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EMpulse Geophysics Ltd.

Adaptive Polarization Stacking

The "transient" approach to AMT was arguably first implemented by Don Hoover of the USGS in the 1970's where a "human computer" located discrete events on a strip chart recorder and calculated scalar impedances with the detected events. Many others have recorded transients in order to maximize signal-to-noise ratio (SNR) including Keeva Vozoff (University of Macquarie) in the 1980's, Ken Paulson (University of Saskatchewan) in the 1980's, Andreas Tzanis and David Beamish (British Geological Survey) in the 1980's and Garner and Thiel in the 1990's.

However, with the exception of Ken Paulson and Peter Kosteniuk, standard processing techniques were used to process the linearly polarized transient data. This can be problematic with a confined distribution of bearings as the polarization diversity of the transient data causes a bias and instability of solution additional to that caused by finite SNR and finite sample size. 

The angle between events is similar to specifying the condition number of a pair of simulataneous linear equations. A narrow angle produces a large condition number and an unstable system, in such as case, small changes of the input data produce large changes on the output (estimated curves).

Ken Paulson and Peter Kosteniuk were unique in the development of a new data processing algorithm that exploited the directional, linearly polarized, properties of the transient waveforms. Their original Polarization Stacking algorithm involved the detection of individual transient events and their classification into two separate "stacks" based on the direction of arrival, stack boundaries oriented N-S and E-W were used.

A key aspect of Polarization Stacking was the further enhancement of SNR through a time domain stacking of the transient waveforms. All recorded events were decomposed into two "super" events contained in the two stacks, Fourier transformation with a simple "two-point" formula yielded impedance tensor and tipper estimates.

Not only were the best possible events used by Paulson and Kosteniuk, they developed a data processing algorithm to work with and exploit the linearly polarized transient data.

EMpulse Geophysics improved the Polarization Stacking algorithm to allow the stack boundaries to be adaptive to the constantly changing bearing and amplitude characteristics of the data, thus our new algorithm is called Adaptive Polarization Stacking or APS. The stack boundaries are chosen in such a way so as to maximize the angle between stacks and to approximately equalize signal energy in each stack. This avoids the problem of all the events going into one stack, as could sometimes happen with the original Polarization Stacking algorithm in times of low source field activity.

A second improvement with APS was the use of weighted averaging, with Polarization Stacking every event in each stack had the same weight of unity. EMpulse found out rather quickly that it only took one or two noisy events to disrupt the stack and estimated impedance/tipper curves. With APS, each event is weighted according to a very simple quality indicator, even with high SNR transient events, it's not uncommon for many events to be downweighted significantly (weight < 0.1) relative to the best event (unity weight). It's very much a "diamond in the rough" type process with typically 20 percent or less of the total number of events possessing significant weight (> 0.3).

A third and critical improvement with APS was the inclusion of error bar estimates, derived through Monte Carlo simulation, that are properly connected to the polarization properties of the source field, the SNR and sample size. The estimated impedance (and admittance) are used to evaluate the residual between predicted and measured electric and magnetic fields, this residual, averaged across all the measured events, is used to obtain an (over) estimate of the noise standard deviation for all five time series  (Hx, Hy, Ex, Ey, Hz). In accord with the noise standard deviations thus found, normally distributed noise is added to the time-series and a "noisy" impedance tensor and tipper estimated and stored, this process is repeated hundreds of times, each time with independent linearly additive noise, to obtain a noisy family of impedance tensor and tipper estimates from which 95 percent confidence limits can be found.

In contrast to conventional Remote-Reference error analysis, our APS error bars typically over-estimate the true error and are thus conservative. Our APS error bars do fail when there are a small number of events, as can happen in the 1 - 5 kHz "dead-band" range. However,  the effect is obvious and simpy requires the manual setting of the error bars in the affected range to a very large level. In this fashion, we avoid the error-bar "guessing game" as is often times required with conventional AMT data, i.e., since the conventional Remote-Reference error bars are usually unrealistically small, it is typical to use several different error bar "guesses" (10 percent uniform error, 20 percent, etc.) ultimately leaving the interpreter with a subjective decision as to which model he/she considers the "best" or most reasonable fit to the measured data.

One downside of Adaptive Polarization Stacking is that since each one of the transient waveforms are in general different, in adiition to cancelling noise, some fine scale signal cancellation also occurs. This is especially so for frequencies less than 50 Hz, where the transient component begins to diminish and the continuing component begins to rise.

In an effort to completely avoid signal cancellation EMpulse also developed a "curve-stacking" algorithm whereby independent pairs of events are chosen based on a quality indicator. Once the events are paired up, an impedance and tipper estimate is found for each pair, thus obtaining a  family of "pair-processed" impedance and tipper curves. These curves are then "stacked"  or averaged in the frequency domain in a weighted sense, either with one global weight per curve or with frequency dependent weights.

EMpulse Geophysics performed an extensive analysis of the bias (and error bar efficiency) of our APS algortithm as well as our frequency domain"curve-stacking" algorithm (and standard Remote-Reference). Real magnetic field data were used in conjunction with a 3D impedance tensor and tipper to make the perfectly matching electric field and vertical magnetic field data. The perfect relation (infinite SNR) was then disrupted through the addition of normally distributed random noise to all five channels. The resulting impedance tensor and tipper estimates were then estimated and stored, this process repeated over many thousands of times permitting an analysis of the bias and error bar performance as a function of frequency for APS, curve-stacking and Remote-Reference.

Even though curve-stacking has absolutely no signal cancellation, we found that APS still provides by far the greatest benefit of accuracy (smallest bias) and error bar efficiency, the smallest error bar that still captures the true underlying curve. Even though some signal cancellation occurs with APS, the benefit of noise cancellation and signal enhancement appear to far outweigh it.

It was futher found that the bias performance of APS is better than standard remote-reference (R.R.) given transients with typical polarization characteristics, please click here to download the paper, presented at the 2001 SEG in San Antonio.

In order to leave no stone turned, we are working on a third algorithm which involves stacking transients as per APS, but only those that have a high degree of self correlation. Therefore, only events with similar time domain shapes are stacked, non-correlating events are completely left out of the time domain stacking process. After stacking as many (weighted) events as possible, we proceed to the frequency domain with the new subset of events and perform curvestacking. This hybrid time/frequency domain processing method may represent the best useage of the data although Monte-Carlo analysis of bias and error bar performance remains to be conducted. Furthermore, the hybrid time/frequency domain algorithm would appear to be adaptable to both auroral and lightning sources, possibly making it useful for sub 1 Hz data.