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

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)