Wave-equation tomography for anisotropic parameters |

where is a focusing operator acting on the background image.

In the least-squares sense, the tomographic objective function can be written as follows:

To perform the WETom for anisotropic parameters, we first need to
extend the tomographic operator from the isotropic medium
(Shen, 2004; Sava, 2004; Guerra et al., 2009) to the anisotropic medium. We
define the image-space wave-equation tomographic operator **T** for anisotropic
parameters as follows:

(9) |

where

(10) |

In this paper, we evaluate the anisotropic tomographic operator in the shot-profile domain.

Both source and receiver wavefields are downward continued in the shot-profile
domain using the one-way wave equations (Claerbout, 1971):

and

where is the source wavefield at the image point with the source located at ; is the receiver wavefield at the image point with the source located at ; is the source signature, and defines the point source function at , which serves as the boundary condition of Equation 11; is the recorded shot gather at , which serves as the boundary condition of Equation 12. Operator is the dispersion relationship for anisotropic wave propogation:

where is the angular frequency, is the slowness at , is the anellipticity at ; is the spatial wavenumber vector. Dispersion relationship 13 can be approximated with a rational function by Taylor series and Padé expansion analysis (Shan, 2009):

where, to the second order, . Using binomial expansion, Equation 14 can be further expanded to polynomials:

The background image is computed by applying the cross-correlation
imaging condition:

where the overline stands for the complex conjugate, and is the subsurface half-offset.

Under the Born approximation, a perturbation in the model parameters
causes a first-order perturbation in the wavefields. Consequently, the
resulting image perturbation reads:

where and are the background source and receiver wavefields computed with the background model , and are the perturbed source wavefield and perturbed receiver wavefield, respectively, which result from the model perturbation .

To evaluate the adjoint tomographic operator
, which
backprojects the image perturbation into the model space, we first
compute the wavefield perturbation from the image perturbation using
the adjoint imaging condition:

The perturbed source and receiver wavefields satisfy the following
one-way wave equations, linearized with respect to slowness and
:

and

where is the row vector , and is the transpose of .

When solving the optimization problem, we obtain the image perturbation by migrating the data with the current background model and performing a focusing operation (Equation 7). Then the perturbed image is convolved with the background wavefields to get the perturbed wavefields (Equation 18). The scattered wavefields are computed by applying the adjoint of the one-way wave-equations 19 and 20. Finally, the model space gradient is obtained by cross-correlating the upward propagated scattered wavefields with the modified background wavefields (terms in the parentheses on the right-hand sides of Equations 19 and 20).

Wave-equation tomography for anisotropic parameters |

2010-05-19