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Wavelet estimation and debubbling using minimum entropy deconvolution and time domain linear inverse… Levy, Shlomo 1979

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WAVELET ESTIMATION AND DEBUBBLING USING MINIMUM ENTROPY DECONVOLUTION AND TIME DOMAIN LINEAR INVERSE METHODS, by SHLOMO LEVY B.Sc,, University of C a l i f o r n i a Los Angeles, 1977 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Geophysics and Astronomy) We accept t h i s thesis as conforming to the required standard: THE UNIVERSITY OF BRITISH COLUMBIA. June, 1979 (c) Shlomo Levy, 1979 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t n f GtOPHyZlCS 4»J> 4<Teo*/nMV The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e APC-- IS " t<\7^ ABSTRACT A new and d i f f e r e n t approach to the solution of the normal equations of minimum entropy deconvolution (MED) i s developed,. This approach which uses singular value decomposition i n the i t e r a t i v e solution of the MED equations increases the si g n a l - t o -noise r a t i o of the deconvolved output and enhances the resolution of MEC* The problem of deconvolution, and in p a r t i c u l a r wavelet estimation, i s formulated as a l i n e a r inverse problem. Both generalized l i n e a r inverse methods and Backus-Gilbert inversion are considered. The proposed wavelet estimation algorithm uses the MED output as a f i r s t approximation to the earth response. The approximated response and the observed seismograms serve as an input to the inversion schemes and the outputs are the estimated wavelets^ The remarkable performance of the l i n e a r inverse schemes f o r cases of highly noisy data i s demonstrated. A debubbling example i s used to show the completeness of the l i n e a r inverse schemes. F i r s t the wavelet estimation part was carried out and then the debubbling problem was formulated as a generalized l i n e a r inverse problem which was solved using the estimated wavelet* This work demonstrates the power of the l i n e a r inverse schemes when dealing with highly noisy data. TABLE OF CONTENTS ABSTRACT ....... TABLE OF CONTENTS , i i i LIST OF TABLES v LIST OF FIGURES v i ACKNOWLEDGEMENTS i x CHAPTER 1: I n t r o d u c t o r y Remarks , .....1 CHAPTER 2: MED Using Matrix S p e c t r a l Decomposition I n t r o d u c t i o n ....5 Theory 6 Examples: .....11 1. Example A: Tapered Ricker wavelet, 2.5% random noise-2. Example B: 25% white n o i s e C o n c l u s i o n .............20 CHAPTER 3: Wavelet E s t i m a t i o n as a L i n e a r Inverse Problem I n t r o d u c t i o n ......22 Theory: C o n v o l u t i o n as a Linear Inverse Problem 26 1. General L i n e a r i n v e r s e 2. B a c k u s - G i l b e r t i n v e r s i o n P r a c t i c a l Notes 38 Examples: 40 i 1. Example A: Good q u a l i t y MED output 2. Example B: Low q u a l i t y MED output 3. Example C: S y n t h e t i c seismogram Con c l u s i o n .......59 CHAPTER 4: Debubbling as a Generalized Linear Inverse Problem. Introduction ....62 Source Signature Estimation and Debubbling ............65 Examples: .....72 1. Example A: Synthetic seismogram 2. Example B: F i e l d example Conclusion ..85 REFERENCES 88 L I S T O P T A B L E S V I Model c h a r a c t e r i s t i c s f o r c a l c u l a t i o n of s y n t h e t i c seismograms. .........72 v i LIST OF FIGURES CHAPTER 1 1. Three liguid. layer model. 2 2. Source wavelet, impulse response and the corresponding seismogram. .. 3 CHAPTER 2 1. Input for the f i r s t MED example. 13 2. Output of the f i r s t MED example. 14 3. Inverse operators of the f i r s t MED example f i l t e r s . ...15 4. Input for the second MED example.1 .................. 17 5. Output of the second MED example. 18 6. Inverse operators of the second MED example f i l t e r s 19 CHAPTER 3 1. Input and output of MED. 24 2. Input matrix for a discrete convolution problem. ...27 3. MED output for the smallest model inversion ...32 4. Kernels of the smallest model inversion. 32 5. MED output and the kernels for f l a t t e s t model inversion. ..................<............ ..36 6. Input traces for the f i r s t l i n e a r inverse schemes example , 41 7. MED output for the f i r s t example. . . 42 8. F i r s t estimated wavelets of the f i r s t example 44 9- Second estimated wavelets of the f i r s t example. ....46 10. Noisy input trace (second example) and i t s MED output. 48 11. Plots of error versus number of eigenvalues l e f t out of the solution of the second example. 49 12. Wavelets which were estimated by the generalized l i n e a r inverse f o r the second example. 52 13. Wavelets which were estimated by the f l a t t e s t model inversion for the second example. ......53 14. Model for synthetic seismogram generation. ........54 15. Source wavelet, synthetic seismogram and the MED output 55 16. Plots of error versus number of eigenvalues l e f t out of the solution of the synthetic seismogram example. 57 17. Output wavelets of the inversion schemes for the synthetic seismogram example. 58 CHAPTEE 4 1. Flow diagram of the debubbling procedure , 65 2. Basic pulse, compound wavelet signature and the output of a convolution of a short MED f i l t e r and the compound wavelet. ...69 3. Sketch of bubble pulse events.. ..................... 71 4. Sketch of r e f l e c t i o n events. 71 5. Layer model for synthetic seismogram ca l c u l a t i o n s . .74 6. Source wavelet signature for synthetic seismogram generation -. 74 7. Synthetic seismograms based on the model of Figure 5 75 8. Debubbled synthetic seismograms. ................... 77 9. Six traces from a r e f l e c t i o n p r o f i l e recorded o f f Vancouver Island. 79 10. MED output of the traces of Figure 9. 80 11. Basic pulse signatures and example of source signature used in the debubbling scheme. ........... 82 12. The traces of Figure 9 after debubbling.. 84 ix ACKNOWLEDGEMENTS I would l i k e to express my gratitude to my supervisor Dr. R.M. Clowes for his encouragement, help and constructive remarks during every stage of the work. For his most stimulating course on l i n e a r inverse methods which triggered the larger part of thi s work as well as for his most helpful suggestions and remarks I wish to express my thanks to Dr. D.W. Oldenburg. My deepest thanks to Dr..T.J. Ulrych for his stimulating discussions and suggestions which led to a large part of t h i s work. Special thanks go to my friend Mr. C.J. Walker f o r his help and suggestions. Synthetic seismograms were generated by computer program STPSYN written by E.A. Wiggins and D.V. Helmberger. Funding f o r thi s project was provided through operating grant A7707 of the Natural Sciences and Engineering Research Council Canada and research grant from the Dniversity of B r i t i s h Columbia. Additional funds were donated by Mobil O i l Canada Ltd., Shell Canada Resources Ltd. and Chevron Standard Ltd. 1 CHAPTER 1 I n t r o d u c t o r y Remarks Seismic methods are commonly used t o a i d the study of the earth's i n t e r i o r . Employing the assumption of a l a y e r e d e a r t h , one uses the f a c t t h a t on p a s s i n g from one l a y e r t o another the energy of an e l a s t i c wave i s separated i n t o a t r a n s m i t t e d and r e f l e c t e d component. The amount of energy r e f l e c t e d or r e f r a c t e d i s b a s i c a l l y a f u n c t i o n of the angle of i n c i d e n c e o f the wave at the i n t e r f a c e and the a c o u s t i c impedance ( d e n s i t y times v e l o c i t y ) of both l a y e r s . A geophone placed on the e a r t h ' s s u r f a c e responds t o t h i s r e f l e c t e d and r e f r a c t e d energy and i t s output, a f t e r a m p l i f i c a t i o n , i s recorded (the seismogram). The task o f the s e i s m o l o g i s t i s t o i n t e r p r e t the seismogram. That i s , the s e i s m o l o g i s t i s l o o k i n g f o r the g e o l o g i c a l model of the subsurface which can be i n f e r r e d from the seismogram. To make t h i n g s c l e a r e r , c o n s i d e r the f o l l o w i n g example; Assume a t h r e e l a y e r model as shown i n F i g u r e 1 . For s i m p l i c i t y l e t these be l i q u i d l a y e r s (no shear waves), with d e n s i t y and v e l o c i t y as shown i n F i g u r e 1 , and the c o n d i t i o n ^ V j < > ^ s v a . Neglect the d i r e c t wave and assume t h a t only r e f l e c t e d energy corresponding t o the rays 1 and 2 i n F i g u r e 1 has been recorded on the seismogram. Define two time s e r i e s : 1 ) the source wavelet, which re p r e s e n t s the shape of the s i g n a l before i t was t r a n s f e r r e d 2 F i g u r e 1 Three l i q u i d l a y e r model t h r o u g h t h e ground; 2) the i m p u l s e response s e r i e s . The l a t t e r i s a time s e r i e s which c o n s i s t s o f z e r o s everywhere e x c e p t f o r the t i m e s which c o r r e s p o n d e x a c t l y t o the a r r i v a l t i m e s o f r e f l e c t e d o r r e f r a c t e d energy. For t h e s e p a r t i c u l a r t i m e s , the i m p u l s e r e s p o n s e becomes a s p i k e w i t h an a m p l i t u d e which r e f l e c t s t h e energy l e f t i n the s i g n a l a f t e r t r a v e l l i n g t h e path s p e c i f i e d by the p a r t i c u l a r r a y . Assume t h a t the s o u r c e w a v e l e t i s a P i c k e r w a v e l e t , and u s i n g the model o f F i g u r e 1 , t h e c o r r e s p o n d i n g i m p u l s e response can be d e r i v e d . F i g u r e 2 shows t h i s time s e r i e s t o g e t h e r w i t h 3 the seismogram which i s the c o n v o l u t i o n of the impulse response and the source wavelet. I f ve assume v e r t i c a l i n c i d e n c e T i, ro Figure 2 (a) Impulse response f o r the model of Figure 1. (b) Ricker wavelet as source waveform. (c) The seismogram (convolution of (a) and (b)). r e f l e c t i o n i n t h i s example, then the two-way r e f l e c t i o n t r a v e l t i m e s to the two l a y e r s are t,=2'h,/V, and t A = 2 h , /V, • 2 h i / V 4 . The example shows t h a t the impulse response s e r i e s r e f l e c t s the s u b s u r f a c e geology and hence i t i s the immediate t a r g e t of the s e i s m o l o g i s t . I f a seismogram i s assumed t o be the 4 c o n v o l u t i o n of a source wavelet with the e a r t h ' s response, then i d e n t i f y i n g and d e r i v i n g the impulse response f u n c t i o n i s termed "d e c o n v o l u t i o n " . This work i s devoted t o t h e de c o n v o l u t i o n problem i n ge n e r a l and, i n p a r t i c u l a r , to the debubbling problem. T h i s i s a s p e c i a l case of the g e n e r a l deconvolution problem which a r i s e s under p a r t i c u l a r p h y s i c a l c o n d i t i o n s . To s o l v e these problems, minimum entropy d e c o n v o l u t i o n and l i n e a r i n v e r s i o n t e c h n i q u e s are developed. 5 CHAPTER 2 M i n i m u m E n t r o p y D e c o n v o l u t i o n U s i n g , M a t r i x S p e c t r a l D e c o m p o s i t i o n ^ I N T R O D U C T I O N The c o n c e p t o f m i n i m u m e n t r o p y d e c o n v c l u t i o n w h i c h was i n t r o d u c e d b y W i g g i n s (1977) p r e s e n t s a u s e f u l t o o l t o a i d s e i s m i c i n t e r p r e t a t i o n . T h i s d e c o n v o l u t i o n t e c h n i q u e s e a r c h e s f o r a f i l t e r F w h i c h , when c o n v o l v e d w i t h a n i n p u t s i g n a l X , w i l l c o n v e r t t h a t s i g n a l t o an o u t p u t Y w h i c h h a s a " s i m p l e " a p p e a r a n c e . (1) Y = X * F w h e r e * d e n o t e s c o n v o l u t i o n . W i g g i n s (1977) d e f i n e d " ' s i m p l e * t o mean t h a t e a c h d e s i r e d s i g n a l c o n s i s t o f a f e w l a r g e s p i k e s o f u n k n o w n s i g n o r l o c a t i o n s e p a r a t e d b y n e a r l y z e r o t e r m s " . A l t h o u g h i t i s a g o o d m e t h o d , MED p r o c e s s i n g t e n d s t o e n h a n c e l a r g e a m p l i t u d e i m p u l s e s c o m p a r e d t o s m a l l e r a m p l i t u d e o n e s a n d t h e r e b y c a u s e s seme l o s s o f v a l u a b l e i n f o r m a t i o n . A m o d i f i c a t i o n o f MED h a s b e e n p r o p o s e d b y Ooe. a n d U l r y c h (1979) who i n t r o d u c e d t h e e x p o n e n t i a l t r a n s f o r m a t i o n w i t h a m o d i f i e d s i m p l i c i t y n o r m . T h e y h a v e s h o w n t h a t t h e i r a p p r o a c h p r o v i d e s a b a l a n c e b e t w e e n t h e n o i s e s u p p r e s s i o n e f f e c t o f MED a n d t h e a b i l i t y t o r e c o v e r s m a l l 6 amplitude impulses. In t h i s work we take the o r i g i n a l version of MED (Wiggins 1977) and improve i t s performance by using a d i f f e r e n t method of solving the normal equations. In p a r t i c u l a r we assume that the noisy components of the solution are mainly associated with the small eigenvalues of the autocorrelation matrix. We write the solution as a weighted summation of the eigenvectors of the autocorrelation matrix and r e j e c t those components of the solution which are associated with the smaller eigenvalues. THEORY Following the approach of Wiggins (1977), we write equation (1) i n i t s time series representation : (2) y . . * S r. • x. . . where i i s the trace index (i=1,, # /Ns), j i s the time index (j=1#w#* Nt) , Nf i s the length of the f i l t e r , Ns i s the number of trace segments, and Nt i s the number of time samples i n a trace. The varimax norm V given i n eguation (3) below i s a norm which measures the s i m p l i c i t y of a s i g n a l . The larger V i s the more "simple" i s the signal. For a detailed discussion on s i m p l i c i t y norms the reader i s referred to Harman (1960). 7 Since we are looking for the f i l t e r which w i l l maximize the varimax (simplest possible appearance of the output t r a c e ) ; we proceed by taking the gradient of V, set t i n g i t egual to zero and solving : 0 " I §*; -from eguation (2) we have: substituting eguation (5) into ( 4 ) we get: 8 •» * J which can be written in a matrix form as (7) A-r*£ where A i s a weighted autocorrelation matrix, G i s a weighted crosscorrelation vector, and F i s the vector of desired f i l t e r c o e f f i c i e n t s . Equation (7) i s highly nonlinear but i t can be solved by an i t e r a t i v e scheme. An i n i t i a l f i l t e r F" i s proposed and convolved with the input series to give Y°. Through eguations ( 3 ) , (4a) and (6), the l a t t e r enables c a l c u l a t i o n of the weighted crosscorrelation G°, and the weighted autocorrelation matrix A°. Having G° and A°, we can solve for an updated f i l t e r F 1 . Then the procedure can be repeated u n t i l the varimax norm changes very l i t t l e with subseguent i t e r a t i o n s . Wiggins (1977) used the Levinson recursion algorithm f o r the solution of (7) in order to implement t h i s i t e r a t i v e procedure. In t h i s paper, we take a different approach. The set of eguations (7) i s solved by decomposition of matrix A into i t s spectral components (Lanczos 1961), and finding i t s inverse. For t h i s procedure A i s decomposed as shown below : (8) A s K A R r where R i s a matrix whose columns consist of the eigenvectors of matrix A. 9 A i s the diagonal matrix consisting of the eigenvalues of A i n decreasing order. A =diag £ * „ 0 ^ , . . . ^ w i t h ft,^*^... and E* i s the transpose of R. A i s a symmetric, positive d e f i n i t e , Toeplitz matrix.. The inverse of A i s given by : (9) A M««A"*ft T Substituting eguation (9) into (7) we obtain : do) J * «A"'* TS writing equation (10) i n summation notation we have N The e s s e n t i a l aspect of this method i s to l e t N take on dif f e r e n t values and then compare the d i f f e r e n t solutions for the following equation : d 2 ) y" = F%X That i s , we choose to retain only a cert a i n percentage of the eigenvalues of A, rejecting the smaller ones. For p r a c t i c a l purposes, we consider only those values of N which are between 10 0.2N^ to 0.8N^ where N^ . i s the f i l t e r length. We consider only these values since the varimax does not converge for very small N and the solution of (12) looks very si m i l a r to the o r i g i n a l MED output for large N. When N=N^, the solution of (12) i s the same as the o r i g i n a l MED solution of Wiggins (1S77). In applying the method we l e t N (the number of eigenvalues used i n the solution) change over predetermined i n t e r v a l s , take the solution of (12) with the smallest N value as our reference, and plot the variance of a l l the other possible solutions versus N . By variance we mean here the sguare of the difference (Y* - Y*), where Y* i s the solution of (12) corresponding to the smallest N value used. The advantage of such a plot i s that i t t e l l s us how many possible solutions we have that are s i g n i f i c a n t l y d i f f e r e n t . For short f i l t e r lengths, i t i s probably best to plot a l l the output solutions Y and examxne how those solutions change as a function of N. 11 EXAMPLES The examples which follow use single channel traces. These are especially appropriate to our approach since matrix spectral decomposition i s expensive computationally. For single channel problems we may write eguation (6) as * J •> or A*- * = &* Since A* i s now independent of Y we w i l l have to f i n d the spectral components of A* only once with consequent saving of computer time. The examples discussed below were constructed i n such a way that a complete recovery of the generating spike sequence by the o r i g i n a l MED version was not possible, as w i l l be shown. Example A. The wavelet, spike t r a i n and the r e s u l t i n g trace with 2.5% added white noise i s shown i n Figure 1. The spacing of the spikes compared to the wavelet length makes t h i s example a p a r t i c u l a r l y d i f f i c u l t deconvolution problem. As stated in the l a s t section, i t i s often useful to plot the variance of a l l possible solutions versus N. Figure 2(a) shows such a plot but 12 i t i s not of much help i n t h i s p a r t i c u l a r example because the f i l t e r length i s short and the number of solution models i s small. As shown i n Figure 2(b) the o r i g i n a l version of MED was unable to recover the generating spike sequence. The spikes become increasingly v i s i b l e as we rej e c t more and more eigenvalues i n the construction of the f i l t e r solution (equation (11)). Note the development of the output from Figure 2(c) to Figure 2 (e). The l a t t e r does include a l l the input spikes shown in Figure 1(b) but i t i s evident that the high freguency content of the output decreases as more eigenvalues are dropped. As shown i n Chapter 3 , the inverse of the MED f i l t e r i s the wavelet i f i t i s assumed that the MED output i s the impulse response. We t r i e d to recover the wavelet by finding the inverse operator of the MED f i l t e r . For t h i s purpose we used the optimum spiking algorithm of T r e i t e l and Robinson (1966). The results are shown i n Figure 3 . The inverse of the MED f i l t e r of the o r i g i n a l version did not give an acceptable description of the input wavelet (compare Figure 3(a) with Figure 1(a)). This r e s u l t was not surprising i n view of the poor spike recovery of that f i l t e r . A s l i g h t improvment was achieved by inverting the MED f i l t e r with N=8 (Figure 3 ( b ) ) , but the estimated wavelet corresponding to the MED f i l t e r with N=5 i s t o t a l l y unacceptable even though we may consider Y s to be our best output (compare Figure 2(e) with 1(b)). This i s e a s i l y understood i n view of the clear differences i n freguency content of the output sequences shown i n Figures 2(e) and 1(b). 1 3 \ i ( , — . p— , 0.0 20.0 40.0 SO.O 80.0 100. TIME F i g u r e 1 (a) The g e n e r a t i n g wavelet W of example A . (b) The s p i k e trace S from which the i n p u t t r a c e X of example A Was d e r i v e d . (c) The i n p u t t r a c e X that i s the r e s u l t of the c o n v o l u t i o n of W with S plus 2.5% White n o i s e : X=K*S + 2.5% white noise . 15a Figure 2 (a) A plot of the variance versus N (the number of eigen-values used i n the so l u t i o n ! . The f i l t e r length i s 14. (b) The solution corresponding to N 14, which i s i d e n t i c a l with the o r i g i n a l MED solution of Wiggins (1977). (c) The solution corresponding to N 13. (d) The solution corresponding to N 8". ( e ) The solution corresponding to N 5. I I 1 1 0.0 20.0 40.0 60.0 TIKE Figure 3. (a) The inverse of the MED f i l t e r corresponding to (b) The inverse of the MED f i l t e r corresponding to N=8. (c) The inverse of the MED f i l t e r corresponding to N=5. 16 Example B. In t h i s example we used a somewhat more complicated wavelet (Figure 4(a)) to generate the synthetic input trace. This wavelet was convolved with the spike sequence of Figure 4(b) and 25% white noise was added. The resultant input trace i s shown in Figure 4(c). Although there i s wavelet overlap i n t h i s example the main obstacle i s the high percentage of white ncise. The plot of the variance versus the number of eigenvalues used i n the construction of the MED f i l t e r (Figure 5(a)) proved useful i n t h i s example i n which we l e t N take the values N=15+2n, n=0,1,....r30. It i s clear from the plot that there are only three s i g n i f i c a n t l y d i f f e r e n t solutions to eguation (12) for t h i s example. These correspond to the f l a t regions between the pairs of arrows. As shown i n Figure 5(b), the o r i g i n a l MED solution did not give a complete recovery of the generating spike seguence. However as we rej e c t more and more eigenvalues i n the construction of the MED f i l t e r a cle a r improvement i s v i s i b l e . The output Y 2 9 (Figure 5(d)) does include a l l the generating spikes although i t s frequency content i s much lcwer than that of the generating spike sequence, as noted before. The r e l a t i v e amplitudes of the recovered sequence shown i n Figure 5(d), match very well with those of the input spike seguence shown i n Figure 4(b) . Figure 6 shows the inverses of the f i l t e r s used to derive Figures 5(b), (c) and (d) . When compared with Figure 4(a), i t i s clear that the source wavelet was not reproduced. This i s due to two causes : (1) when large numbers of eigenvalues are 17 r -0,0 40.0 (I) ~1 I 80.0 120.0 T I M E ISO.O I 2 0 0 . 0 F i g u r e 4 . (a) The i n p u t w a v e l e t H o f e x a m p l e B. (b) The g e n e r a t i n g s p i k e s e q u e n c e S o f e x a m p l e B . (c) The i n p u t t r a c e x i . e. t h e r e s u l t o f t h e c o n v o l u t i o n o f W w i t h S p l u s 25% w h i t e n o i s e ! X=W*S + 25% w h i t e n o i s e . 1 8 -CD ' CL. c c 0 on 0 . 0 " B O . O TJME ~l 1 2 0 . 0 U J o 1 i 1 1 5 . 0 2 7 . 0 3 9 . 0 5 1 . 0 * OF E I G . USED ~T 1 B 0 . 0 —I 2 0 0 . 0 to 6 3 . 0 7 5 . 0 F i g u r e 5. (a) V a r i a n c e v e r s u s N. (b) The s o l u t i o n c o r r e s p o n d i n g to 11 = 75 ( t h e o r i g i n a l MED s o l u t i o n ) -(c) The s o l u t i o n c o r r e s p o n d i n g to N=*)9-(d) The s o l u t i o n c o r r e s p o n d i n g to N=29. 20 used, the exact spike seguence has not been reproduced; and (2) when small numbers of eigenvalues are used, the freguency content of the output i s much lower than that cf the true spike seguence (although Figure 5(d) c l e a r l y shows the impulse response) . CONCLUSION (a) Singular value decomposition applied to the i t e r a t i v e solution of the MED eguations increases the signal-to-noise r a t i o of the deconvolved output and enhances the resolution of the MED approach. (b) It i s commonly known that those eigenvectors associated with the small eigenvalues have more zero crossings than those which are associated with the large eigenvalues (Wiggins et a l 1976). Hence we expect that for a smaller N we would get a lower freguency f i l t e r F and hence the output Y w i l l contain lower frequencies. This point i s readily observed i n examples A and B. (c) The minimum value of N i s determined by the varimax i n the sense that for F to be an acceptable f i l t e r , the varimax must converge. Then the smallest possible N i s the smallest N for which the varimax i n the i t e r a t i v e solution described above does converge . (d) The f i l t e r length i s of e s s e n t i a l importance i n MED problems. The best f i l t e r length for our MED algorithm i s the one that gives the most acceptable r e s u l t i n the o r i g i n a l MED 21 version. An approach to t h i s discussed by Olrych et a l (1979). (e) Inversion of the MED algorithm w i l l not i n general estimate. problem has recently been f i l t e r r e s u l t i n g from our give an acceptable wavelet 2 2 CHAPTER 3 Wavelet Estimation as a Linear Inverse Problem INTRODUCTION The problem of wavelet estimation i s one of ess e n t i a l importance i n seismic deconvolution. It i s commonly assumed that a seismogram can be modelled as the convolution of a seismic wavelet with the earth response- Since the earth response r e f l e c t s the subsurface geology i t i s one of the icain targets of seismic data processing. Given a seismic wavelet, one may deconvolve the seismogram to get the earth's response. Variations of the deconvolution problem are common i n land and marine data processing. An example i s the debubbling problem (Wood et a l 1978) which we w i l l treat i n Chapter 4. Lines and Ulrych (1977) have summarized the current approaches to wavelet estimation. Such technigues include the Weiner-Levinson double inverse method and Wold-Kolmogorov f a c t o r i z a t i o n , both of which use the assumptions of an impulse response that i s a white noise seguence and of a minimum phase wavelet. A more recent approach - wavelet estimation by homomorphic deconvolution - does not use these assumptions. However, i t does assumes that the wavelet cepstrum i s read i l y separable from the cepstrum of the seismic trace. (The cepstrum i s defined as the inverse transform of the logarithm of the time sequence's Fourier transform.) This work approaches the problem of wavelet estimation 23 using a combination of the techniques which are commonly applied in linear inversion and the method of minimum entropy deconvolution or MED (see Chapter 2 ) . The application of MED to an input trace X w i l l y i e l d an output which consists of: a. the spike sequence Y which i s the primary MED output and b. the MED f i l t e r (operator) F. The time series represented by X, Y and F are related to each other through the following eguation: (1) X*F=Y where * denotes convolution. Figure 1 gives an example of three time series which follow the r e l a t i o n described in equation (1) . By convolving both sides of equation (1) with F-1=Wr we get: X*F*W=Y*W or (2) W*Y=X If we assume that the time series Y i s a true representation of the earth response then from eguation (2) , W i s the wavelet. Eguation (2) indicates that i n order to find the wavelet we need to f i n d the inverse of the MED operator. One approach to doing t h i s i s to use the algorithm SPIKER given by Robinson (1967). This algorithm i s designed to fi n d the optimum inverse operator of a given time series ( T r e i t e l and Robinson 1966). It gives good res u l t s on noiseless traces as shown by Figure 1, (d) and (e). However, the wavelet also can be estimated by formulating 2<f 0 . 0 D . l CO 4-0 . 2 0 . 3 0 . 4 TIME ( S E C ) 0 . 5 Figure 1 (a) The input trace X. (b) The KED f i l t e r F. (c) The spike sequence Y. (d) The recovered wavelet using SPIKER. (e) The input wavelet. 25 eguation (2) as a generalized l i n e a r inverse problem or solving the i n t e g r a l equivalent of eguation (2) using the Backus-Gilbert approach. We assume here that the spiked minimum entropy deconvolution output i s a f i r s t estimate of the earth response, and use i t together with the observed seismic trace to extract the wavelet. The generalized l i n e a r inverse approach i s a parametric method and the solution wavelet w i l l be represented by a set of parameters corresponding to discrete time values. In the Backus-Gilbert approach the earth response, the observed seismic trace, and the output wavelet are a l l continuous functions of time . Both technigues allow the user t c control the accuracy of the solution (see the section on P r a c t i c a l Notes), a property which i s very important when dealing with noisy data. In addition the Backus-Gilbert approach also allows us to incorporate boundary conditions into the problem. Since we assume that the MED output represents the earth response, the guality of the estimated wavelet depends on the quality of the computed impulse response ( i . e. MED output). However, we w i l l show i n example B that by using l i n e a r inverse methods a useful wavelet estimation can be extracted from seemingly useless MED output. 26 THEORY : CONVOLUTION AS A LINEAR INVERSE PROBLEM A. General Linear Inverse We w i l l present here only those aspects of the general l i n e a r inverse theory which are relevant to our problem. For additional d e t a i l s concerning t h i s approach, the reader i s referred to Wiggins (1972), Jackson (1972) or Wiggins et a l (1976). The generalized linear inverse development follows Wiggins et a l (1976). Writing the convolution i n t e g r a l i n i t s time series form, we have: Equation (3) i s a set of equations in which m unknown parameters W are related to n observations X . The matrix equivalent of (3) i s (H) X = A - W where 5 i s a vector of length n which contains the observations (input trace) ; W i s a vector of length m which contains the desired parameters (wavelet parameters) ; A i s an nxm matrix shown i n Figure 2; k i s the length of the spike sequence Y; and the r e l a t i o n n=m+k-1 holds. We may express the elements of A as (5) A(i,j) = Y(i-j+1) j = 1....m ; i=1. n with the condition that 27 A* 0 0 y. y. 0 y, y< y. 0 0 0 0 y< y*. y * , y * * 0 y * y«:, y*. a 0 o 0 0 0 0 0 0 y < . ^ i y y*-*».j yc 0 y« F i g u r e 2 The i n p u t m a t r i x A of equation (4). Y (i-j+1) = 0 for i-j+1 < 1 and i-j+1 > k The system i n equation (3) i s an overdetermined one ( i . e the number of equations i s l a r g e r then the number of unknowns) and i t can be s o l v e d by minimizing AX i n a l e a s t squares sense, where AX i s d e f i n e d by (6) A X i = Xi - X i X i r e f e r s t o the <• eft. ;1K X i r e f e r s to the o b s e r v a t i o n i n the i n p u t v e c t o r , and ' ~ c a l c u l a t e d o b s e r v a t i o n . Using s i n g u l a r value decomposition (Lanczos 1961), A i s decomposed i n t o i t s row and column e i g e n v e c t o r s and the a s s o c i a t e d e i g e n v a l u e s . The g e n e r a l i z e d i n v e r s e o f A i s d e f i n e d as (Tnman et a l 1973): 28 <7> H^s A | Ul where UT i s the transpose of 0 which consists of q eigenvectors of length n associated with the column vectors of A ( i . e . the data eigenvectors). V consists of g eigenvectors of length m associated with the row vectors of A (i.e the parameter eigenvectors). A*' i s the inverse of A which consists of the g eigenvalues of A i n descending order. A =diag £ fc,^...« a n <^ q i s the rank of A. The solution to equation (4) i s then (8) W*=H-X W* i s the smallest solution (least euclidian length) that minimizes IAX J 2 . The r e l a t i o n of W* to W can be obtained by substituting equation (4) into (8) which gives (9) W*= (H • A) • W=R. ? The matrix E i s referred to as the resolution matrix (Inman et a l 1973). If q=m, E i s the i d e n t i t y matrix; but i f q<m, the resolution matrix i s no longer the i d e n t i t y matrix and %* i s a weighted summation of the parameter eigenvectors used to describe the wavelet. The calculated observations < X C A L are determined from (10) XCAt=A.W* W* w i l l not be egual to W except for an i d e a l case ; hence the vector AX = ( X°*S - X c A l ) w i l l not be the zero vector. As a matter of f a c t , i n noisy cases we are net interested i n reproducing the observations exactly since i f we do so the 29 errors i n the observations and/or the spike trace w i l l propagate d i r e c t l y into the solution (wavelet). The power of l i n e a r inversion i n t h i s p a r t i c u l a r problem i s that i t gives us the a b i l i t y to control the accuracy of the solution. For example, i n the case of a known noise l e v e l of ^  % we are interested i n a l l the models (wavelets) i n which : (11) Max £ A X J ^ < <* % • Max £ X j ^ j=1 n Thus the exact solution i s c e r t a i n l y not the only solution that interests us and i t i s l i k e l y that i t w i l l not provide the best estimate of the wavelet . B. Backus-Gilbert Inversion The basic assumptions required for the generalized l i n e a r inverse problem - (1) MED output i s a true representation of the earth response, and (2) the seismogram can be modelled as a convolution of the earth response with the input wavelet - also are applicable to the Backus-Gilbert inversion. That i s , we assume that the observed seismic trace can be expressed d i r e c t l y by the convolution i n t e g r a l m (12) X(tj) = ^ W(t) Y(tj-t) dt where X(tj) i s the datum corresponding to time t j . W(t) i s the wavelet . Y (t j-t) i s the spike trace (MED output) ; Y i s ccmmonly referred to as the kernel. In the case where the functions X(t) and/or Y(t) i n 30 equation (12) contain a certain amount of noise, we have some f l e x i b i l i t y . Linear inverse theory allows us to choose a solution model W (t) which w i l l reproduce the data X(t) to within a desired standard deviation instead of solving (12) exactly. This a b i l i t y i s the key which enables us to treat high noise problems succesfully. Define W(t) such that : W(t) = W(t) t g (0,T) W(t) = 0 t j (Q,T) That i s , the wavelet has a f i n i t e length T. Using the d e f i n i t i o n of the wavelet we can rewrite equation (12) as T (13) X(Tj) = ^ W(t) Y(Tj-t) dt 9 The solution to eguation (13) can be constructed by minimizing either t (14) (W(t) ,W(t))= ^(W(t))2 dt or I (15) (W« (t) ,W« (t))= ^ (dW(t) /dt) 2 dt OL The solutions which we get by (14) and (15) are commonly referred to as the smallest and the f l a t t e s t models, respectively. Since the f l a t t e s t model discriminates against 31 steep g r a d i e n t s , we expect i t w i l l ensure good behaviour of the model i n the r e g i o n of i n t e r e s t . 1. S m a l l e s t Model The k e r n e l s of eguation (12) are generated by the MED output. Using the nomenclature Gj f o r the k e r n e l s we o b t a i n (16) G j ( t ) = Y(tj-t+1) 1 ^tj-t+1< T G j ( t ) = 0 1 >tj-t+1 and tj-t+1> T Since the k e r n e l s must be a t l e a s t piecewise-continuous to allow i n t e g r a t i o n , i t i s convenient t o assume t h a t Y i s c o n s t r u c t e d from a s e r i e s of box c a r s of width At (otherwise a s p l i n e curve f i t t i n g approximation should be used to give a proper continuous approximation of the MED o u t p u t ) . For example, suppose we have the MED output shown i n F i g u r e 3: Assume the wavelet i s of l e n g t h 4 (T=4) . Then G1 (t) =Y (1-t+1) t £ (0,4) G2 (t) =Y(2-t + 1) t t (0,4) G3 (t) =Y(3-t+1) t E(0,4) G4(t) =Y(4-t+1) t £(0,4) e t c . F i g u r e s 3 and 4 i l l u s t r a t e t h e r e l a t i o n between the k e r n e l s Gj and the MED output Y as s p e c i f i e d i n eguation (16). We assume t h a t the wavelet i s a l i n e a r combination of the k e r n e l s : 3 2 F i g u r e 3 Example of MED output f o r s m a l l e s t model i n v e r s i o n . I X 3 H 5 * 1 -1 1 r -0+-- G & 1 2 5 H TIME F i g u r e 4 The k e r n e l s o f the s m a l l e s t model i n v e r s i o n g e n e r a t e d from the MED output of F i g u r e 3 . 33 d7) _W(4L-5! 0,tu-y»i tai The c o e f f i c i e n t s J&£ must be found. Substitute (17) into (13) to get: ( V T 0 The problem now i s reduced to the form: The inner product matrix /"* i s defined by : r (20) r i # j a ^ G(.«) *j<4). it For the above example we have H,* =0. 5=/^ ( etc. After c a l c u l a t i n g the inner product matrix, we f i n d i t s spectral components which are given by the following equation : (21) p * *ARr where A i s the diagonalized eigenvalue matrix; A =diag ^A t >5k A...^ with *,» 4>... -34 B i s the eigenvector matrix , and E T i s the transpose of E. To compute the model we use A / ( 2 2 ) _wi;»_j&'jLT» ( R A Y X J . M ^ ^ ) ; ""V b *^ &^ , the basis functions, and o(j , the c o e f f i c i e n t s of the basis functions, are given below . (23) and (24) Since the kernels can be described by a sum of step functions the model also w i l l be a sum of step functions . 2. F l a t t e s t Model To derive the " f l a t t e s t " model, the kernels (the MED output Y) are assumed to be a series of functions . In t h i s case, eguation (16) takes the form: (16a) 0,t i s the d i g i t i z a t i o n i n t e r v a l of the MED output. Note the 35 difference i n the spike sequence representation. Here we use a J* function rather then the step function used i n the smallest model case. Integrating equation (15) by parts we get * r (25) X(Tj)=Hj (t) . W(t) - ^ H j ( t ) . W (t) • dt where 0 » (26) Hj (t) = I G j ( t ) . dt ; W«=dW/dt By the boundary conditions imposed by physical considerations, W(0)=W(T)=0, we have T Hj(t)«W(t) = 0 and the new i n t e g r a l eguation i s T (27) -X(Tj)= ^Hj(t) W» (t) dt 0 We proceed by solving equation (27) exactly as we did i n the smallest model case except for the different representation of the kernels. A simple example may help to c l a r i f y the technigue. Suppose that we have the same MED output as before and assume that the wavelet length i s now T=5 (change of wavelet length i s needed because of the d i f f e r e n t kernel representation). Then the MED output Y, the f i r s t four functions Gj(t) (j=1..4) and the corresponding new kernels Hj(t) are shown i n Figure 5. We have made use of the fact that U> Li (28) \ a t n jit e i 36 - y M (Hr) H. H , 0 t- I 2 2 H S 6 7 TIME 0 »-H , T I M E Figure 5 (a) An example of the d e l t a function representation of the MED output Y as was used i n the f l a t t e s t model i n v e r s i o n . (b) the smallest model kernels G generated from the spike sequence Y shown i n (a) and the corresponding f l a t t e s t model kernels H {G r e l a t e to Y through eguation 16(a), and H r e l a t e to G through eguation (26)). 37 Again we compute the inner product matrix by r (29) ^H.^Hyhli o For the above example we have: r , . = 5 . / |^  3.— f^fi etc* Having the inner product matrix we proceed to find i t s spectral components which are defined by eguation (21). Writing the solution i n terms of the c o e f f i c i e n t s oC; and the base functions and integrating to get the wavelet model we have: t i • i ' where the constant of integration i s defined by the boundary conditions. , the base functions, and 0(« , the c o e f f i c i e n t s of the base functions, are defined by eguations (23) and (24) , respectively (with H replacing G) . 38 C. P r a c t i c a l Notes (a) The MED output might have a p o l a r i t y which i s the reverse of the re a l spike sequence and hence the wavelet p o l a r i t y might also be reversed (Wiggins 1977). (b) The wavelet i s represented by the following expressions : 1. Generalized l i n e a r inverse or following Wiggins et a l (1976) a (30) _w** ^ V 0 A ? V * ; 2 . Backus-Gilbert inversion AJ u, What we are actually looking at are the d i f f e r e n t models W* corresponding to d i f f e r e n t values of Q i n the generalized l i n e a r inverse approach or N for the Backus-Gilbert inversion. (c) Since there are Q° models which can be generated by the generalized l i n e a r inverse method and N° models which can be generated by the Backus-Gilbert inversion (Q° i s the rank of A and N° i s the rank of P ) , we w i l l have to analyse many models. Fortunately those models can be grouped i n t o a small number of groups. The following methods enable a grouping 39 of the models without the necessity of examining every one of them. 1. Generalized l i n e a r inverse: A plot of (Q°-Q) versus ftUiX^/AX^ (Q<> i s the rank of matrix A) , has an en echelon l i k e shape and allows an easy detection of the groups. The en echelon shape can be explained by eguation (30) which states that the solution model i s a l i n e a r combination of the parameter eigenvectors. It i s commcnly known that the more " r e l i a b l e " contributions to the solution come from those eigenvectors associated with the larger eigenvalues. Summation of these contributions forms the large amplitude part of the signal. Removal of those eigenvectors which are associated with the smaller eigenvalues w i l l not change s i g n i f i c a n t l y the shape of the derived signal. 2. Backus-Gilbert inversion A plot of (NO-N) versus standard deviation also has an en echelon shape (N° i s the rank of the inner product matrix). (d) A l l the models are multiplied by a cosine b e l l to ensure zero values at both ends. (e) For good quality MED output we expect the best models to be those that use Q1 and N 1 terms i n the summations (30) and (31) respectively, since these include a l l the r e l i a b l e contributions i n the solution. Q 1 i s the pr a c t i c a l rank of A i n the generalized l i n e a r inverse and N1 i s the p r a c t i c a l rank of the inner product matrix P i n the Backus-Gilbert inversion. The p r a c t i c a l rank i s determined by the number of eigenvalues which s a t i s f y the following condition 40 11 > K A T I O . F o r d o u b l e p r e c i s i o n c o m p u t e r c a l c u l a t i o n s we t a k e RATIO t o b e 1 0 - 1 O . E X A M P L E S E x a m p l e A (Good q u a l i t y MED o u t p u t ) The i n p u t t r a c e X s h o w n i n F i g u r e 6 (a) i s a r e s u l t o f t h e c o n v o l u t i o n o f t h e w a v e l e t s h o w n i n F i g u r e 6 (c) a n d t h e s p i k e s e g u e n c e o f F i g u r e 6 ( b ) . T h i s t r a c e h a s b e e n c o n v o l v e d w i t h t h e MED f i l t e r a n d t h e r e s u l t a n t s p i k e t r a c e Y° i s s h o w n i n F i g u r e 7. T h e s p i k e t r a c e Y° was t r u n c a t e d f r o m b o t h s i d e s s o t h a t i t s l e n g t h w i l l f o l l o w t h e r e l a t i o n " l e n g t h o f o b s e r v a t i o n s = l e n g t h o f s p i k e s e q u e n c e + l e n q t h o f w a v e l e t - 1" a n d t h e r e s u l t a n t t r a c e i s t h e s p i k e s e q u e n c e Y c h o s e n a s t h e p a r t o f t h e t r a c e b e t w e e n t h e a r r o w s i n F i g u r e 7. T r u n c a t i o n i s b a s e d on i n t e r p r e t a t i o n o f t h e n u m e r i c a l v a l u e s o f t h e t r a c e Y ° . I n t h i s e x a m p l e i t i s c l e a r w h e r e Y° s h o u l d be t r u n c a t e d , s i n c e i t i s d e s i r a b l e t h a t t h e s p i k e s i g n a l i n c l u d e s t h e l a r g e r s p i k e s o f t h e MED o u t p u t . N o t e t h a t Y ( t h e t r u n c a t e d t r a c e ) i n c l u d e s o n l y t h e s e c t i o n o f Y ° t h a t c o n t a i n s t h e m a j o r s p i k e s . T h a t i s , t h e i n t e r p r e t a t i o n h a s a s s u m e d t h a t s m a l l a m p l i t u d e ' b u m p s ' a r e n o i s e a n d c a n b e r e j e c t e d . U s i n g t h e i n v e r s i o n s c h e m e s d e s c r i b e d p r e v i o u s l y , w i t h t h e i n p u t X a n d Y a s s p e c i f i e d a b o v e , we g e t t h e r e s u l t s s h o w n i n F i g u r e 8. T h e i n v e r s e o p e r a t o r o f t h e MED f i l t e r ( F i g u r e TIME F i g u r e 6 (a) The i n p u t t r a c e X which i s the c o n v o l u t i o n o f the s p i k e seguence shown i n (b) with the source wavelet shown i n ( c ) . (b) The g e n e r a t i n g s p i k e sequence. (c) The source wavelet. H2 r o. 2D. AQ. TIKE eo. 80. 1 0 0 . F i g u r e 7 The MED o u t p u t Y? That i s , t h e r e s u l t o f t h e c o n v o l u t i o n of t h e MED f i l t e r w i t h t r a c e X o f F i g u r e 6 ( a ) . Arrows mark", t h e s e c t i o n Y used i n t h e i n v e r s i o n schemes. fl(d)) i s an acceptable d e s c r i p t i o n of the generating wavelet although i t s s i d e lobes are exaggerated. T h i s wavelet e s t i m a t i o n i s l i m i t e d i n the sense that e r r o r s i n the MED f i l t e r propagate d i r e c t l y i n t o the estimated wavelet. The s o l u t i o n wavelet of the s m a l l e s t model scheme. Figure 0 ( c ) , does show the general f e a t u r e s of the generating wavelet, but i t a l s o i n c l u d e s an unacceptable amount of noise ( o s c i l l a t o r y parts i n both ends of the recovered s i g n a l ) . The f l a t t e s t model and the g e n e r a l i z e d l i n e a r i n v e r s e s o l u t i o n s F i g u r e 8(b) and (a), are good d e s c r i p t i o n s of the g e n e r a t i n g wavelet shown i n F i g u r e 43 6(c). Except for p o l a r i t y reversals, which r e s u l t from p o l a r i t y reversal of the MED output, the estimated wavelets shown i n Figure 8 are acceptable descriptions of the source wavelet. However, the correct amplitude r e l a t i o n of the small lobes which precede and follow the large amplitude part of the s i g n a l has been l o s t . The deviation of the estimated wavelets from the generating wavelet i s a r e s u l t of the small numerical noise that contaminates the estimated spike sequence (the MED output) . As mentioned i n the section P r a c t i c a l Notes, paragraph (e), we expect to get the best results when Q1 and N1 assume the value 5 9 , the assumed length of the wavelet. The generalized li n e a r inverse model (Figure 8 ( a ) ) , and the f l a t t e s t model (Figure 8 ( b ) ) , behaved i n t h i s manner but the smallest model (Figure 8(c)) did not. I t i s possible that either the smallest model scheme i s more sensitive to noise or that the box car representation of the kernels i s inadequate, or both. The spike trace can be treated further by setting Yi 0 . 1 (32) Yj = 0 lyj < o.i This s p e c i f i c a t i o n i s j u s t i f i e d by the assumption that the small values i n the MED output are actually ncise which has been generated by the MED algorithm. The v a l i d i t y of t h i s assumption can be checked by synthetic examples. Comparison of Figure 6(b) r 1 1 1 r 0.0 20. 40, 60. gO. T I M E F i g u r e 8 (a) The estimated wavelet using general l i n e a r inverse approach (b) the estimated wavelet using the f l a t t e s t model ( N = 5 9 ) . (c) the smallest model wavelet with N=10. (d) The wavelet produced by i n v e r t i n g the MED f i l t e r -Note that the derived wavelets have reversed p o l a r i t y compared with the input wavelet of Figure 6(c) (see P r a c t i c a l Notes ( a ) ) . 45 with Figure 7 shows that the MED output does include some small •bumps' which were not included i n the generating spike sequence. Equation (32) excludes those 'bumps' from the input spike sequence Y. The spike trace Y as defined i n equation (32) and trace X of Figure 6(a) are processed with the inversion procedures. The results are shown i n Figure 9. The smallest model wavelet Figure 9(a) does not reproduce the source wavelet to within the expected standard. For t h i s reason, we w i l l not show ad d i t i o n a l examples using the smallest model calcu l a t i o n s . Note that the models of Figure 9(b) and (c) reproduce the o r i g i n a l wavelet very well. The exact reproduction of the input wavelet by the f l a t t e s t model (Figure 9(b)), and the generalized l i n e a r inverse model (Figure 9 (c)), i s expected, since i n t h i s p a r t i c u l a r example eguation (32) converts the problem to an almost noiseless one. i \ 1 J r 0 . 0 2 0 . 40. SO. 8 0 . T I M E F i g u r e 9 (a) The smallest model wavelet (with N=10). (b) the f l a t t e s t model wavelet (with N=59). (c) the general l i n e a r inverse model wavelet (with Q=60). 47 Example B (Low q u a l i t y MED output) This example was designed to demonstrate the performance of the different approaches i n cases of low q u a l i t y spike traces. We w i l l show that the l i n e a r inversion schemes are capable of extracting useful information out of seemingly meaningless spike traces. In Figure 10(a) and (b) we show the input trace X, b a s i c a l l y the one shown in Figure 6(a) plus 2.5 % white noise, and -the "spike" trace Y<> which i s the re s u l t of the convolution of the MED f i l t e r with X. In t h i s case, the MED operator f a i l e d to f u l f i l l i t s function of simplifying the observations and deriving the correct spike sequence. Compare Figure 10(b) to Figure 6(b), the l a t t e r showing the spike sequence that has been used to generate t h i s example. F i r s t Y° i s truncated to obtain Y (the arrows i n Figure 10(b) mark the truncation). In t h i s example, i t i s not easy to determine which part of Y° should be used and we simply cheese Y to be that section of Y° which includes the large spike at i t s center. Traces X and Y described above form the input for the inversion schemes. The results are shown i n Figure 12 and 13. Since the spike input trace i n t h i s example i s of low guality ( i . e. contaminated by errors), we den't expect that the model that w i l l give the smallest standard deviation w i l l be our best model. In f a c t , the model which w i l l reproduce the observations exactly w i l l necessarily include erroneous components which are due to the errors in the spike sequence. In order to decrease the weight of the errors i n the spike HQ I 1 1 J 1 1 0. 20. 40. 60. 80. 100. TIME F i g u r e 10 (a) The input trace X of example B. This trace i s b a s i c a l l y the one shown i n Figure 6(a) plus 2.5% white noise. o (b) The M ED output Y used in example B. This trace i s the convolution of the tr a c e i n (a) and tne MED f i l t e r . Arrows mark, the section used i n the i n v e r s i o n . 49 Co a o Figure 11 (a) A p l o t o f s t a n d a r d d e v i a t i o n v e r s u s (N°-N) f o r t h e f l a t t e s t n o d e l . (b) A p l o t o f s t a n d a r d d e v i a t i o n v e r s u s (Q°-Q) f o r t h e g e n e r a l l i n e a r i n v e r s e p r o b l e m . * (c) A p l o t of (Q°-Q) v e r s u s Max £lAX,-lj shows t h e same g e n e r a l b e h a v i o r as i s n o t i c e d i n ( b ) . 50 sequence we have to allow some 'errors' i n the reproduced observations. Hence we w i l l have to use the.plots of standard deviation, and maximum error, in order to analyse the d i f f e r e n t possible models (see section P r a c t i c a l Notes, paragraph (c)). In Figure 11(a), the standard deviation for the f l a t t e s t model versus (N°-N) i s plotted. In t h i s example N1 (practical rank of P ) i s 55. Note the f l a t region corresponding to 45<N°-N<90. The significance of t h i s region i s that a l l models i n which 10<N<55 reproduce the observations to within the same standard deviation and hence they can be represented by two models, one for each end of t h i s region. A plot of the standard deviation and the maximum error versus Q°-Q (Figures 11(b) and (c)) for the generalized l i n e a r inverse model shows a f l a t region corresponding to Q°-Q<16. Here we w i l l represent a l l those models i n which 40<Q<56 by cne model, corresponding to the center of t h i s f l a t region. The solution wavelets that represent the d i f f e r e n t groups are shown in Figure 12 f o r the generalized l i n e a r inverse and Figure 13 for the f l a t t e s t model solutions. Note the s i m i l a r i t y i n the general features of the d i f f e r e n t models. The f l a t t e s t model solution corresponding to N=10 (Figure 13(b)) i s the best estimated wavelet. This i s an excellent estimation considering the quality of the input spike sequence. Compare Fiqures 13 (b) and 6 (c), and consider the fact that we used the spike sequence shown in Figure 10(b) which poorly resembles the true spike seguence of Figure 6(b). Note that the large amplitude part of the signal does not change much from one model to another. This type of behavior 51 has been discussed in the P r a c t i c a l Notes, paragraph (c). Example C (Synthetic seismogram) In the following example we w i l l show that the assumption that a seismogram can be modelled as the convolution of a wavelet with the earth's response where the response function i s represented by the MED output, i s useable. This assumption enables a representative wavelet to be computed. A synthetic seismogram using generalized ray theory (Wiggins and Helmberger 1974) has been computed from the model shown i n Figure 14. The following rays were considered : 1. Leaving the source as a P wave and received as a P wave ; a r r i v a l time 0.745 sec; r e l a t i v e amplitude 0.33. 2. Leaving the source as P and received as S ; a r r i v a l time 0.928 sec for the reflected energy and 0.927 sec for the refracted energy; r e l a t i v e amplitude for the combined response i s 0..385. 3. Leaving the source as a S wave and received as S ; a r r i v a l time 1.119 sec; r e l a t i v e amplitude 1.0. 4. Leaving the source as P, r e f l e c t i n g frcm surface B, r e f l e c t i n g from surface A, bouncing again frcm surface B and received as P ; a r r i v a l time 1.374 sec; r e l a t i v e amplitude 0.09. Figure 15(a) shows the synthetic trace computed from these ray paths. The model was constructed such that i t w i l l show strong interference. In such a case, the MED algorithm cannot 52 (£•) r 0 . T r 20. 40, TIME B O . F i g u r e 12 The r e s u l t s o f t h e g e n e r a l l i n e a r i n v e r s i o n : (a) The model c o r r e s p o n d s t o Q=4«J (Q°-Q=12), s t a n d a r d d e v i a t i o n s . 1 and Max =0. 22. (b) The model c o r r e s p o n d s t o Q= 36 (Q°-Q=20), s t a n d a r d d e v i a t i o n = 0 . 13 and ,1ax ^ /iX,/^ =0. 32. (c) The model c o r r e s p o n d i n g t o Q=32 (Q°-Q=24), s t a n d a r d d e v i a t i o n s . 1 5 and Max ^J4X;|^ =0. 42. 53 I 1 1 1 0. 80. 40. 50. T I M E F i g u r e 13 The r e s u l t s of t h e f l a t t e s t n o d e l i n v e r s i o n : s (a) The model which corresponds to N=45 ( N ° - N = 5 5 ) , Standard deviatiou=0. 0 1. (bv) -the model which corresponds to N=10 ( N ° - N = 9 0 ) # standard deviation=0. 0 1. ' 54 H A L F S P A C E - SOURCE SURFACE 6 O— RECEIVE/? Figure 14 The model from which the synthetic seismogram of Figure 15(a) was generated. a c h i e v e a c o n p l e t e r e c o v e r y o f t h e r e f l e c t i v i t y s e q u e n c e . The s o u r c e v a v e l e t and t h e r e c o v e r e d r e f l e c t i v i t y s e q u e n c e a r e shown i n F i g u r e s 15(b) and (c) , r e s p e c t i v e l y . E x a m i n i n g t h e r e c o v e r e d r e f l e c t i v i t y s e q u e n c e we f i n d t h a t t h e s p i k e o f r e l a t i v e a m p l i t u d e 0.2 w h i c h a p p e a r s a t a b o u t 14 t i m e u n i t s i s n o i s e g e n e r a t e d by t h e MED a l g o r i t h m . S i n c e ve g e n e r a l l y do n o t h a v e t h i s i n f o r m a t i o n , we i n c l u d e t h a t s p i k e i n o u r i n p u t s p i k e t r a c e Y. The s p i k e s w h i c h c o r r e s p o n d t o t h e r a y s s p e c i f i e d a b o v e a r e w e l l l o c a t e d b u t c o n t a i n s e r i o u s a m p l i t u d e d i s c r e p a n c i e s . The MED o u t p u t Y° was t r u n c a t e d from b o t h ends a s i n d i c a t e d by t h e 5 5 (I) 0. ~T r~ 20. 40. T I M E (c) 60, T 1 1 > 0 . 0 . 7 lA 2.1 S E C . Figure 15 (a) The input trace X which i s the synthetic seismogram of the model i n Figure 1/+.^  ( b ) The source wavelet. ( c ) The MED output, the trace Y, which i s the r e s u l t of the convolution of the MED f i l t e r with X. Arrows mark the section used i n the i n v e r s i o n . 56 arrows in Figure 15(c). The resultant spike trace Y and the input trace X (Figure 15(a)) served as input to the l i n e a r inversion schemes. A plot of the standard deviation versus N°-N for the f l a t t e s t model was calculated and i s plotted i n Figure 16(a). In t h i s example the p r a c t i c a l rank of the inner product matrix i s 36, and N°=58. Note the f l a t region corresponding t c 22<N°-N<46. Since a l l those models which corresponds to 36>N>12 reproduce the observations to within the same standard deviation, we w i l l represent a l l those models by two models corresponding to the ends of the region. These are shown i n Figure 17(a) and (b). Plots of the maximum error and standard deviation (Figure 16(b) and (c)) for the generalized l i n e a r inverse procedure also have been prepared.. In Figure 16(b), we can distinguish two regions where the maximum error i s approximately constant: Q°-Q<12 and 12<0_o-Q<26. In Figure 16 (c) the d i s t i n c t regions where the standard deviation changes comparatively slowly are: Q0-Q<19 and 19<Q0-Q<26. Since the regions determined from Figures 16(b) and (c) do not coincide, we w i l l have to represent the generalized l i n e a r inverse models by three models corresponding to the di f f e r e n t regions given above. These representative models are shown i n Figure 17(c) to ( f ) . The models shown i n Figures 17(a) and (c) correspond to the f l a t t e s t model with N=34, and the generalized l i n e a r inverse model with Q=33, respectively. These are models which are composed of nearly a l l positive values and hence are unphysical S7 o . a CO o a ( a ) 0. ?0. 40. GO. tf OF EIGENVALUES NOT USED o a • j o cr. 0. 20. 40. » QF EIGENVALUES NOT USED a a t 0. , 20. 40. * OF E I G E N V A L U E S NOT U S E D F i g u r e 1 6 (a) A p l o t of N°-N versus standard deviation f o r the f l a t t e s t model. (b) A p l o t of Q° - Q versus Hax^/AV-/^, f o r the general l i n e a r inverse. (c) A p l o t of Q°-Q versus standard deviation , for the general l i n e a r inverse. 58 TIKE F i g u r e 17 (a) The f l a t t e s t model with N=34. (b) The f l a t t e s t model with N-16. (c) The general l i n e a r inverse model with Q-33. (d) The general l i n e a r inverse model wdth Q=21. (e) The general l i n e a r inverse raodel with Q*15. (f) The general l i n e a r inverse model with Q^7. (g) The input wavelet reversed i n p o l a r i t y . 59 because we expect the wavelet to include positive and negative values. That i s , we expect tensional stresses to fellow compressional stresses or positive displacements to be followed by negative ones. Therefore these models are abandoned on physical grounds. The remaining model wavelets shown i n Figure 17 describe the source wavelet of Figure 15(b) with varying degrees of accuracy. A l l of these models provide a consistent description of the general features of the source wavelet i n the region 7 to 21 time units. However the wavelet of (e) which i s the general l i n e a r inverse model with Q=15 and the wavelet of (f) corresponds to the drop i n Max£|4Xi|^ i n Figure 16(b) (Q=7) , represent the true wavelet more closely and either could be chosen as an appropriate representation. CONCLUSION In t h i s chapter we have shown that the l i n e a r inversion schemes are capable of extracting a great deal of information from a wide variety of input data. We concentrated p a r t i c u l a r l y on seemingly unreliable data since that i s where we could show the f u l l power of the l i n e a r inverse technigues. The smallest model inversion has proved i n f e r i o r for t h i s p articular problem, probably because we represented the spike sequence Y as a series of box cars. The f l a t t e s t model approach performed s l i g h t l y better on example B, while the general l i n e a r inverse scheme gave somewhat 60 better r e s u l t s i n example C. Both techniques determined the main features of the input wavelet (in the case of example C, the region between time units 10 to 20 i n the solution models). Since two integrations are involved i n the Backus-Gilbert f l a t t e s t model c a l c u l a t i o n , we expect that the error w i l l be smoothed out. This w i l l happen only i f the spike input Y i s a "high freguency" trace ; i n other words only i f the wavelet length i s large compared to the dominant period i n the spike trace. I t i s probable that the f l a t t e s t model didn't perform as well i n example C, because of the predominant lew frequency of the spike trace Y of that example. Past experience with MED output shows that, in general, we can expect the dominant period i n the MED output to be small compared to the wavelet length. Hence we can expect substantial smoothing of the error i n the f l a t t e s t model scheme. Since the generalized l i n e a r inverse i s a parametric approach i t does not require continuous representation for the MED output. Also i t does not involve integrations and hence seems to be computationaly simpler. However, the method reguires decomposition of an nXm matrix which needs more computer time then the square matrix decomposition reguired by the Backus-Gilbert approach. For Q equal to the p r a c t i c a l rank of A and N equal to the p r a c t i c a l rank of /"* , the models generated by the generalized l i n e a r inverse and f l a t t e s t model are almost i d e n t i c a l . We demonstrated the importance of the plots of maximum error and standard deviation versus the number of eigenvalues not used in the solution , and showed how such plots can be used 61 to reduce the p r a c t i c a l number of models. At t h i s time, i t i s not possible to suggest an a n a l y t i c a l way to enable the user to choose the best model out of the given group's representative set. The user w i l l have to t r y each group's representative model as a possible estimated wavelet. However, one might get some i n d i c a t i o n on the quality of the estimated wavelet, by formulating a new l i n e a r inverse problem of the form: x=w**s where * denotes convolution and X i s the observation vector, — Ok. Wl i s the • estimated wavelet vector, and _ • . * i S* i s the . ( recovered spike sequence. Then one solves for the spike sequence SA. Having the set of spike sequences S 1 one calculates the associated set of varimaxes V . The "best" estimated wavelet i s the one which i s « associated with the highest varimax trace S*. 62 CHAPTEB 4 Debubblinq as a Generalized Linear Inverse Problem INTRODUCTION The bubble pulse problem i s common to many marine energy sources and has been discussed in d e t a i l by Kramer et a l (1S68). The problem i s caused by successive o s c i l l a t i o n s of the gas bubble generated by the energy source. Each cycle of the o s c i l l a t i n g bubble corresponds to a signal propagating outward. The source wavelet as recorded on a seismic trace i s then a t r a i n of the wavelets generated by the i n d i v i d u a l cycles. The number of pulses and the i r periods are primarily a function of detonation depth and energy released during creation of the bubble. The length of the compound wavelet depends on the number of expand-collapse cycles of substantial energy generated by the bubble, u n t i l i t has completely collapsed inward or vented remaining energy to the atmosphere. Duration of the compound wavelet often exceeds 0.5 sec. . This excesive length creates severe interference problems and causes masking of events, e s p e c i a l l y on those parts of seismograms following large amplitude r e f l e c t i o n s . The aim of debubbling schemes i s to eliminate the bubble puls€ e f f e c t s , and to compress the compound wavelet signature to one that i s simple and of short duration. Mateker ( 1 9 7 1 ) suggested a debubbling scheme which shapes the recorded source 63 wavelet into a pulse at the time of the i n i t i a l pulse. This process can be ca r r i e d out through the use of the Weiner shaping f i l t e r s . Recently, Wood et a l (1978) suggested two clos e l y related methods. The f i r s t can be described by the following stages : 1. Crosscorrelate the data and the known source signature. The resultant trace has a better signal tc noise r a t i o , and i t s wavelet i s the autocorrelation of the source signature. The new wavelet i s a zero phase one. 2. Using Weiner f i l t e r s ( T r e i t e l and Robinson, 1 966), shape the autocorrelated source wavelet i n t o a short duration desired s i g n a l . When shaping the autocorrelated wavelet use the fact that i t i s a zero phase wavelet, one of the advantages of the crosscorrelation of step 1. 3. Apply the shaping f i l t e r of step 2 to the crosscorrelated data. The second i s s i m i l a r and can be described b r i e f l y as follows: 1. Crosscorrelation of the known source signature with the data (as in step 1 above). 2. Compute the zero delay Weiner inverse f i l t e r of the wavelet and apply i t to the data. 3. Crosscorrelate the zero delay Weiner inverse f i l t e r of step 2 with the data. As seen from the above discussion, currently used debubbling methods employ Weiner shaping f i l t e r s i n their algorithms. These depend c r i t i c a l l y on the choice of the length of the f i l t e r (an unknown parameter), and cannot overcome problems which a r i s e due to errors i n the recorded wavelet. 64 In t h i s work, a new and diff e r e n t approach to the debubbling problem i s presented. I t makes use cf the th e o r e t i c a l concepts embodied i n generalized l i n e a r inversion (Wiggins et a l 1976; also see Chapter 3). This technigue i s applied f i r s t to an estimation of the wavelet and secondly to the debubbling of the seismic trace using the estimated wavelet. Debubbling as a generalized l i n e a r inverse problem i s described by the following steps, and shown schematically i n the flow diagram of Figure 1. 1. Apply minimum entropy deconvolution (Wiggins 1977; also see Chapter 2) to the observed data to get the deconvolved spike trace. 2. Ose the deconvolved spike trace and the data as an input to the generalized l i n e a r inverse procedure to compute the estimated bubble pulse wavelet. 3. Use the information acquired i n steps 1 and 2 to construct the compound source signature ( i . e . the i n i t i a l pulse plus appropriate bubble o s c i l l a t i o n s ) . 4. Use the compound source signature and the observed data as input to the generalized l i n e a r inverse scheme to determine the estimated earth impulse response. 65 Figure 1 1. Data. 2. MED. 3. Deconvolved spike t r a c e . 4 . Generalized l i n e a r inverse. 5. Estimated wavelet. 6. Construction of the compound signature. 7 . Debubbled trace Flow diagram of the proposed debubbling procedure. SOURCE S I A TUT? 7=: ESTIMATION AND DE3U3BLING The assumption that a seismogram can be modelled as the convolution of the earth's impulse response with a source wavelet provides the b a s i s f o r t h i s work. Mathematically the statement i s X=S*w • (1) where 7. i s the observed seismogram, S i s the earth impulse response and W i s the source wavelet. In summation n o t a t i o n equation (1) takes the form 66 AM j=1 n, n i s the length of the observed data; k=1.,. ...m, m i s the length of the wavelet; and the r e l a t i o n n=l+m-1 must hold, where 1 i s the length of the earth's impulse response. Having the observed data (the seismogram) and either the source signature or an estimate of the earth's response, one can solve for the unknown time series. For the sake of the discussion l e t us assume that we have the seismogram X and an estimate of the source signature W. Writing eguation (2) i n i t s matrix form we get : X=A*S (3) where A( i , j) =W ( i - j + 1) j=1 1 ; i=1....n (4) with W(i-j + 1)=0 for i - j + 1<0 and i - j + 1>m Eguation (3) represents an overdetermined system i n which n>l, i . e. the number of eguations i s larger then the number t of unknowns. We solve t h i s system of eguations by minimizing the error vector AX=X° a s- X C A L i n a l e a s t squares sense. For a more detailed discussion the reader i s referred to Chapter 3. The least squares solution of (3) i s given by : 67 or i n summation notation A.. where 0 T i s the transpose of U which consists of g eigenvectors of length n associated with the columns of A; these vectors are commonly referred to as the data eigenvectors. V consists of g eigenvectors of length 1 associated with the row vectors of A; these are commonly referred to as the parameter eigenvectors. A"' i s the inverse of A which consists of the q eigenvalues of A in descending order. A = d i a g w h e r e q i s the rank of A. Equation (6) states that the desired earth response i s a weighted summation of the parameter eigenvectors. It i s advantageous to use equation (6) as our solution, p a r t i c u l a r i l y when one i s dealing with noisy data or when the estimated source signature i s contaminated by noise. In both cases one could choose to r e j e c t the smaller and less r e l i a b l e eigenvalues before proceeding with the c a l c u l a t i o n s (see Chapter 3). This i s e a s i l y done by l e t t i n g Q i n equation (6) assume smaller and smaller values. I t i s clear that the euclidian length of the error vector AX grows as we reject more and more of the smaller eigenvalues, but at the same time we eliminate the l e s s (5) (6) 68 r e l i a b l e components of the solution. A useful t o o l i n the analysis of the d i f f e r e n t solutions (corresponding to d i f f e r e n t Q values) i s the plot of the standard deviation (|4X|Z) versus the number of eigenvalues rejected (see Chapter 3 ) . In the alternative case where one has the seismogram X and the estimated earth response S, the only change i n the above eguations occurs i n the d e f i n i t i o n of A. Then A i s given by : A ( i , j) =S ( i - j + 1) j = 1. m ; i = 1 n (7) where S(i-j+1)=0 for i-j+1<0 and i-j+1>l . Eguations (5) and (6) s t i l l hold with W replacing S. The flow diagram of Figure 1 outlines the procedure which i s followed to obtain a debubbled seismic trace. Let us examine the various steps. Having the f i e l d trace, we aim f i r s t at an intermediate target - estimation of the wavelet. To estimate the wavelet we require a rough estimate of the earth's response. To f i n d that estimate we apply MED to the data. This leads to step 1 i n the computation procedure. 1. Apply minimum entropy deconvolution (Wiggins 1977 ; also see Chapter 2) to the f i e l d data. The MED output serves as a f i r s t approximation to the earth's impulse response. A troublesome point i n step 1 concerns the choice of the length of the MED f i l t e r . Long MED f i l t e r s tend to compress much of the energy i n a trace into one large spike, t h i s 69 o v e r s p i k j. ng b e i n g an u n d e s i r a b l e f e a t u r e . S h o r t M E D f i l t e r s w i l l r e c o g n i z e o n l y t h e b a s i c b u b b l e p u l s e s i g n a t u r e as t h e s o u r c e s i g n a t u r e and h e n c e e a c h e v e n t w i l l be r e p r e s e n t e d by t h e s p i k e s a s s o c i a t e d w i t h t h a t s i g n a t u r e . F i g •ire 2 i l l u s t r a t e s t h e l a t t e r s t a t e m e n t . M l t h e s h o r t MED f i l t e r h as done i s t o " s p i k e up" t h e a p p e a r a n c e o f the b u b b l e w a v e t r a i n . However, t h i s s i t u a t i o n i s p r e f e r r e d t o one i n which i n f o r m a t i o n m i g h t be l o s t . Then i n g e n e r a l a s h o r t M E D f i l t e r i s u s e d . Figure 2 (a) Primary bubble pulse signature. (b) Compound wavelet signature (bubble pulse o s c i l l a t i o n s ) . (c) The output of the convolution of a short MED f i l t e r with the compound wavelet. H a v i n g a r o u g h e s t i m a t e o f t h e e a r t h ' s r e s p o n s e we a r e now i n a p o s i t i o n t o c o n t i n u e t o s t e p 2 - e s t i m a t i n g t h e w a v e l e t . 2. U s i n g t h e o b s e r v e d d a t a X and t h e s h o r t f i l t e r MED o u t p u t 5 a s an i n p u t t o t h e g e n e r a l i z e d l i n e a r i n v e r s e method, we e s t i m a t e t h e b a s i c b u b b l e p u l s e s i g n a t u r e W . 70 Note that i n stage 2 we prefer to use a short MED f i l t e r since a long f i l t e r w i l l overspike the data and w i l l cause a s i g n i f i c a n t loss of information. The use of a short MED f i l t e r r e s u l t s i n an estimated wavelet signature corresponding to one in d i v i d u a l basic bubble waveform. As described i n step 3, t h i s wavelet serves as the basic form from which the bubble pulse wave t r a i n i s constructed. 3. Examine the MED output and the observed data and ide n t i f y those events which have been generated by the bubble pulse. Bubble pulse events on a T-X diagram do not show d i f f e r e n t i a l moveout as the distance changes- One then looks for p a r a l l e l hyperbolas corresponding to the primary and the bubble pulses as sketched in Figure 3. Reflection events w i l l show d i f f e r e n t i a l moveout re l a t i v e to the bubble o s c i l l a t i o n s . On a T-X diagram they w i l l appear as converging hyperbolas as shown in Figure 4. Using the MED output and the data one can measure the bubble periods A T ; and estimate the r e l a t i v e amplitudes. Accurate estimation of the bubble pulse periods ATj i s important; however, r e l a t i v e amplitude estimates are not as important and w i l l not influence c r i t i c a l l y the re s u l t s . From the bubble o s c i l l a t i o n periods, the corresponding r e l a t i v e amplitudes, and the basic pulse wavelet, the r e p e t i t i v e bubble pulse wave t r a i n can be constructed. This leads to step 4. 4. Using the observed data X and the compound wavelet signature W as an input to the generalized l i n e a r inverse scheme, we obtain an estimate of the earth's impulse response. This response does not include any of the secondary bubble 7 1 . . I , . ^ ' X, X, x Figure 3 - Bubble pulse events do not show d i f f e r e n t i a l moveout - the branches corresponding to such events w i l l be p a r a l l e l . T Figure 4 - R e f l e c t i o n events on a T-X diagram show d i f f e r e n t i a l moveout. 72 o s c i l l a t i o n s ; the o r i g i n a l complex bubble pulse signature collapses to a single 'spike 1 i n the earth response. Note that step 4 i s simply the reverse of step 2. EXAME1ES A. Synthetic Seismogram The f i r s t example that w i l l be shown demonstrates only step 4,. Steps 1 and 2 were discussed i n d e t a i l i n Chapters 2 and 3, and step 3 w i l l be demonstrated i n the next example. Synthetic seismograms for the simple model shown i n Figure 5 have been generated using generalized ray theory (Wiggins and Helmterger 1974). This model i s a s i m p l i f i c a t i o n of a s i t u a t i o n one may encounter when exploring the depth of the sediments i n a deep water region. Model c h a r a c t e r i s t i c s are summarized i n Table I. TABLE I Model ch a r a c t e r i s i c s for c a l c u l a t i o n of synthetic seismograms. Thickness Depth P -vel o c i t y Density (km) (km) (km/sec) (gm/cm3) Water 1.5 1.5 1.5 1. 0 Sediment 0.75 2. 25 3.0 2.0 Basalt half half 6.0 3. Q layer space space 73 In t h i s example, the primary P r e f l e c t i o n s with v e r t i c a l t r a v e l times of 2 sec for the sea bottom r e f l e c t i o n and 2.5 sec for the r e f l e c t i o n from the base of the sediments w i l l be considered. The compound wavelet which has been used (Figure 6) was constructed to simulate an energy source which sends out a primary signal followed by two bubble pulses. Relative amplitudes and the corresponding bubble periods are shown i n Figure 6. Total length of the compound wavelet i s 0^6 sec. These wavelet c h a r a c t e r i s t i c s were chosen because of the s i m i l a r i t y with the expected c h a r a c t e r i s t i c s of the f i e l d example to be described next. Eight synthetic traces were generated corresponding to offsets between 0.2 km and 1.6 km with receiver separation of 0.2 km. The traces were normalized such that the largest amplitude i n each trace was unity (Figure 7 ) . Relative amplitudes for r e f l e c t i o n s from the base of the sediments range from 0.22 for the near v e r t i c a l r e f l e c t i o n s to 0.1 f o r the widest angle r e f l e c t i o n s . Numerical noise l e v e l i n the synthetic seismograms increases with the o f f s e t as a re s u l t of longer c a l c u l a t i o n i n t e r v a l s . Note that the type cf behavior shown i n Figures 3 and 4 i s c l e a r l y observed i n t h i s synthetic example. In p a r t i c u l a r , the r e f l e c t i o n from the base of the sediments i s hidden by the bubble pulse events on traces 1 to 4 while i t can be observed from traces 5 to 8. The synthetic seismograms of Figure 7 and the bubble o s c i l l a t i o n wavelet of Figure 6 formed the input to the generalized l i n e a r inverse scheme as discussed i n step 4. The debubbled seismograms determined from application of the inverse 7 4 water sediments b a s a l t i c l a y e r \.G KM O . U « - * | | < -X O B — - Q & 1 - O - 0 " I. V " / c c £ _ .2 .0 ^ V c c -SA^r-^ace sea f l o o r Figure 5 - The l a y e r model used in the construction of the synthetic seisnogram. Figure 6 - Signature of the compound source wavelet used t o generate the synthetic seismogram. Figure 7 - Synthetic seismogram (generalized ray theory) based on the model of Figure 5. The source wavelet i s shown i n Figure 6. 76 procedure to each trace i s shown in Figure 8. Two major e f f e c t s are noted. F i r s t , the extended bubble pulse o s c i l l a t i o n wave tr a i n has been reduced to just one pulse, and further t h i s pulse has been compressed to have a greater ' s p i k e - l i k e 1 character. Also, as the numerical noise l e v e l increases, the solution becomes more o s c i l l a t o r y as evidenced by traces 7 and 8. On these traces, the r e f l e c t i o n from the base of the sediments shows only marginally above the background noise l e v e l . Considering that the r a t i o of the smaller amplitude r e f l e c t i o n to the larger one i n traces 7 and 8 i s about one to ten and that these traces include noise with a maximum amplitude of about 50% of the smaller r e f l e c t i o n amplitude, the r e s u l t s of Figure 8 form a good output. B. FIELD EXAMPLE Clowes et a l (1978) and Thorleifson (1978) have described a marine deep seismic sounding program carried out i n Winona Basin, a deep water (2 km) sedimentary basin located west of northern Vancouver Island, B r i t i s h Columbia. Signals from six i n d i v i d u a l hydrophones separated by 90m and suspended at 45m depth from a 600m cable t r a i l e d behind the receiving ship were recorded d i g i t a l l y while the shooting ship detonated explosive charges ranging i n size from 2.2 kg to 200 kg at distances from less then 1 to 95 km. Clowes (1977) has described the system i n 77 78 d e t a i l . Seismograms from 2.2 kg shots were detonated at 7m and recorded at distances up to about 6 km to form s u b - c r i t i c a l incidence r e f l e c t i o n p r o f i l e s . Thorleifson (1978) has described the interpretation of seme of these. Six traces from one shot have been extracted from one p r o f i l e and are shown i n Figure 9. For the purposes of t h i s study, these traces have been normalized such that the maximum amplitude i n each trace has a r e l a t i v e amplitude of 1.0. The data are bandpass f i l t e r e d from 5 to 3C hz. The most prominent a r r i v a l s are the water bottom r e f l e c t i o n with i t s associated bubble o s c i l l a t i o n s as indicated on the figure by W, W1 and W2. By car e f u l l y looking at d i f f e r e n t i a l moveout between d i f f e r e n t phases f o r many correlatable cycles, Thorleifson (1978) was able to identi f y some primary r e f l e c t i o n s from within the sediments of Winona Basin. His i d e n t i f i c a t i o n of these events i s shown by the l e t t e r s A, B and C on Figure 9. We note that the bubble pulse wave t r a i n shows the ch a r a c t e r i s t i c s described i n Figure 3, while the primary r e f l e c t i o n s show those depicted i n Figure 4. From Figure 9, i t i s clear that the source wavelet with bubble o s c i l l a t i o n s can be constructed from the primary pulse followed by two bubble pulses. The f i r s t step i n our debubbling-pulse compression procedure i s to estimate the basic bubble pulse signature in every trace. We do not expect only one basic bubble signature to hold for the d i f f e r e n t traces since each of these corresponds to d i f f e r e n t angles of incidence. The phase s h i f t problem might be circumvented by use 79 F i g u r e 9 - S i x t r a c e s from a r e f l e c t i o n p r o f i l e recorded o f f Vancouver I s l a n d . Traces are at d i s t a n c e s from 1.0 to 1.5 km. Each t r a c e i s n ormalized t o u n i t amplitude. 8 0 Figure 10 — Output traces from the single channel minimum entropy deconvolution procedure applied to the seismogram of Figure 9 . 81 of the envelopes of the analytic s i g n a l (Clayton et a l 1976). However, i n t h i s work we prefer to calculate the actual wavelets from which the i n d i v i d u a l traces are generated. Applying the "short" length MED f i l t e r to the data shown i n Figure 9, one gets the spiked outputs of Figure 10 which serves as a f i r s t approximation to the earth's response. Because phase s h i f t s are expected, different MED f i l t e r s were applied to dif f e r e n t traces. The parts of the spiked traces marked by arrows i n Figure 10 with the corresponding parts of the data traces marked by arrows i n Figure 9 were used to estimate the basic bubble signatures of Figure 11 as discussed i n step 2. Di s t i n c t differences are e a s i l y observed between the di f f e r e n t bubble pulse signatures. Note i n p a r t i c u l a r the wavelet associated with the f i f t h trace corrresponding to offset of 1.4 km. The basic bubble pulse signatures shown i n Figure 11, together with the periods and amplitude information acquired from the MED output and the o r i g i n a l data, are used to construct the compound wavelets as discussed i n step 3. The bubble periods were estimated to be 0.26 sec for the f i r s t bubble pulse, and 0.22 sec f o r the second with basic pulse signature length of 0.12 sec. The t o t a l compound signature length i s 0.6 sec, and the r e l a t i v e amplitudes that were used are: 1.0 f o r the primary pulse, 1.0 to 0.8 f o r the f i r s t bubble pulse and about 0.7 for the second. As mentioned e a r l i e r , the r e l a t i v e amplitude does not carry as much weight as the estimated bubble period, and w i l l not af f e c t the solution as c r i t i c a l l y . The compound bubble pulse wavetrain for trace 1 i s shown i n Figure 8 2 (a) l 1 1 > 0 0 0-2 D.M 0.6 T I K E S E C . Figure 1 1 - (a) The basic bubble signatures associated with the d i f f e r e n t t r a c e s . Note i n p a r t i c u l a r the phase s h i f t e d wavelet associated with the 5 t r a c e . (b) Source wavelet for trace 1 . 83 11(b). The remaining 5 compound wavelets have s i m i l a r appearance a f t e r the construction which used the basic wavelets of Figure 11 (a). The compound wavelet signatures can be used together with the desired section of the recorded data, as described in- step 4, i n order to get the generalized l i n e a r inverse outputs i . e . the desired earth's response. The computed debubbled traces are shown in Figure 12. It i s easy to i d e n t i f y the f i v e r e f l e c t i o n events which dominate the section; these are marked i n Figure 12 by W for the water bottom r e f l e c t i o n , A°, A, B, and C. Figure 12 i s a considerable improvement over the observed data shown i n Figure 9. The ambiguity concerning the authenticity of some events has been removed and the output traces are considerably more spiky. In p a r t i c u l a r , one can now observe without any d i f f i c u l t y the events marked by A° and A which were completely or p a r t i a l l y masked by the bubble pulses. There are some differences i n the l e v e l of performance of the scheme for the d i f f e r e n t traces. For example, consider event C. Traces 1,4,5,and 6 are of better quality then traces 2 and 3. However, for the most part, the events marked on Figure 12 are clear and continuous. In the solution of t h i s example, we did not r e j e c t any of the smaller eigenvalues, since at each stage cf the operation we f e l t that the degree of r e l i a b i l i t y of our r e s u l t s was high. Convolution of the estimated wavelets with the estimated impulse responses gave calculated seismograms which d i f f e r from the f i e l d seismograms by a standard deviation of 0.15 to 0.20 when 83a F i g u r e 12 - The estimated impulse response set (debubbled seismograms) c a l c u l a t e d by the g e n e r a l i z e d l i n e a r i n v e r s e scheme. The p o l a r i t y r e v e r s a l of the output response i n the f i f t h t r a c e (offset, i s about 1.4 km) i s due t o the phase s h i f t e d wavelet. R e f l e c t i o n events are marked b y W, A°, A, B and C. 84 85 normalized to a maximum amplitude of 1. We f e e l that such a standard deviation i s acceptable and the debubbled traces are a better representation of the earth's response function, thereby enabling a better interpretation. CONCLUSION The generalized l i n e a r inverse scheme has been shown to be a useful t o o l for wavelet estimation and debubbling. We estimated the wavelet and used t h i s estimate i n cur debubbling-signature compression procedure. An attempt to deconvolve a trace given an error-contaminated wavelet by using Weiner f i l t e r s or by d i v i d i n g the trace's Fourier transform by that of the given wavelet w i l l often re s u l t i n erroneous outputs. We believe that the method described i n t h i s work increases the processor's a b i l i t y to deal with noisy wavelets. An important advantage of the generalized l i n e a r inverse scheme i s that one does not (and can not) solve for the impulse response exactly. Rather, one gets the l e a s t sguares solution. Since a l l of the eguations i n the deconvolution problem are weighted egually, the presence of errors i n some of them w i l l not necessarily re s u l t i n erroneous solutions. Furthermore, using the properties discussed in Chapter 3, we are able to solve the deconvolution problem to within a prespecified standard deviation, provided that t h i s standard deviation i s 86 consistent with the given problem (one cannot get out of the data more then i t has). The smallest standard deviation (S.D)min i s achieved when a l l the eigenvalues are kept i n the solution of eguation (6). However, i f (S.D) min i s small compared to the estimated noise l e v e l , the mechanism of small eigenvalues rejec t i o n can be used. This mechanism allows us to solve problems with highly noisy observations, or highly ncisy wavelets (see Chapter 3) . The proposed scheme treats debubbling and signature compression i n one step and thus reduces the r i s k of accumulated error. (Although the singular value decomposition process i s a time consuming one, i t seems to be reasonably stable). Since the wavelet signature within the same trace i s time dependent, one may expect that an inverse f i l t e r , calculated for a ce r t a i n time i n t e r v a l , may not perform acceptably i n another time i n t e r v a l along the trace. The generalized l i n e a r inverse scheme i s more f l e x i b l e to errors i n the wavelet and hence we expect that given a ce r t a i n wavelet, the inversion scheme w i l l be able to be used successfully over a longer time i n t e r v a l . An important consideration i n our scheme i s the singular value decomposition process, which i s a time consuming one and hence increases the cost of the inversion. In order to reduce the size of the problem, one has to increase the d i g i t i z a t i o n i n t e r v a l , which i n turn might reduce the solution's guality. However, i n many cases the data are d i g i t i z e d at smaller i n t e r v a l s than what i s r e a l l y necessary. In such cases, increasing the d i g i t i z a t i o n i n t e r v a l w i l l r e s u l t i n considerable saving without any reduction i n solution quality. In the f i e l d 87 example d i s c r i b e d above, we i n c r e a s e d the d i g i t i z a t i o n i n t e r v a l from 0.0032 sec to 0.0096 sec, i n c l u d e d 2.0 sec of observed data with a 0.6 sec wavelet and obtained a response output of l e n g t h 1.4 sec. Even with t h i s data decimation, a matrix of the c r d e r of 200 by 140 had to be i n v e r t e d . Our work and others (D.W. Oldenberg, p e r s o n a l comunication, 1979), suggests l i n e a r i n v e r s e methods may be q u i t e s u c c e s s f u l i n t r e a t i n g d e c o n v o l u t i o n problems i n which nois e i s p r e v a l e n t . Then, i n cases where r e g u l a r time s e r i e s a n a l y s i s technigues do not g i v e a c c e p t a b l e r e s u l t s , the more co m p u t a t i o n a l l y expensive i n v e r s i o n a l g o r i t h m s could be a p p l i e d . In t h i s study we have approached the debubbling procedure as a g e n e r a l i z e d l i n e a r i n v e r s e problem. As d i s c u s s e d i n Chapter 3, an a l t e r n a t i v e approach i n which the problem can be formulated i s the B a c k u s - G i l b e r t technique. The a p p l i c a t i o n of t h i s procedure might g i v e even b e t t e r r e s u l t s than those shown i n t h i s study. 8 8 REFERENCES Clayton R.Wr McClary 5., and Wiggins E.A., 1976. E u l l . Seism. Soc. Am. , 66, 325-326. Clowes B.M., 1977. A marine deep seismic sounding system, Can. J. Earth S c i . , 14, 1276-1285. Clowes E. M., Thorleifson A.J., and Lynch S., 1978. Interpretation of a marine deep seismic sounding survey i n Winona Basin off the West Coast of Canada, Gecl. Surv. Can. Paper 78-1C, 29-34. Harman H.H., 1960. Modern factor analysis, Dniv. Of Chicago Press, Chicago. Inman J.E., Eyu J . , and Ward S.H., 1973. B e s i s t i v i t y inversion, Geophysics, 38, 1088-1108. Jackson D. D., 1972. Interpretation cf inaccurate, i n s u f f i c i e n t and inconsistent data, Geophys. J. B. Astr. Soc., 28, Kramer F.S., Peterson B.A., and Walters W.C, 1968. Seismic energy sources 1968 handbook. United Geophysical Corporation, 57p. Lanczos C., 1961. Linear d i f f e r e n t i a l operators, D. Van Nostrand Co., London. Lines L.E., and Ulrych T.J., 1977. The old and the new i n seismic deconvolution and wavelet estimation, Geophys. Prosp., 25, 512-540. Mateker E.J., 1971. Big benefits seen with maxipulse system use. O i l and Gas J . , 69, 116-120. 89 Ooe M.r and Ulrych T.J., 1979. Minimum entropy deconvolution with an exponential transformation, Geophys. Prosp. (in press). Robinson E.A., 1967. Multichannel time series analysis with d i g i t a l computer programs, Holden-Day, San Francisco. Thorleifson A.J., 1978. A marine deep seismic sounding survey over Winona Basin, M. Ap. Sc.. Thesis, Univ. Of B r i t i s h Columbia, 70p. T r e i t e l S., and Robinson E.A., 1966. The design of high resolution d i g i t a l f i l t e r s , IEE Trans., Geosci. Electronics GE-4, 25-38. Ulrych T.J., Walker C.J., and Jurkevics A.J., 1978. A f i l t e r length c r i t e r i o n for minimum entropy deconvolution, J . Can. Soc. Expl. Geophys.,14, 21-28. Wiggins R.A., Larner K.L., and Wisecup R.D., 1976. Residual s t a t i c s analysis as a general l i n e a r inverse problem, Geophysics, 41, 922-933. Wiggins R.A., 1977. Minimum entropy deconvolution, Proc. Int. Symp. computer aided seismic analysis and discrimination, Falmouth, Mass., IEEE Computer Society, 7-14. Wiggins R.A., Helmberger D.V., 1974. Synthetic seismograms by expansion i n generalized rays, Geophys. J . R. Astr. S o c , 37, 73-90. 

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