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\begin{document}

\title{A tool for designing digital filters for the Hankel and Fourier
transforms in potential, diffusive, and wavefield modeling}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\ms{GEO-2018-0069.R2}

\address{
\footnotemark[1]TU Delft,
Building 23,
Stevinweg 1 / PO-box 5048,
2628 CN Delft,
E-mail: \href{mailto:D.Werthmuller@tudelft.nl}{D.Werthmuller@tudelft.nl};
\href{mailto:E.C.Slob@tudelft.nl}{E.C.Slob@tudelft.nl};
\footnotemark[2]\emph{formerly at the} Instituto Mexicano del Petróleo,
Eje Central Lázaro Cárdenas Norte 152,
Col. San Bartolo Atepehuacan C.P. 07730,
Ciudad de México, México,
\footnotemark[3]Lamont-Doherty Earth Observatory, Columbia University,
305C Oceanography,
61 Route 9W, PO Box 1000,
Palisades NY 10964-8000 US,
E-mail: \href{mailto:KKey@ldeo.columbia.edu}{KKey@ldeo.columbia.edu}.
}


\author{%
Dieter Werthmüller\footnotemark[1]\footnotemark[2], %
Kerry Key\footnotemark[3], and %
Evert C.\ Slob\footnotemark[1]%
}

\footer{}
\lefthead{Werthmüller et al.}
\righthead{Digital filter designing tool}

\maketitle

%%fakesection ===    ABSTRACT    ===
\begin{abstract} % 1-2 sentence(s) each
%
% (1) PRINCIPAL OBJECTIVES AND SCOPE OF THE WORK
  The open-source code \texttt{fdesign} makes it possible to design digital
  linear filters for the Hankel and Fourier transforms used in potential-,
  diffusive-, and wavefield modeling. Digital filters can be derived for any
  electromagnetic method, such as methods in the diffusive limits (direct
  current, controlled-source electromagnetics) as well as methods using higher
  frequency content (ground-penetrating radar, acoustic and elastic wavefields).
%
% (2) METHODOLOGY
  The direct matrix inversion method is used for the derivation of the filter
  values, and a brute-force inversion is carried out over the defined spacing
  and shifting values of the filter basis. Included or user-provided
  theoretical transform pairs are used for the inversion. Alternatively, one
  can provide layered subsurface models that will be computed with a precise
  quadrature method using the electromagnetic modeler \texttt{empymod} to
  generate numerical transform pairs.
%
% (3) RESULTS
  The comparison of the presented 201\,pt filter with previously presented
  filters shows that it performs better for some standard controlled-source
  electromagnetic cases. The derivation of a longer 2001\,pt filter for a
  ground-penetrating radar example with 250\,MHz center frequency proves that
  the filter method works also for wave phenomena, not only for diffusive
  electromagnetic fields.
%
% (4) CONCLUSIONS
  The presented algorithm provides a tool to create problem specific digital
  filters. Such purpose-built filters can be made shorter and can speed up
  consecutive  potential-, diffusive-, and wavefield inversions.
%
\end{abstract}

\section{Introduction}

In his Ph.D. thesis, \cite{PhD.70.Ghosh} proposed a linear filter method for
the numerical evaluation of Hankel transforms that subsequently revolutionized
the computation of electromagnetic (EM) responses in the field of geophysical
exploration. If you use a code that calculates EM responses in the
wavenumber-frequency domain and transforms them to the space-frequency domain,
chances are high that it uses the \emph{digital linear filter} (DLF) method.
Most practical 1D EM modeling codes rely on the DLF method for rapid
computations; these codes are not only used for standalone simulations of EM
fields in layered 1D media, but they are also commonly embedded within 3D
modeling codes for computing the primary fields in scattered-field formulations
or for the background fields required by integral equation methods. Thus, the
DLF method is an important component of many commonly used modeling codes for
EM geophysical data.

\cite{GP.71.Ghosh} states that the DLF idea is based on suggestions made four
decades earlier by \cite{PHY.33.Slichter} and \cite{GEO.40.Pekeris}, in that
``\emph{the kernel function is dependent only on the layer parameters, and an
expression relating it to the field measurements can be obtained by
mathematical processes.}'' However, until the introduction of DLF, these
suggestions found no widespread use, likely because of the missing computer
power to calculate the filter coefficients. He further states that credit goes
to \cite{BK.68.Koefoed, GP.70.Koefoed}, who retook the task of direct
interpretational methods with the introduction of raised kernel functions. DLF
is, as such, an improvement of that approach, providing a faster and simpler
method.

The DLF technique is sometimes referred to as the \emph{fast Hankel transform}
(FHT), popular because of the similarity of the name to the well known
\emph{fast Fourier transform} (FFT), although the algorithms for these
techniques are completely different. The FHT name was likely inspired by the
title of a paper by \cite{GP.79.Johansen}. However, the name FHT can be
misleading as it has \emph{Hankel} in the name, whereas the DLF approach can
more generally be applied to other linear transforms, for example Fourier sine
and cosine transforms.

The introduction of linear filters to EM geophysics, in parallel with rising
computational power, initiated a wealth of investigations, leading to the
development of computer programs that extended and improved the DLF method, and
to numerous publications. These publications fall broadly into one or several
of three categories: (1) new applications that extend the usage of DLF to other
EM measurement techniques; (2) filter improvements that provide either new
filters or improved methods for the determination of filter coefficients; and
(3) computational tools that compute EM responses using DLF techniques. Here,
we briefly review the most relevant publications, without claiming
completeness.

\subsubsection{(1) New applications}

Ghosh used the method originally for the computation of type curves for
Schlumberger and Wenner resistivity soundings: \cite{GP.71.Ghosh} derives a
resistivity model from given Schlumberger or Wenner sounding curves; and
\citet{GP.71.Ghoshb} provides filters for the inverse operation, deriving
resistivity sounding curves from a given resistivity model. The method was next
applied to electromagnetic soundings with horizontal and perpendicular coils
\citep{GP.72.Koefoed}, to vertical coplanar coil systems \citep{GP.73.Verma},
to dipoles and other two electrode systems \citep{GP.74.Das, GP.74.Dasb,
GP.80.Das, GEO.94.Sorensen}, and to vertical dikes, hence vertical instead of
horizontal layers \citep{GEO.75.Niwas}. The first filters were specific to a
particular resistivity sounding type and its transform; later publications used
the method to get one type curve from another type curve \citep{GP.77.Kumar,
GP.78.Kumar}, or generalized the method to be applicable to a wider set of
problems \citep{EXG.80.Davis, GXP.81.Das, GEO.84.Das, GP.84.ONeill}.
Eventually, it passed from pure layered modeling to primary-secondary field
formulations for 3D problems, where DLF is used to compute the spatial
Fourier-Hankel transforms in a horizontally layered background medium and to
compute transient responses from frequency domain computations
\citep{GJI.81.Das, GJI.82.Das, GEO.84.Anderson, GEO.86.Newman,
MGS.17.Kruglyakov}. Other publications delved into the theory of the method,
analyzing the oscillating behaviour of the filters and trying to estimate the
error of DLF \citep{GP.72.Koefoedb, GP.76.Koefoedb, GP.79.Johansen,
GP.90.Christensen}.

\subsubsection{(2) Filter improvements}

\cite{PhD.70.Ghosh} derived the filter coefficients in the spectral domain by
dividing the output spectrum by the input spectrum followed by an inverse
Fourier transform. Improvements to the determination of filter coefficients
were provided by \cite{EXG.75.ONeill, GEO.77.Nyman, GEO.82.Das}, or
specifically for the Fourier transform by \cite{GP.86.Nissen}. A direct
integration method was used by \cite{GP.76.Bichara} and \cite{GP.78.Bernabini}.
\cite{GP.79.Koefoed} proposed a Wiener-Hopf least-squares method, which was
further improved by many authors \citep{GP.82.Guptasarma, GEO.1982.Murakami,
GP.97.Guptasarma}. \cite{GP.07.Kong} proposed a direct matrix inversion method
to solve the convolution equation, which requires only the input and output
sample values. To evaluate the filter's effectiveness, he defines a good filter
as one that recovers small or weak diffusive EM fields. This method was also
used by \cite{GEO.09.Key, GEO.12.Key} to create a suite of filters accurate for
marine EM data. Most works have published filters for the Hankel transform with
$J_0$ and $J_1$ Bessel functions (or $J_{-1/2}$, $J_{1/2}$ if applied to the
Fourier sine/cosine transform), as all higher Bessel functions can be
rewritten, via recurrence relations, using only these two. \cite{GP.94.Mohsen}
is one of the rare cases which provides $J_2$ filter weights.

\subsubsection{(3) Computational tools}

The most well-known codes are likely Anderson's freely available ones.
\cite{USGS.73.Anderson} extends the method to transient responses, applying DLF
not only to the Hankel transform, but also to the Fourier transform. A
transient signal can therefore be obtained be applying twice a digital filter
to the wavenumber-frequency domain calculation. \cite{USGS.75.Anderson,
GEO.79.Anderson} presents improved filters for both Fourier and Hankel
transforms, introducing measures to significantly speed-up the calculation,
such as the lagged convolution or using the same abscissae for $J_0$ and $J_1$.
\cite{TMS.82.Anderson} and the included 801\,pt filter became the de facto
industry standard, to which subsequent filters were compared.
\cite{GEO.89.Anderson} presents a hybrid solution that uses either the DLF or a
much slower but highly accurate adaptive-quadrature approach, which can also be
used to measure the accuracy of the DLF. \cite{GEO.12.Key} presented codes for
comparing the DLF approach with a speed optimized quadrature method called
quadrature-with-extrapolation (QWE); despite its optimizations, QWE is still
not as fast as DLF, but it remains valuable as an independent tool for testing
the accuracy of particular filters for the DLF method. Other examples include
the codes by \cite{GP.75.Johansen}, an interactive system for interpretation of
resistivity soundings, and a tool to calculate filter coefficients by
\cite{GP.90.Christensen}. The latter is available upon request and was used,
for instance, in all the open-source modeling and inversion routines of CSIRO
in the Amira Project 223 \citep{ASEG.07.Raiche}.

All mentioned publications have in common that they were derived for direct
current methods (DC) or low frequency methods, such as time-domain shallow EM
methods (TEM), or controlled-source electromagnetics (CSEM), but not for high
frequency methods such as ground-pe\-ne\-tra\-ting radar (GPR). Generally it
was even thought that the filter method works only in the diffusive limit where
the quasistatic approximation is valid \citep[e.g.,][]{GEO.15.Hunziker}.
Nevertheless, \cite{GEO.17.Werthmuller} tested DLF for modeling 250 MHz
center-frequency GPR data using a 401 pt filter that was designed for diffusive
EM, and found that it was orders of magnitude faster than quadrature
approaches. However, good quality results were only obtained for the first
arrivals at short time-offsets and the later arrivals had poor quality. Yet,
this promising initial test suggests it might be possible to specifically
design a special filter for GPR frequencies where the EM fields propagate as
waves rather than by diffusion.

While there are several versions of digital filters that are freely available,
there are no open-source filter designing tools that readily available; they
are either only described methods in articles or the codes are only available
upon request by the author. Here, we address this gap by presenting the code
\texttt{fdesign}, which is an open-source tool for designing general or
purpose-built filters using the direct matrix inversion method. The code is
completely open-source and can be run using freely available programming
environments, thus there are few barriers to obtaining and running the code.

In the following sections we give a brief overview of the methodology, the
theory, and the code, and then show examples of its usage, as well as
presenting new filters for both CSEM and GPR data. The current article falls
therefore into all three mentioned categories by the new application of DLF to
wave phenomena, which means that the DLF can probably be used to model
seismic wavefields as well, and by providing new filters with the accompanying
code. The algorithm and many more examples of its usage can be found on
\href{https://github.com/empymod/article-fdesign}{github.com/empymod/article-fdesign}.
The examples can be used as templates to design new filters.


\section{Methodology}

The algorithm \texttt{fdesign} is a filter designing tool using the direct
matrix inversion method as described by \cite{GP.07.Kong} and based on scripts
developed by \cite{GEO.12.Key}. The tool is an add-on to the EM modeler
\texttt{empymod} \citep{GEO.17.Werthmuller}, written in Python, and hosted on
GitHub, which should foster interaction and enable anyone to toy around,
improve, and extend it. It can be used to derive digital linear filters for the
Hankel and the Fourier transforms for potential, diffusive, and wavefield
modeling (hence from DC to GPR).

Theoretically, it can be used to derive linear filters for any linear
transform, as long as it is supplied with the inputs and outputs of a transform
pair. Previous works have relied on using theoretical transform pairs commonly
found in tables of integral transforms. Here we follow a similar approach, but
we also propose a new method where filters could be derived from accurately
computed numerical transform pairs, such as those computed using quadrature
approaches. For example, the EM modeler \texttt{empymod} could be used to
compute transform pairs for a particular class of model and data parameters
(for example, a specific EM method and model of interest) using a slow but
highly accurate quadrature method. A good filter derived from the resulting
transform pairs could then be used for faster modeling of similar model and
data parameters.

The main features of \texttt{fdesign} and some differences between it and the
methods it is based upon are:
\begin{itemize}
  \item The algorithm computes many different filters for various spacing and
    shift values (brute-force), with an optional minimization routine for
    the best outcome of the brute-force step.
  \item \cite{GP.07.Kong} and \cite{GEO.12.Key} optimize a filter for the $J_0$
    filter, and then use the obtained shift and spacing value to calculate the
    $J_1$ values. The presented algorithm can optimize either $J_0$, $J_1$, or
    both in the optimization process. Five transform-pairs for each $J_0$ and
    $J_1$, and three transform pairs for each sine and cosine of often used
    functions are included in the routine, but any other transform pair can be
    provided as input.
  \item More complex or more specific numerical transform  pairs  can be
    provided instead of theoretical transform pairs by using the modeler
    \texttt{empymod}.
  \item \cite{GP.07.Kong} defines a \emph{good} filter as one that recovers
    \emph{weak} diffusive EM fields. In  \texttt{fdesign}, you can define a
    relative error level that defines up to what error an obtained result is
    good or not. The optimization seeks to find the smallest amplitude field
    that is accurately modeled (i.e., minimizing the amplitude), but it has
    additionally another mode, where it will maximize the output abscissae $r$
    of the transform pair instead of minimizing the amplitude. The two modes
    yield the same result in the general, simple case of fast decaying
    transform pairs. However, for some complex models and specifically for very
    high frequencies, maximizing $r$ will yield much better and more consistent
    results.
  \item The algorithm can solve equal- and over-determined
    linear systems.
  \item The real or the imaginary part can be used for the inversion of complex
    transform pairs.
\end{itemize}

The quality of a filter depends heavily on the model chosen for comparison. An
obtained filter might be very good for one model or data acquisition setup, but
not that good for another one, which is no different for this designing tool.
There is no way to estimate the error of a result obtained with a certain
filter if you apply it to any other model, unless you calculate this other
model with another method for comparison. All results presented here should
therefore be taken with a certain care. Further, when applying a new filter to
a significantly different model and/or data space than what it has previously
been tested on, we recommend first verifying its accuracy in that space by a
comparison with results computed with a quadrature method such as QWE, which is
also available in \texttt{empymod}.

\subsection{Theory}

Most of the articles mentioned in the review have detailed derivations of the
digital filter method. In this article we focus on the algorithm, and summarize
the theory only very briefly by following \cite{GEO.12.Key}. In
electromagnetics we often have to evaluate integrals of the form
%
\begin{equation}
  F(r) = \int^\infty_0 f(l)K(l r)\mr{d}l \ ,
  \label{eq:HankelInt}
\end{equation}
%
where $l$ and $r$ denote input and output evaluation values, respectively, and
$K$ is the kernel function. In the specific case of the Hankel transform $l$
corresponds to wavenumber, $r$ to offset, and $K$ to Bessel functions; in the
case of the Fourier transform $l$ corresponds to frequency, $r$ to time, and
$K$ to sine or cosine functions. In both cases it is an infinite integral which
numerical integration is very time-consuming because of the slow decay of the
kernel function and its oscillatory behaviour.

By substituting $r = e^x$ and $l = e^{-y}$ we get
%
\begin{equation}
  e^x F(e^x) = \int^\infty_{-\infty} f(e^{-y})K(e^{x-y})e^{x-y}\mr{d}y\ .
  \label{eq:filtint}
\end{equation}
%
This can be re-written as a convolution integral and be approximated for an
$N$-point filter by
%
\begin{equation}
  F(r) \approx \sum^N_{n=1} \frac{f(b_n/r) h_n}{r}\ ,
  \label{eq:filtapprox}
\end{equation}
%
where $h$ is the digital linear filter, and the logarithmically spaced filter
abscissae is a function of the spacing $\Delta$ and the shift $\delta$,
%
\begin{equation}
  b_n = \exp\left\{\Delta(-\lfloor{(N+1)/2\rfloor}+n) + \delta\right\} \ .
  \label{eq:base}
\end{equation}
%
From equation~\ref{eq:filtapprox} it can be seen that the filter method
requires $N$ evaluations at each $r$. For example, to calculate the frequency
domain result for 100 offsets with a 201\,pt filter requires 20,100 evaluations
in the wavenumber domain. This is why the DLF often uses interpolation to
minimize the required evaluations, either for $F(r)$ in what is referred to as
\emph{lagged convolution DLF}, or for $f(l)$, which we here call \emph{splined
DLF}.

In the direct matrix inversion method for solving for the digital filter
coefficients, equation~\ref{eq:filtapprox} is cast as a linear system,
where $\Delta$ and $\delta$ are preassigned scalar values and
$r$ is a range of $M$ preassigned values $r_m$. The filter base coefficients
$b_n$ are computed using equation~\ref{eq:base} and an array of values for $l$
are computed using $l_{mn} = b_n/r_m$. The linear system's left hand side (LHS)
matrix $\bf{A}$ has dimensions $M \times N$ with coefficients
$A_{mn} = f(l_{mn})/r_m$ and the right hand side (RHS)
vector $\bf{v}$ has elements $v_m = F(r_m)$. The
resulting linear system
%
\begin{equation}
  \bf{A}\bf{h} = \bf{v}
  \label{eq:,linearSystem}
\end{equation}
%
can be solved by direct matrix inversion, or any other matrix inversion
routine, to obtain the vector of filter coefficients $\bf{h}$. For a given
filter length $N$, there are many subjective choices that go into the practical
implementation of this method, including the choice of the transform pairs
$F(r)$ and $f(l)$, as well as the value for $M$ and the particular range and
spacing of the values $r_m$. Once values are chosen for these variables, an
optimal filter can be found by a grid search over $\Delta$ and $\delta$ for
values that produce a high quality filter. The choice of metric for what
constitutes a high quality filter is also subjective and is further discussed
below.

\subsection{Pseudo-code for \texttt{fdesign} }

The main input variables are the filter length ($N$), the spacing ($\Delta$)
and shift ($\delta$) ranges over which to loop, and the transform pairs for the
inversion ($f_\mr{I}$) and the check of quality ($f_\mr{C}$). If $f_\mr{C}$ is
not provided, then $f_\mr{I}$ is used for both. There are additional, optional
input parameters, for instance to adjust how the RHS abscissae for the
inversion $r_\mr{I} = p(b, N)$ are calculated, where $b$ is the filter base.
Additionally, ranges $r$ for evaluating the check function need to be
specified.

The basic steps are as follows:
\begin{enumerate}
  \item Evaluate RHS of check-function $f_\mr{C}$:\newline
    $d_\mr{R} = f_\mr{C}^\mr{rhs}(r)$
  \item Loop over each value $\Delta_i$, $\delta_j$ (brute force;
      parameters on the left-hand side are overwritten where $i$, $j$
      are not mentioned):
    \begin{enumerate}
      \item Calculate filter base:\newline
        $b_n = \exp\left\{\Delta_i(-\lfloor{(N+1)/2\rfloor}+n) +
      \delta_j\right\}$
      \item Get required RHS and LHS evaluation points:\newline
        $r_\mr{I} = p(b, N)$\newline
        $l_\mr{I} = b/r_\mr{I}$
      \item Invert for filter coefficients:\newline
        $J = \mr{solve}\left[f_\mr{I}^\mr{lhs}(l_\mr{I}),
        f_\mr{I}^\mr{rhs}(r_\mr{I})\right]$
    \item Calculate numerically RHS of check-function $f_\mr{C}$ with current
      filter $J$:\newline
      $d_\mr{F} = f_\mr{C}^\mr{lhs}(b/r)\cdot J\ /\ r$
      \item Store minimum recovered amplitude or maximum $r$ where relative
        error is less than the provided error:\newline
        $\chi_{ij} = g\left[
        \mr{argmin}\left(|(d_\mr{F}-d_\mr{R})/d_\mr{R}| >
        \mr{error}\right) -1\right]\ $, where $g$ is either $d_\mr{R}$
        or $1/r$.
    \end{enumerate}
  \item Re-calculate (steps 2a-2c) filter base and
    coefficients which yield minimum amplitude or maximum $r$ (a local
    minimization can be run to polish the brute-force result):\newline
    $i_\mr{best}$, $j_\mr{best}$ = $\mr{argmin}(\chi)$\newline
    recalculate $b$, $J$ with $i_\mr{best}$, $j_\mr{best}$\newline
    \newline
  \item Return filter base and coefficients and error matrix (for visual
    QC):\newline
    return $b$, $J$, $\chi$\newline
\end{enumerate}

To solve step 2(c) we implemented the QR-decomposition method in
  \texttt{fdesign}.

\subsection{Minimization criteria}
Figure~\ref{fig:Figure-1-bw} shows the differences between the different
minimization approaches; in (a) for a conventional, fast decaying transform
pair (using the $J_0$ kernel of a fullspace with vertical distance between
source and receiver of 50\,m, frequency $f=1\,$Hz, and resistivity
$\rho=1\,\Omega\,$m); and in (b) for a more complex, high frequency layered
model (using the $J_0$ kernel for $f=200\,$MHz for the GPR model described
in the section \emph{Ground-penetrating radar}, with receiver located at 1\,m
depth). The circles show the minimum amplitude used in previous approaches.
This criterion is not ideal, as it is subject to some random fluctuations and
depends on the choice of $r$. The squares show the minimum amplitude given a
certain acceptable error, and the diamonds show the maximum $r$ given a certain
acceptable error (the inversion is a minimization process, it therefore
minimizes $1/r$, not $r$). In simple cases the minimum amplitude given an
acceptable error and the maximum $r$ given an acceptable error will yield the
same result. However, in complex cases the maximum $r$ is more consistent and
therefore a much better criterion.
%
\plot*{Figure-1-bw}{width=.9\textwidth}{(a) A regular right-hand-side (RHS)
  curve of a transform pair with a purely decaying function. Minimum amplitude
  or maximum $r$ yield the same result in this scenario. Using a relative error
  criteria is more stable than just the absolute minimum amplitude. (b) A 1D
  model for $f=200\,$MHz. Here, the maximum $r$ provides a better criteria for
  the inversion.}
%

While the $\chi$ array is used to identify the optimal values of $\Delta$ and
$\delta$, the value of $\mr{argmin}(\chi)$ will depend on the choice of the
check function $f_C$ and can vary from one transform pair to another; thus it
should not be used to compare the quality of filters created with different
transform pairs.


\section{Numerical examples}

The numerical examples are focused on the Hankel transform, although
\texttt{fdesign} can be used to design digital linear filters for both Hankel
and Fourier transforms. As there is no difference in the procedure of the two,
this should be sufficient to demonstrate the algorithm.

\subsection{Design}
Figures~\ref{fig:Figure-2-bw} and~\ref{fig:Figure-3-bw} show the solution spaces
of seven consecutive inversion runs using the following theoretical transform
pairs \citep[e.g.,][]{USGS.75.Anderson},
%
\begin{align}
  \int^\infty_0\,l \exp\left(-al^2\right) J_0(rl)\,dl &=
  \frac{1}{2a} \exp\left(\frac{-r^2}{4a}\right)\ , \\
  \int^\infty_0\,l^2 \exp\left(-al^2\right) J_1(rl)\,dl &=
  \frac{r}{4a^2} \exp\left(-\frac{r^2}{4a}\right)\ ,
  \label{eq:j01}
\end{align}
%
where $a$ was set to 5. In the algorithm you can provide one pair
for the inversion, and a different pair to get the minimum amplitude or the
maximum $r$. In this case the same transform pair was used for both the
inversion and the check of quality. The acceptable relative error was set to
1\,\%. The RHS evaluation parameter for the inversion was set so  that an
over-determined system was evaluated with $M = 2N$ equations, where $r_\mr{I}$
was logarithmically spaced from $\log_{10}[1/\max(b)] - 1$ to
$\log_{10}[1/\min(b)] + 1$, where $b$ are the filter abscissae as given in
equation~\ref{eq:base}. This is accomplished in \texttt{fdesign} by setting
parameter $r_\mr{def} = (1, 1, 2)$.

Figure~\ref{fig:Figure-2-bw} shows three different inversion runs for $N=101$,
201, and 401 for a wide range of spacing and shift values. From this one has to
decide for a filter length. Longer filters tend to give
more precise results, but their transform is computationally more expensive as
more values have to be evaluated. The length of the filter does not has to be
an odd number, but many published filters do have an odd number of
coefficients. The integral runs from $-\infty$ to $+\infty$ and hence an odd
point filter comes naturally by truncating the integral to a finite sum from
$-(N-1)/2$ to $(N-1)/2$. However, this is only true if the shift value
$\delta = 0$. If it differs from zero then there is no advantage in having an
odd point filter.
%
\plot*{Figure-2-bw}{width=\textwidth}{%
  Test of different filter lengths. Filter length is a trade-off between
    precision and speed, as shorter filters are faster but less precise, and
  longer filters are more precise but computationally more expensive. The red
  square in subplot (b) is the starting point of
  Figure~\ref{fig:Figure-3-bw}.}%
%

Figure~\ref{fig:Figure-3-bw} shows four consecutive inversion runs for $N=201$,
where each run is a focus on a subsection of the previous run, indicated by the
square. From the low resolution overview runs (a) and (b) it looks like this is
a standard, straight-forward minimization problem. However, from the more
detailed results in (c) and (d) it becomes obvious that it is a minimization
problem that has to be solved stochastically, as there are solutions with 2
orders of magnitude difference apparently randomly next to each other.
%
\plot*{Figure-3-bw}{width=\textwidth}{%
  Solution spaces of four consecutive inversion runs for a 201\,pt filter. Each
  consecutive run zooms into a portion of the previous solution space,
  indicated by the square. The more in detail we obtain the solution space, the
  more random appears to be the distribution.}%
%

Figure~\ref{fig:Figure-4-bw} shows in (a) the filter values for $J_0$ and $J_1$
of the best 201 pt filter obtained this way, and in (b) its right-hand-side
solution. The black dots indicate negative values, from which it can be seen
that adjacent values are often alternating between positive and negative
contributions.
%
\plot*{Figure-4-bw}{width=.9\textwidth}{(a) Filter values of the best obtained
  201\,pt filter with the corresponding check of quality in (b). Black points
  indicate negative values, which shows that adjacent values have often
  opposite signs.}
%

Designing filters is to a large extent trial and error. All input variables
influence the outcome, and often an even better filter is found by sheer luck
of choosing the right starting parameters, as can be seen in
Figure~\ref{fig:Figure-4-bw} (h). However, once you are in this zone every
filter is a very good filter. Each of the input parameters has an effect on the
outcome. It also depends heavily on the transform pairs $f_\mr{I;C}$; functions
that decay rapidly are generally better, as noted by earlier authors
\citep[e.g.][]{USGS.75.Anderson}. It also depends on the filter length, and
obviously heavily on the spacing and shift values. Another important point is
how you define the right-hand evaluation points of the inversion
($r_\mr{def}$). Evaluating corresponding transform pairs separately or jointly
also leads to different filter coefficients ($J_0$, $J_1$, or $J_0$ \& $J_1$,
or equally sine, cosine, or sine \& cosine); also if the real or the imaginary
part is used when complex transform pairs or \texttt{empymod} is used; and if
it is inverted for the minimum amplitude or for the maximum $r$.

\subsection{CSEM}
In this section we compare the 201\,pt filter derived in the previous section
to marine CSEM models as used in \cite{GP.07.Kong}, in \cite{GEO.12.Key},
and for a land EM case.

Figure~\ref{fig:Figure-5-bw} compares the derived 201\,pt filter with the two
half-spaces case used by \cite{GP.07.Kong} in his Figure~5. The model consists
of a $0.3125\,\Omega\,$m sea layer of infinite thickness above a subsurface
$1\,\Omega\,$m half-space. The signal of a $1\,$Hz frequency $x$-directed
electric source 50\,m above the interface is measured at an $x$-directed
electric receiver at the interface.
%
\plot*{Figure-5-bw}{width=\textwidth}{(a) Results of different filters and QWE
  for the model of Figure~5 from \cite{GP.07.Kong} with the relative errors
  shown in (b), using the QWE result. The relative error is meaningless from
  about 15.5\,km onwards, as QWE failed as well for these very low amplitudes.}
%
In (a) it can be seen that the new 201\,pt filter is able to recover smaller
amplitudes than the 241\,pt filter from \cite{GP.07.Kong}, the 201\,pt filter
from \cite{GEO.12.Key}, and the 801\,pt filter from \cite{TMS.82.Anderson}
(Wer201, Kong241, Key201, and And801, respectively). It behaves equally well as
the quadrature with extrapolation (QWE), for which we used a 51\,pt quadrature
with relative and absolute tolerance of $10^{-12}$ and $10^{-30}$,
respectively. The relative error is shown in (b), where the QWE result was
taken as \emph{truth}. For offsets greater than roughly 15.5\,km the relative
error becomes meaningless, as the QWE fails itself; this part is greyed out in
the figure.

Figure~\ref{fig:Figure-6-bw}(a) considers the canonical CSEM model as used in
\cite{GEO.12.Key} in his Figure~5: a $0.303\,\Omega\,$m sea with depth of
2\,km, a background resistivity of $1\,\Omega\,$m, in which a $100\,\Omega\,$m
target layer of 100\,m thickness is embedded at 1000\,m below the seafloor.
The source is located at 1990\,m and receivers are positioned on the seafloor.
%
\plot*{Figure-6-bw}{width=\textwidth}{(a) The canonical marine CSEM model of
  \cite{GEO.12.Key} and (b) a land model with source and receiver at the
  surface. The new filter shows relative errors that are generally several
  orders of magnitude lower than for the other filters.
}
%

Figure~\ref{fig:Figure-6-bw}(b) is a land case, with a background resistivity
of $10\,\Omega\,$m, in which a $500\,\Omega\,$m target of 100\,m thickness is
embedded at 1000\,m below the surface. The source depth is 0.5\,m and the
receiver depth is 0.8\,m. In both cases the new filter \emph{Wer201} has
generally a relative error which is orders of magnitude lower than the other
filters. The relative error of the real part is given in black, whereas the
relative error of the imaginary part is given in grey. It is interesting to
note that the real and imaginary parts have very similar errors in
\emph{Wer201}, \emph{Key201}, and \emph{And801}, but that the imaginary part of
\emph{Kong241} seems to behave considerably better for most part than its real
counterpart. To compare the real and the imaginary parts is insofar interesting
as the digital filters are purely real valued.

It is very important to note again that other scenarios might yield very
different error plots. Although the new 201\,pt filter proves to be very
accurate for these three models, it might not be the best filter in other
cases. To verify the applicability of the filter we run two different tests
over a wide range of scenarios. Figure~\ref{fig:Figure-7} shows the relative
errors of DLF using the filter \emph{Wer201} compared to QWE for a deep water
model (2\,km water depth), a shallow water model (400\,m water depth), and a
land model. The models consist of air (upper halfspace), a water layer in the
first two examples, and a subsurface halfspace. Source depth is 1990\,m, 10\,m,
and 0.5\,m in the deep, shallow, and land case, respectively. Receiver depth is
2000\,m, 100\,m, and 0.8\,m. The subsurface halfspace resistivity varies from 1
to 10, 100, and 1000\,$\Omega\,$m from the left to the right column. The error
plots cover offsets from 50\,m to 20\,km on the x-axis, and 0.01\,Hz to 10\,Hz
on the y-axis. It can be seen in the result that the filter is sufficiently
precise for all practical CSEM cases, hence has generally an error far below
1\,\% ($10^{-2}$) in all cases. A relative noise of less than 1\,\% is
generally considered really good for real data. The exceptions are in the
shallow water case at large offsets with relatively high frequencies, e.g.,
10\,km offset with 8\,Hz; this zone has a very high error for subsurface
resistivities of 1\,$\Omega\,$m and 10\,$\Omega\,$m (yellow zones). However,
this is due to the fact that the amplitudes of the signal for these offsets and
frequencies are in the order of $10^{-25}\,$V/m. The grey contour lines in the
plot indicate the power of the amplitude of the signal.
%
\plot*{Figure-7}{width=\textwidth}{Relative error plots comparing the DLF
  (\emph{Wer201}) with QWE for the three cases deep water (top row), shallow
  water (middle row), and land (bottom row). The subsurface halfspace
  resistivity varies from 1 to 1000\,$\Omega\,$m from the left column to the
  right column; offsets vary from 50\,m to 20\,km, and frequencies vary from
  0.01\,Hz to 10\,Hz. The DLF provides sufficiently accurate results for all
  practical CSEM applications. The exception is in the shallow water case for
  $\rho = 1\,\Omega\,$m and $10\,\Omega\,$m for large offsets and high
  frequencies (yellow zones). However, the amplitudes in these regions are in
  the order of $10^{-25}\,$V/m, and therefore several orders of
  magnitude below the noise levels of current EM instrument systems.
  Contour lines show the power of the amplitude of the signal.}
%

Another error test is shown in Figure~\ref{fig:Figure-8}. It shows in (a) the
amplitude of the analytical fullspace solution for a wide range of
resistivities and frequencies: $\rho=$ $10^{-10}\,\Omega\,$m to
$10^{10}\,\Omega\,$m, and $f =10^{-10}$\,Hz to $10^{14}$\,Hz. In this case, the
horizontal offset is set to the skin depth, and the vertical source-receiver
separation to a tenth of it, where the skin depth is given by \citep[e.g., ][
equation~1.49]{B.SEG.88.Ward}
%
\begin{equation}
  z_\mr{skin} = \left\{\frac{\omega^2 \varepsilon\mu}{2}
    \left[ \sqrt{1 + \frac{1}{\left(\omega\varepsilon\rho\right)^2}} -1\right]
  \right\}^{-1/2} \, , \\
  \label{eq:skin}
\end{equation}
%
where we used in the example $10\,\varepsilon_0$ for the electric permittivity
and $\mu_0$ for the magnetic permeability; $\omega = 2\pi f$. The grey contour
lines in the figure show the magnitude of the offset: the line with 0 is where
the offset is 1\,m, the line with 3 where the offset is 1\,km, etc. The contour
lines of the offset have a knee, after which they become almost independent of
frequency. This is where the quasi static approximation breaks down, and wave
phenomena become dominant. The red line indicates where $\omega\varepsilon\rho
= 1$. Subplot (b) shows the relative error of the DLF using the filter
\emph{Wer201} compared to the analytical solution. It shows nicely that the
filter can be used for all cases that are in the quasi static region, but not
outside.
%
\plot*{Figure-8}{width=.8\textwidth}{(a) Amplitude of the analytical
  fullspace solution for a wide range of resistivities and frequencies. The
  contour lines show the power of the offset used for the calculation, where
  the offset is set to the skin depth. (b) Relative error of the DLF compared
  with the analytical solution. The filter \emph{Wer201} works well in the
  diffusive limit, but fails when wave phenomena become dominant. The red
 line shows where $\omega\varepsilon\rho = 1$.}
%

We also comment that while it is tempting to design very short length
filters (much less than about $N=100$) since shorter filters can greatly reduce
EM simulation times, in our experience, shorter filters tend to be less general
purpose than longer filters and we highly recommend thoroughly testing their
accuracy for the desired range of data and model parameters.


\subsection{Ground-penetrating radar}

Figure~\ref{fig:Figure-9} shows the GPR example as calculated in
\cite{GEO.15.Hunziker} and \cite{GEO.17.Werthmuller}; the model parameters are
given in subplot (a). The filter used for this example for the Hankel transform
is a 2001\,pt filter, derived with the fullspace solution with $f=500\,$MHz for
the inversion and the check of quality. The whole space solution used
resistivity $200\,\Omega\,$m, relative permittivity $\varepsilon_\mr{r}=10$,
and permeability $\mu_\mr{r}=1$. Using Equation~\ref{eq:skin} yields a skin
depth of roughly 3.6\,m for these parameters. The source and receiver vertical
separation is 1\,m. For the Fourier transform a 4096\,pt FFT was used with
regularly spaced frequencies from 0.5\,MHz to 850\,MHz and then zero-padded up
to 2048\,MHz. The frequency result is multiplied with a Ricker wavelet with a
center frequency of 250\,MHz, and a gain function ($1 + |t^3|, t$ in ns) is
applied. Subplot (b) shows the result when using the adaptive quadrature (QUAD)
option of \texttt{empymod}, which uses the QAGSE routine from the Fortran
QUADPACK library. Subplot (c) shows the result using DLF with the new 2001\,pt
digital filter. The calculation with DLF took under 9 minutes and is therefore
roughly 80 times faster than the calculation with QUAD, which took roughly 11
hours and 27 minutes. Calculating the same model with QWE took 7 hours and 20
minutes. However, QWE uses in roughly 1/3 of the calculation internally QUAD in
this example, see \cite{GEO.17.Werthmuller}. Note that DLF was run in parallel
using 4 threads at once, taking effectively only 2 minutes and 10 seconds to
calculate.
% EMmod \citep{GEO.15.Hunziker} took for the same model about 18 hours 55
% minutes, using a 61\,pt Gauss-Kronrod integration routine.
The lagged convolution version of DLF and the splined version of QWE were used
in this comparison.
%
\plot*{Figure-9}{width=\textwidth}{GPR example for the model given in (a) using
  \texttt{empymod} with (b) adaptive quadrature (QUAD) or with (c) a
  2001\,pt digital linear filter. QUAD took about 11\,h 27\,min to calculate,
  whereas DLF took 2\,min 10\,s in parallel on 4 nodes, hence under 9\,min in
  total.}
%

Figure~\ref{fig:Figure-10-bw} shows in (a)-(c) the real and imaginary
parts of the frequency domain results and in (d)-(f) the time domain results
for offsets of 0.2\,m, 2.0\,m, and 3.0\,m.
%
\plot*{Figure-10-bw}{width=\textwidth}{The real and imaginary parts of the
  frequency-domain (a-c) and the time-domain (d-f) responses for
  horizontal offsets of (a/d) 0.2\,m, (b/e) 2.0\,m, and (c/f) 3.0\,m.}
%

The 2001\,pt filter was derived with the fullspace solution of a medium which
is very similar to the layer in which source and receiver reside in our GPR
example. To show that this filter can also be applied to different layer
parameters we run a test where we swapped layers one and two of the model in
Figure~\ref{fig:Figure-9} (a). The resulting real and imaginary parts for the
offsets 0.2\,m, 2.0\,m, and 3.0\,m are given in Figure~\ref{fig:Figure-11-bw}.
%
\plot*{Figure-11-bw}{width=\textwidth}{The real and imaginary parts of the
  frequency-domain response for the model given in Figure~\ref{fig:Figure-9}
  (a) with swapped first and second layers for horizontal offsets of (a)
  0.2\,m, (b) 2.0\,m, and (c) 3.0\,m.}
%

These examples show clearly that the filter method can indeed be applied to
high frequency EM modeling and therefore wave phenomena. We are convinced that
with further tests and analysis much better filters could be achieved, and
various concepts could be checked. One approach is to derive a filter for each
frequency band, say one for 1\,MHz--10\,MHz, one for 10\,MHz--100\,MHz, and one
for 100\,MHz--1\,GHz. Another approach could be to derive distinct filters for
$J_0$ and $J_1$, with different spacing and shift values. The first idea would
roughly triple the calculation cost, the second idea double them; however, they
both would still be very fast compared to standard quadrature methods. One
could also apply the digital filter method to acoustic and elastic wavefields
as well.

\section{Conclusions}

The presented, free and open-source code \texttt{fdesign} can be used to design
digital linear filters for the Hankel and Fourier transforms (and more
generally for any linear transform) using either analytical transform pairs or
1D subsurface models together with the EM-modeler \texttt{empymod}. The code is
available from GitHub as part of \texttt{empymod}; the version corresponding to
this publication is \texttt{v1.8.1}.

The presented 201\,pt filter achieves more precise results in the three
presented CSEM cases than other filters, and is included in \texttt{empymod}
from version 1.4.5 onwards. However, as with any digital filter, the quality
depends heavily on the model, and this new filter might or might not behave
that well for other models or data parameters.

The GPR example shows that the digital linear filter method can also be used
for wave phenomena, not only for the diffusive approximation limit of low
frequency EM modeling.

We see \texttt{fdesign} as being useful for at least 3 scenarios:
\begin{enumerate}
  \item Providing a fast method to design problem-specific filters. For bigger
    inversion projects (such as, for instance, massive stochastic CSEM
    inversions) a purpose designed, short filter might save much more time than
    it costs to design it. This could even be integrated into inversion codes,
    as an optional pre-inversion step.
  \item Extending the filter method to new areas, namely higher frequencies and
    acoustic and elastic wavefields.
  \item Raising  interest for digital linear filters in geophysics by making it
    very easy for anyone to play around, create their own filters, and get a
    better understanding of the relative simplicity of the method. We are sure
    there are many great filters that are yet to be discovered.
\end{enumerate}

\section{Acknowledgment}

D.W. would like to thank the electromagnetic research group of A. Mousatov at
the IMP for fruitful discussions, and J. Hunziker for valuable feedback which
improved this manuscript considerably. We thank the assistant editor J. Etgen,
the associate editor J. Dellinger, and the reviewers C. Weiss and R. Mittet
for valuable inputs not only regarding the manuscript, but also regarding the
code and its documentation.


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\end{thebibliography}

\end{document}