Frequency multiscale fullwaveform velocity inversion
ZHANG WenSheng;LUO Jia;TENG JiWen;Academy of Mathematics and Systems Science,Institute of Computational Mathematics and Scientific/Engineering Computing, State Key Laboratory of Scientific and Engineering Computing,Chinese Academy of Sciences;Institute of Geology and Geophysics,Chinese Academy of Sciences;
Inverse problems arise naturally in oil geophysical exploration.In order to detect the structures of oil/gas objects,one of the most important step in seismic data processing is that we need to know the velocity information of subsurface.This belongs to scope of seismic inversion.There are various seismic inversion methods to achieve the goal,for example the seismic tomography based on ray tracing and the conventional velocity analysis based on CDP stack.Currently,the popular method is the fullwaveform inversion(FWI)method based on wave equations.The FWI method uses the prestack data rather than the poststack data as an input and has high accuracy to image complicated velocity structures.The inversion can be implemented in the time domain or in the frequency domain.While there are many works of frequencydomain FWI methods,here we focus on the timedomain FWI method and try to inverse complicated velocity structures.Mathematically,the prestack data can be obtained by the forward modeling of wave equations.The seismic waves generated by a Ricker wavelet source propagate through the underground media.When the waves encounter the interfaces or inhomogeneous objects with different physical parameters,they are reflected back to the surface.These reflected waves are received by the geophones on the surface and used for inversion or other seismic data processing.This process is usually modeled by wave equations such as the acoustic wave equation if we assume the underground media is acoustic.In this paper,we investigate the fullwaveform velocity inversion based on the 2D acoustic wave equation and the corresponding computations are completed in the time domain.The forward problem is to solve the acoustic wave equation numerically and obtain the prestack data.Many numerical methods can be used to solve the forward problem,for example,the finite element method and the finite volume method and so on.In this paper we adopt the finite difference method because of easy programming and its high efficiency.Since the computational domain is a bounded rectangular domain and boundary reflections may devastate inversion results,it is necessary to use the absorbing boundary conditions and eliminate boundary reflections.The boundary conditions we adopted are the secondorder Clayton′s absorbing boundary conditions.The computational discrete schemes for the equation and boundary conditions are of secondorder accuracy both in time and space.The inversion is required to solve a nonlinear leastsquare problem.It is an iterative minimization process between the synthetic data and the observed data.Aiming at the difficulty which the waveform inversion is easy to fall into local extreme points,we propose the"stepwise inversion"strategy.First we use the wavefield on the lower frequency scale to obtain a reasonable initial model.Then we use the wavefield on the other high frequency scales to do inversion step by step.In order to use the wavefield information fully,the information at large scale contain that at low scale.Moreover,the inversion result at the previous lowfrequency scale is chosen as the initial model for the next inversion at the largefrequency scale.The optimization method in inversion is the LBFGS method.The fullwaveform inversion is a typical largescale scientific computational problem and the computations are implemented based on the MPI parallel algorithm on the PC cluster.The theoretical formulae and algorithms,including the finitedifference forward modeling,velocity model correction,gradient computation and corresponding algorithm,are expounded and derived in detail.In numerical computations,the fullwaveform inversion based on the MPI parallel computations for the Marmousi model are completed.The Marmousi model is a complicated benchmark 2D model which is usually used for testing the ability of migration and inversion methods.In the process of"stepwise inversion",the data at five different scales,i.e.0~5Hz,0~10 Hz,0~25 Hz,0~35 Hz,0~60 Hz,are selected.The initial model at first scale is a linear velocity model which is reasonable in practical case.The iteration inversion results at the first scale(i.e.0~5Hz)show that the inversion gets better as the iteration number increases till some iteration number such as the 50 th iteration.This phenomenon can be observed by the variations of object function.Then we stop the inversion and the corresponding inversion result is used for the initial velocity model for the inversion at the second cale(i.e.0~10Hz).The following procedure is similar till the inversion at the fifth scale(i.e.0~60Hz)is completed.At each scale,the inversion iteration number may be set 50 as more inversion iterations can not decline the value of object function.For comparisons,the inversion result without using the stepwise strategy is also given and shows that the iteration inversion converges to a wrong local extreme result.The detail comparisons between numerical and exact results at a fixed CDP are presented.The computations are completed on the PC cluster.The parallel efficiency is high and the scalability is about 0.9.The full waveform inversion is an iterative process of residual minimization between synthetic data and the known records.The inversion is easy to fall into local minimum points.We develop the sequential inversion method based on the inversion with the data at different frequency scales.The main idea is that the inversion result at low frequency scale is chosen as the initial guessing model for the inversion at the next high frequency scale.This strategy effectively solves the problem of inversion divergence when the initial value is far from the true solution.The detailed descriptions including theoretical formula and the corresponding algorithms are given or derived in this paper.Numerical computations for the complex structure model named Marmousi model are carried out.Relative good inversion results are yielded.Many computations show that the method is effective and has high robustness to the initial model.The fullwaveform inversion based on wave equations is a typical large scale scientific computational problem. The implementation based on the MPI algorithm improves the computational efficiency greatly,which provides the basis for further application to real data.


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