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

TL;DR

Problem addressed: reconstruct smooth surfaces conditioned on sparse measurements using a discrete biharmonic model $\Delta^2 u = q$ on a domain $\Omega$ with unknown boundary data and interior constraints. Main approach: model the load as $q = \sum_j p_j q_j$ with Gaussian kernels centered at data points, determine weights by constrained least squares and regularize via truncated SVD, and recover missing boundary conditions from inner curves through a discrete inverse problem solved with a Ritz–VEM framework. Key contributions: integrated PDE-based surface construction methodology for irregular domains with few measurements, including $L$-curve based regularization and demonstration on soil thickness and bedrock topography with an open software workflow. Significance: enables robust, data-efficient terrain modeling in geoscience and provides a generalizable framework for similar problems.

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.

Figures (10)

• Figure 1: A part of the Norwegian region Øvre Eiker with the subregion marked for which the soil thickness is determined, norgeskarta region: EU89 UTM33 (6639960 N, 212750 E). The position of the wells (point data) is indicated by blue circles and the exposed bedrock $\Omega_{\mathrm{e}}$ is indicated by slightly grayed areas.
• Figure 2: Two alternatives to define the underlying domain together with an automatically generated grid. Left: An irregular domain defined by a rectangular map segment where regions with exposed bedrock are excluded. The location of wells is marked by magenta circles and blue stars. Right: The same rectangular map segment including regions with exposed bedrock (shaded).
• Figure 3: Distribution of the point data. Exact soil thickness (equality constraints) $\color{blue} \blacklozenge$. Lower bound of soil thickness (inequality constraints) $\color{magenta} \bullet$.
• Figure 4: Left: Surface produced from the well data. Right: Measured and simulated point data, seen from the north-south axis and seen from the east-west axis: Given soil thickness $\color{blue} \blacklozenge$, well depths (lower bound of soil thickness) $\color{magenta} \bullet$, simulated soil thickness $\circ$.
• Figure 5: Domain $\Omega$ with an inner curve $\Gamma_{e_1}$, forming a square of side length $2r$, discretized with 32 points. The background grid is $n_x \times n_x$ with $n_x = 8$.