# Code¶

## model – Model EM-responses¶

EM-modelling routines. The implemented routines might not be the fastest solution to your specific problem. Use these routines as template to create your own, problem-specific modelling routine!

Principal routines:
• bipole
• dipole

The main routine is bipole, which can model bipole source(s) and bipole receiver(s) of arbitrary direction, for electric or magnetic sources and receivers, both in frequency and in time. A subset of bipole is dipole, which models infinitesimal small dipoles along the principal axes x, y, and z.

Further routines are:

• analytical: Calculate analytical fullspace and halfspace solutions.
• dipole_k: Calculate the electromagnetic wavenumber-domain solution.
• gpr: Calculate the Ground-Penetrating Radar (GPR) response.

The dipole_k routine can be used if you are interested in the wavenumber-domain result, without Hankel nor Fourier transform. It calls straight the kernel. The gpr-routine convolves the frequency-domain result with a wavelet, and applies a gain to the time-domain result. This function is still experimental.

The modelling routines make use of the following two core routines:
• fem: Calculate wavenumber-domain electromagnetic field and carry out
the Hankel transform to the frequency domain.
• tem: Carry out the Fourier transform to time domain after fem.
empymod.model.bipole(src, rec, depth, res, freqtime, signal=None, aniso=None, epermH=None, epermV=None, mpermH=None, mpermV=None, msrc=False, srcpts=1, mrec=False, recpts=1, strength=0, xdirect=False, ht='fht', htarg=None, ft='sin', ftarg=None, opt=None, loop=None, verb=2)[source]

Return the electromagnetic field due to an electromagnetic source.

Calculate the electromagnetic frequency- or time-domain field due to arbitrary finite electric or magnetic bipole sources, measured by arbitrary finite electric or magnetic bipole receivers. By default, the electromagnetic response is normalized to to source and receiver of 1 m length, and source strength of 1 A.

Parameters: src, rec : list of floats or arrays Source and receiver coordinates (m): [x0, x1, y0, y1, z0, z1] (bipole of finite length) [x, y, z, azimuth, dip] (dipole, infinitesimal small) Dimensions: The coordinates x, y, and z (dipole) or x0, x1, y0, y1, z0, and z1 (bipole) can be single values or arrays. The variables x and y (dipole) or x0, x1, y0, and y1 (bipole) must have the same dimensions. The variable z (dipole) or z0 and z1 (bipole) must either be single values or having the same dimension as the other coordinates. The variables azimuth and dip must be single values. If they have different angles, you have to use the bipole-method (with srcpts/recpts = 1, so it is calculated as dipoles). Angles (coordinate system is left-handed, positive z down (East-North-Depth): azimuth (°): horizontal deviation from x-axis, anti-clockwise. dip (°): vertical deviation from xy-plane downwards. Sources or receivers placed on a layer interface are considered in the upper layer. depth : list Absolute layer interfaces z (m); #depth = #res - 1 (excluding +/- infinity). res : array_like Horizontal resistivities rho_h (Ohm.m); #res = #depth + 1. Alternatively, res can be a dictionary. See the main manual of empymod too see how to exploit this hook to re-calculate etaH, etaV, zetaH, and zetaV, which can be used to, for instance, use the Cole-Cole model for IP. freqtime : array_like Frequencies f (Hz) if signal == None, else times t (s); (f, t > 0). signal : {None, 0, 1, -1}, optional Source signal, default is None: None: Frequency-domain response -1 : Switch-off time-domain response 0 : Impulse time-domain response +1 : Switch-on time-domain response aniso : array_like, optional Anisotropies lambda = sqrt(rho_v/rho_h) (-); #aniso = #res. Defaults to ones. epermH, epermV : array_like, optional Relative horizontal/vertical electric permittivities epsilon_h/epsilon_v (-); #epermH = #epermV = #res. Default is ones. mpermH, mpermV : array_like, optional Relative horizontal/vertical magnetic permeabilities mu_h/mu_v (-); #mpermH = #mpermV = #res. Default is ones. msrc, mrec : boolean, optional If True, source/receiver (msrc/mrec) is magnetic, else electric. Default is False. srcpts, recpts : int, optional Number of integration points for bipole source/receiver, default is 1: srcpts/recpts < 3 : bipole, but calculated as dipole at centre srcpts/recpts >= 3 : bipole strength : float, optional Source strength (A): If 0, output is normalized to source and receiver of 1 m length, and source strength of 1 A. If != 0, output is returned for given source and receiver length, and source strength. Default is 0. xdirect : bool or None, optional Direct field calculation (only if src and rec are in the same layer): If True, direct field is calculated analytically in the frequency domain. If False, direct field is calculated in the wavenumber domain. If None, direct field is excluded from the calculation, and only reflected fields are returned (secondary field). Defaults to False. ht : {‘fht’, ‘qwe’, ‘quad’}, optional Flag to choose either the Digital Linear Filter method (FHT, Fast Hankel Transform), the Quadrature-With-Extrapolation (QWE), or a simple Quadrature (QUAD) for the Hankel transform. Defaults to ‘fht’. htarg : dict or list, optional Depends on the value for ht: If ht = ‘fht’: [fhtfilt, pts_per_dec]: fhtfilt: string of filter name in empymod.filters or the filter method itself. (default: empymod.filters.key_201_2009()) pts_per_dec: points per decade; (default: 0) If 0: Standard DLF. If < 0: Lagged Convolution DLF. If > 0: Splined DLF If ht = ‘qwe’: [rtol, atol, nquad, maxint, pts_per_dec, diff_quad, a, b, limit]: rtol: relative tolerance (default: 1e-12) atol: absolute tolerance (default: 1e-30) nquad: order of Gaussian quadrature (default: 51) maxint: maximum number of partial integral intervals (default: 40) pts_per_dec: points per decade; (default: 0) If 0, no interpolation is used. If > 0, interpolation is used. diff_quad: criteria when to swap to QUAD (only relevant if opt=’spline’) (default: 100) a: lower limit for QUAD (default: first interval from QWE) b: upper limit for QUAD (default: last interval from QWE) limit: limit for quad (default: maxint) If ht = ‘quad’: [atol, rtol, limit, lmin, lmax, pts_per_dec]: rtol: relative tolerance (default: 1e-12) atol: absolute tolerance (default: 1e-20) limit: An upper bound on the number of subintervals used in the adaptive algorithm (default: 500) lmin: Minimum wavenumber (default 1e-6) lmax: Maximum wavenumber (default 0.1) pts_per_dec: points per decade (default: 40) The values can be provided as dict with the keywords, or as list. However, if provided as list, you have to follow the order given above. A few examples, assuming ht = qwe: Only changing rtol: {‘rtol’: 1e-4} or [1e-4] or 1e-4 Changing rtol and nquad: {‘rtol’: 1e-4, ‘nquad’: 101} or [1e-4, ‘’, 101] Only changing diff_quad: {‘diffquad’: 10} or [‘’, ‘’, ‘’, ‘’, ‘’, 10] ft : {‘sin’, ‘cos’, ‘qwe’, ‘fftlog’, ‘fft’}, optional Only used if signal != None. Flag to choose either the Digital Linear Filter method (Sine- or Cosine-Filter), the Quadrature-With-Extrapolation (QWE), the FFTLog, or the FFT for the Fourier transform. Defaults to ‘sin’. ftarg : dict or list, optional Only used if signal !=None. Depends on the value for ft: If ft = ‘sin’ or ‘cos’: [fftfilt, pts_per_dec]: fftfilt: string of filter name in empymod.filters or the filter method itself. (Default: empymod.filters.key_201_CosSin_2012()) pts_per_dec: points per decade; (default: -1) If 0: Standard DLF. If < 0: Lagged Convolution DLF. If > 0: Splined DLF If ft = ‘qwe’: [rtol, atol, nquad, maxint, pts_per_dec]: rtol: relative tolerance (default: 1e-8) atol: absolute tolerance (default: 1e-20) nquad: order of Gaussian quadrature (default: 21) maxint: maximum number of partial integral intervals (default: 200) pts_per_dec: points per decade (default: 20) diff_quad: criteria when to swap to QUAD (default: 100) a: lower limit for QUAD (default: first interval from QWE) b: upper limit for QUAD (default: last interval from QWE) limit: limit for quad (default: maxint) If ft = ‘fftlog’: [pts_per_dec, add_dec, q]: pts_per_dec: sampels per decade (default: 10) add_dec: additional decades [left, right] (default: [-2, 1]) q: exponent of power law bias (default: 0); -1 <= q <= 1 If ft = ‘fft’: [dfreq, nfreq, ntot]: dfreq: Linear step-size of frequencies (default: 0.002) nfreq: Number of frequencies (default: 2048) ntot: Total number for FFT; difference between nfreq and ntot is padded with zeroes. This number is ideally a power of 2, e.g. 2048 or 4096 (default: nfreq). pts_per_dec : points per decade (default: None) Padding can sometimes improve the result, not always. The default samples from 0.002 Hz - 4.096 Hz. If pts_per_dec is set to an integer, calculated frequencies are logarithmically spaced with the given number per decade, and then interpolated to yield the required frequencies for the FFT. The values can be provided as dict with the keywords, or as list. However, if provided as list, you have to follow the order given above. See htarg for a few examples. opt : {None, ‘parallel’}, optional Optimization flag. Defaults to None: None: Normal case, no parallelization nor interpolation is used. If ‘parallel’, the package numexpr is used to evaluate the most expensive statements. Always check if it actually improves performance for a specific problem. It can speed up the calculation for big arrays, but will most likely be slower for small arrays. It will use all available cores for these specific statements, which all contain Gamma in one way or another, which has dimensions (#frequencies, #offsets, #layers, #lambdas), therefore can grow pretty big. The module numexpr uses by default all available cores up to a maximum of 8. You can change this behaviour to your desired number of threads nthreads with numexpr.set_num_threads(nthreads). The value ‘spline’ is deprecated and will be removed. See htarg instead for the interpolated versions. The option ‘parallel’ only affects speed and memory usage, whereas ‘spline’ also affects precision! Please read the note in the README documentation for more information. loop : {None, ‘freq’, ‘off’}, optional Define if to calculate everything vectorized or if to loop over frequencies (‘freq’) or over offsets (‘off’), default is None. It always loops over frequencies if ht = 'qwe' or if opt = 'spline'. Calculating everything vectorized is fast for few offsets OR for few frequencies. However, if you calculate many frequencies for many offsets, it might be faster to loop over frequencies. Only comparing the different versions will yield the answer for your specific problem at hand! verb : {0, 1, 2, 3, 4}, optional Level of verbosity, default is 2: 0: Print nothing. 1: Print warnings. 2: Print additional runtime and kernel calls 3: Print additional start/stop, condensed parameter information. 4: Print additional full parameter information EM : ndarray, (nfreq, nrec, nsrc) Frequency- or time-domain EM field (depending on signal): If rec is electric, returns E [V/m]. If rec is magnetic, returns H [A/m]. However, source and receiver are normalised (unless strength != 0). So for instance in the electric case the source strength is 1 A and its length is 1 m. So the electric field could also be written as [V/(A.m2)]. In the magnetic case the source strength is given by $$i\omega\mu_0 A I^e$$, where A is the loop area (m2), and $$I^e$$ the electric source strength. For the normalized magnetic source $$A=1m^2$$ and $$I^e=1 Ampere$$. A magnetic source is therefore frequency dependent. The shape of EM is (nfreq, nrec, nsrc). However, single dimensions are removed.

fem
Electromagnetic frequency-domain response.
tem
Electromagnetic time-domain response.

Examples

>>> import numpy as np
>>> from empymod import bipole
>>> # x-directed bipole source: x0, x1, y0, y1, z0, z1
>>> src = [-50, 50, 0, 0, 100, 100]
>>> # x-directed dipole source-array: x, y, z, azimuth, dip
>>> rec = [np.arange(1, 11)*500, np.zeros(10), 200, 0, 0]
>>> # layer boundaries
>>> depth = [0, 300, 1000, 1050]
>>> # layer resistivities
>>> res = [1e20, .3, 1, 50, 1]
>>> # Frequency
>>> freq = 1
>>> # Calculate electric field due to an electric source at 1 Hz.
>>> # [msrc = mrec = True (default)]
>>> EMfield = bipole(src, rec, depth, res, freq, verb=4)
:: empymod START  ::
~
depth       [m] :  0 300 1000 1050
res     [Ohm.m] :  1E+20 0.3 1 50 1
aniso       [-] :  1 1 1 1 1
epermH      [-] :  1 1 1 1 1
epermV      [-] :  1 1 1 1 1
mpermH      [-] :  1 1 1 1 1
mpermV      [-] :  1 1 1 1 1
frequency  [Hz] :  1
Hankel          :  DLF (Fast Hankel Transform)
> Filter      :  Key 201 (2009)
> DLF type    :  Standard
Kernel Opt.     :  None
Loop over       :  None (all vectorized)
Source(s)       :  1 bipole(s)
> intpts      :  1 (as dipole)
> length  [m] :  100
> x_c     [m] :  0
> y_c     [m] :  0
> z_c     [m] :  100
> azimuth [°] :  0
> dip     [°] :  0
> x       [m] :  500 - 5000 : 10  [min-max; #]
:  500 1000 1500 2000 2500 3000 3500 4000 4500 5000
> y       [m] :  0 - 0 : 10  [min-max; #]
:  0 0 0 0 0 0 0 0 0 0
> z       [m] :  200
> azimuth [°] :  0
> dip     [°] :  0
Required ab's   :  11
~
:: empymod END; runtime = 0:00:00.005536 :: 1 kernel call(s)
~
>>> print(EMfield)
[  1.68809346e-10 -3.08303130e-10j  -8.77189179e-12 -3.76920235e-11j
-3.46654704e-12 -4.87133683e-12j  -3.60159726e-13 -1.12434417e-12j
1.87807271e-13 -6.21669759e-13j   1.97200208e-13 -4.38210489e-13j
1.44134842e-13 -3.17505260e-13j   9.92770406e-14 -2.33950871e-13j
6.75287598e-14 -1.74922886e-13j   4.62724887e-14 -1.32266600e-13j]

empymod.model.dipole(src, rec, depth, res, freqtime, signal=None, ab=11, aniso=None, epermH=None, epermV=None, mpermH=None, mpermV=None, xdirect=False, ht='fht', htarg=None, ft='sin', ftarg=None, opt=None, loop=None, verb=2)[source]

Return the electromagnetic field due to a dipole source.

Calculate the electromagnetic frequency- or time-domain field due to infinitesimal small electric or magnetic dipole source(s), measured by infinitesimal small electric or magnetic dipole receiver(s); sources and receivers are directed along the principal directions x, y, or z, and all sources are at the same depth, as well as all receivers are at the same depth.

Use the functions bipole to calculate dipoles with arbitrary angles or bipoles of finite length and arbitrary angle.

The function dipole could be replaced by bipole (all there is to do is translate ab into msrc, mrec, azimuth’s and dip’s). However, dipole is kept separately to serve as an example of a simple modelling routine that can serve as a template.

Parameters:
src, rec : list of floats or arrays

Source and receiver coordinates (m): [x, y, z]. The x- and y-coordinates can be arrays, z is a single value. The x- and y-coordinates must have the same dimension.

Sources or receivers placed on a layer interface are considered in the upper layer.

depth : list

Absolute layer interfaces z (m); #depth = #res - 1 (excluding +/- infinity).

res : array_like

Horizontal resistivities rho_h (Ohm.m); #res = #depth + 1.

Alternatively, res can be a dictionary. See the main manual of empymod too see how to exploit this hook to re-calculate etaH, etaV, zetaH, and zetaV, which can be used to, for instance, use the Cole-Cole model for IP.

freqtime : array_like

Frequencies f (Hz) if signal == None, else times t (s); (f, t > 0).

signal : {None, 0, 1, -1}, optional
Source signal, default is None:
• None: Frequency-domain response
• -1 : Switch-off time-domain response
• 0 : Impulse time-domain response
• +1 : Switch-on time-domain response
ab : int, optional

electric source magnetic source
x y z x y z

electric

x 11 12 13 14 15 16
y 21 22 23 24 25 26
z 31 32 33 34 35 36

magnetic

x 41 42 43 44 45 46
y 51 52 53 54 55 56
z 61 62 63 64 65 66
aniso : array_like, optional

Anisotropies lambda = sqrt(rho_v/rho_h) (-); #aniso = #res. Defaults to ones.

epermH, epermV : array_like, optional

Relative horizontal/vertical electric permittivities epsilon_h/epsilon_v (-); #epermH = #epermV = #res. Default is ones.

mpermH, mpermV : array_like, optional

Relative horizontal/vertical magnetic permeabilities mu_h/mu_v (-); #mpermH = #mpermV = #res. Default is ones.

xdirect : bool or None, optional
Direct field calculation (only if src and rec are in the same layer):
• If True, direct field is calculated analytically in the frequency domain.
• If False, direct field is calculated in the wavenumber domain.
• If None, direct field is excluded from the calculation, and only reflected fields are returned (secondary field).

Defaults to False.

ht : {‘fht’, ‘qwe’, ‘quad’}, optional

Flag to choose either the Digital Linear Filter method (FHT, Fast Hankel Transform), the Quadrature-With-Extrapolation (QWE), or a simple Quadrature (QUAD) for the Hankel transform. Defaults to ‘fht’.

htarg : dict or list, optional
Depends on the value for ht:
• If ht = ‘fht’: [fhtfilt, pts_per_dec]:

• fhtfilt: string of filter name in empymod.filters or
the filter method itself. (default: empymod.filters.key_201_2009())
• pts_per_dec: points per decade; (default: 0)
• If 0: Standard DLF.
• If < 0: Lagged Convolution DLF.
• If > 0: Splined DLF
• If ht = ‘qwe’: [rtol, atol, nquad, maxint, pts_per_dec,

• rtol: relative tolerance (default: 1e-12)
• atol: absolute tolerance (default: 1e-30)
• maxint: maximum number of partial integral intervals
(default: 40)
• pts_per_dec: points per decade; (default: 0)
• If 0, no interpolation is used.
• If > 0, interpolation is used.
• diff_quad: criteria when to swap to QUAD (only relevant if opt=’spline’) (default: 100)
• a: lower limit for QUAD (default: first interval from QWE)
• b: upper limit for QUAD (default: last interval from QWE)
• limit: limit for quad (default: maxint)
• If ht = ‘quad’: [atol, rtol, limit, lmin, lmax, pts_per_dec]:

• rtol: relative tolerance (default: 1e-12)
• atol: absolute tolerance (default: 1e-20)
• limit: An upper bound on the number of subintervals used in the adaptive algorithm (default: 500)
• lmin: Minimum wavenumber (default 1e-6)
• lmax: Maximum wavenumber (default 0.1)
• pts_per_dec: points per decade (default: 40)

The values can be provided as dict with the keywords, or as list. However, if provided as list, you have to follow the order given above. A few examples, assuming ht = qwe:

• Only changing rtol:
{‘rtol’: 1e-4} or [1e-4] or 1e-4
{‘rtol’: 1e-4, ‘nquad’: 101} or [1e-4, ‘’, 101]
{‘diffquad’: 10} or [‘’, ‘’, ‘’, ‘’, ‘’, 10]
ft : {‘sin’, ‘cos’, ‘qwe’, ‘fftlog’, ‘fft’}, optional

Only used if signal != None. Flag to choose either the Digital Linear Filter method (Sine- or Cosine-Filter), the Quadrature-With-Extrapolation (QWE), the FFTLog, or the FFT for the Fourier transform. Defaults to ‘sin’.

ftarg : dict or list, optional
Only used if signal !=None. Depends on the value for ft:
• If ft = ‘sin’ or ‘cos’: [fftfilt, pts_per_dec]:

• fftfilt: string of filter name in empymod.filters or
the filter method itself. (Default: empymod.filters.key_201_CosSin_2012())
• pts_per_dec: points per decade; (default: -1)
• If 0: Standard DLF.
• If < 0: Lagged Convolution DLF.
• If > 0: Splined DLF
• If ft = ‘qwe’: [rtol, atol, nquad, maxint, pts_per_dec]:

• rtol: relative tolerance (default: 1e-8)
• atol: absolute tolerance (default: 1e-20)
• maxint: maximum number of partial integral intervals
(default: 200)
• pts_per_dec: points per decade (default: 20)
• a: lower limit for QUAD (default: first interval from QWE)
• b: upper limit for QUAD (default: last interval from QWE)
• limit: limit for quad (default: maxint)
• If ft = ‘fftlog’: [pts_per_dec, add_dec, q]:

• pts_per_dec: sampels per decade (default: 10)
• q: exponent of power law bias (default: 0); -1 <= q <= 1
• If ft = ‘fft’: [dfreq, nfreq, ntot]:

• dfreq: Linear step-size of frequencies (default: 0.002)
• nfreq: Number of frequencies (default: 2048)
• ntot: Total number for FFT; difference between nfreq and
ntot is padded with zeroes. This number is ideally a power of 2, e.g. 2048 or 4096 (default: nfreq).
• pts_per_dec : points per decade (default: None)

Padding can sometimes improve the result, not always. The default samples from 0.002 Hz - 4.096 Hz. If pts_per_dec is set to an integer, calculated frequencies are logarithmically spaced with the given number per decade, and then interpolated to yield the required frequencies for the FFT.

The values can be provided as dict with the keywords, or as list. However, if provided as list, you have to follow the order given above. See htarg for a few examples.

opt : {None, ‘parallel’}, optional
Optimization flag. Defaults to None:
• None: Normal case, no parallelization nor interpolation is used.
• If ‘parallel’, the package numexpr is used to evaluate the most expensive statements. Always check if it actually improves performance for a specific problem. It can speed up the calculation for big arrays, but will most likely be slower for small arrays. It will use all available cores for these specific statements, which all contain Gamma in one way or another, which has dimensions (#frequencies, #offsets, #layers, #lambdas), therefore can grow pretty big. The module numexpr uses by default all available cores up to a maximum of 8. You can change this behaviour to your desired number of threads nthreads with numexpr.set_num_threads(nthreads).
• The value ‘spline’ is deprecated and will be removed. See htarg instead for the interpolated versions.

loop : {None, ‘freq’, ‘off’}, optional

Define if to calculate everything vectorized or if to loop over frequencies (‘freq’) or over offsets (‘off’), default is None. It always loops over frequencies if ht = 'qwe' or if opt = 'spline'. Calculating everything vectorized is fast for few offsets OR for few frequencies. However, if you calculate many frequencies for many offsets, it might be faster to loop over frequencies. Only comparing the different versions will yield the answer for your specific problem at hand!

verb : {0, 1, 2, 3, 4}, optional
Level of verbosity, default is 2:
• 0: Print nothing.
• 1: Print warnings.
• 2: Print additional runtime and kernel calls
• 3: Print additional start/stop, condensed parameter information.
• 4: Print additional full parameter information
Returns:
EM : ndarray, (nfreq, nrec, nsrc)
Frequency- or time-domain EM field (depending on signal):
• If rec is electric, returns E [V/m].
• If rec is magnetic, returns H [A/m].

However, source and receiver are normalised. So for instance in the electric case the source strength is 1 A and its length is 1 m. So the electric field could also be written as [V/(A.m2)].

The shape of EM is (nfreq, nrec, nsrc). However, single dimensions are removed.

bipole
Electromagnetic field due to an electromagnetic source.
fem
Electromagnetic frequency-domain response.
tem
Electromagnetic time-domain response.

Examples

>>> import numpy as np
>>> from empymod import dipole
>>> src = [0, 0, 100]
>>> rec = [np.arange(1, 11)*500, np.zeros(10), 200]
>>> depth = [0, 300, 1000, 1050]
>>> res = [1e20, .3, 1, 50, 1]
>>> EMfield = dipole(src, rec, depth, res, freqtime=1, verb=0)
>>> print(EMfield)
[  1.68809346e-10 -3.08303130e-10j  -8.77189179e-12 -3.76920235e-11j
-3.46654704e-12 -4.87133683e-12j  -3.60159726e-13 -1.12434417e-12j
1.87807271e-13 -6.21669759e-13j   1.97200208e-13 -4.38210489e-13j
1.44134842e-13 -3.17505260e-13j   9.92770406e-14 -2.33950871e-13j
6.75287598e-14 -1.74922886e-13j   4.62724887e-14 -1.32266600e-13j]

empymod.model.analytical(src, rec, res, freqtime, solution='fs', signal=None, ab=11, aniso=None, epermH=None, epermV=None, mpermH=None, mpermV=None, verb=2)[source]

Return the analytical full- or half-space solution.

Calculate the electromagnetic frequency- or time-domain field due to infinitesimal small electric or magnetic dipole source(s), measured by infinitesimal small electric or magnetic dipole receiver(s); sources and receivers are directed along the principal directions x, y, or z, and all sources are at the same depth, as well as all receivers are at the same depth.

In the case of a halfspace the air-interface is located at z = 0 m.

You can call the functions fullspace and halfspace in kernel.py directly. This interface is just to provide a consistent interface with the same input parameters as for instance for dipole.

This function yields the same result if solution='fs' as dipole, if the model is a fullspace.

Included are:
• Full fullspace solution (solution='fs') for ee-, me-, em-, mm-fields, only frequency domain, [HuTS15].
• Diffusive fullspace solution (solution='dfs') for ee-fields, [SlHM10].
• Diffusive halfspace solution (solution='dhs') for ee-fields, [SlHM10].
• Diffusive direct- and reflected field and airwave (solution='dsplit') for ee-fields, [SlHM10].
• Diffusive direct- and reflected field and airwave (solution='dtetm') for ee-fields, split into TE and TM mode [SlHM10].
Parameters:
src, rec : list of floats or arrays

Source and receiver coordinates (m): [x, y, z]. The x- and y-coordinates can be arrays, z is a single value. The x- and y-coordinates must have the same dimension.

res : float

Horizontal resistivity rho_h (Ohm.m).

Alternatively, res can be a dictionary. See the main manual of empymod too see how to exploit this hook to re-calculate etaH, etaV, zetaH, and zetaV, which can be used to, for instance, use the Cole-Cole model for IP.

freqtime : array_like

Frequencies f (Hz) if signal == None, else times t (s); (f, t > 0).

solution : str, optional
Defines which solution is returned:
• ‘fs’ : Full fullspace solution (ee-, me-, em-, mm-fields); f-domain.
• ‘dfs’ : Diffusive fullspace solution (ee-fields only).
• ‘dhs’ : Diffusive halfspace solution (ee-fields only).
• ‘dsplit’ : Diffusive direct- and reflected field and airwave
(ee-fields only).
• ‘dtetm’ : as dsplit, but direct fielt TE, TM; reflected field TE, TM,
and airwave (ee-fields only).
signal : {None, 0, 1, -1}, optional
Source signal, default is None:
• None: Frequency-domain response
• -1 : Switch-off time-domain response
• 0 : Impulse time-domain response
• +1 : Switch-on time-domain response
ab : int, optional

electric source magnetic source
x y z x y z

electric

x 11 12 13 14 15 16
y 21 22 23 24 25 26
z 31 32 33 34 35 36

magnetic

x 41 42 43 44 45 46
y 51 52 53 54 55 56
z 61 62 63 64 65 66
aniso : float, optional

Anisotropy lambda = sqrt(rho_v/rho_h) (-); defaults to one.

epermH, epermV : float, optional

Relative horizontal/vertical electric permittivity epsilon_h/epsilon_v (-); default is one. Ignored for the diffusive solution.

mpermH, mpermV : float, optional

Relative horizontal/vertical magnetic permeability mu_h/mu_v (-); default is one. Ignored for the diffusive solution.

verb : {0, 1, 2, 3, 4}, optional
Level of verbosity, default is 2:
• 0: Print nothing.
• 1: Print warnings.
• 3: Print additional start/stop, condensed parameter information.
• 4: Print additional full parameter information
Returns:
EM : ndarray, (nfreq, nrec, nsrc)
Frequency- or time-domain EM field (depending on signal):
• If rec is electric, returns E [V/m].
• If rec is magnetic, returns H [A/m].

However, source and receiver are normalised. So for instance in the electric case the source strength is 1 A and its length is 1 m. So the electric field could also be written as [V/(A.m2)].

The shape of EM is (nfreq, nrec, nsrc). However, single dimensions are removed.

If solution='dsplit', three ndarrays are returned: direct, reflect, air.

If solution='dtetm', five ndarrays are returned: direct_TE, direct_TM, reflect_TE, reflect_TM, air.

Examples

>>> import numpy as np
>>> from empymod import analytical
>>> src = [0, 0, 0]
>>> rec = [np.arange(1, 11)*500, np.zeros(10), 200]
>>> res = 50
>>> EMfield = analytical(src, rec, res, freqtime=1, verb=0)
>>> print(EMfield)
[  4.03091405e-08 -9.69163818e-10j   6.97630362e-09 -4.88342150e-10j
2.15205979e-09 -2.97489809e-10j   8.90394459e-10 -1.99313433e-10j
4.32915802e-10 -1.40741644e-10j   2.31674165e-10 -1.02579391e-10j
1.31469130e-10 -7.62770461e-11j   7.72342470e-11 -5.74534125e-11j
4.61480481e-11 -4.36275540e-11j   2.76174038e-11 -3.32860932e-11j]

empymod.model.gpr(src, rec, depth, res, freqtime, cf, gain=None, ab=11, aniso=None, epermH=None, epermV=None, mpermH=None, mpermV=None, xdirect=False, ht='quad', htarg=None, ft='fft', ftarg=None, opt=None, loop=None, verb=2)[source]

THIS FUNCTION IS EXPERIMENTAL, USE WITH CAUTION.

It is rather an example how you can calculate GPR responses; however, DO NOT RELY ON IT! It works only well with QUAD or QWE (quad, qwe) for the Hankel transform, and with FFT (fft) for the Fourier transform.

It calls internally dipole for the frequency-domain calculation. It subsequently convolves the response with a Ricker wavelet with central frequency cf. If signal!=None, it carries out the Fourier transform and applies a gain to the response.

For input parameters see the function dipole, except for:

Parameters: cf : float Centre frequency of GPR-signal, in Hz. Sensible values are between 10 MHz and 3000 MHz. gain : float Power of gain function. If None, no gain is applied. Only used if signal!=None. EM : ndarray GPR response
empymod.model.dipole_k(src, rec, depth, res, freq, wavenumber, ab=11, aniso=None, epermH=None, epermV=None, mpermH=None, mpermV=None, verb=2)[source]

Return the electromagnetic wavenumber-domain field.

Calculate the electromagnetic wavenumber-domain field due to infinitesimal small electric or magnetic dipole source(s), measured by infinitesimal small electric or magnetic dipole receiver(s); sources and receivers are directed along the principal directions x, y, or z, and all sources are at the same depth, as well as all receivers are at the same depth.

Parameters:
src, rec : list of floats or arrays

Source and receiver coordinates (m): [x, y, z]. The x- and y-coordinates can be arrays, z is a single value. The x- and y-coordinates must have the same dimension. The x- and y-coordinates only matter for the angle-dependent factor.

Sources or receivers placed on a layer interface are considered in the upper layer.

depth : list

Absolute layer interfaces z (m); #depth = #res - 1 (excluding +/- infinity).

res : array_like

Horizontal resistivities rho_h (Ohm.m); #res = #depth + 1.

freq : array_like

Frequencies f (Hz), used to calculate etaH/V and zetaH/V.

wavenumber : array

Wavenumbers lambda (1/m)

ab : int, optional

electric source magnetic source
x y z x y z

electric

x 11 12 13 14 15 16
y 21 22 23 24 25 26
z 31 32 33 34 35 36

magnetic

x 41 42 43 44 45 46
y 51 52 53 54 55 56
z 61 62 63 64 65 66
aniso : array_like, optional

Anisotropies lambda = sqrt(rho_v/rho_h) (-); #aniso = #res. Defaults to ones.

epermH, epermV : array_like, optional

Relative horizontal/vertical electric permittivities epsilon_h/epsilon_v (-); #epermH = #epermV = #res. Default is ones.

mpermH, mpermV : array_like, optional

Relative horizontal/vertical magnetic permeabilities mu_h/mu_v (-); #mpermH = #mpermV = #res. Default is ones.

verb : {0, 1, 2, 3, 4}, optional
Level of verbosity, default is 2:
• 0: Print nothing.
• 1: Print warnings.
• 2: Print additional runtime and kernel calls
• 3: Print additional start/stop, condensed parameter information.
• 4: Print additional full parameter information
Returns:
PJ0, PJ1 : array
Wavenumber-domain EM responses:
• PJ0: Wavenumber-domain solution for the kernel with a Bessel function of the first kind of order zero.
• PJ1: Wavenumber-domain solution for the kernel with a Bessel function of the first kind of order one.

dipole
Electromagnetic field due to an electromagnetic source (dipoles).
bipole
Electromagnetic field due to an electromagnetic source (bipoles).
fem
Electromagnetic frequency-domain response.
tem
Electromagnetic time-domain response.

Examples

>>> import numpy as np
>>> from empymod.model import dipole_k
>>> src = [0, 0, 100]
>>> rec = [5000, 0, 200]
>>> depth = [0, 300, 1000, 1050]
>>> res = [1e20, .3, 1, 50, 1]
>>> freq = 1
>>> wavenrs = np.logspace(-3.7, -3.6, 10)
>>> PJ0, PJ1 = dipole_k(src, rec, depth, res, freq, wavenrs, verb=0)
>>> print(PJ0)
[ -1.02638329e-08 +4.91531529e-09j  -1.05289724e-08 +5.04222413e-09j
-1.08009148e-08 +5.17238608e-09j  -1.10798310e-08 +5.30588284e-09j
-1.13658957e-08 +5.44279805e-09j  -1.16592877e-08 +5.58321732e-09j
-1.19601897e-08 +5.72722830e-09j  -1.22687889e-08 +5.87492067e-09j
-1.25852765e-08 +6.02638626e-09j  -1.29098481e-08 +6.18171904e-09j]
>>> print(PJ1)
[  1.79483705e-10 -6.59235332e-10j   1.88672497e-10 -6.93749344e-10j
1.98325814e-10 -7.30068377e-10j   2.08466693e-10 -7.68286748e-10j
2.19119282e-10 -8.08503709e-10j   2.30308887e-10 -8.50823701e-10j
2.42062030e-10 -8.95356636e-10j   2.54406501e-10 -9.42218177e-10j
2.67371420e-10 -9.91530051e-10j   2.80987292e-10 -1.04342036e-09j]

empymod.model.fem(ab, off, angle, zsrc, zrec, lsrc, lrec, depth, freq, etaH, etaV, zetaH, zetaV, xdirect, isfullspace, ht, htarg, use_ne_eval, msrc, mrec, loop_freq, loop_off, conv=True)[source]

Return the electromagnetic frequency-domain response.

This function is called from one of the above modelling routines. No input-check is carried out here. See the main description of model for information regarding input and output parameters.

This function can be directly used if you are sure the provided input is in the correct format. This is useful for inversion routines and similar, as it can speed-up the calculation by omitting input-checks.

empymod.model.tem(fEM, off, freq, time, signal, ft, ftarg, conv=True)[source]

Return the time-domain response of the frequency-domain response fEM.

This function is called from one of the above modelling routines. No input-check is carried out here. See the main description of model for information regarding input and output parameters.

This function can be directly used if you are sure the provided input is in the correct format. This is useful for inversion routines and similar, as it can speed-up the calculation by omitting input-checks.

empymod.model.wavenumber(src, rec, depth, res, freq, wavenumber, ab=11, aniso=None, epermH=None, epermV=None, mpermH=None, mpermV=None, verb=2)[source]

## kernel – Kernel calculation¶

Kernel of empymod, calculates the wavenumber-domain electromagnetic response. Plus analytical full- and half-space solutions.

The functions wavenumber, angle_factor, fullspace, greenfct, reflections, and fields are based on source files (specified in each function) from the source code distributed with [HuTS15], which can be found at software.seg.org/2015/0001. These functions are (c) 2015 by Hunziker et al. and the Society of Exploration Geophysicists, http://software.seg.org/disclaimer.txt. Please read the NOTICE-file in the root directory for more information regarding the involved licenses.

empymod.kernel.wavenumber(zsrc, zrec, lsrc, lrec, depth, etaH, etaV, zetaH, zetaV, lambd, ab, xdirect, msrc, mrec, use_ne_eval)[source]

Calculate wavenumber domain solution.

Return the wavenumber domain solutions PJ0, PJ1, and PJ0b, which have to be transformed with a Hankel transform to the frequency domain. PJ0/PJ0b and PJ1 have to be transformed with Bessel functions of order 0 ($$J_0$$) and 1 ($$J_1$$), respectively.

This function corresponds loosely to equations 105–107, 111–116, 119–121, and 123–128 in [HuTS15], and equally loosely to the file kxwmod.c.

[HuTS15] uses Bessel functions of orders 0, 1, and 2 ($$J_0, J_1, J_2$$). The implementations of the Fast Hankel Transform and the Quadrature-with-Extrapolation in transform are set-up with Bessel functions of order 0 and 1 only. This is achieved by applying the recurrence formula

$J_2(kr) = \frac{2}{kr} J_1(kr) - J_0(kr) \ .$

Note

PJ0 and PJ0b could theoretically be added here into one, and then be transformed in one go. However, PJ0b has to be multiplied by factAng later. This has to be done after the Hankel transform for methods which make use of spline interpolation, in order to work for offsets that are not in line with each other.

This function is called from one of the Hankel functions in transform. Consult the modelling routines in model for a description of the input and output parameters.

If you are solely interested in the wavenumber-domain solution you can call this function directly. However, you have to make sure all input arguments are correct, as no checks are carried out here.

empymod.kernel.angle_factor(angle, ab, msrc, mrec)[source]

Return the angle-dependent factor.

The whole calculation in the wavenumber domain is only a function of the distance between the source and the receiver, it is independent of the angel. The angle-dependency is this factor, which can be applied to the corresponding parts in the wavenumber or in the frequency domain.

The angle_factor corresponds to the sine and cosine-functions in Eqs 105-107, 111-116, 119-121, 123-128.

This function is called from one of the Hankel functions in transform. Consult the modelling routines in model for a description of the input and output parameters.

empymod.kernel.fullspace(off, angle, zsrc, zrec, etaH, etaV, zetaH, zetaV, ab, msrc, mrec)[source]

Analytical full-space solutions in the frequency domain.

$\hat{G}^{ee}_{\alpha\beta}, \hat{G}^{ee}_{3\alpha}, \hat{G}^{ee}_{33}, \hat{G}^{em}_{\alpha\beta}, \hat{G}^{em}_{\alpha 3}$

This function corresponds to equations 45–50 in [HuTS15], and loosely to the corresponding files Gin11.F90, Gin12.F90, Gin13.F90, Gin22.F90, Gin23.F90, Gin31.F90, Gin32.F90, Gin33.F90, Gin41.F90, Gin42.F90, Gin43.F90, Gin51.F90, Gin52.F90, Gin53.F90, Gin61.F90, and Gin62.F90.

This function is called from one of the modelling routines in model. Consult these modelling routines for a description of the input and output parameters.

empymod.kernel.greenfct(zsrc, zrec, lsrc, lrec, depth, etaH, etaV, zetaH, zetaV, lambd, ab, xdirect, msrc, mrec, use_ne_eval)[source]

Calculate Green’s function for TM and TE.

$\tilde{g}^{tm}_{hh}, \tilde{g}^{tm}_{hz}, \tilde{g}^{tm}_{zh}, \tilde{g}^{tm}_{zz}, \tilde{g}^{te}_{hh}, \tilde{g}^{te}_{zz}$

This function corresponds to equations 108–110, 117/118, 122; 89–94, A18–A23, B13–B15; 97–102 A26–A31, and B16–B18 in [HuTS15], and loosely to the corresponding files Gamma.F90, Wprop.F90, Ptotalx.F90, Ptotalxm.F90, Ptotaly.F90, Ptotalym.F90, Ptotalz.F90, and Ptotalzm.F90.

The Green’s functions are multiplied according to Eqs 105-107, 111-116, 119-121, 123-128; with the factors inside the integrals.

This function is called from the function kernel.wavenumber.

empymod.kernel.reflections(depth, e_zH, Gam, lrec, lsrc, use_ne_eval)[source]

Calculate Rp, Rm.

$R^\pm_n, \bar{R}^\pm_n$

This function corresponds to equations 64/65 and A-11/A-12 in [HuTS15], and loosely to the corresponding files Rmin.F90 and Rplus.F90.

This function is called from the function kernel.greenfct.

empymod.kernel.fields(depth, Rp, Rm, Gam, lrec, lsrc, zsrc, ab, TM, use_ne_eval)[source]

Calculate Pu+, Pu-, Pd+, Pd-.

$P^{u\pm}_s, P^{d\pm}_s, \bar{P}^{u\pm}_s, \bar{P}^{d\pm}_s; P^{u\pm}_{s-1}, P^{u\pm}_n, \bar{P}^{u\pm}_{s-1}, \bar{P}^{u\pm}_n; P^{d\pm}_{s+1}, P^{d\pm}_n, \bar{P}^{d\pm}_{s+1}, \bar{P}^{d\pm}_n$

This function corresponds to equations 81/82, 95/96, 103/104, A-8/A-9, A-24/A-25, and A-32/A-33 in [HuTS15], and loosely to the corresponding files Pdownmin.F90, Pdownplus.F90, Pupmin.F90, and Pdownmin.F90.

This function is called from the function kernel.greenfct.

empymod.kernel.halfspace(off, angle, zsrc, zrec, etaH, etaV, freqtime, ab, signal, solution='dhs')[source]

Return frequency- or time-space domain VTI half-space solution.

Calculates the frequency- or time-space domain electromagnetic response for a half-space below air using the diffusive approximation, as given in [SlHM10], where the electric source is located at [0, 0, zsrc], and the electric receiver at [xco, yco, zrec].

It can also be used to calculate the fullspace solution or the separate fields: direct field, reflected field, and airwave; always using the diffusive approximation. See solution-parameter.

This function is called from one of the modelling routines in model. Consult these modelling routines for a description of the input and solution parameters.

## transform – Hankel and Fourier Transforms¶

Methods to carry out the required Hankel transform from wavenumber to frequency domain and Fourier transform from frequency to time domain.

The functions for the QWE and DLF Hankel and Fourier transforms are based on source files (specified in each function) from the source code distributed with [Key12], which can be found at software.seg.org/2012/0003. These functions are (c) 2012 by Kerry Key and the Society of Exploration Geophysicists, http://software.seg.org/disclaimer.txt. Please read the NOTICE-file in the root directory for more information regarding the involved licenses.

empymod.transform.fht(zsrc, zrec, lsrc, lrec, off, factAng, depth, ab, etaH, etaV, zetaH, zetaV, xdirect, fhtarg, use_ne_eval, msrc, mrec)[source]

Hankel Transform using the Digital Linear Filter method.

The Digital Linear Filter method was introduced to geophysics by [Ghos70], and made popular and wide-spread by [Ande75], [Ande79], [Ande82]. The DLF is sometimes referred to as the Fast Hankel Transform FHT, from which this routine has its name.

This implementation of the DLF follows [Key12], equation 6. Without going into the mathematical details (which can be found in any of the above papers) and following [Key12], the DLF method rewrites the Hankel transform of the form

$F(r) = \int^\infty_0 f(\lambda)J_v(\lambda r)\ \mathrm{d}\lambda$

as

$F(r) = \sum^n_{i=1} f(b_i/r)h_i/r \ ,$

where $$h$$ is the digital filter.The Filter abscissae b is given by

$b_i = \lambda_ir = e^{ai}, \qquad i = -l, -l+1, \cdots, l \ ,$

with $$l=(n-1)/2$$, and $$a$$ is the spacing coefficient.

This function is loosely based on get_CSEM1D_FD_FHT.m from the source code distributed with [Key12].

The function is called from one of the modelling routines in model. Consult these modelling routines for a description of the input and output parameters.

Returns: fEM : array Returns frequency-domain EM response. kcount : int Kernel count. For DLF, this is 1. conv : bool Only relevant for QWE/QUAD.
empymod.transform.hqwe(zsrc, zrec, lsrc, lrec, off, factAng, depth, ab, etaH, etaV, zetaH, zetaV, xdirect, qweargs, use_ne_eval, msrc, mrec)[source]

Quadrature-With-Extrapolation was introduced to geophysics by [Key12]. It is one of many so-called ISE methods to solve Hankel Transforms, where ISE stands for Integration, Summation, and Extrapolation.

Following [Key12], but without going into the mathematical details here, the QWE method rewrites the Hankel transform of the form

$F(r) = \int^\infty_0 f(\lambda)J_v(\lambda r)\ \mathrm{d}\lambda$

as a quadrature sum which form is similar to the DLF (equation 15),

$F_i \approx \sum^m_{j=1} f(x_j/r)w_j g(x_j) = \sum^m_{j=1} f(x_j/r)\hat{g}(x_j) \ ,$

but with various bells and whistles applied (using the so-called Shanks transformation in the form of a routine called $$\epsilon$$-algorithm ([Shan55], [Wynn56]; implemented with algorithms from [Tref00] and [Weni89]).

This function is based on get_CSEM1D_FD_QWE.m, qwe.m, and getBesselWeights.m from the source code distributed with [Key12].

In the spline-version, hqwe checks how steep the decay of the wavenumber-domain result is, and calls QUAD for the very steep interval, for which QWE is not suited.

The function is called from one of the modelling routines in model. Consult these modelling routines for a description of the input and output parameters.

Returns: fEM : array Returns frequency-domain EM response. kcount : int Kernel count. conv : bool If true, QWE/QUAD converged. If not, might have to be adjusted.
empymod.transform.hquad(zsrc, zrec, lsrc, lrec, off, factAng, depth, ab, etaH, etaV, zetaH, zetaV, xdirect, quadargs, use_ne_eval, msrc, mrec)[source]

Hankel Transform using the QUADPACK library.

This routine uses the scipy.integrate.quad module, which in turn makes use of the Fortran library QUADPACK (qagse).

It is massively (orders of magnitudes) slower than either fht or hqwe, and is mainly here for completeness and comparison purposes. It always uses interpolation in the wavenumber domain, hence it generally will not be as precise as the other methods. However, it might work in some areas where the others fail.

The function is called from one of the modelling routines in model. Consult these modelling routines for a description of the input and output parameters.

Returns: fEM : array Returns frequency-domain EM response. kcount : int Kernel count. For HQUAD, this is 1. conv : bool If true, QUAD converged. If not, might have to be adjusted.
empymod.transform.ffht(fEM, time, freq, ftarg)[source]

Fourier Transform using the Digital Linear Filter method.

It follows the Filter methodology [Ande75], using Cosine- and Sine-filters; see fht for more information.

The function is called from one of the modelling routines in model. Consult these modelling routines for a description of the input and output parameters.

This function is based on get_CSEM1D_TD_FHT.m from the source code distributed with [Key12].

Returns: tEM : array Returns time-domain EM response of fEM for given time. conv : bool Only relevant for QWE/QUAD.
empymod.transform.fqwe(fEM, time, freq, qweargs)[source]

It follows the QWE methodology [Key12] for the Hankel transform, see hqwe for more information.

The function is called from one of the modelling routines in model. Consult these modelling routines for a description of the input and output parameters.

This function is based on get_CSEM1D_TD_QWE.m from the source code distributed with [Key12].

fqwe checks how steep the decay of the frequency-domain result is, and calls QUAD for the very steep interval, for which QWE is not suited.

Returns: tEM : array Returns time-domain EM response of fEM for given time. conv : bool If true, QWE/QUAD converged. If not, might have to be adjusted.
empymod.transform.fftlog(fEM, time, freq, ftarg)[source]

Fourier Transform using FFTLog.

FFTLog is the logarithmic analogue to the Fast Fourier Transform FFT. FFTLog was presented in Appendix B of [Hami00] and published at <http://casa.colorado.edu/~ajsh/FFTLog>.

This function uses a simplified version of pyfftlog, which is a python-version of FFTLog. For more details regarding pyfftlog see <https://github.com/prisae/pyfftlog>.

Not the full flexibility of FFTLog is available here: Only the logarithmic FFT (fftl in FFTLog), not the Hankel transform (fht in FFTLog). Furthermore, the following parameters are fixed:

• kr = 1 (initial value)
• kropt = 1 (silently adjusts kr)
• dir = 1 (forward)

Furthermore, q is restricted to -1 <= q <= 1.

The function is called from one of the modelling routines in model. Consult these modelling routines for a description of the input and output parameters.

Returns: tEM : array Returns time-domain EM response of fEM for given time. conv : bool Only relevant for QWE/QUAD.
empymod.transform.fft(fEM, time, freq, ftarg)[source]

Fourier Transform using the Fast Fourier Transform.

The function is called from one of the modelling routines in model. Consult these modelling routines for a description of the input and output parameters.

Returns: tEM : array Returns time-domain EM response of fEM for given time. conv : bool Only relevant for QWE/QUAD.
empymod.transform.dlf(signal, points, out_pts, filt, pts_per_dec, kind=None, factAng=None, ab=None, int_pts=None)[source]

Digital Linear Filter method.

This is the kernel of the DLF method, used for the Hankel (fht) and the Fourier (ffht) Transforms. See fht for an extensive description.

For the Hankel transform, signal contains 3 complex wavenumber-domain signals: (PJ0, PJ1, PJ0b), as returned from kernel.wavenumber. The Hankel DLF has two additional, optional parameters: factAng, as returned from kernel.angle_factor, and ab. The PJ0-kernel is the part of the wavenumber-domain calculation which contains a zeroth-order Bessel function and does NOT depend on the angle between source and receiver, only on offset. PJ0b and PJ1 are the parts of the wavenumber-domain calculation which contain a zeroth- and first-order Bessel function, respectively, and can depend on the angle between source and receiver. PJ0, PJ1, or PJ0b can also be None, if they are not used.

For the Fourier transform, signal is a complex frequency-domain signal. The Fourier DLF requires one additional parameter, kind, which will be ‘cos’ or ‘sin’.

empymod.transform.qwe(rtol, atol, maxint, inp, intervals, lambd=None, off=None, factAng=None)[source]

This is the kernel of the QWE method, used for the Hankel (hqwe) and the Fourier (fqwe) Transforms. See hqwe for an extensive description.

This function is based on qwe.m from the source code distributed with [Key12].

empymod.transform.get_spline_values(filt, inp, nr_per_dec=None)[source]

Return required calculation points.

empymod.transform.fhti(rmin, rmax, n, q, mu)[source]

Return parameters required for FFTLog.

## filters – Digital Linear Filters¶

Filters for the Digital Linear Filter (DLF) method for the Hankel [Ghos70]) and the Fourier ([Ande75]) transforms.

To calculate the dlf.factor I used

np.around(np.average(dlf.base[1:]/dlf.base[:-1]), 15)


The filters kong_61_2007 and kong_241_2007 from [Kong07], and key_101_2009, key_201_2009, key_401_2009, key_81_CosSin_2009, key_241_CosSin_2009, and key_601_CosSin_2009 from [Key09] are taken from DIPOLE1D, [Key09], which can be downloaded at http://marineemlab.ucsd.edu/Projects/Occam/1DCSEM (1DCSEM). DIPOLE1D is distributed under the license GNU GPL version 3 or later. Kerry Key gave his written permission to re-distribute the filters under the Apache License, Version 2.0 (email from Kerry Key to Dieter Werthmüller, 21 November 2016).

The filters anderson_801_1982 from [Ande82] and key_51_2012, key_101_2012, key_201_2012, key_101_CosSin_2012, and key_201_CosSin_2012, all from [Key12], are taken from the software distributed with [Key12] and available at http://software.seg.org/2012/0003 (SEG-2012-003). These filters are distributed under the SEG license.

The filter wer_201_2018 was designed with the add-on fdesign, see https://github.com/empymod/article-fdesign.

class empymod.filters.DigitalFilter(name, savename=None, filter_coeff=None)[source]

Simple Class for Digital Linear Filters.

Parameters: name : str Name of the DFL. savename = str Name with which the filter is saved. If None (default) it is set to the same value as name. filter_coeff = list of str By default, the following filter coefficients are checked: filter_coeff = ['j0', 'j1', 'sin', 'cos'] This accounts for the standard Hankel and Fourier DLF in CSEM modelling. However, additional coefficient names can be provided via this parameter (in list format).
fromfile(self, path='filters')[source]

Load filter base and filter coefficients from ascii files in the directory path; path can be a relative or absolute path.

Examples

>>> import empymod
>>> # Create an empty filter;
>>> # Name has to be the base of the text files
>>> filt = empymod.filters.DigitalFilter('my-filter')
>>> filt.fromfile()
>>> # This will load the following three files:
>>> #    ./filters/my-filter_base.txt
>>> #    ./filters/my-filter_j0.txt
>>> #    ./filters/my-filter_j1.txt
>>> # and store them in filt.base, filt.j0, and filt.j1.

tofile(self, path='filters')[source]

Save filter values to ascii-files.

Store the filter base and the filter coefficients in separate files in the directory path; path can be a relative or absolute path.

Examples

>>> import empymod
>>> filt = empymod.filters.wer_201_2018()
>>> # Save it to pure ascii-files
>>> filt.tofile()
>>> # This will save the following three files:
>>> #    ./filters/wer_201_2018_base.txt
>>> #    ./filters/wer_201_2018_j0.txt
>>> #    ./filters/wer_201_2018_j1.txt

empymod.filters.kong_61_2007()[source]

Kong 61 pt Hankel filter, as published in [Kong07].

Taken from file FilterModules.f90 provided with 1DCSEM.

empymod.filters.kong_241_2007()[source]

Kong 241 pt Hankel filter, as published in [Kong07].

Taken from file FilterModules.f90 provided with 1DCSEM.

empymod.filters.key_101_2009()[source]

Key 101 pt Hankel filter, as published in [Key09].

Taken from file FilterModules.f90 provided with 1DCSEM.

empymod.filters.key_201_2009()[source]

Key 201 pt Hankel filter, as published in [Key09].

Taken from file FilterModules.f90 provided with 1DCSEM.

empymod.filters.key_401_2009()[source]

Key 401 pt Hankel filter, as published in [Key09].

Taken from file FilterModules.f90 provided with 1DCSEM.

empymod.filters.anderson_801_1982()[source]

Anderson 801 pt Hankel filter, as published in [Ande82].

Taken from file wa801Hankel.txt provided with SEG-2012-003.

empymod.filters.key_51_2012()[source]

Key 51 pt Hankel filter, as published in [Key12].

Taken from file kk51Hankel.txt provided with SEG-2012-003.

empymod.filters.key_101_2012()[source]

Key 101 pt Hankel filter, as published in [Key12].

Taken from file kk101Hankel.txt provided with SEG-2012-003.

empymod.filters.key_201_2012()[source]

Key 201 pt Hankel filter, as published in [Key12].

Taken from file kk201Hankel.txt provided with SEG-2012-003.

empymod.filters.wer_201_2018()[source]

Werthmüller 201 pt Hankel filter, 2018.

Designed with the empymod add-on fdesign, see https://github.com/empymod/article-fdesign.

empymod.filters.key_81_CosSin_2009()[source]

Key 81 pt CosSin filter, as published in [Key09].

Taken from file FilterModules.f90 provided with 1DCSEM.

empymod.filters.key_241_CosSin_2009()[source]

Key 241 pt CosSin filter, as published in [Key09].

Taken from file FilterModules.f90 provided with 1DCSEM.

empymod.filters.key_601_CosSin_2009()[source]

Key 601 pt CosSin filter, as published in [Key09].

Taken from file FilterModules.f90 provided with 1DCSEM.

empymod.filters.key_101_CosSin_2012()[source]

Key 101 pt CosSin filter, as published in [Key12].

Taken from file kk101CosSin.txt provided with SEG-2012-003.

empymod.filters.key_201_CosSin_2012()[source]

Key 201 pt CosSin filter, as published in [Key12].

Taken from file kk201CosSin.txt provided with SEG-2012-003.

## utils – Utilites¶

Utilities for model such as checking input parameters.

This module consists of four groups of functions:
1. General settings
2. Class EMArray
3. Input parameter checks for modelling
4. Internal utilities
class empymod.utils.EMArray[source]

Subclassing an ndarray: add amplitude <amp> and phase <pha>.

Parameters: realpart : array Real part of input, if input is real or complex. Imaginary part of input, if input is pure imaginary. Complex input. In cases 2 and 3, imagpart must be None. imagpart: array, optional Imaginary part of input. Defaults to None.

Examples

>>> import numpy as np
>>> from empymod.utils import EMArray
>>> emvalues = EMArray(np.array([1,2,3]), np.array([1, 0, -1]))
>>> print('Amplitude : ', emvalues.amp)
Amplitude :  [ 1.41421356  2.          3.16227766]
>>> print('Phase     : ', emvalues.pha)
Phase     :  [ 45.           0.         -18.43494882]

Attributes: amp : ndarray Amplitude of the input data. pha : ndarray Phase of the input data, in degrees, lag-defined (increasing with increasing offset.) To get lead-defined phases, multiply imagpart by -1 before passing through this function.
empymod.utils.check_time_only(time, signal, verb)[source]

Check time and signal parameters.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: time : array_like Times t (s). signal : {None, 0, 1, -1} Source signal: None: Frequency-domain response -1 : Switch-off time-domain response 0 : Impulse time-domain response +1 : Switch-on time-domain response verb : {0, 1, 2, 3, 4} Level of verbosity. time : float Time, checked for size and assured min_time.
empymod.utils.check_time(time, signal, ft, ftarg, verb)[source]

Check time domain specific input parameters.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: time : array_like Times t (s). signal : {None, 0, 1, -1} Source signal: None: Frequency-domain response -1 : Switch-off time-domain response 0 : Impulse time-domain response +1 : Switch-on time-domain response ft : {‘sin’, ‘cos’, ‘qwe’, ‘fftlog’, ‘fft’} Flag for Fourier transform. ftarg : str or filter from empymod.filters or array_like, Only used if signal !=None. Depends on the value for ft: verb : {0, 1, 2, 3, 4} Level of verbosity. time : float Time, checked for size and assured min_time. freq : float Frequencies required for given times and ft-settings. ft, ftarg Checked if valid and set to defaults if not provided, checked with signal.
empymod.utils.check_model(depth, res, aniso, epermH, epermV, mpermH, mpermV, xdirect, verb)[source]

Check the model: depth and corresponding layer parameters.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: depth : list Absolute layer interfaces z (m); #depth = #res - 1 (excluding +/- infinity). res : array_like Horizontal resistivities rho_h (Ohm.m); #res = #depth + 1. aniso : array_like Anisotropies lambda = sqrt(rho_v/rho_h) (-); #aniso = #res. epermH, epermV : array_like Relative horizontal/vertical electric permittivities epsilon_h/epsilon_v (-); #epermH = #epermV = #res. mpermH, mpermV : array_like Relative horizontal/vertical magnetic permeabilities mu_h/mu_v (-); #mpermH = #mpermV = #res. xdirect : bool, optional If True and source and receiver are in the same layer, the direct field is calculated analytically in the frequency domain, if False it is calculated in the wavenumber domain. verb : {0, 1, 2, 3, 4} Level of verbosity. depth : array Depths of layer interfaces, adds -infty at beginning if not present. res : array As input, checked for size. aniso : array As input, checked for size. If None, defaults to an array of ones. epermH, epermV : array_like As input, checked for size. If None, defaults to an array of ones. mpermH, mpermV : array_like As input, checked for size. If None, defaults to an array of ones. isfullspace : bool If True, the model is a fullspace (res, aniso, epermH, epermV, mpermM, and mpermV are in all layers the same).
empymod.utils.check_frequency(freq, res, aniso, epermH, epermV, mpermH, mpermV, verb)[source]

Calculate frequency-dependent parameters.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: freq : array_like Frequencies f (Hz). res : array_like Horizontal resistivities rho_h (Ohm.m); #res = #depth + 1. aniso : array_like Anisotropies lambda = sqrt(rho_v/rho_h) (-); #aniso = #res. epermH, epermV : array_like Relative horizontal/vertical electric permittivities epsilon_h/epsilon_v (-); #epermH = #epermV = #res. mpermH, mpermV : array_like Relative horizontal/vertical magnetic permeabilities mu_h/mu_v (-); #mpermH = #mpermV = #res. verb : {0, 1, 2, 3, 4} Level of verbosity. freq : float Frequency, checked for size and assured min_freq. etaH, etaV : array Parameters etaH/etaV, same size as provided resistivity. zetaH, zetaV : array Parameters zetaH/zetaV, same size as provided resistivity.
empymod.utils.check_hankel(ht, htarg, verb)[source]

Check Hankel transform parameters.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: ht : {‘fht’, ‘qwe’, ‘quad’} Flag to choose the Hankel transform. htarg : str or filter from empymod.filters or array_like, Depends on the value for ht. verb : {0, 1, 2, 3, 4} Level of verbosity. ht, htarg Checked if valid and set to defaults if not provided.
empymod.utils.check_opt(opt, loop, ht, htarg, verb)[source]

Check optimization parameters.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: opt : {None, ‘parallel’} Optimization flag; use numexpr or not. loop : {None, ‘freq’, ‘off’} Loop flag. ht : str Flag to choose the Hankel transform. htarg : array_like, Depends on the value for ht. verb : {0, 1, 2, 3, 4} Level of verbosity. use_ne_eval : bool Boolean if to use numexpr. loop_freq : bool Boolean if to loop over frequencies. loop_off : bool Boolean if to loop over offsets.
empymod.utils.check_dipole(inp, name, verb)[source]

Check dipole parameters.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: inp : list of floats or arrays Pole coordinates (m): [pole-x, pole-y, pole-z]. name : str, {‘src’, ‘rec’} Pole-type. verb : {0, 1, 2, 3, 4} Level of verbosity. inp : list List of pole coordinates [x, y, z]. ninp : int Number of inp-elements
empymod.utils.check_bipole(inp, name)[source]

Check di-/bipole parameters.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: inp : list of floats or arrays Coordinates of inp (m): [dipole-x, dipole-y, dipole-z, azimuth, dip] or. [bipole-x0, bipole-x1, bipole-y0, bipole-y1, bipole-z0, bipole-z1]. name : str, {‘src’, ‘rec’} Pole-type. inp : list As input, checked for type and length. ninp : int Number of inp. ninpz : int Number of inp depths (ninpz is either 1 or ninp). isdipole : bool True if inp is a dipole.
empymod.utils.check_ab(ab, verb)[source]

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: ab : int Source-receiver configuration. verb : {0, 1, 2, 3, 4} Level of verbosity. ab_calc : int Adjusted source-receiver configuration using reciprocity. msrc, mrec : bool If True, src/rec is magnetic; if False, src/rec is electric.
empymod.utils.check_solution(solution, signal, ab, msrc, mrec)[source]

Check required solution with parameters.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: solution : str String to define analytical solution. signal : {None, 0, 1, -1} Source signal: None: Frequency-domain response -1 : Switch-off time-domain response 0 : Impulse time-domain response +1 : Switch-on time-domain response msrc, mrec : bool True if src/rec is magnetic, else False.
empymod.utils.get_abs(msrc, mrec, srcazm, srcdip, recazm, recdip, verb)[source]

Get required ab’s for given angles.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: msrc, mrec : bool True if src/rec is magnetic, else False. srcazm, recazm : float Horizontal source/receiver angle (azimuth). srcdip, recdip : float Vertical source/receiver angle (dip). verb : {0, 1, 2, 3, 4} Level of verbosity. ab_calc : array of int ab’s to calculate for this bipole.
empymod.utils.get_geo_fact(ab, srcazm, srcdip, recazm, recdip, msrc, mrec)[source]

Get required geometrical scaling factor for given angles.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: ab : int Source-receiver configuration. srcazm, recazm : float Horizontal source/receiver angle. srcdip, recdip : float Vertical source/receiver angle. fact : float Geometrical scaling factor.
empymod.utils.get_azm_dip(inp, iz, ninpz, intpts, isdipole, strength, name, verb)[source]

Get angles, interpolation weights and normalization weights.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: inp : list of floats or arrays Input coordinates (m): [x0, x1, y0, y1, z0, z1] (bipole of finite length) [x, y, z, azimuth, dip] (dipole, infinitesimal small) iz : int Index of current di-/bipole depth (-). ninpz : int Total number of di-/bipole depths (ninpz = 1 or npinz = nsrc) (-). intpts : int Number of integration points for bipole (-). isdipole : bool Boolean if inp is a dipole. strength : float, optional Source strength (A): If 0, output is normalized to source and receiver of 1 m length, and source strength of 1 A. If != 0, output is returned for given source and receiver length, and source strength. name : str, {‘src’, ‘rec’} Pole-type. verb : {0, 1, 2, 3, 4} Level of verbosity. tout : list of floats or arrays Dipole coordinates x, y, and z (m). azm : float or array of floats Horizontal angle (azimuth). dip : float or array of floats Vertical angle (dip). g_w : float or array of floats Factors from Gaussian interpolation. intpts : int As input, checked. inp_w : float or array of floats Factors from source/receiver length and source strength.
empymod.utils.get_off_ang(src, rec, nsrc, nrec, verb)[source]

Get depths, offsets, angles, hence spatial input parameters.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: src, rec : list of floats or arrays Source/receiver dipole coordinates x, y, and z (m). nsrc, nrec : int Number of sources/receivers (-). verb : {0, 1, 2, 3, 4} Level of verbosity. off : array of floats Offsets angle : array of floats Angles
empymod.utils.get_layer_nr(inp, depth)[source]

Get number of layer in which inp resides.

Note: If zinp is on a layer interface, the layer above the interface is chosen.

This check-function is called from one of the modelling routines in model. Consult these modelling routines for a detailed description of the input parameters.

Parameters: inp : list of floats or arrays Dipole coordinates (m) depth : array Depths of layer interfaces. linp : int or array_like of int Layer number(s) in which inp resides (plural only if bipole). zinp : float or array inp[2] (depths).
empymod.utils.printstartfinish(verb, inp=None, kcount=None)[source]

Print start and finish with time measure and kernel count.

empymod.utils.conv_warning(conv, targ, name, verb)[source]

Print error if QWE/QUAD did not converge at least once.

empymod.utils.set_minimum(min_freq=None, min_time=None, min_off=None, min_res=None, min_angle=None)[source]

Set minimum values of parameters.

The given parameters are set to its minimum value if they are smaller.

Parameters: min_freq : float, optional Minimum frequency [Hz] (default 1e-20 Hz). min_time : float, optional Minimum time [s] (default 1e-20 s). min_off : float, optional Minimum offset [m] (default 1e-3 m). Also used to round src- & rec-coordinates. min_res : float, optional Minimum horizontal and vertical resistivity [Ohm.m] (default 1e-20). min_angle : float, optional Minimum angle [-] (default 1e-10).
empymod.utils.get_minimum()[source]

Return the current minimum values.

Returns: min_vals : dict Dictionary of current minimum values with keys min_freq : float min_time : float min_off : float min_res : float min_angle : float For a full description of these options, see set_minimum.
empymod.utils.spline_backwards_hankel(ht, htarg, opt)[source]

Check opt if deprecated ‘spline’ is used.

Returns corrected htarg, opt. r

class empymod.utils.Report(add_pckg=None, ncol=3, text_width=80, sort=False)[source]

Print date, time, and version information.

Use scooby to print date, time, and package version information in any environment (Jupyter notebook, IPython console, Python console, QT console), either as html-table (notebook) or as plain text (anywhere).

Always shown are the OS, number of CPU(s), numpy, scipy, empymod, sys.version, and time/date.

Additionally shown are, if they can be imported, numexpr, IPython, and matplotlib. It also shows MKL information, if available.

All modules provided in add_pckg are also shown.

Parameters: add_pckg : packages, optional Package or list of packages to add to output information (must be imported beforehand). ncol : int, optional Number of package-columns in html table (no effect in text-version); Defaults to 3. text_width : int, optional The text width for non-HTML display modes sort : bool, optional Sort the packages when the report is shown

Examples

>>> import pytest
>>> import dateutil
>>> from emg3d import Report
>>> Report()                            # Default values
>>> Report(pytest)                      # Provide additional package
>>> Report([pytest, dateutil], ncol=5)  # Set nr of columns

class empymod.utils.Versions(add_pckg=None, ncol=3)[source]

New name is Report, here for backwards compatibility.

empymod.utils.versions(mode=None, add_pckg=None, ncol=4)[source]

Old func-way of class Report, here for backwards compatibility.