Analysis of multi-scale correlation and wavelet variance of time series of precipitation and flow of Khorramabad River

Despite the of river flow affected by the amount and nature of precipitation that occurs in the river watershed, the relationship between these two hydro-climatic variables has complex dynamics in terms of frequency and intensity, especially in mountainous areas. This complex dynamic is not only due to the involvement of many natural and human components, each of which has a different multi-scale behavior, but also due to the change of these components over time. along with the change of these components, the intensity of their interaction and impact in the watershed is changed and the behavioral dynamics of the river flow is also changed. After all, precipitation is considered as the most important driving factor of the river flow, which is affected by different mechanisms, each of which is active in a different time horizon. Such different mechanisms have caused the rainfall time series to have a complex and unstable behavior and a multi-scale behavior. So that the intensity and response time of the river flow in relation to the amount of precipitation does not follow a general and linear pattern, but shows different patterns in different time scales. With these interpretations, the time series of precipitation and river flow contain small and large fluctuations. Therefore, the purpose of this study is 1- to identify the fluctuations of precipitation and flow of Khorram Abad river in different time scales. 2- Revealing the contribution of each of the small-large time horizons in determining the variance of river flow and daily precipitation.

In this research, two types of data were used, including the daily precipitation data of Khorramabad synoptic station (1961-2016) and the daily discharge data of Chamanjir discharge measuring station 1961-2016). Also, in order to carry out this research, a modified version of discrete wavelet transforms (DWT) under the name of maximum overlapping discrete wavelet transform (MODWT) was used, so in this study, for wavelet multi-representation analysis (fluctuation analysis), wavelet variance analysis and identification of dynamic relationship and correlation Multiscale between precipitation signal and daily flow of Khorramabad river was used using discrete wavelet approach with maximum overlap based on Symlet 4 (sym4) mother wavelet function. Thus, the time series of precipitation and river flow were divided into 11 scale levels using MODWT. It is worth noting that from signal analysis by MODWT, two types of coefficients including wavelet coefficients (details (D)) and scale coefficients (generalities (S)) are obtained. The scaling coefficients represent the wavelet transform coefficients with high resolution (low frequency) that show the general and smooth trend in the time series of precipitation and river flow. On the other hand, wavelet coefficients (high frequency) provide detailed information of certain trends and intervals of hydrological time series. Each component D represents a specific time period or time scale.

Multispectral analysis of precipitation and river flow signal from one to 13 levels of wavelet decomposition shows a very specific appearance of semi-annual fluctuations compared to other time scales. Also, despite the fact that the precipitation signal is more fluctuating in daily to bimonthly time scales than the river flow signal, some of their peak points correspond to each other. These peaks indicate floods and heavy rains that occurred on a daily scale: on the other hand, very distinct fluctuations with 11- and 22-year cycles were detected in their signal, which correspond to the known cycles of sunspot number and Hill cycle, respectively. The results of analyzing the variance of these two phenomena up to 11 levels of wavelet analysis show the general similarity of their wavelet variance behavior in daily to multi-year time scales, but the rate of change of river flow variance for increasing the time scale is different from the rate of change of precipitation. On the other hand, the highest rainfall variability is assigned to the daily scale (85% of the total variance). If the maximum variability of the river flow is in the form of an absolute limit flow in the annual cycle. Therefore, fast and medium fluctuations, respectively, have a very important contribution to the variability of the precipitation signal and the daily flow of Khorramabad River. Wavelet correlation between precipitation and river flow in different time scales shows a significant positive relationship at the 99% confidence level between these two variables.

With the wavelet analysis of the total variance of the rainfall signal and the flow of Khorramabad river, it was found that the intensity of variability of these two phenomena changes strongly according to the change of time scale. So that the most extreme fluctuations of precipitation and river flow occur on a daily-monthly and annual scale, respectively. Therefore, although the fluctuations of the river flow are affected by the precipitation fluctuations, they are strongly affected by the mechanisms that operate in the time scales of 6-8 months with the annual cycle. Also, in multi-year time scales, points of high values were revealed in the fluctuation of precipitation and river flow, which show the sinusoidal cycle. These fluctuation points exist in periods of 11 and 22 years. In general, the results of this study showed that the precipitation and flow of Khorramabad River have daily, weekly, monthly, annual and several-year fluctuations, and these fluctuations play an important role in the relationship between precipitation and They play a role in the flow of the river. As the relationship between the precipitation and the flow of this river has a multi-scale pattern, if their signals are analyzed, the cycles of these fluctuations can be extracted, which are sometimes noisy and irregular in the overall structure of the signal, and in this way, while identifying them, the mechanism that creates them can also be extracted.