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The expressions of
the derivative of with respect to arbitrary perturbations
of individual velocity components (i.e. *V*_{V}, *V*_{H}, and *V*_{N})
are slightly more complex than with respect to because the wavefronts are deformed when the velocity
components are unevenly perturbed.
These derivatives can be expressed as:
| |
(40) |

| (41) |

| (42) |

The expressions for the derivatives of the slowness function
with respect to the perturbation parameters depend
on the particular form chosen to approximate
the slowness function.
Appendix C derives these derivative
for the VTI group slowness function approximation
expressed in equation 6,
which I used for the numerical experiments shown in this paper.

The partial derivatives of the RMO function are directly derived from the partial derivatives
of , taking into account
that for flat reflectors only the vertical velocity
component *V*_{V}
influences the image depth of normal incidence.
The derivatives of can thus be written as follows:

| |
(43) |

| (44) |

| (45) |

Figures and
show examples of the application of the generalized
RMO functions expressed
in equations 44-46.
As in
Figures - ,
I show the ADCIGs for three different anisotropic rock types,
but, differently from the previous figures, not for the isotropic case.
The order of the rock types is the same as in
Figures - ;
that is: panels a) correspond to Taylor Sand,
panels b) to Mesa Clay Shale, and
panels c) to GreenLight River Shale.
Furthermore, as in
Figures - ,
one figure (Figure )
shows the ADCIG obtained with a smaller perturbation
than the ADCIGs shown in the other figure
(Figure ).
The ADCIGs shown in Figure
were obtained by performing *isotropic* migration
on the synthetic data modeled assuming *anisotropic* velocity.
The ADCIGs shown in Figure
were computed by scaling by .25 the parameter perturbations used
to compute Figure .
The lines superimposed onto the images are the RMO functions
computed by using
the correct values for (solid lines),
and by using in place of (dashed lines).
The predicted RMO functions accurately track the actual RMO functions
when the parameter perturbations are sufficiently
small to be within the range of accuracy of the
linearization at the basis of the derivation of
equation 40 (Figure ).
But even when the perturbations are large
(Figure ) and
cause a substantial RMO (up to 30% of the reflector depth)
the predicted RMO functions are excellent approximations
of the actual RMO functions.

The RMO functions associated with the two strongly unelliptical rocks
(Taylor Sand and GreenLight River Shale) exhibit
a characteristic oscillatory behavior;
the events at narrow-aperture angles are imaged deeper than
the normal incidence event, whereas the events at wide-aperture angles
are imaged shallower.
This oscillatory behavior is well predicted
by the analytical RMO function introduced
in equations 44-46.

In contrast,
the approximation of the group angles with the phase angles
(dashed lines in the figures)
seriously deteriorates the accuracy of the predicted RMO functions.
Notice that, in contrast with the uniform perturbation case
illustrated in
Figures - ,
the dashed lines are different among the panels,
because the derivatives
of the slowness function with respect to
the perturbation parameters depend on the anisotropic parameters
of the background medium.

**Trio_Aniso-iso_overn
**

Figure 9
ADCIGs obtained when data modeled with an *anisotropic* velocity have
been migrated using an *isotropic* velocity.
The anisotropic data were modeled assuming
three rock types:
a) Taylor Sand, b) Mesa Clay Shale,
and
c) GreenLight River Shale.
Superimposed onto the images are the RMO functions
computed using equation 40.
The solid line was computed when
was derived from by applying equation 1,
whereas the dashed line was computed
by approximating as equal to .

**Trio_Aniso-scaled_overn
**

Figure 10
ADCIGs obtained when data modeled with an *anisotropic* velocity have
been migrated using a *less anisotropic* velocity;
that is, with anisotropic parameters obtained
by scaling by .25 the parameter perturbations used
to compute Figure .
The anisotropic data were modeled assuming
three rock types:
a) Taylor Sand, b) Mesa Clay Shale,
and
c) GreenLight River Shale.
Superimposed onto the images are the RMO functions
computed using equation 40.
The solid line was computed when
was derived from by applying equation 1,
whereas the dashed line was computed
by approximating as equal to .

** Next:** Conversion of depth errors
** Up:** Anisotropic residual moveout for
** Previous:** Uniform scaling of velocity
Stanford Exploration Project

5/3/2005