Note
Go to the end to download the full example code.
8. Magnetic source using duality#
Computing the \(E\) and \(H\) fields from a magnetic source using the duality principle.
We know that we can get the magnetic fields from the electric fields using Faraday’s law, see 6. Magnetic field due to an el. source.
However, what about computing the fields generated by a magnetic source? There are two ways we can achieve that:
using the duality principle, which is what we do in this example, or
creating an electric loop source, see 7. Magnetic source using an electric loop.
emg3d solves the following equation,
where \(\eta = \sigma - \mathrm{i}\omega\varepsilon\), \(\zeta = \mathrm{i}\omega\mu\), \(\sigma\) is conductivity (S/m), \(\omega=2\pi f\) is the angular frequency (Hz), \(\mu=\mu_0\mu_\mathrm{r}\) is magnetic permeability (H/m), \(\varepsilon=\varepsilon_0\varepsilon_\mathrm{r}\) is electric permittivity (F/m), \(\mathbf{\hat{E}}\) the electric field in the frequency domain (V/m), and \(\mathbf{\hat{J}}^e_s\) source current.
This is the electric field due to an electric source. One can obtain the magnetic field due to a magnetic field by substituting
\(\eta \leftrightarrow -\zeta\) ,
\(\mathbf{\hat{E}} \leftrightarrow -\mathbf{\hat{H}}\) ,
\(\mathbf{\hat{J}}^e_s \leftrightarrow \mathbf{\hat{J}}^m_s\) ,
which is called the duality principle.
Carrying out the substitution yields
which is for a magnetic dipole. Changing it for a loop source adds a term \(\mathrm{i}\omega\mu\) to the source term, resulting in
see Dipoles and Loops for more information.
emg3d is not ideal for the duality principle. Magnetic permeability is
implemented isotropically, and discontinuities in magnetic permeabilities can
lead to first-order errors in contrary to second-order errors for
discontinuities in conductivity. However, we can still abuse the code and use
it with the duality principle, at least for isotropic media.
The actual implemented equation in emg3d is a slightly modified version of
Equation (1), using the diffusive approximation
\(\varepsilon=0\),
We therefore only need to interchange \(\sigma\) with \(\mu_\mathrm{r}^{-1}\) or \(\rho\) with \(\mu_\mathrm{r}\) to get from (4) to (3).
This is what we do in this example, for an arbitrarily rotated loop in a
homogeneous, isotropic fullspace. We compare the result 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 rotated magnetic loop#
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 = 25
elevation = 10
# Source coordinates, frequency, and strength
source = emg3d.TxElectricDipole(
coordinates=[0, 0, -300, 10, 70], # [x, y, z, azimuth, elevation]
strength=np.pi, # A
)
frequency = 0.77 # Hz
# Model parameters
h_res = 2. # Horizontal resistivity
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.
# Collect common input for empymod.
inp = {
'src': np.r_[source.coordinates[:2], -source.coordinates[2],
source.coordinates[3], -source.coordinates[4]],
'depth': [],
'res': h_res,
'strength': source.strength,
'freqtime': frequency,
'htarg': {'pts_per_dec': -1},
}
# Compute e-field
epm_e = -empymod.loop(
rec=[frx, fry, -rz, azimuth, -elevation], mrec=False, verb=3, **inp
).reshape(np.shape(rx))
# Compute h-field
epm_h = empymod.loop(
rec=[frx, fry, -rz, azimuth, -elevation], **inp
).reshape(np.shape(rx))
:: empymod START :: v2.3.1
depth [m] :
res [Ohm.m] : 2
aniso [-] : 1
epermH [-] : 1
epermV [-] : 1
mpermH [-] : 1
mpermV [-] : 1
> MODEL IS A FULLSPACE
direct field : Comp. in wavenumber domain
frequency [Hz] : 0.77
Hankel : DLF (Fast Hankel Transform)
> Filter : key_201_2009
> DLF type : Lagged Convolution
Loop over : Frequencies
Source(s) : 1 dipole(s)
> x [m] : 0
> y [m] : 0
> z [m] : 300
> azimuth [°] : 10
> dip [°] : -70
Receiver(s) : 65536 dipole(s)
> x [m] : -2550 - 2550 : 65536 [min-max; #]
> y [m] : -2550 - 2550 : 65536 [min-max; #]
> z [m] : 400
> azimuth [°] : 25
> dip [°] : -10
Required ab's : 14 15 16 24 25 26 34 35 36
:: empymod END; runtime = 0:00:00.104950 :: 8 kernel call(s)
:: empymod END; runtime = 0:00:00.090320 :: 9 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
Abuse the parameters to take advantage of the duality principle#
See text at the top. We set here \(\rho=1\) and \(\mu_\mathrm{r} = 1/\rho\) to get:
(in the diffusive regime), and
:: emg3d START :: 15:40:48 :: 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 96 => 884,736 cells
Coarsest grid : 3 x 3 x 3 => 27 cells
Coarsest level : 5 ; 5 ; 5
[hh:mm:ss] rel. error [abs. error, last/prev] l s
h_
2h_ \ /
4h_ \ /\ /
8h_ \ /\ / \ /
16h_ \ /\ / \ / \ /
32h_ \/\/ \/ \/ \/
[15:40:50] 3.240e-02 after 1 F-cycles [3.094e-07, 0.032] 0 0
[15:40:52] 2.986e-03 after 2 F-cycles [2.851e-08, 0.092] 0 0
[15:40:55] 3.054e-04 after 3 F-cycles [2.916e-09, 0.102] 0 0
[15:40:57] 3.300e-05 after 4 F-cycles [3.151e-10, 0.108] 0 0
[15:40:59] 3.965e-06 after 5 F-cycles [3.786e-11, 0.120] 0 0
[15:41:01] 8.931e-07 after 6 F-cycles [8.529e-12, 0.225] 0 0
> CONVERGED
> MG cycles : 6
> Final rel. error : 8.931e-07
:: emg3d END :: 15:41:01 :: runtime = 0:00:13
Plot function#
def plot(epm, e3d, title, vmin, vmax):
# Start figure.
a_kwargs = {'cmap': "viridis", 'vmin': vmin, 'vmax': vmax,
'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
unit = "(V/m)" if "E" in title else "(A/m)"
fig.colorbar(cf0, ax=axs[0, :], label=r"$\log_{10}$ Amplitude "+unit)
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(title, y=1, fontsize=20)
print(f"- Source: {source}")
print(f"- Frequency: {frequency} Hz")
rtype = "Electric" if "E" in title else "Magnetic"
print(f"- {rtype} receivers: z={rz} m; θ={azimuth}°, φ={elevation}°")
Compare the magnetic field generated from the magnetic source#
- Source: TxElectricDipole: 3.1 A;
center={0.0; 0.0; -300.0} m; θ=10.0°, φ=70.0°; l=1.0 m
- Frequency: 0.77 Hz
- Magnetic receivers: z=-400.0 m; θ=25°, φ=10°
Compare the electric field generated from the magnetic source#
get_magnetic_field gets the \(H\)-field from the \(E\)-field with
Faraday’s law,
Using the substitutions introduced in the beginning, and using the same function but to get the \(E\)-field from the \(H\)-field, we have to multiply the result by
Compute electric field \(E\) from the magnetic field#
efield = emg3d.get_magnetic_field(model, hfield)
efield.field *= efield.smu0
e3d_e = efield.get_receiver((rx, ry, rz, azimuth, elevation))
plot(epm_e, e3d_e, r'Diffusive Fullspace $E$', vmin=-17, vmax=-10)
- Source: TxElectricDipole: 3.1 A;
center={0.0; 0.0; -300.0} m; θ=10.0°, φ=70.0°; l=1.0 m
- Frequency: 0.77 Hz
- Electric receivers: z=-400.0 m; θ=25°, φ=10°
Total running time of the script: (0 minutes 15.390 seconds)
Estimated memory usage: 175 MB