Convergence of Rain Process Models to Point Processes

A moisture process with dynamics that switch after hitting a threshold gives rise to a rainfall process. This rainfall process is characterized by its random holding times for dry and wet periods. On average, the holding times for the wet periods are much shorter than the dry. Here convergence is shown for the rain fall process to a point process that is a spike train. The underlying moisture process for the point process is a threshold model with a teleporting boundary condition. This approximation allows simplification of the model with many exact formulas for statistics. The convergence is shown by a Fokker-Planck derivation, convergence in mean-square with respect to continuous functions, of the moisture process, and convergence in mean-square with respect to generalized functions, of the rain process.


I. INTRODUCTION
Rain processes are often studied and tested as single column models in the atmosphere [6,10]. 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) [4,17,22]. One type of rainfall model that has been successful is a Renewal process ( [5,9,11,19]). A renewal process is a continuous-time Markov process that is defined by its holding times [8]. For example, a simple rain renewal process, σ(t ), may be defined by its holding times τ d when σ(t ) = 0, defined to be that the column of air is dry, and τ r when σ(t ) = 1, defined to be that the column of air is raining. Thus a dry (or rain) event will last τ d (or τ r ) time where the distribution of the (possibly) random times are given.
The model presented here was previously studied in [1,13,20,21]. The underlying process is a one dimensional continuous-time stochastic process modeling the moisture q(t ) ∈ (−∞, ∞) for t ≥ 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 σ(t ) an indicator function for rain. One choice of process for q(t ) is the hysteresis dynamics studied in [13]. There the q(t ) process is governed by the stochastic differential equations (SDE) d q(t ) = m d t + D 0 dW t for σ(t ) = 0 −r d t + D 1 dW t for σ(t ) = 1 , q(0) = 0, σ(0) = 0, 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 σ(t ) switch from 0 to 1 when q(t ) reaches a fixed threshold b. That is (q(0), σ(0)) = (0, 0) and σ(t ) = 1 for t = inf{t ≥ 0, q(t ) = b}. Then σ(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 [13]. A realization of the processes q(t ) and σ(t ) are shown in Figure 1. In a renewal process perspective, σ(t ) is modeled by the holding times τ d and τ 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 [1]. In a long time view, the dry events dominate the rain events. In Figure 2 the top panel shows σ(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 σ(t ) resembles a point process.
The main result of the paper is to define and show convergence of the threshold model above as r → ∞. For example, on the level of renewal processes, τ r → 0 and thus σ(t ) converges to a process that is zero everywhere and has spikes at infinity at random times τ d . However, σ(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 σ(t ) is shown to converge rigorously with  of [1] with permission from the authors. Panels (c) and (d) are realizations of the model of the rain rate process σ (t ) with finite rain rate r and σ(t ) the point process, respectively.
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 τ 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. [7,23]), and using threshold triggers in financial models [16]. The strongest connection is with neuron stochastic integrate and fire models (see [18] 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 [2] where the refractory time was constant. Here, the refractory time is random and coincides with the rain duration time τ 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 A. Rigorous convergence of the moistening process E to E is shown with respect to L 2 in Section III B and the rain process σ is shown to converge to the sum of delta distributions σ with respect to generalized functions in Section III C. The results are summarized in Section IV.

II. MODEL DESCRIPTION
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 with the limit as → 0 in mind.
The moisture process q (t ) ∈ R is defined as the solution to the stochastic differential equation (SDE), where m and r / are the moistening and rain rates, and 0 < D 0 ≤ D 1 are the fluctuations of moisture during the respective states. The rain process, σ (t ) ∈ {0, r / } are as follows: since . This process repeats up to an arbitrary final time T .
The associated processes, as → 0, are defined as q(t ) and σ(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 τ d Then the process starts over using the dynamics of (3) until τ d 2 = inf{t > τ d 1 |q(t ) = b}, and the process repeats. The stopping times τ d i are the dry event times of the process. These dynamics arise from the heuristic Fokker-Planck derivation in the next section (see Section III A).
Examples of the processes are shown in Figure 2. The processes with finite rain rate r / for > 0 are shown in panels (a) and (b). Panel (a) is the moisture process q (t ) defined is equation (2). The rain rate process is shown in (d) and takes value r / when q (t ) reaches level b for the first time (panel (a) in black) and resets to zero when q (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 σ(t ) takes an infinite value and the moisture process is reset at zero.
From the definition of τ d i above, the rain point process σ(t ) is defined as 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 q loses moisture at a rate of r / per time, on average. The moisture process q(t ) loses all the moisture built up (b) instantaneously. Note that q (t ) has continuous paths while q(t ) has jump discontinuities. Thus any mode of convergence between q and q with an associated metric (e.g. uniform or Skorohod) will fail. There is another way to define both q and q in which convergence with respect to L 2 with the uniform metric on the space of continuous functions (C [0, T ]) can be shown. To do so, q (t ) is decomposed into an evaporation process, E (t ), and precipitating process P (t ). These processes are defined as, Thus the moisture process q (t ) is written as In the limit, the jumps will be captured in the P process. In the following section it will be shown (see Section III B) that Furthermore, the σ process is defined as above in (5) where τ d i = inf{t > 0|E (t ) = kb, k ∈ N}, i.e. the first passage time of Brownian motion with drift to kb.  (2) with > 0 and (c) q(t ) and (d) σ(t ) defined in equation (3) and (5) respectively.

III. CONVERGENCE TO A POINT PROCESS
In this section convergence is shown both heuristically (e.g. Section III A) and rigorously (e.g. Sections III C and III B). Note that the simplest ideas of convergence break down when considering path-wise convergence of q to q and σ to σ. This is because q is a continuous process (σ is left continuous and has right hand limits) for all > 0 and q is a process with jumps (σ no longer is left continuous). Thus, there is no topology with associated metric d such that q → q with respect to d . However, one can show that q converges in a notion weaker than the Skorohod topology. See [15] for these conditions. This convergence happens in a topology which does not have an associated metric (see [14]). 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 A the Fokker-Planck equation for q 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 B convergence in paths is shown for E to E with respect to the uniform metric for continuous functions on [0, T ]. In Section III C convergence is shown for σ 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 the "natural" convergence to consider when analyzing the errors between using σ and a point process (σ) in, for example, a GCM.

A. Fokker-Planck Equation
In this section, the derivation for the Fokker-Planck equation of equation (3) is shown. The Fokker-Planck equation for the (D2) process (see [13]) is composed of two densities. These densities are denoted ρ 0 and ρ 1 for the dry and rain states respectively. These densities follow the following Fokker-Planck equations where the fluxes f i are defined as and with the following conditions, The proposed limit as → 0 for the Fokker-Planck equation is with the following conditions, The analysis follows asymptotic matching conditions from [3] (Chpt. 9). Consider two regions [0, ] and [ , ∞). Let ρ 1,B be the density in the first, boundary layer region. For this equation, define the rescaled variableq = 1 q. This yields the equation Let ρ 1,B have the asymptotic expansion of the form, Substituting this expansion into equation (17) The solution to the order −2 equation (18a)and applying the absorbing boundary condition atq = 0 yields 6 The order −1 equation (18b) has the same solution as above, after applying the absorbing boundary,

Now consider the interval away from the boundary [O( ), ∞). Let ρ 1,A be the density in this region. The equation in this region is
Let ρ 1,A have the asymptotic expansion Note that the δ term acts on f 0 which is a function of ρ 0 . The asymptotic analysis is for ρ 1 only, thus the density ρ 0 is an order one term. Substituting the expansion into equation (21) gives the following equations, separated into their orders of , The order −1 equation (22a) has the solution Note that ρ 1,A is a density and thus ρ 0 1,A must be integrable on [O( ), ∞). Thus C 3 (t ) = 0 and The order one equation (22b), by substituting in ρ 0 1,A = 0, gives the solution, Note that the constant of integration in each interval of b must be the same. Otherwise, the magnitude of the δ function in (22b) would not be correct. The density ρ 1 1,A must be integrable which implies that The first equation implies that C 1 (t ) = 0 and ρ 0 1,B = ρ 0 1,A = 0. In the limit as → 0 the second equation yields Thus the densities are and Note the flux of ρ 1 at q = 0 is, Using the asymptotic expansion yields, Thus the Fokker-Planck type equation for q(t ) is with the following conditions,

B. Pathwise Convergence
For this section and next, a useful lemma is first stated and proved. (3). The probability of that the number of events is n decays exponentially as n tends to infinity, i.e. for 0 < s < min{r b/ D 2 2 , mb/D 2 1 }

Lemma 1. Let N (T ) be the number of rain events for the q process defined in
Proof. Note that the process N (T ) is a renewal process. It is defined by the interarrival times, where τ d , i (τ r, i ) is the duration for the i th dry (rain) event of the σ process. The distributions of τ d i and τ d , i are the same independent of , while τ r, i depends on epsilon. To estimate the sum in (53), the probability of having N rain events in time T is estimated using the Central Limit Theorem. Consider the probability P N (T ) = N = P (S 1 + S 2 + · · · S N ≤ T, S 1 + S 2 + · · · + S N + S N +1 > T ) . (37) The probability on the right hand side is estimated crudely by only considering one of the two events. Note that S 1 , S 2 , S 3 , ..., S n are IID random variables with E [S 1 ] = E [τ d , + τ r, ], and σ 2 = Var(S 1 ) < ∞, thus P (N (T ) = N ) ≤ P (S 1 + S 2 + · · · + S N ≤ T ) .
The above probability is estimated by using a variant of the Chernoff bound [12]. That is, for s > 0, where E [e −sS i ] = M S i (s) is the moment generating function for the random variable S i . The moment generating function factors due to independence of τ r, i and τ d , i , These moment generating functions are computed explicitly from the distributions found in [13]. They are, (41) where s < min{r b/ D 2 2 , mb/D 2 1 }. Thus Chernoff's bound yields, With this lemma, convergence from E to E is shown in L 2 (Ω) with respect to the uniform metric on the space of continuous functions C [0, T ].

Theorem 1. Let q t be defined as
where E t , P t are solutions to the SDEs in (6). Furthermore let E t be defined as the solution to (7). Then Proof. To begin, note that the SDEs for E and E (see (6)) only differ when σ (t ) = 1. Thus, the solutions to the SDEs give the formula where N (T ) is the number of rain events for T < ∞ and > 0 fixed. The number of rain events is conditioned to be N . By Lemma 1, the sum in (53) converges due to the fast decay of P (N (T ) = N ) as N → ∞. Note that m > 0 and the stochastic integral is a martingale and Doob's maximal inequality yields, Applying the Cauchy-Schwarz inequality to the sum and the Itô isometry to the stochastic integral yields This sum converges due to the fast decay of P (N (T ) = n) as shown in Eq.(45). Tonelli's theorem allows the limit as → 0 to exchange with the infinite sum.
To finish the theorem the following moments of τ r, i are used. The integrals can be computed exactly using the densities for τ r, i found in [13]. They are (50)

C. Distributional Convergence
In this subsection L 2 (Ω) of σ to σ is shown with respect to a generalized function norm. This norm is the one considered due to the nature of the delta function. It is also a natural norm to consider as it is an integrated error. That is, this norm considers the accumulation of errors after running the model for time T > 0.
. Let σ(t ) and σ (t ) be defined in (5). Then where Proof. To prove the theorem, the expectation is conditioned on the number of events N (t ), as is done in the previous section. Thus the expectation is where N (T ) is the number of dry events for the σ(t ) process up to time T . Again, because of the decay of P (N (T ) = N ) as N → ∞ given in Lemma 1, the infinite sum converges.
To estimate the quantity in (53), one rain event is considered and the Cauchy-Schwarz bound will be used. Consider the i th rain event, The function φ(t ) is smooth on [0, T ] and thus is locally Lipschitz. Let the Lipschitz constant be K > 0. Then, along with the Cauchy-Schwarz inequality, With the last inequality resulting from t − τ d , i being an increasing function on [τ d , Using the inequality above, along with the Cauchy-Schwarz inequality, the quantity in (53) is bounded by where all expectations are conditional on N (T ) = N .
To finish the theorem the following moments of τ r, i are used Thus the first term in (64) is The second term in (64) is where the expectation turns into a product because τ r, i and τ d , The third term of (64), is zero because τ d , i and τ d i have the same distribution. For the last "remainder" term, it is written as a comparison between N (T ) and N (T ), If N (T ) < N (T ), then the processes E t and E t must be at least b units apart. Thus From theorem 1, this quantity tends to zero as → 0. Thus, the sum in (53) converges due to the fast decay of P (N (T ) = n) as shown in Eq.(45). Tonelli's theorem allows the limit as → 0 to exchange with the infinite sum. This along with Theorem 1 completes the proof.

IV. CONCLUSIONS
In this paper a threshold model for moisture and rain was shown to converge to interesting processes for various modes of convergence. The original threshold model processes, defined in equation (2), originated from [21] and were studied in [13].
There, exact formulas were derived for various quantities of interest such as stationary distributions and expected rainfall. In [1] a connection was made between the rain process from the threshold model and the point process defined in equation (5).
Here convergence for the moisture processes were defined and shown for the Fokker-Planck equation as well as the paths of the processes. Furthermore, the convergence of the rain process were 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 exactly in the case of point processes as shown in [1]. Furthermore, point processes have been studied extensively in the neural science literature [18] and many statistics have been derived.
The proofs shown here are revealing on their own. The Fokker-Planck derivation in Section III A shows the density for the moisture in the rain state tends to zero while the flux term remains allowing for the "teleporting" boundary condition that arises for 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.