1. Introduction
Precipitation is a key variable for climate, but it is difficult to deal with because it is not a continuous variable. Intermittency is a core characteristic of precipitation and yet is frequently ignored. Often only precipitation amounts are examined, perhaps as monthly means and at best using daily means. There are few attempts to properly address the full characteristics of precipitation: type (e.g., rain or snow), intensity when it precipitates, frequency, duration, amount, and distribution in space and time (Mindling 1918; Trenberth et al. 2003; Trenberth 1998, 2011). There are strong diurnal and seasonal cycles in many places that are also often neglected.
In observational analyses, most in situ data are daily (e.g., Zolina et al. 2010, 2013; Zolina 2014), from a once-per-day reading of a rain gauge that produces accumulated amount, and tipping-bucket or recording instruments that provide higher-frequency information are less commonly available. Remote sensing, whether from ground-based radar or spaceborne sensors, instead produces a measurement related to the rate of precipitation at the time of the scan, and assumptions are required to reconcile these with accumulated amounts. Yet, it is obvious that rain rates continually vary and that most of the time, in most places, it does not rain or snow.
The frequency of rainfall locally varies wildly from 0% to over 50% (e.g., Trenberth 1998; Dai 2001) but depends on the area and threshold used to detect precipitation. Dai (2001) ingeniously used synoptic data of “present weather” to determine the frequency of different kinds of precipitation. The diurnal cycle in precipitation over the United States occurs mainly in frequency rather than intensity (Dai et al. 1999). Dry spells in precipitation mean that time averages inevitably have lower intensity. Similarly, areal averages are likely to feature some places with no precipitation, and hence areal measurements have a higher frequency but lower intensity than point measurements (Trenberth 1998). It is therefore not appropriate to compare grid square averages (from a model) with point measurements, but a number of studies are flawed in this regard (some given below).
A measure of the intermittency of precipitation depends on the sampling interval and the precipitation amount to be exceeded for an interval to count as wet. The commonly used definitions of precipitation characteristics include: frequency F (defined as the percentage of time it rains or snows) and intensity I (the mean precipitation rates averaged over the precipitating time). Also, a common variant is to examine the frequency contingent on precipitation in the previous time interval. Precipitation duration often refers to the time period it rains or snows, and hence a mean precipitation duration D is the mean length averaged over all precipitation events. These all require definition of a precipitation event, often as a time period (1 h for hourly data or 3 h for 3-hourly data) with precipitation rate P exceeding a given threshold. The latter can vary.
Intermittency of precipitation has been addressed to some extent in a few other studies. One approach is to treat precipitation as an on or off process; see Deni et al. (2010) for a discussion of various probabilistic models describing the distribution of dry and wet spells. The tendency for rain events to have a finite lifetime, associated with storms or cold fronts, for instance, means that rain hours tend to cluster, perhaps modeled by a Poisson cluster model or a Markov renewal process (Foufoula-Georgiou and Lettenmaier 1987). Intermittency has also been explored using “interamount” times, which is the waiting time required between successive amounts of precipitation, and “burstiness,” which is a normalized measure of the dispersion of interamount times (Schleiss and Smith 2016). These have the advantage (or disadvantage) of not requiring arbitrary thresholds to be set, but they do not easily relate to amounts. Here we focus on hourly to daily intervals and do not fit the data to theoretical distributions.
The most common way of measuring intermittency is to look at the statistical distribution of dry and wet periods. Major quantities of interest in this approach are the transition probabilities between dry and wet states and the length of dry–wet spells, that is, the number of consecutive days during which the precipitation amount remains below or above a certain threshold (e.g., Zolina et al. 2013). A common threshold for observational data at stations is that it counts as a rain day if precipitation is >1 mm day−1. This excludes light rain but accounts for the limitations of rain gauges, evaporation effects, and so forth in measuring tiny amounts. Some studies then apply the same threshold to define dry days as being <1 mm day−1 amount, but Zolina et al. (2013) use 0 mm day−1, leaving small amounts of rain not accounted for.
Here we are interested in global scales and necessarily in gridded analyzed products. The gridded products have a further set of issues related to the averaging and size of grid squares and the density of the observations and how remote sensing observations are included. Because most of the time during a day it does not precipitate, a daily mean inevitably consists of periods of precipitation at varying rates and times of no precipitation at all. Hourly averages are practicable and viable and now typically include several time steps in models, whereas shorter times are apt to contain numerical noise in models and physical noise in observations associated with individual cloud cells.
Gutowski et al. (2003) noted for a central area of the United States that there was lack of agreement between precipitation in models and observations for 6- and 12-h accumulation periods. Dai and Trenberth (2004), Sun et al. (2006), and Dai (2006) analyzed precipitation characteristics in climate models and found that the frequency of precipitation was too high while the intensity was not large enough. These findings were reinforced by Stephens et al. (2010), who noted the “dreary state” of climate modeling with regard to such characteristics when compared with CloudSat data. Models from phase 5 of the Coupled Model Intercomparison Project (CMIP5) have been examined for extremes confined to daily precipitation amounts (Toreti et al. 2013), a limitation common in many studies, and Herold et al. (2016) noted that CMIP5 models and reanalyses tend to oversimulate wet days.
Observational datasets have evolved and the latest generation of Precipitation Estimation from Remotely Sensed Information Using Artificial Neural Networks–Climate Data Record (PERSIANN-CDR), the Climate Prediction Center morphing technique (CMORPH), and Tropical Rainfall Measurement Mission (TRMM) 3B42 have all been bias corrected using ground-based stations or Global Precipitation Climatology Project (GPCP; Huffman et al. 2009) products, generally with great improvements (Maggioni et al. 2016; Gehne et al. 2016). The new and much improved CMORPH global high-resolution precipitation estimates (Xie et al. 2017) begin in 1998. Issues remain in winter over land because of snow, but careful evaluations show a consistent superior performance over the TRMM 3B42 dataset (Xie et al. 2017). The datasets used are discussed in section 2.
In numerical models of climate or even weather prediction, until recently, the time steps were often from 20 min to an hour (or more), and numerical noise was an important factor in interpreting results at high frequency. Often the phenomena responsible for the precipitation were not explicitly represented in the model because the grid was too big. Convection and clouds have been parameterized, but many phenomena also not well resolved in models are not parameterized in any form. For instance, often there has been no attempt to represent mesoscale convective systems, easterly waves, tropical storms, and so forth in global models.
Accordingly, in the past it made little sense to try to evaluate model hourly or higher-frequency precipitation data both because the observations were lacking and the models were not expected to be realistic. This has changed as increased focus has occurred on extremes of weather and climate, because of their importance for society in terms of impacts and damage, and because they are expected to change as the climate changes (Trenberth et al. 2003). Moreover, both global models and observations have advanced to begin to make this possible.
In this paper, we focus especially on hourly amounts of precipitation. An hour is reasonably consistent with the time scales of convective events; it may be longer than that for individual cells but is more in step with the precipitation events. It is also close to being doable observationally and should become possible almost globally as the Global Precipitation Mission (GPM) products are developed (cf. Kidd et al. 2016). By examining both observations and models, we hope to be able to advance the knowledge from both and hopefully contribute to ways of improving models.
Section 2 discusses the data used, and the methods and metrics used are outlined in section 3. Results are presented in section 4 and discussed in section 5.
2. Data
Because of the small spatial scales associated with precipitation, especially convective precipitation in summer, a high density of stations is required to produce reliable analyses. For instance, Trenberth et al. (2014) and Dai and Zhao (2017) revealed important differences among datasets arising mostly from the number of stations with data included in the analysis. Although interested in datasets that have 3-hourly or hourly temporal resolution, we also make use of daily precipitation products because they have been used for climate purposes up to now. There are several 3-hourly near-global products of precipitation (see Dai et al. 2007). With GPM, high-quality 3-hourly global precipitation products are likely to become realistic, and methods already being exercised in PERSIANN and CMORPH should enable hourly products to become viable.
Substantial issues exist on how to go from an individual site measurement to a gridded area analysis, and differences exist among analyses because of this issue (Herold et al. 2016). The latter suggest that approximately 30 gauges are required to accurately estimate precipitation intensity over a 200 km × 200 km area, but generally this number is not available. Using fewer gauges means that the intensity tends to be higher and the frequency lower than in reality. Here we are interested only in gridded products at 1° resolution or better, but analytic methods matter and can be responsible for differences.
The issue of how to compare instantaneous rainfall rates from sensors, such as those on TRMM, with ground data was addressed by Wolff and Fisher (2008). Sampling errors related to the 1–3 estimates per day from satellite explain on the order of 10% of variance of the differences. Substantial issues occur over coastal regions where land versus ocean effects come into play. Saturation of channels brings in biases at high rain rates, making for large differences during heavy rain events. Over land the TRMM Microwave Imager (TMI) cannot resolve rain rates <8 mm h−1, but over ocean TMI can resolve rates of greater than 0.02 mm h−1, while saturating at greater than 20 mm h−1 rates.
Zhou et al. (2008) evaluated hourly and 3-hourly precipitation satellite products from PERSIANN and TRMM 3B42 over China and found overestimates of rainfall frequency and underestimates of intensity, but recent bias-corrected products have improved many of those aspects. Prat and Nelson (2015) evaluated daily precipitation estimates over the continental United States from satellite, radar, and gauge measurements and found large biases in the bias-corrected TRMM 3B42 product over the western United States for high accumulation rates (>5 mm day−1). However, they did not adequately address the issues of scales (e.g., point vs area averages). Using daily data, Herold et al. (2016) found large discrepancies among observed estimates of intensities, but often this was because of the failure to properly account for point versus area averages.
Maggioni et al. (2016) focused on TRMM and noted that light precipitation and warm rain events are typically underestimated while orographic enhancement of rainfall is not explicitly resolved by the algorithms in mountainous regions, areas further complicated by complex terrain and snow cover. Winter precipitation from lighter rain events, snow, and mixed-phase precipitation showed large biases. TRMM 3B42, version 7 (v7), was shown to be superior to other products evaluated, in part because it was bias corrected. However, it suffers from low detection in many areas.
Gehne et al. (2016) explored several global high-resolution (1°) precipitation products, including versions of the two observational datasets used here, except they are 3 hourly. The analyses are often produced for different purposes and differences can be quite large, indicating that substantial observational uncertainties exist. Gehne et al. (2016) show figures of frequency versus intensity as a function of season, making use of a logarithmic scale to accommodate the more frequent light rain events. We follow this approach here. Correlations among the products were low at daily and annual time scales but higher at monthly time scales, suggesting that large-scale biases are large compared with interannual variability, while random errors are large on a daily time scale.
PERSIANN (Hsu et al. 1997) comes from a neural network, trained by precipitation from TMI (2A12) and other satellites. The PERSIANN-CDR version (Ashouri et al. 2015) is gauge bias corrected but available only daily. PERSIANN, being a near-real-time product, is not a bias-adjusted estimation from gauge observations and its variability on annual time scales is very different from TRMM 3B42, the bias-corrected CMORPH, and PERSIANN-CDR (Gehne et al. 2016), and thus it was excluded from the present analysis.
New estimates of CMORPH (Joyce et al. 2004; Joyce and Xie 2011) have been made at about 8 km resolution from 60°N to 60°S at 30-min temporal resolution since 1998 [CMORPH, version 1.0 (v1.0), bias corrected (CRT)] (Xie et al. 2017). The first version contained substantial biases, and comparisons with surface observations (Dai et al. 2007) showed that the high-resolution satellite products were useful for studying precipitation frequency, intensity, and diurnal cycle, although errors for individual precipitation events were large. The biases have been greatly reduced in CMORPHv1.0, which is bias corrected against a daily rain gauge analysis over land and GPCP pentad data over the ocean. For example, over land, values are first scaled to the climatological values and the second step accounts for the year-to-year variability using a running 31-day window over a radius of 0.5° around each grid point. Detailed comparisons (Xie et al. 2017) show that it is much better at representing daily and 3-hourly precipitation than TRMM 3B42 over many regions of the globe and especially over the United States, where many other (e.g., radar) datasets exist. Gehne et al. (2016) also find that the new CMORPH product is much improved over the original version in terms of biases and spurious trends. Nevertheless, while performance was reasonably good in the tropics, CMORPH gave an overestimate of summertime convective precipitation while underestimating winter precipitation in midlatitudes. We have obtained an hourly and 0.25° resolution version of CMORPHv1.0 from 1998 to 2013 for our analysis.
TRMM 3B42v7 data are available 3 hourly on a 0.25° grid from 50°N to 50°S and from 1998 to the present, although only data through 2013 are used here. They are a merged product based mainly on microwave sensors infilled with infrared estimates, while calibrated to the TRMM sensor precipitation estimates (Huffman et al. 2007). The monthly means of the 3-hourly microwave-calibrated IR rainfall estimates are combined with the Global Precipitation Climatology Centre (Schneider et al. 2014) monthly rain gauge analysis to generate a monthly satellite–gauge combination (TRMM 3B43). Each 3-hourly field then uses pdf matching to rescale the corresponding monthly satellite–gauge field, making for possible discontinuities at the beginning/end of each month. Both observational datasets have large uncertainties in light precipitation, and zero values are not corrected by bias correction.
Substantial issues occur with differences in resolution of both model and observational datasets. These can be examined using datasets with high resolution that are degraded in either space or time. Note that 3 hourly is the highest time frequency in CMIP5 output. Dai and Trenberth (2004) evaluated the diurnal cycle in an earlier version of the NCAR Community Climate System Model and Covey et al. (2016) explored the diurnal cycle of precipitation in CMIP5 models and 3-hourly TRMM data. Future plans included exploring the diurnal cycle in the other datasets analyzed here, but in this paper the focus is on more general aspects of intermittency.
We have used model output from the NCAR Community Earth System Model, version 1 (CESM1), Large Ensemble (CESM-LE; Kay et al. 2015) that now consists of 40 runs using coupled atmosphere, ocean, land, and sea ice model components. The simulations at a resolution of 0.9° × 1.25° span from 1920 through 2100 using estimated observed historical (through 2005) and RCP8.5 projected external forcings from the CMIP5. A detailed analysis of runs from 2000 to 2014 is given in Trenberth et al. (2015). For this project, two special runs (members 34 and 35) were made with output of hourly precipitation data; we use only member 35. We have focused the processing on the same time interval available from observations, namely 1998–2013. All observations at 0.25° resolution were averaged to the model grid using the NCAR Command Language (NCL) Earth System Modeling Framework (ESMF) regridding tool with the “conserve” algorithm, which acts to preserve the integral of the data between grids. This is an issue only for the north–south grid.
3. Methods: Hourly rainfall data
We wish to understand intermittency in precipitation in both the model and observed datasets. Central questions pursued here are, using hourly data, can we understand 3-hourly averages? Another way to think of this is that we wish to assess how much information is lost by averaging to longer time scales. Further, can we use the hourly and 3-hourly data to better understand the daily values that are most widely available and analyzed? In turn, there are even more analyses of monthly mean amounts available.
Hence, if we average the hourly into daily values, then dry spells reduce the hourly rates in ways that depend on the duration and covariability among the values, including diurnal cycle aspects [see Covey et al. (2016) for the latter]. One might expect that if it rains in one hour, the odds are high that it rains in the hour before and after. But maybe the odds are also high that it does not rain, say, 4 h after, because the rain event is over.
Frequency distributions are highly skewed, with frequent small rainfall rates. As in Gehne et al. (2016), we compensate by using logarithmic-sized bins, and results have been plotted both with a linear and logarithmic scale for the rate, but the latter is used for most examples presented here. The integral under the curve for the amount versus rate is equal to the total precipitation amount. As the number of bins is 100, with a constant logarithmic base-10 bin length between 10−4 and 10 mm h−1, the bin length is 0.05. Rates below the lowest bin, including zero, are included in that bin.
We explore several metrics related to the duration of events. The first is a simple count of the frequency that the precipitation exceeds a certain predefined threshold for various times. We focus on three thresholds: 0.02 mm h−1; 0.2 mm h−1 as an intermediate rate; and 2 mm h−1 as a heavy rate, but not so heavy that it excludes many regions. The choice of threshold is based on the consideration that observations from TMI do not resolve rates less than 0.02 mm h−1 (Wolff and Fisher 2008), very light rain rates are not well known, and values less than 0.02 mm h−1 are excluded from later computations to determine the duration of events. Moreover, a trace is defined as less than 0.01 in. = 0.25 mm and is not measured in rain gauges.
The second metric is the count conditional on precipitation at the previous times, which gives the duration of precipitation for the different thresholds, as described further below. This metric averages over all rates, but is quite revealing.
These expectations may be upset if there is no precipitation, and, given that we compute these for a specific range of rates, such as 0.2–2 mm h−1, there can be leakage to or from outside that range, which allows values to be outside the expected range. For example, consider very high hourly rates that do not last long, and thus the daily rates are never that high. One effect of time averaging is that there is a shift in the frequency distribution toward higher frequencies for hourly and 3-hourly data compared with daily. In that frequency band the values can then be greater than 24 h. Similarly at very low rain rates (drizzle), if we set a low threshold, averaging can push the rainfall rates to values below the threshold, and hence we have used different thresholds for the hourly versus 3-hourly versus daily data. For observations neither of these issues arose, but for CESM both problems showed up in a few spots (illustrated later in Fig. 16).
These metrics are readily computed and mapped from histograms of the frequency of various precipitation rates. For presentation purposes, the graphs of frequency and amount versus rate are smoothed over the bins using a 1–3–4–3–1 smoother (effectively giving the x axis 20 bins rather than 100), and the frequency is implicitly that per bin.
Two regions have been examined in considerable detail (Fig. 1), as shown for four 10° × 10° regions 30°–40°–50°N, 90°–100°–110°W over the United States and 10°N–0°–10°S, 140°–150°–160°W straddling the equator in the central Pacific. The red dots show the first 10 grid points starting at the southwestern corner (featured later in Fig. 4), and the blue dots are two arbitrarily chosen grid points used to illustrate results in the southeastern United States at 31.5°N, 95°W and in the tropical Pacific at 9.1°S, 145°W (see Fig. 1). We begin by showing results for two seasons and mainly for the U.S. points.
Two regions have been examined in considerable detail as shown for 20° × 20° boxes over North America and the tropical Pacific. The red dots show the first 10 grid points (featured in Fig. 4), and the blue dot is a single grid point where most in-depth analysis occurred.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
4. Results
a. Basic fields
The mean precipitation amounts for December–February (DJF) from January 1998 to December 2013 (Fig. 2) complement those of Gehne et al. (2016) and are representative of other seasons. The differences among the DJF means (Fig. 2) are small between TRMM 3B42 and CMORPH, with TRMM slightly higher in the tropics and lower in the extratropical ocean storm-track regions. However, the differences with CESM are much larger (see also Trenberth et al. 2015) in all seasons, highlighting that the observational differences are relatively small while CESM has too much precipitation amount in general, except near New Guinea and parts of the South Pacific convergence zone (SPCZ). These differences are similar in all seasons (not shown). However, the standard deviations of the 3-hourly values are much larger (Fig. 2), by a factor of 2 or more, in both observational datasets than in CESM, especially in the SPCZ and also in the monsoon regions more generally.
For 3-hourly values in DJF from 1998 to 2013, the (top) mean precipitation amounts (mm h−1) for CESM, CMORPH, and TRMM; (middle) std devs (mm h−1) for CESM, CMORPH, and TRMM; and (bottom) differences in mean precipitation amounts (mm h−1) between (left) CESM and CMORPH, (center) CESM and TRMM, and (right) CMORPH and TRMM.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
Correlations between the observational 3-hourly precipitation total values for the 16 years (43 200 values; 5% significance level is 0.01) are about 0.7–0.9 over most of the oceans and are very similar for the correlations of the values with the mean annual cycle, as defined by the first four harmonics, removed (Fig. 3). The reason is that the variance associated with the annual cycle is small compared with the variability. Correlations are less than 0.5 only in regions where the total amounts of precipitation and standard deviations are very small (low signal to noise; e.g., Fig. 2).
Correlations between CMORPH and TRMM 3-hourly rates of precipitation with first four annual cycle harmonics removed for (top) DJF and (bottom) JJA 1998–2013.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
Figure 4 illustrates the hourly time series of CESM precipitation for the 10 points shown in the grid square over the southeastern United States (Fig. 1, the 10 red dots) for 110°–97.75°W along 30°N, each 1.25° apart. One season is shown and the axis is in hours starting in December 2004; 400 h is 16.67 days. Note the difference in vertical scales between DJF (winter) versus JJA (summer). Immediately apparent is the episodic nature of the rain events in winter, usually with some signature at all points, but also with large variations among the points. Long dry spells of over a week are evident. In contrast, in summer a strong diurnal cycle is evident and dry spells are much fewer at these locations. Several times rain events extend to only half of the grid points, suggesting smaller spatial scales in summer. Plots for observations have similar character but with fewer events and longer dry spells. The power spectra and autocorrelations (Fig. 5) show that most of the variability is at lower frequencies longer than 12-h periods, but this mainly reflects the long dry spells rather than the duration of rain events. Autocorrelations drop to 0.2–0.5 by 12 h but rebound somewhat in summer as part of the diurnal cycle.
Sample time series (mm h−1) from CESM for 10 grid points (see Fig. 1) from left to right at 30°N and from top (110°W) to bottom (97.75°W) for (left) DJF 2004/05 and (right) JJA 2005. Each plot is offset by one major tick mark on the y axis.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
From CESM for the first 10 grid points (Fig. 4), the (left) autocorrelation at lags up to 24 h and (right) power spectra are shown; colors as in Fig. 4.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
There are simple connections between autocorrelation and the conditional probability of a wet period (e.g., day or hour). If a first-order Markov chain is assumed, the autocorrelation function decays at a geometric rate. The kth-order autocorrelation coefficient can be expressed as the difference in conditional probability of a wet period given the previous period was wet and the conditional probability of a wet period given the previous period was dry, to the power of k.
b. Duration of events
The duration of rain events has been determined for nine thresholds (0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1, 2, and 5 mm h−1) using 3-hourly data. This is a simple frequency count of the number of values exceeding that threshold, given that there was precipitation at previous times. An example of the duration of precipitation events illustrating these aspects (Fig. 6) for all three datasets, using the 3-hourly data at 31.5°N, 95°W, is typical for all grid points examined and all sets of thresholds. The conclusions also apply for hourly data. In Fig. 6 (left), the frequency is the fraction of the total number of hours, thereby including the dry spells, and the number of hours of rain is given in the inset. Figure 6 (right) shows the same results for the wet spells only, and the inset values were used to normalize the results. Here the inset values show the total number of events. In this case, rain hours can count multiple times; for instance, a 5-h duration event includes two 4-h events, three 3-h events, and five 1-h events.
Histograms of percentage frequency of total precipitation events from January 1998 to December 2013 as a function of their duration using 3-hourly data (conditional probability of precipitation, given precipitation the previous times) from CESM, TRMM, and CMORPH for threshold rates of 2, 0.2, and 0.02 mm h−1 at the grid point 31.5°N, 95°W. (left) The frequency is given as function of total time (h), with the inserts in each panel giving the total rain hours. (right) The frequency of rain hours is given, with the inserts giving the total number of events of each duration.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
The much greater number of wet events in CESM than observed affects all times for the lower rain rates (Fig. 6). The frequency of precipitation from CMORPH is somewhat greater than TRMM at the lower rates (0.02, 0.2 mm h−1) and about the same at 2 mm h−1, but both differ considerably from CESM. Given precipitation for 1 h (or 3 h), CESM has much lower conditional probability that the next time interval will also be wet for 2 mm h−1. But as the duration is extended, the observed values fall off much faster than for CESM. After about 6–9 h, CESM has greater probability of an event lasting that long, and some precipitation events last at least twice as long as observed, especially for light precipitation rates. There are many more light rain events that last days in CESM but fewer heavy rains, even though the heavy rains that do occur last longer. This is consistent with the idea of “perpetual drizzle” in many models, and while the drizzle may not be continuous, it does occur sometime within the 3-h window.
These results are expanded globally for 3-, 9-, and 15-h durations for DJF in Fig. 7 (for 0.02 mm h−1) and Fig. 8 (for 2 mm h−1). At the light rain rates (Fig. 7), CESM stands out over the oceans and monsoon areas over land with the high durations, often exceeding 70% of the time throughout the tropics and subtropics. CMORPH values are somewhat higher than for TRMM in the ITCZ and SPCZ regions, with a clear falloff over time from 0- to 12-h durations. Zonal averages over the oceans and land separately (Fig. 9a) confirm that CMORPH and TRMM agree quite well over the oceans, and it is mainly over land in the monsoons that CMORPH exceeds TRMM for all durations considered. But both are substantially different than for CESM. Results are similar for other seasons except that the monsoons move into the Northern Hemisphere.
Percentage of time for precipitation exceeding 0.02 mm h−1 for various durations of 3, 9, and 15 h for (left) CESM, (center) TRMM, and (right) CMORPH using 3-hourly data. Note that the color bar is not linear.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
As in Fig. 7, but for precipitation exceeding 2 mm h−1.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
Zonal averages for 50°N–50°S for DJF of values of the percentage of precipitation exceeding certain durations as given by the values on each panel (3, 6, 9, 12, 15, 18, 21, 24, or 27 h). Solid lines are for land and dashed lines are for ocean for CESM (red), CMORPH (green), and TRMM (blue). Shown is the threshold of (a) 0.02 (from Fig. 7), (b) 0.2, and (c) 2 mm h−1 (from Fig. 8). In (a), the values are a fraction from 0 to 1. In (b), the values are a fraction from 0 to 0.6. In (c), the values are a percentage from 0% to 4% for hours 3, 6, and 9 and from 0% to 2% for hours 12, 15, and 18.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
In Figs. 8 and 9c for 2 mm h−1 thresholds, it is the other way around. TRMM and CMORPH again agree moderately well over the oceans, and TRMM tends to be substantially larger over land in the tropics, but both observationally based datasets have much larger values than for CESM, especially at time 0–3 h. Not surprisingly, the durations for this threshold fall off rapidly by more than half from 3- to 6-h durations, and by more than half again by 9-h duration. The scale is doubled in Fig. 9c for durations from 9–12 to 15–18 h, and values are so small after 18 h that they are not included. However, at the longer durations of 9–18 h, the CESM fraction has increased to be comparable to or exceed the observations. Certain areas, such as the Andes and the southwestern Indian Ocean, stand out as problems in CESM (Fig. 8).
Figure 9b shows the zonal mean results for the intermediate threshold of 0.2 mm h−1, and it is much closer to the result for 0.02 mm h−1. Here the observational results are in closer agreement while again CESM has much larger values.
To introduce the application of the theory outlined in section 3, we again examine the single grid point in the southeastern United States. The frequency versus rate diagram (Fig. 10) is even more revealing than Fig. 6. In general there is a drop off in frequency from daily to 3 hourly and again to hourly values, highlighting the intermittency. The differences are larger for very light rates (<10−2 mm h−1) for TRMM for 3 hourly versus daily (but not for CMORPH), suggesting lack of reliability at these rates (not unexpected). However, the differences between hourly and 3 hourly narrow somewhat for CMORPH at these light rates.
For the grid point at 31.5°N, 95°W, the frequency (%) as a function of precipitation rate (mm h−1) for CMORPH (green), TRMM (blue), and CESM (red) is shown for hourly (solid), 3 hourly (dotted), and daily (dashed) data in (top) DJF and (bottom) JJA.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
The comparisons among the datasets (Fig. 10) reveal that in DJF (winter) CESM has much lower frequency of precipitation above 1–2 mm h−1. This depends on season, and in summer (JJA) CESM is much lower for daily amounts over 0.5 mm h−1. There is reasonable agreement for the daily values for most rates, although CESM is higher for about 0.02–0.2 mm h−1 in summer. The differences among all datasets become more evident for 3-hourly and hourly data. In winter CESM is only slightly higher than CMORPH above about 0.02–1 mm h−1 for both time scales in DJF, but much higher in summer. TRMM 3-hourly values are considerably lower than for both CMORPH and CESM.
To generalize these results, we have composited all grid points for 30°–40°N, 90°–100°W (the southeastern box over the United States; Fig. 1) for the four seasons (Fig. 11), and the noise is greatly reduced (cf. Fig. 10). Figure 11a includes the standard error for each estimate, computed from the variance of each season about the long-term mean, and shows that most differences are statistically significant. Similar results apply for other seasons. The frequency versus rate and amount versus rate show the same information in different ways. The amount versus rate demonstrates the importance of the intensities above 0.1 mm h−1 to the accumulated amount, and the differences between the model and the observations are readily apparent, with far too much light precipitation in the model and not enough intense precipitation, with the transition about 1–2 mm h−1. The logarithmic scales emphasize the importance of the rates greater than about 1 mm h−1 to the total amount, although for the model in summer it is the range 0.1–1 mm h−1 that matters most.
For the grid square 30°–40°N, 100°–90°W, the (a)–(d) frequency (%) and (e)–(h) amount as a function of precipitation rate (mm h−1) for CMORPH (green), TRMM (blue), and CESM (red) of hourly (solid), 3-hourly (dotted), and daily (dashed) data for (from left to right) DJF, MAM, JJA, and SON. In (a), error bars are included as plus/minus one std error of the mean for each curve. (i)–(l) The ratio of the values in the top panel (frequency) as the durations 
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
The results for the ocean points are quite similar relatively; for example, Fig. 12 gives results for the southeastern grid square 0°–10°S, 150°–140°W over the central Pacific Ocean. In all seasons, the CESM rains far too frequently at light rates and not enough at heavy rates, even though the model has far too much total precipitation amount (Fig. 2).
For the grid square 0°–10°S, 150°–140°W, the (a) frequency (%) and (b) amount (mm), as a function of precipitation rate (mm h−1) for CMORPH (green), TRMM (blue), and CESM (red) of hourly (solid), 3-hourly (dotted), and daily (dashed) data for DJF. (c) The ratio of the values in (a) (frequency) as the durations 
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
The frequency increases for nearly all rates as the time series go from hourly to 3 hourly to daily (Figs. 11, 12). Because the observational results have no sound basis for rates less than 0.02 mm h−1, we emphasize the ratios for 0.02–2 mm h−1 to estimate the parameters 
Integrated duration of precipitation (h) based on 
In both locations, the 
The results of simply taking the duration of precipitation above a certain threshold, computed by taking the amount and dividing by the number of hours 
The implied duration if the rate was constant, given the total amount and the rate exceeding the threshold of (a) 0.02 and (b) 2 mm h−1 for northern winter (DJF) with (top) hourly, (middle) 3-hourly, (bottom) daily data for CESM, TRMM, and CMORPH.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
The duration of events, as revealed by the ratios of the hourly to 3 hourly to daily for DJF (Fig. 14), confirm that CESM rains most of the time over the ocean (results are similar for other seasons). Here the rates used are averaged up to 2 mm h−1, but with a low threshold that varies depending on the sampling. The latter was initially fixed at 0.02 mm h−1, which works fine for the observations and most of the domain for CESM, but not everywhere, as discussed below (with Fig. 16). The 24-hourly to hourly ratio allows for events longer than 3 h to come into play and is the main focus for discussion, although the 3-hourly and hourly to daily ratios are very similar. Figure 15 presents the zonal means over land and ocean as summaries.
The implied average duration of events (h) for precipitation rates of 0.01 (daily), 0.08 (3 hourly), and 0.24 (hourly) to 2 mm h−1 based on the ratios of (top) hourly to 3-hourly (i.e., 
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
Zonal average inferred duration of events (h) for precipitation rates of 0.1 (daily), 0.08 (3 hourly), and 0.24 (hourly) to 2 mm h−1 for land (solid) and ocean (dashed) based on the ratios of (top) hourly to 3-hourly (i.e., 
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
For 
The results in Fig. 14 are more definitive than Fig. 13 in terms of duration of events, but prove to be quite sensitive to the thresholds used in a few places. Although no problem points were encountered in the observational data, a few areas gave unrealistic results for CESM. These were points along or close to the equator in what should be the equatorial dry zone (in the absence of El Niño conditions) and in the southeastern Pacific, west of South America, an area of subsidence and persistent high pressure. To illustrate, Fig. 16 shows one point in the Pacific near the equator for one DJF season of 90 days with hourly values plotted alongside daily values. The result is typical except for two years when El Niño events occurred. It is far from a dry zone, but rather there is an almost perpetual drizzle that nonetheless undergoes a distinct diurnal cycle (see inset in Fig. 16). The rates are mostly less than 0.05 mm h−1, but dry spells of an hour or more are rare (most notable near hour 600 and in the last 3 days). These amounts qualify as only a trace every 12 h and are not measureable in the real world. The daily rate seldom exceeds 0.02 mm h−1 and, as a result, the statistics here are interfered with by the selection of thresholds, such as those in Fig. 14. Such problem points could be masked by insisting on mean amounts of, say, 0.01 mm h−1 or more.
For CESM at 0.47°S, 165°W in DJF 2010/11, the hourly (red) and daily (black) precipitation rates (mm h−1). The mean diurnal cycle for this period is given in the inset in 10−3 mm h−1; local noon is at 1300 UTC. The mean rate is 0.011 mm h−1.
Citation: Journal of Hydrometeorology 18, 5; 10.1175/JHM-D-16-0263.1
5. Discussion and conclusions
Intermittency is a core characteristic of precipitation, not well described by data and very poorly modeled. The absence of quality hourly or higher-resolution precipitation datasets has been a major impediment to adequately defining aspects of intermittency in precipitation. We exploit a new hourly observational almost global dataset (60°N–60°S) and the CESM1 to examine many aspects of intermittency. We develop diagnostics related to the precipitation characteristics that help understand the climate system and that are useful for evaluating models.
We define several metrics that together form an indicator of intermittency by distinguishing between continuous precipitation on hourly, 3-hourly, or daily averages. Average duration of precipitation events are derived for the same time averages and three thresholds: 0.02, 0.2, and 2 mm h−1. The former is about the low threshold observable in various ways, whereas 2 mm h−1 is close to the upper limit in many places and near where the average greatest amounts occur.
Conditional probabilities for events lasting longer than certain times (e.g., Fig. 6) can be useful but depend a lot on the thresholds that are set. They provide the fraction of time it precipitates above the threshold for various durations. In observations, the ITCZ, SPCZ, and monsoon rains stand out as regions with most frequent rains that exceed 50% at light rates and 5% at 2 mm h−1 for up to 6 h or so, but falling off exponentially at longer durations. CMORPH frequencies are higher than for TRMM, which tends to be substantially larger over land in the tropics, but both are mostly dwarfed by CESM. At light rain rates CESM has double or more the frequency of the observed, but at heavy rates CESM frequency is much less than observed throughout the tropics.
These results are reinforced by simple evaluations of the implied duration of events computed from the total amount divided by the time that it precipitates above given rates (Fig. 13), which is quite revealing. TRMM values are very similar to but slightly less than CMORPH, and CESM is a different world, with daily durations exceeding 20 h in many places over oceans yet not enough heavy rains.
We also proposed some new integrated metrics of duration of events 
CESM has too much rain amount, and it rains far too often but with insufficient intensity. The picture that emerges is that the model has precipitation at light rain rates that seldom stops: perpetual drizzle and far too frequent light rain events. Yet there are not enough heavy events. This is consistent with the idea that the atmospheric model releases instability far too readily, which prevents convective available potential energy from building up to produce heavy rains (see Ma et al. 2013). No doubt much of this stems from the lack of simulation of the relevant phenomena, especially in the tropics, where the model uses subgrid-scale parameterization of convection and precipitation and fails to adequately represent many organized transient disturbances, although higher-resolution (0.25°) grids are depicting more realistic tropical storms (Bacmeister et al. 2014).
It is readily apparent for the CESM1 climate model that the simulated precipitation is really from another planet. Newer convective parameterizations are now included in the latest model, but it is unlikely that major progress will occur in modeling precipitation unless the full characteristics of precipitation are appreciated and used for evaluations. Even getting the mean amount about right would be quite insufficient because of the importance of the associated latent heat release in the atmosphere and the forcing of transient or quasi-stationary atmospheric waves. It is essential to improve the diurnal cycle and timing of precipitation events (Covey et al. 2016) as well as the frequency, intensity, and duration of events. There is a need to properly represent the phenomena and processes responsible for precipitation either explicitly or implicitly (parameterized) in order to better depict precipitation intermittency in models.
Acknowledgments
We thank John Fasullo for help with the datasets and Aiguo Dai for comments. This research is partially sponsored by the DOE Grant DE-SC0012711. The datasets are discussed in section 2 and those used are public domain and available: GPCP (
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