3D Forward Simulation on a Tree Mesh#

Here we use the module SimPEG.electromagnetics.frequency_domain to simulate the FDEM response for an airborne survey using an OcTree mesh and a conductivity/resistivity model. To limit computational demant, we simulate airborne data at a single frequency for a vertical coplanar survey geometry. This tutorial can be easily adapted to simulate data at many frequencies. For this tutorial, we focus on the following:

  • How to define the transmitters and receivers

  • How to define the survey

  • How to define the topography

  • How to solve the FDEM problem on OcTree meshes

  • The units of the conductivity/resistivity model and resulting data

Please note that we have used a coarse mesh to shorten the time of the simulation. Proper discretization is required to simulate the fields at each frequency with sufficient accuracy.

Import modules#

from discretize import TreeMesh
from discretize.utils import mkvc, refine_tree_xyz

from SimPEG.utils import plot2Ddata, surface2ind_topo
from SimPEG import maps
import SimPEG.electromagnetics.frequency_domain as fdem

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

try:
    from pymatsolver import Pardiso as Solver
except ImportError:
    from SimPEG import SolverLU as Solver

save_file = False

# sphinx_gallery_thumbnail_number = 2

Defining Topography#

Here we define surface topography as an (N, 3) numpy array. Topography could also be loaded from a file. Here we define flat topography, however more complex topographies can be considered.

xx, yy = np.meshgrid(np.linspace(-3000, 3000, 101), np.linspace(-3000, 3000, 101))
zz = np.zeros(np.shape(xx))
topo_xyz = np.c_[mkvc(xx), mkvc(yy), mkvc(zz)]

Create Airborne Survey#

For this example, the survey consists of a uniform grid of airborne measurements. To save time, we will only compute the response for a single frequency.

# Frequencies being predicted
frequencies = [100, 500, 2500]

# Defining transmitter locations
N = 9
xtx, ytx, ztx = np.meshgrid(np.linspace(-200, 200, N), np.linspace(-200, 200, N), [40])
source_locations = np.c_[mkvc(xtx), mkvc(ytx), mkvc(ztx)]
ntx = np.size(xtx)

# Define receiver locations
xrx, yrx, zrx = np.meshgrid(np.linspace(-200, 200, N), np.linspace(-200, 200, N), [20])
receiver_locations = np.c_[mkvc(xrx), mkvc(yrx), mkvc(zrx)]

source_list = []  # Create empty list to store sources

# Each unique location and frequency defines a new transmitter
for ii in range(len(frequencies)):
    for jj in range(ntx):

        # Define receivers of different type at each location
        bzr_receiver = fdem.receivers.PointMagneticFluxDensitySecondary(
            receiver_locations[jj, :], "z", "real"
        )
        bzi_receiver = fdem.receivers.PointMagneticFluxDensitySecondary(
            receiver_locations[jj, :], "z", "imag"
        )
        receivers_list = [bzr_receiver, bzi_receiver]

        # Must define the transmitter properties and associated receivers
        source_list.append(
            fdem.sources.MagDipole(
                receivers_list,
                frequencies[ii],
                source_locations[jj],
                orientation="z",
                moment=100,
            )
        )

survey = fdem.Survey(source_list)

Create OcTree Mesh#

Here we define the OcTree mesh that is used for this example. We chose to design a coarser mesh to decrease the run time. When designing a mesh to solve practical frequency domain problems:

  • Your smallest cell size should be 10%-20% the size of your smallest skin depth

  • The thickness of your padding needs to be 2-3 times biggest than your largest skin depth

  • The skin depth is ~500*np.sqrt(rho/f)

dh = 25.0  # base cell width
dom_width = 3000.0  # domain width
nbc = 2 ** int(np.round(np.log(dom_width / dh) / np.log(2.0)))  # num. base cells

# Define the base mesh
h = [(dh, nbc)]
mesh = TreeMesh([h, h, h], x0="CCC")

# Mesh refinement based on topography
mesh = refine_tree_xyz(
    mesh, topo_xyz, octree_levels=[0, 0, 0, 1], method="surface", finalize=False
)

# Mesh refinement near transmitters and receivers
mesh = refine_tree_xyz(
    mesh, receiver_locations, octree_levels=[2, 4], method="radial", finalize=False
)

# Refine core mesh region
xp, yp, zp = np.meshgrid([-250.0, 250.0], [-250.0, 250.0], [-300.0, 0.0])
xyz = np.c_[mkvc(xp), mkvc(yp), mkvc(zp)]
mesh = refine_tree_xyz(mesh, xyz, octree_levels=[0, 2, 4], method="box", finalize=False)

mesh.finalize()

Defining the Conductivity/Resistivity Model and Mapping#

Here, we create the model that will be used to predict frequency domain data and the mapping from the model to the mesh. Here, the model consists of a conductive block within a more resistive background.

# Conductivity in S/m (or resistivity in Ohm m)
air_conductivity = 1e-8
background_conductivity = 1e-2
block_conductivity = 1e1

# Find cells that are active in the forward modeling (cells below surface)
ind_active = surface2ind_topo(mesh, topo_xyz)

# Define mapping from model to active cells
model_map = maps.InjectActiveCells(mesh, ind_active, air_conductivity)

# Define model. Models in SimPEG are vector arrays
model = background_conductivity * np.ones(ind_active.sum())
ind_block = (
    (mesh.gridCC[ind_active, 0] < 100.0)
    & (mesh.gridCC[ind_active, 0] > -100.0)
    & (mesh.gridCC[ind_active, 1] < 100.0)
    & (mesh.gridCC[ind_active, 1] > -100.0)
    & (mesh.gridCC[ind_active, 2] > -275.0)
    & (mesh.gridCC[ind_active, 2] < -75.0)
)
model[ind_block] = block_conductivity

# Plot Resistivity Model
mpl.rcParams.update({"font.size": 12})
fig = plt.figure(figsize=(7, 6))

plotting_map = maps.InjectActiveCells(mesh, ind_active, np.nan)
log_model = np.log10(model)

ax1 = fig.add_axes([0.13, 0.1, 0.6, 0.85])
mesh.plot_slice(
    plotting_map * log_model,
    normal="Y",
    ax=ax1,
    ind=int(mesh.h[0].size / 2),
    grid=False,
    clim=(np.log10(background_conductivity), np.log10(block_conductivity)),
)
ax1.set_title("Conductivity Model at Y = 0 m")

ax2 = fig.add_axes([0.75, 0.1, 0.05, 0.85])
norm = mpl.colors.Normalize(
    vmin=np.log10(background_conductivity), vmax=np.log10(block_conductivity)
)
cbar = mpl.colorbar.ColorbarBase(
    ax2, norm=norm, orientation="vertical", format="$10^{%.1f}$"
)
cbar.set_label("Conductivity [S/m]", rotation=270, labelpad=15, size=12)
Conductivity Model at Y = 0 m

Simulation: Predicting FDEM Data#

Here we define the formulation for solving Maxwell’s equations. Since we are measuring the magnetic flux density and working with a conductivity model, the EB formulation is the most natural. We must also remember to define the mapping for the conductivity model. If you defined a resistivity model, use the kwarg rhoMap instead of sigmaMap

simulation = fdem.simulation.Simulation3DMagneticFluxDensity(
    mesh, survey=survey, sigmaMap=model_map, solver=Solver
)

Predict and Plot Data#

Here we show how the simulation is used to predict data.

# Compute predicted data for a your model.
dpred = simulation.dpred(model)

# Data are organized by frequency, transmitter location, then by receiver. We nFreq transmitters
# and each transmitter had 2 receivers (real and imaginary component). So
# first we will pick out the real and imaginary data
bz_real = dpred[0 : len(dpred) : 2]
bz_imag = dpred[1 : len(dpred) : 2]

# Then we will will reshape the data for plotting.
bz_real_plotting = np.reshape(bz_real, (len(frequencies), ntx))
bz_imag_plotting = np.reshape(bz_imag, (len(frequencies), ntx))

fig = plt.figure(figsize=(10, 4))

# Real Component
frequencies_index = 0
v_max = np.max(np.abs(bz_real_plotting[frequencies_index, :]))
ax1 = fig.add_axes([0.05, 0.05, 0.35, 0.9])
plot2Ddata(
    receiver_locations[:, 0:2],
    bz_real_plotting[frequencies_index, :],
    ax=ax1,
    ncontour=30,
    clim=(-v_max, v_max),
    contourOpts={"cmap": "bwr"},
)
ax1.set_title("Re[$B_z$] at 100 Hz")

ax2 = fig.add_axes([0.41, 0.05, 0.02, 0.9])
norm = mpl.colors.Normalize(vmin=-v_max, vmax=v_max)
cbar = mpl.colorbar.ColorbarBase(
    ax2, norm=norm, orientation="vertical", cmap=mpl.cm.bwr
)
cbar.set_label("$T$", rotation=270, labelpad=15, size=12)

# Imaginary Component
v_max = np.max(np.abs(bz_imag_plotting[frequencies_index, :]))
ax1 = fig.add_axes([0.55, 0.05, 0.35, 0.9])
plot2Ddata(
    receiver_locations[:, 0:2],
    bz_imag_plotting[frequencies_index, :],
    ax=ax1,
    ncontour=30,
    clim=(-v_max, v_max),
    contourOpts={"cmap": "bwr"},
)
ax1.set_title("Im[$B_z$] at 100 Hz")

ax2 = fig.add_axes([0.91, 0.05, 0.02, 0.9])
norm = mpl.colors.Normalize(vmin=-v_max, vmax=v_max)
cbar = mpl.colorbar.ColorbarBase(
    ax2, norm=norm, orientation="vertical", cmap=mpl.cm.bwr
)
cbar.set_label("$T$", rotation=270, labelpad=15, size=12)

plt.show()
Re[$B_z$] at 100 Hz, Im[$B_z$] at 100 Hz

Optional: Export Data#

Write the true model, data and topography

if save_file:

    dir_path = os.path.dirname(fdem.__file__).split(os.path.sep)[:-3]
    dir_path.extend(["tutorials", "assets", "fdem"])
    dir_path = os.path.sep.join(dir_path) + os.path.sep

    # Write topography
    fname = dir_path + "fdem_topo.txt"
    np.savetxt(fname, np.c_[topo_xyz], fmt="%.4e")

    # Write data with 2% noise added
    fname = dir_path + "fdem_data.obs"
    bz_real = bz_real + 1e-14 * np.random.rand(len(bz_real))
    bz_imag = bz_imag + 1e-14 * np.random.rand(len(bz_imag))
    f_vec = np.kron(frequencies, np.ones(ntx))
    receiver_locations = np.kron(np.ones((len(frequencies), 1)), receiver_locations)

    np.savetxt(fname, np.c_[f_vec, receiver_locations, bz_real, bz_imag], fmt="%.4e")

    # Plot true model
    output_model = plotting_map * model
    output_model[np.isnan(output_model)] = 1e-8

    fname = dir_path + "true_model.txt"
    np.savetxt(fname, output_model, fmt="%.4e")

Total running time of the script: ( 0 minutes 13.211 seconds)

Estimated memory usage: 334 MB

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