# Determination of Seasonal Indices for the Regionalization of Low Flows in the Upper Vistula River Basin

^{*}

## Abstract

**:**

## 1. Introduction

_{95}analysis for a catchments in Switzerland. In the United States [21] analyzed river headwaters and noted an identified distinct variability pattern in the frequency of low flow days. Based on their analysis, they identified three clusters in which low flows occurred mainly during summer and early autumn. They found that in the western case, a precipitation deficit played a role in the occurrence of low flows, while in the central and western regions, which are characterized by dense vegetation, low flows occurred mainly during summer and autumn. This was related to increased evaporation during the summer season [21]. Another example of low flow analysis is the study by Dingman and Lawlor [22], conducted for the Vermont and New Hampshire region. Fangman and Haberlandt [23] studied low flow seasonality in the federal state of Lower Saxony, situated in northwestern Germany. A detailed analysis of the seasonality of low flows was carried out by Laaha [24] and Laaha and Blöschl [25,26]. They conducted such an analysis for low flows q

_{95}in 325 small catchments, located in Austria. They used seasonality histograms (SHs), seasonality index (SI) and seasonality ratio (SR) in their analysis. As a result of this study, they divided the catchments into 10 groups and classified them into summer or winter subregions. Similar analysis was made by Vezza et al., 2010 [12], who used, similarly to Laaha and Blöschl [25], three seasonality indices for low flow regionalization for catchments located in the northwestern Italy. Vlach et al. [17] analyzed the long-term variability and seasonality of low flows and streamflow droughts in fifteen headwater catchments of three regions within Central Europe. In their analysis, SI and SR indices were used.

_{95}flows in ungauged catchments and also, in practice, for modernization and design of new water intakes for small local water supply systems. Additionally, it should be added that the presented analyses are innovative, both with regard to Poland but also with reference to international reports, which are limited.

## 2. Materials and Methods

_{95%}flow quantile, i.e., a flow that is equaled or exceeded on 95% of all days within the observation period. This characteristic was chosen because it is relevant to many aspects of water management, such as design of water supply systems, and is widely used in Europe. Then, Q

_{95%}was subsequently standardized by the catchment area, obtaining unit specific low flow discharge q

_{95}dm

^{3}∙s

^{−1}∙km

^{- 2}. A map of specific low flow discharge q

_{95}dm

^{3}∙s

^{−1}∙km

^{−2}for the analyzed catchments of the Upper Vistula river basin is presented in Figure 2.

^{2}and the largest is 2034 km

^{2}. Additionally, the mean slope of the catchments varies from 0.002 for the Pszczynka and Łęg catchments to 0.091 for the Biała catchment (Table 1). The selected catchments are also diversified in terms of land use. They are dominated by arable land, which ranges from 7.4% for the Łęg to 87% for the Szreniawa, and forests, which constitute from 3% (Szreniawa) to 67.60% (Skawica).

_{95}: a seasonality ratio (SR), a cyclic seasonality index (SI) and seasonality histograms (SHs).

_{95s}) to winter low flow characteristics (q

_{95w}) is given in Equation (1) [26]. First, daily discharge data were divided into summer discharge series, from 1 May to 31 October, and winter discharge series, from 1 November to 30 April, in order to differentiate summer low flows caused by precipitation deficit and winter low flow events caused by snow accumulation and frost in highland and mountain areas [17]. Then, from summer and winter discharge time series data, the characteristic values of q

_{95s}and q

_{95w}were calculated for each catchment.

_{95}. The parameter Θ represents a measure of the average seasonality of low flows by the average day of low flow occurrence in radians. The value of the parameter ranges from 0 to 2π, where Θ = 0 relates to 1 January, $\raisebox{1ex}{$\mathsf{\pi}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ relates to 1 April, π relates to 1 July and $\raisebox{1ex}{$3\mathsf{\pi}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ relates to 1 October. For each catchments, the days when the flow was less than Q

_{95}were taken for further analysis and then were transformed into Julian dates D

_{j}[17,26].

_{j}is the day of the occurrence, when flows are lower than Q

_{95}.

_{θ}and y

_{θ}are the arithmetic mean of Cartesian coordinate of a total of n single days j and are calculated as Equation (4) [26]:

_{95}[26].

_{95}= β

_{0}+ β

_{1}· x

_{1}+ β

_{2}· x

_{2}+⋯+ β

_{p−1}· x

_{p−1}

_{i}—morphoclimatic parameters of a catchment,

_{i}—regression coefficient.

_{p}coefficient as an optimality criterion [25].

^{2}for significance level = 0.05 and the adjusted coefficient of determination R

^{2}

_{adj}(Equation (8)) were calculated in order to determine the consistency of calculated values with observed values.

^{2}—coefficient of determination,

^{2}

_{cv}was determined using leave-one-out cross-validation, which is described in Equation (9). This was applied in order to avoid the so-called error of the third kind and to select the best forecasting model [25].

_{cv}—is the average residual square error and is calculated with Equation (10) [25],

_{95})—is the spatial variance of the observed specific low flow.

_{cv}) was also calculated. RMSE values equal to 0 indicate a perfect fit.

## 3. Results and Discussion

_{95}for Slovak catchments. The study included 198 small and medium-sized catchments from across the country. Similarly to the study carried out in the paper, summer lows were observed mainly from August to October. However, low flows for August were dominant in catchments located in the mountainous parts of Slovakia. The magnitude of the seasonal concentration index (r) was higher than that obtained by the authors of the paper. For summer low flows, it ranged from 0.8 to 0.95 in 84% of the analyzed catchments. On the other hand, winter low flows were observed from December to March for the catchments located in the central and eastern part of Slovakia. Of these low flows, those occurring in January were predominant. Additionally, for this type of low flow, high values of r were observed, which ranged from 0.8 to 0.9 for most of the catchments.

_{95}, for 41 catchments in northwestern Italy. For this area and the three seasonality indices used, they found that, for Italian catchments, the division of low flows into two seasonality groups (summer and winter) best reflected the nature of the area analyzed. Summer low flows were observed in the Apennine–Mediterranean catchments, while winter low flows were found in the catchments of the Alpine region. The coefficient of determination which they obtained for the global model (without taking flow seasonality into account) had, as in the study of the authors of this paper, the lowest value. In a study of the Rhine river basin, Demirel et al. [16], noted that, on the basis of the SR index, the Alpine catchments have low flows in the winter half-year, while the others have low flows in the summer half-year. Based on the WMOD (which is equivalent to the Θ parameter analyzed in the authors’ work), they found that summer half-year low flows occurred in September and October. As for the low flows of the winter half-year, the alpine catchments were characterized by their occurrence mainly in January and February [16]. Similar results were obtained for low flows for the catchments analyzed in this paper.

_{95}, which has a positive sign for the summer seasonality of low flow and a negative sign for the mixed group. This is related to the location of the catchment in the Upper Vistula river basin.

^{2}= 41% (R

^{2}

_{adj}= 39%) was obtained for the global regression model. This is the lowest value of the coefficient, compared to the values calculated for the groups using the K-means method. Calculated values of RMSE and RMSE

_{cv}(Table 2) for the global model were above 0.7, which proves that the determination of the characteristics of low flow, taking into account the seasonality of its occurrence in the case of catchments located in the Upper Vistula basin, allows for its more accurate estimation. Due to the small number of catchments (four catchments), the regression model was not applied to separate three clusters and catchments characterized by the seasonal character of low flow in winter period. The analysis carried out allowed to conclude that separation of the mixed group gave much better fitting of the regression model in comparison to the two groups of seasonality. It is evidenced, inter alia, by the value of coefficient of determination, determined by the cross-validation method for the catchments characterized by summer low flow seasonality, which, in the case of the three clusters, was 66% and was more than twice as high as in the case of the two clusters (R

^{2}

_{cv}= 30%), and in the case of the global model its value was only 23% (Table 2). For the three cluster groups, RMSE and RMSE

_{cv}values below 0.5 were obtained, which confirms the better fit of the regression model compared to the two seasonality groups. Therefore, further analysis was conducted for the three groups.

## 4. Conclusions

_{95}were determined. The regression relations for the global model (without clustering) and for two groups (winter and summer low flow seasonality) were based on catchment slope, which had a positive designation. For the three groups, the factor influencing q

_{95}was soils (cambisols), which had a positive designation in the case of summer low flow seasonality and a negative designation in the case of the mixed group. This is related to the location of the catchment in the Upper Vistula river basin. The mixed group includes catchments located in upland, Carpathian foothills and mountain climates. In terms of water circulation, the catchments of this group are characterized by poorly permeable and impermeable soils. On the other hand, catchments characterized by summer low flow seasonality are located in upland climate and sub-Carpathians and are characterized by occurrence of moderately and easily permeable soils.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The location of the Upper Vistula river basin and analyzed catchments (points show the location of the catchment areas selected for the analysis).

**Figure 2.**Specific low flow discharge q

_{95}for analyzed catchments of the Upper Vistula basin, where: 1—Dłubnia; 2—Opatówka; 3—Biała Tarnowska; 4—Szreniawa; 5—Wieprzówka; 6—Łeg; 7—Tanew; 8—Biała; 9—Pszczynka; 10—Skawa; 11—Łososinka; 12—Biała Nida; 13—Trzebośnica; 14—Czarna; 15—Soła; 16—Wisła; 17—Dunajec; 18—Koprzywianka; 19—Skawica; 20—Czarna Nida; 21—Wschodnia; 22—Ropa; 23—Jasiołka; 24—Solinka; 25—Osława; 26—Stupnica; 27—Mleczka; 28—Łubinka; 29—Grabinianka; 30—Wielkopolka; 31—Wisłoka; 32—Breń.

**Figure 4.**Map of SI (

**a**) and seasonality histograms (

**b**) for analyzed catchments; the x-axis is in the hydrological year for analyzed catchments and y-axis in all cases is from 0 to 400.

**Figure 6.**Differences between observed and calculated q

_{95}(dm

^{3}∙s

^{−1}∙km

^{−2}) for (

**a**) cluster 3_2 and (

**b**) cluster 3_3.

Variable | Variable Description | Units | Min. | Mean | Max. |
---|---|---|---|---|---|

A | Catchment area | km^{2} | 66.3 | 475.73 | 2034.0 |

L | Length of the watercourse | km | 8.8 | 34.33 | 72.0 |

P | Mean annual precipitation | mm | 574 | 729.28 | 1033 |

I | Mean catchment slope | - | 0.002 | 0.021 | 0.091 |

H_{me} | Mean catchment altitude | m a.s.l. | 191.00 | 388.86 | 836.00 |

LU4 | Forests | % | 3.00 | 31.29 | 67.60 |

LU3 | Grassland | % | 0.00 | 8.29 | 32.00 |

LU2 | Arable land | % | 7.40 | 47.61 | 86.00 |

LU1 | Built-up area | % | 0.00 | 6.55 | 29.88 |

S1 | Fluvisols | % | 0.00 | 15.93 | 28.20 |

S2 | Cambisols | % | 0.00 | 54.48 | 100.00 |

S3 | Arenosols | % | 0.00 | 10.34 | 65.00 |

Group | Number of Catchments | Type of Seasonality | Model | R^{2} | R^{2}_{adj} | R^{2}_{cv} | RMSE | RMSE_{cv} |
---|---|---|---|---|---|---|---|---|

global | 32 | - | q_{95} = 1.24 + 27.08∙I | 41 | 39 | 23 | 0.71 | 0.81 |

Two clusters | ||||||||

2_1 | 23 | summer | q_{95} = 1.86 + 20.31∙I − 0.042∙S1 | 52 | 47 | 30 | 0.56 | 0.74 |

2_2 | 9 | winter | q_{95} = 3.61 + 25.64∙I − 0.041∙LU2 | 88 | 84 | 49 | 0.44 | 0.66 |

Three clusters | ||||||||

3_2 | 16 | mixed | q_{95} = 0.92−0.02∙S_{2}−0.02∙LU5 + 0.007∙H_{me} | 76 | 70 | 67 | 0.46 | 0.49 |

3_3 | 12 | summer | q_{95} = −2.66 + 0.009∙S2 + 0.03∙LU5 + 0.004∙P | 88 | 83 | 66 | 0.21 | 0.36 |

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**MDPI and ACS Style**

Cupak, A.; Kaczor, G. Determination of Seasonal Indices for the Regionalization of Low Flows in the Upper Vistula River Basin. *Water* **2023**, *15*, 246.
https://doi.org/10.3390/w15020246

**AMA Style**

Cupak A, Kaczor G. Determination of Seasonal Indices for the Regionalization of Low Flows in the Upper Vistula River Basin. *Water*. 2023; 15(2):246.
https://doi.org/10.3390/w15020246

**Chicago/Turabian Style**

Cupak, Agnieszka, and Grzegorz Kaczor. 2023. "Determination of Seasonal Indices for the Regionalization of Low Flows in the Upper Vistula River Basin" *Water* 15, no. 2: 246.
https://doi.org/10.3390/w15020246