Optimizing Finite-Difference Operator in Seismic Wave Numerical Modeling
Round 1
Reviewer 1 Report
The paper shows an interesting way to find weights for optimal FD. I think it is well written; the subject is attractive and it has a lot of potential. There are two minor comments that I would like to make with the aim of helping to improve the manuscript.
The authors used three nature-inspired optimization algorithms: PSO, CDPSO and KH to compute the weights for the FD. It is interesting that the three methods provide similar weights and I do not know if all of them tend to the same minimum a why would that be possible? If there was a unique solution that could be an explanation, but if not, maybe authors could elaborate on theories that try to explain this behavior.
On Figure 5, I did not find (I could have missed) what the source is (images on the fourth column). I believe it is ok to show this figures. In addition this these figures it could be more illustrative to show some single traces and compare the seismograms of these traces for the four methods.
On line 44 it is written: "In the past decades, are some new window functions were also reported in the literature".
I believe that this sentence has to be re-written.
Once these comments are taken into account the paper could be considered for publication.
Author Response
Comment 1.1 It is interesting that the three methods provide similar weights and I do not know if all of them tend to the same minimum and why would that be possible? If there was a unique solution that could be an explanation, but if not, maybe authors could elaborate on theories that try to explain this behavior.
Response to comment 1.1: The difference between the weights of three methods is minor (from two decimal places to four decimal places). Even though, the weights significantly impact on the modeling precision. To make the difference more obvious, we add more numerical simulations to verify that the KH algorithm outperforms PSO and CDPSO (now Fig.5). Moreover, as per the reviewer’s suggestion, we added the single traces to clearly demonstrate the differences in the final results (now Fig.7).
Comment 1.2 On Figure 5, I did not find (I could have missed) what the source is (images on the fourth column). I believe it is ok to show this figures.
Response to comment 1.2: 'Source' is the data from Wang's Chebyshev auto-convolution combined window. He published the finite-difference coefficient in his paper. In this paper, we also used it as the source. It is our fault that we did not mention it in the last version of manuscript. We added text regarding the source at the beginning of Section 3.4 (line 217).
Comment 1.3 In addition this these figures it could be more illustrative to show some single traces and compare the seismograms of these traces for the four methods.
Response to comment 1.3: According to your suggestion, we added a tracer in the numerical simulation, and compared the obtained trace graphs of the four methods, as shown in Fig.7.
Comment 1.4 On line 44 it is written: "In the past decades, are some new window functions were also reported in the literature". I believe that this sentence has to be re-written.
Response to comment 1.4: We corrected it as ”In the past decades, some new window functions were also reported in the literature“ (line 43).
Reviewer 2 Report
This paper presents an optimization approach to approximate the finite-difference operators in the context of seismic wave numerical modeling.
There are numerous grammatical errors that make reading difficult.
In my opinion, the article is not clear, details are missing and from a theoretical point of view it is weak.
Just to mention some issues, in section 2.1 the sinc function considered is not the usual one, but the normalized sinc function. This should have been mentioned.
When you truncate the summation in (1), taking N+1 terms, what is a measure of the error committed?
Later, after taking the first derivative, this summation of N+1 terms is reduced to a summation with N terms (because the term for n=0 is not defined), but there is no theoretical justification.
It is not clear how you take the first derivative of f(x) at x=0.
If you take the first derivative in the rhs of (1) with respect to x the term f(n\delta x) is not affected. You have included the coefficient $\omega_n^N$, but this has not been defined.
Later it is not clear how you perform the Fourier transform on Eq.4 and Eq.5. The wave number $k_x$ has not been defined.
You say that (6) is the first derivative. This is not clear.
The truncation error is defined as the difference of the two terms in (6). But since (6) is an equality, you would obtain that E_1=0.
Supposedly you want to minimize the truncation errors, but it is not clear how this is done.
In my opinion the paper has not enough quality to be published.
Author Response
Comment 2.1 There are numerous grammatical errors that make reading difficult. In my opinion, the article is not clear, details are missing and from a theoretical point of view it is weak.
Response to comment 2.1: We reorganized the content of the article (Chapter 2.1, 2.2 and Chapter 3). We also expanded the 2.1 for more details in appendix. We strived to explain all the details as clearly as possible.
Comment 2.2 In section 2.1 the sinc function considered is not the usual one, but the normalized sinc function. This should have been mentioned.
Response to comment 2.2: As per your suggestion, we have modified it to "normalized sinc function" in the revised manuscript (line 79).
Comment 2.3 When you truncate the summation in (1), taking N+1 terms, what is a measure of the error committed?
Response to comment 2.3: The error is caused by truncating infinity to [-N/2,N/2]. When the summation on the right side of the formula is used to replace the differentiation on the left side of the formula, the error is the difference between the two sides of the equal sign. We changed the "=" in Eq.2 and Eq.3 (now Eq.4 and Eq.5) to " " to avoid confusion. We also derived formulas that can measure the error (now Eq.11 and Eq.12).
Comment 2.4 Later, after taking the first derivative, this summation of N+1 terms is reduced to a summation with N terms (because the term for n=0 is not defined), but there is no theoretical justification.
Response to comment 2.4: We thank the reviewer point out this. Indeed, n=0 is a singularity. In the revised manuscript, we corrected it by putting back the point n=0 in the formula (now Eq.4 and Eq.5), and discussed the at line 102. The finite-difference coefficients and contains the parameter before . When , .
Comment 2.5 It is not clear how you take the first derivative of f(x) at x=0.
Response to comment 2.5: The first derivative at x equals 0 is the fractional derivative. We added the result of derivation at x=0 (now Eq.2 and Eq.3), and also added more details of formula derivation to Appendix A.1.
Comment 2.6 If you take the first derivative in the rhs of (1) with respect to x the term f(n\delta x) is not affected. You have included the coefficient , but this has not been defined.
Response to comment 2.6: We thank the reviewer point out this. We have rewritten the derivation. In the revised manuscript, should be kept. The coefficient is the window function in the previous version (previous line 78, now line 86). To avoid confusion, we used the instead of in the revised manuscript.
Comment 2.7 Later it is not clear how you perform the Fourier transform on Eq.4 and Eq.5. The wave number has not been defined.
Response to comment 2.7: According to your suggestion, we have added the details of Fourier transform to Appendix A.3. This process is calculated by merging the Fourier transform formula of f(x) into the original equation. The wave number is the variable in the frequency-wavenumber (F-k) domain. The sesmic data can be converted to that of the F-k domain by 2D Fourier transform. For each , there is a truncation error. The range of in the simulation is discussed in section 3.1 (line 185~187).
Comment 2.8 You say that (6) is the first derivative. This is not clear.
Response to comment 2.8: Eq.6 (previous Eq.6, now Eq.9) is the first derivative in the F-k domain. Eq.6 is obtained by Fourier transform of Eq.4 (previous Eq.4, now Eq.7) , which transforms the first-order derivative formula from spatial domain to F-K domain. We have written the detailed Fourier transform formula in the appendix A.3.
Comment 2.9 The truncation error is defined as the difference of the two terms in (6). But since (6) is an equality, you would obtain that E_1=0.
Response to comment 2.9: We have changed the "=" to " " for strictness. Eq.6 (now Eq.9) is derived from Eq.4 (now Eq.7), which is actually a reduced equation from Taylor's expansion with a approximate error. We added the detailed derivation in appendix A.3. Yes, the truncation error is the infinitesimality.
Comment 2.10 Supposedly you want to minimize the truncation errors, but it is not clear how this is done.
Response to comment 2.10: We used the derived error as the optimization objective. And in section 3.1, the cost function value (CFV) (Eq.25) is proposed to optimize the algorithm. By using NIO algorithms, we find the coefficient optimal that minimizes CFV, and thus we obtain the minimal . We explained the method of minimizing error in the last part of subsection 2.1 (line 109 to 113) and formulated the problem by Eq.13.
Reviewer 3 Report
Relevent topic, but the document needs some revisions to clarify its content and basic assumptions of the study.
- figure 1 should be further discussed, no definition is given for colours, etc
- a list of symbols would be beneficial for the reader
- all the abbreviations (also in abstract) must be defined in extended form
- section 3 and 3.1; the text introduces "numerical" analysis, but section 3.1 is for "experimental". Did the authors carried out both? Experimental for me has another meaning
- a weak aspect of the document is the lack of appropriate discussion to introduce the readers to the topic, to emphasize what is the novelty of this study and how the authors obtained these new findings. Please spend some efforts in this direction
Author Response
Comment 3.1 figure 1 should be further discussed, no definition is given for colours, etc.
Response to comment 3.1: According to the suggestion, we explained the colors in the introduction of Figure 1. Different colors in the model represent different media and the speed of wave propagation is related to the media.
Comment 3.2 a list of symbols would be beneficial for the reader.
Response to comment 3.2: According to the suggestion, we added a list of the symbols in Table 1.
Comment 3.3 all the abbreviations (also in abstract) must be defined in extended form.
Response to comment 3.3: We have given the extended form of all abbreviations where it first appears.
Comment 3.4 section 3 and 3.1; the text introduces "numerical" analysis, but section 3.1 is for "experimental". Did the authors carried out both? Experimental for me has another meaning.
Response to comment 3.4: We only performed the numerical analysis. To avoid confusion, we changed the word "experimental" to "simulation" in the revised manuscript.
Comment 3.5 a weak aspect of the document is the lack of appropriate discussion to introduce the readers to the topic, to emphasize what is the novelty of this study and how the authors obtained these new findings. Please spend some efforts in this direction.
Response to comment 3.5: The topic of this paper is using NIO algorithms to optimize finite difference method. Through the comparison of convergence and stability between algorithms, it can be observed that KH algorithm has good stability and outperforms other NIOs mentioned in this paper. The optimized coefficients is then applied to seismic wave numerical simulation, and is compared with the previous optimized window function. It is observed that NIO algorithms have a good suppression effect of spectrum, among which KH is the best. According to the suggestion, we have added discussion on the article in the Introduction (line 62 to 68), simulation (Section 3) and conclusion (Section 4), and highlighted our findings (line 194~198, 206~214, 224~227, 235~243).
Round 2
Reviewer 2 Report
In my previous review I said that the paper should not be published. In the new version some modifications and explanations have been included, but my impression is the same.
There are still grammatical errors. From a mathematical point of view there are many weaknesses.
What supports the fact that if a function f(x) is approximated in some way h(x), this implies that f'(x) can be approximated by h'(x)? This has been used in the paper, but it has not been demonstrated.
There are two sections about Convergence analysis and Stability analysis. In numerical analysis, these two concepts are of crucial importance. Nevertheless, those sections do not consider the theoretical developments, only some numerical realizations are presented. This is not acceptable in a paper submitted to a journal devoted to the analysis of algorithms.
You say that b_0=0. To support this it is said that "we can get that b_{-n}=-b_n", but this have not been proven even for n different from zero. How this would be applied for n=0 is still a mystery.
In my opinion the paper has not quality enough to be accepted in Algorithms.
Author Response
Reviewer 2:
We thank the review for the comments.
comment 1 There are still grammatical errors. From a mathematical point of view there are many weaknesses.
Response to comment 1: According to the suggestion, we have revised the English of the whole manuscript carefully and attempted to avoid any grammar or syntax error. In addition, we also have asked two colleagues who are skilled authors of English language papers to check the English. We hope that the language is now acceptable. We also try our best to improve the mathematical rigorousness.
comment 2 What supports the fact that if a function f(x) is approximated in some way h(x), this implies that f'(x) can be approximated by h'(x)? This has been used in the paper, but it has not been demonstrated.
Response to comment 2: Maybe the confusion arose because of some grammatical error. As a matter of fact, we did not make the approximation of f(x) and h(x). We just made the approximation of f'(x) and h'(x). There are two approximations in this manuscript. One is truncating the derivative of an infinite range to a finite interval using a window function (Formula 4 and 5). The other is the difference scheme of the derivative derived from Taylor's expansion (Eq. 7 and 8). The reason we measure the approximation of f'(x) and h'(x) is that the wave propagation equation is in differential form, and this equation is not easy to solve directly. So, we use the difference form instead, which is the finite difference method we use. Since the substitution brings numerical dispersion, we made a Fourier transform to convert it into the wave number domain (Eq. 9 and 10). To make the substitution clear, we add a sentence at line 108, "The FD method replaces the differential in the wave equation with difference, namely, the difference form of the derivative obtained above."
comment 3 There are two sections about Convergence analysis and Stability analysis. In numerical analysis, these two concepts are of crucial importance. Nevertheless, those sections do not consider the theoretical developments, only some numerical realizations are presented. This is not acceptable in a paper submitted to a journal devoted to the analysis of algorithms.
Response to comment 3: Thanks for this comment. Indeed, it is only numerical evidence. To avoid misleading readers, we change the title “Convergence analysis“ and ”Stability analysis” to "Coefficients convergence" and "Stability".
comment 4 You say that b_0=0. To support this it is said that "we can get that b_{-n}=-b_n", but this have not been proven even for n different from zero. How this would be applied for n=0 is still a mystery.
Response to comment 4:
We obtain the FD coefficients from the Taylor's expansion (Appendix A.2). The N-th order FD operator has the relation and , hence, . We have added the equation at line 100 and line 280.
Author Response File: 
Author Response.pdf
Reviewer 3 Report
The revised document is suitable for publication
Author Response
We thank the review for the comments.
Round 3
Reviewer 2 Report
In numerical analysis, the concepts of convergence and stability are of crucial importance. Nevertheless, the authors have not considered the corresponding theoretical developments. Only some numerical evidence in the words of authors is presented. This is not acceptable in a paper submitted to a journal devoted to algorithms and their analyses.
Author Response
comment 1: In numerical analysis, the concepts of convergence and stability are of crucial importance. Nevertheless, the authors have not considered the corresponding theoretical developments. Only some numerical evidence in the words of authors is presented. This is not acceptable in a paper submitted to a journal devoted to algorithms and their analyses..
Response to comment 1: We agree that convergence and stability are of crucial importance for an algorithm. However, it is very hard to perform convergence analysis for meta-heuristics such as the krill herd algorithm. As mentioned in the paper (https://doi.org/10.1007/978-1-4419-1306-7_6), “At the moment, convergence results for such algorithms seem to be yet unavailable.”
