In order to exploit Hessian information in Full Waveform Inversion (FWI), the matrix-free truncated Newton method can be used. In such a method, Hessian-vector product computation is one of the major concerns due to the huge memory requirements and demanding computational cost. Using the adjoint-state method, the Hessian-vector product can be estimated by zero-lag cross-correlation of the first-order/second-order incident wavefields and the second-order/first-order adjoint wavefields. Different from the implementation in frequency-domain FWI, Hessian-vector product construction in the time domain becomes much more challenging as it is not affordable to store the entire time-dependent wavefields. The widely used wavefield recomputation strategy leads to computationally intensive tasks. We present an efficient alternative approach to computing the Hessian-vector product for time-domain FWI. In our method, discrete Fourier transform is applied to extract frequency-domain components of involved wavefields, which are used to compute wavefield cross-correlation in the frequency domain. This makes it possible to avoid reconstructing the first-order and second-order incident wavefields. In addition, a full-scattered-field approximation is proposed to efficiently simplify the second-order incident and adjoint wavefields computation, which enables us to refrain from repeatedly solving the first-order incident and adjoint equations for the second-order incident and adjoint wavefields (re)computation. With the proposed method, the computational time can be reduced by 70% and 80% in viscous media for Gauss-Newton and full-Newton Hessian-vector product construction, respectively. The effectiveness of our method is also verified in the frame of a 2D multi-parameter inversion, in which the proposed method almost reaches the same iterative convergence of the conventional time-domain implementation.
Spectral properties of the equation (∇ + ieA) × u = ±mu
