.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "content/user-guide/examples/05-fdem/plot_inv_fdem_loop_loop_2Dinversion.py"
.. LINE NUMBERS ARE GIVEN BELOW.
.. only:: html
.. note::
:class: sphx-glr-download-link-note
:ref:`Go to the end <sphx_glr_download_content_user-guide_examples_05-fdem_plot_inv_fdem_loop_loop_2Dinversion.py>`
to download the full example code.
.. rst-class:: sphx-glr-example-title
.. _sphx_glr_content_user-guide_examples_05-fdem_plot_inv_fdem_loop_loop_2Dinversion.py:
2D inversion of Loop-Loop EM Data
=================================
In this example, we consider a single line of loop-loop EM data
at 30kHz with 3 different coil separations [0.32m, 0.71m, 1.18m].
We will use only Horizontal co-planar orientations (vertical magnetic dipole),
and look at the real and imaginary parts of the secondary magnetic field.
We use the :class:`simpeg.maps.Surject2Dto3D` mapping to invert for a 2D model
and perform the forward modelling in 3D.
.. GENERATED FROM PYTHON SOURCE LINES 14-33
.. code-block:: Python
import numpy as np
import matplotlib.pyplot as plt
import time
import discretize
from simpeg import (
maps,
optimization,
data_misfit,
regularization,
inverse_problem,
inversion,
directives,
Report,
)
from simpeg.electromagnetics import frequency_domain as FDEM
.. GENERATED FROM PYTHON SOURCE LINES 34-39
Setup
-----
Define the survey and model parameters
.. GENERATED FROM PYTHON SOURCE LINES 39-50
.. code-block:: Python
sigma_surface = 10e-3
sigma_deep = 40e-3
sigma_air = 1e-8
coil_separations = [0.32, 0.71, 1.18]
freq = 30e3
print("skin_depth: {:1.2f}m".format(500 / np.sqrt(sigma_deep * freq)))
.. rst-class:: sphx-glr-script-out
.. code-block:: none
skin_depth: 14.43m
.. GENERATED FROM PYTHON SOURCE LINES 51-53
Define a dipping interface between the surface layer and the deeper layer
.. GENERATED FROM PYTHON SOURCE LINES 54-76
.. code-block:: Python
z_interface_shallow = -0.25
z_interface_deep = -1.5
x_dip = np.r_[0.0, 8.0]
def interface(x):
interface = np.zeros_like(x)
interface[x < x_dip[0]] = z_interface_shallow
dipping_unit = (x >= x_dip[0]) & (x <= x_dip[1])
x_dipping = (-(z_interface_shallow - z_interface_deep) / x_dip[1]) * (
x[dipping_unit]
) + z_interface_shallow
interface[dipping_unit] = x_dipping
interface[x > x_dip[1]] = z_interface_deep
return interface
.. GENERATED FROM PYTHON SOURCE LINES 77-87
Forward Modelling Mesh
----------------------
Here, we set up a 3D tensor mesh which we will perform the forward
simulations on.
.. note::
In practice, a smaller horizontal discretization should be used to improve
accuracy, particularly for the shortest offset (eg. you can try 0.25m).
.. GENERATED FROM PYTHON SOURCE LINES 87-119
.. code-block:: Python
csx = 0.5 # cell size for the horizontal direction
csz = 0.125 # cell size for the vertical direction
pf = 1.3 # expansion factor for the padding cells
npadx = 7 # number of padding cells in the x-direction
npady = 7 # number of padding cells in the y-direction
npadz = 11 # number of padding cells in the z-direction
core_domain_x = np.r_[-11.5, 11.5] # extent of uniform cells in the x-direction
core_domain_z = np.r_[-2.0, 0.0] # extent of uniform cells in the z-direction
# number of cells in the core region
ncx = int(np.diff(core_domain_x) / csx)
ncz = int(np.diff(core_domain_z) / csz)
# create a 3D tensor mesh
mesh = discretize.TensorMesh(
[
[(csx, npadx, -pf), (csx, ncx), (csx, npadx, pf)],
[(csx, npady, -pf), (csx, 1), (csx, npady, pf)],
[(csz, npadz, -pf), (csz, ncz), (csz, npadz, pf)],
]
)
# set the origin
mesh.x0 = np.r_[
-mesh.h[0].sum() / 2.0, -mesh.h[1].sum() / 2.0, -mesh.h[2][: npadz + ncz].sum()
]
print("the mesh has {} cells".format(mesh.nC))
mesh.plot_grid()
.. image-sg:: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_001.png
:alt: plot inv fdem loop loop 2Dinversion
:srcset: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_001.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-script-out
.. code-block:: none
/home/vsts/work/1/s/examples/05-fdem/plot_inv_fdem_loop_loop_2Dinversion.py:100: DeprecationWarning:
Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
/home/vsts/work/1/s/examples/05-fdem/plot_inv_fdem_loop_loop_2Dinversion.py:101: DeprecationWarning:
Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
the mesh has 34200 cells
<Axes3D: xlabel='x1', ylabel='x2', zlabel='x3'>
.. GENERATED FROM PYTHON SOURCE LINES 120-125
Inversion Mesh
--------------
Here, we set up a 2D tensor mesh which we will represent the inversion model
on
.. GENERATED FROM PYTHON SOURCE LINES 125-130
.. code-block:: Python
inversion_mesh = discretize.TensorMesh([mesh.h[0], mesh.h[2][mesh.cell_centers_z <= 0]])
inversion_mesh.x0 = [-inversion_mesh.h[0].sum() / 2.0, -inversion_mesh.h[1].sum()]
inversion_mesh.plot_grid()
.. image-sg:: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_002.png
:alt: plot inv fdem loop loop 2Dinversion
:srcset: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_002.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-script-out
.. code-block:: none
<Axes: xlabel='x1', ylabel='x2'>
.. GENERATED FROM PYTHON SOURCE LINES 131-137
Mappings
---------
Mappings are used to take the inversion model and represent it as electrical
conductivity on the inversion mesh. We will invert for log-conductivity below
the surface, fixing the conductivity of the air cells to 1e-8 S/m
.. GENERATED FROM PYTHON SOURCE LINES 137-153
.. code-block:: Python
# create a 2D mesh that includes air cells
mesh2D = discretize.TensorMesh([mesh.h[0], mesh.h[2]], x0=mesh.x0[[0, 2]])
active_inds = mesh2D.gridCC[:, 1] < 0 # active indices are below the surface
mapping = (
maps.Surject2Dto3D(mesh)
* maps.InjectActiveCells( # populates 3D space from a 2D model
mesh2D, active_inds, sigma_air
)
* maps.ExpMap( # adds air cells
nP=inversion_mesh.nC
) # takes the exponential (log(sigma) --> sigma)
)
.. GENERATED FROM PYTHON SOURCE LINES 154-158
True Model
----------
Create our true model which we will use to generate synthetic data for
.. GENERATED FROM PYTHON SOURCE LINES 158-170
.. code-block:: Python
m_true = np.log(sigma_deep) * np.ones(inversion_mesh.nC)
interface_depth = interface(inversion_mesh.gridCC[:, 0])
m_true[inversion_mesh.gridCC[:, 1] > interface_depth] = np.log(sigma_surface)
fig, ax = plt.subplots(1, 1)
cb = plt.colorbar(inversion_mesh.plot_image(m_true, ax=ax, grid=True)[0], ax=ax)
cb.set_label(r"$\log(\sigma)$")
ax.set_title("true model")
ax.set_xlim([-10, 10])
ax.set_ylim([-2, 0])
.. image-sg:: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_003.png
:alt: true model
:srcset: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_003.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-script-out
.. code-block:: none
(-2.0, 0.0)
.. GENERATED FROM PYTHON SOURCE LINES 171-175
Survey
------
Create our true model which we will use to generate synthetic data for
.. GENERATED FROM PYTHON SOURCE LINES 175-209
.. code-block:: Python
src_locations = np.arange(-11, 11, 0.5)
src_z = 0.25 # src is 0.25m above the surface
orientation = "z" # z-oriented dipole for horizontal co-planar loops
# reciever offset in 3D space
rx_offsets = np.vstack([np.r_[sep, 0.0, 0.0] for sep in coil_separations])
# create our source list - one source per location
source_list = []
for x in src_locations:
src_loc = np.r_[x, 0.0, src_z]
rx_locs = src_loc - rx_offsets
rx_real = FDEM.Rx.PointMagneticFluxDensitySecondary(
locations=rx_locs, orientation=orientation, component="real"
)
rx_imag = FDEM.Rx.PointMagneticFluxDensitySecondary(
locations=rx_locs, orientation=orientation, component="imag"
)
src = FDEM.Src.MagDipole(
receiver_list=[rx_real, rx_imag],
location=src_loc,
orientation=orientation,
frequency=freq,
)
source_list.append(src)
# create the survey and problem objects for running the forward simulation
survey = FDEM.Survey(source_list)
prob = FDEM.Simulation3DMagneticFluxDensity(mesh, survey=survey, sigmaMap=mapping)
.. GENERATED FROM PYTHON SOURCE LINES 210-215
Set up data for inversion
-------------------------
Generate clean, synthetic data. Later we will invert the clean data, and
assign a standard deviation of 0.05, and a floor of 1e-11.
.. GENERATED FROM PYTHON SOURCE LINES 215-263
.. code-block:: Python
t = time.time()
data = prob.make_synthetic_data(
m_true, relative_error=0.05, noise_floor=1e-11, add_noise=False
)
dclean = data.dclean
print("Done forward simulation. Elapsed time = {:1.2f} s".format(time.time() - t))
def plot_data(data, ax=None, color="C0", label=""):
if ax is None:
fig, ax = plt.subplots(1, 3, figsize=(15, 5))
# data is [re, im, re, im, ...]
data_real = data[0::2]
data_imag = data[1::2]
for i, offset in enumerate(coil_separations):
ax[i].plot(
src_locations,
data_real[i :: len(coil_separations)],
color=color,
label="{} real".format(label),
)
ax[i].plot(
src_locations,
data_imag[i :: len(coil_separations)],
"--",
color=color,
label="{} imag".format(label),
)
ax[i].set_title("offset = {:1.2f}m".format(offset))
ax[i].legend()
ax[i].grid(which="both")
ax[i].set_ylim(np.r_[data.min(), data.max()] + 1e-11 * np.r_[-1, 1])
ax[i].set_xlabel("source location x (m)")
ax[i].set_ylabel("Secondary B-Field (T)")
plt.tight_layout()
return ax
ax = plot_data(dclean)
.. image-sg:: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_004.png
:alt: offset = 0.32m, offset = 0.71m, offset = 1.18m
:srcset: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_004.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-script-out
.. code-block:: none
/home/vsts/work/1/s/simpeg/base/pde_simulation.py:490: DefaultSolverWarning:
Using the default solver: Pardiso.
If you would like to suppress this notification, add
warnings.filterwarnings('ignore', simpeg.utils.solver_utils.DefaultSolverWarning)
to your script.
Done forward simulation. Elapsed time = 22.24 s
.. GENERATED FROM PYTHON SOURCE LINES 264-279
Set up the inversion
--------------------
We create the data misfit, simple regularization
(a least-squares-style regularization, :class:`simpeg.regularization.LeastSquareRegularization`)
The smoothness and smallness contributions can be set by including
`alpha_s, alpha_x, alpha_y` as input arguments when the regularization is
created. The default reference model in the regularization is the starting
model. To set something different, you can input an `mref` into the
regularization.
We estimate the trade-off parameter, beta, between the data
misfit and regularization by the largest eigenvalue of the data misfit and
the regularization. Here, we use a fixed beta, but could alternatively
employ a beta-cooling schedule using :class:`simpeg.directives.BetaSchedule`
.. GENERATED FROM PYTHON SOURCE LINES 279-293
.. code-block:: Python
dmisfit = data_misfit.L2DataMisfit(simulation=prob, data=data)
reg = regularization.WeightedLeastSquares(inversion_mesh)
opt = optimization.InexactGaussNewton(maxIterCG=10, remember="xc")
invProb = inverse_problem.BaseInvProblem(dmisfit, reg, opt)
betaest = directives.BetaEstimate_ByEig(beta0_ratio=0.05, n_pw_iter=1, random_seed=1)
target = directives.TargetMisfit()
directiveList = [betaest, target]
inv = inversion.BaseInversion(invProb, directiveList=directiveList)
print("The target misfit is {:1.2f}".format(target.target))
.. rst-class:: sphx-glr-script-out
.. code-block:: none
The target misfit is 264.00
.. GENERATED FROM PYTHON SOURCE LINES 294-298
Run the inversion
------------------
We start from a half-space equal to the deep conductivity.
.. GENERATED FROM PYTHON SOURCE LINES 298-305
.. code-block:: Python
m0 = np.log(sigma_deep) * np.ones(inversion_mesh.nC)
t = time.time()
mrec = inv.run(m0)
print("\n Inversion Complete. Elapsed Time = {:1.2f} s".format(time.time() - t))
.. rst-class:: sphx-glr-script-out
.. code-block:: none
Running inversion with SimPEG v0.23.1.dev27+g1148ef1e9
simpeg.InvProblem will set Regularization.reference_model to m0.
simpeg.InvProblem will set Regularization.reference_model to m0.
simpeg.InvProblem will set Regularization.reference_model to m0.
simpeg.InvProblem will set Regularization.reference_model to m0.
simpeg.InvProblem will set Regularization.reference_model to m0.
simpeg.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
***Done using same Solver, and solver_opts as the Simulation3DMagneticFluxDensity problem***
model has any nan: 0
============================ Inexact Gauss Newton ============================
# beta phi_d phi_m f |proj(x-g)-x| LS Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
0 3.20e+00 1.42e+04 0.00e+00 1.42e+04 3.21e+03 0
1 3.20e+00 1.48e+03 5.42e+00 1.50e+03 4.61e+02 0
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.4191e+03
1 : |xc-x_last| = 6.2345e+00 <= tolX*(1+|x0|) = 1.3056e+01
0 : |proj(x-g)-x| = 4.6099e+02 <= tolG = 1.0000e-01
0 : |proj(x-g)-x| = 4.6099e+02 <= 1e3*eps = 1.0000e-02
0 : maxIter = 20 <= iter = 2
------------------------- DONE! -------------------------
Inversion Complete. Elapsed Time = 239.33 s
.. GENERATED FROM PYTHON SOURCE LINES 306-309
Plot the predicted and observed data
------------------------------------
.. GENERATED FROM PYTHON SOURCE LINES 309-314
.. code-block:: Python
fig, ax = plt.subplots(1, 3, figsize=(15, 5))
plot_data(dclean, ax=ax, color="C0", label="true")
plot_data(invProb.dpred, ax=ax, color="C1", label="predicted")
.. image-sg:: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_005.png
:alt: offset = 0.32m, offset = 0.71m, offset = 1.18m
:srcset: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_005.png
:class: sphx-glr-single-img
.. rst-class:: sphx-glr-script-out
.. code-block:: none
array([<Axes: title={'center': 'offset = 0.32m'}, xlabel='source location x (m)', ylabel='Secondary B-Field (T)'>,
<Axes: title={'center': 'offset = 0.71m'}, xlabel='source location x (m)', ylabel='Secondary B-Field (T)'>,
<Axes: title={'center': 'offset = 1.18m'}, xlabel='source location x (m)', ylabel='Secondary B-Field (T)'>],
dtype=object)
.. GENERATED FROM PYTHON SOURCE LINES 315-318
Plot the recovered model
------------------------
.. GENERATED FROM PYTHON SOURCE LINES 318-350
.. code-block:: Python
fig, ax = plt.subplots(1, 2, figsize=(12, 5))
# put both plots on the same colorbar
clim = np.r_[np.log(sigma_surface), np.log(sigma_deep)]
# recovered model
cb = plt.colorbar(
inversion_mesh.plot_image(mrec, ax=ax[0], clim=clim)[0],
ax=ax[0],
)
ax[0].set_title("recovered model")
cb.set_label(r"$\log(\sigma)$")
# true model
cb = plt.colorbar(
inversion_mesh.plot_image(m_true, ax=ax[1], clim=clim)[0],
ax=ax[1],
)
ax[1].set_title("true model")
cb.set_label(r"$\log(\sigma)$")
# # uncomment to plot the true interface
# x = np.linspace(-10, 10, 50)
# [a.plot(x, interface(x), 'k') for a in ax]
[a.set_xlim([-10, 10]) for a in ax]
[a.set_ylim([-2, 0]) for a in ax]
plt.tight_layout()
plt.show()
.. image-sg:: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_006.png
:alt: recovered model, true model
:srcset: /content/user-guide/examples/05-fdem/images/sphx_glr_plot_inv_fdem_loop_loop_2Dinversion_006.png
:class: sphx-glr-single-img
.. GENERATED FROM PYTHON SOURCE LINES 351-354
Print the version of SimPEG and dependencies
--------------------------------------------
.. GENERATED FROM PYTHON SOURCE LINES 354-357
.. code-block:: Python
Report()
.. raw:: html
<div class="output_subarea output_html rendered_html output_result">
<table style='border: 1.5px solid;'>
<tr>
<td style='text-align: center; font-weight: bold; font-size: 1.2em; border: 1px solid;' colspan='6'>Fri Mar 21 09:07:54 2025 UTC</td>
</tr>
<tr>
<td style='text-align: right; border: 1px solid;'>OS</td>
<td style='text-align: left; border: 1px solid;'>Linux (Ubuntu 22.04)</td>
<td style='text-align: right; border: 1px solid;'>CPU(s)</td>
<td style='text-align: left; border: 1px solid;'>2</td>
<td style='text-align: right; border: 1px solid;'>Machine</td>
<td style='text-align: left; border: 1px solid;'>x86_64</td>
</tr>
<tr>
<td style='text-align: right; border: 1px solid;'>Architecture</td>
<td style='text-align: left; border: 1px solid;'>64bit</td>
<td style='text-align: right; border: 1px solid;'>RAM</td>
<td style='text-align: left; border: 1px solid;'>6.8 GiB</td>
<td style='text-align: right; border: 1px solid;'>Environment</td>
<td style='text-align: left; border: 1px solid;'>Python</td>
</tr>
<tr>
<td style='text-align: right; border: 1px solid;'>File system</td>
<td style='text-align: left; border: 1px solid;'>ext4</td>
</tr>
<tr>
<td style='text-align: center; border: 1px solid;' colspan='6'>Python 3.10.16 | packaged by conda-forge | (main, Dec 5 2024, 14:16:10) [GCC 13.3.0]</td>
</tr>
<tr>
<td style='text-align: right; border: 1px solid;'>simpeg</td>
<td style='text-align: left; border: 1px solid;'>0.23.1.dev27+g1148ef1e9</td>
<td style='text-align: right; border: 1px solid;'>discretize</td>
<td style='text-align: left; border: 1px solid;'>0.11.2</td>
<td style='text-align: right; border: 1px solid;'>pymatsolver</td>
<td style='text-align: left; border: 1px solid;'>0.3.1</td>
</tr>
<tr>
<td style='text-align: right; border: 1px solid;'>numpy</td>
<td style='text-align: left; border: 1px solid;'>2.1.3</td>
<td style='text-align: right; border: 1px solid;'>scipy</td>
<td style='text-align: left; border: 1px solid;'>1.15.2</td>
<td style='text-align: right; border: 1px solid;'>matplotlib</td>
<td style='text-align: left; border: 1px solid;'>3.10.1</td>
</tr>
<tr>
<td style='text-align: right; border: 1px solid;'>geoana</td>
<td style='text-align: left; border: 1px solid;'>0.7.2</td>
<td style='text-align: right; border: 1px solid;'>libdlf</td>
<td style='text-align: left; border: 1px solid;'>0.3.0</td>
<td style='text-align: right; border: 1px solid;'>pydiso</td>
<td style='text-align: left; border: 1px solid;'>0.1.2</td>
</tr>
<tr>
<td style='text-align: right; border: 1px solid;'>numba</td>
<td style='text-align: left; border: 1px solid;'>0.61.0</td>
<td style='text-align: right; border: 1px solid;'>dask</td>
<td style='text-align: left; border: 1px solid;'>2025.2.0</td>
<td style='text-align: right; border: 1px solid;'>sklearn</td>
<td style='text-align: left; border: 1px solid;'>1.6.1</td>
</tr>
<tr>
<td style='text-align: right; border: 1px solid;'>pandas</td>
<td style='text-align: left; border: 1px solid;'>2.2.3</td>
<td style='text-align: right; border: 1px solid;'>sympy</td>
<td style='text-align: left; border: 1px solid;'>1.13.3</td>
<td style='text-align: right; border: 1px solid;'>plotly</td>
<td style='text-align: left; border: 1px solid;'>6.0.1</td>
</tr>
<tr>
<td style='text-align: right; border: 1px solid;'>memory_profiler</td>
<td style='text-align: left; border: 1px solid;'>0.61.0</td>
<td style='text-align: right; border: 1px solid;'>choclo</td>
<td style='text-align: left; border: 1px solid;'>0.3.2</td>
<td style= border: 1px solid;'></td>
<td style= border: 1px solid;'></td>
</tr>
</table>
</div>
<br />
<br />
.. GENERATED FROM PYTHON SOURCE LINES 358-369
Moving Forward
--------------
If you have suggestions for improving this example, please create a `pull request on the example in SimPEG <https://github.com/simpeg/simpeg/blob/main/examples/07-fdem/plot_loop_loop_2Dinversion.py>`_
You might try:
- improving the discretization
- changing beta
- changing the noise model
- playing with the regulariztion parameters
- ...
.. rst-class:: sphx-glr-timing
**Total running time of the script:** (4 minutes 30.693 seconds)
**Estimated memory usage:** 3161 MB
.. _sphx_glr_download_content_user-guide_examples_05-fdem_plot_inv_fdem_loop_loop_2Dinversion.py:
.. only:: html
.. container:: sphx-glr-footer sphx-glr-footer-example
.. container:: sphx-glr-download sphx-glr-download-jupyter
:download:`Download Jupyter notebook: plot_inv_fdem_loop_loop_2Dinversion.ipynb <plot_inv_fdem_loop_loop_2Dinversion.ipynb>`
.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: plot_inv_fdem_loop_loop_2Dinversion.py <plot_inv_fdem_loop_loop_2Dinversion.py>`
.. container:: sphx-glr-download sphx-glr-download-zip
:download:`Download zipped: plot_inv_fdem_loop_loop_2Dinversion.zip <plot_inv_fdem_loop_loop_2Dinversion.zip>`
.. only:: html
.. rst-class:: sphx-glr-signature
`Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_