Efficient 3D Gravity and Magnetic Modeling
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1 Efficient 3D Gravity and Magnetic Modeling X. Li Fugro Gravity & Magnetic Services Inc., Houston, Texas, USA Summary There are many different 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, gravity gradient and total magnetic intensity (TMI) responses; and invert one field or gradient alone or any combination of these 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 (Li and Chouteau, 1998). 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 is a linear inversion as in seismic tomography. However, the structure inversion is a much-preferred choice in petroleum exploration uses of gravity and magnetic data. Explorationists use such an inversion to determine a structure, i.e., the depth variation of a boundary such as the basement or base of salt, when there is not enough seismic data or its quality is poor. The structure inversion is nonlinear as in seismic depth imaging. The popular approach is to linearize a nonlinear 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 often gets huge for a field project: 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 thus 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. Gravity and magnetics are low-cost methods. A computer cluster is now widely used for seismic processing and interpretation while a laptop computer is still a popular machine for gravity and
2 magnetic modeling. At the same time, users don t want to wait hours for a modeling result. For this reason, we have to deviate from the conventional approaches to forward and inverse modeling and seek solutions in the wavenumber domain. Methodology In petroleum exploration, we 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 forms of density variations within a layer. First, the simplest form is a constant and this is widely used when variable density information is not available. Second, densities of sediments increase with depth due to compaction. Density changes have thus been represented by a linear, exponential, quadratic, parabolic, or hyperbolic function of depth. A depth-density pair may be used to describe these functions. Third, the density in a sedimentary basin may also change horizontally, e.g., the density may decrease basinward from a major fault. A density grid is a general form for describing such a horizontal variation. Fourth, a combination of a horizontal density grid and a depth-density pair allows both vertical and horizontal variations. However, this combination involves one horizontal grid and one single depth-density pair and is not applicable to the most complex and the most general cases. Fifth, such cases can be defined only by a voxet. In practice, a velocity voxet may be available when gravity and/or gravity gradient data is used to help interpret a horizon that seismic encounters difficulties (e.g., base salt). We can convert this velocity voxet into a density voxet using a simple velocity-density relationship or a geostatistical analysis. Evidently, the first four forms are special cases of this last one. A voxet may be divided vertically into a number of laminas (thin sheets). Each lamina is equivalent to a surface density distribution σ on a horizontal plane. The gravity effect on the top surface of a lamina can be written in the wavenumber domain as F[ g( K )] = 2πγF[ σ ( K) ] (1) where γ is the gravitational constant, K the wavenumber vector, and F stands for the Fourier transform. Applying an upward continuation, we obtain the gravity effect on a horizontal plane at altitude z: K z F[ g( K) ] = 2 πγe F[ σ ( K) ] (2) Summing up the effects of all laminas produces the total gravity effect. Applying operations in the wavenumber domain further computes the gravity gradient components TXX, TXY, TXZ, TYY, TYZ, T, T = ( T T 2 and the magnetic effect Δ T. ZZ UV YY XX ) For inversion, Bott (1960) proposed a very simple update formula for interpretation of gravity anomalies over a sedimentary basin. The gravity effect due to an infinite horizontal slab is a constant independent of the vertical position (i.e., burial depth) and the horizontal location of observation. g = 2πγρt (3) where ρ is the bulk density (contrast) of the slab, and t the thickness of the slab. Rearranging equation (3) produces g t = (4) 2πγρ This equation can be used to compute the depth change of a horizon, point by point. The depth inversion is accomplished with an iterative process.
3 More sophisticated algorithms for forward and inverse modeling are needed in practice. For example, equation (4) cannot be used to update depth when the magnetic anomaly or a gravity gradient component is inverted. We have developed advanced and efficient algorithms that can invert not only a field or gradient component individually but also a selection of any number of field and gradient components simultaneously. Below I present a joint inversion example. Figure 1. (a) The base of salt and (b) the density distribution of the starting model used to test a joint inversion of gravity gradients T and T. XY UV Figure 2. The model (a) before and (b) after a joint inversion of TXY. Only TXY grids are shown. One example Our test model consists of a seawater layer, sediments, a salt body and a basement (see Figure 3a). Each of the depth grids is quite large: 960 rows by 960 columns. The sediments have a complex density variation, defined by a voxet. Figure 1a shows the depth to the base of salt and Figure 1b depicts the density distribution of the initial 3D model. Figure 2a displays maps and east-west crosssections of this starting model. We simulate TXY, the curvature gravity gradient components that are directly measured by the Falcon TM airborne gravity gradiometer system. We display T XY
4 responses only although a joint inversion of TXY is performed to make changes to the base of salt so that the calculated gradients match observed gradients. Figure 2b demonstrates the inverted results. We have a seismic section in this test and its location is shown as the NW-SE diagonal line in map windows of Figure 2b. Figure 3 illustrates the cross-section along this diagonal profile. In particular, density variations overlie seismic section in Figure 3b. The algorithms converge fast: it takes 10 iterations (13 minutes) to complete this joint inversion of TXY on a desktop computer (Dell Precision T3400n Workstation with 64-bit Intel Core TM 2 Duo E8500 processor, 3.16 GHz and 8 GB RAM). Figure 3. A cross-section along a seismic profile. The NW-SE diagonal line in maps of Figure 2b indicates the profile location. (a) Structure and (b) densities (in color) overlaid on seismic. Conclusions We have developed wavenumber-domain algorithms for 3D gravity and magnetic modeling. It works effectively with the complexity of petroleum exploration problems and efficiently for the size of a field project. The inversion is always nonunique and depends strongly on the starting model and constraints. An efficient inversion allows users to test different reasonable assumptions and thus to derive a reliable final model. Acknowledgments Many people at Fugro Gravity & Magnetic Services Inc. have contributed to the development of this efficient 3D modeling tool. I also thank Fugro Gravity & Magnetic Services Inc. for permission to publish this work.
5 References Bott, M. H. P., 1960, The use of rapid digital computing methods for direct gravity interpretation of sedimentary basins: Geophysical Journal of the Royal Astronomical Society, 3, Li, X., and Chouteau, M., 1998, Three-dimensional gravity modeling in all space: Surveys in Geophysics, 19,
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