1. Introduction
While we have developed an impressive ability to describe climate and hydrologic systems from both dynamic and thermodynamic perspectives, for practical purposes, we do not yet have the ability to analyze and describe the upper bounds of the intensity of many types of extremes based on physical reasoning. In the case of extreme precipitation, current knowledge of storm mechanisms remains insufficient to allow a precise evaluation of limiting values and very rare extreme precipitation. Nevertheless, estimates of such rare extremes are needed for engineering practice, for example, in dam spillway design. Probabilistic approaches using statistical frequency analysis offer a plausible alternative for estimating extremes for a given return period. These approaches involve the fitting of probability distribution models to recorded storm precipitation amounts and extrapolating the tails of these models to very low exceedance probabilities. These approaches have been criticized (Klemeš 1986, 1987, 2000), for example, on the basis that very long return periods (e.g., for 1000 or 10 000 years) can only be estimated from available 50- or 100-yr observational records with very high uncertainty, which may make these estimates unsuitable for engineering applications. Engineers therefore seek additional information to overcome the limitations of deterministic physical reasoning and probabilistic approaches. Hence, a rational concept called probable maximum precipitation (PMP) is commonly used to estimate a possible magnitude of an extreme having a very high return period that is judged to have a negligible risk of exceedance.
The World Meteorological Organization (WMO; WMO 1986), defines PMP as “the greatest depth of precipitation for a given duration meteorologically possible for a design watershed or a given storm area at a particular location at a particular time of year, with no allowance made for long-term climatic trends.” PMP is commonly used for estimating the probable maximum flood (PMF), which is defined as the largest flood that could occur at a given hydrological basin and is a key parameter used for the design of a given project with high safety requirements. Based on our hydrologic and meteorological knowledge, extremes are the results of many components interacting in a complex way. Typically, PMP is computed as a combination of the maximum of component values, where the rationale is that this combination is unlikely to be exceeded.
Theoretically, PMP is an unknown upper limit for precipitation, which could be very high relative to the largest extreme that might be experienced over a fixed period of time, or might even be unbounded, whereas operationally, PMP is a rational engineering solution (meaning not purely based on scientific knowledge) to provide a possible magnitude of extreme precipitation values that can be used by engineers as a practical upper limit where scientific knowledge does not provide the desired guidance. Hence, whether the theoretical upper limit exists or not, an operational PMP can be obtained by engineers to provide guidance for design decisions.
The operational PMP estimate must be clearly recognized for what it is and not be confused with what it is not, namely, a physical upper limit. Furthermore, WMO (2009) hinted that one must distinguish between the “theoretical PMP” and the “operational PMP” (Salas et al. 2014). In fact, when the operational and theoretical PMP concepts are confused, and when a rational concept is recognized as science, inconsistencies and methodological gaps arise, reducing its credibility and usefulness (Klemeš 1993). Unfortunately, this confusion has led several statistical hydrologists to consider the operational PMP concept to be one of the biggest failures in hydrology (Yevjevich 1968; Papalexiou and Koutsoyiannis 2006), despite its continued heavy use. In reality, this is not a failure; when current scientific knowledge does not allow engineers to make a decision, a rational approach coupled with careful judgement must be used and can be appropriate, but should be recognized as such and not as science (Klemeš 1993). Hence, one must distinguish between the theoretical and rational operational PMP concepts and recognize that the latter has an origin that is reasonable, but not necessarily scientific. Hereafter, to avoid any confusion, we will use the term PMP to refer to the “operational PMP.”
WMO (2009) describes a variety of methods to derive PMP estimates depending on the basin characteristics (size, location, and topographies), the amount and the quality of available data, and storm types producing extreme precipitation. Most of these methods involve comprehensive meteorological analysis, and only one is based on statistics, as was proposed by Hershfield (1961). As mentioned in WMO (2009), PMP calculations depend on data and should always be presented as approximations, since the value depends fundamentally on the amount and quality of the data available and the depth of analysis. Given the considerable uncertainties that may influence PMP estimates, providing a range of PMP values by evaluating the uncertainty of values, that is, the range of values that might be possible under equivalent conditions, rather than relying only on single point estimates is necessary and more suitable, but is often ignored in the literature. Only a few studies have dealt with this subject. For instance, Salas et al. (2014) provides an uncertainty estimate using a statistical method, where the expected value and variance of the calculated PMP value are obtained by assuming that the annual maximum of 24-h precipitation is Gumbel distributed. The expected value and variance estimates are used in combination with Chebyshev’s inequality to obtain probability bounds and risks. Salas and Salas (2016) extended this approach assuming a log-Gumbel rather than Gumbel distribution. Micovic et al. (2015) used an alternative approach based on an uncertainty analysis in which judgment is used to integrate the uncertainty of the different factors involved in computing PMP and showed that PMP should always be presented as a range of values to characterize the impacts of the significant uncertainties that are involved in the calculation.
While a variety of PMP calculation methods are described by WMO (2009), the so-called moisture maximization method remains the most representative and is commonly used by engineers (Rakhecha and Clark 1999). A PMP value obtained with the moisture maximization method may be considered a worst-case scenario estimate in which extreme precipitation efficiency (PE) and extreme precipitable water (PW) occur simultaneously. PW is defined as the depth of water that would be produced at a given location if all the water in the atmospheric column above that location was precipitated as rain, whereas PE is defined as the ratio of actual precipitation amount to the actual PW. Because extreme precipitation almost always involves moisture transport from elsewhere, PE is usually larger than one, with the result that PMP is usually larger than PW. While this is a rational operational approach for obtaining a PMP value, the PMP value that is obtained should not be interpreted as the physical upper limit. This is because the calculation involves two maximization operations, once for PW and again for PE, with a PMP value being obtained as the product of the two maxima. A physical interpretation would require an assumption that PMP events simultaneously have maximum moisture availability and maximum precipitation efficiency (Chen and Bradley 2006). This strong assumption may lead to overestimation of a likely upper bound if the PW and PE extremes are not fully dependent, even if maximization provides plausible, physically reasonable limits for PW and PE individually.
The fact that PMP calculation via moisture maximization involves the maxima of observed time series of PE and PW suggests the use of the extreme value theory, a well-developed statistical discipline (Coles et al. 2001). The probabilistic description of extremes through extreme value theory is generally developed through either the block maxima approach or the peaks-over-threshold approach. The former leads to the use of the generalized extreme value (GEV) distribution to describe the probability distribution of the intensity of block maxima, whereas the latter leads to the use of the generalized Pareto distribution to model excesses over a high threshold. Further details about the extreme value theory can be found in Coles et al. (2001). We consider the block maximum approach in this study since the maximum values of PE and PW over a defined period record can be considered to be the maxima of a series of annual maxima.
Accounting for the dependence structure of exceptional PE and PW events is of practical importance, as noted, and can be accomplished via bivariate extreme value analysis. In particular, a copula function can be used to extend univariate extreme value analysis to the bivariate case. Copula functions provide a way to describe the dependence structure independently of the marginal distributions and thus can use different marginal distributions at the same time without any transformations. The application of copulas in hydrology and climatology has grown rapidly during the past decade. Introductions to copula theory are provided in Joe (1997) and Nelsen (2007a), and a detailed review of the development and applications of copulas in hydrology, including frequency analysis, simulation, and geostatistical interpolation, can be found in Salvadori et al. (2007). In recent years, copula functions have been widely used to describe the dependence structure of climate variables and extremes (e.g., AghaKouchak et al. 2010a; Ben Alaya et al. 2014, 2016; Guerfi et al. 2015; Mao et al. 2015). In this study, an extreme value copula (Salvadori et al. 2007) is used to extend the univariate block maxima approach to the bivariate case.
The aim of this study is to propose a probabilistic framework for PMP estimation using the moisture maximization approach. The proposed approach takes advantage of probabilistic bivariate extreme value analysis to address the limitations of operational PMP estimates obtained via moisture maximization by 1) enabling assessment of the sensitivity of the PMP value to one particular observation, the maximum of the entire sample; 2) allowing an evaluation of the uncertainty and thus providing a range of PMP values; and 3) providing clarification of the impact of the assumption that a PMP event occurs under conditions of maximum moisture availability. The proposed approach is illustrated by using output from the Canadian Centre for Climate Modelling and Analysis Regional Climate Model (CanRCM4) to estimate PMP over North America. The remainder of the paper is organized as follows: the datasets and the proposed method are introduced in section 2. Results and discussions are presented in section 3, and conclusions are given in section 4.
2. Materials and methods
a. Data
Physically based numerical atmospheric models developed over the past three decades play an important role in climate research. In the case of PMP estimation, numerical models are useful since they are able to simulate three-dimensional data representative of long periods. Several previous studies have used numerical climate models to study some aspects of PMP. Abbs (1999) employed a numerical model to evaluate some of the assumptions used in PMP estimation. Ohara et al. (2011) used a numerical model to estimate PMP for the American River watershed in California. Beauchamp et al. (2013) evaluated a warm season PMP estimate based on moisture maximization under recent past climate conditions and then applied it under a future projected climate using output from the Canadian Regional Climate Model (CRCM). In the same way, Rousseau et al. (2014) used CRCM output to develop a methodology to estimate PMP based on moisture maximization accounting for changing climate conditions for the southern region of the province of Quebec, Canada.
Output from the CanRCM4 regional climate model is used in this study over the period 1951–2000 and covering the North American region with 0.44° spatial horizontal resolution (155 × 130 grid points). A more detailed description of CanRCM4 is provided in Scinocca et al. (2016), von Salzen et al. (2013), and Diaconescu et al. (2015). CanRCM4 is a participant in the Coordinated Regional Climate Downscaling Experiment (CORDEX) framework (Giorgi et al. 2009) and is developed by the Canadian Centre for Climate Modelling and Analysis (CCCma) to make quantitative projections of future long-term climate change. The CanRCM4 simulation used in this study is driven by the Second Generation Canadian Earth System Model (CanESM2) and accounts for historical changes in anthropogenic and natural external forcing. From the numerous variables, available in the CCCma archives, we used total precipitation and precipitable water (vertically integrated water vapor through the atmospheric column), both at a 6-hourly temporal resolution.
For this study, we assume that the properties of the PW and PE extremes do not vary substantially over time; that is, stationarity is assumed. Nevertheless, it is recognized that climate change will alter climatic extremes and that the stationarity assumption will be increasingly difficult to justify as the climate continues to warm. A subsequent paper will therefore describe an extension of the proposed methodology to nonstationary situations and will apply the extension to projections of future nonstationary climates. The application of the method in this paper will consider a period, 1950–2000, in which there is still only relatively weak evidence of nonstationarity in the behavior of precipitation extremes, and thus for simplicity, this first application will continue to make the assumption of stationarity. In contrast to the method that will be introduced below, traditional PMP estimation methods, which interpret maxima of finite records as upper bounds, are not suitable for circumstances in which those bounds might change.
b. Methodology
1) Operational PMP calculation using moisture maximization
The operational moisture maximization approach can be considered a rational approach, even if the precise likelihood of exceedance of the calculated PMP value is not known, because it would be rational to consider it unlikely that 
2) Probabilistic characterization of PMP
A suitable probabilistic framework for the estimation of probable maximum precipitation can be constructed to answer these questions and 1) enable an assessment of the sensitivity of the PMP value to one particular observation, the maximum of the entire sample; 2) allow an evaluation of the uncertainty and thus provide a range of PMP values; and 3) provide clarification of the impact of the assumption that a PMP event occurs under conditions of maximum moisture availability. In this section, we describe how a bivariate extreme value analysis can be used to achieve these ends.
The first objective is addressed by applying a block maximum approach separately to PE and PW using the GEV distribution. The series maxima that are used in the traditional approach can both be considered as the mth-order statistics of 
The approach is based on treating the occurrence of extreme precipitation as a compound event, where the distribution of its extremes can be synthesized using the bivariate extreme value distribution of PW and PE. The block maximum approach, which is presented in the online supplemental material (Coles et al. 2001; Embrechts et al. 2013), is used to derive the individual distributions of PW and PE, whereas the extreme value copula function, which is presented in section 2b(2)(i), is used to describe their joint probabilistic behavior. These tools are subsequently applied as described in section 2b(2)(ii) to obtain a range of PMP values that correspond to a specified likelihood. All parameters of the bivariate model are estimated using the maximum likelihood method.
(i) Extreme value copula
There is no finite dimensional parametric family for the Pickands dependence function (Tawn 1988) to guide the choice of dependence function. Nevertheless, the existing literature presents various parametric and nonparametric dependence function estimators [see section 9.3 of Beirlant et al. (2006) for a review]. A classical nonparametric estimator of 
A potential difficulty with the Pickands estimator is that it may not be convex. Although a number of approaches have been proposed to ensure convexity, such methods often result in the joint distribution being singular and nondifferentiable. Since, in most environmental applications, singular distributions do not occur, parametric differentiable models are more suitable. In addition, for simulation purposes, most general sampling algorithms require the first and second derivatives of the function 
(ii) PMP characterization using bivariate extreme values analysis
First, similar to the traditional operational PMP estimation described above, annual PE maxima are pooled using 
Once the univariate GEV distributions for PE and PW have been separately fitted and evaluated for each season, the extreme value copula can be used to connect the two GEV models to obtain an estimated joint distribution for seasonal extremes of PE and PW. The fitted bivariate distribution can then be used to estimate quantiles and return periods in the bivariate setting, including for levels beyond those that have been observed.
Our final objective is to derive information about the product of PE and PW, which implies obtaining the distribution of the product extremes of PE and PW for each season. Our approach estimates PMP by considering the greatest value of these products over a chosen period of time (
Draw four samples of m pairs of PE and PW seasonal maxima (one sample of m pairs for each season) from available data.
Fit the four bivariate GEV models to each of the four samples drawn in step 1, and hence one bivariate GEV model is available for each season.
Draw a sample of m independent pairs of PE and PW extremes from each of the four bivariate distributions fitted in step 2.
Compute the products of the simulated
Keep the largest of the products computed in step 4; this value represents one realization of PMP estimated via moisture maximization based on an m-year sample. The use of four bivariate GEV models corresponding to each season ensures that the simulated PMP value is obtained using PW and PE values occurring within the same season, as it is commonly recommended in practice.
Repeat steps 1–5 to simulate the expected variation in PMP estimates that would occur in repeated analyses of independent m-year periods under the same hydroclimatic conditions.
In addition to providing a probabilistic setting for the interpretation of PMP, the proposed approach offers more flexibility to designers than the operational moisture maximization calculation. Indeed, the algorithm above can be easily adapted to obtain the probability distribution of any order statistic, even those for periods longer than that represented by the available observed record. The mth-order statistics are only used here for comparison purposes with the operational approach.
3. Results and discussion
a. Bivariate extreme value distribution of PE and PW
Univariate GEV models are first fitted separately to the annual maxima of PE and PW at each CanRCM4 grid box and for each season. The Kolmogorov–Smirnov goodness-of-fit test is used at each grid box to assess the differences between the empirical and the fitted GEV cumulative distributions. These tests, which were performed at the 5% significance level, indicate that a GEV distribution reasonably approximates the distribution of annual extremes for both PE and PW over all CanRCM4 grid boxes and for each season.
The shape parameter of the GEV distribution 
Estimated PW and PE GEV model parameters for the winter season (DJF).
Citation: Journal of Hydrometeorology 19, 4; 10.1175/JHM-D-17-0110.1
Extreme precipitation will be bounded if both of its two components, PW and PE, are bounded. An estimate of the bound above for a given component in these circumstances is 
We next proceed to estimating the extreme value copula for each CanRCM4 grid box using the procedure described in section 2. Traditional PMP estimates via moisture maximization are based on the assumption that extreme PE and PW occur simultaneously. In terms of extreme value copulas, this assumption would suggest that the comonotone copula should be the appropriate extreme value copula. Figure 2 shows the Pickands dependence function estimated using the two nonparametric estimators as well as the Gumbel copula for all grid points over North America during the winter (similar figures for each of the other seasons are presented in the supplemental material). Thin gray lines indicate dependence function estimates at an individual grid point. For each estimator, the red curves represent the means and selected quantiles of all gridbox dependence functions over North America, while the solid black lines correspond to the comonotone copula. The profile of the Pickands function appears to be symmetric and very similar for the three estimators. As we can see, the assumption that a PMP event has maximum moisture availability does not appear to be satisfied in practice in the climate simulated by CanRCM4. Indeed, examination of the Pickands dependence functions over North America demonstrates a significant departure of the dependence structure from that of the total dependence (comonotone) copula.
Pickands dependence function of PW and PE over North America during DJF.
Citation: Journal of Hydrometeorology 19, 4; 10.1175/JHM-D-17-0110.1
The dependence (UTD) coefficient is calculated for each CanRCM4 grid box and the corresponding maps obtained using the three estimators 
UTD coefficient of PE and PW estimated over North America during DJF.
Citation: Journal of Hydrometeorology 19, 4; 10.1175/JHM-D-17-0110.1
UTD coefficient estimates for all CanRCM4 grid points over North America during DJF using the Gumbel copula vs the estimates using (left) the Pickands estimator and (right) the CFG estimator.
Citation: Journal of Hydrometeorology 19, 4; 10.1175/JHM-D-17-0110.1
b. Probable maximum precipitation results
It is desirable to quantify how taking into account the dependence structure between extreme PE and PW could affect PMP estimates. Thus, the algorithm proposed in section 2 was used to sample 1000 values in the PMP distribution at each CanRCM4 grid point using the extreme value Gumbel copula and the comonotone copula. Figure 5 shows maps of the mean PMP values for the CanRCM4 climate obtained using the two copulas and that obtained using the traditional single-value PMP estimate. As expected, our estimates based on the assumption that extreme PW occurs simultaneously with extreme PE will typically lead to larger PMP estimates (Figs. 5a,b) while taking a realistic dependence structure into account leads to somewhat smaller PMP estimates (Fig. 5c). Comparing between Figs. 5a and 5b and Fig. 5c suggests that the assumption of total dependence overestimates the mean PMP value for CanRCM4-simulated 6-hourly precipitation by an average of about 15% over North America. Since PMP should not be underestimated for safety reasons nor overestimated for economic reasons, the proposed approach takes advantage of a more realistic representation of the dependence structure during the moisture maximization step that should give a more reliable maximization of precipitation events. Moreover, it should also be noted that the PMP estimates based on the bivariate extreme value analysis exhibit substantially less spatial noise than the corresponding traditional estimates, indicating greater reliability and consistency between locations.
(b) Single-value PMP estimates for CanRCM4 simulated 6-h accumulations via traditional moisture maximization and mean PMP values obtained using the bivariate GEV model via (a) the comonotone copula and (c) the Gumbel copula.
Citation: Journal of Hydrometeorology 19, 4; 10.1175/JHM-D-17-0110.1
It is important to reiterate that PMP estimates, whether traditional or probabilistic, should not be interpreted as absolute upper precipitation bounds. These are numbers that are calculated on the basis of a finite record and are therefore subject to considerable sampling uncertainty. The probabilistic approach has the benefit of providing a quantification of sampling uncertainty in the PMP estimate within a defined statistical framework and thus an opportunity to quantify the likely range of PMP estimates that could arise due to sampling variability. Through further research, this framework could also be further developed to enable a quantification of the likelihood that an observed precipitation event might exceed an estimated PMP value over a defined period, taking into account both the uncertainty in the estimated PMP value and the stochastic nature of extreme precipitation. Figure 6 shows maps of the range of plausible PMP values corresponding to the 10th, 50th, and 90th percentiles using both the estimated Gumbel copula and the comonotone copula. As expected, PMP estimates based on 50 years of records have large uncertainties as indicated by the 80% confidence intervals shown in Fig. 6. To check whether bias in percentiles at a given level has a specific spatial pattern, Fig. 7 shows the maps of the ratios between PMP estimates via the Gumbel copula and the estimates via the comonotone copula at the 10%, 50%, and 90% percentile levels. There is some evidence to suggest that ratios are uniformly less than one for the smaller percentiles, and possibly less than one for all percentiles at higher elevations over western North America.
Estimated PMP percentiles using the bivariate GEV model via (left) the extreme value Gumbel copula and (right) the comonotone copula.
Citation: Journal of Hydrometeorology 19, 4; 10.1175/JHM-D-17-0110.1
Maps of the ratio 
Citation: Journal of Hydrometeorology 19, 4; 10.1175/JHM-D-17-0110.1
In this study, 
Histogram of the PIT based on single-value PMP estimates over all CanRCM4 grid boxes using the bivariate GEV model via (a) the comonotone copula and (b) the estimated extreme value Gumbel copula.
Citation: Journal of Hydrometeorology 19, 4; 10.1175/JHM-D-17-0110.1
Accounting for seasonality in the maximization step is common practice when computing PMP. To this end, the proposed bivariate GEV model was fitted separately for each season. Nevertheless, results of the 10%, 50%, and 90% of PMP percentiles using a single bivariate GEV model fitted to annual maxima of PE and PW do not differ greatly from results obtained by including seasonality (see Fig. S10). Moreover, the use of a single bivariate model fitted to the annual component-wise maxima can be helpful to simplify the analysis and makes the proposed approach more practical and possibly facilitates an extension of the proposed methodology to future nonstationary climate.
We have illustrated the proposed probabilistic approach to PMP estimation using historical change simulations produced with the CanRCM4 regional climate model. The application of PMP estimates derived from these simulations, and the use of projections of future PMP values based on CanRCM4 simulations under future forcing conditions, would require careful evaluation of the model and the derived PMP estimates that is beyond the scope of this paper but will be the subject of future research. Nevertheless, we briefly compare PMP estimates based on CanRCM4 and ERA-Interim reanalysis products. Figure 9 shows maps of the mean PMP estimates via the proposed probabilistic approach using CanRCM4 and the ERA-Interim reanalysis (at 0.44° horizontal spatial resolution interpolated from its 0.75° native resolution) calculated for the period (1979–2005). Although the CanRCM4 is able to well represent the spatial pattern of the PMP, it exhibits a positive bias (~16% on average) relative to ERA-Interim over North America. While the comparison with ERA-Interim is far from perfect given that precipitation is considered to be a type-C reanalysis variable (i.e., only weakly constrained by observations; Kalnay et al. 1996), the results strongly suggest that the further study of the probabilistic PMP estimates is warranted and that RCMs may provide a path toward projecting future change in PMP.
Maps of the PMP estimates via the bivariate GEV model using the Gumbel copula from CanRCM4 and the ERA-Interim reanalysis over North America for the period 1979–2005.
Citation: Journal of Hydrometeorology 19, 4; 10.1175/JHM-D-17-0110.1
Our use of component-wise seasonal PE and PW maxima can be considered as a stepping stone toward a stochastic approach that restricts the analysis to high PE values that coincide with high absolution precipitation amounts and is therefore closer to an approach that is often used by practitioners. A challenge, however, is that asymptotic distribution of extreme high PE values within a block that is restricted to high absolute rainfall accumulation events may not be necessarily be GEV, and thus the bivariate GEV model is not applicable. Heffernan and Tawn (2004) proposed a conditional multivariate extreme value model that may be suitable in such instances, but its application to PMP is not straightforward, and the interpretation of findings, particularly in a nonstationary climate, may be challenging. Implementation would require 1) constructing an asymptotic PE model conditional on extreme precipitation and 2) combining this conditional PE distribution with a univariate extreme value model of PW. Despite the potential interpretation challenges, we plan to attempt such an approach as the next step in the development of a probabilistic framework for PMP estimation. In the interim, the approach considered here, which is based on seasonal elementwise maximization, serves as a stepping stone toward a more comprehensive probabilistic framework for PMP estimation.
Storm transposition concepts have been applied in a number of different ways in traditional single-value PMP estimation approaches (Foufoula‐Georgiou 1989). Generally, these methods access additional information from nearby locations within homogeneous areas, leading to an increase in the effective sample size for a given record period of 
4. Conclusions
The proposed probabilistic method for PMP estimation, as well as providing for the explicit representation of the dependence between PE and PW extremes and allowing evaluation of the uncertainty inherent in PMP estimates, also permits the estimation of PMP for periods longer than the data record period, which is not possible using traditional approaches (see Fig. S12, where PMP estimates are provided over a period of 100 years based on our 50-yr data record). The probabilistic approach also naturally allows the determination of an exceedance probability that, in effect, quantifies what is meant by “probable.” To finish, even if we have the knowledge required to describe key facets of climate and hydrologic systems from first principles, for practical purposes, we do not have the ability to analyze and describe the upper bounds of the intensity of many types of extremes based on physical reasoning. In the case of extreme precipitation, current knowledge of storm mechanisms remains insufficient to allow a precise evaluation of limiting values. In a practical sense, PMP is not deterministically predictable, and thus probabilistic approaches are useful to evaluate the uncertainties. Despite all the considerable uncertainties that may influence PMP estimates using moisture maximization, PMP estimates are still presented as single values. While this is rational from a practical perspective, it is useful to be able to quantify both the uncertainty of these estimates in terms of their expected variation in repeated calculation under statistically equivalent conditions and to be able to estimate the likelihood of future exceedance of the estimated value. This is why this study attempts to take advantage of recent development in probabilistic extreme value analysis to explore the uncertainty and to provide ranges of PMP values with a known likelihood of coverage. This should ultimately lead to more reliable information for design purposes.
Acknowledgments
We gratefully acknowledge Dr. Kharin Slava from Environment and Climate Change Canada for providing the model output used in this work, which is available from the Canadian Centre for Climate Modelling and Analysis (CCCma) upon request.
REFERENCES
Aas, K., C. Czado, A. Frigessi, and H. Bakken, 2009: Pair-copula constructions of multiple dependence. Insur. Math. Econ., 44, 182–198, https://doi.org/10.1016/j.insmatheco.2007.02.001.
Abbs, D. J., 1999: A numerical modeling study to investigate the assumptions used in the calculation of probable maximum precipitation. Water Resour. Res., 35, 785–796, https://doi.org/10.1029/1998WR900013.
AghaKouchak, A., A. Bárdossy, and E. Habib, 2010a: Conditional simulation of remotely sensed rainfall data using a non-Gaussian v-transformed copula. Adv. Water Resour., 33, 624–634, https://doi.org/10.1016/j.advwatres.2010.02.010.
AghaKouchak, A., G. Ciach, and E. Habib, 2010b: Estimation of tail dependence coefficient in rainfall accumulation fields. Adv. Water Resour., 33, 1142–1149, https://doi.org/10.1016/j.advwatres.2010.07.003.
Alexander, G., 1963: Using the probability of storm transposition for estimating the frequency of rare floods. J. Hydrol., 1, 46–57, https://doi.org/10.1016/0022-1694(63)90032-5.
Beauchamp, J., R. Leconte, M. Trudel, and F. Brissette, 2013: Estimation of the summer‐fall PMP and PMF of a northern watershed under a changed climate. Water Resour. Res., 49, 3852–3862, https://doi.org/10.1002/wrcr.20336.
Beirlant, J., Y. Goegebeur, J. Segers, and J. Teugels, 2006: Statistics of Extremes: Theory and Applications. John Wiley & Sons, 514 pp.
Ben Alaya, M., F. Chebana, and T. B. Ouarda, 2014: Probabilistic Gaussian copula regression model for multisite and multivariable downscaling. J. Climate, 27, 3331–3347, https://doi.org/10.1175/JCLI-D-13-00333.1.
Ben Alaya, M., F. Chebana, and T. Ouarda, 2016: Multisite and multivariable statistical downscaling using a Gaussian copula quantile regression model. Climate Dyn., 47, 1383–1397, https://doi.org/10.1007/s00382-015-2908-3.
Capéraà, P., A.-L. Fougères, and C. Genest, 1997: A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567–577, https://doi.org/10.1093/biomet/84.3.567.
Chen, L. C., and A. A. Bradley, 2006: Adequacy of using surface humidity to estimate atmospheric moisture availability for probable maximum precipitation. Water Resour. Res., 42, https://doi.org/10.1029/2005WR004469.
Coles, S., J. Bawa, L. Trenner, and P. Dorazio, 2001: An Introduction to Statistical Modeling of Extreme Values. Springer, 224 pp.
Diaconescu, E. P., P. Gachon, J. Scinocca, and R. Laprise, 2015: Evaluation of daily precipitation statistics and monsoon onset/retreat over western Sahel in multiple data sets. Climate Dyn., 45, 1325–1354, https://doi.org/10.1007/s00382-014-2383-2.
Diebold, F. X., and R. S. Mariano, 1995: Comparing predictive accuracy. J. Bus. Econ. Stat., 13, 253–263, https://doi.org/10.1080/07350015.1995.10524599.
Embrechts, P., C. Klüppelberg, and T. Mikosch, 2013: Modelling Extremal Events: For Insurance and Finance. Springer, 648 pp.
Falk, M., and R. Reiss, 2003: Efficient estimation of the canonical dependence function. Extremes, 6, 61–82, https://doi.org/10.1023/A:1026229314063.
Falk, M., and R. Reiss, 2005: On Pickands coordinates in arbitrary dimensions. J. Multivar. Anal., 92, 426–453, https://doi.org/10.1016/j.jmva.2003.10.006.
Foufoula‐Georgiou, E., 1989: A probabilistic storm transposition approach for estimating exceedance probabilities of extreme precipitation depths. Water Resour. Res., 25, 799–815, https://doi.org/10.1029/WR025i005p00799.
Frahm, G., 2006: On the extremal dependence coefficient of multivariate distributions. Stat. Probab. Lett., 76, 1470–1481, https://doi.org/10.1016/j.spl.2006.03.006.
Frahm, G., M. Junker, and R. Schmidt, 2005: Estimating the tail-dependence coefficient: Properties and pitfalls. Insur. Math. Econ., 37, 80–100, https://doi.org/10.1016/j.insmatheco.2005.05.008.
Genest, C., and J. Segers, 2009: Rank-based inference for bivariate extreme-value copulas. Ann. Stat., 37, 2990–3022, https://doi.org/10.1214/08-AOS672.
Ghoudi, K., A. Khoudraji, and E. L. P. Rivest, 1998: Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles. Can. J. Stat., 26, 187–197, https://doi.org/10.2307/3315683.
Giorgi, F., C. Jones, and G. R. Asrar, 2009: Addressing climate information needs at the regional level: The CORDEX framework. WMO Bull., 58 (3), 175–183.
Gneiting, T., F. Balabdaoui, and A. E. Raftery, 2007: Probabilistic forecasts, calibration and sharpness. J. Roy. Stat. Soc., 69B, 243–268, https://doi.org/10.1111/j.1467-9868.2007.00587.x.
Guerfi, N., A. A. Assani, M. Mesfioui, and C. Kinnard, 2015: Comparison of the temporal variability of winter daily extreme temperatures and precipitations in southern Quebec (Canada) using the Lombard and copula methods. Int. J. Climatol., 35, 4237–4246, https://doi.org/10.1002/joc.4282.
Hamill, T. M., 2001: Interpretation of rank histograms for verifying ensemble forecasts. Mon. Wea. Rev., 129, 550–560, https://doi.org/10.1175/1520-0493(2001)129<0550:IORHFV>2.0.CO;2.
Heffernan, J. E., and J. A. Tawn, 2004: A conditional approach for multivariate extreme values (with discussion). J. Roy. Stat. Soc., 66B, 497–546, https://doi.org/10.1111/j.1467-9868.2004.02050.x.
Hershfield, D. M., 1961: Estimating the probable maximum precipitation. J. Hydraul. Div., 87 (5), 99–116.
Joe, H., 1997: Multivariate Models and Multivariate Dependence Concepts. CRC Press, 424 pp.
Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77, 437–471, https://doi.org/10.1175/1520-0477(1996)077<0437:TNYRP>2.0.CO;2.
Kharin, V. V., and F. W. Zwiers, 2005: Estimating extremes in transient climate change simulations. J. Climate, 18, 1156–1173, https://doi.org/10.1175/JCLI3320.1.
Kharin, V. V., F. W. Zwiers, X. Zhang, and G. C. Hegerl, 2007: Changes in temperature and precipitation extremes in the IPCC ensemble of global coupled model simulations. J. Climate, 20, 1419–1444, https://doi.org/10.1175/JCLI4066.1.
Klemeš, V., 1986: Dilettantism in hydrology: Transition or destiny? Water Resour. Res., 22, 177S–188S, https://doi.org/10.1029/WR022i09Sp0177S.
Klemeš, V., 1987: Hydrological and engineering relevance of flood frequency analysis. Hydrologic Frequency Modeling, Springer, 1–18, https://doi.org/10.1007/978-94-009-3953-0_1.
Klemeš, V., 1993: Probability of extreme hydrometeorological events—A different approach. IAHS Publ., 213, 167–176.
Klemeš, V., 2000: Tall tales about tails of hydrological distributions. I. J. Hydrol. Eng., 5, 227–231, https://doi.org/10.1061/(ASCE)1084-0699(2000)5:3(227).
Leadbetter, M. R., G. Lindgren, and H. Rootzén, 2012: Extremes and Related Properties of Random Sequences and Processes. Springer, 336 pp.
Mao, G., S. Vogl, P. Laux, S. Wagner, and H. Kunstmann, 2015: Stochastic bias correction of dynamically downscaled precipitation fields for Germany through copula-based integration of gridded observation data. Hydrol. Earth Syst. Sci., 19, 1787–1806, https://doi.org/10.5194/hess-19-1787-2015.
Micovic, Z., M. G. Schaefer, and G. H. Taylor, 2015: Uncertainty analysis for probable maximum precipitation estimates. J. Hydrol., 521, 360–373, https://doi.org/10.1016/j.jhydrol.2014.12.033.
Nelsen, R. B., 2007a: An Introduction to Copulas. Springer, 272 pp.
Nelsen, R. B., 2007b: Extremes of nonexchangeability. Stat. Pap., 48, 329–336, https://doi.org/10.1007/s00362-006-0336-5.
Ohara, N., M. Kavvas, S. Kure, Z. Chen, S. Jang, and E. Tan, 2011: Physically based estimation of maximum precipitation over American River watershed, California. J. Hydrol. Eng., 16, 351–361, https://doi.org/10.1061/(ASCE)HE.1943-5584.0000324.
Papalexiou, S., and D. Koutsoyiannis, 2006: A probabilistic approach to the concept of probable maximum precipitation. Adv. Geosci., 7, 51–54, https://doi.org/10.5194/adgeo-7-51-2006.
Pickands, J., 1981: Multivariate extreme value distributions (with a discussion). Bull. Int. Stat. Inst., 49, 859–878, 894–902.
Rakhecha, P., and C. Clark, 1999: Revised estimates of one‐day probable maximum precipitation (PMP) for India. Meteor. Appl., 6, 343–350, https://doi.org/10.1017/S1350482799001280.
Requena, A., L. Mediero Orduña, and L. Garrote de Marcos, 2013: A bivariate return period based on copulas for hydrologic dam design: Accounting for reservoir routing in risk estimation. Hydrol. Earth Syst. Sci., 17, 3023–3038, https://doi.org/10.5194/hess-17-3023-2013.
Rousseau, A. N., I. M. Klein, D. Freudiger, P. Gagnon, A. Frigon, and C. Ratté-Fortin, 2014: Development of a methodology to evaluate probable maximum precipitation (PMP) under changing climate conditions: Application to southern Quebec, Canada. J. Hydrol., 519, 3094–3109, https://doi.org/10.1016/j.jhydrol.2014.10.053.
Salas, J. D., and F. R. Salas, 2016: Estimating the uncertainty of the PMP. XXVII IAHR Latin American Congress of Hydraulics, Lima, Peru, IAHR.
Salas, J. D., G. Gavilán, F. R. Salas, P. Y. Julien and J. Abdullah, 2014: Uncertainty of the PMP and PMF. Handbook of Engineering Hydrology, S. Eslamian, Ed., CRC Press, 575–603.
Salvadori, G., and C. De Michele, 2004: Analytical calculation of storm volume statistics involving Pareto‐like intensity‐duration marginals. Geophys. Res. Lett., 31, L04502, https://doi.org/10.1029/2003GL018767.
Salvadori, G., C. De Michele, N. T. Kottegoda, and R. Rosso, 2007: Extremes in Nature: An Approach Using Copulas. Springer, 292 pp.
Scinocca, J., and Coauthors, 2016: Coordinated global and regional climate modeling. J. Climate, 29, 17–35, https://doi.org/10.1175/JCLI-D-15-0161.1.
Serinaldi, F., 2008: Analysis of inter-gauge dependence by Kendall’s τK, upper tail dependence coefficient, and 2-copulas with application to rainfall fields. Stochastic Environ. Res. Risk Assess., 22, 671–688, https://doi.org/10.1007/s00477-007-0176-4.
Shiau, J., 2003: Return period of bivariate distributed extreme hydrological events. Stochastic Environ. Res. Risk Assess., 17, 42–57, https://doi.org/10.1007/s00477-003-0125-9.
Stephens, M. A., 1970: Use of the Kolmogorov-Smirnov, Cramér-Von Mises and related statistics without extensive tables. J. Roy. Stat. Soc., 32B, 115–122, http://www.jstor.org/stable/2984408.
Tawn, J. A., 1988: Bivariate extreme value theory: Models and estimation. Biometrika, 75, 397–415, https://doi.org/10.1093/biomet/75.3.397.
von Salzen, K., and Coauthors, 2013: The Canadian Fourth Generation Atmospheric Global Climate Model (CanAM4). Part I: Representation of physical processes. Atmos.–Ocean, 51, 104–125, https://doi.org/10.1080/07055900.2012.755610.
WMO, 1986: Manual for estimation of probable maximum precipitation. 2nd ed. Operational Hydrology Rep. 1, WMO-332, 269 pp., https://library.wmo.int/pmb_ged/wmo_332.pdf.
WMO, 2009: Manual on estimation of probable maximum precipitation (PMP). Tech. Rep. WMO-1045, 257 pp., http://www.wmo.int/pages/prog/hwrp/publications/PMP/WMO%201045%20en.pdf.
Yevjevich, V., 1968: Misconceptions in hydrology and their consequences. Water Resour. Res., 4, 225–232, https://doi.org/10.1029/WR004i002p00225.
Yue, S., 2001: A bivariate extreme value distribution applied to flood frequency analysis. Hydrol. Res., 32, 49–64.