## 2 Materials and Methods
### Data and Methods
#### 2.2.3 Multi-Source Time Series Remote Sensing Image Fusion Method Based on Space-Time Features
The Mahalanobis distance algorithm is:  
\[D^{2}_{math}=\left(X_{i}-\overline{X}_{r}\right)^{\top}\sum\
olimits_{\left(X _{i}-\overline{X}_{r}\right)}^{\top} \tag{2}\]  
where \(X_{i}\) indicates the value of pixel \(i\) which is to be evaluated, \(\overline{X}_{r}\) indicates the average pixel value of category \(r\), \(\Sigma\) is the covariance matrix and \(D^{2}_{math}\) reflects the value dissimilarity of the pixels \(i\) and the remaining pixels of the category which they belong to. Because the Mahalanobis distance assumes that the data is multivariate normal, the distance values above approximate follow chi-square distribution. Here, a 90% quantile point is selected as the criterion for evaluating whether a pixel value is abnormal or not. If a pixel value is greater than  
Figure 2: Data processing and analysis flow chart. the quantile point value, it is regarded as abnormal. In this case, the pixel value is replaced with the average pixel value of the category which it belongs to. * Mann-Kendall and Theil Sen Slope Analysis  
The Theil Sen Slope combination Mann-Kendall detection method is an important trend analysis approach [25], which has been increasingly applied to the study of vegetation change [26; 27; 28]. In the Mann-Kendall trend test, the following assumptions are made: (1) the null hypothesis \(H_{0}\) is: the time series data to be detected \((X_{1},X_{2},\ldots,X_{n})\) are randomly with no significant trend; (2) the alternative hypothesis \(H_{1}\) is: for all \(k\), \(j\leq n\), and \(k\
eq j\), the distribution of \(X_{k}\) and \(X_{j}\) has a tendency to increase or decrease. The calculation of test statistics \(S\) is shown as:  
\[S=\sum_{k=1}^{n=1}\sum_{j=k+1}^{n}\text{Sgn}\big{(}X_{j}-X_{k}\big{)} \tag{3}\]  
\[\text{Sgn}\big{(}X_{j}-X_{k}\big{)}=\left\{ \begin{array}{cc}+1&X_{j}-X_{k}>0 \\ 0&X_{j}-X_{k}=0\\ -1&X_{j}-X_{k}<0\end{array} \right. \tag{4}\]  
When \(n>10\), \(S\) approximately follows standard normal distribution with its mean being 0 and its variance expressed as \(V_{ar}(S)=n(n-1)(2n+5)/18\). The standard normal distribution variable \(Z\) is the test statistic for the trend test and is computed as:  
\[Z= \begin{cases}\frac{S-1}{\sqrt{Var(S)}}&s>0\\ 0&s=0\\ \frac{S+1}{\sqrt{Var(S)}}&s<0\end{cases}  \tag{5}\]  
In the bilateral trend test, for a given significance level \(a\), if \(-Z_{1-a/2}\leq Z\leq Z_{1-a/2}\), the null hypothesis \(H_{0}\) is considered to be reliable, indicating that there is no apparent change trend in the time series. If \(|Z|>Z_{1-a/2}\), \(H_{1}\) is considered more reliable, implying that the sequence has a significant increase or decrease trend. For the test statistic \(Z\), \(Z>0\) means that the time series has an increasing trend, while \(Z<0\) suggests a decreasing trend. In the significance test, when the confidence is 90%, \(Z_{0.95}\) is equal to 1.64; when the confidence is 95%, \(Z_{0.975}\) is equal to 1.96. Figure 3: Original and interpolated hypothesized time series (adapted from Figure 4 on page 281 in the paper by [PERSON] et al. [11]). When \(n<10\), a bilateral trend test is performed based on the test statistic \(S\). With a given significance level \(a\), if \(|S|\leq S_{a/2}\), we accept the null hypothesis. It shows that the time series has no statisticalobvious change trend. Otherwise, we reject the null hypothesis and accept the alternative hypothesis, concluding that the time series is significantly increasing or decreasing. For the test statistic \(S\), if \(S>0\), it indicates that the time series has an upward trend. If \(S=0\), it indicates that there is no apparent trend. If \(S<0\), it means that the time series has a decreasing trend. The Theil Sen slope [29; 30] is obtained by taking the median value of the ratio of the gray value (or exponent value) of different years to the time difference as shown in Equation (6). The median value can effectively avoid the noise disturbance. There is no significance test for this value.