Solution of the Linear Inverse Problem in Magnetic Interpretation with Application to Oceanic Magnetic Anomalies

The problem of interpreting a magnetic anomaly usually reduces to either (1) determining the distribution of magnetization given the shape of the body and the direction of magnetization, or (2) determining the shape of one interface, given the magnetization and other interfaces. These involve solution of an integral equation, linear in case (1) and non-linear in case (2). The paper gives two solutions of the linear problem applicable either to gravity or magnetic interpretation, which may also be used as a starting point for the iterative solution of the non-linear problem. Method (1): The Fourier convolution theorem has been used to derive a two-dimensional magnetic version of the ‘equivalent layer’ theorem, which is the simplest case of the inverse problem. This enables a given magnetic anomaly to be replaced by a coating of magnetic moment per unit area of specified direction over the horizontal plane of the measured anomalies. The ‘equivalent layer’ can be continued downwards by existing methods. A computer program applicable to the method is available at Durham. Method (2): A more versatile approach is to approximate the linear integral equation by a summation, resulting in a set of linear equations. If the number of observations are equal to the unknown magnetization parameters the equations are solved directly; if there are more observations than unknowns, least squares is used. In either case, the matrix schemes available on most computers provide the main tool for the method. Method (2) has been applied to the interpretation of oceanic magnetic anomalies in terms of two-dimensional rectangular blocks confined between two specified depths, the magnetization varying laterally from block to block. The matrix inversion needs to be done once only for specified depths, block width and direction of magnetization. It provides a weighting function for the observed anomaly and repeated application of the convolution gives the underlying distribution of magnetization. Applications of the method to the Juan de Fuca Ridge, suggests that unacceptably high contrasts in magnetization are required if the main source is below the oceanic layer 2, supporting the view that rocks in layer 2 cause a substantial part of the anomalies.