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):

and

where

Equations 1 and 2 can be summarized in matrix forms as follows:

and

where

(6) |

(7) |

and

(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:

where is a shifting operator which shifts the wavefield in the direction. Notice that . Equations 4, 5 and 9 are state equations, and , and are the state variables.

To evaluate the accuracy of the subsurface model, we use a DSO objective function (Shen, 2004):

where is the subsurface offset. In practice, other objective functions (linear transformations of the image) can be used rather than DSO. To derive the DSO objective function with respect to and , we follow the recipe provided by Plessix (2006). First, we form the Lagrangian augmented functional:

Then the adjoint-state equations are obtained by taking the derivative of with respect to state variables , and :

Equation 13, 14, and 15 are the adjoint-state equations. Parameters , and are the adjoint-state variables, and can be calculated from the adjoint-state equations.

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:

It is straightforward to understand that the gradients for the model parameters are the crosscorrelations of the perturbed source / receiver wavefield and the scattered background receiver / source wavefield.

Wave-equation migration velocity analysis for anisotropic models on 2-D ExxonMobil field data |

2012-05-10