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Biondi, B. L., 1999, 3-D Seismic Imaging: http://sepwww.stanford.edu/sep/bion do/Lectures/index.html.

Biondi, B., 2001, Narrow-azimuth migration: Analysis and tests in vertically layered media: SEP-108, 105-118.

Bleistein, N., and Handelsman, R. A., 1975, Asymptotic expansions of integrals: Rinehart Winston.

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A

This appendix presents a derivation of the expression for the weighting Jacobian in the case of variable velocity.

The dispersion relation

$$k_z= {\rm SSR}{k}$$ (43)

can be approximated using a Taylor series expansion around the constant reference slowness (s₀):

$$k_z\approx {k_z}_0+ \left. \frac{dk_z}{d s}\right\vert _{s=so}\left (s-s_0\right ).$$ (44)

We then take the derivative of k_z with respect to the frequency $\omega$

$$\frac{dk_z}{d\omega} = \frac{d{k_z}_0}{d\omega} + \frac{d}... ...mega} \left [\frac{d{k_z}_0}{d s} \right ]\left (s-s_0\right ),$$

and if we note that

$$\frac{d{k_z}_0}{ds} = \frac{\omega^2s_0}{{k_z}_0}$$

we obtain

$$\frac{dk_z}{d\omega} = \frac{d{k_z}_0}{d\omega} + \frac{d}... ...left [\frac{\omega^2s_0}{{k_z}_0} \right ]\left (s-s_0\right ).$$ (45)

We continue by evaluating the derivatives with respect to $\omega$ on the right hand side. With little algebra, we obtain

$$\frac{d}{d\omega} \left [\frac{\omega^2s_0}{{k_z}_0} \right ... ...\left (2- \left [\frac{\omega s_0}{{k_z}_0}\right ]^2 \right ).$$

therefore

$$\frac{dk_z}{d\omega} = \frac{\omega s_0}{{k_z}_0} s_0+ \... ...ac{\omega s_0}{{k_z}_0}\right ]^2 \right )\left (s-s_0\right ).$$ (46)

The prestack weighting Jacobian is:

$$\bold W_k_h= \left [ \frac{\omega s_0}{{k_{\rm zs_0}}} s_0+... ...{k_z}_0}\right ]^2 \right )\left (s_r-s_0\right ) \right ]^{-1}$$ (47)

which, in constant slowness, takes the simple form

$$\bold W_k_h= \left [ \frac{\omega s_0}{{k_{\rm zs_0}}} s_0 + \frac{\omega s_0}{{k_{\rm zr_0}}} s_0 \right ]^{-1}.$$ (48)