TM/TE split#
This addon for empymod adjusts [HuTS15] for TM/TEsplit. The development was initiated by the development of emsig/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 empymod.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 == ‘dlf’
htarg == ‘key_201_2012’
Options ft, ftarg, and loop 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.
This routine removes the nonphysical part. It is basically a heavily simplified version of empymod with the following limitations outlined above.
It 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 this file):
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
Functions#

Return the electromagnetic field due to a dipole source. 