1D Inversion of Time-Domain Data for a Single Sounding#

Here we use the module simpeg.electromangetics.time_domain_1d to invert time domain data and recover a 1D electrical conductivity model. In this tutorial, we focus on the following:

How to define sources and receivers from a survey file

How to define the survey

Sparse 1D inversion of with iteratively re-weighted least-squares

For this tutorial, we will invert 1D time domain data for a single sounding. The end product is layered Earth model which explains the data. The survey consisted of a horizontal loop with a radius of 6 m, located 20 m above the surface. The receiver measured the vertical component of the magnetic flux at the loop’s centre.

Import modules#

import os
import tarfile
import numpy as np
import matplotlib.pyplot as plt

from discretize import TensorMesh

import simpeg.electromagnetics.time_domain as tdem

from simpeg.utils import mkvc, plot_1d_layer_model
from simpeg import (
    maps,
    data,
    data_misfit,
    inverse_problem,
    regularization,
    optimization,
    directives,
    inversion,
    utils,
)

plt.rcParams.update({"font.size": 16, "lines.linewidth": 2, "lines.markersize": 8})

# sphinx_gallery_thumbnail_number = 2

Download Test Data File#

Here we provide the file path to the data we plan on inverting. The path to the data file is stored as a tar-file on our google cloud bucket: “https://storage.googleapis.com/simpeg/doc-assets/em1dtm.tar.gz”

# storage bucket where we have the data
data_source = "https://storage.googleapis.com/simpeg/doc-assets/em1dtm.tar.gz"

# download the data
downloaded_data = utils.download(data_source, overwrite=True)

# unzip the tarfile
tar = tarfile.open(downloaded_data, "r")
tar.extractall()
tar.close()

# path to the directory containing our data
dir_path = downloaded_data.split(".")[0] + os.path.sep

# files to work with
data_filename = dir_path + "em1dtm_data.txt"

Downloading https://storage.googleapis.com/simpeg/doc-assets/em1dtm.tar.gz
   saved to: /home/vsts/work/1/s/tutorials/08-tdem/em1dtm.tar.gz
Download completed!

Load Data and Plot#

Here we load and plot the 1D sounding data. In this case, we have the B-field response to a step-off waveform.

# Load field data
dobs = np.loadtxt(str(data_filename), skiprows=1)

times = dobs[:, 0]
dobs = mkvc(dobs[:, -1])

fig = plt.figure(figsize=(7, 7))
ax = fig.add_axes([0.15, 0.15, 0.8, 0.75])
ax.loglog(times, np.abs(dobs), "k-o", lw=3)
ax.set_xlabel("Times (s)")
ax.set_ylabel("|B| (T)")
ax.set_title("Observed Data")

Text(0.5, 1.0, 'Observed Data')

Defining the Survey#

Here we demonstrate a general way to define the receivers, sources, waveforms and survey. For this tutorial, we define a single horizontal loop source as well a receiver which measures the vertical component of the magnetic flux.

# Source loop geometry
source_location = np.array([0.0, 0.0, 20.0])
source_orientation = "z"  # "x", "y" or "z"
source_current = 1.0  # peak current amplitude
source_radius = 6.0  # loop radius

# Receiver geometry
receiver_location = np.array([0.0, 0.0, 20.0])
receiver_orientation = "z"  # "x", "y" or "z"

# Receiver list
receiver_list = []
receiver_list.append(
    tdem.receivers.PointMagneticFluxDensity(
        receiver_location, times, orientation=receiver_orientation
    )
)

# Define the source waveform.
waveform = tdem.sources.StepOffWaveform()

# Sources
source_list = [
    tdem.sources.CircularLoop(
        receiver_list=receiver_list,
        location=source_location,
        waveform=waveform,
        current=source_current,
        radius=source_radius,
    )
]

# Survey
survey = tdem.Survey(source_list)

Assign Uncertainties and Define the Data Object#

Here is where we define the data that are inverted. The data are defined by the survey, the observation values and the uncertainties.

# 5% of the absolute value
uncertainties = 0.05 * np.abs(dobs) * np.ones(np.shape(dobs))

# Define the data object
data_object = data.Data(survey, dobs=dobs, standard_deviation=uncertainties)

Defining a 1D Layered Earth (1D Tensor Mesh)#

Here, we define the layer thicknesses for our 1D simulation. To do this, we use the TensorMesh class.

# Layer thicknesses
inv_thicknesses = np.logspace(0, 1.5, 25)

# Define a mesh for plotting and regularization.
mesh = TensorMesh([(np.r_[inv_thicknesses, inv_thicknesses[-1]])], "0")

Define a Starting and Reference Model#

Here, we create starting and/or reference models for the inversion as well as the mapping from the model space to the active cells. Starting and reference models can be a constant background value or contain a-priori structures. Here, the starting model is log(0.1) S/m.

Define log-conductivity values for each layer since our model is the log-conductivity. Don’t make the values 0! Otherwise the gradient for the 1st iteration is zero and the inversion will not converge.

# Define model. A resistivity (Ohm meters) or conductivity (S/m) for each layer.
starting_model = np.log(0.1 * np.ones(mesh.nC))

# Define mapping from model to active cells.
model_mapping = maps.ExpMap()

Define the Physics using a Simulation Object#

simulation = tdem.Simulation1DLayered(
    survey=survey, thicknesses=inv_thicknesses, sigmaMap=model_mapping
)

Define Inverse Problem#

The inverse problem is defined by 3 things:

Data Misfit: a measure of how well our recovered model explains the field data

Regularization: constraints placed on the recovered model and a priori information

Optimization: the numerical approach used to solve the inverse problem

# Define the data misfit. Here the data misfit is the L2 norm of the weighted
# residual between the observed data and the data predicted for a given model.
# The weighting is defined by the reciprocal of the uncertainties.
dmis = data_misfit.L2DataMisfit(simulation=simulation, data=data_object)
dmis.W = 1.0 / uncertainties

# Define the regularization (model objective function)
reg_map = maps.IdentityMap(nP=mesh.nC)
reg = regularization.Sparse(mesh, mapping=reg_map, alpha_s=0.01, alpha_x=1.0)

# set reference model
reg.reference_model = starting_model

# Define sparse and blocky norms p, q
reg.norms = [1, 0]

# Define how the optimization problem is solved. Here we will use an inexact
# Gauss-Newton approach that employs the conjugate gradient solver.
opt = optimization.ProjectedGNCG(maxIter=100, maxIterLS=20, maxIterCG=30, tolCG=1e-3)

# Define the inverse problem
inv_prob = inverse_problem.BaseInvProblem(dmis, reg, opt)

Define Inversion Directives#

Here we define any directiveas that are carried out during the inversion. This includes the cooling schedule for the trade-off parameter (beta), stopping criteria for the inversion and saving inversion results at each iteration.

# Defining a starting value for the trade-off parameter (beta) between the data
# misfit and the regularization.
starting_beta = directives.BetaEstimate_ByEig(beta0_ratio=1e2)

# Update the preconditionner
update_Jacobi = directives.UpdatePreconditioner()

# Options for outputting recovered models and predicted data for each beta.
save_iteration = directives.SaveOutputEveryIteration(save_txt=False)

# Directives for the IRLS
update_IRLS = directives.UpdateIRLS(max_irls_iterations=30, irls_cooling_factor=1.5)

# Updating the preconditionner if it is model dependent.
update_jacobi = directives.UpdatePreconditioner()

# Add sensitivity weights
sensitivity_weights = directives.UpdateSensitivityWeights()

# The directives are defined as a list.
directives_list = [
    sensitivity_weights,
    starting_beta,
    save_iteration,
    update_IRLS,
    update_jacobi,
]

Running the Inversion#

To define the inversion object, we need to define the inversion problem and the set of directives. We can then run the inversion.

# Here we combine the inverse problem and the set of directives
inv = inversion.BaseInversion(inv_prob, directives_list)

# Run the inversion
recovered_model = inv.run(starting_model)

Running inversion with SimPEG v0.23.1.dev10+gf697d2455

                    simpeg.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
                    ***Done using the default solver Pardiso and no solver_opts.***

model has any nan: 0
=============================== Projected GNCG ===============================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
   0  1.13e+04  4.01e+03  0.00e+00  4.01e+03    8.04e+02      0
   1  5.63e+03  7.79e+02  7.18e-02  1.18e+03    4.57e+02      0
   2  2.82e+03  3.04e+02  1.12e-01  6.21e+02    7.48e+02      0
   3  1.41e+03  1.03e+02  1.14e-01  2.65e+02    4.97e+02      0
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 2.7280329011114577
   4  1.41e+03  3.09e+01  1.18e-01  1.97e+02    3.12e+02      0   Skip BFGS
   5  8.79e+02  3.73e+01  1.22e-01  1.45e+02    4.64e+02      0   Skip BFGS
   6  1.33e+03  1.78e+01  1.22e-01  1.79e+02    2.97e+02      0   Skip BFGS
   7  2.14e+03  1.52e+01  1.35e-01  3.05e+02    2.19e+02      0   Skip BFGS
   8  3.07e+03  2.01e+01  1.51e-01  4.84e+02    1.72e+02      0   Skip BFGS
   9  4.39e+03  2.87e+01  1.65e-01  7.53e+02    1.81e+02      0   Skip BFGS
  10  2.62e+03  4.16e+01  1.68e-01  4.83e+02    1.31e+02      0
  11  1.57e+03  3.39e+01  1.65e-01  2.94e+02    9.92e+01      0
  12  9.40e+02  2.95e+01  1.56e-01  1.76e+02    3.50e+01      0
  13  5.62e+02  2.85e+01  1.44e-01  1.09e+02    6.63e+01      0
  14  3.36e+02  2.90e+01  1.32e-01  7.35e+01    1.19e+02      0
  15  2.01e+02  2.83e+01  1.23e-01  5.31e+01    9.65e+01      0   Skip BFGS
  16  2.51e+02  2.65e+01  1.17e-01  5.58e+01    5.20e+01      0   Skip BFGS
  17  3.07e+02  2.74e+01  1.05e-01  5.96e+01    3.73e+01      0   Skip BFGS
  18  3.75e+02  2.93e+01  9.29e-02  6.42e+01    4.07e+01      0
  19  4.59e+02  3.17e+01  8.28e-02  6.96e+01    5.55e+01      0
  20  2.96e+02  3.41e+01  7.49e-02  5.63e+01    1.40e+01      0
Minimum decrease in regularization. End of IRLS
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 4.0084e+02
1 : |xc-x_last| = 6.9516e-03 <= tolX*(1+|x0|) = 1.2741e+00
0 : |proj(x-g)-x|    = 1.3951e+01 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 1.3951e+01 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =     100    <= iter          =     21
------------------------- DONE! -------------------------

Plotting Results#

# Load the true model and layer thicknesses
true_model = np.array([0.1, 1.0, 0.1])
true_layers = np.r_[40.0, 40.0, 160.0]

# Extract Least-Squares model
l2_model = inv_prob.l2model
print(np.shape(l2_model))

# Plot true model and recovered model
fig = plt.figure(figsize=(8, 9))
x_min = np.min(
    np.r_[model_mapping * recovered_model, model_mapping * l2_model, true_model]
)
x_max = np.max(
    np.r_[model_mapping * recovered_model, model_mapping * l2_model, true_model]
)

ax1 = fig.add_axes([0.2, 0.15, 0.7, 0.7])
plot_1d_layer_model(true_layers, true_model, ax=ax1, show_layers=False, color="k")
plot_1d_layer_model(
    mesh.h[0], model_mapping * l2_model, ax=ax1, show_layers=False, color="b"
)
plot_1d_layer_model(
    mesh.h[0], model_mapping * recovered_model, ax=ax1, show_layers=False, color="r"
)
ax1.set_xlim(0.01, 10)
ax1.grid()
ax1.set_title("True and Recovered Models")
ax1.legend(["True Model", "L2-Model", "Sparse Model"])
plt.gca().invert_yaxis()

# Plot predicted and observed data
dpred_l2 = simulation.dpred(l2_model)
dpred_final = simulation.dpred(recovered_model)

fig = plt.figure(figsize=(7, 7))
ax1 = fig.add_axes([0.15, 0.15, 0.8, 0.75])
ax1.loglog(times, np.abs(dobs), "k-o")
ax1.loglog(times, np.abs(dpred_l2), "b-o")
ax1.loglog(times, np.abs(dpred_final), "r-o")
ax1.grid()
ax1.set_xlabel("times (Hz)")
ax1.set_ylabel("|Hs/Hp| (ppm)")
ax1.set_title("Predicted and Observed Data")
ax1.legend(["Observed", "L2-Model", "Sparse"], loc="upper right")
plt.show()