Potential-Field Continuation and Transformation
Measurement of gravitational and magnetic fields plays an important role in geophysical exploration. These potential fields can be mathematically transformed from the actual observation point to a fictitious observation point to enhance or reduce features in the data. For example, the effect of near-surface sources can be diminished by upward continuation of the observed values measured on the ground surface to a plane above the surface. On the other hand, if we want to enhance the effect of a shallow source, we perform downward continuation, which determines the fields at some depth below the surface.
In normal magnetic and gravity surveys, only one component of the field is measured. However, other components are often required for interpretation purposes. To derive additional information about the geological structure from the observed data, the derivatives of the field are often used. The conversion of one component of the observed data into other components or the calculation of the derivatives is called field transformation.
There are many methods of field continuation and transformation, but most of them are derived assuming that the data are measured on horizontal profiles or planes. However, if the topography is undulating, numerical methods such as the equivalent source technique (Dampney, 1969; Emilia, 1973; Bhattacharyya and Chan, 1977) and the finite harmonic series (Henderson and Cordell, 1971) are needed to process the data.
In this chapter we apply the BEM to the problem of 2-D and 3-D potential-field continuation and transformation.
Figures & Tables
The Boundary Element Method in Geophysics
The boundary element method (BEM) divides only the boundaries of the region under investigation into elements, so it diminishes the dimensionality of the problem, e.g., the 3D problem becomes a 2D problem, and the 2D problem becomes a 1D problem. This simplifies inputting the model into a computer and greatly reduces the number of algebraic equations. The advantage of this is even more evident for some 3D and infinite regional problems that often are encountered in geophysics. Originally published in China, this well-organized book is likely the most comprehensive work on the subject of solving applied geophysical problems. Basic mathematical principles are introduced in Chapter 1, followed by a general yet thorough discussion of the BEM in Chapter 2. Chapters 3 through 7 introduce the applications of BEM to solve problems of potential-field continuation and transformation, gravity and magnetic anomalies modeling, electric resistivity and induced polarization field modeling, magnetotelluric modeling, and various seismic modeling problems. Finally, in Chapter 8, a brief discussion is provided on how to incorporate the BEM and the finite-element method (FEM) together. In each chapter, detailed practical examples are given, and comparisons to both analytic and other numerical solutions are presented. This is an excellent book for numerically oriented geophysicists and for use as a textbook in numerical-analysis classes.