• This paper presents some common absorbing boundary conditions used in numerical simulations of wave propagation, commonly used as part of full-waveform inversion, across academia and industry. The work is quite relevant and valuable.
• High-level questions:
  • Could you add a clear "Contributions" section to the paper that highlights (bullet points) what is being claimed as new in this paper?
  • Can you comment how the choice of velocity models affects the results presented? e.g. why does the SEG/EAGE model behave so differently from the others? Or why all ABCs behave so similarly in the adjoint stage?
  • Can you comment on how different physics (e.g. elastic/viscouacoustic) might affect your results?
  • Does the adjoint field include reflection errors from the forward propagation? i.e. was the d_sim used the reference one or the one that potentially had reflection errors from the forward prop?
  • Are there limitations in the computational implementations (in your/Devito's code) that affect any of the results presented here? e.g. All boundary conditions augment the PDE. Are you solving the same augmented PDE in the entire domain (instead of just the boundary region)? If you could solve a simpler PDE in the physical domain and only solve the augmented PDE in the boundary regions, is it possible that the increase in computational time becomes negligible for all boundary conditions? Similarly for memory - for the fields added because of the ABCs, do you allocate memory over the entire domain?
  • If I understand correctly, the reference problem in Sections 6.1 and 6.2 are using a domain much bigger than the versions with ABCs, in order to get a version with no reflections.
    • If this same reference problem is used as a baseline for Section 6.3, this would be an unfair comparison. I don't think you're doing that.
    • However, if the reference/baseline problem has changed between sections, could you please make that more clear. e.g. by giving it another name - computational reference problem/reference problem B.
  • You are using subsampling in your gradient calculations for FWI. Could you justify the use of subsampling, as well as the chosen subsampling factors?
• Minor:
  • Line 7: DSL is written as DLS (also Line 600)
  • Line 8: Devito is primarily used for seismic modelling problems. Maybe not best to say it targets them.
  • Line 18: Psychical instead of physical
  • Line 47: Devito provides simple examples. Not appropriate to call them defaults.
  • Line 208-209: Combination of sponge layer and ABC is ambiguous since ABC is defined here to encompass all methods discussed, and all methods discussed so far in the paper use sponge layers.
  • Line 269-270: The paper mixes the use of dampening and damping. Please double-check that it means what you want it to mean.
  • Line 357: Missing reference
  • Figure 9: b and c look blank. Are these supposed to be initialised at constant values of 2.5 and 3? Could you choose a colour scheme that makes this less ambiguous?
  • Line 412: underling -> underlying
  • Line 437: Could you highlight which C compiler was used?

The authors present an interesting study for comparing different types of absorbing boundary conditions (ABCs) for the numerical implementations of the full waveform inversion (FWI) problems. They have carefully derived the ABCs in their adjoint form, and have designed and performed numerical examples that demonstrated their conclusions. The paper is overall written, and should be of interest to a wide audience in the seismology community. 

Main points:

1. One of my main concerns is that, in my humble opinion, this paper in its current form contains too many equations (a total of 51). I understant that it is important and necessary to have many of the equations. On the other hand, I think the derivations of the ABCs of the adjoint form can be put into an appendex without damaging the main message of this paper. 

2. I would like to see FWI results for the Marmousi model for all the ABCs that this paper has compared. 

3. Line 168-178. The authors here defined staggered grid in the space and temporal domain. The staggered grid has been widely used for the numerical implementation of FWI, but can be tricky to understand for people without the background of seismic simulation. Can you please add a figure here showing the locations of the staggered grid for different variables in your equation?

1. Line 18, “psychical” Please correct this typo, I suppose it should be physical
2. Line 27: “where the difference between the observed and synthetic data is back propagated in time from the receivers to the source of the waves”. The text here is related to the definition of the so-called “adjoint source” for the adjoint-state method. The adjoint source is not necessarily the difference between the observed data and synthetic data, but depending on what the misfit function is. This statement is only true when the misfit function being the norm-2 of the waveform difference.
3. Lines 28-29: “The back propagation requires saving the wave equation solution in every computational time step, thus meaning a high memory usage to solve a FWI problem. “ This statement is simply wrong. Any FWI software for relatively-scale problem will not save the entire wavefield history in the memory, instead, a in-the-fly wavefield reconstruction technique is usually used.