Airborne Gravity 2010 - Geoscience Australia

Airborne Gravity 2010
Summary
A practical software tool for 3D gravity and
magnetic modeling
Xiong Li 1
1 Fugro Gravity & Magnetic Services Inc. (XLi@fugro.com)
There are a variety of spatial-domain algorithms for 3D gravity and magnetic forward and inverse
modeling. However, an efficient modeling tool for petroleum exploration needs algorithms in the
wavenumber domain. We have developed, tested and applied such a tool, for both forward and
inverse modeling, over two decades. Our algorithms work for density and susceptibility variations of
any complex form; compute gravity, seven gravity gradient components, and total magnetic intensity
(TMI) responses; and invert any combination of these nine field and gradient components
simultaneously. It takes only minutes, not many hours, to complete a joint inversion for structure of a
practically sized project on a personal computer.
Introduction
Spatial-domain closed-form or numerical computation formulae for forward gravity and magnetic
modeling have been extensively studied. People often represent an isolated body by simple
geometries: an ellipsoid, sphere, cylinder, thin sheet, etc. A complex body or structure is expressed by
a combination of right rectangular prisms, polygonal prisms, or polyhedrons. Researchers have also
developed formulae that allow variable densities and susceptibilities within a prism or polyhedron.
These formulae are accurate but inefficient. For example, a widely used formula for computation of the
gravity due to a right rectangular prism with a constant density contains 24 terms: 16 logarithms and 8
arctangents (Li and Chouteau, 1998, p. 344).
Gravity or magnetic data may be inverted for either physical property or structure. In a property
inversion, we often divide the subsurface into cells and invert for the constant density or susceptibility
values of the cells. This process is a linear inversion as is the case with seismic tomography.
However, structure inversion is a much-preferred choice in petroleum exploration applications of
gravity and magnetic data. Explorationists expect such an inversion to resolve a structure, i.e., the
depth variation of a boundary such as the basement or the base of salt, particularly when there is
insufficient seismic data or its quality is poor. Structure inversion is a nonlinear process as is the case
with seismic depth imaging. The popular approach for the solution of a nonlinear inversion is to
linearize the problem and then solve the linear system in a least-squares sense. This approach
requires many iterations of forward computation and solution of a linear system. The size of this
system is often very large for a field project: in matrix form, its number of rows is the total number of
data points and its number of columns is the total number of unknown parameters. Researchers have
designed many advanced mathematical solutions, with a focus on two aspects: (a) transforming a
dense matrix into a sparse one (with many zero elements in the matrix), e.g., by the wavelet
compression technique, and (b) using an effective solver of a sparse matrix, e.g., the conjugate
gradient or LSQR method. Unfortunately, products from such great efforts are far from efficient, and it
is still common to take many hours to run a 3D inversion.
Computer clusters are now widely used for seismic processing and interpretation but a laptop
computer remains the popular machine for gravity and magnetic modeling. However, users do not
want to wait hours for a modeling result. For this reason, we have deviated from the conventional
approaches for forward and inverse modeling and sought solutions in the wavenumber domain.
Methodology
In petroleum exploration, we by and large deal with layered structures as well as isolated bodies such
as igneous intrusions or salt emplacements. The density or susceptibility within a body, particularly a
layer, may vary both vertically and horizontally. An advanced modeling tool should allow different
125