AVO of Multiple Reflections

Even after application of the water-bottom reflection coefficient, the AVO response of the pseudo-primary section created by equation () does not match that of the corresponding NMO-corrected primary section. Refer to Figure 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 (22#22 and 16#16, respectively), and the source-receiver offset x. In constant velocity, the multiple and primary legs of the raypath are similar triangles:  

 23#23 (6)

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:  

 24#24 (7)

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, 25#25, which when applied to a small window of dimension 26#26around the NMO-corrected water-bottom reflection, 27#27, optimally resembles the NMO-corrected [equation ()] first-order water-bottom multiple reflection, 28#28. To achieve this, 25#25 is perturbed to minimize the following quadratic functional.  

 29#29 (8)

25#25 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 25#25 from ``useful'' offsets only.