step 2

Forward time modeling of the particle-displacement ${\bf u}$ and particle-velocity ${\bf \dot{u}}$ wavefields, up to the end time of the shot record. For each time step the following arrays are computed for use in the third step:

• Scalar downgoing particle-displacement field

  $$u(x,z,t) = \mbox{sign}(\dot{u}_z) \sqrt{\dot{u}_x^2 + \dot{u}_z^2}$$

• Unit vector in the group propagation direction of the downgoing wavefield

  $${\bf \hat{i}}(x,z,t).$$

This step also computes the total ``energy," at each grid point

$$\;\;\; E(x,z) = \int u^2(x,z,t) \, dt. \;\;\;$$

The modeling is implemented with the elastic modeling scheme described in another paper included in this report (Cunha, 1991). Because the imaging step requires access to the time slices generated in this step in the reverse time sequence, this step cannot be performed simultaneously with step 3. Since the three modeling cubes are too large to be held in core memory, we are left with two options for solving this problem: to store the cubes in an auxiliary file to be read in reverse order in step 3, or to keep only the last time slice and do reverse time propagation simultaneously with the third step. In the scheme presented here, I use the first option not only because it will have a better performance if a data vault is available, but also because it will not be affected by the absorbing boundary or the extra dispersion caused by the backward propagation.