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We apply state-of-the-art computational geometry methods to the problem ofreconstructing a time-varying sea surface from tide gauge records. Our workbuilds on a recent article by Nitzke et al.~(Computers \& Geosciences,157:104920, 2021) who have suggested to learn a triangulation $D$ of a givenset of tide gauge stations. The objective is to minimize the misfit of thepiecewise linear surface induced by $D$ to a reference surface that has beenacquired with satellite altimetry. The authors restricted their search tok-order Delaunay ($k$-OD) triangulations and used an integer linear program inorder to solve the resulting optimization problem.  In geometric terms, the input to our problem consists of two sets of pointsin $\mathbb{R}^2$ with elevations: a set $\mathcal{S}$ that is to betriangulated, and a set $\mathcal{R}$ of reference points. Intuitively, wedefine the error of a triangulation as the average vertical distance of a pointin $\mathcal{R}$ to the triangulated surface that is obtained by interpolatingelevations of $\mathcal{S}$ linearly in each triangle. Our goal is to find thetriangulation of $\mathcal{S}$ that has minimum error with respect to$\mathcal{R}$.  In our work, we prove that the minimum-error triangulation problem is NP-hardand cannot be approximated within any multiplicative factor in polynomial timeunless $P=NP$. At the same time we show that the problem instances that occurin our application (considering sea level data from several hundreds of tidegauge stations worldwide) can be solved relatively fast using dynamicprogramming when restricted to $k$-OD triangulations for $k\le 7$. Inparticular, instances for which the number of connected components of theso-called $k$-OD fixed-edge graph is small can be solved within few seconds.

Minimum-Error Triangulations for Sea Surface Reconstruction