Statistical pulse compression

The seismic method in petroleum exploration is an echo-location technique to detect interfaces between the subsurface sedimentary layers of the earth. The received seismic reflection record (field trace), in general, may be modeled as a linear time-varying (LTV) system. However, in order to make the problem tractable, we do not deal with the entire field trace as a single unit, but instead subdivide it into time gates. For any time gate on the trace, there is a corresponding vertical section of rock layers within the earth, such that the primary (direct) reflections from these layers all arrive within the gate. Each interface between layers is characterized by a local (or Fresnel) reflection coefficient, which physically must be less than unity in magnitude. Under the hypothesis that the vertical earth section has small reflection coefficients, then within the corresponding time gate the LTV model of the seismic field trace reduces to a linear time-invariant (LTI) system. This LTI system, known as the convolutional model of the seismic trace, says that the field trace is the convolution of a seismic wavelet with the reflection coefficient series. If, in addition, the reflection coefficient series is white, then all the spectral shape of the trace within the gate can be attributed to the seismic wavelet. Thus the inverse wavelet can be computed as the prediction error operator (for unit prediction distance) by the method of least squares. The convolution of this inverse wavelet with the field trace yields the desired reflection coefficients. This statistical pulse compression method, known as predictive deconvolution with unit prediction distance, is also called spike deconvolution. Alternatively, predictive deconvolution with greater prediction distance can be used, and it is known as gapped deconvolution. Other pulse compression methods used in seismic processing are signature deconvolution, wavelet processing, and minimum entropy deconvolution.