1D Inversion of for a Single Sounding#

Here we use the module simpeg.electromangetics.frequency_domain_1d to invert frequency 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 frequency domain data for a single sounding. The end product is layered Earth model which explains the data. The survey consisted of a vertical magnetic dipole source located 30 m above the surface. The receiver measured the vertical component of the secondary field at a 10 m offset from the source in ppm.

Import modules#

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

from discretize import TensorMesh

import simpeg.electromagnetics.frequency_domain as fdem
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/em1dfm.tar.gz”

# storage bucket where we have the data
data_source = "https://storage.googleapis.com/simpeg/doc-assets/em1dfm.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 + "em1dfm_data.txt"

Downloading https://storage.googleapis.com/simpeg/doc-assets/em1dfm.tar.gz
   saved to: /home/vsts/work/1/s/tutorials/07-fdem/em1dfm.tar.gz
Download completed!

Load Data and Plot#

Here we load and plot the 1D sounding data. In this case, we have the secondary field response in ppm for a set of frequencies.

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

# Define receiver locations and observed data
frequencies = dobs[:, 0]
dobs = mkvc(dobs[:, 1:].T)

fig, ax = plt.subplots(1, 1, figsize=(7, 7))
ax.loglog(frequencies, np.abs(dobs[0::2]), "k-o", lw=3)
ax.loglog(frequencies, np.abs(dobs[1::2]), "k:o", lw=3)
ax.set_xlabel("Frequency (Hz)")
ax.set_ylabel("|Hs/Hp| (ppm)")
ax.set_title("Magnetic Field as a Function of Frequency")
ax.legend(["Real", "Imaginary"])

Magnetic Field as a Function of Frequency

<matplotlib.legend.Legend object at 0x7fe5ac4096f0>

Defining the Survey#

Here we demonstrate a general way to define the receivers, sources and survey. The survey consisted of a vertical magnetic dipole source located 30 m above the surface. The receiver measured the vertical component of the secondary field at a 10 m offset from the source in ppm.

source_location = np.array([0.0, 0.0, 30.0])
moment = 1.0

receiver_location = np.array([10.0, 0.0, 30.0])
receiver_orientation = "z"
data_type = "ppm"

# Receiver list
receiver_list = []
receiver_list.append(
    fdem.receivers.PointMagneticFieldSecondary(
        receiver_location,
        orientation=receiver_orientation,
        data_type=data_type,
        component="real",
    )
)
receiver_list.append(
    fdem.receivers.PointMagneticFieldSecondary(
        receiver_location,
        orientation=receiver_orientation,
        data_type=data_type,
        component="imag",
    )
)

# Define source list
source_list = []
for freq in frequencies:
    source_list.append(
        fdem.sources.MagDipole(
            receiver_list=receiver_list,
            frequency=freq,
            location=source_location,
            orientation="z",
            moment=moment,
        )
    )

# Survey
survey = fdem.survey.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, noise_floor=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/or Reference Model and the Mapping#

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 = fdem.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)

# Define the regularization (model objective function)
reg_map = maps.IdentityMap(nP=mesh.nC)
reg = regularization.Sparse(mesh, mapping=reg_map, alpha_s=0.025, alpha_x=1.0)

# reference model
reg.reference_model = starting_model

# Define sparse and blocky norms p, q
reg.norms = [0, 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=50, 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)

# Directive 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.dev1+g60e0c3a5d
/home/vsts/work/1/s/simpeg/simulation.py:197: 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.


                        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  2.44e+02  2.81e+02  0.00e+00  2.81e+02    7.91e+01      0
   1  1.22e+02  4.29e+01  2.20e-01  6.98e+01    3.68e+01      0
Reached starting chifact with l2-norm regularization: Start IRLS steps...
irls_threshold 2.7313354083832295
   2  1.22e+02  7.72e+00  2.27e-01  3.55e+01    7.92e+00      0
   3  1.64e+02  7.39e+00  2.16e-01  4.29e+01    1.23e+01      0   Skip BFGS
   4  2.21e+02  1.09e+01  1.89e-01  5.25e+01    1.92e+01      0
   5  1.22e+02  1.62e+01  1.67e-01  3.65e+01    5.69e+00      0
   6  7.20e+01  1.39e+01  1.83e-01  2.70e+01    5.15e+00      0
   7  4.57e+01  1.15e+01  1.79e-01  1.97e+01    4.67e+00      0
   8  2.90e+01  9.05e+00  1.60e-01  1.37e+01    4.68e+00      0
   9  4.15e+01  6.47e+00  1.51e-01  1.27e+01    7.70e+00      0
  10  6.15e+01  5.98e+00  1.18e-01  1.33e+01    1.39e+01      0
  11  8.85e+01  6.39e+00  8.52e-02  1.39e+01    2.13e+01      0
  12  1.22e+02  7.02e+00  5.77e-02  1.41e+01    2.85e+01      0
  13  1.63e+02  7.50e+00  3.71e-02  1.35e+01    3.26e+01      0
  14  2.14e+02  7.72e+00  2.41e-02  1.29e+01    3.02e+01      0
  15  2.80e+02  7.77e+00  1.60e-02  1.23e+01    2.60e+01      0   Skip BFGS
  16  3.67e+02  7.75e+00  1.08e-02  1.17e+01    2.39e+01      0   Skip BFGS
  17  4.83e+02  7.71e+00  7.33e-03  1.12e+01    2.33e+01      0   Skip BFGS
  18  6.37e+02  7.66e+00  4.96e-03  1.08e+01    2.30e+01      0   Skip BFGS
  19  8.43e+02  7.62e+00  3.35e-03  1.04e+01    2.29e+01      0   Skip BFGS
  20  1.12e+03  7.58e+00  2.26e-03  1.01e+01    2.29e+01      0   Skip BFGS
  21  1.49e+03  7.54e+00  1.52e-03  9.79e+00    2.28e+01      0   Skip BFGS
  22  1.98e+03  7.51e+00  1.01e-03  9.52e+00    2.28e+01      0
  23  2.64e+03  7.47e+00  6.76e-04  9.26e+00    2.28e+01      0
  24  3.54e+03  7.44e+00  4.49e-04  9.03e+00    2.27e+01      0
  25  4.75e+03  7.40e+00  2.97e-04  8.81e+00    2.27e+01      0
  26  6.40e+03  7.35e+00  1.95e-04  8.60e+00    2.26e+01      0
  27  8.64e+03  7.31e+00  1.28e-04  8.42e+00    2.24e+01      0   Skip BFGS
  28  1.17e+04  7.27e+00  8.46e-05  8.26e+00    2.23e+01      0   Skip BFGS
  29  1.59e+04  7.23e+00  5.58e-05  8.12e+00    2.22e+01      0   Skip BFGS
  30  2.16e+04  7.20e+00  3.70e-05  8.00e+00    2.21e+01      0   Skip BFGS
  31  2.94e+04  7.18e+00  2.45e-05  7.90e+00    2.20e+01      0   Skip BFGS
Reach maximum number of IRLS cycles: 30
------------------------- STOP! -------------------------
1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 2.8196e+01
1 : |xc-x_last| = 5.6033e-02 <= tolX*(1+|x0|) = 1.2741e+00
0 : |proj(x-g)-x|    = 2.2006e+01 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 2.2006e+01 <= 1e3*eps       = 1.0000e-02
0 : maxIter   =      50    <= iter          =     32
------------------------- DONE! -------------------------

Plotting Results#

# Load the true model and layer thicknesses
true_model = np.array([0.1, 1.0, 0.1])
true_layers = np.r_[20.0, 40.0, 160.0]

# Extract Least-Squares model
l2_model = inv_prob.l2model

# 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.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=(11, 6))
ax1 = fig.add_axes([0.2, 0.1, 0.6, 0.8])
ax1.loglog(frequencies, np.abs(dobs[0::2]), "k-o")
ax1.loglog(frequencies, np.abs(dobs[1::2]), "k:o")
ax1.loglog(frequencies, np.abs(dpred_l2[0::2]), "b-o")
ax1.loglog(frequencies, np.abs(dpred_l2[1::2]), "b:o")
ax1.loglog(frequencies, np.abs(dpred_final[0::2]), "r-o")
ax1.loglog(frequencies, np.abs(dpred_final[1::2]), "r:o")
ax1.set_xlabel("Frequencies (Hz)")
ax1.set_ylabel("|Hs/Hp| (ppm)")
ax1.set_title("Predicted and Observed Data")
ax1.legend(
    [
        "Observed (real)",
        "Observed (imag)",
        "L2-Model (real)",
        "L2-Model (imag)",
        "Sparse (real)",
        "Sparse (imag)",
    ],
    loc="upper left",
)
plt.show()