Velocity inversion method with reflection travel-time based on Gaussian packet

The scattering Gaussian packet theory tells that scattered waves can be calculated using the Gaussian packet operator with the perturbed model' s spatial distribution of Gabor frames, and the precondition is that the signal of incident waves should be of short-duration in the time domain and broad-band in the frequency domain. Here a velocity inversion approach is proposed by fitting the travel-times of reflected waves based on the scatterring Gaussian packet theory, in which the travel-time residual is expressed by the cross-correlation of observed and simulated local seismic wave-forms, and then the objective function is defined as the L2-norm of travel-time residuals. Thus the velocity inversion can be implemented by the optimization of objective funcion. The key of the proposed inversion method is how to deduce the gradient of objective function with respect to the velocity field based on the first-order Born-approximation. The deduced gradient of the objective function includes two parts: one is the cross-correlation between forward-propagated background wave of source side and the backward-propagated scattered Gaussian packet, and the other is the cross-correlation between backward-propagated background wave of receiver side and the forward-propagated scattered Gaussian packet. Both of the background wave-field and scattered wave-field in the expression of gradients are simulated by the Gaussian packet operator. It is the key points that how to estimate the perturbed model and then to simulate the scattered wave-field, and how to calculate travel-time residual by considering the flexibility of Gaussian packet in the implementation of the proposed velocity inversion approach, which is discussed in details in this paper. The velocity inversion strategy is further illustrated by numerical experiments. The results show that the theory and algorithm of the reflection travel time velocity inversion method based on Gaussian packet are effective and feasible, which can achieve good inversion of data.