A High-Precision Elastic Reverse-Time Migration for Complex Geologic Structure Imaging in Applied Geophysics

Round 1

Reviewer 1 Report

The manuscript describes a rather new implementation for numerically solving velocity-stress elastic wave equation through a decoupled P- and S-wave representation. The paper is generally clear and readable. My main concern is in the development of the main equations in which there exists some inaccuracies (typos I presume) and missing or late definitions of variables. I point them out in the uploaded annotated version of the paper. I also point out numerous unclear statements in the paper. The results look promising, but the Figures seem to have some labeling problems. Also, in Figure 13, the velocities I assume are for the field data, bit the foothills. Also, in Figure 1, it is hard to see the difference beyond the profile, so I suggest the authors add two more profiles. The discussion, as opposed to the rest of the paper, section is poorly written, with many unclear statements.

I include additional corrections and suggestions in the uploaded annotated version of the paper. I encourage the authors to address them and make the necessary revision to incorporate the suggested corrections.

Comments for author File: Comments.pdf

Author Response

We are grateful that you gave us important guidance and careful review. We truly benefit a lot from you and do improve our manuscript significantly. We have carefully revised them per your guidance. The detailed replies and revised traces are in the uploaded "author-coverletter-20492885.v1.pdf". Thanks again.

Author Response File: Author Response.pdf

Reviewer 2 Report

Brief summary:

The manuscript deals with reflection seismology, using a method called elastic reverse time migration. In this method, the partial derivatives in the wavefield equation are approximated using finite differences. The manuscript applies the finite difference approximations presented by Tan and Huang (2014), which are accurate to the fourth order in the time coordinate and to an arbitrary even order in the space coordinates. The P and S waves are decoupled as in the authors' previous paper (Chen et al. 2016b). Using synthetic and real data, it is shown that high quality images can be obtained with a larger discretization time step than if one uses a less accurate approximation for the partial derivatives.

General comments:

1) The author's claim that the quasi-stress-velocity equation is new is not correct, because the authors have already used it in (Chen et al. 2016b) and (Fang et al. 2021), including the decomposition of P and S waves in (Chen et al. 2016b).

2) The authors' claim that the finite difference scheme is new (it reads in the abstract: "a new finite-difference scheme is employed...") is not correct, because Tan and Huang (2014) have already presented exactly the same finite difference scheme.

3) The experiments look good.

Detailed comments:

Abstract: It reads that "current wavefield simulations using the time-domain finite-difference (FD) approach in ERTM have the second-order temporal accuracy", but Tan and Huang (2014) have presented a high-order temporal FD method for Eq. 1. 

Section 1: Please clarify what is the difference between RTM and ERTM? Is it that ERTM uses both P and S waves and RTM only P waves?

Section 1: The sentence "use the dispersion relations to obtain FD coefficients with high-order temporal accuracy" is misleading, because the different approximations are evaluated and compared by studying their dispersions from the true value while the FD coefficients for each approximation are computed independently of the dispersion analysis.

Section 1: It reads that "ERTM with a high-order temporal FD method has not been reported", but Tan and Huang (2014) have presented a high-order temporal FD method for Eq. 1. 

Equation 1: {bf sigma} should be {bf tau}. Vector {bf varphi} is badly composed, because {bf v} and {bf tau} should be row vectors instead of column vectors.

Section 2.1: The quasi-stress-velocity equation has been used in (Chen et al. 2016b) and (Fang et al. 2021), so it is not new.

Equation 3: Vector {bf tilde varphi} is badly composed, because {bf tilde v} and {bf tilde tau} should be row vectors instead of column vectors.

Equation 4: It is not clear why there are alpha-indexed derivatives with respect to beta-indexed velocities (and the other way round) in the lower left block of {bf tilde E} when operated on {bf tilde varphi}. Has it something to do with PS wave simulation?

Section 2.1: The sentence "v_x^alpha and v_z^alpha are calculated by the derivatives of partial_j^alpha" does not make sense. The velocities are differentiated.

Section 2.1: In the sentence "Regarding v_x^beta and v_z^beta, these are wavefields...", the word "wavefields" should be "particle velocities".

Section 2.2: The sentence "...a space-to-time precision conversion based on dispersion relation" is not clear. How is space precision converted to time precision using dispersion and what kind of dispersion relation is there between the velocity and wavelength? Do you refer to the CFL stability condition instead of the dispersion relation?

Equation 5 gives an approximation for partial u_r^chi instead of partial u_r^alpha.

Section 2.2: In the sentence "We use a new FD coefficient derived from a Taylor series expansion and a dispersion relation...", it is not clear what dispersion relation you mean.

Equations 5 and 6 have already been presented in (Tan and Huang 2014), so the reference to the authors' paper (Chen et al. 2016) is not appropriate.

Figure 1 caption: There are mistaken figure labels in the caption, when the intent is to refer to subfigures (b) and (c).

Figure 2c caption: "...extracted from (a), (c), and Fig. 1c" should be "...extracted from (a), (b), and Fig. 1d".

Section 3.1: The units of the RMSEs are missing throughout the section.

Figures 1-4, 7-9, and 14-16: The units of the gray values (particle velocities) are missing.

Section 3.2: The grid size is obviously 800 x 400 and not 400 x 800 in Fig. 4.

Discussion: Where did you suggest this: "We suggested that the small value was 10^{-5} to 10^{-3} times the maximum value of the wavefield"? Do you suggest it here in discussion or have you suggested it in some other paper?

Conclusions: Regarding sentences "This study introduced a novel quasi-stress–velocity equation as a simulation kernel for ERTM. The P- and S-wave decomposition form of the equation was developed for elastic migration imaging.", the quasi-stress-velocity equation and the P and S wave decomposition are not new, because they have already been used in (Chen et al. 2016b).

Author Response

Thanks for your extremely exact and professional comments, including the guidance on the equation, language, and Figures. All of these are extremely important to improve our manuscript. We also thank you for your recognition of numerical results. Thank you for spending so much time and effort helping us improve the manuscript. The detailed replies and revised traces are in the uploaded "author-coverletter-20563506.v1.pdf". Thanks again.

Author Response File: Author Response.pdf

Reviewer 3 Report

Please develop the conclusions. Conclusions should be clearly stated and supported by the quantitative analyses of presented results. Please also highlight the most important practical consequences of presented results.

Author Response

Reply to Reviewer 3

Comments and Suggestions for Authors

Please develop the conclusions. Conclusions should be clearly stated and supported by the quantitative analyses of the presented results. Please also highlight the most important practical consequences of the presented results.

Reply: Thanks for your significant comments and suggestions, which are very important to improve our manuscript. Following your guidance, we have rewritten the conclusions as below and hope you are satisfied.

Conclusions: We have developed a new high-accuracy ERTM for seismic imaging in complex media. The FD schemes with 2Nr-order accuracy in space and fourth-order accuracy in time ensure the high-precision numerical simulation for the ERTM. The proposed ERTM workflow can achieve effective suppression of the temporal dispersion occurring in conventional temporal second-order-accuracy FD schemes with a relatively large time step, ensuring accurate image locations and in-focus diffraction waves. Wavefield simulations validate that the used FD schemes can improve the simulation accuracy and efficiency due to the temporal fourth-order accuracy schemes and relaxed CFL conditions. In conventional ERTM tests, travel time changes and waveform distortions in wavefields caused by temporal dispersion cause that depth of the imaging horizon appears deeper in layered media and the diffracted waves cannot focus at the scattering points of the complex model. The problem can be effectively overcome by our proposed ERTM as demonstrated in the proposed ERTM tests. Furthermore, the proposed method is applied to field data and the expected results are obtained. The proposed high-accuracy ERTM workflow facilitates the integration of the most advanced imaging or inversion techniques into this computational framework.

Round 2

Reviewer 1 Report

The authors have addressed my concerns and made the necessary revision to the paper. I find the paper acceptable. I have very minor typo corrections in the uploaded annotated version of the paper.

Comments for author File: Comments.pdf

Reviewer 2 Report

The authors have replied very well to the comments of the reviewer and revised the manuscript correspondingly.