DOUBLE ELLIPTIC 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  

$$\omega^2 = \frac{(W_x k_x^2 )^3 + (W_x k_x^2)^2 (2W_z + W_{z... ...+ W_{xnmo}) k_x^2 + (W_z k_z^2)^3}{(W_x k_x^2 + W_z k_z^2)^2} ,$$ (1)

where $\omega$ 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_z².

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.