Python programs to apply regularized derivatives in the magnetic tilt derivative and gradient intensity data processing: A graphical procedure to choose the regularization parameter

Abstract

The Tikhonov regularization parameter is a key parameter controlling the smoothness degree and oscillations of a regularized unknown solution. Usual methods to determine a proper parameter (L-curve or the discrepancy principle, for example) are not readily applicable to the evaluation of regularized derivatives, since this formulation does not make explicit a set of model parameters that are necessary to implement these methods. We develop a procedure for the determination of the regularization parameter based on the graphical construction of a characteristic “staircase” function associated with the $L_2$-norm of the regularized derivatives for a set of trial regularization parameters. This function is independent of model parameters and presents a smooth and monotonic variation. The regularization parameters at the upper step (low values) of the ''staircase'' function provide equivalent results to the non-regularized derivative, the parameters at the lower step (high values) leading to over-smoothed derivatives. For the evaluated data sets, the proper regularization parameter is located in the slope connecting these two flat end-members of the staircase curve, thus balancing noise amplification against the amplitude loss in the transformed fields. A set of Python programs are presented to evaluate the regularization procedure in a well-known synthetic model composed of multiple (bulk and elongated) magnetic sources. This numerical approach also is applied in gridded aeromagnetic data covering high-grade metamorphic terrains of the Anápolis-Itauçu Complex in the Brasília Fold Belt central portion of Tocantins Province, central Brazil, characterized by multiple magnetic lineaments with different directions and intersections which are associated with shear zones, geologic faults, and intrusive bodies. The results obtained from the regularization procedure show efficiency in improving the maps of filtered fields, better tracking the continuity of magnetic lineaments and general geological trends. The results from the application in the Brasília Fold Belt enhance the importance and broader coverage of the Pirineus Zone of High Strain.