Algorithm 1 contains pseudocode which could be used to create a variational picking algorithm. Algorithm 2 shows how a variational picking algorithm may be used in a continuation framework. Here,
and
are strictly positive numbers denoting the minimum and maximum possible step sizes at each iteration,
and
are positive early termination parameters,
is the maximum number of iterations at each continuation level,
is a velocity model,
is a starting velocity model,
is the functional gradient, and
is the search direction.
is the number of continuation levels,
is a vector holding the triangle smoothing radius for each spatial dimension at continuation level
, and
is the semblance scaling factor at continuation level
. We assume that scaling and smoothing increase with increasing
.
is the least-smoothed semblance-like volume, and
denotes a semblance-like volume. These algorithms assume one has the following methods defined:
COST
Returns the cost,
according to Equation 4, associated with model
for semblance-like volume
using regularization parameters
and
. The returned object is a scalar.
GRADIENT
Returns the functional gradient,
according to Equation 5, associated with model
for semblance-like volume
using regularization parameters
and
. The returned object is a function with the same domain as
.
UPDATESEARCHDIRECTION
Returns an updated search direction using gradient
according to a quasi-Newton scheme, for example
-BFGS. In this method,
is an abstract state vector for the quasi-Newton scheme. The object
contains hyperparameters related to the method as well as gradient or search direction information from previous iterations which is used in the construction of the updated search direction. Application of the method is assumed to update abstract state
without returning it as an output. The returned object is a function with the same domain as
.
LINESEARCH
Returns the step size
between minimum and maximum step sizes
and
using a method of one's choosing such that
(8)
The returned object is a scalar.
SMOOTH
Applies triangle smoothing to
using smoothing radii for each dimension of
contained in
. The returned object is a smoothed function with the same domain as
A variational approach for picking optimal surfaces from semblance-like panels