# Manual¶

## Theory¶

The code is principally based on

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_2019]. 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 3.5 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 (empymod.scripts). These add-ons provide some very specific, additional functionalities. Some of these add-ons have additional, optional dependencies for other modules such as matplotlib. See the Add-ons-section for their documentation.

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 empymod 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:

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)
> DLF type    :  Standard
Kernel Opt.     :  None
Loop over       :  None (all vectorized)
Source(s)       :  1 bipole(s)
> intpts      :  1 (as dipole)
> length  [m] :  100
> x_c     [m] :  0
> y_c     [m] :  0
> z_c     [m] :  100
> azimuth [°] :  0
> dip     [°] :  0
> x       [m] :  500 - 5000 : 10  [min-max; #]
:  500 1000 1500 2000 2500 3000 3500 4000 4500 5000
> y       [m] :  0 - 0 : 10  [min-max; #]
:  0 0 0 0 0 0 0 0 0 0
> z       [m] :  200
> azimuth [°] :  0
> dip     [°] :  0
Required ab's   :  11
~
:: empymod END; runtime = 0:00:00.005536 :: 1 kernel call(s)
~
>>> print(EMfield)
[  1.68809346e-10 -3.08303130e-10j  -8.77189179e-12 -3.76920235e-11j
-3.46654704e-12 -4.87133683e-12j  -3.60159726e-13 -1.12434417e-12j
1.87807271e-13 -6.21669759e-13j   1.97200208e-13 -4.38210489e-13j
1.44134842e-13 -3.17505260e-13j   9.92770406e-14 -2.33950871e-13j
6.75287598e-14 -1.74922886e-13j   4.62724887e-14 -1.32266600e-13j]


### Hook for user-defined calculation of $$\eta$$ and $$\zeta$$¶

In principal it is always best to write your own modelling routine if you want to adjust something. Just copy empymod.dipole or empymod.bipole as a template, and modify it to your needs. Since empymod v1.7.4, however, there is a hook which allows you to modify $$\eta_h, \eta_v, \zeta_h$$, and $$\zeta_v$$ quite easily.

The trick is to provide a dictionary (we name it inp here) instead of the resistivity vector in res. This dictionary, inp, has two mandatory plus optional entries:

• res: the resistivity vector you would have provided normally (mandatory).

• A function name, which has to be either or both of (mandatory)

• func_eta: To adjust etaH and etaV, or
• func_zeta: to adjust zetaH and zetaV.
• In addition, you have to provide all parameters you use in func_eta/func_zeta and are not already provided to empymod. All additional parameters must have #layers elements.

The functions func_eta and func_zeta must have the following characteristics:

• The signature is func(inp, p_dict), where

• inp is the dictionary you provide, and
• p_dict is a dictionary that contains all parameters so far calculated in empymod [locals()].
• It must return etaH, etaV if func_eta, or zetaH, zetaV if func_zeta.

Dummy example

def my_new_eta(inp, p_dict):
# Your calculations, using the parameters you provided
# in inp and the parameters from empymod in p_dict.
# In the example line below, we provide, e.g.,  inp['tau']
return etaH, etaV


And then you call empymod with res={'res': res-array, 'tau': tau, 'func_eta': my_new_eta}.

Have a look at the example 2d_Cole-Cole-IP in the example-notebooks repository, where this hook is exploited in the low-frequency range to use the Cole-Cole model for IP calculation. It could also be used in the high-frequency range to model dielectricity.

## 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.

## Tests and benchmarks¶

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 pytest-mpl
> # 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 --mpl


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.

There is also a benchmark suite using airspeed velocity, located in the empymod/asv-repository. The results of my machine can be found in the empymod/bench, its rendered version at empymod.github.io/asv.

## Transforms¶

Included Hankel transforms:

• Digital Linear Filters DLF

Included Fourier transforms:

• Digital Linear Filters DLF
• Logarithmic Fast Fourier Transform FFTLog
• Fast Fourier Transform FFT

### Digital Linear Filters¶

The module empymod.filters comes with many DLFs for the Hankel and the Fourier transform. If you want to export one of these filters to plain ascii files you can use the tofile-routine of each filter:

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


Similarly, if you want to use an own filter you can do that as well. The filter base and the filter coefficient have to be stored in separate files:

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


The path can be adjusted by providing tofile and fromfile with a path-argument.

### 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.

Parts of the kernel can run in parallel using numexpr. This option is activated by setting opt='parallel' (see subsection Parallelisation). It is switched off by default.

### 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).
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', six (*) 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


This parallelisation will make empymod faster (by using more threads) 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.)

(*) These statements are (following the notation of [Hunziker_et_al_2015]): $$\Gamma$$ (below eq. 19); $$W^{u, d}_n$$ (eq. 74), $$r^\pm_n$$ (eq. 65); $$R^\pm_n$$ (eq. 64); $$P^{u, d; \pm}_s$$ (eq. 81); $$M_s$$ (eq. 82), and their corresponding bar-ed versions provided in the appendix (e.g. $$\bar{\Gamma}$$). In big models, more than 95 % of the calculation is spent in the calculation of these six equations, and most of the time therefore in np.sqrt and np.exp, or generally in numpy-ufuncs which are implemented and executed in compiled C-code. For smaller models or if transforms with interpolations are used then all the other parts also start to play a role. However, those models generally execute comparably fast.

### Lagged Convolution and Splined Transforms¶

Both Hankel and Fourier DLF have three options, which can be controlled via the htarg['pts_per_dec'] and ftarg['pts_per_dec'] parameters:

• pts_per_dec=0 : Standard DLF;
• pts_per_dec<0 : Lagged Convolution DLF: Spacing defined by filter base, interpolation is carried out in the input domain;
• pts_per_dec>0 : Splined DLF: Spacing defined by pts_per_dec, interpolation is carried out in the output domain.

Similarly, interpolation can be used for QWE by setting pts_per_dec to a value bigger than 0.

The Lagged Convolution and Splined options should be used with caution, as they use interpolation and are therefore less precise than the standard version. However, they can significantly speed up QWE, and massively speed up DLF. Additionally, the interpolated versions minimizes memory requirements a lot. Speed-up is greater if all source-receiver angles are identical. Note that setting pts_per_dec to something else than 0 to calculate only one offset (Hankel) or only one time (Fourier) will be slower than using the standard version. Similarly, the standard version is usually the fastest when using the parallel option (numexpr).

QWE: Good speed-up is also achieved for QWE by setting maxint as low as possible. Also, the higher nquad is, the higher the speed-up will be.

DLF: Big improvements are achieved for long DLF-filters and for many offsets/frequencies (thousands).

Warning

Keep in mind that setting pts_per_dec to something else than 0 uses interpolation, and is therefore not as accurate as the standard version. Use with caution and always compare with the standard version to verify if you can apply interpolation to your problem at hand!

Be aware that QUAD (Hankel transform) always use the splined version and always loops 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 ht = 'FHT' and pts_per_dec!=0 (Lagged Convolution or Splined Hankel DLF). 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 and xdirect¶

Calculating the direct field in the wavenumber-frequency domain (xdirect=False; the default) is generally faster than calculating it in the frequency-space domain (xdirect=True).

However, using xdirect = True can improve the result (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.

### Time-domain land CSEM¶

The derivation, as it stands, has a near-singular behaviour in the wavenumber-frequency domain when $$\kappa^2 = \omega^2\epsilon\mu$$. This can be a problem for land-domain CSEM calculations if source and receiver are located at the surface between air and subsurface. Because most transforms do not sample the wavenumber-frequency domain sufficiently to catch this near-singular behaviour (hence not smooth), which then creates noise at early times where the signal should be zero. To avoid the issue simply set epermH[0] = epermV[0] = 0, hence the relative electric permittivity of the air to zero. This trick obviously uses the diffusive approximation for the air-layer, it therefore will not work for very high frequencies (e.g., GPR calculations).

This trick works fine for all horizontal components, but not so much for the vertical component. But then it is not feasible to have a vertical source or receiver exactly at the surface. A few tips for these cases: The receiver can be put pretty close to the surface (a few millimeters), but the source has to be put down a meter or two, more for the case of vertical source AND receiver, less for vertical source OR receiver. The results are generally better if the source is put deeper than the receiver. In either case, the best is to first test the survey layout against the analytical result (using empymod.analytical with solution='dhs') for a half-space, and subsequently model more complex cases.

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- 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, PB242800; pubs.er.usgs.gov/publication/70045426.
 [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.
 [Chave_and_Cox_1982] Chave, A. D., and C. S. Cox, 1982, Controlled electromagnetic sources for measuring electrical conductivity beneath the oceans: 1. forward problem and model study: Journal of Geophysical Research, 87, 5327–5338; DOI: 10.1029/JB087iB07p05327.
 [Ghosh_1970] Ghosh, D. P., 1970, The application of linear filter theory to the direct interpretation of geoelectrical resistivity measurements: Ph.D. Thesis, TU Delft; UUID: 88a568bb-ebee-4d7b-92df-6639b42da2b2.
 [Guptasarma_and_Singh_1997] Guptasarma, D., and B. Singh, 1997, New digital linear filters for Hankel J0 and J1 transforms: Geophysical Prospecting, 45, 745–762; DOI: 10.1046/j.1365-2478.1997.500292.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] (1, 2) 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.
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