Regularized Derivatives - Revisited

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The methods used for differentiation of potential field data play an important role in interpretation, as various derivatives are included in many of the transformations often used when interpreting such data (e.g.: Analytical Signal, Tilt Angles, Total Gradient and others). However, the evaluation of horizontal and vertical derivatives is an unstable process. The possible (partial) answer to this instability is the Tikhonov regularization based on incorporating an additional property representing maximum smoothness, which is typically obtained by cascading low-pass filtering - managed by means of a regularization parameter. Here we present and compare several ways to evaluate regularized differentiation operators and compare their properties. Among them, a newly introduced form (named as General Form) plays a very important role. As is typical for regularization algorithms, the crucial step is the setting of a quasi-optimal value of the regularization parameter, which gives the best possible solution. Several methods of its estimation (focused on the utilization of Lp norms) are presented and the resulting utility became the keystone in valuation of the presented regularized operators.