## Abstract

This paper presents an in-depth analysis of the variability and trend detection of water discharge in the Hieu River Basin over the period from 1/1/1961 to 31/12/2020. Firstly, the wavelet analysis is implemented to investigate the flow variability at different time scales. Using time series of daily flow at Quy Chau and Nghia Khanh, the periodicities of flow are examined, showing that periodic oscillations in flow mainly occurred at 1 year, from 2 to 4 years, and from 4 to 8 years. Secondly, discharge variability is investigated, revealing an increase from January to September (excluding June) and a decrease from October to December. At Quy Chau, flow increased seasonally and annually by 0.19 and 0.06 m^{3}/s, respectively, while its values at Nghia Khanh decreased of up to −0.50 m^{3}/s. Thirdly, temporal trends of flow are assessed using Mann–Kendall and Sen's slope estimator, and sequential Mann–Kendall test. Results show flow decreases during 1961–1969, 1973–1976, and 1996–2020 and increases in 1969–1973 and 1976–1996. Temporal patterns of flow at Quy Chau and Nghia Khanh demonstrated synchronization across different time scales. Relationship between flow and rainfall is discussed to investigate insights into their relationship.

## HIGHLIGHTS

Wavelet analysis is applied to examine wavelet power spectrum of flow at different time scales.

Flow increases from January to September (excluding June) and a decrease from October to December at the Quy Chau station.

Water discharge decreases during 1961–1969, 1973–1976, and 1996–2020 and increases in 1969–1973 and 1976–1996.

Flow at Nghia Khanh has a stronger relationship with rainfall than those at Quy Chau.

## INTRODUCTION

Water discharge, a volume of water flowing through a given river section over a specific duration, is a critical hydrological quantity that impacts various aspects of the environment, ecology, society, and economy in river basins. In terms of flood prediction and management, for example, by analyzing historical data on water discharge, different hydrological models can be developed to forecast potential flood events, allowing communities to take preventive measures, evacuate residents, and deploy resources to minimize damage and loss of life (Toonen 2015). Water discharge also impacts the availability of water resources for (i) drinking water, (ii) irrigation and drainage, (iii) hydropower generation, (iv) infrastructure such as dams, bridges, and levees, (v) river navigation, waterways, and transportation, and (vi) biodiversity, aquatic habitats and ecosystems as well (Sen 2009; Wang *et al.* 2013; Safaria *et al.* 2020). Climate change can alter precipitation patterns, leading to shifts in discharge variability (e.g., Lui *et al.* 2012). Investigating changes in water discharge over time provides insights into how climate change is affecting hydrological cycles and helps to project future impacts on water resources and ecosystems. On the other hand, water quality related to sediment transport, nutrient levels, and contaminant distribution is also affected by water discharge, requiring a detailed understanding of how flow interacts with water quality for managing and maintaining drinking water supplies (Zheng *et al.* 2019). Thus, studying variability in water discharge is essential for informed decision-making, sustainable resource management, disaster preparedness, ecosystem preservation, and understanding the complex interactions between water, climate, and human activities in river basins.

Different methods such as statistical techniques, time series analysis, and Fourier analysis can be implemented to examine the patterns and trends of time series data of water discharge. The statistical techniques including various measure quantities such as mean, median, range, variance, standard deviation, skewness, kurtosis, and coefficient of variation commonly provide basic information about the central tendency, initial overview of the time series data of water discharge as well as its variability (Sen 2009; Meals *et al.* 2011). Time series analysis is also used to explore patterns and trends over time of the water discharge (Lui *et al.* 2012). Spectral analysis, which is often applied in hydrology, decomposes time series data into frequency components. Such method can help to identify periodic patterns and dominant frequencies in water discharge variability (Grinsted *et al.* 2004; Sen 2009). This method together with Fourier analysis is also used to identify seasonal patterns and long-term fluctuations in discharge in the Huangfuchuan Basin, China (Chen *et al.* 2016) or in England and Wales (Sen 2009). Thus, using a combination of these methods can provide a comprehensive understanding of variability in water discharge.

Regarding trend detection, several approaches such as regression analysis, non-parameter tests, and wavelet analysis are commonly used for investigating the variable trend of time series data in general and particularly of time series of water discharge (Torrence & Compo 1998; Sen 2009). Regression analysis including linear, nonlinear, and multiple regression models can be used to quantify the impact of precipitation on discharge variability (Safaria *et al.* 2020). Non-parametric tests like the Mann–Kendall test and Sen's slope estimator are useful for detecting trends in time series data of water discharge without assuming a specific distribution (Kendall 1975; Cong *et al.* 2010; da Silva *et al.* 2015). Continuous wavelet transforms and the Mann–Kendall test are also applied to identify (i) spatial and temporal variation patterns and (ii) multi-scale trends and abrupt changes in water discharge in the Yellow river basin, China (Lui *et al.* 2012). These examples suggest that regression analysis, non-parameter tests, and wavelet analysis can be used to examine trend detection of water discharge in the present study.

Variability in water discharge at seasonal, annual, and monthly scales has been extensively investigated in many studies by employing parametric and non-parametric tests (e.g., Kuriqi *et al.* 2020; Sönmez & Kale 2020; Ashraf *et al.* 2021; Waikhom *et al.* 2023). Sönmez & Kale (2020) applied the Mann–Kendall and Spearman's *ρ* tests to examine temperature and rainfall effects on annual stream flow in the Filyos river basin, Turkey. Kuriqi *et al.* (2020) also used the Mann–Kendall method to investigate the seasonal shift and streamflow variability in the Godavari river basin, India. Ashraf *et al.* (2021) investigated the variations of monthly, seasonal, and annual stream flow time series at 20 stations over the the Upper Indus River Basin by using Mann–Kendall and Spearman's *ρ* tests. Waikhom *et al.* (2023) analyze the monthly precipitation and streamflow to assess the variability in trends of precipitation over the Narmada basin, Brazil using the Mann-Kendall and Spearman's *ρ* tests.

The main objectives of the present study are three-fold: (i) to examine the periodicity of the monthly, seasonal, and annual series of water discharge using the wavelet analysis, (ii) to identify the variability of water discharge at different time scales such as monthly, seasonal, and annual, and (iii) to investigate temporal variability of water discharge at different time scales (monthly, seasonal, and annual) using the sequential Mann–Kendall test. The daily time series of meteo-hydrological data (e.g., rainfall, water discharge) at Quy Chau and Nghia Khanh locations collected in the period from 1961 to 2020 in the Hieu river basin, Vietnam are used for detailed investigations mentioned above.

## STUDIED RIVER BASIN AND DATA COLLECTION

### Studied river basin

^{2}, with the river basin area from its source to the Nghia Khanh hydrological station encompassing about 4,020 km

^{2}. Comprising numerous tributaries like Nam Quang, Nam Giai, Ke Coc – Khe Nha, Chang, Dinh, Khe Nghia, and Khe Da, the Hieu River is characterized by gravel materials and exhibits narrow and shallow features in its upstream region (from the source to Nghia Dan). During the dry season (from December to July), the water depth ranges from 0.5 to 2.0 m in this area. As the river flows downstream, it undergoes substantial geomorphic changes due to flow variations and meandering tendencies. In the downstream section, the river's width widens significantly, spanning from 150 to 280 m during the flood season (from August to November) and from 100 to 120 m in the dry season. These varying characteristics of the Hieu River influence its hydrological behavior, making it an essential component of the Ca River system in the north-central region of Vietnam.

Regarding weather characteristics, the annual evapotranspiration ranges from 700 to 940 mm, while the annual moisture content varies between 85 and 94%. Monthly rainfall exhibits considerable variation, ranging from 13.8 to 322.0 mm, contributing to an annual rainfall between 1,170 and 2,110 mm. The monthly temperature at Quy Chau fluctuates between 17.4 and 28.5 °C. The average annual flow at the Quy Chau station measures approximately 77.5 m^{3}/s. However, the daily flow demonstrates substantial variability, spanning from as low as 6 m^{3}/s during the dry season to as high as 3,690 m^{3}/s during the flood season. Remarkably, significant floods often occur in September and October, whereas the three driest months are typically observed from February to April. The total water volume during the flood season amounts to 55–75% of the annual water volume, indicating the substantial impact of the flood period on the overall water balance in the region.

The Hieu river basin has an important role in providing the water supply for Que Phong, Quy Chau, Quy Hop, Nghia Dan, and Tan Ky districts in Nghe An province. In detail, the Chang and Dinh tributaries, among the different tributaries mentioned above, are the two largest tributaries that supply water for different users in the Que Phong district. It is thus necessary to determine quantitatively the daily flow in the river basin to allow for achieving a good assessment of the water supply as well as to enhance biodiversity and river ecosystems. Thus, by obtaining accurate daily flow data, effective water resource management and conservation efforts can be implemented, benefitting both the local communities and the ecological health of the river basin.

### Data collection

Name . | Stations . | Statistical properties . | Quantity . | Data collected period . | |||||
---|---|---|---|---|---|---|---|---|---|

Longitude . | Latitude . | Range . | Mean . | Standard deviation . | Skewness . | Kurtosis . | |||

Quy Chau | 105°06′00″ | 19°34′00″ | 0–304.1 | 4.58 | 14.58 | 6.92 | 77.44 | Rainfall (mm) | 1962–2020 |

Quy Chau | 105°12'05″ | 19°33'50″ | 6.7–3,690 | 77.41 | 114.68 | 10.40 | 196.85 | Water discharge (m^{3}/s) | 1961–2020 |

Nghia Khanh | 105°12′00″ | 19°26′00″ | 8.0–4,800 | 123.0 | 220.78 | 8.26 | 100.68 | Water discharge (m^{3}/s) | 1969–2020 |

Name . | Stations . | Statistical properties . | Quantity . | Data collected period . | |||||
---|---|---|---|---|---|---|---|---|---|

Longitude . | Latitude . | Range . | Mean . | Standard deviation . | Skewness . | Kurtosis . | |||

Quy Chau | 105°06′00″ | 19°34′00″ | 0–304.1 | 4.58 | 14.58 | 6.92 | 77.44 | Rainfall (mm) | 1962–2020 |

Quy Chau | 105°12'05″ | 19°33'50″ | 6.7–3,690 | 77.41 | 114.68 | 10.40 | 196.85 | Water discharge (m^{3}/s) | 1961–2020 |

Nghia Khanh | 105°12′00″ | 19°26′00″ | 8.0–4,800 | 123.0 | 220.78 | 8.26 | 100.68 | Water discharge (m^{3}/s) | 1969–2020 |

## METHODS

### Wavelet analysis

Wavelet analysis is a powerful signal analysis method that has been applied in various applications including environmental flow analysis. The method is widely used to examine wavelet spectrum power of time series data such as precipitation or water discharge at specific locations of interest. This method helps to identify dominant modes of variability and to understand how these modes change over different time scales by decomposing time series data. The continuous wavelet transform is the key tool used in wavelet analysis. It projects the time series of data (e.g., water discharge data) into an analytic wavelet space by convolving the time series data with a dilated and translated mother Morlet wavelet.

**x**is defined as Equation (1).where

*ν*is the scale parameter and

*ψ*(

*t*) is the complex conjugate of the Morlet wavelet that is given by:with

*f*

_{o}is the central frequency of the Morlet wavelet and its value of 0.85 is chosen in the present study because this value is often applied in practical applications (Addison 2002).

**x**and

**y**with different wavelet transform

*w*(

_{x}*ν*,t) and

*w*(

_{y}*ν*,t), , the wavelet coherence for two-time series is given by Equation (3).where 〈·〉 represents a localized smoothing operation in both time and wavelet scale performed on the constituent transform components.

Wavelet spectrum and wavelet coherence are two components when using wavelet analysis. The wavelet spectrum highlights regions in the time-frequency plane with significant power, revealing dominant modes of variability. Conversely, wavelet coherence quantifies the strength of the covariance between two time series in time-frequency space, providing insights into their correlations. To assess significance, the wavelet spectrum can be compared with a red noise spectrum, distinguishing meaningful features from random noise. On the other hand, the significant level of wavelet coherence is determined through the Monte Carlo method, employing surrogate datasets to establish a threshold for meaningful relationships between the time series. These methods play a crucial role in uncovering complex patterns and associations within time series data, particularly in environmental flow analysis where periodicity and changes in dominant modes of variability at different time scales may be crucial for understanding the hydrological behavior of a river basin or water network system.

### Trend detection and analysis

#### The Mann–Kendall trend test

The Mann–Kendall, which is well-known as a non-parametric test or statistical method, is often used for determining the trend of time series of water discharge because of the simplicity of the method. Indeed, this method does not require the data to be aligned to any statistical distribution and is flexible to outliers in the data. The method assumes a null hypothesis of no monotonic trend in the data series while its alternative hypothesis assumes that there is the presence of a monotonic trend in the time series data.

*X*=

*x*

_{1}, x_{2},*…*

*x*, the Mann–Kendall test statistic

_{N}*S*is given by Kendall (1975):where

*N*is the number of data point in the time series data,

*x*

_{j}and

*x*

_{i}are the data values in the time series

*j*and

*i*(

*j*>

*i*), respectively, sign(

*x*–

_{j}*x*) is the sign function as indicated in Equation (5).

_{i}*S*and the assumpltion that the time series data are independent and identically distributed, the statisitics

*S*approximately normally distributied when

*N*is greater than 8 (Liu

*et al.*2012), with the variance Var(

*S*) is computed by using Equation (6).with

*m*being the number of tied groups and

*t*is the

_{p}*p*th group.

A positive value of *Z* shows an increasing trend, while a negative value of *Z* indicates a decreasing trend. The trend is significant if *Z* is greater than the standard normal variate *Z _{α}*

_{/2}, where

*α*% is the significant level. In this study, a significant or critical value of

*α*= 0.05 is used.

#### The sequential Mann–Kendall test

The progressive values *u*(*t*) of the Mann–Kendall test were determined in order to see the change of trend with time. Similar to the *Z* value, *u*(*t*) is a standardized variable with zero mean, unit standard deviation, and sequentially fluctuating behavior around the zero level (Partal & Kucuk 2006).

#### The Sen's slope estimator test

*N*being pairs of series data.

An increasing trend is corresponding to a positive value of *β*, a decreasing trend is corresponding to a negative value of *β*.

## RESULTS

### Results of water discharge periodicity

As observed from Figure 4(a) for the monthly series of water discharge at Quy Chau, annual oscillations are presented for the periods of 1962–1964, 1970–1974, 1978–1981, 1986–1988, 1995–1997, 2005–2012, 2018–2019, respectively, with a significance at 95% confidence level. The wavelet power is much weaker in the periods 1964–1970, 1974–1978, 1981–1986, 1988–1995, 1997–2005, and 2012–2018, respectively. The inter-annual oscillations at 1–3 years are seen in the period from 1972 to 1982, with the oscillation center at the year 1978. Indeed, annual oscillation at 5–11 years is presented in the period from 1972 to 2008, with an oscillation center at the year 1990. These inter-annual oscillations are not significant at a 95% confidence level. The other oscillations are even weaker.

At Nghia Khanh (Figure 4(b)), the results show that the 1-year oscillations are obtained in the period of 1970–1975, 1977–1983, 1986–1990, 1996–1998, 2005–2015, and 2018–2019, with oscillation center at 1972, 1981, 1989, 1997, 2010, and 2018, respectively, all of which are significant at 95% confidence level, but these are become less appear in the period from 1998 to 2005. Inter-annual oscillations at 2–4 years are seen in the period from 1970 to 1983, and are significant at 95% confidence level during 1973–1981, with an oscillation center in 1978. Inter-annual oscillation at 4–8-year scales can also be seen in the period from 1979 to 2010 with oscillation center at 1995, but there is no significance at a 95% confidence level.

In terms of the seasonal time scale, the water discharge for the flood season (from August to November) and dry season (from December to July) are firstly calculated based on the daily time series of water discharge at locations of interest. The wavelet analysis was then applied to examine the wavelet power spectrum of the seasonal water discharge as shown in Figure 5. The inter-annual oscillations at 2–3 years are seen in the period from 1974 to 1982, with the oscillation center at the year 1979 for the Quy Chau (Figure 5(a)). Annual oscillations at 5–12 years are presented in the period from 1970 to 2006, with oscillation center at year 1985. At Nghia Khanh (Figure 5(b)), similar pattern is also observed for the seasonal water discharge in the period from 1969 to 2020, with the inter-annual oscillation at 2–3 years and at 4–12 years are observed in the period 1974–1983 and 1979–2015, respectively.

Figure 6 shows the results of wavelet power spectrum of annual water discharge at two locations of interest, revealing that the inter-annual oscillations at 2–3 years are seen in the period from 1974 to 1982, with the oscillation center at year 1979 for the Quy Chau (Figure 6(a)). Annual oscillations at 5–12 years are presented in the period from 1970 to 2006, with oscillation center at year 1985. At Nghia Khanh (Figure 6(b)), similar pattern is also observed for the seasonal water discharge in the period from 1969 to 2020, with the inter-annual oscillation at 2–3 years and at 4–12 years are observed in the period 1974–1983 and 1979–2015, respectively.

### Results of water discharge variability

^{3}/s/month (Table 2). In addition, the linear trend and Sen's slope estimator show that the monthly flow tends to increase in periods from January to May and from July to September, while it decreases in June and from October to December (see Figure 8).

Month Properties . | Jan . | Feb . | Mar . | Apr . | May . | Jun . | Jul . | Aug . | Sept . | Oct . | Nov . | Dec . |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Quy Chau | ||||||||||||

Min (m^{3}/s) | 15.4 | 11.9 | 10.3 | 13.5 | 23.5 | 25.5 | 25.1 | 49.3 | 47.2 | 42.1 | 28.7 | 21.2 |

Max (m^{3}/s) | 149.2 | 115.0 | 77.2 | 77.7 | 174.8 | 157.4 | 290.7 | 312.0 | 425.0 | 518.6 | 265.2 | 184.2 |

Mean (m^{3}/s) | 39.5 | 33.3 | 29.4 | 30.1 | 55.8 | 74.4 | 86.2 | 124.0 | 173.1 | 150.8 | 80.1 | 50.1 |

Coefficient of variation (m^{3}/s/month) | 0.50 | 0.50 | 0.44 | 0.39 | 0.50 | 0.42 | 0.63 | 0.42 | 0.48 | 0.65 | 0.54 | 0.48 |

Linear trend | 0.03 | 0.05 | 0.07 | 0.07 | 0.27 | −0.21 | 0.31 | 0.79 | −0.02 | −0.13 | −0.13 | −0.06 |

S | 94 | 114 | 110 | 164 | 224 | −144 | 84 | 318 | 139 | −62 | −224 | −78 |

Z (Mann–Kendall) | 0.59 | 0.72 | 0.70 | 1.04 | 1.42 | −0.91 | 0.53 | 2.02 | 0.88 | −0.39 | −1.42 | −0.49 |

p-value | 0.05 | 0.047 | 0.049 | 0.03 | 0.015 | 0.036 | 0.06 | 0.04 | 0.038 | 0.070 | 0.015 | 0.062 |

Sen's slope | 0.04 | 0.05 | 0.04 | 0.06 | 0.21 | −0.23 | 0.17 | 0.68 | 0.35 | −0.20 | −0.33 | −0.05 |

Nghia Khanh | ||||||||||||

Min (m^{3}/s) | 20.4 | 16.2 | 13.7 | 16.8 | 25.4 | 18.9 | 30.2 | 64.2 | 51.9 | 60.6 | 31.5 | 27.2 |

Max (m^{3}/s) | 148.1 | 138.0 | 143.3 | 90.0 | 168.1 | 264.6 | 466.6 | 489.3 | 806.0 | 776.2 | 426.9 | 183.0 |

Mean (m^{3}/s) | 53.4 | 44.6 | 39.8 | 40.5 | 75.8 | 104.8 | 124.7 | 200.9 | 318.8 | 280.0 | 123.0 | 66.5 |

Coefficient of variation (m^{3}/s/month) | 0.39 | 0.43 | 0.49 | 0.42 | 0.49 | 0.51 | 0.75 | 0.47 | 0.60 | 0.69 | 0.63 | 0.42 |

Linear trend | −0.19 | −0.12 | −0.06 | −0.13 | 0.08 | −0.36 | 0.16 | 0.61 | −1.25 | −1.38 | −0.48 | −0.33 |

S | −122 | −72 | −18 | −128 | 20 | −116 | 32 | 96 | 52 | −119 | −170 | −192 |

Z (Mann–Kendall) | −0.95 | −0.56 | −0.13 | −1.00 | 0.15 | −0.91 | 0.24 | 0.75 | 0.40 | −0.93 | −1.33 | −1.51 |

p-value | 0.034 | 0.058 | 0.089 | 0.032 | 0.088 | 0.036 | 0.081 | 0.045 | 0.069 | 0.035 | 0.018 | 0.013 |

Sen's slope | −0.17 | −0.08 | −0.01 | −0.10 | 0.06 | −0.45 | 0.11 | 0.65 | 0.51 | −1.75 | −0.59 | −0.32 |

Month Properties . | Jan . | Feb . | Mar . | Apr . | May . | Jun . | Jul . | Aug . | Sept . | Oct . | Nov . | Dec . |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Quy Chau | ||||||||||||

Min (m^{3}/s) | 15.4 | 11.9 | 10.3 | 13.5 | 23.5 | 25.5 | 25.1 | 49.3 | 47.2 | 42.1 | 28.7 | 21.2 |

Max (m^{3}/s) | 149.2 | 115.0 | 77.2 | 77.7 | 174.8 | 157.4 | 290.7 | 312.0 | 425.0 | 518.6 | 265.2 | 184.2 |

Mean (m^{3}/s) | 39.5 | 33.3 | 29.4 | 30.1 | 55.8 | 74.4 | 86.2 | 124.0 | 173.1 | 150.8 | 80.1 | 50.1 |

Coefficient of variation (m^{3}/s/month) | 0.50 | 0.50 | 0.44 | 0.39 | 0.50 | 0.42 | 0.63 | 0.42 | 0.48 | 0.65 | 0.54 | 0.48 |

Linear trend | 0.03 | 0.05 | 0.07 | 0.07 | 0.27 | −0.21 | 0.31 | 0.79 | −0.02 | −0.13 | −0.13 | −0.06 |

S | 94 | 114 | 110 | 164 | 224 | −144 | 84 | 318 | 139 | −62 | −224 | −78 |

Z (Mann–Kendall) | 0.59 | 0.72 | 0.70 | 1.04 | 1.42 | −0.91 | 0.53 | 2.02 | 0.88 | −0.39 | −1.42 | −0.49 |

p-value | 0.05 | 0.047 | 0.049 | 0.03 | 0.015 | 0.036 | 0.06 | 0.04 | 0.038 | 0.070 | 0.015 | 0.062 |

Sen's slope | 0.04 | 0.05 | 0.04 | 0.06 | 0.21 | −0.23 | 0.17 | 0.68 | 0.35 | −0.20 | −0.33 | −0.05 |

Nghia Khanh | ||||||||||||

Min (m^{3}/s) | 20.4 | 16.2 | 13.7 | 16.8 | 25.4 | 18.9 | 30.2 | 64.2 | 51.9 | 60.6 | 31.5 | 27.2 |

Max (m^{3}/s) | 148.1 | 138.0 | 143.3 | 90.0 | 168.1 | 264.6 | 466.6 | 489.3 | 806.0 | 776.2 | 426.9 | 183.0 |

Mean (m^{3}/s) | 53.4 | 44.6 | 39.8 | 40.5 | 75.8 | 104.8 | 124.7 | 200.9 | 318.8 | 280.0 | 123.0 | 66.5 |

Coefficient of variation (m^{3}/s/month) | 0.39 | 0.43 | 0.49 | 0.42 | 0.49 | 0.51 | 0.75 | 0.47 | 0.60 | 0.69 | 0.63 | 0.42 |

Linear trend | −0.19 | −0.12 | −0.06 | −0.13 | 0.08 | −0.36 | 0.16 | 0.61 | −1.25 | −1.38 | −0.48 | −0.33 |

S | −122 | −72 | −18 | −128 | 20 | −116 | 32 | 96 | 52 | −119 | −170 | −192 |

Z (Mann–Kendall) | −0.95 | −0.56 | −0.13 | −1.00 | 0.15 | −0.91 | 0.24 | 0.75 | 0.40 | −0.93 | −1.33 | −1.51 |

p-value | 0.034 | 0.058 | 0.089 | 0.032 | 0.088 | 0.036 | 0.081 | 0.045 | 0.069 | 0.035 | 0.018 | 0.013 |

Sen's slope | −0.17 | −0.08 | −0.01 | −0.10 | 0.06 | −0.45 | 0.11 | 0.65 | 0.51 | −1.75 | −0.59 | −0.32 |

^{3}/s/month when using the linear trend and from −0.01 to −1.75 m

^{3}/s/month when using Sen's slope estimator test (Table 2). The monthly flow rises in July and August, with a value ranging between 0.11 and 0.65 m

^{3}/s/month. The coefficient of variation of monthly flow changes between 0.39 and 0.75 m

^{3}/s/month (Table 2). Detailed values of Mann–Kendall and Sen's slope are shown in Figure 10 for both considered locations.

^{3}/s in the dry season, while its value rises between 0.13 and 0.19 m

^{3}/s in the flood season (Table 3). These values are about −0.10 and −0.50 m

^{3}/s in dry and flood seasons, respectively, for Nghia Khanh. The coefficient variation of seasonal flow changes slightly between seasons, ranging from 0.33 to 0.40 m

^{3}/s/season (Table 3).

Season Properties . | Flood . | Dry . | Annual . | Flood . | Dry . | Annual . |
---|---|---|---|---|---|---|

Quy Chau . | Nghia Khanh . | |||||

Min (m^{3}/s) | 56.98 | 26.93 | 37.68 | 66.72 | 30.40 | 54.63 |

Max (m^{3}/s) | 241.35 | 109.65 | 134.63 | 514.58 | 161.42 | 242.59 |

Mean (m^{3}/s) | 131.99 | 49.83 | 77.21 | 230.67 | 68.75 | 122.72 |

Coefficient of variation (m^{3}/s/season) | 0.33 | 0.34 | 0.28 | 0.40 | 0.36 | 0.32 |

Linear trend | 0.13 | 0.07 | 0.09 | −0.62 | −0.12 | −0.29 |

S | 48.0 | 62.0 | 64.0 | −80.0 | −52.0 | −122.0 |

Z (Mann–Kendall) | 0.30 | 0.39 | 0.40 | −0.62 | −0.40 | −0.95 |

p-value | 0.076 | 0.070 | 0.069 | 0.053 | 0.069 | 0.034 |

Sen's slope | 0.19 | 0.04 | 0.06 | −0.50 | −0.08 | −0.33 |

Season Properties . | Flood . | Dry . | Annual . | Flood . | Dry . | Annual . |
---|---|---|---|---|---|---|

Quy Chau . | Nghia Khanh . | |||||

Min (m^{3}/s) | 56.98 | 26.93 | 37.68 | 66.72 | 30.40 | 54.63 |

Max (m^{3}/s) | 241.35 | 109.65 | 134.63 | 514.58 | 161.42 | 242.59 |

Mean (m^{3}/s) | 131.99 | 49.83 | 77.21 | 230.67 | 68.75 | 122.72 |

Coefficient of variation (m^{3}/s/season) | 0.33 | 0.34 | 0.28 | 0.40 | 0.36 | 0.32 |

Linear trend | 0.13 | 0.07 | 0.09 | −0.62 | −0.12 | −0.29 |

S | 48.0 | 62.0 | 64.0 | −80.0 | −52.0 | −122.0 |

Z (Mann–Kendall) | 0.30 | 0.39 | 0.40 | −0.62 | −0.40 | −0.95 |

p-value | 0.076 | 0.070 | 0.069 | 0.053 | 0.069 | 0.034 |

Sen's slope | 0.19 | 0.04 | 0.06 | −0.50 | −0.08 | −0.33 |

^{3}/s in year are observed for the coefficient of variation at Quy Chau and Nghia Khanh, respectively. Linear method, Mann–Kendall (via

*Z*quantity), and Sen's slope show an increase in annual flow at Quy Chau, while an opposite trend is observed at Nghia Khanh. This result consists of the seasonal flow at these studied locations.

### Results of sequential Mann–Kendall test

## DISCUSSION

### Wavelet coherence between water discharge and rainfall

In each panel of the figures, the horizontal axis represents the observed time period, while the vertical axis represents the period. To account for potential edge effects, the cone of influence, depicted in a lighter shade, indicates the regions where the results might be influenced. The solid black line contours outline the areas of significant coherence (with *p* < 0.05) between rainfall and water discharge. The direction of arrows in the figures illustrates the phase difference (denoted as *θ*) between rainfall and water discharge, with *θ* = 0 and *π* corresponding to arrows pointing left and right, respectively. The coherence is akin to a correlation measure, indicating the degree of association between rainfall and water discharge. A coherence value of one signifies a strong positive correlation, implying that water discharge and rainfall are highly synchronized. Conversely, a coherence value of zero indicates no correlation, implying that there is no discernible relationship between water discharge and rainfall. This wavelet coherence analysis provides valuable insights into the temporal and frequency-based linkages between rainfall and water discharge, aiding in the understanding of the hydrological dynamics in the studied river basin.

A significant coherence relationship between rainfall and water discharge is evident at both of the studied locations, as shown in Figures 14–16. The analysis reveals that the coherence between rainfall and water discharge exhibits intricate scale-dependent characteristics. The coherence remains significant across longer time scales, ranging from 1 to 10 time scales. The strongest and most consistent multi-scale coherence stability is observed for the seasonal and annual time scales, indicating a robust relationship between seasonal and yearly rainfall patterns and water discharge variations. However, the coherence relationship between monthly rainfall and water discharge appears to exhibit more complex scale dependencies, possibly indicating a higher variability in the monthly rainfall contributions to water discharge fluctuations. These findings highlight the dynamic and multi-scale nature of the connection between rainfall and water discharge, underscoring the importance of considering different time scales in understanding the hydrological processes in the studied river basin.

### Relationship between water discharge and rainfall

^{3}/s, a remarkable correlation between rainfall and flow is observed, with a determination coefficient value of 0.46. In flood seasons, where high flow rates are prevalent, the determination coefficient decreases to 0.22. This decrease can be attributed to the fact that high flow is influenced not only by rainfall but also by different flow components (e.g., overland, root zone, and groundwater contributions) as well as characteristics of the river basin (e.g., moisture content and river slope). Human activities can also influence flow patterns during flood seasons additionally. Regarding annual flows, the determination coefficient between annual flow and rainfall is 0.27, showing a moderate correlation between annual rainfall and flow. On the other hand, a hysteresis scatter is observed in the flow hydrographs at different time scales, showing different determination coefficient values for the rising and falling limbs of the hydrographs. This behavior suggests that the relationship between rainfall and water discharge is influenced by the dynamic interactions between rainfall inputs and the hydrological response, which can vary throughout the year. These findings highlight the complex and dependent nature of the rainfall–flow relationship at Quy Chau, emphasizing the need to consider multiple factors and time scales to model the hydrological dynamics in the studied river basin.

The detailed analysis of seasonal fluctuations, long-term trends, and short-term variability of water discharge believes to help in gaining a comprehensive understanding of the patterns and dynamics of water discharge in the Vietnamese Hieu river basin. The later will allow for (i) development of early warning systems of flood and drought, (ii) assessing the impact of human activities, and (iii) examining water discharge changing resulting from climate change in the next step. Furthermore, the flow has a more robust and stronger relationship with rainfall at Nghia Khanh than at Quy Chau in the studied river basin, revealing the importance of local conditions and geographical factors in shaping the flow response to rainfall inputs. Understanding such relationships is crucial for effective water resource management and flood forecasting in the river basin when using rainfall–runoff models.

## CONCLUSION

In this study, variability and trend detection of water discharge over the past 60 years in the Vietnamese Hieu River Basin were investigated using the wavelet analysis, Mann–Kendall, Sen's slope estimator, and sequential Mann–Kendall test. The main remarks of the study can be summarized as:

Using time series of daily water discharge at Quy Chau and Nghia Khanh in the period from 01/01/1961 to 31/12/2020, the periodicities of flow are examined using the wavelet analysis for monthly, seasonal, and annual time scales. The results showed that periodic oscillations in flow mainly occurred at 1 year, from 2 to 4 years, and from 4 to 8 years.

Water discharge increased from January to September (except June), while it decreased from October to December. At Quy Chau, water discharge increased during flood seasons (with a value ranging from 0.13 to 0.19 m

^{3}/s) and dry seasons (with a value varying between 0.04 and 0.07 m^{3}/s). At Nghia Khanh, water discharge decreased in both flood and dry seasons with a value changing from −0.10 to −0.50 m^{3}/s. An increase in the annual flow at Quy Chau, while an opposite trend is observed at Nghia Khanh.A decreasing trend in the fluctuation of water discharge was observed as the time scales increased (e.g., monthly, seasonal, and annual). The flow decreased in the periods 1961–1969, 1973–1976, and 1996–2020 and increased in the periods 1969–1973 and 1976–1996. The water discharge at Quy Chau and Nghia Khanh showed a consistent temporal pattern of variation for different time scales, revealing synchronized hydrological behaviors or shared influencing factors in the two locations.

In terms of relationship between rainfall and water discharge in the studied river basin, water discharge at Nghia Khanh has a stronger relationship with rainfall than those at Quy Chau, revealing the importance of local conditions and geographical factors in shaping the flow response to rainfall inputs.

The findings in seasonal fluctuations, long-term trends, and short-term variability of water discharge will be helpful for water resource management, development of early warning systems of flood and drought, assessing the impact of human activities, and modeling changes in water discharge resulting from climate change in the Vietnamese Hieu River Basin in future.

## ACKNOWLEDGEMENTS

The authors would like to thank the North Central Regional Hydro-Meteorological Center for sharing rainfall and water discharge data used for different calculations in the present study. The author would like to thank the two anonymous reviewers for their valuable comments.

## DATA AVAILABILITY STATEMENT

All relevant data are available from an online repository or repositories: http://thuyloivietnam.vn/home#quantractonghop.

## CONFLICT OF INTEREST

The author declares there is no conflict.