Tilting Mapping Function (TMF) is a tropospheric mapping function to scale the slant tropospheric delays from various elevation and azimuth angles to the zenith direction. Based on the theory of tilting troposphere, TMF can represent the neutral atmosphere's asymmetry more accurately than traditional continued fraction mapping functions.
1. Introduction
Tropospheric delay refers to the effect caused by the propagation of the radio signals among the neutral atmosphere, which can be divided into a hydrostatic part and a wet part
[1]. Many regional or global tropospheric delay models have been built to reduce the tropospheric delay error, which can be divided into two categories, depending on whether meteorological factors are needed or not
[2]. Models of the first category use pressure, temperature, and humidity as their input parameters, such as Hopfield
[3], Saastamoinen
[4], Davis
[5], Baby
[6], Ifadis
[7], Askne, Nordius
[8], and MSAAS
[9]. If in-situ meteorological observations are not available, the standard atmosphere
[10][11][12][10,11,12] or empirical meteorological models
[13][14][15][13,14,15] may also be used in many GNSS data processing applications. The second category doesn’t rely on meteorological measurements, such as UNB
[16], MOPS
[17], TropGrid
[18], ITG
[19], IGGTrop
[20], and SHAtropE
[21].
However, due to the irregular spatial and temporal distribution of water vapor, it is challenging to precisely model the wet part of the tropospheric delay. Thus it has been a commonly used strategy for Global Navigation Satellites Systems (GNSS) data processing to estimate the tropospheric zenith delay
[22][23][24][22,23,24], especially for high precision applications
[25][26][25,26]. The estimated Zenith Wet Delay (ZWD) can be converted into Precipitable Water Vapor (PWV)
[27][28][29][27,28,29], and therefore GNSS meteorology has gradually become a fundamental and effective method for sounding the atmosphere under any weather condition. Barriot, et al.
[30] proposed an approach based on perturbation theory, with the ability to separate eddy-scale variations of the wet refractivity.
The mapping function has been used to scale the slant delays from various elevation angles to the zenith direction. Consequently, the mapping function’s accuracy has significant and direct impacts on the determination of the ZWD and station coordinate. Since Marini first proposed the continued fraction form
[31], almost all modern mapping functions, including Ifadis
[7], MTT
[32], NMF
[33], IMF
[34], UNBabc
[35], VMF
[36], GMF
[37], VMF1
[38], GPT2
[39], and VMF3/GPT3
[40], have taken it as their model expression. Each mapping function has two subtypes: the hydrostatic part and the wet part. The main difference among various mapping functions is the specific value of each coefficient.
However, the Marini concept mapping functions were built on the assumption of the neutral atmosphere’s spherical symmetry
[41][42][43][41,42,43], which can be clearly seen from the expression being independent on azimuth (will be discussed in
Section 2). This assumption holds only approximately even for the troposphere’s normal state, mainly due to the atmospheric bulge, high variation of tropospheric meteorological parameters such as water vapor and temperature. Therefore, such mapping functions would degrade the estimated ZWD and station height in the GNSS data processing. The tropospheric delay’s horizontal gradients, including a North-South and an East-West component, have been used to model the tropospheric delay’s anisotropy
[32][44][45][46][32,44,45,46]. The inclusion of gradient models can significantly improve the accuracy of slant delays
[43], station positions
[44][47][48][49][50][44,47,48,49,50], zenith delays
[51][52][51,52], and PWV
[27][53][27,53]. Nevertheless, only total gradients
[54][55][56][54,55,56] can be estimated in the GNSS data processing, since the hydrostatic and wet gradient mapping functions are very similar. Spherical harmonics were used by Zhang
[57] (using ray-traced delays) and Zhang, et al.
[58] (using GPS-derived delays) to replace the mapping function and gradients. However, many more unknown parameters have to be fitted for those approaches.
To overcome the shortcomings due to the assumption that atmospheric refractivity is spherically symmetric, we tested a new mapping function—TMF—where a concept of tilting the tropospheric zenith by an angle introduced by Gardner
[59], Herring
[32], Chen, et al.
[44], Meindl, et al.
[49] is utilized in this study. The TMF takes not only the elevation but also the azimuth as its input parameter. Ray-tracing
[60] through Numerical Weather Model (NWM) is one of the most accurate approaches to obtaining tropospheric delays. Hence, ERA5 data
[61] of the highest spatio-temporal resolution provided by the European Centre for Medium-range Weather Forecast (ECMWF) was adopted for computing ray-traced delays, using the software WHURT programmed in FORTRAN and developed by Zhang
[2]. In the second part, we discuss some critical algorithms for ray-tracing. A detailed definition of the TMF is given. In the third part, we firstly investigate the asymmetry of the slant tropospheric delays. Then the coefficients of TMF are fitted by the Levenberg–Marquardt nonlinear least-squares method, using ray-traced tropospheric delays. Four fitting schemes were compared, with a different spatial resolution of NWM and different sampling on elevations and azimuths. The performance of TMF against mapping functions based on the VMF3 concept, without or with an estimation of gradient parameters, is presented in the results and discussion section. The summaries and conclusions are given in the last part.
2.TMF: A GNSS Tropospheric Mapping Function for the Asymmetrical Neutral Atmosphere
2.1. Tropospheric Delay Asymmetry
The tropospheric delays’ asymmetry can be assessed visually by skyplots with the removal of the average value over all azimuths on each elevation angle. Due to space limitation, only a few of them are present here exemplarily to demonstrate the spatio-temporal variability.
Figure 17 and
Figure 28 are the IGS station SHAO results on 21 July and 26 December 2018, respectively. The epoch of the left and right panel is 0:00 UTC and 5:00 UTC, respectively.
Figure 17. The asymmetry of ray-tracing slant delays calculated by removal of the mean value of each elevation on all azimuths, for the IGS station SHAO located in Shanghai on 21 July 2018 (a) SHDs at 00:00 UTC, (b) SHDs at 05:00 UTC, (c) SWDs at 00:00 UTC, (d) SWDs at 05:00 UTC. Please note the difference between the bound of the colour bar on the right side of each subfigure.
Figure 28. The asymmetry of ray-tracing slant delays calculated by removal of the mean value of each elevation, for the IGS station SHAO located in Shanghai on 26 December 2018. (a) SHDs at 00:00 UTC, (b) SHDs at 05:00 UTC, (c) SWDs at 00:00 UTC, (d) SWDs at 05:00 UTC. Please note the difference between the bound of the colorbar.
Figure 17d shows much more significant anisotropy (please note the disparity in the bounds of the colour bars) than
Figure 17c, which means that there was a quick variation of SWD from 0:00 to 5:00 UTC. This may be due to the fast-changing distribution of humidity, the typical summer weather conditions at Shanghai, where the SHAO station is located.
The situation is a little different on 26 December. As shown in
Figure 28, although the elevation-dependent pattern is similar to
Figure 27, SHD shows more spatial variability than SWD this time. The SHD range can reach up to ~10 cm at 5° elevation, while the SWD range is no more than several centimeters. This result may be caused by the fact that there is much less water vapor in winter than in summer in Shanghai. However, a comparison between
Figure 28c,d shows that the temporal variations of SWD are still more complicated than SHD.
In order to get a more quantitative investigation on the results at low elevations, we summarise the statistical result of slant delays at four specific elevations: 5°, 10°, 15° and 20° in
Table 13 and
Table 24. There is much in common between the two tables. Firstly, the SHD and SWD and their range and RMS all tend to increase positively as elevation angle decreases. However, the SHD is always 6–15 times as large as the SWD. Secondly, the RMS of SHD and SWD at an elevation above 15° are mainly at the level of several millimetres. Furthermore, the RMS and range at the 5° elevation are always ten times as lager as that at the 20° elevation, both for SHD and SWD. Results above indicate that both the SHD and the SWD may present decametric asymmetry at low elevations.
Table 13. Representative ray-tracing slant delays for SHAO located in Shanghai on 21 July 2018.
UTC |
Elevation
Angle |
SHD (m) |
SWD (m) |
Mean |
Range |
RMS |
Mean |
Range |
RMS |
RMS |
---|
Mean |
Range |
RMS |
0:00 |
5° |
23.135 |
0.040 |
0.012 |
3.515 |
0.119 |
0.037 |
2.2. TMF Fitting
The results of the four fitting schemes introduced in
Table 1 are listed in
Table 35, in which elevations are divided into two bands: low elevation
(3°≤θ≤15°) and high elevation
(15°<θ<90°). As shown in
Table 35, there is no apparent difference for bias and RMS between the two elevation and azimuth angle selection strategies (1 vs. 2, or 3 vs. 4). However, the horizontal resolution of ERA5 has a significant impact on the results. The RMS of the fitted SWDs based on ERA5 with 1° × 1° horizontal resolution is four and 7~9 times larger than that of the 0.25° × 0.25°, at low and high elevation angle bands, respectively. Results for SHD are similar but a little better. Hence, we use Scheme 2 in
Table 1 (ERA5 with a horizontal resolution of 0.25° × 0.25°, at 18 selected elevations and 24 azimuths) to implement ray-tracing in the following research, which aims to keep a balance between the computational accuracy and the efficiency.
Table 35. Statistic result of TMF-derived slant delays, which are the product of the TMF and the ray-traced zenith delays. The meaning of the postfix numbers is listed in Table 2.
Elevation
Angle | ΔSHD (cm) |
---|
bias1 | RMS1 | bias2 | RMS2 | bias3 | RMS3 | bias4 | RMS4 |
|
5° |
23.525 |
0.100 |
0.034 |
1.632 |
0.075 |
0.026 |
0:00 |
10° |
12.896 |
0.035 |
0.012 |
3°–15° | 0.0 | 0.3 | 0.0 | 0.3 | −0.3 | 0.7 | −0.3 | 0.7 |
10° |
12.706 |
0.015 |
0.004 |
1.850 |
0.029 |
0.007 |
0.858 |
0.024 |
0.008 |
15°–89° | 0.0 | 0.0 | 0.0 | 0.1 | −0.2 | 0.3 | −0.2 | 0.3 |
15° |
8.703 |
0.007 |
0.002 |
1.255 |
0.018 |
0.004 |
15° |
8.828 |
0.017 |
0.006 |
0.581 |
0.012 |
0.004 |
20° |
6.636 |
0.004 |
0.001 |
0.953 |
|
20° |
6.730 |
0.010 |
0.003 |
0.440 |
0.007 |
Elevation
Angle | ΔSWD (cm) | 0.012 |
0.002 |
bias1 | 0.003 |
RMS1 | bias2 | RMS2 | bias3 | RMS3 | bias4 | RMS4 |
5:00 |
5° |
23.102 |
0.039 |
0.011 |
3.894 |
0.678 |
|
5° |
23.492 |
0.100 |
0.035 |
1.540 |
0.022 |
0.007 |
3°–15° | 0.0 | 1.9 | 0.0 | 2.0 | 0.0 | 8.00.245 |
0.0 | 7.9 |
10° |
12.689 |
0.015 |
0.004 |
2.078 |
5:00 |
10° |
12.876 |
0.0360.245 |
0.080 |
0.012 |
0.806 |
0.005 |
0.001 |
15°–89° | 0.0 | 0.3 | 0.1 | 0.4 | 0.4 | 2.8 | 0.3 | 2.8 |
15° |
8.691 |
0.007 |
0.002 |
1.410 |
0.118 |
0.037 |
20° |
6.627 |
0.004 |
0.001 |
1.071 |
0.066 |
0.021 |
Table 24. Representative ray-tracing slant delays for SHAO located in Shanghai on 26 December 2018.
UTC |
Elevation
Angle |
SHD (m) |
SWD (m) |
Mean |
Range |
|
15° |
8.814 |
0.019 |
0.006 |
0.545 |
0.003 |
0.001 |
|
20° |
6.719 |
0.011 |
0.003 |
0.414 |
0.002 |
0.000 |