A Time Domain Full Waveform Inversion Based On Spectral Conjugate Gradient Method

Full waveform inversion(FWI)continually modifies the model parameters of the initial model by setting up and solving a nonlinear least squares problem with a optimization method;and it makes the data residual between the synthetic data and the observed data decrease continuously.In turn,the initial model gradually approximating the actual underground structure,and finally we can obtain a high resolution underground velocity distribution.We can see that full waveform inversion is essentially an optimization problem.At present,gradient-type local optimization method has been widely used in full waveform inversion,which mainly includes steepest descent method,conjugate gradient method,and so on.The steepest descent method has simple structure and small storage requirement but with a low convergence rate.The conjugate gradient method has overcome the weakness of slow convergence of the steepest descent method,it requires small calculation amount and storage space.Based on the above advantages,the conjugate gradient method has become a more widely used gradient-type optimization method in full waveform inversion optimization algorithms.In order to improve the computational efficiency of the conjugate gradient method,and overcome the defect of conjugate gradient method to generate ascending search direction,in this paper,spectral conjugate gradient method is applied to full waveform inversion.The spectral conjugate gradient method is an algorithm obtained by combination of spectral gradient method and conjugate gradient method.And it was proved that the search direction at each iteration is a descent direction of objective function even without relying on any line search method.Under the line search method,the spectral conjugate gradient method can realize global convergence.Compared with the conjugate gradient method,the spectral conjugate gradient methodcan provide one more spectral coefficient in calculation of the search direction.If the spectral coefficient is 1,the spectral conjugate gradient method coincides with the conjugate gradient method.As such,the spectral conjugate gradient method is characterized with all the advantages of the conjugate gradient method.Moreover,due to the introduction of spectral coefficient and access to more information of the objective function,the spectral conjugate gradient method shows better numerical performance than the conjugate gradient method does.We give a comparison of the algorithm principles by steepest descent method,conjugate gradient method and spectral conjugate gradient method,and then apply these methods to full waveform inversion for numerical tests on Marmousi model.After comparison and analysis of this inversion results,we can know that compared to the other two methods,spectral conjugate gradient method can do better in improvement of the inversion effects,and the full waveform inversion results by spectral conjugate gradient method are closer to the actual underground conditions.And then,the convergence curves of the objective functions which after iterative 40 times of full waveform inversion that based on above three optimization algorithms are compared.In the convergence curves of the objective functions,we can see that the objective function value after iterative 40 times of the spectral conjugate gradient method is obvious less than the objective function value after iterative 40 times of the other two optimization algorithms,and when the objective function value of the other two optimization algorithms trends already tend to be horizontal,the objective function value of the spectral conjugate gradient method still has a significantly downward trend.Thus,we can know that the full waveform inversion based on the spectral conjugate gradient method compared to the other two methods is not only the inversion results more close to the actual underground conditions,but also the objective function value converges faster.