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Generating ADCIGs for forward-scattered wavefields requires specifying
reflectivity as a function of either the source-side, , or
receiver-side, , reflection angles. Either choice, though,
requires isolating one angle from a system with 6 free parameters:
and . Hence,
solving for, say, requires specifying 6 constraint equations.

Three constraint equations are specified by geometric relationships
(c.f. Figure ). The first constraint equation
is a local conservation of reflection angle given by,

| |
(8) |

The second and third constraint equations derive from a global
conservation of reflection angle arguments that relate the *S* and *R*
plane-wave angles, geologic dip, and the source- and receiver-side
reflection angles through Biondi (2005),
| |
(9) |

and,
| |
(10) |

Snell's Law provides a fourth physical constraint equation by relating
the source- and receiver-side reflection angles with the local
propagation slownesses,
| |
(11) |

which can be rewritten using Equation (10) as,
| |
(12) |

Constraint equations (8-11) do not incorporate
physical observables measured from the generated image volume.
However, we can calculate image-space dips in both the
horizontal subsurface half-offset, , and midpoint, ,
directions. Thus, the final two constraint equations relating
measured dips to free parameters can be obtained by taking the
appropriate partial derivatives of the parametric hyper-plane surface
in Equation (7),

| |
(13) |

and
| |
(14) |

We rewrite Equations (13) and (14)
using the trigonometric angle addition and subtraction rules,
| |
(15) |

and,
| |
(16) |

which we rearrange to yield,
| |
(17) |

where is a ``normalized difference'' of slownesses given by,
. Solving for and
leads to,
| |
(18) |

where the parameters held constant during partial differentiation are
no longer explicitly written. These two expressions can be manipulated to
specify independent equations for reflection angle, ,
| |
(19) |

and true geologic dip, ,
| |
(20) |

When source and receiver propagation slownesses are equal (i.e.,
), these quadratic equations reduce to,
| |
(21) |

which is similar to the expressions derived for the backscattered case
save for a phase rotation (i.e., ). Finally, the solution for the receiver-side
reflection angle, , is obtained from angle through the relation
specified in Equation (12).
In Equations (13) and (14), we
differentiated with respect to variables, *x* and *h*_{x}. This choice
was one step in the development of horizontal ADCIGs.
Equally, we can create vertical ADCIGs by developing two constraint
equations from partial derivatives with respect to vertical variables,
*z* and *h*_{z}, holding horizontal variables *x* and *h*_{x} constant.
Vertical ADCIGs are then generated through introduction of these
functions into Equations (15-20).
Biondi and Symes (2004) detail situations where it is
more advantageous to use vertical ADCIGs than their horizontal
counterparts. In particular, vertical ADCIGs provide better spatial
resolution for scenarios where the wavefield propagation direction is oriented at
steep angles to the geologic dip-field.

** Next:** Numerical Examples
** Up:** From ODCIGs to ADCIGs
** Previous:** From ODCIGs to ADCIGs
Stanford Exploration Project

5/3/2005