Convergence of Rain Process Models to Point Processes

Rain processes are often studied and tested as single column models in the atmosphere Betts and Miller, (1986); Frierson et al., (2004). These single column models are used to “trigger” a transition to convection or rainfall. Furthermore, these rainfall models are simple and have uses in global climate models (GCMs) Lin and Neelin, (2000); Suhas and Zhang, (2014); Bergemann et al., (2017). One type of rainfall model that has been successful is a Renewal process (Bernardara et al., (2007); Foufoula-Georgiou and Lettenmaier, (1987); Green, (1964); Schmitt et al., (1998)). A renewal process is a continuous-time Markov process that is defined by its holding times Cox, (1962). For example, a simple rain renewal process, $\sigma(t)$, may be defined by its holding times $\tau^{d}$ when $\sigma(t)=0$, defined to be that the column of air is dry, and $\tau^{r}$ when $\sigma(t)=1$, defined to be that the column of air is raining. Thus a dry (or rain) event will last $\tau^{d}$ (or $\tau^{r}$) time where the distribution of the (possibly) random times are given.

The model presented here was previously studied in Stechmann and Neelin, (2011, 2014); Hottovy and Stechmann, (2015); Abbott et al., (2016). The underlying process is a one dimensional continuous-time stochastic process modeling the moisture $q(t)\in(-\infty,\infty)$ for $t\geq 0$, typically in cm, for a parcel of air. This quantity is vertically integrated and averaged over a square domain to give the units of length. Here the rain process is modeled as $\sigma(t)$ an indicator function for rain. One choice of process for $q(t)$ is the hysteresis dynamics studied in Hottovy and Stechmann, (2015). There the $q(t)$ process is governed by the stochastic differential equations (SDE)

where $m$ and $r$ are the moistening and rain rates respectively, and $D_{0}$ and $D_{1}$ are the fluctuations of moisture during the respective states. The dynamics of $\sigma(t)$ switch from 0 to 1 when $q(t)$ reaches a fixed threshold $b$. That is $(q(0),\sigma(0))=(0,0)$ and $\sigma(t)=1$ for $t=\inf\{t\geq 0,q(t)=b\}$. Then $\sigma(t)$ switches back to zero when $q(t)$ reaches the threshold $q(t)=0$. This model is referred to as the deterministic trigger, two threshold (D2) model in Hottovy and Stechmann, (2015). A realization of the processes $q(t)$ and $\sigma(t)$ are shown in Figure 1. In a renewal process perspective, $\sigma(t)$ is modeled by the holding times $\tau_{d}$ and $\tau_{r}$. These times are the first passage times for Brownian motion with drift to travel a distance of $b$ units.

Models of triggering precipitation are used for convective parameterizations. Convective parameterizations are useful in global climate and circulation models (GCMs). In GCMs, the models are run for long times to examine potential effects, for example, of climate change. In long times, the model (2) was studied in Abbott et al., (2016). In a long time view, the dry events dominate the rain events. In Figure 2 the top panel shows $\sigma(t)$ process is plotted up to time $T=...$, and in the bottom panel an example of the rain process in observations (OBS DETAILS). Thus for long times, $r$ is a much larger rate than $m$, and $\sigma(t)$ resembles a point process.

The main result of the paper is to define and show convergence of the threshold model above as $r\to\infty$. For example, on the level of renewal processes, $\tau^{r}\to 0$ and thus $\sigma(t)$ converges to a process that is zero everywhere and has spikes at infinity at random times $\tau^{d}$. However, $\sigma(t)$ is right continuous and has left hand limits, where as 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 define (in Section II) and convergence is shown both heuristically (for the Fokker-Planck equation) and rigorously.

There are many novel aspects of this work. 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. Finally, the rain process $\sigma(t)$ is shown to converge rigorously with respect to the the generalized function space. This proof shows convergence of a renewal process to a delta process. Further more, the proof shows what kinds of bounds the rain event times $\tau^{r}$ need 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 phenomena. The connections to rain models was made above. In addition, there has been much work in queuing theory to approximate point processes with renewal processes (e.g. Whitt, (1982); Bhat, (1994)), and using threshold triggers in financial models Lejay and Pigato, (2019). The strongest connection is with neuron stochastic integrate and fire models (see Sacerdote and Giraudo, (2013) 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 Albano et al., (2008) where the refractory time was constant. Here, the refractory time is random and coincides with the rain duration time $\tau^{r}$. Thus the work here is applicable to understanding the differences in using a model without refractoriness versus a model with a short, possible random, refractory time. The structure of the paper is as follows. The processes for moisture and rain are defined in Section II. The modes of convergence are discussed in Section III. The heuristic derivation of the Fokker-Planck equation is shown in Section III.1. Rigorous convergence of the moistening process $E^{\epsilon}$ to $E$ is shown with respect to $L^{2}$ in Section III.2 and the rain process $\sigma^{\epsilon}$ is shown to converge to the sum of delta distributions $\sigma$ with respect to generalized functions in Section III.3. The results are summarized in Section IV.

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 $\epsilon$ with the limit as $\epsilon\to 0$ in mind.

The moisture process $q^{\epsilon}(t)\in\mathbb{R}$ is defined as the solution to the stochastic differential equation (SDE),

where $m$ and $r/\epsilon$ are the moistening and rain rates, and $0<D_{0}\leq D_{1}$ are the fluctuations of moisture during the respective states. The rain process, $\sigma^{\epsilon}(t)\in\{0,r/\epsilon\}$ are as follows: since $\sigma^{\epsilon}(0)=0$, let $\tau_{1}^{d,\epsilon}\equiv\inf\{t>0|q^{\epsilon}(t)=b\}$. Then $\sigma^{\epsilon}(t)=0$ for $t\in[0,\tau_{1}^{d})$. Next let $\tau_{1}^{r}\equiv\inf\{t>\tau_{1}^{d}|q^{\epsilon}(t)=0\}$, and $\sigma^{\epsilon}(t)=r/\epsilon$ for $t\in[\tau_{1}^{d},\tau_{1}^{r})$. This process repeats up to an arbitrary final time $T$.

The associated processes, as $\epsilon\to 0$, are defined as $q(t)$ and $\sigma(t)$ for the moisture and rain processes. The moisture process is the solution to the SDE,

with the unusual boundary condition as follows: Let the usual stopping time be as follows $\tau_{1}^{d}=\inf\{t>0|q(t)=b\}$. Then at time $t>\tau_{1}^{d}$ the process $q(t)$ jumps or “teleports” to $q=0$. Thus

Then the process starts over using the dynamics of (3) until $\tau_{2}^{d}=\inf\{t>\tau_{1}^{d}|q(t)=b\}$, and the process repeats. The stopping times $\tau_{i}^{d}$ are the dry event times of the process. These dynamics arise from the heuristic Fokker-Planck derivation in the next section (see Section III.1).

Examples of the processes are shown in Figure 2. The processes with finite rain rate $r/\epsilon$ for $\epsilon>0$ are shown in panels (a) and (b). Panel (a) is the moisture process $q^{\epsilon}(t)$ defined is equation (2). The rain rate process is shown in (d) and takes value $r/\epsilon$ when $q^{\epsilon}(t)$ reaches level $b$ for the first time (panel (a) in black) and resets to zero when $q^{\epsilon}(t)$ 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 equation (3) and panel (d) shows the rain process defined in equation (5). The moisture process is a Brownian motion with positive drift until reaching level $b$. When $q(t)=0$, the process $\sigma(t)$ takes an infinite value and the moisture process is reset at zero.

From the definition of $\tau_{i}^{d}$ above, the rain point process $\sigma(t)$ is defined as

where $\mathcal{N}(T)$ 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 $q^{\epsilon}$ loses moisture at a rate of $r/\epsilon$ per time, on average. The moisture process $q(t)$ loses all the moisture built up ($b$) instantaneously.

Note that $q^{\epsilon}(t)$ has continuous paths while $q(t)$ has jump discontinuities. Thus any mode of convergence between $q^{\epsilon}$ and $q$ with an associated metric (e.g. uniform or Skorohod) will fail. There is another way to define both $q^{\epsilon}$ and $q$ in which convergence with respect to $L^{2}$ with the uniform metric on the space of continuous functions ($\mathcal{C}[0,T]$) can be shown. To do so, $q^{\epsilon}(t)$ is decomposed into an evaporation process, $E^{\epsilon}(t)$, and precipitating process $P^{\epsilon}(t)$. These processes are defined as,

Thus the moisture process $q^{\epsilon}(t)$ is written as

In the limit, the jumps will be captured in the $P^{\epsilon}$ process. In the following section it will be shown (see Section III.2) that $E^{\epsilon}\rightarrow E$, where $E(t)$ is defined as the solution to the SDE

Furthermore, the $\sigma$ process is defined as above in (5) where $\tau_{i}^{d}=\inf\{t>0|E(t)=kb,\;k\in\mathbb{N}\}$, i.e. the first passage time of Brownian motion with drift to $kb$.

In this section convergence is shown both heuristically (e.g. Section III.1) and rigorously (e.g. Sections III.3 and III.2).

Note that the simplest ideas of convergence break down when considering path-wise convergence of $q^{\epsilon}$ to $q$ and $\sigma^{\epsilon}$ to $\sigma$. This is because $q^{\epsilon}$ is a continuous process ($\sigma^{\epsilon}$ is left continuous and has right hand limits) for all $\epsilon>0$ and $q$ is a process with jumps ($\sigma$ no longer is left continuous). Thus, there is no topology with associated metric $d$ such that $q^{\epsilon}\rightarrow q$ with respect to $d$. However, one can show that $q^{\epsilon}$ converges in a notion weaker than the Skorohod topology. See Kurtz, (1991) for these conditions. This convergence happens in a topology which does not have an associated metric (see Jakubowski et al., (1997)). This is not done here as it is technical and does not give any insight to the model or approximation.

The following three subsections prove convergence of the various processes introduced in Section II. In Section III.1 the Fokker-Planck equation for $q^{\epsilon}$ 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 Section III.2 convergence in paths is shown for $E^{\epsilon}$ to $E$ with respect to the uniform metric for continuous functions on $[0,T]$. In Section III.3 convergence is shown for $\sigma^{\epsilon}$ to $\sigma$ with respect to generalized functions. This norm is necessary because $\sigma$ is a sum of dirac delta functions. In addition, this convergence is the “natural” convergence to consider when analyzing the errors between using $\sigma^{\epsilon}$ and a point process ($\sigma$) in, for example, a GCM.