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Unlike in the previous paper (Li and Biondi, 2011), we derive the anisotropic WEMVA gradients using a Lagrangian augmented functional instead of perturbation theory. The derivation is hence neat and straightforward, and the interpretations of the adjoint-state equations suggest the same implementation as perturbation theory suggests.

We parameterize the VTI subsurface using NMO slowness $ s_n$ , and Thomson parameters $ \eta $ and $ \delta $ (Thomsen, 1986). In the shot-profile domain, both source wavefields $ {\bf D}$ and receiver wavefields $ {\bf U}$ are downward continued using the following one-way wave equation and boundary condition (Shan, 2009):

$\displaystyle \left\{ \begin{array}{l}
\left(\frac{1}{\sqrt{1+2\delta}} \frac{\...
... \\
D(x,y,z=0,{\bf x}_s) = {f_s\delta({\bf x}-{\bf x}_s)}
\end{array} \right.,$     (1)

$\displaystyle \left\{ \begin{array}{l}
\left( \frac{1}{\sqrt{1+2\delta}} \frac{...
... x}_s) = 0 \\
U(x,y,z=0,{\bf x}_s) = Q(x,y,z=0,{\bf x}_s)
\end{array} \right.,$     (2)


$\displaystyle \Lambda = \omega s_n\sqrt{1-\frac{\vert{\bf k}\vert ^2}{\omega ^2 s_n^2 - 2 \eta \vert{\bf k}\vert ^2}}.$ (3)

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

$\displaystyle {\bf L} {\bf D} = {\bf f}$ (4)


$\displaystyle {\bf L^*} {\bf U} = {\bf d},$ (5)


$\displaystyle {\bf f} = f_s \delta({\bf x}-{\bf x}_s, z),$ (6)

$\displaystyle {\bf d} = d_{{\bf x}_s} \delta({\bf x}-{\bf x}_r, z),$ (7)


$\displaystyle {\bf L} = \frac{1}{\sqrt{1+2\delta}} \frac{\partial}{\partial z}-i\Lambda .$ (8)

It is well known that parameter $ \delta $ is the least constrained by surface seismic data because of the lack of depth information. Therefore, we assume $ \delta $ 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 $ s_n$ and $ \eta $ .

We use an extended imaging condition (Sava and Formel, 2006) to compute the image cube with subsurface offsets:

$\displaystyle I_{\bf h} = ({\bf S}_{+{\bf h}} {\bf p})^* ({\bf S}_{-{\bf h}} {\bf q}),$ (9)

where $ {\bf S}_{+{\bf h}}$ is a shifting operator which shifts the wavefield $ +{\bf h}$ in the $ {\bf x}$ direction. Notice that $ ({\bf S}_{+{\bf h}})^* = {\bf S}_{-{\bf h}}$ . Equations 4, 5 and 9 are state equations, and $ {\bf D}$ , $ {\bf U}$ and $ I_{\bf h}$ are the state variables.

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

$\displaystyle J = \frac{1}{2} \sum_{\bf h} \langle {\bf h} I_{\bf h}, {\bf h} I_{\bf h} \rangle.$ (10)

where $ {\bf h}$ 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 $ s_n$ and $ \eta $ , we follow the recipe provided by Plessix (2006). First, we form the Lagrangian augmented functional:
$\displaystyle {{\mathcal L}({\bf D},{\bf U},I_{\bf h};{\bf\lambda},{\bf\mu},\gamma_{\bf h};s_n, \epsilon) = }$
    $\displaystyle \frac{1}{2} \sum_{\bf h}\left < {\bf h}~ I_{{\bf h}},~ {\bf h}~ I_{{\bf h}} \right >$  
    $\displaystyle + \left < \bf\lambda,~ {\bf f} - {\bf L}(s_n,\epsilon) {\bf D} \right >$  
    $\displaystyle + \left < \bf\mu,~ {\bf d} - {\bf L^*}(s_n,\epsilon) {\bf U} \right >$  
    $\displaystyle + \sum_{\bf h} \left < \gamma_{\bf h},~ ({\bf S}_{+{\bf h}} {\bf D})^*({\bf S}_{-{\bf h}}
{\bf U}) - I_{\bf h} \right > .$ (12)

Then the adjoint-state equations are obtained by taking the derivative of $ {\mathcal L}$ with respect to state variables $ {\bf D}$ , $ {\bf U}$ and $ I_{\bf h}$ :

$\displaystyle \frac{\partial {\mathcal L}}{\partial {\bf D}} = - {\bf L^*} (s_n...
...{\bf h} ({\bf S}_{+{\bf h}})^* ({\bf S}_{-{\bf h}} {\bf U}) \gamma_{\bf h} = 0,$ (13)

$\displaystyle \frac{\partial {\mathcal L}}{\partial {\bf U}} = -{\bf L} (s_n,\e...
..._{\bf h} ({\bf S}_{-{\bf h}})^* ({\bf S}_{+{\bf h}} {\bf D})\gamma_{\bf h} = 0,$ (14)

$\displaystyle \frac{\partial {\mathcal L}}{\partial I_{\bf h}} = -\gamma_{\bf h} + {\bf h}^2 I_{\bf h} = 0, \forall ~{\bf h}.$ (15)

Equation 13, 14, and 15 are the adjoint-state equations. Parameters $ {\bf
\lambda}$ , $ {\bf\mu}$ and $ \gamma_{\bf h}$ 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, $ \gamma_{\bf h}$ , 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 $ {\bf
\lambda}$ at location $ {\bf x}$ depends on the image at $ ({\bf x} - {\bf h}, {\bf h})$ and the background receiver wavefield $ {\bf U}$ at $ {\bf x} - 2{\bf h}$ . The same rule applies to the perturbed receiver wavefield $ {\bf\mu}$ .

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 $ {\mathcal L}$ with respect to the model variables $ s_n$ and $ \epsilon$ as follows:

$\displaystyle \nabla_{s_n} J$ $\displaystyle =$ $\displaystyle \left < \bf\lambda,~ -\frac{\partial {\bf L}}{\partial s_n} {\bf ...
... >
+ \left < \bf\mu,~ -\frac{\partial {\bf L}^*}{\partial s_n} {\bf U} \right >$ (16)

$\displaystyle \nabla_{\epsilon} J$ $\displaystyle =$ $\displaystyle \left < \bf\lambda,~ -\frac{\partial {\bf L}}{\partial \epsilon} ...
...left < \bf\mu,~ -\frac{\partial {\bf L}^*}{\partial \epsilon} {\bf U} \right >.$ (17)

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.

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Next: Field data tests Up: Li: Anisotropic MVA Previous: Introduction