# Kriging Interpolation in Modelling Tropospheric Wet Delay
## 3 Experiments in Sparse and Dense Network  
A sparse network with 8 reference stations and a dense network with 19 reference stations were selected in the CORS network of the Netherlands, as can be seen in Figures 1 and 2, respectively. Distances between two nearby reference stations were varying, from 60 km to 120 km for the sparse network and 20 km to 60 km for the dense network. Another 15 user stations were also selected to evaluate the accuracy of the predicted zenith wet delays by comparing with the estimated delays at each user station. The total user stations are divided into three blocks: 5 stations (LWRD, HOOG, LELY, ADR2 and LITH) were inside the networks, 5 stations (TERS, DZYL, OLDE, HHOL and WARM) were on the edge of the networks and 5 stations (VLIS, RIL2, SASG, KERK and MSTR) were outside the networks. Therefore, the impacts of the locations can be investigated since the Kriging mainly depends on the distances between known and unknown points. The IF combination-based Position and Navigation Data Analysis software (PANDA) [40; 41; 42] was used in this study to generate the zenith wet delay estimates at reference and user stations in the use of PPP data processing, and the configuration is shown in Table 1. Since the reference and user receivers were stationary, the static positioning mode was applied in this study with precise a priori coordinates so that the influence of the position errors would be attenuated. The dual-frequency GPS data were used in estimating the zenith wet delay. In addition, except for the coordinates, the dynamic models  
Figure 1: Sparse global navigation satellite system (GNSS) network with 8 reference stations and 15 user stations. Figure 2: Dense GNSS network with 19 reference stations and 15 user stations. were also applied to the tropospheric wet delay and troposphere gradient parameters with the spectral density of 0.02 m\({}^{2}\)/h and 0.001 m\({}^{2}\)/h, respectively, based on their temporal properties. This study applied precise satellite orbit and clock corrections provided from the international GNSS service (IGS) so that these two error sources were not considered in the positioning model. The interval of data processing was set to 60 s by taking into account the computational efficiency and the troposphere property that was temporally stable within a short time span. Since this was the post-processed PPP, forward and backward Kalman filter was applied; and thereby there was no convergence time at the beginning of data processing. Elevation-dependent weighing strategy was implemented due to the fact that signals at low elevation angles were likely to be affected by large model errors. The zenith hydrostatic delay has been corrected by the Saastamoinen model [8], and only the zenith wet delay was estimated. The ambiguities were always kept float in this study. One can refer to [43; 44; 45] for more information of fixing the ionosphere combined ambiguities. The data covering the entire year of 2019 were chosen to evaluate the performance of the Kriging interpolation, and the sample variogram of the sparse network and the dense network were estimated and presented in Figures 3 and 4, respectively. Note that the fitted variogram models were determined at every epoch and used in Ordinary Kriging to predict the zenith wet delay at 15 user stations. RMSEs of the 5 user stations inside the network between the predicted zenith wet delays and estimated delays are shown in Figure 5. The RMSEs of both the sparse network and the dense network during summer (July, August and September) period were larger than the rest of the months. This was reasonable since the water vapor content would change rapidly due to the rise of temperature, and, thus, the zenith wet delays may differ even in a small region. However, the dense network had a better performance in summer period than the sparse network by taking the advantage of the numerous stations. RMSEs exceed 1.5 cm with the sparse network were 1.75% of the sample, while with the dense network it is 0.33%.