# Analysis of Hyetographs for Drainage System Modeling

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Drainage System Modeling

#### 1.2. Reference Hyetographs

^{2}prairie area, in the eastern part of the state of Illinois (USA). He chose precipitation for durations from 1 to 48 h. Based on preliminary analyses, Huff divided precipitation events into four groups, called quartiles. This determines which part of the duration of precipitation its maximum intensity occurred (each group determines another 25% of the duration of precipitation). Based on this division he created for each group, the quartile charts illustrating changes in the amount of precipitation over time are characteristic of these groups. Huff curves are, therefore, a probabilistic representation of the ratio of cumulative precipitation heights to the corresponding cumulative durations (expressed as dimensionless) in the form of probability isopleths. Huff curves have found widespread use to analyze the variability of precipitation over time, performed, among others by Bonta and Rao [37], Pani and Haragan [38], Terranova and Iaquinta [39], Elfeki et al. [40], and Pan et al. [41]. Bonta at work [42] from 2004, presented in detail the methodology for creating Huff mass curves. Bonta also analyzed other factors affecting the construction and use of Huff curves, including the sensitivity of the shape of the curves to the data sampling interval, the impact of minimum rainfall heights selected for analysis, the minimum sample size, or the shape of the curves depending on the season.

## 2. Method of Analysis

- ▪
- location of the interval Δt with the cut off (t
_{peak}) peak of maximum precipitation h_{max}(Δt), - ▪
- location of the interval Δt with the cut off (t
_{cg}) of the center of gravity of the hyetograph P_{c}/2,

_{max}(Δt)), t

_{peak}(h

_{max}(Δt))—the time of peak interval precipitation, min, T—total duration of rainfall, min, r

_{cg}—hyetograph center of the gravity position indicator, t

_{cg}(P

_{c}÷ 2)—time of the center of gravity of the hyetograph (for P

_{c}÷ 2), min, P

_{c}—cumulative (total) rainfall amount (in time T), mm.

- ▪
- m
_{1}—as the cumulative ratio of precipitation height (mass) for the time from t = 0 to t = t_{peak}(before the cut off peak of maximum rainfall h_{max}(Δt)), to the cumulative amount of precipitation for the time from t = t_{peak}down t = T (by peak):$${m}_{1}=\frac{{{\displaystyle \sum}}_{0}^{{t}_{peak}}{h}_{i}}{{{\displaystyle \sum}}_{{t}_{peak}}^{T}{h}_{i}}$$ - ▪
- m
_{2}—as the ratio of the maximum interval height (mass) of precipitation to the total height:$${m}_{2}=\frac{{h}_{max}\left(\Delta t\right)}{{P}_{c}\left(T\right)}$$ - ▪
- m
_{3}—as the ratio of the accumulated height (mass) of precipitation for the time from t = 0 to t = 0.33T, to the total height.$${m}_{3}=\frac{{{\displaystyle \sum}}_{0}^{0.33T}{h}_{i}}{{P}_{c}\left(T\right)}$$ - ▪
- m
_{4}—as the ratio of the accumulated height (mass) of precipitation for the time from t = 0 to t = 0.3T, to the total height.$${m}_{4}=\frac{{{\displaystyle \sum}}_{0}^{0.3T}{h}_{i}}{{P}_{c}\left(T\right)}$$ - ▪
- m
_{5}—as the ratio of the accumulated height (mass) of precipitation for the time from t = 0 to t = 0.5T, to the total height.$${m}_{5}=\frac{{{\displaystyle \sum}}_{0}^{0.5T}{h}_{i}}{{P}_{c}\left(T\right)}$$_{1}, m_{2}, m_{3}, m_{4}, and m_{5}—mass distribution ratios on a hyetograph (sum of rainfall amounts), h_{i}—instantaneous height of rainfall (for Δt = 1 min), mm, h_{max}(Δt)—maximum interval (Δt) precipitation height, mm, P_{c}(T)—total rainfall height (in time T), mm, 0.33T—first one-third of the total rainfall duration (T), min, 0.3T—the first 30% of the total rainfall duration (T), min, and 0.5T—the first 50% of the total rainfall duration (T), min.

_{i}—indicator of rainfall intensity irregularity over time, i

_{max}(Δt)—maximum interval (Δt) rainfall intensity, mm/min, and i

_{m}(T)—mean rainfall intensity (over time T), mm/min.

_{cg}, m

_{1}, m

_{2}, m

_{4}, and m

_{5}) are especially dedicated to the analysis of similarity of mass hyetographs in dimensionless systems (variable value Δt = 0.1T) for the DVWK pattern. The remaining two indicators (m

_{3}and n

_{i}) are dedicated to comparative analyses of shapes of real precipitation hygrographs in dimensional systems (Δt = const = 5 min). In particular, the indicator m

_{3}, according to Formula (5), is intended for analyses of the similarity of convective precipitation hyetographs to a Euler type II pattern.

## 3. Study Area and Data Used

## 4. Results and Discussion

#### 4.1. Grouping Precipitation for Statistical Analysis

#### 4.1.1. Huff’s Method

_{i}/P

_{c}≡ P/P

_{c}—for cumulative durations t

_{i}/T. First, the Huff methodology was used to analyze the similarity of the shapes of the examined hyetographs. Huff curves are based on the division of precipitation into four groups—quartiles depending on the part of the duration, which occurred as the largest amount (mass gain). According to this division method, precipitation increases in each of the 25% of the rainfall duration, which were determined for each of the analyzed precipitation. The rainfall were assigned to four quartile groups on this basis.

#### 4.1.2. Cluster Analysis Using the Ward Method

_{i}and y

_{i}are the values of cumulative rainfall amounts of the examined rainfall x and y, and p is the number of rainfalls tested.

_{i}—value of the cumulative rainfall amount being the segmentation criterion for i rainfall, k—number of objects in focus.

#### 4.1.3. Cluster Analysis Using the Method of k-Means

_{i}/P

_{c}= 0.575—almost six times bigger than the average h

_{i}/P

_{c}= 0.1). In turn, that makes it similar in this respect to the Euler model precipitation (as a model for modeling stormwater drainage).

_{i}/P

_{c}= 0.23), No. 3 (h

_{i}/P

_{c}= 0.27), and No. 4 (h

_{i}/P

_{c}= 0.575) with their relative position: r = t

_{i}/T ≈ (0.6–0.7) for cluster No. 1, r ≈ (0.2–0.3) for cluster No. 3, and r ≈ 0.2 for cluster No. 4. Whereas, for cluster No. 2, the most aligned course of relative rainfall heights (on an average level: h

_{i}/P

_{c}= 0.1) during (T). Method k-means, therefore, gives qualitatively better results of rainfall grouping compared to the Ward method.

#### 4.2. Models Verification

#### 4.2.1. Verification of Euler Type II Pattern

_{cg}∈ [0.12, 0.38], m

_{1}∈ [0.25, 2.73], m

_{2}∈ [0.27, 0.65], m

_{3}∈ [0.32, 0.96], and n

_{i}∈ [2.16, 6.36].

_{cg}do not exceed 0.38. Therefore, they slightly exceed 0.33, which is characteristic for the Euler model. Indicator value m

_{1}is extremely 2.73, which means that the mass of precipitation before the peak to the mass after the peak has a maximum ratio of 2.73:1. The ratio of the maximum interval precipitation to the total height does not exceed the value m

_{2}= 0.65, which means it can be up to 6.5 times the average (h

_{i}/P

_{c}= 0.1). The indicator values m

_{3}are of particular interest (the ratio of accumulated precipitation mass for one-third of the initial time to the total mass). The indicator values m

_{3}change in the range from 0.32 (even rainfall) up to 0.96 (highly uneven precipitation). Similarly, the n

_{i}indicator values are characteristic. They range from n

_{i}= 2.16—low rainfall unevenness (maximum range values are about twice the average), up to even n

_{i}= 6.36—high rainfall unevenness (maximum range values are over six times higher than the average).

_{i}/P

_{c}) are depicted using box charts that include the pictogram information on the location, dispersion, and shape of the empirical distribution of the studied size. The shape of the box (its range) is shown by the shape of the graph, covering the entire range of data (from the lowest to the highest value). The length of the box is equal to the quarter (quartile) range, i.e., the difference between the first and third quartiles. The quartile is, therefore, one of the measures of the location of given observation values. The first quartile (bottom edge of the box) contains 25% of observations. The second quartile divides the observation collection in half, which corresponds to the median. The third quartile (upper edge of the box) divides the observation dataset into two parts, respectively, with 75% located below this quartile and 25% located above. In Figure 10, the whiskers are limited to the 10% and 90% percentile of the data set (which are identified in the literature by Bonta [42] and Huff [36,48,49], with confidence intervals at 10% and 90%, respectively).

_{i}/P

_{c}, according to sum curves. It falls from the level of approximately 0.29 to approximately 0.25 by interval increments. There are no differences in the peak location itself: r = 0.2t

_{i}/T, i.e., in one-fifth of the duration of rainfall.

_{3}as the ratio of accumulated precipitation for the time from t = 0 to t = 0.33T to total height over time t = T (according to Formula (5)). Mass distributions by indicator m

_{3}proved to be almost identical, independent of T and C. The average value of the indicator was m

_{3}= 0.714. The “model” differentiation of unevenness in the intensity of Euler’s model rainfall by the n

_{i}indicator was also analyzed, as the ratio of the maximum intensity of the rainfall interval (Δt = 5 min) to mean intensity over time T (according to Formula (8)). The results of the calculations are given in Table 3.

_{i}varies (from 3.47 to 12.03) in individual durations T = 30–180 min, but independent of precipitation (C) frequency of occurrence. Average indicator value n

_{i}= 6.96 (Table 3). The value of the peak position ratio in Euler type II models is, on average, r = 0.285 changes in the range of 0.25 for T = 30 min to 0.32 for T = 180 min, regardless of C.

_{3,}and n

_{i}. They are dedicated especially to quantitative assessments of the similarity of the shape of real precipitation histograms to the Euler type II standard. For the analyzed precipitation (15 C and 10 F with C ∈ [1.0, > 50] years), characteristic (real—referred to time T) values of indicators are: r ∈ [0.07, 0.39], mean value: 0.22, m

_{3}∈ [0.30, 0.96], mean value: 0.65, n

_{i}∈ [2.16, 9.49], mean value: 4.92.

_{c}= 14.8 mm, T = 38 min, and C = 1.7 years) requires an extension of the actual duration T = 38 min to model one T’ = 45 min. Then penultimate interval Δt, between the 35th and 40th minute, it will contain precipitation from the last 3 min, while the last interval Δt between 40 and 45 min will be empty. On this basis, a re-comparative analysis of real precipitation histograms with Euler’s model histograms was performed for model time T’. Three indicators were again used for quantitative analyzes: r, m

_{3}, and n

_{i}and were interpreted. For example, r’—as the ratio of the position of the interval Δt = 5 min with the peak cut off (t

_{peak}) maximum height h

_{max}(Δt)—until model time T’ (instead of to time T as in Formula (1)). Similarly, the definitions of other indicators were modified. Characteristic for the model time for the analyzed precipitation T’ indicator values are: r’ ∈ [0.06, 0.38], mean value: 0.21, m

_{3′}∈ [0.36, 0.97], mean value: 0.69, n

_{i}′ ∈ [2.22, 9.51], mean value: 5.23.

_{3′}and 0.31 for n

_{i}′, compared to the calculated for the actual duration of precipitation T. However, the results of these analyses allow drawing methodically correct conclusions regarding the comparison of 25 dimensional hyetographs of precipitation from Jelenia Góra with 28 Euler type II models for this station.

- ▪
- Peak position indicator values of maximum height versus time T’, include: r’ ∈ [0.06, 0.38], with an average value r’ = 0.21. The value of this indicator in Euler type II models is on average r = 0.285-changes: r ∈ [0.25, 0.32] for the range T = T’ ∈ [30, 180] min. Both peaks occur in the first, one-third rainfall duration T = T’.
- ▪
- Mass distributions on 25 dimensional histograms were variable within: m
_{3′}∈ [0.36, 0.97], however average value: m_{3′}= 0.69 is very close to the constant value m_{3′}= 0.714 for 28 Euler type II models. In both cases, the main precipitation mass is located in the first, one-third of the duration T = T’. - ▪
- Rainfall irregularity over 25 histograms was significant within limits n
_{i}’∈ [2.22, 9.51], on average n_{i}’ = 5.23 in Euler type II standard precipitation. The unevenness was similar within the limits n_{i}∈ [3.47, 12.03], on average n_{i}= 6.96. These values should also be considered similar.

#### 4.2.2. DVWK Pattern Verification

_{cg}, m

_{1}, m

_{2}, m

_{4}, m

_{5}, and n

_{i}—dedicated especially to quantitative assessments showing the similarity of the real precipitation hyetographs’ shape to the DVWK pattern. The characteristic values of indicators calculated for the analyzed 13 precipitation rates are: r ∈ [0.05, 0.55], mean value: 0.34, r

_{cg}∈ [0.27, 0.49], mean value: 0.38, m

_{1}∈ [0.10, 2.29], mean value: 0.92, m

_{2}∈ [0.18, 0.46], mean value: 0.31, n

_{i}∈ [1.80, 4.61], mean value: 3.05, m

_{4}∈ [0.07, 0.61], mean value: 0.37 (median 0.38), m

_{5}∈ [0.57, 0.93], and mean value: 0.74 (median 0.71).

_{5}mass distribution (i.e., for t

_{i}/T = 0.50)—with median value: h

_{i}/P

_{c}= 0.71 for the analyzed rainfall and h

_{i}/P

_{c}= 0.70 for the DVWK pattern. However, for the mass distribution indicator m

_{4}(for t

_{i}/T = 0.30), there are already discrepancies in median values: h

_{i}/P

_{c}= 0.38 for the analyzed rainfall and h

_{i}/P

_{c}= 0.20 for the DVWK pattern. The average value of unevenness during the intensity of the analyzed precipitation: n

_{i}= 3.05 is practically equal to the value n

_{i}= 3.00 for the DVWK pattern. For the tested precipitation, according to the course of the sum curve, for the first 20% of the duration (T) will occur in 15% of its total amount (P

_{c}) after a time of 60% T appears as 85% P

_{c}and the remaining 15% P

_{c}will occur at 40% T.

## 5. Discussion and Conclusions

_{3′}= 0.69 is very close to the constant value m

_{3}= 0.71—for 28 Euler type II models. This means that the main rainfall mass is located in the first one-third of the duration T = T’. The value of the peak height indicator of the maximum height h

_{max}(Δt) was, on average, r’ = 0.21, and, in Euler type II models, r = 0.285. Therefore, both peaks occur in the first, one-third rainfall duration T = T’. Generally, it should be stated that the Euler type II standard is suitable for the description of precipitation from the IMGW-PIB mountain station in Jelenia Góra. The tested discrepancies fall within the accuracy class of hydrological measurements and calculations related to random phenomena.

_{5}where median values: h

_{i}/P

_{c}= 0.71 for the analyzed precipitation and h

_{i}/P

_{c}= 0.70 for the DVWK pattern. For the second indicator m

_{4}, there are already some discrepancies in median values: h

_{i}/P

_{c}= 0.38—for the analyzed precipitation and h

_{i}/P

_{c}= 0.20 for the DVWK pattern. The distribution of mass of precipitation in Jelenia Góra is as follows: for 20% T—15% P

_{c}, for 60% T—85% P

_{c}(remaining 15% P

_{c}—in 40% T). The average value of unevenness during the intensity of the tested precipitation: n

_{i}= 3.05 is practically equal: n

_{i}= 3.00—for the DVWK standard. Generally speaking, it should be stated that the DVWK standard is approximate in the class of the accuracy of measurements and hydrological calculations of random phenomena.

- ▪
- Euler type II and DVWK model rainfall patterns are similar to real precipitation (in the case of the analyzed station), so they can be used for hydrodynamic modeling of stormwater drainage.
- ▪
- Development of model rainfall scenarios for hydrodynamic modeling should be based on local DDF/IDF curves (developed for a given location, based on many years of measurements).
- ▪
- In order to obtain reliable hydrodynamic modeling results, both Euler type II and DVWK model rainfall should be used in parallel with real precipitation models. This approach will increase the number of variants and, thus, the certainty of simulations.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Typical rainfall intensity distributions over time, according to DVWK [16]: (

**a**) constant intensity precipitation (i.e., block precipitation), (

**b**) precipitation with a maximum intensity at the beginning, (

**c**) precipitation with a maximum intensity at the center, and (

**d**) precipitation with maximum intensity at the end.

**Figure 2.**Diagram of Euler type II model precipitation [12].

**Figure 3.**Recommended distribution of rainfall sums according to DVWK [44].

**Figure 5.**Dimensionless rainfall curves with a median curve for 4 Huff quartile groups: (

**a**) I quartile, (

**b**) II quartile, (

**c**) III quartile, and (

**d**) IV quartile.

**Figure 7.**Dimensionless precipitation mass curves with a median for four clusters according to the Ward method: (

**a**) Cluster No. 1, (

**b**) cluster No. 2, (

**c**) cluster No. 3, and (

**d**) cluster No. 4.

**Figure 8.**Dimensionless rainfall mass curves with a median for four clusters separated by the k-means method: (

**a**) Cluster No. 1, (

**b**) cluster No. 2, (

**c**) cluster No. 3, and (

**d**) cluster No. 4.

**Figure 9.**Dimensionless mass curves with a median for 15 convective (C) and 10 frontal (F with T ≤ 180 min) rainfall from clusters No. 3 and 4 (for k = 4).

**Figure 10.**Dimensionless “reference” hyetograph for 25 precipitation (with T ≤ 180 min) with box charts with measurement results.

**Figure 11.**Box plot hyetograph of 10 frontal and three low-pressure precipitation (solid line) together with the DVWK pattern (dashed line).

Classification | Number of Rainfall (Percent) | ||||
---|---|---|---|---|---|

Duration | t ≤ 120 min | ≤60 min | 10 (12%) | 23 (29%) | 80 (100%) |

(60, 120] min | 13 (16%) | ||||

t ∈ (120, 720] min | (120, 180] min | 13 (16%) | 39 (49%) | ||

(180, 360] min | 12 (15%) | ||||

(360, 720] min | 14 (18%) | ||||

t > 720 min | (720, 1440] min | 11 (14%) | 18 (22%) | ||

>1440 min | 7 (9%) | ||||

Frequency of Occurrence | C ∈ [1, 2) years | 26 (33%) | 80 (100%) | ||

C ∈ [2, 5) years | 24 (30%) | ||||

C ∈ [5, 10) years | 9 (11%) | ||||

C ≥ 10 years | 21 (26%) |

Exceedance Classes C | Quartile Groups | Total | |||
---|---|---|---|---|---|

I | II | III | IV | ||

C ∈ [1, 2) years | 14 | 5 | 5 | 2 | 26 (33%) |

C ∈ [2, 5) years | 8 | 10 | 5 | 1 | 24 (30%) |

C ∈ [5, 10) years | 4 | 2 | 2 | 1 | 9 (11%) |

C ≥ 10 years | 4 | 10 | 7 | 0 | 21 (26%) |

Total | 30 (37%) | 27 (34%) | 19 (24%) | 4 (5%) | 80 (100%) |

Frequency of Rainfall Occurrence | Rainfall Duration T, Min | Mean | ||||||
---|---|---|---|---|---|---|---|---|

30 | 45 | 60 | 75 | 90 | 120 | 180 | ||

C = 1 year | 3.47 | 4.60 | 5.60 | 6.56 | 7.43 | 9.10 | 12.05 | 6.97 |

C = 2 years | 3.46 | 4.59 | 5.61 | 6.53 | 7.44 | 9.05 | 11.99 | 6.95 |

C = 5 years | 3.47 | 4.59 | 5.61 | 6.55 | 7.43 | 9.06 | 12.04 | 6.96 |

C = 10 years | 3.47 | 4.60 | 5.61 | 6.55 | 7.44 | 9.07 | 12.02 | 6.97 |

Mean | 3.47 | 4.60 | 5.61 | 6.55 | 7.43 | 9.07 | 12.03 | 6.96 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wartalska, K.; Kaźmierczak, B.; Nowakowska, M.; Kotowski, A.
Analysis of Hyetographs for Drainage System Modeling. *Water* **2020**, *12*, 149.
https://doi.org/10.3390/w12010149

**AMA Style**

Wartalska K, Kaźmierczak B, Nowakowska M, Kotowski A.
Analysis of Hyetographs for Drainage System Modeling. *Water*. 2020; 12(1):149.
https://doi.org/10.3390/w12010149

**Chicago/Turabian Style**

Wartalska, Katarzyna, Bartosz Kaźmierczak, Monika Nowakowska, and Andrzej Kotowski.
2020. "Analysis of Hyetographs for Drainage System Modeling" *Water* 12, no. 1: 149.
https://doi.org/10.3390/w12010149