Sea-floor consistent multiple suppression

Erratic time shifts from trace to trace have long been dealt with by the so-called surface-consistent statics model. Using this model you fit the observed time shifts, say, t(s,g), to a regression model $t(s,g) \ \approx$ $t_s (s) \,+\, t_g (g)$.The statistically determined functions t_s (s) and t_g (g) can be interpreted as being derived from altitude or velocity variations directly under the shot and geophone. Taner and Coburn [1980] introduced the closely related idea of a surface-consistent frequency response model that is part of the statics problem. We will be interpreting and generalizing that approach. Our intuitive model for the data $P(s,g, \omega )$ is  

$$P(s,g, \omega ) \ \ \ \approx\ \ \ {1 \over 1 \ +\ c_s \, e... ... \ {1 \over 1 \ +\ c_g \, e^{ i \omega \tau (g) } } \ \ \times$$ (5)

$$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ e^{ i \, {\... ...,+\, 4h^2 } } \ H(h, \omega ) \, Y(y, \omega ) \, F ( \omega )$$

The first two factors represent the split Backus filter. The next factor is the normal moveout. The factor $H(h, \omega )$ is the residual moveout. The factor $Y(y, \omega )$ is the depth-dependent earth model beneath the midpoint y. The last factor $F( \omega )$ is some average filter that results from both the earth and the recording system.

One problem with the split Backus filter is a familiar one--that the time delays $\tau (s)$ and $\tau (g)$enter the model in a nonlinear way. So to linearize it the model is generalized to  

$$P' (s,g, \omega ) \ \ \ \approx\ \ \ S(s, \omega ) \ G(g, \omega ) \ H(h, \omega ) \ Y(y, \omega ) \ F( \omega )$$ (6)

Now S contains all water reverberation effects characteristic of the shot location, including any erratic behavior of the gun itself. Likewise, receiver effects are embedded in G. Moveout correction was done to P, thereby defining P'.

Theoretically, taking logarithms gives a linear, additive model:  

$$\ln P' (s,g, \omega ) \ \approx \ \ln S(s, \omega ) + \ln G... ...\ln H(h, \omega ) + \ln Y(y, \omega ) + \ln F ( \omega ) \ \ \\ end{displaymath}$$ (7)

The phase of P', which is the imaginary part of the logarithm, contains the travel-time information in the data. This information loses meaning when the data consists of more than one arrival. The phase function becomes discontinuous, even though the data is well behaved. In practice, therefore, attention is restricted to the real part of (7), which is really a statement about power spectra. The decomposition (7) is a linear problem, perhaps best solved by iteration because of the high dimensionality involved. In reconstructing S and G from power spectra, Morley used the Wiener-Levinson technique, explicitly forcing time-domain zeroes in the filters S and G to account for the water path. He omitted the explicit moveout correction in (5), which may account for the fact that he only used the inner half of the cable.