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## 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 hp+hm is simply the amplitude of the primary at offset hp, scaled by the negative water-bottom reflection coefficient. Still, the question remains: what are hm and hp? For the case of constant velocity, we can use trigonometry to derive hm and hp as a function of the zero offset traveltimes of the primary reflection and water bottom ( and , respectively), and the source-receiver offset x. In constant velocity, the multiple and primary legs of the raypath are similar triangles:
 (3)
Also, for a first-order water-bottom multiple,

hp + hm = x.

These two independent equations can be solved and simplified to give expressions for hp and hm:
 (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, , which when applied to a small window of dimension around the NMO-corrected water-bottom reflection, , optimally resembles the NMO-corrected [equation (1)] first-order water-bottom multiple reflection, . To achieve this, is perturbed to minimize the following quadratic functional.
 (5)
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 from useful'' offsets only.

Next: Least-squares imaging of multiples Up: methodology Previous: NMO for Multiple Reflections
Stanford Exploration Project
6/10/2002