## 3 Materials and Methods  
### Methodology  
#### 3.1.1 The FUKSS Model  
The FUKSS model is based on a universal Kriging state-space model and introduces the observed data \(Y_{t}\) with length \(n_{t}\) of different quantities and distributions into the cyclic calculation process. In the calculation process, we maintain a constant state vector \(\mathbf{Z}_{t}\) with length \(N\) and make corrections through different observation vectors. The FUKSS model is described by the following state and measurement equations:  
\[\mathbf{Z}_{t}=\mathbf{Z}_{t-1}+\mathbf{F}^{T}\mathbf{\beta}_{t}+\mathbf{\epsilon}_{t}, \tag{1}\]  
\[\mathbf{Y}_{t}=\mathbf{W}_{t}\mathbf{Z}_{t}+\mathbf{v}_{t}, \tag{2}\]  
where \(\mathbf{Z}_{t}=[Z_{t}(x_{1}),\ldots,Z_{t}(x_{n})]^{T}\) are the state values at \(N\) sensor nodes positions, and \(\mathbf{v}_{t}\) is a vector of observation noise \((n_{t}*1)\), as determined by the measurement error of the sensor. We can determine that \(\mathbf{v}_{t}=[\mathbf{v}_{t}(x_{1}),...,\mathbf{v}_{t}(x_{n_{t}})]\) is white noise so that \(E[\mathbf{v}_{t}\mathbf{v}_{t}^{T}]=\mathbf{R}_{t}\), and it is uncorrelated with all past noise and data. \(\mathbf{F}^{T}\mathbf{\beta}_{t}\) is the expected value, where \(\mathbf{F}=[\mathbf{f}(x_{1}),\ldots,\mathbf{f}(x_{n})]\), \(\mathbf{\beta}_{t}=[\mathbf{\beta}_{t1}\ldots\mathbf{f}_{tr}]^{T}\) and \(\mathbf{f}(x)=[\mathbf{f}_{1}(x)\ldots\mathbf{f}_{r}(x)]^{T}\). Among them \(\mathbf{f}(x)\) denoting the \(r\) known drift functions, which often used in universal Kriging [33; 34] estimation, and \(\mathbf{\beta}_{t}\) is unknown drift coefficients vector \((r*1)\). \(\mathbf{\epsilon}_{t}=[\mathbf{\epsilon}_{t}(x_{1}),\ldots,\mathbf{\epsilon}_{t}(x_{n})]\), \(\mathbf{\epsilon}_{t}(x)\) denotes the spatially correlated errors, with zero mean and known covariance function:  
\[E[\mathbf{\epsilon}_{t}(x_{a})\mathbf{\epsilon}_{t}(x_{b})]=k(x_{a},x_{b}). \tag{3}\]  
In the calculation process, we introduce the selection matrix \(\mathbf{W}_{t}\) with size \(n_{t}*N\) and rank \(n_{t}\). For a sensor network with \(N\) sensor nodes, all sensors are numbered to obtain a vector for facilitate calculation. Therefore, the T rounds data collection will form a measurement matrix, as shown in Figure 1. Here, each row of \(\mathbf{W}_{t}\) has only one 1 and the rest of the entries are 0, where \(\mathbf{W}_{t}(i,j)=1\) representing the \(i\)th data \(Y_{t}(x_{i})\) of \(Y_{t}\) in the coordinate of measurement matrix is \(j\) at time \(t\). The expression is:  
\[\mathbf{W}_{t}(i,j)=\left\{\begin{array}{l}1,\text{The $i$th node of $\mathbf{Y}_{t}$ in the coordinate of sensor network is $j$.}\\ 0,\text{otherwise}\end{array}\right. \tag{4}\]  
The essence of the FUKSS model is to maintain a fixed state variable \(\mathbf{Z}_{t}\), and use a different number of observation variables \(\mathbf{Y}_{t}\) to calibrate and estimate \(Z_{t}(x)\), in order to realize estimation of the whole process. In this model, the current state, \(Z_{t}(x)\), is determined by the previous state \(Z_{t-1}(x)\) and the stochastic model \(\mathbf{f}^{T}(x)\mathbf{\beta}_{t}+\mathbf{\epsilon}_{t}(x)\) used in universal Kriging. Therefore, we can obtain the estimated for a generic spatial site \(x\), as follows:\[\begin{split}\widehat{Y}_{t}(x)=\widehat{Z}_{t}(x)\\ =\widehat{Z}_{t-1}(x)+\mathbf{f}^{T}(x)\mathbf{\beta}_{t}+\varepsilon_{t}( x).\end{split} \tag{5}\]