## 3 Methods
### Innovative Trend Analysis  
Innovative trend analysis (ITA) is a non-parametric method used for identifying trends in time series data [17] regardless of data distribution and length and allows for a quick graphical understanding of trends. In this method, observational data are divided into two equal lengths starting from the beginning date and sorted from smallest to largest; one of the key advantages of the ITA method is its capability to interpret the trend behaviors of low-, medium-, and high-valued data [84]. The first sub-series is placed on the x-axis and the second sub-series on the y-axis on the Cartesian coordinate plane. If the data fall on or very close to the 1:1 (45\({}^{\circ}\)) line, it is practically understood that there is no significant trend in the series being examined. If the data fall above (or below) the 1:1 (45\({}^{\circ}\)) line and continue to deviate from the straight line in an increasing (or decreasing) manner, the trend is monotonically increasing (or decreasing). Series that do not show a continuous increase (or decrease) at smaller values but exhibit an increasing (or decreasing) character as the values increase are referred to as series with a non-monotonic increasing (or decreasing) trend.  
The steps of obtaining the confidence interval limits used to determine the significance of the trend values produced by the ITA method are described below.  
The trend slope and indicator of the ITA are calculated using Equations (1) and (2).  
\[s=\frac{2(\overline{x}-\overline{y})}{n} \tag{1}\]  
where \(s\) is the trend slope, \(n\) is the number of data points available, \(\overline{x}\) is the mean of the first part of the series, and \(\overline{y}\) is the mean of the second part of the series.  
\[D=\frac{1}{n}\sum_{i=1}^{n}\frac{10(y_{i}-x_{i})}{\overline{x}} \tag{2}\]  
\(D\) is a trend indicator (a positive value indicates an increasing trend and a negative value indicates a decreasing trend), n is the number of data points in each subseries, \(\overline{x}\) is the mean value of the first subseries, and \(x_{i}\) and \(y_{i}\) are the observed data values in the first and second subseries, respectively.  
The significance of the ITA is estimated using a probability distribution function. The confidence interval of a standard normal distribution function with zero mean and standard deviation of the ITA trend slope (\(\sigma_{\rm s}\)) at the significance level \(\alpha\) is calculated using the confidence limit of the trend slope (\(CL\)):  
\[CL_{(1-\alpha)}=0\mp s_{\sigma_{\rm s}} \tag{3}\]  
When the estimated ITA trend slope (i.e., \(s\)) is greater than the critical value, the null hypothesis of no slope is rejected. Therefore, if the data points fall outside the specified confidence limit, s calculated from the data series is considered statistically significant. In this study, a 95% confidence level is considered. An example of a graphical representation of the ITA method is presented in Figure 3.  
Figure 3: ITA example.  
Among the areas where the ITA method is most frequently used is the examination of hydrological and meteorological variables. In this context, considering the strong link between carbon cycles and water cycles [85], we seek to investigate the utility of ITA GPP trend analysis. For the ITA analysis, the trendchange package [86] in R was utilized [87].