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:

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

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

3. 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=1e1)

# 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.Update_IRLS(
    max_irls_iterations=30, minGNiter=1, coolEpsFact=1.5, update_beta=True
)

# 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 v0.22.0

                        simpeg.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
                        ***Done using same Solver, and solver_opts as the Simulation1DLayered problem***

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  1.14e+03  4.01e+03  0.00e+00  4.01e+03    8.04e+02      0
   1  5.69e+02  1.11e+03  1.28e-01  1.18e+03    9.91e+02      1
   2  2.84e+02  6.33e+02  1.33e-01  6.71e+02    1.15e+03      1
   3  1.42e+02  4.04e+02  1.45e-01  4.25e+02    1.66e+03      0   Skip BFGS
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 2.654608696171035
   4  7.11e+01  1.34e+01  1.22e-01  2.21e+01    2.03e+02      0   Skip BFGS
   5  3.96e+02  3.39e+00  1.24e-01  5.24e+01    8.55e+01      0   Skip BFGS
   6  1.68e+03  4.80e+00  1.29e-01  2.21e+02    1.74e+02      0   Skip BFGS
   7  4.48e+03  9.25e+00  1.32e-01  6.01e+02    3.26e+02      0
   8  7.19e+03  2.57e+01  1.34e-01  9.92e+02    3.67e+02      0
   9  4.56e+03  5.99e+01  1.46e-01  7.24e+02    6.87e+01      0
  10  2.76e+03  6.71e+01  1.56e-01  4.98e+02    8.41e+01      0   Skip BFGS
  11  1.71e+03  6.37e+01  1.55e-01  3.29e+02    8.34e+01      0   Skip BFGS
  12  1.19e+03  4.80e+01  1.44e-01  2.20e+02    5.57e+01      0
  13  8.94e+02  4.13e+01  1.40e-01  1.67e+02    3.05e+02      0
  14  6.74e+02  4.09e+01  1.30e-01  1.29e+02    3.93e+02      0
  15  5.17e+02  3.95e+01  1.17e-01  1.00e+02    3.84e+02      0   Skip BFGS
  16  4.15e+02  3.62e+01  1.04e-01  7.94e+01    2.64e+02      0   Skip BFGS
  17  4.15e+02  3.38e+01  9.24e-02  7.22e+01    8.15e+01      0   Skip BFGS
  18  3.43e+02  3.44e+01  8.30e-02  6.29e+01    1.29e+01      0   Skip BFGS
  19  2.83e+02  3.44e+01  7.78e-02  5.64e+01    1.67e+01      0   Skip BFGS
  20  2.83e+02  3.38e+01  7.56e-02  5.52e+01    1.25e+01      0
  21  2.83e+02  3.34e+01  7.41e-02  5.43e+01    1.20e+01      0   Skip BFGS
  22  2.83e+02  3.32e+01  7.30e-02  5.38e+01    1.24e+01      0
  23  2.83e+02  3.30e+01  7.22e-02  5.34e+01    1.20e+01      0   Skip BFGS
  24  2.83e+02  3.29e+01  7.17e-02  5.31e+01    1.10e+01      0   Skip BFGS
Minimum decrease in regularization.End of IRLS
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 4.0084e+02
1 : |xc-x_last| = 1.9934e-03 <= tolX*(1+|x0|) = 1.2741e+00
0 : |proj(x-g)-x|    = 1.0990e+01 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 1.0990e+01 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =     100    <= iter          =     25
------------------------- 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()

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Estimated memory usage: 9 MB