Wave-equation tomography for anisotropic parameters

WETom for anisotropic parameters

Anisotropic WETom is a non-linear inversion process that aims to find the anisotropic model that minimizes the residual field $\Delta {\bf I}$ in the image space. The residual image is derived from the background image ${\bf I}$ , which is computed with current background model. In general, the residual image is defined as (Biondi, 2008)

$$\Delta {\bf I} = {\bf I} - {\bf F}({\bf I}),$$ (7)

where ${\bf F}$ is a focusing operator acting on the background image.

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

$$J = \frac{1}{2}\vert\vert{\Delta {\bf I}}\vert\vert _2 = \frac{1}{2}\vert\vert{\bf I}-{\bf F}({\bf I})\vert\vert^2.$$ (8)

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:

$${\bf T} = \left.\frac{\partial{\bf I}}{\partial{\bf m}}\right\vert _{{\bf m}=\widehat{\bf m}}$$

$$= \left.\frac{\partial{\bf I}}{\partial{\bf s}}\right\vert _{{\bf s... ...\partial{\bf I}}{\partial{\bf \eta}}\right\vert _{{\bf\eta}=\widehat{\bf\eta}},$$ (9)

where m is the anisotropy model, which in this case includes vertical slowness ${\bf s}$ and anellipticity parameter ${\bf\eta}$ ; $\widehat{\bf m}$ is the background anisotropy model, consisting of the background slowness $\widehat{\bf s}$ and background anellipticity $\widehat{\bf \eta}$ ; I is the image. This WETom operator T is a linear operator that relates the model perturbation ${\Delta {\bf m}}$ to the image perturbation ${\Delta {\bf I}}$ as follows:

$$\Delta {\bf I} = {\bf T} \Delta {\bf m}.$$ (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):

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\Lam... ... \\ D(x,y,z=0,{\bf x}_s) = {f_s\delta({\bf x}-{\bf x}_s)} \end{array} \right.,$$ (11)

and

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}-i\Lam... ... x}_s) = 0 \\ U(x,y,z=0,{\bf x}_s) = Q(x,y,z=0,{\bf x}_s) \end{array} \right.,$$ (12)

where $D({\bf x},{\bf x}_s)$ is the source wavefield at the image point ${\bf x} = (x,y,z)$ with the source located at ${\bf x}_s = (x_s,y_s,0)$ ; $U({\bf x},{\bf x}_s)$ is the receiver wavefield at the image point ${\bf x}$ with the source located at ${\bf x}_s$ ; $f_s$ is the source signature, and $f_s\delta({\bf x}-{\bf x}_s)$ defines the point source function at ${\bf x_s}$ , which serves as the boundary condition of Equation 11; $Q(x,y,z=0,{\bf x}_s)$ is the recorded shot gather at ${\bf x}_s$ , which serves as the boundary condition of Equation 12. Operator $\Lambda$ is the dispersion relationship for anisotropic wave propogation:

$$\Lambda = \omega s({\bf x})\sqrt{1 - \frac{\vert{\bf k}\vert ^2}{\omega ^2 s^2({\bf x}) - 2 \eta({\bf x}) \vert{\bf k}\vert ^2}},$$ (13)

where $\omega$ is the angular frequency, $s({\bf x})$ is the slowness at ${\bf x}$ , $\eta({\bf x})$ is the anellipticity at ${\bf x}$ ; ${\bf k}=(k_x,k_y)$ is the spatial wavenumber vector. Dispersion relationship 13 can be approximated with a rational function by Taylor series and Padé expansion analysis (Shan, 2009):

$$\Lambda = \omega s({\bf x}) (1 - \frac{a \vert{\bf k}\vert ^2}{\omega ^2 s^2({\bf x}) - b \vert{\bf k}\vert ^2})$$ (14)

where, to the second order, $a = 0.5, b = 2\eta + 0.25$ . Using binomial expansion, Equation 14 can be further expanded to polynomials:

$$\Lambda = \omega s({\bf x}) - \frac{a}{\omega s^2({\bf x})} \vert{\bf k}\vert ^2 - \frac{3ab}{\omega^3 s^4({\bf x})} \vert{\bf k}\vert^4.$$ (15)

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

$$I({\bf x},{\bf h}) = \sum_{{\bf x}_s}\sum_{\omega} \overline{D({\bf x}-{\bf h},{\bf x}_s)} U({\bf x}+{\bf h},{\bf x}_s),$$ (16)

where the overline stands for the complex conjugate, and ${\bf h}=(h_x,h_y,h_z)$ 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:

$$\Delta I({\bf x},{\bf h}) = \sum_{{\bf x}_s}\sum_{\omega} \left( \overline{\Delta D({\bf x}-{\bf h},{\bf x}_s)} {\widehat U}({\bf x}+{\bf h},{\bf x}_s) + \right.$$

$$\left. \overline{{\widehat D}({\bf x}-{\bf h},{\bf x}_s)} \Delta U ({\bf x}+{\bf h},{\bf x}_s) \right),$$ (17)

where ${\widehat D}({\bf x}-{\bf h},{\bf x}_s)$ and ${\widehat U}({\bf x}+{\bf h},{\bf x}_s)$ are the background source and receiver wavefields computed with the background model ${\widehat m}({\bf x})$ , $\Delta D({\bf x}-{\bf h},{\bf x}_s)$ and $\Delta U({\bf x}+{\bf h},{\bf x}_s)$ are the perturbed source wavefield and perturbed receiver wavefield, respectively, which result from the model perturbation $\Delta m({\bf x})$ .

To evaluate the adjoint tomographic operator ${\bf T}^{\ast}$ , which backprojects the image perturbation into the model space, we first compute the wavefield perturbation from the image perturbation using the adjoint imaging condition:

$$\Delta D({\bf x},{\bf x}_s) = \sum_{\bf h} \Delta I({\bf x},{\bf h}) {\widehat U}({\bf x}+{\bf h},{\bf x}_s)$$

$$\Delta U({\bf x},{\bf x}_s) = \sum_{\bf h} \Delta I({\bf x},{\bf h}) {\widehat D}({\bf x}-{\bf h},{\bf x}_s).$$ (18)

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

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\Lam... ...\bf m}^{\ast}({\bf x})\\ \Delta D(x,y,z=0,{\bf x}_s) = 0 \end{array} \right. ,$$ (19)

and

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}-i\Lam... ...\bf m}^{\ast} ({\bf x})\\ \Delta U(x,y,z=0,{\bf x}_s) = 0 \end{array} \right.,$$ (20)

where ${\bf m}$ is the row vector $[{\bf s}  {\bf\eta}]$ , and ${\bf m}^{\ast}$ is the transpose of ${\bf m}$ .

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