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Binary
Inversion of Gravity Data
Presented
at SEG 2002
When
a nil zone is present in the subsurface salt structure,
it effectively creates an annihilator of density contrast
that gives rise to zero gravity response on the surface.
As a result, part of the salt structure is invisible to
the surface data and inversion algorithms often have difficulties
in recovering the salt structure correctly. We develop a
binary inversion technique in which the density contrast
is restricted to being one of the two possibilities: either
zero or the value expected at a given depth (represented
by ones). The binary condition places a strong restriction
on the admissible models so that the non-uniqueness caused
by nil zones can be resolved. In this presentation, we will
outline the formulation, discuss the solution strategy,
and illustrate it with numerical examples.
Presented
at SEG 2004
We
have developed a hybrid optimization algorithm for inversion
of gravity data using a binary formulation. The new algorithm
utilizes the Genetic Algorithm (GA) as a global search tool,
while implementing Quenched Simulated Annealing (QSA) intermittently
for local search. The hybrid has significantly decreased
computational cost over GA or Simulated Annealing (SA) alone
and has allowed for successful inversion of more realistic
gravity problems. We illustrate the algorithm using a large
2½D model derived from the SEG/EAGE 3D salt model,
which has a complex background density profile and a pronounced
nil zone.
Ph.D.
Thesis
I
present a binary inversion algorithm for inverting gravity
data in salt imaging. The density contrast is restricted
to being either zero or one, where one represents the value
of density contrast of salt at a given depth. I develop
this method to overcome the difficulties associated with
interface-based inversion and density-based inversion while
attempting to draw from the strengths of both existing approaches.
The interface inversion specifies the known density contrast
of salt, but its parameterization can overly restrict the
model from the outset. The density inversion, on the other
hand, affords great flexibility in its model representation,
but cannot directly utilize the known density information.
Binary inversion uses a similar model representation as
in continuous-density inversion by defining a density distribution
as a function of spatial position, but restricts the model
values to those corresponding to two lithologic units as
does the interface inversion.
I formulate the binary inversion using Tikhonov regularization
in which the inverse solution is obtained by minimizing
a weighted sum of a data misfit and a model objective function.
The model objective function serves to stabilize the solution
and to incorporate any prior information that is independent
of gravity data. Because of the discrete nature of the problem,
commonly used minimization techniques are no longer applicable.
I therefore investigate the use of genetic algorithm, quenched
simulated annealing, and a hybrid method based on these
two as potential solvers for the minimization problem associated
with the binary inversion. The use of Tikhonov regularization
is well understood in continuous-variable inversion, but
its application in binary problems has yet to be explored.
I investigate this aspect and conclude that Tikhonov regularization
plays a similar role in discrete inversion, and the corresponding
Tikhonov curve behaves in a similar manner. Thus the commonly
used approaches for determining the level of regularization
is equally applicable in both types of inversions. Finally,
appraisal of solution is a necessary component of inversion,
in which one attempts to understand the uncertainties in
the recovered model and to identify features of high confidence.
I explore the model space of binary inversion, evaluate
the modality of the objective function for this purpose,
and illustrate the improved reliability of interpretation
in the process.
I illustrate binary inversion with synthetic models in 2D
and 3D generated from the SEG/EAGE salt model. As sought
in development of binary inversion, the method incorporates
density information while providing a sharp contact for
the subsurface. It also allows for flexibility in model
representation while solving for density distribution as
a function of spatial position. The binary condition places
a strong restriction on the admissible models so that the
non-uniqueness caused by nil zones might be resolved.
Journal
publications to come soon!
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