Study of full waveform inversion based on L-BFGS algorithm

Transcription

1 Global Geology doi /j. issn Article ID Study of full waveform inversion based on L-BFGS algorithm DENG Wubing HAN Liguo ZHANG Bo HUANG Fei and HAN Miao College of Geo-Exploration Science and Technology Jilin University Changchun China Abstract Full waveform inversion is mainly used to obtain high resolution velocity models of subsurface. The size of full waveform inversion will lead to a gigantic computation cost. Under the available computer resource and the limitation of full waveform inversion the authors propose L-BFGS algorithm as the optimization method to solve this problem. In order to demonstrate the flexibility of the method three different numerical experiments have been done to analyze the properties of full waveform inversion based on L-BFGS. Key words L-BFGS numerical optimization acoustic wave equation initial model full waveform inversion 1 Introduction Although FWI was already introduced in 1980s Tarantola 1984 seismology geophysicists put gigantic efforts into FWI to get higher resolution images of the subsurface's structure due to the huge size of the inversion problem and the limitation of the computer technology. In order to gain a good inversion result and reduce the computation cost lots of strategies such as multi-scale grids Zhang 2011 and different optimization algorithm have been applied to FWI. In this paper we introduce L-BFGS optimization algorithm to full waveform inversion. In order to testify the flexibility of our method numerical experiments are processed on Marmousi model. BP model and noise contained seismic data. BFGS is named after Broyden-Fletcher-Goldfarb- Shanno who first introduced this algorithm to numerical optimization methods Nocedal 1980 Liu & Nocedal 1989 Brdy et al BFGS was regarded as the most efficient algorithm among the quasi- Newton algorithms. However it has a high requirement on memory. Thus L-BFGS was introduced to save the memory which makes BFGS more efficient to perform. L-BFGS not only inherits the advantages of BFGS it overcomes the problem of high requirement on storage and improves the calculation speed as well. Different from BFGS L-BFGS only needs to store a few implicit vector pairs last m iteration results rather than store the whole vector pairs Nocedal Therefore it is always used to solve the inverse problem such as inverse spectral decomposition or combine with other algorithm Deng et al Unlike BFGS i-1 former iteration results are needed L-BFGS can perform very well using only m former iteration results and smaller memory is required. In order to cut the computation cost without reducing the resolution L-BFGS was applied to full waveform inversion. 2 The theory of full waveform inversion based on L-BFGS Generally seismic inversion is aim to get the information about subsurface structure or underground physical properties via processing the data gathered by the acquisition system. Different from other inversion Received 8 January 2012 accepted 10 February 2012

2 162 Deng W. B. Han L. G. Zhang B. et al. methods full waveform inversion method gets an accurate subsurface model by minimizing the differences between the synthetic wavefield and observed wavefield without any other processing strategies if the forward method can match the real acquisition system perfectly Virieux & Operto The forward modeling is based on frequency domain acoustic wave equation in this paper 2 + k 2 W r = - f r 1 where 2 is the Laplace operator k 2 is the wavenumber W r and f r are the wavefield and seismic source at position r respectively. Equation 1 can be simplified as follow Ax = b 2 with A = 2 + k 2 which is solved by two dimensional nine points 2D-9P finite difference scheme Jo 1996 x = W r b = - f r. According to equation 2 the objective function of the inverse problem here can be described as min 1 2 b - Ax λ x 1 3 with λ the regularization parameter. It is a tradeoff between L1-norm and L2-norm Brossier et al We can summarize the procedure of FWI briefly as Fig. 1 As clearly showed in Fig. 1 there are 7 major steps in FWI and step1 to step6 is the iterative part. The iteration only stops when the stopping criterions satisfied. Among these 7 steps we are here to focus our efforts on dealing with step 3 calculating the gradient of the objective function. There are numerous ways to solve step3 such as CG conjugate gradient PCG preconditioned CG and LSQR least square QR decomposition. However quasi-newton algorithms are chosen instead of Newton methods because equation 3 is a nonlinear optimization problem. And L-BFGS is the best among all the quasi- Newton algorithms. Thus L-BFGS algorithm is utilized to calculate the gradient g k through formula 4. B k d k = - g k 4 where B k and d k refer to the approximate Hessian matrix and research direction respectively. Fig. 1 Brief outline of procedure of FWI L-BFGS is a kind of iterative algorithm. It has some good properties which Newton algorithms haven't. Such as quadratic termination fast local convergence rate and low computation cost. More importantly it is little affected by inexact line searches which are obtained from step 4. The L-BFGS update formula is B k +1 = B k + y k +1y T k +1 y T k +1s k +1 + g kg T k g T k d k 5 s k = x k - x k -1 y k = g k - g k -1 6 k denotes the k-th iteration. The demonstrations of these two equations are omitted here. 3 Numerical examples The subsurface structure of the Earth is so complex that we can only get an approximation model of it. However the quality of the inversion result suffers from many aspects such as the accuracy of the initial model the noise level in real seismic data. But we believe that these two aspects can be overcome if a suit-

3 Study of L-BFGS algorithm based full waveform inversion 163 able optimization method is chosen. In order to demonstrate this opinion we have done a series of numerical experiments on Marmousi model and BP model Inversion result obtained by different optimization methods In this section two different algorithms L-BFGS and LSQR are applied to full waveform inversion on BP model. The BP model is discretized on grids with grid spacing 25 m. Acquisition geometry consists of 300 shots distributed in a line near the surface of the model and 300 geophones as well. The inversion results showed in Fig. 2 are obtained under the same computation situation except that different numerical optimization methods are used. Fig. 2a is the hard BP model Fig. 2b is the initial model which is obtained by low wavenumber pass filter. The position encircled by the ellipse is so steep that it is hard to get a high - resolution result through inversion or migration. However this part has been successfully reconstructed by FWI based on L-BFGS which can be seen from the inversion result showed in Fig. 2c. Unfortunately LSQR algorithm failed to recover the region showed in Fig. 2d. So this experiment demonstrates that L-BFGS is much better than LSQR algorithm towards the application in FWI Inversion starts from different initial models The quality of an initial model is of crucial importance to full waveform inversion. In this secttion we have used different initial model to test how FWI based on L-BFGS affected by initial model. Fig. 3a is the same as Fig. 2b which is a low wavenumber passed initial model and its inversion result is showed in Fig. 3c while Fig. 3b is a linear initial model and its inversion result is showed in Fig. 3d. Although the low wavenumber passed initial model has a less difference to the hard model than the linear initial model the inversion results between the two are alomost the same. From Fig. 3e we can see that the difference between the two inversion results below 300 m /s and even approximate to zero in most area except the bright area. So L-BFGS algorithm can reconstruct the subsurface very well with simple structure that the affection caused by the initial model can be neglected. Unfortunately L-BFGS still cannot overcome the affection caused by initial model in the situation of high complexity structures. Fig. 2 Comparison between L-BFGS algorithm and LSQR algorithm on BP model

4 164 Deng W. B. Han L. G. Zhang B. et al. Fig. 3 Comparison of inversion results from different initial models 3. 3 Property of anti-noise of L-BFGS Noise cannot be avoided in the real case. And some of the noise even cannot be eliminated. Thus lots of work has been done to deal with the inherent noise. In this part numerical experiment is processed on Marmousi model to test how L-BFGS performed with noise contained seismic data. The Marmousi model is discretized on grids with grid spacing 25 m. The acquisition consists of 384 shots distributed in a line near the surface of the model and 384 geophones as well. Fig. 4a and Fig. 4b are the hard model and initial model respectively. Two different kinds of noise were added to the seismic records. We group these two noise and related results into two groups. In the first group noise of average equals zero covariance equals is added to the seismic data showed in Fig. 4c which is the seismic wavefield snap at 5Hz in frequency domain so as Fig. 4d. We got the inversion result in this group is showed in Fig. 4e. The reservoir encircled by an ellipse in the model has been successfully inverted. We increase the noise level to covariance equals 0. 5 in group two. We can see from Fig. 4d the seismic data is badly damaged by the noise but the reservoir still can be reconstructed successfully which is showed in Fig. 4f. This experiment successfully demonstrates that FWI based on L-BFGS has a strong ability to deal with noise. However the inversion result becomes blurry as the noise level increased.

5 Study of L-BFGS algorithm based full waveform inversion Fig Inversion results obtained from two different noise contained seismic data Conclusions Through the study of L-BFGS based full waveform inversion with three different numerical examples we get some conclusion as following 1 It is valid and efficient to apply L-BFGS algorithm to full waveform inversion as it can help us to get an accurate solution than some other algorithm small storage is needed 2 Initial model has less affection on L-BFGS based full waveform inversion This is because the way to calculate the gradient of the objective function using by L-BFGS is much more efficient than other algorithm In the next step we are looking forward to modify the L-BFGS's update formula equation 5 and 6 to improve L-BFGS's stability and convergence rate etc References Brossier R Operto S Virieux J 2009 Robust elastic frequency-domain full-waveform inversion using the L1 norm Geophyscial Research Letter L20310 Byrd R H Lu P H Nocedal J et al 1994 A limited memory algorithm for bound constrained optimization SIAM J Sci Comput Deng W B Han L G Li X et al 2011 Apply the combination of L-BFGS and SPGL1 to full waveform inversion / / The 27th Chinese Geophysical Society Symposium 2011 Hefei University of Science and Technology of China Press 714 in Chinese Jo C H Shin C Suh J H 1996 An optimal 9-point finitedifference frequency-space 2-D scalar wave extrapolator Geophysics