** Next:** MODELING AND IMAGING
** Up:** Karrenbach: Double elliptic scalar
** Previous:** APPROXIMATING THE DISPERSION RELATION

The derivation of the double elliptic paraxial approximation is described by
Dellinger et al. (1991).
The resulting dispersion relation in two dimensions is

| |
(1) |

where is the temporal and *k*_{x},*k*_{z} are the spatial frequencies and
W is squared velocity.
The top row in Figure represents the ``phase domain''; the bottom row shows the corresponding ``group domain''. The curves are calculated from
actual stiffness constants, which are used below for modeling and migration.
I approximated in phase domain,
since I show only implementations of
phase domain algorithms in this paper.
**compvel
**

Figure 1 compvel

Exact and approximated phase and group velocities are compared. In general, the approximation in the ``phase domain'' works very well, suggesting that it is more accurate than the approximation in the ``group domain''.
In general, the curve produced by the double elliptic approximation lies within the
true velocity curves. In the phase domain the two curves are in very good agreement, while in the group domain, the disagreement is larger. In particular,
triplications can be modeled using the double elliptic
approximation in the phase domain;
but not in the group domain.
Modeling of triplications is possible
in the ``phase domain'', since the rational polynomial can
produce concavities in the phase slowness surface. Furthermore,
the elliptic parameters govern both phase and group domains;
thus each domain can
be easily converted into the other.
Equation (1) is a very good approximation to the
exact dispersion relation, when Fourier-domain algorithms, such as the
Stolt or phase-shift method are used.
For modeling and its transpose operation,
migration, the dispersion relation must be evaluated and solved
for an unknown *k*_{z}. Both operations can be done quite efficiently using
equation (1).
The evaluation of the dispersion relation is straightforward. And when finding
the *k*_{z} root, we must explicitly solve a third-order polynomial in *k*_{z}^{2}.

The foregoing procedure must be seen in contrast to solving eigenvalue
problems numerically or
finding roots for
higher order polynomials. This is the procedure that has to be followed if
the dispersion relation is evaluated exactly for anisotropic media.
In this context the double elliptic approximation is an efficient way to
solve the problem approximately.

** Next:** MODELING AND IMAGING
** Up:** Karrenbach: Double elliptic scalar
** Previous:** APPROXIMATING THE DISPERSION RELATION
Stanford Exploration Project

12/18/1997