A variety of stochastic models have been used to describe time series of precipitation or rainfall. Since many of these stochastic models are simplistic, it is desirable to develop connections between the stochastic models and the underlying physics of rain. Here, convergence results are presented for such a connection between two stochastic models: (i) a stochastic moisture process as a physics-based description of atmospheric moisture evolution and (ii) a point process for rainfall time series as spike trains. The moisture process has dynamics that switch after the moisture hits a threshold, which represents the onset of rainfall and thereby gives rise to an associated 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 ones, and, in the limit of short wet periods, the rainfall process converges to a point process that is a spike train. Also, in the limit, the underlying moisture process becomes a threshold model with a teleporting boundary condition. To establish these limits and connections, formal asymptotic convergence is shown using the Fokker–Planck equation, which provides some intuitive understanding. Also, rigorous convergence is proved 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.

Time series of precipitation or rainfall
display highly irregular behavior,
as illustrated in Fig.

Sample precipitation time series from observations at

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)

To be more specific, a point-process model of rainfall can be viewed as a spike train, as in Fig.

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

The

The threshold behavior of Eq. (

The main purpose of the paper is to define and show convergence of the threshold model in Eq. (

Some of the novel aspects of this work are as follows. The limit jump process

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 structure of the paper is as follows. The processes for moisture and rain are defined in Sect.

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

The moisture
process

The associated processes, as

Example time series of the processes are shown in Fig.

Realizations are plotted of the processes

From the definition of

Note that

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

Note that the simplest ideas of convergence break down when considering pathwise convergence of

Instead, we pursue convergence in the following senses.
The next
three subsections prove convergence of the various processes introduced in Sect.

In this section, we derive the Fokker–Planck equation of Eq. (

The Fokker–Planck equation for Eq. (

To obtain Eqs. (

One interesting property of these Fokker–Planck equations
is the appearance of Dirac delta source terms, which represent transitions between the dry state and the rain state. For instance, in Eq. (

The proposed limit as

The convergence of the time-dependent Fokker–Planck Eqs. (

Rigorous mathematical convergence is now considered.
For this section and the next, a useful lemma is first stated and proved. In essence, the lemma states that,
for a finite time interval

Let

The proof of the lemma is contained in the Appendix. This lemma shows that the probability of

This theorem shows that the moistening process

In this subsection

The technical details of the proof are given in the Appendix. The procedure is similar to the proof of Theorem 1. However, here the case for different numbers of rain events in the time interval

In this section, important statistics and applications of the processes

Here the analytical solutions to the stationary Fokker–Planck equation are given. The stationary Fokker–Planck equation for the process

Stationary probability density functions (pdfs) for the

Another statistic studied here is the event duration probability density. This density gives information on the probability of a dry/rain event lasting time

These densities are plotted in Fig.

Event duration pdfs for dry events (

The average cloudiness is the fraction of time that the stationary process is in the rain state

For an example of using the results of Theorem 2, consider calculating the total rainfall at a specific grid point of a global circulation model. Define the rainfall time series at a point as

Theorem 2 states that
the integrated difference of the total rainfall tends to zero. That is,

The issue of how to use a finite-event-duration model to inform parameter selection is discussed in this subsection. There are two potential points of concern with the point-process model. One is that the point-process model

There are many potential solutions for the issue of zero-event times for the point-process model. One example is to modify the dry-duration pdf to account for the small but finite size of rain events. That is, let

Another potential modification is the definition of rain amount. For the model with finite

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

The proofs shown here are revealing on their own, and they demonstrate further details of the convergence. The Fokker–Planck derivation in Sect.

The rigorous mathematical proofs and formal asymptotic analysis for the results presented in Sect.

The Fokker–Planck equation for the process

First, to show

Thus, the Fokker–Planck-type equation for

Note that the process

To begin, note that the SDEs for

To finish the proof, the following moments of

To prove the theorem, the expectation is conditioned
on the number of events

To estimate the quantity in Eq. (

To finish the theorem, the following moments of

For the last “remainder” term in Eq. (

Code to produce the figures is available from the authors on request.

All of the data used in this paper are publicly available from the references listed.

Both the authors contributed to the final draft of the work. Additionally, SH contributed to the formal analysis and writing of the original draft preparation, and SNS contributed to the conceptualization and writing, review, and editing.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the Division of Mathematical Sciences (grant no. 1815061).

This paper was edited by Balasubramanya Nadiga and reviewed by three anonymous referees.