Some Remarks About Forward and Inverse Modelling in Hydrology, Within a General Conceptual Framework

Round 1

Reviewer 1 Report (Previous Reviewer 3)

Comments and Suggestions for Authors

The manuscript has been revised with minor corrections. The article's primary goals, conclusions, and organizational framework are all still in place. I disagree with the author's optimistic assessment of the likelihood of slight alterations to the text. The manuscript's organization and content both have a serious flaw. The content is essentially the author's observations on a variety of hydrology modeling-related topics. There are no radically novel concepts or solutions to hydrological modeling problems put forth. It appears to be more of an excellent and practical lecture guide to acquaint the reader with the topic. However, I don't see any scientific findings here. As such, I am unable to suggest that this manuscript be published. I let the Special Issue's editor make the ultimate call.

Author Response

See the attached file

Author Response File: Author Response.pdf

Reviewer 2 Report (New Reviewer)

Comments and Suggestions for Authors

This is a hearty discussion piece by one of the primary workers in inverse problems in geophysics.  The manuscript differs from the usual technical paper in that it proposes a framework for pondering inverse problems.  I find this of great value because there was a time (1990's-90's) when many were working in this field and many students were aware of the basic "inverse problem" elements of model calibration.  That is not the case today.

My only general comments are:

- the schema presented are also relevant to the "900-pound gorilla in the room", that is physics-informed machine-learning (PIML) calibration methods, used for model calibration (not to mention non-informed ML, that offers a replacement of physics-based mathematical modeling.  I have found that many new proponents are unaware of the fundamental mathematical aspects of PIML.  The author might point out that basically everything stated is relevant to the design and performance of PIML.

- One of the most important issues in model calibration in my opinion is non-uniqueness of the inverse problem.  The author does not go there and states why, but I do wish  this one was not in the set of of things not visited.  I am still wondering why recent modelers invented the new term "equifinality" when I think but am not sure that it is identical to "non-uniqueness."

 

Specific comments.

Abstract: "embed" by which I think the au's mean, embed into model formulation (in order to prepare it for inversion or calibration) ?

21-6 not all math models require numerical solution.  Closed-form solutions although often loaded with assumptions are of great practical value.

33 "...extent, quality,... and scale..."

41 with all due respect to Condon et al that citation is wholly focused on *global* gw models and has little to do with inverse (the word appears twice and only in passing) or calibration ("is outside the scope)."

81 I do worry about our young people: I don't think as many know what is a Lagrangian derivative as they used to.

equ. (4) Questrion: can f include any differential operators ?  My answer is yes, when space but not time is discretized.  Or perhaps the "stationarity" assumption is to be used throughout this discussion ?  I wish not as by doing so the authors are eliminating a small but significant fraction of inverse problems.

153 item (2). as above, I personally do not believe that inverse problems necessarily involve discretization of time paradigmatically.

174-193. Also 212.  I was to complain about the absence of the property of "scale" (of d, of p, of the numerical approach).  I see that the au's cover the scale of numerical method at line 188 but it might be useful to emphasize a bit more the issue of scale discrepancy between and among d and p.  The cases are well-known: commercial gw model parameters are often defined as averages over numerical cells sometimes 100km on a side while data on such things pertains either to much smaller measures of piezometric head or stratigraphy through borehole cuttings, or pumping tests.  It seems to me that it might be relatively straightforward to work this notion into equation (8).

Equ. 9. p^(cal) is an argument of the left-hand side but does not appear on the right-hand side.

358-364 - citation (71) is very appropriate here; as noted above I do still wish that uniqueness got a bit more attention here.

435 it might be useful to note Laplace's principle of insufficient reason here.

453 at the risk of exposing my bias, I prefer the phrase "...a *potentially* powerful tool..."

 

Comments on the Quality of English Language

My mother was an editor in an English language country and she said we use "which' way too often.  She proposed that we replace it with "that" as much as possible.

69.  "proposal of fine-tuned the terminology" is a bit unclear.

On line 471 "seeked" should be "sought."

Author Response

See the attached file

Author Response File: Author Response.pdf

Reviewer 3 Report (New Reviewer)

Comments and Suggestions for Authors

Problem 1 - A definition of $P$ is required.

line 190 - This part should be more comprehensive. What is the solution to a DIP?

192 - Please correct the value of $N_{(c)}$ in this line.

260 - How should be understood the last item, the equality of functions with different domains of definition? How is this related to Eq.(4)?

Eq.(11) - The formulation is too vague. $P$ was not defined, '$\arg ...$' should be defined or replaced by a more common description.

314 -  ... with the the values ...

Paragraph 4.2 is difficult to comprehend. The role of the FP is not sufficiently shown. Formulations like e.g. "build $\bt$ in a straightforward way"
and "to populate $y$ with the values of the potential at the measurement points" are too vague.

Only experienced readers can understand.

It seems that only the reader that \uve{already knows} / specialists can
be able to go through.

I suggest to write the presented example similarly to the FP, Eq. (1) and (2). It should start by giving a clearly formulated inverse problem and its goal. Subsequently, one of the four cases should be demonstrated in details.

359-362 - I agree, discussing the concepts given in these lines is not necessary. On the other hand, the paper of this generality that is supposed to help with the orientation in the field of inverse problems, it could be helpful to bring brief definitions of these notions.

 

Author Response

see the attached file

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report (Previous Reviewer 3)

Comments and Suggestions for Authors

Thank you for your comment

This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

 

The manuscript presents a discussion on forward and inverse modeling in hydrology, focusing on the paradigmatic example of the diffusion equation. The distinction between model results, forecasts, and predictions is emphasized, providing a conceptual framework. The paper introduces various calibration methods such as CMM, DCM, DSM, and Bayesian approaches. While the content is generally valuable, there are areas that need clarification, improvement in consistency, and detailed explanations of methods.

 

1. The distinction between model results, forecasts, and predictions is valuable but should be consistently maintained throughout the manuscript. Ensure consistent use of terminology to prevent confusion. 

 

2. Some symbols and formulas are complex and need clearer explanations. Provide more intermediate steps and results to enhance reader understanding.

 

3. References to other literature are made but lack detailed discussion. Provide more context and analysis of these references for better reader comprehension.

 

4. Methods like CMM, DCM, DSM, and Bayesian approaches are mentioned but lack detailed explanations. Elaborate on each method, including steps, advantages, and limitations.

 

 

Comments on the Quality of English Language

While the language is generally clear, some sentences are lengthy and challenging to comprehend. Edit for clarity and conciseness to enhance overall readability.

Author Response

Thank you for your comments. Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

In my opinion, due to its structure, this manuscript appears more suitable for a journal of Applied Mathematics or similar, and it however seems a review about forward and inverse modelling…

I suggest rejection for Hydrology journal. On the contrary, author should make significant efforts to clarify a lot of discussed concepts also for readers not so experts in mathematics.

I know that the described topic is a “macro-topic”, very difficult to synthesize in 12-15 pages, but publication in Hydrology journal should at least require:

·        An Appendix in which all the adopted symbols are briefly explained;

·        A detailed description of at least two more examples (Rainfall-Runoff modelling? Water management from a reservoir? Flood nowcasting?) in which inverse modelling can be applied;

·        More figures (only one, like in the present version, is not so suitable for this journal) which could help the understanding for non-mathematicians readers.

 

Comments on the Quality of English Language

none

Author Response

Thank you for your comments. Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The article has an unusual structure. The author's criticisms about terminology in hydrological modeling are unsubstantiated. Many results are highly subjective and do not relate to contemporary hydrological concepts or development prospects. Unfortunately, comments on assessing climate impacts on water resources do not correlate to the opinions of thousands of scientists and are highly presumptuous. The author's proposed paradigm is not unique; in particular, Bayesian estimating has been employed for many decades. Methods for more rigorous statistical evaluation of physically based hydrological models must be developed. 

I believe that this manuscript cannot be recommended for publishing in Hydrology Journal because it has several inconsistencies with present conceptions of hydrology while also containing no new valid approaches for the growth of hydrology. Further revisions are unlikely to bring the manuscript into publishable shape. 

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

In this work:

 "Remarks about Forward and Inverse Modelling in Hydrology"

 The originality and scientific novelty of the paper are not more serious. The length of the text is correct. I think in the Introduction sections more international references should be cited to introduce other areas to be compared with.

 

 This paper does not review mathematical modelling and inverse problems theory and application.

 This paper is based on the conceptual framework proposed by the author, but some remarks are not innovative (was published with some of his coworkers). Can you please explain and separate the already published frameworks?

 

The specific characteristics of this work are related to:

1. the introduction of proper terminology...

2. the discussion of some properties of the DIP, by making use of a paradigmatic example of non-linear model;

3. the family of methods of solution to the DIPs based on explicit solutions is considered

 

Several researchers dealt with applications in groundwater hydrology and soil physics or pedology. Several other papers provide reviews of inverse problems in other subfields of geophysics.

 

This article can be very useful to cast the FP and the IP properly. In particular, some model calibration methods, e.g., CMM, DCM,  and the Bayesian approach, only partially fit that framework. It can be helpful to cast even some of those methods in more general cases.

 

It is also important to publish such types of articles in scientific journals.

 

General comment:

 

- some literature links are not functional

Author Response

Thank you for your comments. Please see the attachment.

Author Response File: Author Response.pdf