# Detecting Annual and Seasonal Hydrological Change Using Marginal Distributions of Daily Flows

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

## 3. Methodology

#### 3.1. Daily Flows as a Stochastic Process

_{t}; t ≥ 0} is mathematically formulated [35] by the time function x

_{t}= f(t; α, β, γ,…), where t is time, and α, β, γ, etc. are the parameters of a multidimensional distribution describing the temporal structure of the process. Hydrologic daily time series are a close approximation to the continuous stochastic processes, where the process x

_{t}is represented by the observed daily flows x

_{τ,i}(with i = 1, 2, …, N—number of years, and τ = 1, 2, …, 365—number of days). Hence, the x

_{τ,i}series is a set of N process realizations over the [0, 365 days] interval.

_{τ,i}at each station is arranged in a matrix, in which each row represents one date (i.e., the ordinal number of a day) within a year, and each column represents one year:

_{τ}, standard deviation s

_{τ}, and coefficient of skewness C

_{sτ}(τ = 1, 2, …, 365).

#### 3.2. Periodicity Analysis

_{τ,per}may be expressed using the Fourier series [35,37]:

_{τ}, h is the number of significant harmonics, A

_{j}and B

_{j}are the Fourier coefficients, f

_{j}= j/ω is the frequency of the j-th harmonic, ω is the base period of 365 days, and τ = 1, 2, …, 365. The Fourier coefficients are calculated as follows:

#### 3.3. Marginal Distributions

_{t}in matrix (1), i.e., the vectors

#### 3.4. Hydrological Condition Zones

#### 3.5. Estimating Change in Hydrological Regime

- Relative change in runoff volume, ∆Vp (%):$$\u2206Vp=\frac{{Vp}_{1991-2020}-{Vp}_{1961-1991}}{{Vp}_{1961-1991}}\xb7100$$
- Time shift of the centroid of the area below the MDDF quantile line, ∆t (in days):$$\u2206t={Cp}_{1991-2020}-{Cp}_{1961-1990}$$

## 4. Results

#### 4.1. Daily Flow Statistics and Their Periodicity

_{τ}), standard deviation (s

_{τ}), and coefficient of skewness (c

_{sτ}). Figure 4 shows the seasonal variation in the daily statistics for HS3 in all three considered periods, together with their corresponding periodic functions (smooth lines). The mean and standard deviation both exhibit distinct seasonal patterns which agree in phase. The highs occur in the spring and the lows in the summer (Figure 4a,b). This pattern is characteristic for all but two considered stations (HS1 and HS6). The skewness does not exhibit a distinct seasonal pattern like the mean and standard deviation, but a complex one with more highs and lows. It is also worth noting that there is a significant positive skewness over the entire annual cycle at all stations, as shown for HS3 in Figure 4c.

_{yτ}), standard deviation (s

_{yτ}), and skewness (c

_{syτ}), as well as their corresponding periodic functions. Expectedly, logarithmic transformation suppresses the variability in the daily statistics compared with the ones in the original space. The distinct seasonal pattern is still visible for the mean, but not for the standard deviation, for which the amplitude of the intra-annual oscillation is significantly smaller. The logarithmic transformation reduced the skewness and even produced negative values in both the daily skewness series and the estimated periodic function during certain parts of the year. This was noticed at all stations in some part of the year for at least one period considered. This further affects the parameters and the shape of the LPT3 marginal distributions. The statistics and their periodic functions for all stations are given in the Supplementary Materials (Figures S1 and S2).

#### 4.2. Periodic Parameters of the Marginal LPT3 Distributions

_{yτ,per}, s

_{yτ,per}, and c

_{syτ,per}. Figure 6 shows the seasonal variation in the three LPT3 distribution parameters at HS3. The LPT3 parameters highly depend on the value and sign of the skew, c

_{sy.}Therefore, a comparison of periodical functions of statistics (Figure 5) with LPT3 distribution parameters (Figure 6) expectedly shows that the differences between studied periods are transferred from the statistics (especially the skew) to the distribution parameters in accordance with the expressions in Equation (8). The results for all stations are given in the Supplementary Materials (Figure S3).

#### 4.3. Marginal Distributions of Daily Flows

_{τ,per}and γ

_{τ,per}(Figure 6), the MDDF quantiles are represented by smooth lines. For days with a negative skew of log-transformed data (c

_{syτ,per}< 0) for 1961–1990 (green line in Figure 5c), marginal LPT3 distributions are bounded from above (Equation (9), also, e.g., [42]). The values of the upper bound were computed for all such cases and it was found that they do not cause underestimation of daily flow upper-tail quantiles for any non-exceedance probability of interest in hydrologic applications.

#### 4.4. The Zones of Hydrological Conditions

#### 4.5. Annual and Seasonal Hydrologic Condition Changes

## 5. Discussion

#### 5.1. Long-Term Changes in Hydrological Regime

#### 5.2. Probabilistic Annual Runoff Cycle as an Indicator of Hydrological Conditions

## 6. Conclusions

- The seasonal runoff pattern changed from one period to another in terms of temporal shift and the occurrence of more extreme flows. However, the general pattern of seasonal runoff remained the same. The prevailing pattern is simple and unimodal, while the less present mixed regime is bimodal.
- In most of the catchments, runoff volume has decreased in the recent 1991–2020 period at both the annual and seasonal scales. The critical season is summer for dry and average conditions, with volume reduction in all catchments.
- The most pronounced shift in runoff timing is found on the annual scale. Dry and average conditions occur earlier at this scale. The change in runoff timing is found to be insignificant for all seasons and zones, except for wet conditions, which occur earlier in spring.

## Supplementary Materials

_{τ,per}, (b) scale parameter β

_{τ,per}, (c) location parameter γ

_{τ,per}; Figure S4: The diagrams of MDDF at all HS for the three periods showing the quantiles (a) for a full range of probabilities p(x), from 0.01 to 0.99, and (b) for low to median probabilities. Solid lines represent the latest 1991–2020 period, the dashed lines represent the earlier 1961–1990 period, and the dotted lines represent the whole 1961–2020 period.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Results (p-values) of the tests applied to the annual flow series for the period 1961–2020.

HS# | Z ^{1} | F ^{2} | L ^{3} | M-W ^{4} | W-W ^{5} | M-K ^{6} |
---|---|---|---|---|---|---|

1 | 0.792 | 0.888 | 0.946 | 0.792 | 0.520 | 0.444 |

2 | 0.507 | 0.469 | 0.732 | 0.584 | 0.367 | 0.329 |

3 | 0.150 | 0.430 | 0.563 | 0.092 | 0.899 | 0.211 |

4 | 0.402 | 0.344 | 0.442 | 0.382 | 0.700 | 0.506 |

5 | 0.206 | 0.809 | 0.420 | 0.152 | 0.896 | 0.367 |

6 | 0.374 | 0.673 | 0.619 | 0.393 | 0.053 | 0.415 |

7 | 0.358 | 0.089 | 0.215 | 0.184 | 0.520 | 0.255 |

8 | 0.111 | 0.729 | 0.793 | 0.084 | 0.367 | 0.154 |

9 | 0.254 | 0.968 | 0.588 | 0.262 | 0.520 | 0.293 |

10 | 0.601 | 0.654 | 0.390 | 0.516 | 0.700 | 0.354 |

^{1}Z-test;

^{2}Fisher’s F-test;

^{3}Levene’s test;

^{4}Mann–Whitney test;

^{5}Wald–Wolfowitz test;

^{6}Mann–Kendall test.

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**Figure 1.**Locations of the hydrological stations used in this study on the hydrographic map of Serbia (

**left**) and a schematic of the river network and stations (

**upper right**panel).

**Figure 3.**Hydrological condition zones based on probabilistic threshold levels for (

**a**) flows, (

**b**) runoff volumes.

**Figure 4.**Seasonal variation in daily flow statistics (peaky lines) and their corresponding periodic functions (smooth lines) at HS3 estimated for the three periods: (

**a**) mean, (

**b**) standard deviation, and (

**c**) coefficient of skewness. Modified from [41].

**Figure 5.**Seasonal variation in log-transformed daily flow statistics (peaky lines) and their corresponding periodic functions at HS3 estimated for the three periods: (

**a**) mean, (

**b**) standard deviation, and (

**c**) coefficient of skewness. Modified from [41].

**Figure 6.**Periodical parameters of LPT3 marginal distribution of daily flows at HS3: (

**a**) shape parameter α

_{τ,per}, (

**b**) scale parameter β

_{τ,per}, (

**c**) location parameter γ

_{τ,per}. Modified from [41].

**Figure 7.**The diagrams of MDDFs at HS3 for the three periods showing the quantiles (

**a**) for a full range of probabilities p(x), from 0.01 to 0.99, and (

**b**) for low to median probabilities. The solid lines represent the later 1991–2020 period, the dashed lines represent the earlier 1961–1990 period, and the dotted lines represent the whole 1961–2020 period.

**Figure 8.**Hydrological condition zones for all stations defined by the quantiles of marginal daily flow distributions for selected probabilities. The zones for the 1961–2020 period are shaded, while the zones for two subperiods are indicated by their upper threshold lines.

**Figure 9.**The centroids of the area below the upper thresholds of the hydrologic condition zones for (

**a**) annual and (

**b**) seasonal scales. Red dots refer to the recent 1991–2020 period, and green ones to the earlier 1961–1990 period.

**Figure 10.**The boxplots of changes in the annual and seasonal runoff volumes in 1991–2020 compared with 1961–1990 at all stations for the wetness condition zones defined using the upper quantiles of probability p(x). The lines in the boxplots indicate the median value and x indicates the mean, while the whiskers correspond to 1.5 × IQR, where IQR is the interquartile range, with the outliers outside IQR displayed.

**Figure 11.**The boxplots of changes in the annual and seasonal runoff timing in 1991–2020 compared with 1961–1990 at all stations for the wetness condition zones defined by the upper quantiles of probability p(x). The lines in the boxplot indicate the median value and x indicates the mean, while the whiskers correspond to 1.5 × IRQ, where IRQ is the interquartile range, with the outliers outside IQR displayed.

**Figure 12.**The maps of relative change in annual runoff volume at ten stations for dry, average, and wet condition zones, represented by upper quantiles of probability p(x). Red circles denote volume reduction in 1991–2020 compared with 1961–1990, and blue circles indicate volume increase. The circle size corresponds to the amount of change.

**Figure 13.**The maps of changes in timing of annual runoff at ten stations for dry, average, and wet condition zones, represented by upper quantiles of probability p(x). Red circles denote earlier runoff occurrence in 1991–2020 compared with 1961–1990, and blue circles indicate later timing. The circle size corresponds to the amount of change.

HS# | Station Name | River | Catchment Area [km^{2}] | Watershed |
---|---|---|---|---|

1 | Sremska Mitrovica | Sava | 87,996 | Sava |

2 | Valjevo | Kolubara | 340 | Kolubara |

3 | Bagrdan | Velika Morava | 33,446 | Velika Morava |

4 | Ljubičevski most | Velika Morava | 37,320 | Velika Morava |

5 | Jasika | Zapadna Morava | 14,721 | Zapadna Morava |

6 | Ušće | Studenica | 540 | ZapadnaMorava |

7 | Grdelica | Južna Morava | 3782 | Južna Morava |

8 | Mojsinje | Južna Morava | 15,390 | Južna Morava |

9 | Doljevac | Toplica | 2052 | Južna Morava |

10 | Niš | Nišava | 3870 | Južna Morava |

**Table 2.**Relative change in runoff volume (%) in the recent 1991–2020 period compared with the 1961–1990 period. The first column indicates the probabilities p(x) of the upper thresholds of the hydrologic condition zones. Changes are shown for annual (Ann.) and seasonal scales (Autumn, Winter, Spring, and Summer), and colored in blue (increase) and red (decrease).

p(x) | Season | HS1 | HS2 | HS3 | HS4 | HS5 | HS6 | HS7 | HS8 | HS9 | HS10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.05 | Ann. | −8 | 11 | −1 | −2 | 5 | 7 | −8 | −15 | 9 | −23 |

Aut. | −3 | −9 | −10 | −3 | 7 | −2 | −24 | −25 | 0 | −37 | |

Win. | 14 | 34 | −9 | −13 | −1 | 6 | −16 | −28 | 2 | −33 | |

Spr. | −22 | 2 | 16 | 11 | 16 | 17 | 8 | 3 | 28 | −12 | |

Sum. | −16 | −13 | −10 | −7 | −4 | −2 | 0 | −17 | −15 | −13 | |

0.3 | Ann. | −5 | −1 | −6 | −5 | −2 | 10 | −5 | −11 | 1 | −18 |

Aut. | 12 | 4 | −2 | 3 | 5 | 8 | −7 | −3 | −5 | −8 | |

Win. | 5 | 15 | −5 | −6 | −4 | 13 | −11 | −14 | −2 | −16 | |

Spr. | −19 | −16 | −5 | −4 | −1 | 12 | 2 | −12 | 10 | −23 | |

Sum. | −19 | −13 | −14 | −10 | −7 | 2 | 1 | −8 | −10 | −15 | |

0.5 | Ann. | −4 | −5 | −7 | −5 | −5 | 10 | −4 | −10 | −2 | −15 |

Aut. | 15 | 9 | 0 | 4 | 3 | 11 | −2 | 3 | −8 | 4 | |

Win. | 3 | 6 | −4 | −4 | −6 | 15 | −9 | −9 | −2 | −11 | |

Spr. | −16 | −20 | −11 | −8 | −7 | 9 | −2 | −16 | 3 | −25 | |

Sum. | −19 | −11 | −15 | −10 | −8 | 3 | −1 | −6 | −9 | −13 | |

0.7 | Ann. | −2 | −8 | −8 | −5 | −7 | 9 | −5 | −10 | −4 | −12 |

Aut. | 15 | 11 | 1 | 5 | 1 | 12 | −1 | 6 | −11 | 12 | |

Win. | 2 | −2 | −4 | −3 | −7 | 15 | −6 | −7 | −1 | −7 | |

Spr. | −11 | −23 | −14 | −10 | −10 | 6 | −6 | −19 | −3 | −26 | |

Sum. | −17 | −7 | −16 | −10 | −9 | 3 | −4 | −6 | −9 | −8 | |

0.99 | Ann. | 5 | −6 | −11 | −6 | −12 | −2 | −15 | −19 | −5 | −8 |

Aut. | −3 | 12 | −11 | −2 | −8 | −1 | −21 | −20 | −19 | −4 | |

Win. | 5 | −26 | −8 | −9 | −13 | 5 | 3 | −17 | 14 | −14 | |

Spr. | 15 | −9 | −12 | −5 | −14 | −10 | −28 | −21 | −18 | −15 | |

Sum. | 3 | 29 | −13 | −2 | −7 | 2 | −24 | −21 | −13 | 38 | |

Legend: ΔV (%) |

**Table 3.**The changes in runoff timing expressed as the shift in the centroid date (in days) in the recent 1991–2020 period compared with 1961–1990. The first column indicates the probabilities p(x) of the upper threshold of the hydrologic condition zones. Changes are shown for annual (Ann.) and seasonal scales (Autumn, Winter, Spring, and Summer), and colored in blue (later date) and red (earlier date).

p(x) | Season | HS1 | HS2 | HS3 | HS4 | HS5 | HS6 | HS7 | HS8 | HS9 | HS10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.05 | Ann. | −8 | −4 | 4 | 3 | 0 | 1 | 9 | 8 | 3 | 12 |

Aut. | 0 | −1 | −1 | −1 | −2 | 1 | 2 | −1 | 0 | −1 | |

Win. | 0 | 2 | 2 | 1 | 1 | −1 | −1 | 1 | 2 | 1 | |

Spr. | 0 | −2 | 1 | 1 | 1 | 1 | 0 | 1 | 2 | 3 | |

Sum. | 0 | 2 | −2 | −1 | −1 | −1 | −1 | −2 | −5 | −4 | |

0.3 | Ann. | −11 | −7 | −3 | −2 | −2 | −1 | 4 | −1 | 1 | −3 |

Aut. | 0 | −2 | −1 | −1 | −3 | −1 | 2 | −1 | −2 | −2 | |

Win. | −1 | 2 | 1 | 1 | 2 | 1 | 1 | 0 | 1 | −1 | |

Spr. | 0 | −2 | −1 | −1 | −1 | 0 | 0 | 1 | 0 | 2 | |

Sum. | 1 | 1 | 2 | 2 | 2 | 0 | −3 | 0 | 1 | 0 | |

0.5 | Ann. | −11 | −8 | −4 | −4 | −3 | −2 | 1 | −3 | 1 | −6 |

Aut. | 0 | −2 | −1 | −1 | −3 | −1 | 2 | −1 | −2 | −2 | |

Win. | −1 | 2 | 1 | 1 | 2 | 1 | 1 | 0 | 1 | −2 | |

Spr. | −1 | −1 | −2 | −2 | −1 | −1 | 0 | 0 | −1 | 1 | |

Sum. | 1 | 1 | 3 | 2 | 2 | 1 | −3 | 1 | 3 | 2 | |

0.7 | Ann. | −10 | −7 | −5 | −4 | −3 | −3 | −1 | −5 | 0 | −8 |

Aut. | 0 | −3 | −1 | −1 | −2 | −1 | 2 | 0 | −2 | −2 | |

Win. | −1 | 2 | 1 | 1 | 2 | 2 | 2 | 0 | 2 | −2 | |

Spr. | −1 | −1 | −3 | −2 | −2 | −1 | −1 | −1 | −2 | 0 | |

Sum. | 2 | 2 | 3 | 3 | 2 | 1 | −2 | 2 | 3 | 3 | |

0.99 | Ann. | 4 | 4 | −1 | 0 | 0 | −2 | −6 | −2 | −2 | 4 |

Aut. | −1 | −8 | 4 | 3 | 1 | 2 | 4 | 5 | 4 | 4 | |

Win. | −1 | 2 | 3 | 2 | 3 | 2 | 2 | 1 | 6 | −1 | |

Spr. | −1 | 0 | −5 | −4 | −2 | −2 | −9 | −8 | −4 | −5 | |

Sum. | 4 | 7 | −1 | 0 | −3 | −2 | 7 | 6 | −5 | 4 | |

Legend: Δt (days) |

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## Share and Cite

**MDPI and ACS Style**

Blagojević, B.; Mihailović, V.; Bogojević, A.; Plavšić, J.
Detecting Annual and Seasonal Hydrological Change Using Marginal Distributions of Daily Flows. *Water* **2023**, *15*, 2919.
https://doi.org/10.3390/w15162919

**AMA Style**

Blagojević B, Mihailović V, Bogojević A, Plavšić J.
Detecting Annual and Seasonal Hydrological Change Using Marginal Distributions of Daily Flows. *Water*. 2023; 15(16):2919.
https://doi.org/10.3390/w15162919

**Chicago/Turabian Style**

Blagojević, Borislava, Vladislava Mihailović, Aleksandar Bogojević, and Jasna Plavšić.
2023. "Detecting Annual and Seasonal Hydrological Change Using Marginal Distributions of Daily Flows" *Water* 15, no. 16: 2919.
https://doi.org/10.3390/w15162919