Time series of precipitation or rainfall display highly irregular behavior, as illustrated in Fig. , and many valuable models have been based on stochastic processes. A variety of different stochastic models have been used, including renewal processes, Markov chains, Poisson processes, and point processes . The many applications of these models include weather forecasting, stochastic weather generation, climate impact assessment, climate model downscaling, hydrological modeling, ecological modeling, and agricultural modeling.

Commonly, stochastic models for rainfall are empirical – i.e., based mainly on fitting the model behavior to match observational rainfall data – rather than based mainly on the underlying physical laws. Nevertheless, it is desirable to relate the stochastic models to physical principles, to the extent possible. Here, we investigate such a relation.

In particular, the goal of the present paper is to prove a connection between (i) a point-process description of rainfall time series and (ii) a physics-based model for the stochastic evolution of moisture. At first glance, the point-process model appears to be somewhat disconnected from basic physical laws based on mass, momentum, and energy. However, the point-process model can be seen to arise from the underlying evolution of moisture (which is the mixing ratio of water vapor in the air) . Here, this connection is demonstrated via formal asymptotics on the Fokker–Planck equation and proved rigorously in the mean-square sense.

To be more specific, a point-process model of rainfall can be viewed as a spike train, as in Fig. d, where a rainfall event is an instantaneous spike. The point process could be defined and characterized by the random waiting time, τd, of the duration of the “dry spell” in between rain events. As an empirical model of rainfall, one could estimate the probability density function (pdf) of τd based on observational data . For such an empirical approach, one could use data of rainfall time series alone, without appealing to any physical laws or any other type of observational data (humidity, wind speed, etc.). Similarly, beyond point processes, one could use a renewal process as a model of rainfall time series, as in Fig. c, by introducing a finite (and possibly random) time τr for the duration of the rain event. Again, as in the case of a point process, one could use a renewal process as an empirical model, based on data of rainfall time series alone, without appealing to any physical laws or any other type of observational data. However, it would be desirable to show that the point-process and renewal-process models can also arise from more physically based underpinnings.

Here, as mentioned above, a point-process model of precipitation will be linked to the evolution of moisture to provide a more physically based foundation of the point-process model. The moisture model used here is a continuous-time stochastic process for q(t), which represents the amount of water vapor in a column of the atmosphere, as an anomaly from a baseline level, at time t . For example, the anomaly q(t)=0 corresponds to 62 mm of moisture in the column, and q(t)=b=3 mm will be an upper threshold of 65 mm of moisture in the column.

The q(t) process is governed by the stochastic differential equations (SDEs) 1dq(t)=mdt+D0dWtfor σ(t)=0-rdt+D1dWtfor σ(t)=r,q(0)=0,σ(0)=0, where m and r are the moistening and rain rates, respectively, and D0 and D1 are the constant diffusion coefficients which capture the fluctuations of moisture during the respective states and Wt is a standard Wiener process. The quantity σ(t) is an indicator function for rain, and the dynamics of σ(t) switch from 0 to r when q(t) reaches a fixed threshold b>0. For instance, supposing that (q(0),σ(0))=(0,0), then σ(t)=0 until the time t1=inf⁡{t≥0:q(t)=b}, at which time the value of σ switches to σ(t)=r. Then σ(t) switches back to zero at a later time when the moisture has been depleted to the lower threshold (q(t)=0). Figure a, b show a realization of the processes q(t) and σ(t). The process σ(t) can be viewed as a renewal process, with random durations τd and τr of dry spells and rain events, respectively, although σ(t) is not just a standalone renewal process, since it arises from the underlying dynamics of moisture q. The moisture in the column can potentially reach negative values, i.e., q(t)<-62, on very rare occasions. This effect is due to the idealized nature of the model.

The threshold behavior of Eq. () is a fundamental feature of the moisture–rainfall relationship that is seen in nature , and it is a basic aspect of many more complex moisture models and convective parameterizations as well . Sometimes the threshold is also called a trigger . The threshold can be viewed as the threshold for the release of moist convective instability, and the moisture q is used as the physical quantity that governs the onset of moist convection. In this way, Eq. () is a physically based model of atmospheric moisture, and, from it, one can obtain a rainfall time series as a secondary or auxiliary quantity.

The main purpose of the paper is to define and show convergence of the threshold model in Eq. () as r→∞ to a point-process model of rainfall. For example, on the level of renewal processes, τr→0, and thus σ(t) converges to a process that is zero everywhere and that has spikes at infinity after random durations of length τd. That is, σ(t) converges to a Dirac delta process. However, σ(t) is right-continuous and has left-hand limits, whereas the spike train is not. Thus, the mode of convergence is not clear. For q(t) the limit is also unclear but will be redefined in a way to show convergence with respect to the topology on continuous functions with the uniform metric. In this study, the limiting processes are defined (in Sect. ), and convergence is shown both heuristically (for the Fokker–Planck equation) and rigorously.

Some of the novel aspects of this work are as follows. The limit jump process q(t) has an associated Fokker–Planck equation that is derived using a matched asymptotic method. The resulting Fokker–Planck equation has a peculiar boundary flux condition which defines a “teleporting” boundary condition of q(t). The processes are decoupled into evaporating and precipitating processes. Only after this decoupling can convergence of the evaporation processes be shown rigorously with respect to the uniform metric on the space of continuous functions. Also, the rain process σ(t) is shown to converge rigorously with respect to the generalized function space. This proof shows convergence of a renewal process to a delta process. Furthermore, the proof shows what kinds of bounds are needed for the rain event times τr in order for integrated convergence to hold.

The convergence results shown here have the potential to impact various other fields. Many fields of study use similar renewal processes to model different types of phenomena . The connections to rain models were made above. In addition, there has been work in queuing theory to approximate point processes with renewal processes (e.g., ) and using threshold triggers in financial models . Thresholds arise in many applications of piecewise dynamical systems where the threshold marks a change in the dynamics, as in Fillipov dynamics and hybrid switching diffusions . The limiting process is similar to a stochastic resetting process studied in and . Here the process stochastically resets to q=0 after a random hitting time τd which depends on the process. Another interesting connection is with neuron stochastic integrate and fire models (see , for a review). The moisture process with a finite rain rate is similar to a Wiener process model of a single neuron with refractoriness. A similar model was studied in , where the refractory time was constant. Here, the refractory time is random and coincides with the rain duration time τr. Thus, the work here is applicable to understanding the differences in using a model without refractoriness versus a model with a short, possibly random, refractory time.

The structure of the paper is as follows. The processes for moisture and rain are defined in Sect. . The modes of convergence are discussed in Sect. . The heuristic convergence with the Fokker–Planck equation is shown in Sect. . Rigorous convergence of the moistening process Eϵ to E is shown with respect to L2 in Sect. , and the rain process σϵ is shown to converge to the sum of delta distributions σ with respect to generalized functions in Sect. . Some important statistics as well as an analysis of the differences when using the two processes are shown in Sect. . The results are summarized in Sect. . Technical details of the proofs and derivations are given in the Appendix.

In this section the moisture and precipitation processes are defined. First, the underlying moisture process of the renewal rain process is defined. The processes are defined with a small parameter ϵ with the limit as ϵ→0 in mind.

The moisture process qϵ(t)∈R is defined as the solution to the SDE, 2dqϵ(t)=mdt+D0dWtfor σϵ(t)=0-rϵdt+D1dWtfor σϵ(t)=rϵ,qϵ(0)=0,σϵ(0)=0, where m and r/ϵ are the moistening and rain rates, and 0<D0≤D1 are the fluctuations of moisture during the respective states. Here the small parameter ϵ is essentially the ratio of moistening and rain rates (times a constant O(1) factor). In other words, ϵ is the value which makes m/r order 1. In the tropics, the rain rate is often seen to have values in excess of 15 mm h-1 (e.g., see Fig. a and b). See Fig. 7a in for an estimate of a typical moistening rate in the tropics. There the moistening rate is roughly 0.4 mm h-1. The rain process σϵ(t)∈{0,r/ϵ} is defined as follows: since σϵ(0)=0, let T1ϵ≡inf⁡{t>0|qϵ(t)=b}. Then σϵ(t)=0 for t∈[0,T1ϵ). Next, let T2ϵ≡inf⁡{t>T1ϵ|qϵ(t)=0} and σϵ(t)=r/ϵ for t∈[T1ϵ,T2ϵ). This process repeats up to an arbitrary final time T. Define the time intervals τid,ϵ and τir,ϵ as 3aτ1d,ϵ=T1ϵ-0,3bτ1r,ϵ=T2ϵ-T1ϵ,3cτ2d,ϵ=T3ϵ-T2ϵ,3dτ2r,ϵ=T4ϵ-T3ϵ, and so on. These are the duration times for dry and rain events, respectively.

The associated processes, as ϵ→0, are defined as q(t) and σ(t) for the moisture and rain processes. (It would perhaps be appropriate to denote the limiting processes as q0(t) and σ0(t), to indicate that they arise from qϵ(t) and σϵ(t) in the limit ϵ→0. However, we will drop the superscript 0 from q0(t) and σ0(t) to ease notation.) The moisture process is the solution to the SDE 4dq(t)=mdt+D0dWt,q<b,q(0)=0, with the unusual boundary condition as follows: let the usual stopping time be T1=inf⁡{t>0|q(t)=b}. Then, at time t=T1, the process q(t) jumps or “teleports” to q=0. However, the function is defined as both 0 and b at T1. For convention, the process is defined as cadlag (continuous from the right with left-hand limits), i.e., 5lim⁡t→(T1)-q(t)=b,lim⁡t→(T1)+q(t)=0,q(T1)=0. Then the process starts over using the dynamics of Eq. () until T2=inf⁡{t>T1|q(t)=b}, and the process repeats. The time intervals 6τid=Ti+1-Ti are the dry event durations. The rain event duration, on the other hand, is not defined for this limiting process, since rain events are instantaneous in the intense-rain-rate limit of ϵ→0.

Example time series of the processes are shown in Fig. . The processes with finite rain rate r/ϵ for ϵ>0 are shown in panels (a) and (b). Panel (a) is the moisture process qϵ(t) defined in Eq. (). The rain rate process is shown in panel (b) and takes the value r/ϵ when qϵ(t) reaches level b for the first time (panel a in black) and resets to zero when qϵ(t) reaches zero (panel a in gray). This process repeats. The limiting processes are shown in panels (c) and (d). Panel (c) shows the limiting moisture process q(t) defined in Eq. (), and panel (d) shows the rain process defined in Eq. (). The moisture process is a Brownian motion with positive drift until reaching level b. When q(t)=b, the process σ(t) takes an infinite value, and the moisture process is reset to zero.

From the definition of τid above, the rain point process σ(t) is defined as 7σ(t)=b∑i=1N(T)δ(t-Ti), where N(T) is the random variable of the number of times the process q(t) reaches b in time T. The quantity b arises because the moisture process qϵ loses moisture at a rate of r/ϵ per time, on average. The moisture process q(t) loses all the moisture built up (which is an amount b) instantaneously.

Note that qϵ(t) has continuous paths, while q(t) has jump discontinuities. Thus, any mode of convergence between qϵ and q with an associated metric (e.g., uniform or Skorohod) will fail . Nevertheless, there is another way to define both qϵ and q in which convergence with respect to L2 with the uniform metric on the space of continuous functions (C[0,T]) can be shown. To do so, qϵ(t) is decomposed into an evaporating process, Eϵ(t), and precipitating process, Pϵ(t). These processes are defined as 8dEtϵ=mdt+D0dWtfor σtϵ=00for σtϵ=r/ϵ, and dPtϵ=0for σtϵ=0-rϵdt+D1dWtfor σtϵ=r/ϵ. Thus, the moisture process qϵ(t) is written as qϵ(t)=Eϵ(t)+Pϵ(t). In the limit, the jumps will be captured in the Pϵ process. In the following section it will be shown (see Sect. ) that Eϵ→E, where E(t) is defined as the solution to the SDE 9dE(t)=mdt+D0dWt,E(0)=0. Furthermore, the spike times of the σ process, which was defined above in Eq. (), could now also be defined in terms of the E(t) process as Ti=inf⁡{t>0|E(t)=ib,i∈N}, i.e., the first passage time of Brownian motion with drift to ib.

In this section convergence is shown both heuristically (e.g., Sect. ) and rigorously (e.g., Sect.  and ).

Note that the simplest ideas of convergence break down when considering pathwise convergence of qϵ to q and σϵ to σ. This is because qϵ is a continuous process for all ϵ>0, whereas q is a process with jumps, σϵ is left-continuous with right-hand limits, and σ is no longer left-continuous. Thus, there is no topology with the associated metric d such that qϵ→q with respect to d . However, one could try to show that qϵ converges in a notion weaker than the Skorohod topology; see for these conditions. Such convergence would happen in a topology which does not have an associated metric (see ). This approach is not pursued here as it is technical and does not give any insight into the model or approximation.

Instead, we pursue convergence in the following senses. The next three subsections prove convergence of the various processes introduced in Sect. . In Sect.  the Fokker–Planck equation for qϵ is shown to converge (formally) to a Fokker–Planck equation for q. This derivation gives rise to an interesting partial differential equation (PDE) with unusual “teleporting” boundary conditions. In Sect.  convergence in paths is shown for Eϵ to E with respect to the uniform metric for continuous functions on [0,T]. In Sect.  convergence is shown for σϵ to σ with respect to generalized functions. This norm is necessary because σ is a sum of Dirac delta functions. In addition, this convergence is natural to consider for applications where the errors are analyzed between using σϵ and a point process (σ) in, for example, a climate model or as a model for observational time series.

In this section, important statistics and applications of the processes (qϵ,σϵ) and (q,σ) are presented. These statistics show the differences between the processes and give motivation for the approximations. These include the stationary Fokker–Planck solution, the rain and dry event distributions, the rain fraction, and an application of Theorem 2.

In this paper, a threshold model for moisture and rain was shown to converge to a point process and related processes and to converge for various modes of convergence. By demonstrating this type of convergence, the simple ideas of a point-process model of rainfall, which at first may appear to be only an empirical model, can be linked with underlying physical processes and the evolution of moisture.

Here convergence for the moisture processes was defined and shown for the Fokker–Planck equation as well as the paths of the processes. Furthermore, the convergence of the rain process was shown in mean square difference with respect to the space of generalized functions.

Using a point process to approximate rainfall allows simplification for computation and exact formulas. For example, the autocorrelation function is known in the case of point processes as shown in . Furthermore, point processes have been studied extensively in the neural science literature , and many statistics have been derived. Some examples of exact statistics are shown. These examples have interesting characteristics as the small parameter ϵ tends to zero.

The proofs shown here are revealing on their own, and they demonstrate further details of the convergence. The Fokker–Planck derivation in Sect.  shows that the density for the moisture in the rain state tends to zero, while the flux term remains nonzero, allowing for the “teleporting” boundary condition that arises for the limiting moisture process. For the convergence of paths of moisture shown in Theorem 1, the moisture process must first be decoupled into a moistening and precipitating process. Then the moistening process is shown to converge (Theorem 1), while the precipitating process contains all of the discontinuities. Finally, the proof of convergence of the rain processes in Theorem 2 gives estimates that would be useful for determining the error rates for using the point-process approximation. This is done in an example of total rainfall.