Reconstructing the Dynamic 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 (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 $k=2$.