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.
So far fluid mechanical Nambu brackets have mainly been given on an intuitive basis. Alternatively an algorithmic construction of such a bracket for the two-dimensional vorticity equation is presented here. Starting from the Lie–Poisson form and its algebraic properties it is shown how the Nambu representation can be explicitly constructed as the continuum limit from the structure preserving Zeitlin discretization.
We address the minimization of the Canham-Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures.With respect to previous contributions, no symmetry of the minimizers is here assumed.Correspondingly, the problem is reformulated and solved in the weaker frame of oriented curvature varifolds. We present a lower semicontinuity result and prove existence of singleand multiphase minimizers under area and enclosed-volume constrains. Additionally, we discuss regularity of minimizers and establish lower and upper diameter bounds.
We investigate the equilibrium configurations of closed planar elastic curves of fixed length, whose stiffness, also known as the bending rigidity, depends on an additional density variable. The underlying variational model relies on the minimisation of a bending energy with respect to shape and density and can be considered as a one-dimensional analogue of the Canham–Helfrich model for heterogeneous biological membranes. We present a generalised Euler–Bernoulli elastica functional featuring a density-dependent stiffness coefficient. In order to treat the inherent nonconvexity of the problem, we introduce an additional length scale in the model by means of a density gradient term. We derive the system of Euler–Lagrange equations and study the bifurcation structure of solutions with respect to the model parameters. Both analytical and numerical results are presented.
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