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A novel method for tropospheric delay mapping function vertical modeling

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In high-precision space geodetic techniques data processing, the mapping function (MF) is a key factor in mapping the radio waves from the zenith direction down to the signal incoming direction. Existing MF products, either site-wise Vienna Mapping Function (VMF1 and VMF3) or grid-wise VMF1 and VMF3, are only available at the Earth surface. For overhead areas, height correction is always required, which is becoming increasingly important with growing airborne aircraft activity. In this contribution, we introduce a novel method aimed at providing a large number of MFs to the user in a simple and efficient manner, while minimizing the loss of precision. The approach effectively represents the vertical profile of the MFs from the Earth's surface up to altitudes of 14 km. In addition, the new model corrects for height in the assessment using the fifth generation of the European Centre for Medium-Range Weather Forecasts ReAnalysis (ERA5) ray tracing calculations for a global 5° × 5° grid with 54 layers in the vertical direction, a total of 8 azimuths in the plane, and 7 elevation angles, for each day in 2021. Specifically, for both polynomial and exponential model of order 2 and 3, the relative residuals are < 0.3% for the hydrostatic delay MF coefficient \(a_{{\text{h}}}\), and < 1% for the wet delay MF coefficient \(a_{{\text{w}}}\). The precision of the new model on the Earth’s surface is evaluated using site-wise VMF1 and VMF3 GNSS (Global Navigation Satellite System) products from Technische Universität Wien. The root mean square error of slant hydrostatic delay and slant wet delay at a 3° elevation angle is approximately 4–5 cm and 2–5 cm, respectively.

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Data availability

The ERA5 datasets analyzed during the current study are available from!/dataset/reanalysis-era5-pressure-levels?tab=form (Hersbach et al. 2023; accessed on 25-Jun-2022). The VMF1/VMF3 data are available from (Böhm et al. 2006; Landskron and Böhm 2018; accessed on 16-Nov-2022). The model parameters proposed in this study and the output results are hosted within the GitHub repository (


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The authors acknowledge that the open-source software RADIATE is released by TUW ( This work was supported by the National Natural Science Foundation of China (No. 11673050), the Key Program of Special Development Funds of Zhangjiang National Innovation Demonstration Zone (Grant No. ZJ2018-ZD-009), the National Key R&D Program of China (No. 2018YFB0504300) and the Key R&D Program of Guangdong Province (No. 2018B030325001).

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All authors contributed to the study conception and design. JC and JW proposed the idea; JD developed the software, designed the experiment and wrote the manuscript; JC, JW and YZ contributed to discussion of the idea and helped with writing. All authors reviewed the manuscript.

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Correspondence to Junping Chen.

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Ding, J., Chen, J., Wang, J. et al. A novel method for tropospheric delay mapping function vertical modeling. J Geod 98, 37 (2024).

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