Three-Dimensional Limited-Memory BFGS Inversion of Magnetic Data Based on a Multiplicative Objective Function
Round 1
Reviewer 1 Report
Abstract start with statement of magnetic inversion method development stage and its efficiency. However it is not described what field it is applied in it is only mention in introduction that it is method to perform geophysical research. I suggest to put a statement about research field it to the abstract, so one could understand the context of the study.
Low efficiency of traditional magnetic three-dimensional inversion serves as a reason for current study. What is “efficiency” for current study should be described in the introduction. Is it accuracy or slow computation or something else. It might be a good idea to provide some numeric estimate of resources that method require. For now statements about suggested method: “requires less memory, performs only one quasi-forward simulation and is fast and efficient” seems to qualitative and does not provide sense of the problem. Converting qualitative statement into quantitative estimate might be a good idea.
Fix Figure 1 or its description to make notation in Figure and in text agree with each other. For example the is no “s (x, y, z)” in the Figure. Figure 1 attempts to illustrates formula (1) however it is so simple so it doesn’t contribute to understanding what is going on letter. Please, consider to put some additional information on the Figure 1 so it could be a proper illustration.
Provide a reference that contains equation (1) derivation or provide derivation in the manuscript. Equation (1) contains ξ (Xi) and η (Eta) however they are not described in the text.
Transform between (3) and (4) is not clear from the manuscript.
Author should rework method description. For example, the form equation (1) appears in the manuscript doesn’t allow to appreciate the how it is derved. Many point become much clear if spherical coordinates systems, as it is done [Zhang, L., Lu, G., Zhu, Z., & Cao, S. (2022). An improved 3d magnetization inversion based on smoothness constraints in spherical coordinates. Magnetochemistry, 8(11), 157.].
From my point of view there several key points in the manuscript: forward problem; inversion problem statement and method inversion problem solution. Authors should make clear division between these points and describe those separately and describe approaches for every point (for example [Utsugi, M. (2019). 3-D inversion of magnetic data based on the L1–L2 norm regularization. Earth, Planets and Space, 71, 1-19. ] and [Li, Yaoguo.; D. W. Oldenburg. Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method. J. Geophys. 2003, 152, 251-265] use different method to solve inversion problem and I think there are other method to approach it). I suggest to add subsection devoted to previous works and significantly increase number of references.
Authors propose new method eq (8-10) however it is not clear why equation (8) should look like it provided in the manuscript. Is it due to design of L-BFGS approach. Please provide reasoning to choose and make more descriptive since just putting equation does not allow a reader to appreciate method.
Check wording, below ONLY some issues that I have noted.
Line 113:
Consider use different notation for model vector, since “m” is already used for direction cosine.
Line 132:
“m is the permeability κ”
What is the reason to use m instead of conventional μ (Mu) as it is done in equation (1)?
What is κ?
Line 153:
“In addition, δ in Equation (8) is a constant.”
It is appear in equation (10).
Line 175:
What is “ethe”?
Line 214:
“the inclination and declination are 90° and 0°”
Is a word missing?
Lines: 412-413:
The reference is not used in the text.
Author Response
Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted/in track changes in the re-submitted files.
Point-by-point response to Comments and Suggestions for Authors.
Comments 3: [Fix Figure 1 or its description to make notation in Figure and in text agree with each other. For example the is no “s (x, y, z)” in the Figure. Figure 1 attempts to illustrates formula (1) however it is so simple so it doesn’t contribute to understanding what is going on letter. Please, consider to put some additional information on the Figure 1 so it could be a proper illustration] |
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Response 3: Thank you for pointing this out. Regarding Figure 1, I have corrected the inconsistencies. Because it is only a forward element, if too many factors are designed, the formula of this part will not be clearly expressed. |
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Comments 4: [Provide a reference that contains equation (1) derivation or provide derivation in the manuscript. Equation (1) contains ξ (Xi) and η (Eta) however they are not described in the text.] |
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Response 4: Thank you for pointing this out. The derivation of this part of the formula refers to the following literature: Guan, Zhining. Geomagnetic field and magnetic exploration. M. Geological Publishing House,2005,119-121. I have added it to the references. |
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Comments 5: [Transform between (3) and (4) is not clear from the manuscript.] |
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Response 5: Thank you for pointing this out. Formula 3 is the derivation of formula 4, the sum of multiple units is derived forward to formula 3, and the formula mapped to the entire inversion is formula 4. I will include the necessary notes in this section of the article. |
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Comments 6: [Author should rework method description. For example, the form equation (1) appears in the manuscript doesn’t allow to appreciate the how it is derved. Many point become much clear if spherical coordinates systems, as it is done [Zhang, L., Lu, G., Zhu, Z., & Cao, S. (2022). An improved 3d magnetization inversion based on smoothness constraints in spherical coordinates. Magnetochemistry, 8(11), 157.].] |
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Response 6: Thank you for pointing this out. I'll give you a reference for equation 1. |
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Comments 7: [From my point of view there several key points in the manuscript: forward problem; inversion problem statement and method inversion problem solution. Authors should make clear division between these points and describe those separately and describe approaches for every point (for example [Utsugi, M. (2019). 3-D inversion of magnetic data based on the L1–L2 norm regularization. Earth, Planets and Space, 71, 1-19. ] and [Li, Yaoguo.; D. W. Oldenburg. Fast inversion of large-scale magnetic data using wavelet transforms and a logarithmic barrier method. J. Geophys. 2003, 152, 251-265] use different method to solve inversion problem and I think there are other method to approach it). I suggest to add subsection devoted to previous works and significantly increase number of references] |
Response 7: Thank you for pointing this out. I will try to be as clear as possible about the descriptions of forward and inversion problems. I have increased the number of references.
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Comments 8: [Authors propose new method eq (8-10) however it is not clear why equation (8) should look like it provided in the manuscript. Is it due to design of L-BFGS approach. Please provide reasoning to choose and make more descriptive since just putting equation does not allow a reader to appreciate method] |
Response 8: Thank you for pointing this out. I will provide new references here. The establishment of the objective function has nothing to do with L-bfgs, and has nothing to do with inversion iteration, and has nothing to do with inversion method. References are Xie Haijun, Tan Handong, Ma Fengqun. Three-dimensional resistivity inversion based on product type objective function [J]. Geoscience,2023,37(01):67-73.
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Comments 9: [Consider use different notation for model vector, since “m” is already used for direction cosine] |
Response 9: I agree with you. I'm going to use something else here to represent the direction cosine.
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Comments 10: [“m is the permeability κ” What is the reason to use m instead of conventional μ (Mu) as it is done in equation (1)? What is κ?] |
Response 10: Thank you for pointing this out. In the manuscript, u represents the weight of the logarithmic obstacle term, k represents the magnetic susceptibility, and m represents the model vector for the inversion of the objective function.
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Comments 11: [ “In addition, δ in Equation (8) is a constant.” It is appear in equation (10).] |
Response 11: Thank you for pointing this out. This was an oversight on my part and I have corrected the problem.
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Comments 12: [What is “ethe”?] |
Response 12: Thank you for pointing this out. I have amended the original text to read’ I have amended the original text to read’.
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Comments 13: [“the inclination and declination are 90° and 0°”Is a word missing?] |
Response 13: Thank you for pointing this out. There are no missing words here. They represent each other.
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Comments 14: [The reference is not used in the text.] |
Response 14: Thank you for pointing this out. This is my mistake, I have fixed the problem
Reviewer 2 Report
Manuscript:
Three-Dimensional Limited-Memory BFGS Inversion of Magnetic Data Based on a Multiplicative Objective Function
submitted by:
Shuaishuai Liu, Handong Tan, Miao Peng, and Yanxing Li
Magnetic exploration is a kind of geophysical exploration method that uses the magnetic field changes (magnetic anomalies) caused by the magnetic differences between various rocks in the crust to find useful mineral resources and study the underground structure. In reviewed paper, based on the limited-memory BFGS method, the three-dimensional magnetic inversion of a multiplicative objective function was realized. The results demonstrate that the limited-memory BFGS method significantly enhances the inversion efficiency and yields superior inversion outcomes compared to traditional magnetic three-dimensional inversion methods.
The structure of the manuscript is well-organized and clear. In the introduction, the background and comprehensive review of the problem's literature were presented. The Authors describe Three Dimensional Forward Modelling of the Magnetic Method and Magnetic Three-Dimensional Inversion mathematically. Experiments and results of the research are presented in graphic forms. Conclusions, on the basis of the research, are comprehensive and clear.
I have no significant substantive comments. Specific suggestions are following:
1. The number of references needs to be increased there are only 15 references
2. Reference [4] was lost in the text
3. "x" is used for vector multiplication. I suggest to write 12.5×108 rather than 12.5 × 108
4. The coordinate system, used for presentation figure 12 should be described
Author Response
Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted/in track changes in the re-submitted files.
Point-by-point response to Comments and Suggestions for Authors.
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Comments 1: [The number of references needs to be increased there are only 15 references.] |
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Response 1: Thank you for pointing this out. I have increased the number of references. |
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Comments 2: [Reference [4] was lost in the text] |
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Response 2: Thank you for pointing this out. I have fixed the error. The correct result can be found on line 37. Comments 3: ["x" is used for vector multiplication. I suggest to write 12.5×108 rather than 12.5 × 108] Response 3: I agree with you. I have fixed this type error on lines 230 and 291. Comments 4: [The coordinate system, used for presentation figure 12 should be described] Response 4: Thank you for pointing this out. Figure 12 uses the Chinese geodetic coordinate system. I have explained this in the article. This description can be found on line 346. |
Reviewer 3 Report
Dear Authors,
I think that the technically well-written manuscript could be suitable for possible publication in the journal. Therefore, I suggest some clarifications in the revision that will benefit the readers of the journal.
Best regards.
Comments
1. You should express all the parameters given in Equation 1.
2. Please explain why the iteration numbers (i.e., 3-20) change to approximate the Hessian matrix. Also, how do you obtain the Jacobian matrix?
3. How do you determine the initial model to invert the data?
4. Are there criteria for determining the initial step size before you perform a linear search to find the relatively best iteration step size?
5. Please provide a clearer label for each plot in Figure 4.
6. I am interested in how you get RMS values that are much lower than the amount of noise added to the synthetic data. Is there no unit for the RMS values?
7. I think you should carefully consider the following sentence because it does not mean that the optimization problem is linear.
“The forward model studied in this paper does not consider remanence and demagnetization, so the forward model is treated as a linear problem.”
Author Response
Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted/in track changes in the re-submitted file.
Point-by-point response to Comments and Suggestions for Authors.
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Comments 1: [You should express all the parameters given in Equation 1.] |
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Response 1: Thank you for pointing this out. I agree with this comment. Therefore, I have supplemented all the data in Equation 1. |
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Comments 2: [Please explain why the iteration numbers (i.e., 3-20) change to approximate the Hessian matrix. Also, how do you obtain the Jacobian matrix?] |
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Response 2: Thank you for pointing this out. L-BFGS uses the objective function gradient of the first (3-20) iterations to approximate the Hessian matrix will be more productive. Because the magnetic method studied in this paper is an analytical solution problem, the Jacobian matrix can be solved directly. |
Comments 3: [How do you determine the initial model to invert the data?]
Response 3: Thank you for pointing this out. I set the initial model of the inversion to the same value as the surrounding rock. I have included that part in the manuscript. This section can be found on lines 233 and 234 of the manuscript.
Comments 4: [Are there criteria for determining the initial step size before you perform a linear search to find the relatively best iteration step size?]
Response 4: Thank you for pointing this out. I have added to this section in the article that the initial iteration step size has selection criteria. Formulas 19 and 20 in the manuscript describe this part.
Comments 5: [Please provide a clearer label for each plot in Figure 4]
Response 5: Thank you for pointing this out. This is my oversight, I have modified Figure 4.
Comments 6: [ I am interested in how you get RMS values that are much lower than the amount of noise added to the synthetic data. Is there no unit for the RMS values?]
Response 6: Thank you for pointing this out. I will answer you this question: when the result of the inversion is exactly the same as the true value, the data fit difference rms is 1, and the last iteration will continue to reduce the objective function, so the value of Rms will be less than 1. Because Rms is a ratio, it has no units.
Comments 7: [I think you should carefully consider the following sentence because it does not mean that the optimization problem is linear “The forward model studied in this paper does not consider remanence and demagnetization, so the forward model is treated as a linear problem.”]
Response 7: I agree with your comment. I have modified this to ‘’The forward model studied in this paper does not consider remanence and demagnetization, so forward inference can therefore be regarded as an analytic solution problem.‘‘This section can be found on line 78 of the manuscript.
Round 2
Reviewer 3 Report
Dear Authors,
According to the revised version of the manuscript, you have submitted a satisfactory revision based on my suggestions. However, it would be nice if you would refer to previously published 3D inversion work on this topic in the introductory part of the paper. I congratulate you on your efforts to improve the original version of the paper.
Best regards.
Author Response
Thank you for taking the time to read my manuscript again. The following is my revision to your new review comments.
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Point-by-point response to Comments and Suggestions for Authors |
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Comments 1: [According to the revised version of the manuscript, you have submitted a satisfactory revision based on my suggestions. However, it would be nice if you would refer to previously published 3D inversion work on this topic in the introductory part of the paper. I congratulate you on your efforts to improve the original version of the paper.] |
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Response 1: Thank you for pointing this out. I agree with your comment. Therefore, I have added examples about the application of multiplicative objective function and the limited-memory BFGS (L-BFGS) method in three-dimensional inversion, and the application of multiplicative objective function in three-dimensional magnetic inversion is not yet available. This section can be found on lines 59-61 of the manuscript |
