** Next:** Dip misfit by crossproduct
** Up:** Plane-wave misfit
** Previous:** Plane-wave misfit

A three-dimensional plane-wave function
is a projection of a one-dimensional waveform *f*(*s*):

where is normal to the constant planes.
Additionally, is non-zero ().
Such a projection spreads
the value *f*(*s*) along the plane perpendicular
to the normal direction . For any given point ,
the projection
yields the argument of the function *f*
based on the geometric interpretation of the vector product.
Given an input image ,
plane-wave estimation
finds an approximation of normal vector
and a waveform estimation *f*(*s*).
The naive formulation
of the plane-wave estimation problem

is nonlinear and highly non-convex ().
Alternatively, the problem
can be reformulated into
two well-behaved estimations.
The estimation
finds the best-fitting vector
that is normal to *g*'s gradient.
The subsequent estimation
uses the normal estimated by the first minimization
to find the waveform *f*(*s*).
The formulation of the cross-product minimization
stems from the fact that
an image volume
is a plane-wave volume
with normal (),
if and only if at all

If *g* is indeed a plane-wave function
with a normal proportional to , then
and, consequently,
Similarly,
any image that satisfies equation (12)
can be shown to be a plane-wave volume
as defined in equation (8).
The constraint of a nonzero normal vector
can be implemented in various ways.
The constraint
leads to a symmetric formulation
in all spatial variables *x*,*y*, and *z*.
However,
for mathematical convenience,
I choose the constraint *p*_{z} = 1,
which excludes vertical plane waves
and leads to an asymmetry between the spatial variables.
In practice, this particular constrain
does not limit the processing
of ordinary seismic images
since image features are rarely vertical.

The minimization of equation (12)
can be solved analytically.
Let be the vector of partial *x*-derivatives of
discretized *g*.
Each element represents the derivative at a given location .The vectors and are the corresponding
partial *y*- and *z*-derivatives.
Given the constraint *p*_{z} = 1,
equation (12) can be expressed in matrix form as

The minimization's corresponding normal equation is
The variables
estimate the gradient slope in the *x* and *y* direction of *g*.
Substitution
of the gradient expressions (14)
into the normal equation (13) yields
Finally,
the equation system can be solved for
the *x*- and *y*-component of the plane wave's normal vector:
Claerbout 1992 shows
how the gradient slopes *q*_{x} and *q*_{y}
can be computed from the input image
using finite-difference approximations of
the partial derivative expressions
,
and
.The solution fully specifies the normal vector since
the vector's *z* component *p*_{z} is constrained to 1.
If we set the variables
in the *y* dimension - *q*_{y} and *p*_{y} - to zero,
the solution simplifies to

which is identical to Claerbout's 1992
two-dimensional dip picking scheme.
My original cross-product formulation (12)
expresses the natural symmetry of the
plane-wave detection problem that
Claerbout's Lomoplan formulation lacks.
However, my constraint for the nonzero normal vector
(*p*_{z} = 1) introduces the same asymmetry later.
Stacking of the image volume along planes
orthogonal to the estimated normal vector
yields a first estimate of the unknown
plane-wave waveform *f*(*s*).
Symes formulates a similar, but more sophisticated,
iterative estimation scheme of the normal vector and
the waveform *f*(*s*).

** Next:** Dip misfit by crossproduct
** Up:** Plane-wave misfit
** Previous:** Plane-wave misfit
Stanford Exploration Project

3/8/1999