TRENDS IN PRECIPITATION ON THE WETTEST DAYS OF THE YEAR ACROSS THE CONTIGUOUS USA

INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 24: 1873 1882 (2004) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/joc.1102 TRENDS IN PRECIPITATION ON THE WETTEST DAYS OF THE YEAR ACROSS THE CONTIGUOUS USA PATRICK J. MICHAELS, a,b PAUL C. KNAPPENBERGER, c, * OLIVER W. FRAUENFELD d and ROBERT E. DAVIS b a Cato Institute, Washington, DC, USA b Department of Environmental Sciences, University of Virginia, Charlottesville, VA, USA c New Hope Environmental Services, 5 Boar s Head Lane, Charlottesville, VA 22903, USA d Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO, USA Received 5 November 2003 Revised 9 August 2004 Accepted 9 August 2004 ABSTRACT Over the course of the 20th century, average annual precipitation for the contiguous USA has increased by nearly 10%. This increase has been described as being dominated by disproportionate increases in extreme precipitation events. However, methodological constraints have confounded detailed interpretation of such results. Here, we briefly describe those limitations and re-evaluate the nature of the observed precipitation changes using a method that allows for a more accurate examination of changes in the proportion of precipitation delivered in extreme daily events. We focus our analysis only on the trends in precipitation on the 10 wettest days of the year and compare the trends observed on those days with the trend in overall precipitation. When averaged across the USA, we find that the precipitation trends on the 10 wettest days of the year are not significantly different from the trend in total overall precipitation. On a regional level, in the northeast and southeast there is some evidence that the rate of precipitation increase on the wettest days exceeds that of total precipitation, whereas in the rest of the country the precipitation on the wettest days is increasing at a rate less than the increase in total precipitation. Copyright 2004 Royal Meteorological Society. KEY WORDS: precipitation; rainfall; climate change; extremes; United States 1. INTRODUCTION During the past 100 years, the annual average total precipitation for the 48 states of the contiguous USA has risen by approximately 10% (National Climatic Data Center, 1994). Many researchers (e.g. Karl et al., 1995; Karl and Knight, 1998; Groisman et al., 2001, Kunkel et al., 2003) have examined US precipitation histories in an attempt to ascertain the nature of the observed precipitation increase. The research that is most cited on this topic is that by Karl and Knight (1998). Using a fixed percentiles approach one in which event definitions remain constant through time they found that the greatest increases, due to both an increase in the number of events and an increase in the intensity of events, were contained in the highest percentiles of daily precipitation. This finding led them to conclude that there had been a disproportionate increase in precipitation derived from heavy and extreme daily precipitation events. Their results have been cited in both national (National Assessment Synthesis Team, 2000) and international (Houghton et al., 2001) assessments of climate change as evidence that the precipitation climate of the USA has been trending toward a more extreme state during a period of increasing atmospheric concentrations of greenhouse gases. However, owing to the nature of the distribution of daily rainfall, the results from a fixed-percentile approach alone cannot be fairly used to address the disproportionality of observed changes. Here, we examine more * Correspondence to: Paul C. Knappenberger, New Hope Environmental Services, 5 Boar s Head Lane, Charlottesville, Virginia 22903, USA; e-mail: pck4s@nhes.com Copyright 2004 Royal Meteorological Society

1874 P. J. MICHAELS ET AL. closely the trends in extreme daily precipitation events outside of the confines of fixed-bins methodologies. We rank precipitation events within each year, and then assess the trends on the wettest days of the year (i.e. the days on which the greatest amount of liquid-equivalent precipitation falls) in comparison with the trends in overall precipitation. Through our analysis, we are able to provide a better characterization of the observed precipitation increases across the USA. 2. LIMITATIONS OF FIXED-BIN APPROACHES Because the characteristic distribution of daily rainfall at a given station is dominated by light events (closely approximated by a gamma distribution (Groisman et al., 1999)), increases in total rainfall over time will generally have a greater contribution from extreme events (events with the greatest daily precipitation amounts). As such, this behaviour represents the expectations against which observed changes should be judged. A simple observation of a greater contribution to total precipitation from events with higher amounts should, therefore, not be interpreted as a disproportionate increase if the contribution has not been compared with the expected value. For example, if precipitation events are defined in fixed-amount bins (e.g. Karl et al., 1995; Groisman et al., 2001), or fixed-percentile bins (e.g. Karl and Knight, 1998; Groisman et al., 2001), there exist a number of scenarios in which uniform (i.e. not disproportionate) changes in precipitation characteristics (frequency or intensity) produce results in which the bins containing the heaviest daily precipitation amounts preferentially are the ones that show the greatest increases. From the results of a fixed-bin analysis alone, therefore, it is impossible to determine the nature of the underlying changes in precipitation unambiguously and, thus, they cannot be used to determine reliably whether disproportionate changes have occurred to the underlying distribution of daily precipitation events. To demonstrate this fact, we provide a set of examples in which total precipitation increases from changes that would not be described as disproportionate produce what appear to be disproportionate changes in extreme events when analysed using a fixed-bin approach. To demonstrate that this type of approach is inadequate, we do not need to demonstrate that the technique fails in all cases, merely that it fails in some instances that are not physically unreasonable. To this end, we run a series of simulations using randomly generated data, which change in prescribed ways, and analyse the results using a fixed-bin (fixed-percentiles) approach. First, we assume that a given, hypothetical station has a rainfall probability density function (f (x)) approximated by a gamma distribution f(x)= (x/β)γ 1 e x/β βɣ(γ) (1) where γ is the shape parameter, β is the scale parameter, and Ɣ(γ) is the gamma function, defined as Ɣ(γ) = 0 t γ 1 e t dt (2) for x, γ, β>0. Because light precipitation events are common and heavy precipitation events are rare, the assumption that γ = 1 provides a reasonable approximation for the distribution of daily precipitation events over the course of a year in the USA. The scale parameter effectively redistributes the weight for a given x under the constraint that the integrated area under the probability density function is equal to unity. A standard gamma distribution, with γ = β = 1, simplifies to the exponential distribution: f(x)= e x (3) Second, we randomly draw 10 000 values from this distribution, with replacement, and these values (which represent daily events ) are sorted and organized into 20 bins, each of which contains 5% of all observations.

US WETTEST DAY PRECIPITATION TRENDS 1875 Total precipitation within each bin P b is calculated as the sum of all events n b P b = P bj for b = 1, 2, 3,...,20 (4) j=1 where n b is the number of observations within bin b. The total precipitation is the sum over all 20 bins. The bin definitions and totals from the original 10 000 random draws represent the baseline conditions. We then examine the impact of a 10% increase in total precipitation occurring from three different scenarios: (1) a simple 10% increase in precipitation frequency (events drawn from the same underlying distribution as the baseline); (2) a simple uniform 10% increase in precipitation intensity; (3) an equal combination of an increase in frequency and in intensity. We simulate a 10% increase in total precipitation occurring only from changes in the frequency of precipitation events by randomly drawing an additional 1000 events (for a total of 11 000) from the same standard gamma distribution and assigning each new event to the appropriate baseline bin. The contribution to the total precipitation change made by changes within each bin, P b, is calculated as P b = P (inc,b) P (base,b) P (inc) P (base) (5) where P (inc,b) is the amount of total precipitation P (inc) within bin b in the 11 000 case sample and P (base,b) is the amount of the total precipitation P (base) within each bin in the 10 000 case baseline sample. The 10% increase in total precipitation is thus apportioned across all 20 bins. This procedure, starting with the baseline draw, is repeated 200 times and the averaged difference within each of the 20 five-percentile bins is presented in Figure 1(a). We simulate the impact of a uniform 10% increase in precipitation intensity by simply multiplying each of the original 10 000 (baseline) events by 1.1. These new values are then assigned into the appropriate original (baseline) percentile bins and the relative increase in precipitation is again calculated as in Equation (5). The average of 200 simulations is presented in Figure 1(b). Finally, we simulate a combination of increasing precipitation event frequency and intensity by randomly drawing 10 500 observations (with replacement) and multiplying each value by 1.048 to produce a total precipitation increase of 10% when summed across all bins. These new values are again assigned into the original (baseline) bins defined in the 10 000 member sample and within-bin percentage changes are calculated as in Equation (5). The mean results from 200 repetitions are presented in Figure 1(c). In each case, we simulate a change in precipitation that is not disproportionately biased to the extreme events, and yet in each case the results from using a fixed-bin approach appear to indicate that the increase has been primarily manifest in the highest bins (i.e. the ones containing the most extreme events). This clearly demonstrates the weaknesses and limitations of a fixed-bin approach in assessing precipitation changes the true nature of the underlying change is obscured by the analysis. Therefore, conclusive statements about the proportionality (or disproportionality) of the observed changes cannot be reliably made. Again, we are not suggesting that the fixed-bin approaches are not sensitive to changes in the underlying distribution of daily precipitation events that are truly disproportionate across bins, but simply that these techniques cannot differentiate between proportionate and disproportionate precipitation changes in all cases. To assess the proportionality of observed changes accurately, a different analysis technique must be employed. 3. EXAMINATION OF PRECIPITATION TRENDS ON RANKED DAYS The results from our above example suggest that a determination of proportionality through a simple comparison of the relative change across fixed bins does not accurately reflect the true nature of the precipitation changes. Although the precipitation changes may prove to be disproportionate across bins, such observations do not prove the changes to be disproportionate to expectations. Therefore, a different method of

1876 P. J. MICHAELS ET AL. Figure 1. (a) Simulated changes in the percentage change of precipitation within 20 five-percentile bins from a 10% increase in the number of events; (b) same as in (a), except for a 10% increase in the intensity of each event; (c) same as is (a), except for a combination of more events and greater intensity events analysis must be employed, one in which the observed changes in precipitation events are considered within the scope of the evolved changes in total precipitation. We must determine if the rate of change of rainfall delivered by extreme precipitation events is greater than or less than the rate of change of total precipitation. This is the approach that we take in this study. We examine trends in the amount of precipitation delivered on the 10 wettest days of the year and then compare these changes with the overall change in precipitation. Through this method, we are able to determine whether there have been changes in the percentage of total precipitation that is accounted for by the wettest (most extreme ) days of each year (i.e. whether the changes in precipitation on the wettest days of the year are

US WETTEST DAY PRECIPITATION TRENDS 1877 disproportionate to changes in total precipitation). Our results are not confused by the use of statically defined bins, but instead reflect dynamic changes in the precipitation delivery over time. We feel it is appropriate to limit an examination of extreme precipitation to the 10 wettest days, because days below this rank are typically not associated with significant flooding (Dunne and Leopold, 1978; Changnon, 1983), and the larger concerns of the environmental and planning communities are indeed for flood-producing high-intensity events rather than more moderate rainfall events. 4. DATA AND METHODS We used an update through 2001 of the same set of 182 high-quality daily precipitation stations from the United States Historical Climate Network as used by Karl and Knight (1998) and also replaced missing values using a methodology similar to that of Karl and Knight (1998). Specifically, for each station and for each month, we determine both the chance of precipitation and the distribution of daily precipitation amounts on days with observed precipitation using data from the complete period of record for that month. Then, for each missing value, we first randomly determine whether precipitation occurred (from a binomial distribution based upon the prior probability of occurrence); if so, we then draw randomly from the observed distribution of rain days to determine the missing precipitation amount. Stations with more than 5% missing data are discarded. This procedure reduced the original 182-station data set to 129 stations (Figure 2). For each station and each year we then rank all daily precipitation amounts from highest to lowest. We confine our subsequent analyses to only the top 10 wettest days of each year at each station. Our choice of 10 days is driven by our desire to have consistency across all regions, as well as our desire to focus on the heaviest precipitation events. As many stations in the western USA had some years that did not have many more than 10 days with measurable precipitation, we were constrained by this least common denominator. To illustrate the climatology of long-term changes in precipitation across the USA at national and regional scales more easily, we divide the USA into seven regions of approximately equal area. Precipitation for each of the top 10 ranked days is averaged across all years both nationally and by region (i.e. the heaviest precipitation day is averaged across all stations, the second heaviest precipitation day, etc.). Although the ranked data could be examined on a station-by-station basis, the inherently high variability of rainfall, particularly with respect Figure 2. The station locations and regions used in this study

1878 P. J. MICHAELS ET AL. (a) (b) Figure 3. (a) Average daily precipitation (inches) occurring on each of the 10 wettest days of the year. (b) Average percentage of total annual precipitation occurring on each of the 10 wettest days of the year

US WETTEST DAY PRECIPITATION TRENDS 1879 to heavy events, reduces our ability to perform robust statistical analyses on data for individual stations. At both the regional and national scales, our procedure results in averaging precipitation that arose from different events (i.e. that occurred on different days), but our goal is to examine the long-term climatology of heavy precipitation events in general and not changes in specific storm types. For each region we perform an ordinary least-squares regression analysis through the 92 year time series (1910 2001) of the precipitation amount for the top 10 ranked daily precipitation days, i.e. the time series defined by the wettest day each year, the second wettest day each year, to the 10th wettest day each year. Additionally, we perform the same regression analysis through the 92 year time series of the ranked precipitation events expressed as percentages of the total annual precipitation. To calculate trends at the national scale, we first average the ranked precipitation values for each of the seven regions for each year and then subsequently perform the same regression analyses mentioned above. We conduct tests for significance of the temporal trends under the standard null hypothesis that the trend is not significantly different from zero (p <0.05). Strictly speaking, this means that, within the parameters of the test, any statements about increasing trends are only defensible when the null hypothesis of a regression slope equal to zero is rejected. 5. RESULTS AND DISCUSSION Figure 3(a) shows the average total precipitation on each of the 10 wettest days and Figure 3(b) shows the average percentage of the total annual precipitation on each of those days, for each region, and can be considered the reference climatology. A greater amount of precipitation falls during the 10 wettest days in the regions along the Atlantic and Gulf coasts than in the western regions, but this pattern is reversed somewhat when considering the percentage of total annual precipitation accounted for by the wettest days. Averaged across the USA, the wettest day of the year produces 2.50 inches of precipitation and accounts for 8.31% of the annual precipitation total. The 10th wettest day, by comparison, averages 0.85 inches of precipitation and contributes 2.70% to the annual total. Taken together, 43.58% (13.37 inches) of the total annual precipitation falls during the 10 wettest days of the year. Figure 4(a) shows the trend in precipitation on each of the ranked days over the period from 1910 to 2001. In general, all changes have been towards increasing precipitation, but significant changes are found only in the Northeast and the High Plains (all days, ranked one through ten), and in four of the wettest 5 days in the Midwest. In the remaining four regions, only 4 days show significant increases. One of these increases is on the heaviest day (Northwest), while the other three are confined to the relatively low classes of eighth heaviest day and lower. When averaged together nationally, significant increases are found on each of the 10 wettest days of the year. To assess whether the changes depicted in Figure 4(a) reflect disproportionate increases in precipitation on the wettest days, we must examine these changes compared with changes in total annual precipitation in each region. Figure 4(b) is the change in the percentage (proportion) of the total annual contribution of precipitation by each of 10 wettest days across the period of record. Positive values indicate an increasing fraction of annual precipitation occurring on the day in question, and negative values indicate a decline in the percentage contribution to annual precipitation. There are few significant changes in the proportion of annual precipitation delivered on the wettest days. In the Northeast and Southeast regions the values are positive, but only significant for ranked days 4 through to 10 in the Southeast. The values are negative throughout most of the rest of the country, with significant declines on ranked days 8 through to 10 in the Northwest and ranked days 1 and 10 in the Southwest. When taken together, the national average shows little change in the proportion of precipitation falling on the wettest days of the year. As an example of these results for a single ranked day, we consider the wettest day of the year in the national average. Figure 5(a) shows that the average heaviest daily precipitation amount has increased at a rate of 0.26 inches/100 years during our period of study. This represents an approximate 10% increase in the precipitation delivery since the beginning of the record in 1910 on the day characterized by the most extreme precipitation. Figure 5(b), however, shows that there has been no significant change in the proportion of annual precipitation that is contributed by the heaviest precipitation day. This indicates that, although there

1880 P. J. MICHAELS ET AL. (a) (b) Figure 4. (a) Average trend in daily precipitation (inches/100 years) occurring on each of the 10 wettest days of the year. Significant trends (p <0.05) are indicated by solid black bars. (b) Average trend in the percentage of total annual precipitation occurring on each of the 10 wettest days of the year (%/100 years). Significant trends (p <0.05) are indicated by solid black bars

US WETTEST DAY PRECIPITATION TRENDS 1881 Figure 5. (a) Time series of the wettest day of the year averaged across the seven regions. (b) Time series of the percentage of total annual precipitation contributed by the wettest day, averaged across the seven regions has been an increase in the amount of precipitation that falls during extreme daily events, this increase has occurred in direct proportion to the overall increase in total annual precipitation. 6. CONCLUSIONS Our results support the contention that, where changes are significant, there is an increase in the amount of rain occurring on heavy rain days. However, our results provide no support for the argument that the increase in total annual rainfall observed across the USA is disproportionately occurring on the wettest days a contention that may have arisen from methodological constraints rather than true changes in the nature of precipitation delivery. After allowing for the total rain increases within each of our seven regions, we find no consistent national behaviour in the US precipitation record. Increases are indeed disproportionate for ranked days 4 through to 10 in the Southeast, but there is a balancing disproportionate decrease in the Northwest and in the Pacific Southwest.

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