11: TRUE AMPLITUDE MIGRATION

Our task will be to choose a kernel $w({\bf r},r)$ that supplies the proportionality of image amplitude and reflection coefficient.

We shall neglect: 1) directional characteristics of the source, 2) refraction coefficients, 3) influence of the free boundary, and 4) variation of the density $\rho$. With these restrictions the amplitude of a wave on the surface $\Sigma$ is:  

$$A_0 (r) = {{c \cdot R} \over {\sqrt{J(s,r)}}} \sqrt{{v(s)} \over {v(r)}}$$ (116)

where s and r are the source and receiver locations, $\sqrt{J(s,r)}$describes the geometrical spreading of a wave propagating along the ray $s{\bf r}r$ (see Figure 1), and R is the reflection coefficient at the point ${\bf r}$.

Equation (116) is good for marine surveys. For land surveys we usually measure the vertical component

$$\stackrel{\sim}{A_0} (r) = A_0 (r) \cos \theta.$$