Comment on: “Application of the variable projection scheme for frequency-domain full-waveform inversion” (M. Li, J. Rickett, and A. Abubakar, Geophysics, 78, no. 6, R249–R257)

First of all, we would like to thank the authors for their interest in our paper. The detailed discussions and numerical tests help us to further understand the variable projection method and its application to seismic full-waveform inversion (FWI). We also thank the authors for adding references on this topic (Aravkin and van Leeuwen, 2012).

As we described in our paper (Li et al., 2013), the residual vector lies in the null space of the Jacobian matrix of the calibration coefficient. Therefore, we can still maintain the accuracy of the inversion algorithms, such as the nonlinear conjugate gradient method, in which we only need gradient information (product of the Jacobian matrix with the residual vector). We agree with the authors in the discussion that the gradient of the data misfit function in Aravkin et al. (2012) is accurate. We apologize for the potentially misleading information to the readers in this regard.

Moreover, both Aravkin and van Leeuwen (2012) and we agree that the approximated Jacobian without the correction term will affect the accuracy of other algorithms requiring more than the gradient information. The effect needs further investigation. On the other hand, computing the full Jacobian matrix needs few additional resources in the frequency-domain FWI algorithm. Therefore, we choose to use the full Jacobian matrix in our computation. As a comparison, we rerun the Marmousi data inversion using Jacobian with and without this approximation. We use the same settings as in our paper (Li et al., 2013). Note that we added 10% of error to the data. First, we inverted the data using the preconditioned nonlinear conjugate gradient (P-NLCG) method. The results are shown in Figure 5. We observe that the inverted model using the approximated Jacobian is close to the one using the full Jacobian. This verifies our guess on the accuracy of gradient-based inversion algorithms. Compared with the true model, the inverted model misfits are 0.12584 using the approximated Jacobian and 0.12282 using the full Jacobian. The inverted model using the full Jacobian matches the true model slightly better than the one using the approximated Jacobian. This is because the P-NLCG method requires an approximated Hessian matrix to construct a preconditioner. Using the full Jacobian matrix can construct a better preconditioner to accelerate the convergence of inversion. We also plot the data misfit at each iteration in the inversion of the first frequency in Figure 6 and observe a faster convergence by using the full Jacobian to construct the Hessian.

We also run the Marmousi data inversion using the Gauss-Newton (GN) inversion algorithm. First, we run a single-frequency data inversion at 6 Hz. The initial model is shown in Figure 7a. We add 5% noise to the data. The other settings are the same as in Li et al. (2013). The inverted models using the full and approximate Jacobian are shown in Figure 7b and 7c with model misfit 0.11486 and 0.13284, respectively. The inverted model using the full Jacobian is slightly better than the one using the approximate Jacobian. Figure 8 shows the data misfit at every iteration. We also observe a fast convergence using the full Jacobian.

We then run the Marmousi data inversion using the GN method with all the data as in Li et al. (2013). Note that here we added 10% noise to the data. We set the maximum number of inversion iterations to five for every frequency. The inverted model using the full Jacobian is shown in Figure 9a. The inversion process reconstructs the Marmousi model very well with a model misfit of 0.11942. The inverted model using the approximated Jacobian is shown in Figure 9b with a larger model misfit of 0.18245. From this comparison, we think that increasing the accuracy in the Jacobian could increase the robustness of the GN inversion because it leads to a better approximation of the Hessian.

In conclusion, we observe that the gradient vector is still exact with the approximated Jacobian in the variable projection method. Therefore, the nonlinear conjugate gradient method will maintain its accuracy. For other inversion methods requiring more than gradient information (P-NLCG or GN method), the convergence of the inversion could be changed by the approximation. The model inverted using the full Jacobian can have a better accuracy than the model inverted using the approximate Jacobian. The data misfit between the measured and simulated data is also lower. Moreover, when the data noise level increases, having an accurate Jacobian can increase the robustness of the inversion. Therefore, we think using a full Jacobian is helpful in the variable projection method if the computational cost is affordable. In the end, we would like to thank the authors in the discussion again. We look forward to more studies to improve the accuracy and efficiency of this scheme.

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