# Synthetic seismogram with multiples

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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 8 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Previously, we discussed the synthetic trace without multiples, and we have just looked at multiple water reverberations in the topmost water layer. We now turn to the more general case of a synthetic seismogram for a model which can produce multiples in any of its layers. The details of generating a synthetic trace with multiples were given in Robinson and Treitel (1978), who also described an essential mathematical simplification that holds for the case of small reflection coefficients. The term small is defined in a relative manner that depends on the circumstances at hand. In some cases, small might mean a magnitude less than 0.2, whereas in other cases, it might mean a magnitude less than 0.05. Here, we present a method of generating a synthetic trace with multiples in such a way as to bring out what the mathematics is doing, rather than in terms of a strict mathematical derivation. The key concepts are feedforward and feedback.

We start with the case of only two interfaces (namely, the surface and the first subsurface interface). They are separated by a vertical distance with two-way traveltime T = 1. The reflectivity is ($\varepsilon _{0}$ , $\varepsilon _{1}$ ). Its one-sided autocorrelation (with the zero-lag term set equal to unity) is (1, $\varepsilon _{0}$ , $\varepsilon _{1}$ ), which we denote by (1, $g_{1}$ ) where $g_{l}={\varepsilon }_{0}{\varepsilon }_{l}$ ….

We recognize that the current problem is the same as the reverberation problem treated above, but now with two-way time T = 1. Thus, the impulse response h (with multiples) in the case of two interfaces has the Z-transform

$H\left(Z\right)={\varepsilon }_{1}Z+{\varepsilon }_{1}\left(-{\varepsilon }_{0}{\varepsilon }_{1}\right)Z^{2}+{\varepsilon }_{1}{\left(-{\varepsilon }_{0}{\varepsilon }_{1}\right)}^{2}Z^{2}+$ {\begin{aligned}={\varepsilon }_{1}Z\left[{1}+\left(-{\varepsilon }_{0}{\varepsilon }_{1}\right)Z+{\left(-{\varepsilon }_{0}{\varepsilon }_{1}\right)}^{2}Z^{2}+\dots \right]={\frac {{\varepsilon }_{1}Z}{{1}+{\varepsilon }_{0}{\varepsilon }_{1}Z}}.\end{aligned}} (22)

We recall that the impulse response (without multiples) is generated by a purely feedforward system (with the reflection coefficients on the feedforward loops). Here, the expression for the Z-transform shows that the impulse response (with multiples) is generated by a feedforward-feedback system (with the reflection coefficients on the feedforward loops and the autocorrelation coefficients on the feedback loops). Equation 22, which is for the case of one layer, is exact. In other words, equation 22 gives the Z-transform of the impulse response for the dynamic model (equation 1).

Next let us consider the case of three interfaces. What is the impulse response (with multiples) for three interfaces? The reflectivity is then $\left({\varepsilon }_{0}{,\ }{\varepsilon }_{1}{,\ }{\varepsilon }_{2}\right)$ . Its one-sided autocorrelation (with the zero-lag term set equal to unity) is

 {\begin{aligned}\left({1,\ }g_{1},g_{2}\right)=\left({1,\ }{\varepsilon }_{0}{\varepsilon }_{1}+{\varepsilon }_{1}{\varepsilon }_{2}{,\ }{\varepsilon }_{0}{\varepsilon }_{2}\right).\end{aligned}} (23)

By analogy with the result for two interfaces given above, the synthetic impulse response h (with multiples) in the case of three interfaces has the Z-transform

 {\begin{aligned}H\left(Z\right)={\frac {{\varepsilon }_{1}Z+{\varepsilon }_{2}Z^{2}}{{1}+g_{1}Z+g_{2}Z^{2}}}.\end{aligned}} (24)

However, expression 24, obtained by analogy, is not exact but instead is an approximation to the true dynamic model. Equation 24 leads to the synthetic trace with multiples (equation 2).

Now let us consider the case of four interfaces. What is the impulse response (with multiples) for four interfaces? The reflectivity is then $\left({\varepsilon }_{0}{,\ }{\varepsilon }_{1}{,\ }{\varepsilon }_{2}{,\ }{\varepsilon }_{2}\right)$ . Its one-sided autocorrelation (with the zero-lag term set equal to unity) is

 {\begin{aligned}\left({1,\ }g_{1},g_{2},g_{3}\right)=\left({1,\ }{\varepsilon }_{0}{\varepsilon }_{1}+{\varepsilon }_{1}{\varepsilon }_{2}+{\varepsilon }_{2}{\varepsilon }_{3}{,\ }{\varepsilon }_{0}{\varepsilon }_{2}+{\varepsilon }_{1}{\varepsilon }_{3}{,\ }{\varepsilon }_{0}{\varepsilon }_{3}\right).\end{aligned}} (25)

By analogy, the synthetic impulse response (with multiples) in the case of four interfaces has the Z-transform

 {\begin{aligned}H\left(Z\right)={\frac {{\varepsilon }_{1}Z+{\varepsilon }_{2}Z^{2}+{\varepsilon }_{3}Z^{3}}{{1}+g_{1}Z+g_{2}Z^{2}+g_{3}Z^{3}}}.\end{aligned}} (26)

However, expression 26, obtained by analogy, is not exact but is an approximation to the true dynamic model. Expression 26 can be diagrammed as the feedforward-feedback system shown in Figure 10.

By analogy, the synthetic impulse response (with multiples) for N interfaces has the Z-transform

 {\begin{aligned}H\left(Z\right)={\frac {{\varepsilon }_{1}Z+{\varepsilon }_{2}Z^{2}+\dots {.+.}{\varepsilon }_{N}Z^{N}}{{1+}g_{1}Z+g_{2}Z^{2}+.+g_{N}Z^{N}}}.\end{aligned}} (27)

As before, this expression is an approximation to the dynamic model. Let us define E(Z) and G(Z) as

 {\begin{aligned}E\left(Z\right)={\varepsilon }_{1}Z+{\varepsilon }_{2}Z^{2}+\dots +{\varepsilon }_{N}Z^{N}\mathrm {\ and\ } G\left(Z\right)={1}+g_{1}Z+g_{2}Z^{2}+\dots +g_{N}Z^{N}.\end{aligned}} (28)

Equation 27 can be written as

 {\begin{aligned}H\left(Z\right)={\frac {E\left(Z\right)}{G\left(Z\right)}}=E\left(Z\right)M\left(Z\right)\mathrm {\ where\ } M\left(Z\right)={\frac {1}{G\left(Z\right)}}=1+m_{1}Z+m_{2}Z^{2}+\ldots .\end{aligned}} (29)

For equation 29 to be true, the denominator polynomial G(Z) must be a minimum-delay polynomial. In such a case, we have the approximation $h=m*\varepsilon$ , which gives $x=s*m*\varepsilon$ . Thus, equation 2 holds for the synthetic seismogram with multiples.

Let us now discuss the approximation that is required. It is called the small-reflection-coefficient approximation, and it was introduced by Robinson and Treitel (1978) and further developed by Robinson (1982, 1999). The approximation requires that the reflection coefficients be small enough to make the denominator polynomial G(Z) a minimum-delay polynomial. Thus, before this approximation can be used, the denominator polynomial must be tested for minimum delay. If the denominator polynomial is minimum delay, polynomial division can be used to find the coefficients $m=(1,m_{1},m_{2},m_{2},....)$ of $M\left(Z\right)$ . If the denominator polynomial is not minimum delay, the approximation fails, and the exact dynamic expression (i.e., the expression obtained without the small-reflection-coefficient approximation) must be used. In fact, it is always prudent to use the exact dynamic model (equation 1) in computations (Robinson, 1999).

If the small-reflection-coefficient approximation holds, then the approximation $H\left(Z\right)=E\left(Z\right)M\left(Z\right)$ holds, and the multiples $M\left(Z\right)$ can be removed by ordinary (i.e., linear time-invariant) deconvolution. In other words, the small-reflection-coefficient approximation justifies the use of ordinary deconvolution. For this reason, a sequence of time gates is chosen on the actual seismic trace. The choice is made on the supposition that within each gate, the small-reflection-coefficient approximation holds. A deconvolution operator is computed for each gate. The deconvolved trace is made up of all the deconvolved gates along with appropriate interpolation between any two adjacent gates.