Post-Stack Residual Migration

In an earlier article Fomel (1994), I found the integral solution of the boundary problem for the velocity continuation partial differential equation Claerbout (1986)  

$${{\partial^2 P}\over{\partial v\,\partial z}} + v\,t\,{{\partial^2 P}\over{\partial x^2}} = 0$$ (73)

with the boundary conditions $\left.P\right\vert _{v=v_0} = P_0$ and $\left.P\right\vert _{z \rightarrow \infty} = 0$. The solution has the form of the stacking operator (1), with the model M replaced by P₀, the summation path  

$$\widehat{\theta}(y;z,x) = \sqrt{z^2+{{(x-y)^2}\over {v^2-v_0^2}}}\;,$$ (74)

the weighting function  

$$w_{(-)}(y;z,x) = {1\over{\left(2\,\pi\right)^{m/2}}} \, {1 \over {v^m\,t^{m/2}}}\;,$$ (75)

which is coincident with (68), and the correction filter $\left(\mbox{sign}(v_0-v)\,{d \over {d t}}\right)^{m/2}$. Comparing equations (74) and (63), we can see that this solution is equivalent kinematically to residual migration with the velocity $v_r=\sqrt{v^2-v_0^2}$ Rothman et al. (1985). The reverse operator is the solution of equation (73) with the boundary condition on v and has the reciprocal form of the summation path

$$\theta(x;t,y) = \sqrt{t^2+{{(x-y)^2}\over {v_0^2-v^2}}} = \sqrt{t^2-{{(x-y)^2}\over {v^2-v_0^2}}}\;,$$ (76)

the weighting function

$$w_{(+)}(x;t,y) = {1\over{\left(2\,\pi\right)^{m/2}}} \, {1 \over {v^m\,z^{m/2}}}\;,$$ (77)

and the correction filter $\left(\mbox{sign}(v-v_0)\,{d \over {d\,z}}\right)^{m/2}$. The derivative filters are connected by the simple asymptotic relationship

$$\left(\pm\,{d \over {d\,z}}\right)^{m/2} = \left(\pm\,{d \o... ...d \over {d\,t}}\right)^{m/2}\, \left({z\over t}\right)^{m/2}\;,$$ (78)

which transforms the reversed velocity continuation operator to the familiar form (31) with the weighting function equal to (75). According to formula (69), these two operators are seen to be asymptotically inverse.

To obtain the velocity continuation operator completely equivalent to residual migration with the weighting function (66), we can divide the continued wavefield by the time t, which is equivalent to transforming equation (73) to the form  

$${{\partial^2 P}\over{\partial v\,\partial t}} + v\,t\,{{\par... ...ial x^2}} + {1 \over t}\,{\partial P \over {\partial v}} = 0\;.$$ (79)

The reverse continuation in this case has the weighting function (70).

Analogously, one can obtain the pseudo-unitary residual migration with the weighting functions (71) and (72) by dividing the wavefield by $\sqrt{t}$. This leads to the equation

$${{\partial^2 P}\over{\partial v\,\partial t}} + v\,t\,{{\par... ...^2}} + {1 \over {2\,t}}\,{\partial P \over {\partial v}} = 0\;.$$ (80)

It is apparent that the operators of forward and reverse continuation with equation (73) become adjoint to each other if the definition of the dot product is changed according to formulas (28) and (29) with the model weight W_M(z)=z and the data weight W_S(t)=t. Analogously, the solutions of equation (79) are adjoint if $W_M(z)={1 \over z}$ and $W_S(t)={1 \over t}$. This is a simple example of how the arbitrarily chosen definition of the dot product can affect the basic properties of the inverted operators.