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.

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Transformation based on downward continuation of potential fields is an important tool in their interpretation – depths of shallowest important sources can be determined by means of stable downward continuation algorithms. We analyse here selected properties of one from these algorithms (based on Tikhonov’s regularization approach) from the scope of two most important discretization parameters – dimensions of the areal coverage of the interpreted field and the sampling interval size. Estimation of the source depth is based on the analysis of computed LP-norms for various continuation depths. A typical local minimum of these norms disappears at the source depth. We show on several synthetic bodies (sphere, horizontal cylinder, vertical rod) and also real-world data-sets (results from a magnetic survey for unexploded ordnance detection) that there is a need for relatively large surroundings around the interpreted anomalies. Beside of this also the sampling step plays its important role – grids with finer sampling steps give better interpretation results, when using this downward continuation method. From this point of view, this method is more suitable for the interpretation of objects in near surface and mining geophysics (anomalies from cavities, unexploded ordnance objects and ore bodies). Anomalies should be well developed and separable, and densely sampled. When this is not valid, several algorithms of interpolation and extrapolation (grid padding methods) can improve the interpretation properties of studied downward continuation method.

Solutions to the direct problem in gravimetric interpretation are well-known for wide class of source bodies with constant density contrast. On the other hand, sources with non-uniform density can lead to relatively complicated formalisms. This is probably why analytical solutions for this type of sources are rather rare although utilization of these bodies can sometimes be very effective in gravity modeling. I demonstrate an analytical solution to that problem for a spherical shell with radial polynomial density distribution, and illustrate this result when applied to a special case of 5th degree polynomial. As a practical example, attraction of the normal atmosphere is calculated.

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