Applications of Hilbert transform and Newton's root finding algorithm in delineating buried fracture from magnetic anomaly data

Abstract

I propose a simple and sufficiently robust data‐based semi‐automated interpretation of a total magnetic intensity profile to delineate an isolated fracture or a fractured zone embedded within a homogeneous host medium. The interpretation technique, which is based on the premise of induced magnetization, uses a model of a thin, two‐dimensional dipping sheet embedded in a homogeneous half‐space. The proposed quantitative interpretation uses a set of closed form formulae in delineating model parameters; in those the Hilbert transform, the first‐ and second‐order derivatives of the local amplitude of the zero‐order analytical signal are the essential components. The locations of zero crossings of the first‐ and second‐order derivatives of analytical signal and the Hilbert transform remain the essential keys in estimating structural parameters of the model. The magnetic susceptibility contrast is delineated from the peak amplitude of the analytical signal. An appropriately designed Savitzky–Golay derivative filter and Hilbert–Noda transformation matrix are used as robust and efficient data‐processing tools. An additional tool, Newton's root finding algorithm, is proposed for locating the zero crossings. The reconstructed data fit is validated using a statistical appraisal, and the estimated model parameters are defined with an uncertainty measure using the 95% confidence limit. The proposed technique has been applied to synthetic data generated from a model depicting a realistic example, and also on field data from two examples, one of which closely resembles the example of synthetic data and the other as a benchmark example. Numerical experiments validate the applicability of the method.