Generalized spline-approximation problem formulation for spatial data modeling in geosciences

Abstract

Abstract The discussed spline approximation in spatial data modeling for geosciences implies formulation of the variational problem in terms of functional minimization and allows simultaneous inversion for several surfaces. This modeling employs the following basic elements: stabilizers to define the common properties of unknown surfaces; differential operators to describe the unknown surfaces and their relation with the known fields; data specified locally at test points; partial differential equations similar to equations of mathematical physics for the properties of the surfaces of interest; elements of regression analysis, with the regression coefficients being calculated while solving the principal modeling problem; arbitrary amounts of direct or indirect information which is incorporated additionally into the functional on the basis of approximate conditions using weight coefficients as control parameters. The suggested generalized formulation includes the concepts of global and local equations, and strict and nonstrict relationships. This formulation, realized in the GST software, may apply to many surface modeling problems to be solved using second-order partial differential equations, with multiple criteria optimization of results and with the use of different auxiliary datasets.