Nonhyperbolic reflection moveout of -waves: An overview and comparison of reasons |

does not change along any given ray (Snell's law). This fact leads to the explicit parametric relationships

where

Straightforward differentiation of parametric equations (19) and (20) yields the first four coefficients of the Taylor series expansion

in the vicinity of the vertical zero-offset ray. Series (22) contains only even powers of the offset because of the reciprocity principle: the pure-mode reflection traveltime is an even function of the offset. The Taylor series coefficients for the isotropic case are defined as follows:

where

Equation (24) shows that, at small offsets, the reflection moveout has a hyperbolic form with the normal-moveout velocity equal to the root-mean-square velocity . At large offsets, however, the hyperbolic approximation is no longer accurate. Studying the Taylor series expansion (22), Malovichko (1978) introduced a three-parameter approximation for the reflection traveltime in vertically heterogeneous isotropic media. His equation has the form of a shifted hyperbola (Castle, 1988; de Bazelaire, 1988):

If we set the zero-offset traveltime equal to the vertical
traveltime , the velocity equal to , and the *parameter of heterogeneity* equal to , equation
(30) guarantees the correct coefficients , , and
in the Taylor series (22). Note that the parameter
is related to the variance of the squared velocity
distribution, as follows:

As follows from the definition of the parameters [equations (29)] and the Cauchy-Schwartz inequality, the expression (32) is always nonnegative. This means that the shifted-hyperbola approximation tends to overestimate traveltimes at large offsets. As the offset approaches infinity, the limit of this approximation is

Equation (33) indicates that the effective horizontal
velocity for Malovichko's approximation (the slope of the shifted
hyperbola asymptote) differs from the normal-moveout velocity. One
can interpret this difference as evidence of some *effective*
depth-variant anisotropy. However, the anisotropy implied in
equation (30) differs from the true anisotropy in a
homogeneous transversely isotropic medium [see equation (1)].
To reveal this difference, let us substitute
the effective values
,
,
, and
into equation (30). After eliminating the
variables and , the result takes the form

Figure 3 illustrates the difference between the VTI model and the effective anisotropy implied by the Malovichko approximation. The differences are noticeable in both the shapes of the effective wavefronts (Figure 3a) and the moveouts (Figure 3b).

nmofrz1,nmofrz2
Comparison of the
wavefronts (a) and moveouts (b) in the VTI (solid) and vertically
inhomogeneous isotropic media (dashed). The values of the effective
vertical, horizontal, and NMO velocities are the same
in both media and correspond to Thomsen's parameters
and
.
Figure 3. |
---|

In deriving equation (35), we have assumed the correspondence

The difference between equations (36) and (37) is an additional indicator of the fundamental difference between homogeneous VTI and vertically heterogeneous isotropic media. The three-parameter anisotropic approximation (12) can match the reflection moveout in the isotropic model up to the fourth-order term in the Taylor series expansion if the value of is chosen in accordance with equation (37). We can estimate the error of such an approximation with an equation analogous to (32):

The difference between the error estimates (32) and (38) is

For usual values of , which range from to , the expression (39) is positive. This means that the anisotropic approximation (12) overestimates traveltimes in the isotropic heterogeneous model even more than does the shifted hyperbola (30) shown in Figure 3b. Below, we examine which of the two approximations is more suitable when the model includes both vertical heterogeneity and anisotropy.

Nonhyperbolic reflection moveout of -waves: An overview and comparison of reasons |

2014-01-27