next up previous [pdf]

Next: Angle-domain image and illumination Up: Tang and Biondi: Angle-dependent Previous: introduction

Subsurface-offset-domain image and illumination

Linearized modeling (Born modeling) from a prestack image parameterized as a function of subsurface offset can be described as follows:

$\displaystyle d({\bf x}_r,{\bf x}_s,\omega) = \sum_{{\bf x}}\sum_{{\bf h}}L_{h}({\bf x},{\bf h},{\bf x}_r,{\bf x}_s,\omega)m_h({\bf x},{\bf h}),$     (1)

where $ d({\bf x}_r,{\bf x}_s,\omega)$ is the seismic data with a source located at $ {\bf x}_s=(x_s,y_s,z_s=0)$ and a receiver located at $ {\bf x}_r=(x_r,y_r,z_r=0)$ ; $ \omega$ is the angular frequency; $ m_h({\bf x},{\bf h})$ is the prestack image located at $ {\bf x}=(x,y,z)$ for a half subsurface offset $ {\bf h}=(h_x,h_y,h_z)$ ; $ L_{h}$ is the sensitivity kernel defined as follows:
$\displaystyle L_{h}({\bf x},{\bf h},{\bf x}_r,{\bf x}_s,\omega) = \omega^2f_s(\omega)G({\bf x}-{\bf h},{\bf x}_s,\omega)G({\bf x}+{\bf h},{\bf x}_r,\omega),$     (2)

where $ f_s(\omega)$ is the source signature, and $ G({\bf x},{\bf x}_s,\omega)$ and $ G({\bf x},{\bf x}_r,\omega)$ are the Green's functions connecting the source and receiver, respectively, to the image point $ {\bf x}$ .

Reconstruction of the prestack image $ m({\bf x},{\bf h})$ can be posed as an inverse problem by minimizing the following objective function defined in the data space:

$\displaystyle F({\bf m}) = \frac{1}{2}\sum_{\omega}\sum_{{\bf x}_s}\sum_{{\bf x}_r}\vert W({\bf x}_r,{\bf x}_s)r({\bf x}_r,{\bf x}_s,\omega)\vert^2,$     (3)

where $ r({\bf x}_r,{\bf x}_s,\omega)=d({\bf x}_r,{\bf x}_s,\omega)-d_{\rm obs}({\bf x}_r,{\bf x}_s,\omega)$ is the data residual and $ W({\bf x}_r,{\bf x}_s)$ is the acquisition mask operator, which contains unity values where we record data and zeros where we do not. The gradient of the objective function $ F$ reads
$\displaystyle I_{h}({\bf x},{\bf h}) = \sum_{\omega} \sum_{{\bf x}_s} \sum_{{\b...
...}^{*}({\bf x},{\bf h},{\bf x}_r,{\bf x}_s,\omega)r({\bf x}_r,{\bf x}_s,\omega),$     (4)

where $ ^{*}$ denotes complex conjugation. Equation 4 is similar to the prestack shot-profile migration formula and it produces migrated reflectivity images defined in the subsurface offset domain (Rickett and Sava, 2002).

The Hessian can be obtained by taking the second-order derivatives of $ F$ with respect to the model parameters as follows:

$\displaystyle H_{h}({\bf x},{\bf x}',{\bf h},{\bf h}') =\sum_{\omega} \sum_{{\b...
...\bf x}_r,{\bf x}_s,\omega) L_{h}({\bf x}',{\bf h}',{\bf x}_r,{\bf x}_s,\omega).$     (5)

When $ {\bf x}={\bf x}'$ and $ {\bf h}={\bf h}'$ , we obtain the diagonal elements of the Hessian operator
$\displaystyle H_{h}({\bf x},{\bf h}) = \sum_{\omega}\sum_{{\bf x}_s}\sum_{{\bf ...
...x}_r,{\bf x}_s)
\vert L_{h}({\bf x},{\bf h},{\bf x}_r,{\bf x}_s,\omega)\vert^2.$     (6)

The diagonal of the Hessian is often known as the illumination map of the subsurface, it contains illumination contribution from both sources and receivers for a given acquisition configuration.

next up previous [pdf]

Next: Angle-domain image and illumination Up: Tang and Biondi: Angle-dependent Previous: introduction