Technical Notes - Helicopter Electromagnetics

Introduction to Inversions

Basics

The inversion process starts with a set of data from 1 or more frequencies and phases, and a starting model. From the starting model, it Forward models the EM response that would be measured over that model, and then calculates the error between that solution and the input data. It then repeatedly varies each of the forward model parameters in turn to try and minimise the error between the modeled data and the input data, until it arrives at a solution for which the error is less than the allowable fitting error set in the input parameter file.

Types of Inversions

We have two types of inversion algorithms, SVD (singular value decomposition) and Occam. Inversions use all input channels to match to all output parameters. Sengpiel and Differential sections are calculations of the depth and resistivity for each channel separately, not inversions. The graph at left shows each type of inversion and section for the geological model (in black).

The SVD, or multi-layer inversion models the data with a specific number of discrete layers, for each of which the program will change the thickness and resistivity until a best fit is achieved. (See USING SVD / Weights for notes on controlling changes in parameters.) SVD inversion tends toward a stepped model with sharply defined layers and resistivities. This type of inversion applies best where the geologic model is that of discrete layers with discrete conductivities (water and bottom, overburden and bedrock). The layers may change thickness (overburden) or resistivity (fresh water) but the number of layers should remain constant. In theory, if the number of layers is set to the maximum expected in section, then when the number of layers in the true geologic section are reduced, two layers of the model should appear to have the same resistivity and add up to the correct thickness. In practice an excess number of layers can lead to unstable results, with the algorithm changing first one layer then the next to match a middle layer. DBSVD1D02 runs the same inversion algorithm but allows the range of the output parameters to be limited.

The Occam inversion uses a constant number of layers and only varies the resistivity. Thick layers are matched by assigning a number of the thin layers the same resistivity. This allows a better fit in areas where the number of layers may change along the same section. Occam inversions tend to model towards a smoothly changing model, and so are not as definitive in areas where an exact fit to a layer is needed. It is possible to vary the thickness of the layers, to provide for higher resolution near surface, lower resolution at depth.

RUNNING INVERSIONS

Number of channels
Increasing the number of EM channels increases the number of input controls which the output model must fit, and increases the accuracy and precision of the output. Higher frequency channels improve the accuracy of the result near surface, lower frequencies improve it deeper. Use as many channels as you can, if they contain useful information.

However, in our imperfect world, adding channels can also add noise sources. While higher frequencies may have more effect near surface, and low frequencies deeper, noise in any channel can affect the entire range of resistivities and depths, making the inversion unstable. Do not use channels if they are particularly noisy, or if they do not significantly affect the resolution of the system at the depth you are trying to model.

Example 1a: For the DSTO Sydney Harbour survey we used the CP330, CX900, and CP7200. The CP56k was of such high frequency that it didn't penetrate enough to provide any useful information about the bottom.

If two coil pairs have nearly the same frequency, such as CX900 and CP900, the algorithm must fit a smooth curve closely around these two very near points, which can make it unstable (like pushing the short end of a lever). Do not use these matched channels.

Example 1b: For the DSTO Sydney Harbour survey the CX900 was far enough in frequency from the CP330 that it did not de-stabilise the inversion.

When inverting for the top and bottom of a thick layer, it may be best to use the lower frequencies to model the bottom, and then use this model as input to the inversion for the top, which would use the higher frequencies only. We cannot currently use the results of one model at input to the next, but are planning to do this.

(Sengpiel and differential sections are different: Noise in any channel affects only that channel on Sengpiel sections, and that channel and the immediately following channel on differential sections. It also affects the colour distribution on the section between the bad channel(s) and the two adjacent ones.)

Station Spacing and Filters
Inversions assume a layered earth, unchanging in the horizontal directions. Because of this they are not useful in conductor mapping, or steeply dipping geology. Because they are less stable than Sengpiel or Differential sections, they will also be difficult to use in low-signal conditions.

Inversion calculations are slow, and because they apply when the geology is changing slowly in the horizontal direction, they do not need to be applied to every 1/10 second sample, but can be applied to every 5th or 10th sample. This will allow the use of some fairly heavy filters, if necessary.

Dealing With Noise
Inversions are very susceptible to noise, which causes them to become unstable, resulting in a "patchwork" section on SVD inversions, and conductive layers from the Occam inversion. If the noise is short-wavelength, it will cause each individual inversion along a line to see a different array of signals from the adjacent calculation, so the result will change dramatically from station to station. By inverting every 5th sample and using a fairly heavy filter before running the inversions (11 point median?), the noise can be smoothed out. Bear in mind that these are layered earth inversions, so any feature than about 100m does not fit the model, and will not return a true geologic model.

Drift is noise also, of very long wavelength. Because it is not changing the data from station to station, it may not create the "hash" look of short wavelength noise, but can still cause the inversion to be unstable by creating "impossible" data for a geologic model. The best way to reduce this problem (other than reducing the drift) is to increase the allowable error. This allows the inversion "more room" in trying to fit to impossible data, and should produce a smoother result.

Noise can cause the data to no longer fit the response which could come from a true geologic model, which will cause the inversion to become unstable. The forward modeling algorithm cannot find a geological solution which would produce data that matches the input. The result is that the inversion "fit" to the data is good on some channels, poor on others. For the next data point, it may have a similar fit, but instead of fitting channels 1 and 3 best, it fits 2 and 4. The overall fit quality could be the same, but the model vastly different.

Non-leveled channels, like drift, will also cause the inversion to become unstable, if the allowed misfit is too low.

USING SVD INVERSIONS

These are run from the command line, with a standard control file.

Control File
The control file has three parts: the inversion controls (fit errors, etc.), the input survey parameters with input parameter weights, and the starting model parameters and weights. Beside the usual system parameters (field names, frequencies, etc.) the survey parameters also include standard deviations, which define how much weight the inversion will apply to each of the input channels. The layer inputs define the starting model and controls on the output parameters.

Weights
Weighting on model parameters controls how much this parameter will be changed to fit the model, relative to the other parameters. A weight of 1 is freely changing, 0 is fixed. If a parameter has a weight less than another, the model will try to change the higher-weighted parameter more.

The factor is changes geometrically: a reduction from 1 to 0.1 has the same effect, relatively as from 0.1 to 0.01. I have found that the weight has to be reduced (from 1) to less than 0.1 before a significant change in the variability of the parameter is noticeable. A weight of 0.01 keeps the variable from changing too much, but allows some change.

GENERAL NOTES on SVD
Success with this inversion seems to come from controlling as many parameters as possible in some situations, and in others by letting all parameters go free. If the geologic section is close to a halfspace, then the best results come from holding as many parameters constant as practical. This limits the choices that the model has to change to those which are geologically practical. Without controls, the subtle changes in EM signal due to small variations from the halfspace will be accommodated by the program by changing different parameters at each station.

If the geologic model is accurately known and relatively constant, and the derived starting resistivity model is a good fit to the data then the inversion will perform best tightly controlled. However, if the geologic model is changing or uncertain, attempting to control the inversion by reducing weights or fixing parameters will force unsuitable models, resulting in unstable results. It is better to let the inversion go with a unrestricted parameters.

Number of Layers
Dr. Sengpiel proposed the technique of increasing the number of layers until the fit error "bottomed out". The error decreased significantly with each layer added, until it reached a relatively constant level. The number of layers at which it first reached this constant level is presumed to be the best fit to the data. It should be observed that as the layers are increased past the ideal number, they modify the original model only slightly, rather than markedly change it. However, this approach of having many layers can lead to unstable inversions along a survey line, particularly where the geology is close to a halfspace. The many degrees of freedom allow the program to change different parameters at adjacent stations to match very slight differences in the data. For example, it may change parameters 1 and 3 at this point, and 2 and 4 at the next, making the results very inconsistent.

Dboverb
This is a specialised version of SVD for low signal, two layer cases, where the lower layer is very resistive, the upper conductive. It matches the signal strength for each input channel against a threshold. If the data is below the threshold, then the channel is dropped from the input parameter list, and the model is simplified to keep the number of output parameters down. This is done with geophysical logic. If no data from any frequency is above the threshold, the program assumes that there is no overburden and assigns the bedrock resistivity to the upper layer (halfspace case). If less than two frequencies are above the threshold, the layer resistivities are kept constant, and it inverts for the depth of the top layer only. Otherwise, it inverts for the top layer thickness and the bedrock resistivity.

USING OCCAM

The Occam inversion is slower than the SVD inversion, but has the advantage of being able to change the geologic model completely along the line of data. The result is a smoother fit to the data. It is run from the command line, with a standard control file.

LAYERS
The number of layers is user specified. The more layers, the slower the inversion will run. The total depth of all the layers should be below the depth of interest. Normal values would be 20 layers of 5 - 10m thickness. They need not be all the same thickness. If more detail is needed near surface, it is possible to have the first layers thinner than the rest. In conductive ground where near-surface layers are important, it is possible to use 1m, 2m, 3m and 5m layers. Thin layers near surface reduce the amount of oscillation when there is an error in the altimeter.

GENERAL NOTES on OCCAM

Fit Errors
The Occam inversion tends to be very unstable if the RMS misfit (REQTOL) is set too low. This is seen in the data as a tendency to oscillate about the proper resistivity. This oscillation will appear in the section as a series of conductive bands (generally 2) very continuous across the length of the section. The error should be raised to eliminate these oscillations. When the number is very slightly too low, there will be a conductive band in the bottom layer of the model section.

If the REQTOL is too high, it allows the model fit to be too wide, and the layers shown in the section start to blend into one another - the section becomes too smooth. To achieve the best results, start with a best guess number, and if the model seems to be oscillating, then raise the number. If the result is smooth, then lower the number until the model starts to oscillate, then back it off.

In the example shown here, the inversion with a REQTOL set to 4 is too smooth. REQTOL equal to 2 is better, and 1 is best of all. When REQTOL equals 0.5, the error is too small, and the output oscillates.

Errors in the altimeter (which appear as part of the high frequency depth) will cause the inversion to be much more unstable. Adding a few thin layers near surface seems to smooth this out. Increasing the allowed misfit will also give the routine more room to maneuver in fitting the data.

If the ground is conductive and the high frequency apparent resistivities are close to the same, then it can be assumed that the high frequency is seeing the ground as a halfspace. The high-frequency pseudo-layer depth can then be assumed to be equal to the altimeter error. Adding this value to the altimeter may serve to correct the altimeter, and improve the inversion. (Nippon Engineering is using their 137kHz data as altimeter.)

In areas which are close to a halfspace resistivity, the Occam inversion will work much more easily than the multi-layer inversion. It will not necessarily produce better results than a perfectly tuned multi-layer inversion (at least on model data), but it will produce good results with much less effort and less geologic knowledge. In these conditions (near halfspace) the smoothly varying option is best.

EXAMPLES:

Bathymetry - SVD
Because the model is precisely known (two layer case, salt water and bottom), this puts strong constraints on the possible results. There is no question about which model is correct, but the results must be very consistent. An accurate measure of the water conductivity is essential, although the final fixed resistivity of the first layer used to get an accurate inversion was slightly different than this.

For the DSTO work, the inversion was two layer, and the best results came with the upper layer fixed at 0.265 ohm-m, although the measured resistivity was 0.23 ohm-m. The bottom layer had to be severely restrained, with a weight of 0.01, or the inversion would use the bottom layer resistivity instead of the depth to adjust the results, resulting in wildly varying depths.

Permafrost - SVD
This was applied to try and map the depth of the water over the river, the depth to the top of the permafrost, and the bottom of the permafrost. The inversions could not model a thin, conductive layer near surface to represent the active layer, but this layer may have been too thin to be measurable. Instead of a conductive layer at surface, the "open" inversion put a resistive layer there. This was interpreted to be the thickness of the permafrost, although there was concern that it was a leveling error between the 56k and the 7200.
If the inversion was controlled to put a conductive layer at surface (by using an automatic first layer resistivity - which would equal the 56k res) then a stable result and a decent looking section was possible. However, the modeled thickness of this layer was several metres, greater than would be reasonable for an active layer. (The "active zone" depth map given to Komex by others showed an active layer thickness of up to 8m - very unlikely.)

Permafrost - OCCAM
This was used on the tie lines, which crossed islands, rivers and land, to match the changing geology. The Occam inversion showed a very similar result to the SVD, which gave some confidence in the results.

Greg Hodges, Chief Geophysicist, 2000