Tropospheric delay is an important error affecting GNSS high-precision navigation and positioning, which will decrease the precision of navigation and positioning if it is not well corrected. Actually, tropospheric delay, especially in the zenith direction, is related to a series of meteorological parameters, such as temperature and pressure. To estimate the zenith tropospheric delay (ZTD) as accurately as possible, the paper proposes a new fused model using the least squares support vector machines (LSSVM) and the particle swarm optimization (PSO) to improve the precision and temporal resolution of meteorological parameters in global pressure and temperature 2 wet (GPT2w). The proposed model uses the time series of meteorological parameters from the GPT2w model as the initial value, and thus, the time series of the residuals can be obtained between the meteorological parameters from meteorological sensors (MS) and the GPT2w model. The long time series of meteorological parameters is the evident periodic signal. The GPT2w model describes its dominant frequency (harmonics), and the residuals thus can be seen as the short-period signal (nonharmonics). The combined PSO and LSSVM model (PSO-LSSVM) is used to predict the specific value of the short-period signal. The new GPT2w model, in which the meteorological parameter value is obtained by combining the estimated meteorological parameters residuals and the GPT2w-derived meteorological parameters, can be acquired. The GNSS network stations in Hong Kong throughout 2017-2018 are processed by the GNSS Processing and Analysis Software (GPAS), which is developed by the Chinese Academy of Surveying & Mapping, to estimate the zenith tropospheric delay and station coordinates using the new GPT2w model. Statistical results reveal that the accuracy of the new GPT2w model-derived ZTD was improved by 60% or more compared with that of the GPT2w-derived ZTD. In addition, the positioning accuracy of the GNSS station has been effectively improved up to 44.89%. Such results reveal that the new GPT2w model can greatly reduce the influence of nonharmonic components (short-period terms) of the meteorological parameter time series and achieve better accuracy than the GPT2w model.

Tropospheric delay is an important error that affects the positioning accuracy of the global navigation satellite system (GNSS). However, it is also an important parameter to calculate tropospheric delay for GNSS meteorology. Usually, the value can reach tens of meters [

According to the relationships between ZTD and meteorological parameters obtained from the ground, the commonly ZTD models are established including Hopfield model [

Thus, the ZTD empirical models are proposed that only rely on the location and observation time without the need of any auxiliary information. For example, the UNB3 model [

However, the ZTD empirical models are not applicable to some regions limited by the spatiotemporal resolution. Fortunately, the Continuously Operating Reference Station (CORS), whose ZTD products have high accuracy and high temporal resolution, provides an opportunity for establishing new ZTD models with higher accuracy. This paper proposes a new model combining the particle swarm optimization algorithm with the least squares support vector machine (PSO-LSSVM) model to improve the GPT2w model. First, the GPT2w is used to calculate meteorological parameters that as the initial value. Second, the time series of meteorological parameters residuals can be obtained as the difference between meteorological parameters from meteorological sensors (MS-derived) and GPT2w-derived meteorological parameters over GNSS stations. Third, the PSO-LSSVM model is used to predict meteorological parameters residuals. So the meteorological parameter value can be acquired by combining the estimated meteorological parameters and GNSS-derived meteorological parameters. Finally, the zenith tropospheric delay (ZTD), zenith hydrostatic delay (ZHD), and station coordinates can be obtained by GNSS Processing and Analysis Software (GPAS) using the meteorological parameters, which is developed by the Chinese Academy of Surveying & Mapping.

The Global Pressure and Temperature (GPT) series model is an empirical model to provide the global temperature and pressure at any GNSS station in the world. These models include GPT [

In equation (

In equations (

Support vector machine (SVM) is a machine learning method with a perfect theoretical system, which is different from general statistical methods. It avoids the process from induction to deduction and thus realizes the inference and estimation from training samples to forecast samples and obtains the simplification of regression analysis and other problems [

The LSSVM can be explained as follows: for the training sample

In equation (

In equations (

The equation (

In equation (

Let

In equation (

In equation (

In equation (

There are two important parameters (

For the optimization problem of the PSO, the solution can be regarded as a bird in the search space, which has its own initial velocity, position, and fitness to a certain position. Finding the optimal solution is to find the position where the particle has the best fitness value from the starting position to the current position. The update formulas for the velocity and position of each particle in the particle swarm are as equations (

In equations (

To demonstrate the advantage of the LSSVM and the PSO clearly, the optimization process is shown in Figure

Initialize the particle swarm and the parameters of the LSSVM.

Calculate the fitness of each particle. The fitness of the current particle is compared with the fitness of the individual’s optimal position and the historical optimal position. If the fitness of the current particle is optimal, the position is replaced; otherwise, the original optimal position is maintained.

Update the position and velocity of the particles by the maximum number of evolutions.

Judgment of termination conditions. When the error requirements or the maximum number of evolutions are met, the process will end. Otherwise, the process will repeat Step

The parametrization workflow of PSO-LSSVM.

The GNSS Processing and Analysis Software (GPAS) is developed by Chinese Academy of Surveying & Mapping to obtain the zenith tropospheric delay (ZTD), zenith hydrostatic delay (ZHD), and station coordinates. The process of GPAS is shown in Figure

GNSS processing of GPAS software.

The Hong Kong Survey and Mapping Office (SMO) of the Lands Department builds a local satellite positioning reference station network (SatRef). It consists eighteen CORSs and six of them are chosen considering the continuity and completeness of observation data, which is shown in Figure

The distribution of selected (GNSS) stations in Hong Kong SatRef.

The ground weather stations obtain the temperature, air pressure, relative humidity, and other meteorological parameters at the site through meteorological sensors (MS). The air pressure detection accuracy can reach

In equation (

The time series of pressure and temperature residuals between GPT2w and MS at stations (HKOH, HKKT, HKPC, HKSC, HKSS, and HKST) are given in Figure

Time series of pressure from MS and GPT2w at stations. (a) HKOH. (c) HKKT. (e) HKPC. (g) HKSC. (i) HKSS. (k) HKST. Time series of temperature from MS and GPT2w at stations. (b) HKOH. (d) HKKT. (f) HKPC. (h) HKSC. (j) HKSS. (l) HKST. Throughout 2017-2018.

Taking the HKOH as example, Figure

Time series of (a) pressure and (b) temperature residuals between MS and GPT2w at HKOH.

In Figure

The LSSVM is a nonparametric model, which means that it does not require any prior information about the underlying data. Thus, this paper uses the past 24 hours of historical meteorological parameters residuals (nonharmonic part) to train the model. To determine the appropriate forecast range, the meteorological parameters residuals of the next 1 h, 2 h, 4 h, 6 h, 8 h, 10 h, and 12 h are, respectively, predicted. The RMS values of the model meteorological parameters and the measured meteorological parameters are calculated to evaluate the performance of different prediction models. It can also be seen from Figure

RMS of meteorological parameters under different forecasting modes at HKOH.

Figure

Time series of (a) pressure and (b) temperature residuals of the observed and proposed model prediction at HKOH.

Thus, the predicted meteorological parameters are obtained by adding the predicted nonharmonic components and the harmonic components estimated by the GPT2w model and then the GPAS software is used to output ZHD and ZTD.

It can be seen from Figure

Time series of ZHD of the GPT2w, MS and PSO-LSSVM at stations. (a) HKKT. (b) HKOH. (c) HKPC. (d) HKSC. (e) HKSS. (f) HKST.

The paper selects the data from August 10th, 2018, to August 27th, 2018, when the weather of Hong Kong was rainy continuously to validate the proposed model. The right picture in Figure

Time series of ZHD of the GPT2w, MS, and PSO-LSSVM at station HKOH in 2018 (left) and Time series of ZHD of the GPT2w, MS, and PSO-LSSVM at station HKOH throughout August 10, 2018-August 27, 2018 (right).

The left pictures of Figure

Frequency histogram of the ZTD difference using the GPT2w model at selected stations. (a) HKKT. (c) HKOH. (e) HKPC. (g) HKSC. (i) HKSS. (k) HKST. Frequency histogram of the ZTD difference using the PSO-LSSVM model at selected stations. (b) HKKT. (d) HKOH. (f) HKPC. (h) HKSC. (j) HKSS. (l) HKST.

It can be seen from Table

Comparison of ZTD between GPT2w and PSO-LSSVM at selected stations.

GNSS stations | GPT2w-ZTD RMS (mm) | PSO-LSSVM-ZTD RMS (mm) | Percentage of improvement |
---|---|---|---|

HKKT | 1.4514 | 0.4989 | 65.62% |

HKOH | 1.3078 | 0.3231 | 75.29% |

HKPC | 1.4006 | 0.3320 | 76.29% |

HKSC | 1.3983 | 0.3141 | 77.54% |

HKSS | 1.3845 | 0.3014 | 78.23% |

HKST | 1.3074 | 0.3036 | 76.78% |

Finally, the proposed model is applied to the GPAS and compared with the GPT2w model. From Table

Comparison of coordinate accuracy between GPT2w and PSO-LSSVM at selected stations.

GNSS stations | GPT2w-coordinate RMS (mm) | PSO-LSSVM-coordinate RMS (mm) | Percentage of improvement | ||||||
---|---|---|---|---|---|---|---|---|---|

HKKT | 0.4366 | 0.7504 | 0.3188 | 0.2936 | 0.4117 | 0.1757 | 32.75% | 45.14% | 44.89% |

HKOH | 0.3499 | 0.5935 | 0.2735 | 0.2324 | 0.3495 | 0.1483 | 33.58% | 41.11% | 45.78% |

HKPC | 0.2678 | 0.5316 | 0.2585 | 0.2216 | 0.3654 | 0.1555 | 17.25% | 31.26% | 39.85% |

HKSC | 0.4832 | 0.5391 | 0.2514 | 0.2975 | 0.3768 | 0.1425 | 38.43% | 30.11% | 43.32% |

HKSS | 0.2610 | 0.5217 | 0.2543 | 0.2439 | 0.3792 | 0.1917 | 6.55% | 27.31% | 24.62% |

HKST | 0.2766 | 0.5170 | 0.2462 | 0.2569 | 0.4627 | 0.2189 | 7.12% | 10.50% | 11.09% |

To improve the accuracy of estimating ZTD, the new meteorological parameters model based on GPT2w model and PSO-LSSVM model is proposed. Based on the advantages of MS-derived meteorological parameters and GPT2w model, the time series of meteorological parameters is divided into two categories: harmonic and nonharmonic. Then, the PSO-LSSVM model is used to estimate the nonharmonic and GPT2w is applied to fit the harmonic. Then, the improved meteorological parameters are obtained. Finally, these parameters are input into GPAS which can process GNSS data; the ZTD and coordinates of stations are obtained accordingly. The results show that the proposed model has higher temporal resolution and higher accuracy than GPT2w model. Also, the proposed model is robust even though the weather is rainy.

The future work will focus on the following aspects. First, the past 24 h of historical meteorological parameter residuals (nonharmonic part) are used to train the PSO-LSSVM model. A shorter time should also be considered, such as 12 h. Second, the estimation and forecasting of meteorological parameters are a complicated process. This paper only considers the correlation of its one-dimensional time series. In the following work, more relevant variables can be considered to improve the accuracy of model.

The codes of GPT2w model can be downloaded at

The authors declare no conflict of interest.

At the end of the paper, I would like to extend my sincere gratitude to those who have provided help in my process of writing this paper. First, I would like to appreciate Xu who has given me guidance and has helped me to improve this paper many times patiently. Second, I would like to thank Dang who has provided favorable environments and working conditions. Last but not least, Chinese Academy of Surveying & Mapping is acknowledged for providing GPAS software and I would like to thank the research group of Advanced Geodesy of TU Vienna and the Hong Kong Geodetic Survey Services for providing the GPT2w model and meteorological data, respectively. This research is funded by the Fundamental Research Funds for the Central Public Research Institutes (AR2004, AR2005) and Research on Regional Ground Surface Dynamic Environmental Impact Model and the Overall Refinement Method of Reference Frame (No.41974010).