# Time-varying convolutional model

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 10 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Hubral et al. (1980)[1] introduced the sum autoregressive representation of a seismic trace. This representation displays the reflection response as the convolution of the reflectivity with a time-varying sequence of wavelets. In other words, the sum autoregressive representation is a time-varying convolutional model of the trace and includes all multiple reflections. Each time-varying wavelet in the sequence is associated with one and only one interface. Mathematically, a single interface is associated with each discrete instant on the trace. That is, each interface is associated with its own time-varying wavelet, which is called either the generalized primary or the interface wavelet. Each generalized primary (or interface wavelet) is autoregressive, with its denominator given by the product of two successively longer prediction-error operator Z-transforms so that the generalized primary is minimum phase. The interface wavelet of a layer depends only on the reflection coefficients at and above that layer. Further treatment is given in Robinson (1999)[2].

Suppose we are dealing with a marine prospect that (at depth 0) has a water-surface normal-incidence reflection coefficient equal to 0.9. Let the water-bottom reflection coefficient (at relative depth 4) be equal to 0.8. Add two major reflectors at depth, the first at a relative depth of 15 with a reflection coefficient equal to -0.5 and the second at a relative depth of 25 with a reflection coefficient equal to 0.4. The water layer is homogeneous and contains no physical interfaces between the water surface and the water bottom, so the interface wavelets at the discrete depths within the water layer are merely spikes resulting from the impulsive source. The water bottom has its own interface wavelet, as does each interface below the water bottom.

Figure 1.  Zero-value reflection coefficients between major interfaces.

Suppose first that all the reflection coefficients between the major interfaces are zero. The interface wavelets for this case are shown in Figure 1. Black traces correspond to interfaces with nonzero reflection coefficients, and gray traces correspond to interfaces with zero reflection coefficients. The trace at interface n depends on the reflections coefficients at n and above n but not on any below n. In the sum autoregressive formula, each trace is weighted by its reflection coefficient. Because the reflection coefficient corresponding to each gray trace is zero, it follows that the gray traces do not contribute to the sum autoregressive formula.

Now suppose that all the reflection coefficients between the major interfaces are random and white and lie in the range of -0.05 to +0.05. These are minor reflection coefficients. The interface wavelets for this case are shown in Figure 2.

Figure 2.  Small reflection coefficients between major interfaces.

Next suppose that all the reflection coefficients between major interfaces are random and white and lie in the range of -0.20 to +0.20. These are major reflection coefficients. The interface wavelets for this case are shown in Figure 3.

Figure 3.  Large reflection coefficients between major interfaces.

These three figures demonstrate the fact that between major reflectors, the interface wavelets are approximately constant. Thus, between major reflectors, the time-varying convolutional model reduces to the time-invariant convolutional model within each window, as defined by the major interfaces. The message is that deconvolution-design windows should be chosen between major reflectors, so the time-invariant model approximately holds within each window.

The net result is that each of the wavelets is a minimum-delay wavelet. The problem of deconvolution in seismic data processing can be stated simply as follows: Given the reflection seismogram (which is observed at the surface of the earth), we must find the reflectivity function (which provides stratigraphic information as a function of depth). To find a solution to this problem, we use the signature-free convolutional model ${\displaystyle z=m*\varepsilon }$, discussed above, which we assume holds within in each given window of the seismogram. The multiple wavelet m is minimum delay.

Despite the presence of strongly reflecting interfaces interspersed in the geologic column, significant sections remain whose interfaces are characterized by small and random reflection coefficients. Thus, on corresponding sections of the reflection seismogram, the reflectivity function can be assumed to be white. In our careful selection of the time windows on a reflection seismogram, we could pick out sections for which we could assume that the reflectivity function is a white-noise sample. Let us recall that our convolutional model of the reflection seismogram states that the seismogram is the convolution of the reflectivity with a minimum-delay wavelet. This wavelet is the appropriate interface wavelet, which (approximately) holds for all interfaces within the window. With proper choices of the time windows, the input ${\displaystyle \varepsilon }$ can be viewed as a white-noise series.

## References

1. Hubral, P., S. Treitel, and P. R. Gutowski, 1980, A sum autoregressive formula for the reflection response: Geophysics, 45, 1697-1705.
2. Robinson, E. A., 1999, Seismic inversion and deconvolution, dual sensor technology: Elsevier.