Exploring the sensitivity of river flow patterns to meteorological drought using multifractal and cross-correlation applications

The carbon emissions from crops (Abbas et al. 2023c), livestock (Elahi et al. 2024), and fish and industrial sectors (Abbas et al. 2022a; Elahi & Khalid 2022) are the main reasons for extreme weather events. In particular, in hydrology and climatology, drought is commonly acknowledged as an important and impactful event. There are many different types of droughts, and each has its unique characteristics. A substantial drought that affects water supply is known as a meteorological drought (Xu et al. 2022; Kheyruri et al. 2023; Qiu et al. 2023).

Meteorological drought is a natural phenomenon in which there is less than average accessible water in a given place for a protracted period. A meteorological drought may be brought on by human or natural factors. The primary natural reasons are 1. insufficient precipitation, 2. variability in the weather, 3. geographical elements, and 4. evaporation and transpiration (Marino et al. 2017; Omer et al. 2020; Parvizi et al. 2022). An extended period of low precipitation or snowfall is one of the main causes of meteorological drought. Rainfall that is significantly less than the average lowers groundwater levels, river flow, and soil moisture. Natural climate variability, such as El Niño and La Niña occurrences, can modify precipitation patterns and contribute to the development of meteorological droughts (Forootan et al. 2019; Rambal et al. 2020).

Topography, altitude, and closeness to water are examples of geographic characteristics that could impact the occurrence and severity of meteorological drought (Zhang et al. 2018; Munyaev et al. 2021). Due to increased water loss from the soil and vegetation, high rates of transpiration and evaporation might worsen meteorological droughts (Bai et al. 2018; Lintunen et al. 2021). The amount of evaporation and transpiration depends on a number of factors, including temperature, wind speed, humidity, and vegetation (Davarzani et al. 2014; Choi & Kim 2018). The frequency and intensity of meteorological droughts can be influenced by human activities in addition to natural causes, such as land use and water extraction (Ji et al. 2023).

Droughts caused by water might have a broad effect on several ecosystems and sectors. The extent of these effects is determined on the severity and duration of the drought (Meresa et al. 2023; Qiu et al. 2023; Hisdal et al. 2024). Drought has a major effect on rivers, which are among the most vital suppliers of water. A number of regional studies on drought and climate change can be mentioned, including Abbas et al. (2021, 2022a, 2022b, 2023a, 2023b). Numerous investigations have demonstrated the nonlinear relationship between river flow and drought. Therefore, an evaluation tool is needed to describe and analyze their behavior. In this case, multifractal analysis and chaos theory are two useful techniques.

Geometric patterns that show self-similarity at various sizes are called fractals. They have significant consequences for the management of water resources and are a fundamental idea in chaos theory. Fractal patterns may be seen in the temporal variability of meteorological processes, the branching patterns of watersheds, and the geographical distribution of river networks in the context of water resource systems. Water resource systems are fractal in nature, and understanding this fact can help us better understand their resilience, interconnectedness, and structure. Conversely, scaling describes the connection between a system's behaviors at various sizes. Managers are better able to anticipate changes and disruptions and respond to them when they are aware of the scaling features of water resource systems. The following studies investigated drought patterns from a fractal viewpoint: Azizi & Azizi (2024), Azizi & Azizi (2022), Seebocus et al. (2021), Ogunjo (2021), Li et al. (2020), and Yao et al. (2020).

In order to address two main questions, this study attempted to investigate the impact of meteorological drought on river flow (time series of river discharge) using two powerful tools: multifractal and chaos theory. Does meteorological drought have anything connection with nonlinear river flow patterns? The fact that drought has a wide range of effects, some of which are hidden in the complex dynamics of river flow, explains the importance of these responses. As a vital water resource, river flow may also be predicted and managed with the aid of this influence. Lastly, it provides a nonlinear dynamic viewpoint on drought's effects on other climatical and geophysical processes, which is helpful for water management and engineering.

The objectives of this investigation were to examine the impact of meteorological drought on long-term river flow conduct as earlier studies failed to address these issues and provide answers to the aforementioned question. To this end, multifractal detrended fluctuation analysis (MF-DFA), standard precipitation index (SPI), and cross-correlation (CC) were used to achieve this goal.

This study represents a novel investigation into the relationship between changes in meteorological drought and wetness in a watershed, and the multifractality of river flow, the sensitivity of the system governing river flow to large and small changes in flow, and the correlated behavior of river flow. In previous studies, this methodology has been employed to examine the statistical interrelationship between diverse parameters. However, in this study, the relationship between the dynamic behavior of the river flow and changes in meteorological drought and wetness will be elucidated. This represents a novel approach that has not previously been employed. Moreover, the relationship between the dynamic parameters of river flow and changes in meteorological drought and wetness has not been previously analyzed from this perspective.

The study methodology and methods will be explained in the next section, which will be followed by the presentation of the findings and a discussion. Finally, the study's findings will be highlighted.

Rainfall and river discharge data were from 1966 to 2016 and were collected from the Iranian Water Resources Management Organization. Prior to analysis, the data were prepared by removing outlier data and then replacing the missing data with values calculated by the mathematical technique of mode. The approaches employed will be described in the following sections.

In Equation (1), c and d are constants, and t is computed using the PDF and the GD.

It is important to note that prior to the application of the CC technique, the time series of the data were subjected to de-noising in order to prevent the generation of any erroneous results.

The opening length of the f(α) vs. α curve represents the intensity of the time-series multifractality. The higher the Δα((α_max − α_min)), the stronger the time-series multifractality. In other words, the time series becomes increasingly multifractal. Assuming Δf(α) = f(α₂) − f(α₁), if Δf(α) > 0, the time-series structure is not affected by significant local variations. If Δf(α) < 0, the time-series structure is unaffected by tiny local changes. There are two possibilities for α(0) values: >0.5 and <0.5. The time-series behavior is correlated in the first example but uncorrelated in the second.

Δα, Δf(α), and α(0) values were computed for a 20-year time series of river discharge, and their relationship to the SPI was explored using the CC method.

The time series was not subjected to the removal of the trend and seasonality. The aforementioned specifications in the nonlinear dynamic analysis represent latent characteristics that, by removing them, may result in erroneous outcomes in the dynamic analysis.

MATLAB was employed for modeling and computation.

The cross-correlation graphs for the aforementioned parameters (Δα, Δf(α), α(0), and SPI) are presented below, along with their results and interpretations. It should be noted that the correlation above the criterion line (black - - line) shown in the graph should be considered.

It should be noted that the CC technique reveals the existence of the relationship between the two parameters in question. Nevertheless, it does not reveal how and why the two parameters affect each other. Consider Figure 9 as an example, it depicts the relationship between the multifractal strength that governs river flow and the values regarding meteorological drought and wetness. In other words, this shows that the meteorological drought after 19 years has the most impact on the multifractal behavior of the flow. In the event of a significant drought or wetness, a pronounced shift in the multifractal behavior of river flow is likely to be observed after 19 years. It is important to note that the most pronounced impact can be expected after 19 years, with smaller effects being revealed in the same year in which severe drought or wetness occurs. Furthermore, the multifractal patterns of the river flow will be affected in the years following the severe meteorological drought or wetness. However, the greatest impact on the multifractal patterns of the river flow will be observed 19 years after the significant event in the catchment area. This inference process is applicable to all cross-correlation graphs. Consequently, the CC diagram does not permit the determination of the manner and cause the meteorological drought's impact on the multifractal behavior of the river flow.

The foregoing data demonstrated that as meteorological drought increased, the susceptibility of the river discharge's time-series structure to substantial changes diminished. In addition, the river flow became more correlated. Moreover, the river flow behavior exhibited more multifractal characteristics. However, the timing of the meteorological drought's influence on nonlinear river flow metrics varied.

Suman et al. (2023), Adarsh et al. (2019), and Zhang et al. (2016) conducted studies on meteorological drought using the multifractal approach. These investigations found that meteorological drought altered the dynamic patterns of river flow. These results are consistent with the findings of this research. Młyński et al. (2021) and Wu et al. (2018) conducted investigations that demonstrated how meteorological drought affected the correlation of time series of river flow, and their conclusions are identical to those of this study.

Other studies have been conducted on the multifractal properties of river flow and droughts, including Zhan et al. (2023), Rahmani & Fattahi (2021), Hou et al. (2018), Zhang et al. (2016), and Zhang et al. (2008).

However, none of the aforementioned studies examined the influence of meteorological drought on river flow behavior from a multifractal perspective or used a singularity spectrum and cross-correlation diagrams.

The significance of this study was that meteorological drought has several dimensions, and its influence on hydrological and geophysical phenomena is critical. As a result, we attempted to investigate the impact of meteorological drought on river flow behavioral patterns over a long period (long time scale). To the best of the authors' knowledge, no study has been conducted on the impact of meteorological drought on river flow behavior. In addition, the behavior patterns of rivers in the Hamoon Basin were not studied using multifractal analysis.

The findings of this research indicate that major meteorological droughts and wetnesses exert a significant impact on the dynamic behavior of river flow patterns, with effects ranging from the short term (less than 1 year) to the long term (19 years). Consequently, in order to make optimal use of the river flow, it is of the utmost importance to consider the changes in the river flow patterns in both the short and long term. Additionally, the findings indicated that the frequency of droughts is projected to increase in the coming years, which will result in significant alterations to the river flow patterns. Consequently, it is recommended that rigorous policies be implemented to regulate the consumption of river water and the underground water (which is influenced by the river flow in the basin) of the catchment area.

One of the research's limitations is its failure to consider the short-term time scale, which can be considered in future studies and compared to the river's long-term behavior (which was investigated in this study). Another limitation is that the present approach does not permit the determination of the manner and cause the meteorological drought's impact on the dynamic behavior of the river flow patterns.

The goals of this study were to investigate the influence of meteorological dryness on long-term river flow behavior, as previous research had not been successful in addressing these issues. To do this, researchers employed MF-DFA, SPI, and CC. Rainfall and discharge data regarding the Hamoon Basin from 1966 to 2016 were studied. The findings indicated that the meteorological drought increase led to:

• A reduction in the susceptibility of the river discharge's time-series structure to substantial changes.

• More correlated behavior of the river flow.

• Exhibiting more multifractal characteristics by river flow.

However, the timing of the meteorological drought's influence on nonlinear river flow metrics varied. In other words, the meteorological drought exerted its maximum effect on the correlated behavior of river flow within a period of less than 1 year. After 5 years, the meteorological drought had the greatest impact on the sensitivity of the system to both large and small changes in river flow. Finally, the meteorological drought had its maximum impact on the multifractal patterns of the river flow after 19 years.

The significance of this study was that meteorological drought has several dimensions, and its influence on hydrological and geophysical phenomena is critical. As a result, it was attempted to investigate the impact of meteorological drought on river flow behavioral patterns over a long period (long time scale).

Future studies can consider the short-term time scale for analysis to compare the results with the river's long-term behavior explored in the present study. Moreover, it is recommended that other types of droughts (e.g., hydrological or agricultural) and their interactions with meteorological drought be investigated. Furthermore, climate projections can be considered to offer insights into future river flow patterns under changing climatic conditions.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data cannot be made publicly available; readers should contact the corresponding author for details.

The authors declare there is no conflict.