AVO of Multiple Reflections

Even after application of the water-bottom reflection coefficient, the AVO response of the pseudo-primary section created by equation (1) does not match that of the corresponding NMO-corrected primary section. Refer to Figure 2 and note that for constant-AVO water-bottom reflection (and a free surface reflection coefficient of -1), the amplitude of the water-bottom multiple at offset h_p+h_m is simply the amplitude of the primary at offset h_p, scaled by the negative water-bottom reflection coefficient. Still, the question remains: what are h_m and h_p? For the case of constant velocity, we can use trigonometry to derive h_m and h_p as a function of the zero offset traveltimes of the primary reflection and water bottom ($\tau$ and $\tau^*$, respectively), and the source-receiver offset x. In constant velocity, the multiple and primary legs of the raypath are similar triangles:  

$$\frac{\tau v}{h_p} = \frac{\tau^* v}{h_m}.$$ (3)

Also, for a first-order water-bottom multiple,

h_p + h_m = x.

These two independent equations can be solved and simplified to give expressions for h_p and h_m:  

$$h_p = \frac{\tau}{\tau+\tau^*}x \hspace{0.2in} \mbox{ and } \hspace{0.2in} h_m = \frac{\tau^*}{\tau+\tau^*}x.$$ (4)

I omit the general form of the expression for orders of multiple higher than one, although it is straightforward to derive.

 

avo Figure 2 Assuming a constant AVO water-bottom reflection and constant velocity, we can write the AVO of water-bottom multiples with offset hp+hm as a function of the AVO of the primary recorded at a shorter offset, hp.

To obtain an estimate of the water-bottom reflection coefficient, I solve a simple least squares problem to estimate a function of location, $\bold a(x)$, which when applied to a small window of dimension $nt \times nx$around the NMO-corrected water-bottom reflection, $\bold p(t,x)$, optimally resembles the NMO-corrected [equation (1)] first-order water-bottom multiple reflection, $\bold m(t,x)$. To achieve this, $\bold a(x)$ is perturbed to minimize the following quadratic functional.  

$$\mbox{min} \; \left( \sum_{j=1}^{nx} \sum_{i=1}^{nt} a(j)*p(i,j) - m(i,j) \right)^2$$ (5)

$\bold a(x)$ may not be reliable at far offsets, due to either NMO stretch or non-hyperbolicity, so in practice, an estimate of the single best-fitting water-bottom reflection coefficient is made using the $\bold a(x)$ from ``useful'' offsets only.