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Why increasing transmitter-receiver coil separation in EM survey (EM31, also here) increases depth of penetration?

Suppose there is a horizontal flat conductive layer underneath a flat resistive layer. Why receiver coil records stronger signals at 4 m transmitter-receiver coil spacing compared to 2 m transmitter-receiver coil spacing? What us the physical explanation of this?

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As described in the question there are two similar coils, one radiates the AC magnetic field ("transmitter") and the other picks up the weak radiated AC magnetic field produced by subsurface electrical currents induced by the primary field.

Sensitivity vs. Depth

There are two modes of operation. The following two figures are from Section 4.1 of GEONICS Technical note #6; Electromagnetic Terrain Conductivity Measurement at Low Induction Numbers. (more technical notes here) In both figures the two coils are separated by a horizontal distance $s$ which is typically a few meters.

In Horizontal mode the axes of the two coils point in the horizontal plane. In this orientation the sensitivity always peaks near the surface, and falls off to about 30% at a depth of $0.4 s$.

Electromagnetic Terrain Conductivity Measurement

In Vertical mode the axes of the two coils point "up" or vertically. In this orientation the sensitivity is low near the surface and peaks at depth of $0.4 s$.

Electromagnetic Terrain Conductivity Measurement

In either case, if $s$ is twice as big (the coils are twice as far apart) these two shapes will be stretched by the same amount. If $s$ is 2 meters, the vertical method will have peak sensitivity at 0.8 meters. If $s$ is 5 meters, vertical sensitivity peaks at 2.0 meters.

The two plots can be seen on the right side of Figure 1. in The Application of EM38: Determination of Soil Parameters, Selection of Soil Sampling Points and Use in Agriculture and Archaeology Kurt Heil and Urs Schmidhalter, Sensors 2017, 17(11), 2540; https://doi.org/10.3390/s17112540

That article also links to Comparison of the EM38 and EM38-MK2 electromagnetic induction-based sensors for spatial soil analysis at field scale which may cite further references but it's paywalled.

Electromagnetic Terrain Conductivity Measurement

Figure 1. (Left) Relative cumulative contribution vs depth for vertically (RV(z)) and horizontally (RH(z)) orientated dipoles; (Right) Comparison of the relative responses for vertically (FV(z)) and horizontally (FH(z)) oriented dipoles.

Yes, that's all fine and dandy... But Why?

I can't come up with a simple model for why this is true, so I'll have to read further. It's possible someone else will be able to provide a better answer. I think it's related to how the phase of the signals are process, but I'm not sure about that yet.

However, the Computers and Electronics in Agriculture paper Accuracy issues in electromagnetic induction sensing of soil electrical conductivity for precision agriculture (also available here) K.A.Sudduth, S.T.Drummond and N.R.Kitchen, Volume 31, Issue 3, May 2001, Pages 239-264 https://doi.org/10.1016/S0168-1699(00)00185-X happens to give equations for these shape, and a clue to where these are from.

enter image description here

Those equations are close but $\color{red}{wrong!!}$ they are missing a division "/" sign.

Fig. 2. Relative response of EM38 sensor as a function of distance (adapted from McNeill, 1992).

The equations in the figure have obviously been typed incorrectly, but if I make some simple adjustments they work nicely

$$\phi_V(z) = \frac{4z}{(4z^2+1)^{3/2}} $$

$$\phi_H(z) = 2 - \frac{4z}{(4z^2+1)^{1/2}} $$

where z has to be unitless, so is almost certainly given by:

$$z \equiv \frac{\text{depth}}{s}$$

where $s$ is the separation between the two coils as before.

plot of equations

Possibly helpful links:

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    $\begingroup$ Note that the simple linear model presented here becomes very inaccurate in the presence of higher conductivity soils. More complicated nonlinear models are available. $\endgroup$ – Brian Borchers Jul 29 at 18:02
  • $\begingroup$ @BrianBorchers I'd love to know more! While I was able to find a source for these equations, I don't understand them yet. Any further reading/references are welcome. Your point is well taken; as soon as the soil starts interacting significantly with the field all bets are off. $\endgroup$ – uhoh Jul 29 at 22:34
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    $\begingroup$ THis is explained in great detail in Wait's Geoelectromagnetism book. $\endgroup$ – Brian Borchers Jul 30 at 19:28
  • $\begingroup$ @BrianBorchers do you mean this one? elsevier.com/books/geo-electromagnetism/wait/978-0-12-730880-7 $\endgroup$ – uhoh Jul 30 at 21:31
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    $\begingroup$ Yes, that's the book. See my implementation in MATLAB and a reference to a paper about the model at euler.nmt.edu/~brian/nonlinem38.html $\endgroup$ – Brian Borchers Jul 30 at 23:04

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