empymod¶

Version: 1.5.2 ~ Date: 26 April 2018

The electromagnetic modeller empymod can model electric or magnetic responses due to a three-dimensional electric or magnetic source in a layered-earth model with vertical transverse isotropic (VTI) resistivity, VTI electric permittivity, and VTI magnetic permeability, from very low frequencies (DC) to very high frequencies (GPR). The calculation is carried out in the wavenumber-frequency domain, and various Hankel- and Fourier-transform methods are included to transform the responses into the space-frequency and space-time domains.

See https://empymod.github.io/#features for a complete list of features.

More information¶

For more information regarding installation, usage, add-ons, contributing, roadmap, bug reports, and much more, see

Website: https://empymod.github.io,
Documentation empymod: https://empymod.readthedocs.io,
Documentation add-ons: https://empyscripts.readthedocs.io,
Source Code: https://github.com/empymod,
Examples: https://github.com/empymod/example-notebooks.

Citation¶

If you publish results for which you used empymod, please give credit by citing this article:

Werthmüller, D., 2017, An open-source full 3D electromagnetic modeler for 1D VTI media in Python: empymod: Geophysics, 82(6), WB9–WB19; DOI: 10.1190/geo2016-0626.1.

All releases have a Zenodo-DOI, provided on the release-page. Also consider citing Hunziker et al. (2015) and Key (2012), without which empymod would not exist.

License information¶

Licensed under the Apache License, Version 2.0. See the LICENSE- and NOTICE-files or the documentation for more information.

Manual¶

Theory¶

The code is principally based on

[Hunziker_et_al_2015] for the wavenumber-domain calculation (kernel),
[Key_2012] for the DLF and QWE transforms,
[Slob_et_al_2010] for the analytical half-space solutions, and
[Hamilton_2000] for the FFTLog.

See these publications and all the others given in the references, if you are interested in the theory on which empymod is based. Another good reference is [Ziolkowski_and_Slob], which will be published in late 2018. The book derives in great detail the equations for layered-Earth CSEM modelling.

Installation¶

You can install empymod either via conda:

conda install -c prisae empymod

or via pip:

pip install empymod

Required are Python version 3.4 or higher and the modules NumPy and SciPy. The module numexpr is required additionally (built with Intel’s VML) if you want to run parts of the kernel in parallel.

The modeller empymod comes with add-ons (empyscripts). These add-ons provide some very specific, additional functionalities. To install them just follow the instructions for empymod, replacing empymod with empyscripts in the command. You can find more information regarding the add-ons directly on github.com/empymod/empyscripts.

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. If you install [Anaconda](https://www.anaconda.com/download). If you install Anaconda you can simply start the Anaconda Navigator, add the channel prisae and both empymod and empyscripts will appear in the package list and can be installed with a click.

Warning

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.

The structure of empymod is:

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 Digital Linear Filters method DLF (Hankel and Fourier transforms).

Usage/Examples¶

A good starting point is [Werthmuller_2017b], and more information can be found in [Werthmuller_2017]. There are a lot of examples of its usage available, in the form of Jupyter notebooks. Have a look at the following repositories:

Example notebooks: https://github.com/empymod/example-notebooks,
Geophysical Tutoriol TLE: https://github.com/empymod/article-tle2017, and
Numerical examples of [Ziolkowski_and_Slob]: https://github.com/empymod/csem-ziolkowski-and-slob.

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          :  DLF (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
   Receiver(s)     :  10 dipole(s)
     > 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]

Contributing¶

New contributions, bug reports, or any kind of feedback is always welcomed! Have a look at the Roadmap-section to get an idea of things that could be implemented. The best way for interaction is at https://github.com/empymod. If you prefer to contact me outside of GitHub use the contact form on my personal website, https://werthmuller.org.

To install empymod from source, you can download the latest version from GitHub and either add the path to empymod to your python-path variable, or install it in your python distribution via:

python setup.py install

Please make sure your code follows the pep8-guidelines by using, for instance, the python module flake8, and also that your code is covered with appropriate tests. Just get in touch if you have any doubts.

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.

> pip install pytest coveralls pytest-flake8
> # 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 --cov=empymod --flake8

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.

Transforms¶

Included Hankel transforms:

Digital Linear Filters DLF
Quadrature with Extrapolation QWE
Adaptive quadrature QUAD

Included Fourier transforms:

Digital Linear Filters DLF
Quadrature with Extrapolation QWE
Logarithmic Fast Fourier Transform FFTLog
Fast Fourier Transform FFT

FFTLog¶

FFTLog is the logarithmic analogue to the Fast Fourier Transform FFT originally proposed by [Talman_1978]. The code used by empymod was published in Appendix B of [Hamilton_2000] 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_1988] proposed a logarithmic Fourier transform (abbreviated by the authors as LFT) for electromagnetic geophysics, also based on [Talman_1978]. 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_1982], and Haines and Jones cite [Anderson_1979]. 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_1975]). 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_1975] or [Key_2012]. 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

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

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 DLF 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 DLF 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!

Memory¶

By default empymod will try to carry out the calculation in one go, without looping. If your model has many offsets and many frequencies this can be heavy on memory usage. Even more so if you are calculating time-domain responses for many times. If you are running out of memory, you should use either loop='off' or loop='freq' to loop over offsets or frequencies, respectively. Use verb=3 to see how many offsets and how many frequencies are calculated internally.

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).¶
	source bipole			receiver bipole
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 package numexpr has to be built with Intel’s VML, otherwise it won’t be used. You can check if it uses VML with

>>> import numexpr
>>> numexpr.use_vml

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
>>> numexpr.set_num_threads(4)

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 DLF (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 DLF. (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 DLF, big improvements are achieved for long DLF-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.

DLF: 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 Fourier transforms FFTlog, QWE, and FFT always use interpolation too, either in the frequency or in the time domain. With the DLF Fourier transform (sine and cosine transforms) you can choose between no interpolation and interpolation (splined or lagged).

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.

For a graphical explanation of the differences between standard DLF, lagged convolution DLF, and splined DLF for the Hankel and the Fourier transforms see the notebook 7a_DLF-Standard-Lagged-Splined in the example-notebooks repository.

Looping¶

By default, you can calculate many offsets and many frequencies all in one go, vectorized (for the DLF), 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.

License¶

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

http://www.apache.org/licenses/LICENSE-2.0

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- and NOTICE-files on GitHub for more information.

Note

This software was initially (till 01/2017) developed with funding from The Mexican National Council of Science and Technology (Consejo Nacional de Ciencia y Tecnología, http://www.conacyt.gob.mx), carried out at The Mexican Institute of Petroleum IMP (Instituto Mexicano del Petróleo, http://www.gob.mx/imp).

References ¶

[Anderson_1975]

(1, 2) 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_1979]

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_1982]

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.

[Ghosh_1971]

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_1988]

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_2000]

(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_2015]

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(1), F1–F18; DOI: 10.1190/geo2013-0411.1; Software: software.seg.org/2015/0001.

[Key_2009]

Key, K., 2009, 1D inversion of multicomponent, multifrequency marine CSEM data: Methodology and synthetic studies for resolving thin resistive layers: Geophysics, 74(2), F9–F20; DOI: 10.1190/1.3058434. Software: marineemlab.ucsd.edu/Projects/Occam/1DCSEM.

[Key_2012]

(1, 2) Key, K., 2012, Is the fast Hankel transform faster than quadrature?: Geophysics, 77(3), F21–F30; DOI: 10.1190/geo2011-0237.1; Software: software.seg.org/2012/0003.

[Kong_2007]

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_1955]

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_2010]

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_1978]

(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_2000]

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_1989]

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_2017]

Werthmüller, D., 2017, An open-source full 3D electromagnetic modeler for 1D VTI media in Python: empymod: Geophysics, 82(6), WB9-WB19; DOI: 10.1190/geo2016-0626.1.

[Werthmuller_2017b]

Werthmüller, D., 2017, Getting started with controlled-source electromagnetic 1D modeling: The Leading Edge, 36, 352-355; DOI: 10.1190/tle36040352.1.

[Wynn_1956]

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.

[Ziolkowski_and_Slob]

(1, 2) Ziolkowski, A., and E. Slob, 2018, Introduction to Controlled-Source Electromagnetic Methods: Cambridge University Press; expected to be published late 2018.

Roadmap¶

A collection of ideas of what could be added or improved in empymod. Please get in touch if you would like to tackle one of these problems!

Additional modelling routines
- tdem (TEM) [empymod#8]: Issues that have to be addressed: ramp waveform, windowing, loop integration, zero-offset (coincident loop).
  - in-loop
  - coincident loop
  - …
- Ramp waveform [empymod#7]
- Arbitrary waveform [empymod#7]
- Improve the GPR-routine [empymod#9]
- Load and save functions to easily store and load model information (resistivity model, acquisition parameters, and modelling parameters) together with the modelling data (using pickle or shelve).
Inversion [empyscripts#1]: Inversion routines, preferably a selection of different ones.
Additional (semi-)analytical functions (where possible)
- Complete full-space (electric and magnetic source and receiver); space-time domain
- Extend diffusive half-space solution to magnetic sources and receivers; space-frequency and space-time domains
- Complete half-space
Fourier transform
- Include the method outlined by Mulder et al., 2008, Geophysics (piecewise-cubic Hermite interpolation with a FFT).
- Change fft to use discrete sine/cosine transforms instead, as all other Fourier transforms
- If previous step is successful, clean up the internal decisions (utils.check_time) when to use sine/cosine transform (not consistent at the moment, some choice only exists with ffht impulse responses, fqwe and fftlog use sine for impulse, and all three use sine for step-on responses and cosine for step-off responses)
Hankel transform
- Add the fht-module from FFTLog for the Hankel transform.
Extend examples (example-notebooks)
- Add different methods (e.g. DC)
- Reproduce published results
A cython, numba, or pure C/C++ implementation of the kernel and the transform modules. Maybe not worth it, as it may improve speed, but decrease accessibility. Both at the same time would be nice. A fast C/C++-version for calculations (inversions), and a Python-version to tinker with for interested folks. (Probably combined with default parallelisation, removing the numexpr variant.)
Abstraction of the code.
GUI.
Add a benchmark suite, e.g. http://asv.readthedocs.io, in addition to the testing suite.
Add some clever checks, e.g. as in Key (2012): abort loops if the field is strongly attenuated (more relevant if once an inversion is implemented).
Move empymod from channel ‘prisae’ to ‘conda-forge’ (pros/cons?).

Changelog¶

v1.5.2 - 2018-04-25¶

DLF improvements:
- Digital linear filter (DLF) method for the Fourier transform can now be carried out without spline, providing 0 for pts_per_dec (or any integer smaller than 1).
- Combine kernel from fht and ffht into dlf, hence separate DLF from other calculations, as is done with QWE (qwe for hqwe and fqwe).
- Bug fix regarding transform.get_spline_values; a DLF with pts_per_dec can now be shorter then the corresponding filter.

v1.5.1 - 2018-02-24¶

Documentation:
- Simplifications: avoid duplication as much as possible between the website (empymod.github.io), the manual (empymod.readthedocs.io), and the README (github.com/empymod/empymod).
  - Website has now only Features and Installation in full, all other
    
    information comes in the form of links.
  - README has only information in the form of links.
  - Manual contains the README, and is basically the main document for all
    
    information.
- Improvements: Change some remaining md-syntax to rst-syntax.
- FHT -> DLF: replace FHT as much as possible, without breaking backwards compatibility.

v1.5.0 - 2018-01-02¶

Minimum parameter values can now be set and verified with utils.set_minimum and utils.get_minimum.
New Hankel filter wer_201_2018.
opt=parallel has no effect if numexpr is not built against Intel’s VML. (Use import numexpr; numexpr.use_vml to see if your numexpr uses VML.)
Bug fixes
Version of manuscript submission to geophysics for the DLF article.

v1.4.4 - 2017-09-18¶

[This was meant to be 1.4.3, but due to a setup/pypi/anaconda-issue I had to push it to 1.4.4; so there isn’t really a version 1.4.3.]

Add TE/TM split to diffusive ee-halfspace solution.
Improve kernel.wavenumber for fullspaces.
Extended fQWE and fftlog to be able to use the cosine-transform. Now the cosine-transform with the real-part frequency response is used internally if a switch-off response (signal=-1) is required, rather than calculating the switch-on response (with sine-transform and imaginary-part frequency response) and subtracting it from the DC value.
Bug fixes
Version of CSEM book.

v1.4.2 - 2017-06-04¶

Bugfix: Fixed squeeze in model.analytical with solution='dsplit'.
Version of final submission of manuscript to Geophysics.

v1.4.1 - 2017-05-30¶

[This was meant to be 1.4.0, but due to a setup/pypi/anaconda-issue I had to push it to 1.4.1; so there isn’t really a version 1.4.0.]

New home: empymod.github.io as entry point, and the project page on github.com/empymod. All empymod-repos moved to the new home.
- /prisae/empymod -> /empymod/empymod
- /prisae/empymod-notebooks -> /empymod/example-notebooks
- /prisae/empymod-geo2017 -> /empymod/article-geo2017
- /prisae/empymod-tle2017 -> /empymod/article-tle2017
Modelling routines:
- New modelling routine model.analytical, which serves as a front-end to kernel.fullspace or kernel.halfspace.
- Remove legacy routines model.time and model.frequency. They are covered perfectly by model.dipole.
- Improved switch-off response (calculate and subtract from DC).
- xdirect adjustments:
  - isfullspace now respects xdirect.
  - Removed xdirect from model.wavenumber (set to False).
Kernel:
- Modify kernel.halfspace to use same input as other kernel functions.
- Include time-domain ee halfspace solution into kernel.halfspace; possible to obtain direct, reflected, and airwave separately, as well as only fullspace solution (all for the diffusive approximation).

v1.3.0 - 2017-03-30¶

Add additional transforms and improve QWE:
- Conventional adaptive quadrature (QUADPACK) for the Hankel transform;
- Conventional FFT for the Fourier transform.
- Add diff_quad to htarg/ftarg of QWE, a switch parameter for QWE/QUAD.
- Change QWE/QUAD switch from comparing first interval to comparing all intervals.
- Add parameters for QUAD (a, b, limit) into htarg/ftarg for QWE.
Allow htarg/ftarg as dict additionally to list/tuple.
Improve model.gpr.
Internal changes:
- Rename internally the sine/cosine filter from fft to ffht, because of the addition of the Fast Fourier Transform fft.
Clean-up repository
- Move notebooks to /prisae/empymod-notebooks
- Move publications/Geophysics2017 to /prisae/empymod-geo2017
- Move publications/TheLeadingEdge2017 to /prisae/empymod-tle2017
Bug fixes and documentation improvements

v1.2.1 - 2017-03-11¶

Change default filter from key_401_2009 to key_201_2009 (because of warning regarding 401 pt filter in source code of DIPOLE1D.)
Since 06/02/2017 installable via pip/conda.
Bug fixes

v1.2.0 - 2017-02-02¶

New routine:
- General modelling routine bipole (replaces srcbipole): Model the EM field for arbitrarily oriented, finite length bipole sources and receivers.
Added a test suite:
- Unit-tests of small functions.
- Framework-tests of the bigger functions:
  - Comparing to status quo (regression tests),
  - Comparing to known analytical solutions,
  - Comparing different options to each other,
  - Comparing to other 1D modellers (EMmod, DIPOLE1D, GREEN3D).
- Incorporated with Travis CI and Coveralls.
Internal changes:
- Add kernel count (printed if verb > 1).
- numexpr is now only required if opt=='parallel'. If numexpr is not found, opt is reset to None and a warning is printed.
- Cleaned-up wavenumber-domain routine.
- theta/phi -> azimuth/dip; easier to understand.
- Refined verbosity levels.
- Lots of changes in utils, with regards to the new routine bipole and with regards to verbosity. Moved all warnings out from transform and model into utils.
Bug fixes

v1.1.0 - 2016-12-22¶

New routines:
- New srcbipole modelling routine: Model an arbitrarily oriented, finite length bipole source.
- Merge frequency and time into dipole. (frequency and time are still available.)
- dipole now supports multiple sources.
Internal changes:
- Replace get_Gauss_Weights with scipy.special.p_roots
- jv(0,x), jv(1,x) -> j0(x), j1(x)
- Replace param_shape in utils with _check_var and _check_shape.
- Replace xco and yco by angle in kernel.fullspace
- Replace fftlog with python version.
- Additional sine-/cosine-filters: key_81_CosSin_2009, key_241_CosSin_2009, and key_601_CosSin_2009.
Bug fixes

v1.0.0 - 2016-11-29¶

Initial release; state of manuscript submission to geophysics.

Credits¶

Thanks to

Jürg Hunziker, Kerry Key, and Evert Slob for answering all my questions regarding their codes and publications (Hunziker et al., 2015, Key, 2009, Key, 2012, Slob et al., 2010).

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 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’: [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 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’: [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. If <1, no interpolation is used at all. (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 DLF/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 DLF/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

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 (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.

`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 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 [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 Digital Linear Filter method.

The Digital Linear Filter method was introduced to geophysics by [Ghosh_1971], and made popular and wide-spread by [Anderson_1975], [Anderson_1979], [Anderson_1982]. 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 [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 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 [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 DLF, 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)¶

Hankel Transform using Quadrature-With-Extrapolation.

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 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 ([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, <htarg> 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, <htarg> might have to be adjusted.

empymod.transform.ffht(fEM, time, freq, ftarg)¶

Fourier Transform using the Digital Linear Filter method.

It follows the Filter methodology [Anderson_1975], 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 [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)¶

Fourier Transform using Quadrature-With-Extrapolation.

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, <ftarg> 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.dlf(signal, points, out_pts, filt, lagged, splined, kind=None, factAng=None, ab=None)¶

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 requires two additional 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 DO depend on the angle between source and receiver.

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)¶

Quadrature-With-Extrapolation.

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 Linear Filters¶

Filters for the Digital Linear Filter (DLF) method for the Hankel [Ghosh_1971]) and the Fourier ([Anderson_1975]) 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 [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 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 [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 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)¶: Simple Class for Digital Linear Filters.

empymod.filters.anderson_801_1982()¶

Anderson 801 pt Hankel filter, as published in [Anderson_1982].

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

License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_101_2009()¶

Key 101 pt Hankel filter, as published in [Key_2009].

Taken from file FilterModules.f90 provided with 1DCSEM.

License: Apache License, Version 2.0,.

empymod.filters.key_101_2012()¶

Key 101 pt Hankel filter, as published in [Key_2012].

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

License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_101_CosSin_2012()¶

Key 101 pt CosSin filter, as published in [Key_2012].

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

License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_201_2009()¶

Key 201 pt Hankel filter, as published in [Key_2009].

Taken from file FilterModules.f90 provided with 1DCSEM.

License: Apache License, Version 2.0,.

empymod.filters.key_201_2012()¶

Key 201 pt Hankel filter, as published in [Key_2012].

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

License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_201_CosSin_2012()¶

Key 201 pt CosSin filter, as published in [Key_2012].

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

License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_241_CosSin_2009()¶

Key 241 pt CosSin filter, as published in [Key_2009].

Taken from file FilterModules.f90 provided with 1DCSEM.

License: Apache License, Version 2.0,.

empymod.filters.key_401_2009()¶

Key 401 pt Hankel filter, as published in [Key_2009].

Taken from file FilterModules.f90 provided with 1DCSEM.

License: Apache License, Version 2.0,.

empymod.filters.key_51_2012()¶

Key 51 pt Hankel filter, as published in [Key_2012].

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

License: http://software.seg.org/disclaimer.txt.

empymod.filters.key_601_CosSin_2009()¶

Key 601 pt CosSin filter, as published in [Key_2009].

Taken from file FilterModules.f90 provided with 1DCSEM.

License: Apache License, Version 2.0,.

empymod.filters.key_81_CosSin_2009()¶

Key 81 pt CosSin filter, as published in [Key_2009].

Taken from file FilterModules.f90 provided with 1DCSEM.

License: Apache License, Version 2.0,.

empymod.filters.kong_241_2007()¶

Kong 241 pt Hankel filter, as published in [Kong_2007].

Taken from file FilterModules.f90 provided with 1DCSEM.

License: Apache License, Version 2.0,.

empymod.filters.kong_61_2007()¶

Kong 61 pt Hankel filter, as published in [Kong_2007].

Taken from file FilterModules.f90 provided with 1DCSEM.

License: Apache License, Version 2.0,.

empymod.filters.wer_201_2018()¶

Werthmüller 201 pt Hankel filter, 2018.

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

License: Apache License, Version 2.0,.

`utils` – Utilites¶

Utilities for model such as checking input parameters.

This module consists of four groups of functions:

General settings
Class EMArray
Input parameter checks for modelling
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.

Returns:

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.

Returns:

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.

Returns:

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.

Returns:

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.

Returns:

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.

Returns:

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.

Returns:

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.

Returns:

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)¶

Check source-receiver configuration.

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.

Returns:

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.

Returns:

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.

Returns:

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.

Returns:

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.

Returns:

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.

Returns:

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.

empymod.utils.set_minimum(min_freq=None, min_time=None, min_off=None, min_param=None, min_angle=None)¶

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_param : float, optional

Minimum aniso, [m/e]perm[H/V] [-] (default 1e-20).

min_angle : float, optional

Minimum angle [-] (default 1e-10).

empymod.utils.get_minimum()¶

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_param : float

min_angle : float

For a full description of these options, see set_minimum.

empymod¶

More information¶

Citation¶

License information¶

Manual¶

Theory¶

Installation¶

Usage/Examples¶

Contributing¶

Transforms¶

FFTLog¶

Notes on Fourier Transform¶

Note on speed, memory, and accuracy¶

Memory¶

Depths, Rotation, and Bipole¶

Parallelisation¶

Spline interpolation¶

Looping¶

Vertical components¶

License¶

References ¶

Roadmap¶

Changelog¶

v1.5.2 - 2018-04-25¶

v1.5.1 - 2018-02-24¶

v1.5.0 - 2018-01-02¶

v1.4.4 - 2017-09-18¶

v1.4.2 - 2017-06-04¶

v1.4.1 - 2017-05-30¶

v1.3.0 - 2017-03-30¶

v1.2.1 - 2017-03-11¶

v1.2.0 - 2017-02-02¶

v1.1.0 - 2016-12-22¶

v1.0.0 - 2016-11-29¶

Credits¶

Code¶

model – Model EM-responses¶

kernel – Kernel calculation¶

transform – Hankel and Fourier Transforms¶

filters – Digital Linear Filters¶

utils – Utilites¶

`model` – Model EM-responses¶

`kernel` – Kernel calculation¶

`transform` – Hankel and Fourier Transforms¶

`filters` – Digital Linear Filters¶

`utils` – Utilites¶