A Geometric Calibration Method of Hydrophone Array Based on Maximum Likelihood Estimation with Sources in Near Field

Round 1

Reviewer 1 Report

Major problems:

In the paper there is not "Methods" paragraph. Some of the methods are described in 3rd paragraph, but without any details about e.g. optimization parameters, values rated later in results paragraph etc. In 3rd par. Authors described ML-GC, ML-GAC and MLM-GC, while no information is given about the reference method (EV-GC). 

It was shown, that MLM-GC is much better than EV-GC - another reference should be used that includes the multipath environment. 

Comparison between ML-GC and EV-GC is rather poor. The only described advantage of ML-GC is its better results for high SNR. Please note, that for high SNR, all methods give very small error, so in my opinion it is not so important. For small SNR, EV-GC can give better results. Maybe Authors can describe some another feature of ML-GC (calculation speed, complexity, ease of use, required number of elements, critical distances....).

In all manuscript Authors used visual inspection of the results (e.g. higher, more...). Please suggest some criterions for evaluating the results and give some values in comparing methods/results.

Minor problems:

between value and unit there should be space (1 m/s instead of 1m/s). Unit can not be written in italics (like 1 m/s). 

In graphs, unit is given after slash ("/") - in some cases it is problematic, because it can mean also division (Y-RMSE/d). Please change to comma or square brackets ([m]).

All other minor and major problems are marked in pdf comments in attached file. 

Comments for author File: Comments.pdf

Author Response

Response to Reviewer 1 Comments

Thank you very much for your kind comments, and they make me change our paper better.

Point 1: In the paper there is not "Methods" paragraph. Some of the methods are described in 3rd paragraph, but without any details about e.g. optimization parameters, values rated later in results paragraph etc. In 3rd par. Authors described ML-GC, ML-GAC and MLM-GC, while no information is given about the reference method (EV-GC).

 Response 1: Yes, the method is not sufficiently described. We change the title of the 3rd paragraph into “Methods”, and add some main derivation of our estimator.  The optimization parameters are given too. The reference method (EV-GC) is a kind of classical calibration method. In section 4.1.1, we add its basic principle where it first appeared. All the changes are highlighted in red in the paper, and it’s the same below.

Point 2: It was shown, that MLM-GC is much better than EV-GC - another reference should be used that includes the multipath environment.

Response 2: In section 4.1.3, we add EVM-GC to compare with MLM-GC. It’s an improved EV-GC for multipath environment. We achieve a simulation result that MLM-GC has a higher estimation accuracy than EVM-GC with coefficient error.

Point 3: Comparison between ML-GC and EV-GC is rather poor. The only described advantage of ML-GC is its better results for high SNR. Please note, that for high SNR, all methods give very small error, so in my opinion it is not so important. For small SNR, EV-GC can give better results. Maybe Authors can describe some another feature of ML-GC (calculation speed, complexity, ease of use, required number of elements, critical distances....).

Response 3: The accuracy of ML-GC is better than EV-GC in high SNR or more snapshots. On one hand, the SNR is usually high in the near-field calibration scene and at this time ML-GC is better than EV-GC in terms of accuracy. On the other hand, ML-GC is unbiased estimation, but EV-GC is not. Based on this, I change some sentences on Line 310. I regret to say, in terms of calculation speed, complexity, and ease of use, ML-GC is not more prominent than EV-GC. The required number of elements and critical distances of ML-GC and EV-GC are also the same. However, due to the performance, the suppression of multipath and the tolerance of multipath error, this method is still meaningful.

Point 4: In all manuscript Authors used visual inspection of the results (e.g. higher, more...). Please suggest some criterions for evaluating the results and give some values in comparing methods/results.

Response 4: We add some quantitative analysis and give some values in comparing methods/results, such as Line 266, 275, 319, 331, 347, 360, 385, 386, 402, and so on.

 Point 5: between value and unit there should be space (1 m/s instead of 1m/s). Unit can not be written in italics (like 1 m/s).

Response 5: We change all of them in our paper. Because this problem is involved in many places, the modification is not marked in red.

Point 6: In graphs, unit is given after slash ("/") - in some cases it is problematic, because it can mean also division (Y-RMSE/d). Please change to comma or square brackets ([m]).

Response 6: We change all to comma brackets in our paper.

Point 7: All other minor and major problems are marked in pdf comments in attached file.

Response 7: We have made detailed amendments according to the attached documents. The revised parts are marked with red in the paper. Some points need to be explained:

• Some figure numbers are missing, I don’t know why, but we add them again.
• We add the definitionof snapshot under formula (8).
• Figure 6shows the estimated error of element position, so Y axis is not amplitude.
• Under Figure 6, ‘when the orientations of the two auxiliary sources coincide, the position of the elements is unmeasurable, and the estimation error tends to infinity.’ In Figure 6, it’s only 0.2 to 0.3. It’s because that, when the orientations of the two auxiliary sources coincide, there is no solution, so we directly connect the points on both sides of the peak.
• In the paragraph under Figure 6, there is a comment ‘please give limits - maximum angle’. I’m sorry, we can’t give limits-maximum angle, that because the larger it is the better, but the auxiliary source should beaway from the axial direction of the line array. And the maximum angle is related to the array aperture.

Reviewer 2 Report

In the paper, a new geometric calibration method with near field sources is proposed. The method is based on the non-linear optimization problem solved by the DE. Numerical results demonstrate that the proposed approach is more accurate and versatile than its alternatives (e.g. the eigenvector method). The paper is well written and the obtained results are solid. My only comments are as follows:

1. Is it possible to have N = M in eq. (8)? and if so, how would the objective function be defined?
2. The results are compared with the EV-GC method in terms of accuracy, but I imagine that DE optimization would be significantly more expensive in terms of computational resources. Can you comment on the computational cost of the proposed approach?   
3. In the context of the given problem, does DE have any advantage over other optimization methods? 

Author Response

Response to Reviewer 2 Comments

Thank you very much for your kind comments, and they make me change our paper better.

Point 1: Is it possible to have N = M in eq. (8)? and if so, how would the objective function be defined?

 Response 1: Base on another expert’s comments, we detailed the derivation of our method. Eq. (8) changes to eq. (20). The ML-GC, ML-GAC and MLM-GC methods are not suitable for the condition of N=M, we add the condition to the variance as the limit in eq. (20), eq. (21) and eq. (33).

Point 2: The results are compared with the EV-GC method in terms of accuracy, but I imagine that DE optimization would be significantly more expensive in terms of computational resources. Can you comment on the computational cost of the proposed approach?

Response 2: Yes, our method uses DE to solve a non-linear problem, and its computational resources is more expensive. We add the computation analysis in Line 229 to Line 234, like this:

The computation of this method is mainly decided by the population size , the number of variables and times of iterations. Firstly, we calculate the data covariance matrix, and just do once, so it’s small amount of calculation. Its amount of calculation is dimension matrix multiplication. Secondly, population initialization, we randomly generate  dimension matrix, where is the length of . Variation, crossing and selection are all based on this dimension matrix to complete some simple logic and numerical operations. This process will repeat 500 times.

Point 3: In the context of the given problem, does DE have any advantage over other optimization methods?

Response 3: We give the advantages in Line 201 to 205, like this:

In the later simulation, we use the DE to solve the above optimization problem. Its characteristic is that it uses the local information of individual and the global information of the group to search together, and has strong universality. It can be directly applied to auxiliary calibration and self-calibration without changing the basic model of the algorithm. Its calculation process is simple, but the amount of computation of this kind of evolutionary algorithms is a little large.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

After correction, I have only one objection to the manuscript: please delete space between number and degree symbol. I asked before to put space between number and unit, but it does not apply to % and °