Reconstructing The Time-Variable Sea Surface From Tide Gauge Records Using Optimal Data-Dependent Triangulations

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 (here: sea surface anomalies). In particular, min-error criteria have been suggested to construct triangulations that approximate a given surface. 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 anomalies are continuously monitored at a sparse set of stations and, in addition, observations of a control 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 control data that is available, 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. Moreover, we combine our approach of min-error triangulation with k-order Delaunay triangulation to stabilize the triangles geometrically. We show that this approach is in particular advantageous for the reconstruction of the sea surface by combining tide gauge measurements (which are sparse in space but cover a long period back in time) with data of modern satellite altimetry (which have a high spatial resolution but cover only the last decades). We show how to learn a min-error triangulation and a min-error k-order Delaunay triangulation using an exact algorithm based on integer linear programming. We confront our reconstructions against the Delaunay triangulation which had been proposed earlier for sea-surface modeling and find superior quality. With real data for the North Sea we show that the min-error triangulation outperforms the Delaunay method substantially 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. With a running time of less than one second our approach would be applicable to areas with far greater extent than the North Sea.