Design Rainfall Uncertainty For Hydrologic Engineering Applications

Published Date: 02 Nov 2017

Yeou-koung Tung1 and Chi-leung Wong2

1 Department of Civil & Environmental Engineering, HKUST, Clearwater Bay, Kowloon, Hong Kong, China. Tel - ++852 2358 8764, Fax - ++852 2358 2534, [email protected]

2 Drainage Services Department, the Government of the Hong Kong Special Administrative Region, China

Abstract: Hydrologic engineering designs and analyses often require the specification of design storm which involves rainfall amount, duration and hyetograph. In practice, the determination of design rainfall in hydrologic engineering applications involves frequency analysis of extreme rainfalls of different durations and establishment of rainfall hyetograph for the design event under consideration. Sampling errors exist in the magnitudes of estimated rainfall quantiles from frequency analysis, which will be transmitted in the process of determining the design rainfall hyetograph. This paper presents a practical methodological framework based on the bootstrap resampling scheme to assess uncertainty associated with the magnitude of estimated rainfall quantiles and the corresponding design hyetographs. For demonstration purpose, the commonly used IDF model with Chicago rainstorm profile is applied. Bootstrap resampling scheme is modified to handle unequal record period of annual maximum rainfall series of different durations and to account for their intrinsic correlations. By the rainfall IDF model, the design rainfall hyetograph is a function of IDF model coefficients. Due to the intrinsic correlation among rainfall quantiles of different durations, the IDF coefficients are found to be strongly related in a nonlinear fashion which should not be ignored in the establishment of the design hyetographs.

Key Words: Design rainfall, Rainfall IDF, Uncertainty analysis

Introduction

Rainstorm characteristics pertinent to drainage related studies consist of rainfall amount, duration, and temporal profile (hyetograph). At the design stage, the amount of design rainstorm is generally determined for a specified frequency and duration through a rainfall frequency analysis. On the other hand, the shape of design rainstorm hyetograph is often determined according to either a simple geometric shape or averaged pattern of representative historical rainstorm events. Examples of the former are Chicago rainfall hyetograph and its variation (Keifer and Chu 1957; Chen 1976) and triangular hyetograph (Yen & Chow 1980, 1983) whereas of the latter are those of Huff (1967) and Wu et al. (2006).

Typical design rainfall profile used in Hong Kong Territory for drainage system design and study follows the Chicago type along with the following rainfall intensity-duration-frequency (IDF) relationship (Drainage Services Department 2000)

(1)

in which = averaged rainfall intensity of design return period T and duration t; and aT, bT, and cT are model coefficients. Based on the IDF relationship, such as Eq. 1, along with a simple geometric assumption about the shape of rainfall hyetograph, the design rainfall hyetograph can be established. For example, the well-known Chicago storm (Kiefer and Chu 1957) with symmetric shape defines hyetograph, according to Eq. 1, as

(2)

in which is the instantaneous rainfall intensity.

In practical applications of engineering hydrology, rainfall IDF model coefficients, such as aT, bT, and cT in Eq. 1, are estimated by suitable curve fitting technique using estimated rainfall quantiles of different durations. However, due to the use of finite number of rainfall data in frequency analysis, sampling error induces uncertainty in the estimated rainfall quantiles and such uncertainty, in turn, is transmitted to the rainfall IDF coefficients and, eventually, to the rainfall amount and hyetograph. As design runoff characteristics (e.g., peak discharge and hydrograph) are very much dependent on those of the design rainstorm, uncertainties in design rainstorm will be transmitted to the corresponding runoff which will affect the design of hydrosystem infrastructure and its performance reliability.

Conventionally, the values of rainfall intensity quantiles of different return periods used in determining rainfall IDF model coefficients are estimated separately duration by duration without considering the intrinsic dependence of annual maximum rainfalls of different durations in the analysis. Such intrinsic dependence causes statistical correlation among rainfall IDF model coefficients, such as aT, bT, and cT in Eq. 1. The main objective of this paper is to present a framework to assess and quantify uncertainty features associated with design rainstorm hyetograph following a parametric IDF model, such as Eq. 1, by considering intrinsic correlation of annual maximum rainfall data of different durations.

2. Uncertainty of Rainfall Intensity/Depth Quantiles

For an annual maximum rainfall series of specified duration, uncertainty associated with the estimated quantiles of different frequencies is a function of underlying probability distribution, sample size, and method of estimating parameters. Ideally, uncertainty feature of an estimated quantile can be expressed in terms of its sampling distribution. For some distributions, sampling distribution corresponding to an estimated quantile can be analytically derived. For example, sampling distribution of estimated quantiles for normal random variables is non-central t-distribution (Stedinger et al. 1993). Alternatively, uncertainty associated with an estimated quantile can be expressed in terms of its standard error which can be estimated parametrically or non-parametrically. Expressions for asymptotic standard error of sample quantile under different distributions can be found elsewhere (Kite 1988; Rao and Ahmed 2000). However, these parametric or asymptotic approaches for assessing quantile uncertainty are univariate in nature because it is not easy to extend such approaches to a multivariate domain by considering correlation of rainfall quantiles of different durations.

In this study, a non-parametric approach, called bootstrap re-sampling technique, is applied to numerically estimate the statistical features associated with estimated rainfall depth quantiles of different durations. The bootstrap technique was first proposed by Efron (1979a, b) to deal with variance estimation of sample statistics. The technique intends to be a more general and versatile procedure for sampling distribution problems without having to rely heavily on the normality condition on which classical statistical inferen­ces are based. In fact, it is not uncommon to observe non-normal data in hydrosystem engineering problems. Although the bootstrap technique is computationally intensive – a price to pay to break away from the dependence of the normality theory – such concern is diminishing as the calculating power of the computers increases (Diaconis and Efron 1983). A summary on the variations of bootstrap techniques and other re-sampling procedures can be found elsewhere (Efron 1982; Efron and Tibshirani 1993). In hydrosystems engineering, bootstrap procedures have been applied to assess the uncertainty associated with the distributional parameters in flood frequency analysis (Tung and Mays 1981; Tung and Yen 2005).

To assess uncertainty features of estimated rainfall depth quantiles of various durations, one has to be aware of the intrinsic positive correlations among annual maximum rainfall data series of different durations, i.e., rainfall depth quantiles of different durations are not statistically independent, but correlated. Table 1 shows correlation coefficients of annual maximum rainfalls of different durations at Hong Kong Observatory (HKO). It can be seen that quite significant positive correlations exist between annual maximum rainfall depths of different durations. Furthermore, there is a tendency that the value of correlation between the two durations decreases as the relative time difference increases. The correlation that exists in the raw rainfall data of different durations expectedly will induce correlation among rainfall quantiles of the same frequency among different durations.

To preserve the intrinsic correlation in annual maximum rainfalls of different durations, re-sampling should be taken in a year-by-year manner with all durations included, rather than separately by duration. At HKO, rainfall data are further complicated the fact that annual maximum rainfall data of different durations do not have equal record length as shown below

15-sec: 1952-1990;

15-, 30-, 60- & 120-min: 1947-1951; 1952-1990;

4-, 6-, 8-, 12-, 18- & 24-hr: 1884-1939; 1947-1951; 1952-1990

Because the record length varies among different rainstorm durations, a straightforward re-sampling over the entire record period of 1884-1939, 1947-1990 by including those no-record period for some shorter durations will result in varying sample size for 15-sec ~ 120-min from one bootstrap sample to another. This implies that the degree of uncertainty associated with estimated rainfall quantiles for those durations will not be homogeneous among bootstrap sample repetitions. To maintain the same sample size for different durations in the bootstrap samples as those in the original sample, re-sampling of data can be proceeded in three time blocks as shown in Table 2.

Note that the stratified re-sampling can potentially cause reduction in variation within rainfall data series. This effect is assessed by taking the annual maximum rainfall data of 30-min (2 time blocks of 44 years) and 4-hr (3 time blocks of 100 years) and conducting bootstrap re-sampling with and without stratification. Furthermore, three re-sampling repetitions, namely, B=1000, 2000, and 3000, were tested to investigate the effect of bootstrapped sample size on the estimated uncertainty features of the mean, standard deviation, skew coefficient of the 30-min and 4-hr annual maximum rainfalls. The results for 4-hr rainfalls, for illustration, are summarized in Table 3. It was found that the effect of sampling variation reduction in estimated statistics due to stratification is practically nil and a sample size of B=2000 is adopted in the study.

In each bootstrap re-sampling repetition, annual maximum rainfall depth vectors of different lengths (according to rainfall durations of available record) from each time block are randomly drawn. That is, each bootstrap sample consists of 39 rainfall depth vectors with all durations from time block-C (1952-1990); 5 rainfall depth vectors with durations 15-min ~ 8-hr from time block-B (1947-1951); and 56 rainfall depth vectors with durations 4-hr~8-hr from time block-A (1884-1939). In doing so, the resulting bootstrapped annual maximum rainfall data will not only have identical sample size as the original data, but also maintain the same sample size from one bootstrap sample to another. Specifically, the stratified bootstrap procedure developed for preserving intrinsic correlations of annual maximum rainfall data of different durations with unequal record lengths is outlined below:

From HKO, collect available annual maximum rainfall depths of different durations, say, 15-sec, 15-min to 24-hr,

for i = 1, 2, …, n

in which di = vector containing rainfall depth data of different durations in year-i; di,t = annual maximum rainfall depth for year-i with duration t; n = record length in year.

To preserve intrinsic correlation in rainfall depths among different rainstorm durations and to maintain identical record length as the original data in bootstrap samples, rainfall depth vectors from the original data set are randomly drawn within each time block separately before they are combined to form a bootstrap sample, i.e., in which

for i = 1, 2, …, n.

Conduct Gumbel-based (Gumbel is the distribution model adopted in the Stormwater Drainage Manual (SDM) (Drainage Services Department 2000) frequency analysis on the bootstrap samples to determine T-yr rainfall depth quantiles for the various durations considered, i.e., .

Apply suitable optimization technique to determine optimal coefficients in the rainfall intensity-duration equation according to. Following the SDM of Hong Kong, two sets of IDF coefficients () are established in the study for two rainstorm duration ranges: (1) t 4-hr and (2) t ≥4-hr.

Repeating Step (2)-(3) B times (say, B=2000), one obtains vectors of for each of the two duration ranges and each vector contains B values of IDF model coefficients, i.e.,

;

;

.

From B sets of bootstrapped samples of statistical properties of aT, bT, and cT (e.g., their statistical moments, and correlations) can be assessed.

For illustration, Table 4 lists the uncertainty features of 50-yr rainfall depths of different durations derived from the 2000 samples through the stratified bootstrapping procedure following Gumbel-based rainfall frequency analysis procedure in Technical Note 86 (HKO 1990). The uncertainty features include mean, standard deviation (StDev), skewness coefficient (Skew), as well as lower and upper bounds at 95% probability levels. Table 4 shows that estimated 50-yr rainfall depths do not appear to have very high variability and are relatively symmetric because of small skewness coefficient (see histograms shown in Figures 1). In addition, Table 5 shows the correlation among estimated 50-yr rainfall depths of different durations. It clearly shows a positive correlation among 50-yr design rainfall depths and the magnitude of this correlation is dependent on the relative difference in time scale of rainfall depth of the two durations under consideration.

3. Uncertainty Features of Rainfall IDF Coefficients

Design rainfall hyetograph specified in the SDM follows Chicago storm profile along with rainfall IDF model shown in Eq. 1. Hence, the shape of design rainfall hyetograph is determined by the IDF coefficients. Table 6 lists IDF coefficients in Eq. 1 in the SDM for rainfall duration t ≤ 4-hr. Although there is no such information provided in the SDM for rainstorm duration t ≥ 4-hr, the IDF Eq. 1 is adopted herein for t ≥ 4-hr to facilitate generating the 8-hr rainfall hyetograph. The rainfall IDF coefficients in Eq. 1 are determined, separately, for the two duration segments (i.e., t ≤ 4-hr and t ≥ 4-hr) by the least squares criterion as follows:

(3)

in which t1,…, tm define the rainfall durations considered in fitting the IDF equation. The constraint is needed to ensure the value of (t+bT) being positive in the process of searching for the optimal IDF coefficients.

Due to the sampling error in the estimated rainfall quantile, the best-fit IDF model coefficients are also subject to uncertainty. The effect of such uncertainty on the design rainfall hyetograph is accounted for through the quantification of uncertainty of the IDF model coefficients. Hence, based on the 2000 bootstrap re-sampled rainfall quantiles of different durations the corresponding best-fit IDF coefficients are determined and their uncertainty features assessed. Table 6 summarizes the uncertainty features of IDF coefficients sets (a1,50, b1,50, c1,50) and (a2,50, b2,50, c2,50) for the 50-yr design rainfall event corresponding to the two duration segments t ≤ 4-hr and t ≥ 4-hr at HKO.

Table 6 indicates that the degree of uncertainty associated with coefficients a50 and b50 are relatively larger than that of c50. Between the two duration segments, the uncertainties of IDF coefficients for the longer duration segment are significant higher than those of the shorter duration segment. Furthermore, the values of "Skew" indicate that all IDF coefficients are positively skewed which are shown in Figures 2 and 3.

Due to inherent correlation among rainfall quantiles of different durations, it is expected that the best-fit IDF coefficients will be intrinsically correlated. For t ≤ 4-hr, Table 7(a) shows the existence of strong positive correlation among IDF coefficients of 50-yr rainfall event which is also evidenced in Figure 4. Correlation of IDF coefficients between the two duration segments (i.e., t ≤ 4-hr & t > 4-hr) is relatively weak. For t > 4-hr, correlation between b2,50 and c2,50 is very strong (0.979). However, correlations between a2,50 and other two coefficients are moderate but positive. Scatter plots shown in Figure 5 clearly indicate strong but definite nonlinear relationships between a2,50 and other two coefficients. As shown in the scatter plots of ranked data (Figure 6), clear linear relationships appear and the associated rank correlation values (see Table 7(b)) are expected to be significantly higher than those of Pearson correlation coefficients shown in Table 7(a).

4. Uncertainty of Design Rainfall Hyetograph

Following the guideline of SDM, uncertainty features of design rainfall hyetograph of a chosen frequency/duration can be assessed according to the adopted IDF equation. This can be done by utilizing bootstrapped random samples of IDF model coefficients in conjunction with Eq. 2. For illustration, random 50-yr/8-hr design rainfall hyetographs are generated. It is interesting to find that the correlations of 50-yr rainfall depths of various durations back-calculated by the 2000 bootstrapped IDF coefficients (see Table 8(a)) are relatively higher than those directly from the 2000 bootstrapped rainfall quantiles shown in Table 5, especially when the rainstorm durations are shorter than 2-hr or across the two duration segments. The discrepancies in correlations among 50-yr rainfall depths of different durations can arise from (1) non-linear transformation between rainfall depth and IDF coefficients through the adopted IDF equation in which smoothening of the data occurs and correlation among the data is increased at the same time, and (2) separately fitting IDF coefficients for the two duration segments in which a common rainfall intensity of 4-hr is applied. Also shown in Table 8(b) are the correlation coefficients of incremental rainfall depths between two different durations, t2>t1 , defined as

in which a, b, c represent generically IDF model coefficients in Eq. 1. As expected the correlations between incremental rainfalls exist due to sharing of common rainfall IDF model parameters. These correlation relationships are important features to consider in the generation of IDF model-based design rainfall hyetographs.

Before the design hyetograph corresponding to the rainfall IDF coefficients (a1,50, b1,50, c1,50) and (a2,50, b2,50, c2,50) is constructed, a check of consistency of rainfall depths and intensities around the time instants at t=120-min and t=360-min is made. The inconsistency arises from the occurrence of mismatch of 4-hr average rainfall intensity corresponding to the two sets of rainfall IDF equation coefficients associated with the two duration segments, that is,

The use of two different sets of IDF coefficients, for some generated 50-yr/8-hr rainfall scenarios, can result in a situation that in which is the 50-yr rainfall depth corresponding to t-hr duration estimated by the IDF coefficients from the k-th duration segment (k=1,2) and  is a small time increment. Then, the corresponding average rainfall intensity (or depth) hyetograph will have negative average rainfall intensity (or depth) ordinates around the time instants t=120-min from the hyetograph peak with rainstorm duration of t=240-min. Under this circumstance, corrections were made to reduce such inconsistency and discontinuity by the following two steps:

Step (1) - Set average 4-hr rainfall intensity equal to the mean of the average intensities computed by the two sets of IDF equation coefficients, i.e.,

(4)

Step (2) - Use polynomial interpolation for average intensities around time interval (80-min, 140-min) to smooth out the discontinuity due to fitting IDF equation for two duration segments.

(5)

where i() = average rainfall intensity at time instant . The four (J=4) reference time instants used are (, , , )=(80-min, 90-min, 120-min, 140-min) for interpolating the average intensities at = 100- and 110-min. Note that average intensities at i(80-min) and i(90-min) in Eq. 5 are obtained using IDF coefficients (a2,50, b2,50, c2,50) with durations t=320-min and 300-min, respectively; i(120-min) = average intensity computed by Eq. 4; and i(140-min) is the average intensity obtained using IDF coefficients (a1,50, b1,50, c1,50) with duration t=200-min.

For illustration, three sample hyetographs corresponding to low, medium, and high 50-yr/8-hr rainfall depths (i.e., 295mm, 328mm, and 410mm) are shown in Figure 7(a) and 10 sample rainfall mass curves of 50-yr/8-hr design rainstorm events are shown in Figure 7(b). For some design hyetographs generated, the above procedure cannot completely remove the discontinuity in the rainfall hyetograph around ±120-min from the center as shown in Figure 7(a). Since most stormwater drainage design studies are interested in the performance of the system responses under the maximum water surface elevation and peak discharge, the shape of design rainfall hyetograph in the tail parts of hyetograph will not expect to have significant effect on the maximum water level or peak discharge.

5. Summary and Conclusions

In the design of hydrosystem infrastructures, it generally requires the specification of design rainfall hyetograph for deriving the corresponding runoff hydrograph. This paper presents a framework for integrated uncertainty analysis of design rainfall hyetograph starting from frequency analysis of extreme rainfalls, derivation of rainfall IDF model, and generation of design rainfall hyetograph. The uniqueness of this framework is to apply stratified bootstrap re-sampling method for assessing uncertainty features of design rainstorm hyetograph by taking into account of the intrinsic correlation and unequal record length of annual maximum rainfall depths (or intensities) of varying durations, and those of coefficients in the adopted rainfall IDF equation. The framework is straightforward and can be implemented relatively easily. Knowing uncertainty features of design rainstorm hyetograph allows further assessment of runoff hydrograph uncertainty and reliability of the hydrosystem infrastructure under design.

Acknowledgments

Materials presented in this paper are parts of the study on "Uncertainty and Sensitivity Analysis of Mike-11 Model for Shenzhen River Basin." sponsored by the Drainage Services Department of Hong Kong Special Administrative Region.