Forward And Radial Inverse Modeling Of Potential Fields Produced By A 2D Homogeneous Source

Gravity method and magnetic method, which are usually refereed as a whole as potential field methods, are two important branches of applied geophysics. The forward and inverse modeling methods of potential fields have always been the central issues for the geophysicists. In this thesis the author starts from the integrals of Newtonian potential and scalar magnetic potential to derive the integrals for the gravity anomaly, gravity anomaly gradients, magnetic anomaly and magnetic anomaly gradients, and then transforms these areal integrals into linear integrals with respect to the source boundary, by which the author discusses the continuity of these quantities inside, outside and on the boundary of the source.The integrals of the potential fields against an arbitrary cross-section are very hard to evaluate. Therefore, the author uses a model with a polygonal cross-section to approximate an arbitrary cross-section, and the integrals against such geometry are easier to evaluate. Many people have discussed the formulae for calculating the gravity anomaly and the magnetic anomaly produced by a source with polygonal cross-section. The formulae for calculating the gravity anomaly gradient and magnetic anomaly gradient have been mentioned by some authors but without detailed derivation process, thus the author of this thesis completes this job. These new formulae are applicable inside and outside the source. The gravity anomaly is continuous at the vertices of the polygon, but the others will approach to infinity there. This singularity is due to the hypothesis of continuous media, which is the mathematical prerequisite upon which the integrals of Newtonian potential and scalar magnetic potential, which serves as the bases of all the derivations in this thesis, stand. The singularity is attributed to the deviation of the hypothesis of continuous media from the objective laws of the nature. Polygonal model have sharp vertices at which the curvatures are infinitely large, but in reality abstract geometric entity as such does not exist at all because no material in this world is continuous. This means that the gradient of the gravity anomaly, the magnetic anomaly and the gradient of the magnetic anomaly with infinite magnitude are impossible. The gradients of gravity and magnetic fields are not among the fundamental forces fields, so they cannot produce mechanic effect. They cannot be measured through any physical means. In practice, such quantities must be estimated through finite differences of measured potential fields along certain directions. Therefore, in any practical survey, the gradients must be finite. However, one should be cautious with this singularity when writing computer programs to implement these formulae to prevent unrealistic values or floating number overflow.The radial inversion method is proposed by Brazil geophysicists, Silva and Barbosa. This method is based on a radial polygon, of which the vertices can only move along specified directions. This property could prevent the edges of the polygon from intersecting with each other. The parameters of radial inversion method are the radii of the vertices with respect the center. Silva and Barbosa inverted gravity data using this method. The author of this thesis expanded its usage to invert the gradients of gravity anomaly, magnetic anomaly and the gradients of magnetic anomaly. The workflow in this thesis allows the regularizing factors associated with each constraint to be adjusted according the the condition number of the matrix of the system of linear equations, while the original procedures proposed by Silva and Barbosa do not allow this. The control on the condition number could keep the inversion process stable.As an example, the author invert the potential fields and their gradients produced by a laccolithic model by applying the constraints of absolute proximity, relative proximity and concentration of physical property along preferred directions. The relative proximity constraint will enforce the parameters to be equal with each other; the absolute constraint will enforce the parameters to be close to the initial values; the concentration constraint will enforce the parameters to concentrate along the preferred directions. The matrices associated with the absolute proximity constraint and the concentration constraint are full-ranked, therefore they can reduce the condition number of the matrix associated with the inversion problem, and, as a result, could be used alone. The absolute proximity constraint is only a special case of the concentration constraint. The matrix associated with the relative proximity constraint is not full-ranked, and it cannot stabilize the inversion process, therefore cannot be used along, and must be used in combination with other constraints. To demonstrate the stability of the inversion processes, the author compared the results of inversion against precise data and data contaminated with random noises. The results shows that random noises has very little affect on gravity anomaly inversion, but make the inversion results of the others present tiny undulations. In general, the affect of random noises on the inversion results are quite limited. The concentration constraint is the most effective one compared with the others because it introduces in semi-quantitative information a bout the distribution of the source. Therefore the major advantage of radial inversion method is that it could transform semi-quantitative information about the source into a quantitative description about its distribution.The deficiencies of the radial inversion method include: (1) The representation ability of the radial polygon is limited, and it cannot be used to invert potential data produced by sources with too complex shapes; (2) The center of the radial polygon must be located inside the source, which requires that the approximate location of the source should be known in advance.