3.2.2 Multi-temporal correlation
The negative values of the multi-temporal components were eliminated with the help of equivalent substitution. Subsequently, double cumulative curves were plotted for the multi-temporal components of runoff and sediment discharge to study their multi-temporal correlations, detailed evolution, and structural breaks. The double cumulative curves of the multi-temporal components are shown below.
—————————————place Figure 5 here——————————————-
From Figure 5:
(1) Higher goodness of fit is noted in the runoff-sediment discharge double cumulative curves of the IMF1 and IMF2 components. The corresponding R2 values are 0.9996 and 0.9995, respectively. The curve of IMF3 shows structural breaks in 1978, 1989, 2001, and 2005, and that in 2005 was the most prominent. Similarly, structural breaks were noted in 1994, 1997, and 2001 for the curve of IMF4, and the most prominent occurred in 2001. Furthermore, the double cumulative curve of the RES component demonstrated that the runoff and sediment discharge showed consistent variation macroscopically. In short, the runoff-sediment discharge double cumulative curves were different at different time scales. At some time scales, more distinct structural breaks were present, indicating significant variation in the runoff-sediment discharge relationship.
(2) The slopes of the trendlines for the multi-temporal components declined gradually. This suggests that, microscopically, the components had different amplitudes and fluctuations. Local characteristics of the runoff and sediment discharge variation became observable only when the components were analyzed separately. Thus, different slopes were found for the trendlines of different double cumulative curves and the runoff-sediment discharge correlations varied at different time scales. For the IMF1 and IMF2 components, higher goodness of fit was observed in their double cumulative curves. However, lower goodness of fit was noted for the curves of the IMF3 and IMF4 components. This indicated that medium- and high-frequency components exhibited stronger runoff-sediment discharge correlations, whereas low-frequency ones demonstrated weaker correlations. Additionally, the RES component reflected the macroscopic variation of water and sediment systems. Reasonable goodness of fit was observed for its double cumulative curve, demonstrating a relatively strong runoff-sediment discharge correlation at the macroscopic scale. Hence, the characteristics of the macroscopic variations in the runoff and sediment discharge reflected by the RES component should be examined to study the runoff-sediment discharge relationship and its evolution.
(3) Overall, some points of structural breaks on the microscopic scale in the runoff-sediment discharge double cumulative curves of the multi-temporal components appeared as ordinary points on the macroscopic scale in the curve of the raw series. Hence, accurate identification of structural break points is closely related to the time scales employed during the analysis. When the raw series was examined, characteristics of the macroscopic scale may bury or neutralize those of the microscopic scale. Thus, when structural breaks are concerned, more attention should be paid to the details. For relatively complex raw series, medium- and/or high-frequency modal components can be utilized to study the runoff-sediment discharge relationships and their detailed evolution. Research on runoff-sediment discharge relationships can focus on relevant short- and/or medium-term observations. When the relationships show change and information (for example, years), structural breaks have to be determined and low-frequency components can be employed.