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

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Formulation for 1 dimension

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

 (7)

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:

 (8)

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:

 (9)

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

Next: Application of spectral factorization Up: Explicit Vs. Implicit Finite Previous: Explicit Vs. Implicit Finite

2010-05-19