# Kriging Interpolation in Modelling Tropospheric Wet Delay
## 2 Kriging Interpolation in Modelling tropospheric Wet Delay  
One of the commonly used GNSS positioning model, the ionosphere-free (IF) combination model [36], is applied in this study to estimate the zenith wet delay at network and user stations. The IF combination has removed the effects of ionospheric refractions in phase and code observables and, therefore, is appropriate for the data processing experiments since this study focuses on the other atmospheric refraction, the tropospheric delay. The linearized observation equations read as  
\[ \begin{array}{c}E\left[\Delta\phi_{r}^{s}\right]=g_{r}^{sT}\Delta x_{r}+dt_{ r}+m_{w}\tau_{r}+m_{g}\cdot\cot(e)\cdot\left(g_{n}\cdot\cos(\alpha)+g_{e} \cdot\sin(\alpha)\right)+\lambda_{IF}a_{r}^{s}\\ E\left[\Delta p_{r}^{s}\right]=g_{r}^{sT}\Delta x_{r}+dt_{r}+m_{w}\tau_{r}+m_{ g}\cdot\cot(e)\cdot\left(g_{n}\cdot\cos(\alpha)+g_{e}\cdot\sin(\alpha)\right) \end{array}  \tag{1}\]  
where \(E\{\cdot\}\) is the expectation operator; \(\Delta\phi_{r}^{s}\) and \(\Delta p_{r}^{s}\) are the linearized IF observed-minus-computed phase and code observations from satellite \(s\) to receiver \(r\), in meters. The precise satellite orbit and clock corrections have been applied in the observables. \(g_{r}^{s}\) denotes the line-of-sight unit vector from the satellite to the receiver; \(\Delta x\) is the increment of the receiver position and \(dt_{r}\) denotes the receiver clock offset; \(\tau_{r}\) denotes the zenith wet delay and \(m_{w}\) its corresponding mapping function which introduces an elevation-dependent scaling factor for each satellite; \(g_{n}\) and \(g_{e}\) are the troposphere gradients in north-south and east-west direction, respectively, and \(m_{g}\) their corresponding mapping function. These gradients are to account for the effects due to the unsymmetrical troposphere [37]. In such a case, the tropospheric delay at a specific elevation \(e\) would slightly vary with the azimuth angle \(\alpha\). \(\lambda_{IF}\) denotes the wavelength of the IF combined phase observable and \(a_{r}^{s}\) is the ambiguity. Note that the IF ambiguity does not preserve the integer property, and ambiguities will always be float in the data processing in this study. One of the most common ways of Kriging interpolation is the so-called ordinary Kriging. It assumes that the attribute under consideration has a constant mean over the entire spatial domain, but the mean is unknown. It also assumes that the predictor of the unknown point is a weighted linear function of all observed data. The covariance of ordinary Kriging can be described by one covariance function, which is valid for the entire spatial domain under consideration. The covariance function describes the variation of the observed data, implying that values at two nearby positions are largely correlated, whereas two positions far away from each other will have little dependence. A further simplification of ordinary Kriging is simple Kriging, in which it is assumed that the mean value of the attribute is known. Besides, if the mean value is an unknown linear combination instead of an unknown constant, it is called universal Kriging. This study implemented ordinary Kriging because it is reasonable to assume a constant-mean wet delay in a local area. The zenith wet delays can be estimated in PPP data processing with Equation (1). Then the tropospheric estimations at the network stations are regarded as the known values \(z_{i}\) of the Kriging interpolation. Then the main procedure is to calculate the unknown variable \(z_{0}\) (zenith wet delay at user stations) as a linear combination of the known values \(z_{i}\), as shown in Equation (2). \[z_{0}=\sum_{i}w_{i}z_{i} \tag{2}\]where \(w_{i}\) is the ordinary Kriging weight of the zenith wet delay at the ith network station. In practice, the observations close to \(z_{0}\) have larger weights than those further away. The weight can be calculated from the following linear equations:  
\[