Elastic and anelastic adjoint tomography with and full Hessian kernels
Authors: Xie Yujiang , Rychert Catherine A , Harmon Nicholas 
Published: March 2023

The elastic and anelastic structures of the Earth offer fundamental constraints for understanding its physical and chemical properties. Deciphering small variations in the velocity and amplitude of seismic waves can be challenging. Advanced approaches such as full-waveform inversion (FWI) can be useful. We rewrite the anelastic Fréchet kernel expression of Fichtner & van Driel using the displacement–stress formulation. We then derive the full Hessian kernel expression for viscoelastic properties. In these formulations, the anelastic Fréchet kernels are computed by the forward strain and a shift of the adjoint strain. This is complementary to the quality factor Q (i.e., inverse attenuation) Fréchet kernel expressions of Fichtner & van Driel that are explicit for the velocity–stress formulation. To reduce disk space and I/O requirements for computing the full Hessian kernels, the elastic full Hessian kernels are computed on the fly, while the full Hessian kernels for Q are computed by a combination of the on-the-fly approach with the parsimonious storage method. Applications of the Fréchet and full Hessian kernels for adjoint tomography are presented for two synthetic 2-D models, including an idealized model with rectangular anomalies and a model that approximates a subduction zone, and one synthetic 3-D model with an idealized geometry. The calculation of the full Hessian kernel approximately doubles the computationally cost per iteration of the inversion; however, the reduced number of iterations and fewer frequency stages required to achieve the same level of convergence make it overall computationally less expensive than the classical Limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) FWI for the 2-D elastic tested models. We find that the use of full Hessian kernels provides comparable results to the L-BFGS inversion using the improved anelastic Fréchet kernels for the 2-D anelastic models tested for the frequency stage up to 0.5 Hz. Given the computational expense of the Q full Hessian kernel calculation, it is not advantageous to use it in Q inversions at this time until further improvements are made. For the 3-D elastic inversion of the tested model, the full Hessian kernel provides similar image quality to the L-BFGS inversion for the frequency stage up to 0.1 Hz. We observe an improved convergence rate for the full Hessian kernel inversion in comparison to L-BFGS at a higher frequency stage, 0.1–0.2 Hz, and we speculate that at higher frequency stages the use of full Hessian kernels may be more computationally advantageous than the classical L-BFGS for the tested models. Finally, we perform 3-D elastic and Q L-BFGS inversions simultaneously using the rederived Q kernels, which can reduce the computational cost of the inversion by about 1/3 when compared to the classical anelastic adjoint tomography using the additionally defined adjoint source. The recovered Q model is smeared when compared to the recovered elastic model at the investigation frequencies up to 0.5 Hz. Q inversion remains challenging and requires further work. The 2-D and 3-D full Hessian kernels may be used for other purposes for instance resolution analysis in addition to the inversions.