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Assume that anisotropic velocities have a vertical axis of symmetry,
like the transversely isotropic (TI) media described in
Thomsen 1986. Although that paper describes
``weak'' anisotropy, the same equations
can be applied to very strong anisotropy Tsvankin and Thomsen (1994).
Three of Thomsen's parameters, V_{z}, , and ,are defined by the elastic constants of a general TI medium.
These constants can be used to specify three different
effective velocities at a single point in the model.
V_{z} is the velocity of a wave traveling vertically along
the axis of symmetry. The velocity in any horizontal direction
is V_{x} determined by
 
(19) 
and a ``normal moveout velocity'' (NMO) velocity V_{n} defined by
 
(20) 
Phil Anno of Conoco has shown that if
the TI properties represent the equivalent medium of
many isotropic layers Backus (1962); Schoenberg and Muir (1989),
then the above inequalities can
be expected to hold. (One additional assumption is
that the V_{s}/V_{p} ratio and V_{s} have a positive correlation.)
Notice that
and
, so that .
For convenience, researchers at the Colorado School of Mines
Alkhalifah and Tsvankin (1994); Tsvankin and Thomsen (1994) have also defined a constant
 
(21) 
Many combinations of three of these parameters can be used to describe
a TI medium. An approximation has already dropped a fourth constant
to which compressional P waves are very insensitive.
The exact equations for TI phase velocity as a function of
angle are rather clumsy, and no explicit form is available
for group velocity. Alternative approximate equations can
used which fit almost the same family of curves
as the original correct equations Michelena et al. (1993).
I use an approximate equation for group velocity
which appears to emulate closely the exact curves for
large ranges of positive and negative .Since I aim to estimate these anisotropic velocities from
noisy measurements, I expect our estimated curves to have larger errors
than introduced by these approximations.
I choose approximate curves with the three velocities defined above.
Let be the group angle of a raypath from the vertical.
Then the group velocity can be expressed as
 
(22) 
Greg Lazear of Conoco found that
a good approximation of the phase velocity as a function of the phase angle takes a similar form, but with reciprocals of velocities:
 
(23) 
The NMO velocity also turns out to have a physical
interpretation. Imagine an experiment on a homogeneous
and anisotropic medium (or imagine a small scale experiment
on a smooth model). Measure the traveltime t_{0} between two
points placed on a vertical line, separated by a vertical
distance V_{z} t_{0}. Now displace the
upper point a distance h along a horizontal line
and measure the new traveltime t_{h}.
Then according to equation (22) the traveltime t_{h}
as a function of offset h is exactly
 
(24) 
When then the value of t_{h} in this
``moveout equation'' is
controlled by the NMO velocity V_{n} rather than V_{x}. In
the other case , the raypath is almost horizontal
and V_{x} dominates.
I find it convenient to define a stacking velocity V_{h}(h)
as a function of the offset h for a fixed vertical distance:
 
(25) 
Thus, we can construct a equation which describes
the best fitting hyperbola to traveltimes at zero offset
and at a single finite offset h:

t^{2} = t_{0}^{2} + h^{2} / V_{h}(h)^{2} .

(26) 
Note that this stacking velocity covers the range
, increasing in value as h increases.
(To use twoway reflection times in 26
we need only replace the half offset
h by the full offset.)
Theoretically, three measurements of traveltimes at three
different offsets h should uniquely determine the three
velocity constants V_{z}, V_{x}, V_{n}. However, the
traveltimes are much more sensitive to V_{n}, which determines
moveouts at small offsets, and to V_{x}, which determines
moveout at larger offsets. The vertical velocity V_{z} affects
only the rate at which the stacking velocity changes from
one limit to the other. As long as V_{z} has roughly the
correct magnitude, then we can fit all measured traveltimes
very well. Remember that we expect
for equivalent layered media.
For imaging data in time, we can set V_{z} = V_{n} and simplify
our equations even further. To image surface data in depth,
we can focus images very well with good values for V_{x} and V_{n},
then adjust imaged depths to tie wells with V_{z}, holding
the other two velocities constant.
Stacking velocity analysis can be optimized with V_{n} because
it is relatively close to V_{h}.
If the maximum offset equals the depth, and if V_{x} = 1.1 V_{n},
then V_{h} = 1.02 V_{n} for a flat reflection, which is small
enough difference for such a large anisotropy.
This anisotropy model, although certainly not the most general,
describes the most important properties of transversely
isotropic velocities. The
very simple form allows for easy optimization and inversion.
A tomographic algorithm which builds on such a model will
include the necessary dependence of velocities on angle. Any
refinements in the anisotropic behavior will be easy to introduce
without major alterations of the computer program.
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Stanford Exploration Project
11/12/1997