Time of concentration estimated of overland flow

Time of concentration (Tc) is a crucial aspect in the hydrological model, especially for determining flood discharge. Time concentration is the amount of time required for water to go from a watershed’s farthest point to its outflow. Although it is an important value, however, Tc does not have a universal equation that can be used as a reference. Based on the literature review, there are several concentration time calculation methods. Each method has its parameters and approach to produce varied values. Several experimental methods were applied to calculate concentration time at sites, among others are Kirpich, SCS Lag, and FAA. The calculation results are displayed in a dendrogram with group division by using the Euclidean approach. The calculation results show that the difference in Tc values can reach up to 500%, hence Grimaldi calls it a paradox in modern hydrology.


Introduction
The Time of Concentration, or Tc, is a critical parameter in the hydrological sciences that defines how a watershed reacts to excessive rainfall across its surface [1].A major basin parameter called period of concentration indicates how long it takes to move from a watershed's hydraulically farthest point to its outlet.The precision of the anticipated time of concentration affects the assessment of peak discharge.Tc has been heavily included in well-liked hydrological design methodologies including the rational approach and unit hydrograph theory, especially in the context of daily engineering applications.
Nevertheless, the literature presents a wide array of definitions and estimating methodologies [2] [3], Despite the existence of significantly varied design values of tc, it continues to be regarded as one of the most enigmatic and uncertain ideas within contemporary hydrology, thereby presenting a paradox [4].McQueen in his research provides several kinds of definitions related to concentration time.The definitions provided encompass various temporal intervals within the context of hydrograph analysis.These include the duration from the termination of effective rainfall to the inflection point on the hydrograph, the duration from the center of excess rainfall to the peak of direct runoff, the duration from the peak intensity of rainfall to the peak discharge, the duration from the effective center of mass of precipitation to the center of mass of direct runoff, the duration from the centroid of excess rainfall to the peak of total runoff, and the duration from the initiation of runoff to the peak discharge of the total runoff.In conjunction with the aforementioned six computational definitions, the technical literature also uses the subsequent theoretical concepts [2] including the amount of time from the end of effective rainfall to the end of direct runoff that it takes for a drop of precipitation to reach the outlet portion of a watershed (DAS) from its most hydraulically remote point.
The various definitions above have encouraged hydrologists to develop several hydrologic models connected to the focus period.This value is significantly influenced by meteorological variables (rainfall and runoff) as well as watershed characteristics (land cover, slope, soil type, drainage density) and river channel geomorphology.In practice, the empirical method is most often used to estimate the concentration-time values of these variables.Nearly every technique created has an empirical foundation [3] and represents constant properties for specific watersheds.Multiple regression analysis is the most widely used empirical method to relate temporal parameters and catchment characteristics considering surface runoff [5], [6], [7] or main channel runoff [8], [9], [10], [11] or both [12], [13], [14].
Based on the literature review, it is known that there are many theories for calculating concentration time.Estimation of concentration time has many empirical equations or methods [15].Different methods give different results from one another up to 500% [4].The difference in results is a paradox in hydrology, where the concentration-time should give an accurate value but vice versa.Therefore, in this study trying to examine more deeply the time of concentration.A total of seventeen empirical equations were chosen for estimate.These equations were then applied to the dataset obtained from the basin, which served as the subject of the case study.The researchers employed hierarchical cluster analysis, namely the Cluster method, to assess the level of similarity between the chosen methodologies.In-depth research, especially to determine the most dominant parameters at the time of concentration.As well as knowing the effect of soil type concentration time value.

Material and Method
The seventeen items used to time watershed concentrations were selected after complicated literature boundaries.The selection methodology takes priority on rainfall intensity along with the watershed's physical characteristics.

Study Area
The research location was carried out in watersheds or sub-watersheds in the Brantas watershed.The selected watersheds are those with rain gauge stations and water estimation posts so that calibration and verification can be carried out.Brantas Watershed is the second largest was in Java Island which is located in East Java Province.The Brantas Watershed itself is divided into several sub-watersheds, one of which is the Brantas Hilir Watershed which has coordinates 7°11'34.165"S-7°46'50.277"Sand 111°58'45.276"E-112°52'28.046"E(Figure 1).Brantas Hilir watershed is crossed by several rivers, including the Gunting River, Turibaru River, Beng River, Brangkal River, Sadar River, Bongkok River, Porong River, and Mas.Brantas Hilir itself has an area of ±3.5 km 2 and an average maximum rainfall of 122.5 mm.Brantas Hilir watershed covers several areas in the East Java region, namely Surabaya City, Sidoarjo Regency, Mojokerto Regency, and Jombang Regency.This area is classified as an area that has a mild slope (0-8%), the length of the main river is 92 km, and land use in this area is divided into several, namely, field, pond, mangroveSoil classification in this area is prioritized by sandy loam with 58.69% (Figure 2).

Hierarchic Cluster Analysis (Cluster)
Hierarchical cluster analysis (HCA), sometimes referred to as hierarchical clustering, is a widely utilized method in the field of big data research and data mining.Its primary objective is to establish a hierarchical structure of clusters [16] [17] [18].Therefore, HCA seeks to group subjects who have similar characteristics into groups.In the field of healthcare administration (HCA), two distinct tactics are commonly employed: agglomeration and strategic division.The agglomerative clustering method, which involves merging the "leaves" towards the "roots" of the cluster tree, is commonly referred to as a "bottom-up" technique [19].Partial clustering is seen as a "top-down" methodology that progresses from the roots to the leaves.Initially, all observations are regarded as a singular cluster, and then, the process of conversion is executed iteratively as the cluster progresses downwards in the hierarchy.Clinical studies frequently exhibit heterogeneity among patient populations, even when employing extensive lists of inclusion and exclusion criteria [20] [21].
For example, sepsis and/or septic shock is often considered an organic disease in clinical trials.However, there is significant heterogeneity between septic patients in terms of site of infection, comorbidities, inflammatory response, and duration of treatment [22] [23].Confounding factors have historically been seen as influential variables and can be effectively accounted for through the utilization of multivariate regression analysis [24].Nevertheless, this approach primarily emphasizes the prediction and adjustment aspects, lacking the ability to effectively classify a diverse population into more homogeneous subgroups.The objective of clustering analysis is to categorize diverse populations into more cohesive groups by utilizing the given attributes.Every cluster possesses a unique signature for providing aid [24].Researchers may have a vested interest in examining the relationship between physiologic detection and the prediction of subacute events, such as sepsis, hemorrhage, and intubation, within the distinct context of the intensive care unit (ICU).For instance, a study revealed that there were comparable physiological indicators of bleeding in both surgical and critical care patients, indicating a degree of similarity between these two specific subpopulations.The objective of this article is to present fundamental information on the utilization of Hierarchical Cluster Analysis (HCA) and its visualization using dendrograms and heat maps.
In this study the analysis was carried out using an agglomeration hierarchy with the steps for forming agglomeration groups (ascending): Step 1: involves handling every data point as though it were a distinct cluster.Step 2: Eliminate the "outsiders" from the least cohesive cluster at the end of each iteration.Step 3: If a group of all samples has already been created, stop; if not, move on to step 2.Moreover, the categorization is established using the Euclidean Distance.The Euclidean distance (Figure 3.) between two points in Euclidean space is the length of the line segment that connects them.Because it can be calculated from the Cartesian coordinate system using the Pythagorean theorem, this distance is frequently referred to as the Pythagorean distance.[25].

Result and Discussion
The equation for determining the value of concentration time has been found by many experts.Of the many existing empirical equations, in this study, a number of 17 equations were determined to be compared.The following are the 17 equations used in this study.Which is summarized in Table 1.(2,000 hectares) where: Tc: Time of concentration (h); L: Stream length (km); S: Average slope (m/m); i: Rainfall intensity (mm/h); n: manning roughness coefficient (m-1/3 .s);A: Watershed area (km 2 ); N: coefficient of retardation (adm); CN: Curve number (scs method) (adm); H: Difference between main canal ends (m); C: runoff coefficient of rational method (adm); Hm: average height in the basin (m); Sscs: maximum capacity of storage (mm) The use of the Kerby-Hatheway equation can be applied to watersheds with different characteristics [26].South African National Roads Authority Limited [27] recommends using the Kerby equation [6] which at first was created for small flat basins with a preponderance of inland flow.However, the Kerby equation can be used extensively when discussing urban stormwater in the United States (e.g., paved parking lots, roads, commercial and industrial areas, residential areas, etc.).
According to [28] the kinetic wave equation is based on the theory of kinetic waves, treating surface runoff as a fairly wide channel and accounting for the presumption of steady rainfall and turbulent flow.This equation is sufficient for basins where surface runoff predominates.
The authors suggest that the FAA is often used in estimating urban runoff because the rational method flow coefficient (C) is included [7] [26].Kirpich recommends applying his adjustment curve only to rural watersheds serving an area between 0.0040 and 0.8094 km 2 .SCS Lag is used in small catchment areas in rural areas where shallow flows are common.The original Simas-Hawkins equation deals with the delay time, adding the multiplier to get the proper focusing time formula.In the kinetic hysteresis wave equation, FAA, Kirpich, and SCS, no significant differences were found in the reference bibliographies, although the versions differ according to the units of measure chosen for the variables.The original Vent Te Chow formula is an enhanced formula [29] [30], with a choice of individual units.In the publication of Prusky and Silva, the Ven Te Chow peak time formula is considered as the concentration time formula, without applying a correction factor.
Giandotti's equation is widely used in Europe, especially in Italy.Hence, different authors obtained consistent results applying this method to the Italian basin.López et al. showed that its use is suitable for mountain basins.The original formula denominator is 0.8H, where H is the difference in height between the exit point and the average hip height.MOPU simplifies this by replacing H by multiplying the slope by the length.However, replacing H with LS involves an error, because it makes more sense to consider 2H = LS, the total slope.Another Italian formula, by Pasini and Venture, was published in Brazil [31].There is a slight difference in the coefficients of the Pasini formula (0.107 versus 0.108 for the Italian Internet) which is irrelevant.Greppi proposed to apply the Pasini equation to a basin with a gentle slope.Luino et al. used this equation in flood studies following the work of Pasini.Ventura is defined as a rural watershed.According to Mata-Lima et al., the equation was developed using agricultural catch data.The Picking equation has its primary reference in Brazil [32].
The ASCE equation is only proposed for L<0.09 km of the basin, but Kang et al. stated that the equation proved to work well in large basin studies.Haktanir & Sezen developed their method using regression analysis using data from the Türkiye watershed.Papadakis & Kazan was developed using USDA Agricultural Research Service data from 84 small rural watersheds in 22 US states (A<5 km 2 ).The NRCS technique is based on the design force relationship between rainfall intensity and time and was first created by Welle & Woodward [33] to refrain from using the original kinetic wave equation repeatedly [34] and is based on the design force relationship between time and rainfall intensity.

Watershed Physiographic Features
In carrying out concentration time analysis in a watershed area or basin, several parameters are needed according to the needs of each determined empirical formula.Table 2. provides an overview of the physiography and variables used to calculate the time of concentration in the examined basin: From the table above it can be seen the parameters of the Brantas Watershed along with their values, which will then be analyzed using the 17 empirical equations to calculate the concentration time.The analysis is carried out by entering the parameter values that have been obtained into 17 predetermined empirical equations.The following are examples of calculations for each empirical equation.The results obtained by applying the selected equations are presented in Table 3 As we can see a relative difference of 196% between the results obtained.The creation of hierarchical clustering typically results in its generation as an output.The primary application of dendrograms is in the determination of optimal object allocation to clusters.The present study employed the utilization of a dendrogram to ascertain clusters or groups that would exhibit the degree of similarity among 17 empirical equations utilized for the computation of time of concentration (Tc).The dendrogram will produce several groups based on distance in the Euclidean method and the dendrogram will be presented in Figure 4.This dendrogram uses a single Euclidean clustering method which groups data based on the similarity of the closest distance between points.So, adjacent points in Euclidean space will tend to belong to the same group in the dendrogram.From the dendrogram, each group formed in the dendrogram will produce a different level of grouping from each other.This means that there is heterogeneity and homogeneity between the groups formed.The groups formed are grouped because they display similar characteristics, for example, physical characteristics or areas or locations that have similarities in each method.
From the dendrogram results in Figure 4, it can be seen that the 17 equations are divided into 2 classes.Class 1 is the class that has the highest distance, meaning it has no level of similarity with other similarities.Consists of 3 equations, namely NRCS, SCS Lag, and Ventura.Class 1 has an average value from the dendrogram calculation, namely 1, which means that the further away from 0, the lower the level of similarity between the other equations.Based on the data these 3 equations have different physical characteristics and are located on different continents.
Class 2 is a class consisting of 14 equations Pickering, CHPW, Kinematic Wave, ASCE, Kerby Hathaway, Ven Te Chow, Kirpich, FAA, Giandotti, Haktanir and Sezen, Papadakis and Kazan, McCuen, Simas-Hawkin, and Pasini.Each equation in class 2 has a distance at the same level is at the lowest level and has an average value from the dendrogram calculation that is close to 0, which means this group has a high level of similarity between one equation and another.Therefore, it can be concluded that the equation that has a high level of similarity or it can be said that the equation that is most suitable for use in the location area under consideration is the equation that is included in class 2. Table 4 for more detailed results from the dendrogram.

Conclusion
There are behavioral differences between the studied method and available methods for developing concentrations over time, which could lead to various numerical values (the relative likelihood of achieving is 196%).To determine the best technique for estimating time estimates of concentration in the basin, hydrological monitoring must be conducted on a regular and thorough basis.Among the analyzed methods it can be seen that the 17 equations are divided into 2 classes.Class 1 consists of 3 equations NRCS, SCS Lag, and Ventura.Class 2 consists of 14 equations Pickering, CHPW, Kinematic Wave, ASCE, Kerby Hathaway, Ven Te Chow, Kirpich, FAA, Giandotti, Haktanir and Sezen, Papadakis and Kazan, McCuen, Simas-Hawkin, and Pasini.

Figure 1 .Figure 2 .
Figure 1.Location of the studied area

Figure 4 .
Figure 4. Dendrogram resulting from the methodologies grouping process

Table 1 .
A list of techniques for estimating concentration times . ( √ ) .

Table 2 .
Physiographic features and parameters of WS Brantas

Table 3 .
Tc values that were obtained after applying the chosen equations

Table 4 .
The class between dendrogram result equations