Wave-equation migration velocity analysis for anisotropic models on 2-D ExxonMobil field data |
We parameterize the VTI subsurface using NMO slowness
, and Thomson parameters
and
(Thomsen, 1986).
In the shot-profile domain, both source wavefields
and receiver wavefields
are
downward continued using the following one-way wave equation and boundary condition
(Shan, 2009):
Equations 1 and 2 can be summarized in matrix forms as follows:
(6) |
(7) |
(8) |
It is well known that parameter is the least constrained by surface seismic data because of the lack of depth information. Therefore, we assume can be correctly obtained from other sources of information, such as check shots and well logs. In this paper, we are going to invert for NMO slowness and .
We use an extended imaging condition (Sava and Formel, 2006) to compute the image cube with subsurface offsets:
To evaluate the accuracy of the subsurface model, we use a DSO objective function (Shen, 2004):
Then the adjoint-state equations are obtained by taking the derivative of with respect to state variables , and :
The physical interpretation of the adjoint-state equations offers better understanding of the physical process and provides insights for implementation. Clearly, the solution to equation 15, , is the perturbed (residual) image at a certain subsurface offset. Equations 13 and 14 define the perturbed source and receiver wavefields, respectively. Notice the perturbed source wavefield at location depends on the image at and the background receiver wavefield at . The same rule applies to the perturbed receiver wavefield .
With the solutions to the equations above, we can now derive the gradients of the objective function
10 by taking the derivative of the augmented functional
with respect to the
model variables
and
as follows:
Wave-equation migration velocity analysis for anisotropic models on 2-D ExxonMobil field data |