Addons¶
fdesign
– Digital Linear Filter (DLF) design¶
The addon fdesign can be used to design digital linear filters for the Hankel or Fourier transform, or for any linear transform ([Ghos70]). For this included or provided theoretical transform pairs can be used. Alternatively, one can use the EM modeller empymod to use the responses to an arbitrary 1D model as numerical transform pair.
More information can be found in the following places:
 The article about fdesign is in the repo https://github.com/empymod/articlefdesign
 Example notebooks to design a filter can be found in the repo https://github.com/empymod/empymodexamples
This filter designing tool uses the direct matrix inversion method as described
in [Kong07] and is based on scripts by [Key12]. The whole project of
fdesign
started with the Matlab scripts from Kerry Key, which he used to
design his filters for [Key09], [Key12]. Fruitful discussions with Evert Slob
and Kerry Key improved the addon substantially.
Note that the use of empymod to create numerical transform pairs is, as of now, only implemented for the Hankel transform.
Implemented analytical transform pairs¶
The following tables list the transform pairs which are implemented by default. Any other transform pair can be provided as input. A transform pair is defined in the following way:
from empymod.scripts import fdesign
def my_tp_pair(var):
'''My transform pair.'''
def lhs(l):
return func(l, var)
def rhs(r):
return func(r, var)
return fdesign.Ghosh(name, lhs, rhs)
Here, name
must be one of j0
, j1
, sin
, or cos
, depending
what type of transform pair it is. Additional variables are provided with
var
. The evaluation points of the lhs
are denoted by l
, and the
evaluation points of the rhs
are denoted as r
. As an example here the
implemented transform pair j0_1
def j0_1(a=1):
'''Hankel transform pair J0_1 ([Ande75]_).'''
def lhs(l):
return l*np.exp(a*l**2)
def rhs(r):
return np.exp(r**2/(4*a))/(2*a)
return Ghosh('j0', lhs, rhs)
Implemented Hankel transforms¶
j0_1
[Ande75]\[\int^\infty_0 l \exp\left(al^2\right) J_0(lr) dl = \frac{\exp\left(\frac{r^2}{4a}\right)}{2a}\]j0_2
[Ande75]\[\int^\infty_0 \exp\left(al\right) J_0(lr) dl = \frac{1}{\sqrt{a^2+r^2}}\]j0_3
[GuSi97]\[\int^\infty_0 l\exp\left(al\right) J_0(lr) dl = \frac{a}{(a^2 + r^2)^{3/2}}\]j0_4
[ChCo82]\[\int^\infty_0 \frac{l}{\beta} \exp\left(\beta z_v \right) J_0(lr) dl = \frac{\exp\left(\gamma R\right)}{R}\]j0_5
[ChCo82]\[\int^\infty_0 l \exp\left(\beta z_v \right) J_0(lr) dl = \frac{ z_v (\gamma R + 1)}{R^3}\exp\left(\gamma R\right)\]j1_1
[Ande75]\[\int^\infty_0 l^2 \exp\left(al^2\right) J_1(lr) dl = \frac{r}{4a^2} \exp\left(\frac{r^2}{4a}\right)\]j1_2
[Ande75]\[\int^\infty_0 \exp\left(al\right) J_1(lr) dl = \frac{\sqrt{a^2+r^2}a}{r\sqrt{a^2 + r^2}}\]j1_3
[Ande75]\[\int^\infty_0 l \exp\left(al\right) J_1(lr) dl = \frac{r}{(a^2 + r^2)^{3/2}}\]j1_4
[ChCo82]\[\int^\infty_0 \frac{l^2}{\beta} \exp\left(\beta z_v \right) J_1(lr) dl = \frac{r(\gamma R+1)}{R^3}\exp\left(\gamma R\right)\]j1_5
[ChCo82]\[\int^\infty_0 l^2 \exp\left(\beta z_v \right) J_1(lr) dl = \frac{r z_v(\gamma^2R^2+3\gamma R+3)}{R^5}\exp\left(\gamma R\right)\]
Where
Implemented Fourier transforms¶
sin_1
[Ande75]\[\int^\infty_0 l\exp\left(a^2l^2\right) \sin(lr) dl = \frac{\sqrt{\pi}r}{4a^3} \exp\left(\frac{r^2}{4a^2}\right)\]sin_2
[Ande75]\[\int^\infty_0 \exp\left(al\right) \sin(lr) dl = \frac{r}{a^2 + r^2}\]sin_3
[Ande75]\[\int^\infty_0 \frac{l}{a^2+l^2} \sin(lr) dl = \frac{\pi}{2} \exp\left(ar\right)\]cos_1
[Ande75]\[\int^\infty_0 \exp\left(a^2l^2\right) \cos(lr) dl = \frac{\sqrt{\pi}}{2a} \exp\left(\frac{r^2}{4a^2}\right)\]cos_2
[Ande75]\[\int^\infty_0 \exp\left(al\right) \cos(lr) dl = \frac{a}{a^2 + r^2}\]cos_3
[Ande75]\[\int^\infty_0 \frac{1}{a^2+l^2} \cos(lr) dl = \frac{\pi}{2a} \exp\left(ar\right)\]

empymod.scripts.fdesign.
design
(n, spacing, shift, fI, fC=False, r=None, r_def=(1, 1, 2), reim=None, cvar='amp', error=0.01, name=None, full_output=False, finish=False, save=True, path='filters', verb=2, plot=1)[source]¶ Digital linear filter (DLF) design
This routine can be used to design digital linear filters for the Hankel or Fourier transform, or for any linear transform ([Ghos70]). For this included or provided theoretical transform pairs can be used. Alternatively, one can use the EM modeller empymod to use the responses to an arbitrary 1D model as numerical transform pair.
This filter designing tool uses the direct matrix inversion method as described in [Kong07] and is based on scripts by [Key12]. The tool is an addon to the electromagnetic modeller empymod [Wert17]. Fruitful discussions with Evert Slob and Kerry Key improved the addon substantially.
Example notebooks of its usage can be found in the repo github.com/empymod/empymodexamples.
Parameters:  n : int
Filter length.
 spacing: float or tuple (start, stop, num)
Spacing between filter points. If tuple, it corresponds to the input for np.linspace with endpoint=True.
 shift: float or tuple (start, stop, num)
Shift of base from zero. If tuple, it corresponds to the input for np.linspace with endpoint=True.
 fI, fC : transform pairs
Theoretical or numerical transform pair(s) for the inversion (I) and for the check of goodness (fC). fC is optional. If not provided, fI is used for both fI and fC.
 r : array, optional
Righthand side evaluation points for the check of goodness (fC). Defaults to r = np.logspace(0, 5, 1000), which are a lot of evaluation points, and depending on the transform pair way too long r’s.
 r_def : tuple (add_left, add_right, factor), optional
Definition of the righthand side evaluation points r of the inversion. r is derived from the base values, default is (1, 1, 2).
 rmin = log10(1/max(base))  add_left
 rmax = log10(1/min(base)) + add_right
 r = logspace(rmin, rmax, factor*n)
 reim : np.real or np.imag, optional
Which part of complex transform pairs is used for the inversion. Defaults to np.real.
 cvar : string {‘amp’, ‘r’}, optional
If ‘amp’, the inversion minimizes the amplitude. If ‘r’, the inversion maximizes the righthand side evaluation point r. Defaults is ‘amp’.
 error : float, optional
Up to which relative error the transformation is considered good in the evaluation of the goodness. Default is 0.01 (1 %).
 name : str, optional
Name of the filter. Defaults to dlf_+str(n).
 full_output : bool, optional
If True, returns best filter and output from scipy.optimize.brute; else only filter. Default is False.
 finish : None, True, or callable, optional
If callable, it is passed through to scipy.optimize.brute: minimization function to find minimize best result from bruteforce approach. Default is None. You can simply provide True in order to use scipy.optimize.fmin_powell(). Set this to None if you are only interested in the actually provided spacing/shiftvalues.
 save : bool, optional
If True, best filter is saved to plain text files in ./filters/. Can be loaded with fdesign.load_filter(name). If full, the inversion output is stored too. You can add ‘.gz’ to name, which will then save the full inversion output in a compressed file instead of plain text.
 path : string, optional
Absolute or relative path where output will be saved if save=True. Default is ‘filters’.
 verb : {0, 1, 2}, optional
 Level of verbosity, default is 2:
 0: Print nothing.
 1: Print warnings.
 2: Print additional time, progress, and result
 plot : {0, 1, 2, 3}, optional
 Level of plotverbosity, default is 1:
 0: Plot nothing.
 1: Plot bruteforce result
 2: Plot additional theoretical transform pairs, and best inv.
 3: Plot additional inversion result
 (can result in lots of plots depending on spacing and shift) If you are using a notebook, use %matplotlib notebook to have all inversion results appear in the same plot.
Returns:  filter : empymod.filter.DigitalFilter instance
Best filter for the input parameters.
 full : tuple
Output from scipy.optimize.brute with full_output=True. (Returned when
full_output
is True.)

empymod.scripts.fdesign.
save_filter
(name, filt, full=None, path='filters')[source]¶ Save DLFfilter and inversion output to plain text files.

empymod.scripts.fdesign.
load_filter
(name, full=False, path='filters')[source]¶ Load saved DLFfilter and inversion output from text files.

empymod.scripts.fdesign.
plot_result
(filt, full, prntres=True)[source]¶ QC the inversion result.
Parameters:   filt, full as returned from fdesign.design with full_output=True
  If prntres is True, it calls fdesign.print_result as well.
 r

empymod.scripts.fdesign.
print_result
(filt, full=None)[source]¶ Print best filter information.
Parameters:   filt, full as returned from fdesign.design with full_output=True

class
empymod.scripts.fdesign.
Ghosh
(name, lhs, rhs)[source]¶ Simple Class for Theoretical Transform Pairs.
Named after D. P. Ghosh, honouring his 1970 Ph.D. thesis with which he introduced the digital filter method to geophysics ([Ghos70]).

empymod.scripts.fdesign.
j0_4
(f=1, rho=0.3, z=50)[source]¶ Hankel transform pair J0_4 ([ChCo82]).
Parameters:  f : float
Frequency (Hz)
 rho : float
Resistivity (Ohm.m)
 z : float
Vertical distance between source and receiver (m)

empymod.scripts.fdesign.
j0_5
(f=1, rho=0.3, z=50)[source]¶ Hankel transform pair J0_5 ([ChCo82]).
Parameters:  f : float
Frequency (Hz)
 rho : float
Resistivity (Ohm.m)
 z : float
Vertical distance between source and receiver (m)

empymod.scripts.fdesign.
j1_4
(f=1, rho=0.3, z=50)[source]¶ Hankel transform pair J1_4 ([ChCo82]).
Parameters:  f : float
Frequency (Hz)
 rho : float
Resistivity (Ohm.m)
 z : float
Vertical distance between source and receiver (m)

empymod.scripts.fdesign.
j1_5
(f=1, rho=0.3, z=50)[source]¶ Hankel transform pair J1_5 ([ChCo82]).
Parameters:  f : float
Frequency (Hz)
 rho : float
Resistivity (Ohm.m)
 z : float
Vertical distance between source and receiver (m)

empymod.scripts.fdesign.
empy_hankel
(ftype, zsrc, zrec, res, freqtime, depth=None, aniso=None, epermH=None, epermV=None, mpermH=None, mpermV=None, htarg=None, verblhs=0, verbrhs=0)[source]¶ Numerical transform pair with empymod.
All parameters except
ftype
,verblhs
, andverbrhs
correspond to the input parameters toempymod.dipole
. See there for more information.Note that if depth=None or [], the analytical fullspace solutions will be used (much faster).
Parameters:  ftype : str or list of strings
Either of: {‘j0’, ‘j1’, ‘j2’, [‘j0’, ‘j1’]}
 ‘j0’: Analyze J0term with ab=11, angle=45°
 ‘j1’: Analyze J1term with ab=31, angle=0°
 ‘j2’: Analyze J0 and J1terms jointly with ab=12, angle=45°
 [‘j0’, ‘j1’]: Same as calling empy_hankel twice, once with ‘j0’ and
 one with ‘j1’; can be provided like this to fdesign.design.
 verblhs, verbrhs: int
verbvalues provided to empymod for lhs and rhs.
 Note that ftype=’j2’ only works for fC, not for fI.
tmtemod
– Calculate up and downgoing TM and TE modes¶
This addon for empymod
adjusts [HuTS15] for TM/TEsplit. The development
was initiated by the development of
https://github.com/empymod/csemziolkowskiandslob ([ZiSl19]).
This is a strippeddown version of empymod
with a lot of simplifications
but an important addition. The modeller empymod
returns the total field,
hence not distinguishing between TM and TE mode, and even less between up and
downgoing fields. The reason behind this is simple: The derivation of
[HuTS15], on which empymod
is based, returns the total field. In this
derivation each mode (TM and TE) contains nonphysical contributions. The
nonphysical contributions have opposite signs in TM and TE, so they cancel
each other out in the total field. However, in order to obtain the correct TM
and TE contributions one has to remove these nonphysical parts.
This is what this routine does, but only for an xdirected electric source with
an xdirected electric receiver, and in the frequency domain (src and rec in
same layer). This version of dipole
returns the signal separated into TM++,
TM+, TM+, TM–, TE++, TE+, TE+, and TE– as well as the direct field TM and
TE contributions. The first superscript denotes the direction in which the
field diffuses towards the receiver and the second superscript denotes the
direction in which the field diffuses away from the source. For both the
plussign indicates the field diffuses in the downward direction and the
minussign indicates the field diffuses in the upward direction. It uses
empymod
wherever possible. See the corresponding functions in empymod
for more explanation and documentation regarding input parameters. There are
important limitations:
ab
== 11 [=> xdirected el. source & el. receivers]signal
== None [=> only frequency domain]xdirect
== False [=> direct field calc. in wavenrdomain]ht
== ‘fht’htarg
== ‘key_201_2012’ Options
ft
,ftarg
,opt
, andloop
are not available. lsrc
==lrec
[=> src & rec are assumed in same layer!] Model must have more than 1 layer
 Electric permittivity and magnetic permeability are isotropic.
 Only one frequency at once.
Theory¶
The derivation of [HuTS15], on which empymod
is based, returns the total
field. Internally it also calculates TM and TE modes, and sums these up.
However, the separation into TM and TE mode introduces a singularity at
\(\kappa = 0\). It has no contribution in the spacefrequency domain to the
total fields, but it introduces nonphysical events in each mode with opposite
sign (so they cancel each other out in the total field). In order to obtain the
correct TM and TE contributions one has to remove these nonphysical parts.
To remove the nonphysical part we use the file tmtemod.py
in this
directory. This routine is basically a heavily simplified version of
empymod
with the following limitations outlined above.
So tmtemod.py
returns the signal separated into TM++, TM+, TM+, TM–,
TE++, TE+, TE+, and TE– as well as the direct field TM and TE contributions.
The first superscript denotes the direction in which the field diffuses towards
the receiver and the second superscript denotes the direction in which the
field diffuses away from the source. For both the plussign indicates the field
diffuses in the downward direction and the minussign indicates the field
diffuses in the upward direction. The routine uses empymod
wherever
possible, see the corresponding functions in empymod
for more explanation
and documentation regarding input parameters.
Please note that the notation in [HuTS15] differs from the notation in [ZiSl19]. I specify therefore always, which notification applies, either Hun15 or Zio19.
We start with equation (105) in Hun15:
Ignoring the incident field, and using \(J_2 = \frac{2}{\kappa r}J_1  J_0\) to avoid \(J_2\)integrals, we get
From this the TM and TEparts follow as
Equations (108) and (109) in Hun15 yield the required parameters \(\tilde{g}^{tm}_{hh;s}\) and \(\tilde{g}^{te}_{zz;s}\),
The parameters \(P^{u\pm}_s\) and \(P^{d\pm}_s\) are given in equations (81) and (82), \(\bar{P}^{u\pm}_s\) and \(\bar{P}^{d\pm}_s\) in equations (A8) and (A9); \(W^u_s\) and \(W^d_s\) in equation (74) in Hun15. This yields
where \(d^\pm\) is taken from the text below equation (67). There are four terms in the righthand side, two in the first line and two in the second line. The first term in the first line is the integrand of TE+, the second term in the first line corresponds to TE++, the first term in the second line is TE+, and the second term in the second line is TE–.
If we look at TE+, we have
and therefore
We can compare this to equation (4.165) in Zio19, with \(\hat{I}^e_x=1\) and slightly rearranging it to look more alike, we get
The notation in this equation follows Zio19.
The difference between the two previous equations is that the first one contains nonphysical contributions. These have opposite signs in TM+ and TE+, and therefore cancel each other out. But if we want to know the specific contributions from TM and TE we have to remove them. The nonphysical contributions only affect the \(J_1\)integrals, and only for \(\kappa = 0\).
The following lists for all 8 cases the term that has to be removed, in the
notation of Zio19 (for the notation as in Hun15 see the implementation in
tmtemod.py
):
Note that in the first and fourth equations the correction terms have opposite sign as those in the fifth and eighth equations because at \(\kappa=0\) the TM and TE mode correction terms are equal. Also note that in the second and third equations the correction terms have the same sign as those in the sixth and seventh equations because at \(\kappa=0\) the TM and TE mode reflection responses in those terms are equal but with opposite sign: \(R^\pm_{V;1}(\kappa=0) = R^\pm_{V;1}(\kappa=0)\).
Hun15 uses \(\phi\), whereas Zio19 uses \(x\), \(y\), for which we can use

empymod.scripts.tmtemod.
dipole
(src, rec, depth, res, freqtime, aniso=None, eperm=None, mperm=None, verb=2)[source]¶ Return the electromagnetic field due to a dipole source.
This is a modified version of
empymod.model.dipole()
. It returns the separated contributions of TM–, TM+, TM+, TM++, TMdirect, TE–, TE+, TE+, TE++, and TEdirect.Parameters:  src, rec : list of floats or arrays
Source and receiver coordinates (m): [x, y, z]. The x and ycoordinates can be arrays, z is a single value. The x and ycoordinates must have the same dimension.
Sources or receivers placed on a layer interface are considered in the upper layer.
Sources and receivers must be in the same 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 : float
Frequency f (Hz). (The name
freqtime
is kept for consistency withempymod.model.dipole()
. Only one frequency at once. aniso : array_like, optional
Anisotropies lambda = sqrt(rho_v/rho_h) (); #aniso = #res. Defaults to ones.
 eperm : array_like, optional
Relative electric permittivities epsilon (); #eperm = #res. Default is ones.
 mperm : array_like, optional
Relative magnetic permeabilities mu (); #mperm = #res. Default is ones.
 verb : {0, 1, 2, 3, 4}, optional
 Level of verbosity, default is 2:
 0: Print nothing.
 1: Print warnings.
 2: Print additional runtime and kernel calls
 3: Print additional start/stop, condensed parameter information.
 4: Print additional full parameter information
Returns:  TM, TE : list of ndarrays, (nfreq, nrec, nsrc)
Frequencydomain EM field [V/m], separated into TM = [TM–, TM+, TM+, TM++, TMdirect] and TE = [TE–, TE+, TE+, TE++, TEdirect].
However, source and receiver are normalised. So the source strength is 1 A and its length is 1 m. Therefore the electric field could also be written as [V/(A.m2)].
The shape of EM is (nfreq, nrec, nsrc). However, single dimensions are removed.