It is widely accepted that spatial dependencies have to be acknowledged appropriately in data that are spatially aligned. However, most spatial models still assume that the dependence structure does not vary over space, i.e., it is stationary. While assuming stationarity considerably facilitates estimation, it is often too restrictive when describing atmospheric phenomena such as precipitation. Nonetheless, the applicability of non-stationary models is often hindered, as their use reveals to be cumbersome and improvements over stationary models can be hard to identify. The stochastic partial differential equation (SPDE) approach to spatial modelling allows for flexible specifications of non-stationary models. In particular, given the German orographic diversity, it makes sense to investigate potential non-stationarity in precipitation patterns. Taking such potential non-stationarities into account may, in particular, leads to improved smoothing of the observed precipitation pattern taken on a finite set of measurement stations, and therefore to improved inputs to hydrological models. We suggest an SPDE-based model where the mean and dependence structure are allowed to vary with elevation, as well as a more flexible non-parametric alternative based on multivariate B-Splines. As factors such as wind may cause higher dependence in a given direction, we include anisotropy in the model. Results show that, according to the widely applicable Bayesian information criterion, a non-stationary model provides a better fit to the data. Taking German monthly precipitation as a motivation, we set up a simulation study to explore the ability of the elevation and spline-based models to correctly identify non-stationarity.

众所周知，必须在空间对齐的数据中适当确认空间依赖性。但是，大多数空间模型仍然假设依赖结构在空间上不发生变化，即它是固定的。尽管假设平稳性极大地促进了估算，但在描述诸如降水之类的大气现象时通常过于严格。但是，非平稳模型的适用性通常会受到阻碍，因为它们的使用显示很麻烦，并且很难确定对平稳模型的改进。用于空间建模的随机偏微分方程（SPDE）方法允许灵活地指定非平稳模型。特别是，考虑到德国的地形多样性，研究降水模式中潜在的非平稳性是有意义的。考虑到这种潜在的非平稳性，尤其可以导致改善在有限的一组测量站上观测到的降水模式的平滑度，从而改善对水文模型的输入。我们建议基于SPDE的模型，其中均值和依存结构允许随海拔高度而变化，以及基于多元B样条的更灵活的非参数替代方案。由于诸如风之类的因素可能导致在给定方向上的较高依赖性，因此在模型中包括各向异性。结果表明，根据广泛适用的贝叶斯信息准则，非平稳模型可以更好地拟合数据。以德国每月降水为动力，