Copula-based modeling of dependence structure in geodesy and GNSS applications: case study for zenith tropospheric delay
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ORIGINAL ARTICLE
Copula‑based modeling of dependence structure in geodesy and GNSS applications: case study for zenith tropospheric delay in complex terrain Roya Mousavian1,2 · Christof Lorenz2 · Masoud Mashhadi Hossainali1 · Benjamin Fersch2 · Harald Kunstmann2,3 Received: 9 October 2019 / Accepted: 9 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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. Keywords Zenith tropospheric delay · Copulas · GNSS · Dissimilarity measures · Atmospheric modeling · Correlation
Introduction Knowing and modeling the dependence structure of different variables, including the statistical distributions of and relationships between geophysical quantities, is mandatory for a wide range of applications in geosciences. For example, * Roya Mousavian [email protected]; [email protected] 1
Faculty of Geodesy and Geomatics Engineering, K. N. Toosi University of Technology, No. 1346, ValiAsr Street, Mirdamad Cross, Tehran, Iran
2
Institute of Meteorology and Climate Research (IMK‑IFU), Campus Alpin, Karlsruhe Institute of Technology, Kreuzeckbahnstr. 19, 82467 Garmisch‑Partenkirchen, Germany
3
Institute of Geography, University of Augsburg, Alter Postweg 118, 86159 Augsburg, Germany
the dependence between different observations (Tiberiu
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