Finite-difference datuming

Finite-difference wave extrapolation is performed by transforming the dispersion relation into a differential equation (Claerbout, 1985). In 2-D, the dispersion relationship

$$k_z \;=\; \frac{\omega}{v}\,\sqrt{1 - \frac{v^2 k_x^2}{\omega^2}},$$

is converted to a differential equation by substituting $ik_z \rightarrow \partial /\partial z$, to get

$$\frac{\partial P}{\partial z}\;=\;\frac{i\omega}{v}\, \sqrt{1 - \frac{v^2}{\omega^2}k_x^2}\, P.$$

Muir's continued fraction expansion approximation to the square root and the substitution $ik_x \rightarrow \partial /\partial x$yields the $15^{\circ}$ equation  

$$\frac{\partial P}{\partial z}\;=\; \frac{i\omega}{v(x,z)}P\, + \frac{v(x,z)}{-i2\omega}\,\frac{\partial^2 P}{\partial x^2}.$$ (22)

Depth stepping proceeds by solution of equation () in two stages. The first stage,

$$\frac{\partial P}{\partial z}\;=\;\frac{i\omega}{v}P,$$

is solved analytically by

$$P(z-\Delta z)\;=\;P(z)e^{\frac{i\omega}{v}\Delta z}.$$

The second stage,

$$\frac{\partial P}{\partial z}\;=\; \frac{v}{-i2\omega}\,\frac{\partial^2 P}{\partial x^2},$$

is solved by use of some method of finite differencing. For a given depth step, $\Delta z$, the finite-difference extrapolation is given by the product D_i L_i, where D_i is a matrix representation of the finite-difference solution method. In principle this could be any solver, but in practice the examples in this dissertation are solved using the Crank-Nicholson implicit method (Claerbout, 1985).

For the geometry of Figure , L_i is the matrix

$$\left[ \begin{array} {cccccc} e^{i\frac{\omega}{v(x_1,z_i)... ...e^{i\frac{\omega}{v(x_6,z_i)} \Delta z} \\ \end{array}\right].$$

Therefore, the upward continuation operator can be written as  

$$\displaystyle{ \left[ \begin{array} {ccc} D_3 L_3 D_2 L_2 D... ...] \left[ \begin{array} {c} P(x,z_s,\omega)\end{array}\right], }$$ (23)

The adjoint algorithm is found by transposing each matrix and reversing the multiplication order, as follows:  

$$\displaystyle{ \left[ \begin{array} {ccc} A & B & C \end... ... \begin{array} {c} P(x,z_{\rm dat},\omega)\end{array}\right]. }$$ (24)

Prestack finite-difference datuming can be performed by extrapolation of shot and receiver gathers separately or by implementation of the double square root equation,

$$\frac{\partial P}{\partial z}\;=\; \left[ \sqrt{ \left(\frac... ...ac{\partial^2}{\partial x_s^2} } \,\right]\, P(x_s,x_g,\omega),$$

and alternation between shot x_s and group x_g coordinates for every depth step $\Delta z$.