Modeling a Smooth Surface by a Constrained Biharmonic Equation with Application in Soil Science

TL;DR

This paper introduces a regularized biharmonic equation-based method for modeling smooth surfaces from limited empirical data, with applications in soil science and geological mapping.

Contribution

It develops a novel approach for surface modeling using constrained biharmonic equations and regularization to handle ill-posed boundary data problems.

Findings

Effective surface modeling with limited data

Application to soil thickness and geological maps

Regularization improves boundary data stability

Abstract

This paper presents a method for mathematical modelling of surfaces conditioned on empirical data. It is based on solving a discrete biharmonic equation over a domain with given inner point and inner curve data. The inner curve data is used to model boundary values while the inner point data is used for modeling a load vector with the goal to generate a smooth surface. The construction of boundary data is an ill-posed problem, for which a special regularization approach is suggested. The method is designed for surface construction problems with a very limited amount of measured data. In the paper we apply the method by using empirical data of soil thickness and geological maps indicating exposed bedrock regions.