Mondaic - Full Waveform Solutions

This tutorial is presented as Python code running inside a Jupyter Notebook, the recommended way to use Salvus. To run it yourself you can copy/type each individual cell or directly download the full notebook, including all required files.

Download Jupyter Notebook [129.89 kB]

Reference solution: Gar6more2D
Physics: Coupled acoustic/elastic wave equation

In this notebook will use Gar6more2D to generate a set of of semi-analytic solutions to the 2-D coupled acoustic-elastic wave equation, and then compare these solutions to those computed within Salvus. To make things a bit more interesting, we will consider a domain with one stress-free (free-surface) boundary, in an analogue to Lamb's problem in elastic media.

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%config Completer.use_jedi = False
%matplotlib inline

# Stdlib packages
import os
import pathlib
import shutil
import subprocess

# Third party packages
import matplotlib.pyplot as plt

%matplotlib inline
import numpy as np
import obspy
import pyasdf

# Import things from SalvusFlow
from salvus.flow import api

# Configuration helpers from SalvusFlow.
from salvus.flow import simple_config as sc

# And some helper functions to run the integration tests.
from integration_test_mesh_helper import (
    get_mesh,
    Physics,
    AnalyticCode,
    read_gar6more,
)

# Number of processes SalvusCompute will run with.
# Get it from the environment or default to 4.
MPI_RANKS = int(os.environ.get("NUM_MPI_RANKS", 4))
# Choose on which site to run this.
SALVUS_FLOW_SITE_NAME = os.environ.get("SITE_NAME", "local")

First we must compile and run Gar6more2D. As the Gar6more2D git repository is included here as a submodule, as long as the module has been pulled the following paths should work on your machine. The next cell should compile Gar6more2D in the default location (if it has not been compiled yet).

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# Set up the paths for the Gar6more2D data and binaries.
gar6more2d_base_path = pathlib.Path("gar6more2d")
gar6more2d_build_path = gar6more2d_base_path / "build"
gar6more2d_bin = gar6more2d_build_path / "Gar6more2D.out"
gar6more2d_par_file_path = gar6more2d_build_path / "Gar6more2D.dat"

# Compile Gar6more2D
os.makedirs(gar6more2d_build_path, exist_ok=True)
if not os.path.exists(gar6more2d_bin):
    assert (
        subprocess.run(["cmake", ".."], cwd=gar6more2d_build_path).returncode
        == 0
    )
    assert subprocess.run(["make"], cwd=gar6more2d_build_path).returncode == 0

-- The C compiler identification is GNU 9.4.0
-- The CXX compiler identification is GNU 9.4.0
-- Check for working C compiler: /usr/bin/cc
-- Check for working C compiler: /usr/bin/cc -- works
-- Detecting C compiler ABI info
-- Detecting C compiler ABI info - done
-- Detecting C compile features
-- Detecting C compile features - done
-- Check for working CXX compiler: /usr/bin/c++
-- Check for working CXX compiler: /usr/bin/c++ -- works
-- Detecting CXX compiler ABI info
-- Detecting CXX compiler ABI info - done
-- Detecting CXX compile features
-- Detecting CXX compile features - done
-- Looking for sgemm_
-- Looking for sgemm_ - not found
-- Looking for pthread.h
-- Looking for pthread.h - found
-- Performing Test CMAKE_HAVE_LIBC_PTHREAD
-- Performing Test CMAKE_HAVE_LIBC_PTHREAD - Failed
-- Looking for pthread_create in pthreads
-- Looking for pthread_create in pthreads - not found
-- Looking for pthread_create in pthread
-- Looking for pthread_create in pthread - found
-- Found Threads: TRUE  
-- Looking for sgemm_
-- Looking for sgemm_ - found
-- Found BLAS: /usr/lib/x86_64-linux-gnu/libblas.so  
-- Looking for cheev_
-- Looking for cheev_ - not found
-- Looking for cheev_
-- Looking for cheev_ - found
-- A library with LAPACK API found.
-- The Fortran compiler identification is GNU 9.4.0
-- Check for working Fortran compiler: /usr/bin/gfortran
-- Check for working Fortran compiler: /usr/bin/gfortran  -- works
-- Detecting Fortran compiler ABI info
-- Detecting Fortran compiler ABI info - done
-- Checking whether /usr/bin/gfortran supports Fortran 90

-- Checking whether /usr/bin/gfortran supports Fortran 90 -- yes
-- Found: /tmp/tmpntygrf9r/gar6more2d/mod/m_const.F90;/tmp/tmpntygrf9r/gar6more2d/mod/m_num.F90;/tmp/tmpntygrf9r/gar6more2d/mod/m_phys.F90;/tmp/tmpntygrf9r/gar6more2d/mod/m_result.F90;/tmp/tmpntygrf9r/gar6more2d/mod/m_sismo.F90;/tmp/tmpntygrf9r/gar6more2d/mod/m_source.F90
-- Found: /tmp/tmpntygrf9r/gar6more2d/lib/libacousacous/acousacous.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacousacous/sub_incid_aa.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacousacous/sub_reflex_aa.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacousacous/sub_reflex_free_aa.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacousacous/sub_reflex_wall_aa.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacousacous/sub_transmit_aa.F90
-- Found: /tmp/tmpntygrf9r/gar6more2d/lib/libacouselasto/acouselasto.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacouselasto/sub_incid_ae.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacouselasto/sub_reflex_ae.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacouselasto/sub_transmitp_ae.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacouselasto/sub_transmits_ae.F90
-- Found: /tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/calccoeffP_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/calccoeffS_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/elastoelasto.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_incidP_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_incidS_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexPP_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexPP_free_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexPP_wall_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexPS_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexPS_free_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexPS_wall_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexSP_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexSP_free_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexSP_wall_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexSS_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexSS_free_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_reflexSS_wall_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_transmitPP_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_transmitPS_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_transmitSP_ee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libelastoelasto/sub_transmitSS_ee.F90
-- Found: /tmp/tmpntygrf9r/gar6more2d/lib/libgeneral/arrivaltime.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libgeneral/derivee.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libgeneral/path.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libgeneral/source.F90
-- Found: /tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/calccoefff_free_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/calccoefff_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/calccoefff_wall_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/calccoeffs_free_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/calccoeffs_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/calccoeffs_wall_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/poroporo.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_incid_pp_f.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_incid_pp_s.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexff_free_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexff_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexff_wall_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexfpsi_free_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexfpsi_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexfpsi_wall_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexfs_free_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexfs_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexfs_wall_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexsf_free_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexsf_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexsf_wall_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexspsi_free_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexspsi_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexspsi_wall_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexss_free_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexss_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_reflexss_wall_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_transmitff_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_transmitfpsi_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_transmitfs_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_transmitsf_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_transmitspsi_pp.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libporoporo/sub_transmitss_pp.F90
-- Found: /tmp/tmpntygrf9r/gar6more2d/lib/libacousporo/acousporo.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacousporo/calccoeff_ap.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacousporo/sub_incid_ap.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacousporo/sub_reflex_ap.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacousporo/sub_transmitf_ap.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacousporo/sub_transmitpsi_ap.F90;/tmp/tmpntygrf9r/gar6more2d/lib/libacousporo/sub_transmits_ap.F90
-- Configuring done
-- Generating done
-- Build files have been written to: /tmp/tmpntygrf9r/gar6more2d/build
Scanning dependencies of target mod
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Now we generate the parameter file to pass to Gar6more2D. We choose a medium which is infinite on 3 sides, and which has a free-surface at the bottom boundary. We use a ricker source-time function to generate the acoustic wavefield, with a center frequency of

100

Hz

. We place this source

250

m

above the free surface boundary

(0, 250)

, and place 5 receivers

50

m

above the same boundary, spaced equidistantly from

x = -200 m

x = +200 m

. We compute the solution between

0

and

0.1

seconds, with a sampling interval of

1 \times 10^{-4}

seconds, for a total of

1000

time samples. We also choose an acoustic wave velocity of

5800

m/s

and a density of

2600

kg/m^3

, which corresponds to a bulk modulus

\mu

8.7464 \times 10^{10}

Copy

# Generate and write the parameter file.
amplitude = 1e2
time_step = 1e-4
center_frequency = 100.0
gar6more2d_par_file = """4 infinite medium (1), free surface (2), wall boundary (3) or bilayered medium(4)
0 first layer : acoustic (0) elastodynamic (1), poroelastic (2)
1d2 Central frequency of the source
1d2 Amplitude of the source
2d-2 Delay of the source
125d0 Height of the source
{:f}d0 Height of the line of receivers
-200d0 Abscissa of the first receiver
200d0  Abscissa of the last receiver
5 Numbers of receivers
0 Start time
1d-1 Stop time
1e-4 Time step
1000 Number of intervals for the numerical computation of the convolution
87464000000 2600 mu and rho
1 second layer : acoustic (0) elastodynamic (1), poroelastic (2)
41600000000 4264000000 2600 mu, lambda and rho
"""

Now, we generate the semi-analytic pressure solution, and read the results into an obspy stream object.

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# Run code.
for rec_depth in [-75.0, +75.0]:
    if rec_depth == -75.0:
        test_file = gar6more2d_build_path / "Ux_safe.dat"
        test_file_y = gar6more2d_build_path / "Uy_safe.dat"
    else:
        test_file = gar6more2d_build_path / "P_safe.dat"

    if not os.path.exists(test_file):
        with open(gar6more2d_par_file_path, "w") as fh:
            fh.write(gar6more2d_par_file.format(rec_depth))
        assert (
            subprocess.run(
                ["./Gar6more2D.out"], cwd=gar6more2d_build_path
            ).returncode
            == 0
        )

        if "Ux" in str(test_file):
            shutil.copy(gar6more2d_build_path / "Ux.dat", test_file)
            shutil.copy(gar6more2d_build_path / "Uy.dat", test_file_y)
        else:
            shutil.copy(gar6more2d_build_path / "P.dat", test_file)

    # Read data.
    if rec_depth == -75.0:
        gar6more2d_data_file = test_file
        gar6more2d_data_ux = obspy.Stream(read_gar6more(gar6more2d_data_file))
        gar6more2d_data_file = test_file_y
        gar6more2d_data_uy = obspy.Stream(read_gar6more(gar6more2d_data_file))
    else:
        gar6more2d_data_file = test_file
        gar6more2d_data_p = obspy.Stream(read_gar6more(gar6more2d_data_file))

 Do you want
 - An infinite homogeneous medium (1);
 - A semi-infinite homogeneous media with a free surface boundary condition at it bottom (2):
 - A semi-infinite homogeneous media with a wall boundary condition at it bottom (3):
 - A bilayered infinite media with a plane interface (4):
           4
 Is the first medium acoustic (0), elastodynamic (1) or porous (2)?
           0
 Frequency of the source?
   100.00000000000000     
 Amplitude of the source?
   100.00000000000000     
 delay of the source (f(t)=0 if t>2tdelay)?
   2.0000000000000000E-002
 Height of the source?
   125.00000000000000     
 Height of the line of receivers?
  -75.000000000000000     
 abscissa of the first receiver?
  -200.00000000000000     
 abscissa of the last receiver?
   200.00000000000000     
 How many receivers?
           5
 dx=   100.00000000000000     
 At which time shall the seismo begin?
   0.0000000000000000     
 At which time shall the seismo stop?
  0.10000000000000001     
 What is the time-step?
   1.0000000000000000E-004
        1001
 How many intervals are required for the numerical computation of the time convolution?
        1000
 mu and rho?
   87464000000.000000        2600.0000000000000     
 V1=   5800.0000000000000     
 Is the second medium acoustic (0) elastodynamic (1) or porous (2)?
           1
 mu,lambda,rho
   41600000000.000000        4264000000.0000000        2600.0000000000000     
 Vp2, Vs2   5800.0000000000000        4000.0000000000000     
 Computation of the transmitted P wave
 Computation of the transmitted S wave
 Do you want
 - An infinite homogeneous medium (1);
 - A semi-infinite homogeneous media with a free surface boundary condition at it bottom (2):
 - A semi-infinite homogeneous media with a wall boundary condition at it bottom (3):
 - A bilayered infinite media with a plane interface (4):
           4
 Is the first medium acoustic (0), elastodynamic (1) or porous (2)?
           0
 Frequency of the source?
   100.00000000000000     
 Amplitude of the source?
   100.00000000000000     
 delay of the source (f(t)=0 if t>2tdelay)?
   2.0000000000000000E-002
 Height of the source?
   125.00000000000000     
 Height of the line of receivers?
   75.000000000000000     
 abscissa of the first receiver?
  -200.00000000000000     
 abscissa of the last receiver?
   200.00000000000000     
 How many receivers?
           5
 dx=   100.00000000000000     
 At which time shall the seismo begin?
   0.0000000000000000     
 At which time shall the seismo stop?
  0.10000000000000001     
 What is the time-step?
   1.0000000000000000E-004
        1001
 How many intervals are required for the numerical computation of the time convolution?
        1000
 mu and rho?
   87464000000.000000        2600.0000000000000     
 V1=   5800.0000000000000     
 Is the second medium acoustic (0) elastodynamic (1) or porous (2)?
           1
 mu,lambda,rho
   41600000000.000000        4264000000.0000000        2600.0000000000000     
 Vp2, Vs2   5800.0000000000000        4000.0000000000000     
 Computation of the incident wave
 Computation of the reflected wave

Now, we will run a fully numerical simulations in Salvus, and attempt to replicate the semi-analytic seismograms. For the sake of brevity, we've left the specifics of the mesh and parameter file generation to the IntegrationTestMesh.py module, which is shared between all integration test instances. Feel free to peak inside to see how the meshes are made -- or alternatively check out the meshing tutorials for a more in-depth explanation.

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# Adjust the scaling of the source term to be equivalent to what is
# used in GAR6MORE.
bulk_modulus = 8.7464e10
gar6more_scale = bulk_modulus / (2 * np.pi**2 * center_frequency**2)
source_amplitude = amplitude / gar6more_scale

Use the helper objects in SalvusFlow to generate sources, receivers, and boundary conditions.

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source = sc.source.cartesian.ScalarPoint2D(
    x=500.0,
    y=375.0,
    f=source_amplitude,
    source_time_function=sc.source.stf.Ricker(
        center_frequency=center_frequency
    ),
)

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# Generate 5 receivers in the acoustic part.
receivers_acoustic = [
    sc.receiver.cartesian.Point2D(
        station_code=str(i), x=x, y=325.0, fields=["phi"]
    )
    for i, x in enumerate(range(300, 701, 100))
]

# And 5 in the elastic part
receivers_elastic = [
    sc.receiver.cartesian.Point2D(
        station_code=str(i + 5), x=x, y=175.0, fields=["displacement"]
    )
    for i, x in enumerate(range(300, 701, 100))
]

# Combine to a single list.
receivers = receivers_acoustic + receivers_elastic

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boundary = sc.boundary.Absorbing(
    side_sets=["y1"], width_in_meters=0.0, taper_amplitude=0.0
)

Finally run the simulations for a number of different shape mapping orders.

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output_dirs = []

# Run for a number of different orders.
for order in [1, 2, 4, 7]:
    # Setup the mesh for this simulation.
    mesh = get_mesh(
        dimension=2,
        analytic_code=AnalyticCode.Gar6more2D,
        physics=Physics.COUPLED,
        n_elem_per_wavelength=2,
        polynomial_order=max(4, order),
        shape_order=min(4, order),
    )

    # Unique job name.
    job_name = f"GAR6MORE2D_COUPLED_ORDER_{order}"
    output_dirs.append(pathlib.Path(job_name) / "output")

    # Configure Salvus
    w = sc.simulation.Waveform()
    w.set_mesh(mesh)

    w.physics.wave_equation.start_time_in_seconds = -2e-2
    w.physics.wave_equation.end_time_in_seconds = 8e-2
    w.physics.wave_equation.time_step_in_seconds = time_step

    w.add_sources(source)
    w.add_receivers(receivers)
    w.add_boundary_conditions(boundary)

    # The input files can optionally be already validated.
    w.validate()

    api.run(
        site_name=SALVUS_FLOW_SITE_NAME,
        output_folder=output_dirs[-1],
        input_file=w,
        ranks=MPI_RANKS,
        get_all=True,
        overwrite=True,
    )

SalvusJob `job_2403151019962754_d24fe55971` running on `local_f64` with 4 rank(s).
Site information:
  * Salvus version: 0.12.16
  * Floating point size: 64

* Downloaded 161.5 KB of results to `GAR6MORE2D_COUPLED_ORDER_1/output`.
* Total run time: 1.47 seconds.
* Pure simulation time: 1.27 seconds.
SalvusJob `job_2403151020417667_dccf3826ed` running on `local_f64` with 4 rank(s).
Site information:
  * Salvus version: 0.12.16
  * Floating point size: 64

* Downloaded 161.5 KB of results to `GAR6MORE2D_COUPLED_ORDER_2/output`.
* Total run time: 1.65 seconds.
* Pure simulation time: 1.42 seconds.
SalvusJob `job_2403151020119974_200a0f605d` running on `local_f64` with 4 rank(s).
Site information:
  * Salvus version: 0.12.16
  * Floating point size: 64

* Downloaded 161.5 KB of results to `GAR6MORE2D_COUPLED_ORDER_4/output`.
* Total run time: 1.90 seconds.
* Pure simulation time: 1.57 seconds.
SalvusJob `job_2403151020072340_73cfddf3d7` running on `local_f64` with 4 rank(s).
Site information:
  * Salvus version: 0.12.16
  * Floating point size: 64

* Downloaded 161.5 KB of results to `GAR6MORE2D_COUPLED_ORDER_7/output`.
* Total run time: 1.91 seconds.
* Pure simulation time: 1.57 seconds.

Finally, we'll now read in the seismograms from Salvus, and plot them overtop the semi-analytic solutions we generated in Gar6more2D. If you've used all the default settings, things should match up exactly. However, feel free to play with some parameters to see how the accuracy can be increased or decreased. Note that, since these are our actual analytic tests, a reduction in accuracy may cause the np.testing.assert_allclose line to fail. If you encounter this, and would still like to see the effect of your changes on the seismograms, feel free to comment out that line.

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# Setup the figure.
f, ax = plt.subplots(5, 2, figsize=(15, 30), sharex=True)
f.suptitle("Integration Test (Gar6more2D // COUPLED [Acoustic / Elastic])")

# Read in the data produced by Salvus.
for output_dir in output_dirs:
    with pyasdf.ASDFDataSet(output_dir / "receivers.h5") as dataset:
        # Loop over the receivers in both Gar6more2D and Salvus,
        # and plot them overtop of one another.
        above = [dataset.waveforms[i] for i in dataset.waveforms.list()[:5]]
        below = [dataset.waveforms[i] for i in dataset.waveforms.list()[5:]]

        for _i, (a, sa, se, gx, gy, ga) in enumerate(
            zip(
                ax,
                above,
                below,
                gar6more2d_data_ux,
                gar6more2d_data_uy,
                gar6more2d_data_p,
            )
        ):
            analytic_ux = gx.copy().differentiate().data[:750]
            salvus_ux = -se.displacement[0].data[:750]
            analytic_uy = gy.copy().differentiate().data[:750]
            salvus_uy = -se.displacement[1].data[:750]
            analytic_p = ga.copy().data[:600]
            salvus_p = sa.phi[0].data[:600]

            valid_p = np.where(np.abs(analytic_p) > 1e-3)[0]
            valid_ux = np.where(np.abs(analytic_ux) > 1e-7)[0]
            valid_uy = np.where(np.abs(analytic_uy) > 1e-7)[0]

            assert len(valid_p) > 0
            assert len(valid_uy) > 0

            # Nodal line.
            if _i != 2:
                assert len(valid_ux) > 0

            # Plot (should deploy these to some server).
            a[1].plot(analytic_ux, label="Gar6more2D_ux", ls="solid")
            a[1].plot(salvus_ux, label="Salvus_ux", ls="dashed")
            a[1].plot(analytic_uy, label="Gar6more2D_uy", ls="solid")
            a[1].plot(salvus_uy, label="Salvus_uy", ls="dashed")
            a[0].plot(analytic_p, label="Gar6more2D_p", ls="solid")
            a[0].plot(salvus_p, label="Salvus_p", ls="dashed")
            a[0].set_xlabel("Time sample")
            a[0].set_ylabel("Amplitude")
            a[0].legend()
            a[1].legend()

            np.testing.assert_allclose(
                analytic_ux[valid_ux],
                salvus_ux[valid_ux],
                rtol=1e-1,
                atol=1e-10,
            )
            np.testing.assert_allclose(
                analytic_uy[valid_uy],
                salvus_uy[valid_uy],
                rtol=1e-1,
                atol=1e-10,
            )
            np.testing.assert_allclose(
                analytic_p[valid_p], salvus_p[valid_p], rtol=1e-1, atol=1e-5
            )