## Depth migration by an unconditionally stable explicit finite-difference method

#### Jun Ji and Biondo Biondi

We develop an unconditionally stable, explicit depth migration method.
The downward continuation operator derived by a finite-difference
approximation of the one-way equation is given by the exponential
of a banded matrix.
We approximate this exponential by decomposing the banded matrix
into block diagonal matrices, of which the
exponential can be computed analytically.
The derived downward continuation operator
is explicit and unconditionally stable,
and thus it may be efficiently implemented on either
vector or parallel computers.

First, we apply this algorithm to a 15-degree migration.
To increase the accuracy at steep dip, we add
more leading terms to the Taylor-series expansion of the square-root operator.
However, the improvement in accuracy gained by adding more terms to
the Taylor series is at the expense of higher computational cost.

#### 61st Ann. Internat. Mtg., Soc., Expl. Geophys., Expanded abstracts,
1122-1125, (1991).