1. Introduction
The troposphere is the lowest portion of the Earth’s atmosphere, and tropospheric delay due to the neutral atmosphere is one of the main error sources of the Global Navigation Satellite System. This delay can cause up to 2.5 m at zenith direction of the Global Navigation Satellite System (GNSS) signal transmission and over 20 m when satellites are at low elevation angles; e.g., below 10 degree [
1,
2]. The tropospheric delay is commonly expressed with the following model [
3]
where
e and
are respectively the elevation and the azimuth angle of a specific satellite. The total tropospheric slant delay
T between receiver and satellite at an elevation angle
e is the sum of three portions: a hydrostatic portion, a wet portion and a gradients portion.
and
are the zenith hydrostatic delay and zenith wet delay, respectively.
and
are the mapping functions for the zenith hydrostatic and wet delay, respectively.
and
are the gradients which account for the azimuthally inhomogeneous troposphere in north–south and east–west directions with the corresponding gradient mapping function
.
The hydrostatic delay due to the refractivity of the dry gases in the troposphere can be corrected by the conventional models such as Saastamoinen [
1] and Hopfield [
4], which can model the hydrostatic delay at the millimeter level in the zenith direction [
5]. Collins and Langley [
6] proposed a neutral atmosphere model designed for Wide Area Augmentation System (WAAS) users, which is the so-called UNB model series (UNB1 through UNB4) and has been assessed for the use in North America [
7,
8], Europe [
9] and Japan [
10]. Li et al. [
11,
12] developed a multi-dimensional grid model, IGGtrop, to provide tropospheric delay corrections for the users of the BeiDou Navigation Satellite System (BDS) and the area augmentation system based on BDS in China. Although the models mentioned above can correct the wet delay to some extent, the accuracy varies from centimeter to decimeter level, which is still insufficient for high precision positioning and navigation. In addition, using the empirical atmospheric information obtained from the profile of global pressure and temperature may reduce the accuracy of the troposphere models due to the high spatial and temporal variability of water vapor [
13,
14]. Therefore, the zenith wet delay is usually estimated as an unknown parameter at each epoch or within a certain time span.
When the zenith tropospheric delays are estimated or provided, the slant delays to each visible satellite are obtained by assuming a specific relation between the zenith and slant direction in which the troposphere is assumed to be symmetrical about the vertical direction of the receiver. The relation between zenith and slant delay can be modeled by a so-called mapping function, as already shown in Equation (
1), for which a wide range of mapping functions have been developed in the past. The Niell mapping function (NMF) [
15] and the global mapping function (GMF) [
16] consist of easy-to-handle formulae which only need the input parameters of approximate latitude, height and day of year. On the other hand, the isobaric mapping function (IMF) [
17] and the Vienna mapping function 1 (VMF1) [
18] provide support for mapping functions derived from numerical weather models (NWM) by applying the ray-tracing technique and/or climatological data. The crucial variable in mapping functions is the elevation angle. Most mapping functions are azimuth-independent, which reveals the underlying assumption that the troposphere is azimuthally homogeneous. A successful application of an azimuthally inhomogeneous tropospheric delay modeling in GPS geodesy and very long baseline interferometry (VLBI) was proposed by MacMillan [
19] and Chen and Herring [
20], in which the so-called horizontal gradients are considered in addition to a mapping of the zenith to slant delays. In this way, a linear asymmetry of the troposphere is accounted for by introducing a tilted direction instead of the zenith direction. For an extensive review of the troposphere model and mapping function, see Teunissen and Montenbruck [
21].
Positioning in severe weather conditions has received more attentions in recent years. Yasyukevich et al. [
22] investigated the influence of solar flares on the GNSS and high-frequency propagation. Luo et al. [
23] analyzed the performance of double and single-frequency base PPP during three typical geomagnetic storms. As for the tropospheric delay, the standard troposphere model is capable of estimating the tropospheric delay with centimeter accuracy in normal weather conditions [
24,
25]; however, it should be investigated how positioning results can be improved if the residuals of the tropospheric delay caused by weather events are taken into account. The issue is that satellites at the same elevation angle would be compensated by almost the same tropospheric delay correction based on the standard mapping function approach. However, the symmetrical troposphere about the zenith direction of the receiver is not realistic when it suffers from the complex weather situation. The performance of the horizontal gradients is also limited, because they can only consider a linear asymmetry of the troposphere around the geodetic site [
26]. Kleijer [
27] analyzed that significant biases can be introduced in the estimated ZWD when the atmosphere is not symmetrical. However, the suggestion of using an accurate wet mapping function is still limited by the assumption of the homogeneous troposphere. Li et al. [
28] assessed the impacts of the tropospheric biases on the integer ambiguity resolution and gave the recommendations of under which conditions the tropospheric biases can be ignored. However, only zenith tropospheric biases are taken into account, without considering the biases caused by the inhomogeneous troposphere. Hobiger et al [
29] proposed a method to combine the mesoscale and fine-mesh numerical weather model to provide the ray-traced tropospheric slant delay during a typhoon passage. The result shows that the height repeatability is improved up to
compared to standard data processing. However, this could still be insufficient for high-precision positioning, and it is not possible to provide the fine-mesh numerical weather model to worldwide users in (near) real-time.
The detection, identification and adaptation (DIA) procedure was first demonstrated by Baarda [
30] and Teunissen [
31,
32]. Teunissen [
33] introduced this method into GNSS to detect, identify and adapt the mismodeled errors, and then it was applied in a wide range of GNSS applications; for example, kinematic GNSS surveying [
34], permanent station resolution [
35] and observation quality control [
36,
37]. In this contribution, a real-time recursive DIA procedure is implemented to detect the model errors which have the same effects on both phase and code observables, and once the errors are identified, additional parameters will be added to the functional model to account for the measurement residuals. One of the applications of this approach is to detect model errors caused by the tropospheric delay; therefore it was evaluated with GPS data during two different rainfall events in Darwin, Australia, proving the usefulness of compensated residual slant tropospheric delay for positioning results. Comparisons with the standard approach show that the precision of the up component is improved significantly during the period of the weather events, and the precision of the horizontal component is also improved.
This article is organized as follows.
Section 2 reviews the standard functional model for PPP data processing and the theory of DIA and the construction of the improved functional model, which takes into account the model errors.
Section 3 analyzes the performance of the proposed procedure via two case studies during a weather event.
Section 4 contains the summary and conclusions.
3. Case Studies and Results
Two case studies will be presented in which we know there was severe weather during the observation period, so as to evaluate the capability of the proposed approach to account for associated model errors due to the asymmetrical behavior of the tropospheric delays during such events. The data of the first event is from the Australian Continuous Operational Reference Station 00NA in Darwin on 14 November 2017. In the sequence, it is referred to as Event 1. The second data set is from 24 March 2018 of an IGS permanent station DARW which is also located in the same region; in the sequence it is referred to as Event 2. In both cases there was heavy rainfall with thunderstorms on the specific days. Temperature and humidity for both days were obtained from
https://www.wunderground.com/ and are shown in
Figure 2 and
Figure 3, respectively. The GPS data and IGS products are used to ensure highly precise orbit and clock corrections. Since this study focuses on the model error detection, the final orbit and clock products are applied in the data processing to eliminate associated errors as much as possible. Configuration of the data processing for the real-time PPP can be seen in
Table 1, in which the significance level defines the critical region where the value for test statistic lies in the null hypothesis is rejected.
As for Event 1 of
Figure 2, the sun rises at UTC 21:30 (6:00 local time) and then temperature increases while humidity decreases; rainfall appears from 5:00 to 8:00 (UTC) with the temperature dropping 10 degrees within 1 h. A similar effect can also be seen in the change of humidity of that day. For Event 2 of
Figure 3, the weather event appears from 2:00 to 5:00 (UTC). The area between the red lines shows the period of the weather event during which a significant influence on the positioning is present. The shadow highlights the period of the dramatic change of temperature and humidity. The hydrostatic delay depends only on the total density of the air, and the change of temperature and humidity would somehow affect the density; thus, with a high probability, the rapid shift in temperature and humidity will impact the hydrostatic delay. However, the inaccuracy of the zenith hydrostatic delay would not be a problem for the proposed model because the residuals of the hydrostatic delay will be lumped into the wet delay. In this case, DIA is to identify the model errors caused by the combined wet delay and residuals of the hydrostatic delay.
Figure 4 shows the statistics of the LOM test exceeding the threshold are mostly concentrated at the beginning of the event when the front is passing through, which causes significant spatial and temporal gradients in the integrated water vapor above the receiver. The number of subsequently identified model errors is shown as well, and here, at most two model errors are identified at one epoch, which means only one or two satellites are affected by the event at the same time. Besides, it can be seen that the statistics of the LOM test are below the threshold after the DIA procedure, indicating that there is no indication for remaining undetected model errors.
Similar behavior of the LOM test and identified model errors can also be seen in
Figure 5 for Event 2; the rejected LOM test and identified model errors are mostly concentrated at the beginning of the weather event. However, outside the period of this event, there is one satellite detected to be biased at
UTC. Although it is difficult to prove that these model errors are caused by the tropospheric delay, the results of the up component around
UTC are also significantly improved, which means the proposed approach is suitable to adapt for model errors which have the same effects on both phase and code measurement.
With the proposed adapted model of Equation (
16) during the weather event, the additional parameters which represent the residuals of the slant wet delay are considered to account for the model errors due to the tropospheric delay.
Table 2 and
Table 3 are the mean and root mean squared error (RMS) of the phase and code residuals of satellites being identified with the model error during the weather events of Event 1 and Event 2, respectively. As can be seen in these two tables, residuals of the phase observations of the affected satellites are mostly reduced because the adjusted functional model is more reliable with the additional parameters accounting for the model errors. As for the residuals, the improvement of the phase observation is more significant than that of the code observation, because the value of the additional parameter mainly depends on the phase observation due to its much higher weight compared to the code observation.
Figure 6 illustrates the results of the up component and the horizontal component within the time span from 5:00 to 8:00 UTC of Event 1. The pattern of the up component positioning error with the standard approach represented as a blue line shows a typical trend affected by the tropospheric delay. On the contrary, the vertical positioning errors with the proposed method are reduced because the residual slant wet delays have been compensated by the additional parameters. Although there is still a systematic error in the east direction, the performance of the proposed method in horizontal displacement is better than that with the standard approach during the weather event. As can be seen in the skyplot of
Figure 7, most of the affected satellites are located in the east part of the skyplot, which leads to a partially biased horizontal component after adjusting.
Table 4 summarizes the improvement of the results obtained from the proposed method compared to the standard approach. Significant improvement can be seen in the up component, since it is known that the tropospheric delay is one of the main error sources in the vertical direction due to high correlation. The horizontal precision of the proposed method is also improved by about
. For Event 2, most of the satellites affected by the weather event are located in the east part of the skyplot (Figure 9).
The distribution of the influenced satellites is shown in the skyplot of
Figure 7.
Similarly, the positioning errors of the up component of Event 2 in
Figure 8 are also reduced, since the effects of the weather event have been removed. This approach also works well for the aforementioned model errors identified outside the period of the weather event at around
UTC, indicating that the model errors can be compensated if they have the same influence on the phase and code observables.
Table 5 shows a significant improvement of the up component, which is the same as Event 1. Meanwhile, a system error still exists in the east component, though the precision of the horizontal component is also improved. From the skyplot of
Figure 9, one can see that most of the influenced satellites are concentrated in the west part of the site, which partly causes the east-west bias of the horizontal component.
Occurrences of the model’s errors during Event 1 are shown in
Figure 7 as functions of time and azimuth, and elevation and azimuth, respectively. The blue lines show the trajectories of the satellites, and the red points indicate at which epochs a model error was identified.Most of the identified model errors are concentrated within the azimuth angle range
degrees; i.e., the east part of the skyplot. At the beginning of the weather event, two satellites, PRN19 and PRN6 at around 150 degrees, were affected by the event. Both of them are at low elevation angles, as shown in the skyplot of
Figure 7. When PRN2 and PRN12 approached this area, they also identified with model errors, which means signals are affected by an extra tropospheric delay in this direction compared to any other azimuth angles. Then, the front moved from 150 degrees to 60 degrees, and thus satellite PRN5 and PRN20 were affected, after which it dissipated. It is worth noting that wrong identification might be present; e.g., PRN6 is affected by the weather event for a long period of time and among which PRN2 at almost the same azimuth angle as PRN6 is identified with the model error for several epochs.
Figure 9 shows the distribution of the identified model errors in Event 2, which resembles the distribution of Event 1 in which the model errors are concentrated within a certain range of azimuth angle; i.e.,
–300 degrees. This property may indicate the heading direction of the front. The event keeps affecting PRN15 for a long time, which is not at a low elevation angle. Satellites close to this range of azimuth angle are also detected with model errors at some epochs.
Figure 10 shows values of the estimated additional parameters which are due to the unmodeled slant wet delays caused by the weather events for this specific case. For Event 1 on the left side, the model error of PRN19 reaches up to more than 40 cm, as this satellite goes down to a low elevation angle. It seems reasonable, since the wet delay may lead to a delay of up to several meters at a low elevation angle. The mean and RMS phase residuals of PRN19 shown in
Table 2 reduce, respectively, to −0.01 mm and 0.04 mm when compensated by the estimated additional parameters. The negative values of the additional parameters are because of the mismodeled hydrostatic delay. The mean and RMS phase residual of PRN6 also drop to 0.31 mm and 7.3 mm, respectively.
As for Event 2, values of PRN15 change rapidly from +20 cm to –10 cm, and they are stable for a long time span; even these additional parameters are considered to be epoch independent. This implies a further implementation of the global overall model test which takes into account the test statistics over a period of time rather than a single epoch. As can be seen in
Table 3, the mean and RMS phase residuals of PRN15 reduce to –2.24 mm and 14.73 mm, respectively, which still show significant improvements compared to the standard PPP without the DIA procedure.
4. Conclusion
In this contribution, a DIA procedure was implemented to identify the model errors which have the same impact on both the phase and code observables; one of the applications is to account for model errors caused by tropospheric delays. An improved functional model was proposed with the additional parameters to account for the model errors. Although precise orbit and clock products were applied in the data testing to avoid any other model errors, this troposphere identification model can be easily implemented in real-time PPP, and the DIA procedure can be processed in real-time. This procedure was evaluated by two case studies of weather events, during which the tropospheric delay might be azimuthal asymmetric around the receiver, and thus the model errors due to the inhomogeneous troposphere can be detected by this procedure. The phase residuals of the satellites identified with model errors are compared to the standard approach during the weather events, since the unmodeled wet delay can at least be partly absorbed in the additional parameters. The positioning results are also improved during the events, and the improvement is most significant for the up component ( and improvement of RMS for two weather events) since the tropospheric delay is one of the main error sources in the vertical direction. The positioning performance of the horizontal component obtained from the proposed method is also improved (more than improvement of RMS) compared to the standard PPP. The values of the additional parameters indicate the model errors due to the tropospheric delay can reach 40 cm when the satellite is at a low elevation angle.
At most two model errors are identified at one epoch in the two case studies, which indicates that not too many satellites are affected by the asymmetrical troposphere, even during the weather events. Despite the complexity of extreme weather, the identified model errors are concentrated at the beginning of both heavy rainfall processes when the front causes significant spatial and temporal gradients of the integrated water vapor above the receiver. Besides, the satellites affected by the events are concentrated within a certain range of azimuth angle, which is related to the path of the front. This proposed procedure can also be used in monitoring severe weather. If the outliers detected by this method increase dramatically, it may indicate the front line of weather event is passing through. More testing shows that the proposed procedure may not always bring a very large improvement, but as least it does not deteriorate the positioning solutions; meanwhile, it does prevent severe impacts in some cases.