Modeling and understanding the statistical relationships between geophysical quantities is a crucial prerequisite for many geodetic applications. While these relationships can depend on multiple variables and their interactions, commonly used scalar methods like the (cross) correlation are only able to describe linear dependencies. However, particularly in regions with complex terrain, the statistical relationships between variables can be highly nonlinear and spatially heterogeneous. Therefore, we introduce Copula-based approaches for modeling and analyzing the full dependence structure. We give an introduction to Copula theory, including five of the most widely used models, namely the Frank, Clayton, Ali-Mikhail-Haq, Gumbel and Gaussian Copula, and use this approach for analyzing zenith tropospheric delays (ZTDs). We apply modeled ZTDs from the Weather and Research Forecasting (WRF) model and estimated ZTDs through the processing of Global Navigation Satellite System (GNSS) data and evaluate the pixel-wise dependence structures of ZTDs over a study area with complex terrain in Central Europe. The results show asymmetry and nonlinearity in the statistical relationships, which justifies the application of Copula-based approaches compared to, e.g., scalar measures. We apply a Copula-based correction for generating GNSS-like ZTDs from purely WRF-derived estimates. Particularly the corrected time series in the alpine regions show improved Nash–Sutcliffe efficiency values when compared against GNSS-based ZTDs. The proposed approach is therefore highly suitable for analyzing statistical relationships and correcting model-based quantities, especially in complex terrain, and when the statistical relationships of the analyzed variables are unknown.

建模和理解地球物理量之间的统计关系是许多大地测量应用的关键先决条件。尽管这些关系可能取决于多个变量及其相互作用，但常用的标量方法（如（互）相关）只能描述线性相关性。但是，特别是在地形复杂的区域中，变量之间的统计关系可能是高度非线性的并且在空间上是异质的。因此，我们介绍了基于Copula的方法来对完整的依赖结构进行建模和分析。我们介绍了Copula理论，包括五个最广泛使用的模型，即Frank，Clayton，Ali-Mikhail-Haq，Gumbel和Gaussian Copula，并使用此方法来分析天顶对流层延迟（ZTD）。我们应用了来自天气和研究预报（WRF）模型的建模ZTD，并通过处理全球导航卫星系统（GNSS）数据估算了ZTD，并评估了中欧地形复杂的研究区域内ZTD的像素依赖性结构。结果表明统计关系中的不对称性和非线性，这证明了与例如标量度量相比，基于Copula的方法的应用是正确的。我们应用基于Copula的校正方法，从纯WRF得出的估算值中生成类似GNSS的ZTD。与基于GNSS的ZTD相比，特别是在高山地区经过校正的时间序列显示出更高的Nash–Sutcliffe效率值。因此，建议的方法非常适合分析统计关系并校正基于模型的数量，