基于最优数据相关三角剖分的验潮站动态海面重建
Reconstructing the Dynamic Sea Surface from Tide Gauge Records Using Optimal Data-Dependent Triangulations
摘要：
在卫星高度计时代之前，海平面的重建通常来自验潮记录;然而，大多数算法假设海平面变化的模式是平稳的，这在几十年内是不正确的。在这里，我们提出了一种方法，是基于优化的数据依赖三角测量站的网络。数据相关三角测量是点集的三角测量，不仅依赖于2D点位置，还依赖于附加数据（例如高程、异常）。在这篇文章中，我们将展示如何使用具有最小误差标准的数据依赖三角测量来重建在较长时间内的海面异常的2D地图，假设高度异常在一组稀疏的站被连续监测，此外，在较短的时间内提供参考表面的观测。在我们的方法的核心是学习一个最小误差三角测量的基础上可用的参考数据，并使用学习三角测量随后计算分段线性表面模型的时代，其中只有从监测站的观测。我们将最小误差三角剖分方法与$k$阶德洛内三角剖分相结合，以几何稳定三角形。我们表明，这种方法是有利的海面重建验潮站测量数据与现代卫星测高相结合。我们展示了如何使用整数线性规划学习最小误差三角剖分和最小误差$k$阶德洛内三角剖分。我们用德洛内三角剖分进行重建。与北海的实际数据，我们表明，最小误差三角测量优于Delaunay方法显着重建的时间回到18年，和$k$阶Delaunay最小误差三角测量，甚至高达21年的$k=2$。
英文摘要：
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 $k=2$.
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