Reconstructing the Dynamic Sea Surface Using Optimal Data-Dependent Triangulations

Data dependent triangulations are triangulations of point sets in the plane that are computed under consideration not only of the points' $x$- and $y$-coordinates but also of additional data (e.g., elevation). In particular, min-error criteria have been suggested to compute triangulations that approximate a given surface. In this article, we show how data dependent triangulations with min-error criteria can be used to reconstruct a dynamic surface over a longer time period if height values are continuously monitored at a sparse set of stations and, additionally for few epochs or a single one an accurate reference surface is given. The basic idea is to learn an height-optimal triangulation based on the available reference data and to use the learned triangulation to compute piece-wise linear surface models for epochs in which only the observations of the monitoring stations are given. In addition, we combine our approach of height-optimal triangulation with $k$-order Delaunay triangulation to stabilize the triangles geometrically. We consider these approaches particularly useful for reconstructing the historical evolution of a sea surface by combining tide-gauge measurements with data of modern satellite altimetry. We show how to learn a height-optimal triangulation and a $k$-order height-optimal triangulation using an exact algorithm based on integer linear programming and evaluate our approaches both with respect to its running times and the quality of the reconstructions in contrast to a solution by a Delaunay triangulation which has earlier been used for sea-surface modeling. For existing data of the North Sea we show that we can better reconstruct the dynamic sea surface for about 156 months back in time using the height-optimal triangulation than the Delaunay triangulation and even for about 180 months using the $k$-order height-optimal triangulation for $k=2$.