Asymptotic pseudounitary stacking operators

3. Common-offset migration

In the case of common-offset migration in a general variable-velocity medium, the weighting function (46) cannot be simplified to a different form, and all its components need to be calculated explicitly by dynamic ray tracing (Cerveny and de Castro, 1993). In the constant-velocity case, we can differentiate the explicit expression for the summation path

$$\widehat{\theta}(y;z,x) = z + {{\rho_s(x,y) + \rho_r(x,y)} \over v}\;,$$ (50)

where $\rho_s$ and $\rho_r$ are the lengths of the incident and reflected rays:

$$\rho_s(y,x) = \sqrt{x_3^2 + (x_1 - y_1 + h_1)^2 + (x_2 - y_2 + h_2)^2}\;,$$ (51)

$$\rho_r(y,x) = \sqrt{x_3^2 + (x_1 - y_1 - h_1)^ 2+ (x_2 - y_2 - h_2)^2}\;.$$ (52)

For simplicity, the vertical component of the midpoint $y_3$ is set here to zero. Evaluating the second derivative term in formula (46) for the common-offset geometry leads, after some heavy algebra, to the expression

$$\left\vert{{\partial^2 T\left(s(y),x\right)} \over {\partial x \... ...{{\rho_s + \rho_r} \over {v \rho_s \rho_r}}\right)^{m-1} \cos{\alpha(x)}\;.$$ (53)

Substituting (53) into the general formula (46) yields the weighting function for the common-offset true-amplitude constant-velocity migration:

$$\widehat{w}_{CO}(y;z,x) = {1\over{\left(2 \pi\right)^{m/2}}} ... ...+ \rho_r)^{m-1} (\rho_s^2 + \rho_r^2)} \over {v (\rho_s \rho_r)^{m/2+1}}}\;.$$ (54)

Equation (54) is similar to the result obtained by Sullivan and Cohen (1987). In the case of zero offset $h=0$ , it reduces to equation (49). Note that the value of $m=1$ in (54) corresponds to the two-dimensional (cylindric) waves recorded on the seismic line. A special case is the 2.5-D inversion, when the waves are assumed to be spherical, while the recording is on a line, and the medium has cylindric symmetry. In this case, the modeling weighting function (42) transforms to (Deregowski and Brown, 1983; Bleistein, 1986)

$$w(x;t,y) = {1\over{\left(2 \pi\right)^{1/2}}} {\sqrt{v} {C\left(s(y),x,r(y)\right)} \over {\sqrt{\rho_s \rho_r (\rho_s + \rho_r)}}}\;,$$ (55)

and the time filter is $\left({\partial \over {\partial z}}\right)^{1/2}$ . Combining this result with formula (53) for $m=1$ , we obtain the weighting function for the 2.5-D common-offset migration in a constant velocity medium (Sullivan and Cohen, 1987):

$$\widehat{w}_{CO;2.5D}(y;z,x) = {1\over{\left(2 \pi\right)^{1/2}}... ... + \rho_r} (\rho_s^2 + \rho_r^2)} \over {\sqrt{v} (\rho_s \rho_r)^{3/2}}}\;.$$ (56)

The corresponding time filter for 2.5-D migration is $\left(- {\partial \over {\partial t}}\right)^{1/2}$ .

In the common-offset case, the pseudo-unitary weighting is defined from (47) and (53) as follows:

$$w^{(-)}_{CO}(y;z,x) = {1\over{\left(2 \pi v\right)^{m/2}}} {... ...r 2} \sqrt{\rho_s^2 + \rho_r^2}} \over {(\rho_s \rho_r)^{{m+1} \over 2}}}\;,$$ (57)

where

$$\cos{\alpha} = \left({ {(x - y)^2 + \rho_s \rho_r - h^2} \over {2 \rho_s \rho_r}} \right)^{1/2}\;.$$ (58)

Asymptotic pseudounitary stacking operators