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The chain rule for partial differentiation gives

| |
(66) |

In our usual notation
the Fourier representation of the
time derivative
is .Likewise, and the spatial
derivatives are associated
with *k*_{g} , *k*_{s} , *k*_{z} ).
Using these Fourier variables in the vectors of (66) and
differentiating
(58), (59), (60), and (61)
to find the indicated elements in the
matrix of (66), we get
| |
(67) |

Let *S* be the sine of the takeoff angle at the source and let *G* be
the sine of the emergent angle at the geophone.
If the velocity *v* is known, then these angles will be directly
measurable as stepouts on common-geophone gathers and common-shot gathers.
Likewise, on a constant-offset section or a slant stack, observed
stepouts relate to an apparent dip *Y*,
and on a linearly moved-out common-midpoint gather, stepouts
measure the apparent stepout *H*'.
The precise definitions are

| |
(68) |

| (69) |

With these definitions the second and third rows of (67) become
| |
(70) |

| (71) |

The familiar offset stepout angle *H* is related to the
LMO residual stepout angle *H*' by *H*' = *H* -*pv*.
Setting *H*' equal to
zero means setting *k*_{h' } equal to zero,
thereby indicating integration over *h*' , which in turn
indicates slant stacking data with slant angle *p*.
Small values of *H*' /*v* or refer
to stepouts near to *p*.

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** Up:** INTERVAL VELOCITY BY LINEAR
** Previous:** Common-midpoint Snell coordinates
Stanford Exploration Project

10/31/1997