the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 09 Feb 2024
Prognostic Assumed-PDF (DDF) Approach: Further Generalization and Demonstrations
Abstract. A methodology for directly predicting the time evolution of the assumed parameters for the distribution densities based on the Liouville equation, as proposed earlier, is extended to multi–dimensional cases as well as when the systems are constrained by integrals over a part of the variable range. The extended methodology is tested against a convective energy cycle system as well as the Lorenz’s stranger attractor. As a general tendency, the variance tends to collapse to a vanishing value over a finite time regardless of the chosen assumed distribution form. This general tendency is likely due to the common cause as collapse of the variance commonly found in ensemble–based data assimilation.
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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.
- Preprint
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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-2024-287', Anonymous Referee #1, 03 Apr 2024
This paper investigates a method to predict the evolution of the parameters of assumed probability density functions (PDF), which is applied to different dynamical systems for which an exact solution of the Liouville equation is available. In this paper the method is extended to cases in which constraints are defined over subdomains, the distribution takes different forms in different subdomains and to multidimensional cases. Overall, the method fails to reproduce the evolution of the mean and variances provided by the solution of the Liouville equation.
General comments:
Although the main objective of the paper is relevant and the discussion is supported by a rigorous mathematical analysis, it is difficult to understand the general motivation of this study, how it relates to other works in the field, as well as the justification for the particular choices made in the different examples used to illustrate the application of the method, and what is the main message the author wants to convey. The abstract and the introduction in their current form are insufficient to understand the relevance of this paper.
Furthermore, the structure of the paper could be modified to improve the clarity of the discussion. In particular, some choices are not adequately explained. Adding some references between sections would be helpful to improve the coherence of the paper.
Therefore, my recommendation is to reconsider this paper after a major revision to address these issues and the specific comments listed below.
Specific comments:
L4-6: More context is needed to understand this sentence.
L8: What does "the common cause" mean?
L17: It would be interesting to mention weather forecasting as well.
L50: Please clarify the meaning of σl.
L104: Please add a brief description of principle proposed in YLP.
L109: It is not clear at this point why the distribution is only defined differently depending on the value of x. It would be helpful to indicate that the example is extended in the next subsection.
Section 3: A figure would be useful to visualise the distribution.
L195-196: The choice of σ is not clearly explained.
L255: Why are these values of σ chosen?
L293-294: What would be a possible solution to improve the solution with an assumed PDF?
L308: What is the computational cost of the numerical computations?
L311: Is it a “minor disadvantage” that the solution covers unphysical values? Please clarify what is meant by this sentence.
L348: If “this case is not actually attempted”, what is the purpose of section 4.3?
L540: This study applies the assumed-PDF approach to dynamical systems for which it is possible to compute the solution of the Liouville equation, but the method generally fails to reproduce the exact solution. What would be the appropriate procedure for cases where the exact solution is not available?
L566-567: This sentence is somewhat redundant.
Technical corrections:
L70: Is the + in λl in the second integral correct?
L134: subdevide -> subdivide
Caption in Figs. 3 and 4: Please remove “To Be Done: Captilize titley”
L481: wehn -> when
L527: extend -> extends
L530: have -> has
L532: have applied -> have been applied; system -> systems
L535: performances -> performance; have -> has been
L583: prevent,t -> prevent
Citation: https://doi.org/10.5194/egusphere-2024-287-RC1 - AC1: 'Reply on RC1', Jun-Ichi Yano, 15 Apr 2024
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RC2: 'Comment on egusphere-2024-287', Anonymous Referee #2, 08 Apr 2024
This is a difficult review, as the manuscript provides a careful and well-done analytical analysis of the Liouville equations and the assumed Pdf approach, but at the same time is likely of rather limited practical benefit as also noted by the author.
In general I agree with the assessment by Reviewer 1 and do not have much to add, apart-abstract and conclusion: the authors claim that their finding of vanishing variance is "likely the common cause of collapse in variance found in ensemble-based data assimilation". This is not founded here and partly misleading as ensemble based data assimilation systems introduce spread/variance through perturbations to the observations and the model; please rewrite with proper references or remove.
- the analysis is accurately done, I wasn't able to verify all equations but only probes with seemed to be correct- there has already been a Technical Memorandum in EGUsphere by the authors on the subject, but this time also integration over limited domains has been added and practical solutions using Gauss distributions and Gauss and Gamma distributions with the contstraints of mean and variance have been provided for the Lorenz and convective energy cycle systems. I liked the example/solution part, but it gets too long without benefit. I strongly suggest to drop 5.2 and 5.3 Lorenz Model II and III and also 4.3.
Typos:-l12 "as in values themselves" ???? rewrite
-l24 "into"->"to"
-l46 "Evolution"
-l224 "dimensinoal"->dimensional-l 287 Caption Figure 3 "To Be Done ..."
-Figures 7,8: (a) (a) ->(a) (b)Citation: https://doi.org/10.5194/egusphere-2024-287-RC2 - AC2: 'Reply on RC2', Jun-Ichi Yano, 15 Apr 2024
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2024-287', Anonymous Referee #1, 03 Apr 2024
This paper investigates a method to predict the evolution of the parameters of assumed probability density functions (PDF), which is applied to different dynamical systems for which an exact solution of the Liouville equation is available. In this paper the method is extended to cases in which constraints are defined over subdomains, the distribution takes different forms in different subdomains and to multidimensional cases. Overall, the method fails to reproduce the evolution of the mean and variances provided by the solution of the Liouville equation.
General comments:
Although the main objective of the paper is relevant and the discussion is supported by a rigorous mathematical analysis, it is difficult to understand the general motivation of this study, how it relates to other works in the field, as well as the justification for the particular choices made in the different examples used to illustrate the application of the method, and what is the main message the author wants to convey. The abstract and the introduction in their current form are insufficient to understand the relevance of this paper.
Furthermore, the structure of the paper could be modified to improve the clarity of the discussion. In particular, some choices are not adequately explained. Adding some references between sections would be helpful to improve the coherence of the paper.
Therefore, my recommendation is to reconsider this paper after a major revision to address these issues and the specific comments listed below.
Specific comments:
L4-6: More context is needed to understand this sentence.
L8: What does "the common cause" mean?
L17: It would be interesting to mention weather forecasting as well.
L50: Please clarify the meaning of σl.
L104: Please add a brief description of principle proposed in YLP.
L109: It is not clear at this point why the distribution is only defined differently depending on the value of x. It would be helpful to indicate that the example is extended in the next subsection.
Section 3: A figure would be useful to visualise the distribution.
L195-196: The choice of σ is not clearly explained.
L255: Why are these values of σ chosen?
L293-294: What would be a possible solution to improve the solution with an assumed PDF?
L308: What is the computational cost of the numerical computations?
L311: Is it a “minor disadvantage” that the solution covers unphysical values? Please clarify what is meant by this sentence.
L348: If “this case is not actually attempted”, what is the purpose of section 4.3?
L540: This study applies the assumed-PDF approach to dynamical systems for which it is possible to compute the solution of the Liouville equation, but the method generally fails to reproduce the exact solution. What would be the appropriate procedure for cases where the exact solution is not available?
L566-567: This sentence is somewhat redundant.
Technical corrections:
L70: Is the + in λl in the second integral correct?
L134: subdevide -> subdivide
Caption in Figs. 3 and 4: Please remove “To Be Done: Captilize titley”
L481: wehn -> when
L527: extend -> extends
L530: have -> has
L532: have applied -> have been applied; system -> systems
L535: performances -> performance; have -> has been
L583: prevent,t -> prevent
Citation: https://doi.org/10.5194/egusphere-2024-287-RC1 - AC1: 'Reply on RC1', Jun-Ichi Yano, 15 Apr 2024
-
RC2: 'Comment on egusphere-2024-287', Anonymous Referee #2, 08 Apr 2024
This is a difficult review, as the manuscript provides a careful and well-done analytical analysis of the Liouville equations and the assumed Pdf approach, but at the same time is likely of rather limited practical benefit as also noted by the author.
In general I agree with the assessment by Reviewer 1 and do not have much to add, apart-abstract and conclusion: the authors claim that their finding of vanishing variance is "likely the common cause of collapse in variance found in ensemble-based data assimilation". This is not founded here and partly misleading as ensemble based data assimilation systems introduce spread/variance through perturbations to the observations and the model; please rewrite with proper references or remove.
- the analysis is accurately done, I wasn't able to verify all equations but only probes with seemed to be correct- there has already been a Technical Memorandum in EGUsphere by the authors on the subject, but this time also integration over limited domains has been added and practical solutions using Gauss distributions and Gauss and Gamma distributions with the contstraints of mean and variance have been provided for the Lorenz and convective energy cycle systems. I liked the example/solution part, but it gets too long without benefit. I strongly suggest to drop 5.2 and 5.3 Lorenz Model II and III and also 4.3.
Typos:-l12 "as in values themselves" ???? rewrite
-l24 "into"->"to"
-l46 "Evolution"
-l224 "dimensinoal"->dimensional-l 287 Caption Figure 3 "To Be Done ..."
-Figures 7,8: (a) (a) ->(a) (b)Citation: https://doi.org/10.5194/egusphere-2024-287-RC2 - AC2: 'Reply on RC2', Jun-Ichi Yano, 15 Apr 2024
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Jun-Ichi Yano
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(3943 KB) - Metadata XML