Thus, the frequency representation of the complex seismic trace can be expressed explicitly in terms of X (), the Fourier amplitude spectrum of the real stacked seismic trace. The calculation of z (t) is simply the operation of taking the inverse Fourier transform of Z (). Once z (t) is constructed, the amplitude, phase, and frequency behavior of the complex trace can be calculated by the equations:

 

This mathematical procedure is summarized in Figure 3.

 

Figure 3. The principal numerical attributes that are derived from the complex seismic trace are the instantaneous amplitude, a (t), the instantaneous phase, ø (t), and the instantaneous frequency, (t) which are shown here in a schematic form. These attributes are referred to as instantaneous measures because they are calculated at each datum point along the time axis.

The computation of the complex seismic trace can be accomplished by performing two Fourier transforms — a forward transform to create X (), and then an inverse transform of the complex trace spectrum, Z (), which is created by manipulating X () as indicated in Equation 6.

 

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