files...

🚀  Submitted job_2606220914215938_cfa7f7c9b7@local
[2026-06-22 09:14:49,881] INFO: Submitting job ...
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🚀  Submitted job_2606220914887037_d44dcb5b96@local
[2026-06-22 09:14:50,987] INFO: Submitting job ...
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🚀  Submitted job_2606220914993909_fd1091e88b@local
Note that both blob_fmax runs are essentially identical, while the homogeneous run is very different. The small differences in the two blob runs are expected due to the different mesh resolutions and the broadband nature of the source wavelet.
p.viz.waveforms(
    event="event_0000",
    receiver_name="XX.XX",
    data=["blob_fmax", "blob_fmax_1.5", "homo_fmax_1.5"],
    receiver_field="phi",
)
[]
Add 3-D "plume" like structures. Add positive vp perturbations to the crust, negative vp perturbations to the mantle.
# Spherical coordinates
lat = np.linspace(-5.0, 5.0, 101)
lon = np.linspace(-5.0, 5.0, 101)
dep = np.linspace(0.0, 660e3, 101)

# Gaussian
xx, yy, zz = np.meshgrid(lat, lon, dep, indexing="xy")
g = np.zeros_like(xx)
for lats in lat[25:-25:25]:
    for lons in lon[::25]:
        g += np.exp(-(((xx - lats) ** 2 + (yy - lons) ** 2) / (2 * 0.5**2)))

# Xarray dataset
ds = xr.Dataset(
    coords={"latitude": lat, "longitude": lon, "depth": dep},
    data_vars={
        "vp": (["latitude", "longitude", "depth"], g * 20, {"units": "%"}),
    },
    attrs={
        "geospatial_lon_units": "degrees",
        "geospatial_lat_units": "degrees_north",
        "geospatial_vertical_units": "m",
    },
)

ds_mantle = ds.copy()
ds_mantle["vp"] *= -1

# Salvus wrapper.
mc = sn.model.volume.seismology.CrustalModel(name="crust", data=ds)
mm = sn.model.volume.seismology.MantleModel(name="mantle", data=ds)

# Plot depth slice.
mm.ds.vp.isel(depth=0).plot()
<matplotlib.collections.QuadMesh at 0x737800440490>
d = sn.domain.dim3.SphericalChunkDomain(
    lat_center=0.0,
    lon_center=0.0,
    lat_extent=5.0,
    lon_extent=5.0,
    radius_in_meter=6371e3,
)

# Add mantle and crustal model.
p = sn.Project.from_domain("proj_3d", d, True)
for m in [mc, mm]:
    p.add_to_project(m, overwrite=True)
s = sn.simple_config.source.seismology.SideSetVectorPoint3D(
    fr=1e10,
    ft=0.0,
    fp=0.0,
    latitude=-2.5,
    longitude=-2.5,
    depth_in_m=0.0,
    side_set_name="r1",
)

r = sn.simple_config.receiver.seismology.SideSetPoint3D(
    latitude=2.5,
    longitude=2.5,
    fields=["velocity"],
    station_code="XX",
    network_code="XX",
    side_set_name="r1",
)

p.add_to_project(sn.EventCollection.from_sources(sources=s, receivers=r))
p_min = 50.0

# Common event configuration.
ec = sn.EventConfiguration(
    wavelet=sn.simple_config.stf.Ricker(center_frequency=1 / (2 * p_min)),
    waveform_simulation_configuration=sn.WaveformSimulationConfiguration(
        end_time_in_seconds=600.0
    ),
)

# Heterogeneous - meshed at f_max.
s0 = sn.SimulationConfiguration(
    name="blobs",
    tensor_order=2,
    max_depth_in_meters=660e3,
    min_period_in_seconds=p_min,
    event_configuration=ec,
    model_configuration=sn.ModelConfiguration(
        background_model=sn.model.background.one_dimensional.BuiltIn(
            name="prem_iso_one_crust"
        ),
        volume_models=["mantle", "crust"],
    ),
    elements_per_wavelength=2.0,
)

# 1D only - meshed at a slightly lower resolution.
s1 = sn.SimulationConfiguration(
    name="1d",
    tensor_order=2,
    max_depth_in_meters=660e3,
    min_period_in_seconds=p_min,
    event_configuration=ec,
    model_configuration=sn.ModelConfiguration(
        background_model=sn.model.background.one_dimensional.BuiltIn(
            name="prem_iso_one_crust"
        ),
    ),
    elements_per_wavelength=1.5,
)
for s in [s0, s1]:
    p.add_to_project(s, overwrite=True)
p.viz.nb.simulation_setup("blobs")
[2026-06-22 09:14:53,445] INFO: Creating mesh. Hang on.
Interpolating model: mantle.
Interpolating model: crust.
<unstructured_mesh.unstructured_mesh.UnstructuredMesh object at 0x7377d3d64c90>
Interpolate the "plume" model from the "blobs" mesh to the "1d" mesh. Note that in this case we set use_layers and use_1d_vertical_coordinate to True. This ensures that the interpolation only happens between layers are consistent with each other (i.e. discontinuities are preserved) and that any topography and bathymetry will be flattened before the interpolation occurs.
# Interpolate.
m2m = interpolate_mesh_to_mesh(
    p.simulations.get_mesh("blobs"),
    p.simulations.get_mesh("1d"),
    use_layers=True,
    use_1d_vertical_coordinate=True,
)

# Assert mesh is different.
assert m2m.nelem != p.simulations.get_mesh("blobs").nelem

# Visualize.
m2m
[2026-06-22 09:14:57,621] INFO: Creating mesh. Hang on.
<unstructured_mesh.unstructured_mesh.UnstructuredMesh object at 0x7377d4062a50>
p.add_to_project(
    sn.UnstructuredMeshSimulationConfiguration(
        name="blobs_reinterp", unstructured_mesh=m2m, event_configuration=ec
    )
)
for name in ["blobs", "blobs_reinterp", "1d"]:
    p.simulations.launch(
        simulation_configuration=name,
        events=p.events.list(),
        site_name=SALVUS_FLOW_SITE_NAME,
        ranks_per_job=1,
    )
    p.simulations.query(block=True)
[2026-06-22 09:15:00,884] INFO: Submitting job ...
Uploading 1 files...

🚀  Submitted job_2606220915889332_b741e67f0e@local
[2026-06-22 09:15:22,218] INFO: Submitting job ...
Uploading 1 files...

🚀  Submitted job_2606220915225580_c855093809@local
[2026-06-22 09:15:31,495] INFO: Submitting job ...
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🚀  Submitted job_2606220915500379_2b5be23a9b@local
Note that both runs with 3-D models are essentially identical, while the homogeneous run is very different. The small differences in the two 3-D runs are expected due to the different mesh resolutions, the broadband nature of the source wavelet, and the heterogeneity of the model.
p.viz.waveforms(
    event="event_0000",
    receiver_name="XX.XX",
    data=["blobs", "blobs_reinterp", "1d"],
    receiver_field="velocity",
)
[]
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