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J_ID: CHAOEH DOI: 10.1063/1.4921719 Date: 21-May-15

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Scaling of extreme rainfall areas at a planetary scale Naresh Devineni,1,2,a) Upmanu Lall,3,4,b) Chen Xi,5,6 and Philip Ward7,8

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Department of Civil Engineering, The City University of New York (City College), New York, New York 10031, USA 2 NOAA-Cooperative Remote Sensing Science and Technology Center, The City University of New York (City College), New York, New York 10031, USA 3 Department of Earth and Environmental Engineering, Columbia University, New York, New York 10027, USA 4 Columbia Water Center, The Earth Institute, Columbia University, New York, New York 10027, USA 5 Bureau of Hydrology, Changjiang Water Resources Commission, Wuhan 430010, China 6 State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China 7 Institute for Environmental Studies (IVM), VU University, Amsterdam, De Boelelaan 1087, 1081 HV Amsterdam, The Netherlands 8 Amsterdam Global Change Institute (AGCI), VU University Amsterdam, De Boelelaan 1087, 1081 HV Amsterdam, The Netherlands

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(Received 2 March 2015; accepted 13 May 2015; published online xx xx xxxx)

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Event magnitude and area scaling relationships for rainfall over different regions of the world have been presented in the literature for relatively short durations and over relatively small areas. In this paper, we present the first ever results on a global analysis of the scaling characteristics of extreme rainfall areas for durations ranging from 1 to 30 days. Broken power law models are fit in each case. The past work has been focused largely on the time and space scales associated with local and regional convection. The work presented here suggests that power law scaling may also apply to planetary scale phenomenon, such as frontal and monsoonal systems, and their interaction with local moisture recycling. Such features may have persistence over large areas corresponding to extreme rain and regional flood events. As a result, they lead to considerable hazard exposure. A caveat is that methods used for empirical power law identification have difficulties with edge effects due to finite domains. This leads to problems with robust model identification and interpretability of the underlying relationships. We use recent algorithms that aim to address some of these issues in a principled way. Theoretical research that could explain why such results may emerge across the world, as analyzed for the first time in this paper, is C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4921719] needed. V

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Power laws may be ubiquitous in nature. Their identification and underlying cause is debated, and in many cases has led to new theoretical developments and thinking. However, studies that empirically identify power laws from data have been shown to be sensitive to the methods used. Scaling of rainfall magnitude, intermittence, and areal coverage has been pursued in the past through an analysis of event data or satellite based images over relatively short durations and areas. Likewise, the relation of flood peak discharge to contributing drainage area has been demonstrated. Here, we look at a rainfall variable that has not previously been analyzed and possible scaling behavior over long event durations is demonstrated. These findings lead to the question as to how the climate system organizes over these scales, overcoming the substantial apparent heterogeneity in process dynamics.

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I. INTRODUCTION

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Floods and extreme rainfall are a significant hazard to society. Understanding the structure and dynamics of such

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a)

Email: [email protected] Email: [email protected]

b)

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events in space and time has long been a subject of scientific interest. Since such events impact specific regions, and the physical processes associated with rainfall and with flooding vary considerably by location, most of the research on these topics, whether it is based on deterministic or statistical methods, tends to be focused on specific mechanisms, regions or environments, and on the time scales of the typical events, which may be of the order of hours to a day. Even the research on these phenomenon that would be of interest to applied mathematicians and physicists has had a similar focus. Thus, fractal and multifractal descriptions have been developed for rainfall intensity fields, for flood peaks as a function of the contributing drainage area, and applications of multiplicative cascades and self organized criticality have been demonstrated for rainfall and flood dynamics in certain contexts (Arakawa, 2006; Dickman, 2003; Gupta et al., 1996; Gupta et al., 1994; Gupta and Waymire, 1993; Gupta, 2004; Lima and Lall, 2009, 2010; Malamud and Turcotte, 2006; Menabde and Sivapalan, 2001; Tsonis and Elsner, 2007; Pandey et al., 1998; Peters et al., 2001; Peters and Neelin, 2006; Rajagopalan and Tarboton, 1993; Schertzer and Lovejoy, 1987; Svensson et al., 1996; Singh et al., 2009; Sapozhnikov and Foufoula-Georgiou, 2007; and Lashermes and Foufoula-Georgiou, 2007).

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In our review of this extensive literature, we did not find research that looked at the areas covered by extreme rainfall for a specified duration. Specifically, we have considered rainfall events, such that the event rainfall for that duration exceeds the 90th percentile of all annual maximum rainfall events at that location for that duration. The choice of the 90th percentile was made to ensure that a reasonable sample size is achieved. We have considered durations of 1, 5, 10, and 30 days, and used a gridded global daily rainfall data set, to identify the annual maximum value for each duration, and its 90th percentile, for each global grid. The contiguous area (adjoining grid cells) associated with simultaneous exceedance of the 90th percentile is then recorded. For each rainfall duration, we are interested in exploring whether or not a power law scaling model can explain the frequency distribution of contiguous areas associated with an exceedance of the 90th percentile of annual maximum rainfall for that duration. The interest in the contiguous area covered by extreme rainfall for these long durations was motivated by the observation that the 2010 and 2011 floods in Thailand and on the Mississippi river were longer than 100 days in terms of their disruption of socio-economic activity, and we wondered how unusual it was to get floods of this nature. This led us to the Colorado Flood Observatory’s flood data set (http://floodobservatory.colorado.edu/) that has recorded flood inundation events using satellite sources and media verification since 1985, and currently has over 4200 entries with approximate location of the center of the area flooded, the dates and duration of flooding and notes as to societal impacts. This is the only global data set of this kind. The locations of the floods recorded as well as the associated duration are shown in Figure 1. Since these floods represent inundations that were visible from satellite images over a period of repeated satellite visits, they represent events that are censored to include larger floods, and not flash floods with a very short duration. Approximately, 30% of the events have an inundation duration greater than 10 days, and approximately 10% have a duration longer than 30 days. The inundated area typically increases with inundation duration. The correlation between

inundated area and duration of the flooding event is 0.31 (or 0.48 in log-log space), which is significantly different from 0 at the 1e16 level for this sample size. An examination of several of these longer duration and larger area events suggests that persistent moisture transport into the area that manifest as repeated large precipitation events over the multi-day duration are associated with the large areaduration flooding. Such events would then be expected to be related to a different mechanism than, for instance, extreme 1 day, small area events that may be related to local moisture sources, and/or to local convection. This observation leads to the global analysis presented here. The following questions are explored in this paper:

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(1) Across the planet, does extreme rainfall have spatial organization that is manifest as a scaling relationship with contiguous area over which extreme rainfall occurred, for a specified duration longer than 1 day? (2) If yes, then do the parameters of the scaling law vary systematically by the duration considered, the latitude belt, and by Koppen climate zones reflecting differences in tropical, sub-tropical, and extra-tropical climate mechanisms?

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This paper presents an exploration of these questions, as a precursor to the development of models and theory to explain the results. Developing a theoretical explanation for empirically observed power laws is often a challenge, especially when one analyzes global climate data, where we know that the underlying processes are expected to be quite different in different places. The data used for rainfall are described in Sec. II. The methods and results are presented in Secs. II–IV.

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II. DATA AND EXPERIMENTAL DESIGN

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A. Data


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Daily rainfall data, at a resolution of 0.5 0.5 latitude and longitude from the CPC Unified Gauge Global Analysis (Chen et al. 2008a; 2008b) is obtained from International Research Institute for Climate and Society’s Data Library

FIG. 1. Locations and durations of floods recorded at the Colorado Flood Observatory from 1985 to January 2015.

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For each rainfall duration j, the contiguous area Aij, for the ith event where each of the contiguous grid boxes exceeded the R90 for that grid box, and the latitude and longitude of the corresponding grid boxes were recorded. The frequency distribution of Aij for each j was then explored for possible evidence of scaling of the number of exceedances of R90 versus the contiguous area associated with the

FIG. 2. Schematic representation of extreme rainfall area identification. Each grid is indexed with 1 (if annual maximum rainfall > R90) or 0 (if annual maximum rainfall < R90). Contiguous extreme rainfall area is then identified as independent cluster of conjoined grids with an index of 1.

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(http://iridl.ldeo.columbia.edu/SOURCES/.NOAA/.NCEP/ .CPC/.UNIFIED_PRCP/.GAUGE_BASED/.GLOBAL/ .v1p0/.RETRO/.rain/). These global gridded (land only) rainfall data are developed based on the optimal interpolation, inverse-weighting algorithm, and Shepard’s algorithm on about 30000 stations obtained from GTS, COOP, and other national and international agencies. The station observations are subjected to quality control with validation from historical climatology, nearest stations, concurrent radar, and satellite observations. Station data are interpolated to 0.5 0.5 grids while accounting for orographic effects (Xie et al., 2007). Chen et al., (2008a) report a less than 1% bias resulting from the interpolation methods globally over the land grids. The data comprise a retrospective version from 1979 to 2005 and a real-time version from 2006 to current. While the retrospective analysis is based on 30 000 stations, the real-time version is based on 17 000 stations. In this study, we used 34-year data from 1979 to 2012. These data were further processed as follows. (1) The 1-day, 5-day, 10-day, and 30-day annual maximum rainfall events were identified for each grid box from this data set, by considering each possible consecutive n-day period and its annual maximum rainfall. (2) The 90th percentile, R90, of the annual maximum rainfall was identified at each grid box, and for each of the 4 durations (1-day, 5-day, 10-day, and 30-day) considered. (3) The rainfall data for each rainfall duration were then processed to identify the rainfall for the specified duration in a grid box that exceeds the estimated 90th percentile annual maximum value, R90 for that duration, using a moving window over the period of record, and that grid box. This process can potentially identify multiple events across the globe in a given year which met these criteria. (4) For each such event, how the contiguous area over which the T-year event was exceeded was identified is illustrated in Figure 2. An example of the resulting spatial distribution of contiguous areas and events that exceeded the R90 for each duration in 2010 is provided in Figure 3. We see that as the duration increases, the contiguous area also increases, and that there is correspondence between most of the long duration rainfall events and the longer duration floods recorded in the Colorado Flood Observatory data set.

FIG. 3. Spatial distribution of area under rainfall extremes for the year 2010. Figure 3(a) shows the distribution of the grids greater R90 for annual maximum 1-day rainfall. Figures 3(b) and 3(c) represent the distribution of the grids greater R90 for annual maximum 10-day rainfall and 30-day rainfall respectively. Figure 3(d) presents the regions reported to have experienced floods in year 2010 (Colorado Flood Observatory). Only those floods that have duration greater than 10 days are shown here. The primary cause is also presented.


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FIG. 4. Latitude bands and Koppen climate zones used in the study.

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exceedance over the entire period of record. These analyses were further organized by (a) global (60 N–60 S latitudes), (b) latitude bands (tropical grids (20 N–20 S latitudes), subtropical grids (20 N–40 N and 20 S–40 S latitudes), extratropical grids (40 N–60 N and 40 S–60 S latitudes), and (c) Koppen climate zones (Peel et al. 2007) as illustrated in Figure 4. For each duration j, and each spatial domain (groups within latitude bands and Koppen zones), we explore a scaling law of the form: aj

f ðAj Þ / Aj

for Aj > Amin;j j ¼ f1; 5; 10; 30g days; (1)

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where Amin,j, and aj are a threshold, and scaling exponent to be estimated from the data for each rainfall duration j for each domain, and f(.) refers to the probability distribution for (.).

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B. Methods

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Several authors have pointed to the challenges with estimating and justifying power laws purely on empirical grounds. Many methods ranging from log-log regression of event counts vs. scale, or least squares or maximum likelihood estimation of the parameters of the power law probability distribution with or without binning of the data have been used. Stumpf and Porter (2012), Clauset and Shalizi (2009), and Deluca and Corral (2013) discuss the performance of several of these methods. Rather than reproducing this discussion in detail, we refer the reader to these papers for a discussion of the main issues with fitting such models to data. The key points are that edge effects due to finite sampling domains can have a strong influence on the fit and interpretation, and that there are conditions under which each method has problems. One direction that is consistent between a statistical and a physics based view of the problem

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is that a Pareto tail may be reasonable, i.e., only values of the process exceeding some minimum threshold may follow a scaling law. After a review of the literature, and initial exploration with log-log fits of the counts of events with a certain area exposed vs. the area, we noticed that our data were subject to edge effects on both the upper and lower tails, but that these effects were potentially more pronounced at the lower tail, reflecting both the discussion in the literature, and our expectation that the climatic and scaling processes associated with short duration rainfall extremes are likely to be different from those associated with rainfall extremes over longer durations covering larger areas. Consequently, we chose to follow the approach developed by Clauset and Shalizi (2009), who discuss the shortcoming of several of the traditional methods and propose a formal procedure based on hypothesis testing to (a) select a lower bound for the applicability of a power law, (b) to estimate the scaling exponent given this threshold using maximum likelihood, and (c) to perform likelihood ratio tests based on resampling strategies to assess the probability that a power law distribution or other candidate distributions may be reasonable for a given data set. Their exploratory approach to identify potential suitability, rather than assert that a power law holds has led to a re-examination of many of the “power laws” that were discovered using earlier methods. Gillespie (2014) provides an implementation of these ideas in the open source software R, in the package “poweRlaw” which can be downloaded from within R. Given that the data we use are gridded, the contiguous areas are comprised of a number of contiguous grids. Consequently, we use the discrete version of the algorithms provided by Gillespie. The key points are summarized below. Letting x represents the contiguous area associated with a 90th percentile rainfall exceedance event for a duration of


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interest, the probability mass function of X contiguous grid boxes being observed can be defined as xa Pð X ¼ xÞ ¼ ; fða; xmin Þ

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where fða; xmin Þ ¼ ðn þ xmin Þa is the generalised zeta n¼0 function. The exponent a and xmin are selected by drawing bootstrap samples from the original data, and for each sample, and for each candidate value of xmin, a is estimated using an approximation to the maximum likelihood solution: ^ a ¼1þn

n X i¼1

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(2)

!1

ln

xi xmin 0:5

:

(3)

For each bootstrap sample, the appropriate values of xmin and the corresponding value of â are selected based on a Kolmogrov-Smirnov distance (KSd) between the fitted power law distribution, and the empirical cumulative distribution function. In our experiments with this approach, we noted that often the optimal choices of xmin are clustered close to the value selected as the optimal from the bootstrap procedure, but sometimes a dramatic reduction in the sample size n can take place. To address such parameter variability, and its

adverse effects, the significance with which the null hypothesis that the chosen parameters are indeed reasonable is assessed through a bootstrap approach. Under the null hypothesis that the estimated xmin and the corresponding value of â are the best estimates for a power law, data are simulated such that each sample has n1 values (equal to the size of the subsample smaller than xmin in the original data) from a uniform distribution from 1:xmin and (n n1) values greater than xmin are simulated from a power law distribution with parameter â . For each bootstrap sample that is generated in this way, the Kolmogorov-Smirnov distance between the sample and the original data is computed. The percentage of cases in which the KSd is greater than the corresponding distance for the bootstrap sample with respect to the original data is computed and is used as an estimate of the probability that the model is a good fit to the data. The mean and standard deviation of xmin and â from the bootstrap samples are also reported. In our analyses, we used the default of 5000 bootstrap samples.

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III. RESULTS

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The results of the investigations are presented for each rainfall duration, and for each spatial domain using the format illustrated in Figure 5 for the 1-day duration. The power laws fitted for the global and the latitude belts are illustrated

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FIG. 5. Results from scaling analysis for 1-day rainfall events that exceed the 90th percentile of annual maximum rainfall. (a) Fitted power laws for global and latitude belt analyses with associated p-values from the bootstrap (in inset table), and median xmin as dashed vertical lines; (b) the same information for the Koppen climate zones; (c) boxplots with the variation of the scaling exponent corresponding to (a); and (d) boxplots corresponding to (b). Note that each grid is 0.5 latitude and longitude, and hence has an average area of about 2500 km2.


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FIG. 6. Same as in Figure 5, but for a 5-day duration.

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on the left, and for the Koppen climate zones on the right. The upper half in each case has a plot of the cumulative distribution function of the data, the p-value for the significance of the power law fit for each case, and the median value of xmin selected. The lower half has the box-plots of the scaling exponent as selected across the bootstrap samples. A p-value greater than 0.1 is taken to indicate that the power law may be admissible for that data set, consistent with Clauset and Shalizi (2009) and Gillespie (2014). The higher the p-value, the higher the plausibility. Note that for all the non-global analyses, and especially for the Koppen climate zones, the domains are much smaller, and bounded, and hence there is potential censoring of the largest areas that are admitted in the analysis. For each zone, we only count the contiguous areas in that zone, and hence boundary effects can proliferate. Recall, that our analysis uses rainfall over land, and not over the oceans. This contributes to the edge effects in estimation. Consequently, the global analysis is least constrained by boundary effects, while the arid and warmtemperate zones of the sub-sets are most constrained based on their area and degree of fragmentation (see Figure 4 for the spatial distribution of these zones). Note that the results are presented in terms of the number of contiguous grids, each of which is 0.5 0.5 or about 2500 km2 on average. Most past analyses of rainfall scaling are at sub-grid scales relative to what we present here.

For the 1-day duration, from Figure 5, we see that the power law appears to be admissible (p > 0.1) for all the latitude bands and the global analysis, with xmin varying from 25 to 40 grid cells, with the strongest results for the tropics and sub-tropics. The tropics have the highest values (>3) of â , but also the highest uncertainty. The median values of â range from 2.5 to 3.5 across the different cases. The global and extra-tropics results are comparable, while the subtropics appear to have the lowest value for â . For the Koppen climate zones, only the equatorial and the snow zones appear to have admissible power laws, and the associated exponents appear to be comparable to those for the tropics, and the sub-tropics, respectively. We note the rather significant departure of the power law from the data for the Arid zone, and the corresponding p-value of 0. In this case, visually one may choose a higher xmin, resulting in a much higher â . However, the dramatic decrease in the sample size coupled with the assumption of a uniform distribution for x < xmin in Gillespie’s approach makes such a choice numerically untenable. From Figure 6, we note that for a 5-day duration, based on the p-values only, the equatorial climate zone appears plausible for a power law, with an exponent that is comparable to but slightly lower than the corresponding one for the 1-day duration. The range of xmin values is similar to what was selected for the 1-day duration, but all estimated scaling


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FIG. 7. As in Figure 5, but for a 10-day duration.

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exponents are generally smaller, but with reduced uncertainty. For the 10-day duration (Figure 7), the global, equatorial, and warm temperate regions appear to have plausible power laws. The xmin values are in a similar range as before, and the scaling exponents in the significant cases appear to decrease relative to the 1-day values, with lower uncertainty. The ordering of the exponents for the cases that are significant is also similar, with the equatorial exponent being the largest. For the 30-day duration (Figure 8), we note that the power law appears to be plausible for the extratropics, tropics, equatorial, and snow zones. The range and largest values of xmin have increased, perhaps reflecting that if there is a power law structure at these longer durations, it may be at larger spatial scales. The scaling exponent for the tropics and the equatorial regions appears to have decreased confirming the trend we noted earlier with increasing duration. A similar, but weaker assertion, can be made for the other zones that demonstrate significance with respect to a power law hypothesis.

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IV. DISCUSSION

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While there has been much interest in fitting and interpreting power laws to empirical data, and such work has generated some excitement, many of the initial claims have been disputed and their strength has been questioned. Often, and

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especially for planetary scale processes, a theoretical underpinning for such results is hard to come by at first glance. Consequently, one has to approach all such analyses in an exploratory spirit, with the goal of formulating potential hypotheses to falsify or support using theoretical arguments following the data analysis, rather than as an evidence of some universal behavior. This is the spirit in which we embarked on the present exercise. The presence of limited scales of data, and spatial boundary effects, as well as limited temporal sampling with respect to the exceedance of extreme rainfall thresholds, contribute to our trepidation in interpreting the results of such an analysis, and making grand claims. With these caveats in mind, let us review what we have learned from this experiment.

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(1) We considered the potential scaling of contiguous gridded areas, where rainfall during the year simultaneously exceeded the 90th percentile of the annual maximum rainfall for each of the adjoining grids, for a specified duration. This is a threshold exceedance exercise for a reasonably high quantile of the distribution of the process. Seasonal factors were not explicitly considered, but we hope that over most of the earth, there is a reasonably well defined season for extreme rainfall, and hence for the climatic processes driving exceedances of such a threshold. (2) The longer durations (e.g., 10 and 30 days) considered here will most likely translate into rainfall extremes at

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FIG. 8. As in Figure 5, but for a 30-day duration. 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441

the 90th percentile of annual maxima that are associated with persistent and recurrent rainfall events over that time window, or with a very intense single (few) day event that has a high probability of being localized, rather than present over the range of areas we consider here (note that an xmin of 15–40 grids 3.75 104 to 105 km2, and hence the power law being explored would apply to areas larger than that). Sub-seasonal scales of precipitation extreme organization are consequently being explored. (3) We anticipate that the typical events for a 1-day rainfall exceeding R90 may belong to a different population than the longer duration events, and that more of these events may be concentrated over smaller areas, i.e., if there is a scaling relationship, generally, it may have a higher exponent. Given the typical time scales of 3–7 days (Blackmon et al. 1984; Dickinson, 1968; and Peixoto and Oort, 1992) associated with baroclinic waves and disturbances that are involved in frontal and cyclonic precipitation in the mid-latitudes, we may expect that especially outside the tropics or equatorial zones, large scale organization may be weaker over a 5-day duration, but may re-emerge over the longer 10–30 day durations, as averaging over multiple waves leads to potential persistent and recurrent exposure over the same areas. On the other hand, tropical convection is known to exhibit organization (Madden and Julian, 1972) over time scales

that are typically 20–60 days, i.e., a 30-day duration would approach a sampling time scale similar to the bounds presented by the 5 and 10 day durations for the synoptic waves that have a 4–7 day duration. Thus, there is reason to expect that both the significance of power law scaling, the xmin, and the scaling exponent associated with large areas of extreme rainfall may vary in response to the underlying mechanisms across these durations, and across the geographical regions. The global results are a complex aggregation of these subregional results. (4) As argued above, and expected, our results suggest that for the cases that show significance with respect to a power law, the scaling exponents typically decrease as event duration increases from 1 to 30 days, suggesting a higher probability of tail events where large areas encounter events that exceed the R90 for that duration. This has significance for potential impacts on flooding, depending on how the terrestrial hydrologic system, i.e., the soils and the associated drainage network, organizes the runoff from the persistent and recurrent, large area coverage rainfall events. (5) Similarly, we see that especially for the shorter durations, the tropical and equatorial regions have much stronger support for power law scaling, but with relatively higher exponents and also variability in those exponents. The consistency is a bit higher for the


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equatorial Koppen climate zone than for the tropical latitude belt, perhaps reflecting the mix of humid and arid regions in the tropics which are separated by the climate zones into different categories. This difference is most striking for the 5- and 10-day durations. (6) The evidence for power law scaling in the extratropics and subtropics, and snow climate zones is higher for the longer durations, consistent with the argument that persistence in the mid-latitude jet stream and the associated recurrent, delivery of extreme moisture to the same geographies over relatively large, contiguous areas becomes more likely as the longer duration is considered. The spatial scales associated with mid-latitude and extratropical cyclones are typically larger than for regional convection in those areas. The scaling exponents are also typically lower than those for the tropics and equatorial regions, supporting the idea that the longer durations and larger spatial scales of impact become relatively more likely. Interestingly the extratropics, subtropics, and the snow climate zone do offer evidence for the significance of power laws for the 1-day and then for the 30-day durations, but not at the 5 and 10-day levels, consistent with our speculation as to the role the mid-latitude atmospheric dynamics may play. This paper presents an initial investigation of planetary scale organization in the area coverage of extreme rainfall at selected durations from 1 to 30 days. We do not offer claims of power laws dominating such a process as a conclusion. However, we do identify interesting aspects of organization of the system that provoke thinking and systematic investigation as to how such processes may work and whether the underlying mechanisms can suggest, at least qualitatively, that the patterns we find here can be explained. Application to understanding whether the current generation of global circulation models adequately represent the underlying mechanisms and organization in planetary precipitation, and how they could improve could then be pursued. Following on the empirical analysis, one can explore whether phenomenon such as the El Nino Southern Oscillation and the Pacific Decadal Oscillation translates into patterns of organization in the tropics and extratropics that lead to larger areas and longer durations associated with extreme rainfall, depending on their phase, with a specified geographic character. This would lead to the next step of investigating how such behavior maps to land hydrology, and hence to societal impact. We are also curious as to how our results may change if rainfall over land and ocean were considered, thus reducing the edge effects induced by the geometry associated with the current sub-regions. Data limitations may be the primary challenge in this regard.

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ACKNOWLEDGMENTS

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This work was partially supported by American International Group (AIG) under the “Climate Informed Global Flood Risk Assessment” project. U. Lall was also supported by an IPA from the U.S. Army Corps of Engineers.

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