# 7. Magnetic source using an electric loop#

Computing the $$E$$ and $$H$$ fields generated by a magnetic source

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:

We create a “magnetic dipole” through an electric loop perpendicular to the defined dipole in a homogeneous VTI fullspace, 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 loop dipole#

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#

emg3d.TxMagneticDipole creates an electric square loop perpendicular to the defined dipole, where the area of the square loop corresponds to the length of the dipole.

# 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.TxMagneticDipole(
coordinates=[-0.5, 0.5, -0.3, 0.3, -300.5, -299.5],  # [x1,x2,y1,y2,z1,z2]
strength=np.pi,  # A
)
frequency = 0.77  # Hz

# Model parameters
h_res = 1.              # Horizontal resistivity
aniso = np.sqrt(2.)     # Anisotropy
v_res = h_res*aniso**2  # Vertical 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[:4], -source.coordinates[4:]],
'depth': [],
'res': h_res,
'aniso': aniso,
'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  ::  v2.2.0

depth       [m] :
res     [Ohm.m] :  1
aniso       [-] :  1.41421
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 bipole(s)
> intpts      :  1 (as dipole)
> length  [m] :  1.53623
> strength[A] :  3.14159
> x_c     [m] :  0
> y_c     [m] :  0
> z_c     [m] :  300
> azimuth [°] :  30.9638
> dip     [°] :  -40.6129
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 = 0:00:00.180658 :: 8 kernel call(s)

:: empymod END; runtime = 0:00:00.177053 :: 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

 MESH EXTENT CELL WIDTH FACTOR dir nC TensorMesh 512,000 cells x 80 -4,155.20 4,378.72 40.00 223.51 1.04 y 80 -4,155.20 4,378.72 40.00 223.51 1.04 z 80 -4,678.72 3,855.20 40.00 223.51 1.04

# Define the model
model = emg3d.Model(
grid, property_x=h_res, property_z=v_res, mapping='Resistivity')

# Compute the electric field
efield = emg3d.solve_source(model, source, frequency, verb=4, plain=True)

:: emg3d START :: 21:47:42 :: v1.8.0

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  :  80 x  80 x  80     => 512,000 cells
Coarsest grid  :   5 x   5 x   5     => 125 cells
Coarsest level :   4 ;   4 ;   4

[hh:mm:ss]  rel. error                  [abs. error, last/prev]   l s

h_
2h_ \                  /
4h_  \          /\    /
8h_   \    /\  /  \  /
16h_    \/\/  \/    \/

[21:47:45]   4.451e-03  after   1 F-cycles   [4.617e-09, 0.004]   0 0
[21:47:48]   1.643e-04  after   2 F-cycles   [1.704e-10, 0.037]   0 0
[21:47:51]   8.138e-06  after   3 F-cycles   [8.443e-12, 0.050]   0 0
[21:47:54]   4.803e-07  after   4 F-cycles   [4.983e-13, 0.059]   0 0

> CONVERGED
> MG cycles        : 4
> Final rel. error : 4.803e-07

:: emg3d END   :: 21:47:54 :: runtime = 0:00:12


### Plot function#

def plot(epm, e3d, title, vmin, vmax):

# Start figure.
a_kwargs = {'cmap': "viridis", 'vmin': vmin, 'vmax': vmax,

e_kwargs = {'cmap': plt.cm.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}°")

fig.show()


## Compare the electric field generated from the magnetic source#

e3d_e = efield.get_receiver((rx, ry, rz, azimuth, elevation))
plot(epm_e, e3d_e, r'Diffusive Fullspace $E$', vmin=-17, vmax=-10)

- Source: TxMagneticDipole: 3.1 A;
e1={-0.5; 0.8; -300.0} m; e2={-0.5; -0.3; -299.3} m
- Frequency: 0.77 Hz
- Electric receivers: z=-400.0 m; θ=25°, φ=10°


## Compare the magnetic field generated from the magnetic source#

# Get the magnetic field :math:H from the electric field
hfield = emg3d.get_magnetic_field(model, efield)
e3d_h = hfield.get_receiver((rx, ry, rz, azimuth, elevation))
plot(epm_h, e3d_h, r'Diffusive Fullspace $H$', vmin=-13, vmax=-8)

- Source: TxMagneticDipole: 3.1 A;
e1={-0.5; 0.8; -300.0} m; e2={-0.5; -0.3; -299.3} m
- Frequency: 0.77 Hz
- Magnetic receivers: z=-400.0 m; θ=25°, φ=10°

emg3d.Report()

 Wed Aug 31 21:47:57 2022 CEST OS Linux CPU(s) 4 Machine x86_64 Architecture 64bit RAM 15.5 GiB Environment Python File system ext4 Python 3.9.12 | packaged by conda-forge | (main, Mar 24 2022, 23:22:55) [GCC 10.3.0] numpy 1.22.4 scipy 1.9.0 numba 0.55.2 emg3d 1.8.0 empymod 2.2.0 xarray 2022.6.0 discretize 0.8.2 h5py 3.7.0 matplotlib 3.4.3 tqdm 4.64.0 IPython 8.4.0 Intel(R) oneAPI Math Kernel Library Version 2022.0-Product Build 20211112 for Intel(R) 64 architecture applications

Total running time of the script: ( 0 minutes 16.924 seconds)

Estimated memory usage: 99 MB

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