Note

Go to the end to download the full example code

# 3D Forward Simulation on a Cylindrical Mesh#

Here we use the module *SimPEG.electromagnetics.frequency_domain* to simulate the
FDEM response for a borehole survey using a cylindrical mesh and radially symmetric
conductivity model. For this tutorial, we focus on the following:

How to define the transmitters and receivers

How to define the survey

How to solve the FDEM problem on cylindrical 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 CylindricalMesh
from discretize.utils import mkvc
from SimPEG import maps
import SimPEG.electromagnetics.frequency_domain as fdem
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
write_file = False
# sphinx_gallery_thumbnail_number = 2
```

## Create Airborne Survey#

Here we define a x-offset borehole survey that consists of a single vertical line of source-receiver pairs which measred the secondary magnetic flux density

over a range of frequencies.

```
# Frequencies being predicted (10 Hz to 10000 Hz)
frequencies = np.logspace(1, 4, 16)
# Defining transmitter locations
xtx, ytx, ztx = np.meshgrid([0], [0], np.linspace(0, -500, 21))
source_locations = np.c_[mkvc(xtx), mkvc(ytx), mkvc(ztx)]
ntx = np.size(xtx)
# Define receiver locations
xrx, yrx, zrx = np.meshgrid([100], [0], np.linspace(0, -500, 21))
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(ntx):
# Define receivers of different types at each location. Real and imaginary
# measurements require separate receivers. You can define the orientation of
# the transmitters and receivers for different survey geometries.
bzr_receiver = fdem.receivers.PointMagneticFluxDensitySecondary(
receiver_locations[ii, :], "z", "real"
)
bzi_receiver = fdem.receivers.PointMagneticFluxDensitySecondary(
receiver_locations[ii, :], "z", "imag"
)
receivers_list = [bzr_receiver, bzi_receiver] # must be a list
for jj in range(len(frequencies)):
# Must define the transmitter properties and associated receivers
source_list.append(
fdem.sources.MagDipole(
receivers_list, frequencies[jj], source_locations[ii], orientation="z"
)
)
# Define the survey
survey = fdem.Survey(source_list)
```

## Create Cylindrical Mesh#

Here we create the cylindrical mesh that will be used for this tutorial 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)

## Create 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. The model consists of several layers. For this example, we will have only flat topography.

```
# Conductivity in S/m (or resistivity in Ohm m)
air_conductivity = 1e-8
background_conductivity = 1e-1
layer_conductivity_1 = 1e0
layer_conductivity_2 = 1e-2
# Find cells that are active in the forward modeling (cells below surface)
ind_active = mesh.cell_centers[:, 2] < 0
# Define mapping from model to active cells
model_map = maps.InjectActiveCells(mesh, ind_active, air_conductivity)
# Define the model
model = background_conductivity * np.ones(ind_active.sum())
ind = (mesh.cell_centers[ind_active, 2] > -200.0) & (
mesh.cell_centers[ind_active, 2] < -0
)
model[ind] = layer_conductivity_1
ind = (mesh.cell_centers[ind_active, 2] > -400.0) & (
mesh.cell_centers[ind_active, 2] < -200
)
model[ind] = layer_conductivity_2
# Plot Conductivity Model
mpl.rcParams.update({"font.size": 14})
fig = plt.figure(figsize=(5, 6))
plotting_map = maps.InjectActiveCells(mesh, ind_active, np.nan)
log_model = np.log10(model)
ax1 = fig.add_axes([0.14, 0.1, 0.6, 0.85])
mesh.plot_image(
plotting_map * log_model,
ax=ax1,
grid=False,
clim=(np.log10(layer_conductivity_2), np.log10(layer_conductivity_1)),
)
ax1.set_title("Conductivity Model")
ax2 = fig.add_axes([0.76, 0.1, 0.05, 0.85])
norm = mpl.colors.Normalize(
vmin=np.log10(layer_conductivity_2), vmax=np.log10(layer_conductivity_1)
)
cbar = mpl.colorbar.ColorbarBase(
ax2, norm=norm, orientation="vertical", format="$10^{%.1f}$"
)
cbar.set_label("Conductivity [S/m]", rotation=270, labelpad=15, size=12)
```

## Simulation: Predicting FDEM Data#

Here we define the formulation for solving Maxwellâ€™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 the given model.
dpred = simulation.dpred(model)
# Data are organized by transmitter location, then component, then frequency. We had 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.
bz_real = np.reshape(bz_real, (ntx, len(frequencies)))
bz_imag = np.reshape(bz_imag, (ntx, len(frequencies)))
# Plot secondary field along the profile at f = 10000 Hz
fig = plt.figure(figsize=(5, 5))
ax1 = fig.add_axes([0.2, 0.15, 0.75, 0.75])
frequencies_index = 0
ax1.plot(bz_real[:, frequencies_index], receiver_locations[:, -1], "b-", lw=3)
ax1.plot(bz_imag[:, frequencies_index], receiver_locations[:, -1], "b--", lw=3)
ax1.set_xlabel("Bz secondary [T]")
ax1.set_ylabel("Elevation [m]")
ax1.set_title("Response at 10000 Hz")
ax1.legend(["Real", "Imaginary"], loc="upper right")
# Plot FEM response for all frequencies
fig = plt.figure(figsize=(5, 5))
ax1 = fig.add_axes([0.2, 0.15, 0.75, 0.75])
location_index = 0
ax1.semilogx(frequencies, bz_real[location_index, :], "b-", lw=3)
ax1.semilogx(frequencies, bz_imag[location_index, :], "b--", lw=3)
ax1.set_xlim((np.min(frequencies), np.max(frequencies)))
ax1.set_xlabel("Frequency [Hz]")
ax1.set_ylabel("Bz secondary [T]")
ax1.set_title("Response at Smallest Offset")
ax1.legend(["Real", "Imaginary"], loc="upper left")
```

```
<matplotlib.legend.Legend object at 0x7f2dec857310>
```

**Total running time of the script:** ( 0 minutes 10.174 seconds)

**Estimated memory usage:** 63 MB