## 1. Introduction

The shape of the distribution of temporally averaged precipitation intensity has a long history of being represented by a gamma distribution since at least Thom (1958). For sufficiently short averaging periods (e.g., daily average intensities), the typical shape of the distribution has probability falling slowly over a few orders of magnitude, summarizing the wide range of precipitation intensities experienced in a given region. Is there a fundamental explanation for this behavior in terms of simple physical processes? The goals of this paper are to (i) explain why temporally averaged distributions have a gamma-like distribution shape using a simple model based on the moisture equation [e.g., Neelin and Zeng 2000, their (2.2); Sobel and Maloney 2013, their (1)], and (ii) provide theory for how the gamma distribution parameters depend on physical processes and averaging interval (e.g., 3-hourly, daily, monthly precipitation). To anchor the discussion, we use daily precipitation statistics as the leading example, but results apply to other averaging intervals, as expanded in the final part of the paper.

*k*is the shape parameter,

*θ*the scale parameter, and

*x*represents temporally averaged precipitation. For daily precipitation, the shape parameter controls the probability of light and moderate daily precipitation totals, while the scale parameter is a useful metric to track changes of the extremes (Groisman et al. 1999; Wilby and Wigley 2002; Watterson and Dix 2003; Martinez-Villalobos and Neelin 2018b, hereafter MN18). As such, understanding the processes that control these parameters has important societal value.

In this paper we show that two main ingredients can be used to explain daily (or other temporal average) precipitation statistics:

Knowledge of the distribution of precipitation accumulations, defined as the amount of precipitation integrated over an event from start to end of precipitation. The theory for this is outlined below.

Knowledge of the distribution of the number of times it precipitates within the temporal average scale of interest

*t*_{avg}. If*t*_{avg}is equal to 1 day this is referred to as the “daily number of events distribution,” and similar for other*t*_{avg}.

This is shown schematically in Fig. 1. Here the *y* axis shows the instantaneous precipitation rate (for brevity “instantaneous rate” will be simply referred as “rate” in what follows), and the accumulation is the total amount from precipitation start to termination. There are three relatively small accumulation events on day 1 (wet day), zero events on day 2 (dry day), and one big accumulation event on day 3 (wet day), with the daily precipitation totals being the summation of the accumulations in each day. Here we show only three days, but in general there is a distribution of the number of times it precipitates in a day (daily number of events distribution), and a distribution of the total amount that it rains each time (accumulation distribution). The interplay between these two distributions shapes the resulting daily precipitation statistics.

In contrast to daily precipitation, the fundamental physical processes that shape accumulation distributions are reasonably well understood. This distribution by definition is only affected by processes occurring in the wet regime (from precipitation onset to termination). At the most fundamental, accumulation distributions can be understood using a few observationally constrained ingredients, based on the observed relationship between precipitation and column water vapor *q* (Raymond 2000; Bretherton et al. 2004; Peters and Neelin 2006; Neelin et al. 2009; Muller et al. 2009). These include (i) a fundamental climate equation (the column water vapor equation), (ii) a threshold for precipitation onset, which reflects the observation that precipitation tend to start when column water vapor exceeds a certain threshold, given by convective instability or large-scale saturation, and (iii) a similarly defined threshold for precipitation termination. Using these simple ingredients, Stechmann and Neelin (2014, hereafter SN14) derive the fundamental shape of accumulation distributions, with qualitative success in explaining observed distributions (Peters et al. 2001, 2010; Deluca and Corral 2014; MN18). Importantly, the parameters of the SN14 derived accumulation distribution can be directly related to processes occurring in the wet regime, including a dependence of the probability of the most extreme accumulations on column water vapor (Neelin et al. 2017, hereafter N17), with important expected consequences under global warming.

The second important point is the distribution of the number of times it precipitates in a given time interval of interest. This number of events distribution encapsulates the effects of intermittency (Schleiss 2018) on the resulting time-averaged precipitation distributions. Here we show that this distribution depends on both the dry (i.e., between accumulation events; see Fig. 1) and wet (within precipitation accumulation events) regimes, as opposed to accumulation distributions that only depend on wet regime physics. This allows a conceptual separation between dry and wet regime effects on time-averaged precipitation statistics.

In this paper we show how both these processes occurring in the wet and dry regimes combine to give shape to the observed precipitation distributions. Section 2 gives a brief overview of accumulations and daily precipitation distributions in observations. Section 3 presents two simple stochastic prototypes that are used to model the dry and wet regimes with simplified physics, and that provide a representation of daily (or longer averages) precipitation distributions. While reality is considerably more complex than these models, we argue that their simplicity is their main strength, as important insight into this problem can be gained that can be used to interpret observed temporally averaged precipitation distributions. Section 4 provides an explanation of why daily precipitation distributions are well fitted by gamma-like distributions. We derive analytical approximations for the gamma distribution parameters as a function of key physical processes on both wet and dry regimes in section 5. Section 5 also exemplifies how the shape of the daily precipitation distribution respond to changes of wet and dry regime dynamics. Section 6 explains how distributions change as a function of averaging interval, both subdaily and longer-than-daily precipitation averages. Finally, we conclude and discuss results on section 7, with particular emphasis on global warming implications.

## 2. Accumulation and daily precipitation distributions

The main goal of this paper is to provide an explanation of why precipitation distributions can be well fitted by gamma-like distributions. We note that there are several other distributions that are also used to describe precipitation (e.g., Woolhiser and Roldán 1982; Cho et al. 2004; Papalexiou and Koutsoyiannis 2013; O’Gorman 2014; Kirchmeier-Young et al. 2016). It is not the intention of this study to distinguish the often subtle differences in fit among these distributions. For our purposes we employ the gamma distribution because parameters can be easily interpreted, provides a good enough fit in most cases, and we can track distribution changes quantitatively. In addition, as elaborated below, gamma distributions resemble accumulation distributions in mathematical form, so the similarities and differences between accumulation and daily precipitation distributions can be made more quantitative. To highlight parallels between daily precipitation (or other averaging intervals) with accumulations measured in millimeters, in the rest of the paper we look at the distribution of daily precipitation totals measured in millimeters, but a conversion to daily precipitation intensities measured in millimeters per day is straightforward.

*P*, we prefer a representation of the form

*τ*

_{P}[

*τ*

_{P}= 1 −

*k*in (1),

*τ*

_{P}< 1] can be regarded as a power-law exponent governing the rate of decay of the distribution or probability density function (PDF) in the power-law range, and

*P*

_{L}[

*P*

_{L}=

*θ*in (1)] can be regarded as a daily precipitation cutoff scale, where the probability drops sharply.

*t*is the event duration, and

*R*(

*t*′) is the precipitation rate at time

*t*′ after the precipitating event has started. Accumulation distributions are well fitted in observations (Peters et al. 2010; Deluca and Corral 2014; MN18) and models (SN14; N17) by an expression of the form

*τ*is a power-law exponent (usually > 1) governing the rate of decay of

*p*

_{s}in its power-law range,

*s*

_{L}is a cutoff scale for which the probability of extreme accumulation events decay sharply, and

*B*is a normalization factor. An expression resembling (4) has been derived analytically under simplifying assumptions by SN14, and will be further discussed in section 3a. We note that after (2) has been rearranged, daily precipitation and accumulation distributions formulas look similar, with the main difference being the sharper power-law exponent for accumulations (

*τ*> 1 and

*τ*

_{P}< 1). In addition, both (2) and (4) highlights the importance of the cutoffs

*P*

_{L}and

*s*

_{L}in controlling the probability of extremes. For

*τ*> 1 there must a change in form for very small values of

*s*, as discussed below, but this is smaller than can typically be observed.

There are different approaches used in the literature to fit distributions to data. To fit accumulations and daily precipitation distributions we use a simple linear regression technique (see appendix A), with the estimated parameters being correlated with estimations using maximum likelihood (Thom 1958; Husak et al. 2007), and the method of moments (Fig. S1 in the online supplemental material). As notation, parameters estimated using the method of moments are denoted with a hat symbol (^), and unadorned parameters are estimated using the regression technique.

Figure 2a illustrates the key features of accumulation and daily precipitation distributions in observations. Here, accumulation and daily precipitation values are calculated using 1 min precipitation values from a DOE ARM site station located in Manus Island in the western tropical Pacific (Gaustad and Riihimaki 1996; Holdridge and Kyrouac 1997). A gamma distribution fit, (2), is overlaid on the daily precipitation distribution, and a fit given by (4) is overlaid on the accumulation distribution. The difference in the power-law exponent is immediately apparent, with accumulations decaying faster in the power-law range. The effect of the cutoff in limiting the probability of the largest events is also clear, especially compared to the respective dashed lines, which shows what the probability would be without the cutoffs. These main features are not restricted to this particular dataset. Similar features can be seen across a range of meteorological regimes for accumulations in (Peters et al. 2010; Deluca and Corral 2014), and for accumulation and daily intensity comparison in MN18.

## 3. Modeling accumulation and daily precipitation distributions

### a. Model setup

*q*reaches a certain threshold

*q*

_{c}, as occurs in observations (Peters and Neelin 2006; Neelin et al. 2009; Ahmed and Schumacher 2015; Schiro et al. 2016; Kuo et al. 2018) and general circulation models (Sahany et al. 2012, 2014). Similarly, precipitation termination occurs when

*q*has decreased below another threshold

*q*

_{np}=

*q*

_{c}−

*b*, with

*b*a hysteresis parameter. The model is given by

*E*is a positive mean evaporation source,

*R*(

*q*) is the precipitation rate. Fluctuations in moisture convergence are parameterized by

*D*

_{E}

*η*in the dry regime, and

*D*

_{P}

*η*in the wet regime, with

*η*being Gaussian white noise [⟨

*η*(

*t*)⟩ = 0, ⟨

*η*(

*t*)

*η*(

*t*′)⟩ =

*δ*(

*t*−

*t*′)], with ⟨⋅⟩ denoting expectation value), and

*D*

_{E}and

*D*

_{P}the noise amplitude in the respective regimes. For analytical simplicity we set

*R*(

*q*). We derive analytically the effect that this makes—essentially a slight modification to the cutoff scale

*s*

_{L}of order

*δ*≪ 1—and show numerically its consequences in section S1 in the supplemental material. Another simplification of the model is that the moisture equation is independent of the large-scale flow. This implies that aspects related to the spatial organization of precipitation are not explicitly treated here. However, key features of the observed PDF of spatial clusters (Quinn and Neelin 2017) can be captured in models related to the one used here (Hottovy and Stechmann 2015; Ahmed and Neelin 2019). Despite simplifications, the essential elements explaining accumulation distributions—the interplay between moisture convergence fluctuations and moisture loss by precipitation—are retained in the model.

*R*(

*q*) in the above model, for brevity denoted “on–off precipitation” and “ramp precipitation” given by

*s*), with

*τ*= 1.5. Here

*s*

_{L}is the accumulation cutoff given by

*t*

_{L}is a similarly defined precipitating event duration cutoff (discussed below). From (11) we can see that

*s*

_{L}(with same units as

*q*) is proportional to the amplitude of moisture convergence fluctuations (∝

*D*

_{P}) in the wet regime. Under increasing moisture availability, these fluctuations are expected to scale with moisture (although locally dynamical effects can be important), which implies an extension of the power-law range and a large increase in probabilities for extremes under a global warming scenario (N17; Norris et al. 2019a). The ramp precipitation case provides a more realistic scenario for the precipitating regime, where precipitation acts as a negative feedback opposing further moisture increases. While an analytical solution for accumulation distributions in this case is not available, section S2 shows that (11) also holds numerically.

The physical mechanisms and mathematical derivation of why the accumulation distribution in the on–off precipitation case is given by (9) are discussed by SN14 and N17, and why more generally (4) should hold as a good approximation for observed accumulation distributions is discussed by N17 (see also N17 Fig. 4). For the reader’s convenience we repeat the derivation of (9) in section S3 and summarize its main points here. In the on–off case the relation between column water vapor *q* and accumulation *s* can be made clearer by rewriting (6) as *dq* = −*ds* + *D*_{s}*η*_{s}*ds*, with *η*_{s} white noise in the *s* coordinate. This captures the main physical mechanisms governing the column water vapor equation within a precipitating event—the moisture converged to/diverged from the column (*D*_{s}*η*_{s}*ds*) and the loss of moisture by precipitation accumulation (−*ds*). If precipitation were not a moisture sink, there would be many trajectories where *q*(*s*) lingers above the critical threshold for event termination, yielding a long tail for the probability of event accumulation *p*_{s} ∝ *s*^{−1.5} in this case. In reality, moisture loss by precipitation limits the long incursions of *q*(*s*) above the threshold for event termination, as captured by the exponential term in (9) exp[−(*s*/*s*_{L})]. While there is no analytical solution for *p*_{s} available in the ramp precipitation case, the two competing processes (fluctuations in moisture convergence and moisture loss due to precipitation) also occur, which yield a similar form for the accumulation distribution numerically, although with a modified power-law exponent.

By similar means as before, an analytical solution for the distribution of wet-spell durations can be calculated for the model with on–off precipitation (see appendix B). This solution also contains a cutoff for long durations *t*_{L}, which is relevant to certain approximations for understanding time-averaged intensities, discussed in section 4. Similarly, the dry-spell duration distribution also has an analytical solution (appendix B), although it should be noted that this solution is unrealistic for low *q* values. In integrating the model (5) we simply set a rigid boundary at *q* = 1 mm, with *q* restored to the value at the previous time step if the boundary is reached. A more realistic treatment of the dry regime at low moisture values is implemented in a forthcoming paper. Nevertheless, the main way in which the dry regime affects daily precipitation distribution, namely in the different daily number of events distributions (section 4b) for mean moisture convergent

### b. Daily precipitation and accumulation distributions in the model

As an example of the accumulations and daily precipitation distributions arising from the setup in (5) and (6), we integrate the model, using both on–off and ramp precipitation variants, for 100 years, using the Euler–Maruyama stochastic integration scheme (Gardiner 2009; Ewald and Penland 2009), with a time step of 0.6 s. Parameters in the wet regime (see caption) are chosen to generate similar accumulation and duration moment ratios (see section S4) compared to observations. Parameters in the dry regime are similar to the ones chosen by SN14 and Abbott et al. (2016), with the value of *q* in (5) and (6) is calculated, from which it is decided whether it is precipitating (*q* > *q*_{c}) or not (*q* < *q*_{np}), and from which a precipitation rate can be calculated using (7) and (8) as appropriate. Then accumulations and daily precipitation values can be calculated from the generated precipitation-rate time series.

Figures 2b and 2c show the resulting accumulation and daily precipitation distributions, in the on–off precipitation case and the ramp precipitation case, respectively. Gamma distribution fits are overlaid on the simulated daily precipitation distributions. With some small differences, the resulting distributions show features much like what is seen in observations in both cases (Fig. 2a). Specifically, both daily precipitation and accumulation distributions have a cutoff scale, evident when comparing to the dashed lines indicating no cutoff, and the power-law range in the daily precipitation distributions is less steep than for accumulations. Given the relative simplicity of the setup, these results are encouraging and suggest that the physics included in the models is adequate to explain fundamental processes underlying these distributions. For the sake of obtaining analytical expressions, we use the on–off precipitation parameterization in most of what follows. The ramp precipitation model produces results that are qualitatively similar to the on–off precipitation model (see supplementary information).

## 4. What explains the shape of the daily precipitation distribution

In this section we provide a rationale for how daily precipitation distributions get their shape. The two ingredients are (i) the distribution of accumulations and (ii) the daily number of events distribution. Point (i) depends entirely on wet regime properties, while point (ii) depends on both wet and dry regime dynamics. This allows a conceptual separation of mechanisms between wet and dry regime processes controlling daily precipitation distributions. We focus on daily precipitation here, with section 6 showing other averaging intervals.

The main requirement for this partition to work is that *t*_{avg} ≫ *t*_{L}, with *t*_{avg} the averaging interval (1 day in this case), and *t*_{L} the wet-spell duration cutoff [*R*_{0}) or large moisture convergence fluctuations in the wet regime (large *D*_{P}). This requirement is put in place so the whole accumulation distribution can be sampled during *t*_{avg}, and that boundary effects from when a day starts and ends are small. This requirement also implies that the number of events and accumulation distributions are asymptotically independent. In this section and in the derivation of analytical results in section 5 we assume that we are in a regime where *t*_{avg} ≫ *t*_{L} holds perfectly. We test deviations from this requirement for *t*_{avg} = 1 day numerically in section 5. In practice, subdaily precipitation distributions are the most affected. However, some insight can still be gained for these distributions, as discussed in section 6a.

### a. Distribution of daily precipitation from the accumulation distribution

*n*. Denoting

*P*

_{n}as the total precipitation in days with

*n*events, then

*P*

_{n}values, here denoted as the “conditional daily precipitation distribution”

*p*

_{n}is given by

*p*

_{n}, (13), is obtained from knowledge of the accumulation distribution, (9). See appendix C for details of this derivation. This distribution

*p*

_{n}has the same shape as the accumulation distribution

*p*

_{s}in (9), but with mean

*p*

_{n}only depends on processes occurring while precipitating. Figure 3a shows examples of

*p*

_{n}distributions for different

*n*values. Since it typically rains a larger daily precipitation amount in days with many accumulation events, the

*p*

_{n}distributions, (13), show lower probability for small daily precipitation amounts and higher probability for large daily precipitation amounts for increasing

*n*. Consequently, as

*n*gets large the resulting

*p*

_{n}distributions look less asymmetric, that is, evolve to be less skewed (skewness decreases with

*n*as

*n*

^{−1/2}), with mean increasing proportionally to

*n*, and with fixed

*s*

_{L}, although it should be noted that the interpretation of

*s*

_{L}as a distinct cutoff in

*P*

_{n}is clear only for low enough

*n*.

### b. Daily number of events distribution

*p*

_{n}distribution, the daily precipitation distribution can be calculated as the mixture

*w*

_{n}(the weights of the mixture;

*fraction*of wet days with

*n*events. For this particular choice of parameters (see caption) it rains most often once per day (~18% of rainy days), and the probability decreases monotonically for larger

*n*. This quantity depends on both wet and dry regimes as shown in section 5b. An analytical solution for this distribution is not presently available, although it can be readily calculated from observations or from model integrations.

### c. Resulting daily precipitation distribution

Figure 3c shows graphically how daily precipitation distributions get their shape as we increase *n*_{max} in (14). Only considering *n* = 1 the resulting distribution is equal to the accumulation distribution. As we increase *n*_{max} we observe that small size values lose probability, and large daily precipitation values increase in probability, as less asymmetric *p*_{n} distributions (Fig. 3a) are incorporated. This process flattens out the resulting distribution, generally resulting in a daily precipitation power-law exponent *τ*_{P} that is *smaller* than the accumulation distribution power-law exponent. This implies that the resulting daily precipitation distribution can *often* be fitted by gamma distributions (*τ*_{P} < 1). We note that the apparent power-law range in the gamma distribution is not precisely a power law as in the accumulation solution. The power-law approximation holds well because a true scale free range is being modified by a procedure that does not introduce any dominant scale, thus leaving a range that remains essentially scale free. As all *w*_{n}*p*_{n} are included, the distribution resulting from (14) is very similar to the one calculated directly from the integration. This leads to a dependence of the daily precipitation distribution on the parameters of the underlying accumulation distributions, as further elaborated below. This simple model thus provides a rationale for how daily precipitation PDFs arise, and why they can be fitted by gamma distributions.

## 5. Analytical approximation of daily precipitation distribution parameters

### a. Analytical approximations

*τ*

_{P}and cutoffs

*P*

_{L}can be obtained using the method of moments (see appendix D) as

We should note that maximum likelihood estimates of *τ*_{P} and *P*_{L} will generally differ from the estimates calculated above, but they will be proportional (see Fig. S1). As previously stated, *t*_{avg} of interest. On the other hand, *s*_{L} and *t*_{avg}. That is, *s*_{L} contribute the same to daily or monthly precipitation statistics, with the difference between statistics for different averaging intervals being accounted by

### b. Exploring the parameter space

In this section we investigate the influence of both wet and dry regimes in the daily precipitation distribution power-law exponent and cutoff. In each case we use a number of 500-yr integrations of (5) and (6) for different parameters in the dry and wet regimes, using an integration time step of one minute for speed. Analytical formulas (17) and (18) are used to interpret the results.

#### 1) Dependence on wet regime

*s*

_{L}values. As shown in (11),

*s*

_{L}is proportional to the amplitude of moisture converge fluctuations (∝

*D*

_{P}) in the wet regime. Figure 4a shows that the accumulation and daily precipitation cutoffs are very well correlated. That is, increases in the accumulation cutoff are associated with increases in the daily precipitation cutoff. This relation between cutoffs is also reproduced in the ramp precipitation case (Fig. S4a). This dependence can be explained from (17) by noting that the expression

*P*

_{L}is set by the wet regime with

Unlike the accumulation power-law exponent *τ*, which always has a value of 1.5 in the on–off precipitation case, the daily precipitation power-law exponent *τ*_{P} exhibits an important dependence on model parameters. Figure 4c shows that *s*_{L}) all else being equal. Similar behavior is found in the ramp model (Fig. S4c), although the range of variation of

The left column of Fig. 5 shows the distributions associated with the left column of Fig. 4 for values *D*_{P} = 10, 15, 20 mm h^{−1/2} (*R*_{0} fixed), corresponding to *s*_{L} = 20, 45, 80 mm, respectively, with dry regime parameters fixed *D*_{P}) are associated with fewer total events (Fig. S6a), and to a number of events distribution more weighted toward fewer events per day (Fig. 5c). This apparently occurs because for increasing *D*_{P} the system spends more time precipitating, which results in larger but fewer events. This implies a larger contribution from more asymmetric conditional daily precipitation distributions *p*_{n} (13) in (14), which results in a steeper daily precipitation power-law range for larger *D*_{P} (or *s*_{L}) (Fig. 5e), which agrees with numerical and analytical results (Fig. 4c). It can also be seen that increases in *s*_{L} translates to similar increases in the daily precipitation cutoff (Fig. 5e), in agreement with (19).

#### 2) Dependence on dry regime

In this section we explore the dependence of the gamma distribution parameters for variations in the dry regime. Keeping the wet regime fixed (to *s*_{L} = 45 mm), Figs. 4b and 4d show the dependence of *τ*_{P} does have a slight dependence on

The right panel in Fig. 5 shows the distributions associated with the right panel in Fig. 4 for *s*_{L} = 45 mm). Since the wet regime is fixed, the three cases considered here have the same accumulation distribution (Fig. 5b). Consequently, the dry regime may only affect daily precipitation distributions through its effect on daily number of events distributions (Fig. 5d). As expected, the number of events distribution is weighted toward fewer events per day for mean moisture divergent conditions, as raining events are rarer in that case. This implies that for the *p*_{n}, (13), that make up the daily precipitation distribution, (14). This results in a steeper power-law range in this case, in agreement with numerical and analytical results in Fig. 4d.

#### 3) Summary

In summary, the wet regime controls the daily precipitation cutoff *P*_{L} (*P*_{L} ∝ *s*_{L}). Both wet and dry regimes have influence on the power-law exponent *τ*_{P}—steeper power law for larger *D*_{P} in the wet regime and/or decreasing *τ*_{P} (over the range considered), but the presence of the dry regime is essential to setting the difference between *τ*_{P} and the exponent for accumulations *τ*.

#### 4) Caveat on analytical approximations

It should be noted that the analytical approximations (17) and (18) provide a good understanding of the numerical values shown in Fig. 4, but the comparison is not perfect. This occurs because as *s*_{L} increases in Figs. 4a and 4c so does *t*_{L}, and the condition *t*_{avg} ≫ *t*_{L} implies that (17) and (18) progressively becomes a worse approximation. For the parameters in Fig. 4, when *s*_{L} is equal to 80 mm, *t*_{L} is equal to 8 h, a value not that well separated from 1 day. The analytical approximations hold better as *t*_{avg} increases as can be seen in Fig. S7. An implication of the *t*_{avg} ≫ *t*_{L} requirement is the asymptotic independence between the accumulation and number of events distributions. This is a simplification, as short events tend to preferentially occur in days with many events, and longer events to preferentially occur in days with few events. Despite this, leading-order effects are well captured by (17) and (18), and can be used to provide insight into how gamma distribution parameters respond to dry and wet regime physics.

## 6. How precipitation distributions change as a function of averaging interval

As can be seen in (17) and (18), there are two main effects that explain precipitation distributions over a fixed averaging interval. The first one, which may be thought as a fundamental effect arising from the lifetime of individual storms, is the distribution of accumulations, which impacts *s*_{L}. This contribution is completely independent from averaging considerations. The second effect, which may be thought as arising from the temporal aggregation of individual storms within the fixed averaging time of interest, will include the effects from averaging. Here those effects are encapsulated by the mean

Before proceeding, we point out some practical consequences that the resolution of observational data has on observed distributions. In the model world it is possible to generate precipitation data at high temporal resolution. In contrast, observational data is often only available through already discretized accumulated values (e.g., 1 h precipitation). The model with on–off precipitation generates an accumulation power-law exponent *τ* = 1.5 and cutoff *s*_{L} = 2*D*^{2}/*R*_{0} when precipitation is calculated using instantaneous values generated every *dt* interval (with *dt* small). Using coarser temporal resolution, as in observations, results in power-law exponents <1.5 (e.g., 1.3, when averaging from 1 to 15 min; see Fig. S8) and also in somewhat smaller values of *s*_{L}. This effect occurs for reasons analogous to the changes in time-averaged intensities: coarse graining the data before computing event accumulations tends to cause some small events that, if observed at high resolution, are interrupted by short dry spells to instead be counted as larger events. This should be taken into account when evaluating distributions in observations.

### a. Subdaily precipitation distributions

The study of subdaily precipitation statistics is an active area of research (e.g., Lenderink and van Meijgaard 2008; Westra et al. 2014; Barbero et al. 2017; Prein et al. 2017), so it seems useful to test to what extent the insight gained at daily time scales can be translated to subdaily scales as well. As previously stated, expressions (17) and (18) are approximations to gamma distribution parameters under the assumption that the averaging interval *t*_{avg} is much longer than the local storm duration cutoff *t*_{L}; see (B2). This assumption becomes progressively worse as the averaging interval is decreased. This occurs because long events, that preferentially contribute to the tail of the accumulation distribution, cannot be fully sampled in a short averaging interval. Despite this, qualitative statements based on how precipitation distributions arise (see section 4) may still provide insight.

Figure 6a shows the accumulation, 3-h, 12-h, and daily precipitation distributions calculated from almost 15 years of 1 min data from Manus Island station. It can be seen that a power law with a cutoff is a good fit in all cases. All the time-averaged distributions have a gentler power-law range decay compared to accumulations, and the power-law exponent decreases as we increase the averaging interval. The cutoff scale (as would be expected) increases with averaging interval, although the increase is relatively slow.

Figure 6a may be compared to Fig. 6c generated by the model with on–off precipitation. We integrate the model for 100 years (for consistent statistics) using a time step of 0.6 s (necessary to resolve short time variations and small precipitation increments) from which 1-min totals are calculated, to make the output more consistent with observations in computing the distributions. The parameters (see caption) were chosen to give similar accumulation and duration moment ratios as observations (⟨*s*^{2}⟩/⟨*s*⟩ and ⟨*t*^{2}⟩/⟨*t*⟩, respectively), and a value of *τ*_{P} and *P*_{L} parameters calculated, Fig. 6c shows good qualitative agreement with observations. Specifically, all the time-averaged distributions are well fitted by gamma distributions, with the correct ranking of power-law exponents simulated, and an (albeit faster) increase in the cutoff with averaging interval.

### b. Longer-than-daily precipitation distributions

A comparison between observations and model for longer averages can also be made. In this case we use 64 years of hourly precipitation data from the Miami Airport station available from the NOAA/NCEI Climate Data Online system. The minimum instrumental resolution is 0.254 mm h^{−1} (compared to 0.1 mm min^{−1} in Manus Island) and observations are given in multiples of 0.254 mm h^{−1}. Figure 6b shows this station’s accumulation, daily, 2-day, and 5-day precipitation distributions. Some of the same features as the subdaily precipitation distributions are seen. All time-averaged distributions have a smaller power-law exponent than accumulations, and the power-law exponents decrease in magnitude as the averaging interval is increased. In addition, the cutoffs increase slowly with averaging interval. Note that the accumulation distribution power-law exponent being smaller (in magnitude) than 1 most likely occurs due to the observations being given at 1 h intervals (see Fig. S8; also see section S4).

Noting that both Manus Island and Miami stations are located in regions with plentiful convection, it is worth asking whether accumulation and temporally averaged precipitation distributions behave similarly in regions and seasons dominated by frontal precipitation. We repeat the analysis leading to Fig. 6b for two other stations, one located in the northeastern United States and the other in Southern California, for both annual and the extended winter (November–April) season. In all cases the main features seen in Fig. 6b—the general shape of the distributions, the sharper power-law exponent for accumulation compared to daily precipitation distributions, and the decrease of *τ*_{P} for increasing *t*_{avg}—are also present in the other locations and season analyzed (Fig. S9). This suggests the robustness of these results to geographical location and main type of precipitation.

As before, we integrate the model with on–off precipitation with parameters chosen such as to generate similar accumulation and duration moment ratios compared to Miami Airport observations (see caption), and we use a value of

The decrease in power-law exponent with increasing averaging interval *t*_{avg} can be explained by revisiting section 4, as well as by inspecting (18). As *t*_{avg} increases the number of events distribution (analogous to the daily number of events distribution in Fig. 3b) is weighted toward more events per interval (simply reflecting the fact that there are more precipitating events in a month than in a day), which increases the contribution of the less asymmetric conditional precipitation distributions *p*_{n}, (13), in (14) *τ*_{P} value, and even a negative *τ*_{P} for long enough averaging interval. At this point it would be more convenient to return to the usual gamma distribution definition in (1), but for consistency with the rest of the paper we continue using *τ*_{P} and *P*_{L} (keeping in mind the translation between (1) and (2), *θ* = *P*_{L}, and *k* = 1 − *τ*_{P}). The relatively small increase of *P*_{L} with averaging interval in the model occurs because the ratio *P*_{L} for different *t*_{avg} in (17)] increases slowly with increasing *t*_{avg}. This is discussed in more detail below.

#### Asymptotic dynamics for long averages

*s*

_{L}, regions characterized by mean moisture convergence will tend to satisfy the inequality faster. This can be observed in Figs. 7a,b where we show the values of

^{−1}, respectively, all else being equal. For the two cases shown here the inequality is satisfied for a 26 day average in the mean moisture divergent case, and for a 4 day average in the mean moisture convergent case (Fig. 7b). That is, we expect roughly similar shapes of the distribution for averaging intervals on the order of a month in dry areas compared to averaging intervals on the order of days in wet areas.

*t*

_{avg}increases (Fig. S10). This implies that

*t*

_{avg}, which according to (17) implies that

*t*

_{avg}are very well anticorrelated. This explains how fast

*τ*

_{P}decreases for the mean moisture convergent region case (Fig. 7b). We note that as

While these conclusions have been made in basis of the simple model, it should be emphasized that they have corresponding behavior in observations. Figures 7c and 7d show similar plots for *t*_{avg} (although this occurs more slowly with *t*_{avg} than in the model). The parameter

## 7. Conclusions and discussion

There is a long tradition of using gamma-like distributions to represent temporally averaged precipitation distributions. There has been little justification for their use beyond being a distribution bounded by zero that can represent skewed data (Ropelewski et al. 1985; Wilks and Eggleston 1992). Here we present a more fundamental view on how gamma-like distributions arise as a good fit to represent precipitation PDFs. To address this question, we use two simple stochastic models that, despite their simplicity, condense what is arguably the most important aspect that explains observed precipitation distributions—the competition in the moisture budget between fluctuations by moisture convergence and dissipation by moisture loss due to precipitation. Under this simplified framework, gamma-like distributions arise by the interplay between the distribution of storm accumulations (from event onset to termination) and the distribution of the number of these storms (number of events distribution) in the averaging interval of interest. The distribution of accumulations can be physically derived from the moisture equation, a fundamental equation of atmospheric models (SN14; N17), with the distribution shape consisting of a relatively sharp power-law range with an exponential cutoff *s*_{L} for large sizes. Here we extend the insight gained from the study of accumulations to temporally averaged precipitation distributions by noting that the total precipitation falling in the temporal average of interest (for instance a day) basically consists in the addition of different accumulation events occurring within this one day period (or other *t*_{avg}). This leads to higher probabilities of larger values and smaller probabilities for smaller values with respect to accumulations, yielding power-law exponents for daily precipitation distributions that are strictly smaller than the underlying accumulations. In addition, the resulting daily precipitation distribution also features an exponential cutoff, with properties inherited from the underlying accumulations. This results in a daily precipitation PDF that can be well fitted by gamma distributions.

There are several implications arising from this framework. From previous research (SN14; N17) we know that the accumulation distribution cutoff is proportional to the size of moisture convergence fluctuations in the precipitating regime (*s*_{L} ∝ *D*_{P}). Here we show that this statement can also be made for daily precipitation (or similar *t*_{avg}). That is, the daily precipitation cutoff is also proportional to the size of moisture convergence fluctuations (*P*_{L} ∝ *s*_{L} ∝ *D*_{P}). The proportionality between *s*_{L} and *P*_{L} has been observed to occur over the United States (MN18), and this framework provides explanation for this observational result.

The importance of shifts in the accumulation cutoff has been shown in general circulation models (N17) and observations (MN18). These shifts in cutoff, proportional to changes in *D*_{P}, involve increases in moisture (thermodynamic contribution) and changes in convergence (dynamic contribution) (Pfahl et al. 2017; Norris et al. 2019b)—note that both effects enter into the same parameter of the PDF. In most regions this implies an increase in *s*_{L}, which extends the accumulation power-law range, yielding approximately exponential increases in the probability of the largest accumulations (N17; Norris et al. 2019a). One of the main conclusions of this work is that a similar increase in probability of daily (or similar averages) precipitation extremes occurs as moisture increases. In this case the changes in probability will be slightly more complicated than for accumulations, as the power-law exponent will also change as *D*_{P} increases [see (18)], and also depends on changes of the dry regime dynamics.

To illustrate this, we consider the effect that a postulated increase in the amplitude of fluctuations of moisture convergence under a global warming scenario has on daily precipitation distributions. Figure 8 shows the risk ratio (Otto et al. 2012; N17), defined as the ratio of the probability (conditioned on event occurrence) of daily precipitation larger than a certain amount (*x* axis in Fig. 8) in the warmer compared to current conditions, for two different cases calculated using long runs of the ramp precipitation model. In the first case (red) we consider a 21% increase of *D*_{P} and *D*_{E}, which would correspond to a Clausius–Clapeyron scaling with 3°C warming (increase of 3 × 7%) in the amplitude of moisture convergence fluctuations. In the second case (blue), only *D*_{P} scales up. This case can provide insight into the effect of changes in vertical velocity (which are linked to changes in the dynamic contribution to moisture convergence via the continuity equation) that are asymmetric for ascending and descending regimes that have been suggested to occur under global warming (Pendergrass and Gerber 2016). In both cases, increases in *D*_{P} yield increases in the daily precipitation cutoff *P*_{L} with resulting exponential increases in the probability of the largest daily precipitation values, much as it occurs for accumulations. Similar increases in risk ratio for extreme daily precipitation have been observed in the United States during recent decades (MN18). The main difference between the two cases is in how the power-law exponents adjust, which points to the role of the dry regime dynamics. In the more symmetric first case—with increases in both *D*_{P} and *D*_{E}—the power-law exponents are similar in current and warmer conditions (as is the case for accumulations), resulting in exponential increases in risk ratio starting in the moderate daily precipitation range. In the second case *τ*_{P} adjusts, increasing its value for warmer conditions. This results in a reduction in the probability of moderate daily precipitation, with exponential increases in risk ratio starting for larger values. We argue that the simple arguments laid out here may account for changes in the occurrences of extremes in the daily precipitation distribution that have already been observed (e.g., Kunkel et al. 2013; MN18) or that have been projected to occur for climate warming scenarios (Fischer and Knutti 2016; Pendergrass 2018).

The simple scaling argument (*P*_{L} ∝ *D*_{P}) also provides explanation for other relations regarding daily precipitation extremes found in the literature. Several studies have shown the scale parameter (our *P*_{L}) of gamma distributions to be a useful indicator of changes in daily precipitation extremes (Groisman et al. 1999; Wilby and Wigley 2002; Watterson and Dix 2003; MN18). Our analytical results, (17), provide a more formal justification of this, and suggest that changes in *P*_{L} may also be used to track changes in extremes for other relatively short averages. Another aspect that may be explained by this simple framework is the increase in daily precipitation variability observed in global warming model projections (Pendergrass et al. 2017). Not only does *P*_{L} scale with moisture availability in our model, but our results also imply a similar scaling for daily precipitation variance

Overall, for daily precipitation (or other relatively short temporal average, as elaborated below) the wet regime controls to a great extent the resulting time-average intensity distribution. Changes in the dry regime (with wet regime fixed) modify the resulting distribution to a secondary extent, with slight adjustments to the power-law exponent. This dependence on the dry regime, encapsulated in changes in mean moisture convergence *D*_{E}), may be relevant to explain changes in the distribution under global warming in subtropical–midlatitude transition zones (e.g., Garreaud et al. 2017; Swain et al. 2018), that may occur in association to a poleward expansion of subtropical high pressure regions (Lu et al. 2007; Frierson et al. 2007; Kang and Lu 2012; Levine and Schneider 2015). All else fixed, regions characterized by mean moisture convergence have a gentler power-law range compared to regions characterized by mean moisture divergence in the model.

Although we focus on daily precipitation, the framework presented here applies to other averaging intervals as well. For shorter, subdaily averaging intervals, caution is required when there is not a good separation between the averaging interval *t*_{avg} and the event duration cutoff *t*_{L}. The computation using mixture distributions that explains the resulting gamma-like distributions in section 4, and the analytical approximations for *t*_{avg} ~ *t*_{L}). However, the numerical stochastic model generates subdaily distributions that resemble observations (Figs. 6a,c), and this suggests a smooth evolution of behavior with averaging interval. In all cases there is a tendency for lower values of *t*_{avg} (3-, 12-, 24-h) increases. Thus, the qualitative rationale for the relationship of the gamma distribution to the accumulation distribution continues to work over some range of subdaily time scales (which may have a regional dependence because *t*_{L} can vary).

For longer averaging intervals, *t*_{avg} ≫ *t*_{L} (i.e., much longer than the precipitation duration cutoff, which tends to occur for daily or longer averages), we can demonstrate that the daily precipitation power-law exponent is strictly smaller in magnitude than the accumulation power-law exponent. Because it typically rains more than once per averaging interval (Fig. 3b), the precipitation intensity distribution, (14), as a mixture of the conditional distributions given by (13), will have less probability than accumulations for small values, and more probability for larger values, yielding a less steep precipitation intensity power-law range than for accumulations. Similarly, longer averages will have a number of events distribution weighted to a larger number of events within *t*_{avg}, yielding power-law exponents that eventually change sign for sufficiently long averages. This argument leads us to conclude that *t*_{avg} (giving allowance to sampling variations in observations). In fact, for long enough *t*_{avg}, we show that *t*_{avg}.

For regions with plentiful moisture supply, the paradigm of thinking about a power law and cutoff for the time-average precipitation distribution applies up to an averaging interval measured in days, with *P*_{L} scaling with moisture. For much longer averaging interval, the resulting distribution begins to have properties not much different from a Gaussian, in which case changes in the mean and variance, (15) and (16), may be more useful. In mean moisture divergence regions the power-law and cutoff paradigm may be valid for longer *t*_{avg} (on the order of a month), as the power-law exponent decreases slowly due to the mean number of precipitating events

## Acknowledgments

This research was supported by National Science Foundation Grant AGS-1540518 and by National Oceanic and Atmospheric Administration Grant NA18OAR4310280. Manus Island precipitation data are courtesy of the U.S. Department of Energy Atmospheric Radiation Measurement (ARM) Climate Research Facility Tropical West Pacific field campaign. We thank K. Schiro for assistance with this dataset. Miami Airport, Hartford Airport and Fullerton Dam precipitation data are courtesy of the NOAA/NCEI Climate Data Online system (https://www.ncdc.noaa.gov/cdo-web/search?datasetid=PRECIP_HLY\#). We thank F. Zwiers for a postseminar question that helped motivate this work and J. Meyerson for graphical assistance. A portion of this work has previously been presented at an American Physical Society meeting (Martinez-Villalobos and Neelin 2018a) and at an American Geophysical Union meeting (Martinez-Villalobos and Neelin 2018c).

## APPENDIX A

### Estimation of Accumulation and Temporally Averaged Precipitation Distribution Parameters

*x*here). The parameters are calculated by assuming a relationship between log(

*p*) and functions of

*x*as follows:

*c*

_{i}coefficients. The parameters are then given as

*τ*

_{x}= −

*c*

_{2}, and

*x*

_{L}= −(1/

*c*

_{3}).

This way to calculate parameters is consistent for both accumulations and daily precipitation distributions and generally give more consistent fits for variation in data resolution. We note that this methodology has a small dependence on the binning scheme used. The parameters calculated in this way are proportional to parameters calculated using either maximum likelihood or the method of moments (Fig. S1).

## APPENDIX B

### Distribution of Wet- and Dry-Spell Durations

*p*

_{t}, that informs the range of validity of the analytic approximations, is derived in SN14 for the case of on–off precipitation as a rescaling of the accumulation distribution, (9):

*t*the event duration. The mean duration

*t*

_{L}are given by

The analytical solution for the distribution of event durations in the ramp precipitation case can be adapted from similar equations in the finance literature (Yi 2010), discussed in section 2. An important point of this solution is that *t*_{L} = 1/*α*, which is used to numerically validate (11) in this case.

The solution for the distribution of dry-spell duration (SN14) has the same shape as (B1), but with mean dry-spell duration

## APPENDIX C

### Analytical Formulas for *p*_{n} Distributions and Moments

*p*

*s*

_{i}from the same

*p*

_{s}distribution

*n*times, then the variable

*p*

_{s}(

*nμ*,

*n*

^{2}

*λ*), which is a particular case of a more general additive property of inverse Gaussians (Tweedie 1957; Folks and Chhikara 1978). Going back to the original variables, we find that

*P*

_{n}is distributed as in (13).

*P*

_{n}⟩ and variance

*p*

_{n}distribution are given by

## APPENDIX D

### Derivation of Gamma Distribution Parameter Formulas

*m*th moment about zero ⟨

*P*

^{m}⟩ of a distribution

*p*

_{p}is given by

*p*

_{p}is a temporally averaged precipitation distribution. Using (14), this can be rewritten in terms of the moments of the

*p*

_{n}distributions, (13), which we denote as

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