Implicit finite difference in time-space domain with the helix transform |

The two-way acoustic wave equation in one dimension reads:

In order to formulate it as an implicit finite-difference approximation, I first looked to the formulation of the implicit finite-difference operator of the 1-way wave equation in frequency-space domain in Claerbout (2009) Chapter 9. The Crank-Nicolson differencing method is used to create an implicit finite-difference approximation for downward-continuation of a wavefield:

The Crank-Nicolson method achieves better accuracy by balancing the 2nd derivative between the current wavefield values (at depth ) and at the next as-of-yet unknown wavefield values (at depth ). Hence the division by in the denominators of Equation 8. Borrowing from this methodology, I attempted to formulate a scheme which balances the values of the wavefield at known and unknown locations. Since Equation 7 has a second derivative on the left hand side (as opposed to first derivative in Equation 8), it implies that this balancing must be done over three time ``locations''. In one dimension, these considerations led me to the following approximation:

Note that the division by effectively averages the spatial derivative between the three time steps: , and . In order to propagate the wavefield, the values of the the wavefield at time must be equated to the values at times and .

Implicit finite difference in time-space domain with the helix transform |

2010-05-19