Least-Squares Reverse Time Migration of Primary and Internal Multiple Predicted by the High-Order Born Modeling Method
Aifei Bian
Round 1
Reviewer 1 Report
General Comments:
Is the paper new, technically correct, and relevant?
Yes, the paper is new and technically sounds. Results somehow does support the methodology, but needed to be more cleared by the author in case of properties of the data.
Is the paper well organized?
The paper is properly organized, good literature review, suitable motivation and clear explanation on results are positive points to that.
Is the abstract concise?
Yes, but I think it needs to be rephrased after revision to add some comments about any artifacts or negative points in the method, if exist.
Is the introduction motivating?
Yes, Introduction section is motivating.
Are the methodology, results, and conclusions completely developed?
No, they need to be modified and developed according to the technical comments.
Are there language, mathematics, reference, or style errors? There is no mathematical, reference or style error.
Technical Comments:
Are the codes available for this research? As I found, there is no code available for this study, e. g. in Github. If the authors could make the codes available, the manuscript could be much better evaluated, not only for reviewers, but also for possible readers. When it is not possible to upload the code for public access, such as in Github, could they be provided for reviewer for better assessment of the study?
The study is comprehensive and requires large time to be read carefully and being reviewed. The theoretical background has been well explained in details, and the experiments and related models are presented. But a simple flowchart that shows the trend of the experiments and analysis is missing. It could be added at the end of section 1, as figure 1.
The result comparison parts are well organized and presented. The display way is good. But qualitative and quantitative evaluation is somehow too much that one can get lost in that. I think it would be better that you add more explanation to that.
How did you evaluate the final result? How did you consider to finally selection a methodology for the most complicate problem, I mean data with various types of multiples, like as peg-leg, long path or WBM?
What about when the models are more geometrically complex and there is strong multiple energy in data?
The introduction section is a nice one. It is architected very beautifully, while written fully academic and comprehend. I assume that any change in the introduction section is not necessary, but one of the important tasks after publishing a study is to increase its chance to be seen by the most possible number of researchers, so I would like to give two recommendations. First, to get your published study in the list of searched for papers based on keywords, I propose to increase variety of your keywords. In my viewpoint, they do not cover the whole topic of the study and are not widely searched words. I propose to add at least the keyword “seismic data analysis”. Second, one of the methods in the publisher’s website that brings a publication on to the researchers, is based on the similar publications that they have read before. So, the more you cite similar publication, the more the chance that the search engine in the publisher website propose your paper to the researcher. Besides of that, it will also complete your introduction section. As another advantage, it rises new ideas to the researchers by combining various methods, or resolving drawback of one seen paper by reading the similar one, or extending the methodology to a fully automatic one. So, based on these points, I would like to ask to cite to the following similar publication in the manuscript which used PCA and feature selection for deep learning, but in different field of study. The first proposed publication is: Moradpouri, F. Morad-Zadeh, A., Pestana, R.C., and Soleimani, M., (2016), Seismic reverse time migration using a new wave field extrapolator and a new imaging condition. Acta Geophysica, 64(5), 1673-1690. The second publication for citation is: Moradpouri, F. Moradzadeh, A., Pestana, R.N.C., Ghaedrahmati, R., and Soleimani, M., (2017), An improvement in wave-field extrapolation and imaging condition to suppress RTM artifacts, Geophysics, 82(6), S403-S409., and the other publication could be: Soleimani, M. (2017), Challenges of seismic imaging in complex media around Iran, from Zagros overthrust in the southwest to Gorgan Plain in the northeast. The Leading Edge, 36(6), 499–506.
The abstract focusses mainly on the general problem and ignores the other items of the abstract such as the methodology, good introduction, results and conclusion.
The authors should explain what limitations did they find out about the proposed method.
Best Regard
Comments for author File: 
Comments.pdf
Author Response
Dear Editors and Reviewers:
Our manuscript entitled “Least-squares reverse time migration of primary and internal multiple predicted by the high-order Born modeling method” has been resubmitted to the Applied Sciences. The original version of this paper was reviewed by three experts and encouraged to revise in terms of the comments. We have carefully revised this manuscript according to the comments and suggestions of experts. The revised mode in Microsoft Word is provided for the comparison of the modifications, and the detailed point-by-point responses to issues are listed in the following part.
We would like to express our sincere appreciation to the editors and the reviewers for their hard work. Looking forward to hearing from you soon.
Kind regards,
Ruiding Chen
Summary of the revisions
Thanks for the helpful suggestions from the reviewers and editors. We have revised our manuscript carefully according to the comments of four reviewers. We have labeled every modification in our revised manuscript by using the revised mode in the Word format. The item-by-item responses to the issues are as follows:
Reviewer 1
Point 1、Are the codes available for this research? As I found, there is no code available for this study, e. g. in Github. If the authors could make the codes available, the manuscript could be much better evaluated, not only for reviewers, but also for possible readers. When it is not possible to upload the code for public access, such as in Github, could they be provided for reviewer for better assessment of the study?
Response 1 Thanks for your advices. It’s a pity that the code is not available because of our rules of laboratory. Our laboratory maybe upload our code for public access in the future. I’m sorry for that..
Point 2、The study is comprehensive and requires large time to be read carefully and being reviewed. The theoretical background has been well explained in details, and the experiments and related models are presented. But a simple flowchart that shows the trend of the experiments and analysis is missing. It could be added at the end of section 1, as figure 1.
Response 2 Thanks for your advice, I have add a table for that in line 250-251:
“Table 1. the main content and conclusion of the numerical test
|
The model |
Three-layer Model |
Two-dimensional Salt Hill Model |
|
The data and multiple of single shot |
1. the 30th shot; 2. the internal multiple can be predicted by Born modeling |
1. The 35th shot; 2. the internal multiple can be predicted by Born modeling |
|
Compare of LSRTM with or without internal multiple |
1. The artifacts can be eliminated by the LSRTM with or without internal multiple |
1. The artifacts can be eliminated by the LSRTM with or without internal multiple; 2. the useful structure is damaged in LSRTM without internal multiple while it is compensated in LSRTM with internal multiple |
”
Point 3、The result comparison parts are well organized and presented. The display way is good. But qualitative and quantitative evaluation is somehow too much that one can get lost in that. I think it would be better that you add more explanation to that.
Response 3 Thanks for your advice. We have add some more explantation:
Line 277-286
“Here we Choose the data of the 30th shot. Figure 4a shows the shot data of the model. Figure 4b shows the internal multiples predicted by high-order Born modeling of the reflectivity model after five iterations. Figure 4c shows the data residual, Figure 4d shows the predicted data residual of the primary after using the weighting matrix of high-order Born modeling after five iterations. Figure 4e,f shows the data residual of the primary and internal multiple after using the weighting matrix of high-order Born modeling after five iterations, respectively. Due to the amplitude of internal multiple is too small, the boundary reflection is still exist in the shown data. The internal multiple is about 1.5s when the x-axis is 0km. And we can find the internal multiple in Figure 4a,b,c,d,f, shows that it have been estimated, and is separated from the data residual.
”
line 298-305
“Figure 5 shows the results of the LSRTM and RWLSRTM images of the primary and the RWLSRTM image of the primary and internal multiples. As seen in Figure 5a, the artifacts are at a depth of about 1.4 km, and they are suppressed in Figure 5b,c. Figure 5c shows the results of the RWLSRTM image of the primary and internal multiples. Comparing Figure 5b and 5c, there are no other artifacts introduced. Figure 5d,e is the compare of 150th trace and its extract of the depth of 1.25km and 1.875km.We can see the artifacts at a depth of about 1.4km is only exist in the LSRTM and is eliminated when internal multiple is separated.
”
Line 318-329
“Figure 6a are the velocity model and Figure 6b is the reflectivity model as the compare of the LSRTM image. Figure 7a shows the common shot gather. Figure 7b shows the predicted internal multiples of high-order Born modeling of the reflectivity model after five iterations. Figure 7c shows the data residual of the primary after using the weighting matrix of high-order Born modeling after five iterations. Figure 7d shows the data residual of the internal multiples after using the weighting matrix of the high-order Born modeling after five iterations. And we can find the internal multiple in Figure 4a,b,d, shows that it have been estimated, and is separated from the data residual. Figure 4e is waveform comparison of data residual and data residual of primary and internal multiple. Through comparison, we can find the multiple exist in about 0.8s, the energy of multiple is low when the energy of primary is strong, and the energy of multiple is strong when the energy of primary is weak. So we can say that the primary and multiple separated in data residual.
”
Line 374-380
“We choose the 290th trace and 320th trace to compare their image, as figures 9 a and b show. In figure 9a, the useful structure is about in the 1.6km. Only the LSRTM without the internal multiple didn’t image here, shows that the internal multiples contain some information of useful structure and can be a good complement in image. In figure 9b, the artifacts is about in 1.4km. Only the LSRTM contains the artifacts, shows that separating the internal multiples can reduce the artifacts of internal multiple.
”
Point 4、How did you evaluate the final result? How did you consider to finally selection a methodology for the most complicate problem, I mean data with various types of multiples, like as peg-leg, long path or WBM?
Response 4 Thanks for your advice. The internal multiple we discuss is divided with the difference of reflection interface. This method can predict the internal multiple that satisfied these condition: (1) the reflection interface is all underground. (2) the reflection happens three times (3) the reflection interface satisfied the second reflection in swallower than another two reflection interface. (4) reflection interface is not too close. We didn’t considered the impact multiple have differences in propagation paths. That’s may be next step of our research. Thanks for your advice.
Point 5、What about when the models are more geometrically complex and there is strong multiple energy in data?
Response 5 Thanks for your advice. We have no enough time and better computer to compute the model that is more complex. I think it will have effect for it is also have a strong multiple energy in data of the two-dimensional Salt Hill model.
Point 6、The introduction section is a nice one. It is architected very beautifully, while written fully academic and comprehend. I assume that any change in the introduction section is not necessary, but one of the important tasks after publishing a study is to increase its chance to be seen by the most possible number of researchers, so I would like to give two recommendations. First, to get your published study in the list of searched for papers based on keywords, I propose to increase variety of your keywords. In my viewpoint, they do not cover the whole topic of the study and are not widely searched words. I propose to add at least the keyword “seismic data analysis”. Second, one of the methods in the publisher’s website that brings a publication on to the researchers, is based on the similar publications that they have read before. So, the more you cite similar publication, the more the chance that the search engine in the publisher website propose your paper to the researcher. Besides of that, it will also complete your introduction section. As another advantage, it rises new ideas to the researchers by combining various methods, or resolving drawback of one seen paper by reading the similar one, or extending the methodology to a fully automatic one. So, based on these points, I would like to ask to cite to the following similar publication in the manuscript which used PCA and feature selection for deep learning, but in different field of study. The first proposed publication is: Moradpouri, F. Morad-Zadeh, A., Pestana, R.C., and Soleimani, M., (2016), Seismic reverse time migration using a new wave field extrapolator and a new imaging condition. Acta Geophysica, 64(5), 1673-1690. The second publication for citation is: Moradpouri, F. Moradzadeh, A., Pestana, R.N.C., Ghaedrahmati, R., and Soleimani, M., (2017), An improvement in wave-field extrapolation and imaging condition to suppress RTM artifacts, Geophysics, 82(6), S403-S409., and the other publication could be: Soleimani, M. (2017), Challenges of seismic imaging in complex media around Iran, from Zagros overthrust in the southwest to Gorgan Plain in the northeast. The Leading Edge, 36(6), 499–506.
Response 6 Thanks for your advice. I have added the keyword of “seismic data analysis”. And I have added these papers and the serial number is 29-32.
Line 60
‘Reverse time migration is proposed by Whitmore and developed a lot [28-32].’
Point 7、The abstract focusses mainly on the general problem and ignores the other items of the abstract such as the methodology, good introduction, results and conclusion.
Response 7 Thanks for your advice. I have change the abstract add the propose, results and coclusion .
Line 7-28
“Multiples contain information about underground structures; however, they cause artifacts in imaging. In imaging, if the internal multiple is not been eliminated, it can cause the artifacts. However, if the internal multiple is been eliminated, the information contained by the internal multiple could be eliminated and degrade the image quality of the some useful structure. If the multiples and the primary can be separated from the recorded seismic data for imaging, the information contained by the multiples can be used and the artifacts can be attenuated. Here we developed a method to separate the primary and internal multiples and use them in least squares reverse time migration ( LSRTM). This method first separates the primary and the internal multiples in the data residual and predicts the wavefield of the primary and internal multiples in a forward-propagation wavefield. The internal multiples are predicted by a method based on the high-order Born modeling method, which predicts the internal multiples in a forwarding process in the time domain. In the internal multiple prediction process, we get the wavefield of the primary and internal multiples in the forward-propagation wavefield. Then, by introducing the weighting matrix and using the primary and the internal multiples in the data residual and wavefield of the primary and internal multiples in the forward-propagation wavefield, we established the objective function for imaging of the primary and the internal multiples separately. In this method, the multiple prediction process provided the internal multiples to suppress the artifacts, and LSRTM constructed the model for the multiple prediction process. Finally, we performed numerical tests using synthetic data, and the results indicated that the LSRTM without the internal multiple can suppress the artifacts of internal multiples but some useful structures below the salt dome may be suppressed in the same time, and LSRTM with primary and internal multiple can suppress the artifact of internal multiples and the useful construction below the salt dome is compensated. ”
Point 8、The authors should explain what limitations did they find out about the proposed method.
Response 8 Thanks for your advice. We have discuss it in the conclusion, numeric test and appendix.
Line 397-402
“The method still has disadvantages. The overlap of the artifacts and the actual structure was also suppressed. This caused the actual structure to have some discontinuity points. Second, owing to the property of internal multiple prediction using high-order Born modeling, the internal multiples caused by two close layers may not be well predicted. Therefore, the resolution of the prediction of the small structure internal multiples needs to be improved.
”
Line461-462
“This can separated the overlap better but the internal multiple that the reflection interface is close couldn’t be predicted.
”
Line 355-357
‘Because the artifacts were suppressed, the overlap between the actual structure and the artifact was also suppressed, which made the actual structure discontinuous, as indicated in the bottom half by the letter A.
“
Best regards.
Author Response File: Author Response.pdf
Reviewer 2 Report
Comments in attachment.
Comments for author File: 
Comments.doc
Author Response
Dear Editors and Reviewers:
Our manuscript entitled “Least-squares reverse time migration of primary and internal multiple predicted by the high-order Born modeling method” has been resubmitted to the Applied Sciences. The original version of this paper was reviewed by three experts and encouraged to revise in terms of the comments. We have carefully revised this manuscript according to the comments and suggestions of experts. The revised mode in Microsoft Word is provided for the comparison of the modifications, and the detailed point-by-point responses to issues are listed in the following part.
We would like to express our sincere appreciation to the editors and the reviewers for their hard work. Looking forward to hearing from you soon.
Kind regards,
Ruiding Chen
Summary of the revisions
Thanks for the helpful suggestions from the reviewers and editors. We have revised our manuscript carefully according to the comments of four reviewers. We have labeled every modification in our revised manuscript by using the revised mode in the Word format. The item-by-item responses to the issues are as follows:
Point 1. In the section of Abstract:
In the manuscript, only one sentence is used to describe the result. It is too easy.
Response 1. Thanks for your advice. The Abstract have been changed, here is the latest version:
Line 24-28
“Finally, we performed numerical tests using synthetic data, and the results indicated that the LSRTM without the internal multiple can suppress the artifacts of internal multiples but some useful structures below the salt dome may be suppressed in the same time, and LSRTM with primary and internal multiple can suppress the artifact of internal multiples and the useful construction below the salt dome is compensated. “
Point 2. Key words: “least squares” ---> “least-squares”
Response 2. Thanks for your advice. The Key words have been changed from “least squares” to “least-squares”
Point 3. In the section of Introduction:
Conclude and discuss the existing methods instead of just list them.
Response 3. Thanks for your advice. I have discuss the existing methods, now the abstract becomes:
Line 45-59
“IME methods predict internal multiples by convolving and correlating the recorded data at the surface. This method isn’t rely on the model and predicts internal multiples layer-by-layer. The ISS is derived from the Lippmann–Schwinger equation, and its third term can be used to predict the internal multiples in the wavenumber domain. The ISS can predict the multiple precisely but need do lots of Fourier transforms which make computational cost is high. In addition, Marchenko multiple elimination, another data-driven method, was recently proposed, which uses the Marchenko equation to estimate the internal multiples and image [23]. The Marchenko multiple elimination can estimate the surface multiple and internal multiple, and use them in imaging at the same time, but it needs to be improved when face in complex structure. There are two main model-driven methods: one method was published by Pica, which predicts the internal multiples by using several extrapolations of the data [24], and the other method is full wave migration (FWM) [25-27], which estimates the internal multiples in the forwarding process and uses the primary, surface multiples, and internal multiples to image by one-way wave equation.”
Point 4. In the section of Method:
I suggest a calculation flow instead of just given simple steps.
Response 4. Thanks for your advice. I have add the flow as figure 2.
Line 243-246
Thus, we can conduct the LSRTM process using the weighting matrix to realize the internal multiples as Figure 3 shows.
Figure 2. Flow chart of the LSRTM image of primary and internal multiples
Point 5. In the section of Internal multiple prediction using high-order Born modeling:
I suggest that you’d better transfer the derivation to the Appendix.
Response 5. Thanks for your advice, I have transfer the derivation to the Appendix A.
Line 412-462
“Appendix A
The scattering wavefield can be written as the series:
. (A.1)
is the portion of that is ith order. And we can write equation (10) as:
. (A.2)
Use as an approximate replacement of . substitute into Equation (8) and substitute Equation (8) into Equation (9), and we can have:
. (A.3)
So we can get:
. (A.4)
Do approximate replacement and substitute again, we can have:
. (A.5)
The represents the wavefield experienced scattering for three times. The first-order internal multiple is also the wavefield experienced scattering for three times and contained in the . The scatter point of have the depth condition of , where , , is the depth of first, second, third scatter point, indicates that the propagation of the first-order internal multiple is down-up-down, so Equation (13) need to be separate the wave field into up- and downgoing components:
. (A.6)
The subscripts and represent the down- and upgoing components. Subscripts 1 and 2 indicate that the iteration ordinals are 1st and 2nd when computing scatter wavefields.
Figure A1. Components of the wave field caused by virtual sources
In Figure A1 is the wavefield, the subscripts , , and indicate that the direction of propagation is downward, upward, and rightward, and is generated by according to Equation (15). Through Figure 2 we see that the existence of causes the virtual source of to exist in . In other words, there is overlap of and and the depth condition can only reach . To solve this problem, we need not only to split the up- and downgoing wavefields but also to introduce the zeroing matrix to remove the overlap of and . The internal multiples are then changed as follows:
. (A.7)
The zeroing matrix is expressed as:
. (A.8)
where is a threshold associated with . In the overlap of and , the wave after the ith reflection contributes most of the energy, and the other is the wavefield after a reflection greater than i. Therefore, the threshold setting should separate the wavefield with the stronger energy and the others, while , which we need, cannot be removed. Thus, we set:
. (A.9)
where is the threshold parameter. Equation (18) indicates that a threshold is associated with the max of and is then multiplied by the reflectivity model. If the value of is set too high, the overlap may not removed clearly; if the value of is set too low, the wavefield we need maybe removed. So, we need to set a suitable value. Here we can set a relatively higher value, and about one-quarter to half a cycle before and after the wavefield part that meets the threshold is reset to zero. Thus, we get:
. (A.10)
This can separated the overlap better but the internal multiple that the reflection interface is close couldn’t be predicted.
”
Point 6. In the section of Numerical tests:
Please increase the font size of the coordinate axis in Figures 3-8.
Response 6. Thanks for your advice, I have increase the font size in Figures 3-8.
Point 7. In the first example of Numerical tests:
The imaging result of LSRTM (Figure 5a) is poor: a. much low-frequency noise; b. low resolution; c. unbalanced amplitude. In conventional LSRTM, the result should be better than yours. In the imaging result from RWLSRTM of primary (Figure 5b) and internal multiples (Figure 5c), where have they improved? You should give more details, such as residual curve, demigration shot, partial enlarged drawing, special instructions, etc.
In the internal multiples predicted by high-order Born modeling of the reflectivity model after 5 iterations, there are some noise surround the predicted internal multiple reflection. Please give an explanation.
Response 7. Thanks for your advice. I guess that reason of these flaws of my imaging result of LSRTM is we just use Laplacian filter to eliminate the low-frequency noise and the iteration number is few. We do the test that if increase the iteration to the 10 iteration, the amplitude and noise can be improved a little. So I guess that it may be better if we use more time to calculate much more iteration. However, limitation of our computer makes it hard to do these test.
We give a single trace compare to show the improvement of Figure 5b and Figure 5c.
The amplitude of internal multiple of three-layer model is small. We give 100 layers of PML, the boundary reflection still have a similar amplitude of internal multiple. But the three-layer model is simple, so I think it can be good for analysis of our method.
Line 291-297
Line 283-286
“Due to the amplitude of internal multiple is too small, the boundary reflection is still exist in the shown data. The internal multiple is about 1.5s when the x-axis is 0km. And we can find the internal multiple in Figure 4a,b,c,d,f, shows that it have been estimated, and is separated from the data residual.”
Point 8. In the reflectivity shown in Figure 3b and 6b, what is the use of the reflectivity, please clarify it.
Response 8.Thanks for your advice. We didn’t use the reflectivity in our method, we just give it as a stand answer of the image. We show them to compare with the LSRTM image. We have add these words to mark their function.
Line 253-254
“Figure 3a illustrates the velocity model and Figure 3b illustrates the reflectivity model as the compare of the LSRTM image”
Line 318-319
“Figure 6a are the velocity model and Figure 6b is the reflectivity model as the compare of the LSRTM image.”
Point 9. Figure 7. I suggest you supply a waveform comparison.
Response 9. Thanks for your advice. I have add the waveform comparison.
Line 339-345
Figure 7. (a) Shot data of the reference model. (b) The internal multiples predicted by high-order Born modeling of the reflectivity model after 5 iterations. (c) The data residual of the primary after using the weighting matrix of high-order Born modeling after 5 iterations. (d) The data residual of the internal multiple after using the weighting matrix of high-order Born modeling after 5 iterations. (e) waveform comparison of data residual and data residual of primary and internal multiple.
’
Point 10. A field data example should be supplied to support your method.
Response 10. Thanks for your advice. Because the computation cost of the field data is high, and compute of our laboratory can’t afford it. So that’s a pity that we can not use the field data. As long as the conditions are met in the future, we will do field data experiments.
Point 10. English writing needs to be polished by professional editors or native speakers.
Response 11. Thanks for your advice. I have use the English edit.
Best regards.
Author Response File: Author Response.pdf
Reviewer 3 Report
Overall a good example of joint RTM with primary and first order multiples. Major problems includes:
1. problem on structure of manuscript. A discussion part is needed to explain the difference of this research with previous research, importance and limitations of proposed method. Methods in section 2 should be re-arranged in a more logic way for reads to understand. For example first give brief introduction of RTM and LSRTM in general. Technical writing may also be improved.
2. Compared with previous researches, especially your own research in Acta Geophysica (2022), what is the main point and new contribution of this study, what is the major problem solved? Much similarity between theoretic and experiment part of this manuscript and previous one in Acta Geophysica (2022). It is better to rewrite the method part with a more concise way with proper organization and reference of major equation used.
3. Is the comparison between ISS and high-order Born modeling necessary in this research? When we say high order, it is better to show accuracy of prediction for different order and influence of different parameters for each order, and influence of each order to RTM.
4. take carefully with explanation of each equation and each parameter when they were first appeared, especially the physical meaning of each parameters.
5. Other confusing expression problems are marked yellow in review PDF.
Comments for author File: Comments.pdf
Author Response
Dear Editors and Reviewers:
Our manuscript entitled “Least-squares reverse time migration of primary and internal multiple predicted by the high-order Born modeling method” has been resubmitted to the Applied Sciences. The original version of this paper was reviewed by three experts and encouraged to revise in terms of the comments. We have carefully revised this manuscript according to the comments and suggestions of experts. The revised mode in Microsoft Word is provided for the comparison of the modifications, and the detailed point-by-point responses to issues are listed in the following part.
We would like to express our sincere appreciation to the editors and the reviewers for their hard work. Looking forward to hearing from you soon.
Kind regards,
Ruiding Chen
Summary of the revisions
Thanks for the helpful suggestions from the reviewers and editors. We have revised our manuscript carefully according to the comments of four reviewers. We have labeled every modification in our revised manuscript by using the revised mode in the Word format. The item-by-item responses to the issues are as follows:
Point 1. problem on structure of manuscript. A discussion part is needed to explain the difference of this research with previous research, importance and limitations of proposed method. Methods in section 2 should be re-arranged in a more logic way for reads to understand. For example first give brief introduction of RTM and LSRTM in general. Technical writing may also be improved.
Response 1. Thanks for your advice, I have put the previous research which is the derivation of the high Born modeling method into the Appendix A. And I have add the derivation of the demigration equation as the basic of high-order Born modeling method and LSRTM.
Line 141-162
“In the theory of scattering series, we first get the wave equation of the actual velocity model and reference velocity model:
,
. (6)
where and is wavefield of actual model and reference model, and are the actual and reference differential operators and satisfy:
,
. (7)
The can be expressed as the wavefield of the reference reflectivity add the scatter wavefield:
. (8)
where is the scattering wavefield. Let the two equations of Equation (7) be subtracted, and let , where is the reflectivity model following, we can have:
. (9)
Equation (8) descript how the scatter wavefield can be computed, However, couldn’t be get, so we use as an approximate replacement of . And we can have:
. (10)
Use the equation to compute . By continuously substitute into Equation (8) and substitute Equation (8) into Equation (9) to compute , we can get the internal multiple predicted by the high-order Born modeling method. and the derivation is showed in Appendix A and the method is expressed as follows:”
Line 412-462
“The scattering wavefield can be written as the series:
. (A.1)
is the portion of that is ith order. And we can write equation (10) as:
. (A.2)
Use as an approximate replacement of . substitute into Equation (8) and substitute Equation (8) into Equation (9), and we can have:
. (A.3)
So we can get:
. (A.4)
Do approximate replacement and substitute again, we can have:
. (A.5)
The represents the wavefield experienced scattering for three times. The first-order internal multiple is also the wavefield experienced scattering for three times and contained in the . The scatter point of have the depth condition of , where , , is the depth of first, second, third scatter point, indicates that the propagation of the first-order internal multiple is down-up-down, so Equation (13) need to be separate the wave field into up- and downgoing components:
. (A.6)
The subscripts and represent the down- and upgoing components. Subscripts 1 and 2 indicate that the iteration ordinals are 1st and 2nd when computing scatter wavefields.
Figure A1. Components of the wave field caused by virtual sources
In Figure A1 is the wavefield, the subscripts , , and indicate that the direction of propagation is downward, upward, and rightward, and is generated by according to Equation (15). Through Figure 2 we see that the existence of causes the virtual source of to exist in . In other words, there is overlap of and and the depth condition can only reach . To solve this problem, we need not only to split the up- and downgoing wavefields but also to introduce the zeroing matrix to remove the overlap of and . The internal multiples are then changed as follows:
. (A.7)
The zeroing matrix is expressed as:
. (A.8)
where is a threshold associated with . In the overlap of and , the wave after the ith reflection contributes most of the energy, and the other is the wavefield after a reflection greater than i. Therefore, the threshold setting should separate the wavefield with the stronger energy and the others, while , which we need, cannot be removed. Thus, we set:
. (A.9)
where is the threshold parameter. Equation (18) indicates that a threshold is associated with the max of and is then multiplied by the reflectivity model. If the value of is set too high, the overlap may not removed clearly; if the value of is set too low, the wavefield we need maybe removed. So, we need to set a suitable value. Here we can set a relatively higher value, and about one-quarter to half a cycle before and after the wavefield part that meets the threshold is reset to zero. Thus, we get:
. (A.10)
”
Point 2. Compared with previous researches, especially your own research in Acta Geophysica (2022), what is the main point and new contribution of this study, what is the major problem solved? Much similarity between theoretic and experiment part of this manuscript and previous one in Acta Geophysica (2022). It is better to rewrite the method part with a more concise way with proper organization and reference of major equation used.
Response 2. Thanks for your advice, I have put the derivation of the high Born modeling method into the Appendix A and make the method part more clearly.
Point 3. Is the comparison between ISS and high-order Born modeling necessary in this research? When we say high order, it is better to show accuracy of prediction for different order and influence of different parameters for each order, and influence of each order to RTM.
Response 3. Thanks for your advices. I have canceled the ISS in this research. The amplitude of internal multiple with the higher order is weak, so we only consider first-order internal multiple. Because here we use the first-order internal multiple, so we only used the three-order scattering. And I add a sentence in the paper.
Line 196-197
‘Due to the amplitude of higher-order internal multiple is low, so we only consider the first-order internal multiple. Ignore higher order terms:”
’
Point 4. take carefully with explanation of each equation and each parameter when they were first appeared, especially the physical meaning of each parameters.
Response 4. Thanks for your advice. I have check these parameter and correct them.
Point 5. Other confusing expression problems are marked yellow in review PDF
Response 5.Thanks for your advice, I have solve these problem in the PDF. They are listed below:
- Clearly point what is the main problem, what new contribution to solve problem.
Add these words in abstracts:
Line 8-line 10
“In imaging, if the internal multiple is not been eliminated, it can cause the artifacts. However, if the internal multiple is been eliminated, the information contained by the internal multiple could be eliminated and degrade the image quality of the some useful structure.”
- Reflectivity
Line 63
I have changed it.
- Can
Line 98
I have changed it.
- I have changed it as :
Line 107-109
is the backward wavefields of the first-order multiple, is the backward wavefields of the second-order multiple.
- both primary and multiples
I have changed it as:
Line 113-116
The first term on the right side of Equation (4) is the cross-correlation between the forward wavefields and the backwards wavefields of the primary and corresponding order internal multiples received by the receiver.
- explain V and V1,V2,...
I have canceled these content about ISS.
- no where to use these z??
I have canceled these content about ISS.
- remove ISS?
I have canceled these content about ISS.
- correct lamda in fig
I have change the lamda in fig:
line 434
- first use, no where to cite before here
I have canceled it. And add figure 2 in line 245
Figure 2. Flow chart of the LSRTM image of primary and internal multiples
- I have changed the wrong place in the equation:
Line 195
- problem?same eq. as 30
To explain these two equation, we add some words:
Line 199-208
In order to meet the requirement of cross-correlation of corresponding-orders internal multiples in the image condition, we introduced the weighting matrix, and the objective function is changed to:
. （16）
where represents the weighted matrix to select the primary and represents the weighted matrix to select the internal multiples. and is designed to separate the primary or multiple by given different weighting according to the observed data and the predicted internal multiple. Here, we set：
, (17)
. (18)
- We canceled the description of quantitative relationship of and ,
“We can have：
. (17)”
These words have been canceled
- Values are always set?
These words now have been changed as:
Line 213-215
“Where the value of is calculated as 1, the useful content is the primary and the internal multiples could be suppressed; where the value of is calculated as 1, the useful content are internal multiples and the primary could be suppressed.”
- which equation?
These words now have been changed as:
Line 234
“Thus, the equation of the gradient should contains two terms related to the internal multiple.”
- add extra compare for single trace to show artifacts
I have add the compare of single trace:
Line 291-297
- the function do not requires born modeling. the multiple prediction is through ISS or born modeling
I have changed this words, now it becomes:
Line 391
“We established the objective function according to the RWLSRTM.”
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
I would like to congratulate the authors for this interesting piece of work. I would to recommend it to be published as it is.
