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For increasing the accuracy at higher dips, we can use more terms
in the Taylor-series expansion. Resulting in a matrix with
thicker bands than the tridiagonal matrix.
Using the second-order term in the expansion, we see that
the approximated square-root operator takes the form
| |
(21) |

where **I ** is the identity matrix, and **T** represents
the tridiagonal matrix that approximates the second partial derivative.
Let
with
and
We split the matrix into five pieces, *M*=*M*_{e}+*M*_{o}+*M*_{t1}+*M*_{t2}+*M*_{t3},
where

and
In the same manner as the tridiagonal matrix, we can approximate the
time-stepping operator as
| |
(22) |

*M*_{t1}, *M*_{t2}, and *M*_{t3} are block diagonal, and
the small block matrix *F* along the diagonal is defined as
Using the series definition of the exponential function, we see that
Exponentiating *M*_{t1}, *M*_{t2}, and *M*_{t3} amounts to
exponentiating *F*. The terms , and are also block diagonal.
Since the eigenvalues of are 1 and with imaginary *c*, it follows that
.As before, to prove the unconditional stability of the algorithm,
we need only show that
. This follows immediately since
each of which is equal to 1 according to the preceding argument.
In Figure , we compared the impulse responses of first-order
approximation with the second-order approximation in the Taylor-series
expansion of the square-root operator.
We also superposed semicircles which is the theoretical solution of
the extrapolation operator on Fig and the higher order
approximation shows better fitting to semicircles than the first-first
order approximation.

With the same manner as showed in this section, we can get more
accurate operator by taking more terms in a Taylor series expansion
of the square-root operator. However, it will produce a matrix
with increasing width of the band and thus will cause to an
increasing in computation cost.

**fig3
**

Figure 3 Impulse response of (a) explicit depth migration with the first-order
approximation for the square-root operator and of (b) explicit depth migration
with the second-order approximation for the square-root operator.

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** Up:** Ji and Biondi: Explicit
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Stanford Exploration Project

12/18/1997