Probabilistic assessment of hydrologic retention performance of green roof considering aleatory and epistemic uncertainties

Green roofs (GRs) are well known for source control of runoff quantity in sustainable urban stormwater management. By considering the inherent randomness of rainfall characteristics, this study derives the probability distribution of rainfall retention ratio R r and its statistical moments. The distribution function of R r can be used to establish a unique relationship between target retention ratio R r , T , achievable reliability AR , and substrate depth h for the aleatory-based probabilistic (AP) GR design. However, uncertainties of epistemic nature also exist in the AP GR model that makes AR uncertain. In the paper, the treatment of epistemic uncertainty in the AP GR model is presented and implemented for the uncertainty quanti ﬁ cation of AR . It is shown that design without considering epistemic uncertainties by the AP GR model yields about 50% con ﬁ dence of meeting R r , T . A procedure is presented to determine the design substrate depth having the stipulated con ﬁ dence to satisfy R r , T and target achievable reliability AR T . epistemic the analysis procedures via a numerical determine GR substrate depth having ﬁ dence to satisfy target retention ratio and reliability.


INTRODUCTION
The use of green roofs (GRs) is becoming popular in sustainable urban stormwater management. Contributions of GRs to urban runoff control and management are primarily attributed to their retention and detention abilities, which not only reduce runoff volume but also delay and attenuate runoff peak discharge ( A commonly used hydrologic indicator for GR's retention performance is the rainfall retention ratio R r : in which v, v rg are the rainfall amount and the corresponding runoff volume from a GR system, respectively. The complimentary performance indicator to retention ratio R r is the runoff production ratio R p : From the runoff control viewpoint, a GR system with a lower runoff production ratio or higher rainfall retention ratio is more desirable.

Basic GR hydrologic model
As the focus of performance herein is hydrologic retention, a simple lumped water balance model (Zhang & Guo ) for a GR system is adopted: where R c is the retention capacity of the GR system; S l is the interception by plants; S c is the capacity of the storage layer; θ fc is the field capacity of the substrate; θ i is the initial soil moisture content at the beginning of each rainstorm event; h is the depth of the substrate. The term, (θ fc À θ i )h, in Equation (3) is the available water holding capacity (WHC) in the substrate during a rainstorm event (Allen et al. ; Fassman & Simcock ). Assuming the substrate is maintained above the plant's wilting point, θ wp , the GR system reaches its maximum retention capacity R c,max when θ i ¼ θ wp as: The term (θ fc À θ wp )h is the maximum WHC of the substrate.
The initial soil moisture θ i at the beginning of a rainfall event depends on the length of antecedence dry period b, evapotranspiration (ET) rate E a , and evapotranspirable water content W i in the GR system at the end of the preceding rainfall event. The runoff volume from a GR system can be obtained as (Zhang & Guo ): From Equation (5), one is able to determine runoff volume v rg from which the GR retention ratio can be calculated by Equation (1).

Uncertainties in GR performance evaluation
Referring to Equation (5), hydrologic retention assessment of a GR system involves uncertainties from various sources, which can be generally categorized into two types: aleatory and epistemic uncertainties. The former is due to the inherent natural randomness of rainfall events such as rainfall depth, duration, inter-event dry period, and temporal pattern. On the other hand, epistemic uncertainties arise from knowledge insufficiency about the rainfall-runoff transformation process in GR systems (i.e., the model), and lack of complete characterization of model parameters associated with the soil-plantclimatic system. Therefore, the assessment of the performance of a GR system in reality cannot be certain.  (Tung ).

Outline of the study
The overall probabilistic analysis of GR retention performance presented herein consists of two stages (see Figure 1).

AP GR MODEL
Consider the aleatory uncertainty due to the natural randomness of rainfall amount V and inter-event dry period B. By taking these two rainfall properties to be statistically independent random variables, each, respectively, has an exponential distribution with the probability density functions (PDFs) defined as: Inter-event dry time (B): Figure 1 | Outline of the probabilistic analysis/design procedure for extensive GRs.
in which ζ ¼ 1=μ V and ψ ¼ 1=μ B are, respectively, exponential distribution parameters relating to the mean values of random rainfall depth μ V and inter-event dry period μ B . Verifications and justifications of exponential distribution models for V and B of individual storm event can be found in numerous analysis of rainfall data (e.g., Eagleson ; Adams et al. Based on Equation (5), along with exponential distributions for rainfall properties, Equations (6) and (7), Zhang & Guo () derived the cumulative distribution function (CDF) and PDF of the GR runoff volume V rg as functions of the rainfall distribution parameters ζ and ψ as: where F Vrg ( Á ) and f Vrg ( Á ) are, respectively, the CDF and PDF of the GR runoff volume.

Probability distribution of rainfall retention ratio
Based on the CDF and PDF of V rg given in Equations (8) and (9), this section presents the distribution functions of retention ratio R r as: where F Rr ( Á ), f Rr ( Á ) are the CDF and PDF of R r , respectively; η 0 is the dummy variable; and Pr(V rg ¼ 0) is the probability that the GR system produces zero runoff, which can be determined by Equation (8) as: A brief description of the mathematical derivation of the CDF and PDF of R r is presented in the Appendix in Supplementary Materials.

Statistical moments of retention ratio
To estimate the mean retention ratio E(R r ), a simple way is by the first-order linear approximation through which the mean values of rainfall amount and runoff volume are used as: in which E( Á ) is the statistical expectation operator. Note that the above approximation assumes that random rainfall amount and runoff volume are statistically independent.
Since R r is non-linearly related to rainfall amount V and runoff volume V rg , and the latter is also affected by the former, this indicates that rainfall amount and runoff volume are correlated. However, the first-order linear approximation, given by Equation (13), does not account for dependence between rainfall amount and runoff volume. By considering the second-order approximation, E(R r ) can be estimated by (Tung & Yen ): f Rr η 0 ð Þ ¼ ζ e À ðζRc;max=η 0 Þ η 02 e ÀðψW i =EaÞ which shows that the information about the variance of rainfall amount and its correlation with the runoff volume, represented by Cov(V rg , V), also play a role in estimating E(R r ). The covariance of V rg and V can be obtained from: where E(V) ¼ 1=ζ defined by Equation (6) and The mean runoff volume E(V rg ) in Equation (15) has been derived by Zhang & Guo () as: From the PDF of R r , Equation (11), the statistical moments of R r of any order m can be presented by: The analytical expression of E(R r ) can be derived as: in which E 1 (θ) is the exponential integral defined as (Abramowitz & Stegun ): Similarly, the analytical expression for the variance of R r can be obtained from Var(R r ) ¼ E[R 2 r ] À E 2 (R r ) in which, For the skew coefficient of R r , it can be derived from the

AP GR design
To quantify the probabilistic performance of a GR system by solely considering aleatory uncertainty, achievable reliability is utilized herein as a performance indicator: where AR(R r,T ; h) is the achievable reliability of meeting the target retention ratio, R r,T , conditioned on substrate depth, h. As shown in Equations (10) and (22), a unique functional relation can be established between the distributional properties of R r (i.e., distribution, statistical moments, and achievable reliability) and h because R c,max , as shown in Equation (4), is a function of h.
However, this unique relation for substrate depth h, target retention ratio R r,T , and achievable reliability AR(R r,T ; h), defined by Equation (22)

INCORPORATING EPISTEMIC UNCERTAINTY IN THE AP GR MODEL
To quantify the overall uncertainty of the GR model, the parameters subject to epistemic uncertainty, in addition to aleatory ones, that affect probabilistic features of GR performance (such as V rg , R r , and AR) should also be analyzed. In this study, the five model parameters subject to epistemic uncertainties are θ fc , θ wp , S l , E a , and initial solid moisture ratio,

Initial soil moisture
In the above AP GR model, Equations (8)-(11), the evapotranspirable water amount W i depends on the initial soil moisture θ i of substrate and other system parameters as: in which the value of θ i should be bounded in [θ wp , θ fc ]. In this study, θ i is represented by the initial soil moisture ratio C i ¼ (θ i =θ fc ) and treated as one of the model parameters.
Since θ i largely depends on the rainfall characteristics, substrate properties, and climatic factors, its value could be highly variable from one rainfall event to another. Physically, C i is bounded within [(θ wp =θ fc ), 1] of which the lower bound of the bounding interval depends on two soil moisture characteristics subject to uncertainty.

Method of UA
When a design is based on model outputs that where x km is the mth generated random variate of the kth random variable X k ; F À1 k ( Á ) is the inverse CDF of the kth random variable; s km is a random permutation of 1 to M for the kth random variable, X k ; and u km is a uniform random variate in [0, 1], i.e., u km ∼ U[0, 1]. For a problem involving N concerned model outputs (e.g., achievable reliability of varying target retention ratios and substrate depths) and K random model parameters, an input-output database can be generated by the LHS scheme as: where g( Á ) is the representation of the model; By the AV technique, the statistical features (e.g., moments) of concerned output, Y, can be estimated by computing the arithmetic average of its two unbiased estimators as: Suppose that AR is a random variable having a CDF F AR (ϑ AR ), defined by its distributional parameters ϑ AR .
Because AR to meet a specified R r,T is uncertain, from design viewpoint, one would wish to determine the substrate depth h, such that the GR system can meet the target reliability AR T with a specified confidence level ω. Referring to Figure 2, the GR system with h having confidence ω of meeting the desired R r,T and AR T is the exceedance probability as shown by the shaded areas. The two solid curves in Figure 2 each represents confidence level, ω 1 and ω 2 (ω 2 > ω 1 ), respectively. As can be seen that, to maintain the same R r,T and AR T with a higher confidence ω, one has to increase substrate depth h. In the context of design, considering aleatory and epistemic uncertainty simultaneously, the design substrate depth h dsgn can be determined by solving the following equation: where ω dsgn is the desired confidence level; F AR(R r,T ,h dsgn ) {AR T jϑ AR } is the CDF of random AR(R r,T , h dsgn ).
Since the value of AR is bounded between 0 and 1, it is reasonable to postulate that the AR follows a standard Beta distribution, i.e., AR ∼ f Beta (μ AR , σ AR ), with its mean and standard deviation that are related to R r,T and h. Namely, μ AR (R t,T , h) and σ AR (R t,T , h) can be explicitly expressed in terms of R r,T and h. Under the condition of standard Beta distribution for AR, the design substrate depth h dsgn having design confidence ω dsgn of meeting R r,T and AR T can be determined by solving: in which, F AR(R r,T , h dsgn ) {AR T jϑ AR } is the CDF of random AR.

ILLUSTRATION
To illustrate the probabilistic performance of an extensive GR system considering aleatory and epistemic uncertainty, data used in Zhang & Guo () are adopted herein. The two rainfall properties at Metro International Airport  To quantify model parameter uncertainty that reflects the on-site condition, it is desirable to analyze the variation of local climatic variables and conduct tests on limited in situ GR substrate samples. It should also be noted that the statistical features of retention ratio (e.g., mean value and achievable reliability) presented in this example should be referenced to the period of analysis of rainfall record (i.e.,

April-October in this example).
Behavior of the AP GR model  Table 1 are used.

Distribution of retention ratio
The PDF and CDF (Equations (10) and (11)) of R r under W i ¼ R c,max are shown in Figure 3, respectively. These two figures clearly show that R r is a random variable with discontinuity at R r ¼ 1 where the GR produces no runoff. As the substrate depth increases, the probability of producing zero runoff at R r ¼ 1 gets higher and the curve corresponding to 0 R r < 1 drops lower as shown in Figure 3.
For a fixed substrate depth, h ¼ 100 mm, Figure 4 shows a comparison of the exceedance probability (1-CDF) of R r under the conservative and optimistic conditions of evapotranspirable water. With regard to rainwater retention performance, W i ¼ R c,max represents the conservative scenario, whereby more water in the substrate is available for ET during the dry period. Under such circumstance, the available WHC in the substrate to accommodate the incoming rainfall would be less, so is the corresponding R r . Hence, the likelihood that the GR system to have R r exceeding a stipulated value would be lower than that of under the optimistic condition of W i ¼ 0. increases with substrate depth because available WHC of the substrate becomes larger. Also, the rate of increase in mean R r is decreasing with substrate depth. Interception loss, S l (mm) Initial soil moisture ratio, C i 0.11 ± 25% (Uniform) 0.232 ± 15% (Uniform) 0.116 ± 20% (Uniform) 2 ± 30% (Uniform) 0.5-1 (Uniform)

Statistical moments of retention ratio
Note: ±% value defines the range of variation about the mean value.
As for the standard deviation of R r (the two red lines), Figure 5 reveals that the variability of R r is lower when the value of W i is smaller. The variability of retention ratio with substrate depth is fairly stable under W i ¼ R c,max , whereas it decreases with substrate depth under W i ¼ 0.
This can be explained that with a low value of W i , the   corresponding initial soil moisture content at the beginning of the incoming rainfall event would be low and the available WHC of the substrate would be high to accommodate the rainfall event. The net effect would result in a higher mean and lower standard deviation of R r .
As presented in the 'Statistical moments of retention ratio' section, the mean R r can be estimated by several ways. Figure 6 shows a comparison of estimated mean R r by the first-order method, Equation (13), the second-order method, Equation (14), and the analytical solution, Equation (19). In comparison with the analytical solution, Figure 6 shows that the first-order method significantly underestimate the mean R r , whereas the second-order method, as expected, provides significant improvement with somewhat over-estimation as the substrate depth increases. This is because that the correlation between the rainfall amount and runoff volume plays an important role in estimating the mean R r .

GR design using the AP model
Based on Equation (10), the CDF of R r can be used to define a unique relationship for AR, substrate depth, h, and R r,T as shown in Figure 7, under the conservative condition W i ¼ R c,max . Clearly, for a given h, the AR of a GR system decreases with increase in R r,T . For a specific R r,T , the performance reliability increases with h. One can also see that by increasing h, the higher R r,T would be expected while maintaining AR at the same.
Since W i ¼ R c,max corresponds to a conservative condition with possible minimum WHC in the substrate, Figure 7 defines the lower bound of AR À R r,T À h relationship for a given h. The upper AR À R r,T À h curve can be obtained under the optimistic condition of W i ¼ 0. Under the normal condition, the reliability would be somewhere between the two curves.

Uncertainty quantification of achievable reliability considering epistemic uncertainties
According to LHS samples of size 50, 100, 200, 300, 500, and 1,000 for the five GR model parameters in Equation (22), it was found that the estimated values of the first three statistical moments of AR did not satisfactorily converge, even under the sample size of 1,000. Hence, the AV technique for variance reduction is incorporated in the LHS scheme to enhance a stable and accurate estimation.
As it turns out that the estimated statistical moments of AR from the AP GR model had a quick and satisfactory convergence with sample size of only 100.
To estimate the uncertainty features of AR(R r,T , h), random variates of the five parameters having epistemic uncertainty are generated by the LHS scheme jointly with the AV technique as: From u 00 : AR(R r,T , h) 00 m ¼ g(E 00 a,m , θ 00 fc,m , θ 00 wp,m , S 00 l,m , C 00 i,m ), m ¼ 1, 2, . . . , M in which u 00 ¼ 1 À u 0 . The LHS/AV-based statistical properties of AR from the AP GR model can be computed  according to Equation (26). For example, the raw moments of any order of AR(R r,T , h) by considering the epistemic uncertainty can be estimated by: where s is the order statistical moment; and Then, the mean of LHS/AV-based estimator of AR(R r,T , h) can be estimated by Equation (31) with s ¼ 1 and the variance with s ¼ 2 as: Other than the statistical moments, it is also important to assess its probability distribution of AR(R r,T , h) for reliability-based analysis and design of GR systems.  of R r,T and h were tested for their goodness-of-fit to the standard Beta distribution. The standard Beta distribution was considered for being theoretically bounded in [0, 1] and versatile in shape. For illustration, Figure 10 shows the quantile-

Reliability-based GR design considering both aleatory and epistemic uncertainties
The reliability-based GR design requires the establishment of functional relationships between the statistical properties of standard Beta distribution and the two design parameters (R r,T and h). The two parameters of standard Beta distribution are related to the first two moments as: in which α AR , β AR are the parameters of standard Beta distribution; μ AR , σ AR are the mean and standard deviation of AR(R r,T , h), respectively. Based on 100 LHS/AV-generated samples, the empirical functional relations between the mean μ AR and standard deviation σ AR of AR(R r,T , h) with R r,T and h are established, respectively, through regression analysis as: μ AR (R t,T , h) ¼ 0:7413 À 0:9883 R t,T þ 0:0325 h þ 0:3871 R 2 r,T À 0:00079 h 2 þ 0:001659R t,T h (37) σ AR (R t,T , h) ¼ 0:02573 þ 0:004612 h À 0:03304R 2 r,T À 0:000223h 2 þ 0:003187R t,Th In Equations (37) and (38), range of data in R r,T and h used are 0.4-0.9 and 5-15 cm, respectively. The coefficient of determination corresponding to the above two empirical equations are both 0.999. Utilizing Equations (37) and (38), the mean and standard deviation of AR(R r,T , h) for a pair of (R r,T , h) can be computed which, in turn, can be used to determine the corresponding Beta distribution parameters defined by Equations (35) and (36). Then, the design substrate depth, h dsgn , corresponding to R r,T , AR T , and design confidence, ω dsgn , can be obtained by solving F AR(Rr,T ,h) {AR T j α(ARjh dsgn , R r,T ), β(ARjh dsgn , R r,T )} where α( Á ) and β( Á ) are the parameters of standard Beta distribution describing random achievable reliability. The design confidence ω dsgn is the probability that the random achievable reliability exceeds the stipulated target AR T . Figure 12 shows an example design diagram obtained from solving Equation (39) that defines the relationship between h dsgn with confidence ω dsgn to meet target R r,T and AR T . It is clear that, for the fixed value of R r,T , h dsgn increases with ω dsgn and AR T .

CONCLUSION AND DISCUSSION
By considering the inherent randomness of rainfall amount of individual rainstorm event and inter-event dry period, this study extends the work of Zhang & Guo () to derive the PDFs of the GR retention ratio R r , based on which the analytical expression for the exact mean and variance of the retention ratio are derived. The analytical expression allows direct calculations of relevant statistical characteristics of R r to rapidly assess the hydrological retention performance of a GR system without intensive simulation.
This study also evaluates the accuracy of estimating the mean R r by two approximations with respect to the exact solution. The simplistic first-order approximation shows under-estimation of the true mean R r . The second-order approximation significantly improves the first-order estimation because the variance of the rainfall amount and its correlation with the runoff volume play an important role. Physically, runoff volume from a GR system is caused by rainfall, and therefore, the two random quantities should be positively correlated. This paper shows that such a correlation indeed is quite strong.
From the distribution function of random R r , the relationship between target retention ratio R r,T , achievable reliability AR, and substrate depth h for the AP GR model can be established. For illustration, the study shows a unique relation between the design substrate depth h dsgn and AR. Nonetheless, there exists non-rain factors describing soil, plant, and climatic properties that affect the rainfall-runoff transformation process in the GR system.
These non-rain factors are model parameters subject to the uncertainty of epistemic nature induced by knowledge deficiency. When epistemic uncertainty is taken into consideration, the AR obtained from the AP model is no longer certain. Thus, in order to have a comprehensive reliability assessment of the GR design, epistemic uncertainties are further incorporated in the AP model.
This study presents a systematic framework to assess the influence of the epistemic uncertainty on the performance of a GR system. Due to highly nonlinearity of the model parameter-output relations, the AV technique was jointly implemented with the LHS scheme in UA to obtain fast, stable, and accurate estimations of the statistical features of AR. Furthermore, the standard Beta distribution was found to fit the distribution of AR satisfactorily. One can easily construct the quantile curves and confidence intervals of AR according to its estimated moments from the LHS/AV procedure.
This study shows that the design of a GR system by considering only aleatory uncertainty due to the natural randomness of rainfall characteristics would roughly have 50% confidence to achieve the desired R r . To determine a substrate depth for achieving the target reliability (AR T ) with 50% confidence or higher, when epistemic uncertainties are considered, one would have to use a thicker substrate depth h. The incremental depth beyond the nominal h (under aleatory uncertainty only) depends on the degree of epistemic uncertainty and the design confidence level (ω dsgn ). This incremental depth can be viewed as the safety margin to account for the presence of epistemic uncertainties. The proposed analysis framework leads to a more comprehensive and complete analysis/design of a GR system.