The Application of Discrete Tikhonov Regularization Inverse Problem in Seismic Tomography

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1 The Application of Discrete Tikhonov Regularization Inverse Problem in Seismic Tomography KAMBIZ TEIMOORNEGAD, NEDA POROOHAN 2, Geology Department Islamic Azad University, Lahijan Branch 2 Islamic Azad University, Lahijan Branch P.O.BOX:66,Shaqayeq St.,Kashef St.,Lahijan,IRAN. tymoornegad@yahoo.com Abstract: - Because of instability, a solution called regularization had been used, considering some constraints in problem solving, that is useful in solving Ill Posed problems.in a cellular model, waves geometrical path was supposed to be straight. For this purpose, svd solution, zero order, first order and second order Tikhonov Regularization were applied. To implement the program, Matlab has been used and the results were obtained as contour map of velocity distribution. To solve Tikhonov inverse problem, the constraint of zero order was applied. Single digonal matrix and attenution parameters were used in different ways in none of which a proper solution was attained. In solving Tikhonov first order inverse problem, smoothing D matrix and regularization parameters of 000, 00, 0 and were used. Parameters of 00 and higher demonstrated good convergence. Key-Words: - Discrete inverse problem, Seismic tomography, Solution instability, Regularization, Svd. Introduction A lot of geophysical problems are mathematically ill-posed since they use incomplete data (Jackson, 972). Regularization is an approach that turns them into well- posed models by adding constraints on their models ( Angl et al.,996; Zhdanov, 2002). That developed by Tikhonov( 963) has been used in many geophysical problems such as travel time tomography(bube and Langan,999;Clapp etal.,2004), migration velocity analysis ( Woodward, et al. 988; Zhou et al.,2003), high resolution radon transform ( Trad et al., 2003), spectral resolution (Portniaguine & Castagna, 2004) and so on. Geophysical data inversion is aimed at taking quantitative information on subsurface structures from some limited indirect observation including noise.the present study tries to learn about structural distribution of wave velocity into a section involving two vertical wells. Any ray through all of the plane s cells and the presence of noise in data vector ( arrival time ) makes equations system coefficient matrix ( Kernel G matrix ) shapes as if it will be impossible to solve inverse problem using the usual method like the least square method in order to estimate model parameter ( i.e. wave path speed). Thus, the problem becomes unstable and some restraints of the problem need to be considered while trying to change it into a stable one. This leads us to use an approach, Tikhonov regularization approach with different solving orders. 2 Methods 2. Tikhonov regularization approach In effort to solve inverse problems, we have to bear in mind that in many such problems small changes in measurements such as a small noise in d= d true + n often brings about relatively big changes in the model estimated. As such, we ll face with unstable inverse problems which need some additional constraints to be considered in order to turn into stable ones. Regularization can provide a useful solution to solve ill-posed problems. Using Tikhonov regularization approach, there will be no need to minimize the following expression min Gm-d 2 (2.) but the expression below is minimized instead:min Gm-d 2 + m 2 (2.2) The first term is the data misfit, and the second is the regularization term. As shownhere, the two aspects of the minimization (make the data misfit small, make the modelnorm small) get equal weight.using regularization solution, λ factor, called Tikhonov parameter, is used to control balance[,2]. The structure of resulting model is dependent in this parameter. min Gm-d 2 + λ Dm 2 (2.3) Given that D n,n=0,,2, ISSN: ISBN:

2 n is a n order discrete difference operator.after that the expression (2.3) has been derived and regarded equal zero, the general expression will be as follows: ( G T G + λ D T D ) m λ = G T d (2.4) The answer to Tikhonov inverse problem which makes a balance between residual vector and model parameter vector will be as follows: m λ = ( G T G + λ D T D ) - G T d (2.5) If D=0 is zero matrix, the approach will be the linear least square. If D=I is (d) identity matrix, the resulting approach will be usual Tikhonov regularization or zero order Tikhonov.If D is (d2) matrix, the approach will be the first- order Tikhonov and if D is (d3) matrix, the approach will be the second- order Tikhonov approach[6]: d=... d2= d3= Singular values decomposition A general approach for solving the linear least square problem, especially unstableproblems and rank deficiency, issingular values decomposition.kernel matrix(g) is decomposed into three U, S, V matrices: G= USV T (2.6) U is an m by m orthogonal matrix with columns that are basis vectors Spanning the model space, R m. S is m x n diagonal matrix, elements of which in the order of value are as follows: S S 2 S 2 S min (m,n) 0 (2.7) V is an n by n orthogonal matrix with columns that are basis vectors Spanning the model space, R n. given that d=gm and singular values decomposition, we will have m T = VS - U T d (2.8) There are one or more zero singular or very small ( S i ) in unstable problems which are as follows: G= [ U p, U o ] [ ] [V p V 0 ] T (2.9) The compact formula of svd of Kernel matrix will be given below: T G= U p S p V p (2.0) The solution will lead to results that are less sensitive to noise data. When Tikhonov regularization approach is used, more constraints must be taken into consideration. The approach suggests that higher weights should be given to bigger singular values and lower weights to smaller singular values of svd[6]. 3 Seismic Tomography of arrival time A section (plane) into the ground involves two wells, sources and receivers. The wells were dug vertically on the ground located to 0 units far from each other. 40 sources and 40 receivers are located in the wells with the distance of 0.4 units. the plane is considered as having cells with 25 long by 25 wide.two primary models have been chosen in the research. One is a brick model of about 0 unit long by 0 unit wide (25 25 cells), background velocity Of V=0, anomaly velocity of V2=9, with 3 unit wide by 3 unit long in the middle ( Fig.).Another model is blob model with the size of 0 unit long by 0 unit wide (25 25 cells), background velocity of V= 0,Velocity anomaly was supposed as a gradual change of velocity from the center of a circle to outwards with V2 =r/3+9 pace. The circle has the diameter about r<3 unit( fig.2). Raypaths have been drawn directly from sources to receivers on the plane. For example, raypaths from 3 sources to all receivers are drawn ( fig.3). Fig. Brick model( background velocity v=0 and anomaly velocity v=9) ISSN: ISBN:

3 Fig.2 Blob model (background velocity v=0 and anomaly velocity v2=r/3+9) Well Raypath RAY PATH Receiver Source Well 2 Fig.3 Raypath, source and receiver location in wells A linear equation system is formed using all rays. t = l s + l 2 s 2... l n s n t 2 = l 2 s + l 22 s 2... l 2n s n t m =l m s + l m2 s 2... l mn s n (3.) t i =arrival time of ith ray s j = slowness of jth cell l ij = section of raypath of ith into the jth cell As follows, the linear equation system in matrix form will be: d=gm (3.2) In above formula, d is observation vector ( arrival time), G is coefficient matrix of Kernel matrix (pieces of raypath ) and m is model parameter s vector ( slowness). t l l... l s 2 n t l l 2... l n s2.,g=....,. d = d= (3..3) m m m m t l l 2... l n s n Generally speaking, any linear problem in is a special form of the following formula: D= d true + n (3.4) n vector is considered the noise of model parameter, d R m, G R m n and m true R n are data, direct discrete operator and actual model s parameters respectively ( m and n are the number of data and model parameter respectively). n stands for noise element of data consisting of two other elements one is related to random noises and another related to nonrandom noises. As is derived from definition, random noises include changes of data which are impossible to regenerate. Nonrandom noises include all laws taken for granted while modeling. It is, in fact, the rupture effect of direct operator. In the present research, it is assumed that all physical laws of problem were not included in the mode, thus, discrete surface is small enough so that the error resulted from discreteness is considerably small. It s why nonrandom element of noise is put aside. When the element is removed, it will be easier to solve equation 4. A problem arises when data include noises. The traveltimes is determined considering two initial velocity model and different sources and receiver coordinates of the two wells. After that using random order in MATLAB, 676 random data (for 676 ray) was generated and then added to the arrivaltime of each ray. Some synthetic data was achieved as observational data (fig. 4). Results of Tikhonov regularization approach is practically less sensitive to noisy data and are used to solve illposed discrete problems. Fig.4 Synthetic data obtained traveltimes and random noise ISSN: ISBN:

4 4 Conclusion Regarding every initial brick and blob models and produced synthetic data and using Tikhonov approach, a program in MATLAB has been used because of instability in solving reverse problem ( seismic tomography).the final results were obtained as distribution maps of velocity among the two wells ( fig. 5,6,7,8,9,0). Applying Tikhonov regularization approaches with different parameters of λ=,0,00,000 the following results were produced. Given all parameters mentioned above and zero-order Tikhonov regularization approach, final solution of estimated model was proved to be different from that of real model so that no appropriate solution was achieved. Using first order Tikhonov regularization parameter of 00 and more showed a well convergence toward the real model. After that second-order tikhonov regularization approach with parameter of 00 was used, distribution model of estimated velocity represented no closer convergence than first-order Tikhonov approach but with the parameter of 000, a rather good convergence was found. 5 Discussion Above results show that we can possess two options to shoes among in attempt to balance between object function or misfit and model parameter. They can either chose the appropriate D matrix with different orders or determine the optimum λ regularization parameter in order to get high convergence. When we tend to use the former method, depending on whether the velocity distribution model is homogenous or heterogeneous, we can decide either of them so that in the present study, first order regularization approach becomes closer to the solution thanks to the absence of velocity contrast. Using the second solution, it was found that for our selected models (brick and blob) in which regular contrast changes of velocity occurred we could approach the real model via increasing regularization parameter. Selection of an optimum matrix and regularization parameter in heterogeneous models with the aim of converging problem solving requires more studies which will be taken into consideration in future studies in details. References: [] A.N. Tikhonov, On the stability of inverse problems, Dokl. Akad Nauk SSSR., Vol.39, No.5, 943, pp [2] A.N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Soviet Math., No.4, 963, pp [3] D.L. Donoho, Nonlinear Solution of Linear Inverse Problems by Wavelet Vaguelette Decomposition, Appl. Compu Harmon., Vol.2, No.2, 995, pp [4] J.A.. Scale, M.L. Smith, Introductory Geophysical inverse theory, Samizdat Press, Colorado School of Mine Denver., 200. [5] P.C. Hansen, Rank-Deficient and Discrete Ill- Posed Problems, Textbook, SIAM. USA, 998. [6] RC. Aster, B. Borchers, C.Thurber, Parameter estimation and inverse problems, Textbook,New Mexico, Tech., [7] S. Fomel,, Shaping regularization in geophysical-estimation problems, Geophysics, Vol.72, No.2, 2007, pp [8] T. Sakamoto, G. Kitagawa,, Akaike information criterion statistics, D. Reidel, Holand., 986. [9] W.P. Gouviea,J.A. Scales, Resolution of seismic waveform inversion, Bayes versus Occams, Inverse Probl., No.3, 997, pp ISSN: ISBN:

5 Zero - order Second - order Fig.5 velocity distribution using zeroorder λ=,0, 00,000 and brick model. Fig.7 velocity distribution using secondorder λ=,0, 00,000 and brick model First - order Fig.6 velocity distribution using firstorder λ=,0, 00,000 and brick model ISSN: ISBN:

6 Zero - order Second - order Fig.8 velocity distribution using zeroorder λ=,0, 00,000 and blob model. First - order Fig.9 velocity distribution using secondorder λ=,0, 00,000 and blob model. Fig.0 velocity distribution using firstorder λ=,0, 00,000 and blob model. ISSN: ISBN: