Preconditioned Conjugate Gradient Algorithm-Based 2D Waveform Inversion for Shallow-Surface Site Characterization

Jianbo Guan,Yu Li,Guohua Liu,Denise-Penelope Kontoni

doi:10.1155/2021/3164358

Abstract

The full-waveform inversion (FWI) of a Love wave has become a powerful tool for shallow-surface site characterization. In classic conjugate gradient algorithm- (CG) based FWI, the energy distribution of the gradient calculated with the adjoint state method does not scale with increasing depth, resulting in diminished illumination capability and insufficient model updating. The inverse Hessian matrix (HM) can be used as a preprocessing operator to balance, filter, and regularize the gradient to strengthen the model illumination capabilities at depth and improve the inversion accuracy. However, the explicit calculation of the HM is unacceptable due to its large dimension in FWI. In this paper, we present a new method for obtaining the inverse HM of the Love wave FWI by referring to HM determination in inverse scattering theory to achieve a preconditioned gradient, and the preconditioned CG (PCG) is developed. This method uses the Love wave wavefield stress components to construct a pseudo-HM to avoid the huge calculation cost. It can effectively alleviate the influence of nonuniform coverage from source to receiver, including double scattering, transmission, and geometric diffusion, thus improving the inversion result. The superiority of the proposed algorithm is verified with two synthetic tests. The inversion results indicate that the PCG significantly improves the imaging accuracy of deep media, accelerates the convergence rate, and has strong antinoise ability, which can be attributed to the use of the pseudo-HM.

Highlights

To alleviate this problem, four different approaches have been proposed and developed: the distance-weighting, layerstripping, and full Hessian matrix (HM) schemes and the approximate-HM strategies. e distance-weighting [19, 20] method adjusts the gradient amplitude across the model space by multiplying the gradient by a function of the distance from the source, and this method balances the energy distribution of the gradient by giving greater weight to the far offset and deep regions
We present a new method to estimate the HM in Love wave full-waveform inversion (FWI) based on the calculation method of the HM in inverse scattering theory, and the preconditioned conjugate gradient algorithm- (CG) (PCG) is developed in combination with the CG. e proposed method uses the stress components of the Love wave wavefield to construct a pseudo-inverse HM to filter, balance, and regularize the gradient and alleviate the influence of nonuniform coverage from source to receiver, including double scattering, transmission, and geometric diffusion, improving the inversion accuracy
The construction of the gradient and the HM are derived in detail, and the implementation procedure of the PCG-based Love wave FWI is described in detail. e trial results from the checkerboard model and the complex structure model indicate that compared with the classical CG, the PCG can clearly rebuild the stratigraphic anomaly boundary and reconstruct the primary stratigraphic anomaly at depth, and it has strong antinoise ability. e advantages of PCG are due to the presence of the pseudo-HM

Summary

Calculate the preconditioned operator

Because of the insufficient illumination and uneven energy distribution of the gradient operators, the normalized descent direction in the first iteration of the CG presents artifacts near the free surface of the model, and its amplitude decreases sharply in the deeper medium (Figure 2(c)), which seriously affects the updating of the deep medium parameters and the stability of the inversion process. Normalized cost function value (misfit) of the two algorithms is close to 0 (Table 1), and the waveforms of the observed data and the inversion results are nearly identical (Figure 3(h)). In order to further illustrate the advantages of the PCG in enhancing the illumination capability and improving the imaging accuracy of deep media, the second test example is a deeper model containing a complex high-velocity anomaly and a fault (Figure 4). In order to further illustrate the advantages of the PCG in enhancing the illumination capability and improving the imaging accuracy of deep media, the second test example is a deeper model containing a complex high-velocity anomaly and a fault (Figure 4). e grid in this model has a size of 81 × 101 in the vertical and horizontal directions with a step length of 0.5 m in either direction, resulting in an actual model size of 40 × 50 m. e model has three layers: the first layer has a velocity of 300 m/s, the second layer has a velocity of 400 m/s and a “boot-like” high-velocity anomaly, and the third layer has a velocity of 500 m/s and a fault (Figure 4(a))

Final value of RMSE

Discussion and Conclusion