  We propose two preconditioned gradient direction for full waveform inversion
(FWI). The first one is using time integral wavefields. The Least square
problem is formulated as the time integral residual wavefields, which can
partially resolve the effect of high-passed filter in the traditional gradient
formula; the convergence rate is greatly accelerated. The other one is
localized offset Hessian inspired by the generalized imaging condition, which
provides another redundancy in the Hessian. We compare the traditional
conjugate gradient scaled by the shot illumination and localized offset Hessian
(actually, only diagonal part is considered here), and contrast their
performance for waveform inversion. The results demonstrate the localized
offset Hessian (diagonal part) can provide much more information in the
subsurface, and is preferred to the layer-strip inversion.
