Note
Go to the end to download the full example code.
9. Magnetic permeability#
The solver emg3d uses the diffusive approximation of Maxwell’s equations;
the relative electric permittivity is therefore fixed at
\(\varepsilon_\rm{r} = 1\). The magnetic permeability \(\mu_\rm{r}\),
however, is implemented in emg3d, albeit only isotropically.
In this example we run the same model as in the example mentioned above: A
rotated finite length bipole in a homogeneous VTI fullspace, but here with
isotropic magnetic permeability, and compare it to the semi-analytical solution
of empymod. (The code empymod is an open-source code which can model
CSEM responses for a layered medium including VTI electrical anisotropy, see
emsig.xyz.)
import emg3d
import empymod
import numpy as np
import matplotlib.pyplot as plt
Full-space model for a finite length, finite strength, rotated bipole#
In order to shorten the build-time of the gallery we use a coarse model.
Set coarse_model = False to obtain a result of higher accuracy.
coarse_model = True
Survey and model parameters#
# Receiver coordinates
if coarse_model:
x = (np.arange(256))*20-2550
else:
x = (np.arange(1025))*5-2560
rx = np.repeat([x, ], np.size(x), axis=0)
ry = rx.transpose()
frx, fry = rx.ravel(), ry.ravel()
rz = -400.0
azimuth = 22
elevation = 13
# Source coordinates, frequency, and strength
source = emg3d.TxElectricDipole(
coordinates=[-50, 50, -30, 30, -320., -280.], # [x1, x2, y1, y2, z1, z2]
strength=2.8, # A
)
frequency = 0.7 # Hz
# Model parameters
h_res = 1. # Horizontal resistivity
aniso = np.sqrt(2.) # Anisotropy
v_res = h_res*aniso**2 # Vertical resistivity
mperm = 2.5 # Magnetic permeability
empymod#
Note: The coordinate system of empymod is positive z down, for emg3d it is positive z up. We have to switch therefore src_z, rec_z, and elevation.
# Compute
epm = empymod.bipole(
src=np.r_[source.coordinates[:4], -source.coordinates[4:]],
rec=[frx, fry, -rz, azimuth, -elevation],
depth=[],
res=h_res,
aniso=aniso,
strength=source.strength,
srcpts=5,
freqtime=frequency,
mpermH=mperm,
htarg={'pts_per_dec': -1},
verb=3,
).reshape(np.shape(rx))
:: empymod START :: v2.3.1
depth [m] :
res [Ohm.m] : 1
aniso [-] : 1.41421
epermH [-] : 1
epermV [-] : 1
mpermH [-] : 2.5
mpermV [-] : 2.5
> MODEL IS A FULLSPACE
direct field : Comp. in wavenumber domain
frequency [Hz] : 0.7
Hankel : DLF (Fast Hankel Transform)
> Filter : key_201_2009
> DLF type : Lagged Convolution
Loop over : Frequencies
Source(s) : 1 bipole(s)
> intpts : 5
> length [m] : 123.288
> strength[A] : 2.8
> x_c [m] : 0
> y_c [m] : 0
> z_c [m] : 300
> azimuth [°] : 30.9638
> dip [°] : -18.9318
Receiver(s) : 65536 dipole(s)
> x [m] : -2550 - 2550 : 65536 [min-max; #]
> y [m] : -2550 - 2550 : 65536 [min-max; #]
> z [m] : 400
> azimuth [°] : 22
> dip [°] : -13
Required ab's : 11 12 13 21 22 23 31 32 33
:: empymod END; runtime = 0:00:00.389484 :: 45 kernel call(s)
emg3d#
if coarse_model:
min_width_limits = 40
stretching = [1.045, 1.045]
else:
min_width_limits = 20
stretching = [1.03, 1.045]
# Create stretched grid
grid = emg3d.construct_mesh(
frequency=frequency,
properties=h_res,
center=source.center,
domain=([-2500, 2500], [-2500, 2500], [-2900, 2100]),
min_width_limits=min_width_limits,
stretching=stretching,
lambda_from_center=True,
lambda_factor=0.8,
center_on_edge=False,
)
grid
# Define the model, with magnetic permeability
model = emg3d.Model(
grid, property_x=h_res, property_z=v_res,
mu_r=mperm, mapping='Resistivity'
)
# Compute the electric field
efield = emg3d.solve_source(model, source, frequency, verb=4, plain=True)
# Get responses at receivers
e3d = efield.get_receiver((rx, ry, rz, azimuth, elevation))
:: emg3d START :: 15:41:03 :: v1.8.3
MG-cycle : 'F' sslsolver : False
semicoarsening : False [0] tol : 1e-06
linerelaxation : False [0] maxit : 50
nu_{i,1,c,2} : 0, 2, 1, 2 verb : 4
Original grid : 96 x 96 x 80 => 737,280 cells
Coarsest grid : 3 x 3 x 5 => 45 cells
Coarsest level : 5 ; 5 ; 4
[hh:mm:ss] rel. error [abs. error, last/prev] l s
h_
2h_ \ /
4h_ \ /\ /
8h_ \ /\ / \ /
16h_ \ /\ / \ / \ /
32h_ \/\/ \/ \/ \/
[15:41:05] 3.578e-02 after 1 F-cycles [2.608e-05, 0.036] 0 0
[15:41:07] 3.810e-03 after 2 F-cycles [2.777e-06, 0.106] 0 0
[15:41:08] 5.278e-04 after 3 F-cycles [3.847e-07, 0.139] 0 0
[15:41:10] 8.297e-05 after 4 F-cycles [6.048e-08, 0.157] 0 0
[15:41:11] 1.414e-05 after 5 F-cycles [1.031e-08, 0.170] 0 0
[15:41:13] 2.595e-06 after 6 F-cycles [1.892e-09, 0.184] 0 0
[15:41:14] 5.524e-07 after 7 F-cycles [4.027e-10, 0.213] 0 0
> CONVERGED
> MG cycles : 7
> Final rel. error : 5.524e-07
:: emg3d END :: 15:41:14 :: runtime = 0:00:11
Plot#
# Start figure.
a_kwargs = {'cmap': "viridis", 'vmin': -12, 'vmax': -6, 'shading': 'nearest'}
e_kwargs = {'cmap': plt.get_cmap("RdBu_r", 8),
'vmin': -2, 'vmax': 2, 'shading': 'nearest'}
fig, axs = plt.subplots(2, 3, figsize=(10, 5.5), sharex=True, sharey=True,
subplot_kw={'box_aspect': 1})
((ax1, ax2, ax3), (ax4, ax5, ax6)) = axs
x3 = x/1000 # km
# Plot Re(data)
ax1.set_title(r"(a) |Re(empymod)|")
cf0 = ax1.pcolormesh(x3, x3, np.log10(epm.real.amp()), **a_kwargs)
ax2.set_title(r"(b) |Re(emg3d)|")
ax2.pcolormesh(x3, x3, np.log10(e3d.real.amp()), **a_kwargs)
ax3.set_title(r"(c) Error real part")
rel_error = 100*np.abs((epm.real - e3d.real) / epm.real)
cf2 = ax3.pcolormesh(x3, x3, np.log10(rel_error), **e_kwargs)
# Plot Im(data)
ax4.set_title(r"(d) |Im(empymod)|")
ax4.pcolormesh(x3, x3, np.log10(epm.imag.amp()), **a_kwargs)
ax5.set_title(r"(e) |Im(emg3d)|")
ax5.pcolormesh(x3, x3, np.log10(e3d.imag.amp()), **a_kwargs)
ax6.set_title(r"(f) Error imaginary part")
rel_error = 100*np.abs((epm.imag - e3d.imag) / epm.imag)
ax6.pcolormesh(x3, x3, np.log10(rel_error), **e_kwargs)
# Colorbars
fig.colorbar(cf0, ax=axs[0, :], label=r"$\log_{10}$ Amplitude (V/m)")
cbar = fig.colorbar(cf2, ax=axs[1, :], label=r"Relative Error")
cbar.set_ticks([-2, -1, 0, 1, 2])
cbar.ax.set_yticklabels([r"$0.01\,\%$", r"$0.1\,\%$", r"$1\,\%$",
r"$10\,\%$", r"$100\,\%$"])
ax1.set_xlim(min(x3), max(x3))
ax1.set_ylim(min(x3), max(x3))
# Axis label
fig.text(0.4, 0.05, "Inline Offset (km)", fontsize=14)
fig.text(0.05, 0.3, "Crossline Offset (km)", rotation=90, fontsize=14)
fig.suptitle(r'Diffusive Fullspace, $E$-field', y=1, fontsize=20)
print(f"- Source: {source}")
print(f"- Frequency: {frequency} Hz")
print(f"- Electric receivers: z={rz} m; θ={azimuth}°, φ={elevation}°")
- Source: TxElectricDipole: 2.8 A;
e1={-50.0; -30.0; -320.0} m; e2={50.0; 30.0; -280.0} m
- Frequency: 0.7 Hz
- Electric receivers: z=-400.0 m; θ=22°, φ=13°
Total running time of the script: (0 minutes 12.498 seconds)
Estimated memory usage: 151 MB