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  • View in gallery

    Return period estimates (years) from the Monte Carlo experiment (described in section 3a) performed on 50 and 50 × 6 samples drawn from the observed 24-h annual maximum precipitation data at stations in the Susquehanna watershed. The red dashed line shows the observed (reference) return period. Bias is defined as the difference between the median return period (black line inside a box) and the reference return period. Estimation uncertainty is defined as the interquartile range (IQR) of the estimated return periods. The numbers in each panel indicate station IDs in the Susquehanna watershed. The blue curve in the last panel shows the difference in the absolute median biases estimated for 50 and 50 × 6 samples over all stations. A positive value along the blue curve indicates that the absolute median bias estimated from 50 samples is bigger than that estimated from 50 × 6 samples.

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    As in Fig. 1, but for the Florida peninsula.

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    Bias (model minus observation) in 24-h mean annual maximum precipitation (MAM; mm day−1). (a) The top-left panel shows the observed MAM over the Susquehanna watershed and uses the color scale along the right edge of the figure. Other panels show bias in the MAM (model minus observation) and use the color scale along the bottom edge of the figure. The MAM is computed for the period 1951–2005 in the observational data and for 1956–2005 in model data. (b) As in (a), but over the Florida peninsula. The polygons in (b) represent the boundary of the Kissimmee–Southern Florida watershed.

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    Taylor diagrams of 24-h MAM comparing station observations and models. (a) For the Susquehanna. Each diagram shows how closely the spatial pattern of the MAM in the observation resembles that in a model. The reference point (observation) is marked as a solid green square. Letters indicate the position of each model. The dashed black lines on the outermost semicircle indicate pattern correlation of MAM between the observation and models. The blue dashed curves indicate the normalized standard deviation (NSD) defined as spatial standard deviation of MAM in models and observation divided by the spatial standard deviation in the observation. The standard deviation is measured as the radial distance from the origin. The green dashed curves show the normalized root mean squared difference (NRMSD) defined as the RMSD between model and observation divided by the standard deviation of the observed MAM. The NRMSD is measured as a distance from the reference point (solid green square). (b) As in (a), but for the Florida peninsula.

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    The ratio (model value over observed value) of interannual IQR of 24-h annual maximum precipitation (AMP). (a) The top-left panel shows the observed IQR of the AMP (mm day−1) for the Susquehanna, and uses the color scale along the right edge of the figure. The other panels show the interannual IQR ratio of the AMP in models and observation and use the color scale along the bottom edge of the figure. The ratio is unitless. (b) As in (a), but for the Florida peninsula. The polygons in (b) represent the boundary of the Kissimmee–Southern Florida watershed.

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    The IVSS as expressed in Eq. (1). The IVSS is a unitless quantity. The closer the IVSS value of a model to zero, the better is the performance of the model in simulating the interannual variability of the observed AMP. The horizontal dashed line indicates the chosen IVSS threshold (=1.13) as defined in section 3. (a) For the Susquehanna. (b) For the Florida peninsula.

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    Scatter diagrams of NRMSD as computed in the Taylor diagram (x axis) against IVSS values (y axis). “Corr” indicates the correlation between NRMSD and IVSS for all models. (a) For the Susquehanna. (b) For the Florida peninsula.

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    Scatter diagrams of the product of correlation skill and NSD (x axis) against IVSS values (y axis). The horizontal and vertical dashed lines are drawn using the selection criteria discussed in section 3. Models in the green shaded region are selected for PIF estimation. (a) For the Susquehanna. (b) For the Florida peninsula.

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    Differences (RCP8.5 minus historical) in 24-h mean MAM (mm day−1) computed from raw (not bias-corrected) historical and RCP8.5 simulations. The MAM is calculated for the period 1956–2005 in the historical simulations and for 2049–2098 in the RCP8.5 simulations. Stippling shows differences significant at the 5% significance level. (a) For the Susquehanna. (b) For the Florida peninsula.

  • View in gallery

    PIF estimates of 24-h precipitation totals in the bias-corrected historical and future simulations. The red curve and yellow shading indicate 24-h PIF estimates and corresponding 90% confidence interval in the bias-corrected historical simulation. The blue curve and green shading indicate 24-h PIF estimates and corresponding 90% confidence interval in the bias-corrected RCP8.5 simulation. The median-all panel shows the median of 24-h PIF estimates from all models, while the median-pooled shows the median of 24-h PIF estimates from models that are used for pooling. Pooled shows 24-h PIF estimates computed from pooling of better performing models. Models that are used for pooling are shown in red letters in the top-left corner of the figures. The x axis indicates return periods in years, and the y axis indicates intensity (mm day−1). (a) For the Susquehanna. (b) For the Florida peninsula.

  • View in gallery

    Changes in 24-h precipitation (mm day−1) for (a) 5-yr and (b) 50-yr return periods in the Susquehanna. Differences significant at the 5% significance level are shown as solid squares and those not significant at 5% are shown as blank circles. The significance is computed from the Z statistic as shown in Eq. (6).

  • View in gallery

    As in Fig. 11, but for the Florida peninsula.

  • View in gallery

    Changes in 24-h precipitation (mm day−1) for 2-, 5- 10- 25-, 50-, and 100-yr return periods computed from pooled models. The differences that are significant at the 5% significance level are shown as solid squares, and those not significant at the 5% level are shown as blank circles. The significance is computed from the Z statistic as shown in Eq. (6). “Signif. stns” shows the percentage of stations at which the differences are significant. (a) For the Susquehanna. Pooled models used: C, E, F, G, H, I, J, K, and L. (b) For the Florida peninsula. Pooled models used: C, E, G, I, J, and K.

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Pooling Data Improves Multimodel IDF Estimates over Median-Based IDF Estimates: Analysis over the Susquehanna and Florida

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  • 1 Department of Land, Air and Water Resources, University of California, Davis, Davis, California
  • 2 Department of Civil Engineering, Boise State University, Boise, Idaho
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Abstract

Traditional multimodel methods for estimating future changes in precipitation intensity, duration, and frequency (IDF) curves rely on mean or median of models’ IDF estimates. Such multimodel estimates are impaired by large estimation uncertainty, shadowing their efficacy in planning efforts. Here, assuming that each climate model is one representation of the underlying data generating process, i.e., the Earth system, we propose a novel extension of current methods through pooling model data: (i) evaluate performance of climate models in simulating the spatial and temporal variability of the observed annual maximum precipitation (AMP), (ii) bias-correct and pool historical and future AMP data of reasonably performing models, and (iii) compute IDF estimates in a nonstationary framework from pooled historical and future model data. Pooling enhances fitting of the extreme value distribution to the data and assumes that data from reasonably performing models represent samples from the “true” underlying data generating distribution. Through Monte Carlo simulations with synthetic data, we show that return periods derived from pooled data have smaller biases and lesser uncertainty than those derived from ensembles of individual model data. We apply this method to NA-CORDEX models to estimate changes in 24-h precipitation intensity–frequency (PIF) estimates over the Susquehanna watershed and Florida peninsula. Our approach identifies significant future changes at more stations compared to median-based PIF estimates. The analysis suggests that almost all stations over the Susquehanna and at least two-thirds of the stations over the Florida peninsula will observe significant increases in 24-h precipitation for 2–100-yr return periods.

Denotes content that is immediately available upon publication as open access.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/JHM-D-20-0180.s1.

Corresponding author: Abhishekh K. Srivastava, asrivas@ucdavis.edu

Abstract

Traditional multimodel methods for estimating future changes in precipitation intensity, duration, and frequency (IDF) curves rely on mean or median of models’ IDF estimates. Such multimodel estimates are impaired by large estimation uncertainty, shadowing their efficacy in planning efforts. Here, assuming that each climate model is one representation of the underlying data generating process, i.e., the Earth system, we propose a novel extension of current methods through pooling model data: (i) evaluate performance of climate models in simulating the spatial and temporal variability of the observed annual maximum precipitation (AMP), (ii) bias-correct and pool historical and future AMP data of reasonably performing models, and (iii) compute IDF estimates in a nonstationary framework from pooled historical and future model data. Pooling enhances fitting of the extreme value distribution to the data and assumes that data from reasonably performing models represent samples from the “true” underlying data generating distribution. Through Monte Carlo simulations with synthetic data, we show that return periods derived from pooled data have smaller biases and lesser uncertainty than those derived from ensembles of individual model data. We apply this method to NA-CORDEX models to estimate changes in 24-h precipitation intensity–frequency (PIF) estimates over the Susquehanna watershed and Florida peninsula. Our approach identifies significant future changes at more stations compared to median-based PIF estimates. The analysis suggests that almost all stations over the Susquehanna and at least two-thirds of the stations over the Florida peninsula will observe significant increases in 24-h precipitation for 2–100-yr return periods.

Denotes content that is immediately available upon publication as open access.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/JHM-D-20-0180.s1.

Corresponding author: Abhishekh K. Srivastava, asrivas@ucdavis.edu

1. Introduction

Studies, using both observational and model data, suggest more intense and more frequent extreme precipitation events may occur over midlatitude land areas in a warming climate (Tebaldi et al. 2006; Kharin et al. 2007, 2013; Collins et al. 2013; Donat et al. 2013; Fischer and Knutti 2016; Rajczak and Schär 2017; Easterling et al. 2017). The intensity and frequency of extreme precipitation events are also projected to increase in many parts of the contiguous United States (Easterling et al. 2017; Prein et al. 2017). Extreme precipitation events pose a significant threat to society and ecosystems with severe implications for human lives, infrastructure, economy, and food production (Rosenzweig et al. 2001; Smith and Katz 2013; Ziegler et al. 2014; Estrada et al. 2015). Therefore, reliable projections of change in the intensity and frequency of the extreme precipitation events are needed for planning and adaptation efforts by stakeholders and government authorities.

Precipitation intensity–frequency (PIF) estimates—commonly referred to as intensity, duration, and frequency (IDF) curves—are the estimates of probable intensity of precipitation associated with different durations and return periods. These curves are widely used for a variety of applications such as storm water and flood management, and design of dams, reservoirs, bridges, and highways (Trefry et al. 2005; Simonovic and Peck 2009; Sugahara et al. 2009; AghaKouchak et al. 2018). Traditionally, IDF curves are estimated assuming temporal stationarity in the intensity and frequency of extreme rainfall (Trefry et al. 2005; Bonnin et al. 2006; Perica et al. 2011, 2013). However, the stationarity assumption is not valid in the face of temporal changes in frequency and intensity of extreme precipitation (Milly et al. 2008; Simonovic and Peck 2009; Katz 2013; Cheng and AghaKouchak 2014; Mondal and Mujumdar 2015). Studies have shown that IDF curves maintaining the stationarity assumption tend to underestimate extreme precipitation events (Cheng and AghaKouchak 2014; Sarhadi and Soulis 2017; Hosseinzadehtalaei et al. 2018). These studies stimulated a number of subsequent studies to adopt nonstationary models for IDF analysis (Villarini et al. 2010; Tramblay et al. 2013; Cheng et al. 2014; Mondal and Mujumdar 2015; Sarhadi and Soulis 2017; Ragno et al. 2018; AghaKouchak et al. 2018; Ganguli and Coulibaly 2019; Ouarda et al. 2019; Schardong and Simonovic 2019; Wehner et al. 2020; Wehner 2020). Notably, some of these studies have used historical observed data for nonstationary analysis assuming that similar trends or nonstationary behavior in extremes continues into the future (Willems and Vrac 2011; Cheng and AghaKouchak 2014; Sarhadi and Soulis 2017; Agilan and Umamahesh 2018).

Recent international collaborative efforts have made available high-resolution regional climate models (RCMs) that are found to provide more credible climate projections than global climate models (GCMs) (Giorgi et al. 2016; Gutowski et al. 2020). RCMs are physically based climate models representing complex components (land, ocean, sea ice) of the Earth system and their interactions at much finer spatial scale than conventional coarse-resolution GCMs. The higher resolution enabled by RCMs improves the representation of local forcings such as topography, coastlines, and complex land structure, as well as anthropogenic forcing such as greenhouse gas concentrations, land-use changes, and aerosols (Giorgi et al. 2009) at local-to-regional scales. Many studies have used RCMs for estimating future IDF curves, providing useful information about changes in IDF estimates (Ragno et al. 2018; AghaKouchak et al. 2018; Ganguli and Coulibaly 2019). However, such estimates also come with some limitations. First, most previous studies do not analyze the historical performance of models and therefore do not exclude poorly performing models that may bias the estimates (Ragno et al. 2018; AghaKouchak et al. 2018; Hosseinzadehtalaei et al. 2018). Evaluation of a model’s performance in simulating the observed variability is a first step toward understanding climate change signals, which renders confidence in quantifying uncertainty in future projections (Giorgi et al. 2004; Knutti et al. 2010; Bukovsky 2012; Rupp et al. 2013; Wang et al. 2015). The reliability of a future projection increases if models are weighed or at least selected based upon their skill (Knutti et al. 2010; Mishra et al. 2018). The Intergovernmental Panel on Climate Change (IPCC) report on the evaluation of climate models mentions that “the spread in climate projections can be reduced by weighting of models according to their ability to reproduce past observed climate” (Flato et al. 2014). Second, most simulations in individual models are not long enough (typically spanning 50–100 years), leading to large uncertainty around the IDF estimates for return periods longer than the sample size (Sadegh et al. 2018). Third, most studies, based upon a multi-ensemble approach, apply a form of averaging (e.g., mean or median) of IDF estimates from individual RCM/GCM models (Ragno et al. 2018; AghaKouchak et al. 2018; Schardong and Simonovic 2019; Padulano et al. 2019). Such median (mean) based estimates produce a lessening of the intensity of probable extremes, and are also impaired by large estimation uncertainty.

In this paper we present a novel extension of conventional methods for computing multimodel IDF estimates based upon pooling data from models that perform reasonably (using our performance criteria defined in section 3b) in simulating the historical observed variability. The pooling of model data is based upon the assumption that each climate model is one representation of the physical processes that govern the Earth system, and data from reasonably performing models represent samples from the observed (true) distribution. Our hypothesis is that pooling (concatenating) information from various models, rather than adopting mean or median, can help reduce the bias (difference between the observed and estimated IDF estimates/return periods) and the uncertainty (90% confidence interval) around the IDF estimates. We test this hypothesis using Monte Carlo simulations on synthetic data derived from the distributions of the observed 1-, 6-, 12-, and 24-h duration annual maximum precipitation, and show that pooling annual model information reduces the biases and uncertainty in the estimated return periods of precipitation across different durations. We then apply our method to the historical and RCP8.5 (future) simulations from a set of 12 RCMs from the North American Coordinated Regional Climate Downscaling Experiment (NA-CORDEX) project to estimate future changes in 24-h precipitation frequency estimates over the Susquehanna watershed and Florida peninsula. The method follows this procedure: first, evaluate the historical performance of RCMs in simulating the spatial and temporal variability of the observed annual maximum precipitation (AMP). Specifically, the spatial and temporal variability of the simulated AMP are evaluated using Taylor diagram and the interannual variability skill score (IVSS), respectively. Second, bias-correct the historical and RCP8.5 AMP data of climate models. Bias correction reduces spatial scale mismatch between station (point) observations and gridded model output (areal average inside a grid box) (Sharma et al. 2007; Turco et al. 2017). Third, pool the bias-corrected historical and RCP8.5 AMP data of selected models. Pooling concatenates data from each model, enhancing the fitting of the extreme value distribution to the data. We then use the fitted distribution to compute nonstationary 24-h PIF estimates in the pooled historical and future simulations using an annual maxima approach.

We select two watersheds with widely different climatology and physical characteristics, and with significant regional importance. The Susquehanna watershed spreads over parts of New York, Pennsylvania, and Maryland. The watershed is important for power production, agriculture, and drinking water supplies, among other uses. It is one of the most flood-prone regions in the United States and has also experienced droughts in parts of the watershed (https://www.srbc.net/our-work/reports-library/technical-reports/state-of-susquehanna-2013/). However, until recently, decision-making has largely relied upon historical records that do not account for climate projections. Consequently, information on future changes in extreme precipitation events has featured prominently in requests from stakeholders, especially water managers. Farther to the south, the Florida peninsula is a diverse ecosystem that includes the Kissimmee–Southern Florida watershed and the Florida Everglades. The key challenges to this region are drinking water management, restoration of natural ecosystems, sea level rise and flooding. Addressing these challenges requires reliable information on changes in the intensity, duration, and frequency of precipitation. However, the geography of Florida is barely resolved in the global climate models (Misra et al. 2011), and so regional stakeholders generally require downscaled information. Mesoscale events on the order of 10–1000 km play a significant role in Florida’s hydroclimate, calling for high-resolution climate models to resolve these processes (Maxwell et al. 2012; Prat and Nelson 2013). In both regions, high-resolution regional climate models are thus necessary to enable estimation of changes in IDF estimates.

We emphasize that the results obtained from the Monte Carlo simulations performed on the observed annual maximum precipitation for different durations (here, 1-, 6-, 12-, and 24-h) show that pooling data is superior to the conventionally used median model selection in reducing bias and uncertainty in estimating return periods. Since subdaily data are not available for most of the NA-CORDEX models, we applied our method only to the 24-h annual maximum precipitation. It is also worth mentioning that changes in 24-h precipitation for return periods up to 100 years are of interest to the stakeholders in both regions (Jagannathan et al. 2021).

The remainder of the paper is summarized as follows. Section 2 describes the observed and model data used in the study. Section 3 describes metrics used for assessing model performance, and the framework for IDF estimates. Section 4 discusses results of the study, and section 5 summarizes the results.

2. Data

In this analysis we have used AMP data calculated for each calendar year. The station-based AMP data are downloaded from the NOAA Atlas 14 website (https://hdsc.nws.noaa.gov/hdsc/pfds/pfds_map_cont.html). Most of the station-based datasets have at least 40 years of data over the period 1951–2005. The model data for the analysis are obtained from the historical (1956–2005) and RCP8.5 (2049–2098) simulations of regional climate models at 0.22° grid spacing in the NA-CORDEX (Mearns et al. 2017). The 12 RCMs analyzed here are run with boundary conditions from four GCM simulations from the fifth phase of the Climate Model Intercomparison Project (CMIP5) archive (Taylor et al. 2012). The list of RCMs with their host institutions is given in Table 1. Detailed information on the RCMs, such as dynamical core, model components, model physics and parameterization schemes can be found at https://na-cordex.org/rcm-characteristics and in the references mentioned therein. The model data are interpolated onto station locations using the nearest neighbor interpolation scheme.

Table 1.

List of NA-CORDEX models analyzed in this study. UQAM is Université du Québec à Montréal. CCCma is Canadian Centre for Climate Modelling and Analysis. NCAR is National Center for Atmospheric Research.

Table 1.

3. Methods

a. Monte Carlo simulations

If we assume that models are distinct realizations of the processes that govern the Earth system, and data from reasonably performing models represent samples from the “true” underlying data generating distribution, then pooling annual maximum data from models theoretically can reduce bias and uncertainty in the IDF estimates. We hypothesize that drawing a higher number of samples from the underlying data generating process promotes a superior distribution fit and thereby more reliable IDF estimates. To test this hypothesis, we perform Monte Carlo simulations on synthetic data derived from the observed distribution. The procedure has the following steps:

  1. Estimate the reference return period for an arbitrarily chosen quantile q by fitting a GEV distribution (reference distribution FO) to the observed annual maximum data. The return period for the reference quantile q is defined as RO = 1/[1 − FO(q)], where FO(q) is the nonexceedance probability.
  2. Generate S samples of annual maximum precipitation from FO.
  3. Fit a GEV distribution FS to the S samples drawn at step 2.
  4. Estimate the return period RS for the reference quantile q from FS, defined as
    RS=1/[1FS(q)]
  5. Repeat steps 2–4 100 times.
  6. Estimate bias (difference between the reference return period and the median of return periods from step 5 and interquartile range (IQR) of the estimated return periods from step 5.
  7. Draw L samples (LS) from the reference GEV distribution.
  8. Fit a GEV distribution FL to the L samples drawn at step 7.
  9. Estimate the return period RL for the reference quantile q from FL, defined as
    RL=1/[1FL(q)].
  10. Repeat steps 7–9 100 times.
  11. Estimate bias and IQR of the estimated return periods from step 10.
  12. Compare biases and IQRs from steps 6 and 11.
If the bias and IQR of the estimated return period from pooled samples (L) are smaller than those derived from S samples, we conclude that pooling enables better fitting of the extreme value distribution to the data, reducing the bias and uncertainty in the return period estimates. It should be noted that we use the pooled data only to fit the GEV distribution, and then, derive the return periods/or PIF estimates using an annual maximum approach.

b. Evaluation of model performance

While there are a number of approaches for evaluating model performance, a reasonable approach is to use a metric that scores models based upon their ability to simulate the mean climatology and temporal variability of the variable of interest. Since only AMP data are required for IDF estimation, we analyze models on the basis of their ability to simulate the spatial and temporal variability of the observed AMP. The selected performance metrics are described below.

1) Taylor diagram

The skill of a model in simulating the spatial pattern of the observed AMP is analyzed using a Taylor diagram (Taylor 2001). For generating a Taylor diagram, the long-term mean of the AMP (hereafter, MAM) is computed at each station location so that there are as many MAM values as the number of stations in a study region. The Taylor diagram provides a concise statistical summary of similarity between a model’s MAM and the observed MAM in terms of their pattern correlation (correlation between the MAM in the observation and that in a model); normalized standard deviation (NSD) (computed by dividing the spatial standard deviation of the MAM in a model by the standard deviation of the MAM in the observation); and normalized root-mean-square difference (NRMSD) (defined as the root-mean-square difference between a model’s MAM and the observed MAM, divided by the standard deviation of the observed MAM). A model perfectly simulating spatial patterns of the observed MAM should have correlation equal to 1, spatial standard deviation equal to that of the observation (i.e., NSD equal to 1), and NRMSD equal to zero. Each model is represented by a single point on the Taylor diagram. Taylor diagram enables to concisely evaluate the simulated spatial pattern with informative details. For example, as is nicely summarized in Taylor (2001), a simple pattern correlation does not tell if the two patterns (reference and model) have similar magnitude of spatial variation. Similarly, a simple RMSD based metric does not convey how much of the error is due to the difference in structure or phase and how much is due to the difference in the magnitude of variation.

2) Interannual variability skill score

The IVSS is a “symmetric” variability measure, similar to that in Gleckler et al. (2008) that scores two models equally if one simulates twice the observed temporal variability and the other simulating half of the observed temporal variability. IVSS is defined as
IVSS=1Nn=1N(IQRmIQRoIQRoIQRm)2,
where IQRm and IQRo are the interquartile ranges of the AMP at a station in a model and the observation, respectively. The variable N is the total number of stations in each study region. IVSS will be zero for a model perfectly simulating the observed IQR. The smaller the IVSS, the better the model performance. We defined IVSS using IQR since IQR is less affected by outliers in the data, and hence considered a more robust statistic than standard deviation. Similar metrics have been used for model evaluation in some previous studies (Chen et al. 2011; Jiang et al. 2015; Srivastava et al. 2020).

3) Selection of models

Models employed for calculating PIF are selected using following criteria:

  • Taylor diagram: spatial correlation ≥ critical value of correlation (2/N), where N is the number of stations in the regions (42 over the Susquehanna and 73 over the Florida peninsula). NSD is between 0.8 and 1.2.
  • IVSS: IVSS ≤ 1.13, assuming that 0.6 ≤ IQRm/IQRo ≤ 1/0.6 for each station in a region.
We note that the performance criteria defined above are somewhat subjective in nature, and one may use an alternative performance criterion for model selection.

c. Bias correction of models

Bias correction improves the usability of models, especially for users interested in impacts. However, as argued in Zhang and Soden (2019), bias correction is only useful in constraining intermodel spread when applied to models that are sufficiently performing in the historical period and are shown to capture the relevant processes. Our down-selection of models based on performance is thus necessary to constrain future projections.

Many existing statistical bias correction methods, such as simple quantile mapping (QM), assume that higher-order statistics of a distribution, such as variance and skewness, remain stationary and only the mean changes. However, this assumption may not hold in a nonstationary climate (Meehl et al. 2004). Therefore, following Wang and Chen (2014) and Ganguli and Coulibaly (2019), we use a bias-correction method called equiratio cumulative distribution function matching (ERCDFM) that allows the possibility of changes in the higher-order moments by incorporating information from the CDF of a model projection. This method is a modified version of the equidistant cumulative distribution function matching (EDCDFM) proposed by Li et al. (2010). The bias-correction is applied on the AMP data. If xh is the “raw” historical AMP time series in a model, then bias-adjusted value of xh can be formulated as
x^h=Fo1[Fh(xh)],
where x^h is the bias-corrected historical AMP data, Fh is the cumulative distribution function (CDF) of xh, and Fo1 is the inverse CDF of the AMP time series in the observation. The future bias-corrected values of AMP time series are obtained as
x^f=xf×Fo1[Ff(xf)]Fh1[Ff(xf)],
where, xf and x^f are raw and bias-corrected future values of AMP data, respectively. The term Fh1 is the inverse of the CDF Fh, and Ff is the CDF of the future AMP data.

d. Pooling of reasonably performing models

We pool the bias-corrected historical and RCP8.5 AMP from the selected models. Pooling increases the sample size, enhancing the fitting of the extreme value distribution to the data. This helps in reducing the bias and uncertainty in the IDF/PIF estimates.

e. Extreme value analysis

IDF or PIF estimates using data in the form of block maxima (e.g., annual maximum data in our case) are generally computed by fitting generalized extreme value (GEV) distribution to the data. The theoretical justification for fitting GEV distribution to the block maxima is described in Coles et al. (2001). The GEV distribution is defined as
G(z)=exp{[1+ζ(zμσ)]1/ζ},
where μ, σ, and ζ are the location, scale, and shape parameters, respectively. In the stationary model of a GEV distribution, parameters μ, σ, and ζ are considered time invariant, i.e., fixed in time. The nonstationary GEV distribution is modeled by introducing a time component as a covariate in the location and/or scale parameters. We used the following linear regression model for incorporating nonstationarity in the location and shape parameters:
μ(t)=μ0+μ1t,
σ(t)=σ0+σ1t,
where μ(t) and σ(t) are the time dependent location and scale parameters, respectively. Parameters of the GEV distribution are estimated by using maximum likelihood estimation (MLE) method (Coles et al. 2001). To choose between stationary and nonstationary models, we adopted the following approach. If no significant trend at the 5% level in the AMP is found using the Mann–Kendall (MK) trend test (Mann 1945), the stationary model is chosen for the IDF analysis. If a significant trend is found, then one of the two nonstationary models described here is adopted: (i) a nonstationary model with time as a covariate in the location parameter as described in Eq. (5a), or (ii) a nonstationary model with time as a covariate in the location and scale parameters as described in Eqs. (5a) and (5b). To choose between nonstationary models (i) and (ii) we use model selection criteria called Akaike information criteria (AIC) (Akaike 1974). The nonstationary model with the lower AIC value is chosen for the GEV analysis. A similar approach has been adopted in Ragno et al. (2018) and AghaKouchak et al. (2018). For the GEV analysis we have used “extRemes2.0” extreme value analysis package (Gilleland and Katz 2016) in R (R Core Team 2018).

f. Metric for estimating significance of change in PIF estimates

To test the significance of difference between RCP8.5 and historical PIF estimates we use the Z statistic as defined in Srivastava et al. (2019). The statistic is defined as
Z=PTRPTHσTR2NR+σTH2NH,
where PTR and PTH are the T-year precipitation estimates in the RCP8.5 and historical simulations; σTR and σTH are the standard deviations of the corresponding estimates; and NR and NH are the number of observations used in calculating PTR and PTH, respectively. The terms in the denominator can be estimated from confidence intervals of the respective T-year estimates. For instance, the (1 − α)% confidence interval of PTH is expressed as
(1α)%CI=PTH±zα/2σTHNH,
where zα/2 is the (1 − α) quantile of the standard normal distribution. For a significance level of α = 0.05, (corresponding to a 95% confidence interval), z5% equals 1.96. If |Z| ≤ 1.96, we say that the null hypothesis cannot be rejected at the 5% significance level—or, in other words, the difference between the two estimates is not significant at the 5% significance level. Similar metrics have been used in Nataraj and Grenney (2005), Madsen et al. (2009), Ganguli and Coulibaly (2019), and Rhoades et al. (2020).

4. Results

a. Monte Carlo simulations

Figure 1 and Table 2 show return periods estimated from Monte Carlo simulations for 50 and 50 × 6 samples drawn from the observed 24-h annual maximum precipitation over all stations in the Susquehanna watershed. Bias in the estimated return period is indicated by the difference between the observed and the median of the estimated return periods. The uncertainty is indicated by the interquartile range. It is apparent that the range of absolute bias in return periods from 50 samples (column “aBias.S50” in Table 2) is 0–0.48 years and that from 50 × 6 samples (column “aBias.S300” in Table 2) is 0–0.17 years. The median absolute bias over the watershed, computed for 50 samples, is 0.15 years and that for 50 × 6 samples is 0.04 years. Also, for more than 71% of the cases, the bias for 50 × 6 samples is smaller than that for 50 samples—this is also clear from the last panel (blue curve) in Fig. 1. Moreover, the uncertainty across the return period estimates obtained from 50 × 6 samples is considerably smaller than that obtained from 50 samples. This indicates that pooling of data reduces both the bias and uncertainty in the estimates.

Fig. 1.
Fig. 1.

Return period estimates (years) from the Monte Carlo experiment (described in section 3a) performed on 50 and 50 × 6 samples drawn from the observed 24-h annual maximum precipitation data at stations in the Susquehanna watershed. The red dashed line shows the observed (reference) return period. Bias is defined as the difference between the median return period (black line inside a box) and the reference return period. Estimation uncertainty is defined as the interquartile range (IQR) of the estimated return periods. The numbers in each panel indicate station IDs in the Susquehanna watershed. The blue curve in the last panel shows the difference in the absolute median biases estimated for 50 and 50 × 6 samples over all stations. A positive value along the blue curve indicates that the absolute median bias estimated from 50 samples is bigger than that estimated from 50 × 6 samples.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

Table 2.

Return period estimates from the Monte Carlo experiment performed on the synthetic data derived from the observed 24-h annual maximum precipitation data at stations in the Susquehanna watershed. StnId: station identifier, Rp.obs: return period in the observed data, Rp.S50: median return period estimated from 50 samples, Rp.S300: median return period estimated from 50 × 6 samples, aBias.S50: absolute difference between Rp.S50 and Rp.obs, aBias.S300: absolute difference between Rp.S300 and Rp.obs. Units are in years.

Table 2.

We draw a similar conclusion from the Monte Carlo simulations performed on synthetic data drawn from the observed 24-h annual maximum precipitation for the Florida peninsula (Fig. 2 and Table 3). Here, too, the range of absolute bias computed from 50 × 6 samples (0–0.25 years; column “aBias.S300” in Table 3) is smaller than that from 50 samples (0.01–0.54 years; column “aBias.S50” in Table 3). The median absolute bias over the peninsula is also smaller for 50 × 6 samples (0.03 years) than for 50 samples (0.13 years), and the absolute bias from 50 × 6 samples remains lower than that from 50 samples about 80% of the time. The estimation uncertainty is smaller for 50 × 6 samples as indicated by IQR in Fig. 2.

Fig. 2.
Fig. 2.

As in Fig. 1, but for the Florida peninsula.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

Table 3.

As in Table 2, but for the Florida peninsula.

Table 3.

To show that the method of pooling model data works for precipitation of different durations, we repeated the Monte Carlo simulations test on 1-, 6-, and 12-h annual precipitation maxima over the Susquehanna. Also, to account for the fact that synthetic data have smaller uncertainty than the observed data, we added a red noise with a ±5% standard deviation to the synthetic data generated at steps 2 and 7 of the Monte Carlo procedure mentioned above. Moreover, to demonstrate that the method of pooling data is applicable for distributions other than the GEV distribution, we fitted five candidate distributions to the data: generalized logistic (GLO), generalized extreme value (GEV), generalized normal (GNO), Pearson type III (PE3), and generalized Pareto (GPA). We select the best fitted distribution using a goodness-of-fit measure proposed by Hosking and Wallis (1997). The goodness-of-fit measure (Zdist) estimates how well the L-kurtosis of the fitted distribution matches with that of the sample data. If a candidate distribution represents the true underlying distribution, its goodness-of-fit measure should have approximately a normal distribution. Hosking and Wallis (1997) suggest that a fit is considered adequate if Zdist is sufficiently close to zero, a reasonable criterion being |Zdist| < 1.645 (corresponding to the 90% confidence interval). However, Hosking and Wallis (1997) clarify that the criterion is a rough indicator of goodness of fit and is not recommended as a formal test of significance. We use the distribution with the smallest |Zdist| satisfying the above criteria to estimate the observed return period. The results are shown in the Tables S1–S3 and Figs. S1–S3 in the online supplemental material. These results confirm that pooling of model data reduces the bias and uncertainty in the return period estimates of precipitation across various durations.

b. Evaluation of AMP in climate models

1) Bias in the mean annual maximum precipitation (MAM)

Figure 3 shows bias (model minus observation) in MAM over the Susquehanna and Florida peninsula. For the Susquehanna (Fig. 3a) the observed MAM over the Susquehanna ranges between 45 and 85 mm day−1 and generally increases from north to south. The maximum precipitation is observed along the southern edge of the watershed. The figure shows that there exists considerable variability in the bias across models. In general, model biases range between −20 and +20 mm day−1 for most of the stations. In particular, CanESM2.CanRCM4 (A), MPI-ESM-LR.CRCM5-OUR (I), and MPI-ESM-LR.CRCM5-UQAM (J) have positive biases in MAM over most parts of the watershed. Whereas, some models [e.g., CanESM2.CRCM5-OUR (B), HadGEM2-ES.WRF (H), and MPI-ESM-LR.WRF (L)] show positive bias in the northern areas of the watershed and negative bias in southern areas.

Fig. 3.
Fig. 3.

Bias (model minus observation) in 24-h mean annual maximum precipitation (MAM; mm day−1). (a) The top-left panel shows the observed MAM over the Susquehanna watershed and uses the color scale along the right edge of the figure. Other panels show bias in the MAM (model minus observation) and use the color scale along the bottom edge of the figure. The MAM is computed for the period 1951–2005 in the observational data and for 1956–2005 in model data. (b) As in (a), but over the Florida peninsula. The polygons in (b) represent the boundary of the Kissimmee–Southern Florida watershed.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

The observed MAM in the Florida peninsula ranges between 90 and 150 mm day−1 (Fig. 3b). The maximum MAM is observed along the southeastern edge of the peninsula. Generally, coastal areas receive higher precipitation than inland areas. The lowest rainfall (<110 mm day−1) is observed along the northern and southeastern edges of the Kissimmee–Southern Florida watershed. Generally, most models show strong negative biases (>20 mm day−1) in MAM across most of the basin. The largest and the most widespread dry biases are observed in CanESM2.CanRCM4 (A), CanESM2.CRCM5-UQAM (C), GFDL-ESM2M.WRF (F), HadGEM2-ES.WRF (H), and MPI-ESM-LR.WRF (L). Noticeably, MPI-ESM-LR.CRCM5-OUR (I) exhibits positive bias over most stations, except those at the southeastern edge of the peninsula.

In summary, for the Susquehanna there is considerable variability in the magnitude, sign, and pattern of biases across models, whereas in the Florida peninsula most models show dry biases throughout. Generally, models exhibit biases of larger magnitude over the Florida peninsula than over the Susquehanna.

2) The Taylor diagram

Figure 4a and 4b show performance of climate models in simulating the spatial pattern of the observed MAM. For the Susquehanna (Fig. 4a), MPI-ESM-LR.CRCM5-OUR (I) agrees best with the observation followed by CanESM2.CRCM5-UQAM (C) and MPI-ESM-LR.CRCM5-UQAM (J). MPI-ESM-LR.CRCM5-OUR (I) has the least centered NRMSD resulting from the highest correlation (~0.85) and the NSD close to 1. Models CanESM2.CanRCM4 (A), GFDL-ESM2M.RegCM4 (E), CanESM2.CRCM5-OUR (B), and GFDL-ESM2M.CRCM5-OUR (D) have similar correlation, but CanESM2.CRCM5-OUR (B) and GFDL-ESM2M.CRCM5-OUR (D) have standard deviation much lower (less than half) than in the observation. This results in the largest RMS error in CanESM2.CRCM5-OUR (B) and GFDL-ESM2M.CRCM5-OUR (D).

Fig. 4.
Fig. 4.

Taylor diagrams of 24-h MAM comparing station observations and models. (a) For the Susquehanna. Each diagram shows how closely the spatial pattern of the MAM in the observation resembles that in a model. The reference point (observation) is marked as a solid green square. Letters indicate the position of each model. The dashed black lines on the outermost semicircle indicate pattern correlation of MAM between the observation and models. The blue dashed curves indicate the normalized standard deviation (NSD) defined as spatial standard deviation of MAM in models and observation divided by the spatial standard deviation in the observation. The standard deviation is measured as the radial distance from the origin. The green dashed curves show the normalized root mean squared difference (NRMSD) defined as the RMSD between model and observation divided by the standard deviation of the observed MAM. The NRMSD is measured as a distance from the reference point (solid green square). (b) As in (a), but for the Florida peninsula.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

The Taylor diagram for Southern Florida is shown in Fig. 4b. MPI-ESM-LR.WRF (L) outperforms other models as it has the correct standard deviation of the MAM (equal to the observation) and the highest correlation skill (nearly 0.7) resulting in the lowest RMS error. MPI-ESM-LR.WRF (L) and MPI-ESM-LR.RegCM4 (K) have about the same standard deviation as in the observation, but MPI-ESM-LR.RegCM4 (K) has much lower correlation skill, which makes its NRMSD higher than MPI-ESM-LR.WRF (L). CanESM2.CanRCM4 (A) and CanESM2.CRCM5-OUR (B) perform poorly compared to other models in this region. Both of these models have negative correlation skill resulting in their NRMSD being larger than the other models in the group.

In summary, models show slightly better skills in simulating the spatial variability of the observed MAM over the Susquehanna than over the Florida peninsula. CanESM2.CanRCM4 (A), CanESM2.CRCM5-OUR (B), and GFDL-ESM2M.CRCM5-OUR (D) perform least well in both regions.

3) Bias in the interannual interquartile range of AMP

Figure 5 shows bias in the IQR of AMP in terms of the ratio of the interannual IQR of AMP in models over that in the observation. The panel “obs” in Fig. 5a shows the interannual IQR of AMP (in mm day−1) in the observation over the Susquehanna. The IQR of the observed AMP varies between 10 and 40 mm day−1. The interannual IQR generally increases from north to south—this pattern is consistent with the pattern of the observed MAM, as noted in Fig. 3, that generally increases from north to south. Noticeably, a majority of the models underestimate the observed interannual variability. In contrast, CanESM2.CanRCM4 (A) and MPI-ESM-LR.CRCM5-UQAM (J) overestimate the observed temporal variability at most of the stations in the watershed. Overall, the IQR of AMP for MPI-ESM-LR.CRCM5-OUR (I) over most of the stations is the closest to that in the observation (ratio within 0.5–1.5 range) indicating that this model best simulates the interannual variability among NA-CORDEX models. Generally, biases vary with the magnitude of the observed interannual IQR.

Fig. 5.
Fig. 5.

The ratio (model value over observed value) of interannual IQR of 24-h annual maximum precipitation (AMP). (a) The top-left panel shows the observed IQR of the AMP (mm day−1) for the Susquehanna, and uses the color scale along the right edge of the figure. The other panels show the interannual IQR ratio of the AMP in models and observation and use the color scale along the bottom edge of the figure. The ratio is unitless. (b) As in (a), but for the Florida peninsula. The polygons in (b) represent the boundary of the Kissimmee–Southern Florida watershed.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

For the Florida peninsula (Fig. 5b), the interannual IQR of the observed AMP varies between 25 and 85 mm day−1 for most of the peninsula except southeastern coastal region. The interannual variability of the observed AMP is generally higher at stations in the coastal areas than at stations that are in the middle of the peninsula. The highest variability is observed along the southeastern edge of the region. The spatial pattern of the observed temporal variability is consistent with that of the observed mean AMP as noted in Fig. 3. A majority of models underestimate the observed interannual variability at a majority of stations [CanESM2.CRCM5-UQAM (C), GFDL-ESM2M.RegCM4 (E), GFDL-ESM2M.WRF (F), HadGEM2-ES.RegCM4 (G), HadGEM2-ES.WRF (H), and MPI-ESM-LR.WRF (L)], whereas some models such as MPI-ESM-LR.CRCM5-OUR (I) and MPI-ESM-LR.CRCM5-UQAM (J) overestimate the observed interannual variability at most of the stations. In nearly all of the models the largest biases in the interannual IQR are observed along the coast, most prominently in the southeastern stations. This pattern suggests that, generally, biases in the IQR vary with the magnitude of the observed IQR.

In summary, models generally underestimate the observed IQR in both the regions. Also, model biases vary with the IQR magnitude of the observed AMP. The magnitude of model biases in the interannual variability is generally larger over the Florida peninsula than over the Susquehanna.

4) Estimation of IVSS

Figure 6 shows IVSS of models. For the Susquehanna (Fig. 6a), models MPI-ESM-LR.CRCM5-OUR (I), HadGEM2-ES.WRF (H), MPI-ESM-LR.CRCM5-UQAM (J), and CanESM2.CRCM5-UQAM (C) have the lowest IVSS values (<0.5) indicating that these models best simulate the observed interannual variability. GFDL-ESM2M.CRCM5-OUR (D) and CanESM2.CanRCM4 (A), on the other hand, have much larger IVSS values (>1).

Fig. 6.
Fig. 6.

The IVSS as expressed in Eq. (1). The IVSS is a unitless quantity. The closer the IVSS value of a model to zero, the better is the performance of the model in simulating the interannual variability of the observed AMP. The horizontal dashed line indicates the chosen IVSS threshold (=1.13) as defined in section 3. (a) For the Susquehanna. (b) For the Florida peninsula.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

Over the Florida peninsula (Fig. 6b) MPI-ESM-LR.RegCM4 (K), MPI-ESM-LR.CRCM5-UQAM (J), CanESM2.CanRCM4 (A), and GFDL-ESM2M.CRCM5-OUR (D) have the smallest IVSS values (<0.5) indicating that these models perform the best in capturing the observed interannual variability. GFDL-ESM2M.WRF (F), HadGEM2-ES.WRF (H), and MPI-ESM-LR.WRF (L) perform poorly compared to other models. Noticeably, the IVSS value of majority of models in the Florida peninsula is spread over a narrow range of IVSS values (0.4–0.75) indicating that these models have comparable skill in simulating the observed interannual variability. Interestingly, models CanESM2.CanRCM4 (A) and GFDL-ESM2M.CRCM5-OUR (D) that perform least well in the Susquehanna are among the best performers in the Florida peninsula. Conversely, model HadGEM2-ES.WRF (H) is the second best performer over the Susquehanna, but the second to the worst performer over the Florida peninsula.

5) Overall model performance

Figure 7 shows a scatter diagram of models’ NRMSD values (x axis) from the Taylor diagram against IVSS values (y axis). The figure shows overall performance of models in simulating the spatial and temporal variability of the observed AMP. As shown in Fig. 7a, for the Susquehanna, models that perform relatively better in simulating spatial variability of the observed AMP (relatively lower NRMSD) also perform better in simulating the temporal variability (relatively lower IVSS) and vice versa. This is also evident from a high positive correlation of 0.7 between NRMSD and IVSS.

Fig. 7.
Fig. 7.

Scatter diagrams of NRMSD as computed in the Taylor diagram (x axis) against IVSS values (y axis). “Corr” indicates the correlation between NRMSD and IVSS for all models. (a) For the Susquehanna. (b) For the Florida peninsula.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

For the Florida peninsula (Fig. 7b), the majority of models that perform relatively better in simulating the observed spatial variability perform relatively poorly in simulating the observed temporal variability and vice versa. This is also evident from a high negative correlation of −0.5 between NRMSD and IVSS values for all models considered.

6) Selection of models

Figure 8 shows the scatter diagram of the product of correlation skill and normalized standard deviation (x axis) against IVSS (y axis). The dashed horizontal and vertical lines are drawn using our selection criteria defined in section 3b(3). The models that are selected by us for further evaluation are located in the green shaded region. Models selected for the Susquehanna are CanESM2.CRCM5-UQAM (C), GFDL-ESM2M.RegCM4 (E), GFDL-ESM2M.WRF (F), HadGEM2-ES.RegCM4 (G), HadGEM2-ES.WRF (H), MPI-ESM-LR.CRCM5-OUR (I), MPI-ESM-LR.CRCM5-UQAM (J), MPI-ESM-LR.RegCM4 (K), and MPI-ESM-LR.WRF (L), and for the Florida peninsula are CanESM2.CRCM5-UQAM (C), GFDL-ESM2M.RegCM4 (E), HadGEM2-ES.RegCM4 (G), MPI-ESM-LR.CRCM5-OUR (I), MPI-ESM-LR.CRCM5-UQAM (J), and MPI-ESM-LR.RegCM4 (K).

Fig. 8.
Fig. 8.

Scatter diagrams of the product of correlation skill and NSD (x axis) against IVSS values (y axis). The horizontal and vertical dashed lines are drawn using the selection criteria discussed in section 3. Models in the green shaded region are selected for PIF estimation. (a) For the Susquehanna. (b) For the Florida peninsula.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

In summary, the analysis presented in section 4b demonstrates the reason for analyzing the historical performance of climate models before using them for quantifying future changes in extreme events. It is evident that none of the models show comparable skill across regions. Selecting models that have reasonable skill in simulating the observed spatial and temporal variability is expected to lead to greater confidence in future projections (Zhang and Soden 2019).

c. Bias correction of models and pooling of reasonably performing models

As noted in the previous section, since models exhibit large biases in simulating the spatial and temporal variability of the observed AMP, we apply bias correction to the historical and future simulations of all models. Further, based upon model evaluation we selected models using the criteria defined in section 3b(3). Finally, AMP data from the bias-corrected historical and RCP8.5 simulations of these models were pooled together for estimating future changes in 24-h precipitation events.

d. Estimation of changes in precipitation extremes

1) Changes in the MAM

Figure 9 shows differences (RCP8.5 minus historical) in the MAM computed from raw (not bias-corrected) historical and RCP8.5 simulations. In the Susquehanna (Fig. 9a) most of the models project an increase, at the 5% significance level, of 5–20 mm day−1 in the MAM across the watershed. The largest increase (>20 mm day−1) is projected in CanESM2.CanRCM4 (A). As for the Florida peninsula (Fig. 9b), although most of the models project an increase in the MAM, only a few of them [CanESM2.CanRCM4 (A), GFDL-ESM2M.RegCM4 (E), and HadGEM2-ES.RegCM4 (G)] project significant increases (at the 5% level) of 10–50 mm day−1 in the MAM throughout the peninsula. MPI-ESM-LR.CRCM5-OUR (I) projects a decrease in the MAM in parts of the peninsula, although the decrease is not significant.

Fig. 9.
Fig. 9.

Differences (RCP8.5 minus historical) in 24-h mean MAM (mm day−1) computed from raw (not bias-corrected) historical and RCP8.5 simulations. The MAM is calculated for the period 1956–2005 in the historical simulations and for 2049–2098 in the RCP8.5 simulations. Stippling shows differences significant at the 5% significance level. (a) For the Susquehanna. (b) For the Florida peninsula.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

In summary, although an increase in the MAM over both the regions is projected in most of the models, a larger variability in both the sign and magnitude of changes in the MAM across models is projected over the Florida peninsula than over the Susquehanna.

2) Changes in 24-h PIF estimates: Individual models vs median vs pooled

Figure 10 shows 24-h PIF estimates in the bias-corrected historical and RCP8.5 simulations at a station within each study region. We arbitrarily selected these stations for presentation purposes, since we cannot show 24-h PIF curves for all stations due to the space limitation. The panel label “median-all” refers to 24-h PIF estimates computed from taking the median of individual 24-h PIF estimates. The label “median-pooled” refers to 24-h PIF estimates computed from taking the median of individual 24-h PIF estimates from models that are involved in pooling. Finally, “pooled” indicates 24-h PIF estimates computed from pooling the reasonably performing models. We used “median” PIF estimates, since, median is less affected by the presence of outlier models. For the Susquehanna, as is evident from Fig. 10a, all models except CanESM2.CRCM5-OUR (B), GFDL-ESM2M.RegCM4 (E), and MPI-ESM-LR.RegCM4 (K), project an increase in 24-h precipitation for all return periods. But, the uncertainty in the projected increase (blue curve values minus red curve values) across models is quite large. Apparent from this figure, MPI-ESM-LR.RegCM4 (K) projects little change in the 24-h PIF estimates, whereas CanESM2.CanRCM4 (A) projects the largest increase among all models for all return periods (for instance, around 75 mm day−1 for the 50-yr return period). GFDL-ESM2M.RegCM4 (E) projects a decrease in 24-h PIF estimates for 20-yr or longer return periods. Another noticeable feature is that a large estimation uncertainty (90% CI around the estimates) exists in the historical and RCP8.5 PIF estimates. This uncertainty is partly due to small sample sizes (50 years) in climate models. The projected changes in both the median-all and median-pooled 24-h PIF estimates suggest increases in the precipitation of comparable magnitude for all return periods, but this increase may not be statistically significant because of large and overlapping estimation uncertainties around the estimates. The pooled 24-h PIF estimates are computed by pooling bias-corrected simulations of CanESM2.CRCM5-UQAM (C), GFDL-ESM2M.RegCM4 (E), GFDL-ESM2M.WRF (F), HadGEM2-ES.RegCM4 (G), HadGEM2-ES.WRF (H), MPI-ESM-LR.CRCM5-OUR (I) and MPI-ESM-LR.CRCM5-UQAM (J), MPI-ESM-LR.RegCM4 (K), and MPI-ESM-LR.WRF (L). For all return periods, the increase in 24-h PIF estimates from pooled models is similar in magnitude to that in the two median cases, although the estimation uncertainty is much smaller. Noticeably, since the 90% confidence intervals around the estimates (yellow shading around red curve and green shading around blue curve) in the pooled case do not overlap, the change in the 24-h PIF estimates is deemed statistically significant for all return periods.

Fig. 10.
Fig. 10.

PIF estimates of 24-h precipitation totals in the bias-corrected historical and future simulations. The red curve and yellow shading indicate 24-h PIF estimates and corresponding 90% confidence interval in the bias-corrected historical simulation. The blue curve and green shading indicate 24-h PIF estimates and corresponding 90% confidence interval in the bias-corrected RCP8.5 simulation. The median-all panel shows the median of 24-h PIF estimates from all models, while the median-pooled shows the median of 24-h PIF estimates from models that are used for pooling. Pooled shows 24-h PIF estimates computed from pooling of better performing models. Models that are used for pooling are shown in red letters in the top-left corner of the figures. The x axis indicates return periods in years, and the y axis indicates intensity (mm day−1). (a) For the Susquehanna. (b) For the Florida peninsula.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

Over the Florida peninsula (Fig. 10b) all models project an increase in 24-h precipitation for all return periods except model MPI-ESM-LR.RegCM4 (K) that projects a decrease in the precipitation for 25-yr or longer return periods. As in the Susquehanna, the uncertainty in projected changes in the 24-h PIF estimates across models is large (e.g., the projected change for 50-yr return period ranges from around −6 mm day−1 [MPI-ESM-LR.RegCM4 (K)] to 150 mm day−1 [HadGEM2-ES.RegCM4 (G)]. Also, the uncertainty is large in all climate models. Both of the median cases project an increase in 24-h PIF estimates for all return periods examined, but the changes do not seem to be statistically significant because confidence intervals from historical and RCP8.5 simulations overlap and also because confidence intervals from one simulation include PIF estimates from the other simulation. Models that are used for pooling are CanESM2.CRCM5-UQAM (C), GFDL-ESM2M.RegCM4 (E), HadGEM2-ES.RegCM4 (G), MPI-ESM-LR.CRCM5-OUR (I), MPI-ESM-LR.CRCM5-UQAM (J), and MPI-ESM-LR.RegCM4 (K). The historical and future 24-h PIF estimates in the pooled cases are similar in magnitude as in the median cases. Both of the median cases and pooled models project an increase of around 50 mm day−1 in 24-h precipitation for the 50-yr return period. However, the changes projected by the pooled models seem to be statistically significant in contrast to the changes projected by the median cases.

Figure 11a shows changes in 5-yr precipitation over the Susquehanna in bias-corrected models. In Fig. 11 (and all subsequent figures) the significance of change is estimated at the 5% significance level as described in section 3f. All models project increases in 24-h precipitation at most of the stations, although both the magnitude of change and its significance vary considerably across models. For instance, CanESM2.CanRCM4 (A) projects the largest changes (>25 mm day−1) at most of the stations, whereas, MPI-ESM-LR.CRCM5-OUR (I) projects an increase (<5 mm day−1) that is not statistically significant at the 5% level. The median-all combination projects significant increases (at the 5% level) of 10–25 mm day−1 at most of the stations except the stations in the southeast corner. The median-pooled combination projects significant increases in the precipitation at stations located in the western half of the watershed. The magnitude of changes in the pooled models is similar to that in both of the median cases; but the pooled models project statistically significant changes (at the 5% level) at all stations across the watershed. For 50-yr precipitation (Fig. 11b), although most of the models project increases in the precipitation, only a few of them such as CanESM2.CanRCM4 (A), GFDL-ESM2M.CRCM5-OUR (D), and HadGEM2-ES.RegCM4 (G) project a statistically significant increase (at the 5% level) at a few stations. Noticeably, some models such as GFDL-ESM2M.RegCM4 (E) and MPI-ESM-LR.CRCM5-UQAM (J) project decreases in extreme precipitation at a few stations, though statistically not significant. Both of the median cases show increases (<50 mm day−1) in the precipitation that are generally not significant. However, pooled models project significant increases (at the 5% level) of less than 50 mm day−1 in the precipitation at most of the stations. Our analysis of 24-h precipitation for other return periods (e.g., for 2-yr return period shown in the Fig. S4) shows that pooled models project a significant increase in the 24-h precipitation at more stations than the two median cases. We do not show the results for individual models beyond the 50-yr return period as by convention the data should not be extrapolated to evaluate return periods longer than the sample size (Sadegh et al. 2018; J. R. M. Hosking 2019, personal communication).

Fig. 11.
Fig. 11.

Changes in 24-h precipitation (mm day−1) for (a) 5-yr and (b) 50-yr return periods in the Susquehanna. Differences significant at the 5% significance level are shown as solid squares and those not significant at 5% are shown as blank circles. The significance is computed from the Z statistic as shown in Eq. (6).

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

Over the Florida peninsula (Fig. 12a), although a majority of models project an increase in 5-yr precipitation at most of the stations, only a couple of them project significant increases at the 5% level throughout the peninsula [CanESM2.CanRCM4 (A), GFDL-ESM2M.RegCM4 (E), and HadGEM2-ES.RegCM4 (G)]. Interestingly, MPI-ESM-LR.CRCM5-OUR (I), one of the pooled models, projects a decrease in the precipitation. Both of the median cases show a significant increase at the 5% level in the precipitation between 10 and 50 mm day−1 at nearly half of the stations, mostly north of 27°N. As noted before, the pooled models project a statistically significant increase (<50 mm day−1) in the precipitation at most of the stations. For 50-yr precipitation (Fig. 12b), except model CanESM2.CanRCM4 (A), most of the models do not show significant changes at most of the locations. Some models that show an increase in the 5-yr precipitation at some stations project a decrease in 50-yr precipitation at those same stations [e.g., CanESM2.CRCM5-UQAM (C), GFDL-ESM2M.WRF (F), and MPI-ESM-LR.RegCM4 (K)]. In contrast, MPI-ESM-LR.CRCM5-OUR (I) projects an increase in 50-yr precipitation at some stations in the Kissimmee–Southern Florida watershed accompanied by a decrease in 5-yr precipitation. Both of the median cases project an increase, though not significant, in 50-yr precipitation. The pooled models show significant increases over most of stations across the peninsula. For 24-h precipitation with 2-yr return period (Fig. S5), pooled models project a statistically significant changes (at the 5% level) in 24-h precipitation at more stations than the two median cases.

Fig. 12.
Fig. 12.

As in Fig. 11, but for the Florida peninsula.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

In summary, changes in 24-h precipitation projected by both the median-all and median-pooled combinations are similar in magnitude and significance. Noticeably, over both study regions, pooled models project statistically significant increases at the 5% level in 24-h precipitation for all return periods examined at more stations than in the two median approaches.

3) Changes in 24-h precipitation in the pooled models

To summarize the results from pooled models we show projected changes in 24-h precipitation from the pooled models for 2-, 5-, 10-, 25-, 50-, and 100-yr return periods over the Susquehanna and Florida peninsula in Fig. 13. For the Susquehanna, Fig. 13a, the pooled models project a significant increase at the 5% level in the precipitation for all return periods at almost all the stations (≥90%). The magnitude of increase in the precipitation increases with increasing return periods. This suggests that, on average, the Susquehanna is expected to observe a statistically significant increase in 24-h precipitation for all return periods examined.

Fig. 13.
Fig. 13.

Changes in 24-h precipitation (mm day−1) for 2-, 5- 10- 25-, 50-, and 100-yr return periods computed from pooled models. The differences that are significant at the 5% significance level are shown as solid squares, and those not significant at the 5% level are shown as blank circles. The significance is computed from the Z statistic as shown in Eq. (6). “Signif. stns” shows the percentage of stations at which the differences are significant. (a) For the Susquehanna. Pooled models used: C, E, F, G, H, I, J, K, and L. (b) For the Florida peninsula. Pooled models used: C, E, G, I, J, and K.

Citation: Journal of Hydrometeorology 22, 4; 10.1175/JHM-D-20-0180.1

For the Florida peninsula, the pooled models project statistically significant increases at the 5% level in 2–10-yr precipitation at most of the stations (≥86%). For 25–100-yr return periods the precipitation is expected to increase for at least two-thirds of the stations over the Florida peninsula.

5. Summary

In this work we propose a novel extension of current methods for computing multimodel IDF estimates. The method is based upon the assumption that each model represents distinct realizations of the underlying Earth system, and data from reasonably performing models represent samples from the true underlying data-generating distribution. Therefore, pooling information from various models can help reduce bias and uncertainty in the IDF estimates. The motivation for this approach is that a larger sample from the true underlying distribution provides more information and enhanced fitting of the extreme value distribution to the data, as compared to lower sample sizes. We employed Monte Carlo simulations to test the proposed pooling method on synthetic data derived from the distributions of the observed 1-, 6-, 12-, and 24-h duration annual maximum precipitation. The simulation results suggest that pooling of annual maxima reduces the bias and uncertainty in the estimated precipitation return periods across various durations, resulting from the enhanced distribution-fitting of the data.

We applied the pooling method to estimate future changes in 24-h precipitation for 2–100-yr return periods over the Susquehanna watershed and Florida peninsula in a suite of regional climate models from the NA-CORDEX project. The method involves the following steps: First, assess the historical performance of models in capturing spatial and temporal variability of the observed annual maximum precipitation (AMP). The model skill for simulating spatial variability of long-term mean of the observed AMP (MAM) is assessed using Taylor diagram, whereas skill for simulating temporal variability of the AMP is assessed using the interannual variability skill score (IVSS). Second, bias correct the historical and future (RCP8.5) annual maximum precipitation (AMP) data. Third, pool each year’s historical and RCP8.5 AMP data of reasonably performing models, and finally, quantify significant future changes in 24-h precipitation for 2–100-yr return periods by fitting a GEV distribution in a nonstationary framework. Our approach aims to address limitations of previous studies that estimate IDF. Through analysis of historical model performance and selection of reasonably performing models, we can enhance the credibility of future projections. This step is also important because model data are inherently uncertain, and the historical evaluation of models ensures that models reasonably (using an objective performance criteria) capture the statistics of the observed data, reducing the uncertainty across models. Pooling model data promotes a superior distribution fit and thereby more reliable IDF estimates. Last, our method avoids traditional mean or median based approaches for computing multimodel IDF estimates, which result into a lessening of the intensity of probable extremes, and are also impaired by large estimation uncertainty around the median (mean) estimates.

Our analysis indicates that most models exhibit negative bias in the mean annual maximum precipitation over the Florida peninsula. In contrast, there exists considerable variability across models in the magnitude, sign, and pattern of biases in the MAM over the Susquehanna watershed. Models generally underestimate the interannual IQR of the observed AMP in both the regions. Detailed analyses using Taylor diagram and IVSS metrics indicate that models do not perform consistently across regions. For the Susquehanna, models that perform well in simulating the spatial pattern of the long-term mean AMP (MAM) also perform well in simulating the observed interannual temporal variability of the AMP and vice versa. But, for the Florida peninsula, models that perform well in simulating the temporal variability of the observed AMP fail to capture the spatial variability of the observed MAM and vice versa. This indicates the importance of carefully selecting models for further analysis.

Using the performance criteria defined in section 3b(3), nine models are selected for the Susquehanna and six models are selected for the Florida peninsula for estimating future changes in 24-h precipitation estimates. Our results show that the 24-h precipitation estimates for 2–50-yr return periods for both the historical and RCP8.5 simulations in pooled models have smaller estimation uncertainty than in individual models and in cases where medians of the 24-h PIF estimates are used. Moreover, over both study regions, pooled models project statistically significant increases at the 5% level in 24-h precipitation for all return periods examined (2–50 years) at more stations than in the median approaches, implying that conventional median-based IDF estimations may underestimate the risks of a changing climate.

Estimation of changes in 24-h precipitation using the pooled models suggests that most of the stations (≥90%) in the Susquehanna are expected to observe a statistically significant (at the 5% level) increase in 24-h precipitation for all return periods examined (2–100 years), whereas at least two-thirds of the stations over the Florida peninsula will observe statistically significant (at the 5% level) increases in 24-h precipitation for all return periods examined.

In this paper we analyze annual maximum precipitation for projecting changes in 24-h precipitation intensity–frequency estimates as generally PIF estimates/IDF curves are constructed using annual maximum precipitation data (e.g., NOAA Atlas 14 volume 1–9 reports). It will be interesting for future studies to analyze changes in seasonal PIF estimates/IDF curves (e.g., those based upon DJF or JJA annual maximum precipitation) as changes in seasonal precipitation extremes can be due to seasonal differences in the mix of precipitation-generating mechanisms. We propose that the proposed method is useful for both scientists and stakeholders (particularly water managers). IDF estimates constructed using this approach have the potential to inform climate policy and adaptation planning. In the future, we intend to extend this method to additional regions across the continental United States.

Acknowledgments

The authors sincerely thank Dr. Timothy DelSole of George Mason University, Dr. Seth McGinnis, and Dr. Eric Gilleland of National Center for Atmospheric Research (NCAR) for their valuable suggestions on the project. This work is supported by the Department of Energy Office of Science Award DE-SC0016605, “A Framework for Improving Analysis and Modeling of Earth System and Intersectoral Dynamics at Regional Scales.” Additional support comes from the USDA National Institute of Food and Agriculture, Hatch project Accession 1001953 and 1010971. The authors declare that they have no known competing financial interests.

Data availability statement

The station-based annual maximum precipitation data used in this study can be downloaded from the NOAA Atlas 14 website (https://hdsc.nws.noaa.gov/hdsc/pfds/pfds_map_cont.html). The model data used in this study are archived at the North American CORDEX Program website (https://na-cordex.org/index.html).

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