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Radiocommunication Study Groups805180317500Received:14 April 2023Reference: Rec. ITU-R P.835-6 ITU-R P.676-13 Document USWP 3J-0314 April 2023United States of AmericaDiscussion document regarding horizontal interpolation methods for gridded atmospheric data1. IntroductionAtmospheric data derived from numerical weather models has the general form of gridded, discrete profiles of meteorological parameters of pressure, temperature, and water vapour density and height. Horizontal interpolation is required for application at geographic locations other than the grid points. Atmospheric profiles can be determined from a bilinear interpolation of the meteorological parameters from the four surrounding grid points, at the specified levels of the data. Propagation effects can then be calculated for the location. Alternatively, integrated quantities such as the path attenuation and the integrated water vapour can be determined by horizontal interpolation of the integrated quantity calculated at each grid point. The issue, then, is which method should be recommended for use. The pertinent question in this regard is which method best represents the data itself. It has been suggested to compare interpolation methods with radiosonde data. While RAOBs are useful as a metric to judge the accuracy of the numerical weather data, they have no particular relation with respect to the data and do not represent the truth. Consequently, such an analysis is irrelevant to answering the question.This document provides an analysis that addresses the issue of which horizontal interpolation method should be recommended. The main problem is to establish the “truth” regarding the data. Here, two approaches are used. The first approach uses a simple gridded data structure that allows for user defined atmospheric profiles. The second approach examines ERA5 data using the atmospheric data at a grid point as the truth. Pressure level data was used in the ERA5 calculations.2. Analysis2.1. User defined gridded atmospheric dataA simplified gridded data structure is shown in Fig. 1. This model has two grid points located at 0 and 1. The ground level of grid 1 is 0 km and the ground level is at H km. The location of interest is at a distance X from grid point 0. Here X is a unit-less parameter. The atmospheric profiles at the grid points are user defined and specified in the analysis.Three cases were analysed. Atmospheric profiles were taken from the ITU-R P.835-6 Annex 1 reference atmospheres. The parameters used are summarized in Table 1. Figures 2-4 compare the two horizontal interpolation methods for the calculation of slant path attenuation. Table 1. Summary of parameters for Figs 2-4CaseFigureFrequency(GHz)Atmosphere at Grid Point 0Atmosphere at Grid Point 1H (km)Elevation Angle (°)1220ITU Stnd. Ref. Atm.Summer Mid-Latitude0902320ITU Stnd. Ref. Atm.Summer Mid-Latitude1903420ITU Stnd. Ref. Atm.ITU Stnd. Ref. Atm.190842010bottomFigure SEQ Figure \* ARABIC 1. Atmospheric data used for analysis00Figure SEQ Figure \* ARABIC 1. Atmospheric data used for analysis4229105662295Figure 4 Comparison of interpolation methods for case 3. See Table 1.00Figure 4 Comparison of interpolation methods for case 3. See Table 1.7181852434590Figure SEQ Figure \* ARABIC 3 Comparison of interpolation methods for case 2. See Table 1.00Figure SEQ Figure \* ARABIC 3 Comparison of interpolation methods for case 2. See Table 1.499110-509905Figure 2 Comparison of horizontal interpolation methods for case 1. See Table 1.00Figure 2 Comparison of horizontal interpolation methods for case 1. See Table 1.2.2. ERA5 data analysiscenter2533650Table SEQ Table \* ARABIC 2 Integrated water vapourLocationLat(deg)Long(deg)H(km)Meteorological(mm)Integrated(mm)Rome, NY43.2-75.47016.10816.10926.7976.771Milano/Linate45.439.28019.25319.25727.3957.4200Table SEQ Table \* ARABIC 2 Integrated water vapourLocationLat(deg)Long(deg)H(km)Meteorological(mm)Integrated(mm)Rome, NY43.2-75.47016.10816.10926.7976.771Milano/Linate45.439.28019.25319.25727.3957.42ITU-R P.835-6 Annex 3 data is derived from ERA15 numerical weather model reanalysis. This is likely to be replace by ERA5 data. The following analysis is with respect to ERA5 annual data. Table 2 compares the two interpolation methods for the calculation of integrated water vapour (IWV) for Rome, NY and Milano, Italy. Here h is the initial height above the sea level. If Zground > h, then integration begins at Zground. These results reflect the observation that generally both methods give similar results.To examine which horizontal interpolation method best represents the data, consider the gridded data structure shown in Fig. 5. Normally, horizontal interpolation for a given location would use the four surrounding grid points. In this analysis the location of interested is chosen to be located at a grid point (black circle) and the horizontal interpolation is with respect to the (blue) grid points at the four corners. The concept is to compare the horizontal interpolation results with that at the centre grid point, which is taken to be the truth with regards to the ERA5 data. Some example results are shown in Table 3. Similar results were observed for other locations. The results marked Meteorological/Zg used the ERA5 ground heights at the centre grid point, which in both cases were lower than the interpolated ground height.8515356033770Figure 5. Gridded structure used for ERA5 data analysis. 00Figure 5. Gridded structure used for ERA5 data analysis. 260985904875Table 3. IWV (mm) Location(43.25, -75.5)(45.5 ,9.25)Height0202Grid Point16.1066.67719.0277.355Meteorological15.5536.77318.0897.406Integrated15.5576.75518.1457.430Meteorological/ Zg16.1746.70219.1577.4100Table 3. IWV (mm) Location(43.25, -75.5)(45.5 ,9.25)Height0202Grid Point16.1066.67719.0277.355Meteorological15.5536.77318.0897.406Integrated15.5576.75518.1457.430Meteorological/ Zg16.1746.70219.1577.41 3. DiscussionThe analysis in section 2.1 used a simple model of gridded atmospheric profiles. Results of cases 1, 2, and 3 indicate that the two interpolation methods generally will give different results, but may give very similar results under some conditions. For the conditions in case 3, the atmosphere must be the same everywhere. Clearly, since the horizontal interpolated meteorological parameters have the same profile, those results should be considered the correct representation of the gridded data. Analysis of the ERA5 data in section 2.2 showed that the two horizontal interpolation methods gave similar results for the integrated quantities. Both interpolation methods differed somewhat from the centre grid point value for the total columnar water vapour. The reason for this is that the interpolated ground height differed from the centre grid point. However, the meteorological interpolation allows for the option of specifying the ground height at the location of interest. In the above examples, the ERA5 ground height at the centre grid point was used and the other parameters were interpolated/extrapolated to the ground height. Consequently, the meteorological interpolation method was able to provide a more accurate IWV with respect to the ERA5 data. In the same way, for arbitrary locations, the use of known ground heights can correct for ERA5 height inaccuracies.4. SummaryA simple two-dimensional gridded data structure allowed to know the truth with regards to a desired location between grid points. In that case horizontal interpolation of meteorological parameters best represented gridded atmospheric data. Analysis of the ERA5 data showed no significant difference between interpolation of meteorological parameters and interpolation of integrated quantities when using ERA5 interpolated data only. Neither method accurately matched the total integrated water vapour at the centre grid point, which provided the truth with respect to the ERA5 data. On the other hand, the meteorological interpolation method gave more accurate results when the known ground height was specified. ................
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