Regularized derivatives of potential fields and their role in semi‐automated interpretation methods

Abstract

A B S T R A C TEvaluation of higher derivatives (gradients) of potential fields plays an important role in geophysical interpretation (qualitative and/or quantitative), as has been demonstrated in many approaches and methods. On the other hand, numerical evaluation of higher derivatives is an unstable process -it has the tendency to enlarge the noise content in the original data (to degrade the signal-to-noise ratio). One way to stabilize higher derivative evaluation is the utilization of the Tikhonov regularization.In the submitted contribution we present the derivation of the regularized derivative filter in the Fourier domain as a minimization task by means of using the classical calculus of variations. A very important part of the presented approach is the selection of the optimum regularization parameter -we are using the analysis of the C-norm function (constructed from the difference between two adjacent solutions, obtained for different values of regularization parameter). We show the influence of regularized derivatives on the properties of the classical 3D Euler deconvolution algorithm and apply it to high-sensitivity magnetometry data obtained from an unexploded ordnance detection survey. The solution obtained with regularized derivatives gives better focused depth-estimates, which are closer to the real position of sources (verified by excavation of unexploded projectiles). I N T R O D U C T I O NSemi-automated interpretation methods in applied gravimetry and magnetometry are based on estimation of source parameters directly from the measured (processed) and/or transformed potential fields. These methods are built on introduction of a priori information on the properties of the desired solution, mainly at the mathematical level. In the majority of cases, the a priori information is based on the recognition of a predefined source type response in the interpreted data. This approach does not fully solve the ambiguity of the inverse problem in potential field's interpretation. By means of this approach, we introduce into the solutions the so-called model error (Dmitriev et al. 1990) -this is connected with the problem of description of the complex real situations by sim- * E-mail: pasteka@fns.uniba.sk ple models. The problem of the instability of inverse problem solutions is manifested in these kinds of methods by a great sensitivity to the noise and errors in the interpreted fields.A large group of these interpretation methods use higher gradients of the interpreted field (e.g., Werner and Euler deconvolution, methods based on analytical signal evaluation, normalized derivative methods and many other approaches, e.g.

Three‐dimensional density modeling of the EGM2008 gravity field over the Mount Paekdu volcanic area

“…Euler source points were calculated from the isostatic residual field with the in-house program REGDER. Pašteka et al [2009] proposed using SI = 2 for calculations of the center of mass source points. The mass points located between 4 and 11 km depth (red-, blue-, and green-filled circles in Figure 10) in the volcanic zone (grey hatched area in Figure 10a) are clustered mainly around Mount Paekdu.…”

“…The Euler deconvolution technique has been applied to potential field data for more than two decades [e.g., Thomson, 1982;Reid et al, 1990;Pašteka et al, 2009] and assists the interpretation of potential fields in areas of simple density distributions (e.g., dike-like or isolated structures). The degree of homogeneity N in the Euler equation can be interpreted using the structural index (SI), which is a measure of the rate of change of a field with distance [Pašteka et al, 2009], including the center of mass point (SI = 2), the center of a dike-like feature (SI = 1), and the position of the top of a dike-like feature or tectonic boundary (SI = 0). Euler source points were calculated from the isostatic residual field with the in-house program REGDER.…”

3D gravity interpretation of the pre-Tertiary basement in the intramontane depressions of the Western Carpathians: a case study from the Turiec Basin

“…For this reason, we evaluated the map of total horizontal gradients (Fig. 9) using the regularized (smoothed) derivatives in the Fourier domain, selecting the optimum regularization (low-pass filter) parameter by means of the C-norm functions analysis (Pašteka et al 2009(Pašteka et al , 2012. In the TB (Figs.…”

High resolution Slovak Bouguer gravity anomaly map and its enhanced derivative transformations: new possibilities for interpretation of anomalous gravity fields

“…We took the concept of the calculation of the regularized derivatives with the use of a choice of optimum parameter on the basis of the analysis of the C-norm (Pašteka et al, 2009), as the standard determination of the numerical derivatives greatly emphasized the noise in the original grid of the CBA. The calculation of transformed fields was implemented in the medium of the program REGTILT (Pašteka et al, 2015), which uses the concept of regularized derivatives with the use of other L p -norms (Pašteka et al, 2012).…”