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Reconstructions of sea level prior to the satellite altimeter era are usuallyderived from tide gauge records; however most algorithms for this assume thatmodes of sea level variability are stationary which is not true over severaldecades. Here we suggest a method that is based on optimized data-dependenttriangulations of the network of gauge stations. Data-dependent triangulationsare triangulations of point sets that rely not only on 2D point positions butalso on additional data (e.g. elevation, anomalies). In this article, we showhow data-dependent triangulations with min-error criteria can be used toreconstruct 2D maps of the sea surface anomaly over a longer time period,assuming that height anomalies are continuously monitored at a sparse set ofstations and, in addition, observations of a reference surface is provided overa shorter time period. At the heart of our method is the idea to learn amin-error triangulation based on the available reference data, and to use thelearned triangulation subsequently to compute piece-wise linear surface modelsfor epochs in which only observations from monitoring stations are given. Wecombine our approach of min-error triangulation with $k$-order Delaunaytriangulation to stabilize the triangles geometrically. We show that thisapproach is advantageous for the reconstruction of the sea surface by combiningtide gauge measurements with data of modern satellite altimetry. We show how tolearn a min-error triangulation and a min-error $k$-order Delaunaytriangulation using integer linear programming. We confront our reconstructionsagainst the Delaunay triangulation. With real data for the North Sea we showthat the min-error triangulation outperforms the Delaunay method significantlyfor reconstructions back in time up to 18 years, and the $k$-order Delaunaymin-error triangulation even up to 21 years for $k=2$.

Reconstructing the Dynamic Sea Surface from Tide Gauge Records Using Optimal Data-Dependent Triangulations