# Kriging Interpolation in Modelling Tropospheric Wet Delay
## 1 Introduction  
The global navigation satellite system (GNSS) positioning technique has been implemented in various applications, e.g., geodetic surveying and mapping [1; 2], vehicle and pedestrian navigation [3; 4], earthquake and deformation monitoring [5] and meteorology [6; 7]. Therefore, many GNSS positioning based studies have been emerging in the last two decades. The tropospheric delay plays an important role in GNSS positioning. On the one hand, it is one of the main error sources impacting the positioning accuracy. As one of the two components of the tropospheric delay, the hydrostatic delay amounts to roughly 2.3 m in zenith direction, whereas the other component, the zenith wet delay is in the range of 0.15 m in the global average. Although the magnitude of the hydrostatic delay is greater than that of the wet delay, it is predicted well by empirical models due to the stability of surface pressure [8; 9; 10; 11]. The wet delay, on the contrary, is less accuratelypredictable from ground-based sensor data due to the high spatial and temporal variability of water vapor. In such a case, the zenith wet delay is considered as an unknown parameter in some highly precise applications by means of GNSS, e.g., precise point positioning (PPP). On the other hand, the zenith wet delay estimates make GNSS available to measure atmospheric water vapor. [PERSON] et al. discussed the possibilities of transforming the zenith wet delay into the precipitable water overlying GNSS receivers [6; 7]. Shortly afterwards, the tropospheric wet delay was implemented in weather forecasting [12] and numerical weather prediction (NWP) models [13; 14] along with the traditional meteorological sensors like radiosonde, light detection and ranging (LIDAR), weather balloons and aircraft. Because of the significance of the tropospheric wet delay to GNSS and meteorology, this contribution aims to predict the zenith tropospheric wet delay through GNSS networks using Kriging interpolation. The predicted wet delays can be first used in strengthening the GNSS positioning models; e.g., with the help of the external zenith wet delay predictions, positioning accuracy can be improved and convergence time to achieve a certain precision-level can be reduced [15; 16; 17]. Second, the predicted wet delays can also be used in weather forecasting and NWP models in the region where the meteorological sensors or GNSS receivers are not available [18; 19; 20; 21]. The Kriging interpolation is a method of using observations taken at nearby locations and presenting predictors in the form of weighted averaging. The weights are chosen such that the corresponding errors are less than any other linear summations. It has been widely used in best linear unbiased prediction [22; 23; 24], geostatistics [25], meteorology [26], cartography [27] and other disciplines [28; 29]. Although the tropospheric wet delay is less accurately predictable than the hydrostatic delay, it is relatively stable in a small region due to the relatively homogeneous water vapor content in the atmosphere [30; 31]. This feature has provided the opportunity of implementing the Kriging interpolation to predict the zenith wet delays at any locations. This is because the weights of the Kriging interpolation depend upon the distances between the unknown points and all available measurements, as well as the covariance reflected in semivariogram. Therefore, many studies have demonstrated the use of the Kriging interpolation in generating the zenith tropospheric delay. [PERSON] [32] confirmed that the Kriging interpolation is more suitable for tropospheric delay prediction as compared to polynomial and spline technique. [PERSON] et al. [33] assessed two approaches of the Kriging interpolation, the direct interpolation and the grid interpolation, and found that only slight differences were presented between the results obtained from these two approaches. [PERSON] et al.