Installation & requirements¶

The easiest way to install the latest stable version of empymod is via conda:

> conda install -c prisae empymod


or via pip:

> pip install empymod


> python setup.py install


Required are python version 3.4 or higher and the modules NumPy and SciPy. If you want to run parts of the kernel in parallel, the module numexpr is required additionally.

Note: Do not use scipy == 0.19.0. It has a memory leak in quad, see github.com/scipy/scipy/pull/7216. So if you use QUAD (or potentially QWE) in any of your transforms you might see your memory usage going through the roof.

If you are new to Python I recommend using a Python distribution, which will ensure that all dependencies are met, specifically properly compiled versions of NumPy and SciPy; I recommend using Anaconda (version 3.x; anaconda.com/download). If you install Anaconda you can simply start the Anaconda Navigator, add the channel prisae and empymod will appear in the package list and can be installed with a click.

Usage¶

The main modelling routines is bipole, which can 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. The model is defined by horizontal resistivity and anisotropy, horizontal and vertical electric permittivities and horizontal and vertical magnetic permeabilities. By default, the electromagnetic response is normalized to source and receiver of 1 m length, and source strength of 1 A.

A simple frequency-domain example, with most of the parameters left at the default value:

>>> 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          :  Fast Hankel Transform
> Filter      :  Key 201 (2009)
> pts_per_dec :  Defined by filter (lagged)
Hankel 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]


Frequency- and time-domain examples can be found in the empymod/example-notebooks-repository.

More information and more examples can be found in the following articles:

Structure¶

• model.py: EM modelling routines.
• utils.py: Utilities for model such as checking input parameters.
• kernel.py: Kernel of empymod, calculates the wavenumber-domain electromagnetic response. Plus analytical, frequency-domain full- and half-space solutions.
• transform.py: Methods to carry out the required Hankel transform from wavenumber to space domain and Fourier transform from frequency to time domain.
• filters.py: Filters for the Fast Hankel Transform FHT [Anderson_19820], and the Fourier Sine and Cosine Transforms [Anderson_19750].

Missing features¶

A list of things that should or could be added and improved can be found in the Roadmap.

Testing¶

The modeller comes with a test suite using pytest. If you want to run the tests, just install pytest and run it within the empymod-top-directory.

> conda install pytest
> # or
> pip install pytest
> # and then
> cd to/the/empymod/folder  # Ensure you are in the right directory,
> ls -d */                  # your output should look the same.
docs/  empymod/  tests/
> # pytest will find the tests, which are located in the tests-folder.
> # simply run
> pytest


It should run all tests successfully. Please let me know if not!

Note that installations of empymod via conda or pip do not have the test-suite included. To run the test-suite you must download empymod from GitHub.

Info¶

Citation¶

Werthmüller, D., 2017, An open-source full 3D electromagnetic modeler for 1D VTI media in Python: empymod: Geophysics, 82, WB9–WB19; DOI: 10.1190/geo2016-0626.1.

Also consider citing [Hunziker_et_al_20150] and [Key_20120], without which empymod would not existk

All releases have a Zenodo-DOI, provided on the release-page.

Notice¶

This product includes software that was initially (till 01/2017) developed at The Mexican Institute of Petroleum IMP (Instituto Mexicano del Petróleo, http://www.gob.mx/imp). The project was funded through The Mexican National Council of Science and Technology (Consejo Nacional de Ciencia y Tecnología, http://www.conacyt.mx). Since 02/2017 it is a personal effort, and new contributors are welcome!

Licensed under the Apache License, Version 2.0 (the “License”); you may not use this file except in compliance with the License. You may obtain a copy of the License at

Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License.

See the LICENSE-file in the root directory for a full reprint of the Apache License.

Note on speed, memory, and accuracy¶

There is the usual trade-off between speed, memory, and accuracy. Very generally speaking we can say that the FHT is faster than QWE, but QWE is much easier on memory usage. QWE allows you to control the accuracy. A standard quadrature in the form of QUAD is also provided. QUAD is generally orders of magnitudes slower, and more fragile depending on the input arguments. However, it can provide accurate results where FHT and QWE fail.

There are two optimisation possibilities included via the opt-flag: parallelisation (opt='parallel') and spline interpolation (opt='spline'). They are switched off by default. The optimization opt='parallel' only affects speed and memory usage, whereas opt='spline' also affects precision!

I am sure empymod could be made much faster with cleverer coding style or with the likes of cython or numba. Suggestions and contributions are welcomed!

Included transforms¶

Hankel transform:

• Fast Hankel Transform FHT ([Gosh_19710])
• Quadrature with Extrapolation QWE ([Key_20120])

Fourier transform:

FFTLog¶

FFTLog is the logarithmic analogue to the Fast Fourier Transform FFT originally proposed by [Talman_19780]. The code used by empymod was published in Appendix B of [Hamilton_20000] and is publicly available at casa.colorado.edu/~ajsh/FFTLog. From the FFTLog-website:

FFTLog is a set of fortran subroutines that compute the fast Fourier or Hankel (= Fourier-Bessel) transform of a periodic sequence of logarithmically spaced points.

FFTlog can be used for the Hankel as well as for the Fourier Transform, but currently empymod uses it only for the Fourier transform. It uses a simplified version of the python implementation of FFTLog, pyfftlog (github.com/prisae/pyfftlog).

[Haines_and_Jones_19880] proposed a logarithmic Fourier transform (abbreviated by the authors as LFT) for electromagnetic geophysics, also based on [Talman_19780]. I do not know if Hamilton was aware of the work by Haines and Jones. The two publications share as reference only the original paper by Talman, and both cite a publication of Anderson; Hamilton cites [Anderson_19820], and Haines and Jones cite [Anderson_19790]. Hamilton probably never heard of Haines and Jones, as he works in astronomy, and Haines and Jones was published in the Geophysical Journal.

Logarithmic FFTs are not widely used in electromagnetics, as far as I know, probably because of the ease, speed, and generally sufficient precision of the digital filter methods with sine and cosine transforms ([Anderson_19750]). However, comparisons show that FFTLog can be faster and more precise than digital filters, specifically for responses with source and receiver at the interface between air and subsurface. Credit to use FFTLog in electromagnetics goes to David Taylor who, in the mid-2000s, implemented FFTLog into the forward modellers of the company Multi-Transient ElectroMagnetic (MTEM Ltd, later Petroleum Geo-Services PGS). The implementation was driven by land responses, where FFTLog can be much more precise than the filter method for very early times.

Notes on Fourier Transform¶

The Fourier transform to obtain the space-time domain impulse response from the complex-valued space-frequency response can be calculated by either a cosine transform with the real values, or a sine transform with the imaginary part,

$\begin{split}E(r, t)^\text{Impulse} &= \ \frac{2}{\pi}\int^\infty_0 \Re[E(r, \omega)] \cos(\omega t)\ \text{d}\omega \ , \\ &= -\frac{2}{\pi}\int^\infty_0 \Im[E(r, \omega)] \sin(\omega t)\ \text{d}\omega \ ,\end{split}$

see, e.g., [Anderson_19750] or [Key_20120]. Quadrature-with-extrapolation, FFTLog, and obviously the sine/cosine-transform all make use of this split.

To obtain the step-on response the frequency-domain result is first divided by $$i\omega$$, in the case of the step-off response it is additionally multiplied by -1. The impulse-response is the time-derivative of the step-response,

$E(r, t)^\text{Impulse} = \frac{\partial\ E(r, t)^\text{step}}{\partial t}\ .$

Using $$\frac{\partial}{\partial t} \Leftrightarrow i\omega$$ and going the other way, from impulse to step, leads to the divison by $$i\omega$$. (This only holds because we define in accordance with the causality principle that $$E(r, t \le 0) = 0$$).

With the sine/cosine transform (ft=’ffht’/’sin’/’cos’) you can choose which one you want for the impulse responses. For the switch-on response, however, the sine-transform is enforced, and equally the cosine transform for the switch-off response. This is because these two do not need to now the field at time 0, $$E(r, t=0)$$.

The Quadrature-with-extrapolation and FFTLog are hard-coded to use the cosine transform for step-off responses, and the sine transform for impulse and step-on responses. The FFT uses the full complex-valued response at the moment.

For completeness sake, the step-on response is given by

$E(r, t)^\text{Step-on} = - \frac{2}{\pi}\int^\infty_0 \Im\left[\frac{E(r,\omega)}{i \omega}\right] \sin(\omega t)\ \text{d}\omega \ ,$

and the step-off by

$E(r, t)^\text{Step-off} = - \frac{2}{\pi}\int^\infty_0 \Re\left[\frac{E(r,\omega)}{i\omega}\right] \cos(\omega t)\ \text{d}\omega \ .$

Depths, Rotation, and Bipole¶

Depths: Calculation of many source and receiver positions is fastest if they remain at the same depth, as they can be calculated in one kernel-call. If depths do change, one has to loop over them. Note: Sources or receivers placed on a layer interface are considered in the upper layer.

Rotation: Sources and receivers aligned along the principal axes x, y, and z can be calculated in one kernel call. For arbitrary oriented di- or bipoles, 3 kernel calls are required. If source and receiver are arbitrary oriented, 9 (3x3) kernel calls are required.

Bipole: Bipoles increase the calculation time by the amount of integration points used. For a source and a receiver bipole with each 5 integration points you need 25 (5x5) kernel calls. You can calculate it in 1 kernel call if you set both integration points to 1, and therefore calculate the bipole as if they were dipoles at their centre.

Example: For 1 source and 10 receivers, all at the same depth, 1 kernel call is required. If all receivers are at different depths, 10 kernel calls are required. If you make source and receivers bipoles with 5 integration points, 250 kernel calls are required. If you rotate the source arbitrary horizontally, 500 kernel calls are required. If you rotate the receivers too, in the horizontal plane, 1‘000 kernel calls are required. If you rotate the receivers also vertically, 1‘500 kernel calls are required. If you rotate the source vertically too, 2‘250 kernel calls are required. So your calculation will take 2‘250 times longer! No matter how fast the kernel is, this will take a long time. Therefore carefully plan how precise you want to define your source and receiver bipoles.

Example as a table for comparison: 1 source, 10 receiver (one or many frequencies).
kernel calls intpts azimuth dip intpts azimuth dip diff. z
1 1 0/90 0/90 1 0/90 0/90 1
10 1 0/90 0/90 1 0/90 0/90 10
250 5 0/90 0/90 5 0/90 0/90 10
500 5 arb. 0/90 5 0/90 0/90 10
1000 5 arb. 0/90 5 arb. 0/90 10
1500 5 arb. 0/90 5 arb. arb. 10
2250 5 arb. arb. 5 arb. arb. 10

Parallelisation¶

If opt = 'parallel', a good dozen of the most time-consuming statements are calculated by using the numexpr package (https://github.com/pydata/numexpr/wiki/Numexpr-Users-Guide). These statements are all in the kernel-functions greenfct, reflections, and fields, and all involve $$\Gamma$$ in one way or another, often calculating square roots or exponentials. As $$\Gamma$$ has dimensions (#frequencies, #offsets, #layers, #lambdas), it can become fairly big.

The module numexpr uses by default all available cores up to a maximum of 8. You can change this behaviour to a lower or a higher value with the following command (in the example it is changed to 4):

>>> import numexpr


This parallelisation will make empymod faster if you calculate a lot of offsets/frequencies at once, but slower for few offsets/frequencies. Best practice is to check first which one is faster. (You can use the benchmark-notebook in the empymod/example-notebooks-repository.)

Spline interpolation¶

If opt = 'spline', the so-called lagged convolution or splined variant of the FHT (depending on htarg) or the splined version of the QWE are applied. The spline option should be used with caution, as it is an interpolation and therefore less precise than the non-spline version. However, it significantly speeds up QWE, and massively speeds up FHT. (The numexpr-version of the spline option is slower than the pure spline one, and therefore it is only possible to have either 'parallel' or 'spline' on.)

Setting opt = 'spline' is generally faster. Good speed-up is achieved for QWE by setting maxint as low as possible. Also, the higher nquad is, the higher the speed-up will be. The variable pts_per_dec has also some influence. For FHT, big improvements are achieved for long FHT-filters and for many offsets/frequencies (thousands). Additionally, spline minimizes memory requirements a lot. Speed-up is greater if all source-receiver angles are identical.

FHT: Default for pts_per_dec = None, which is the original lagged convolution, where the spacing is defined by the filter-base, the transform is carried out first followed by spline-interpolation. You can set this parameter to an integer, which defines the number of points to evaluate per decade. In this case the spline-interpolation is carried out first, followed by the transformation. The original lagged convolution is generally the fastest for a very good precision. However, by setting pts_per_dec appropriately one can achieve higher precision, normally at the cost of speed.

Warning

Keep in mind that it uses interpolation, and is therefore not as accurate as the non-spline version. Use with caution and always compare with the non-spline version if you can apply the spline-version to your problem at hand!

Be aware that QUAD (Hankel transform) always use the splined version and always loop over offsets. The same applies for all frequency-to-time transformations.

The splined versions of QWE check whether the ratio of any two adjacent intervals is above a certain threshold (steep end of the wavenumber or frequency spectrum). If it is, it carries out QUAD for this interval instead of QWE. The threshold is stored in diff_quad, which can be changed within the parameter htarg and ftarg.

Looping¶

By default, you can calculate many offsets and many frequencies all in one go, vectorized (for the FHT), which is the default. The loop parameter gives you the possibility to force looping over frequencies or offsets. This parameter can have severe effects on both runtime and memory usage. Play around with this factor to find the fastest version for your problem at hand. It ALWAYS loops over frequencies if ht = 'QWE'/'QUAD' or if opt = 'spline'. All vectorized is very fast if there are few offsets or few frequencies. If there are many offsets and many frequencies, looping over the smaller of the two will be faster. Choosing the right looping together with opt = 'parallel' can have a huge influence.

Vertical components¶

It is advised to use xdirect = True (the default) if source and receiver are in the same layer to calculate

• the vertical electric field due to a vertical electric source,
• configurations that involve vertical magnetic components (source or receiver),
• all configurations when source and receiver depth are exactly the same.

The Hankel transforms methods are having sometimes difficulties transforming these functions.

References ¶

 [Anderson_19750] (1, 2, 3, 4) Anderson, W.L., 1975, Improved digital filters for evaluating Fourier and Hankel transform integrals: USGS Unnumbered Series; http://pubs.usgs.gov/unnumbered/70045426/report.pdf.
 [Anderson_19790] Anderson, W. L., 1979, Numerical integration of related Hankel transforms of orders 0 and 1 by adaptive digital filtering: Geophysics, 44, 1287–1305; DOI: 10.1190/1.1441007.
 [Anderson_19820] (1, 2) Anderson, W. L., 1982, Fast Hankel transforms using related and lagged convolutions: ACM Trans. on Math. Softw. (TOMS), 8, 344–368; DOI: 10.1145/356012.356014.
 [Gosh_19710] Ghosh, D. P., 1971, The application of linear filter theory to the direct interpretation of geoelectrical resistivity sounding measurements: Geophysical Prospecting, 19, 192–217; DOI: 10.1111/j.1365-2478.1971.tb00593.x.
 [Haines_and_Jones_19880] Haines, G. V., and A. G. Jones, 1988, Logarithmic Fourier transformation: Geophysical Journal, 92, 171–178; DOI: 10.1111/j.1365-246X.1988.tb01131.x.
 [Hamilton_20000] (1, 2) Hamilton, A. J. S., 2000, Uncorrelated modes of the non-linear power spectrum: Monthly Notices of the Royal Astronomical Society, 312, pages 257-284; DOI: 10.1046/j.1365-8711.2000.03071.x; Website of FFTLog: casa.colorado.edu/~ajsh/FFTLog.
 [Hunziker_et_al_20150] Hunziker, J., J. Thorbecke, and E. Slob, 2015, The electromagnetic response in a layered vertical transverse isotropic medium: A new look at an old problem: Geophysics, 80, F1–F18; DOI: 10.1190/geo2013-0411.1; Software: software.seg.org/2015/0001.
 [Key_20090] Key, K., 2009, 1D inversion of multicomponent, multifrequency marine CSEM data: Methodology and synthetic studies for resolving thin resistive layers: Geophysics, 74, F9–F20; DOI: 10.1190/1.3058434. Software: marineemlab.ucsd.edu/Projects/Occam/1DCSEM.
 [Key_20120] (1, 2, 3, 4) Key, K., 2012, Is the fast Hankel transform faster than quadrature?: Geophysics, 77, F21–F30; DOI: 10.1190/geo2011-0237.1; Software: software.seg.org/2012/0003.
 [Kong_20070] Kong, F. N., 2007, Hankel transform filters for dipole antenna radiation in a conductive medium: Geophysical Prospecting, 55, 83–89; DOI: 10.1111/j.1365-2478.2006.00585.x.
 [Shanks_19550] Shanks, D., 1955, Non-linear transformations of divergent and slowly convergent sequences: Journal of Mathematics and Physics, 34, 1–42; DOI: 10.1002/sapm19553411.
 [Slob_et_al_20100] Slob, E., J. Hunziker, and W. A. Mulder, 2010, Green’s tensors for the diffusive electric field in a VTI half-space: PIER, 107, 1–20: DOI: 10.2528/PIER10052807.
 [Talman_19780] (1, 2) Talman, J. D., 1978, Numerical Fourier and Bessel transforms in logarithmic variables: Journal of Computational Physics, 29, pages 35-48; DOI: 10.1016/0021-9991(78)90107-9.
 [Trefethen_20000] Trefethen, L. N., 2000, Spectral methods in MATLAB: Society for Industrial and Applied Mathematics (SIAM), volume 10 of Software, Environments, and Tools, chapter 12, page 129; DOI: 10.1137/1.9780898719598.ch12.
 [Weniger_19890] Weniger, E. J., 1989, Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series: Computer Physics Reports, 10, 189–371; arXiv: abs/math/0306302.
 [Werthmuller_20170] Werthmüller, D., 2017, An open-source full 3D electromagnetic modeler for 1D VTI media in Python: empymod: Geophysics, 82; DOI: 10.1190/geo2016-0626.1.
 [Werthmuller_2017b0] Werthmüller, D., 2017, Getting started with controlled-source electromagnetic 1D modeling: The Leading Edge, 36, 352-355; DOI: 10.1190/tle36040352.1.
 [Wynn_19560] Wynn, P., 1956, On a device for computing the $$e_m(S_n)$$ tranformation: Math. Comput., 10, 91–96; DOI: 10.1090/S0025-5718-1956-0084056-6.

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.
• wavenumber: Calculate the electromagnetic wavenumber-domain solution.
• gpr: Calculate the Ground-Penetrating Radar (GPR) response.

The wavenumber 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=True, ht='fht', htarg=None, ft='sin', ftarg=None, opt=None, loop=None, verb=2)

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. 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, 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. Defaults to True. ht : {‘fht’, ‘qwe’, ‘quad’}, optional Flag to choose either the Fast Hankel Transform (FHT), 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’: [filter, pts_per_dec]: filter: string of filter name in empymod.filters or the filter method itself. (default: empymod.filters.key_201_2009()) pts_per_dec: points per decade (only relevant if spline=True) If none, standard lagged convolution is used. (default: None) 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; only relevant if opt=’spline’ (default: 80) 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 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’: [filter, pts_per_dec]: filter: 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. If none, standard lagged convolution is used. (Default: None) 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’, ‘spline’}, 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). If ‘spline’, the lagged convolution or splined variant of the FHT or the splined version of the QWE are used. Use with caution and check with the non-spline version for a specific problem. (Can be faster, slower, or plainly wrong, as it uses interpolation.) If spline is set it will make use of the parameter pts_per_dec that can be defined in htarg. If pts_per_dec is not set for FHT, then the lagged version is used, else the splined. This option has no effect on QUAD. 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 B [T] (not H [A/m]!). In the case of the impulse time-domain response, the unit is further divided by seconds [1/s]. 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          :  Fast Hankel Transform
> Filter      :  Key 201 (2009)
> pts_per_dec :  Defined by filter (lagged)
Hankel 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=True, ht='fht', htarg=None, ft='sin', ftarg=None, opt=None, loop=None, verb=2)

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.

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, 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. Defaults to True.

ht : {‘fht’, ‘qwe’, ‘quad’}, optional

Flag to choose either the Fast Hankel Transform (FHT), 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’: [filter, pts_per_dec]:

• filter: string of filter name in empymod.filters or
the filter method itself. (default: empymod.filters.key_201_2009())
• pts_per_dec: points per decade (only relevant if spline=True)
If none, standard lagged convolution is used.
(default: None)
• 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; only relevant if opt=’spline’
(default: 80)
• 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 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’: [filter, pts_per_dec]:

• filter: 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. If none, standard lagged
convolution is used. (Default: None)
• 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’, ‘spline’}, 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).
• If ‘spline’, the lagged convolution or splined variant of the FHT or the splined version of the QWE are used. Use with caution and check with the non-spline version for a specific problem. (Can be faster, slower, or plainly wrong, as it uses interpolation.) If spline is set it will make use of the parameter pts_per_dec that can be defined in htarg. If pts_per_dec is not set for FHT, then the lagged version is used, else the splined. This option has no effect on QUAD.

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 B [T] (not H [A/m]!).

In the case of the impulse time-domain response, the unit is further divided by seconds [1/s].

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)

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, [Hunziker_et_al_2015].
• Diffusive fullspace solution (solution=’dfs’) for ee-fields, [Slob_et_al_2010].
• Diffusive halfspace solution (solution=’dhs’) for ee-fields, [Slob_et_al_2010].
• Diffusive direct- and reflected field and airwave (solution=’dsplit’) for ee-fields, [Slob_et_al_2010].
• Diffusive direct- and reflected field and airwave (solution=’dtetm’) for ee-fields, split into TE and TM mode [Slob_et_al_2010].
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).

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).
• ‘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 B [T] (not H [A/m]!).

In the case of the impulse time-domain response, the unit is further divided by seconds [1/s].

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=True, ht='quad', htarg=None, ft='fft', ftarg=None, opt=None, loop=None, verb=2)

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

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 wavenumber
>>> 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 = wavenumber(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_spline, use_ne_eval, msrc, mrec, loop_freq, loop_off, conv=True)

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)

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.

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 [Hunziker_et_al_2015], 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)

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 [Hunziker_et_al_2015], and equally loosely to the file kxwmod.c.

[Hunziker_et_al_2015] 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)

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)

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 [Hunziker_et_al_2015], 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)

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 [Hunziker_et_al_2015], 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)

Calculate Rp, Rm.

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

This function corresponds to equations 64/65 and A-11/A-12 in [Hunziker_et_al_2015], 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)

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 [Hunziker_et_al_2015], 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')

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 [Slob_et_al_2010], 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 routine is not strictly part of empymod and not used by it. However, it can be useful to compare the code to this analytical solution.

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 FHT Hankel and Fourier transforms are based on source files (specified in each function) from the source code distributed with [Key_2012], 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, angle, depth, ab, etaH, etaV, zetaH, zetaV, xdirect, fhtarg, use_spline, use_ne_eval, msrc, mrec)

Hankel Transform using the Fast Hankel Transform.

The Fast Hankel Transform is a Digital Filter Method, introduced to geophysics by [Gosh_1971], and made popular and wide-spread by [Anderson_1975], [Anderson_1979], [Anderson_1982].

This implementation of the FHT follows [Key_2012], equation 6. Without going into the mathematical details (which can be found in any of the above papers) and following [Key_2012], the FHT 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 [Key_2012].

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 FHT, this is 1. conv : bool Only relevant for QWE/QUAD.
empymod.transform.hqwe(zsrc, zrec, lsrc, lrec, off, angle, depth, ab, etaH, etaV, zetaH, zetaV, xdirect, qweargs, use_spline, use_ne_eval, msrc, mrec)

Quadrature-With-Extrapolation was introduced to geophysics by [Key_2012]. It is one of many so-called ISE methods to solve Hankel Transforms, where ISE stands for Integration, Summation, and Extrapolation.

Following [Key_2012], 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 FHT (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 ([Shanks_1955], [Wynn_1956]; implemented with algorithms from [Trefethen_2000] and [Weniger_1989]).

This function is based on get_CSEM1D_FD_QWE.m, qwe.m, and getBesselWeights.m from the source code distributed with [Key_2012].

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, angle, depth, ab, etaH, etaV, zetaH, zetaV, xdirect, quadargs, use_spline, use_ne_eval, msrc, mrec)

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)

Fourier Transform using a Cosine- or a Sine-filter.

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 [Key_2012].

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)

It follows the QWE methodology [Key_2012] 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 [Key_2012].

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)

Fourier Transform using FFTLog.

FFTLog is the logarithmic analogue to the Fast Fourier Transform FFT. FFTLog was presented in Appendix B of [Hamilton_2000] 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)

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.qwe(rtol, atol, maxint, inp, intervals, lambd=None, off=None, factAng=None)

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 [Key_2012].

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

Return required calculation points.

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

Return parameters required for FFTLog.

filters – Digital Filters for FHT¶

Filters for the Fast Hankel Transform (FHT, [Anderson_1982]) and the Fourier Sine and Cosine Transforms [Anderson_1975].

To calculate the fhtfilter.factor I used

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


The filters kong_61_2007 and kong_241_2007 from [Kong_2007], and key_101_2009, key_201_2009, key_401_2009, key_81_CosSin_2009, key_241_CosSin_2009, and key_601_CosSin_2009 from [Key_2009] are taken from DIPOLE1D, [Key_2009], which can be downloaded at marineemlab.ucsd.edu/Projects/Occam/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 [Anderson_1982] and key_51_2012, key_101_2012, key_201_2012, key_101_CosSin_2012, and key_201_CosSin_2012, all from [Key_2012], are taken from the software distributed with [Key_2012] and available at software.seg.org/2012/0003. These filters are distributed under the SEG license.

class empymod.filters.DigitalFilter(name)

Simple Class for Digital Filters.

empymod.filters.anderson_801_1982()

Anderson 801: [Anderson_1982].

Anderson 801 pt filter, as published in [Anderson_1982]; taken from file wa801Hankel.txt from [Key_2012], published by the Society of Exploration Geophysicists; software.seg.org/2012/0003. License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_101_2009()

Key 101 2009: [Key_2009].

Key 101 pt filter, as published in [Key_2009]; taken from file FilterModules.f90 from [Key_2009], available on marineemlab.ucsd.edu/Projects/Occam/1DCSEM. License: Apache License, Version 2.0, http://www.apache.org/licenses/LICENSE-2.0.

empymod.filters.key_101_2012()

Key 101 2012: [Key_2012].

Key 101 pt filter, taken from file kk101Hankel.txt from [Key_2012], published by the Society of Exploration Geophysicists; software.seg.org/2012/0003. License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_101_CosSin_2012()

Key 101 CosSin 2012: [Key_2012].

Key 101 pt filter, taken from file kk101CosSin.txt from [Key_2012], published by the Society of Exploration Geophysicists; software.seg.org/2012/0003. License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_201_2009()

Key 201 2009: [Key_2009].

Key 201 pt filter, as published in [Key_2009]; taken from file FilterModules.f90 from [Key_2009], available on marineemlab.ucsd.edu/Projects/Occam/1DCSEM. License: Apache License, Version 2.0, http://www.apache.org/licenses/LICENSE-2.0.

empymod.filters.key_201_2012()

Key 201 2012: [Key_2012].

Key 201 pt filter, taken from file kk201Hankel.txt from [Key_2012], published by the Society of Exploration Geophysicists; software.seg.org/2012/0003. License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_201_CosSin_2012()

Key 201 CosSin 2012: [Key_2012].

Key 201 pt filter, taken from file kk201CosSin.txt from [Key_2012], published by the Society of Exploration Geophysicists; software.seg.org/2012/0003. License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_241_CosSin_2009()

Key 241 CosSin 2009: [Key_2009].

Key 241 pt filter, as published in [Key_2009]; taken from file FilterModules.f90 from [Key_2009], available on marineemlab.ucsd.edu/Projects/Occam/1DCSEM. License: Apache License, Version 2.0, http://www.apache.org/licenses/LICENSE-2.0.

empymod.filters.key_401_2009()

Key 401 2009: [Key_2009].

Key 401 pt filter, as published in [Key_2009]; taken from file FilterModules.f90 from [Key_2009], available on marineemlab.ucsd.edu/Projects/Occam/1DCSEM. License: Apache License, Version 2.0, http://www.apache.org/licenses/LICENSE-2.0.

empymod.filters.key_51_2012()

Key 51 2012: [Key_2012].

Key 51 pt filter, taken from file kk51Hankel.txt from [Key_2012], published by the Society of Exploration Geophysicists; software.seg.org/2012/0003. License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_601_CosSin_2009()

Key 601 CosSin 2009: [Key_2009].

Key 601 pt filter, as published in [Key_2009]; taken from file FilterModules.f90 from [Key_2009], available on marineemlab.ucsd.edu/Projects/Occam/1DCSEM. License: Apache License, Version 2.0, http://www.apache.org/licenses/LICENSE-2.0.

empymod.filters.key_81_CosSin_2009()

Key 81 CosSin 2009: [Key_2009].

Key 81 pt filter, as published in [Key_2009]; taken from file FilterModules.f90 from [Key_2009], available on marineemlab.ucsd.edu/Projects/Occam/1DCSEM. License: Apache License, Version 2.0, http://www.apache.org/licenses/LICENSE-2.0.

empymod.filters.kong_241_2007()

Kong 241: [Kong_2007].

Kong 241 pt filter, as published in [Kong_2007]; taken from file FilterModules.f90 from [Key_2009], available on marineemlab.ucsd.edu/Projects/Occam/1DCSEM. License: Apache License, Version 2.0, http://www.apache.org/licenses/LICENSE-2.0.

empymod.filters.kong_61_2007()

Kong 61: [Kong_2007].

Kong 61 pt filter, as published in [Kong_2007]; taken from file FilterModules.f90 from [Key_2009], available on marineemlab.ucsd.edu/Projects/Occam/1DCSEM. License: Apache License, Version 2.0, http://www.apache.org/licenses/LICENSE-2.0.

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

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)

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)

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)

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)

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)

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)

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’, ‘spline’} Optimization flag. 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_spline : bool Boolean if to use spline interpolation. 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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

Print start and finish with time measure and kernel count.

empymod.utils.conv_warning(conv, targ, name, verb)

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