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In this appendix I derive the expressions for evaluating
the derivatives of image depth with respect to
the subsurface offset and
the midpoint ;these derivatives are computed along the tangent plane to the
impulse response of the generalized migration operator,
which is defined in equations 1824.
I start by deriving the equation for the vector normal to the
impulseresponse surface,
:
where
, , and
are respectively the unit vectors
along the three dimensions
, , and .
The equation of the tangent plane at the
image point with coordinates
is given by:
 

 
 (32) 
The derivative of the depth with respect o the subsurface offset,
at constant midpoint, is given by:
 
(33) 
and similarly the derivative of the depth with respect to the midpoint,
at constant subsurface offset, is given by:
 
(34) 
To evaluate
equations 3334.
we need to evaluate the following partial derivatives,
obtained by differentiating the expressions in
equations 1820:
 

 
 
 
 
 (35) 
The derivative of path length are evaluated as follows:
 

 (36) 
and
 

 (37) 