Copula-based modeling of dependence structure in geodesy and GNSS applications: case study for zenith tropospheric delay in complex terrain

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.