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

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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.

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Acknowledgements

This study was funded by scholarships from K. N. Toosi University of Technology (Tehran, Iran) and the Karlsruhe Institute of Technology—Institute of Meteorology and Climate Research, Atmospheric Environmental Research (Garmisch-Partenkirchen, Germany). It was enabled additionally by funds from the German Research Foundation (DFG-ATMOWATER, KU 2090/10) and the German Ministry of Science and Education (BMBF)-funded GROW-SaWaM project. We would further like to thank the German Research Center for Geosciences (GFZ) for providing the GNSS data that were used for this work.

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Authors and Affiliations

  1. Faculty of Geodesy and Geomatics Engineering, K. N. Toosi University of Technology, No. 1346, ValiAsr Street, Mirdamad Cross, Tehran, Iran

    Roya Mousavian & Masoud Mashhadi Hossainali

  2. Institute of Meteorology and Climate Research (IMK-IFU), Campus Alpin, Karlsruhe Institute of Technology, Kreuzeckbahnstr. 19, 82467, Garmisch-Partenkirchen, Germany

    Roya Mousavian, Christof Lorenz, Benjamin Fersch & Harald Kunstmann

  3. Institute of Geography, University of Augsburg, Alter Postweg 118, 86159, Augsburg, Germany

    Harald Kunstmann

Authors
  1. Roya Mousavian
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  2. Christof Lorenz
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  3. Masoud Mashhadi Hossainali
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  4. Benjamin Fersch
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  5. Harald Kunstmann
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Correspondence to Roya Mousavian.

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Appendix: The estimation of zenith tropospheric delay

Appendix: The estimation of zenith tropospheric delay

Zenith tropospheric delays (ZTDs) can be expressed in two components:

$${\text{ZTD}} = {\text{ZHD}} + {\text{ZWD}}$$
(29)

where ZHD and ZWD are dry or zenith hydrostatic and wet delays considered from the surface to the top level in the troposphere, respectively. They are computed by (Ghoddousi-Fard 2009)

$${\text{ZHD}} = 10^{ - 6} \sum\limits_{{i = {\text{surface}}}}^{{\text{last level}}} {\left( {k_{1} R_{{\text{d}}} \left( {\frac{{p_{i} - e_{i} }}{{R_{{\text{d}}} T_{i} }} + \frac{{e_{i} }}{{R_{{\text{w}}} T_{i} }}} \right)} \right){\text{d}}h_{i} }$$
(30)
$${\text{ZWD}} = 10^{ - 6} \sum\limits_{{i = {\text{surface}}}}^{{\text{Last level}}} {\left( {k_{2}^{\prime } \frac{{e_{i} }}{{T_{i} }} + k_{3} \frac{{e_{i} }}{{T_{i}^{2} }}} \right){\text{d}}h_{i} }$$
(31)

In these equations, \(T_{i}\) is the temperature in Kelvin, \(p_{i}\) is the total pressure in hPa, and Rd = 287 J/kg k and Rw = 461 J/kg k are gas constants for dry air and water vapor, respectively. Moreover, the index i refers to the level and \(k_{1}\), \(k^{\prime}_{2}\) and \(k_{3}\) are empirically determined constants. Various researchers have proposed different values for these parameters. For example, according to (Bevis et al. 1994):

$$\begin{aligned} k_{1} & = 77.60 \pm 0.05\,{\text{Kh}}\,{\text{Pa}}^{ - 1} \\ k_{2}^{\prime } & = 22.2 \pm 2.2\,{\text{K}}\,{\text{h}}\,{\text{Pa}}^{ - 1} \\ k_{3} & = 3.739 \pm 0.012 \times 10^{5} \,{\text{K}}^{2} \,{\text{h}}\,{\text{Pa}}^{ - 1} \\ \end{aligned}$$
(32)

Finally, \(e_{i}\) is the water vapor pressure in hPa and is computed using the following formula:

$$e_{i} = \frac{{q_{i} p_{i} }}{{\frac{{M_{{\text{w}}} }}{{M_{{\text{d}}} }} + \left( {1 - \frac{{M_{{\text{w}}} }}{{M_{{\text{d}}} }}} \right)q_{i} }}$$
(33)

where qi is specific humidity in kg/kg and Mw and Md are the molar weight of wet and dry air, respectively. These parameters, together with the gas constants, fulfill in the following relation:

$$\frac{{M_{{\text{w}}} }}{{M_{{\text{d}}} }} = \frac{{R_{{\text{w}}} }}{{R_{{\text{d}}} }} = 0.62197 \simeq 0.622$$
(34)

Since there is no bending effect in the zenith direction, the distance traveled by the ray in each layer can be considered the same as user-defined values (integration step size). Hence, ZTD can be calculated simply by the following summation (Ghoddousi-Fard 2009):

$${\text{ZTD}} = 10^{ - 6} \sum\limits_{{i = {\text{surface}}}}^{{\text{last level}}} {\left( {\left( {k_{1} R_{{\text{d}}} \left( {\frac{{p_{i} - e_{i} }}{{R_{{\text{d}}} T_{i} }} + \frac{{e_{i} }}{{R_{{\text{w}}} T_{i} }}} \right) + \left( {k_{2}^{\prime } \frac{{e_{i} }}{{T_{i} }} + k_{3} \frac{{e_{i} }}{{T_{i}^{2} }}} \right)} \right) \times {\text{d}}h_{i} } \right)}$$
(35)

where dhi is the integration step size at level i, and other parameters have already been introduced in (30) to (34). Using (35) one calculates the total amount of ZTD through the troposphere in each position.

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Mousavian, R., Lorenz, C., Mashhadi Hossainali, M. et al. Copula-based modeling of dependence structure in geodesy and GNSS applications: case study for zenith tropospheric delay in complex terrain. GPS Solut 25, 12 (2021). https://doi.org/10.1007/s10291-020-01044-4

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  • DOI: https://doi.org/10.1007/s10291-020-01044-4

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