All,

Thanks to Randal for raising this interesting topic.  The Teager-Kaiser 
Algorithm and Randal's rectification scheme are efforts to find the 
instantaneous amplitude (or envelop) of a seismic record.  Think of a 
short enough piece of a seismic record -- it can be modeled by a 
waveform of a given frequency, phase and amplitude.  Throw away the 
frequency and phase information and you are left with just the amplitude 
as a function of time.  So you can think of a record (very 
approximately) as a collection of unit-amplitude waveforms from an 
appropriate frequency range that has been AM modulated by the envelop.

The FFT of the envelop, then, can be thought of as the Fourier Transform 
of the "modulating" function.

There are different ways of approximating the envelop.  Many texts on 
signal processing discuss the "analytic trace" or "analytic signal" 
method, which is mathematically rigorous.  If the original record is X, 
then the quadrature (90 degree advanced) record, Y, is the Hilbert 
Transform of X.  The envelop of X is SQRT ( X2 + Y2 ).

In the (remote) chance that anyone wants to pursue this, contact me and 
I'll show you an easy way to calculate the Hilbert Transform.


Chuck Burch
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