Fast 3D velocity updates using the pre-stack exploding reflector model

pre-stack-exploding-reflector model

The fundamental idea of PERM is to model data that describes the correct kinematics of an isolated SODCIG. When using Born modeling, since we do not know beforehand which shots contribute to forming the image at a point in the model space, we have to model several shots. Ideally, instead of performing many modeling experiments, we would like to synthesize a small amount of data with the condition that migration has the same kinematics as the initial SODCIG. This can be achieved by extrapolating source and receiver wavefields starting from a pre-stack image computed with wave-extrapolation methods.

The modeling of PERM source $D_{\scriptscriptstyle P}$ and receiver $U_{\scriptscriptstyle P}$ wavefields can be carried out by any wavefield-continuation scheme. Here, we use the following one-way wave equations:

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}-i\sqr... ...riptscriptstyle P}(x,y,z=z_{\rm max},\omega;{\bf x_m}) = 0 \end{array} \right.,$$ (1)

and

$$\left\{ \begin{array}{l} \left( \frac{\partial}{\partial z}+i\sqr... ...riptscriptstyle P}(x,y,z=z_{\rm max},\omega;{\bf x_m}) = 0 \end{array} \right.,$$ (2)

where $I_D({\bf x_m},{\bf h})$ and $I_U({\bf x_m},{\bf h})$ is the isolated SODCIG at the horizontal location ${\bf x_m}$ for a single reflector, suitable for the initial conditions for the source and receiver wavefields, respectively. The subsurface-offset $\bf h$ is parameterized as ${\bf h}=(h_x,h_y)$ . The initial conditions are obtained by rotating the original unfocused SODCIGs according to the apparent geological dip of the reflector (Biondi, 2007). By doing so, image-point dispersal is corrected such that the velocity information needed for migration velocity analysis is maintained.

To decrease the number of modeling experiments, linearity of wave propagation can be used to combine isolated SODCIGs and inject them simultaneously into one single model experiment, using the same modeling equations as above with the initial conditions replaced by the combined SODCIGs. The selection of SODCIGs can be thought of as the multiplication of the pre-stack image by spatial 2D $Comb$ functions, which are shifted laterally to select new set of SODCIGs to initiate the modeling of another pair of combined wavefields. After shifting along one period of the sampling function in the $x$ and $y$ directions, all the points on the reflector are used in the modeling. Consequently, the number of modeling experiments equals the number of lateral shifts of the sampling function.

The choice of the sampling period determines the amount of crosstalk in the migrated image. To obtain a crosstalk-free image, the sampling period must be large enough that wavefields initiated at different SODCIGs do not correlate. PERM wavefields generated from SODCIGs at an interval equals to twice the subsurface-offset range still contribute to the image at the central SODCIG. For the same reason, no crosstalk is generated during migration if the period of the sampling function is larger than that interval. Since the focusing of energy in the SODCIG is velocity-error dependent, so is the the period of the sampling function. Therefore, for small velocity errors a small sampling period can be used and, consequently, a smaller number of combined modeling experiments is needed.

Fast 3D velocity updates using the pre-stack exploding reflector model