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.

卫星高度计时代之前的海平面重建通常来自潮汐测量记录；然而，大多数算法都假设海平面变化的模式是静止的，这在几十年中是不正确的。在这里，我们建议一种方法，该方法基于测量站网络的优化数据相关三角测量。数据相关三角剖分是点集的三角剖分，不仅依赖于 2D 点位置，还依赖于附加数据（此处：海面异常）。特别是，已建议最小误差标准来构建近似给定表面的三角剖分。在本文中，我们展示了如何使用具有最小误差标准的数据相关三角测量来重建较长时间段内海面异常的二维地图，假设在一组稀疏的台站连续监测异常，此外，在更短的时间内提供对控制面的观测。我们方法的核心是基于可用的控制数据学习最小误差三角测量，然后使用学习的三角测量计算分段线性表面模型，其中只有来自监测站的观测数据给。此外，我们将我们的最小误差三角测量方法与 并随后使用学习的三角剖分计算分段线性表面模型，用于仅给出来自监测站的观测值的历元。此外，我们将我们的最小误差三角测量方法与 并随后使用学习的三角剖分计算分段线性表面模型，用于仅给出来自监测站的观测值的历元。此外，我们将我们的最小误差三角测量方法与$克$-order Delaunay 三角剖分以几何方式稳定三角形。我们表明，这种方法通过将潮汐计测量值（空间稀疏但覆盖时间长）与现代卫星测高数据（具有高空间分辨率但仅涵盖过去几十年）。我们展示了如何学习最小误差三角剖分和最小误差$克$使用基于整数线性规划的精确算法的 -order Delaunay 三角剖分。我们将我们的重建与之前提出的用于海面建模的 Delaunay 三角剖分进行比较，并发现了卓越的质量。使用北海的真实数据，我们表明最小误差三角剖分在 18 年以前的重建中大大优于 Delaunay 方法，并且$克$-order Delaunay 最小误差三角剖分甚至长达 21 年 $克=2$. 运行时间不到一秒，我们的方法将适用于范围远大于北海的区域。