We present a partition-enhanced least-squares collocation (PE-LSC) which comprises several modifications to the classical LSC method. It is our goal to circumvent various problems of the practical application of LSC. While these investigations are focused on the modeling of the exterior gravity field the elaborated methods can also be used in other applications. One of the main drawbacks and current limitations of LSC is its high computational cost which grows cubically with the number of observation points. A common way to mitigate this problem is to tile the target area into sub-regions and solve each tile individually. This procedure assumes a certain locality of the LSC kernel functions which is generally not given and, therefore, results in fringe effects. To avoid this, it is proposed to localize the LSC kernels such that locality is preserved, and the estimated variances are not notably increased in comparison with the classical LSC method. Using global covariance models involves the calculation of a large number of Legendre polynomials which is usually a time-consuming task. Hence, to accelerate the creation of the covariance matrices, as an intermediate step we pre-calculate the covariance function on a two-dimensional grid of isotropic coordinates. Based on this grid, and under the assumption that the covariances are sufficiently smooth, the final covariance matrices are then obtained by a simple and fast interpolation algorithm. Applying the generalized multi-variate chain rule, also cross-covariance matrices among arbitrary linear spherical harmonic functionals can be obtained by this technique. Together with some further minor alterations these modifications are implemented in the PE-LSC method. The new PE-LSC is tested using selected data sets in Antarctica where altogether more than 800,000 observations are available for processing. In this case, PE-LSC yields a speed-up of computation time by a factor of about 55 (i.e., the computation needs only hours instead of weeks) in comparison with the classical unpartitioned LSC. Likewise, the memory requirement is reduced by a factor of about 360 (i.e., allocating memory in the order of GB instead of TB).

我们提出了一种分区增强的最小二乘搭配（PE-LSC），它包括对经典 LSC 方法的一些修改。我们的目标是规避 LSC 实际应用中的各种问题。虽然这些研究侧重于外部重力场的建模，但精心设计的方法也可用于其他应用。LSC 的主要缺点和当前限制之一是其高计算成本，随着观测点的数量呈立方增长。缓解此问题的常用方法是将目标区域平铺成子区域并单独求解每个平铺。此过程假定 LSC 核函数具有特定的局部性，通常不会给出该局部性，因此会导致边缘效应。为了避免这种情况，建议对 LSC 内核进行定位，从而保留局部性，并且与经典 LSC 方法相比，估计的方差不会显着增加。使用全局协方差模型涉及计算大量勒让德多项式，这通常是一项耗时的任务。因此，为了加速协方差矩阵的创建，作为中间步骤，我们在各向同性坐标的二维网格上预先计算协方差函数。基于这个网格，并在协方差足够平滑的假设下，然后通过简单快速的插值算法获得最终的协方差矩阵。应用广义多元链式法则，也可以通过该技术获得任意线性球谐函数之间的互协方差矩阵。连同一些进一步的小改动，这些修改在 PE-LSC 方法中实现。新的 PE-LSC 在南极洲使用选定的数据集进行测试，其中总共有超过 800,000 个观测值可供处理。在这种情况下，与经典的未分区 LSC 相比，PE-LSC 将计算时间加快了大约 55 倍（即，计算只需要几小时而不是几周）。同样，内存需求减少了大约 360 倍（即按 GB 而不是 TB 的顺序分配内存）。与经典的未分区 LSC 相比，PE-LSC 将计算时间加快了大约 55 倍（即，计算只需要几小时而不是几周）。同样，内存需求减少了大约 360 倍（即按 GB 而不是 TB 的顺序分配内存）。与经典的未分区 LSC 相比，PE-LSC 将计算时间加快了大约 55 倍（即，计算只需要几小时而不是几周）。同样，内存需求减少了大约 360 倍（即按 GB 而不是 TB 的顺序分配内存）。