WAVE-EQUATION DATUMING ALGORITHM

Wave-equation datuming is the 2-D equivalent of the 1-D static shifting. It takes seismic data from a given surface and transports it to another surface. We hope that in the transportation process nothing is lost, and seismic data will look exactly as if it is recorded on the secondary surface.

We can formulate the datuming algorithm in a similar way we can formulate the migration algorithm being the conjugate transpose of the modeling algorithm. Given the data recorded on a non-level surface, we want to continue the data to a level surface. We can define the direct problem as a wave equation extrapolation from a level surface to an irregular surface. The conjugate transpose to this problem will bring the data from the irregular topographic surface to the level surface.

The direct problem is easier to formulate than the conjugate. Suppose we record a wavefield in a zero-offset experiment on a flat surface. The direct problem is to upward continue the wavefield to the uneven surface, similar to the one in Figure . I propose a scheme in which the wavefield is propagated upward using a phase-shift or PSPI method and the values of the wavefield are extracted at each height corresponding to the datum. The advantage of the method is that it allows us to use simple and fast algorithms to extrapolate the wavefield while the disadvantage is that the algorithm is only an approximation to the exact solution.

 

Datumdraw Figure 1 Data recorded on the flat datum (1) is extrapolated to the datum (2) and then, using the conjugate transpose algorithm it is extrapolated back to the flat datum(3).

The conjugate transpose algorithm starts by downward extrapolating the data from the highest point on the topographic datum. Each time the wavefield reaches the datum, data is inserted into the downward propagated wavefield.

The mathematical formulation of the direct datuming algorithm is simpler in the case of depth variable velocity. However I should note that for computational purposes there is almost no speed difference between a depth varying velocity v(z) medium and a laterally and depth varying velocity v(x,z) medium. The wavefield p(x,z=0,t) is Fourier transformed in time to have $P(x,z=0,\omega)$ and each frequency can be independently upward extrapolated. By Fourier transforming the horizontal space variable, the wave equation can be written as  

$${ {\partial^2 P(k_x,z,\omega)} \over {\partial z^2}}={(k^2_x- {\omega^2 \over {v^2(z)}})} P(k_x,z,\omega)$$ (1)

valid for all values of k_x and $\omega$.For a constant velocity medium we introduce a constant k_z as  

$$k_z={\pm {[{\omega^2 \over v^2}-{k^2_x }]^{1 \over 2}}}.$$ (2)

k_z is constant for two given values of k_x and $\omega$.Equation (1) becomes an ordinary differential equation  

$${\partial^2 P \over \partial z^2}=-k_z^2 P .$$ (3)

For a constant k_z it has the analytic solution  

$$P(k_x,z_0+z,\omega)={P(k_x,z_0,\omega)}e^{ik_z z} .$$ (4)

For depth varying velocity v(z), k_z is considered approximately constant for small depth intervals ($\Delta z$) where the velocity is considered constant. Therefore equation (4) can be written  

$$P(k_x,z=z_0+\Delta z,\omega)={{P(k_x,z=z_0,\omega )}e^{ik_z \Delta z}}.$$ (5)

This form of the equation can be used to downward or upward extrapolate the wavefield for a small depth interval.

Thus the forward datuming algorithm can be summarized in four steps.

• Fourier transform the wavefield $P(x,z=z_0,\omega)$ over the space variable x.
• Upward propagate the wavefield $P(k_x,z=z_0,\omega)$ to the next depth level $P(k_x,z_0-\Delta z,\omega)$.
• Forward transform the wavefield $P(k_x,z=z_0-\Delta z,\omega)$ over the space variable x
• Extract the traces corresponding to a depth level on the topographic surface.