Systematic Assessment of Minimum Inter-Event Time Determination Methods and Precipitation Thresholds for Constructing Design-Critical Huff Hyetographs

Abstract

The primary processing of high-resolution precipitation records (5 min and shorter) is crucial for constructing dimensionless design hyetographs and identifying design-critical precipitation scenarios for urban drainage systems. A key step in this process is separating continuous precipitation records into individual precipitation events, typically based on minimum inter-event time (MIT) and precipitation amount thresholds. This separation directly influences the subsequent analysis steps and the accuracy of the design hyetographs. Building upon this foundation, this study systematically analyses how different MIT determination methods influence the construction of dimensionless Huff hyetographs in a moderately humid continental climate. Three approaches for defining MIT were examined: a fixed MIT method (1–12 h), an autocorrelation-based method (AC), and a kernel density estimation approach (KDE). The analysis also considers the effects of minimum precipitation thresholds (P = 1, 3, and 5 mm) and precipitation duration classes (all durations and short-duration events with $T\le 2$ h), utilising a continuous 10-year series of 5 min precipitation data. The results demonstrate that the choice of MIT substantially affects the identified precipitation events, duration, total amount, and the median Huff curve’s shape, especially for precipitation types with early and late maximum intensity. Specifically, increasing MIT values produces longer and deeper events with steeper Huff curves, while precipitation thresholds mainly filter weaker events rather than impacting peak intensities. The AC method yields results similar to larger fixed MIT values (≈6–9 h), whereas the KDE method corresponds to shorter separations (≈1–3 h). To unify the assessment of design relevance, a composite design index combining Huff curve slope and short-term peak intensities was introduced. Analysis shows that short-duration convective precipitation with an early maximum is the most critical design scenario. However, late-maximum events (events in which peak intensity occurs in the fourth quartile of storm duration, Type 4) can become equally critical when longer MIT values or autocorrelation-based separation are applied. These findings underscore the importance of a transparent and methodologically consistent definition of precipitation event separation criteria when using dimensionless hyetographs in urban drainage design.

1. Introduction

The analysis of precipitation data and the construction of dimensionless hyetographs are essential steps in hydrology, especially for defining design precipitation in urban runoff calculations. Dimensionless hyetographs are widely used in hydrological engineering because they allow the temporal distribution of precipitation to be constructed for any storm duration and total precipitation amount derived from intensity–duration–frequency (IDF) relationships. However, data with higher time resolution (1 h or less) from stations are generally either not sufficiently distributed across all locations of hydrological interest or have relatively short time series. In contrast, spatial data (from reanalyses, meteorological and climate model results) distribute precipitation both temporally and spatially. Notably, on longer time scales (1 day or, especially, a month or a year), these data sources generally provide accurate precipitation values. Still, on shorter time scales (1 h or less), they tend to underestimate actual intensity. As a result, hydrological-hydraulic modelling and dimensioning based solely on recorded precipitation is very demanding if the intention is to cover a wide range of real possibilities in the basin and include analysis of the probability and return periods of individual events. When processing the precipitation record for hyetograph preparation, it is therefore necessary to consider the analysis’s purpose and application. To support this process, dimensionless hyetographs serve as practical tools for describing precipitation timing distributions. Their shape provides insight into the character of precipitation events, distinguishing between short-term, high-intensity convective showers and longer-term, lower-intensity cyclonic precipitation and determining the position of maximum intensity within each event. In engineering practice, various methods are used to create hyetographs, including Huff curves, the average-variability method, the triangular method and the Chicago method [1]. In this paper, particular emphasis is placed on the dimensionless Huff hyetograph, which will be discussed next.

To introduce the primary method of interest, Huff curves are probability isolines of dimensionless curves of accumulated precipitation as a function of time [2]. In this method, each event is divided into four equal time segments and the quarter containing the highest precipitation is identified. Accordingly, precipitation events are classified into four types depending on whether the maximum precipitation intensity occurs in the first, second, third or fourth quartile of the storm duration. Note that individual storm durations are variable; the Huff method normalises each event’s total duration to a dimensionless scale (0–100%), enabling direct comparison of events of different lengths. Numerous studies confirm the utility of Huff curves [3,4,5,6,7]. More recently, an additional fifth type has been introduced to represent time-uniformly distributed precipitation [6].

Huff [8] points out that the median (50%) Huff curve is the most representative and stable among all quartiles, because it retains its relative position when new events are added, unlike the extreme (10% and 90%) curves. Therefore, in engineering practice, the median curve is most often used as a representative design hyetograph. Building on this, Bonta [3] analysed the sensitivity of Huff curves to different precipitation data classification methods and found that precipitation in the first quartile typically lasts less than 6 h. Additionally, in an earlier paper, Bonta [9] recommends creating separate Huff curves for different seasons, since winter precipitation shows less variability in intensity than spring-summer showers, which often include convective events or sub-events due to high humidity. The importance of adequately preparing and categorising precipitation data was also highlighted by Azli and Rao [10] and Bonacci [1]. Furthermore, Bonacci [1] emphasises that, in analysing precipitation and constructing the runoff hyetograph, it is necessary to apply thresholds for minimum precipitation amount or intensity. Bonacci [1] also points out that convective and cyclonic precipitation should be analysed separately because they originate from fundamentally different atmospheric processes, which result in distinct temporal structures and intensity patterns of precipitation. Therefore, Bonacci [1] notes that convective precipitation events are typically short-lived, usually lasting about 1–2 h, which can be used as a practical criterion for distinguishing them from longer-duration cyclonic precipitation.

Separating the continuous precipitation record into individual precipitation events is a key step in constructing a dimensionless Huff hyetograph. Central to this process is the Minimum Inter-event Time (hereinafter, MIT). The choice of MIT values directly affects the number of isolated events, their duration, total precipitation and short-term intensity statistics [11,12,13]. As MIT increases, longer drought breaks are combined into a single event, leading to longer events with higher precipitation. At the same time, the temporal position of the peak precipitation intensity within the event—the quantity that determines the Huff curve type classification—changes as well, which in turn indirectly alters the shape of the resulting dimensionless Huff curves. Establishing an appropriate MIT is thus a critical methodological consideration. The literature reports very different MIT values, ranging from a few tens of minutes to several hours, depending on the climatic region, data time resolution and the analysis purpose [14,15,16]. In practice, MIT values are usually determined in three different ways: 1. by personal assessment, 2. by statistically based methods and 3. by autocorrelation of the rain time series. For example, Huff [8] used a value of 6 h to separate storms in the eastern US, which was later widely accepted in hydrology [12,17]. On the other hand, in the analysis of short-term, intense precipitation with high time resolution, significantly shorter criteria are often applied, on the order of 1–2 h [13]. In this context, intense precipitation refers to precipitation events characterised by high short-duration precipitation intensities, commonly represented by the maximum 30 min precipitation intensity, which is widely used to quantify storm intensity and precipitation erosivity [13]. This wide range suggests that a universal MIT value does not exist, underscoring the importance of carefully selecting event-separation criteria, as these choices can significantly affect precipitation’s statistical characteristics.

Statistically based methods for defining MIT assume that the arrival of independent precipitation events follows a Poisson process, while the times between events follow an exponential distribution. This assumption represents a widely adopted operational simplification rather than a strict physical model of precipitation dynamics—precipitation is inherently non-stationary and may exhibit seasonal clustering or synoptic-scale persistence. The applicability of this assumption for any given MIT–threshold combination is therefore not imposed a priori but verified empirically through statistical tests described below, with combinations that fail the Poisson consistency criterion excluded from further consideration.

The exponential method of Restrepo-Posada and Eagleson [18] determines MIT as the value of the dry period for which the coefficient of variation of successive dry phases equals 1 [17,19]. Alternatively, optimal MIT can be defined as the value that gives the maximum p-value in the exponential distribution compliance test, most commonly the Kolmogorov–Smirnov test [20,21]. As part of these procedures, the Poisson consistency test of the annual number of isolated events is often additionally performed. In the recent literature, statistical tests (Kolmogorov–Smirnov and Poisson) are increasingly combined with subsequent smoothing of the empirical time distribution between events using the Kernel Density Estimation method [21]. In the present paper, this unified approach is referred to as the KDE method. This provides a continuous approximation of the dry period distribution, from which the MIT value is determined based on the intersection of the exponential and empirical distributions.

A third, commonly used, group of approaches is based on the analysis of the autocorrelation of the precipitation series, in which MIT is defined as the time lag at which the autocorrelation becomes statistically insignificant [1,19,21]. This approach directly links the choice of MIT to the internal time structure of the precipitation record and often yields higher MIT values than methods based on the distribution of dry periods.

In addition to the event separation criterion, the minimum total precipitation amount threshold is often used in practice [1,17,21]. This threshold (in millimetres) is used to exclude from the analysis all precipitation events whose total amounts are less than a specified threshold. This way, the analysis focuses on precipitation events that generate runoff. Different minimum precipitation thresholds have been reported in the literature, depending on the analysis purpose and climatic conditions. Bonacci [1] recommends using minimum precipitation thresholds to construct runoff hyetographs, thereby excluding very weak events that have little hydraulic influence. Azli and Rao [10] use a minimum total event precipitation amount of 25.4 mm (accumulated over the entire event duration) to identify significant precipitation events. Some studies further highlight the 5 mm threshold as suitable for design-critical precipitation analyses in urban areas [21]. The application of the threshold changes the number and characteristics of isolated events and affects the shape of the median Huff curves, particularly their slope and the position of the maximum precipitation concentration.

The present study provides a systematic and comprehensive assessment of different minimum inter-event time (MIT) determination approaches and their influence on the construction of dimensionless Huff curves. Unlike previous studies, fixed MIT values, autocorrelation-based (AC) and statistically based (KDE) methods are analysed within a unified framework, together with different precipitation amount thresholds. In addition, a composite design index is introduced to evaluate the design relevance of precipitation events for urban drainage applications and a separate analysis of short-duration events ($T\le 2$ h) is performed to better represent convective precipitation. During the analysis, special attention was paid to the fact that the constructed Huff curves would serve as the basis for the analysis and design of the urban stormwater drainage system.

Section 2 presents the research methodology, including descriptions of the data used, the procedures for primary processing of the precipitation record and the methods used for separating precipitation events and constructing Huff hyetographs.

Section 3 presents the results of the analysis, with an emphasis on the sensitivity of the physical characteristics of precipitation and the shapes of dimensionless hyetographs to the selected processing methods. Section 4 provides a discussion of the findings, including their physical interpretation and implications for engineering practice. In Section 5, the main findings and recommendations for applying the results in urban drainage design are presented.

2. Materials and Methods

2.1. Study Area

The area of analysis includes the city of Osijek, located in the eastern part of the Republic of Croatia, in the region of Slavonia. The climate of the area, according to the Köppen (Köppen-Geiger) classification, belongs to a moderately warm humid continental climate (type C), without a pronounced dry season (type f), with warm to hot summers (on the border between type a and type b) and pronounced seasonal temperature contrasts [22]. In the warm season (May–September), intense convective precipitation is frequent, often in the form of short, heavy showers (often ≥10 mm/h). Such precipitation plays a key role in sizing urban drainage systems due to its short duration and high peak intensities.

The analysis used data from the automatic rain gauge station in Čepin [23], located in the immediate vicinity of the city of Osijek. The station’s position (WGS84) is 45.502250° N, 18.561694° E, at an altitude of 89 m. A continuous series of 5 min precipitation records from 2016 to the end of 2024 was used. The data are uninterrupted and show no significant deficiencies in the record, allowing for reliable identification of precipitation events and statistical analysis of their physical characteristics. The 5 min time resolution enables detailed monitoring of peak intensities and short-term intense showers, which are of particular interest for creating dimensionless hyetographs and estimating design-critical precipitation scenarios in urban drainage.

2.2. Methodology

The analysis of the impact of primary processing of pluviographic records on the construction of dimensionless hyetographs was conducted using our own MATLAB^® code [24]. The methodology is designed as a series of successive steps leading from the original precipitation record to the final comparison of the design criticality of different types of hyetographs. Design criticality is a composite index introduced in this study to compare the analysed hyetograph variants quantitatively in terms of their relevance to urban drainage design. Figure 1 shows the entire analysis process. The initial step involves reviewing and preparing the input 5 min precipitation data. After that, the criteria for separating individual precipitation events are defined, where the minimum time interval between events (MIT) is determined using three different methods:

1.

Fixed MIT value method—the influence of MIT values in the range from 1 h to 12 h is analysed, with a step $\Delta T=1$ h;

2.

Determination of MIT values by the autocorrelation method (hereinafter: AC method);

3.

Statistical method—application of the Kolmogorov–Smirnov and Poisson tests with the Kernel Density Estimation (KDE) method of function smoothing (hereinafter referred to as the KDE method).

Based on the defined MIT, the continuous record is separated into individual precipitation events, after which thresholds for total precipitation amount are applied to exclude weak events from the analysis; three threshold values are evaluated independently: $P=1$ mm, $P=3$ mm and $P=5$ mm. In each variant, events whose total precipitation amount is less than the selected threshold are excluded. The study was carried out separately for precipitation of all durations and for short-duration precipitation ($T\le 2$ h).

The threshold of T ≤ 2 h was selected to allow a separate analysis of short-duration precipitation events, which are typically associated with convective precipitation. This distinction follows the recommendations of Bonacci [1], who emphasises the importance of distinguishing short-lived convective precipitation from longer-duration cyclonic events.

For each isolated precipitation event, basic physical characteristics are determined, events are grouped into types based on the position of maximum intensity within the precipitation duration and dimensionless Huff curves are constructed. Based on the obtained curves and the corresponding physical parameters, the sensitivity analysis of the selected processing methods is conducted and the final comparison of the design criticality across different precipitation and duration classes is presented.

The analysis was structured as a full factorial combination of three dimensions: (i) event-separation method—12 fixed MIT values (1–12 h, step 1 h), the autocorrelation-based method (AC) and the KDE-based method, yielding 14 variants; (ii) precipitation duration class—all isolated events and short-duration events ($T\le 2$ h), yielding 2 variants; and (iii) Huff curve type—Types 1–5, yielding 5 variants. This results in a total of 14 × 2 × 5 = 140 median Huff curve variants evaluated.

2.3. Methods for Determining the MIT

2.3.1. Minimum Inter-Event Time (MIT) Fixed Value Method

MIT’s fixed-value method represents the most straightforward approach to separating precipitation events. The MIT value is chosen in advance based on experience, local climate or literature. Fixed MIT values from 1 h to 12 h, with a 1 h step ($\Delta T=1$ h), were analysed to assess the sensitivity to this parameter systematically.

Events are separated according to the following rule: if the dry period between two consecutive precipitation intervals is shorter than the selected MIT value, they are treated as a single event; otherwise, they are treated as separate events. This method is easy to use, but its main drawback is the subjectivity in selecting MIT values, which can significantly influence the number of detected events, their duration and total precipitation.

2.3.2. Autocorrelation Method (AC)

The autocorrelation method (AC) defines MIT as the time lag for which the autocorrelation function of the precipitation series becomes statistically insignificant [1,19,21]. In the present paper, Spearman’s rank correlation was applied, which is more robust to deviations from normality and to the presence of extreme values in the precipitation series [1].

Let ${\left\{x_i\right\}}_{i=1}^n$ denote the original time series of the 5 min precipitation data. Then the rank transformation is defined as:

$$R_i=\mathrm{rank}\left(x_i\right)$$

(1)

where $R_i$ represents the rank (ordinal position) of the values $x_i$ within the ascending sorted array of all values. Given the frequent occurrence of the same values in a series of precipitation (e.g., $x_i=0$ for dry intervals), the method of tied ranks was applied when assigning ranks, where identical values are assigned with the arithmetic mean of their positions. An autocorrelation function based on ranks is defined as:

$$r_s\left(k\right)={\displaystyle \frac{{\displaystyle \sum _{i=1}^{n-k}(R_i-\overline{R})(R_{i+k}-\overline{R})}}{{\displaystyle \sum _{i=1}^n{(R_i-\overline{R})}^2}}}$$

(2)

where $r_s\left(k\right)[-]$ is the dimensionless Spearman rank autocorrelation coefficient; $R_i[-]$—rank of precipitation values in time step i; $\overline{R}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{R}_i={\displaystyle \frac{n+1}{2}}[-]$—the mean of the rank sequence is a fixed quantity determined solely by the sample size n. Since ranks are assigned as consecutive integers from 1 to n, their sum equals ${\displaystyle \frac{n(n+1)}{2}}$ and their mean is therefore always ${\displaystyle \frac{n+1}{2}}$, irrespective of the underlying precipitation values. $k[-]$—time lag expressed in the number of time steps (dimensionless integer); $n[-]$—total number of measurements (dimensionless integer).

MIT is defined as the shortest time lag k for which Spearman’s rank correlation coefficient $r_s\left(k\right)$ becomes statistically insignificant ($p=0.05$), i.e., it falls within the confidence limits. Alternatively, it is the time of the first local minimum of the autocorrelation function, whichever occurs first (Figure A1) [1]. Before identifying the local minimum, the autocorrelation function was smoothed with a Savitzky–Golay filter [25]. This step removes noise from the coefficient set to improve the reliability of MIT point detection and avoid minor numerical oscillations.

The criterion of the first local minimum serves as a pragmatic operational bound that prevents unrealistically long MIT values when the autocorrelation decays slowly without crossing the significance threshold—a situation typical of prolonged cyclonic precipitation. It is acknowledged, however, that this criterion cannot unambiguously distinguish a genuine inter-system separation from a transient reduction in temporal persistence within a single multi-cell or frontal precipitation system. Consistent with the assessment of Bonacci [1], the rank correlation method is therefore regarded as only partially physically based and the local minimum criterion should be interpreted as an operational approximation rather than a definitive indicator of meteorological event boundaries. Using the rank transformation rather than the original precipitation values is justified by the inhomogeneity of the precipitation series. The data includes many zeros (dry intervals) and occasional extreme values (intense showers). Spearman’s rank correlation maintains robustness with this data structure.

2.3.3. Kernel Density Estimation (KDE) Method

The KDE method is a statistically based approach to determining optimal MIT and precipitation amount thresholds by estimating the probability density function using core functions [21]. Unlike the traditional approach that relies on a subjective assessment of the coefficient of variation ($c_v\approx 1$), the KDE method introduces objective statistical tests.

Statistical Representation of Precipitation Events

For the selected pair of MIT and precipitation amount threshold (P) values, the continuous precipitation record is separated into a series of precipitation event–dry period cycles. Each event is characterised by three variables: precipitation amount (A), event duration (T) and the dry period to the next event (b). These variables are assumed to follow an exponential distribution (Figure A2):

$$\begin{array}{cc}\hfill f_A\left(A\right)& =\zeta e^{-\zeta A},A>0\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill f_T\left(T\right)& =\lambda e^{-\lambda T},T>0\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill f_B\left(b\right)& =\psi e^{-\psi b},b>0\hfill \end{array}$$

(5)

where $\zeta$ [${\mathrm{mm}}^{-1}$], $\lambda$ [${\min }^{-1}$] and $\psi$ [${\mathrm{h}}^{-1}$] are the exponential rate parameters for precipitation amount, event duration and inter-event time, respectively. Consequently, the probability density functions $f_A\left(A\right)$, $f_T\left(T\right)$, and $f_B\left(b\right)$ carry units of ${\mathrm{mm}}^{-1}$, ${\min }^{-1}$ and ${\mathrm{h}}^{-1}$, respectively, ensuring that each integrates into unity over its non-negative domain.

Probability Density Function Estimation Using KDE

The probability density function for each characteristic precipitation event variable is estimated using a KDE. A Gaussian kernel is centred at each observed value and the resulting contributions are summed to produce a continuous density estimate:

$$\widehat{f}\left(x\right)={\displaystyle \frac{1}{nw}}\sum _{i=1}^{n}K\left({\displaystyle \frac{x-x_i}{w}}\right)$$

(6)

where $n[-]$ is the number of samples (dimensionless); w carries the same physical units as $x_i$ (mm for precipitation amount, min for event duration and h for inter-event time); $x_i$ is the individual measured value with the corresponding units; and $K(·)[-]$ is the dimensionless kernel function. The estimated density $\widehat{f}\left(x\right)$ accordingly carries units of [unit of x]⁻¹ (i.e., ${\mathrm{mm}}^{-1}$, ${\min }^{-1}$ or ${\mathrm{h}}^{-1}$, depending on the variable being estimated). In this paper, the Gaussian core function is used:

$$K\left(u\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{e}^{-u^2/2}$$

(7)

The Gaussian kernel $K\left(u\right)[-]$ is dimensionless; its argument $u=(x-x_i)/w[-]$ is also dimensionless by construction, as both the numerator and denominator carry the same units as x.

The bandwidth w controls the smoothness of the KDE estimate: a narrow bandwidth produces a spiky, high-variance estimate, while an excessively wide bandwidth over-smooths the distribution and obscures genuine features. To select w in an objective and reproducible manner, Silverman’s rule of thumb is applied [26]:

$$w={\left({\displaystyle \frac{4}{3n}}\right)}^{1/5}\widehat{\sigma }$$

(8)

where $\widehat{\sigma }$ is the sample standard deviation, carrying the same units as $x_i$ (mm, min or h, respectively). Consequently, the bandwidth w also carries the same units as the input variable, preserving dimensional consistency in Equation (6).

Given that the characteristics of precipitation events are non-negative, a marginal bias correction was applied using the data-mirroring method. The new estimated density function is:

$${\widehat{f}}_n\left(x\right)={\displaystyle \frac{1}{nw}}\left[\sum _{i=1}^{n}K\left({\displaystyle \frac{x-x_i}{w}}\right)+\sum _{i=1}^{n}K\left({\displaystyle \frac{-x-x_i}{w}}\right)\right]$$

(9)

The boundary-corrected estimate ${\widehat{f}}_n\left(x\right)$ carries the same units as $\widehat{f}\left(x\right)$ (i.e., [unit of x]⁻¹).

Statistical Tests

For each combination of MIT and the precipitation threshold, two statistical tests are performed. The Poisson test checks whether the annual number of precipitation events (n) follows the Poisson distribution. The ratio is calculated:

$$r_p={\displaystyle \frac{\mathrm{Var}\left(n\right)}{\mathrm{E}\left(n\right)}}$$

(10)

The ratio $r_p[-]$ represents the dimensionless index of dispersion. If $r_p$ is close to unity (within the critical limits for the selected significance level $\alpha =0.05$), the assumption of the Poisson distribution is accepted. $\mathrm{Var}(·)$ and $\mathrm{E}(·)$ denote the variance and expectation operators, respectively.

The Kolmogorov–Smirnov (K-S) test checks whether the variables A, T and b follow an exponential distribution. The test compares the cumulative distribution function (CDF) obtained from the KDE estimation with the theoretical exponential CDF. The null hypothesis shall be accepted if the maximum deviation is less than the critical value for a given significance level.

Of all the combinations that pass both tests, the one with the lowest relative error $R_r$ is chosen:

$$R_r={\displaystyle \frac{max\left|{\widehat{F}}_n\left(A\right)-F\left(A\right)\right|}{F\left(A^{*}\right)}}\times 100\%$$

(11)

where ${\widehat{F}}_n\left(A\right)[-]$ is the dimensionless empirical CDF obtained from the KDE estimate; $F\left(A\right)=1-exp(-\zeta A)[-]$ is the dimensionless theoretical exponential CDF; $F\left(A^{*}\right)[-]$ is the value of the theoretical CDF at the point of maximum deviation $A^{*}$ [mm]; and $R_r$ [%] is the relative error expressed as a percentage. Both CDFs are dimensionless by definition, as they represent cumulative probabilities bounded between 0 and 1.

In this study, MIT combinations were tested over 1–12 h and with amount thresholds of 0–5 mm. The optimal combination was identified as the one that satisfied all statistical tests and had the minimum $R_r$. In the KDE method, all threshold combinations ranging from 0 to 5 mm were tested, but thresholds of 1, 3 and 5 mm were selected for comparison with other methods.

2.4. Schutz Index for Uniform Precipitation Identification (Type 5)

After separating the continuous record into individual precipitation events, each event is classified into one of five types based on the position of its maximum intensity. In this paper, the modified Huff method proposed by Nguyen and Chen [6] was applied, introducing a fifth precipitation type with a uniform time distribution.

To distinguish uniform precipitation, Type 5 (T5), from other types, the Schutz index (S) was used, which was originally proposed as a measure of income distribution equality in the economy [27]. The Schutz index quantifies the total deviation of values in each time step ($y_i$) from the mean ($y_{\mathrm{mean}}$):

$$S={\displaystyle \frac{1}{2}}·{\displaystyle \frac{{\displaystyle \sum _{i=1}^n|y_i-y_{\mathrm{mean}}|}}{{\displaystyle \sum _{i=1}^n{y}_i}}}$$

(12)

where $y_i$ [mm] is the measured precipitation amount in time step i; $y_{\mathrm{mean}}$ [mm] is the mean precipitation per time step; and $n[-]$ is the total number of time steps within the precipitation event. The Schutz index $S[-]$ is dimensionless, as the units of $y_i$ and $y_{\mathrm{mean}}$ cancel in both the numerator and denominator.

The Schutz index ranges from 0 to 1. When $S=0$, the precipitation in each time step is equal to the mean, indicating a perfectly uniform distribution. When S approaches 1, the precipitation distribution deviates significantly from uniform.

The procedure for classifying a precipitation event is carried out according to the following algorithm (Figure 2):

• Calculation of the Schutz index (S): For each precipitation event, the value of S is calculated according to Equation (12).
• Uniformity check: If $S<0.30$, the precipitation event is classified as Type 5 (uniform distribution).
• Type determination by quartile: If $S\ge 0.30$, the type is determined by the quartile in which the maximum amount of precipitation occurs (Types 1–4).

A value of $S<0.30$ identifies events with a near-uniform temporal distribution, classified as Type 5, consistent with the threshold recommended in the literature [6] (see Figure 2).

2.5. Code Implementation of the Methodological Framework

The entire methodological framework described in the previous chapters is implemented in MATLAB [24], generating user-defined functions for each analysis step. This approach makes the methodological framework very robust, fast and flexible, enabling easy adjustment of analysis parameters and application across different precipitation datasets. Precipitation input data from the Čepin meteorological station are structured by year and month as MATLAB structures, where each structure element contains a time series of precipitation for a particular month.

Although the raw input data were organised on a monthly basis, precipitation events crossing the calendar boundary between two consecutive months were explicitly handled by a dedicated post-processing routine. After-event separation was performed independently for each month, the routine systematically checked all month-to-month transitions: if the dry period between the last event of one month and the first event of the following month was shorter than the selected MIT value, the two segments were automatically merged into a single continuous event prior to further analysis. This procedure ensures that no genuine precipitation event is artificially truncated at a monthly boundary and that the monthly data organisation has no adverse effect on the identified event statistics.

The modular approach to programming allows for easy expansion of the methodological framework with additional methods for determining MIT, alternative statistical tests or different criteria for classifying precipitation events, thereby ensuring the applicability of the developed program code for future research and analysis at other locations.

2.6. Methods of Sensitivity Analysis and Variant Comparison

After the formation of individual precipitation events and the construction of dimensionless Huff curves, the sensitivity of precipitation’s physical characteristics and hyetograph shape to selected primary processing methods was analysed. The analysis was carried out at the level of individual precipitation events and median Huff curves for each precipitation type.

For each isolated precipitation event, the basic physical characteristics relevant to urban drainage design were determined: total precipitation amount, event duration and maximum 10, 15 and 20 min intensities. Subsequently, for each type of Huff curve, the number of precipitation events n, the average precipitation amount $A_{\mathrm{mean}}$, the maximum precipitation amount $A_{max}$ and the maximum 10, 15 and 20 min intensities were determined. The shape of the median Huff curve was quantified by the average slope $S_k$ in the quartile of the maximum precipitation concentration.

The median curve is interpolated to $N=1000$ equidistant points, whereby each of the four quartiles is described by $n_k=250$ segments. The value $N=1000$ was selected on the basis of a preliminary resolution sensitivity test comparing $N=100$, 1000 and 10,000: at $N=100$, the 25-point per-quartile resolution introduced smoothing artefacts in the slope estimates for Types 1 and 4, which exhibit the steepest and most localised intensity peaks; at N = 10,000, no improvement in curve shape, slope or fractile accuracy was observed relative to $N=1000$, while computational cost increased by approximately two orders of magnitude. $N=1000$ therefore represents the optimal balance between numerical fidelity and computational efficiency across the full set of 140 analysed variants. The average slope for type k ($k=1,2,3,4$) is calculated as:

$$S_k={\displaystyle \frac{1}{n_k}}\sum _{i=(k-1)n_k}^{k{n}_k-1}{\displaystyle \frac{P_{i+1}-P_i}{\Delta t}}$$

(13)

where $P_i[-]$ is the dimensionless cumulative normalised precipitation at interpolation point i (expressed as a fraction of total event amount, i.e., in the range $[0,1]$); $\Delta t=0.1$% $[-]$ is the dimensionless normalised interpolation time step (expressed as a fraction of total event duration). The slope $S_k[-]$ is therefore dimensionless, representing the ratio of normalised precipitation increment to normalised time increment (conventionally written as %/% to emphasise the interpretation as a rate of accumulation per unit normalised time).

Higher values of the dimensionless slope $S_k$ indicate a higher concentration of precipitation within the corresponding quartile, a characteristic of design-critical hyetographs with pronounced peak intensities.

To assess the sensitivity of precipitation types to the choice of minimum inter-event time, the coefficient of variation ($c_v$) was applied specifically to the set of fixed MIT values (MIT = 1–12 h, with a step of $\Delta t=1$ h). For each precipitation type and precipitation threshold, $c_v$ was computed across all considered MIT values for key event characteristics, including event count, mean precipitation amount, mean duration and maximum short-term intensity. In this way, the reported $c_v$ values inherently reflect the variability of these characteristics resulting from the use of different MITs.

To unify the assessment of the design criticality of the hyetograph, a composite design index, $I_{\mathrm{crit}}$, was defined that simultaneously accounts for precipitation concentration over time and short-term peak intensities. The design index is defined as the product of the normalised slope of the Huff curve $S_k$ and the arithmetic mean of the normalised maximum 10-, 15- and 20 min intensities, where each quantity is normalised by its respective maximum value across all analysed MIT variants:

$$I_{\mathrm{crit}}={\displaystyle \frac{S_k}{max\left(S_k\right)}}·{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{i_{10,max}}{max\left(i_{10,max}\right)}}+{\displaystyle \frac{i_{15,max}}{max\left(i_{15,max}\right)}}+{\displaystyle \frac{i_{20,max}}{max\left(i_{20,max}\right)}}\right)$$

(14)

All terms in Equation (14) are dimensionless by construction: the slope ratio $S_k/max\left(S_k\right)[-]$ and each intensity ratio $i_{x,max}/max\left(i_{x,max}\right)[-]$ are normalised by their respective maxima across all analysed MIT variants, so that $I_{\mathrm{crit}}\in [0,1][-]$.

Short-duration precipitation intensities (10, 15 and 20 min) were selected because urban catchments typically have short times of concentration and therefore respond primarily to high-intensity precipitation over short durations, which are critical for urban drainage design [28,29]. Higher values of the composite design index indicate precipitation events with more concentrated precipitation in time and higher short-duration intensities, which are particularly critical for urban drainage systems. Such events are more likely to generate rapid runoff and peak flows, making them highly relevant for the analysis of urban flooding.

The analysis was carried out separately for two classes of precipitation duration: all isolated events, regardless of duration and a subset of short-term events with duration $T\le 2$ h. The final comparison between these two duration classes was based on the maximum values of the design index, thereby identifying the dominant precipitation type in the most critical urban drainage design scenario.

3. Results

The results are organised as follows: firstly, the sensitivity of precipitation event characteristics and Huff curve shape to the fixed MIT method is presented (Section 3.1), including the effects of both the precipitation threshold and the MIT value; then, the AC and KDE methods are compared in terms of event statistics, maximum amounts, intensities and curve slopes (Section 3.2); the composite design index is evaluated for all precipitation durations (Section 3.3); short-duration events ($T\le 2$ h) are analysed separately (Section 3.4); and a final comparison of design criticality between the two duration classes is presented (Section 3.5).

3.1. Sensitivity Analysis for the Fixed Minimum Inter-Event Time (MIT) Method

3.1.1. Impact of the Precipitation Threshold Amount, P

The change in precipitation threshold does not affect the maximum precipitation amount or the maximum 10-, 15- and 20 min intensities in the analysed dataset. The influence of the threshold on the slope of median Huff curves is present but not significant: a very slight increase in slope is observed for Types 2, 3 and 4, and a slight decrease for Type 1 (Figure 3 and

Table A1. Effect of precipitation amount threshold on the average slope ($S_k$) of Huff curves (averaged over fixed MIT values).

Threshold	Type 1	Type 2	Type 3	Type 4	Type 5
$P=1$ mm	2.50	1.90	1.97	2.13	$\approx 1.00$
$P=3$ mm	2.48	1.91	2.04	2.17	$\approx 1.00$
$P=5$ mm	2.44	2.00	2.03	2.20	$\approx 1.00$

Notes: The values represent the average slope $S_k$ of the median Huff curve within the quartile of maximum precipitation concentration, averaged over the 12 fixed MIT values (1–12 h) for each precipitation amount threshold. Type 5 consistently shows a near-uniform distribution ($\approx 1.00$) across all thresholds. Abbreviations: $S_k$ —average slope of the median Huff curve in quartile k (dimensionless ratio of cumulative normalised precipitation increment to normalised time increment, expressed as %/%); P—minimum precipitation amount threshold (mm); MIT—minimum inter-event time (h); Types 1–5—Huff curve types classified by the quartile position of maximum precipitation intensity (Types 1–4) or by temporal uniformity of precipitation distribution (Type 5).

). A noticeable influence of the threshold change on the number of precipitation events, average amount and average precipitation duration was observed, as quantified below.

Number of Precipitation Events, n

Regardless of the threshold, Type 1 consistently generates the most precipitation events (on average $n=202$) and this relative ranking remains stable across all threshold variants (

Table A2. Average number of precipitation events (n) by type for MIT values of 1–12 h and percentage reduction in n with increasing precipitation amount threshold.

Threshold	Type 1	Type 2	Type 3	Type 4	Type 5	$\mathbf{\Sigma }$
$P=1$ mm	202	122	108	126	63	621
$P=3$ mm	128	89	84	79	35	415
$P=5$ mm	102	70	71	61	27	331
$P_1\to P_3$ (%)	−36	−27	−22	−37	−44	−33
$P_3\to P_5$ (%)	−20	−21	−15	−22	−23	−20
$P_1\to P_5$ (%)	−49	−42	−34	−51	−57	−46

Notes: Values represent averages across MIT values of 1–12 h. Percentage reductions are computed relative to the event count at the lower threshold (e.g., $P_1\to P_3$ gives the reduction from $P=1$ mm to $P=3$ mm). Abbreviations: n—average number of identified precipitation events per type, averaged across MIT values of 1–12 h; $\mathrm{\Sigma }$—total number of precipitation events summed across all five Huff curve types; P—minimum precipitation amount threshold (mm); $P_1$, $P_3$, $P_5$ denote thresholds of 1, 3 and 5 mm, respectively; MIT—minimum inter-event time (h); Types 1–5—Huff curve types classified by the quartile position of maximum precipitation intensity (Types 1–4) or by temporal uniformity of precipitation distribution (Type 5).

).

Increasing the threshold reduces the number of precipitation events: the transition from $P=1$ mm to $P=3$ mm has a 1.5–2 times greater impact than the transition from $P=3$ mm to $P=5$ mm. Types 2 and 3 show less sensitivity to threshold changes than Types 1, 4 and 5.

Average Amount $A_{\mathrm{mean}}$ and Average Duration, $T_{\mathrm{mean}}$

Increasing the precipitation threshold from 1 to 3 mm results in a significant increase in the average precipitation event amount across all Huff types, ranging from 21.9% (Type 3) to 59.6% (Type 5), with notable increases also observed in Types 1 and 4. A further increase from 3 to 5 mm leads to a more moderate but consistent increase in $A_{\mathrm{mean}}$ of approximately 12–20% for all types (

Table A3. Effect of precipitation amount threshold on the mean precipitation amount $A_{\mathrm{mean}}$ and mean precipitation duration $T_{\mathrm{mean}}$ by event type (averaged over fixed MIT values of 1–12 h). Values for $T_{\mathrm{mean}}$ are reported as mean ± SD.

	Type 1		Type 2		Type 3		Type 4		Type 5
Threshold	${\mathit{A}}_{\mathrm{mean}}$ [mm]	${\mathit{T}}_{\mathrm{mean}}$ [min]	${\mathit{A}}_{\mathrm{mean}}$ [mm]	${\mathit{T}}_{\mathrm{mean}}$ [min]	${\mathit{A}}_{\mathrm{mean}}$ [mm]	${\mathit{T}}_{\mathrm{mean}}$ [min]	${\mathit{A}}_{\mathrm{mean}}$ [mm]	${\mathit{T}}_{\mathrm{mean}}$ [min]	${\mathit{A}}_{\mathrm{mean}}$ [mm]	${\mathit{T}}_{\mathrm{mean}}$ [min]
$P=1$ mm	10.63	$630\pm 243$	9.17	$650\pm 265$	10.84	$602\pm 203$	8.46	$596\pm 220$	8.09	$191\pm 9$
$P=3$ mm	15.50	$797\pm 299$	11.65	$753\pm 298$	13.21	$674\pm 224$	12.24	$759\pm 268$	12.92	$298\pm 16$
$P=5$ mm	18.47	$882\pm 299$	13.71	$844\pm 313$	14.80	$722\pm 224$	14.61	$825\pm 271$	15.49	$359\pm 24$

Notes: Values are averaged over fixed MIT values of 1–12 h and a precipitation amount threshold of $P=1$, 3 and 5 mm, respectively. Standard deviation (SD) reflects inter-annual and inter-seasonal variability. Abbreviations: $A_{\mathrm{mean}}$—mean precipitation amount per event, averaged across MIT values of 1–12 h (mm); $T_{\mathrm{mean}}$—mean precipitation event duration, averaged across MIT values of 1–12 h (min), reported as mean ± SD; SD—standard deviation of precipitation event duration, reflecting inter-annual and inter-seasonal variability (min); P—minimum precipitation amount threshold (mm); MIT—minimum inter-event time (h); Types 1–5—Huff curve types classified by the quartile position of maximum precipitation intensity (Types 1–4) or by temporal uniformity of precipitation distribution (Type 5).

).

By increasing the precipitation threshold from 1 to 3 mm, the average duration $T_{\mathrm{mean}}$ increases by 12–27% in Types 1–4, while the uniform Type 5 records a significantly higher increase of up to 56%. A further increase from 3 to 5 mm results in a more moderate increase of 7–12% for Types 1–4, but a still notable increase of 20% for Type 5. Increasing the threshold from 1 to 3 mm also increases the average standard deviation by type (Table A3); this increase is not observed between $P=3$ and 5 mm.

3.1.2. Impact of the Change in MIT Values

Number of Precipitation Events, n

Table 1. Coefficient of variation $c_v$ of the number of precipitation events for each type and threshold.

Threshold	Type 1	Type 2	Type 3	Type 4	Type 5
$P=1$ mm	0.076	0.159	0.179	0.047	0.581
$P=3$ mm	0.048	0.114	0.137	0.092	0.536
$P=5$ mm	0.073	0.071	0.095	0.153	0.503

Notes: The coefficient of variation $c_v$ is computed across the 12 fixed MIT values (1–12 h) for each Huff curve type. The statistical significance of all reported $c_v$ values was assessed using McKay’s test; all values are highly significant ($p\ll 0.001$), confirming that the observed variability across MIT values differs from zero in all groups. The largest p-values—while still far below the significance threshold of $\alpha =0.05$—are associated with Type 5 across all precipitation amount thresholds, consistent with its characteristically high $c_v$ values relative to the other types. Abbreviations: $c_v$—coefficient of variation (dimensionless ratio of standard deviation to mean); $c_v=0$ indicates no variability across MIT values, while $c_v\to 1$ indicates high variability; P—minimum precipitation amount threshold (mm); MIT—minimum inter-event time (h); Types 1–5—Huff curve types classified by the quartile position of maximum precipitation intensity (Types 1–4) or by temporal uniformity of precipitation distribution (Type 5).

presents the values of the coefficient of variation $c_v$ for each Huff curve type. As the threshold increases, the variability in the number of events across all types, except Type 4, decreases. Type 4 shows minimal variability at $P=1$ mm but transitions to more sensitive behaviour at $P=5$ mm. Type 5 (uniform) is the most sensitive of all types to changes in MIT, while its variance changes only slightly with threshold.

At the threshold $P=1$ mm, for smaller MIT values (1–4 h), the decrease in the number of events with increasing MIT is more pronounced, following an exponential trend. At $P=3$ and 5 mm, for Types 1 and 4, the trend reverses: an increase in MIT is associated with an increase in the number of precipitation events. Figure 3 shows the distribution of precipitation events by type and the median Huff curves for MIT $=1$, 6 and 12 h at a threshold of 1 mm. The corresponding graphs for thresholds of 3 and 5 mm are shown in the Appendix (Figure A3 and Figure A4).

Average Amount $A_{\mathrm{mean}}$ and Average Duration, $T_{\mathrm{mean}}$

Table 2. Coefficient of variation $c_v$ of average amount $A_{\mathrm{mean}}$ and precipitation duration $T_{\mathrm{mean}}$.

	Threshold	Type 1	Type 2	Type 3	Type 4	Type 5
$c_v$ ($A_{\mathrm{mean}}$)	$P=1$ mm	0.142	0.131	0.122	0.243	0.047
	$P=3$ mm	0.090	0.107	0.087	0.190	0.034
	$P=5$ mm	0.072	0.077	0.069	0.159	0.030
$c_v$ ($T_{\mathrm{mean}}$)	$P=1$ mm	0.403	0.408	0.352	0.386	0.049
	$P=3$ mm	0.375	0.396	0.332	0.353	0.053
	$P=5$ mm	0.354	0.387	0.324	0.342	0.069

Notes: The coefficient of variation $c_v$ is computed across the 12 fixed MIT values (1–12 h) for each Huff curve type and precipitation amount threshold. The statistical significance of all reported $c_v$ values was assessed using McKay’s test; all values are highly significant ($p\ll 0.001$), confirming that the observed variability across MIT values differs from zero in all groups. Abbreviations: $c_v$—coefficient of variation (dimensionless ratio of standard deviation to mean; $c_v=0$ indicates no variability across MIT values, while $c_v\to 1$ indicates high variability); $A_{\mathrm{mean}}$—mean precipitation amount per event (mm); $T_{\mathrm{mean}}$—mean precipitation event duration (min); P—minimum precipitation amount threshold (mm); MIT—minimum inter-event time (h); Types 1–5—Huff curve types classified by the quartile position of maximum precipitation intensity (Types 1–4) or by temporal uniformity of precipitation distribution (Type 5).

summarises the coefficients of variation for average precipitation amount and duration. Uniform precipitation events (Type 5) exhibit the lowest variability in average amount ($c_v\approx 0.05$), whereas events with a late maximum (Type 4) exhibit the highest ($c_v\approx 0.24$). Types 1–3 show moderate, mutually similar variability ($c_v\approx 0.12$–$0.14$). Increasing the threshold systematically decreases the coefficient of variation $c_v$ for all types, with Type 4 showing the highest variability across thresholds. For Types 1–3, $c_v$ decreases considerably (e.g., Type 1: $0.142\to 0.090\to 0.072$).

In terms of average duration, Type 5 shows very low variability ($c_v\approx 0.05$–$0.07$), whereas Types 1–4 exhibit high duration variability ($c_v\approx 0.32$–$0.41$), with an approximately linear increase with increasing MIT. Types 1 and 4 show an increase of more than 360% (from 209 to 974 min and from 202 to 935 min, respectively). Type 2 shows an increase of 292% (from 270 to 1056 min) and Type 3 shows an increase of 223% (from 295 to 953 min). For short MIT durations, Types 1 and 4 have the shortest durations, but this relationship inverts as MIT increases. Increasing the precipitation threshold from 1 to 5 mm results in a slight decrease in $c_v$ for Types 1–4, whereas in Type 5, the threshold increase leads to a slight increase in variability.

Maximum Precipitation Amounts, $A_{max}$

Type 1 exhibits the highest maximum precipitation amount (87.1 mm for MIT $>3$ h), followed by Type 2 (80.9 mm for MIT $>6$ h), Type 4 (61.0 mm for MIT $=6$ h), Type 3 (56.2 mm for MIT $>9$ h) and Type 5 (39.6 mm for all MIT values).

The MIT-dependent behaviour of maximum short-term intensities and Huff-curve slope is summarised in

Table 3. Coefficient of variation $c_v$ and breakpoint (change in value with MIT).

Intensity	Type 1	Type 2	Type 3	Type 4	Type 5
$i_{10,max}$	0 (Constant)	0 (Constant)	0 (Constant)	0.379 (MIT $=6$ h)	0.137 (MIT $=6$ h)
$i_{15,max}$	0 (Constant)	0 (Constant)	0.03 (MIT $=2$ h)	0.346 (MIT $=6$ h)	0.323 (MIT $=11$ h)
$i_{20,max}$	0 (Constant)	0.055 (MIT $=6$ h)	0.128 (MIT $=3$ & 9 h)	0.338 (MIT $=4$ & 7 h)	0.332 (MIT $=11$ h)

Notes: A value of $c_v=0$ indicates that the maximum intensity remains constant across all 12 fixed MIT values (1–12 h). Where two MIT values are given in brackets (e.g., MIT = 3 & 9 h), the maximum intensity exhibits two distinct breakpoints across the 1–12 h range, reflecting non-monotonic behaviour. The statistical significance of all reported $c_v$ values was assessed using McKay’s test; all values are highly significant ($p\ll 0.001$), confirming that the observed variability across MIT values differs from zero in all groups. Abbreviations: $c_v$—coefficient of variation (dimensionless ratio of standard deviation to mean; $c_v=0$ indicates no variability across MIT values, while $c_v\to 1$ indicates high variability); $i_{10,max}$, $i_{15,max}$, $i_{20,max}$—maximum 10-, 15- and 20 min precipitation intensities (mm ${\min }^{-1}$); MIT—minimum inter-event time (h); Types 1–5—Huff curve types classified by the quartile position of maximum precipitation intensity (Types 1–4) or by temporal uniformity of precipitation distribution (Type 5).

and

Table 4. Change in the slope of the Huff curve with respect to MIT and the coefficient of variation $c_v$ for $P=1$ mm.

	Type 1	Type 2	Type 3	Type 4
$S_k$ at MIT $=1$ h [%/%]	2.11	1.83	1.85	1.57
$S_k$ at MIT $=12$ h [%/%]	2.61	1.93	2.15	2.27
Increase [%]	24	5	16	45
$c_v$	0.06	0.023	0.053	0.113

Notes: Type 5 is excluded from the slope analysis due to its uniform temporal distribution. The coefficient of variation $c_v$ is computed across all 12 fixed MIT values (1–12 h). The statistical significance of all reported $c_v$ values was assessed using McKay’s test; all values are highly significant ($p\ll 0.001$), confirming that the observed variability across MIT values differs from zero in all groups. Abbreviations: $S_k$—average slope of the median Huff curve in quartile k (dimensionless ratio of cumulative normalised precipitation increment to normalised time increment, expressed as %/%); $c_v$—coefficient of variation (dimensionless ratio of standard deviation to mean; $c_v=0$ indicates no variability across MIT values, while $c_v\to 1$ indicates high variability); P—minimum precipitation amount threshold (mm); MIT—minimum inter-event time (h); Types 1–4—Huff curve types classified by the quartile position of maximum precipitation intensity. Results for $P=3$ mm and $P=5$ mm exhibit the same qualitative patterns and are therefore not repeated here; see Appendix Table A1 for threshold-averaged slope values.

. The results for maximum precipitation amount are presented in

Table 5. Maximum amounts $A_{max}$ [mm] for fixed values of MIT, AC and KDE methods.

Method	Type 1	Type 2	Type 3	Type 4	Type 5
Fixed MIT	87.1 (MIT $>3$ h)	80.9 (MIT $>6$ h)	56.2 (MIT $>9$ h)	61.0 (MIT $=6$ h)	39.6 (constant)
AC	96.3	80.9	33.3	55.2	39.6
KDE	60.1	40.7	35.4	48.1	39.6

Notes: For the fixed MIT method, the stabilised maximum value is reported together with the MIT value at which stabilisation occurs (in brackets). Values are insensitive to changes in the precipitation threshold. Abbreviations: $A_{max}$—maximum precipitation amount recorded across all events of a given type (mm); MIT—minimum inter-event time (h); AC—autocorrelation-based method for MIT determination; KDE—kernel density estimation-based method for MIT determination; Types 1–5—Huff curve types classified by the quartile position of maximum precipitation intensity (Types 1–4) or by temporal uniformity of precipitation distribution (Type 5).

, together with values obtained using the AC and KDE methods. An increase in maximum precipitation amount with increasing MIT is observed across all precipitation types. Beyond a certain MIT threshold, $A_{max}$ stabilises: for Type 1, no further increase is observed for MIT $>3$ h; Type 2 shows a gradual increase up to MIT $=6$ h, followed by a pronounced increase and stable values thereafter; Types 3 and 4 exhibit more complex behaviour, characterised by an increase up to MIT $=6$ h, a decrease between MIT $=6$ and 8 h and a subsequent increase followed by stable values for MIT $\ge 9$ h.

Maximum 10-, 15- and 20 min Intensities, $i_{10,max}$, $i_{15,max}$ and $i_{20,max}$

Changes in maximum short-term intensities occur at different MIT values depending on the type, with the breakpoint often around MIT $=6$ h. Type 1 has the highest intensity values across all analysed durations. Types 1 and 2 maintain constant intensities across the entire MIT range. Type 3 is stable for $i_{10}$ and $i_{15}$, while $i_{20}$ shows slight fluctuations. Types 4 and 5 exhibit considerable oscillation with MIT change; this is particularly pronounced in Type 4 for $i_{10}$ (MIT $=1$ h: $i_{max}=1.15$ mm/min; MIT $\ge 6$ h: $i_{max}=3.17$ mm/min). In Table 3, the bracketed MIT value indicates the MIT at which the change in intensity occurred.

Slope of Huff Curves, $S_k$

Type 5 is excluded from the slope analysis owing to the uniform distribution of precipitation over time. Types 1 and 4 show the highest sensitivity to the MIT value, while Type 2 is almost insensitive ($c_v=0.023$, Table 4). Type 1 consistently has the highest slopes, followed by Type 4, while Types 2 and 3 achieve lower values. For short intervals (MIT $=1$–2 h), Types 2 and 3 have higher slopes than Type 4, but with further increases in MIT, the slopes of Types 1 and 4 increase sharply. A breakpoint is observed around MIT $=3$ h for Type 1 (particularly at threshold $P=1$ mm) and around MIT $=5$ h for Type 4, after which slope values stabilise (Figure 4).

3.2. Comparison of Autocorrelation (AC) and Kernel-Density Estimation (KDE) Methods

The Huff curves for all 5 rain types and both methods (AC and KDE) are presented in the Appendix (Figure A5 and Figure A8). The variability of the Huff curves for both methods is also shown in Figure A6 and Figure A9, while the statistical distribution grouped by type, for both methods, is shown in Figure A7 and Figure A10. The influence of MIT and threshold on the statistical parameters and the shape of the median curves is described below.

3.2.1. Number of Precipitation Events, n

As shown in Figure 5 and Figure 6, the autocorrelation (AC) method tends to identify longer inter-event times due to its sensitivity to temporal dependence in the precipitation series. On the other hand, the KDE-based method is more sensitive to shorter gaps and therefore detects shorter, closely spaced events. Therefore, KDE consistently generates a larger number of events, particularly for Type 5 (Figure 5). The number of events obtained by the AC method is equivalent to fixed MIT $=8$ h ($P=1$ mm) and 9 h ($P=3$ and 5 mm). KDE results correspond to a fixed MIT value of 1–2 h. In Type 4, a trend reversal is observed: whereas KDE generally yields more events than AC, this relationship is reversed at thresholds of $P=3$ and 5 mm, where AC produces more events than KDE. The KDE method shows less seasonal variability in MIT than the AC method, which produces substantially higher values in winter (Figure 6).

Both methods identify Type 1 as the most common and Type 5 as the rarest type across all thresholds. However, the rankings of intermediate types differ: AC assigns Types 2 and 4 as the second-most common across all thresholds, whereas KDE prioritises Types 2 and 3.

3.2.2. Average Amounts $A_{\mathrm{mean}}$ and Average Duration, $T_{\mathrm{mean}}$

AC consistently yields higher average precipitation amounts ($A_{\mathrm{mean}}$) for all types except Type 5, with the most significant difference observed for Type 4 (Figure 7 and

Table A4. Comparison of mean precipitation amount $A_{\mathrm{mean}}$ [mm] and mean precipitation duration $T_{\mathrm{mean}}$ [min] for the AC and KDE methods, by event type and precipitation amount threshold.

		Type 1		Type 2		Type 3		Type 4		Type 5
Threshold	Method	${\mathit{A}}_{\mathrm{mean}}$ [mm]	${\mathit{T}}_{\mathrm{mean}}$ [min]	${\mathit{A}}_{\mathrm{mean}}$ [mm]	${\mathit{T}}_{\mathrm{mean}}$ [min]	${\mathit{A}}_{\mathrm{mean}}$ [mm]	${\mathit{T}}_{\mathrm{mean}}$ [min]	${\mathit{A}}_{\mathrm{mean}}$ [mm]	${\mathit{T}}_{\mathrm{mean}}$ [min]	${\mathit{A}}_{\mathrm{mean}}$ [mm]	${\mathit{T}}_{\mathrm{mean}}$ [min]
$P=1$ mm	AC	10.7	762	10.8	839	9.6	696	10.7	883	7.6	171
	KDE	7.7	304	8.1	358	9.1	352	5.7	337	8.2	211
$P=3$ mm	AC	16.6	1017	13.2	996	12.0	748	14.9	1126	11.4	249
	KDE	12.2	395	10.4	425	11.4	403	9.9	468	13.2	337
$P=5$ mm	AC	19.7	1117	15.6	1156	13.8	805	17.8	1278	13.7	294
	KDE	15.8	449	12.6	498	13.6	441	12.1	522	15.7	400

Notes: Values are computed from precipitation events extracted using the respective MIT determination method (AC or KDE) with a fixed precipitation amount threshold of $P=1$, 3 and 5 mm. Abbreviations: $A_{\mathrm{mean}}$—mean precipitation amount per event for the given method and threshold (mm); $T_{\mathrm{mean}}$—mean precipitation event duration for the given method and threshold (min); P—minimum precipitation amount threshold (mm); AC—autocorrelation-based method for MIT determination; KDE—kernel density estimation-based method for MIT determination; MIT—minimum inter-event time (h); Types 1–5—Huff curve types classified by the quartile position of maximum precipitation intensity (Types 1–4) or by temporal uniformity of precipitation distribution (Type 5).

). Type 3 shows minimal differences between the methods (1–5%). In terms of average duration ($T_{\mathrm{mean}}$), KDE yields significantly shorter values across all types, approximately at the level of fixed MIT $=1$–2 h, whereas AC yields considerably longer durations, comparable to fixed MIT $\approx 8$–9 h. The exception is Type 5, where KDE yields a slightly longer duration than AC.

3.2.3. Maximum Precipitation Amounts, $A_{max}$

The AC method yields significantly higher maximums for Types 1 and 2, with differences of up to approximately 40 mm compared to the KDE method (Table 5). Type 5 remains completely stable regardless of the method chosen. AC results are closely aligned with those obtained for higher fixed MIT values, whereas KDE gives values comparable to fixed MIT $\approx 3$ h. Both methods consistently identify Type 1 as the type with the highest total precipitation amounts and are insensitive to changes in the precipitation threshold.

3.2.4. Maximum 10-, 15- and 20 min Intensities, $i_{10,max}$, $i_{15,max}$ and $i_{20,max}$

The AC and KDE methods yield very similar maximum short-term intensities, with a few observed differences. AC gives higher values for Type 4 across all analysed intensities, while KDE gives higher values for Type 3 for $i_{15,max}$ and $i_{20,max}$. Comparison with the fixed MIT method shows that the intensity values are almost identical (

Table 6. Intensity values $i_{10,max}$ [mm/min] for fixed values of MIT, AC and KDE.

Method	Type 1	Type 2	Type 3	Type 4	Type 5
Fixed MIT	3.28 (3.28)	1.34 (1.30)	2.04 (2.04)	3.17 (2.45)	3.17 (2.74)
AC	3.28	1.30	2.04	3.17	2.44
KDE	3.28	1.30	2.04	1.95	3.17

Notes: The mean values of the maximum intensities obtained for the range of fixed MIT values (1–12 h) are shown in brackets. Values are insensitive to changes in the precipitation threshold. Abbreviations: $i_{10,max}$—maximum 10 min precipitation intensity recorded across all events of a given type (mm ${\min }^{-1}$); MIT—minimum inter-event time (h); AC—autocorrelation-based method for MIT determination; KDE—kernel density estimation-based method for MIT determination; Types 1–5—Huff curve types classified by the quartile position of maximum precipitation intensity (Types 1–4) or by temporal uniformity of precipitation distribution (Type 5).

). A more pronounced difference occurs only in Type 4, where KDE yields lower intensities, comparable to those for fixed MIT $\approx 4$ h. Maximum intensity remained unchanged across different threshold values.

3.2.5. Slope of Huff Curves, $S_k$

The AC method consistently yields higher slope values than the KDE method. For a threshold of $P=1$ mm, the differences are 10–14% for Types 1–3, while for Type 4, the most significant difference of 21.6% is observed. By increasing the precipitation threshold, the differences in slope between the methods generally decrease, except for Type 4, where the difference increases significantly, reaching 41.4% at $P=5$ mm. Comparison with results for fixed MIT values shows that AC corresponds to MIT of the order of 6–9 h, while KDE corresponds to MIT of the order of 1–3 h. The Huff curves obtained with both methods for all three threshold values are shown in Figure 8.

3.3. Design Index for All Precipitation Durations

The composite design index $I_{\mathrm{crit}}$ (Equation (14)) was evaluated for parameters that proved more stable across the MIT range and showed minor sensitivity to threshold changes, namely the slope $S_k$ and the maximum short-term precipitation intensity. Figure 9 presents $I_{\mathrm{crit}}$ values for all precipitation types as a function of fixed MIT, together with comparisons to the AC and KDE methods.

Type 1 consistently achieves the highest $I_{\mathrm{crit}}$ values across the entire MIT range, with the index becoming insensitive to changes beyond MIT $=3$–4 h. Type 4 shows a pronounced increase in $I_{\mathrm{crit}}$ with increasing MIT, particularly up to MIT $=6$ h. The AC method intersects the Type 4 curve in the large-MIT range, whereas KDE intersects it in the low-MIT range. Types 2 and 3 yield lower and more stable index values, with minimal differences between methods. Increasing the threshold does not significantly change the index’s absolute value; the most pronounced increase is observed for Type 4, but the relative ordering of types and the observed trends remain unchanged.

3.4. Analysis for Durations $T\le 2$ h

3.4.1. Number of Precipitation Events, n

An exponential decline in the number of events with increasing MIT was observed across all types, with a notable change in trend around MIT $=4$ h. The most pronounced reduction in the total number of events compared to all durations was observed in Type 1. At all thresholds and MIT values $<3$ h, the smallest reduction was observed in Type 4. Types 3 and 4 have the smallest coefficient of variation across the range of fixed MIT values.

KDE produces approximately twice as many short-term precipitation events as AC across all thresholds. The AC results are equivalent to higher MIT values (typically ∼5–9 h), whereas KDE corresponds to lower MIT values (∼1–3 h).

3.4.2. Average Precipitation Amounts, $A_{\mathrm{mean}}$

The average precipitation amount for events of duration $T\le 2$ h is lower than that for the set of all events. This result is consistent with the physical composition of the two duration classes: the $T\le 2$ h subset is dominated by short-duration convective events (primarily Type 1), which exhibit high peak intensities but limited total accumulation, whereas the all-duration class retains longer cyclonic and mixed events (predominantly Types 2–4) that accumulate substantially greater amounts over extended durations.

The lower average depth for $T\le 2$ h events follows directly from the compositional difference between the two duration classes. The all-duration class retains Types 2, 3, and 4 events—longer-duration cyclonic or mixed systems that accumulate substantial total depths—which systematically inflate the class mean. The $T\le 2$ h class is instead dominated by Type 1 convective events, which, despite high peak intensities, yield limited total accumulation due to their brief duration. This is consistent with the physical distinction between convective and cyclonic precipitation noted by Bonacci [1] and with the findings of Bonta [3]. The pattern is further confirmed by its consistency across all MIT values and all three precipitation thresholds analysed.

As MIT increases, $A_{\mathrm{mean}}$ remains relatively stable for the short-duration class. With increasing MIT, the differences between $A_{\mathrm{mean}}$ for $T\le 2$ h and for all durations become more pronounced. The greatest sensitivity to MIT was observed in Types 2 and 5 across all thresholds. The AC and KDE methods yield very similar $A_{\mathrm{mean}}$ values for this duration class.

With increasing threshold, average amounts increase in both groups ($T\le 2$ h and all durations), while the differences between them decrease slightly. This effect is most pronounced in Type 4, where the differences between the two groups for $P=5$ mm are of the order of 2 mm. For $T\le 2$ h, the KDE method and MIT $<6$ h also yield higher $A_{\mathrm{mean}}$ values than the all-duration set.

3.4.3. Maximum Precipitation Amounts, $A_{max}$

For Types 3, 4 and 5, maximum precipitation amounts are insensitive to changes in the fixed MIT value. For Types 1 and 2, the values are constant up to MIT $=5$ h; only at higher MIT values does $A_{max}$ decrease. In contrast, the maximum amounts for the set of all events increases with increasing MIT. For short-term events, the differences between the AC and KDE methods are negligible, although KDE yields higher Type 1 values (consistent with its shorter MIT). In contrast, AC yields values identical to those for MIT $>5$ h.

3.4.4. Maximum 10-, 15- and 20 min Intensities, $i_{10,max}$, $i_{15,max}$ and $i_{20,max}$

For events of $T\le 2$ h, the values of $i_{10,max}$, $i_{15,max}$ and $i_{20,max}$ at MIT $=1$–5 h are identical to those obtained for the set of all events. However, at MIT $=6$ h, a sharp decrease in maximum intensities is observed in the short-duration class, whereas no such decrease occurs in the all-duration set. KDE preserves the intensities characteristic of small MIT values, whereas AC yields values closer to those of larger MIT values. No change in maximum intensities was observed with varying threshold values.

3.4.5. Slope of Huff Curves, $S_k$

For $T\le 2$ h, Type 1 retains the most significant and most stable slope values ($c_v\approx 0.02$–$0.04$), while Type 4 exhibits the highest sensitivity ($c_v\approx 0.15$–$0.22$) and shows a notable increase in slope with increasing MIT, especially at higher thresholds. For Type 4, AC yields slopes approximately 50% higher than KDE at thresholds of 3 and 5 mm. Type 1 shows minimal differences between AC and KDE at all thresholds, whereas Type 3 systematically yields higher slopes under KDE than under AC.

Comparison with slope values across all durations reveals a systematic increase in slope for $T\le 2$ h across all types and thresholds at lower MIT values. KDE specifically emphasises this increase. The sole exception is Type 4, which shows unstable behaviour in this respect.

3.4.6. Composite Design Index for $T\le 2$ h

The ranking based on $I_{\mathrm{crit}}$ for the short-duration class is presented in Figure 10. Type 1 achieves the highest $I_{\mathrm{crit}}$ at all thresholds, with values approximately twice those of the other types up to MIT $=5$ h. By increasing the threshold from 1 to 3 mm, $I_{\mathrm{crit}}$ for Type 1 increases at MIT $=1\mathrm{and}2$ h; this increase is not observed at the 5 mm threshold. A sudden decrease in $I_{\mathrm{crit}}$ for Type 1 occurs at MIT $=6$ h across all thresholds, with a similar but less pronounced drop evident in Type 2. Types 3, 4 and 5 generate low, stable values with minor differences across methods.

AC and KDE rank the types identically, with KDE assigning higher values to Types 1 and 3 and AC assigning higher values to Types 2 and 4. Increasing the threshold from 1 to 3 mm increases the index values for all types under both KDE and fixed MIT, except for Type 4, where the increase is observed only at higher MIT values. A further increase to 5 mm reverses this trend, with $I_{\mathrm{crit}}$ decreasing in most types.

3.5. Comparison of Design Criticality Between Duration Classes

The analysis of all durations ranks the types according to the design index as T1 > T4 > T3 > T2 > T5. The ranking for events of duration $T\le 2$ h is T1 > T3 > T2 > T4 > T5. Figure 11 shows the change in the design index for Types 1 and 4 for a threshold of $P=1$ mm (for other thresholds, see Figure A11 and Figure A12). For Type 1, the maximum index values for events $T\le 2$ h slightly exceed those obtained from the analysis of all durations, up to MIT $=5$ h. The KDE method for the $T\le 2$ h class yields an index value very similar to the maximum obtained for MIT $=4$ h. The AC method for events with $T\le 2$ h yields low index values, comparable to those obtained with fixed MIT values greater than 6 h. For Type 4, index curves are identical up to MIT $=3$ h. After that, critical index values for duration $T\le 2$ h are lower. Maximum values are gained for MIT > 6 h. The KDE method yields similar results across duration classes. The AC method yields higher index values for all duration analyses, in accordance with MIT, for durations greater than 6 h.

4. Discussion

4.1. Influence of Precipitation Thresholds on Precipitation Event Characteristics

The results demonstrate that precipitation thresholds primarily act as a filter for weaker events without substantially altering the peak characteristics of intense precipitation (Section 3.1.1). The absence of a threshold effect on maximum amounts and short-term intensities confirms that thresholds affect the lower tail of the event distribution rather than the extreme values that govern urban drainage design (Table 3 and Table 5). This finding is consistent with the recommendations of Bonacci [1], who proposed minimum precipitation thresholds to exclude hydraulically insignificant events from runoff hyetograph construction.

The pronounced increase in average event amount and duration when transitioning from $P=1$ mm to $P=3$ mm, compared with the more moderate changes between $P=3$ mm and $P=5$ mm (Table A3), suggests that most weak and very short-duration events are concentrated below the 3 mm threshold. The simultaneous decrease in the coefficient of variation (Table 1 and Table 2) indicates that the retained events form a more homogeneous population, thereby improving the statistical representativeness of the derived Huff curves for engineering design. The particular sensitivity of Type 5 (uniform) to threshold changes (Table A2) stems from the fact that uniform events are typically low-intensity and are therefore disproportionately affected by the exclusion criterion.

Considering both the preservation of event population size and the removal of hydraulically irrelevant events, the threshold $P=3$ mm emerges as an optimal compromise for constructing design-relevant Huff hyetographs in urban drainage applications. This is evidenced by the curve shapes and event counts shown in Figure 3 and Figure A3 and Figure A4 and quantified in Table A1 and is supported by similar recommendations for design-critical precipitation analysis in urban settings reported by Cao et al. [21].

4.2. Sensitivity of Huff Curve Shape and Precipitation Event Statistics to the MIT Value

The strong dependence of event duration, total amount and Huff curve slope on the MIT value (Table 2 and Table 4) underscores the central role of the event separation criterion in shaping the resulting dimensionless hyetographs. The observed exponential decline in the number of events at lower MIT values, transitioning to a more gradual decrease at higher MIT (Table A2; Section 3.1.2), reflects the progressive merging of adjacent events and is consistent with the findings of Dunkerley [11] and Tu et al. [13].

The contrasting sensitivities of different Huff curve types to MIT carry critical physical implications. The high slope sensitivity of Type 1 (early maximum) to MIT at lower values (a breakpoint around MIT $=3$ h; Table 4 and Figure 4) can be explained by the characteristic structure of convective events: short, intense precipitation cores are often preceded or followed by brief, low-intensity phases. At very low MIT values, these phases are treated as separate events, fragmenting what is physically a single convective event. Once MIT exceeds the typical duration of such intermediate pauses, the core event is captured intact and the slope stabilises (Figure 4).

Type 4 (late maximum) shows even more pronounced slope sensitivity, with the breakpoint occurring at MIT $\approx 5$ h and a slope increase of 45% over the full MIT range (Table 4). This behaviour reflects the typical structure of events with late peak intensity, which often begin with an extended period of low-intensity precipitation. At low MIT values, this introductory phase is separated from the main precipitation core and the resulting truncated event may be reclassified as another type (frequently Type 1), as illustrated by the distribution shifts visible in Figure 3, and Figure A3 and Figure A4 in the Appendix. Conversely, at higher MIT values, the full event structure is preserved, revealing the late maximum and increasing the slope of the Type 4 Huff curve (Figure 4).

The near-insensitivity of Type 2 ($c_v=0.023$; Table 4) to MIT variations indicates that events with a mid-early maximum possess a compact temporal structure that is robust to changes in the separation criterion. This stability makes Type 2 hyetographs exceptionally reliable for design applications, as their shape is mainly independent of the primary processing decisions.

The reversal of the event-number trend for Types 1 and 4 at higher thresholds ($P=3$ and 5 mm)—whereby an increase in MIT leads to an increase rather than a decrease in the number of events—suggests a redistribution of events between types (Table A2; Section 3.1.2). As MIT increases, some events previously classified as Types 2, 3 or 5 are merged with adjacent events and reclassified as Types 1 or 4, a process whose rate exceeds that of event loss due to merging.

4.3. Comparison and Physical Interpretation of the AC and KDE Methods

The systematic difference between the AC and KDE methods—with AC yielding MIT values comparable to the fixed MIT $\approx 6$–9 h and KDE yielding fixed MIT $\approx 1$–3 h—reflects fundamentally different approaches to defining event independence (Section 3.2.1, Section 3.2.2, Section 3.2.3, Section 3.2.4 and Section 3.2.5; Figure 5). The autocorrelation method captures the full temporal persistence of the precipitation signal, including the gradual decay associated with synoptic-scale processes, which naturally leads to longer inter-event separations. The KDE method, being based on the statistical distribution of dry periods and the assumption of Poisson-distributed event arrivals, identifies the shortest separation that satisfies the exponential independence criterion (Figure A2).

The pronounced seasonal variability of the AC method, particularly the high winter MIT values (Figure 6), can be attributed to the prevailing long-duration, low-intensity cyclonic precipitation during the cold season. In such conditions, the autocorrelation decays much more slowly than in short-duration convective showers (Figure A1), leading to the identification of longer intermediate periods and, consequently, the merging of precipitation into extended events. The KDE method, by contrast, shows considerably less seasonal variability (Figure 6), which may represent an advantage in applications requiring temporally consistent separation criteria.

The trend reversal observed in Type 4—where the general pattern of KDE yielding more events than AC is reversed at higher thresholds ($P=3$ and 5 mm; Figure 5; Section 3.2.1), with AC producing more events than KDE—merits particular attention. Because Type 4 events at $P=1$ mm have, on average, the shortest durations of all types (Table A4), the low MIT values produced by KDE generate a large number of short events. When the threshold is raised, a disproportionately large fraction of these short, shallow KDE-separated events falls below it, resulting in fewer retained events than under the AC method, which produces longer, deeper events that are more likely to exceed the threshold (Table A4).

The close agreement between the AC and KDE methods for maximum short-term intensities across most types (Table 6; Section 3.2.4) suggests that extreme precipitation is captured robustly irrespective of the separation approach. The notable exception of Type 4, where AC yields substantially higher intensities (Table 6), is consistent with the reclassification mechanism discussed above: at higher MIT values, intense events that would otherwise be classified as Type 1 are absorbed into longer Type 4 events (Figure A5 and Figure A8).

4.4. Design Criticality and the Role of Duration Class

The composite design index $I_{\mathrm{crit}}$ (Equation (14)) provides a unified framework for assessing the relative importance of different precipitation types in urban drainage design. The consistent dominance of Type 1 across both duration classes and all separation methods (Figure 9, Figure 10 and Figure 11) confirms that short-duration convective precipitation with an early intensity maximum represents the most critical design scenario. This finding aligns with established engineering practice, in which design storms for smaller urban catchments are typically characterised by short, intense bursts of precipitation [2,3].

However, the analysis reveals that Type 4 can approach the criticality of Type 1 when longer MIT values or the AC method are applied, particularly across all durations and at the $P=3$ mm threshold (Figure 9 and Figure 11; Section 3.3). This finding carries practical significance for the design of systems with significant retention or storage capacity: if antecedent low-intensity precipitation reduces the available capacity, a subsequent late-maximum event (event in which peak intensity occurs in the fourth quartile of storm duration, Type 4) may generate critical runoff despite its lower instantaneous peak. This observation is consistent with the recommendation of Bonta [9] to consider separate Huff curves for different seasonal and synoptic conditions.

The sharp decline in $I_{\mathrm{crit}}$ for Type 1 at MIT $=6$ h in the $T\le 2$ h class (Figure 10; Section 3.4.6) is a direct consequence of event merging: intense short-duration showers, when combined with adjacent weaker events, produce longer events that no longer qualify as short-duration events, thereby removing them from the analysis. This merging effect is more pronounced than for any other type (compare Figure 9 and Figure 10) and highlights the sensitivity of short-duration analyses to the chosen MIT value.

The finding that maximum $I_{\mathrm{crit}}$ values for $T\le 2$ h events slightly exceed those for all durations (Figure 11; Section 3.5) confirms that the most critical individual design scenarios for urban drainage arise from short-duration convective precipitation with steep Huff curves. Longer events, while more relevant in an averaged sense (e.g., for volume-based design), do not generate the highest peak design intensities. This distinction is essential for practitioners, as it implies that dimensionless hyetographs derived exclusively from all-duration analyses may underestimate the criticality of short-duration convective events (Figure A6 and Figure A9).

5. Conclusions

This paper analyses the influence of primary processing of pluviographic records (precipitation intensity measured over time) on the construction of dimensionless Huff hyetographs (standardised representations of how precipitation intensity varies over the course of an event). Special emphasis is placed on the method for defining the minimum interval between precipitation events (MIT, the smallest dry period separating two rain events) and on the choice of the precipitation threshold (minimum precipitation amount considered an event). Also, the analysis was conducted for two classes of precipitation duration: all durations and durations less than 2 h. Based on a 10 year series of 5 min measurements, a systematic analysis of the sensitivity of precipitation event physical characteristics and the shape of the Huff curves to the applied data-processing methods was conducted.

Increasing the precipitation threshold for selecting events for analysis significantly reduces the number of detected precipitation events, while the average precipitation amount and duration increase. The influence of the threshold on the maximum short-term intensities and maximum precipitation amounts was not observed. In contrast, the slope of the Huff curves proved to be poorly sensitive to changes in the threshold. This confirmed that the thresholds primarily affect the selection of weaker events, but do not change the peak characteristics of intense precipitation relevant for urban drainage design.

A change in the MIT value strongly affects the event duration, total precipitation amount and shape of the resulting Huff curves (Section 3.1.2). Among the five Huff types, those with an early maximum (Type 1) or late maximum (Type 4) exhibited the highest sensitivity to the choice of event-separation criterion, whereas Type 2 (mid-early maximum) showed almost no sensitivity. When compared to the fixed MIT approach, the autocorrelation-based (AC) method corresponds to an equivalent fixed MIT of approximately 6–9 h, whereas the kernel density estimation-based (KDE) method corresponds to approximately 1–3 h. Consequently, the choice of event-separation method has a direct and quantifiable effect on the shape of the resulting dimensionless hyetographs.

This confirmed that the choice of event-separation method can directly affect the resulting dimensionless hyetographs. The KDE method is better suited for identifying short-duration events (e.g., convective storms), whereas the AC method may be more appropriate for capturing broader temporal structures.

The introduction of the composite design index, which combines the slope of the Huff curve and the maximum short-term intensities, enables a unique assessment of the design criticality of different precipitation types. For all durations, it was found that Type 1 consistently represents the most critical form of the hyetograph. However, it should be emphasised that Type 4 can become almost equally critical at higher MIT values and that the autocorrelation method should be applied to the analysis of precipitation of all durations. This confirmed that the late maximum can represent a relevant design scenario in systems where merging events result in longer events, especially when prior precipitation conditions reduce retention capacity.

The analysis of a separate class of short-term events ($T\le 2$ h) showed that the early maximum remains the only extremely critical type of precipitation. In contrast, the late maximum loses significance due to the absence of long-term pre-phases of precipitation. The slopes of the Huff curves for $T\le 2$ h are systematically higher than in the analysis of all durations, which reflects a distinct concentration of precipitation in a short time.

The final comparison of the critical design showed that the maximum values of the design index for events of duration less than 2 h are either equal to or slightly exceed those obtained from the analysis of all durations. This confirmed that the most critical individual design scenarios for urban drainage arise from short-term convective precipitation with extremely steep Huff curves. At the same time, longer events become more relevant in the average sense, but not necessarily in the absolute criticality maximum. Additionally, it was found that the precipitation threshold $P=3$ mm represents an optimal compromise between selecting relevant events and preserving the representative shape of the Huff curves. The lower threshold ($P=1$ mm) includes a large number of very short and weak events that flatten the Huff curves and reduce their slope, while the higher threshold ($P=5$ mm) significantly reduces the number of available events without a significant increase in peak intensities. For the threshold $P=3$ mm, the highest values of the design index are achieved with a stable shape of the Huff curves, which makes it particularly suitable for defining the design storm hyetograph in the context of urban drainage sizing.

The methodology enabled a systematic examination of the impact of primary record processing steps on the final form of dimensionless hyetographs. The results confirm that the choice of event-separation method and precipitation amount threshold has a direct and quantifiable effect on both the hyetograph shape and the identified design-critical scenario; practical guidance on these choices is provided below.

5.1. Implications for Engineering Practice

The results of this study have practical applications for creating and applying Huff hyetographs in the design of urban drainage systems.

First, the choice among the MIT methods—KDE, fixed MIT values and the AC method—should match the main design goal. For analyses focused on the worst rain in small urban areas with short runoff times, both the statistically based KDE method and fixed MIT values of 1–3 h work well (Figure 10; Table 4), as they preserve short, intense storms and give more cautious (steeper) Huff curves for the main Type 1 event. For analyses of larger systems, where antecedent soil moisture conditions and total precipitation volume are relevant, the autocorrelation (AC) method or longer fixed MIT values may be better (Figure 9), as they capture the full precipitation pattern, including the possibly important Type 4 event.

Second, the precipitation threshold should be selected to balance between excluding hydraulically insignificant events and maintaining a sufficiently large sample size. Based on the present analysis (Table A1, Table A2 and Table A3; Figure 3 and Figure A3 and Figure A4), $P=3$ mm is recommended as an effective threshold for urban drainage applications, as it achieves the highest design index values with a stable Huff curve shape (Figure 9 and Figure 10).

Third, the pronounced sensitivity of Type 4 to the separation method (Table 4; Figure 4) implies that engineering assessments should explicitly consider late-maximum events when designing systems with retention capacity or when evaluating scenarios involving sequential precipitation. Omitting Type 4 from the analysis or using a separation method that fragments these events may lead to an underestimation of design-critical conditions (Figure 9 and Figure 11).

Finally, the consistency of results across three fundamentally different MIT determination approaches—fixed, autocorrelation-based and statistically based—demonstrates the robustness of the Huff curve framework (Figure A5 and Figure A8; Table 5 and Table 6). These approaches differ: the fixed and KDE methods focus on short, high-intensity events, while the AC method captures longer, more gradual events. Regardless of method, the results are robust as long as the separation criteria are transparently documented and methodologically justified. The observed sensitivity to event processing emphasises the need to report the adopted MIT method, threshold and duration class alongside any published Huff hyetographs, thereby ensuring reproducibility and comparability of results across studies.

For practical applications in urban drainage design, precipitation events should be separated using the KDE-based approach ($\mathrm{MIT}\approx 1$–3 h), with a precipitation amount threshold of $P=3$ mm. The analysis should focus on short-duration events ($T\le 2$ h), using Type 1 as the primary design hyetograph, supplemented by verification with Type 4 hyetographs obtained with the AC method or fixed MIT $\approx 6$ h.

5.2. Limitations

Several limitations of this study should be acknowledged. The analysis uses data from one station (Čepin) with a 10-year record (Section 2.1). This period is sufficient to identify primary Huff curve sensitivities to processing methods (Figure 9, Figure 10 and Figure 11), but may not fully capture extreme event variability (Table 5). Although the raw input data were organised on a monthly basis, precipitation events crossing calendar-month boundaries were explicitly handled by a dedicated post-processing routine (Section 2.5), ensuring no genuine event was artificially truncated. A residual limitation is that events spanning more than one calendar month—which are extremely rare in the study climate—are still interrupted at the year boundary in the current implementation.

It should be noted that throughout this study, the terms ‘convective’ and ‘cyclonic/frontal’ are used in an operational sense to describe short-duration, high-intensity events (dominated by Type 1) and longer-duration, lower-intensity events (predominantly Types 2–4), respectively, based solely on the temporal structure of the precipitation record at a single station. Without concurrent analysis of synoptic-scale meteorological conditions (e.g., radiosonde observations, reanalysis fields or weather radar data), this attribution should be regarded as indicative rather than definitive; some individual events may exhibit mixed characteristics or arise from processes not adequately captured by the single-station temporal structure alone.

Additionally, the study examines a moderately humid continental climate (Section 2.1). However, the performance of the MIT methods may differ in other climatic regions, especially in arid or tropical settings where precipitation patterns vary greatly. Therefore, future work should extend this analysis across more stations and climatic zones to test whether the observed patterns (Figure 4, Figure 9 and Figure 10) are generalisable.

Finally, the composite design index $I_{\mathrm{crit}}$ (Equation (14); Figure 9, Figure 10 and Figure 11) was introduced as a pragmatic metric for comparing design criticality. Still, it does not account for the frequency of occurrence of each precipitation type nor incorporate hydrological response modelling. For a more comprehensive assessment, the next step would be to couple the derived hyetographs (Figure A5 and Figure A8) with a precipitation–runoff model for a specific catchment, thereby translating the dimensionless curve properties into actual peak flows and system performance indicators.

Notwithstanding these limitations, several findings are expected to be broadly applicable. Although the analysis is based on a single station, several findings are expected to be generally applicable and are consistent with previous studies. The dominant role of Type 1 (early peak) precipitation as the most design-critical scenario reflects the typical behaviour of short-duration convective precipitation reported in the literature. Likewise, the use of a moderate precipitation amount threshold ($P=3$ mm) represents a robust approach for excluding hydraulically insignificant events without substantially affecting the peak characteristics of intense precipitation, which are most relevant for urban drainage design. In contrast, the exact MIT values and their seasonal variability are site-specific and depend on local climatic conditions and precipitation regimes.