# Smooth digital terrain modelling in irregular domain using finite element thin plate splines and adaptive refinement

## TL;DR

This paper evaluates the finite element thin plate spline method for smooth digital terrain modeling in irregular domains, demonstrating its efficiency and accuracy improvements through adaptive refinement and boundary condition approximation.

## Contribution

It introduces an adaptive mesh refinement approach for TPSFEM in irregular domains, enhancing computational efficiency and accuracy for large terrain data sets.

## Key findings

- TPSFEM performs well in irregular domains for terrain modeling.
- Adaptive refinement significantly improves computational efficiency.
- TPSFEM is competitive with other methods in accuracy and cost.

## Abstract

Digital terrain models (DTMs) are created using elevation data collected in geological surveys using varied sampling techniques like airborne lidar and depth soundings. This often leads to large data sets with different distribution patterns, which may require smooth data approximations in irregular domains with complex boundaries. The thin plate spline (TPS) interpolates scattered data and produces visually pleasing surfaces, but it is too computationally expensive for large data sizes. The finite element thin plate spline (TPSFEM) possesses smoothing properties similar to those of the TPS and interpolates large data sets efficiently. This article investigates the performance of the TPSFEM and adaptive mesh refinement in irregular domains. Boundary conditions are critical for the accuracy of the solution in domains with arbitrary-shaped boundaries and are approximated using the TPS with a subset of sampled points. Numerical experiments are conducted on aerial, terrestrial and bathymetric surveys. It is shown that the TPSFEM works well in square and irregular domains for modelling terrain surfaces and adaptive refinement significantly improves its efficiency. A comparison of the TPSFEM, TPS and compactly supported radial basis functions indicates its competitiveness in terms of accuracy and costs.