14  Exercises in Python

14.1 Preparation

We prepare a working environment with Python and some essential packages. Make sure that the following packages are installed:

  • numpy
  • matplotlib
  • scipy
  • sympy

Further, we will use empymod to generate reference responses. It can be installed with the following commands:

conda install pip
pip install --upgrade empymod

14.2 empymod

When in doubt, see the documentation.

The coordinate system is either

  • left-handed, where \(x\) points to East, \(y\) points to North, and \(z\) points downwards, angle \(\theta\) between E and N, \(\phi\) down
  • right-handed, where \(x\) points to East, \(y\) points to North, and \(z\) points upwards, angle \(\theta\) between E and N, \(\phi\) up

14.2.1 VMD in full-space

Show the code 
import empymod
import numpy as np
import matplotlib.pyplot as plt

Survey paramaters:

  • Conductivity \(\sigma = 0.01\) S/m
  • TX at origin, \(z\)-directed
  • RX (magnetic) with offset of \(100\) m, \(z\)-directed
Show the code 
freq = np.logspace(-1, 5, 301)
src = [0, 0, 0, 0, 90]    # z-dir. source at the origin [x, y, z, azimuth, dip]
rec = [100, 0, 0, 0, 90]  # z-dir. receiver 100 m away from source
cond = 0.01

Computation using empymod:

Show the code 
inp = {'src': src, 'rec': rec, 'depth': [], 'res': 1/cond, 'verb': 1}

inp['freqtime'] = freq
inp['mrec'] = True
fmm_dip_dip = empymod.loop(**inp)
Show the code 
fs = 12
fig = plt.figure(figsize=(8, 6), constrained_layout=True)
plt.plot(freq, fmm_dip_dip.real, 'C0-', label='Real')
plt.plot(freq, -fmm_dip_dip.real, 'C0--')
plt.plot(freq, fmm_dip_dip.imag, 'C1-', label='Imag')
plt.plot(freq, -fmm_dip_dip.imag, 'C1--')
plt.xscale('log')
plt.yscale('log')
plt.xlabel('Frequency (Hz)', fontsize=fs-2)
plt.ylabel('Amplitude (A/m)', fontsize=fs-2)
plt.legend()

14.2.2 VMD over layered half-space

Survey parameters for a 3-layer model:

  • Frequency range \(10^2\) to \(10^5\) Hz
  • TX-RX offset range \(1\) to \(10^3\) m
  • Electrical conductivities: \([10^{-2}, 1, 10^{-2}]\) S/m
  • Layer thicknesses \([10, 10]\) m

Make plots for

  • amplitude vs. frequency
  • amplitude vs. offset
  • all three components of \(\mathbf B\)
  • all three components of \(\mathbf E\)
  • for all three possible directions of the magnetic dipole

Notes:

  • Use mrec=False to force the calculation of electric fields
  • Define the orientation of TX and RX with the appropriate choice of the angles \(\theta\) and \(\phi\) (parameters 4 and 5 of src and rec, resp.)

14.2.3 HEM example

Survey parameters for the 5-layer model after Siemon et al. (2009):

  • TX-RX offset \(8\) m
  • Frequencies: \([387, 1820, 8225, 41550, 133200]\) Hz
  • Bird height: \(30\) m above ground
  • Resistivities: \([200, 100, 5, 1000]\) \(\Omega \cdot\)m
  • Thicknesses: \([20, 30, 10]\) m

Calculate the HEM responses in terms of real part \(R\) and quadrature part \(Q\) in ppm!