Wave-equation tomography using image-space phase-encoded data

prestack exploding-reflector modeling

The general idea of prestack exploding-reflector modeling (Biondi, 2006) is to model the data and corresponding source function that are related to only one event in the subsurface. In this case, a single unfocused subsurface-offset-domain common-image gather (SODCIG) containing a single reflector is used as the initial condition for recursive upward continuation with the following one-way wave equations:

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z} - i \... ... h};x_m,y_m) \\ Q_D(x,y,z=z_{\rm max},\omega;x_m,y_m) = 0 \end{array} \right.,$$ (14)

and

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z} - i \... ... h};x_m,y_m) \\ Q_U(x,y,z=z_{\rm max},\omega;x_m,y_m) = 0 \end{array} \right.,$$ (15)

where $I_D({\bf x},{\bf h};x_m,y_m)$ and $I_U({\bf x},{\bf h};x_m,y_m)$ are the isolated SODCIGs at the horizontal location $(x_m,y_m)$ for a single reflector, and are suitable for the initial conditions for the source and receiver wavefields, respectively. As Biondi (2006) discusses, a rotation of the image gathers according to the apparent geological dip must be performed prior to modeling. By collecting the wavefields at the surface, we obtain the areal source data $Q_D(x,y,z=0,\omega;x_m,y_m)$ and the areal receiver data $Q_U(x,y,z=0,\omega;x_m,y_m)$ for a single reflector and a single SODCIG located at $(x_m,y_m)$. $\Lambda$ is the square-root operator defined by

$$\begin{array}{l} \Lambda = \sqrt{\omega ^2 {\hat s}^2({\bf x})-\vert{\bf k}\vert ^2}, \end{array}$$

where $s({\bf x})$ is the slowness at ${\bf x}$ and ${\bf k}=(k_x,k_y)$ is the spatial wavenumber vector.

Since the size of the migrated image volume can be very large in practice, and there are usually many reflectors in the subsurface, modeling each reflector and each SODCIG may generate a data set even larger than the original data set. One strategy to reduce the cost is to model several reflectors and several SODCIGs simultaneously (Biondi, 2006); however, this process generates unwanted crosstalk. As discussed by Guerra and Biondi (2008), random phase encoding can be used to attenuate the crosstalk.

One important characteristic of the prestack exploding reflector modeling is that, for velocity model building, the wavefields can be upward propagated to a certain depth level or depth horizon, provided that the velocity model above is sufficiently accurate. Therefore, a target-oriented strategy can be applied to derive the velocity model below the that depth.

Wave-equation tomography using image-space phase-encoded data