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the Creative Commons Attribution 4.0 License.
| 28 Sep 2022
Rain process models and convergence to point processes
Abstract. 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, 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.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2022-865', Anonymous Referee #1, 20 Oct 2022
Review of "Rain process models and convergence to point processes" by Scott Hottovy and Samuel N. Stechmann
discusses the convergence of two coupled diffusion processes, with different diffusion and drift coefficients,
to a point process, when one drift term diverges to infinity.(1) The identification with a rain process is questionable as the humidity in the model takes negative values.
A few of my question are:
(2) In line 45 the authors state that "D_0 and D_1 are the fluctuations"; no, they are the diffusion coefficients.
Are these coefficients time dependent ?(3) I do not understand the boundary conditions (13), if the densities are zero the process will never reach this states
and there will never be a jump ? (shouldn't the derivative vanish rather than the value?)(4) The pdf should converge to a stationary state, a time independent pdf? I think that it can be obtained analytically.
This important point is not discussed and I do not understand why the time dependence
of the pdf is kept in the calculations of the pdf?There are many points in the paper which I do not understand and the authors should consider sending the paper
to a mathematical journal specialized in stochastic processes.Citation: https://doi.org/10.5194/egusphere-2022-865-RC1 -
RC2: 'Comment on egusphere-2022-865', Anonymous Referee #2, 21 Oct 2022
The authors propose a pathway on how to link between a somewhat physically based diffusion processes and more common empirical/statistical pointwise processes. Even though the diffusion processes suggested are not necessarily the most widely accepted models for rainfall. The idea is very good as it may help in guiding future modeling strategies in terms of being able to combine physical intuition with data. The main results consists of a a) rigorous proof in the L2 norm, and b) and not so rigorous and rather very shaky asymptotic expansion, of the former type of processes to the latter. I think the paper can potentially become a very good publication worthy for the readers of NPG-EGUSphere but I conquer that the paper maybe better read if it was submitted to a more math/stats oriented paper as statisticians/mathematicians from other disciplines my appreciate as well as. Nonetheless, the following points may need to be addressed before the paper can be published.
The most serious suggestion I can make is that the authors could simply forgo the section of the asymptotic expansion because it is a mere distraction that doesn't add anything to the main result. I have a hard time making sense of the asymptotic expansion work which seems to have very serious flaws (see specific comments below). However, I trust that the convergence results in Section 3 are correct although I haven’t gone through all the details, especially I haven’t checked all the references to make sure that results reported in the literature have been applied correctly, namely because it is my specific area. The revised paper may benefit from being reviewed by a theoretician working in the area of statistics/stochastic processes.
Some of them as mis-comprehension/bad notations or typos and some are more serious by they are provided in the order as they appear in the paper.
1. Line 30 - 4: (which is the mixing ratio of water vapor in the air), delete "mass, or mass"
2. Line 45+3: Change "when q(t) reaches a lower threshold" by "when all the moisture is depleted".
3. Line 55+2: Onset of moist convective instability has no meaning. Do you mean to say the onset or convection or the threshold for the release of convective instability --- an example is "sampling" parcels of air that have enough energy (buoyancy) to overcome the CIN energy barrier, e.g. see
Mapes, B.E.: Convective inhibition, subgridscale triggering energy, and “stratiform instability” in a toy tropical wave model. J. Atmos. Sci. 57, 1515–1535 (2000)
Majda, A., and B. Khouider, 2002: Stochastic and mesoscopic models for tropical convection. Proc Natl. Acad. Sci. USA, 99, 1123–1128.
3. Line 60: The main purpose instead of the main result. A definition is not a result.
4. Line 60: convergence to what?
5. Line 61: Spikes at infinity means here: do you mean that sigma(t) become a Dirac delta distribution, i.e, the spikes in sigma are infinite? Also, this view is not consistent with Eq 1 where the the values of sigma are either 0 or 1 not zero or infinity. This is very confusing as to what exactly all this means!
6. Line 60+2: whereas is one word.
7. Equation 2: Eq 2: should the second value of sigma^epsilon be r over epsilon?
8. Line 105-1: These are the duration times for dry and rain events, respectively. Add respectively.
9. Line 110: Should the math expression at the end be t = T_1 instead of t > T_1
10. Line 125-1: to zero instead of at zero.
11. Figure 2: Some labeling of some sort should be added to panel d to illustrate the fact sigma is singular, that the spikes are infinite.
12. Equation 8: sigma^epsilon = 1 —-> sigma^epsilon = r/epsilon; Should one of the D_0’s be D_1 instead?
13. Line 165+1: Having both the subsrcipt/assignment notations q=0 and q=b and the Dirac deltas (delta(q) and delta(q-b) at the end of the rho_0 and rho_1 equations in (10) and (11) is an overkill. One of the two should suffice.
14. Line 175, end of paragraph: I agree with the authors that the equations in (10), (11) and (12) are both unusual and interesting. However, providing a simple reference for their justification is suboptimal. It will be helpful for the readers if some more discussion is provided, especially in terms of why and/or how the singular terms are obtained. Readers who do not have access or do have the time to read that reference should be given enough information to be able accept/trust these equations. Also there are a lot of similarities and also discrepancies between (10) versus (11) and (12a) versus (12b). If I am reading the equations correctly, the singular terms introduce coupling between the two distribution at q=0 and at q=b. However, at rho_1 =0 at q=0 and rho_0=0 at q=b may render these coupling terms obsolete, especially if their 1st and second derivatives follow suit, which is likely the cases if the distributions are smooth enough!
15. Eqn 15-16: It is a bit weird that the finite epsilon Fokker-Plank equations in (10) and (11) are defined for q in (-infinity, + infinity) but the limiting equations in (15) are restricted to (-infinity, b)!!? Some explanation/reconsideration is warranted.
16. Eqns 19-20: I am not sure I understand the goal nor the effectiveness of asymptotic expansion ansatz. Obviously the nondim parameter in Eq. 19 is epsilon^2 but the ansatz stops short at epsilon level. I am not an expert in asymptotic expansion and I give the authors the benefit of the doubt that what they are doing is most likely correct but I think they owe the reader some motivation about this ansatz. Maybe a better expansion should be in terms of even powers of epsilon only. That way the repetition in (20a) and (20b) wont happen! Without mentioning anything about the higher order terms, O(epsilon^2) there is no guarantee that there isn't secular growth.
17. Line 195+1: Why O(epsilon) and not simply epsilon as it is initially set in the definition of the two regions above?
18: Lemma 38 (by the way why this is called lemma 38, this is the only lemma every written in the paper, did I miss something? Same applies for the two theorems. )
This lemma should be reworded so that the main result is the inequality-upper bound of the probability distribution as provided. The fact that it decays exponentially as N tends to infinity follows immediately --as a consequence.
19. Line 270: Should the equation be sigma^epsilon = r/epsilon?
Citation: https://doi.org/10.5194/egusphere-2022-865-RC2 -
RC3: 'Comment on egusphere-2022-865', Anonymous Referee #3, 31 Oct 2022
Review of the paper "Rain process models and convergence to point processes"
by Scott Hottovy and Samuel N. StechmannThis paper establishes a novel connection between a widely used
empirical point process model for rainfall time and stochastic model
for moisture evolution. The authors prove that the moisture model
converges to a point process for large rain rates. This is done by
using formal asymptotic expansion of the Fokker-Planck equation, as
well as, rigorous convergence analysis.Although, I am not an expert on the rigorous analysis and because of
this cannot verify the corresponding part (Sec. 3.2, 3.3) of the
paper, I agree with the authors that the demonstrated connection is
very interesting and revealing. However, the authors should be much
more specific about the possible applications of their results in the
context of constructing physically constrained models of
precipitation. For example, is it possible to make statements about
the limitations of existing purely empirical point process models? In
the concluding section the authors mention error rates for point
processes. Can such estimates be given in the revised paper using some
observational data? Similar estimates will demonstrate the potential
of the results to the NPG-readers. Alternatively, the authors might
consider submitting the paper to a more mathematical journal in order
to access the mathematical interested readers.Some additional major comments:
In eq. 1 the water vapor mass mixing ratio, q, can become negative
which is clearly non-physical. Can the approach be modified to account
for positive values of q only? If not, this limitation should be
discussed in the paper.Eq. 2 and line 97. What are the correct values for the rain process
\sigma(t): {0, 1} or {0,r/\epsilon}? Both values can be found at
various places in the paper (e.g. equation 8), but since \sigma
converges to a Delta function, r/\epsilon should be the correct one.Fig.2. What value for \epsilon was used for Fig.2a and Fig2.b? Axis
tick values and labels are missing. In order to prove convergence
large raining rates are assumed; is this large compared to the
moistening rate m? Is \epsilon defined as m/r ? Can you give an
estimate of \epsilon from real data? What rain and moistening rates
are used in other models, or what are the corresponding dry and rain
event duration \tau^d and \tau^r, respectively?Citation: https://doi.org/10.5194/egusphere-2022-865-RC3 -
AC1: 'Author Comments on egusphere-2022-865', Scott Hottovy, 30 Dec 2022
We would like to thank the reviewer for their comments, which have improved the manuscript in many ways. One main theme of the revisions has been to better emphasize the importance of the results for a geophysical audience. To do this, the paper has been significantly restructured to make this aim more clear. We have moved many of the technical arguments to the Appendix. To highlight the usefulness of the approximations, we have added an entirely new section.
The details of our responses to the reviewer's comments are in the supplement document "HottovyStechmann_Reply.pdf."
Thank you,
Scott Hottovy and Samuel N. Stechmann
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2022-865', Anonymous Referee #1, 20 Oct 2022
Review of "Rain process models and convergence to point processes" by Scott Hottovy and Samuel N. Stechmann
discusses the convergence of two coupled diffusion processes, with different diffusion and drift coefficients,
to a point process, when one drift term diverges to infinity.(1) The identification with a rain process is questionable as the humidity in the model takes negative values.
A few of my question are:
(2) In line 45 the authors state that "D_0 and D_1 are the fluctuations"; no, they are the diffusion coefficients.
Are these coefficients time dependent ?(3) I do not understand the boundary conditions (13), if the densities are zero the process will never reach this states
and there will never be a jump ? (shouldn't the derivative vanish rather than the value?)(4) The pdf should converge to a stationary state, a time independent pdf? I think that it can be obtained analytically.
This important point is not discussed and I do not understand why the time dependence
of the pdf is kept in the calculations of the pdf?There are many points in the paper which I do not understand and the authors should consider sending the paper
to a mathematical journal specialized in stochastic processes.Citation: https://doi.org/10.5194/egusphere-2022-865-RC1 -
RC2: 'Comment on egusphere-2022-865', Anonymous Referee #2, 21 Oct 2022
The authors propose a pathway on how to link between a somewhat physically based diffusion processes and more common empirical/statistical pointwise processes. Even though the diffusion processes suggested are not necessarily the most widely accepted models for rainfall. The idea is very good as it may help in guiding future modeling strategies in terms of being able to combine physical intuition with data. The main results consists of a a) rigorous proof in the L2 norm, and b) and not so rigorous and rather very shaky asymptotic expansion, of the former type of processes to the latter. I think the paper can potentially become a very good publication worthy for the readers of NPG-EGUSphere but I conquer that the paper maybe better read if it was submitted to a more math/stats oriented paper as statisticians/mathematicians from other disciplines my appreciate as well as. Nonetheless, the following points may need to be addressed before the paper can be published.
The most serious suggestion I can make is that the authors could simply forgo the section of the asymptotic expansion because it is a mere distraction that doesn't add anything to the main result. I have a hard time making sense of the asymptotic expansion work which seems to have very serious flaws (see specific comments below). However, I trust that the convergence results in Section 3 are correct although I haven’t gone through all the details, especially I haven’t checked all the references to make sure that results reported in the literature have been applied correctly, namely because it is my specific area. The revised paper may benefit from being reviewed by a theoretician working in the area of statistics/stochastic processes.
Some of them as mis-comprehension/bad notations or typos and some are more serious by they are provided in the order as they appear in the paper.
1. Line 30 - 4: (which is the mixing ratio of water vapor in the air), delete "mass, or mass"
2. Line 45+3: Change "when q(t) reaches a lower threshold" by "when all the moisture is depleted".
3. Line 55+2: Onset of moist convective instability has no meaning. Do you mean to say the onset or convection or the threshold for the release of convective instability --- an example is "sampling" parcels of air that have enough energy (buoyancy) to overcome the CIN energy barrier, e.g. see
Mapes, B.E.: Convective inhibition, subgridscale triggering energy, and “stratiform instability” in a toy tropical wave model. J. Atmos. Sci. 57, 1515–1535 (2000)
Majda, A., and B. Khouider, 2002: Stochastic and mesoscopic models for tropical convection. Proc Natl. Acad. Sci. USA, 99, 1123–1128.
3. Line 60: The main purpose instead of the main result. A definition is not a result.
4. Line 60: convergence to what?
5. Line 61: Spikes at infinity means here: do you mean that sigma(t) become a Dirac delta distribution, i.e, the spikes in sigma are infinite? Also, this view is not consistent with Eq 1 where the the values of sigma are either 0 or 1 not zero or infinity. This is very confusing as to what exactly all this means!
6. Line 60+2: whereas is one word.
7. Equation 2: Eq 2: should the second value of sigma^epsilon be r over epsilon?
8. Line 105-1: These are the duration times for dry and rain events, respectively. Add respectively.
9. Line 110: Should the math expression at the end be t = T_1 instead of t > T_1
10. Line 125-1: to zero instead of at zero.
11. Figure 2: Some labeling of some sort should be added to panel d to illustrate the fact sigma is singular, that the spikes are infinite.
12. Equation 8: sigma^epsilon = 1 —-> sigma^epsilon = r/epsilon; Should one of the D_0’s be D_1 instead?
13. Line 165+1: Having both the subsrcipt/assignment notations q=0 and q=b and the Dirac deltas (delta(q) and delta(q-b) at the end of the rho_0 and rho_1 equations in (10) and (11) is an overkill. One of the two should suffice.
14. Line 175, end of paragraph: I agree with the authors that the equations in (10), (11) and (12) are both unusual and interesting. However, providing a simple reference for their justification is suboptimal. It will be helpful for the readers if some more discussion is provided, especially in terms of why and/or how the singular terms are obtained. Readers who do not have access or do have the time to read that reference should be given enough information to be able accept/trust these equations. Also there are a lot of similarities and also discrepancies between (10) versus (11) and (12a) versus (12b). If I am reading the equations correctly, the singular terms introduce coupling between the two distribution at q=0 and at q=b. However, at rho_1 =0 at q=0 and rho_0=0 at q=b may render these coupling terms obsolete, especially if their 1st and second derivatives follow suit, which is likely the cases if the distributions are smooth enough!
15. Eqn 15-16: It is a bit weird that the finite epsilon Fokker-Plank equations in (10) and (11) are defined for q in (-infinity, + infinity) but the limiting equations in (15) are restricted to (-infinity, b)!!? Some explanation/reconsideration is warranted.
16. Eqns 19-20: I am not sure I understand the goal nor the effectiveness of asymptotic expansion ansatz. Obviously the nondim parameter in Eq. 19 is epsilon^2 but the ansatz stops short at epsilon level. I am not an expert in asymptotic expansion and I give the authors the benefit of the doubt that what they are doing is most likely correct but I think they owe the reader some motivation about this ansatz. Maybe a better expansion should be in terms of even powers of epsilon only. That way the repetition in (20a) and (20b) wont happen! Without mentioning anything about the higher order terms, O(epsilon^2) there is no guarantee that there isn't secular growth.
17. Line 195+1: Why O(epsilon) and not simply epsilon as it is initially set in the definition of the two regions above?
18: Lemma 38 (by the way why this is called lemma 38, this is the only lemma every written in the paper, did I miss something? Same applies for the two theorems. )
This lemma should be reworded so that the main result is the inequality-upper bound of the probability distribution as provided. The fact that it decays exponentially as N tends to infinity follows immediately --as a consequence.
19. Line 270: Should the equation be sigma^epsilon = r/epsilon?
Citation: https://doi.org/10.5194/egusphere-2022-865-RC2 -
RC3: 'Comment on egusphere-2022-865', Anonymous Referee #3, 31 Oct 2022
Review of the paper "Rain process models and convergence to point processes"
by Scott Hottovy and Samuel N. StechmannThis paper establishes a novel connection between a widely used
empirical point process model for rainfall time and stochastic model
for moisture evolution. The authors prove that the moisture model
converges to a point process for large rain rates. This is done by
using formal asymptotic expansion of the Fokker-Planck equation, as
well as, rigorous convergence analysis.Although, I am not an expert on the rigorous analysis and because of
this cannot verify the corresponding part (Sec. 3.2, 3.3) of the
paper, I agree with the authors that the demonstrated connection is
very interesting and revealing. However, the authors should be much
more specific about the possible applications of their results in the
context of constructing physically constrained models of
precipitation. For example, is it possible to make statements about
the limitations of existing purely empirical point process models? In
the concluding section the authors mention error rates for point
processes. Can such estimates be given in the revised paper using some
observational data? Similar estimates will demonstrate the potential
of the results to the NPG-readers. Alternatively, the authors might
consider submitting the paper to a more mathematical journal in order
to access the mathematical interested readers.Some additional major comments:
In eq. 1 the water vapor mass mixing ratio, q, can become negative
which is clearly non-physical. Can the approach be modified to account
for positive values of q only? If not, this limitation should be
discussed in the paper.Eq. 2 and line 97. What are the correct values for the rain process
\sigma(t): {0, 1} or {0,r/\epsilon}? Both values can be found at
various places in the paper (e.g. equation 8), but since \sigma
converges to a Delta function, r/\epsilon should be the correct one.Fig.2. What value for \epsilon was used for Fig.2a and Fig2.b? Axis
tick values and labels are missing. In order to prove convergence
large raining rates are assumed; is this large compared to the
moistening rate m? Is \epsilon defined as m/r ? Can you give an
estimate of \epsilon from real data? What rain and moistening rates
are used in other models, or what are the corresponding dry and rain
event duration \tau^d and \tau^r, respectively?Citation: https://doi.org/10.5194/egusphere-2022-865-RC3 -
AC1: 'Author Comments on egusphere-2022-865', Scott Hottovy, 30 Dec 2022
We would like to thank the reviewer for their comments, which have improved the manuscript in many ways. One main theme of the revisions has been to better emphasize the importance of the results for a geophysical audience. To do this, the paper has been significantly restructured to make this aim more clear. We have moved many of the technical arguments to the Appendix. To highlight the usefulness of the approximations, we have added an entirely new section.
The details of our responses to the reviewer's comments are in the supplement document "HottovyStechmann_Reply.pdf."
Thank you,
Scott Hottovy and Samuel N. Stechmann
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Samuel N. Stechmann
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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(652 KB) - Metadata XML