Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies |

The biggest and most obvious problem with Thomsen's approximations to the wave
speeds generally occurs in
. The key issue is that Thomsen's
approximation for
is completely symmetric around
, while unfortunately
this is generally not true of the actual wave speeds
. This error
may seem innocuous in itself since it is not immediately clear whether it
affects the results for small angles of incidence ()
or not, but it can in fact lead to large over- or under-estimates
of wave speeds in the neighborhood of both the extreme value located at
and also at
. To improve this situation while still making use of a
practical approximation to the wave speed, I reconsider
an approach originally proposed in Berryman (1979).
In particular, notice that the square root formula for
can be conveniently, and exactly, rewritten as:

To simplify this expression, first notice that has an absolute maximum value, which occurs when takes the value determined by

The extreme value of is given by

where

For realistic systems, it is always true that
.
[For example, in the fractured reservoir examples presented later in the paper, the
largest observed value of
. Also, note for all layered media
since
for layered elastic media (Postma, 1955; Backus, 1962; Berryman, 1979).]
So, we can expand the square root in equation 11, keeping only its first order Taylor series correction, which is

and

Note that the only approximation made in arriving at equations 18 and 19 again was the approximation of the square root via equation 17.

Clearly, the analysis is not really restricted in any way to using just the first order
Taylor approximation in equation 17. For example, other authors
(Fowler, 2003; Pederson *et al.*, 2007)
have explored rational approximations to such square roots at length. These approaches
can certainly be useful in many applications as they provide higher order approximations
(not necessarily just first and second order Taylor contributions), while avoiding the
computational complexity of the square root operation. Nevertheless, such efforts are beyond
our current scope and so will not be discussed further here.

Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies |

2007-09-15