3.1. Difference Between Models and Reanalyses
 We start by computing the 3, 5, 10, 30, 50, and 100 year return level precipitation extremes for 6, 12, 18, 24, 36, 48, 72, 120, and 240 h duration at each grid cell using a 30 year moving window. It should be noted that since the five 21st-century simulations are continued from the twentieth century control runs, the 1970–2000 20C3M annual maxima are adopted as the initial values to support the moving window analysis into the 21st century. For clarity, the extreme rainfall estimates are labeled by the ending year of the window (e.g., year 2039 estimate was computed from 2010–2039 annual precipitation maxima).
 The year 1999 estimates (1970–1999 window) are illustrated in Figure 1 to provide a general understanding of precipitation extremes. The 24 h, 30 year precipitation extremes from NCEP1, ERA-40, CCSM3, and CSIRO, associated with their corresponding 10% and 90% bootstrapping uncertainty bounds are illustrated. The 10% and 90% bounds are computed based on a 1000 member bootstrap [Efron and Tibshirani, 1994; Kharin et al., 2007]. The largest difference occurs near the tropics (30S ∼ 30N), and the bootstrapping uncertainty is not as large as the difference across models and reanalyses.
Figure 1. The 24 h duration, 30 year return period precipitation extremes estimated from the 1970–1999 annual maxima. The 10% and 90% uncertainty bounds are computed based on a 1000 member bootstrap.
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 To assist in further interpretation, the ERA-40, CCSM3, and CSIRO estimates shown in Figure 1 are bilinearly interpolated to the NCEP1 resolution for direct comparison. (We note that the interpolated data sets were used only in generating Figure 2, discussed next.) Percentage difference, which is defined as 100*(A − B)/[(A + B)/2] between variables A and B, is calculated for each grid cell and illustrated in Figure 2. The latitudinal averages are also shown. As expected [Kharin et al., 2007; O'Gorman and Schneider, 2009a, 2009b], the relative agreement of the climate model simulations with reanalysis data sets, as well as between the climate models at the extratropics (higher than 30 degree latitudes), contrast with the disagreement in the tropics. The disagreement in the tropics between model-model, model-reanalysis, and reanalysis-reanalysis pairs in Figure 2 suggests that future research may be needed to understand and improve modeling of the physics that drive tropical precipitation extremes. The largest difference in tropical extremes appears to occur between the two reanalysis data sets (this verified in Table 1, which is discussed in the next paragraph), while the two climate models show relatively better agreement with each other. The difference of spatial resolution among data sets may introduce another source of bias in extreme precipitation statistics; however, Chen and Knutson  and Kharin et al.  examined the difference between NCEP2 and ERA-40 and concluded that return values were reduced only by a few percentage points. The majority of variation in the tropics, then, may not be a result of the differences in spatial resolution.
Figure 2. Percentage difference of 24 h duration, 30 year return period precipitation estimated from 1970 to 1999 extremes from model and reanalysis data sets.
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Table 1. Summary of Precipitation Extremes and Their Uncertainty Bounds Compared Across Pairs of Climate Models and Reanalysisa
|Models||Range of Percentage Difference|
|<−50%||−50% ∼ −20%||−20% ∼ 20%||20% ∼ 50%||>50%|
 Following Figure 2, the percentage difference of precipitation extremes between NCEP1, ERA-40, CCSM3, and CSIRO is summarized in Table 1. We divide the percentage difference into five ranges (<−50%, −50% ∼ −20%, −20% ∼ 20%, 20% ∼ 50%, and >50%) and report the percentage of grid cells that lie in each range. The percentage difference is also computed for the 10% and 90% uncertainty bounds shown in Figure 1. Around 40% ∼ 50% of the total grid cells fall in the −20% ∼ 20% range and around 70% ∼ 80% fall in the −50% ∼ 50% range. Perhaps the most interesting insight from Table 1 is that the maximum similarity is found between CSIRO and CCSM3, while the least is between NCEP1 and ERA-40. It may be nonintuitive that the largest disagreement is not between models and reanalysis. This is interesting and may suggest the need for deeper exploration of precipitation physics and parameterization schemes, as well as improved quantitative precipitation estimates for generating more accurate and more consistent observations. Studies on differences between the two families of reanalysis data sets as compared to observational estimates of variables of interest can be found in the works of Trenberth et al.  (tropical temperature) and Zolina et al.  (European precipitation extremes). Further exploration of reanalysis quality, specifically with regard to tropical precipitation extremes, may be of value. With the tropical region excluded (30S ∼ 30N), approximately 55% ∼ 65% of the total grid cells fall in the −20% ∼ 20% range, and 85% ∼ 95% of the total grid cells fall in the −50% ∼ 50% range. The largest observation/model inconsistency for precipitation extremes is in the tropical regions. Though the above illustrations are only based on the 24 h, 30 year precipitation extremes, we note that similar patterns are observed for other durations, recurrence intervals, and temporal windows as well. The annual maxima and the corresponding derived rainfall estimates are used in the following analysis.
3.2. Contribution of Convective Precipitation in Annual Maximum Rainfall
 The reanalysis data sets and climate model outputs provide a unique opportunity to examine the influence of modeled convective precipitation on rainfall extremes, even though they are not direct observations. The percentage contribution of convective rainfall in the AMP series is examined next. These percentage contributions are illustrated in Figure 3, in which the average contribution of convective precipitation in each of 1979–1999 NCEP1, NCEP2, ERA-40, and CCSM3 annual maximum is computed at all latitudinal bands. We note that the CSIRO convective rainfall outputs are not achieved continuously at a daily scale within WCRP's CMIP3 and hence are excluded from this part of analysis.
Figure 3. Contribution of convective precipitation to the annual maxima according to climate model simulations and reanalysis data sets.
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 In Figure 3a, the average percentage contribution of convective rainfall within 6 h annual maxima along latitudinal bands is shown to vary substantially. Convection is generally considered to be the predominant generative mechanism of precipitation extremes, especially for shorter subdaily duration (with exceptions; see Scinocca and McFarlane ). While convective contributions appear to be more dominant in the tropics, the significant differences between reanalysis and models may diminish the value of any generic insights. CCSM3-simulated convective contributions to 6 h extremes are in between NCEP1, NCEP2 (nearly 90%), and ERA-40 (barely 50%). When combined with the uncertainty in the tropics, these results suggest that significant improvements are needed in our understanding of precipitation processes beyond what is suggested in the literature [Allan and Soden, 2008; Lenderink and van Meijgaard, 2008; O'Gorman and Schneider, 2009a, 2009b; Pall et al., 2007; Diffenbaugh et al., 2005; Alexander et al., 2009].
 In Figure 3b, the global area-weighted average of convective rainfall contribution is computed for various durations (note that duration is plotted in log-scale). The fact that NCEP1, ERA-40, and CCSM3 are nearly parallel to each other may suggest different parameterizations with constant bias, while the different behavior (nearly constant versus almost linear growth) of the subdaily versus daily or greater than daily extremes may point to differences in the underlying mechanisms generating shorter and longer duration extremes. The contribution of convection to subdaily duration precipitation extremes appear relatively constant with changing durations (even though the two reanalyses and –CCSM3 suggest different levels), but for longer (than one day) duration extremes, the contributions from convection appear to increase linearly (in log-scale) with duration. This suggests that the underlying mechanisms for shorter and longer duration extremes may be fundamentally different. While convective rainfall contributes the most in NCEP1, its contribution is least in ERA-40. Once again, the maximum difference occurs between two reanalysis data sets rather than between a reanalysis data set and CCSM3; the exact cause for this discrepancy may need to be investigated. Possible causes appear to be differences either in the precipitation parameterization schemes or the actual observations used to drive the forecasting models, which generate the two reanalysis data sets.
 The geographical pattern of convective contribution in 6 h annual maxima is plotted in Figure 4 for four models. The spatial variation is significant, with NCEP2 suggesting predominance of convection in most land areas but ERA-40 and NCEP1 showing convective precipitation mainly over tropical oceans. For NCEP1 (Figure 4a), convective precipitation has the largest contribution in the entire tropical band (30S∼30N, including both land and ocean), while for NCEP2 (Figure 4b), convective precipitation has a larger contribution on the land than on the ocean surface. Compared to NCEP1, the NCEP2 shows more convective activity over land, especially in the extratropics and in the northern hemisphere. The CCSM3 simulations (Figure 4c) appear to exhibit a noticeable discontinuity between land and ocean around continental Africa, south Asia, and northern Australia. The likely causes of this discontinuity could be due to the difference between land and ocean models in the CCSM3. The difference in surface heat capacity between ocean and land may trigger convection in different ways in the CCSM3 cumulus parameterization. Also, there may be a diurnal effect over land that is absent over ocean. ERA-40 (Figure 4d) exhibits the largest precipitation extreme depths, but the smallest contribution from convection. The precipitation extremes in ERA-40 are primarily controlled by a large-scale precipitation mechanism. We note that while Figure 4 shows the 6 h annual maxima results, these patterns are similar for other durations.
Figure 4. Regional variability in the average contribution of convective precipitation to the 6 h annual maxima precipitation extremes.
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3.3. Trend of Rainfall Extremes Under Warming Scenarios
 In order to understand variation of precipitation extremes over time, the previous analysis is performed repeatedly with a 30 year moving window. For each 30 year period, the average temperature (required in equation (4) for computing the CC ratio), GEV parameters, and 30 year rainfall depth are estimated. The year 1999 values (corresponding to the 1970–1999 window) are selected as a baseline for comparison. By setting T1 as the average surface temperature from 1970–1999, the CC ratio is computed for every grid and for all windows. Since the CC ratio represents the theoretical increase/decrease of saturated vapor density due to temperature change, it can be regarded as a theoretical reference of extreme rainfall variation to year 1999. Similarly, the depth ratios, which are defined as extreme rainfall estimates normalized by the corresponding year 1999 baseline values, are computed for comparison. The depth ratios represent the rate of change of extreme rainfall magnitudes that correspond to the same return period but estimated at different temporal windows. In addition, by referring to the year 1999 baseline values, we further compute the corresponding return periods (frequency) for different time periods.
 The analysis is performed for the two reanalyses (NCEP1, ERA-40) and two climate models (CCSM3, CSIRO). NCEP2 is not included here since its data coverage is insufficient for continuous analysis. Taking 24 h duration as an example, the global area-weighted medians of CC ratio, depth ratio, and return period are shown in Figure 5, with insets emphasizing the overlapping period from 1987–2008. While the median is suggested as a proper measure by Kharin et al. , the difference in grid sizes along various latitudes must be adjusted; otherwise the median will be biased toward the extratropics. To make a proper correction, the area-weighted median is computed instead. The area-weighted median [see Yin et al., 1996] is a general form of median in which each grid value is assigned a corresponding areal weight. By sorting the values and using the weights as widths, the 50% percentile is identified to be the area-weighted median. If all grids are equal sizes, the weighted median will be equivalent to the common median.
Figure 5. Intensification of precipitation extremes at global scales based on climate models and reanalysis data sets and the significance of Clausius-Clapeyron relationship.
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 As shown in Figures 5a and 5b, the Clausius-Clapeyron ratio almost exactly mirrors the depth ratio and it provides a theoretical reference of global precipitation extremes intensification owing to atmospheric warming. However, the trends from NCEP1 in the prior decade do not agree with the trends in climate model hindcasts and must be further investigated. The climate model hindcasts suggest that early twentieth century extreme intensities were less intense (about 96% of current) and less frequent (current 30 year intensities, or occurring with a probability of 1/30, were about 40 year then). Climate models project precipitation extremes to be more intense and frequent in the future. The worst case A1FI scenario projects an intensification of 30% with the 30 year rainfall event becoming as frequent as the current 7 year event. We note that the scenarios show considerable difference, suggesting that emissions may heavily influence the intensification and larger frequencies of precipitation extremes in the future. The Commit scenario, which sustains atmospheric concentration at year 1999 levels, shows the intensification in the future owing to both system delay of stabilization and temporal memory in the moving average calculation; the rainfall extreme stabilizes after approximately three decades. The intensity and frequency projections display similar trends but have considerably more uncertainties and variability at regional scales. The intensification and increasing frequencies of precipitation extremes in a warming environment, as well as the correspondence with the CC relation, are clearly illustrated in this approach, which also relies on extreme value statistics.
 Given the potential limitation of inferences regarding precipitation extremes at tropical regions (as discussed in section 3.1), Figure 6 illustrates the area weighted median of extratropical regions (90S ∼ 30S and 30N ∼ 90N). The general trends are similar to Figure 5, but the Clausius-Clapeyron provides a higher ratio in the extratropics. This may appear nonintuitive since the reanalysis and model pairs appear to match better in the extratropics (see Figures 1 and 2, as well as O'Gorman and Schneider [2009a, 2009b]). It suggests that equation (4) and the use of surface temperature may be overly simple. Because this may be the case, regional and local scale trends may not be captured, even though the relationship appears reasonable at a global-average scale. Figure 7 shows the results for Europe. We selected Europe as a case study because, visually, there is a good match between the various reanalysis and climate model pairs. We observe that the intensification of precipitation extreme trends does not appear obvious from NCEP1 but is relatively clear from ERA-40. The general trends agree with Figures 5 and 6, while the variability is larger compared to the others, probably because there are considerably fewer grid cells.
Figure 6. Intensification of precipitation extremes at the extratropical regions based on climate models and reanalysis data sets as well as the significance of Clausius-Clapeyron relationship.
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 The general temporal trends are observed in Figures 5–7, and spatial variability is shown to be large. An example is shown in Figure 8, in which the 6 h duration precipitation extremes from NCEP1 and CCSM3 are illustrated in a more detailed fashion for the extratropical land (90S ∼ 30S, 30N ∼ 90N, land). By referring to the year 1999 30 year return level estimates at each grid cell, Figure 8a displays histograms (spanning all grid cells) of the corresponding return levels of the year 1977 (1948–1977 window) and year 2008 (1979–2008 window) NCEP1 precipitation extremes. Less than half (48%) of the grid cells exhibit precipitation extremes that are less than the 30 year level at year 1977, implying that these extremes were less frequent historically. Conversely, more than half (59%) of the grid cells show precipitation extremes less than 30 year levels during year 2008, implying more frequent extremes. However, it should be noticed that spatial variability is large, and the trend may be flat or opposite within some individual grid cells. Similarly, Figure 8b compares the year 2099 CCSM3 precipitation extremes of five scenarios. Spatially empirical probability density functions are estimated. The degree of intensification follows the order of projected CO2 emissions.
Figure 8. Increasing frequency of 6 h duration 30 year return periods of precipitation extremes over the last several decades as well as for the rest of the 21st century in the extratropical land.
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 To understand the spatial variability, the 30 year return period, 24 h duration CCSM3 and CSIRO depth ratios at year 2099 (2070–2099 window) are illustrated for four emission scenarios (Commit, B1, A1B, and A2) in Figures 9 and 10, respectively. A nine-neighbor grid smoothing is performed for easier visualization and identification of regional patterns. The model-projected intensification trends grow stronger with increases in projected CO2 emissions (as defined in the IPCC-SRES scenarios). Increased intensities of design storms, their large variability across geographical regions, and the spotty nature of the visuals (reflecting the large spatial variability of precipitation and their extremes, as well as the estimation sensitivities) are clear from the maps. While the intensification trends projected by CCSM3 and CSIRO agree relatively well at global and continental scales, the projections are inconsistent for relatively finer resolution regional scales, even after smoothing.
3.4. Intensity-Duration-Frequency Relationships of Rainfall Extremes
 Precipitation intensity-duration-frequency (IDF) curves are frequently used in hydraulic design and water-resources management [Houghtalen et al., 2009; Dunne and Bergere, 1978; Koutsoyiannis et al., 1998]. By plotting the average rainfall intensity (total depth divided by duration) versus duration, it is empirically observed that rainfall intensities with the same frequency are negatively correlated to duration on the log-log scale. We note that the duration here refers to the temporal window used to compute annual maxima instead of the actual storm durations. In other words, IDF curves are the empirical relationship of rainfall extremes across different durations from actual observation. Whether this relationship holds for reanalyses and climate projections is seldom discussed, partially due to the challenge of analyzing rainfall extremes across a wide range of rainfall durations. In order to support the construction of IDF curves, temporally higher resolution data sets must be available.
 Building on the 6, 12, 18, 24, 36, 48, 72, 120, and 240 h extratropical area-weighted median of year 1999 (1970–1999 window) rainfall average intensities, Figure 11 shows the NCEP1, ERA-40, CCSM3, and CSIRO IDF curves for 3, 5, 10, 30, 50, and 100 year return periods (note that the CSIRO data can only support the computation of extremes at durations of a day or more). The linear patterns are preserved in all cases and the curves are more or less parallel to each other, suggesting that the scales of rainfall extremes across different durations seem reasonable. IDF curves can be utilized when investigating precipitation extremes with an arbitrary duration, which has not been encountered before or computed previously. These curves are potentially helpful to understand the behavior of rainfall extremes from subdaily, daily, to multiday scales.
Figure 11. Intensity-Duration-Frequency (IDF) curves of precipitation extremes for various frequencies and durations.
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 Another comparison is shown in Figure 12. Figure 12a compares the year 1999 30 year IDF curves of NCEP1, ERA-40, CCSM3, and CSIRO. The curves appear fairly linear on the log-log scale. The differences between the two climate models are relatively small compared to the differences between the two reanalysis data sets. NCEP1 rainfall intensity is less than CCSM3 in shorter durations, while it becomes larger than CCSM3 for longer durations. Focusing on CCSM3, Figure 12b compares various IDF curves at year 2099 under all emission scenarios. The 30 year IDF curves from the various emission scenarios in the end of the century are parallel to the twentieth century control runs in the log-log graph. Parallel IDF curves in the log-log plots imply a constant ratio, which in turn may translate to a safety factor for engineering design and water-resources management in the context of climate change adaptation. However, the larger differences among climate models and reanalysis (Figure 12a) point to uncertainty which must be characterized and/or ideally reduced prior to making risk informed decisions. The climate change-influenced evolution of IDF curves directly illustrate that water management can no longer assume stationarity [Milly et al., 2008]. Figures 12c and 12d show two examples of IDF curves generated at regional scales, specifically, North America and Europe. While IDF curves could be developed for any region or locale of interest, the relatively low resolutions of extreme precipitation processes within global climate models and the corresponding increase in uncertainty of model-based precipitation extremes projections at higher resolutions limit the credibility of local or even regional IDF curves. Improving the credibility of higher resolution IDF curves may be possible through higher resolution global climate models, and/or through dynamical or statistical downscaling of the global model outputs and/or through improved understanding of extreme precipitation processes. However, the ability of higher resolution global climate models to reduce uncertainty in precipitation extremes remains a hypothesis to be tested (despite reports of initial success [Wehner et al., 2010]) while downscaling may cause additional cascading uncertainties [Schiermeier, 2010].
Figure 12. (a and b) Intensity-Duration-Frequency (IDF) curves for precipitation extremes from climate models and reanalysis data sets. (c and d) IDF curves of precipitation extremes for North America and Europe.
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