125 THE FULL WAVEFORM ACOUSTIC LOG INVERSION PROBLEM by G.H. Garcia and C.H. Cheng Earth Resources Laboratory Department of Earth, Atmospheric, and Planetary Sciences Massachusetts Institute of Technology Cambridge, MA 02139 ABSTRACT This paper presents the full waveform acoustic log inversion problem for a multisource - multireceiver tool logging a cylindrical fluid-filled borehole. The probiem is formulated in the frequency-depth domain. A priori knowledge about the source array spectra and the borehole formation parameters (velocity, attenuation, density) is taken Into account. It is shown that the inversion problem can be formulated as a Separable Nonlinear Least Squares (SNLS) problem with separable nonlinear equality constraints. The case where the source Is known and parameterized is studied with synthetic data and an inversion for Vi' and Pb is implemented and studied. v..,v., INTRODUCTION Full waveform acoustic logging has become a widely used tool in formation evaluation. However, up to now the techniques for determination of formation compressional and shear wave velocities and formation attenuation have been done separately. In "slow" formations shear wave velocities cannot be determined directly and have to be inverted from Stoneley wave velocities. In some tools, the sourcereceiver characteristics prevent one from identifying the Stoneley wave in "slow" formations. Measurements of formation attenuation are complicated by the effects of modal dispersion. The obvious answer to all these problems is the simultaneous determination of the formation velocity, attenuation, and density from inversion of the fUll waveform data using a model of elastic wave propagation in a borehole. In this paper, we address the full waveform inversion problem from a theoretical point of view of a constrained least squares inversion. We introduce the concept of separable least squares to decouple the effects of the source from the responses of the formation. We then present some results from the inversion of synthetic full waveform acoutic logging data and study the sensitivity of the waveform to the variations in each individual parameter. INVERSION THEORY Statisticians usually distinguish two broad kinds of data analysis problems (Parzen, 1978): (1) "Statistical Inference-type 1" or model analysis: Garcia and Cheng 126 Problem: What is the answer to a mathematically well-posed question? Approach: Theorems on the properties of algorithms derived from estimation and optimization theory. (2) "Statistical Inference-type 2" or model synthesis: Problem: What is the question, or equivalently what is the model? Approach: Data analysis procedures to be learned by doing (observations, experimentation, modelling•.•) This paper deals with a "statistical inference-type 1" problem, the full waveform acoustic inversion in a cylindrical borehole geometry. This is a parametric inversion method as opposed to non-parametric inversion methods such as semblance or correlation. Previous work in parametric and non-parametric inversion Parametric modelling is common practice in many applications, the very general Autoregressive Moving Average (ARMA) formulation for time series being one of the better known examples. In ARMA modelling, the spectrum of the process Is parameterized by a rational fraction In the frequency domain. For the cylindrical full waveform inversion problem, one considers a parametric form of the frequency-wavenumber spectrum with a finite number of parameters. Much Is lost in the inference process if the parametric form is ignored, because when the number of parameters is small compared with the total number of observations, the results of the inference will be much stronger than when using a non-parametric approach. Parametric methods involve statements about the probability distribution of the observed data. Parametric Inference presupposes a known form of the statistical distribution of the data being inverted for. Because of the finite amount of data, the class of probability distribution is relatively restricted, usually to the exponential family including Poisson, Gaussian, Binomial, etc., via the probability distribution. The model is above all a mathematical formulation of the knowledge about the tool-borehole-formation system under study. Equation (1), for exampie, reflects an hypothetical link between the observed signal and the unknown signal. In other words, there exists an expected value of the borehole wavefield signai in the form of a model-hypothesis built from physical laws (the wave equation in cylindrical geometry). By comparison, the non-parametric approaches, suoh as semblance or oorrelation methods, do not make any assumption about the statistical distribution of the data. For the determination of P- and S-wave velocities for instance, most of the work has concentrated so far on non-parametric inversion methods (Willis, 1983; Willis and Toksoz, 1982). Non-parametric methods are usually more expedient and robust (i.e., results are practically independent of any statistical distribution of the data) than their parametric counterpart. Recent asymptotic optimality results, however, show that these methods cannot be substantially improved without the more stringent modelling assumptions (Yakowitz, 1985). They are used in situations where other estimators are difficult to implement because of mathematical manipulative difficulties. They often rely, however, on heuristic techniques such as windowing, threshold detection, etc. 6-2 Full Waveform Inversion 127 Moreover, in slow formations, where the shear' wave velocity is lower than the borehole fluid velocity, non-parametric semblance/correlation methods cannot extract the shear wave information -- although present -- directly from the data. In these cases, one is justified in trying the more "fragile" and compute-intensive parametric inversion methods by using models of the tool-borehole system with additional a priori knowledge in the form of constraints. In this paper, the forward model structure assumed is set forth and the method of Maximum Likelihood (ML), one of the most versatile and powerful tools of estimation, will be used for inference on the parameters and the associated estimation error covariance. When there is a priori knowledge on the parameters (for example, data from a density tool, a Caliper Log, or an initial inverse from a non-parametric method), the Maximum A Posteriori (MAP) method will provide (via the Bayes theorem) a way to systematically incorporate these a priori in the inversion. Since the source function is in general not known, this will then lead to the Constrained Separable inversion method which eliminates the scurce function inversion from the formation parameters inversion proper and allows a priori about the source and the formation parameters to be taken into account. THE CYLINDRICAL FORWARD MODEL ASSUMPTIONS The assumptions are illustrated in Figure 1. The pressure frequency response P(r,z,G)) in a fluid filled borehole at an axial distance z and radial distance r from a point isotropic source is well known (Tsang and Rader, 1979; Cheng et al., 1982). it Is given by (1) where X(G)) is the complex Fourier spectrum of the source. The first·term within the brackets represents the direct source term. A(@ll.)' the modal coefficient, is given by A(all.) = g(a,,)K,(f R) -Ko(f R) g(all.)I, (f R) + IoU R) (2) and (3) where all. is the parameter vector: 6-3 Garcia and Cheng 128 (4) and = w is the angular frequency; G is the phase velocity; k wi G is the axial wave number; V. and V are the P and S wave velocity of the formation and the f borehole li'uid veiocity; respectively; R is the borehole radius; Pb and Pf are the formation and fluid density; and 1; and Ki are the modified Bessel functions of the ith. order. The term A(0a,)Io(/r) represents the response of the borehole. ,v. Formation and fluid attenuations appear as the imaginary parts in the complex velocities Vf (Cheng et aI., 1982). . v;, ,v., THE -DISCRETE FORWARD CYLINDRICAL MODEL The one source, one receiver (1S-1R) tool For one source and one receiver separated by a distance z, equation (1) above can be written: P(w,z ,a) = G(w,z ,0) X(w) (5) The observed waveform W at frequency w is modelled as: W(w) = G(w,z) X(w) +N,,(w) (6) where W(r.:J) is the Fourier transform of the observed output field, P(w,z ,a) is the modelled output pressure field, X(w) is the input source and N" (w) is the corrupting observation noise. G(w, z, 0) is the complex frequency response of the tool-borehole formation system at frequency w. It represents the term within brackets in Equation (1). is a parameter vector, a subset of aa, shown in Equation (4). One or more of the parameters in aa, may be fixed or known a priori with some uncertainty. The source function may also be parameterized by a vector ~ a X(w) =X(w,~) 6-4 ( Full Waveform Inversion and ~ 129 may be included in the inversion. The Fourier spectrum of the pressure can be discretized at N" frequencies !(;Ji l, i = 1 ,N", as follows: (This is a valid approximation by the Stone-Weirstrass theorem if the power spectral density matrix of the modelled process is positive definite and continuous) In matrix form the N" equations above become: P(z,B) =G(z,S) (7.a) X where: p(z,e) = !P«;Ji'z,S), i=1, . , . N"J G(z,S) = DiagHG«;Ji'z,S», i=1,··· XW = !X«;Ji'~)' i=1,...,N"l (7.b) N"J (7.c) (7.d) Note that P(z,S), X(~) are complex valued vectors and that G(z,e) is the complex transmittance matrix from the source to the receiver at position z. Multi-source/multi-receiver discrete model In a similar fashion one can derive (see Appendix) the multi-receiverjmultisource discrete model. The equation has the same form as above. For a tooi with N~ receivers, Ns sources and N" sampling points in frequency domain, Equation (7a) can be generalized into the form P(S) =G(e) . X (8) where P is an (N x 1) complex vector, with N = N,,'Ns ·NT ; S is a vector of borehole/formation parameters; G is a N xl.! matrix where M = N,,'Ns with the structure shown in Figure 4; and X is a M x 1 source array vector. The details are given in the Appendix. NUMERICAL EXAMPLES The case where the source characteristics are perfectly known will now be studied on synthetic data. The purpose of this exercise is to find out if the problem as defined above is properly formulated and study its conditioning. In addition it brings information about the sensitivity of the functional being minimized with respect to the parameter being inverted in a controllable manner. The synthetic waveforms have been generated with the following formation parameters: Tj, 4 km/s, V. 2 km/s, VI 1.5 kmjs, Ph (bulk density) 2.3 = = = 6-5 = Garcia and Cheng 130 = = = = gm/cm 3 , P 1.2 gm/cm3 , Qp 60, Q. 60, Qf 20, and the radius is 0.10 m. We used a Kely source (Toksoz et al., 1 983) with a center frequency c.Jo = 10kHz. Figures 6 to 12 show the shape of the residual energy as a function of various formation and borehole parameters. In each of the figures, we have kept all but one parameter constant. We vary that parameter about the actual model and compute the square of the difference in the pressure function (in the frequency domain) between the varied and actual models. Figures 6 and 7 point out the fact that the determination of >j, is not as well conditioned as the determination of Ys. The shape of the residual energy function (the "incoherence") is much flatter for >j, than for Ys. This is due in part to the small P-wave energy in the signai. Furthermore, as pointed out by Cheng et al. (1 982), there is very little P-wave energy in the guided wave packet, which is usually the dominant arrival in a microseismogram. The P-wave velocity is nevertheless determined by the whole waveform, rather than the mere arrival time. It can be shown that this increases the achievable signai-to-noise ratio compared with any other inversion method which does not use the complete waveform. The formation shear wave veiocity, on the other hand, is very well resolved. A small error in the estimation of the shear wave velocity results in a large residual energy. Since a least squares inversion minimizes the residual energy, the formation shear wave velocity is very well constrained. In Figure 8 one sees that a good determination of the borehole fluid velocity is necessary since the shape of the residual energy-function is drastically affected by the value of the fluid velocity. Note the same kind of behavior is observed for the variation of the energy norm as a function of the borehole radius (Figure 10). It appears therefore. useful to include the caliper and independent measurements of the borehole fluid P.-wave velocity as a priori in the inversion with an appropriate errcr covariance to take into account this inherent sensitivity. Figure 11 illustrates the poor resolution of the density parameter Ph' The scale has to be magnified two orders of magnitUde to see the variations as a function c r Ph' This suggests that formation density is not well resolved in the inversion as it is set up at the present time. However, by using an external source of density measurement as a constraint to the problem, we may improve. the resolution of the formation density.. In Figure g variations in the residual energy as a function of the P-wave quality factor Qp are plotted. Qp influences the function in an asymmetrical manner, and the minimum is not as sharp as for the other parameters. This suggests a reparameterization of the inversion. Instead of using Qp as the variable, 1/ Qp would give the problem a better conditioning and avoid the sharp angular features in the residual energy. Physically this makes much more sense since 1/ Q is a direct measure of attenuation (amplitude decay). The variations as a function of the pwave attenuation are representative of the variations for all the 3 attenuation parameters. In Figure 12 the sensitivity to the source center frequency is plotted. One can see that the source parameters do affect the waveform quite drastically and that it is a potential problem since there is plenty of experimental and theoretical evidence of the variability of the source as a function of environmental conditions. The best 6-6 ," Full Waveform Inversion solution would be to measure the source and constrain the problem with that measurement, or to use the Separable Inversion method to circumvent the source inversion problem. Rgure 13 and following illustrate the inversion of synthetic data for 4 parameters: formation p- and S-wave velocity, P-wave velocity in the borehole fluid, and formation density. Figure 13 shows the residual energy function or incoherence as a function of the number of iterations. We can see that the incoherence decreases monotonically as a function of the number of iterations. Note that Vs and Vi are much better behaved during the inversion than Y;, and Ph (Figure 14). This i1fustrates the fact that the introduction of the density and the P-wave velocity creates large excursions in the parameter space. These non-physical excursions do slow down the inversion problem B.nd can cause divergence in the case of large residuals. Again the behavior for the synthetic case suggests the limitations of the tool in real situations, and the use of additional sources of informatior. to "robustify" the problem. One can expect therefore, without any additional a priori knowledge, that there will be comparatively large uncertainties on some of the ML answers. The MAP criterion will have to be used in the real world to obtain more reliable information about the shear veiocity and attenuation. The evolution of the synthetic full waveform that would be generated from the successive values of the 4-parameter vector and their spectra is shown in the last two figures (Figures 15 and 16). One can see the significant changes in the microseismograms and their associated spectra as the parameters converge to the model solution. Note that even between the last two iterations (7 and 11) there is a substantial change in the waveform character despite the small changes in the parameters (Figure 14). SUMMARY In this paper we have formulated the full waveform inversion problem for a simple cylindrical borehole. The probiem is formulated in the frequency-depth demain. We demonstrated our algorithm using synthetic data in a simple one source - one receiver model. The sensitivity and the shape of the residual energy function are studied. The problem is well behaved. It is shown that formation shear wave velocity has the most influence on the residual energy function, while the formatlcn compressional wave velocity and formation density have less influence. This information is vaiuable for the applying of constraints in the inversion of actual field data. ACKNOWLEDGEMENTS This research is supported by the Full Waveform Acoustic Logging Consortium at M.I.T. 6-7 . i ' 131 132 Garcia and Cheng REFERENCES Cheng, C.H., M.N. Toksoz, and MoE. Willis, 1982, Determination of in situ attenuation from full waveform acoustic logs: J. Geophys. Res., 87, 5477-5484. Golub, G.H. and V. Pereyra, 1973, The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate: SIAM J. Numer. Anal., 40,413-432. Parzen, E., 1978, Discussion on histogram splines: J. R. Stat. Soc., B, 40, 1-25. Toksoz, M.N., C.H. Cheng and M.E. Willis, 1983, Wave types in a borehole -- a review: M.I.T. Full Waveform Acoustic Logging Consortium Annual Report, 19S3. Tsang L. and D. Rader, 1979, Numerical evaluation of the transient acoustic waveform due to a point source in a fluid·filled borehole: Geophysics, 44, 1706-1720. Willis, M.E., 19S3, Seismic velocity and attenuation from full waveform acou.stic iogs: Ph.D. Thesis, Massachusetts Institute of Technoiogy, Cambridge, Massachusetts. Willis, M.E. and M.N. Toksoz, 1983, Automatic P and S velocity determination from full waveform digital acoustic logs: Geophysics, 48, 1631·1644. Yakowitz, S.J., 1985, Nonparametric estimation, prediction and regression for Markow sequences: J. Am. Stat. Assoc., 80, 215-221. 6-8 Full Waveform Inversion 133 APPENDIX The multi-receiver forward model For a tool with NT receivers. the observed set of waveforms at each receiver location z,.zZ.....zN Is modelled as T (A-1 ) In matrix form the model of the pressure field for an NT receiver tool is pea) = G(e) (A-2) X(O where: pea) = nP(CVi,Zj.a). G(a) =[(G(z,.a)•.. , i=1 .....N,,; j =1 •...,NT nT G(Zj.a)•... ,G(zN ,e)]T T with G(Zj.a) =DiagnG(cvi,Zj.a), i=1, .... N"n N"n T X(~") = nX(cvi'~)' i=1 •... • As shown in Figure 3. the matrix approximation for NT receivers is obtained by algebraically "stacking" NT models. one for each source-receiver separation, Zj' The transfer matrix G( a) is obtained by also stacking the individual transfer matrices. The vector pea) has length (Nw . NT) • Matrix G has dimensions (NT xN) and the source vector has length N ,,' The multi-source!multi-receiver forward model When the tool has an array of Ns diffecent sources represented by complex vectors X, • Xz • ",X N (the Fourier spectra of each source function). a new matrix s approximation is obtained by stacking Ns models as follows (see also Figure 4): For each source indexed by u the model of the pressure field for Nr receivers is (A-3) where with 6-9 Garcia and Cheng 134 and z}u) is the distance between source (u) and receiver (j). The ."I. models of equation (A-3) "see" the same one equation: p(e,O e and can be aggregated intc = G(e) X(O (A-4) We can re-organize the equations such that at each receiver (u) the ."I. waveforms are stacked. One now has the ."I. equations: (A-5) with the structure shown in Figure 4. This can be written in one matrix form: (A-6) ( including all the receivers amounts to stacking the matrix equations above Nr times into one equation: pee) =G(e) X(O (A-71 where G now has the structure shown in Figure 4. G depends on a vector parameter e, and on the unknown source vector X. The separable inversion method The function to minimize, the residual energy, is a sum of squares of functions. The frequency domain formulation reveals that the problem has more structure than the classical nonlinear least squares because the source vector X appears linearly in the functional E: E(6,X) = I/w-G(e) XI/ 2 , (A-8) The approach to the solution of is to modify the residual energy functional E( a,X) such that consideration of the unknown source parameter X is deferred. The variables and X are separated so that in each iteration the solution for 6 is obtained first and X is obtained in ONE STEP after that. e 6-10 Full Waveform Inversion 135 For any fixed @ , the problem of finding the optimal source vector X corresponding to that 19 is a linear least squares problem. The solution is found by: ~ (A-9) where G+(e)is the pseudo-inverse of Gat e =e Define now a new functional E 2 (0) by (A-10) One sees now that E 2 depends on 19 only and can be minimized with respect to 0 by means of a Gauss-Newton algorithm to give an optimal So, if there is no a priori knowledge, the separable method is 1) mini I (I-G(0) G+(e))WI1 2 gives e e. 9 Golub and Pereyra (1973) had shown that this two-step method is equivalent to the direct inversion, provided G(e) has a constant rank around By taking advantage of this structure, one does not have to minimize the residual energy E with respect to the source parameters X and so the size of the nonlinear problem has been reduced. e. 6-11 Garcia and Cheng 136 sources __ "s (~) Receivers ( Model (6) Source ( f) Figure 1: The Cylindrical Fluid-Filled Borehole Model. 6-12 Full Waveform Inversion 137 one receiver,one source S Vector-Matrix form: .Borehole + Formation FFT Data Source Figure 2: The matrix approximation structure for a one source, one receiver tool. 6-13 138 Garcia and Cheng Multiple Receivers (Nrol them) p = G(e)·S(~) + N r N = (~1'~2 .... ~N) Figure 3: The multireceiver model. 6-14 Full Waveform Inversion 139 Multiple sources (Nrof them) Multiple Receivers eNs of them) Sources vector Data p=G(e).s(~) Figure 4: The multisource, multireceiver model. 6-15 +N Garcia and Cheng 140 MEASUREMENT SPACES (kz,t) !)~/' .. ", (z,t) /"f/ .... , 'Unitary '" . Transforms ,, ,, ,, , ~ \l c.,-<" (z,c.J) THE INVERSE PROBLEM Measurement Space Parameter Space , Estimation Theory 1-"----1 OptU:zation --;;:t' I.. . ----:~~: Figure 5: The unitary transform between the 4 measurement spaces. 6-16 ~.~ Full Waveform Inversion 141 ~ ~-.-------,..---,..-----,..----., ~ ~ ~ -' ~ - : -: : -,-:- """! -:- -;- -; - -; - - - -- - II p _.- - - - w~ J~ S . G( V p) - - - -; - - ... :- ... -';_ ~ -; --- - - ... - - -, - ;.. .. ;- -----;-~~-~-- ;- -~- - . ....:, - , -:_:_ .... _ ..... _:_1..._:_ '- ~ - : - -0 - -, - - - -,- ~ ~ -,---~~-r-;- __ :_ ~ _L. j_~ _:_ - -: - - -: - .... -:- - -:- - - :- ..... - t- -i- ; ,. ,.:;,;;, --;----~--------~-~_.- - - - - - - -' - - - - - - - _:.... ....: - ;... -.- - - - ~ -, - - -,- ~ -:- ~ - ~ ~ - ~ -;..; a.. .- - - - - - -. - - - - - :- -: - _.- _:.._'- - - -'-": -'-....:- ~ -:- - -- - - -,- - - : - - -:- -: ;- .~ ~ -:- ~ ~ ~ --~-~~------'---~~-~-'- gl-~.oo 4.20 3.80 5.00 4.60 VP ~.,..-----------.,.---.,.-.,.-------, - - ,- -; - : -: - .... - - ~ -:- : -: ~ - r -,- --------------~f----c-- II IIp-s.G(Vp ) -=====:= - :... - -,- - :- - - - _.--------------------_.'- ~ ~ -: , " -- - - - -- - - - - :... ..;. -, - - - --- .:.. -'- ;-. ~ ' , , ' - -:-' - - '- ..; - '- - '- -' - :.. -' - ....: - - - - t.:l X (flO o '0 a.. '" o o +-----,----,-.,:",.-""'-,....---.,.-----1 ~.oo J.40 4.20 3.80 4.60 5.00 VP Figure 6: The one source - one receiver (15-1 R) model residual energy as a function of y;,. 6-17 Garcia and Cheng 142 0 0 0 LI> (\J -j , '"0 0 -~ XO --' 0 -, (\J - - ~ P - -. -;- ~ - -' -0 (JJLI> -; >- --' r.=: - ~ - L - , , - ,- -+ - - -,-,- -i Xo G( Vs ) ,- ~ "" -< .,. - ;- .., ;- -; - '- --' - - - ; - , - - , - ;-, - ;- -,- + ! - , , '1" ;.. -I -" ,- ""1' Ir 1 ...;..-- ~ S '0 _0 '- -- - - ~ - , - t- -,- -!' _. --; - ! ! -r- +-. -f, 1 '- -! - i- -1- ,. -~- , i- , -t- ~ - --; ~O "" Do --' 0: 0 -; ZLI> - ,.. -,- .,. c - - - c ~ -' "" '- -' - ~ -i - -, - '- --' - -' -,- -,.. ! , , !.. _i_ ~ - ;- ~ -, 0 0 0 ("-!- l.._1._.. ~' '- .J oj r- -j j .., - '- --' , , ,- - -; , -; ~ 0 I 0 ll. r- 4 - L, ! ;- -; T '- - - (JJ~ ; . 1 00 . 1 40 . 1 80 2. 20 2. 60 VS Figure 7: 18-1 R model residual energy as a function of 6-18 10;,. 3 • 00 , f Full Waveform Inversion o o • o U"l.,..-....,........., l'\J "'0 0 143 ....,........,...-,.._......,......,......,....,..._......,......,....,.......,..._......,....,...--, ! ! ! !- "1 "r" '-j" ··t···-·j·- ·t" . . J . _.. L.-1.- . 1.. _.1__ J.. _.L . J L.. _L_ . 1 . _.1._.. J.. . . _["- 'I ··-t- ·,t" ---1- '"t -c: 1 ; !: , , ii , j , "r- "-r- "T '-'r- .._;- -"r "-r "; "-1 :1:0 . .L.._1._ . 1... i 1 j ··-t·_··+·· ! i o N "T" , s p .+.. _.I-. .+ .J i .-; ! !, , "T"'-T-"';' Iii I i i i G( Vf 2 ) . o (.f)0 o 0... i ! i ... _'.. ; i i . ._+_.+.i 1 1 i i ! . .-.f.. _..t-" ._~.- . 1 ! ! · ·..T-··!"·-r-., . . -!.: .-j' -l-. o o t ; , : ~ ! ...;. i ! ! i 1 ; , '_ , i -.I. .L. '_~"_" i . _.!,_ . 1. _, !__,.J.. ! I! 1 i ·f-··_(--t -1- -t- -'i---t· i I .1-, _ ' . _ -toi .-. i1__ j .:... j ...l. i ! i i f- ~ +_. -[ -". +- ---1 _.-+ -"i-·· ! 1 l ! , ! i !- I" -'r.. _! '-"T -T- "1' '-"i-" 1-~. - . i- -! +- . _.j _. -1, i ; ! I ; ! ; ! 1 T'-T- r'l :-~ -1-.-1 t, i-.. J J. I i t- . -t. J .;.. _;i l l ~ j, 50 VF 1. 60 Figure 8: 1S-1 R model residual energy as a function of V . f 6-19 .. ! ; +-,.. , _!._... , l ! .t- ·-t·i ~ .. ! . _+._. +-.. _+_.. . ---"r" r 1 . --f.. _.. t-. --f--" L- . --l ,L..-+._. , , ,;"" : ;-.--; I j -i- I j,1.J0 ; ,~, I ! !' -'r"J' i :- -r- T 1 , , ~ -+, -~,- .J... , ·T··-1'-"· , ; f- .-: .+- . _,[._ . i ~ . L._l._ . j_.~ i __ L .. J. - t - "t ! "'~"-"T""' i , ; ...--;..- ..t- ··-t·_··· ~."! : -r- i --t-"-i ; ..1- .. , t-···-l.. _·· t· --·1·-- .-t. --··1- -f 1 i l ........i : 1 ··-t-.. t· -"i-' l "!"-r-!, ! ! -·r..··! I . . --t..- .'t-' -;-,..--;..- .+- ·-r-··· L. . J . _; i ! i "r-- ·-t·_··, ~ ._,~-, r.=J :1:0 '-1 ·_·.. t '-1"'-'" L .. J . _...L ._1._.. ~ ! 1 ! ."~ 1. 70 t -1-.. _1._.. : y-·-f·_.. ! ! 1.80 144 Garcia and Cheng o o :;;,---.-----~--O:___:_7""'"'7'""_:_~"'"'7"'"""':"_;___:__:__::__r N ! j .4, _:_ -l I- _:_ . . _1 p-s G( '_ ..... - ~""""t Qp) ,, ,, , .-f. _., j-... -f-' ~ r- , , , ..- ;- , ~ Il. - i- -+- - 'f- -+- -1- + , i - , , - ! , -ii - I - i-- I i. - !- i. I - i.- -I - -i- ., I, .j. - ~, -r-1-' -1, , _.L-J.._.' , _!.l. , J ! ! .i. ~ - ~ , - , , i- ! , ~"1' , .- -r _. - -' -,;- , ~ i , , 'i - _. t-tI I ~' -t·_· ! -'---: : : -r-' L _L_ _. j-1- t _.[- .... j-" , , , 1 -'- - ~-' , , i ~ ~ i ! i i ~-:- ... -i-- - ,- ..; ', , , , ; ..; -f-i--~-+-;- , I , , ! 1 ! ! !- -i !-. -! 1- -i- +- -! - -+ - [- oj - '!. f , - r , , - I -; - ,: ! , , , .l. , - - - j 'I , _.:_.~. -;- i , --1·· r-'~' iii ; , ~. -t - I _. r- -r- t -t _ . . . _1._ ... - "" -. - - - .-.- -_.-.+- ! -,-1 - _. t' -t-· ..J_:"._1._ , , , ~ 90.00 QP -t - l, -'- ..l., i 130.00 Figure 9: 1 8-1 R model residual energy as a function of Qp' 6-20 ;.. - - t- -!.l. r -, -+ ,.. , - '- - -. -i= ~ =~-~ - r \ _ l... _;_ 170.00 210.00 Full Waveform Inversion 145 o o o 0....-....,.......,..._.,..-....,.......,...__.,..-....,.......,...---,_.,.-.,.-__....,.......,.......,.......,..._....,......,......., i ! . .-t..- . t"" .t· ! i -:-"-; 't-,··_!,·_..'t···-t-···t i···_..·t- '-i" -t ·-t-_..· . . J . _.. L.. _1._.. 1. _ 1_ 1. _..L J i .1.. _1_.. J.. .J . L.--1.. _ . L--1.-.. ! i [ ! ! i ! ! iii i i .._+._.+ ··-f·- . -,-- -r- T ,r" ...;......_.;._...-.;.. ' " "1 "-"r "-1"'- '0; "-r-'" : .L..J . _ j __1._.. . .J . _..1.. . _1._.1 _l_. J..._..L .. ! ._J._ i ! ! i i ,or i i j ! 'r i i i i . . . ~ . .-;.._ . +- ··-f·- .+. -;-" -+ .f-.. ! ! ! ! i r -t-··-f·_··· ! ! 0 ~o "'-1"- T "-r-' ~o ",-!., 0. 0:0 0:(0 t:l :K (1')0 o 10 i ~" ! ; ; ~. ! ; ... _'. i :::l' Cl.. !! ! 1 ~·-t· f ! ''-'' 1 [ ! T ; ..";"'.. i -.- !! f..··-f·- -+. 1 i -!- ! 1 -+ ! ! i "r- o'-i 0 0 ._.L_. ..L, .-!.- ! .t-. i '-'"1- "-r . ,! , 0 _ ..!-. ....1.. L.. ! P --i i,- ...;, ! ! S _+. 1-. ~ .. . f- .. .. .. ._.!.__.+.. ; i ! i ! j .! "-'j-' t· ' ' T '-r'-' ! . ...; ,-L ~ 1- _.l.-. -1..! ! ! ! i ! -! t ! i L.. J L ._1 1, _.i_.. 1.. ! ! ! ! ! :- i i- -1 T -'1'- -r _~ : o ._~. 0 0.13 i 0.17 RADIUS Figure 10: 15-1 R model residual energy as a function of R. 6-21 "-r-'" o , ! --1'-- . L ~ "-1'-'" ! "J.. _ . t-.. -!._.. ! ! i ! ! 1 . _+_. + .._ ..j- . -+. +- -+.- ·+··-t·_..· - j.._. i 0.09 , -i z ; .. --! ; 1 o o "'---1 ~ i -i- -,- ! ! i ! r,' ...j J. t· -to ··t -t- i ; . 1.. ! ..-i. ! J. .: 9J.05 .! i ..... ! ! ..--1 i ..--+, ! ; ~ G(r) . -f "": --i i, ...l. i 'i' ! .. ~ .~. . +- ..-i·_.. ; ! i i T- .....1"-. '-r-'" .l. j . - . L . -1._.. ! ! ! ! -! t" ,--.l._. L. _1._. L i i Ii l r' 0 r-' " -r-'T"-r-' ! o 0.21 0.25 Garcia and Cheng 146 o ~~,-.,---,-,---,-,-,--,-,,-.,-,--:--.,......, -: - !'" -; - to - , - - - ,- ..... - .- -; - ~; t= = ~ 1" - ; - + - i - -t - : - -I - f 1~ 1"" - ; - trt = =:= ==!= ==C1= =l = =1= =!= ·· = =,=. - . ........ . - _._.- ~=~=:===llp-s.G(p) 'b :::5 ____ 00 :I:u> 0:- ._~_ - -.; - c -: - - - ,- ~ 0 - ~~ - ;.. -: - II , . , . , ·, . ., ~_:........;-;.._:- ~_!--i_i.._i_ ;- - : - ., - : - ~ - r- -i - ~ -:- -,- ... -;-""'! - t- -!-.,. -;-"'!" -!- "": - r- -: - t'" ; : : , : --..- - :- , - ' -'-.-'- : -'-~- xo ; 010 ~ o ~ ~ ~ ; ~ ~ ; ; ~ ~ 10 ___ . . . _ ... _:_ ... _:_ a... ~ _:_ ..... _;... .... _L_~_ : ' ~' : -~~-~~-~-i-~-~~-~~-~-!" :I: 0 0:0 00 -i - ----.---.-- :_ ~ _,_ .,: Zu> ck.oo -; - ~ .. - : - '7 - ,- "'7 - : - --: - -: - ;'" - : - "- - -:- . . . - ~ -:- ~ -;- 7-:- ~ - ~,- ~ -i. , .. , . . . " " , -----------"----------_.- I- -; - o 0-l- - - --. .------~~---.-.--"":------- ...;-...:.-_..:.-.,.....;.....--'-;._....;.....-l 2,10 2.20 RHO 2.30 2.40 2.50 :i\,---------------.,--....,-....,--, -, -,.. - - r -.-'" -:-"""'! - ... -;-:- -,- -: -;- '1-~ ~ - : -;- . .. ----------,-----------, . , . , .. , . ,. .", "." . "", -~--~---,-~----.---~~-~-'- -- .; - .- -: - ;.. -:- ---'-'--'-'- ----'-:~ - _.- .; _:.... - - ( :.. _:- ~---:----.-.-~------,- x .- 010 o 10 a... '" ~ -' - -._. -"~ , , --~-------- ,. . ------- -.-....; _... ~---,- "2.00 2.10 2.20 RHO 2.30 2.40 Figure 11: 1 S-l R model residual energy as a function of Pb' 6-22 2.50 Full Waveform Inversion 147 0 0 0 , In N .. r- --; ; 0 0 -~ x0 0 N _0 _0 <.::lo x In a3: ~o ..., - - - ,.. -, _. -! - i.. -, - ,.. _. -' ~ --j -; (f)0 -;.!. .:. - ...j ::l:0 0: 0 Oo Zin - ~ -' - ~ t· -i - 1" - -- - t -i-' """! 1 _.1- ; _.i -',-! -i! -,-- "'! --r- t' 2 -; --' C- _. - - i! ! , - --' , ; ,~ -,- -. - ;- r- -1 - t f- -' , , ;- -oj ~ ~ - ;- - -; - , -i- , ,- ~ ! , ! _. ! - - ,.. ,- '-; '- ;- -~- , '- + -1- -j - _.'- ; +- ,. , , , , t -t- ; ; ~ ! L -.- , ; -!' - 1 r- i- -1 _. i - f- i _.i_.. i '- -, - '- -,- -,. , - - ;, ~ --' '- L. ! ....... _. 1-" ! ! !--+.. - ; , p - S . G( CJ o ) , , - , - ,.. .., c- , - , - t -i- t ,- 0- , ~ 0 I 0 - ;- """! , ; -t , - , T ; -; '- t +- -l- r- - - - - -:- :- -,- -'.,. - - - .,. -: - ,.. -, -' - '- -' '- - 0 0 95. 00 7. 00 9. 00 1 1 00 CEN TER F REQUE NC Y 1 3. 00 15. 00 W0 Figure 12: 18-1 R model residual energy as a function of wo ' the source center frequency. 6-23 148 Garcia and Cheng 0 0 0 0 0 - j-..._ ...t-..._ ... -j! II> X .i .1. ; -t. I "T 'r' . . _.. L .L 0 0 co 1 ; .. ~ ; .l.. 1. ot- ..l. i t- +1 J' ·"t ! 0 0 co .L. . .L +-.! . t·i W u zo a:o ; w~ ~. i; c- W:::I' I ~ ; i ::I: +! •..i.... I .1. i ··t·; 'i , i j I I ~. 'ri ..!. zo ; C\l i j- L . . _.. t-.i 0 0 i 1 00 0 .-+ , ! =r=( V ! P - S G( Vs ) 1 ( Vf ) i- "I! "I 1"_..1-. .L. .J. -!._. -1··_· ,; i i ( Pb ) -'i- '-"1- f-" -j -1"-" -1"-' i ; ~ ; i; -; , ; ~ ~ ..- ; '-1"I -i ..;.; I I -+ ~ I I I ..: i; ; I ; ; i -0 '-'!- ,_1 .. _ . i -,; - : ; ._.1._. ! ! ! I I i; t I i ,-1"-,, -.j._. -,,1- ._.. j'-i"-' ; i I; , i -1-- -1,,-- -: - , i ; _.. i_ 7 00 0 . 1a 00 I TERA T I CJ N S TEP Figure 13: The residual energy reduction during a 4 parameter sion. 6-24 ;... I ; ... ; .;.. !j- ,L i ., 'Y-' ; ! 13 00 0 (v." .~ .+- .f! I TO' , 'I' L i -·f-.. --i- I I t- -i-- ; ! TO, ,; . i I ; I '-'1'-' r'; +-: +-; i ; -_.!-_. ..L. ; ; ~ ; - ,~ .1--...- .;... ._+_. -1"-- --!.- . _..!- -_.-!; I i ~ i '-",- I .~. i ! 1 '-r- '-j'-' -l- '-"f-' -'i- r~ -\.- ,-1.-. -.1--. -·l- -·1-. "i i ! -[-- -"i-1 _. -1 - "T" 00 ;,... ._.~- r .,...,; 't t·· ; ; ..l. I l-, I I 0 ; i ._!..- -~ .. lj t- r--j- ._L_.. _.1._. _.1_.. _..J_ ;,... +,.. .1.. ,, ! I I ! : i i ._"!""-1 -[,._. ·_··f- '-"1- 'r- +_. 'r -r'-; ; I i. i .J i"I '1"-' !-_. L L ._.. 1.... _.. 1I i 1 i -t,. .~ -i._ p ) +-. t2 i ! ..1 .-:i ...;. t 1 ,.; 1--1- --j"-"'-,],- '-"1-"'-"1- _.+ 1 .J. ... 0 0 ~ .~ I; ...... 0 1 -1. 1; I + : ..._ ...1- 0 W 0 1 i i i ._.. -.;.-i ....'f I 0 0 I i; '1 ~ .L ! °0 '-'0 ! i t .;... . ,1 ; ; 1 6 00 0 v., vf ' p) inver- Full Waveform Inversion o "''" N,.------------------, ~:.,-------f..\----,--------,--. ~-~-~--i--i--~-~-r-~.. ., ., ,. ., ., ., ., - -:- -:- -:- -:- -:- -;- -:- - ... ~ M _ ... ~-;'i--!--i- __ r:-..~,"-- o o .; - _. __. -';"'- - -:- -;- -;- -!- -i- -:- - i--;"- o o 149 _ ---------~-~&-+------- N _l-.-:-_i-_i-_ . ----------_._-~-------,..- -~-~-~---~--,------~-~-~- - ':" ___ ~ - -; :..._~_~ - --- ~ -: ~ - -;- - - -.- - __ __ .__ ~ ;- - ,... - - ;- _ ~_~_L ------------------.--:--~---,...--- - - - - - "-" - - - __ .... - - - - - -- ~_-_~_~_~_--_. '"> --~-~---~--_~_--~--~--.-_~_--;---- - .,. - - - - - - - - - - - - -'- - ... - - - -- ~-7---~-i-~-~--i--i--~-~-t-t--~-~_"'-~----'--'--l----~-~_~- ... o o ..; ----- -- -- ;- _:... _.<. - -' - -- - - ... - - .,. -. ., -, - -; ... -,- -;- - ,- -.- - -..: - -,- -:- -:- -'- - - -: -'., ... -,- - ,- - - ;- ... :- '- ... In -' '- - ......... - -----_~_~_~_--_._--------- -_~_~_-'--------_:..._-Q o:-I-__~--~--'--.;_--_;....--'-___l 4.00 7.00 !O.OO 13.00 ITERATION STEP o o ... ;- - -'- ... - -:- - -: - -1.00 ... ;--;--~-t ....... - '- - ... - .... ... 7,00 10.00 13.00 ITERRTION STEP 15.00 -:.,.----,---,------,----~ _ ;.. _ ~ _ ~ _ i _ i _ -: _ o -:--i--~--!--;--;....-:'_-;...-;- ~ 11.00 o o o -----------~-~-1--:--j--r-~-L-L... ,.. - ;:.' - .,. - - ... .., - -------. 0-l---'----.----,-__---r--'_ _.;_--___l 16.00 "'"':,.----------------~ '" N .'. ---;--;---_._-,--;--;---- -;--:---:--- '" '- -:.. ... : ' - -,.... o ~.OO ._--_~_~- _._~_ - '" - ..; ... -:- -:- -i- ... ;- ... !- ., i- ... ,.. - ;.. - :- - ':" - j __ !__:__ :__ _:. . _:- _ _ ~ ~ ; - ..; - ...; - -: - -; - - j - -j- -;- - :- - :- - ;.. - - .... - ..: - "1 - - - - -,.. -i - -: - -:- -:- -:- -;- - . . - . . - -, -:- "- - ,.. - .,... ~ -~- -:- ~-i-i-~--i--i--~-r-;--T- _-l ... -i _.j ... _: __ ~ __ ' __ ~ __ ;_ _ ,- ... L_ o o lU~ .... a: a:: -,"' ............. C :I: a:: ----_~_~-~-~-~-_._-,--~--~-~_~_-- "' '" - - -:- - -:- - . - - - - ... - .., ':" ~ -;- -:- -,- -;- -- __ - -. - -: - -,- - ,- - -~-:.._~_~-~--_~-~-_. ., -;- - ~ - - _...: - - - -'. -:- -!- -;- o o -:-- ~_~_~-:.._:..- _~- ,- - ,. - ,... - -t.OO -1.00 10.00 ITERATION STEP _. - - - - - -'-,.. '- - - -;.. - _:- -,- -:...- o 7.00 ~--"----;---------- ..; g:+:-:----"'T--,-.,..-----.:...---T~---~'----.:..---_T',__- .,..-.,__---.:..---_T.----~'------C _q.oO ..: - -!- -"- -;- -;- -;- -'- _:... -;-- ""0 ........................... -~~_'-" ...,;,. - - - - - -- - -i- -i- -:- - '- - - _:..-:- -:- -~-~----;--~-~-~---:-- lU o __ ..,. ... .;. ... ....: ... ...; ... ..: _ ......... ; ... _:_ ... i...... ;... _ :.. ... :. _ 13.00 16.00 0:1-....:..---'_~-'--....:..--...,-'---~....:..-'---.;_--!j.OO 7.00 Figure 14. Residual energy reduction individual contributions 6-25 10.00 ITERRTION STEP 0:;', y", VI ,p). 13.00 16.00 150 Garcia and Cheng - .... 1.e 1 - 0- e.• f\ 1\ e.e -2l!l~- -<.S -.. a~9-f-·-~-1 ._•. e e., ., """-r-r-- i e.• e.e e•• 1.e I Le l 7 e.s I e.e .L.i••• I ••• I e.• I e.• I e.e I I.' , .e e.s II '.e -a.s -<.s -, .•.:j..---r--,--T.-'-'-,-~-"-~-T--,---j a.a if.iI e.. t.. e.' 1.' -I.il.+_~_ 1- --·-"'-T--'-l·-r·--,,-...,.-~ 0.. a.,i It... e.. t., l.. Figure 15. The time series evoiution during inversion. 5-26 ( Full Waveform Inversion 151 . ..,....--------------, -..,...---------------, sa 1M .. , •• ; ! • , -r-r-rrr-Tl-T"T""f!T'-TT" TTTTTT" • ,. IS e. - ., --,,-,-,-j,. I I 'U~~TTTrPTTTTT . .. ~ • , " eo ,. .. . -... 7 / " - e.. ... • J' ,. • , .. rrTT Figure 16. The spectral density evolution during inversion. 6-27 ,. II ..~ • ,. 15 eo . JO ,. 152