Drawing from a NOAA database of hourly precipitation data from 5995 stations in the contiguous United States over the period 1949–2009, the authors investigate possible trends in the variance of the hourly precipitation, averaged over the diurnal and annual cycles and normalized by the square of the mean precipitation at that site. This normalized variance is a measure of storminess, distinct from increases in precipitation attributable to warming. For the 1722 stations surviving quality control with data on at least 80% of days in at least 30 years, the authors compute the rate of change of the logarithm of the normalized variance at each station and set bounds on its mean (over stations) trend. The logarithmic function weights the trends at calm stations equally to those at stormy stations and enhances the statistical power of the mean. The authors find a logarithmic rate of change of the mean normalized variance of yr−1 (). The upper bounds on any continentally averaged trend, increasing or decreasing, are about 0.001 yr−1 (doubling or halving times > 1000 years). It is found that the normalized variance in the Los Angeles basin has increased at a statistically significant rate. This may be attributable to a decrease in the number of aerosol condensation nuclei. Upper bounds are set on any effect of the 11-yr solar cycle.
Surface air temperatures in most or all climate zones, temporally averaged over shorter-term and decadal oscillations, have warmed since the nineteenth century (Jones and Moberg 2003; Menne and Williams 2005; Hansen et al. 2010; Rhode et al. 2013). Other questions remain open. Many studies (Katz and Brown 1992; Karl et al. 1996; Hennessy et al. 1997; Karl and Knight 1998; Trenberth 1998, 1999; Easterling et al. 2000; Milly et al. 2002; Emori and Brown 2005; Groisman et al. 2005; Alexander et al. 2006; Peterson et al. 2008; O’Gorman and Schneider 2009; Min et al. 2011; Utsumi et al. 2011; Trenberth 2011; Groisman et al. 2012; Coumou and Rahmstorf 2012; Sillmann et al. 2013; Villarini et al. 2013; Westra et al. 2013; Wuebbles et al. 2014; Horton et al. 2015; Lehmann et al. 2015; Ning et al. 2015; Stott et al. 2016; Prein et al. 2017) and the IPCC Fifth Assessment Report (Hartman et al. 2013) have suggested that warming has been or will be accompanied by an increased frequency of “extreme weather events.” These include periods of unusually high or low precipitation, intense precipitation, or droughts. Although intense precipitation and droughts are opposites, they both correspond to increased variability of weather. Statistically, they both increase the variance and higher moments of the temporal distribution of precipitation. The purpose of this paper is to evaluate long-term historic trends in precipitation variance, averaging over the diurnal and annual cycles.
Counts and semiquantitative measures of severe storms such as hurricane categories and the Fujita tornado scale have several drawbacks as tools to measure trends in extreme weather events (Holton et al. 2003). Detection of storms has improved as technology has improved, biasing their statistics; an obvious example is that satellite observations record hurricanes that never strike land, whose detection in the presatellite era required their fortuitous encounter with ships. Quantification of a storm by sampled wind speeds is only a rough approximation to the energy contained in its full three-dimensional velocity field and is biased by the development of more extensive observing systems. Perhaps most important, identified discrete storms are few, limiting the statistical power of their data (Befort et al. 2016).
Severe storms are usually accompanied by, or may even be defined by, short periods of intense precipitation that are best manifested in hourly data. Even tornadoes, not themselves sources of intense rain, are produced by strong thunderstorms accompanied by intense precipitation. Precipitation data include events over the entire spectrum of intensity, without arbitrary thresholds such as the distinction between a hurricane and a tropical storm. Precipitation data are available at a large number of stations, at hourly cadence, over several decades, so that their analysis may have great statistical power.
Most previous studies of precipitation statistics (Karl et al. 1996; Karl and Knight 1998; Kunkel et al. 1999; Groisman et al. 2005; Alexander et al. 2006; Min et al. 2011; Utsumi et al. 2011; Balling and Goodrich 2011; Kunkel et al. 2013a,b; Fischer and Knutti 2014; Anderson et al. 2015; Cavanaugh et al. 2015; Lehmann et al. 2015; Powell and Keim 2015) have been concerned with the largest 1-day or few-day (often 5-day) precipitation totals found in an annual or longer period. Other studies (Groisman et al. 2012) have similarly studied the frequency of days with precipitation over a high threshold or in a high range. These data are valuable to civil engineers and planners, who must design storm water management systems, but they do not fully describe the statistics of precipitation, nor do they have the statistical power and homogeneity of lengthy hourly precipitation records that include the information contained in lesser events and dry periods. The statistics of extreme events (Epstein 1948; Johnson 1964; Doremus 1983; Katz 1998, 1999) also do not take advantage of the information present in these more frequent events, nor utilize all the data, and may depend on arbitrary choices of criteria and thresholds. Studies of hourly data that deal only with their extremes (Lenderink and van Meijgaard 2008; Wong et al. 2011; Lenderink et al. 2011; Barbero et al. 2017) do not utilize the information contained in the many hours with significant, but not record-setting, precipitation, and can be affected by a few rare outliers.
These factors also complicate combining information from multiple stations that may be in different climatic regimes in which different definitions of “extreme weather events” might be appropriate; for example, thunderstorms that occur nearly daily on the U.S. Gulf Coast would be extraordinary in the maritime climate of the Pacific Northwest. Utilization of all available data has been a powerful tool for extracting global warming from temperature trends that may be inconsistent among stations that show stochastic local and correlated regional variations (Pan et al. 2004; Meehl et al. 2012; Hartman et al. 2013; Rhode et al. 2013). Therefore, we define metrics that combine information distributed throughout entire time series and that are weighted so that trends at calm and stormy stations contribute comparably to the overall averages. Our metrics utilize information from periods of lower-intensity precipitation as well as from intense precipitation events because trends in either may be manifestations of changing climate.
Trenberth et al. (2003) pointed to the importance of hourly data in studying the effects of climate change on precipitation. For quantification of the “storminess” of climate, hourly data offer information and insights that are lost when precipitation is summed over 24-h periods. A day of steady rain is not the same as a day during which an intense storm produces the same amount of rain but concentrates it in one or a few hours. To take advantage of the full statistical power of the extensive NOAA database containing (allowing for missing hours) about a billion hourly data, this study uses information from more than a thousand stations over the 61 years from 1949 to 2009 to determine continentwide trends.
The increase of water saturation vapor pressure with increasing temperature (according to the Clausius–Clapeyron equation) as the climate warms has been predicted to increase the mean precipitation, the frequency of periods (typically one or a few days) with precipitation above thresholds, and record values of precipitation. There is evidence of such effects in the frequency of extreme precipitation events (Pall et al. 2007; Berg et al. 2013; Kunkel et al. 2013a,b; Westra et al. 2013; Allan et al. 2014; Berg et al. 2014; Lehmann et al. 2015; O’Gorman 2015; Donat et al. 2016; Barbero et al. 2017; Sippel et al. 2017; Tan et al. 2017). In contrast, we distinguish trends in the variance of hourly precipitation from trends in mean precipitation and in the frequency of extreme events. These are independent statistics that describe distinct properties of the temporal distribution (hourly in this study) of precipitation.
To identify trends associated with climate change, we analyze, after filtering for suspect or low-quality data, the entire database, which consists of all sites and the entire period over which the data extend. This precludes identifying regional variations or effects of decadal variability but maximizes the statistical power of our results and avoids any bias introduced by necessarily arbitrary definitions of regions. We find no statistically significant evidence of variability on decadal time scales; possible regional effects may be seen in the figures.
The NOAA database (National Climatic Data Center 2011) contains records of hourly precipitation at 5995 stations in the contiguous United States from 1948 to 2009. There are few data from 1948, so we exclude that year. A master (COOP) file lists the intervals (typically of many years duration) during which each station was nominally collecting data. Individual station files contain information for each day during which there was at least one hour with measured precipitation and flag hours with missing data. An individual station file and the COOP file must be used together to determine the hours when the station was actually collecting data. We ignore three stations for which no information is included in the COOP file.
To obtain uncorrupted and statistically valid time series the raw data must be filtered, as Menne et al. (2012) did in preparing the Global Historical Climatology Network–Daily database and as Barbero et al. (2017) did with the NOAA database. Our cuts are similar to, but were chosen independently of, those made by Barbero et al. (2017). Some of these cuts are necessarily arbitrary, but our results are not sensitive to the thresholds chosen.
Over the time period considered, almost all stations had at least some months or years without data. Indeed, according to the COOP file, only 1051 stations (of those for which there are hourly precipitation data) were collecting data without extended interruption (these stations were still subject to transient interruptions, typically of a few hours duration, indicated in the individual station files but not in the COOP file). Even stations without extended interruptions generally do not have a complete 1949–2009 data series. Many stations also moved small distances (typically a few kilometers) during the period analyzed. We ignore these small moves because a moved station still samples the same macroclimate.
We first addressed the problem of missing data. The individual data files have a code for isolated hours when the station was “down,” and some files used the same code for whole months when that station was not recording data. Hours and days so indicated were removed from the data. More problematic were entries indicated as the cumulative precipitation for an unspecified number of hours ending in the hour of the entry. In such cases the individual hourly values cannot be determined, so we ignored any day with such an indicated hour.
It is also necessary to ensure accurate counts of hours of data collection in order to compute valid averages. The absence of an entry in a station data file may be due to either the station being down or to the absence of precipitation. Unfortunately, the station data collection times obtained from the COOP file are sometimes inconsistent with the precipitation data in the individual station data files. This can be in either direction: 1) precipitation data shown when the COOP file indicates that the station was down that day, or 2) no data in the station files for months and in places where it is very implausible that there was no precipitation although the COOP file indicates the station was up.
We ignore any months within which the COOP file indicates the station was down during a period for which the data file indicates data were collected.
We ignore data for complete years during which the individual station file lists no precipitation (this is very unlikely to be true for any location in the 48 contiguous states). We cannot distinguish individual months for which the COOP file indicates data were collected, but that might be missing data, from months that actually had zero precipitation. We do not exclude these months because many stations, particularly in the Southwest and California, have months without any precipitation that must be included in calculating average precipitation. A complete absence of recorded precipitation in a month in the original dataset is not an indication that a station was down. Fortunately, the value of the average precipitation has little influence on our storminess metric.
A second set of cuts was made to reduce bias introduced by some types of missing data:
We consider a station down on any day with any cumulative data (rather than exclusively hourly data).
We ignore data from a station for a calendar year during which the station was not up at least 80% of the days; the possible preferential absence of data in some seasons could introduce bias.
To study the variation of precipitation statistics as the climate warms, we further restrict consideration to stations that have a long duration of data collection. We require that there be at least 30 years of data spread over at least four 11-yr solar cycles (to minimize any spurious trends resulting from possible solar cycle effects on weather), with at least six valid years (with the station up at least 80% of the days) of data in each cycle after earlier cuts have been made.
Although it is not possible to remove completely the effects of incomplete or biased averaging of precipitation over diurnal or annual cycles (e.g., if data are missing with different frequencies in wet than dry, or stormy than calm, periods), these cuts minimize such effects. If such a bias changed systematically over our 61 years of data, it might introduce a spurious trend, but there is no indication that this occurred.
After subjecting the data in National Climatic Data Center (2011) to these cuts, 1722 stations remain. This gives our results statistical power not found in a preliminary study (Muschinski and Katz 2013) of 13 stations that were considered individually. That earlier study found a nominally very significant () trend in the normalized second moment at one station, a significance that depends on the assumption of a normal distribution of the annual normalized second moment at each station about its mean trend.
The dimensionless normalized nth moment for station j is defined
where the are the measured hourly data, including hours when there was no precipitation; i denotes the date and hour of measurement; T denotes the temporal interval (the year) over which the moment is averaged; is the number of valid data included in the sum in the numerator (missing data are ignored); and J is the set of all valid data (over the entire period 1949–2009), containing elements, for station j. Parameter is generally less than the number of hours in T, and is less than the number of hours in J because some data are missing and some months or years of data are rejected because of inconsistency with the COOP file or to minimize bias from incompletely sampled years. The mean precipitation is
For convenience, we refer to the variance, the normalized second moment for the station j and year T, as its “storminess,” even though storms involve winds as well as precipitation. The variance can be affected by changes in the diurnal or seasonal distributions of precipitation, but is dominated by a few intense events that occur during storms. Normalization distinguishes storminess from wetness or dryness and permits comparison of storminess at wet and dry stations. Wet stations may be either stormy (on the Gulf Coast) or calm (in the Pacific Northwest), while dry stations are generally stormy (in the desert Southwest) because what precipitation they do receive comes in infrequent intense precipitation events.
Normalization by the mean over the entire record minimizes any artifacts resulting from shorter time-scale variations of the mean precipitation. Such phenomena may be aliased into long-term trends if folded into bins such as calendrical decades. By fitting a linear trend to the entire record of annual storminess at each station, we minimize effects of phenomena on decadal or shorter time scales. It is still, with only 61 years of data, not possible to distinguish the effects of phenomena on that, or longer, time scales (e.g., the Little Ice Age) from “genuine” secular trends. This is a well-known problem in all, necessarily finite, geophysical time series (Mandelbrot and Wallis 1969) because they are characterized by noise with a “pink” spectrum in which the spectral power density is higher at low frequencies.
In averaging over the 1722 surviving stations, we have not attempted to allow for the spatial density with which they sample climatic information, in contrast to the work of Rhode et al. (2013) with temperature data. We question the utility of doing so because the spatial scale on which climate varies is very nonuniform. Large areas (such as the U.S. Midwest) may have similar climates, but in mountainous areas and near coasts, especially the U.S. West Coast, sites separated by 1 km may have different, and perhaps independently varying, climates. It is not straightforward to define an unbiased algorithm to account for this. In our earlier study of drought (Finkel et al. 2016) the site-averaged and area-averaged results agreed to about 10% despite a nonuniform distribution of stations.
Figure 1 shows the mean values of the natural logarithms of the storminess at our 1722 stations. They are high in regions (the southwestern desert, Californian coast, and western Great Plains) where precipitation is concentrated both seasonally and into intense storms. They are lower in regions, such as the Appalachian Mountains and the Northeast, where precipitation is frequent, and lowest in the maritime climate of the Pacific Northwest, where precipitation occurs as steady drizzle. There is also a strong east–west gradient of storminess in the Great Plains, dividing humid regions from drier but occasionally stormier regions requiring dry farming.
For each station we make a linear fit to the logarithm of the storminess as a function of time; the slope is the time derivative of . Logarithms are used, in part, because we are interested in the percentage (or fractional) rate of change in storminess. This permits comparison of trends at stations whose storminesses may differ by large factors (as much as 50); a significant trend would be scientifically interesting (and might be practically important) whether it occurred at a stormy or a calm station.
Logarithms are used also because their mean slope is not dominated by those of a few very stormy stations; a 10% change (for example) at a calm station contributes as much to the mean slope (and to our understanding of climate change) as a 10% change at a stormy station. The uncertainty in the mean slope would be about of the standard deviation of the individual (logarithmic) slopes if the stations were uncorrelated, not , even for uncorrelated stations, where , a few dozen, is the number of stormy stations that would contribute most of the variance in the slopes of the raw storminess.
The results are shown in Fig. 2. There is much more spatial heterogeneity than for the mean storminess (Fig. 1), with less correlation among neighboring stations. Individual trends are not significant, but regional averages may be: there are hints that the mean trend in the Pacific Northwest and around Lake Superior is negative and that the mean trend in the Intermountain Basin and the southern Great Plains is positive. More rapidly increasing storminess is found in the Los Angeles area.
Figure 3 shows a histogram of the distribution of slopes (time derivatives) of the logarithms of the storminess. The mean slope is yr−1 (), where the nominal standard deviation of the mean assumes independent random variables, and the median slope is yr−1. The mean slope nominally differs from zero by , which is not significant. The nominal range is yr−1. If this trend were to continue unchanged, the mean (over the area studied) storminess would change by less than 10% in a century with an e-folding time of a millennium or more. The skewness of the distribution is and its excess (compared to a Gaussian) kurtosis is , both of which are, at least nominally, very statistically significant. These suggest strong local or regional trends because independent random variation would imply Gaussian statistics. Such trends are consistent with the existence of a variety of correlated (in space and time) weather variations on all temporal and spatial scales.
Most of the logarithmic slopes are small, with characteristic time scales of variation (defined as the reciprocal of the slope of the logarithm) of centuries. Negative and positive slopes are nearly balanced in number and magnitude, and the mean slope is much less than the magnitudes of the slopes at the stations with the most rapid change. Although it cannot be proven, the large differences among neighboring sites, even in the same climate regimes, are consistent with the hypothesis that the observed slopes are mostly the consequence of stochastic local events rather than systematic trends. However, by averaging over a large number of stations, we can set tight bounds on the mean trend. In the period 1949–2009, during which global warming was comparatively rapid, the mean storminess in the 48 contiguous states changed comparatively slowly, if at all.
Figure 4 plots the logarithmic slope of the storminess at each station against its mean value. There is a weak correlation (Pearson’s correlation coefficient is 0.25) that is nominally significant, but the random scatter is greater than any trend. Storminess is marginally more likely to be increasing at the stormier sites.
Figure 2 shows a striking concentration of positive storminess slopes in the greater Los Angeles area. This is shown in more detail in Fig. 5. The boundaries of this region are necessarily arbitrary, but of the 38 stations inside it, 9 have slopes in the highest vigintile (5%) of slopes of the 1722 stations, 13 in the highest decile and 10 in the second decile. The a priori probabilities, for sites drawn randomly from the 1722 stations, of 9 in the highest vigintile is , of 13 in the highest decile is , and of 23 in the highest quintile is . The regional trend of increasing variance is highly statistically significant.
There are roughly 150 independent regions of similar size in the 48 contiguous states, so the a priori probability of finding such an anomaly somewhere is 150 times as great, but still ≪1%. This geographical concentration of large positive slopes may be described as no more than a correlation among nearby stations, but the fact of such a correlation indicates it is a real regional effect and not an accidental fluctuation.
It may also be useful to look for trends in the mean storminess by first averaging over stations and then fitting a slope to those averages. This inverts the order of analysis shown in Figs. 2–4, in which the slopes are first fitted at each station. In Fig. 6 the mean (over the 1722 stations with sufficient data to be valid) natural logarithm of the storminess is computed for each year. The logarithms of storminess are used, rather than the storminesses themselves, so that the mean trend is not dominated by the stormy stations. The best linear fit is indicated, with a slope of yr−1, consistent with zero. The nominal range of slopes is yr−1, corresponding to characteristic e-folding times of the storminess of greater than 500 years. This result is fully consistent with the upper bounds found by first fitting slopes at each station, as illustrated in Figs. 3 and 4. Figure 6 also shows that the fitted slope would not be sensitive to a different choice of endpoints. Removing the last five years of data would increase the fitted slope by yr−1, only about ⅓ of the uncertainty.
There is a long history (Rind et al. 2008) of searching for effects of the 11-yr solar cycle on climate, with controversial results. We folded the mean (over stations with sufficient data to be valid) natural logarithms of storminess shown in Fig. 6 by year of an assumed exact 11-yr solar cycle (actual times between sunspot maxima or minima have varied by a year or more from their nominal 11-yr period). The results are shown in Fig. 7. The amplitude of the best fit sine wave is 0.039, but the null hypothesis of constant log storminess (equal to 4.362, 10 degrees of freedom) cannot be rejected even at the 75% confidence level ( with 10 degrees of freedom); there is no significant evidence for a solar effect.
4. Discussion and conclusions
The data shown here set upper bounds on any long-term trend in the mean storminess, defined as the normalized second moment, averaged over the contiguous United States during the span of our data 1949–2009. During that period the mean storminess had a characteristic (e-folding) time scale of variation of not less than about 1000 years at the level of significance or a corresponding mean rate of change of not more than 0.1% yr−1. There is no evidence for a rapid increase in storminess, and the mean over stations decreased, though the rate of decrease was not significantly different from zero.
Extrapolation of these conclusions to the long-term future climate depends on the assumption (necessarily uncertain) that the 60-yr period 1949–2009 will be representative of long-term secular trends and was not affected by shorter-term fluctuations analogous to the Little Ice Age. The mean global warming rate over the period 1950–2010 was 0.010°C yr−1 (Goddard Institute for Space Studies 2012). If the natural logarithm of the normalized second moment of precipitation varies in proportion to the warming, the nominal range of their ratio, the sensitivity of to global warming, is in the range from −0.06° to +0.006°C and is consistent with zero. It is not known if this can be extrapolated to the future, for which climate models predict significant additional warming.
We normalized storminess to the mean precipitation (at each station) over the entire period 1949–2009, making no allowance for the mean increase of precipitation of about yr−1 predicted from the mean warming rate and the Clausius–Clapeyron equation. If this assumed mean increase is applied to the normalizing denominator in Eq. (1) (which is included to equalize the weighting of trends from stations in different climate regimes, not to allow for long-term trends in mean precipitation), the resulting predicted trend in this modified mean storminess would be yr−1, with an uncertainty that depends chiefly on the (difficult to estimate) uncertainty in the rate of increase of mean precipitation. Unlike the storminess defined in Eq. (1), this would not be a measure of the absolute size of extreme weather events.
The comparatively rapid and statistically significant increase in storminess in the Los Angeles basin shown in Fig. 5 adds a new dimension to the findings of Rosenfeld (2000) and Ramanathan et al. (2001) that anthropogenic aerosols suppress precipitation. During the period covered by our data, 1949–2009, air pollution in that basin was observed to be dramatically reduced. Particulate smog may serve as condensation nuclei for water vapor; if they are numerous, the supersaturated water vapor content of the atmosphere is divided among many small droplets, producing a gentle drizzle. A possible explanation of our finding is that in cleaner air with fewer condensation nuclei, the supersaturated water vapor condenses as fewer but larger droplets, producing more intense rain and increased storminess.