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 triangulationwhich 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.
Reconstructions of sea level prior to the satellite altimeter era are usually derived from tide gauge records; however most algorithms for this assume that modes of sea level variability are stationary which is not true over several decades. Here we suggest a method that is based on optimized data-dependent triangulations of the network of gauge stations. Data-dependent triangulations are triangulations of point sets that rely not only on 2D point positions but also on additional data (e.g. elevation, anomalies ). In this article, we show how data-dependent triangulations with min-error criteria can be used to reconstruct 2D maps of the sea surface anomaly over a longer time period , assuming that height anomalies are continuously monitored at a sparse set of stations and, in addition, observations of a reference surface is provided over a shorter time period. At the heart of our method is the idea to learn a min-error triangulation based on the available reference data , and to use the learned triangulation subsequently to compute piece-wise linear surface models for epochs in which only observations from monitoring stations are given. We combine our approach of min-error triangulation with k-order Delaunay triangulation to stabilize the triangles geometrically. We show that this approach is advantageous for the reconstruction of the sea surface by combining tide gauge measurements with data of modern satellite altimetry. We show how to learn a min-error triangulation and a min-error k-order Delaunay triangulation using integer linear programming . We confront our reconstructions against the Delaunay triangulation. With real data for the North Sea we show that the min-error triangulation outperforms the Delaunay method significantly for reconstructions back in time up to 18 years, and the k-order Delaunay min-error triangulation even up to 21 years for k=2.
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