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Given the short concentration time in urban watersheds, the design of municipal water infrastructures often requires knowledge of sub-daily precipitation intensity. Sub-daily time series can be directly used in a rainfall–runoff model or to derive intensity–duration–frequency (IDF) curves and calculate the design precipitation. Given that precipitation projections are typically at a daily time scale, temporal disaggregation using techniques of variable complexity is often needed to evaluate the risk/performance of urban infrastructure in the future. This paper proposes a simple steady-state stochastic disaggregation model that generates wet/dry day occurrence using a binomial distribution and precipitation intensity using an exponential distribution. Daily precipitation data from four regional climate models (RCMs) forced with the high-emission scenario representative concentration pathway (RCP 8.5) were downscaled using the quantile mapping (QM) method. The performance of the developed method is compared to widely used temporal disaggregation methods, namely, the multiplicative random cascade model (MRC), the Hurst–Kolmogorov process (HKP), and three versions of the K-nearest neighbour (KNN) model, using the Kolmogorov–Smirnov (KS) test. The six disaggregation techniques were assessed at four stations in the South Nation River Watershed in Eastern Ontario, Canada. Results indicate that, despite its simplicity, the proposed method performed well compared to other temporal disaggregation methods when resampling the observed extreme precipitation.

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  • Climate change impacts on short-duration precipitation extreme events are investigated using different temporal disaggregation methods.

  • A simple steady-state stochastic disaggregation model is introduced to generate future sub-daily precipitation intensities.

  • The developed disaggregation method adequately resamples the observed short-duration extreme precipitation for application in municipal water infrastructures.

Graphical Abstract

Graphical Abstract
Graphical Abstract
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Graphical Abstract
Graphical Abstract
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climate change, FO method, IDF curves RCM, temporal disaggregation
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Changes in the temporal variability of precipitation at all timescales are expected due to global warming. Such changes affect urban water infrastructure by potentially influencing its performance and risk of failure. Unfortunately, there is considerable uncertainty about how hydrological variables will change in the future. While uncertainty is present at all timescales, the climate signal in the daily time series simulated by climate models, for instance, can be estimated with much greater certainty than in the simulated hourly time series (Jakob et al. 2011). Sub-daily (fine-scale) time series are critical in some water resource engineering problems, especially for local hydrological evaluations in a changing climate (Zhu et al. 2012). One way of obtaining hourly or sub-hourly precipitation times in the future is to temporally disaggregate daily time series generated using either weather generators or downscaling climate model outputs. Daily time series can be disaggregated into hourly time steps and converted to intensity–duration–frequency (IDF) curves or be directly fed to a rainfall–runoff model (Alzahrani et al. 2022). For instance, Zhao et al. (2021) proposed a framework for constructing future IDF curves for tropical cities in Southeast Asia. It consists of bias-correcting daily regional climate model (RCM) outputs using a version of the quantile mapping (QM) method, temporally downscaling it, and generating IDF curves. Temporal disaggregation is the conversion of annual, monthly, or daily scale observations into sub-daily scales, such as hourly or minute scales, using a stochastic or machine learning approach. Examples of temporal disaggregation techniques include (i) the K-nearest neighbour disaggregation model (KNN); (ii) the multiplicative random cascade model (MRC); (iii) the microcanonical random cascade model (MRCM) (Olsson 1998; Olsson & Berndtsson 1998; Molnar & Burlando 2005; Licznar et al. 2011; Jebari et al. 2012; Müller & Haberlandt 2015; Garbrecht et al. 2017; Gaur & Lacasse 2018); and (iv) the Hurst–Kolmogorov process (HKP) downscaling model (Hurst 1951; Koutsoyiannis & Cohn 2008; Müller & Haberlandt 2015; Lombardo et al. 2017).

The KNN disaggregation technique is a version of the method of fragments (MF). The MF is a popular nonparametric temporal disaggregation approach in which sub-daily precipitation values are estimated as fractions (fragments) of the daily precipitation amount. In the KNN approach, the fragments are calculated using available daily and hourly precipitation time series that are close (neighbour) to the period of interest based on a distance measure (e.g., distance in time, magnitude, etc.). Park & Chung (2020) applied the KNN model using 3-day patterns and found that the disaggregated hourly rainfall data had similar statistical properties to the observed series. Sharif et al. (2013) applied the KNN model to synthesize and temporally disaggregate daily precipitation under climate change scenarios in the Upper Thames River Basin (UTRB). They found that the model effectively reproduces simulated sequences of future climate scenarios represented and predicted by Global Climate Models (GCMs). Lu et al. (2015) investigated the effects of climate change on heavy regional rainfall by producing a spatially and temporally varying rainfall time series using a hybrid stochastic weather generator (LARS-WG) and a KNN model. Uraba et al. (2019) used a stochastic KNN disaggregation model to convert downscaled GCM scale rainfall production; they found that climate change's impact on rainfall intensity would be noticeable in storms with shorter durations and longer return times. Sharma & Srikanthan (2006) applied KNN combined with the fragment (KNN–MOF) method. Their results showed that the KNN–MOF performed better in producing the descriptive statistics of wet and dry spell distributions than the other models. The main challenge in the KNN model is choosing the optimum value of neighbours for the temporal downscaling process. The number of neighbours should be estimated to synthesize daily time series using the KNN model (Sharif & Burn 2007).

HKP is a stochastic process disaggregation model that uses Gaussian white noise to convert precipitation from coarser to finer scales. This model uses the Hurst exponent (H) to qualify the data behaviour. Hurst (1951) observed randomness in numerous physical time series, and the tendency of natural events was the most significant even though these events were in higher and lower values. The HKP method was intensely used in hydrology and water resource management to generate sub-hourly or sub-daily scales for different purposes in the urban water system. A challenge in the HKP model is determining the H value, which is expensive to estimate. Lombardo et al. (2012) used HKP to convert the daily precipitation into a 5-min time step. However, the model is in the log domain and not all daily time series are appropriate for a lognormal distribution.

The MRC model, first introduced by Olsson (1998), is an invariance scaling method used for temporal downscaling. Gaur & Lacasse (2018) applied the MRC model to multivariate climate variables at several sites within the same region to assess the model's accuracy in performance. Licznar et al. (2011) generated a fine-scale using the multiplicative cascade model and a microcanonical cascade, yielding intermittency on a 5-min scale. Molnar & Burlando (2005) found that the cascade model performed well in terms of rainfall distribution preservation at a 10-min scale; the model could reproduce the growth of the intermittency period. Koutsoyiannis & Onof (2001) developed a stochastic disaggregation model that combines a precipitation simulation with an adjustment process to convert the coarse-scale to a finer-scale. This model's main advantage is preserving descriptive precipitation statistics; however, its main limitation is that the model does not reproduce instantaneous hourly series because of the variance in the generated weights in each cascade level. Müller & Haberlandt (2015) found that a 5-min resolution will lead to missing time steps with dry intervals if the branching of the cascade is constant and this is generally considered a limitation for the cascade model.

Temporal disaggregation techniques have different levels of complexity and unequal performances and the generated time series may not have the same statistical characteristics (Tayşi & Özger 2022). HKP and MRC are mathematically challenging to calibrate, while the KNN methods are straightforward. A natural question is whether more complex temporal disaggregation techniques lead to better results regarding IDF curves and descriptive statistics of the disaggregated time series. In this paper, a simple temporal disaggregation technique called the Fahad–Ousmane (FO) method is introduced. Daily time series of future precipitation are generated by spatially downscaling the outputs of four RCMs using the QM method. It has only two parameters and its calibration is straightforward compared to HKP and MRC. The FO model and five alternative temporally downscaling methods (the MRC, the HKP, and three versions of the KNN) are used to generate hourly precipitation time series in both the historical (1976–2005) and future periods (short-term 2010–2040, medium-term 2041–2070, and long-term 2071–2100). Besides the visual comparison of generated IDF curves, the two-sample Kolmogorov–Smirnov (KS) test is used to compare the distribution of the time series of annual precipitation of durations 1 and 24 h.

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General methodology

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The general methodology of the work, as illustrated in Figure 1, is described as follows:

  • Developing a simple temporal disaggregation technique called the FO method

  • Generating daily precipitation time series by downscaling the nearest grid data of a climate model's precipitation output using the QM method

  • Disaggregating these time series from daily to sub-daily (hourly) timescales by the MRC, the HKP, three versions of the KNN (5, 15, and 30), and the FO model

  • Extracting the maximum annual precipitation intensity for durations of 1–24 h from the disaggregated and observed time series

  • Generating IDF curves from the maximum annual precipitation of the disaggregated and observed time series

  • The disaggregation performance was tested using visual comparisons, main statistics, and the two-sample KS test

Figure 1
Flowchart of the adopted methodology of this work.
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Flowchart of the adopted methodology of this work.

Figure 1
Flowchart of the adopted methodology of this work.
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Flowchart of the adopted methodology of this work.

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Study area and available data

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The study area is the South Nation watershed (approximately 4,000 km2), located southeast of Ottawa in Eastern Ontario, Canada (Figure 2). The South Nation River drains the watershed for 175 km with relatively flat topography (approximately 80 m between its headwaters and output at the Ottawa River), making the area prone to significant flooding risk. The observed daily precipitation (PCP) and hourly precipitation acquired from Environment Canada cover the 1976–2005 period at four stations, as presented in Table 1. Table S1 (Supplementary Materials) presents the available daily outputs of several RCMs from the NA-CORDEX experiment forced with a high Representative Concentration Pathway (RCP 8.5) greenhouse gas emission scenario. The available horizontal spatial resolution, namely the North America Domain (NAM-44) (0.44°) with an approximately horizontal projected grid resolution of 50 km, was used.
Table 1

Weather station information

StationLatitude (N)Longitude (W)Elevation (m)Mean daily PCP (mm)Standard Deviation of daily PCP (mm)
St. Albert 45° 17′ 14″ 75° 03′ 49″ 80 2.77 6.06 
South Mountain 44° 58′ 00″ 75° 29′ 00″ 84.7 2.64 6.07 
Morrisburg 44° 55′ 25″ 75° 11′ 18″ 81.7 2.77 6.17 
Russell 45° 15′ 46″ 75° 21′ 34″ 76.2 2.63 5.97 
StationLatitude (N)Longitude (W)Elevation (m)Mean daily PCP (mm)Standard Deviation of daily PCP (mm)
St. Albert 45° 17′ 14″ 75° 03′ 49″ 80 2.77 6.06 
South Mountain 44° 58′ 00″ 75° 29′ 00″ 84.7 2.64 6.07 
Morrisburg 44° 55′ 25″ 75° 11′ 18″ 81.7 2.77 6.17 
Russell 45° 15′ 46″ 75° 21′ 34″ 76.2 2.63 5.97 
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Figure 2
The meteorological gauges in the South Nation watershed (Alodah & Seidou 2019).
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The meteorological gauges in the South Nation watershed (Alodah & Seidou 2019).

Figure 2
The meteorological gauges in the South Nation watershed (Alodah & Seidou 2019).
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The meteorological gauges in the South Nation watershed (Alodah & Seidou 2019).

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Downscaling RCM outputs at the daily timescale

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The outputs of the RCMs are downscaled using the QM technique. The QM method consists of remapping the probability density function (PDF) of the uncorrected RCM series in combination with the probability of the observed data series. The following formula transforms the corrected RCM to cumulative distribution functions:
(1)
where XCorrected is the projected time series, is the inversed cumulative density probability, FRCM is the cumulative density probability, and XRCM is the RCM data before projection. In practice, Equation (1) is replaced by the following mapping function between GCM outputs and observations:
(2)
where Xobservation is the observed time series and XRCM is the raw time series obtained from the climate model.

The limitation of this approach is that it cannot predict values beyond the range of historical data. The reference period is split between validation and calibration. Equation (2) is fitted to the calibration period and applied to the validation and future periods. To avoid any bias resulting from oscillations in the observed precipitation time series, the calibration data set contains all data series from 1976 to 1991, and the validation period covers the data series for the time range of 1992–2005.

Temporal disaggregation techniques

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The following subsections describe the models and theories of each temporal stochastic disaggregation method. Four temporal downscaling techniques are considered: the KNN disaggregation technique, the MRC model, the HKP downscaling model, and a novel disaggregation technique called the FO model.

KNN disaggregation technique

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KNN is a particular case of the fragment (MF) method, a nonparametric approach commonly used by researchers because of its simplicity. The principle of the method is to generate fragments that are fractions of daily precipitation that occur in each hour of the day and then, the fragments are summed to 1. The general formulation of the MF is given in Equations (3) and (4). Equation (3) contains wi, the weight that will be computed, hi is the data selected to produce fragments from the series, and n is the number of data in the series (24 h).
(3)
Then, each fragment is multiplied by the daily data d in Equation (3) to obtain the new hourly data or disaggregated values.
(4)
The hourly series of data used to estimate fragments could be chosen randomly; however, this would lead to inadequate statistical properties (Srikanthan et al. 2006). Therefore, the best choice is to compare the total disaggregated daily precipitation with the close total daily precipitation before the disaggregation process (Wey 2006). The KNN is a particular case of the fragment method, where the fragment is selected from historical hourly precipitation in a sample of neighbours with similar precipitation amounts. The neighbours are selected in a temporal window to preserve the seasonal characteristics of precipitation. Note that the sliding window is based on the Julian calendar and spans all years in the data. The K neighbours with the most negligible difference in precipitation amount are selected. Then, one neighbour is sampled from the set and used to calculate the fragment. The probability of sampling neighbour j is
(5)
where p(j) is the precipitation amount of neighbour j and P(s) is the precipitation amount on the simulation day.

FO disaggregation model

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FO is a steady-state stochastic disaggregation model that generates wet hours (hours with precipitation above 1 mm) using a binomial distribution with parameter p, and for each wet hour, precipitation intensity is generated using an exponential distribution with parameter α. The parameters are fitted month by month using observed data in the calibration period as follows:

  • (a)

    Estimate all hourly precipitation for month M.

  • (b)

    Estimate p as the ratio of hours with precipitation higher than 1 mm to the total number of hours.

  • (c)

    Create a time series of hours with precipitation above 1 mm.

  • (d)

    Fit the exponential distribution to the above time series to obtain α.

The generation of hourly precipitation for a day with precipitation amount pday is calculated as follows:

  • (a)

    If p= 0, set all hourly values to 0.

  • (b)

    If p> 0,

    • a.

      Generate a series of 0 (dry hours) and 1 (wet hours); repeat until you obtain at least one nonzero value.

    • b.

      For each wet hour h, take a sample from the exponential distribution with the parameter α to obtain ⁠; otherwise, set to 0 for dry hours.

The final precipitation for each hour of the day is set to
(6)

Hurst–Kolmogorov downscaling model

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HKP is a technique that converts coarse-scale data into fine-scale data. This model uses pure downscaling to generate precipitation based on fractional Gaussian noise, called the HKP (Koutsoyiannis 2002). As shown in Figure 3, a dyadic additive cascade is used to disaggregate fractional Gaussian noise; each higher amount of precipitation is disaggregated into a lower-scale level. Then, the disaggregated fractional Gaussian noise is exponentially transformed to derive the actual precipitation. The HKP represents the long-term persistence of the observed time series with a simple and efficient stochastic process. Additionally, it has been demonstrated that this random representation is not solely based on data, as the behaviour of Hurst–Kolmogorov develops from entropy production (Koutsoyiannis et al. 2011). For any integers i and j and any timescale f and i, the HKP could be defined as a stochastic stationary process, as in the following equation:
(7)
where indicates the equality in probability distributions, H is the Hurst coefficient in the range between 0 and 1, is the Gaussian noise, and is the mean value. For the related process, in the following equation is used:
(8)
where ⁠.
Figure 3
A sketch of the disaggregation model steps at the level of k (Lombardo et al. 2012).
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A sketch of the disaggregation model steps at the level of k (Lombardo et al. 2012).

Figure 3
A sketch of the disaggregation model steps at the level of k (Lombardo et al. 2012).
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A sketch of the disaggregation model steps at the level of k (Lombardo et al. 2012).

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The HKP is implemented as follows: consider the to be the depth of daily precipitation aggregated in an enormous timescale (j = 1). is assumed to be a random variable with a mean and standard deviation of the stochastic process. The observed daily precipitation is assumed to be a lognormal distribution. An auxiliary Gaussian random variable of the aggregated HKP on a timescale f with mean and variance is expressed as in the following equations:
(9)
(10)
is disaggregated as a dyadic (b = 2) additive cascade. Then, is split into two (b = 2) Gaussian random variables based on the timescale Δs = f/2. For example, at the first cascade level k = 1, we have ⁠. Similarly, the k cascade level corresponds to Δsk= 2−k f
(11)

Consequently, the equation is adequate to generate and then obtain ⁠.

This is a generic procedure that resembles the well-known interpolation procedure. Therefore, the following linear generation scheme is considered, as shown in Figure 3:
(12)
where ⁠, Θ is a vector of parameters, and V is Gaussian white noise. This equation allows the interpolated value to preserve the total depth of two earlier lower-level variables (level k) and one later higher-level variable (level k − 1) (Koutsoyiannis 2002).
Therefore, in each disaggregation step, two lower-level variables are estimated by
(13)
(14)
where the parameters a2, a1, b0, and b1 and the variance V are estimated in terms of the correlation coefficient ⁠, which is independent of j and k, and of the HKP variance at level k (Koutsoyiannis 2002), as given by the following equations:
(15)
and
(16)
The mean and the variance in the HKP at the k-level of the cascade are
(17)
(18)
where Δsk = 2−kf and H is HKP.
The above step-by-step disaggregation approach was introduced by Koutsoyiannis (2002), who established that it effectively generates fractional Gaussian noise. However, the precipitation process is not a Gaussian distribution. Indeed, specific exponentiation to the HKP is applied by the following equation to make a lognormal domain and preserve the scaling properties:
(19)
HKP is assumed to be a unique process in the untransformed domain and the characteristics of the exponentiation transformation are changed for the transformed domain. Using different characteristics for different disaggregation steps while utilizing the scale-dependence of the daily step will help to transform the Gaussian domain to a lognormal domain by the following equations:
(20)
(21)

The explained model is a disaggregation model only if the random variables are Gaussian processes. However, the hypothesis of lognormal rainfall, a conversion model of a coarser-scale into a small scale, is statistically consistent with the given process Z at the coarser-scale. The Hurst coefficient H is the only parameter of the HKP downscaling model. Therefore, the H value is estimated numerically.

H values have been estimated for each season from the historical time series because the disaggregated values are subject to the range of H values between 0 and 1. If the H value is closer to or beyond 1, the final values will be imagined, negative, or extreme. As a result, the simulated annual maximum precipitation (AMP) would be the effect. Therefore, the results are needed for the calibration procedure. The procedure is composed of thresholds and correction factors, which are estimated numerically using evolutionary optimization, such as a genetic algorithm, to estimate the optimal AMP values. The following equations illustrate the coefficients and thresholds, and these coefficients play the primary role in the results:
(22)
where
(23)
(24)
where
(25)
(26)
(27)
(28)
(29)

Because the discrete data (precipitation to be simulated) are manipulated as samples of an analogue signal with its local maxima and minima of the AMP values, in some time intervals (some seasons), the coefficient performs as a periodic (distorted) signal with some periodicity.

The first poly (pr1) contains a sinusoidal component, the factors play a role in the amplitude and phase (x12 manages the amplitude, and x11 controls the frequency of the sinusoidal component). The second poly (pr2) has a DC value (direct current or constant value) where the simulated date is raised or lowered based on the factor value (x1, x8, and threshold [thr1]). The third poly (pr3) is added to cover the linear component (x4). The fourth poly (pr4) is added to consider a square component where the observed data are squared and multiplied linearly by the factors (x3 and x10). In the last poly, a cubic component of the actual data is applied to cover more possibilities of correlation with the actual data (x2, thr2, x1, and x9).

The MRC model

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The MRC model is a temporal scaling method that converts the time series from a coarse to a fine scale. Also, the conversion is based on branch splitting of each value at a daily or sub-daily scale. Therefore, the best branch splitting occurs when the original value is split into two cells, called a dyadic cascade. Then, the final format of the disaggregated hourly series comprises consecutive tree vertices of each rainy-day event.

In this section, Pr or R is the precipitation average over the timescale at the time origin (j = 1). Pr is assumed to be a random variable with a mean u0 and variance sigma2 of a stochastic process.

The branching number, which estimates the number of branches assigned to the next level with the smaller timescale steps for the distributed rainfall from the coarser-scale time step, is a structural component of the model. Using b= 2 means that the 24 h of the day could be split into day and night hours. Similarly, the day's hours could be split into morning and afternoon hours. The precipitation amount V is multiplied by multiplicative weights W1 and W2 to obtain V at each time step. The total summation of these weights must equal 1 at each level and they are not independent of one another; therefore, there is a probability of how the total amount of precipitation in each cascade level is converted. The following equation illustrates the three cases of each cascade level of branching (b = 2):
(30)

P denotes the probability for each time step in the level. The first case, which is P (1/0), means that the total precipitation is assigned in the first time step (W1= 1), and no precipitation will be allocated to the second time step when (W2= 1 − W1=0). The probability of the P (0/1) disaggregation level is accomplished for the second case. The third case, which is the probability (x/(1 − x) splitting, is to redistribute the rainfall amount over the time step, where x is defined as 0<x< 1, and it represents the relative fraction of precipitation values that is attributed to the first step. The x value is a random variable in all disaggregation steps, and the PDF f(x) can be estimated for each value of x. These cases in Equation (30) provide the following four different types of wet boxes with Pi > 0: the first box is the starting box, including a dry box in the past time step and a wet box in the later time step. Second, the ending box follows a wet box and is followed by the dry box; this can be called a chain box. Next, the isolated box is defined as side-by-side past boxes, and the next steps are dry. Finally, the enclosed box is an isolated box case, but it is for wet boxes.

A reverse scaling procedure achieved these probability cases for three different cases in Equation (29). Backward cascade level branching assumptions lead to highly resolved precipitation time series aggregation. Each cascade level of branches is summed to represent the total volume of precipitation at the preceding higher-level. The statistics (mean, maximum, minimum, and standard deviation) and probabilities are calculated for each branch in the level by dividing the counts of each case by the total number of elements of the higher cascading level time series. Additionally, statistical calculations are needed to evaluate the case P(x/1 − x), where x is the relative weight. Therefore, the relative weight (x) is a value between zero and the 90th percentile, influenced by the mean and standard deviation. This process aggregates the rainfall as follows: 1 → 3 h (3*20 h), 3 → 6 h (3*21 h), 6 → 12 h (3*22 h), and 12 → 24 h (3*23 h). In each aggregation step, a related weight is assigned to probabilities P (0/1), P (1/0), and P (x/(1 − x)) before estimating the averages of all probabilities of all steps (Güntner et al. 2001). As a result, the probability and weight matrices are derived to represent the station's scaling. Notably, an empirical distribution is applied to estimate the parameters, which is done by a random number generator if there is no representation.

Sequentially, the weight matrices are applied to disaggregate the daily time series of precipitation. Tracing the random numbers for each cascade level, which incorporates the probabilities (P (0/1), P (1/0), and P (x/(1 − x))) (they are cumulatively evaluated), defines the branching type in each time step.

When the disaggregation of the daily precipitation model begins, the drawing of the random numbers determines the type of branching that incorporates the probabilities (three cases in the above equations). If the random numbers are in the range P(x/(1 − x)), a similar technique is applied to estimate the weight x value by using another random number.

A numerical simulation of the cascade is carried out if μo= 1 and = 0. Thus, the summary statistics are
(31)
where j = 1, 2, …, bk is the position index in the series at level k; i is the index of the cascade level; and g(i, j) represents a function defining the position in the series at level i, i.e., ⁠, which is a ceiling function (Gaume et al. 2007). For k = 0, we have W1,0 = 1.
(32)
where < > denotes the expected value, which is an average over the independent realizations of the stochastic process. The expected value of Rj,k is given by
(33)
For a microcanonical cascade model, the k-level's mean process equals the process at the 0 level, which means that a further cascade level relationship will hold for every pair of successive aggregations.
(34)
where j= 1,… bk − 1 with k> 0. The following example explains how the microcascade model processes during each cascade level when b= 2:
(35)

Therefore, in the above equation, the weights Wj,k fulfil μW= 1 and W < b.

Let us assume that the branching value is b= 2, and the exponent hj,k(t) represents the top of the tree, excluding the cascading level start that belongs to both simple paths leading to Rj k and Rj+t,k. The hj,k(t) is calculated as follows:
(36)
The following Equation (37) illustrates the computation of the model's exponent ⁠. In the computation, we use this equation: ⁠. The arrows show the links to those variables considered, where Θ[n] is the discrete form of the Heaviside step function, defined for a discrete variable (integer) n as
(37)
(38)
(39)
The mathematical part can be calculated numerically in the following section. The numerical experiment of the MRC model is explained by the following equation:
(40)
The weights W refer to the lognormal distribution defined as in the following equation:
(41)
where Y is a normal random variable N(1,0). Therefore, the variance in the weights is given by the following equation:
(42)
where is a parameter defining the standard random variable of normal distribution. To summarize, as the simulated sub-daily scale is essential for the storm control design system, this section elaborates on the mathematical scheme of the open-source software in which the MRC model was implemented. The model makes valuable contributions to water resource engineering.

Estimation of intensity-duration-frequency curves

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The extreme theoretical value (EV) distribution approach estimates the frequency of extreme precipitation for different durations and return period events. The Gumbel type I distribution is
(43)
where μ is the location and β is the scale parameter. The random variable XT associated with the return period can be estimated as in the following equation:
(44)
where is the mean of the observed or simulated AMP values and S is the standard deviation. KT is the frequency factor calculated with a given return period (T):
(45)

The duration events of 1–24 h are used and the return periods of 2-, 5-, 10-, 50-, and 100-year are chosen to construct the curves.

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The uncorrected outputs of the four RCMs were downscaled using QM at the four climate stations on four arbitrarily chosen periods: a historical period (1979–2005), a short-term period (2011–2040), a medium-term period (2041–2070), and a long-term period (2071–2100). Each of the 64 daily time series was temporally disaggregated using five techniques (HKP, MRC, FO, and KNN using 5, 15, and 30 neighbours). When using HKP and MRC, the daily time series were temporally disaggregated on five levels to obtain a time step of 1/36 h. These time series were linearly interpolated to produce hourly time series. Each hourly time series was afterwards aggregated to obtain precipitation intensity time series at a 2, 3, …., and 24 h time step, which in turn were converted into 24 h time series of annual maximums and an IDF curve. The performance of each disaggregation time series was assessed by (a) comparing the time series of annual maxima to corresponding observations using box plots and the KS test and (b) comparing the IDF curves derived from temporally disaggregated data with the IDF curves generated by observations on the historical period. The results of these comparisons are presented in the following sections:

Comparison of the simulated and observed time series of each temporal downscaling method

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Figure 4 compares the observed and the disaggregated annual maxima resulting from the FO, KNN (5), KNN (15), KNN (30), HKP, and MRC models in the historical period for the 1 and 24 h time series. None of the models perfectly captures the interquartile range of the observed data, as it was generally underestimated by MRC but overestimated by the KNN models, HKP and FO at the 1 h time step. Almost all models overestimated the interquartile range for 24 h time series, while HKP did not mimic the observed values regardless of the RCM used. Results also show that HKP and MRC lead to more severe biases: HKP overestimates precipitation intensity at the 1 h time step, while MRC underestimates it at the 24 h time step. FO performed well compared to other models in terms of bias and interquartile range. The main descriptive statistics for each model, namely the mean, standard deviation, kurtosis, and skewness, are presented in Table 2. All methods have mean and standard deviation values of the historical period close to the observations, while there are deviations in the kurtosis and skewness coefficients. The skewness of the observation and the FO and KNN models (all neighbour sizes) were found to be relatively similar to those of the observed values. MRC and HKP, in contrast, underestimated the skewness and the kurtosis values.
Table 2

Descriptive statistics of observed and disaggregated hourly time series of Russell Station

Disaggregation methodGCM, RCMMeanStandard deviationSkewnessKurtosis
Observed – 0.1 0.6 20.7 760 
HKP CanESM2, CanRCM4 0.1 0.5 7.1 82.4 
EC-EARTH, HIRHAM5 0.1 0.4 8.6 139.8 
EC-EARTH, RCA4 0.1 0.4 7.8 108.1 
GFDL, RegCM4 0.1 0.6 21.9 679.5 
MRC CanESM2, CanRCM4 0.1 0.4 9.9 150.4 
EC-EARTH, HIRHAM5 0.1 0.5 11.9 283.7 
EC-EARTH, RCA4 0.1 0.5 9.9 152.5 
GFDL, RegCM4 0.1 0.5 11.8 244.5 
KNN(5) CanESM2, CanRCM4 0.1 0.6 18.4 5,613 
EC-EARTH, HIRHAM5 0.1 0.4 18.2 526.4 
EC-EARTH, RCA4 0.1 0.6 18.9 516.7 
GFDL, RegCM4 0.1 0.6 22.8 782.5 
KNN(15) CanESM2, CanRCM4 0.1 0.6 21.7 885.2 
EC-EARTH, HIRHAM5 0.1 0.4 18.4 539.7 
EC-EARTH, RCA4 0.1 0.6 18.2 475.6 
GFDL, RegCM4 0.1 0.7 24.8 910.4 
KNN(30) CanESM2, CanRCM4 0.1 0.6 19.5 636.5 
EC-EARTH, HIRHAM5 0.1 0.4 18.3 535.2 
EC-EARTH, RCA4 0.1 0.6 18.8 507.3 
GFDL, RegCM4 0.1 0.8 24.9 756.5 
FO CanESM2, CanRCM4 0.1 0.6 15.6 394 
EC-EARTH, HIRHAM5 0.1 0.4 11.9 196.1 
EC-EARTH, RCA4 0.1 0.6 14.3 264.9 
GFDL, RegCM4 0.1 0.7 27.6 1,112.5 
Disaggregation methodGCM, RCMMeanStandard deviationSkewnessKurtosis
Observed – 0.1 0.6 20.7 760 
HKP CanESM2, CanRCM4 0.1 0.5 7.1 82.4 
EC-EARTH, HIRHAM5 0.1 0.4 8.6 139.8 
EC-EARTH, RCA4 0.1 0.4 7.8 108.1 
GFDL, RegCM4 0.1 0.6 21.9 679.5 
MRC CanESM2, CanRCM4 0.1 0.4 9.9 150.4 
EC-EARTH, HIRHAM5 0.1 0.5 11.9 283.7 
EC-EARTH, RCA4 0.1 0.5 9.9 152.5 
GFDL, RegCM4 0.1 0.5 11.8 244.5 
KNN(5) CanESM2, CanRCM4 0.1 0.6 18.4 5,613 
EC-EARTH, HIRHAM5 0.1 0.4 18.2 526.4 
EC-EARTH, RCA4 0.1 0.6 18.9 516.7 
GFDL, RegCM4 0.1 0.6 22.8 782.5 
KNN(15) CanESM2, CanRCM4 0.1 0.6 21.7 885.2 
EC-EARTH, HIRHAM5 0.1 0.4 18.4 539.7 
EC-EARTH, RCA4 0.1 0.6 18.2 475.6 
GFDL, RegCM4 0.1 0.7 24.8 910.4 
KNN(30) CanESM2, CanRCM4 0.1 0.6 19.5 636.5 
EC-EARTH, HIRHAM5 0.1 0.4 18.3 535.2 
EC-EARTH, RCA4 0.1 0.6 18.8 507.3 
GFDL, RegCM4 0.1 0.8 24.9 756.5 
FO CanESM2, CanRCM4 0.1 0.6 15.6 394 
EC-EARTH, HIRHAM5 0.1 0.4 11.9 196.1 
EC-EARTH, RCA4 0.1 0.6 14.3 264.9 
GFDL, RegCM4 0.1 0.7 27.6 1,112.5 

Bold indicating the reference point for comparison (Observed Data).

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Figure 4
Grouped boxplot for all disaggregation methods and different RCMs in the historical period using (a) 1 h precipitation data, and (b) 24 h precipitation data. The horizontal solid black lines represent the 25th and 75th percentiles while the horizontal solid red line represents the 50th percentile of the observed data. The dashed lines represent the whiskers (extended to the most extreme data points not considered outliers) of the observed data. The outliers are shown as ‘ + ’ marker symbol.
View largeDownload slide

Grouped boxplot for all disaggregation methods and different RCMs in the historical period using (a) 1 h precipitation data, and (b) 24 h precipitation data. The horizontal solid black lines represent the 25th and 75th percentiles while the horizontal solid red line represents the 50th percentile of the observed data. The dashed lines represent the whiskers (extended to the most extreme data points not considered outliers) of the observed data. The outliers are shown as ‘ + ’ marker symbol.

Figure 4
Grouped boxplot for all disaggregation methods and different RCMs in the historical period using (a) 1 h precipitation data, and (b) 24 h precipitation data. The horizontal solid black lines represent the 25th and 75th percentiles while the horizontal solid red line represents the 50th percentile of the observed data. The dashed lines represent the whiskers (extended to the most extreme data points not considered outliers) of the observed data. The outliers are shown as ‘ + ’ marker symbol.
View largeDownload slide

Grouped boxplot for all disaggregation methods and different RCMs in the historical period using (a) 1 h precipitation data, and (b) 24 h precipitation data. The horizontal solid black lines represent the 25th and 75th percentiles while the horizontal solid red line represents the 50th percentile of the observed data. The dashed lines represent the whiskers (extended to the most extreme data points not considered outliers) of the observed data. The outliers are shown as ‘ + ’ marker symbol.

Close modal

Table 3 provides the results of the similarity percentage between observed and downscaled AMP in the calibration and validation periods. The similarity percentage is estimated based on each climate station's KS test results of each temporal downscaling. The null hypothesis was systematically rejected for the MRC, except for the 1 h duration event accepted by the test (the Russell Station, duration = 1 h) with a very small p-value, i.e., close to rejection. HKP had 20 and 70% acceptance rates, while KNN and FO had better acceptance rates. KNN (30) and FO had acceptance rates of 100% except for one station, making them the best disaggregation techniques in this study.

Table 3

Percentage of similarity between observed and downscaled AMP in the calibration (Cal.) and validation (Val.) periods across all climate change models

StationRussell
Morrisburg
St. Albert
South Mountain
ModelCal. (%)Val. (%)Cal. (%)Val. (%)Cal. (%)Val. (%)Cal. (%)Val. (%)
HKP 63 50 65 70 63 60 20 60 
MRC 0 0 0 60 0 10 10 60 
KNN 5 63 88 94 100 60 55 80 100 
KNN 15 65 100 100 100 65 100 100 100 
KNN 30 75 100 100 100 75 100 100 100 
FO 100 100 94 100 100 100 100 100 
StationRussell
Morrisburg
St. Albert
South Mountain
ModelCal. (%)Val. (%)Cal. (%)Val. (%)Cal. (%)Val. (%)Cal. (%)Val. (%)
HKP 63 50 65 70 63 60 20 60 
MRC 0 0 0 60 0 10 10 60 
KNN 5 63 88 94 100 60 55 80 100 
KNN 15 65 100 100 100 65 100 100 100 
KNN 30 75 100 100 100 75 100 100 100 
FO 100 100 94 100 100 100 100 100 
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Comparison of the generated IDF curves of each temporal stochastic downscaling method

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The IDF curves generated using observations and disaggregated time series on the historical period are shown in Figure 5, representing one sample station, namely Russell Station. IDF curves for the same station on the long-term period are shown in Figure 6 (next section). The IDF curves have been generated using AMP generated using all temporal stochastic disaggregation models for each return period of 2-, 5-, 10-, 50-, and 100-year events in the historical period and three future climate periods (short-term: 2011–2040, medium-term: 2041–2070 and long-term: 2071–2100). Figure 5 shows that the IDF curves generated by KNN models and FO are generally closer to the IDF curves of observations than those generated using HKP and MRC. KNN (30) and FO are competing for first place. For two RCMs (CanESM2, CanRCM4, and EC-EARTH HIRHAM5), disaggregation using KNN (30) leads to the most realistic IDF curves. For the two others (EC-EARTH RCA4, GFDL RegCM4), FO seems to be the best.
Figure 5
IDF curves at the Russell Station for all disaggregation methods using different RCMs in the historical period for the return period of 2, 5, 50, and 100 years.
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IDF curves at the Russell Station for all disaggregation methods using different RCMs in the historical period for the return period of 2, 5, 50, and 100 years.

Figure 5
IDF curves at the Russell Station for all disaggregation methods using different RCMs in the historical period for the return period of 2, 5, 50, and 100 years.
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IDF curves at the Russell Station for all disaggregation methods using different RCMs in the historical period for the return period of 2, 5, 50, and 100 years.

Close modal
Figure 6
Grouped boxplot for KNN(30) and FO disaggregation methods and different RCMs in the future periods using (a) 1 h precipitation data in the short-term period; and (b) 1 h precipitation data in the medium-term period; and (c) 1 h precipitation data in the long-term period. (f) 24 h precipitation data in the short-term period, and (e) 24 h precipitation data in the medium-term period, and (d) 24 h precipitation data in the long-term period. The horizontal solid black lines represent the 25th and 75th percentiles while the horizontal solid red line represents the 50th percentile of the observed data. The dashed lines represent the whiskers (extended to the most extreme data points not considered outliers) of the observed data. The outliers are shown as ‘ + ’ marker symbol.
View largeDownload slide

Grouped boxplot for KNN(30) and FO disaggregation methods and different RCMs in the future periods using (a) 1 h precipitation data in the short-term period; and (b) 1 h precipitation data in the medium-term period; and (c) 1 h precipitation data in the long-term period. (f) 24 h precipitation data in the short-term period, and (e) 24 h precipitation data in the medium-term period, and (d) 24 h precipitation data in the long-term period. The horizontal solid black lines represent the 25th and 75th percentiles while the horizontal solid red line represents the 50th percentile of the observed data. The dashed lines represent the whiskers (extended to the most extreme data points not considered outliers) of the observed data. The outliers are shown as ‘ + ’ marker symbol.

Figure 6
Grouped boxplot for KNN(30) and FO disaggregation methods and different RCMs in the future periods using (a) 1 h precipitation data in the short-term period; and (b) 1 h precipitation data in the medium-term period; and (c) 1 h precipitation data in the long-term period. (f) 24 h precipitation data in the short-term period, and (e) 24 h precipitation data in the medium-term period, and (d) 24 h precipitation data in the long-term period. The horizontal solid black lines represent the 25th and 75th percentiles while the horizontal solid red line represents the 50th percentile of the observed data. The dashed lines represent the whiskers (extended to the most extreme data points not considered outliers) of the observed data. The outliers are shown as ‘ + ’ marker symbol.
View largeDownload slide

Grouped boxplot for KNN(30) and FO disaggregation methods and different RCMs in the future periods using (a) 1 h precipitation data in the short-term period; and (b) 1 h precipitation data in the medium-term period; and (c) 1 h precipitation data in the long-term period. (f) 24 h precipitation data in the short-term period, and (e) 24 h precipitation data in the medium-term period, and (d) 24 h precipitation data in the long-term period. The horizontal solid black lines represent the 25th and 75th percentiles while the horizontal solid red line represents the 50th percentile of the observed data. The dashed lines represent the whiskers (extended to the most extreme data points not considered outliers) of the observed data. The outliers are shown as ‘ + ’ marker symbol.

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The increasing intensity and frequency of extreme urban precipitation have been reported in the literature due to a changing environment. Short-duration extreme precipitation, in particular, is expected to become increasingly severe in many areas. As a result, the traditional frequency analysis method's underlying assumption for generating IDF curves has been called into question (Qu et al. 2020; Yan et al. 2020; Cook et al. 2020). Furthermore, the covariate-based nonstationary model can be used to generate future IDF (Yan et al. 2021).

Given that sub-daily, finer-scale precipitation time series are essential for stormwater control and management in urban water systems, an efficient temporal disaggregation method is crucial due to the lack of such data in some locations. The input of disaggregated or projected IDF to the urban water management model is one of the most crucial applications to analyse further the effects of climate change on local urban floods (Xiong et al. 2019). The sub-daily time series can be directly used in rainfall–runoff models or transformed into IDF curves using the rational method. Six different temporal disaggregation models (the KNN disaggregation with neighbour sizes of 5, 15, and 30, the HKP, the MRC, and the proposed FO model) were applied to the downscaled precipitation time series of four climate stations in the South Nation watershed. Various tests were used to assess the performance of the temporal disaggregation techniques. The main findings are recapitulated in Table 4.

Table 4

Summary of results

Temporal disaggregation methodPreservation of descriptive statistics in the generated hourly time seriesComparison of the generated annual maximums to observations using boxplotsPercentage of similarity of annual maximum time series using KS testComparison of IDF curves generated with disaggregated time series to IDF curves of observations
MRC Mean and standard deviation preserved, underestimate skewness and kurtosis Underestimates interquartile range and mean intensity at the 1 h time step Systematic rejection Significantly deviates from the IDF of observations 
HKP Mean and standard deviation preserved, underestimate skewness and kurtosis Underestimates interquartile range and overestimates mean intensity at the 24 h time step Acceptance rate ranging from 20 to 70% Significantly deviates from the IDF of observations 
FO Mean and standard deviation preserved, moderate underestimation of skewness. Kurtosis is generally underestimated (three out of four RCMs) Interquartile range overestimated; reasonable estimation of mean intensity compared to MRC and HKP. Similar to KNN The acceptance rate is generally 100%, except for one station where it is 94% Generally close to the IDF of observations 
KNN Mean and standard deviation preserved, reasonable estimation of skewness and kurtosis Interquartile range overestimated; reasonable estimation of mean intensity compared to MRC and HKP. Similar to FO Acceptance rate increase with neighbourhood. Generally, 100% for a neighbourhood of 30, except for two stations where it is 75% Generally close to the IDF of observations. Agreement increases with the size of the neighbourhood. 
Temporal disaggregation methodPreservation of descriptive statistics in the generated hourly time seriesComparison of the generated annual maximums to observations using boxplotsPercentage of similarity of annual maximum time series using KS testComparison of IDF curves generated with disaggregated time series to IDF curves of observations
MRC Mean and standard deviation preserved, underestimate skewness and kurtosis Underestimates interquartile range and mean intensity at the 1 h time step Systematic rejection Significantly deviates from the IDF of observations 
HKP Mean and standard deviation preserved, underestimate skewness and kurtosis Underestimates interquartile range and overestimates mean intensity at the 24 h time step Acceptance rate ranging from 20 to 70% Significantly deviates from the IDF of observations 
FO Mean and standard deviation preserved, moderate underestimation of skewness. Kurtosis is generally underestimated (three out of four RCMs) Interquartile range overestimated; reasonable estimation of mean intensity compared to MRC and HKP. Similar to KNN The acceptance rate is generally 100%, except for one station where it is 94% Generally close to the IDF of observations 
KNN Mean and standard deviation preserved, reasonable estimation of skewness and kurtosis Interquartile range overestimated; reasonable estimation of mean intensity compared to MRC and HKP. Similar to FO Acceptance rate increase with neighbourhood. Generally, 100% for a neighbourhood of 30, except for two stations where it is 75% Generally close to the IDF of observations. Agreement increases with the size of the neighbourhood. 
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All methods preserved the main descriptive statistics, such as the mean and standard deviation of the precipitation intensity. The three last comparison methods all seem to suggest that FO and KNN (30) are the most appropriate temporal disaggregation methods for the study area. Note that these results cannot be generalized and that different results could have been obtained if other climate stations or RCMs were used. It is also worth noting that the parameters of MRC and HKP were obtained by optimization, and its algorithm could have affected the final performance of the methods. The box plots of the projected precipitation intensities using the two best methods (FO and KNN (30)) are shown on Figure 6. Figure 7 shows the corresponding IDF curves. All figures suggest precipitation intensities will increase in the future in the study area, potentially making urban water infrastructure less safe as the impacts of global warming kick in.
Figure 7
IDF curves of Russell Station curves resulting from KNN 30 and FO disaggregation methods, long period (all climate models); for return periods T = 2, 5, 50, and 100 years.
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IDF curves of Russell Station curves resulting from KNN 30 and FO disaggregation methods, long period (all climate models); for return periods T = 2, 5, 50, and 100 years.

Figure 7
IDF curves of Russell Station curves resulting from KNN 30 and FO disaggregation methods, long period (all climate models); for return periods T = 2, 5, 50, and 100 years.
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IDF curves of Russell Station curves resulting from KNN 30 and FO disaggregation methods, long period (all climate models); for return periods T = 2, 5, 50, and 100 years.

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This study highlights the importance of carefully selecting the method used to generate future IDF curves, as two different methods can yield wildly different results. The resilience of urban infrastructure, public safety, and global economic prosperity in the future are at stake and the blind use of a particular method can lead to maladaptation and costly infrastructure failure.

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Given that precipitation projections are typically at a daily time scale, temporal disaggregation using techniques of variable complexity is often needed to evaluate the risk/performance of urban infrastructure in the future. Temporal disaggregation techniques have different levels of complexity and unequal performances, and the generated time series may not have the same statistical characteristics. In this paper, a simple steady-state stochastic disaggregation model that generates wet/dry day occurrence using a binomial distribution and precipitation intensity using an exponential distribution is proposed and compared to widely used temporal disaggregation methods: the MRC model, the HKP, and three versions of the KNN model using the nonparametric KS test. Daily time series of future precipitation are generated by temporally downscaling the outputs of four RCMs using the QM method. Results show that for the selected RCMs and climate stations, while all methods can preserve descriptive statistics in the hourly time series, they perform very differently when it comes to the annual maximum time series. The annual maximum time series generated using MRC systematically failed the KS test comparing them to observations, while HKP had low (20–70%) acceptance rates. FO and KNN (30) had acceptance rates close to 100% and led to the most realistic annual maximum time series at sub-daily and IDF curves, suggesting that complexity does not necessarily yield the best results. This research emphasizes the necessity of choosing the appropriate approach for creating future IDF curves with caution, as the outcomes can differ significantly depending on the method employed. Furthermore, it should be noted that the findings of this study cannot be extrapolated to other locations and those alternate outcomes would have likely occurred had different climate regimes or RCMs been utilized.

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Data cannot be made publicly available; readers should contact the corresponding author for details.

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The authors declare there is no conflict.

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