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Restoration of random motion degraded sonar images Tseng, David Tai Hee 1986

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RESTORATION OF RANDOM MOTION DEGRADED SONAR IMAGES by DAVID TAI HEE TSENG B.Eng., McMaster U n i v e r s i t y , 1980 THESIS SUBMITTED IN PARTIAL FULFILLMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ELECTRICAL ENGINEERING We accept t h i s t h e s i s as conforming to the r e q u i r e d Standard THE UNIVERSITY OF BRITISH COLUMBIA August 1986 © David T a i Hee Tseng, 1986 In 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 the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the 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 study. I f u r t h e r agree 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 copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or 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 not be allowed without my w r i t t e n p e r m i s s i o n . Department of EI.e:e;trical Engineeriiaeg The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date August 2 6 , 1 9 8 6 i i A b s t r a c t The problem of sonar images degraded by wave-induced random s h i p motion and t h e i r r e s t o r a t i o n by f i l t e r i n g methods i s i n v e s t i g a t e d . The nature of the random motion i s examined i n d e t a i l , and a model i s set up to d e s c r i b e i t s power spectrum i n terms of the sea spectrum and the s h i p ' s receptance. A sonar measurement formula and i t s approximated form i s d e r i v e d . I t i s shown that the approximation r e p r e s e n t s a s i g n a l with a d d i t i v e c o l o u r e d noise p r o c e s s . The s i g n a l i s the measured s e a f l o o r p r o f i l e and i s approximated by a f i r s t - o r d e r Markov p r o c e s s . S e v e r a l f i l t e r s are proposed: Kalman F i l t e r , R e cursive Least Squares I n t e r p o l a t i n g (RLSI) F i l t e r , and Adaptive ARMA F i l t e r . In a d d i t i o n , Fast E s t i m a t i o n A l g o r i t h m and Adaptive A l g o r i t h m are i n t r o d u c e d to determine unknown parameters i n the Kalman F i l t e r . S i m u l a t i o n r e s u l t s are generated using these f i l t e r s . Performances are found to be s t r o n g l y dependent on both s i g n a l and n o i s e c h a r a c t e r i s t i c s , with the e x c e p t i o n of the RLSI F i l t e r , which i s r e l a t i v e l y independent of wind speed, the main nois e parameter. Computational c o m p l e x i t i e s , e s t i m a t i o n d e l a y and convergence r a t e s a s s o c i a t e d with the v a r i o u s f i l t e r s are a l s o examined. F i n a l l y , Extended Kalman F i l t e r and S e l f - T u n i n g F i l t e r are proposed as p o s s i b l e c a n d i d a t e s f o r d e a l i n g with n o n - s t a t i o n a r y , t i m e - v a r y i n g degradation problem. i i i T able of Contents A b s t r a c t i i Table of Contents i i i L i s t of Tables v i L i s t of F i g u r e s v i i Symbols and No t a t i o n s x i Acknowledgement x i i 1 .0 I n t r o d u c t i o n 1 1 .1 O b j e c t i v e s 1 1.2 System D e s c r i p t i o n 3 1.3 Previous Work 8 1.4 O u t l i n e of the T h e s i s 10 2.0 M o d e l l i n g of Random Ship Motion and Bottom-Reflected Sonar Measurement 14 2.1 M o d e l l i n g of Random Ship Motion 14 2.1.1 Ocean Waves and i t s Spectrum 19 2.1.2 P i t c h i n g and Heaving i n Random Sea 25 2.2 M o d e l l i n g of Sonar Measurement E r r o r 28 2.3 M o d e l l i n g of the S e a f l o o r 39 3.0 F i l t e r i n g i n the Presence of Noise S t a t i s t i c s 43 3.1 The F i l t e r i n g Problem 43 3.2 Kalman F i l t e r 49 3.2.1 The St a t e Space Model 51 i v 3.2.2 Measurement-Differencing Kalman F i l t e r 57 n 3.2.3 Fast E s t i m a t i o n A l g o r i t h m 61 3.3 E s t i m a t i o n of Process and Measurement Noise 65 3.3.1 Adaptive Algorithms 65 3.3.2 Innovation C o r r e l a t i o n Method 67 3.4 Adaptive ARMA F i l t e r 73 3.4.1 Adaptive ARMA F i l t e r S t r u c t u r e 73 3.4.2 Design of L i n e a r Phase ARMA P r e - F i l t e r 75 3.4.3 Adaptive IIR Alg o r i t h m 76 4.0 F i l t e r i n g i n the Absence of Noise S t a t i s t i c s 80 4.1 Extended Kalman F i l t e r 80 4.2 S e l f - T u n i n g F i l t e r 82 4.2 Recursive Least-Squares I n t e r p o l a t i n g F i l t e r 84 5.0 S i m u l a t i o n and F i l t e r Performances 90 5.1 S i m u l a t i o n Procedures 90 5.2 D e f i n i t i o n s of Performance Parameters 92 5.3 Performances of Kalman and Adaptive F i l t e r 94 5.4 Performance of RLSI F i l t e r 108 5.5 Performance of LP F i l t e r 111 5.6 Summary of F i l t e r Performances 116 6.0 Computational Complexity 120 7.0 C o n c l u s i o n s 130 7.1 General Observations on System Performances 130 7.2 E r r o r Sources 133 V 7.3 Suggestions f o r F u r t h e r Research 134 B i b l i o g r a p h y 136 Appendices 142 A. Sea and Heave Spectra 142 B. Kalman Algorithms 146 C. Design A l g o r i t h m of L i n e a r Phase ARMA P r e - F i l t e r 154 D. S i m u l a t i o n Algorithm, Data and A n a l y s i s 156 E. S i m u l a t i o n R e s u l t s 174 F. Programs 195 v i L i s t of T a b l e s 2.1 Wind Speed and S i g n i f i c a n t Wave Height 24 2.2 Wave Data 24 3.1 F i l t e r i n g Techniques 49 3.2 M o d e l l i n g Parameters and E r r o r s 54,57 3.3 E s t i m a t i o n E r r o r s e-^ 65 5.1 Optimal (Q,L) f o r RLSI F i l t e r 108 5.2 F i l t e r Performance C h a r a c t e r i s t i c s 119 6.1 Computational Requirements of the Measurement D i f f e r e n c i n g Kalman F i l t e r 122 6.2 Computational Requirements of the Innovation C o r r e l a t i o n A l g o r i t h m - F u l l A d aptation 124 6.3 Averaged Computational Requirements of the Innovation C o r r e l a t i o n A l g o r i t h m 125 6.4 Computational Requirements of the Adaptive ARMA F i l t e r 127 6.5 F i l t e r Computational C o m p l e x i t i e s and Exe c u t i o n Time 128 D.1 S t a t i s t i c a l Parameters of S e a f l o o r P r o f i l e S i m u l a t i o n 161 D.2 S t a t i s t i c a l Parameters of Measurement E r r o r S i m u l a t i o n - Large Ship 161 D.3 S t a t i s t i c a l Parameters of Measurement E r r o r S i m u l a t i o n - Small Ship 161 v i i L i s t of F i g u r e s 1.1 Towing C o n f i g u r a t i o n 2 1.2 Sonar Images 2 1.3 D i g i t a l Sonar System 5 1.4 D e t a i l of Sonar S i g n a l Processor 5 1.5 Tra n s m i t t e d and R e f l e c t e d S i g n a l Records 7 2.1 Body Axes f o r d e f i n i n g Ship Motions 15 2.2 LTI System R e p r e s e n t a t i o n of Ship Motions 15 2.3 Frames of Reference used in Ship Motion A n a l y s i s 17 2.4 A t r a v e l l i n g Harmonic Wave 17 2.5 A 1-dimensional Wave Spectrum 19 2.6 T y p i c a l Frequency Spectrum of V a r i a t i o n i n l e v e l of Sea Surface 20 2.7 Growth of Waves 21 2.8 Neumann Frequency Spectra f o r v a r i o u s Wind Speeds 21 2.9 Contour P l o t s of S h o r t - c r e s t e d and Long-crested sea 22 2.10 T y p i c a l D i r e c t i o n a l Wave Spectrum f o r a Ship t r a v e l l i n g i n S h o r t - c r e s t e d Head Sea 22 2.11 D e f i n i t i o n of £ z 27 2.12 Sonar Paths i n absence of D i s t u r b a n c e s 29 2.13 Sonar Paths i n presence of Di s t u r b a n c e s 29 2.14 1-dimesional Sonar Paths in presence of Di s t u r b a n c e s 31 2.15 2-dimesional Sonar Paths i n presence of Di s t u r b a n c e s 33 2.16 Approximation E r r o r s 36-37 2.17 Sound V e l o c i t y in Water 38 2.18 The Expected Mean R e f l e c t e d S i g n a l as a Fu n c t i o n of kacos0 40 v i i i 2.19 (a) Approximate S e a f l o o r C o r r e l a t i o n (b) Approximate S e a f l o o r S p a t i a l Power Spectrum 42 3.1 A t t e n u a t i o n E f f e c t of Time Delay 45 3.2 T y p i c a l S z ( f ) 45 3.3 Block Diagram of F i l t e r i n g i n the Frequency Domain 46 3.4 Block Diagram of D i s c r e t e - T i m e Wiener F i l t e r 47 3.5 Block Diagrams of E q u i v a l e n t F i l t e r i n g Systems 48 3.6 System Model and D i s c r e t e Kalman F i l t e r 50 3.7 T r a n s f e r Function Approximation of S n ( f ) 52 3.8 Approximating Spectra - Large Ship 55 3.9 Approximating Spectra - Small Ship 56 3.10 Shaping F i l t e r f o r Measurement Noise 58 3.11 Estimated R z ( t ) - Large Ship 63 3.12 Estimated R z ( t ) - Small Ship 64 3.13 Block Diagram of Adaptive Kalman F i l t e r 68 3.14 Test f o r O p t i m a l i t y 70 3.15 Flow Chart of Innovation C o r r e l a t i o n Adaptive A l g o r i t h m 72 3.16 Block Diagram of Adaptive ARMA F i l t e r 74 3.17 Block Diagram of Adaptive IIR F i l t e r 77 4.1 Block Diagram of RLSI F i l t e r 87 4.2 P l o t s of o(n),0(n) 88 5.1 D e f i n i t i o n of SNR 93 5.2 J 2 as a F u n c t i o n of (Q,R) 101-102 5.3 Optimal Q/R f o r Kalman F i l t e r i n g 95 5.4 Kalman F i l t e r Performances 103-104 5.5 S e n s i t i v i t y of J 2 to A L 2 97 5.6 J 2 as a F u n c t i o n of i N 99 ix 5.7 Performances of Adaptive Kalman F i l t e r i n g - F u l l A d aptation 105 5.8 Performances of Adaptive Kalman F i l t e r i n g - P a r t i a l Adaptation 106 5.9 Performances of Adaptive Kalman F i l t e r i n g - Increased NBATCH 107 5.10 Performances of RLSI F i l t e r 110 5.11 T r a n s i t i o n Bandwidth of LP F i l t e r s 112 5.12 Performances of LP F i l t e r 115 5.13 Convergence Curves 117 A1.1-2 Sea Spectrum and Response Amplitude Operator 143 A1.3 Heave and Measurement E r r o r S p e c t r a - Large Ship 144 A1.4 Heave and Measurement E r r o r S p e c t r a - Small Ship 145 D.1 Equal Area P a r t i t i o n of Spectrum S ( f ) 157 D.2 Flow Chart of Si m u l a t i o n A l g o r i t h m 158 D.3-D.7 Si m u l a t i o n R e s u l t s of S e a f l o o r P r o f i l e s 163-167 D.8-D.13 S i m u l a t i o n R e s u l t s of Measurement E r r o r S e r i e s 168-173 E1.1 RMS E r r o r Curves of Kalman F i l t e r i n g 175 E1.2-E1.5 RMS E r r o r Curves of Adaptive Kalman F i l t e r i n g 176-183 E1.6 RMS E r r o r Curves of Adaptive Kalman F i l t e r i n g - P a r t i a l Adaptation 184 E1.7 - E1.8 RMS E r r o r Curves of Adaptive Kalman F i l t e r i n g - Increased NBATCH 185-188 E2.1 Performance Curves of RLSI F i l t e r as a Fun c t i o n of L 189-190 E2.2 Performance Curves of RLSI F i l t e r as a F u n c t i o n of Q 191-192 E3.1 Performance Curves of LP F i l t e r as a Fun c t i o n of X C u t o f f Frequency F c 193 E3.2 Performance Curves of LP F i l t e r as a F u n c t i o n of F i l t e r Length 194 X I Symbols and No t a t i o n s Innovation C o r r e l a t i o n d S e a f l o o r E l e v a t i o n G k Kalman Gain M a t r i x H Observation Matrix L 2 S e a f l o o r C o r r e l a t i o n D i s t a n c e M A P r i o r i E r r o r Covariance M a t r i x P A P o s t e r i o r i E r r o r Covariance Matrix r k Innovation Sequence R ( T . ) A u t o c o r r e l a t i o n F u n c t i o n S(w) Power S p e c t r a l D e n s i t y u Wind Speed V Ship's Speed V Observation Noise Vector X S t a t e Vector X E s t i m a t i o n of State Vector z Noisy Sonar Measurement r System Disturbance D i s t r i b u t i o n Matrix A F i l t e r Time Delay e d T o t a l Measurement E r r o r D e r i v e d Measurement Vector *z Net Heave Pk Normalized C o r r e l a t i o n C o e f f i c i e n t of r ^ T c Roundtrip Time Delay T r a n s i t i o n M a t r i x Ship Encounter Angle w e Wave Encounter Frequency X I 1 Acknowledgement I would l i k e to thank Dr. Mabo R. I t o f o r h i s guidance and academic s u p e r v i s i o n of the p r o j e c t , and f o r h i s h e l p and encouragement i n p r e p a r i n g t h i s t h e s i s . I would a l s o l i k e to thank Drs. R.W.Donaldson and C.S.K.Leung f o r t h e i r h e l p f u l suggestions dur i n g the i n i t i a l phase of t h i s p r o j e c t . F i n a n c i a l support was pro v i d e d by an NSERC Postgraduate S c h o l a r s h i p , as w e l l as a U.B.C. Teaching A s s i s t a n t s h i p . 1 Chapter 1 INTRODUCTION 1.1 O b j e c t i v e s In a marine seismic experiment, an image of the s e a f l o o r or of the sub-bottom r e f l e c t o r s i s c o n s t r u c t e d from the a c o u s t i c echoes. A bottom-scanning sonar i s u s u a l l y employed. The r e f l e c t e d s i g n a l s , p i c k e d up by a hydrophone r e c e i v e r , are processed to generate images as w e l l as other s e a f l o o r r e l a t e d i n f o r m a t i o n , such as r e f l e c t i o n and s c a t t e r i n g c o e f f i c i e n t s . A t y p i c a l measurement setup i s i l l u s t r a t e d i n F i g . 1 . 1 . The accuracy of the image or i n f o r m a t i o n generated i s determined by a host of f a c t o r s , which can be summarised as three c l a s s e s : c h a r a c t e r i s t i c s of the sonar system, the ocean and f l o o r environment, and random p l a t f o r m motion. Sonar c h a r a c t e r i s t i c s i n c l u d e beampatterns, transmit pulse type, shape, l e n g t h , power l e v e l , frequency and sampling r a t e . Environment c o n d i t i o n s i n c l u d e volume and boundary b a c k s c a t t e r i n g s t r e n g t h s , propagation l o s s , and a c o u s t i c a l t r a n s m i s s i o n c h a r a c t e r i s t i c s i n water. F i n a l l y , the t h r e e - d i m e n s i o n a l random motion of the s h i p , c oupled to the t r a n s m i t t e r and sensor, degrades the sonar images by c o n t r i b u t i n g a random component. A t y p i c a l degraded image i s shown i n F i g . 1 . 2 . To compensate f o r the e f f e c t due to s h i p ' s random motion, a number of approaches have been adopted. One way i s to make use of the a d d i t i o n a l i n s t r u m e n t a t i o n s a v a i l a b l e on board, such F i g u r e 1.1 Towing C o n f i g u r a t i o n 6000 t ( sec ) 800.0 600.0 t ( sec ) 600.0 reoo.o F i g u r e 1.2 Sonar Images (a) 0 r i g i n a l (b) Degraded 3 as motion sensors, to p r o v i d e estimates of random motion. The data can be used to a d j u s t p u l s e f i r i n g i n s t a n t s of the t r a n s m i t t e r [13]. Another way i s to set up a two-ship sonobuoy, or to introduce a m u l t i c h a n n e l towed a r r a y to even out the random e f f e c t . In e i t h e r cases, i t i s c l e a r that a d d i t i o n a l i n s t r u m e n t a t i o n s must be employed, r e s u l t i n g i n i n c r e a s e d c o s t or i n c r e a s e d l o g i s t i c a l complexity. T h i s t h e s i s i n v e s t i g a t e s the p o s s i b i l i t y and methods of removing the random component i n degraded images i n the absence of extraneous i n s t r u m e n t a t i o n . A m o d e l - t h e o r e t i c approach i s adopted, i . e . , c o r r e c t i o n i s based s o l e l y on the degraded data and models of the random motion and of the s e a f l o o r . The a p p l i c a b i l i t y of each method r e l a t i v e to the absence or presence of such models w i l l be d i s c u s s e d . Experimental runs w i l l be performed using the methods introduced and the r e s u l t s e v a l u a t e d . P a r t i c u l a r a t t e n t i o n i s p a i d to performance c r i t e r i a such as e r r o r c o r r e c t i o n , computational complexity and e s t i m a t i o n l a g . 1 .2 System D e s c r i p t i o n The sonar system c o n s i s t s of a t r a n s m i t t e r , u s u a l l y mounted on a towed instrument package (Fish) by a taut c a b l e from the s h i p , and a r e c e i v e r , u s u a l l y a towed hydrophone. In some cases, the sonar a r r a y may be mounted d i r e c t l y on the h u l l of the s h i p [21]. The s i g n a l i s l i n e a r FM with t y p i c a l frequency of 6.4 kHz and a c o u s t i c a l power of 50 kW. 4 Let s t be the t r a n s m i t t e d p u l s e . Then s t ( t ) = a ( t ) e i £ J c t where a ( t ) i s the envelope, and co the frequency. The r e c e i v e d p u l s e s r has the s t r u c t u r e s r ( t ) = a ( t - r ) e J ^ c + w d ) ( t _ r ) + n ( t ) where T = 2 R / C i s the round t r i p d e l a y . R i s the range of the t a r g e t , and c the propagation speed of the s i g n a l . u$ i s the Doppler s h i f t i n c a r r i e r frequency given by 2 V . C J C /c=2v/X. The n o i s e n ( t ) i s i n c l u d e d to model bottom r e v e r b e r a t i o n and other d i s t u r b a n c e s and i s u s u s a l l y white a d d i t i v e Gaussian. The r e c e i v e d envelope a has, i n g e n e r a l , a time delay r . In the case of a Range Doppler sonar, the main i n t e r e s t i s uc, which can be used to g i v e r e l a t i v e v e l o c i t y between p l a t f o r m and t a r g e t . In the present case , that of a bottom-scanning sonar, the main i n t e r e s t i s the time delay r present i n the envelope, which i s used to deduce the range R. In Fig.1.3, a d i g i t a l sonar system i s shown. Modular p r o c e s s o r s are used to c o n t r o l and monitor the s i g n a l c o n d i t i o n e r , the t r a n s m i t subsystem, and the s i g n a l p r o c e s s o r . One s e c t i o n of the s i g n a l p r o c e s s o r i s shown in Fig.1.4. A f t e r matched f i l t e r i n g , a time r e s o l u t i o n c o r r e s p o n d i n g to the r e c i p r o c a l of the s i g n a l bandwidth i s o b t a i n e d . The envelope d e t e c t o r r e c o v e r s the phase i n f o r m a t i o n . The time delay present i n each r e c e i v e d sequence i s e x t r a c t e d , and used to r e c o n s t r u c t the range p r o f i l e of the t a r g e t ( s e a f l o o r ) . The output s i g n a l from each block i s shown in Fig.1.5. 5 S.C. ARRAY ^ j ^ . SIGNAL CONDITIONER TRANSMIT SUBSYSTEM T.S. /up BEAMFORMER SONAR SIGNAL PROCESSOR I fa? DISPLAY & CONTROL SYSTEM CONTROLLER F i g u r e 1.3 D i g i t a l Sonar System[2] SONAR SIGNAL PROCESSOR MATCHED FILTER ENVELOPE DETECTOR & TIME DELAY ESTIMATOR MOTION COMPENSATOR Restored Images F i g u r e 1.4 D e t a i l of Sonar S i g n a l Processor 6 As can be seen from Fig.1.4, the l o g i c a l p o s i t i o n of the motion compensator i s at the output of the time delay e s t i m a t o r , where range i n f o r m a t i o n , degraded by random motion of the s h i p , i s a v a i l a b l e . Thus, the compensator f u n c t i o n s as a s o r t of p o s t - d e t e c t i o n i n f o r m a t i o n p r o c e s s o r , at which the random d i s t u r b a n c e s are removed or f i l t e r e d out, based on p r i o r s t a t i s t i c s of the random motion and s e a f l o o r . Bottom-scanning i s used in t o p o g r a p h i c a l mapping and a n a l y s i s of ocean f l o o r data. Other a p p l i c a t i o n s i n c l u d e n a v i g a t i o n , o i l e x p l o r a t i o n , remote sensing, l a y i n g of submarine c a b l e s , and d e t e c t i o n of submerged or b u r i e d o b j e c t s . In the l a t t e r cases, r e a l t i m e o p e r a t i o n may be d e s i r a b l e s i n c e e s t i m a t i o n l a g s c o u l d be a c r i t i c a l f a c t o r . Consequently, time delay i n e s t i m a t i o n should be i n c l u d e d as one of the c o n s t r a i n t s i n d e s i g n i n g f i l t e r i n g a l g o r i t h m . The s h i p coupled with the F i s h t r a v e l s i n s t r a i g h t l i n e at a constant speed v, t r a n s m i t t i n g p u l s e s p e r i o d i c a l l y . A p r o f i l e of the s e a f l o o r i s c o n s t r u c t e d by i n t e r p o l a t i n g between the r e t u r n s i g n a l records ( F i g . 1 . 5 ) . T h i s o p e r a t i o n can be v i s u a l i z e d as a sampling process, i . e . , the s e a f l o o r i s sampled by the t r a v e l l i n g sonar at a sampling r a t e equal to the pulse r e p e t i t i o n r a t e of the sonar. Since the s e a f l o o r s u r f a c e i s a ( s p a t i a l ) s t o c h a s t i c p r o c e s s , to a v o i d a l i a s i n g the pulse r e p e t i t i o n r a t e must obey the Sampling Theorem f o r random pr o c e s s . The m o d e l l i n g of the s e a f l o o r and the c h o i c e of pulse r e p e t i t i o n r a t e w i l l be d i s c u s s e d i n S e c t i o n 2.3. 7 1 T r a n s m i t t e d LFM Pulse T 1 L Received S i g n a l 4 { S i g n a l a f t e r Matched F i l t e r i n g At 2At 3At S i g n a l Alignment a f t e r Time Delay E s t i m a t i o n z ( t ) C o n s t r u c t i o n of Range P r o f i l e of T a r g e t At 2At 3At F i g u r e 1 . 5 T r a n s m i t t e d and R e f l e c t e d S i g n a l Records 8 1.3 Pr e v i o u s Work Recent advancement i n f i l t e r i n g methods and s o l i d - s t a t e technology f a c i l i t a t e s on-board s i g n a l p r o c e s s i n g i n r e a l time. In s e v e r a l e a r l i e r papers [43],[ 4 9 ] , [51],[54], the problem of e s t i m a t i n g or p r e d i c t i n g s h i p motions were addressed, and the s o l u t i o n s were formulated t o i n c o r p o r a t e the above techniques. There are b a s i c a l l y two approaches : a) The design i s based on r e a l i s t i c model of the s h i p motion dynamics. System parameters and input noise v a r i a n c e s are s e l e c t e d to match measured s p e c t r a l d e n s i t i e s of the s h i p dynamics. Using t h i s S t a t e Space r e p r e s e n t a t i o n , a Kalman f i l t e r can be designed to perform e s t i m a t i o n or p r e d i c t i o n of s h i p l i n e a r and angular motions. The m o d e l l i n g of the random motion i t s e l f has been the sub j e c t of i n v e s t i g a t i o n of e a r l i e r papers [42]. Through e x t e n s i v e o b s e r v a t i o n and measurement s e v e r a l forms of the g l o b a l ocean wave spectrum f o r f u l l y developed sea have been obtained s e m i - e m p i r i c a l l y . These proposed s p e c t r a have been widely adopted i n modelling s h i p motion. T h i s t o p i c w i l l be covered i n S e c t i o n 2.1. b) In the absence of s t a t i s t i c a l data on the s h i p ' s motion or wave spectrum, some a d a p t i v e f i l t e r i n g a l g o r i t h m s have been formulated which seek o p t i m a l e s t i m a t i o n based s o l e l y on the observed d a t a [ 4 3 ] . Among the al g o r i t h m s proposed was the Widrow Least Mean Square (LMS) Adaptive F i l t e r [ 5 5 ] . However, l i t t l e i s known about the behaviour of LMS a l g o r i t h m i n a p p l i c a t i o n s r e l a t e d to s h i p motion e s t i m a t i o n , or i t s performance i n n o n s t a t i o n a r y c o n d i t i o n s . 9 Depending on the a v a i l a b i l i t y of p r i o r s t a t i s t i c s , t h e r e f o r e , one of the abcve approaches may be adopted. I t should be p o i n t e d out t h a t , while s h i p motion parameters e s t i m a t i o n i s used i n such a p p l i c a t i o n s as n a v i g a t i o n and strapdown system i n i t i a l alignment, and t h e r e f o r e not a p p a r e n t l y r e l a t e d to the problem of s h i p motion compensation, the two can a c t u a l l y be formulated as r e c i p r o c a l problems 1. Thus, many of the techniques used i n e s t i m a t i n g s h i p motion can be adapted t o sol v e motion compensation problem. In a recent paper by El-Hawary[13], the problem of random-motion compensation was analyzed i n some d e t a i l and a Kalman f i l t e r was proposed to perform o p t i m a l l i n e a r e s t i m a t i o n . F i r s t , an approximate e x p r e s s i o n f o r the random motion component, s p e c i f i c a l l y , heave, was d e r i v e d . A second-order State Space model was then c o n s t r u c t e d to approximate the heave dynamics. Once a State Space model was known, Kalman f i l t e r c o u l d be designed to i d e n t i f y the heave componenet of the r e t u r n s i g n a l r e c o r d . The r e s u l t s were compared with those generated by a d e t e r m i n i s t i c method, namely, low pass f i l t e r i n g . Realtime c a p a b i l i t y of the Kalman f i l t e r was recog n i z e d as one of i t s advantages. Whereas El-Hawary's r e s u l t s proved u s e f u l i n an a c t u a l e x periment 2, s e v e r a l inadequacies can be i d e n t i f i e d : a) The e x p r e s s i o n f o r the heave component was d e r i v e d from a crude model. Neither the c o u p l i n g e f f e c t of other motions ' I l l u s t r a t e d i n S e c t i o n 3.1 2 Sea t r i a l performed i n Outer P l a c e n t i a Bay o f f the coast of Newfoundland. 10 nor the r o u n d t r i p time delay e f f e c t were c o n s i d e r e d . b) In the context of the a c t u a l experiment, the r e s u l t s were a p p l i c a b l e only to short term r e c o r d , s i n c e component due to the major t r e n d of the s e a f l o o r was assumed to be compensated e l e c t r o n i c a l l y d u r i n g f i r i n g of p u l s e . Thus, the r e s u l t i n g s e a f l o o r (trend) was assumed to be h o r i z o n t a l or c o n s i s t i n g of low l e v e l white noise o n l y . A l s o , the method y i e l d e d best r e s u l t s i n shallow sea o n l y . T h i s f o l l o w s from the crude model i n a) (but was not e x p l i c i t l y mentioned i n the paper). c) There were no r e s e a r c h i n the e f f e c t s of v a r y i n g system parameters on r e s t o r a t i o n performance. V a r y i n g parameters such as wind speed, s h i p heading, mean depth of sea, e t c . , can r e s u l t i n n o n s t a t i o n a r i t y , n e c e s s i t a t i n g a d a p t i v e t e c h n i q u e s . We i n t e n d i n t h i s t h e s i s to examine these problems i n a systematic manner, i n t r o d u c i n g new a l g o r i t h m s where necessary. The r e s u l t s r e p o r t e d are o r i g i n a l and are ex t e n s i o n s of pre v i o u s work i n t h i s a r ea. 1.4 O u t l i n e of the T h e s i s Since no bulk data of sonar r e l e c t i o n r e c o r d s were a v a i l a b l e at hand, a d e c i s i o n was made to generate both heave noise and s e a f l o o r data by s i m u l a t i o n , based on e x i s t i n g , r e a l i s t i c models. C r i t i c a l parameters were monitored to ensure simulated data conform to the model. Both s i m u l a t i o n and experiment were done using the main UBC Computing f a c i l i t y , an 11 Amdahl 5850 running under MTS. The programs were w r i t t e n i n FORTRAN; l i b r a r y r o u t i n e s were used e x t e n s i v e l y , Some s i g n a l p r o c e s s i n g software, e s p e c i a l l y those of s p e c t r a l e s t i m a t i o n , were adapted from [45]. The g r a p h i c s were generated using DISSPLA 8.0 v i a QMSPLOT. For the remainder of t h i s t h e s i s , Chapter 2 d e a l s with m o d e l l i n g , which i s r e q u i r e d f o r both data g e n e r a t i o n and f i l t e r d e s i g n . T h i s i s d i v i d e d i n t o 3 s e c t i o n s . In s e c t i o n 1, a model f o r random s h i p motion i s i n t r o d u c e d , together with a d i s c u s s i o n on the nature of ocean wave and i t s spectrum. In the next p a r t , a d e t a i l d e r i v a t i o n of the sonar measurement e r r o r i s presented. S u c c e s s i v e approximation i s brought in at each step and the e r r o r s e v a l u a t e d . F i n a l l y , an e m p i r i c a l model of the s e a f l o o r i s reviewed. In Chapter 3, a g e n e r a l view of the f i l t e r i n g methods r e l a t i n g to the a v a i l a b i l i t y of p r i o r s t a t i s t i c s i s p r o v i d e d . The r e s t of the s e c t i o n t r e a t s the problem of f i l t e r i n g i n the presence of s h i p motion model. A S t a t e Space model r e l a t i n g to system parameters i s d e r i v e d . Approximation by n o n l i n e a r programming i s i l l u s t r a t e d and j u s t i f i e d . Two more f a c t o r s are i d e n t i f i e d : the s e a f l o o r c o r r e l a t i o n d i s t a n c e and n o i s e v a r i a n c e s . I f the l a t t e r i s known, then e s t i m a t i o n can be c a r r i e d out by s t r a i g h t f o r w a r d a p p l i c a t i o n of the measurement-differencing Kalman f i l t e r . Otherwise, an a d a p t i v e Kalman f i l t e r can be employed u s i n g output i n n o v a t i o n c o r r e l a t i o n a l g o r i t h m . As f o r the former f a c t o r , a f a s t e s t i m a t i o n a l g o r i t h m i s presented and i t s a c c u r a r y j u s t i f i e d by 12 some e x p e r i m e n t a l r e s u l t s . A l t e r n a t i v e l y , a s i l l u s t r a t e d i n t h e l a s t s e c t i o n , a d i f f e r e n t a p p r o a c h i s i n t r o d u c e d , u s i n g a d a p t i v e ARMA f i l t e r . F o r t h i s p a r t , o n l y t h e a l g o r i t h m i s p r e s e n t e d , a c t u a l e x p e r i m e n t a t i o n w i l l n o t be c a r r i e d o u t a s t h e c o m p u t a t i o n i s f o u n d t o be more c o m p l e x t h a n t h e K a l m a n f i l t e r s , a n d t h e r e f o r e l e s s e f f i c i e n t . I n C h a p t e r 4 , t h e c a s e i n w h i c h no m o d e l s a r e a s s u m e d i s t r e a t e d . T h r e e a l g o r i t h m s a r e p r e s e n t e d : t h e E x t e n d e d K a l m a n F i l t e r , t h e S e l f - T u n i n g F i l t e r a n d t h e r e c u r s i v e l e a s t - s q u r e s i n t e r p o l a t i n g ( R L S I ) f i l t e r . A s t h e l a s t i s s i m p l e a n d e f f i c i e n t , e x p e r i m e n t s w i l l be p e r f o r m e d u s i n g t h i s f i l t e r i n t h e n e x t C h a p t e r . I n C h a p t e r 5 , e x p e r i m e n t s on s i m u l a t e d d a t a u s i n g t h e v a r i o u s f i l t e r i n g a l g o r i t h m s a r e c a r r i e d o u t a n d t h e r e s u l t s a n a l y z e d . We n e e d , h o w e v e r , a d i s c u s s i o n on t h e s i m u l a t i o n p r o c e s s , a s w e l l a s t o d e v i s e a g r o u p o f m e a s u r e m e n t p a r a m e t e r s so t h a t t h e r e s u l t s c a n be m e a n i n g f u l l y p r e s e n t e d . T h e s e a r e c o v e r e d i n t h e f i r s t two s e c t i o n s . F o r t h e r e s t o f t h i s c h a p t e r , p e r f o r m a n c e s o f t h e f o l l o w i n g f i l t e r s a r e e v a l u a t e d : m e a s u r e m e n t - d i f f e r e n c i n g a n d a d a p t i v e K a l m a n f i l t e r , t h e RLSI f i l t e r , a n d , f o r c o m p a r i s o n p u r p o s e , t h e F I R l o w p a s s f i l t e r . I n e a c h c a s e , s e n s i t i v i t y t o v a r i a t i o n s i n s y s t e m o r f i l t e r p a r a m e t e r s i s i n v e s t i g a t e d . B a s e d on t h e s e r e s u l t s , an i n t e l l i g e n t c h o i c e o f f i l t e r i n g m e t h o d a n d i t s a s s o c i a t e d p a r a m e t e r s c a n be made u n d e r a c o m b i n a t i o n o f d i f f e r e n t e x t e r n a l c o n d i t i o n s . 13 In Chapter 6, the computational complexity i n v o l v e d with each a l g o r i t h m w i l l be i n v e s t i g a t e d . Timing c o s t s based on a c t u a l runs w i l l be compared. E f f i c i e n t and s t a b l e implementation of Kalman F i l t e r w i l l be d i s c u s s e d . The major c o n c l u s i o n s of the t h e s i s are summarised i n Chapter 7. Areas that r e q u i r e f u r t h e r r e s e a r c h w i l l be i d e n t i f i e d . T h i s t h e s i s a l s o i n c l u d e s an Appendix which i s d i v i d e d i n t o 5 s e c t i o n s . They aim at supplementing the main body of the t h e s i s with d e t a i l s of the a l g o r i t h m s and s i m u l a t i o n data used i n the experiments. 14 Chapter 2 MODELLING OF RANDOM SHIP MOTION AND BOTTOM-REFLECTED SONAR MEASUREMENT 2.1 M o d e l l i n g of Random Ship Motion The motion of a s h i p as a r i g i d body i s d e s c r i b e d with r e f e r e n c e to i t s body axes Cx, Cy, and Cz as the three t r a n s l a t i o n a l motions of surge, sway, heave and the three r o t a t i o n a l (angular) motions of r o l l , p i t c h , and yaw ( F i g . 2 . 1 ) . Depending on wave d i r e c t i o n , they are coupled and cannot be t r e a t e d s i n g l y . In beam seas, r o l l and sway motions dominate, w h i l s t i n head seas, p i t c h and heave are the dominant motions. I t i s g e n e r a l l y recognized that the motions induced i n a s h i p are mainly due to wave d i s t u r b a n c e and depend p r i n c i p a l l y upon p r o p e r t i e s of the s h i p (mass, s i z e , e t c . ) and the wave (sho r t or long c r e s t e d n e s s , wind speed, propagation d i r e c t i o n , e t c . ) [ 44]. Ship motions can be modelled as the output of a l i n e a r , time i n v a r i a n t system i n which the wave d i s t u r b a n c e s or sea loads are the input ( F i g . 2 . 2 ) . A complete a n a l y s i s of s h i p dynamics i n random seas must i n c l u d e both the s t e a d y - s t a t e wave-induced response and the t r a n s i e n t - s t a t e slam-induced response. The former i s l i n e a r , t i m e - i n v a r i a n t , zero-mean Gaussian p r o c e s s . The l a t t e r occurs when there are l a r g e p i t c h and heave motions with a proper phase angle between the s h i p and waves. Slamming i s l a r g e l y n o n s t a t i o n a r y and i s modelled as a f i l t e r e d Poisson process, i . e . , as a sequence of random impulses [ 8 ] . In subsequent 15 2 Heave Roll Surge x F i g u r e 2.1 Body A x e s f o r d e f i n i n g S h i p M o t i o n s a n a l y s i s , f o r s i m p l i c i t y o n l y t h e s t e a d y - s t a t e r e s p o n s e w i l l be i n c l u d e d . We s h o u l d r e m i n d o u r s e l v e s , h o w e v e r , t h a t s l a m m i n g i s n o t n e g l i g i b l e a n d s h o u l d be c o n s i d e r e d i n a c o m p l e t e a n a l y s i s . R e f e r r i n g t o F i g . 2 . 2 , t h e e q u a t i o n r e l a t i n g t h e s i x d e g r e e s o f s h i p m o t i o n s x a n d t h e i n p u t ( s t e a d y - s t a t e wave e x c i t a t i o n ) F ( t ) c a n be e x p r e s s e d i n m a t r i x f o r m where M i s a 6x6 m a t r i x d e s c r i b i n g t h e l umped s h i p m a s s , C t h e n a t u r a l d a m p i n g m a t r i x , a n d K t h e s t r u c t u r a l s t i f f n e s s ma t r i x . Mx+C i+Kx=F( t ) ( 2 . 1 ) Ship-Surge ——Sway Wave Wave System input "" —-Yaw F i g u r e 2 .2 L T I S y s t e m R e p r e s e n t a t i o n o f S h i p M o t i o n s 16 Two important assumptions have been made : a) The wave input d i s t r i b u t e d along the s h i p ' s l e n g t h i s r e p l a c e d by an e q u i v a l e n t wave l o a d i n g a c t i n g at the s h i p ' s c e n t r e of g r a v i t y 0*. b) The hydrodynamic p r o p e r t i e s at any s e c t i o n of the s h i p are independent of the hydrodynamic p r o p e r t i e s of adjacent s e c t i o n s . Thus, the 3-dimensional hydrodynamic problem i s reduced to one i n 2 dimensions. T h i s i s known as the ' s t r i p ' theory and s i n c e i t agrees reasonably w e l l with experimental r e s u l t s , i s widely used. [47],[37] F o l l o w i n g a ) , we d e f i n e a set of e q u i l i b r i u m axes 0*X'Y'Z' which i s r e l a t e d to the f i x e d c o o r d i n a t e OXYZ 3 by X'=ut+X Y' =Y Z'=Z ( F i g . 2.3) Consider an i n d i v i d u a l t r a v e l l i n g wave with v e l o c i t y c ( F i g . 2.4). I t s e l e v a t i o n rj i s a f u n c t i o n of X,Y and t : T ? ( X , Y , t )=Tj ocos(kcosM.X+ksinu.Y-cjt+ 0 ) (2.2) where k=27r/X i s the wave number, and <t> the phase angle. Measured with respect to the e q u i l i b r i u m axes, TJ becomes T?(X' ,Y' , t) = j? ocos(kX'cos0+kY' s i n ^ - w e t ) (2.3) where a>e=co-ukcos^ i s the 'frequency of encounter', caused by the r e l a t i v e motion between the s h i p and the wave. ^ i s the angle of encounter between the wave and the s h i p . 30Z i s orth o g o n a l t o the mean sea s u r f a c e d e f i n e d by OXY 17 Ship Velocity v Velocity c F i g u r e 2.3 Frames of Reference used i n Ship Motion A n a l y s i s v ,t) o / ^ F i g u r e 2.4 A T r a v e l l i n g Harmonic Wave 18 In deep water, k=cj 2/g, so that we=cj-uco2cosi///g (2.4) The t r a v e l l i n g wave c o n s i d e r e d so f a r i s d e t e r m i n i s t i c , s i n u s o i d a l wave. I t i s p o s s i b l e , however, to d e s c r i b e the random sea s t a t i s t i c a l l y as the summation of a l a r g e number of r e g u l a r waves having d i f f e r e n t frequency o, phase 4>, and amplitude T} 0. T h i s f o l l o w s from a d i r e c t a p p l i c a t i o n of the C e n t r a l L i m i t Theorem which r e q u i r e s sea waves to be homogeneous, erg o d i c and Gaussian. The l a s t f a c t has been confirmed by Longuet-Higgins in the s i x t i e s who showed that the j o i n t p r o b a b i l i t y d i s t r i b u t i o n of TJ i s given by P ( 7 ? ) = ( 2 7 T T 7 2 ) " 1 / 2 e x p { - t 2 [ 1+m3H3/6+. . . ]/2} where t = r,/( T J 2 ) 1 / 2 , m3= T J 3 / ( T 7 2 ) 3 / 2 and H 3 = t 3 - 3 t . P(T J) i s c l o s e to the i d e a l Gaussian d i s t r i b u t i o n P Q ( T J ) = [ 2 T T 7 J 2 ] - 1 / 2 exp{ - T? 2/2rj 2}, although t h e r e are non-zero skewness and k u r t o s i s , probably due to the tendency of waves to form sharp c r e s t s and shallow troughs [41]. Let S^co) be the one-dimensional wave spectrum. The energy i n an increment 5CJ at c e n t r a l frequency con ( F i g . 2.5) i s pgtS^CunJSu] (2.5) From harmonic wave theory, energy per u n i t s u r f a c e area of the wave i s l/2pgr? 2 (2.6) Equating (2.5) and (2.6), we have 7 j 2 = 2S 7 ?(co n)5cj or T?= ( 2Sn(o3n) Sco) 1 / 2 (2.7) A p p l y i n g the S u p e r p o s i t i o n Theory, the e l e v a t i o n of the sea s u r f a c e TJ as a f u n c t i o n of time can be expressed as the l i m i t 19 F i g u r e 2 . 5 A 1 - d i m e n s i o n a l wave s p e c t r u m o f a sum o f a l a r g e number o f h a r m o n i c wav es r?(t)= l i m Z c o s [ c J n t + <Mwn) ] [ 2 S „ ( c j n ) 6 w ] 1 / 2 b w- » 0 ( 2 . 8 ) 2 . 1 . 1 O c e a n Waves a n d i t s S p e c t r u m An e x a m i n a t i o n o f a g e n e r a l wave s p e c t r u m ( F i g . 2 . 6 ) c o n f i r m s t h a t w i n d g e n e r a t e d waves a r e o f t h e g r e a t e s t i n t e r e s t t o o u r p r o b l e m ; t h e o t h e r s , c a u s e d by e a r t h q u a k e s o r t i d a l f o r c e s , h a v e much l o n g e r p e r i o d s . T h e f o r m a t i o n o f wav es i s e s s e n t i a l l y a p r o c e s s o f e n e r g y t r a n s f e r f r o m a t m o s p h e r i c t u r b u l e n c e ( ' w i n d ' ) t o t h e s e a , a n d d e p e n d s u p o n t h e w i n d s p e e d , t h e a r e a o f s e a i n c o n t a c t w i t h t h e w i n d ( ' f e t c h ' ) , a n d t h e t i m e d u r a t i o n o f t h i s c o n t a c t . T h e g r o w t h o f w a v e s ' i s e x p o n e n t i a l w i t h d u r a t i o n a n d f e t c h u n t i l e q u i l i b r i u m i s r e a c h e d when r a t e o f e n e r g y t r a n s f e r e q u a l s r a t e o f e n e r g y d i s s i p a t i o n t h r o u g h w a v e - b r e a k i n g . A t e q u i l i b r i u m , t h e s e a i s ' U s u a l l y m e a s u r e d by ' S i g n i f i c a n t h e i g h t ' H s , d e f i n e d a s t h e a v e r a g e h e i g h t o f t h e 1/3 h i g h e s t w a v e s . 20 Energy (m2, decade) units Seasonal Principal tides Shallow water ,, Seiches " & shelf ' waves t'i Tsunamis 10"3 10" 10" 1 10 102 103 3 years 100days 10days 24hr 23 hr 15 min 100s — Gravity waves Planetary — —Geostrophk effects-Capillaries 10* 105 Frequency (cycles/day) 10s Is Period Sterk Surface tension F i g u r e 2.6 T y p i c a l Frequency Spectrum of v a r i a t i o n i n l e v e l of Sea Surface[44] s a i d to be ' f u l l y developed' ( F i g . 2.7a,b). Because the high frequency waves develop more r a p i d l y , they w i l l a c h i e v e steady s t a t e e a r l i e r than waves of lower frequency. Thus, the peaks of wave s p e c t r a move towards lower f r e q u e n c i e s as d u r a t i o n i n c r e a s e s ( F i g . 2.7c), and as wind speed i n c r e a s e s ( F i g . 2.8). In f u l l y developed sea, the waves show l i t t l e s p r e a d i n g . T h i s i s r e f e r r e d to as a ' l o n g - c r e s t e d ' sea, and can be adequately r e p r e s e n t e d by a one-dimensional wave spectrum Sjjjjiu). In stormy sea, waves spread out and depend on d i r e c t i o n u ( F i g . 2.9). A complete r e p r e s e n t a t i o n r e q u i r e s a two-dimensional spectrum S T J 7j(w, ri) ( F i g . 2.10). I t i s p o s s i b l e , however*, t o convert a one-dimensional spectrum to d i r e c t i o n a l 21 w (rad s ') (0 F i g u r e 2.7 Growth of Waves [44] a) (rad s" 1 ) F i g u r e 2.8 Neumann Frequency Spectra f o r v a r i o u s Wind Speeds [44] 22 F i g u r e 2.9 Contour p l o t s of F i g u r e 2.10 T y p i c a l D i r e c t i o n a l (a) A S h o r t - c r e s t e d Sea Wave Spectrum of (b) A Long-crested Sea S h o r t - c r e s t e d Sea [44 ] spectrum by approximating S ^ (to,/J) as S 7 7 7 (4O,M)=ST?TJ(U>) .f (u) ( 2 . 9 ) where f ( j u ) i s the spreading f u n c t i o n w i t h a t y p i c a l form [ 2 6 ] f (M) = 2COS2MA (2.10) E m p i r i c a l forms of the one-dimensional wave spectrum Sjjqiu) have been proposed, u s i n g wind v e l o c i t y u as the main parameter. The Neumann Spectrum, i l l u s t r a t e d i n F i g . 2.8, i s given by [ 4 4 ] S 7 ? 7j(w) = (AB/to 6)exp{-B/u) 2} ( 2 . 1 1 ) where A and B are co n s t a n t s depending on u. The Pierson-Moskowitz Spectrum [ 4 2 ] was o b t a i n e d by e x t e n s i v e a n a l y s i s of wave data from f u l l y developed sea c o n d i t i o n s i n 23 the North A t l a n t i c S„„(u) = (0.008lg 2 A> 5)exp{-0.74(g /ua> ) 4 } (2.12) or, expressed i n l i n e a r frequency f S n n ( f ) = (a/f 5)exp{-/3/f4} (2.13) with a=4.99xl0~ 4, /3=4.3794/u. The wave spectrum has dimension m 2s, with u i n ms~ 1. The spectrum i s narrow band with s i g n i f i c a n t p o r t i o n l y i n g i n 0.2<a><2 rad s _ 1 , and dominant frequency wd=0.6 rad s - 1 , or f d=0.095 Hz f o r hig h wind speed. F i g . A1.1 i n Appendix A shows i t s growth with wind speed. For n o n - f u l l y developed sea, the a l t e r n a t i v e form i s given by the JONSWAP ( J o i n t Oceanographic North Sea Wave A n a l y s i s Program) Spectrum [41] where a=0.0323, 7=3.3, f d=0.127 a o=0.07, f < f d =0.09, f > f d The JONSWAP spectrum has a narrower peak and a l a r g e r dominant frequency. T a b l e 2.1 shows the wind speed c l a s s i f i c a t i o n and the s i g n i f i c a n t wave h e i g h t s generated i n f u l l y developed sea. Table 2.2 shows the percentage p r o b a b i l i t y of encountering wave of c o r r e s p o n d i n g height i n the North A t l a n t i c , and world wide. S 7 ? T J ( f ) = ( g 2 a / ( 2 7 r ) 4 f 5 ) exp[-1 . 2 5 ( f d / f ) 4 ] 7 k k = e x p [ - 0 . 5 ( f - f d ) / a o f d ) 2 ] (2.14) 24 Table 2.1 Beaufort D e s c r i p t i o n Wind Speed Wind Speed Hs Number of Wind (knots) (m/s) (m) 0 Calm 0-1 0-0.5 0-0.005 1 L i g h t A i r 2-3 1-1.5 0.02-0.05 2 L i g h t Breeze 4-7 2-3.5 0.085-0.26 3 Gentle Breeze 8-11 4-5.5 0.34-0.65 4 Moderate Breeze 12-16 6-8 0.77-1.37 5 Fresh Breeze 17-21 8.5-10.5 1.54-2.35 6 Strong Breeze 22-27 11-13.5 2.58-3.89 7 Moderate Gale 28-33 14-16.5 4. 1 8-5 . 8 1 Table 2.2 [44] Sea S t a t e D e s c r i p t ion Wave Height North World Code of Sea observed (m) A t l a n t i c Wide 0 Calm ( g l a s s y ) 0 1 Calm ( r i p p l e d ) 0-0.1 8.3103 1 1 .2486 2 Smooth (wavelets) 0.1-0.5 3 S l i g h t 0.5-1.25 28.1996 31 .6851 4 Moderate 1 .25-2.5 42.0273 40.1944 5 Rough 2.5-4 15.4435 12.8005 6 Very Rough 4-6 4.2938 2.0253 7 High 6-9 1.4968 0.9263 25 2 . 1 . 2 P i t c h i n g a n d H e a v i n g i n Random S e a s We s h a l l c o n s i d e r t h e c a s e i n w h i c h t h e s h i p h e a d s d i r e c t l y i n t o a f u l l y d e v e l o p e d s e a . T h e n , o n l y h e a v e £ a n d p i t c h 6 a r e o f s i g n i f i c a n c e . F r o m ( 2 . 1 ) , t h e c o u p l e d e q u a t i o n s o f m o t i o n s f o r h e a v e a n d p i t c h a r e o f t h e f o r m "M 0" + ~A1 1 A 12| [ i ( t ) + B 1 1 B 1 2 l i t ) + c 1 1 c 1 2 m ) 0 I _A21 A22] Bit) _B21 B22_ 9(t) c 2 1 c 2 2 0 ( t ) ( 2 . 1 5 ) M i s t h e s h i p m a s s , I t h e moment o f i n e r t i a a b o u t y - a x i s , 7j i s t h e wave e l e v a t i o n e q u a l t o •q0e^uet. T h e c o e f f i c i e n t s A ' s , B ' s , a n d C ' s a r e d e t e r m i n e d by m o d e l t e s t s o r by a n a l y t i c means [ 4 4 ] , [ 4 7 ] , T h e e x c i t i n g f o r c e s F ^ a n d FQ a r e , i n g e n e r a l , f u n c t i o n s o f co a n d wave a n g l e . T h e y a r e m o d e l l e d a s s e c o n d - o r d e r o r f o u r t h - o r d e r t r a n s f e r f u n c t i o n s o f co [ 5 1 ] . R e w r i t i n g ( 2 . 1 5 ) i n t h e c o n c i s e f o r m M A 3 c ( t ) + B i ( t ) + C x ( t ) = F e 3 " e t ( 2 . 1 6 ) w i t h x(t)= i 2 < t ) F ( t ) = F | ( t ) 6(t) one may d e f i n e t h e t r a n s f e r f u n c t i o n H(CJ 6) H(CJ 6) = [ ( C - o ; e 2 M A ) + j a ) e B ] " 1 ( 2 . 1 7 ) H(a> e) i s known a s t h e ' r e c e p t a n c e ' i n s h i p a n a l y s i s . I n r e a l i t y , rj i s r a n d o m . H e n c e , 77 i n ( 2 . 1 5 ) s h o u l d be r e p l a c e d by e x p r e s s i o n ( 2 . 8 ) . F r o m t h e w e l l - k n o w n r e s u l t o f l i n e a r s y s t e m t h e o r y , t h e s p e c t r a l d e n s i t y o f o u t p u t v e c t o r x i s g i v e n by Sx(coe)=H*(coe) . S n n ^ e ) . H T ( w e ) ( 2 . 1 8 ) 26 or, i n f u l l , S ^ ( c o e ) S$0(u e) In a l o n g - c r e s t e d sea, at constant wind speed and wave encounter, the wave encounter spectrum S ^ ^ g ) i s r e l a t e d to S v r ) ( u ) by S T J 7 ?(a; e)= Svn(u)/\due/6u\ = S n v ( u ) / \ 1-2wocos^/g| (2.19) The f o l l o w i n g s p e c t r a may be d e f i n e d : Heave Spectrum S ^ ( c o e ) = |H^ ( c u e ) | 2 . S 7j 7j(w e) Heave-pitch spectrum S^g (co e ) = H*^ T ? ( co e ) Hg 7 J(w e) S T ? Tj(o; e) Pitch-heave spectrum Sg^(co e ) = H^ 7 J ( co e ) H*gjj (co e ) S J ? 7j ( c j e ) P i t c h Spectrum S e e ( w e ) = | H d r ) ( u e ) | 2 . S 7 J r J(w e) (2.20) In s h o r t - c r e s t e d seas, the spectrum depends on co e as w e l l as 4>. The one-dimensional response spectrum can be ob t a i n e d by i n t e g r a t i n g the two-dimensional spectrum over a l l v a l u e s of \}/. Thus, f o r example, the one-dimensional heave spectrum i s given by S ^ ( c o e ) = J |H£,j(we,i//) | 2 . Sv1}(u>e,\l/) d</< (2.21) SJJJJ^OJQ,^) can be approximated by (2.9), and the JONSWAP spectrum should be used f o r S ^ f c o ) . T r a n s f o r m a t i o n to u>e i s again given by (2.19). Suppose the hydrophone i s l o c a t e d at a d i s t a n c e / from O*. Then, the net v e r t i c a l displacement upward £ from 0* i s £ z(t)=£(t)-/sin0(t) =£(t)-/0(t) s i n c e 6 i s small ( F i g . 2.11) The a u t o c o r r e l a t i o n f u n c t i o n of £ 2 i s R M Z ( t ) = < * z ( T ) * z ( t + T ) > H*eV"e> S T ? T j ^ e ) [ _ H ^ ^ e ) Hr9T 7 ( £ de )J O * L Mean Position of Ship F i g u r e 2.11 D e f i n i t i o n of £ z = < [ * ( t ) - / 0 ( t ) ] U ( t + T ) - / 0 ( t + T ) ] > = R ^ ( T ) + / 2Ree(r)-l [Re^(r)+Rei(-T)] Hence, S ^ z ^ z ( w ) = S ^ ( w ) + / 2 See(co)-2lRe{ Se^{u>)} (2.22) where Re{ } denotes r e a l p a r t of the spectrum. D e f i n e the net v e r t i c a l displacement receptance H ^ z T J ( / , c J e ) as |H*z7?(/,"e>l2=.+>2 |H f l l ? U> e >|»-2MRe{ n^zV(coe) H*erl(ue)} (2.23) Then,(2.22) reduces to S M z ( W e ) = \ a ^ r i ( c o e t l ) \ i . Snn(ae) (2.24) For subsequent experiments, two receptances H^ z 7 ? s i m i l a r to the examples i n Lewis[26] are used. They are f o r a 'small' s h i p ( 250ft.) and a 'long' s h i p ( 500ft.) r e s p e c t i v e l y . The magnitude squared p l o t s are i l l u s t r a t e d i n Fig.A1.2(a) and ( b ) . The v a r i a t i o n s with wind speed u and encounter angle ^ of Sj-Z£z are shown i n F i g . A l . 3 ( a ) , ( c ) and Fig.A1.4(a),(c) f o r l a r g e and small s h i p , r e s p e c t i v e l y . The f o l l o w i n g c h a r a c t e r i s t i c s are noted : 28 a) Small s h i p s p e c t r a are wider and has l a r g e r mean square v a l u e s . T h i s i s expected, as small s h i p s are more s u s c e p t i b l e to random d i s t u r b a n c e s . b) In c o n t r a s t to the o r i g i n a l wave spectrum, the peaks of the heave spectrum show l i t t l e s h i f t i n g with r e s p e c t to frequency as wind speed i n c r e a s e s , though the magnitudes of the s p e c t r a on the whole s t i l l r i s e s h a r p l y . T h i s i s a p r o p e r t y of t r a n s f o r m a t i o n to encounter frequency ue. c) As the encounter angle i n c r e a s e s , i . e . , heading changes g r a d u a l l y from head to f o l l o w i n g sea, the peaks of the heave s p e c t r a occur at lower f r e q u e n c i e s . In the experiments we s h a l l keep 4/ constant at 0°, and vary the wind speed o n l y . The same techniques used i n f i l t e r i n g with v a r y i n g wind speed may be a p p l i e d to deal with problems with v a r y i n g encounter angle. 2.2 M o d e l l i n g of Sonar Measurement E r r o r In the f o l l o w i n g a n a l y s i s , we assume that the beam of sonar s i g n a l d i r e c t e d at the bottom i s narrow, so that the small e n s o n i f i c a t i o n area O s may be approximated to a p o i n t ( F i g . 2 . 1 2 ) . In g e n e r a l , the F r e s n e l zone of e n s o n i f i c a t i o n can be reduced by i n c r e a s i n g frequency. A narrow-beam echosounder u s i n g 12-kHz multibeam a r r a y sonar has been shown to achieve r e s o l u t i o n of c l o s e to 1 metre [4]. Assuming a l s o that the s h i p and towed body t r a v e l s at constant speed u, and that d, the r e a l depth, i s reasonably l a r g e (compared to magnitudes of a l l random motions). The 30 apparent depth d' of O S i s deduced as d i s t a n c e t r a v e l l e d in h a l f of the r o u n d t r i p delay, or TC/2, where c i s the speed of sound i n water. A c t u a l l y , TC/2=/, the one way t r i p from 0^ to 0 S . The r e a l depth d i s given by d = / ( r 2 c 2 - r 2 u 2 ) / 4 = r / ( c 2 - u 2 ) / 4 =r/2 / ( c 2 - u 2 ) ~TC/2 i f v < c (2.25) C l e a r l y , the f a c t that the s h i p has t r a v e r s e d a f i n i t e d i s t a n c e TV d u r i n g T c o n t r i b u t e s to the e r r o r i n measuring d. Thi s e r r o r i s dependent on v, the s h i p ' s speed, as w e l l as r, which i s p r o p o r t i o n a l to the depth d i t s e l f . The s i t u a t i o n i s f u r t h e r complicated by the i n t r o d u c t i o n of random motions of the s h i p . In the presence of random motions, the t r a n s m i t t i n g p o s i t i o n of the s h i p i s d i s p l a c e d from 0<p to O - r ' by a 3-dimensional random v e c t o r r T . S i m i l a r l y , 0 R i s d i s p l a c e d to O R ' by r R ( F i g . 2.13). Both rtj< and r R can be r e s o l v e d i n t o the three t r a n s l a t i o n a l and three angular random motions of the s h i p . The round t r i p d elay i s given by 1 T'+1 R' r = c . (2.26) where lip'=l«p+rip l R ' = l R + r R are the a c t u a l sonar paths and r T = ( x T 2 + y T 2 + 2 r p 2 ) 1 / 2 r R = ( x R 2 + y R 2 + z R 2 ) 1 / 2 are magnitudes of random motions at the t r a n s m i t t i n g and r e c e i v i n g ends about t h e i r mean p o s i t i o n s . 31 O z Y t Z l t ) t Z l t ) £ z ( t + T C ) F i g u r e 2 .14 1 - D i m e n s i o n a l S o n a r P a t h s i n p r e s e n c e o f £ z ( t ) We s h a l l f i r s t e x a m i n e t h e one d i m e n s i o n a l c a s e ( random m o t i o n l i m i t e d t o £ z ) , t h e n p r o c e e d i n g t o h i g h e r d i m e n s i o n a l c a s e s . L e t T c be t h e r o u n d t r i p t i m e d e l a y i n t h e a b s e n c e o f d i s t u r b a n c e s , i . e . , t i m e d e l a y b e t w e e n OR and 0>j>, a n d r t h e a c t u a l t i m e d e l a y b e t w e e n O p ' a n d O ^ ' • T h e n T o e v a l u a t e l ^ ' a n d 1 R ' , c o n s i d e r t h e s i m i l a r t r i a n g l e s A a n d B ( F i g . 2 . 1 4 ) d + * 2 ( t > * z ( t + T c ) - { z ( t ) 1 T ' + 1 R ' c r r c u - 2 r T c u [ d + £ 2 ( t ) ] r= U z ( t+r c )+{ 2 ( t )+2d) ( 2 . 2 7 ) which g i v e s 32 1 T' +1R'= 2 [ r 2 + ( d + £ z ( t ) ) 2 ] 1 / 2 + [ ( r c y - 2 r ) 2 + ( £ z ( t + T c ) - £ z ( t ) ) 2 ] 1 / 2 T C 2 V 2 = 2 ( d + U ( t ) ) / + 1 ' U z ( t + T c ) + * 2 ( t ) + 2 d ) 2 / 2 ( d + * - ( t ) ) " + / T C 2 U 2 ( 1 ) 2 + U z ( t + r c ) - £ z ( t ) ) 2 U z ( t + T c ) + $ 2 ( t ) + 2 d ) ( 2 . 2 8 ) = 2 ( d + £ z ( t ) ) k + / r c 2 i » 2 ( ) 2 + U z ( t + r c ) - £ 2 ( t ) ) 2 £ z ( t + r c ) + £ z ( t ) + 2 d = 2 ( d + $ z ( t ) ) k + U z ( t + T c ) - £ z ( t ) ) k = k [ 2 d + £ z ( t + r c ) + £ z ( t ) ] ( 2 . 2 9 ) / ' a 2 * 2 w h e r e k = / +1 =1 ( 2 . 3 0 ) U z ( t + T c ) + $ z ( t ) + 2 d ) 2 s i n c e d >> £ z a n d c >> v. L e t t h e e s t i m a t e d d e p t h be d . U s i n g ( 2 . 2 5 ) a n d ( 2 . 2 9 ) , d = T c . c / 2 = d + ( £ z ( t + r c ) + £ z ( t ) ) / 2 ( 2 . 3 1 ) H e n c e , t h e e r r o r i n m e a s u r e m e n t i s g i v e n by e d = ( £ z ( t + T c ) + * z ( t ) > / 2 ( 2 . 3 2 ) We now e x t e n d s t h e r e s u l t t o i n c l u d e b o t h n e t h e a v e l z ( t ) a n d n e t sway £ y ( t ) ( F i g . 2 . 1 5 ) . By t h e same a p p r o a c h d + t 2 ( t ) _ 6 z ( t + r c ) - $ z ( t ) r r c i > - 2 r - $ y ( t ) + £ y ( t + T G ) [ T C W - { y ( t ) + t y ( t + T C ) ] [ d + { y ( t ) ] [ £ 2 ( t + r c ) + $ z ( t ) + 2 d ] ( 2 . 3 3 ) I T , +1R'= [ ( d + £ 2 ( t ) ) 2 + r 2 ] 1 / 2 + [ ( r c i > - r - £ y ( t ) + * y ( t + r c ) ) 2 + ( d + £ 2 ( t + r c ) ) 2 ] 1 / 2 / r c u ~ * y ( t ) ~ * y ^ - c = ( d + £ z ( t ) ) / ( -( t + r „ ) - ) 2 + 1 S 2(t+r c)+£ 2(t)+2d o * z ( t ) 33 « Z ( t + T c ) T c » - 2 r - { y . ( t ) + J y ( t + T c ) F i g u r e 2 . 1 5 2 - D i m e n s i o n a l S o n a r P a t h s ( t ) W\rcv-i y ( t ) + * y ( t + T c ) ] ' [ - ( -* z ( t + T c ) + £ 2 ( t ) + 2 d ) + l ] 2 + ( d + { 2 ( t + T c ) ) 2 ( 2 . 3 4 ) A c c o r d i n g t o a s s u m p t i o n , d > > £ 2 ( t ) o r £ 2 ( t + T c ) , t h u s t h e f i r s t r a d i c a l i n ( 2 . 3 4 ) i s a p p r o x i m a t e l y 1, and t h e t e r m a+i 2(t) 1 —as— $ 2 ( t + T c ) + * 2 ( t ) + 2 d 2 so t h a t 1 T ' + 1 R ' = d + £ 2 ( t ) + / [ ( T c i » - t y ( t ) + { y ( t + T c ) ) ] 2 / 4 + ( d + { 2 ( t + r c ) ) 2 ( F i r s t A p p r o x i m a t i o n , 2 . 3 5 ) * d + £ z ( t ) + ( d + * 2 ( t + T c ) ) ( T c u - { y ( t ) + { y ( t + r c ) ) 2 4 ( d + { 2 ( t + T c ) ) 2 * 2 d + £ 2 ( t ) + £ 2 ( t + T c ) ( S e c o n d A p p r o x i m a t i o n , 2 . 3 6 ) where t h e s e c o n d a p p r o x i m a t i o n i s made by o b s e r v i n g t h a t d » | v ( t ) , £ z ( t ) , i . e . , v a l u e o f r a d i c a l i s a p p r o x i m a t e l y 1. 34 Note that t h i s i s s i m i l a r to the r e s u l t i n the 1-dimensional case, the d i f f e r e n c e i n t h i s case being two approximations are r e q u i r e d . In the three-d i m e n s i o n a l case, i n t r o d u c i n g S x ^ ) a n c * $ x ( t + r c ) , the a c t u a l sonar paths 1 T"+1 R" i s given by l T " + l R " = / l T ' 2 + £ x ( t ) 2 + / l R ' 2 + £ x ( t + r c ) 2 = ( d + * z ( t ) ) [ i + ( ) 2+ ( ) 2 ] d+£ z(t) d+£ z(t) + (d+€ z(t + T c ) ) { l + ( — ^ ? E _ ) 2 + ( * 2 )2 ] l/2 2 ( d + $ z ( t + r c ) ) d + £ z ( t + r c ) (2.37) = ( d + £ z ( t ) ) [ l + A 2 + B 2 ] 1 / 2 + ( d + £ z ( t + T c ) ) [ l + C 2 + D 2 ] 1 / 2 Compared to A, B i s more s i g n i f i c a n t s i n c e A=r/(d+| z(t)) equals T C V - « y ( t ) + € y ( t + T c ) i n v o l v e s the term 2d i n i t s denominator. * z < t + r c ) + £ z ( t ) + 2 d S i m i l a r l y , D i s more s i g n i f i c a n t than C. Hence, i n a f i r s t approximation, we ignore A and C, and take the f i r s t two f i r s t two terms i n bino m i a l expansions of [ 1 + B 2 ] 1 / 2 and [ 1 + D 2 ] 1 / 2 : 1 £ x ( t ) 2 S x ( t + r c ) 2 / T " + / R " =2d+£ z(t) + $ z ( t + r c ) + [ + ] 2 d+« z(t) d+$ z(t) 1 € x ( t + r c ) 4 _ i x ( t ) 4 - [ + ]=di 8 ( d + £ z ( t + r c ) ) 3 ( d + { z ( t ) ) 3 (2.38) In the second approximation, a l l expansion terms are ignored, g i v i n g the f a m i l i a r r e s u l t 1 T'+1 R'= 2 d + $ z ( t ) + £ z ( t + r c ) = d 2 (2.39) 35 Define the f i r s t approximation by e^' 1^, and the second by e d ( 2 ) : e d ( 1 (2.41) e d ( 2 ) = ( d 0 - d 2 ) / 2 (2.41) e^^ 1^ and e ^ 2 ) are p l o t t e d a g a i n s t d, using t y p i c a l maximum value s of £ z ( t ) , £ v ( t ) , £ x ( t ) . v i s chosen to be 2 ms~ 1, and c= 1 520 ins"' . In F i g . 2.16(a) and (b) , high frequency random motions are assumed, so that the net magnitudes i n x, y and z d i r e c t i o n s separated by T c have opposite s i g n s . In F i g . 2.16(c), low frequency motions are assumed, and the v a l u e s of £z( t)'£y( f c)>£x( f c) remain unchanged at t + T C . In a l l cases, e^^ 1) and e^/ 2^ i n c r e a s e as d decreases, i n d i c a t i n g t h a t approximations become l e s s v a l i d as d becomes comparable to the £'s and TCI>. A l s o , as expected, e ^ 2 ^ > 1 ^ , and t h e i r magnitudes are small when a b s o l u t e v a l u e s of £'s are s m a l l . In ( c ) , e ^ 1 ) i s q u i t e n e g l i b l e , and e^2^ i s smaller than i n ( a ) . F i n a l l y , as surge i n l a r g e s h i p i s known to be s m a l l , we n e g l e c t the J - X ( t ) ' s i n ( d ) . In t h i s case, compared with ( a ) , while remains approximately the same, e^^ 2^ i s s m a l l e r . Taking (d) as r e p r e s e n t a t i v e of a worst case s c e n a r i o , the r a t i o e d ( 2 ) / £ z ( t ) > 0.01 at d l e s s than 160m. Depending on the system accuracy d e s i r e d , t h i s c o u l d d e f i n e the c r i t i c a l value of d i n employing the approximation e x p r e s s i o n d ( t ) = d ( t ) + U z ( t + r c ) + £ z(t) V/2 (2.42) Note that i n shallow water, T c i s s m a l l , so £ z ( t + r c ) i s u n l i k e l y to be s i g n i f i c a n t l y d i f f e r e n t from £ z ( t ) . I f n \ — 1 \ \ \ \ • f,(t)= 2.0*, | x ( t + r c ) = - 2 . 0 ? y ( t )= 2 . 0 m (y(t + T c ) = - 2 . 0 W £z(t)= 3.0 m £ z(t+T c)=-3.0/* 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 d o © co-rn O o o d o-o O o d 9' o d. 1 ,1 \ \ N \ <•'> w f x ( t )= 1.0 m f x ( t + r c ) = - l . 0 ^ * y ( 0 = LO ^ fy(t + T c ) = - 1 . 0 m £z0)= 1.0 m f z(t+r c)=-1.0w 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 d F i g u r e 2.16 Approximation E r r o r s 38 VELOCITY ( Km/SEC ) F i g u r e 2.17 Sound V e l o c i t y i n Water[l9] fcz(t+Tc)=£z(t), then d(t)=d(t)+£ z(t) (2.43) which i s the measurement formula used by many authors i n c l u d i n g El-Hawary[13]. In (2.42), the estimate d i s a f u n c t i o n of d, as w e l l as r c . But T c i s a f u n c t i o n of d and c, the speed of sound i n water. W h i l s t i n most l i t e r a t u r e c i s taken to be c o n s t a n t ( a t approximately 1520 ms~ 1), c i s a c t u a l l y v a r y i n g , depending on temperature, s a l i n i t y and depth. Fig.2.17 i s a p l o t of c versus depth d. The p r o f i l e s A, B and C are recorded, r e s p e c t i v e l y , at 28°S, 39°S and 43°S. They a l l show c h a r a c t e r i s t i c minima i n the r e g i o n d-1.1 to 1.4 km, known as 39 the deep sound channel. We s h a l l assume d to be slo w l y v a r y i n g , and r c to be given by r c=2d/c, where d i s the .mean value of depth f o r the d u r a t i o n of the experiment,and c i s determined from an a p p r o p r i a t e v e l o c i t y p r o f i l e , s i m i l a r to F i g . 2.17. 2.3 M o d e l l i n g of the S e a f l o o r The formation of the s e a f l o o r was caused by many u n r e l a t e d processes, a l l random i n nature. Thus i t i s not s u r p r i s i n g that knowledge about i t s nature i s scarce and that any s t a t i s t i c a l d e c r i p t i o n of i t i s l i k e l y to be complicated. S e v e r a l recent works [ 1 ] , [ 1 1 ] , [ 4 ] have attempted to a r r i v e at some s t a t i s t i c a l c h a r a c t e r i z a t i o n of the s e a f l o o r based on seismic experiments and p r a c t i c a l models. The wide range of t o p o g r a p h i c a l f e a t u r e s of the s e a f l o o r can roughly be d i v i d e d i n t o three c a t e g o r i e s : a) Gross f e a t u r e s , of the order of tens to hundreds of k i l o m e t e r s , are mainly p h y s i o g r a p h i c f e a t u r e s such as h i l l s , a b y s s a l p l a i n s , trenches, seamounts, e t c . b) Intermediate f e a t u r e s , of the order of hundreds to thousands of meters, are s e c t i o n s of the gross f e a t u r e s , such as l o c a l v a l l e y s and r i d g e s . c) Small f e a t u r e s , of the order of a few c e n t i m e t e r s to a meter, c o n s i s t of r i p p l e s , b o u l d e r s , rocks, e t c . Featur e s (c) are r e s p o n s i b l e f o r s c a t t e r i n g i n the r e f l e c t e d s i g n a l as they are of the same order as the 40 F i g u r e 2.18 The Expected Mean Refected S i g n a l as a F u n c t i o n of kacos0[11] a c o u s t i c a l wavelengths. The beam p a t t e r n s of the s c a t t e r e d s i g n a l at bottoms of v a r y i n g roughness and the co r r e s p o n d i n g r e c e i v e d p a t t e r n s are shown i n F i g . 2.18. To c h a r a c t e r i z e the roughness of the bottom, an index known as rms roughness a i s d e f i n e d as f o l l o w : < P >/ Po = e x p [ - 2 k 2 o 2 c o s 2 0 ] where < p > i s the c o h e r e n t l y r e f l e c t e d s i g n a l , p D i s the e q u i v a l e n t s i g n a l r e f l e c t e d from a smooth s u r f a c e , k i s wavenumber, and 6 the i n c i d e n t angle. As shown i n F i g . 2.18, as a i n c r e a s e s , the r a t i o of c o h e r e n t l y r e f l e c t e d s i g n a l d ecreases, i n d i c a t i n g l a r g e s c a t t e r i n g . The s u b j e c t of s c a t t e r i n g and i t s removal by d e c o n v o l u t i o n has been t r e a t e d by v a r i o u s p apers[7] and w i l l not be d i s c u s s e d here. I t s u f f i c e s to assume that the r e f l e c t e d s i g n a l has been s u f f i c i e n t l y deconvolved ( r e c o v e r i n g the p r e c i s e ranges at a, b, c, etc.) 41 before motion compensation to y i e l d p r e c i s e measurements as i f r e f l e c t e d from a smooth s u r f a c e . Then, the f e a t u r e s most r e l e v a n t to our problem are of the intermediate c a t e g o r y . S t u d i e s on a wide range of s e a f l o o r types i n d i c a t e t h at most s e a f l o o r e l e v a t i o n s have Gaussian d i s t r i b u t i o n s , with a small f r a c t i o n having non-Gaussian d i s t r i b u t i o n [ 4 ] . Based on experimental r e s u l t s , Clay and Leong [11] a r r i v e d at an approximate one-dimensional s p a t i a l a u t o c o r r e l a t i o n f u n c t i o n f o r a gen e r a l s e a f l o o r , which i s e s s e n t i a l l y a f i r s t order Markov p r o c e s s : R(/ ) = o 2exp[-/3|/ | ] (2.44) where o=mean square e l e v a t i o n 0=1/L2/ L 2 = c o r r e l a t i o n d i s t a n c e of the s e a f l o o r d e f i n e d as R ( L 2 ) = l / e . R ( 0 ) . For most s e a f l o o r s , L 2 > 100m r e p r e s e n t s smooth or u n d u l a t i n g p r o f i l e , w h i l s t L2 < 100m i n d i c a t e s the presence of s t r o n g i r r e g u l a r i t i e s such as r i d g e s , f r a c t u r e s , l o c a l v a l l e y s , e t c . The s p a t i a l spectrum of t h i s model i s given by S ( u s ) = 20o- 2/(w s 2 +02) ( F i g . 2.19) (2.45) Conversion from s p a t i a l to time c o r r e l a t i o n i s accomplished by t a k i n g T as T = X / v Then, R(r)=<d(t+r)d(t)> =<d(-+-)d(-)> V V V =R(-) V The c o r r e l a t i o n time i s consequently s c a l e d by 1/u. 42 Re c e n t l y , Berkson and Matthews[4] proposed the a l t e r n a t i v e approximate form of S(o; s) S(w s)=c/w s b (2.46) with b ranging from 1.6 to 5 f o r widely d i f f e r e n t s e a f l o o r types, and mean value of 1.8. f o r North A t l a n t i c , b i s r a r e l y l e s s than 1, i n d i c a t i n g there are very few f e a t u r e s that are t a l l r e l a t i v e t o t h e i r h o r i z o n t a l dimensions. Compared to the Clay-Leong formula, t h i s i n c o r p o r a t e s gross f e a t u r e s as the spectrum p l a c e s s t r o n g emphasis on low s p a t i a l frequency r e g i o n . For higher f r e q u e n c i e s , they are e s s e n t i a l l y the same f o r b=2, s i n c e the c o n t r i b u t i o n of 0 2 i n (2.45) becomes n e g l i g i b l e . Thus, the Clay-Leong formula can be thought of as an approximate model of p r i n c i p a l l y i n t e r m e d i a t e s e a f l o o r f e a t u r e s , and s h a l l be used i n subsequent a n a l y s i s . 43 Chapter 3 FILTERING IN THE PRESENCE OF NOISE STATISTICS 3.1 The F i l t e r i n g Problem In the sonar s i g n a l p r o c e s s i n g system, the deconvolved r e f l e c t e d s i g n a l records c o n s i s t of a sequence of d i s c r e t e measurements degraded by the random motion components. The approximate r e p r e s e n t a t i o n of t h i s degraded s i g n a l has been e s t a b l i s h e d as z(nAt)=d(nAt)+[$ z(nAt)+£ z(nAt+r c)]/2 (3.1) where At i s the sampling p e r i o d . E q u i v a l e n t l y , i n continuous t ime z ( t ) = d ( t ) + U z ( t ) + £ z(t + r c ) ]/2 (3.2) In making t h i s approximation, the f o l l o w i n g assumptions were made: a) The sea i s reasonably deep (to s a t i s f y the c r i t i c a l l i m i t imposed by the r a t i o ea^ 2V£ z(t) as d i s c u s s e d i n S e c t i o n 2.3) . b) The s h i p speed v «: c. c) A narrow-beam sonar i s used. d) The mean wind speed u i s constant d u r i n g experiment In a d d i t i o n , we s h a l l assume T c to be constant f o r the d u r a t i o n of the experiment, i . e . , a constant mean depth can be found. Allowance i s s t i l l made f o r the presence of a s l o w l y - v a r y i n g t r e n d i n s e a f l o o r p r o f i l e . A higher rate of v a r i a n c e , as i n a steep s l o p e , w i l l r e s u l t i n the s i g n a l z ( t ) being n o n s t a t i o n a r y . 44 Equation (3.2) can be w r i t t e n as z ( t ) = d ( t ) + n ( t ) where n ( t ) = [ $ z ( t ) + $ 2 ( t + r c ) ] / 2 , (3.3) r e c o g n i z i n g that d ( t ) and n ( t ) are both zero-mean, Gaussian p r o c e s s e s , and that they are u n c o r r e l a t e d . T h e r e f o r e , we can pose the degradation problem as that of an a d d i t i v e , s t a t i o n a r y n o i s e c o r r u p t i o n problem, with the noise given by n ( t ) . Such problem r e a d i l y lends i t s e l f to treatment by v a r i o u s f i l t e r i n g t e chniques. To gain f u r t h e r i n s i g h t i n t o the nature of d ( t ) and n ( t ) , c o n s i d e r the a u t o c o r r e l a t i o n of z ( t ) , given by Rz(r)=Rd(T)+R^z(r)/2+R^z(T+Tc)/i+R^z(T-Tc)/A ( 3 - 4 ) The corresponding power spectrum i s given by S 2(cu)=S d(co)+S£ z(cu) + [e J £ J 7'cS£ 2(co)+e" : i' J rcS£ 2(a))]/4 = S d(co)+S£ 2(cj) [ 1+C O S C J T c ]/2 (3.5) Thus, the e f f e c t on S£ z(a>) of time delay r c d u r i n g measurement i s to s e l e c t i v e l y a t t e n u a t e the o r i g i n a l spectrum S£2(co) by (1 +coscJ7 c )/2 ( F i g . 3 . 1 ) . A t t e n u a t i o n i n c r e a s e s as r c i n c r e a s e s , and as frequency i n c r e a s e s . Thus, i n a d d i t i o n to a t t e n u a t i o n , there i s a l s o apparent narrowing of the n o i s e power spectrum. Such e f f e c t s are i n agreement with the o b s e r v a t i o n that l a r g e r o u n d t r i p delay tends to average out the n o i s e component. However, s i n c e the s i g n i f i c a n t p o r t i o n of the n o i s e spectrum f o r both small and l a r g e s h i p s l i e w i t h i n 0.1 to 0.2 Hz, a t t e n u a t i o n e f f e c t does not become s i g n i f i c a n t t i l l r c > 0.5s, or, d > 400m. Frequency (hz) F i g u r e 3.1 A t t e n u a t i o n E f f e c t of Time Delay o 8-. P £ 8-o 6. o o d 0.00 f (Hz) F i g u r e 3.2 T y p i c a l S 2 ( f ) (Corresponding to Fig.1.2, f o r the case of small s h i p at u=lOms~ 1, L2=300m) Sn<o>> 46 S d < « > S z(w) H(w) S €(w) F i g u r e 3.3 Block Diagram of F i l t e r i n g i n the Frequency Domain The s t a t i o n a r y , Gaussian nature of (3.3) ensures that Sz(u>) i n (3.5) i s a l s o Gaussian and s t a t i o n a r y . Equation (3.5) can be w r i t t e n as S z iu>) =S d (w) +S n (w) with S n(w)=S$ z(cj) [ ( 1+COSWTc)/2] (3.6) A t y p i c a l Sz(co) spectrum i s shown i n Fig.3.2. T h i s corresponds to a small s h i p heading d i r e c t l y i n developed sea at wind speed u=10 ms~ 1, performing experiments on s e a f l o o r of c o r r e l a t i o n d i s t a n c e L2=300m. We are now i n a p o s i t i o n to s t a t e the f i l t e r i n g problem. Let H(co) be a l i n e a r , t i m e - i n v a r i a n t f i l t e r . The f i l t e r i n g problem i s t o minimize the e r r o r power P E given by The output of the f i l t e r i s d ( t ) , the optimal estimate of d ( t ) . P £ = ( 1 / 2 W ) ; [Se(u>)]du> = (l/27r)r [S y(w)-S d(&))]dw »(1/2»)J [(1/2»)J S 2 ( u ) |H(e3 w) | 2dw-S d(w) ]dcj =(1/2*)/ [ ( 1 / 2 * ) / (S d(a>)+S n(w v) |H(e3 w) | 2du-S d(u)]da> (3.7) n(n) 47 d(n) + z ( n ) h(n) y(n) e(n) F i g u r e 3 . 4 B l o c k D i a g r a m of D i s c r e t e T ime W i e n e r F i l t e r To d e t e r m i n e H (CJ ) , i t i s e s s e n t i a l t h a t b o t h S d ( to) and S n ( w ) a r e known. The s o l u t i o n t o ( 3 . 7 ) i s t h e w e l l known W i e n e r F i l t e r ( F i g . 3 . 4 ) . F o r m u l a t e d i n d i s c r e t e t i m e , t h e W i e n e r F i l t e r m i n i m i z e s E [ e n 2 ] = E [ ( y ( n ) - d ( n ) ) 2 ] = E [ ( I z ( k ) h ( n - k ) - d ( n ) ) 2 ] = Z I E [ z ( l ) z ( m ) ] h ( n - l ) h ( n - m ) + E [ d 2 ( n ) ] Z m - 2 E [ Z z ( k ) d ( n ) h ( n - k ) ] = H R Z Z H T - 2 H C z d + o d 2 ( 3 . 8 ) where R Z Z = R D D ( n ) + R £ Z ( n ) / 2 + R £ Z ( n A t + r c ) / 4 + R £ Z ( n A t - r c ) / 4 i s t h e d i s c r e t e t i m e a u t o c o r r e l a t i o n o f t h e ( i n p u t ) n o i s y s i g n a l z ( t ) , a n d C z d ( n ) i s t h e c r o s s c o r r e l a t i o n be tween z ( t ) and d ( t ) . M i n i m i z i n g , g H n = 2 H T R z z - 2 C z d and s e t t o z e r o g i v e s H o p t ' T_ =R z z - 1 c z d ( 3 . 9 ) The W i e n e r F i l t e r i s n o t v e r y c o n d u c i v e t o i m p l e m e n t a t i o n i n r e a l - t i m e s i n c e t h e o r d e r o f H 0 p t T i s u s u a l l y h i g h , a l t h o u g h i t c a n be i m p l e m e n t e d i n l a t t i c e f o r m i n w h i c h t h e c o m p l e x i t y 48 d ( t ) z ( t ) - d ( t ) z ( t ) H2(u>) "H,(«) n ( t ) F i g u r e 3.5 B l o c k D i a g r a m o f E q u i v a l e n t F i l t e r i n g S y s t e m i s d e t e r m i n e d by t h e a c c u r a c y d e s i r e d . The Ka lman F i l t e r i s a l s o d e r i v e d by m i n i m i z i n g e n 2 , b u t s i n c e i t i s a r e c u r s i v e a l g o r i t h m b a s e d on s t a t e - s p a c e m o d e l s o f b o t h d ( t ) a n d n ( t ) , i t c a n be i m p l e m e n t e d i n much l o w e r o r d e r s . I t h a s t h e f u r t h e r a d v a n t a g e o f e a s y e x t e n s i o n t o c a s e s w h i c h i n v o l v e v a r y i n g w ind s p e e d . S i n c e b o t h d ( t ) a n d n ( t ) a r e u n c o r r e l a t e d G a u s s i a n p r o c e s s e s , i t i s p o s s i b l e t o d e f i n e a f i l t e r w h i c h s e e k s t o m i n i m i z e P e 2 = ( 1 / 2 ? r ) J * £ s e 2 ] d u w i t h S e 2 ^ ) = S y ( t o ) - S n ( t o ) ( 3 . 1 0 ) i . e . , t h e f i l t e r a t t e m p t s t o o p t i m a l l y e s t i m a t e n ( t ) . T h u s , t h e o p t i m a l e s t i m a t e n ( t ) i s g i v e n by The e q u i v a l e n c e b e t w e e n t h e two f i l t e r i n g s y s t e m s i s i l l u s t r a t e d i n F i g . 3 . 5 . S i n c e s t a t i s t i c s o f n ( t ) a r e more r e a d i l y a n d more a c c u r a t e l y a v a i l a b l e t h a n t h a t o f d ( t ) , we s h a l l h e n c e f o r t h a d o p t t h e s e c o n d m o d e l , by t r e a t i n g n ( t ) a s t h e ' s i g n a l ' , a n d d ( t ) t h e ' n o i s e ' . n ( t ) - z ( t ) - d ( t ) (3.11 ) 49 If the s t a t i s t i c s of d ( t ) and n(t) are both unknown, then s t r a i g h t f o r w a r d a p p l i c a t i o n of Kalman F i l t e r i s not p o s s i b l e . Kalman F i l t e r can be used, however, as a parameter e s t i m a t o r , p r i o r to s t a t e e s t i m a t i o n , which i s the idea behind the s e l f - t u n i n g f i l t e r . A l t e r n a t i v e l y , r e c u r s i v e l e a s t - s q u a r e i n t e r p o l a t i n g f i l t e r can be used. The f i l t e r i n g techniques to be used i n each d i f f e r e n t case are summarised i n Table 3.1. Table 3.1 S n (w) Known Unknown Known Wiener F i l t e r -Unknown Kalman F i l t e r with RLSI F i l t e r a u x i l i a r y e s t i m a t i o n of L 2 Extended Kalman S e l f - T u n i n g F i l t e r F i l t e r Adaptive ARMA F i l t e r 3.2 Kalman F i l t e r The process and measurement models of a dynamic system has the f o l l o w i n g g e n e r a l s t a t e - s p a c e form x(t)=Fx('t)+Gw(t) (3.12a) z(t ) = H x ( t ) + v ( t ) (3.12b) Here, x ( t ) i s the system s t a t e v e c t o r , z ( t ) the o b s e r v a t i o n v e c t o r . F i s the process dynamics matrix, and H, the o b s e r v a t i o n m a t r i x . w(t) and v ( t ) are, r e s p e c t i v e l y , the system d r i v i n g n o i s e and o b s e r v a t i o n n o i s e processes, and are both 50 F i g u r e 3.6 S y s t e m M o d e l and D i s c r e t e Ka lman F i l t e r G a u s s i a n ( F i g . 3 . 6 ) . F o l l o w i n g s e c t i o n 3 . 1 , t h e p r o c e s s m o d e l i s c h o s e n t o be n ( t ) . T h e n , v ( t ) m o d e l s d ( t ) . A c c o r d i n g t o l i n e a r d y n a m i c s y s t e m t h e o r y , a s h a p i n g  f i I t e r d r i v e n by s t a t i o n a r y w h i t e G a u s s i a n n o i s e c a n be d e t e r m i n e d so a s t o g e n e r a t e a p r o c e s s h a v i n g i d e n t i c a l s p e c t r a l c h a r a c t e r i s t i c s a s x ( t ) . T h e r e s u l t i n g p r o c e s s i s an i n n o v a t i o n r e p r e s e n t a t i o n o f x ( t ) . I f x ( t ) i s G a u s s i a n , t h e n t h e f i l t e r i s l i n e a r , c a u s a l a n d c a u s a l l y i n v e r t i b l e [ 2 4 ] . I f t h e power s p e c t r a l d e n s i t y o f x ( t ) i s r a t i o n a l , t h e f i l t e r p a r a m e t e r s a r e d e t e r m i n e d d i r e c t l y f r o m S x (co) by s p e c t r a l f a c t o r i z a t i o n [ 2 8 ] . T h e f i l t e r p a r a m e t e r s i n t u r n f u l l y d e t e r m i n e s F , G , a n d H i n t h e s t a t e s p a c e m o d e l . I f t h e power s p e c t r a l d e n s i t y o f x ( t ) i s i r r a t i o n a l , a s i s t h e c a s e o f n ( t ) , t h e n i t s h o u l d be a p p r o x i m a t e d a s c l o s e l y a s d e s i r e d by a r a t i o n a l m o d e l , a n d t h e same p r o c e d u r e f o l l o w s . S i n c e n ( t ) i s s t a t i o n a r y , t h e s h a p i n g f i l t e r i s t i m e - i n v a r i a n t . 51 Let H a(jcj) be the approximation f i l t e r . The approximation procedure i s c a r r i e d out by minimizing the cost f u n c t i o n J=f~(S a(a>)-S n(w) )2dco (3.13) where S a {co) = (1 /2ir) |H( jw) | 2b{co)dco i s the power spectrum generated by the white n o i s e d r i v e n shaping f i l t e r . The f o l l o w i n g must now be decided: a) The t r a n s f e r f u n c t i o n H a ( s ) to be used. b) The form of the c o s t f u n c t i o n J . c) The m i n i m i z a t i o n a l g o r i t h m . I n s p e c t i o n of S n (CJ) (Appendix A1 ) c o n f i r m s that a second-order narrow-band f i l t e r response i s probably a good i n i t i a l approximation. C l o s e r approximation can be achieved by i n c r e a s i n g the order of the f i l t e r . T h i s may be necessary f o r the case of s m a l l s h i p i n high wind speed (u>15 ms~ 1), i n which a second peak at a lower frequency appears i n S n(o>). Two second-order f i l t e r t r a n s f e r f u n c t i o n s , each r e p r e s e n t i n g a peak, can be summed, r e s u l t i n g i n a f o u r t h - o r d e r f i l t e r ( F i g . 3 . 7 b ) . For s i m p l i c i t y and f o r c o n s i s t e n c y in i n t e r p r e t i n g r e s u l t s , we s h a l l adopt a second-order f i l t e r throughout. 3.2.1 The S t a t e Space Model A narrow-band system i s c h a r a c t e r i s e d by two f a c t o r s : the n a t u r a l ( c e n t r e ) frequency f 0 , and the Q f a c t o r d e f i n e d as Q = f 0 / ( f 2 - f 1 ) (Fig.3.7) (3.14) T h e i r r e l a t i o n s to the g e n e r a l second-order system cs N(s) H ( S ) = = s 2+as+b W(s) (3.15) 52 s n ( f ) F i g u r e 3.7 T r a n s f e r F u n c t i o n Approximation of S ( f ) are a= c o 0 / Q b = ( c o 0 ) 2 c = k u > 0 / Q , where k i s a s c a l i n g constant (3.16) A s t a t e - s p a c e model f o r n ( t ) i s obtained by choosing a 2x1 s t a t e v e c t o r x ( t ) , x ( t ) = [ x 1 ( t ) , X 2 ( t ) ] T , and l e t t i n g t h e i r L a p l a c e transforms be X 2 ( s ) = s X 1 ( s ) (3.17a) X, (s)=QN(s)/ko) 0s (3.17b) r e s u l t s i n N(s)=cX2(s). Thus, X2(s) i s a s c a l e d v e r s i o n of N ( s ) , the t r a n s f o r m of n ( t ) . Using (3.17), the st a t e - s p a c e equations are x 1 ( t ) = x 2 ( t ) (3.18a) x 2 ( t ) = - w 0 2 x 1 ( t ) - ( u 0 / Q ) x 2 ( t ) + w ( t ) (3.18b) or, i n v e c t o r - m a t r i x form x , ( t ) "0 1 " i c ^ t ) 0 x 2 ( t ) _""o 2 "wo/Q_ x 2 ( t ) 1 u(t) i . e . , x (t)-Fx(t)+Gw(t) (3.19) (3.20) 53 with F= G= -COr "o" 1 In d i s c r e t e - t i m e , the s t a t e - s p a c e equations become x(k+1)=$x(k)+rw(k) z(k+1)=Hx(k+1)+v(k+1) with T, H given by[38] #(k+1,k) = T(k+1,k)= a (A) s i n (ojdA+i/>) (a (A)/a>0) s i n (co^A) -cj Da (A) s i n (u dA) -a (A) s i n (co^A-i//) 1 exp(-o) 0A/2Q [ 1 -2 /!-(1/4Q 2) -]sin(wdA+<//) exp(-cj QA/2Q) sin(w^A) coQ 1/4Q2) H=(0 coQ/Q) where A = T J C + I - T J c i s the sampling p e r i o d of the system, e x p ( - 6 j Q A / 2 Q ) a(A) = w 0 /1-(1/4Q 2) (3.21a) (3.21b) (3.22a) (3.22b) (3.23a) (3.23b) (3.23c) (3.23), a ud=co0 /1-(1/4Q 2) i//=tan~1 (/4Q2-1 ) S u b s t i t u t i n g the r e l a t i o n s (3.16) i n t o (3.22) to sta t e - s p a c e model can be set up to approximate the given process n ( t ) . The optimal v a l u e s of a, b, c of a second-order system are obtained by min i m i z i n g J i n (3.13). The c o s t f u n c t i o n J i s d e f i n e d i n terms of an a p p r o p r i a t e m o d e l l i n g e r r o r , e m . A p o s s i b l e candidate f o r e m i s the mean square l o g s p e c t r a l 54 f l a t n e s s [ 6 ] : 1 S n ( c j ) log ( — — - ) l 2 d c u 2?r S a(co) (3.24) T h i s e r r o r measure i s chosen because i t p r o v i d e s an i n t e g r a l measure over the e n t i r e bandwidth of approximation. Yhe mi n i m i z a t i o n problem (3.13) can then be r e c a s t i n the form Min [ e m 2 ] su b j e c t to the c o n s t r a i n t s a = 2rrf 0/Q b = ( 2 7 r f 0 ) 2 (3.25) which i s the f a m i l i a r c o n s t r a i n e d n o n - l i n e a r programming problem. To solve t h i s , we choose the d i r e c t search Simplex method. The r e s u l t s are given i n Tables 3.2(a) and (b), f o r l a r g e and smal l s h i p s . The l a s t column i n both t a b l e s l i s t the undimensional q u a n t i t y e m , which i s a measure of the accuracy of o p t i m i z a t i o n . In using the Simplex method, the i n i t i a l value i s e v a l u a t e d by a p p l y i n g the c o n s t r a i n t s . The Simplex method i s used because i t i s p a r t i c u l a r l y s u i t e d to problem with small number of v a r i a b l e s . u(ms 1) a Table 3.2(a) Large b Ship c em 9 0. 1313 0.6168 0.3120 0.07624 1 0 0. 1313 0.6168 0.4420 0. 1 3501 1 1 0. 1313 0.6168 0.5883 0.23924 1 2 0. 1313 0.6168 0.6681 0.29798 13 0. 1313 0.6168 0.7883 0.38763 1 5 0. 1313 0.6168 0.8891 0.46189 1 7 0. 1313 0.6168 0.9452 0.60186 Figure 3.8 Approximating Spectra - Large Ship (a) u=lOms~1 (biu-^ms" 1 (c)u=l5ms (a) u=lO s~' (b)u=l2 s o Ti-I 0.00 0.05 0.10 0.15 0.20 I • '• •• - i 0.25 f (Hz) (a) 0.30 0.35 0.40 0 4 5 0.50 0.00 0.20 0.25 f ( H Z ) 0.30 0.35 0.40 0.45 0.50 (b) o - o . O -O ' If). o o. Figure 3.9 Approximating Spectra - Small Ship, (a) u=6ms"1 (b)u=l2ms_1 (c)u=l5ms .OA 00 0.00 0.05 0.10 0.15 0.20 0.26 f (Hz) 0.30 "o.35 ' 0.40 •* 1 0.45 (a) 0.50 (b) f (Hz) 57 Table 3.2(b) Small Ship u(ms~ 1) a b c em 6 0.1914 1.4252 0.2123 0.02459 8 0.2107 1.3657 0.5496 0.05225 10 0.2128 1.3264 0.7215 0.20019 12 0.2136 1.3207 0.8223 0.27772 13 0.2205 1.3189 0.8431 0.35367 15 0.2214 1.3152 0.8756 0.51280 17 0.2202 1.3141 0.8828 0.69074 Examples of the s p e c t r a and t h e i r approximating f u n c t i o n s are shown i n F i g u r e s 3.8 and 3.9, f o r l a r g e and small s h i p s , at v a r i o u s speeds. As u i n c r e a s e s , approximations become l e s s a c c u r a t e , with the Sn(a>) skewed towards the lower f r e q u e n c i e s . The i n c r e a s i n g approximation e r r o r s are r e f l e c t e d i n the t a b l e s . In g e n e r a l , at high wind speeds (u > 15ms - 1) e m i s gre a t e r f o r small s h i p . T h i s i s expected, as the measurement e r r o r spectrum of small s h i p has a wider bandwidth. With the above approximating technique, one can determine <i>,r, H i n (3.21), given the measurement noise spectrum S n(w). To reduce computation, approximating f u n c t i o n H a ( s ) of the lowest order should be chosen. 3.2.2 Measurement-Differencing Kalman F i l t e r The measurement noise v ( t ) i n the st a t e - s p a c e equation (3.12) models the s e a f l o o r p r o f i l e process d ( t ) , and i s , ac c o r d i n g to S e c t i o n 2.3, a f i r s t - o r d e r Markov p r o c e s s . T h i s has the i n n o v a t i o n r e p r e s e n t a t i o n (Fig.3.10) v ( t ) = - 0 v ( t ) + u ( t ) (3.26) The d i s c r e t e - t i m e v e r s i o n i s d e s c r i b e d by u ( t ) -0 F i g u r e 3.10 S h a p i n g F i l t e r f o r M e a s u r e m e n t N o i s e v ( k + 1 ) = ¥ v ( k ) + u ( k ) (3.27) w i t h t h e t r a n s i t i o n m a t r i x g i v e n by *= (e ^ ) , a n d u ( k ) a d r i v i n g w h i t e n o i s e s e q u e n c e . v ( k ) i s c o n s e q u e n t l y an e x p o n e n t i a l l y c o r r e l a t e d p r o c e s s [ l 6 ] . The c o m p l e t e d i s c r e t e - t i m e m o d e l S^ o f t h e n o i s e - d e g r a d e d m e a s u r e m e n t i s g i v e n by f x ( k+1 ) = * x ( k ) + r w ( k ) (3.28a) z(k+1 )=Hx(k+1 )+v(k+1) (3.28b) { v ( k + 1 ) = ¥ v ( k ) + u ( k ) (3.28c) 1 L e t n be t h e o r d e r o f t h e a p p r o x i m a t i n g f i l t e r . T h e n , s t a t e - v e c t o r x i s n x l , • i s n x n , T i s n x l , H i s i x n , a n d b o t h z a n d v a r e s c a l a r . The p r o c e s s n o i s e w(k ) i s n x l , and m e a s u r e m e n t n o i s e v ( k ) s c a l a r , w a n d u a r e b o t h z e r o - m e a n w h i t e G a u s s i a n s e q u e n c e s h a v i n g c o v a r i a n c e s E { w ( k ) v ( l ) T } = Q (3.29a) E { u ( k ) u ( l ) T } « R 6 ( k - l ) (3.29b) E { w ( k ) u ( l ) T } = 0 (3.29c) The p r o b l e m o f f i l t e r i n g w i t h c o r r e l a t e d m e a s u r e m e n t n o i s e c a l l s f o r a t r a n s f o r m a t i o n w h i c h r e n d e r s m e a s u r e m e n t n o i s e 59 white. S i m i l a r l y , the c o n v e n t i o n a l Kalman A l g o r i t h m i s r e v i s e d based on the transformed model. The t r a n s f o r m a t i o n i s e f f e c t e d by i n t r o d u c i n g a new measurement v e c t o r £, d e f i n e d as £(k)=z(k+1 )-*z(k) (3.30) S u b s t i t u t i n g (3.28a,b) f o r z(k+1), $(k)=H*x(k)+Hrw(k)+*v(k)+u(k)-¥Hx(k)-*v(k) Thus, the transformed model S 2 i s given by • x(k+1)=$x(k)+rw(k) (3.31a) S 2 J |(k)=H'x(k)+v'(k) (3.31b) . v'(k)=u(k)+HTw(k) (3.31c) where H'=H*-*H The nois e c o v a r i a n c e s are given by E{w(k)w(l) T}=Q E{v*(k)v'(l) T}=R+HrQr TH T=R' E { w ( k ) v ' ( l ) T } = Q r T H T = S or, E w(k) V (k) [ w ( l ) T v ' ( l ) T ] Q S S T R' (3.32a) (3.32b) (3.32c) Thus, the t r a n s f o r m a t i o n r e s u l t s i n a new measurement noise v', which i s white (with c o v a r i a n c e R'). But v' i s now c o r r e l a t e d with w ( c o v a r i a n c e S ) . The u s e f u l n e s s of t h i s t r a n s f o r m a t i o n i s e s t a b l i s h e d by n o t i n g t h a t the convergence and s t a b i l i t y c o n d i t i o n s of the optimal f i l t e r designed f o r s i g n a l model S 2 are the same as that designed f o r the o r i g i n a l model S^[12]. 60 In d e v e l o p i n g the Kalman F i l t e r equations f o r model S2, i t should be noted that s i n c e £(k) i n c l u d e s z(k+1), the formal ' p r e d i c t i o n ' problem x k + 1 based on £(k) i s a c t u a l l y the ' e s t i m a t i o n ' x k + 1 based on z(k+1)[5]. For c o r r e l a t e d n o i se, the p r e d i c t i o n equation i s given by xk+1|k=* X k | k - 1 + G k r k (3.33a) where r k=$ k-H' x k | k - i i s the i n n o v a t i o n sequence Gk=D+ (4>-DH)Kk i s the P r e d i c t i o n Gain D=TSR'~1 The update equtions f o r the Kalman g a i n K k, the a p r i o r i e r r o r c o v a r i a n c e matrix M k and the a p o s t e r i o r i e r r o r c o v a r i a n c e matrix P k are given by K k=M kH' T(H'M kH' T+R')~ 1 (3.33b) P k = ( l - K k H ' ) M k ( l - K k H ' ) T + K k R ' K k T (3.33c) M k + 1=(*-DH')P k(*-DH') T +rQr T-DR'D T (3.33d) D e r i v a t i o n of (3.33) i s given i n Appendix B, which a l s o i n c l u d e s a r e a l i z a t i o n f o r the case of n=2. C l e a r l y , the performance of the Kalman F i l t e r depends on the accuracy of the s i g n a l model S2. Assuming the process model has been approximated reasonably a c c u r a t e l y , there remain the f o l l o w i n g parameters which have y e t to be determined: a)The matrix *, i n t h i s case, depends on L 2 , the s e a f l o o r c o r r e l a t i o n d i s t a n c e . Whereas one may have a rough idea of what L2 i s , L2 i s not u s u a l l y assumed a c c u r a t e l y known, p r i o r t o experiment. In the next s u b - s e c t i o n , a f a s t e s t i m a t i o n a l g o r i t h m i s presented which p r o v i d e s a rough estimate of L 2 . F u r t h e r , based on a c t u a l run, i t w i l l be 6 1 shown i n S e c t i o n 5 . 3 t h a t L 2 i s n o t s i g n i f i c a n t on p e r f o r m a n c e u n l e s s i t i s l e s s t h a n a c e r t a i n c r i t i c a l v a l u e . b) T h e n o i s e c o v a r i a n c e s Q a n d R a r e , a g a i n , u n k n o w n . I n many a p p l i c a t i o n s , t h e Q a n d R a r e c h o s e n b a s e d on e x p e r i e n c e , e . g . , by e x a m i n i n g t h e r e l a t i v e m a g n i t u d e s o f s i g n a l a n d n o i s e p r o c e s s e s [ 2 8 ] . S e t s o f ( Q , R ) d a t a a r e f o u n d h e u r i s t i c a l l y i n S e c t i o n 5 . 3 . A t t h e c o s t o f a d d i t i o n a l c o m p u t a t i o n s , a d a p t i v e a l g o r i t h m c a n be i m b e d d e d i n t h e K a l m a n F i l t e r , as d i s c u s s e d i n S e c t i o n 3 . 3 c ) T h e p r o c e s s m o d e l i s d e s i g n e d b a s e d on f i x e d w i n d s p e e d , i . e . , p a r a m e t e r s o f 4>, T , H a r e c a l c u l a t e d u s i n g a s i n g l e e n t r y i n T a b l e 3 . 2 ( a ) o r ( b ) . T h i s i s u s u a l l y a c c e p t a b l e i n c a s e o f s h o r t t r i a l , d u r i n g w h i c h t h e mean w i n d s p e e d i s u n l i k e l y t o v a r y s i g n i f i c a n t l y . T h e s o f t w a r e must t h e n i n c l u d e memory s t o r a g e o f T a b l e 3 . 2 . O t h e r w i s e , t h e p r o c e s s m o d e l i s no l o n g e r s t a t i o n a r y , o r must be c o n s i d e r e d u n k n o w n , b o t h s i t u a t i o n s o f w h i c h a r e t r e a t e d i n C h a p t e r 4 . 3 . 2 . 3 F a s t E s t i m a t i o n A l g o r i t h m The f o l l o w i n g a l g o r i t h m p r o v i d e s a s i m p l e mean o f e s t i m a t i n g t h e a u t o c o r r e l a t i o n f u n c t i o n o f a z e r o - m e a n s t a t i o n a r y G a u s s i a n p r o c e s s . F r o m ( 3 . 4 ) , R z ( r ) = R d ( r ) + R ^ ( r ) , w h e r e R^ ( T ) =R^ z ( T ) /2+ R^(T+TC) /i+ R^(T-TC) S i n c e a d 2 >> 0 £ 2 , i t f o l l o w s | R d ( r ) | >> | R j t ( r ) | , - » < T < » . I n o t h e r w o r d s , | R D ( T ) | « | R z ( T ) | . T h u s , a p p l y i n g t h e a l g o r i t h m t o z ( t ) , a z e r o - m e a n G a u s s i a n p r o c e s s , we c a n o b t a i n an a p p r o x i m a t i o n o f R d ( r ) , a n d c o n s e q u e n t l y L 2 . 6 2 A c c o r d i n g to H e r t z [ 2 3 ] , the estimated a u t o c o r r e l a t i o n R Z i s given by R z ( k ) = C N Z Zisgnfzi+j^) where C N = ( TT/2N2 )Z | ZJ | (3.34) and N, the sample s i z e , should be s u f f i c i e n t l y l a r g e . Note that CN needs to be computed only once, the remaining summations can be computed by a d d i t i o n s o n l y . From R Z the c o r r e l a t i o n time r e i s determined as R z ( T 6 ) = ( 1 / e ) . R z ( 0 ) then, L 2 = u . r e . It i s found that N should be at l e a s t 1500. T h i s method r e q u i r e s , t h e r e f o r e , a p r e - e s t i m a t i o n p e r i o d to be reserved f o r L 2 e s t i m a t i o n before a c t u a l p r o c e s s i n g can begin. At a sampling p e r i o d of 0.5s, t h i s amounts to at l e a s t 13 minutes. F i g u r e s 3.11 and 3.12 show samples of estimated R Z f o r l a r g e and small s h i p s . I t i s found that v a r i a t i o n of wind speed has l i t t l e e f f e c t on the e s t i m a t i o n of L 2 , c o n f i r m i n g the prope r t y | ( T ) | >> | R ^ ( T ) | . T a b l e s 3.38(a) and 3.3(b) l i s t the e s t i m a t i o n e r r o r s a g a i n s t the s e a f l o o r types. Here e-^ i s d e f i n e d as e L = ( L 2 - L 2 ) / L 2 Except f o r L 2=300m and 600m, t h i s method y i e l d s resonably good e s t i m a t e s . Sources of e r r o r i n c l u d e i n t e r p o l a t i o n procedure, as w e l l as s i m u l a t i o n . As e x p l a i n e d i n Appendix D, because of f i n i t e data l e n g t h the simulated p r o f i l e has a lower a than expected. T h i s has the apparent e f f e c t of higher 64 3 .12 E s t i m a t e d R z ( t ) - Small S h i p (a )L2=600m T b ) L 2 c l 5 0 m (c ) L 2 = l 0 0 m 350.0 400.0 (a) 400.0 (b) 100.0 ttO.O 200.0 t (sec) (c) 65 d i s t r i b u t i o n o f e n e r g y i n t h e h i g h e r f r e q u e n c y r e g i o n . C o n s e q u e n t l y L 2 f a l l s s h o r t o f r e a l L 2 . I t i s q u i t e p o s s i b l e t h a t i n a c t u a l a p p l i c a t i o n t h e r e s u l t s w i l l be b e t t e r t h a n t h o s e f r o m s i m u l a t i o n . B a s e d on t h e s e , we c o n c l u d e t h a t t h e F a s t E s t i m a t i o n A l g o r i t h m c a n o p e r a t e t o g e t h e r w i t h t h e K a l m a n F i l t e r i n an a u x i l i a r y m a n n e r , t h e r e b y a v o i d i n g t h e n e c e s s i t y o f u s i n g E x t e n d e d K a l m a n F i l t e r . T a b l e 3 . 3 ( a ) L a r g e S h i p ( b ) S m a l l S h i p L 2 L 2 e L 50 0 . 0 4 0 50 0 . 160 100 0 . 140 100 0 . 0 6 0 1 50 - 0 . 0 6 7 150 - 0 . 106 300 - 0 . 2 5 0 300 - 0 . 2 6 0 600 - 0 . 3 7 7 600 - 0 . 4 6 7 3 . 3 E s t i m a t i o n o f P r o c e s s a n d M e a s u r e m e n t N o i s e s 3 . 3 . 1 A d a p t i v e A l g o r i t h m s T o e n s u r e o p t i m a l p e r f o r m a n c e o f t h e K a l m a n F i l t e r , t h e p r o c e s s a n d m e a s u r e m e n t n o i s e c o v a r i a n c e s , Q a n d R, must be known p r e c i s e l y . I m p r o p e r l y a s s i g n e d v a l u e s l e a d t o s u b o p t i m a l f i l t e r i n g , o r e v e n d i v e r g e n c e . S i n c e i n p r a c t i c e t h e s e q u a n t i t i e s a r e known o n l y w i t h l a r g e d e g r e e o f u n c e r t a i n t y , i t w o u l d be d e s i r a b l e t o e s t i m a t e t h e m s i m u l t a n e o u s l y w i t h e s t i m a t i o n , o r , t o c o n s t a n t l y a d j u s t t h e a s s u m e d n o i s e 66 s t r e n g t h s i n the f i l t e r ' s i n t e r n a l model, based upon i n f o r m a t i o n obtained o n - l i n e from the measurements, so t h a t the f i l t e r i s c o n t i n u a l l y 'tuned' as w e l l as p o s s i b l e . The l a t t e r a l g o r i t h m i s termed adaptive or s e l f - t u n i n g , and w i l l be i n v e s t i g a t e d here. Many d i f f e r e n t a d a p t i v e schemes have been proposed[30]. The Maximum L i k e l i h o o d method develops a f u l l s c a l e estimator capable of e s t i m a t i n g Q and R. For o t h e r s , the key to a d a p t a t i o n l i e s i n the r e s i d u a l s , or i n n o v a t i o n s of the s t a t e e s t i m a t o r , which are the d i f f e r e n c e s between a c t u a l measurements and p r e d i c t i o n s based on the f i l t e r ' s i n t e r n a l model. The degree of mismatch thus p r o v i d e s the i n f o r m a t i o n needed i n a d a p t a t i o n . In p a r t i c u l a r , the Hampton A l g o r i t h m makes use of the o r t h o g o n a l i t y p r o p e r t i e s of the r e s i d u a l s to formulate an e r r o r measure, which i s i n turn s o l v e d by a second-order s t o c h a s t i c approximation a l g o r i t h m [ 2 2 ] . With i n n o v a t i o n c o r r e l a t i o n methods, p r o p e r t i e s of the c o r r e l a t i o n of the i n n o v a t i o n sequence are manipulated. Among the d i f f e r e n t forms of t h i s method, Belanger's approach assumes Q and R can be expressed as l i n e a r f u n c t i o n s of a set of parameters. The f a c t that the c o r r e l a t i o n of the i n n o v a t i o n i s a l s o l i n e a r i n the same set of parameters i s used to d e r i v e an a l g o r i t h m which e f f e c t i v e l y performs a r e c u r s i v e weighted l e a s t - s q u a r e s f i t to a sequence of a c t u a l , measured i n n o v a t i o n c o r r e l a t i o n p r o d u c t s [ 3 ] . Another method i s due to Mehra[29]. In t h i s case, the i n n o v a t i o n c o r r e l a t i o n s are expressed i n terms of known system parameters. E m p i r i c a l c o r r e l a t i o n s are 67 s u b s t i t u t e d i n t h e e q u a t i o n s , a n d t h e p r o c e d u r e l e a d s e v e n t u a l l y t o an e s t i m a t e o f t h e c u r r e n t v a l u e s o f Q a n d R . T h e a d a p t a t i o n p r o c e s s i s i t e r a t i v e , w i t h a r b i t r a r y i n t e r v a l . I n g e n e r a l , f o r t h e p r e s e n t p r o b l e m , we recommend u s i n g i n n o v a t i o n c o r r e l a t i o n m e t h o d b e c a u s e o f i t s g r e a t e r s t a b i l i t y , a n d b e c a u s e o f i t s f l e x i b i l i t y w i t h r e s p e c t t o c h o i c e of i t e r a t i v e i n t e r v a l . 3 . 3 . 2 I n n o v a t i o n C o r r e l a t i o n M e t h o d T h e f o l l o w i n g d e s c r i p t i o n o f t h e i n n o v a t i o n c o r r e l a t i o n m e t h o d i s an e x t e n s i o n o f M e h r a ' s m e t h o d t o t h e c a s e i n w h i c h t h e p r o c e s s a n d m e a s u r e m e n t n o i s e s a r e c o r r e l a t e d , a s i n s i g n a l m o d e l S 2 M 8 ] . T h e m e t h o d o p e r a t e s i n b a t c h - r e c u r s i v e m o d e , i . e . , a d a p t a t i o n i s p e r f o r m e d a t r e g u l a r , p r e - d e t e r m i n e d i n t e r v a l s , b e t w e e n w h i c h a b a t c h o f m e a s u r e d d a t a i s a n a l y z e d a g a i n s t e s t i m a t e d d a t a . A d a p t a t i o n y i e l d s a s y m p t o t i c a l l y u n b i a s e d a n d c o n s i s t e n t e s t i m a t e s o f Q a n d R . T h e v a l u e s o f Q a n d R a r e a d j u s t e d d i r e c t l y i n t h e K a l m a n e q u a t i o n s ( F i g . 3 . 1 3 ) . A t t h e e n d o f e a c h i n t e r v a l , a s u i t a b l e t e s t f o r o p t i m a l i t y i s p e r f o r m e d . I f t h e r e s u l t o f t h e t e s t i n d i c a t e s t h e f i l t e r i s c l o s e t o o p t i m a l o p e r a t i o n , a d a p t a t i o n c a n be t e r m i n a t e d . A l s o , a c h o i c e i s a v a i l a b l e t o t h e d e s i g n e r o f e i t h e r f u l l a d a p a t i o n ( b o t h Q a n d R ) , o r p a r t i a l a d a p t a t i o n ( R o n l y ) . T h e l a t t e r mode c a n r e s u l t i n some s a v i n g o f c o m p u t i n g c o s t when t h e m a g n i t u d e o f d r i v i n g p r o c e s s n o i s e i s k n o w n . C a l c u l a t e 68 E s t i m a t e 4-, x k+1|k * H *k|k -1 F i g u r e 3 .13 B l o c k D i a g r a m o f A d a p t i v e Ka lman F i l t e r To commence f i l t e r i n g , i n i t i a l v a l u e s o f Q, R, S, d e n o t e d Q 0 , R D , S Q , and t h e a d a p t a t i o n i n t e r v a l NBATCH, must be p r o v i d e d . N o t e t h a t , a c c o r d i n g t o s i g n a l t r a n s f o r m a t i o n , S = Q ( H D T . T h e r e f o r e , o n l y Q a n d R n e e d t o be a ' d a p t e d . F o r t h e i n i t i a l b a t c h o f NBATCH p o i n t s , m e a s u r e m e n t - d i f f e r e n c i n g Ka lman F i l t e r i s a p p l i e d , u s i n g Q c , R 0 , S G . A t t h e e n d o f t h e f i r s t i n t e r v a l , a c h e c k i s p e r f o r m e d on t h e f i l t e r t o s e e w h e t h e r i t h a s r e a c h e d s t e a d y - s t a t e ( b y , e . g . , e x a m i n i n g G k ) . T h e r e s i d u a l o r i n n o v a t i o n s e q u e n c e r k i s d e f i n e d a s ' k - t k - H ^ k l k - l ( 3 ' 3 5 ) D e f i n e C K t o be t h e e m p i r i c a l i n n o v a t i o n c o r r e l a t i o n . C k i s c a l c u l a t e d a s C k = ( l / N ) Z r i r i - k T ( 3 . 3 6 ) where N i s t h e s a m p l e s i z e . F o r o b v i o u s r e a s o n , maximum N e q u a l s NBATCH. L e t i N be t h e s a m p l e s i z e o f i n n o v a t i o n 69 c o r r e l a t i o n s , i . e . , a set of { C ^ , C^2t'' D e f i n e an ij^xn matrix A as H'(*-G kH' A= H'(*-G kH') i 2" 1$ CJN^ * s c a l c u l a t e d . (3.37) _H' (4>-GkH' ) i N _ 1 $ _ where G k i s the Kalman Gain at the end of the c u r r e n t i n t e r v a l . Now d e f i n e the nx1 ve c t o r A as C i i A=A# C i 2 +*"1Gk C 0 -IN (3.38) and the nxn matrix fl as n=G kC 0G k T -G kA T* T-$AG k T (3.39) In (3.38), A# i s the pseudoinverse of A, d e f i n e d as A#=(A TA)- 1A T (3.40) Q i s obtained by s o l v i n g the n equations Z H ' * J r Q r T(^-k)T H.T + S T r T ( $ - k - 1 ) T H . T _ H . < i ) k - 1 r s  = A T ( 4 > - k ) T H , T -H'^A-Z H'*30(*3-k)T H'T k=1,...,n (3.41) where S i s s u b s t i t u t e d by Q ( H D T . Equations (3.41) can be expressed i n matrix form, i n the event # and H' are known. Note that the number of unknowns i n Q cannot be g r e a t e r than nx1, e l s e a unique s o l u t i o n cannot be found. T h e r e f o r e , Q should be s t r u c t u r e d i n such a way as to f a c i l i t a t e unique s o l u t i o n of (3.41). 70 0 . 1 J-- 0 . 1 +1 .96/VN -1 .96/ i /N F i g u r e 3 . 1 4 T e s t f o r o p t i m a l i t y ( I n t h i s c a s e , b a n d l i m i t i s s e t a t ± 1 . 9 6 / V N . C o u n t = 3 ) O n c e Q h a s b e e n d e t e r m i n e d , R i s c a l c u l a t e d a s R = C 0 - H ' A - H ' * " 1 r S ( 3 . 4 2 ) F o r p a r t i a l a d a p t a t i o n , Q Q i s u s e d t h r o u g h o u t , o n l y R i s a d a p t e d . C o n s e q u e n t l y , e q u a t i o n s ( 3 . 3 9 ) a n d ( 3 . 4 1 ) c a n be s k i p p e d . T o e v e n o u t t h e e f f e c t o f a d a p t a t i o n , t h e a c t u a l Q a n d R t o be u s e d , d e n o t e d Q k a n d R k a f t e r k t n b a t c h , a r e o b t a i n e d by t h e f o l l o w i n g s t o c h a s t i c a p p r o x i m a t i o n s [ 2 9 ] : Q k = Q k - 1 + [ l / k ] ( Q k - Q k _ 1 ) ( 3 . 4 3 ) R k = R k _ ! + [ 1 / k ] ( R k - R k _ ! ) ( 3 . 4 4 ) D e r i v a t i o n o f t h e a l g o r i t h m i s p r o v i d e d i n A p p e n d i x B . A g a i n , a r e a l i z a t i o n f o r n=2 i s i n c l u d e d . A w e l l - k n o w n p r o p e r t y o f t h e i n n o v a t i o n s e q u e n c e s t a t e s t h a t a n e c c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r a K a l m a n F i l t e r t o be o p t i m a l i s t h a t t h e i n n o v a t i o n s e q u e n c e be w h i t e [ 2 4 ] . I n o t h e r w o r d s , t h e i n n o v a t i o n c o r r e l a t i o n s e q u e n c e 71 C k should converge to a s i n g l e impulse at k = 0. T h i s f a c t i s used to provide some convenient t e s t f o r the o p t i m a l i t y of the f i l t e r . With C k d e f i n e d as i n (3.36), the normalized c o r r e l a t i o n c o e f f i c i e n t p k of the inn o v a t i o n sequence r k i s c a l c u l a t e d as Pk=C k/|C 0| k=0,1,2,... (3.45) The sequence p k i s then examined. A l i m i t band can be d e f i n e d and the number of times [p] l i e o u t s i d e t h i s band counted. Based on t h i s count, one can decide whether r k i s white, and t h e r e f o r e whether f u r t h e r a d a p t a t i o n i s r e q u i r e d ( F i g . 3 . 1 4 ) . I t should be noted that success of the t e s t depends on a la r g e sample s i z e N. I f N i s s m a l l , an a l t e r n a t i v e t e s t , by examining the L i k e l i h o o d F u n c t i o n L(Q,R,k) of Q and R. L(Q,R,k) i s d e f i n e d as L( Q , R , k ) = ( l / N ) L r i T ( H ' M k H ' T + R ) " 1 r j -In|H'MkH'T+R| (3.46) As the Kalman F i l t e r approaches o p t i m a l i t y , L should i n c r e a s e . T h e r e f o r e , the rate of i n c r e a s e of L i s a good i n d i c a t i o n of how c l o s e the f i l t e r i s to optimal o p e r a t i o n . The a l g o r i t h m i s summarised i n Fig.3.15 Q 0 R 0 S D NBATCH i N  .1 I 1 11. Kalman Filtering k = k + l No Calculate C_, C-,,..., CL (3^6) 1 Estimation A of Q. R, S Calculate A,fl (3.38), (3.39) Calculate § , 's (3.41) Calculate *k (3.42) Kalman Filtering for next NBATCH 1 Calculate \ Test for optimality Kalman Filtering for rest qf sequence using Q, R, S Figure 3.15 Flow Chart of Innovation Correlation Adaptive Algorithm 73 3.4 Adaptive ARMA F i l t e r 3.4.1 Adaptive ARMA F i l t e r S t r u c t u r e The f i l t e r s c o n s i d e r e d so f a r r e q u i r e knowledge of both s i g n a l and no i s e models. We s h a l l now propose a design which r e q u i r e s knowledge of the measurement e r r o r o n l y , i . e . , the ' s i g n a l ' model. T h i s design i s motivated by recent development in a c l a s s of f i l t e r s termed Adaptive L i n e Enhancers (ALE), whose o b j e c t i v e i s to r e t r i e v e s i g n a l i n n o i s e . Among the d i f f e r e n t types i s the f i n i t e - i m p u l s e response ALE(FIR-ALE), developed by Widrow, G r i f f i t h s and o t h e r s [ 5 5 ] . By e x p l o i t i n g the coherent nature of the s i g n a l to separate the narrow band s i g n a l from white n o i s e , i t has the a t t r a c t i v e f e a t u r e of bypassing p r i o r m o d e l l i n g . The s e p a r a t i o n i s brought about by a d e c o r r e l a t i o n delay, which i m p l i e s the s i g n a l and noise should have v a s t l y d i f f e r e n t c o r r e l a t i o n times. The FIR-ALE has two main l i m i t a t i o n s : a) Large f i l t e r l e n g t h i s f r e q u e n t l y r e q u i r e d to achieve good performance. b) The wide-band noise i s assumed t o be white. Some improvement to l i m i t a t i o n (a) i s p o s s i b l e by us i n g a form of ALE based on the a u t o r e g r e s s i v e moving average (ARMA) model. The r e s u l t i n g f i l t e r i s termed the ARMALE[14], and i s b a s i c a l l y an IIR f i l t e r with c o e f f i c i e n t s a d a p t i v e l y a d j u s t e d by a Re c u r s i v e Maximum L i k e l i h o o d (RML) A l g o r i t h m . The problem (b) can be s o l v e d by p r e f i l t e r i n g the r e c e i v e d s i g n a l z ( t ) with a whitening f i l t e r H ^ z ) [46]. Since z ( t ) i s 74 H ^ z ) 1 Adaptive 2 IIR F i l t e r F i g u r e 3.16 Block Diagram of Adaptive ARMA F i l t e r g i ven by z ( t ) = d ( t ) + n ( t ) (From (3.3)) The whitening f i l t e r response i s given by (w)=l/S n +(w) (3.47) where S n*(co) i s the c a n o n i c a l s p e c t r a l f a c t o r of S n ( z ) . Thus, r e f e r r i n g to Fig.3.16, the s i g n a l at p o i n t 1 c o n s i s t s of a f i l t e r e d v e r s i o n of d ( t ) , p l u s white n o i s e . The output power i s S n(w) |H, ( e j W ) | 2&u+( 1/2ir)J S d(u)|H«(e ^) \ 2du = ( 1 / 2 T T ) ; W6(w)dw + P d'(u) « W+Pd'(w) (3.48) The output of the whitening f i l t e r i s the input t o the ad a p t i v e ARMA F i l t e r . Let 6<=[a^ , . . . ,&na'c\ ,. .. , c n c ] T be the v e c t o r of f i l t e r c o e f f i c i e n t s . Then, the a d a p t i v e f i l t e r attempts t o minimize W: min P 1 = min W + P d'(w)=P d'(a>) (3.49) 6 6 Thus, the output power at p o i n t 2 should, at optimal c o n d i t i o n , be equal t o Pd'(w). The p o s t - f i l t e r H 2 ( z ) r e s t o r e s the o r i g i n a l P d(w) by having response 75 H 2 ( w ) = S n + ( « ) (3.50) P 3=(l/2»r ) 2 £ s d(w) IH, ( e j t J ) | 2 | H 2 ( e j " ) | 2 (do;) 2 = ( 1/4TT2 ) £ £s d(co)da> 2 (3 . 51 ) 3.4.2 Design of L i n e a r Phase ARMA P r e - F i l t e r The whitening f i l t e r H ^ z ) with response given by (3.47) can e i t h e r be FIR (AR) f i l t e r , or I I R (ARMA) f i l t e r . We tend to p r e f e r the l a t t e r d e s i g n , again because of the e x c e s s i v e f i l t e r length- r e q u i r e d of the FIR f i l t e r . In f a c t , approximations to samples of Sn(a>) from our s i m u l a t i o n r e q u i r e FIR f i l t e r l e n g t h N to be 2 1 3 or h i g h e r . The ARMA(m,n) f i l t e r , however, can manage to achieve s i m i l a r degree of approximation with n+m as low as 32, n,m being the order of the AR and MA p a r t s of the f i l t e r , r e s p e c t i v e l y . A c c o r d i n g to Wold[52], any r e a l , s t a t i o n a r y s t o c h a s t i c process z t can be decomposed i n t o a d e t e r m i n i s t i c component x t , and a n o n - d e t e r m i n i s t i c component v t z t=x t+v t (3.52) x t and v t are s t a t i o n a r y and mutually u n c o r r e l a t e d . z t a l s o has a g e n e r a l r e p r e s e n t a t i o n by an ARMA(m,n) process C ( z " 1 ) z _ e A ( z _ 1 ) (3.53) where e^ i s a white d r i v i n g n o i s e p r o c e s s . The design of a l i n e a r - p h a s e ARMA whitening f i l t e r to y i e l d a given spectrum S n(w) i s borrowed from a method due to Monden[34], and based on techniques developed by Scharf and 76 Luby[48]. The key idea i n t h i s method i s a two-step s u c c e s s i v e approximation to a given spectrum. F i r s t , an AR f i l t e r i s designed to approximate the spectrum. The f i l t e r c o e f f i c i e n t s are then used to generate c o n s i s t e n t u n i t impulse sequence ( h k ) order, time-delayed ARMA f i l t e r approximates the high-order AR f i l t e r by s o l v i n g normal equations i n v o l v i n g {h k} and { r k } . It can be seen t h a t , i n r e t u r n f o r e x t r a complexity in de s i g n i n g the l i n e a r - p h a s e ARMA f i l t e r , the r e s u l t i n g whitening f i l t e r as implemented i n the block s t r u t u r e of Fig.3.16 can be of much lower order than that of FIR f i l t e r . The design a l g o r i t h m i s o u t l i n e d i n Appendix C. 3.4.3 Adaptive IIR A l g o r i t h m The s i g n a l at the output of the whitening f i l t e r , z t ' ' has, a c c o r d i n g to Wold, the general r e p r e s e n t a t i o n where v t i s the white noi s e with power W. The s i g n a l x^ r being d e t e r m i n i s t i c , has the AR r e p r e s e n t a t i o n and c o v a r i a n c e sequence {r^}. In the second step, a lower * t ' = x t + v t (3.54) * t " ! a i x t - i + u t (3.55) where u t i s a white noi s e p r o c e s s . ( u t + A ( z ) v t ) (3.56) where A(z) = 1+a^z~ 1 +...+a n az (3.57) Since u t and v t are independent white n o i s e processes, they can be combined t o a s i n g l e n o i s e process e^. Let F i g u r e 3 .17 B l o c k D i a g r a m o f A d a p t i v e H R F i l t e r u t + A ( z ) v t = C ( z ) e t (3.58) w i t h C ( z ) = 1 + c 1 z " 1 + . . . + c n z ' 1 (3.59) C ( z ) i s c h o s e n so t h a t t h e r e s u l t i n g power s p e c t r a a r e i d e n t i c a l : C ( z ) o e 2 C ( z ~ 1 ) = A ( z ) o v 2 A ( z _ 1 ) + o u 2 ( 3 . 6 0 ) T h u s , z t * i s g i v e n by z t ' = C ( z ) / A ( z ) . e t ( 3 . 6 1 ) In an a d a p t i v e H R f i l t e r , t h e s i g n a l x t i s e s t i m a t e d f r o m z t ' by m i n i m i z i n g t h e e r r o r e t = z t ' # i z t - i ' - | c i e t - i ( 3 . 6 2 ) The f i l t ? r p a r a m e t e r s { a ^ c j } a r e a d a p t i v e l y a d j u s t e d by a s u i t a b l e r e c u r s i v e p a r a m e t e r e s t i m a t i o n a l g o r i t h m , s u c h as t h e RML A l g o r i t h m . 78 With r e f e r e n c e to Fig.3.17, the RML a l g o r i t h m may be summarised as f o l l o w [ l 5 ] : F i l t e r equation z' 1 1 1 _ T = - ^ a j X t _ i + £ b i e t _ i + e t (3.63) Let 6 denotes the v e c t o r of f i l t e r c o e f f i c i e n t s 0 = [ a i , . . . , a n a , C i , . . . , c n g ] T (3.64) $ t denotes the data v e c t o r , * t denotes a f i l t e r e d data v e c t o r * t = [ " z ' t - 1 ' • • • ' z ' t - n a ' e t - 1 ' • • • ' e t - n c ^ T (3.65) * t = f " z ' t - 1 ' • • • ' Z ' t - n a ' ^ t - i , • • • , e t - n c ] T (3.66) The r e c u r s i o n r u l e s f o r a d j u s t i n g the f i l t e r c o e f f i c i e n t s when a new data p o i n t z\ becomes a v a i l a b l e i s given by e t + 1 = z ' t + 1 - * t + 1 T 0 t = p r e d i c t i o n e r r o r (3.67a) P t * t + 1 T * t + 1 p t P t + 1 = [ P t- ( ) 3/X+1 =error c o v a r i a n c e *t+1 + *t+1 TPt*t+1 matrix (3.67b) 6>t+i = 6 1 t+(Pt+i * t +1 ^ e t + 1 = f i l t e r c o e f f i c i e n t s update (3.67c) t+1=output (3.67d) e t + 1 = z t + 1 ' ~ z t + 1 ' = r e s i d u a l (3.67e) with i n i t i a l c o n d i t i o n s P 0 = a l , 1<a<l00 (3.68a) 6 O = 0 (3.68b) X t can e i t h e r be a cons t a n t , with 0^X51, or t i m e - v a r y i n g X t + 1=XX t+(1-X) (3.69) X a c t s an e x p o n e n t i a l weighting on the data. z t ' , e^ are o b t a i n e d by p r e f i l t e r i n g through a f i l t e r 1 / D t ( z " 1 ) , where D t ( z " 1 ) ••  1 +Kc y (t) z" 1 +.. . +Kc n c (t) z " n c . By v a r y i n g the K's, the r a t e of convergence of the a l g o r i t h m can be c o n t r o l l e d . 79 The convergence p r o p e r t i e s of the RML a l g o r i t h m as s t u d i e d by Ljung and others [50][27] are q u i t e p r o m i s i n g . They can be summarised as: a) Asymptotic c o n s i s t e n y , i . e . , fln •• as N • ». b) Asymptotic e f f i c i e n c y , i . e . , P^ - approaches the Cramer-Rao lower bound. Whereas the RML a l g o r i t h m may appear c o m p u t a t i o n a l l y more complex than the LMS a l g o r i t h m employed i n FIR-ALE, the a c t u a l computation can be much lower because the t o t a l order N=n a+n c r e q u i r e d i s u s u a l l y much s m a l l e r . Furthermore, update of P t can be sped up by us i n g one of the many ' f a s t ' a l g o r i t h m s r e c e n t l y d e v i s e d , such as the Fast Kalman Rec u r s i v e Least Square A l g o r i t h m (Ljung, Morf and Fa l c o n e r 1978), or the Rec u r s i v e Square-Root Normalized L a t t i c e A l g orithm(Lee, Morf and F r i e d l a n d e r 1981>[10] . As time p r o g r e s s e s , the ad a p t i v e H R f i l t e r converges to a low-pass f i l t e r . The bulk of the computation i s taken up by the a d a p t a t i o n p r o c e s s . Together with the pre- and p o s t - f i l t e r , the t o t a l computation may be higher than that of the a d a p t i v e Kalman F i l t e r . There i s a l s o an a s s o c i a t e d time delay (equal to two times the minimum phase of the p r e - f i l t e r ) . Thus, t h i s system i s u s e f u l only i n the case where there i s l a r g e u n c e r t a i n t y i n L 2 , or i n the model of the s e a f l o o r p r o f i l e i t s e l f . 80 Chapter 4 FILTERING IN THE ABSENCE OF NOISE STATISTICS There are a number of s i t u a t i o n s i n which the s i g n a l or noise s t a t i s t i c s are not a v a i l a b l e : e i t h e r an a c c u r a t e model i s not known, or e l s e some p r i n c i p a l parameters may be ti m e - v a r y i n g , e.g., the wind speed, the heading angle, or the s e a f l o o r c o r r e l a t i o n d i s t a n c e . For the l a t t e r cases, v a r y i n g wind speed has the e f f e c t of a l t e r i n g the magnitude of the power spectrum of n ( t ) but l e a v i n g i t s shape l a r g e l y unchanged; whereas i n c r e a s i n g heading angle has the e f f e c t of s h i f t i n g and narrowing the spectrum to lower f r e q u e n c i e s (evident from Fig.A1.3 and A1.4) . I n c r e a s i n g L 2 tends to c o n f i n e S n(w) to lower f r e q u e n c i e s . In terms of a second-order system model (Eq. 3.16), they a f f e c t , r e s p e c t i v e l y , c o n l y ; both a, b and c; and 0 i n the t r a n s i t i o n matrix 4.1 Extended Kalman F i l t e r Suppose the c o e f f i c i e n t s a, b, c and 0 can be represented as some d i f f e r e n t i a b l e f u n c t i o n s of a parameter vec t o r f?(k) (e.g., time d u r a t i o n ) , then the Extended Kalman F i l t e r (EKF) approach can be used to estimate the s t a t e x(k) i n f i l t e r i n g model S 2 [ 1 7 ] , The standard procedure i s to append f?(k) to the s t a t e v e c t o r , to form the augmented s t a t e v e c t o r y ( t ) = x(k) 0(k) (4.1 ) The r e s u l t i n g s t a t e equations become 81 y ( k + 1 ) = f ( y ( k ) ) + r w ( k ) 0 ( 4 . 2 a ) £ ( k ) = g ( y ( k ) ) + v ' ( k ) ( 4 . 2 b ) w h e r e f ( y ( k ) ) = $ ( 0 ) x ( k ) 0 ( k ) ( 4 . 3 ) g ( y ( k ) ) = H ' ( 0 ) x ( k ) ( 4 . 4 ) L i n e a r i z i n g t h e s t a t e e q u a t i o n s a b o u t y = y ( k ) g i v e s t h e EKF e q u a t i o n s y ( k + 1 ) = f ( y ( k ) ) + K ( k ) [ y ( k ) - g ( y ( k ) ) ] ( 4 . 5 a ) K ( k ) = [ F ( y ( k ) ) P ( k ) G T ( y ( k ) ) + S ] [ G ( y ( k ) ) P ( k ) G T ( y ( k ) ) + R ] _ 1 ( 4 . 5 b ) P ( k + 1 ) = F ( y ( k ) ) P ( k ) F T ( y ( k ) ) + Q -K ( k ) [ G ( y ( k ) ) P ( k ) G T ( y ( k ) ) + R ] K ( k ) T ( 4 . 5 c ) P ( 0 ) = P ° ( 4 . 5 d ) y ( 0 ) = y ° ( 4 . 5 e ) w h e r e F ( J ( k ) ) = i f ( y ) | y = j ( k ) ( 4 . 6 a ) 3 G ( y ( k ) ) = — g ( y ) | ( 4 . 6 b ) 3 y ' y = y ( k ) I n t h e c a s e o f v a r y i n g L 2 o n l y , c a l c u l a t i o n s c a n be c o n s i d e r a b l y r e d u c e d s i n c e o n l y H ' i s d e p e n d e n t on 0 . T h e d e p e n d e n c e of * on fl may be d r o p p e d . F o r v a r y i n g u o r V/, f u l l e v a l u a t i o n s a r e r e q u i r e d s i n c e b o t h 4> a n d H ' a r e d e p e n d e n t on 8. I n s u c h c a s e s t h e E K F a p p r o a c h u s u a l l y r e q u i r e s h e a v y c o m p u t a t i o n s i n c e r e l i n e a r i z a t i o n i s p e r f o r m e d a t e a c h s t e p . A g a i n , t h e r e i s t h e p r o b l e m o f a p r i o r i i n f o r m a t i o n a b o u t t h e n o i s e c o v a r i a n c e s Q , R a n d S . F u r t h e r m o r e , t h e E K F i s known t o 82 be prone to divergence and b i a s problems. The l a t t e r may be a l l e v i a t e d by a method due to Ljung [17] who r e d e f i n e d F in (4.6a) i n terms of the asymptotic s e n s i t i v i t y of the Kalman gain K ( k ) , and i n t r o d u c e d a p r o j e c t i o n scheme to keep the r e s u l t i n g H' s t a b l e at each ste p . U n f o r t u n a t e l y , f u r t h e r computations are introduced as a r e s u l t . 4.2 S e l f - T u n i n g F i l t e r To circumvent the problems a s s o c i a t e d with the EKF, an a l t e r n a t i v e approach separates the f i l t e r i n t o two stages: f i r s t , the parameters of the system are estimated, then the s t a t e s of the system given the parameters are estimated. The parameters i n t h i s case are based on a v e c t o r d i f f e r e n c e equation (VDE) r e p r e s e n t a t i o n of the system: A ( z _ 1 )m)=D(z - 1 )e(k) (4.7) with A(z~ 1)=1+A 1z" 1+A 2z" 2+...+A n az" n a (4.8a) Dfz" 1)=1+D 1z" 1+D 2z" 2+...+D n dz" n d (4.8b) e(k) i s a white zero-mean sequence with c o v a r i a n c e R e. (4.7) may be r e w r i t t e n as £(k)=¥(k)0(k)+e(k) (4.9) where ¥ ( k ) = [ - k U - 1 ) , . . . , -£(k-na), e ( k - l ) , . . . , e(k-nd)] i s the o b s e r v a t i o n v e c t o r and t?(k) = [A 1 ,... ,A n a,D 1 ,... , D n d ] T i s the (constant) parameter v e c t o r . 0(k+l)=0(k) (4.10) 83 (4.7) and (4.10) together d e f i n e a s t a t e e quation, i n which 0(k) i s the ' s t a t e ' to be estimated. Since *(k) i n ge n e r a l depends on 0 ( . ) , the r e p r e s e n t a t i o n (4.9) i s n o n l i n e a r and EKF must be used. The r e s u l t i n g parameter e s t i m a t i o n a l g o r i t h m i s known as the Panuska method [40]: S(k+1|k)=0(k|k-1)+K(k)[£(k)-¥(k)£(k|k-1)] (4.11a) K ( k ) = P ( k ) G T ( k ) [ G ( k ) P ( k ) G T ( k ) + R e ( k + 1 ) ] _ 1 (4.11b) P(k+1)=P(k)-K(k)[G(k)P(k)G T(k)+R e(k+1)]K T(k) (4.11c) R e(k+1)= R c ( k ) + [ e 2 ( k ) -R e(k)]/k+1 (4.11d) with i n i t i a l c o n d i t i o n s 0 ( O)=0° (4.12a) P(0)= P° (4.12b) R e(0)= R e ° (4.12c) and 9 G<k> . _ m > » | g . j ( | [ ) (4.,3) A A A e(k)=£(k)-•(k)8{k|k-1) i s the computed i n n o v a t i o n sequence. No other assumptions are r e q u i r e d , a s i d e from the i n i t i a l c o n d i t i o n s l i s t e d i n (4.12), and the order of A ( z ~ 1 ) and D ( z ~ 1 ) . A c c o r d i n g to S e c t i o n 3.4.2, a reasonable e s t i m a t i o n fo r our problem would be na+nd^32. Having o b t a i n e d the parameter v e c t o r 8, two d i f f e r e n t approaches may be taken to estimate the s t a t e s of the system. A c c o r d i n g to Nelson and Stear [36], 0(k) can be mapped to the parameter space of and H' of a c a n o n i c a l s t a t e - s p a c e model. A A A Thus, at time k, the r e s u l t i n g model i s [*(k), T ( k ) , H ' ( k ) ] , which can be s u b s t i t u t e d to the Kalman equations to y i e l d the A s t a t e e s t imate x(k+1|k). Another more d i r e c t method was 84 developed by Grimble [33], who formulated the optimal s t a t e e s t i m a t o r i n terms of the parameters 8{k) themselves and i n n o v a t i o n e ( k ) . T h i s e s t i m a t o r was d e r i v e d by i d e n t i f y i n g the r e l a t i o n s h i p between A ( z ~ 1 ) , D ( z ~ 1 ) and F ( z ) , the r e t u r n d i f f e r e n c e matrix of an optimal t i m e - i n v a r i a n t l i n e a r f i l t e r i n the z-domain. F u r t h e r work i s r e q u i r e d to extend t h i s method to the case i n which d r i v i n g (system) noise and measurement noise are c o r r e l a t e d . T h i s approach proves promising, s i n c e s i g n i f i c a n t savings i n computation are p o s s i b l e , e s p e c i a l l y when parameters i n the s i g n a l and n o i s e models are unknown. 4.3 Recursive Least-Squares I n t e r p o l a t i n g F i l t e r In c o n t r a s t to the p r e v i o u s approach i n which v a r i o u s attempts were made to estimate the s t a t e s or parameters of an assumed model of the s i g n a l and noise processes, we now present an a l g o r i t h m which assumes no model at a l l , except f o r the f a c t that the ' s i g n a l ' and 'noise' s p e c t r a be reasonably w e l l separated ( i n other words, the s i g n a l and n o i s e processes have d i s t i n c t l y d i f f e r e n t c o r r e l a t i o n t i m e s ) . Admittedly, t h i s f i l t e r i s suboptimal i n nature, but, as shown i n S e c t i o n 5.4, s i m u l a t i o n r e s u l t s i n d i c a t e i t can y i e l d b e t t e r r e s u l t s than other f i l t e r s i n some ci r c u m s t a n c e s . The f i l t e r i s due to van Schooneveld[53]. I t was o r i g i n a l l y used to process sonar s i g n a l s i n a tone p u l s e echo ranging system. The sonar s i g n a l s in t h i s system were degraded by high frequency r e v e r b e r a t i o n no i s e which was d i f f i c u l t to d e t e c t or model. 85 From (3.8), an optimal f i l t e r minimizes the mean square e r r o r I ( y ( i ) - d ( i ) ) 2 (4.14) max i = 0 with i ranging from 0 to N m a x , the d u r a t i o n of the r e c o r d s . In a suboptimal f i l t e r , we i n t r o d u c e a weighting f u n c t i o n q n _ 1 , which e x p o n e n t i a l l y weights the p r e v i o u s d a t a . The r e s u l t i n g e r r o r i s a weighted l e a s t - s q u a r e e r r o r e n = £ [ z ( i ) - y ( i | n ) ] 2 . q n _ i , 0 < q < 1, (4.15) F u r t h e r , s i n c e l e a s t - s q u a r e s curve f i t t i n g i s known to approximate low-pass f i l t e r i n g [ 5 3 ], we introduce the f i l t e r equation as y ( i | n ) = y D ( n ) + ( i - n ) . y , ( n ) , i=0,..,n (4.16) which i s i n f a c t a f i r s t degree polynomial curve f i t t i n g to the r e c e i v e d data sequence [ z ( 0 ) , . . , z(n ) ]. Let L be the e s t i m a t i o n l a g . Then the time delay estimate obtained by i n t e r p o l a t i n g i n t o the past i s given by y(n-L|n)=y 0(n)+(n-L-n).y,(n) = y 0 ( n ) - L . y 1 ( n ) (4.17) The f i l t e r can thus be termed ' r e c u r s i v e l e a s t - s q u a r e s i n t e r p o l a t i n g ' (RLSI) F i l t e r . D e f i n e the window le n g t h W as W=(1-q)" 1 (4.18) I t w i l l be shown that both L and W has strong e f f e c t s on the convergence and e s t i m a t i o n e r r o r of the f i l t e r . The c o e f f i c i e n t s y Q ( n ) , y,(n) are determined such that e n i s a minimum. Hence, s u b s t i t u t i n g (4.16) t o (4.15) and t a k i n g 86 p a r t i a l d e r i v a t i v e s 9e T 3yc>> 3e =21 [ z t n - k J - J o t n J + k . y , ( n ) ] q k ^ =2kl [ z ( n - k ) + y 0 ( n ) - k . y 1 ( n ) ] q k ay](n) where k=n-i. S e t t i n g (4.19a),(4.19b) to zero equations f o r y 0 , y i : m 0(n) mi m^n) IH21 or, My=z where m-j(n) = (-1 )!*£ k l q k z-: (n) = (-1 ) JE k J q k . z ( n - k ) )] y 0 (n)~ z 0 ( n ) ) i 1 (n) Z ! (n) (4.19a) (4.19b) y i e l d s two l i n e a r (4.20) (4.21a) (4.21b) y can of course be s o l v e d d i r e c t l y i n (4.20) by matrix i n v e r s i o n . However, s i n c e summations over n data p o i n t s are cumbersome a more e f f i c i e n t way i s to f i n d a s o l u t i o n i n r e c u r s i v e form. E x p r e s s i n g mj(n-1), z j ( n - 1 ) i n terms of mj(n), Z j ( n ) m.j(n-1 )=q _ 1 £ (jj) [ m k ( n ) - 0 k ] (4.22a) (4.22b) z-:(n-l)=q- 1 I (?) [ z k ( n ) - z ( n ) . 0 k ] where 0 k =1 k=0 = 0 k*0 S u b s t i t u t i n g the r e l a t i o n s (4.22a),(4.22b) i n t o (4.21a),(4.21b), we have 87 ztn) • y, (n.l) 7 © T - K ^ — % r f , (n ) yolnl L—' F i g u r e 4.1 Block Diagram of RLSI F i l t e r [ 5 3 ] m 0(n-1) mi(n-1) mi (n-1) m 2 ( n - 1 ) y 0 ( n - 1 ) + y } ( n - 1 ) y 1 ( n - 1 ) z i (n) -z(n)+y Q(n-1)+y,(n-1) 0 (4.23) S u b t r a c t (4.23) from (4.20) g i v e s y 0 ( n ) y, (n) _y 1 (n)_ y 0< n> o(n) 0(n) . [ z ( n ) - y 0 ( n ) ] (4.24) where y 0 ( n ) Y1 <n) a(n) Pin) y,(n-1 ) m2 ( n) -mi(n) det M det M = m 0 ( n ) m 2 ( n ) " m l 2 ( n ) (4.25a) (4.25b) (4.25c) Equation (4.24) d e f i n e s the r e c u r s i o n r e l a t i o n by which Yo^")' Y l ( n ) m a Y D e c a l c u l a t e d . Fig.4.1 shows the block diagram of the RLSI F i l t e r . Examination of (4.24) r e v e a l s t h a t y Q ( n ) , y,(n) may be regarded as ' p r e d i c t i o n ' of y 0 ( n - 1 ) , y ^ n - 1 ) . The estimated y 0 ( n ) . 88 y i ( n ) are obtained by t a k i n g the p r e v i o u s p r e d i c t i o n s and adding amounts p r o p o r t i o n a l to the d i f f e r e n c e [ z ( n ) - y 0 ( n ) ] between a c t u a l o b s e r v a t i o n and p r e d i c t i o n . The p r o p o r t i o n a l i t y c o n s t a n t s a ( n ) , 0(n) depend on the f i l t e r c h a r a c t e r i s t i c s (window l e n g t h ) , and are time dependent. Thus, the RLSI f i l t e r o q o = o.7 o = 0.75 * = 0.8 - = 0.65 * = 0.9 0.0 10.0 20.0 30.0 40.0 50 0 60.0 70.0 60.0 90 0 100.0 n o q D = 0.7 o = 0.75 A - 0.8 • = 0.85 x = 0.9 - _. , - -j - - i - - - i • • i • • ' i "1" ; ^ — - , 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80 0 90.0 100.0 n F i g u r e 4.2 P l o t s of a ( n ) , 0(n) 89 i s t i m e - v a r i a n t . Since a ( n ) , /J(n) are independent of r e c e i v e d data z ( n ) , they can be precomputed (by (4.25b), (4.25c) and (4.21a)) and s t o r e d i n ROM. F i g . 4 . 2 ( a ) , ( b ) show p l o t s of a ( n ) , 0(n) a g a i n s t n with v a r i o u s q. A f t e r s u f f i c i e n t e l a p s e d time, both converge to a l i m i t . The r a t e of convergence i n c r e a s e s with d e c r e a s i n g q. In other words, a s h o r t e r window l e n g t h tends to speed up convergence. T h i s i s reasonable, s i n c e a longer window i m p l i e s smoothing over a longer data l e n g t h , and t h e r e f o r e longer time i s r e q u i r e d f o r convergence. In a d d i t i o n to q, the performance of the f i l t e r i s a l s o dependent on L, the e s t i m a t i o n l a g . T h i s can be seen by c o n s i d e r i n g E [ e n 2 ] = E [ z ( n ) 2 ] - 2 E [ y 0 ( n ) J 1 ( n ) ] . L + E [ y i ( n ) 2 ] . L 2 (4.26) As no simple r e l a t i o n s can be w r i t t e n f o r the e s t i m a t i o n e r r o r s i n case both s i g n a l and nois e processes are c o l o r e d , we s h a l l opt f o r a h e u r i s t i c approach t o determine the e f f e c t of (q,L) on system performance. T h i s w i l l be i l l u s t r a t e d i n S e c t i o n 5.4. 90 Chapter 5 SIMULATION AND FILTER PERFORMANCES 5.1 S i m u l a t i o n Procedures The aim of s i m u l a t i o n i s to generate as a c c u r a t e l y as p o s s i b l e , sequences of data r e p r e s e n t i n g sampled v e r s i o n s of the s e a f l o o r p r o f i l e and sonar measurement e r r o r time s e r i e s . The s t a t i s t i c a l i n f o r m a t i o n r e q u i r e d i s t o t a l l y c o n tained i n the c o rresponding power s p e c t r a , d i s c u s s e d i n Chapter 2. There e x i s t a number of methods f o r s i m u l a t i n g a random sequence given i t s power spectrum. Among these, the time domain method i s the most d i r e c t , and can f u r t h e r be d i v i d e d i n t o two d i f f e r e n t approaches: matrix m u l t i p l i c a t i o n method, and f i l t e r e d white n o i s e method. With the former approach, the random v a r i a b l e s are formed by m u l t i p l y i n g the cova r i a n c e matrix with a v e c t o r of pseudonumbers. With the l a t t e r , the sequence i s generated as the output of a f i l t e r ( g e n e r a l l y FIR) which models the shaping f i l t e r of the d e s i g n a t e d spectrum[6]. Both methods s u f f e r from cumbersome c a l c u l a t i o n and/or l a r g e memory space requirement. A much f a s t e r procedure, but more i n d i r e c t , i s frequency domain s i m u l a t i o n . One way i s to simulate the DFT of the time s e r i e s f i r s t , then FFT i s used to r e v e r t to the time domain [32]. Another way, more e f f i c i e n t i n nature, i s the random phase method[39]. The accuracy of t h i s method i s dependent on user c o n t r o l l e d parameters. T h i s w i l l be the method adopted. 91 The scope of t h i s t h e s i s i s l i m i t e d to the case of s t a t i o n a r y time s e r i e s , with the f o l l o w i n g chosen parameters: Ship speed constant at v=2ms~1 Sampling period=0.5s Heading angle constant at \//=0o T o t a l data p o i n t s generated N=2400 (t N=1200s) S t a t i o n a r i t y i s j u s t i f i e d by the f a c t that the d u r a t i o n i s r e l a t i v e l y s h o r t , and t h e r e f o r e parameters are u n l i k e l y to vary s i g n i f i c a n t l y . In the event of ti m e - v a r y i n g parameters, n o n - s t a t i o n a r y s i m u l a t i o n i s r e q u i r e d . S u i t a b l e methods i n c l u d e the ti m e - v a r y i n g AR c o e f f i c i e n t modeling u s i n g mean square e r r o r c r i t e r i o n [ 6 ] , o r, the more e f f i c i e n t minimum AIC methodt 25]. The p r i n c i p a l v a r i a b l e s are, i n the case of measurement e r r o r s i m u l a t i o n , wind speed u, and, i n the case of s e a f l o o r s i m u l a t i o n , the mean c o r r e l a t i o n d i s t a n c e L 2 . Since they have the g r e a t e s t e f f e c t s on the s t o c h a s t i c behaviour of the time s e r i e s , performance r e s u l t s should be presented with r e f e r e n c e to these two v a r i a b l e s . The ranges of u are chosen to be (6,17) f o r smal l s h i p , and (9,17) f o r l a r g e s h i p . R e f e r r i n g to Table 1.1 and 1.2, the m a j o r i t y of s i g n i f i c a n t wave h e i g h t s encountered world wide range from 0.5m to 6m, corresp o n d i n g to wind speeds of 6 to 17 ms~ 1. For s e a f l o o r p r o f i l e s , f i v e s e r i e s w i t h L 2 of 50m, 100m, 150m, 300m, and 600m, and constant a were s i m u l a t e d (Figs.DI t o D5). The f i r s t two repr e s e n t very rough p r o f i l e with l a r g e d i c o n t i n u i t i e s that are l i k e l y to be encountered i n the t r a n s i t i o n r e g i o n from c o n t i n e n t a l s h e l f to 92 a b y s s a l p l a i n . The l a s t two are more smooth, and the one with L2=l50m i s an int e r m e d i a t e f e a t u r e . The reader should c o n s u l t Appendix D on d e t a i l s of s i m u l a t i o n and analyses of the r e s u l t s . 5.2 " D e f i n i t i o n s of Performance Parameters The o b j e c t i v e of s i m u l a t i o n i s to compare the performances of the v a r i o u s proposed f i l t e r i n g a l g o r i t h m s so that one can a r r i v e at an i n t e l l i g e n t c h o i c e based on the f o l l o w i n g c r i t e r i a : 1. E r r o r performances. 2. E s t i m a t i o n time d e l a y . 3. Computational complexity. 4. Memory requirement. 5. Convergence r a t e . 6. S t a b i l i t y of a l g o r i t h m , i n c l u d i n g the degree of degradation with r e s p e c t to u n c e r t a i n t y i n one or more parameters. We are i n t e r e s t e d i n d e f i n i n g two performance c r i t e r i a . The f i r s t , termed , i s the normalized rms e r r o r of the f i l t e r e d data a t steady s t a t e . L et d^ be the true e l e v a t i o n at time i , d j be the estimated e l e v a t i o n , and o d be the standard d e v i a t i o n of the s e a f l o o r p r o f i l e s e r i e s . Then The second c r i t e r i o n , J 2 , i s the percentage c o r r e c t i o n of rms e r r o r of the f i l t e r at steady s t a t e . J 2 i s d e f i n e d i n (5.1) 93 terms of J the i n i t i a l , normalized rms e r r o r of the noisy sonar measurements: J o = ^ d > _ 1 N+1 ( 5 . 2 ) (5.3) I t i s c l e a r that J 1 and J 2 are both f u n c t i o n s of the f i l t e r c h a r a c t e r i s t i c s as w e l l as the two p r i n c i p a l v a r i a b l e s of the system, u and L 2 . As p o i n t e d out p r e v i o u s l y , the f i l t e r performance i s s t r o n g l y a f f e c t e d by the r e l a t i v e s p e c t r a l c h a r a c t e r i s t i c s of the processes d ( t ) and n ( t ) , i t i s d e s i r a b l e to d e f i n e a f u n c t i o n which combines these c h a r a c t e r i s t i c s . A p o s s i b l e c a n d i d a t e i s the r a t i o of the c o r r e l a t i o n d i s t a n c e s ; but, as the c o r r e l a t i o n d i s t a n c e of the narrow band process n ( t ) i s d i f f i c u l t to d e f i n e because of the o s c i l l a t o r y nature of i t s a u t o c o r r e l a t i o n f u n c t i o n , we opt f o r an i n d i r e c t d e f i n i t i o n , i n terms of t h e i r v a r i a n c e s . D e f i n e the ' s i g n a l - t o - n o i s e ' r a t i o (SNR) as S(u) F i g u r e 5.1 D e f i n i t i o n of SNR 94 SNR= j! , , S d ( f ) d f Jj -S n(£)df (5.4) The numerator i s the v a r i a n c e of the ' s i g n a l ' process, d ( t ) , and the denominator, that of the 'noise' process, n ( t ) . R e f e r r i n g to Fig.5.1, the SNR i s j u s t the r a t i o A2/A,. Note that as a r e s u l t of t h i s d e f i n i t i o n , f o r the same s e a f l o o r , a higher wind speed r e s u l t s i n a sm a l l e r SNR, and a lower wind speed r e s u l t s i n a higher SNR. The above d e f i n i t i o n s w i l l be used as the p r i n c i p a l means of p r e s e n t i n g the r e s u l t s of s i m u l a t i o n s . 5.3 Performances of Kalman and Adaptive F i l t e r s Using the techniques o u t l i n e d i n S e c t i o n 3.2 and data from Table 3.2, we set up the measurement-differencing Kalman F i l t e r . In a d d i t i o n , assuming l a r g e i n i t i a l u n c e r t a i n t y with x(0 ) , we c o n s i d e r the e r r o r c o v a r i a n c e matrix P(0) as di a g o n a l with a r b i t r a r i l y l a r g e v a l u e s , e.g., 100. A l s o , the noise c o v a r i a n c e matrix Q i s assumed to be d i a g o n a l with i d e n t i c a l elements. The choice of i n i t i a l v a l u e s f o r P has minimal e f f e c t on f i l t e r performance, s i n c e P 0 as t 0 0, and the convergence i s q u i t e f a s t . I n i t i a l v a l u e s of Q and R, however, have d e f i n i t e e f f e c t on f i l t e r performance as w e l l as convergence. B r i e f l y , i n c r e a s i n g Q p l a c e s g r e a t e r weights on measurements, and i n c r e a s e s the r a t e of growth of the elements of P. I n c r e a s i n g R has the reverse e f f e c t . Thus, the r a t i o Q/R i s the determing f a c t o r of steady s t a t e g a i n , and i s connected to the SNR as d e f i n e d p r e v i o u s l y . The former f a c t i s confirmed by a s e r i e s of J 2 p l o t t e d a g a i n s t (Q,R), f o r v a r i o u s O d CM . or o ; O ih -o ; o o -LU • • — o ; CL o o • d ' 95 A \ \ \ 30.0 35.0 40.0 45.0 50.0 55.0 SNR 60.0 65.0 70.0 (a) o d cr o o d C M . D .io d CL^ ^ o o d \ X V -30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 SNR 70.0 (b) F i g u r e 5.3 Optimal Q/R (a)Large Ship (b)Small Ship combinations of wind speeds and f l o o r t y p e s ( F i g . 5 . 2 ) . It can be seen t h a t i d e n t i c a l peak val u e s of J 2 occur at (approximately) i d e n t i c a l r a t i o s of Q/R. F u r t h e r , J 2 can f a l l below zero, i n d i c a t i n g d i v e r g e n c e , i f Q becomes too l a r g e or s m a l l . 96 Based on such p l o t s , we a r r i v e at a t a b l e of optimal Q/R r a t i o s f o r f i l t e r i n g , which i s i l l u s t r a t e d g r a p h i c a l l y i n Fig . 5 . 3 . T h i s confirms the c o n j e c t u r e that optimal Q/R i n c r e a s e s as u i n c r e a s e s , but remains approximately the same as L 2 v a r i e s , s i n c e the v a r i a n c e s of the simulated s e a f l o o r p r o f i l e s are c o n s t a n t . With these e m p i r i c a l optimal Q/R, we o b t a i n the o v e r a l l performance curves i n Fig.5.4. Samples of the i n d i v i d u a l rms e r r o r convergence curves and the r e s u l t i n g f i l t e r e d s i g n a l s are i n c l u d e d i n F i g . E l . 1 , i n Appendix E. A r e l a t i o n i s seen to e x i s t between convergence r a t e and performance: i f J 2 i s c l o s e to 1 ( f i l t e r performs w e l l ) then convergence i s f a s t ( t c < 40s), but i f J 2 i s s m a l l e r , as i n F i g . E 1 . 1 ( b ) , then convergence i s slow, with t c c l o s e to 100s or l a r g e r . In Fig.5.4(a) and ( c ) , J , i n c r e a s e s as SNR decreases, and as L 2 decreases. As SNR decreases, wind speed i n c r e a s e s , implying g r e a t e r i n i t i a l J c as w e l l as modeling e r r o r s . The J 2 c u r v e s , however, remain l a r g e l y the same f o r i d e n t i c a l f l o o r type, i n the case of l a r g e s h i p . In the case of small sh i p , there are c h a r a c t e r i s t i c s drops i n J 2 towards e i t h e r extremes i n SNR : i n the low SNR r e g i o n , t h i s i s due to the l a r g e r modeling e r r o r s f o r small s h i p ; i n the high SNR r e g i o n , t h i s i s mainly due t o the small i n i t i a l J 0 , so that improvement by f i l t e r i n g i s comparatively s m a l l e r . The l a t t e r f a c t i s confirmed by the J , curves i n ( c ) , which behaves approximately the same as i n ( a ) , a l b e i t with higher magnitudes. In both (b) and (d) i t i s observed t h a t f o r L 2=300m and 600 m, J 2 > 0.5; 97 d -o -Z.2. ' 3ocm d -o d . | 1 1 1 | 1 1 , , 1 -100.0 - 8 0 . 0 - 6 0 . 0 - 4 0 . 0 - 2 0 . 0 0.0 20.0 40.0 60.0 BOO 100.0 F i g u r e 5 . 5 S e n s i t i v i t y o f t o A L 2 f o r more r o u g h f l o o r t y p e s , J 2 < 0 . 5 , b u t s t i l l g r e a t e r t h a n z e r o . T h i s shows t h e u s e f u l n e s s o f K a l m a n f i l t e r i n g , e s p e c i a l l y f o r s m o o t h e r f l o o r t y p e . T o s t u d y t h e e f f e c t o f v a r i a t i o n i n L 2 on p e r f o r m a n c e , we o b t a i n a s e r i e s o f J 2 c u r v e s w i t h d e v i a t i o n s i n L 2 ( F i g . 5 . 5 ) . We f i n d t h a t a b o v e a c e r t a i n c r i t i c a l v a l u e , p e r f o r m a n c e i s r e l a t i v e l y i n s e n s i t i v e t o d e v i a t i o n i n L 2 . T h i s c r i t i c a l v a l u e v a r i e s b e t w e e n -50% t o -55% f o r a l l s e a f l o o r s , e x c e p t f o r t h e s m o o t h e s t f l o o r , w h i c h h a s a c r i t i c a l v a l u e o f a b o u t -70%. I n g e n e r a l , J 2 i s l e s s t o l e r a n t t o u n d e r e s t i m a t i o n i n L 2 > a n d t o l e r a n c e d e c r e a s e s a s L 2 d e c r e a s e s ( a s f l o o r g e t s m o r e r o u g h ) . B u t , a s t h e f L i n e s t i m a t i n g L 2 u s i n g t h e f a s t a l g o r i t h m a l l f a l l w i t h i n t h e s e c r i t i c a l v a l u e s ( c f . T a b l e 3 . 3 ) , we c a n u s e t h e m e t h o d i n c o n j u n c t i o n w i t h K a l m a n F i l t e r w i t h some c o n f i d e n c e , t h u s a v o i d i n g t h e n e c e s s i t y o f u s i n g a m o r e c o m p l e x 98 f i l t e r . Proceeding to the Adaptive Kalman F i l t e r , again the Q matrix i s assumed to be di a g o n a l i n order to s a t i s f y the unique s o l u t i o n requirement of (3.41). To maximize range of a d a p t a t i o n , we allow the two elements i n Q, and q 2 to vary. The r e s u l t s of i n d i v i d u a l s e r i e s of f i l t e r i n g are shown i n F i g . E1.2 to E l . 5 . The examples taken a r e , r e s p e c t i v e l y , l a r g e s h i p with u=13ms _ 1, and L2=600m, 150m, and 50m, and small s h i p with u=13ms _ 1, L 2=150m. In each case, f u l l a d a p t a t i o n i s used with 6 iterations(NBATCH=400), and i N=80. The rms e r r o r performances are comparable to t h a t of measurement-differencing Kalman F i l t e r . The convergence of Q, R and L i k e l i h o o d F u n c t i o n s are c l e a r l y demonstrated i n these r e s u l t s . Convergence are f a i r l y independent of i n i t i a l v a l u e s of Q and R, although o p t i m a l i t y i s l i k e l y to be reached i n a s h o r t e r time i f the chosen Q c and R Q are c l o s e to optimal Q and R. Thus, the r e s u l t s i n Fig.5.3 can pro v i d e a u s e f u l guide to choosing Q 0 and R 0. The L i k e l i h o o d F u n c t i o n seems to provide b e t t e r o p t i m a l i t y t e s t than the p k count method (the l i m i t was taken to be ±7.8//NBATCH), which y i e l d s no a p p r e c i a b l e d e c l i n e i n o f f - l i m i t count of p with i n c r e a s i n g i t e r a t i o n s . In most cases, there i s very l i t t l e improvement i n L(Q_,R) with the l a s t i t e r a t i o n , i n d i c a t i n g a d a p t a t i o n can be terminated. Next, we examine the r e s u l t s of p a r t i a l a d a p t a t i o n ( F i g . E 1 . 6 ) . The rms e r r o r performance i s s l i g h t l y i n f e r i o r , and the convergence i n R i s swinging r a t h e r than asymptotic as with f u l l a d a p t a t i o n . A l s o , the L i k e l i h o o d f u n c t i o n values are 99 20.0 30.0 40.0 50.0 60.0 '0.0 80.0 90.C 100.0 F i g u r e 5 . 6 J 2 a s a F u n c t i o n o f i N s m a l l e r , a n i n d i c a t i o n t h a t t h e r e i s room f o r f u r t h e r i m p r o v e m e n t i n a d a p t a t i o n . T h i s f a c t i s a l s o r e f l e c t e d i n t h e s l i g h t l y h i g h e r c o r r e l a t i o n o f t h e p k s e r i e s a t t h e l a s t i t e r a t i o n . T h e s i z e o f NBATCH i s now i n c r e a s e d t o 8 0 0 . T h e m a i n e f f e c t i s on t h e a c c u r a c y i n e s t i m a t i n g C k , a n d t h e r e f o r e Q a n d R . T h i s i s c o n f i r m e d i n t h e two c a s e s shown i n F i g . E l . 7 a n d E 1 . 8 , d e s p i t e t h e f i n i t e d a t a l e n g t h a v a i l a b l e ( i . e . , o n l y 3 i t e r a t i o n s a r e p o s s i b l e i n t h e e x p e r i m e n t s ) . T h e r e s u l t s show some i m p r o v e m e n t i n J^. T h e s i z e o f i j j h a s s t r o n g e f f e c t s on f i l t e r p e r f o r m a n c e . A s shown i n F i g . 5 . 6 , t h e r e e x i s t s a c r i t i c a l i ^ b e l o w w h i c h t h e f i l t e r d e g r a d e s r a p i d l y . F o r L2=600m, t h e c r i t i c a l i N i s a b o u t 4 5 ; f o r o t h e r f l o o r t y p e s , t h i s v a r i e s f r o m 55 t o 60 , 100 i n c r e a s i n g with rougher s e a f l o o r . These val u e s are f a i r l y t y p i c a l r e g a r d l e s s of s h i p type or wind speeds. Thus a value of 80, used throughout i n the experiments, may be regarded as t y p i c a l or 'moderate' s i z e of i ^ . Reducing t o , f o r example, 50, r i s k s divergence, whereas f u r t h e r i n c r e a s e beyond 80 r e s u l t s i n very l i t t l e improvement i n f i l t e r performance. The o v e r a l l performance curves are given i n Fig.5.7 to 5.9. For f u l l a d a p t a t i o n , the r e s u l t s c l o s e l y resemble those by measurement-differencing Kalman F i l t e r , except f o r rough s e a f l o o r s (L2=50m, 100m), the J 2 ' s are i n f e r i o r . P a r t i a l a d a p t a t i o n , however, c l e a r l y g i v e s i n f e r i o r r e s u l t s i n the case of l a r g e s h i p , e s p e c i a l l y with L2=50-I50m. For small s h i p , the d i f f e r e n c e i s s m a l l , but the curves a s s o c i a t e d with L2=300m and 600m are f l a t t e n e d . T h i s means i n f e r i o r r e s u l t s f o r medium wind speeds but improvement f o r high or low wind speeds. I n c r e a s i n g i ^ to 800 improves the performance s l i g h t l y i n the case of l a r g e s h i p . In the case of small s h i p t h i s e f f e c t i s more n o t i c e a b l e , e s p e c i a l l y with smooth f l o o r s , i n which there are o v e r a l l i n c r e a s e i n J 2 by as much as 0.1. T h i s manifests i t s e l f as a ' l i f t ' of the L2=600m and 300m curves. Thus, we are faced with a number of c h o i c e s . If the f l o o r s are smooth, and computational c o s t i s not of major concern, then i t i s d e s i r a b l e to i n c r e a s e NBATCH. E l s e there i s d e f i n i t e s aving i n computation by p a r t i a l a d a p t a t i o n . I f memory spaces are a v a i l a b l e , there i s s t i l l f u t h e r saving by measurement-differencing Kalman F i l t e r a l o n e . 101 o (a) m c i F i g u r e 5.2 J 2 as a F u n c t i o n of (Q,R) - Large Ship (a)u=l5ms~ 1, L 2=600m (b)u=9ms \L 2=l50m 102 in 6--3 ° ID d-o = r = 0.5 o = r = 1.0 A = r = 2.0 * = r = 4.0 p 7" (c) 0.0 - 1 — 2.0 4.0 6.0 8.0 10.0 0 12.0 14.0 16.0 18.0 20.0 in d - i -3 ° in d -i D = r = 0.5 o = r = 1.0 A = r = 2.0 * = r = 4.0 p 0.0 I — 2.0 4.0 6.0 - 1 — 8.0 10.0 0 12.0 14.0. 16:0 18.0 20.0 (d) F i g u r e 5.2 J 2 as a F u n c t i o n of (Q,R) ; Small Ship (c)u=8ms _ 1, L 2=600m (d)u=l3ms ,L2=50m 103 o 6" eg 6 ' 6" 6" ^ o' oo o o d" o d" o I i 30.0 35 .C 4 0 0 * 5 . 0 50.0 SNR ( d b ) 55 0 60.0 65.0 70 0 (a) SNR ( d b ) F i g u r e 5.4 Kalman F i l t e r Performances - Large Ship 1 04 45.0 5 0 0 SNR (db) 55.0 60.0 65.0 70.0 (c) Seafloor L2 D = 50m o = 100m A = 150m • = 300m x = 600m 45.0 50.0 SNR (db) 55.0 60.0 65.0 70.0 (d) F i l t e r Performances - Small Ship 105 CM -3 m b -l 30.0 35.0 40.0 45.0 50.0 SNR (db) 55.0 60.0 65.0 70.0 (a) F i g u r e 5.7 Performances of Adaptive Kalman F i l t e r i n g - F u l l A d aptation (a) Large Ship (b) Small Ship 106 e 5.8 P e r f o r m a n c e s o f A d a p t i v e Ka lman F i l t e r i n g - P a r t i a l A d a p t a t i o n (a ) L a r g e S h i p (b) S m a l l 107 F i g u r e 5.9 Performances of Adaptive Kalman F i l t e r i n g - Increased NBATCH (a) Large Ship (b) Small Ship 108 5.4 Performance of RLSI F i l t e r The RLSI F i l t e r has two v a r i a b l e s : q and L. A complete h e u r i s t i c search f o r optimal performances would i n c l u d e these two v a r i a b l e s as w e l l as the two system v a r i a b l e s : u and L 2 . To reduce computations, we f i r s t note that among the system v a r i a b l e s , L 2 has the g r e a t e r e f f e c t on performance. In other words, optimal q and L chosen f o r a s p e c i f i c s e a f l o o r type are v a l i d f o r a wide range of wind speeds. Proceeding along t h i s l i n e of argument, we o b t a i n a s e r i e s of performance curves with v a r y i n g L ( F i g . E 2 . l ) , and with v a r y i n g q ( F i g . E 2 . 2 ) . From Fig.E2.1, i t can be seen that L i n c r e a s e s with L 2 , f o r optimal performance. For smooth f l o o r , L i s approximately 11, and f a l l s to 2 f o r the roughest p r o f i l e . Using these v a l u e s of L, we o b t a i n the curves i n Fig.E2.2, from which the optimal q's are determined. In g e n e r a l , rough s e a f l o o r has sma l l e r q ( i . e . , s h o r t e r window l e n g t h W). T h i s i s expected, as the RLSI F i l t e r should smooth over a s h o r t e r d i s t a n c e . Table 5.1 summarises the optimal (q,L) f o r d i f f e r e n t f l o o r t y pes. Table 5.1 (a)Large Ship (b)Small Ship L 2 (m) <3opt L o p t L 2 (m) Sopt L o p t 50 0.783 2 50 0.700 2 100 0.843 3 100 0.788 2 1 50 0.850 3 150 0.802 2 300 0.830 1 1 300 0.784 9 600 0.825 1 1 600 0.779 9 109 The o v e r a l l performance curves are given in Fig.5.10. Comparing to those of the Kalman F i l t e r s , the most s t r i k i n g f e a t u r e i s t h a t the J 2 curves are f l a t t e r and more widely spaced, f o r both l a r g e and small s h i p s . In other words, RLSI F i l t e r i s more s e n s i t i v e to s e a f l o o r c h a r a c t e r i s t i c s but q u i t e i n s e n s i t i v e t o v a r i a t i o n i n wind speeds ( i . e . , SNR). The l a t t e r p r o p e r t y can be a t t r i b u t e d to the absence of m o d e l l i n g e r r o r problems i n RLSI F i l t e r . The f l a t t e r J 2 curves imply that at e i t h e r extremes of SNR range, i . e . , at very high or low wind speeds, the performance of RLSI F i l t e r exceed that of Kalman F i l t e r . T h i s i s e s p e c i a l l y t r u e f o r the case of low wind speeds, where the Kalman F i l t e r performance curves d i p s i g n i f i c a n t l y . However, for rougher s e a f l o o r s ( L 2 = 500m to 150m) the RLSI F i l t e r i s i n f e r i o r to Kalman F i l t e r , except f o r the lowest wind speed as noted above. The e s t i m a t i o n time delay a s s o c i a t e d with the RLSI F i l t e r i s j u s t equal to L. As expected, L i s p r o p o r t i o n a l to the c o r r e l a t i o n d i s t a n c e of the s i g n a l ( i . e . , s e a f l o o r p r o f i l e ) , with the r e s u l t t h at time delay encountered i n p r o c e s s i n g rough f l o o r s i s q u i t e i n s i g n i f i c a n t ( L equals 2 ), but becomes p r o g r e s s i v e l y l a r g e r as the s e a f l o o r becomes smoother. 110 30.0 35.0 * 0 . 0 45.0 50.0 SNR ( d b ) 55.0 60.0 65.0 70.0 (a) 30.0 35.0 40.0 45.0 50.0 SNR ( d b ) 55.0 60.0 65.0 70.0 (b) F i g u r e 5.10 Performances of RLSI F i l t e r (a)Large Ship (a)Small Ship 111 5 . 5 Performance of LP F i l t e r For the purpose of comparison we s h a l l examine the r e s u l t s of a simple p r o c e s s i n g method, that of low-pass f i l t e r i n g . Most n o t a b l e f e a t u r e s of t h i s method are the l a r g e time delay i n v o l v e d , and the f a c t that complete knowledge of the s p e c t r a of s i g n a l and noise i s r e q u i r e d . The l a t t e r l i m i t a t i o n can be r e l a x e d i n case of c l e a r s e p a r a t i o n of the two s p e c t r a . T h i s f a c t w i l l be supported by some experimental r e s u l t s . We s h a l l use the FIR d i g i t a l f i l t e r because i t has a constant phase. A simple and e f f i c i e n t design method i s adopted, namely, the Window method [ 4 5 ] . An a p p r o p r i a t e window f u n c t i o n i s int r o d u c e d i n the time domain to y i e l d a f i n i t e f i l t e r time s e r i e s h(n) and to C o n t r o l i t s convergence. Thus, i f w(n) i s a window f u n c t i o n , the r e s u l t i n g f i l t e r frequency response i s the c i r c u l a r c o n v o l u t i o n of H(a>) and W(CJ) . As f a r as f i l t e r performance i s concerned , there are four determining f a c t o r s : f i l t e r l e n g t h , . c u t o f f frequency, t r a n s i t i o n bandwidth, and s i d e l o b e r i p p l e of the window. In p r a c t i c a l a p p l i c a t i o n , i t i s d e s i r a b l e t o e l i m i n a t e the number of v a r y i n g f a c t o r s to as small as p o s s i b l e . We s h a l l see i f t h i s can be ac h i e v e d by c o n s i d e r i n g three commonly used window f u n c t i o n s : r e c t a n g u l a r , Hamming and K a i s e r . For o p t i m a l performance, the f i l t e r should have a small t r a n s i t i o n bandwidth and l a r g e s i d e l o b e r i p p l e a t t e n u a t i o n . In the order as l i s t e d above, the windows have i n c r e a s i n g t r a n s i t i o n bandwidth, and i n c r e a s i n g s i d e l o b e a t t e n u a t i o n . F i g u r e 5.11 T r a n s i t i o n B a n d w i d t h o f L P F i l t e r s H o w e v e r , a p p l i c a t i o n s o f f i l t e r s d e s i g n e d w i t h t h e s e window t y p e s show t h a t t h e f o r m e r f a c t o r h a s f a r g r e a t e r e f f e c t on p e r f o r m a n c e , so t h a t , w i t h a l l o t h e r v a r i a b l e s k e p t c o n s t a n t , t h e r e c t a n g u l a r - w i n d o w e d LP F i l t e r h a s t h e b e s t r e s u l t . F i g .5.11 shows t h e t r a n s i t i o n b a n d w i d t h s o f v a r i o u s window t y p e s a s a f u n c t i o n o f f i l t e r l e n g t h . In g e n e r a l , r e c t a n g u l a r window h a s t h e s m a l l e s t b a n d w i d t h ; b e s i d e s , b a n d w i d t h i s d e c r e a s e d by i n c r e a s i n g f i l t e r l e n g t h . A d o p t i n g t h e r e c t a n g u l a r w indow, we s h a l l c o n s i d e r t h e p e r f o r m a n c e a s a f u n c t i o n o f F c , t h e f i l t e r c u t o f f f r e q u e n c y , a n d f i l t e r l e n g t h . F u r t h e r , we a r e a l s o i n t e r e s t e d i n o b t a i n i n g a u n i v e r s a l L P f i l t e r t h a t i s c a p a b l e o f a p p l i c a t i o n 1 1 3 i n a wide range of f l o o r types and wind speeds. In F i g . E3.1, performance as a f u n c t i o n of F c f o r l a r g e and smal l s h i p s are shown. I t can be seen t h a t , i n both cases, maxima occur at approximately F c=0.05Hz, a f a c t c o n s i s t e n t with the shape of Sziu>) (e.g., c o n s u l t F i g . 3 . 2 ) . The wind speed i n the example i s 12 ms" 1, but the r e s u l t s are f a i r l y t y p i c a l f o r other wind speeds. As expected, performance improves as L 2 i n c r e a s e s . Thus, we can f i x the c u t o f f frequency at 0.05 Hz. With t h i s v alue of F c , the performance curves as a f u n c t i o n of N, the f i l t e r l e n g t h , are obtained ( F i g . E3.2). I n c r e a s i n g N reduces the main lobe width of the f i l t e r , which i n turn reduces the t r a n s i t i o n bandwidth, and t h e r e f o r e should give b e t t e r performance. T h i s i s confirmed i n the curves, which e x h i b i t c r i t i c a l t u r n i n g p o i n t s at around N=25-30, below which J 2 f a l l s r a p i d l y , but beyond which there are l i t t l e f u r t h e r improvements i n performance. Based on t h i s , we a r r i v e at the c o n c l u s i o n that N=50 i s the minimum f i l t e r l e n g t h f o r c l o s e to optimal performance. Note t h a t , f o r FIR f i l t e r , the time delay i s given by A=[(N-1)/2] where [x] denotes the s m a l l e s t i n t e g e r 2 x. Thus, f o r N=50, A=25. The o v e r a l l performance curves are obtained by f i l t e r i n g with the u n i v e r s a l LP f i l t e r , ' t e n t a t i v e l y d e f i n e d as rectangular-windowed, having F c=0.05 Hz, and N=50 (Fig.5.12). The shapes of the curves are g e n e r a l l y convex, being more s h a r p l y peaked i n the case of small s h i p . In a d d i t i o n , 1 14 p e r f o r m a n c e s o f d i f f e r e n t f l o o r t y p e s t e n d t o c o n v e r g e a t low SNR. E l s e w h e r e , f i l t e r i n g w i t h r o u g h f l o o r s g i v e l o w e r J 2 . T h i s i s due t o t h e f i x e d F c i n a u n i v e r s a l f i l t e r , w h i c h a t t e n u a t e s p o r t i o n o f t h e l o w p a s s s i g n a l i n t h e f i l t e r i n g p r o c e s s . H o w e v e r , t h e s e p a r a t i o n o f t h e s i g n a l and n o i s e s p e c t r a e n s u r e s t h a t f i l t e r i n g by t h i s method s t i l l y i e l d s a c c e p t a b l e r e s u l t s , w i t h t h e e x c e p t i o n o f t h e c a s e o f s m a l l s h i p w i t h l o w e s t w i n d s p e e d , u=6ms~ 1 , and L2=50m ( J 2 = - 0 . 0 6 , r e n d e r i n g t h e f i l t e r u s e l e s s ) . C o m p a r i n g t o t h e r e s u l t s by Ka lman f i l t e r i n g , t h e LP f i l t e r i s s u p e r i o r e x c e p t f o r s m a l l s h i p w i t h l o w e s t w i n d s p e e d s ( u = 6 m s ~ 1 ) . C o m p a r i n g t o t h e RLSI f i l t e r , h o w e v e r , s i t u a t i o n i s more c o m p l i c a t e d . The f o l l o w i n g o b s e r v a t i o n s c a n be made : a ) F o r l a r g e s h i p , a t low SNR ( h i g h w i n d s p e e d s ) , RLSI f i l t e r p e r f o r m s b e t t e r w i t h smooth s e a f l o o r ( L 2 > 300m), and LP f i l t e r p e r f o r m s b e t t e r w i t h r o u g h s e a f l o o r ( L 2 < 300m). A t h i g h e r SNR, b o t h f i l t e r s a r e r o u g h l y e q u a l w i t h smooth f l o o r s , b u t LP f i l t e r i s b e t t e r w i t h r o u g h f l o o r s . b) F o r s m a l l s h i p , a t low SNR t h e a b o v e b e h a v i o u r s h o l d . A t medium SNR ( 4 5 - 5 0 d B , where p e a k i n g o c c u r s f o r b o t h f i l t e r s ) , LP f i l t e r s a r e s u p e r i o r w i t h a l l t y p e s o f s e a f l o o r . F o r i n s t a n c e , a maximum J 2 o f 0 .88 i s r e a c h e d f o r L 2 =600m a n d 0 .67 f o r L2=50m. H o w e v e r , a t h i g h e r SNR, a s n o t e d b e f o r e t h e LP f i l t e r d e g r a d e s r a p i d l y , a n d i n t h i s c a s e (SNR >> 6 2 . 5 dB) t h e RLSI f i l t e r i s t o be p r e f e r r e d . 1 15 d -in d -i i - — - 1 • 1 • 1 • 1 • 1 • — - i • 1 • 1 (a) 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0 SNR (db) o Seaf loo r L2 • = 50m o = 100m A = 150m • = 300m x = 600m ID O -I (b) 30.0 35.0 40.0 45.0 50.0 55.0 60.0 65.0 70.0 SNR (db) F i g u r e 5.12 Performances of LP F i l t e r (a)Large Ship (a)Small Ship 1 16 5.6 Summary of F i l t e r Performances S i m u l a t i o n r e s u l t s ( J ^ curves) i n Fig.5.13 i l l u s t r a t e the convergence r a t e s of the v a r i o u s f i l t e r i n g a l g o r i t h m s . They show that the Kalman F i l t e r s have the f a s t e s t convergence r a t e s of about 25-30 sec (50-60 i t e r a t i o n s ) . The convergence curves of the v a r i o u s adaptive Kalman f i l t e r s , together with those of the measurement-differencing Kalman F i l t e r , are q u i t e c l o s e to each o t h e r . G e n e r a l l y , a d a p t i v e f i l t e r with i n c r e a s e d NBATCH (equals 800) g i v e s the best r e s u l t s , f o l l o w e d by f u l l and p a r t i a l a d a p t a t i o n . Convergence r a t e s a l s o vary i n the same or d e r . On the other hand, the RLSI F i l t e r converges more slow l y , i n about 80 sec (160 i t e r a t i o n s ) . The LP F i l t e r i s the slowest, with convergence i n 100 sec or more. The convergence r a t e i s a l s o dependent on the s i g n a l and n o i s e c h a r a c t e r i s t i c s , i . e . , on u and L 2 . Of the two, L 2 has the g r e a t e r e f f e c t . T h i s can be seen i n case (d) of Fig.5.13, i n which the s e a f l o o r processed i s rough (L 2=50m). The convergence r a t e s of a l l f i l t e r s are slowed down: f o r the Kalman f a m i l y convergence time i s about 50 sec; f o r RLSI F i l t e r , about 100 sec; f o r LP f i l t e r , about 250 sec (extending beyond i l l u s t r a t e d a r e a ) . T h i s a l s o v e r i f i e s the f a c t that slower convergence i s a s s o c i a t e d with lower performance as p o i n t e d out i n S e c t i o n 5.3. Note that i n t h i s case, the RLSI F i l t e r i s i n f e r i o r t o the Kalman F i l t e r s i n performance (J 2=0.35 versus 0.37). LI I 1 1 8 The s i m u l a t i o n r e s u l t s presented above enable us to respond to the c r i t e r i a as set out i n S e c t i o n 5.2. With computational and memory requirement aspects d i s c u s s e d i n the next chapter, the f o l l o w i n g can be s t a t e d : 1. E s t i m a t i o n Time Delay - A l l Kalman F i l t e r s are r e a l - t i m e a l g o r i t h m s . The RLSI F i l t e r has time delay ranging from 2 to 11 f o r l a r g e s h i p , and 2 to 9 f o r small s h i p . The LP F i l t e r has the l a r g e s t time delay; f o r u s e f u l r e s u l t s the delay i s at l e a s t 50 5 . 2. Convergence Rate - Dependent on s e a f l o o r c h a r a c t e r i s t i c s . In g e n e r a l , the rougher the s e a f l o o r , the slower the convergence. For a f i x e d f l o o r type and wind speed, Kalman F i l t e r s have the f a s t e s t convergence, f o l l o w e d by RLSI F i l t e r and LP F i l t e r . 3. S t a b i l i t y of A l g o r i t h m - Performances are more s t a b l e for l a r g e than f o r sma l l s h i p . The RLSI F i l t e r i s the l e a s t s e n s i t i v e t o v a r i a t i o n i n SNR, but more s e n s i t i v e to s e a f l o o r c h a r a c t e r i s t i c s . At low SNR, the LP F i l t e r i s i n s e n s i t i v e t o both, but at h i g h SNR, performance degrades as f l o o r becomes more rough. The Kalman F i l t e r s are s e n s i t i v e t o both f a c t o r s , and, together with LP F i l t e r , e x h i b i t s peaking at medium SNR. In other words, both Kalman and LP f i l t e r s are more s u i t a b l e f o r moderate wind speeds, i n case of u n c e r t a i n t y i n m o d e l l i n g heave motion. In the f o l l o w i n g t a b l e , the f i l t e r s are l i s t e d in descending order of performances c o r r e s p o n d i n g to low SNR ( < 5The numbers quoted are i n number of i t e r a t i o n s . 119 42dB ), medium SNR ( 42-50 dB ), and high SNR ( >50 dB) : Table 5.2 SNR Ship S i z e Low Medium High LP LP LP Large Kalman Kalman RLSI RLSI RLSI Kalman LP LP RLSI Small RLSI Kalman Kalman Kalman RLSI LP The t a b l e above summarizes the gen e r a l behaviours of the f i l t e r s with respect to v a r y i n g wind speeds (SNR). For i n d i v i d u a l f i l t e r response to v a r y i n g s e a f l o o r c h a r a c t e r i s t i c s , one should c o n s u l t F i g u r e s 5.4 to 5.11. 120 C h a p t e r 6 COMPUTATIONAL C O M P L E X I T Y F r o m t h e i m p l e m e n t a t i o n a l p o i n t o f v i e w , c o m p u t a t i o n a l e f f i c i e n c y i s an i m p o r t a n t f a c t o r i n d e t e r m i n i n g w h i c h a l g o r i t h m i s b e s t . H o w e v e r , no p r e c i s e c h a r a c t e r i z a t i o n o f c o m p u t a t i o n a l e f f i c i e n c y e x i s t s so f a r , b e c a u s e i t i s known t o d e p e n d on many o t h e r f a c t o r s , s u c h a s o p e r a t i o n a l c o m p u t e r p a r a m e t e r s ( i n s t r u c t i o n s e t , w o r d l e n g t h , CPU t i m e , e t c . ) , p r o g r a m m i n g m e t h o d s ( c o d e , f i l e a n d d a t a s t r u c t u r e , l i b r a r y r o u t i n e s , s i n g l e o r d o u b l e p r e c i s i o n , e t c . ) , a n d , o f c o u r s e , t h e c o m p u t a t i o n a l c o m p l e x i t y o f t h e a l g o r i t h m s . I t i s o u r o b j e c t i v e h e r e t o p r o v i d e a n o r d e r - m a g n i t u d e a p p r o x i m a t i o n on c o m p u t a t i o n a l r e q u i r e m e n t s o f t h e v a r i o u s f i l t e r i n g a l g o r i t h m s c o n s i d e r e d , w i t h o u t r e f e r e n c e t o o t h e r m a c h i n e - r e l a t e d f a c t o r s . T o g e t h e r w i t h t h e c o m p a r i s o n s t u d i e s i n C h a p t e r 5 , t h e d e s i g n e r w i l l be a b l e t o p e r f o r m a u s e f u l c o s t - e f f e c t i v e n e s s t r a d e - o f f a n a l y s i s . By c o m p u t a t i o n a l r e q u i r e m e n t s we mean b o t h c o m p u t a t i o n a l t i m e i n number o f o p e r a t i o n s a n d memory r e q u i r e m e n t . S i n c e o n l y a p p r o x i m a t e a s s e s s m e n t i s s e e k , t h e f o l l o w i n g a s s u m p t i o n s a r e m a d e : a ) A g r o u p o f p a r a m e t e r s a r e a s s u m e d t o be g i v e n , i . e . , e i t h e r p r e c o m p u t e d ( e . g . T , H , e t c . ) o r m e a s u r e d ( e . g . , z k ) ( c o l l e c t e d u n d e r G r o u p I i n T a b l e 6 . 1 ) . I n o t h e r w o r d s , a s t r i c t l y s t a t i o n a r y s y s t e m i s a s s u m e d . I n t h e c a s e o f a d a p t i v e ARMA f i l t e r , t h e ARMA p r e - a n d p o s t - f i l t e r s a r e 121 assumed known. b) L o g i c time i s excluded. No d i s t i n c t i o n i s made between m u l t i p l i c a t i o n and d i v i s i o n (M/D), or a d d i t i o n and s u b t r a c t i o n (A/S). c) E x t r a c t i o n of square root of a s c a l a r i s assumed to be accomplished i n q m u l t i p l i c a t i o n s . The Kalman F i l t e r s i n the Measurement-Differencing and Adaptive forms w i l l be examined f i r s t , f ollowed by the Adaptive ARMA F i l t e r . Then comparisons with the RLSI and LP F i l t e r s w i l l be given based on the implementation in t h i s t h e s i s , i . e . , dimension of s t a t e v e c t o r n=2. For Kalman F i l t e r , the computer time and memory requirements are given i n terms of n,m and t , the dimensions of the s t a t e v e c t o r , measurement v e c t o r and measurement nois e v e c t o r , r e s p e c t i v e l y . Of course, i n our case, both m and t equals 1, and n equals the order of the approximating f u n c t i o n to S n ( f ) . Furthermore, a l l matrix i n v e r s i o n s are assumed to be performed v i a the Cholesky f a c t o r i z a t i o n r o u t i n e , which r e q u i r e s [n 3/2+3n 2/2+nq] o p e r a t i o n s and [n 2/2+n/2] storage l o c a t i o n s . Table 6.1 g i v e s the computational requirements of the Measurement-Differencing Kalman F i l t e r . The v a r i a b l e s are d i v i d e d i n t o two groups: Group I are " f i x e d " with respect to each i t e r a t i o n , r e q u i r i n g only storage; those i n Group II are updated at each i t e r a t i o n , t h e r e f o r e both computations and storage are r e q u i r e d . Note t h a t update of the P matrix i n 122 (3.33c) uses the s t a b i l i s e d (or Joseph) form, which i s b e t t e r c o n d i t i o n e d and l e s s s e n s i t i v e to computer round-off e r r o r s than the c o n v e n t i o n a l form. Table 6.1 Computational Requirements of the Measurement-Differencing Kalman F i l t e r Computation Sequence Operations Dimension (M/D) (A/S) Storage Group I $ r H' D * 2k+1 z k Q R' $-DH' rgrT DRD T Group I U k = z k + 1 - * z k r k=| k-H'x k| k_ 1 G k r k X k + I | k = * x k | k - 1 + G k r k MkH'T H'MkH,T+R' (H'M kH' T+R')~ 1 K k=M kH' T(H'M kH , T+R') _ 1 I-K kH' ( I - K k H ' ) M k (I-K kH' )M k(I~K kH' ) T P k=(l-KuH' ) M k ( l - K k H ' ) +K kR'K kT (*-DH')Pk M k + 1 = (*-DH' )Pk(f>-DH' ) T +rQrT-DR'DT k nxn n 2 nxn n 2 1 xn n nx 1 n 1x1 1 1x1 1 1x1 1 nxn n 2 1x1 1 nxn n 2 nxn n 2 nxn n 2 1x1 1 1 1 1x1 n n 1 nx 1 n 2+n n n nx 1 n 2 n 2 n nx 1 n 2 n 2 -n n 1x1 n 2 n 2 -n+1 1 1x1 1 nx 1 n n nxn n 2 n 2 n 2 nxn n 3 n 2 (n-1) nxn n3 n 2 (n-1 ) nxn 2n 2 n 2 n 2 nxn n3 n 2 (n-1) n 2 nxn n3 2n 2 ( n - D n 2 1 23 T o t a l M/D Operations=4n 3+7n 2+3n+2 T o t a l A/S Operations=5n 3+2 T o t a l Storage=8n 2+6n+7 The bulk of the computation i s taken up by c o v a r i a n c e matrix updating, as w e l l as matrix i n v e r s i o n i n updating Kalman Gain. Both these computations are much reduced by using a low-order f i l t e r , and f u r t h e r by having only one measurement, m=1. Likewis e , storage s i z e i s l a r g e l y determined by the s i z e of the co v a r i a n c e m a t r i c e s , which i s i n turn dependent on the order of the f i l t e r . In the adaptive form, Q,R',D,*-DH', rQT T and DR'DT have to be updated a f t e r each batch of NBATCH p o i n t s processed. To a f i r s t - o r d e r approximation, these e x t r a computations can be incop o r a t e d i n t o the t o t a l computation count of the Innovation C o r r e l a t i o n A l g o r i t h m . For the l a t t e r , p r e v i o u s n o t a t i o n s are used, i . e . , N i s the sample s i z e , ijj the f i r s t dimension of the matrix A, NBATCH the s i z e of each batch. Table 6.2 g i v e s the requirements f o r f u l l a d a p t a t i o n . 124 Table 6.2 Computational Requirements of the Innovation C o r r e l a t i o n A l g o r i t h m - F u l l Adaptation Computation Operat ions Sequence Dimension (M/D) (A/S) Storage D=TSR'~~1 nx 1 n 2+n n $-DH' nxn n 2 n 2 n 2 nxn 2n3 n 2 DR'DT nxn 2n 2 n 2 {C i i , . . , C i N } ( i N + 1 ) N + i N 2 / 2 i N . N + i N 2 / 2 *N + i N / 2 A i N x n n 2 ( i N - l ) + ( n 3 ( 2 n 2 + n ) i N n . i N + n 2 ) i N A# n x i N 2 n 2 i v , + ( n 3 2n 2 i N - n 2 - n i N n . i M + ( n 2 +3n 2)/2+nq +n)/2 A nx 1 i N 2 n + n 2 + n n i N + n 2 - n n *A nx 1 n 2 n 2-n n * A G k T nxn n 2 n 2 G k c o G k T nxn n 2+n n 2 £2 nxn 2n 2 n 2 z:H'#JrQrT(*J - k ) T H t T 2n 4+2n 3-n 2 2n 4-5n 2/2+3n/2 2n 3 3-0 +£k.2n 2 +fk(2n 2-2n) 2n 3-2n ( Q ( H D T ) T r T ( * - k - 1 ) T H ' T H 'd,k-1 r(Q( H r) T ) A T ( * - k ) T H , T 2n 3+2n 2 2n 2n 3 2n 3 2n 3-2n 2 2n 3-2n 2 n n H - $ k A 2 n 3 2n 3-2n 2 n SH'#30(*D-k) T H,T I 2 k n 2 2k(2n 2-2n) +(n 2+n)/2 n p-1 nxn (n 3+3n 2)/2 (n 2+n)/2 Q=F~1/3 + n q nxn n 3 n 2(n-1) n 2 H ' * - 1 r s 1x1 n 3+n 2+n (n 2+n+1)(n-1) 1 H'A 1x1 n n-1 1 R 1x1 2 1 For p a r t i a l a d a p t a t i o n , c a l c u l a t i o n s of A and Q are skipped. Since a d a p t a t i o n s are performed i n i n t e r v a l s , the t o t a l o p e r a t i o n count should be averaged by NBATCH. Table 6.3 125 p r o v i d e s the averaged computational requirements of the two modes of a d a p t a t i o n ( i n a d d i t i o n to those l i s t e d i n Table 6.1). Table 6.3 Averaged Computational Requirements of the Innovation C o r r e l a t i o n A l g o r i t h m Operat ions Adaptation (M/D) (A/S) Storage P a r t i a l [(3n 3+13n 2)/2+4n [n 3+n 2+n+i N 2/2 5(n 2+n)/2+(2n + n 3 i K + 4 n 2 i N + i N 2 n + i N / 2 + N i N + 4 n 2 i N +1)i N+3=M 1 +i N 2/2+i N/2+i N/N+N +ni N]/NBATCH=N 2 +nq]/NBATCH=N1 F u l l N 1+[2n 4+27n 3/2 N 2+[2n 4+19n 3/2-2n 2-n M!+2n 3+9n 2/2 +11n2/2+n+nq +2tk(2n2-2n)]/NBATCH +15n/2 +2fk.2n 2]/NBATCH *" Some p r e l i m i n a r y remarks may be made r e g a r d i n g Table 6.3. For moderate valu e s of n (e.g., n < 10 ), the o p e r a t i o n s count of p a r t i a l a d a p t a t i o n are dominated by i ^ and N. At t y p i c a l v a l u e s of i N and N ( 60 < i N < 100, 200 < N < 1000), N, and N 2 are of comparable order of magnitude to the o p e r a t i o n s count of the Measurement-Differencing Kalman F i l t e r . The q u a n t i t y of a d d i t i o n a l o p e r a t i o n s with f u l l a d a p t a t i o n are q u i t e marginal, u n l e s s n > 10, s i n c e NBATCH i s l a r g e . Thus, the main i n c r e a s e i n computations l i e s not so much i n ALU o p e r a t i o n s as i t i s i n storage, e s p e c i a l l y M,, where the i n c r e a s e can be of the order of 10, s i n c e M, i s dominated by the product n . i N . One problem commonly encountered i n implementing Kalman F i l t e r i s t h a t of numerical i n s t a b i l i t y . F i n i t e wordlength causes accumulation of roundoff e r r o r s , which may e v e n t u a l l y 126 r e s u l t i n P l o s i n g i t s p o s i t i v e d e f i n i t e n e s s . T h i s i s e s p e c i a l l y true when measurements are very a c c u r a t e , so that the e i g e n v a l u e s of R are small r e l a t i v e to those of P. Since t h i s i s ap p a r e n t l y not the case with our problem, we encounter no i n s t a b i l i t y i n implementing the Kalman F i l t e r , a lthough some c r i t i c a l cases do occur d u r i n g a d a p t a t i o n . To combat t h i s problem, a s i z e a b l e number of techniques have been formulated, based on Square Root Covariance A l g o r i t h m [ 9 ] . The idea i s to propagate and update e r r o r c o v a r i a n c e square root i n s t e a d of the c o v a r i a n c e , thereby improving numerical p r e c i s i o n . One convenient form, the C a r l s o n F i l t e r [28], r e q u i r e s only modest a d d i t i o n a l computations but improves p r e c i s i o n and speed. Another form, the U-D Covariance F a c t o r i z a t i o n F i l t e r , can be e a s i l y adapted to the c o n v e n t i o n a l form, as i n our case [20]. T h i s f i l t e r i s designed based on the f a c t t h a t P can be expressed as P=UDUT, where U i s a u n i t a r y upper t r i a n g u l a r matrix and D i s d i a g o n a l . The cov a r i a n c e update can be f a c t o r e d i n the same manner, which i n turn l e a d s t o the Kalman Gain d e f i n e d i n terms of the f a c t o r e d U and D. T h i s method has the advantage of p r e s e r v i n g accuracy and e f f i c i e n t r e c u r s i o n i n c a l c u l a t i n g U and D. The ad a p t i v e ARMA F i l t e r a d j u s t s i t s parameters v i a the RML a l g o r i t h m i n case of unknown L 2 , whereas the ARMA pre- and p o s t - f i l t e r s e c t i o n s are designed based on known S n ( f ) . Table 6.4 g i v e s i t s computational requirements, using p r e v i o u s l y adopted n o t a t i o n s : (n,m) i s the order of the ARMA pre - and p o s t - f i l t e r , p=n a+n c i s the t o t a l order of the ad a p t i v e IIR 127 s e c t i o n . The f i l t e r e d estimate z t ' i s of dimension 1x1. Table 6.4 Computational Requirements of the Adaptive ARMA F i l t e r Computation Operations Sequence (M/D) (A/S) Storage ARMA Pre- & P o s t - F i l t e r 2(n+m) 2(n+m+2) n+m z ' t | t - 1 P p+2 P et+1 = 2't+1 _*t+1 et P 1 1 n t = p t ( * T * ) p t p 2+2p 3 2p 2(p-1) P 2 7=* TP t*+X t + 1 P 2 + P p 2(p-1)+(p-1)+1 1 p t + 1 = [ p t - n t / 7 ] / x + i 2P 2 P 2 P 2 e t + 1 = t ? t + ( P t t 1 * ) e t + 1 P 2+P (p+1)(p-1)+p P z t + i ' : * t + i T ^ t + i p p-1 1 e t + i = 2 t + i , - z t + i ' 1 1 X t + 1=XX t+(1-X) 1 2 1 T o t a l M/D Operations=2(n+m)+2p 3+5p 2+5p+1 T o t a l A/S Operations=2(n+m)+3p 3+p 2+p+8 T o t a l Storage=(n+m)+2p 2+2p+5 As can be seen, even with moderate, t y p i c a l values of p, the t o t a l number of o p e r a t i o n s amounts to the thousands, which i s q u i t e p r o h i b i t i v e . The bulk of the computations l i e i n the updating of the P matrix, which i n v o l v e s p 3 term. A number of f a s t a l g o r i t h m s o u t l i n e d i n [10] reduce t h i s to p 2 , with the r e s u l t t h a t the t o t a l counts become comparable t o those of the adaptive Kalman F i l t e r . Thus, the Adaptive ARMA F i l t e r i s u s e f u l i n the case of unknown L 2 , or r a p i d l y v a r y i n g L 2 . The a c t u a l computational c o m p l e x i t i e s of the f i l t e r s as implemented i n the s i m u l a t i o n experiments are assessed i n Table 128 6.5. The exe c u t i o n time, i n CPU seconds, i s the average time to process a sequence of 2400 p o i n t s , i n double p r e c i s i o n , on Amdahl 5850. Again, i t must be s t r e s s e d t h a t the timings given are only approximate, but there i s s u f f i c i e n t c o r r e l a t i o n between t h e o r e t i c a l complexity estimates and a c t u a l timings to deduce the r e l a t i v e complexity between the a l g o r i t h m s . Table 6.5 F i l t e r Computational C o m p l e x i t i e s and Execution Time Execution Time F i l t e r Type Operat ions (M/D) (A/S) Storage Large Ship Small Ship M-D KF 68 42 51 0.566 0.6242 ADPKF-Full Adapt. 114.53 1 34 518 0.651 0.737 ADPKF-Partial Adapt. 1 14 1 33.7 469 0.613 0.6632 ADPKF-Full Adapt. 92.03 • 128.16 518 0.608 0.6534 NBATCH=800 RLSI F i l t e r 10 9 3 0.240 0.246 LP F i l t e r 50 49 50 0.497 0.514 n = 2 i N=80 N=50 NBATCH=400 unless otherwise noted The RLSI F i l t e r i s , without doubt, the l e a s t c o s t l y method, with both low o p e r a t i o n counts and small storage. For low order Kalman F i l t e r , the computations are roughly the same as the LP F i l t e r . At n=2, a d a p t a t i o n i n c r e a s e s M/D count by roughly 2, and A/S count by roughly 3, with the l a r g e s t i n c r e a s e being i n storage (by 10 t i m e s ) . These p r o p e r t i e s agree with the remarks made about Table 6.3. The execution time, however, i n c r e a s e s by a much l e s s p r o p o r t i o n . Perhaps the more unexpected f e a t u r e s are to be found i n p a r t i a l 1 29 a d a p t a t i o n , where there are n e g l i g i b l e savings i n ALU o p e r a t i o n s . The main saving i s i n memory. N e v e r t h e l e s s , there are s t i l l a p p r e c i a b l e r e d u c t i o n i n exe c u t i o n time compared to f u l l a d a p t a t i o n . In g e n e r a l , t h i s i s true f o r low order f i l t e r (n < 10 ). I f n > 10, then the s i t u a t i o n i s rev e r s e d , i . e . , main savings are i n ALU o p e r a t i o n s . In the case of a d a p t a t i o n using i n c r e a s e d NBATCH (of 800), there are some f u r t h e r savings i n computations. T h i s i s due to the averaging e f f e c t of NBATCH, d e s p i t e the corres p o n d i n g i n c r e a s e i n N. Thus, although storage requirement remains unchanged, the execution time i s reduced. Since i n c r e a s i n g NBATCH a l s o improves performance s l i g h t l y ( S e c t i o n 5.3), t h i s i s probably the p r e f e r a b l e form of a d a p t a t i o n . 130 Chapter 7 CONCLUSIONS 7.1 General Observations on System Performances The problem of r e s t o r a t i o n of random motion degraded sonar images has been formulated as a l i n e a r f i l t e r i n g problem. z ( t ) , the r e c e i v e d , degraded s i g n a l was approximated as d ( t ) + n ( t ) , i . e . , as s i g n a l with a d d i t i v e n o i s e . The nature of n(t ) was an a l y z e d in d e t a i l . In f u l l y developed head sea heave and p i t c h were the p r i n c i p a l mode of motions, and an ex p r e s s i o n was obtained d e s c r i b i n g the heave spectrum i n terms of the wave encounter spectrum and the s h i p ' s receptance. A n a l y s i s was a l s o made on a 3-dimensional sonar r e f l e c t i o n system, which, a f t e r two s u c c e s s i v e approximations, l e d to n ( t ) being approximated by [£ z ( t ) + £ z ( t + r c ) ] / 2 . Thus, the noise was e s s e n t i a l l y n o n - s t a t i o n a r y . But, slowly v a r y i n g s e a f l o o r p r o f i l e ensures T c to be roughly c o n s t a n t , ren d e r i n g the noise approximately s t a t i o n a r y . The e f f e c t of T c was found to be s e l e c t i v e l y a t t e n u a t i n g and s l i g h t l y narrowing the heave spectrum. An approximate s p a t i a l spectrum, a f i r s t - o r d e r Markov spectrum, was a l s o proposed f o r d ( t ) , the s e a f l o o r p r o f i l e , based on p r e v i o u s e m p i r i c a l r e s e a r c h . The p r i n c i p a l parameters were a2 and j3, which was r e l a t e d to L 2 , the s e a f l o o r c o r r e l a t i o n d i s t a n c e . For S n(co) the parameters were v, u and \p, which a f f e c t e d the wave spectrum. Among these parameters, u and L 2 were chosen t o be the v a r y i n g parameters f o r 131 experimental study, mainly because they are most l i k e l y to change i n a p r a c t i c a l environment, and because the others can be e a s i l y kept constant (e.g., v and \p), or normalized (e.g., a2). Two performance measures were d e f i n e d : J i and J 2 . As f i l t e r performance depends on s t a t i s t i c s of both d ( t ) and n ( t ) , i t was convenient to d e f i n e the v a r i a b l e parameter SNR as the r a t i o of s e a f l o o r to heave noi s e s p e c t r a . S e v e r a l f i l t e r s were proposed : Kalman F i l t e r , RLSI F i l t e r , Adaptive ARMA F i l t e r , and LP F i l t e r . T h e i r c h a r a c t e r i s t i c s may be summarised as f o l l o w : a) The Kalman f i l t e r s operate i n r e a l - t i m e , i . e . , there i s zero time d e l a y . They a l s o have the f a s t e s t convergence r a t e , which v a r i e d between 25-50 sec, depending on the s e a f l o o r c h a r a c t e r i s t i c s . Performance depended on both s e a f l o o r c h a r a c t e r i s t i c s and wind speed. For l a r g e s h i p , J 2 ranged from 0.35 to 0.7. For small s h i p , performance was best f o r medium SNR, with J 2 ranging from 0.43 to 0.7, but d e t e r i o r a t e d at e i t h e r very high or very low wind speeds. In a l l cases, performance improved as s e a f l o o r became smoother ( L 2 i n c r e a s e d ) . b) The RLSI F i l t e r has v a r y i n g time delay, ranging from 2 to 11 f o r l a r g e s h i p , and 2 to 9 f o r small s h i p . The performance curves were more f l a t , but l e s s c l o s e l y packed. T h i s meant that RLSI F i l t e r was q u i t e i n s e n s i t i v e to v a r i a t i o n i n SNR, but more s e n s i t i v e to s e a f l o o r c h a r a c t e r i s t i c s . For t h i s reason, the RLSI F i l t e r was b e t t e r than Kalman F i l t e r with 1 32 small s h i p at low SNR, and was the best of a l l at high SNR (J2=0.1 to 0.72). Convergence was much slower than Kalman F i l t e r , ranging from 80 sec to more than 100 sec, but s t i l l f a s t e r than LP F i l t e r . RLSI F i l t e r has the advantage that no a p r i o r i m o d e l l i n g was r e q u i r e d , and c o u l d operate i n n o n s t a t i o n a r y c o n d i t i o n s . In a d d i t i o n , i t has the a t t r a c t i v e f e a t u r e of r e q u i r i n g the l e a s t amount of computations. c) The LP F i l t e r has the l a r g e s t time delay (at l e a s t 50) and slowest convergence r a t e (100 to more than 250 sec) and r e q u i r e d a c curate m o d e l l i n g of both s e a f l o o r and heave s p e c t r a . N a t u r a l l y , i t was unsuited to problems i n v o l v i n g t i m e - v a r y i n g parameters. But f o r reasonably w e l l - s e p a r a t e d s p e c t r a , an optimum F c and minimum f i l t e r l e ngth N c o u l d be found, about 0.05 Hz and 50, r e s p e c t i v e l y . The r e s u l t i n g performances were good, p a r t i c u l a r l y f o r l a r g e s h i p . The performance curves tend to converge at low SNR, peaking at medium SNR ( J 2 ranging from 0.65 to about 0.8), then dropping at high SNR. Computationally, the LP F i l t e r was roughly e q u i v a l e n t to the Kalman F i l t e r . In implementing Kalman F i l t e r , knowledge of L 2 and the noise c o v a r i a n c e s Q and R were necessary. The techniques f o r s o l v i n g these problems combined both adaptive and h e u r i s t i c o n - l i n e methods. A Fast E s t i m a t i o n method was shown to y i e l d s a t i s f a c t o r y approximation of the mean L 2 , e s p e c i a l l y f o r rough s e a f l o o r . A d d i t i o n a l l y , the Kalman F i l t e r was shown to be i n s e n s i t i v e to o v e r e s t i m a t i o n of L 2 , e s p e c i a l l y f o r smooth 133 s e a f l o o r . The combination of these two s e t s of r e s u l t s p r o v i d e d a v i a b l e way of d e a l i n g with the problem of e s t i m a t i n g the s e a f l o o r c o r r e l a t i o n d i s t a n c e . To estimate the optimum Q and R, e i t h e r h e u r i s t i c or ad a p t i v e methods c o u l d be used, the l a t t e r y i e l d i n g s l i g h t l y b e t t e r r e s u l t s i n f u l l a d a p t a t i o n mode. There were s e v e r a l parameters inherent to the i n n o v a t i o n a d a p t i v e a l g o r i t h m : NBATCH, ijq and N. Performance improved as e i t h e r one or combinations of these were i n c r e a s e d . The a d d i t i o n a l computations r e q u i r e d with a d a p t a t i o n was modest, i n the order of 2 or 3 times, with the main i n c r e a s e being i n storage r e q u i r e d . 7.2 E r r o r Sources S e v e r a l e r r o r sources can be i d e n t i f i e d , common to a l l f i l t e r s : a) The approximation e r r o r c d 2 i n the measurement formula. For t y p i c a l , moderate magnitudes of random motions, t h i s was shown to be < 0.01 at sea deeper than 160m. b) For Kalman F i l t e r , the approximation e r r o r e m i n choosing a low-order t r a n s f e r f u n c t i o n to approximate the spectrum S n(a>). The e r r o r has been shown to be i n c r e a s i n g with l a r g e wind speed, and of l a r g e r magnitudes f o r s m a l l s h i p . c) F i l t e r i n g e r r o r s due to c h o i c e of Q, R through h e u r i s t i c or a d a p t i v e means, which were suboptimal to begin with. d) For RLSI F i l t e r , the f i l t e r i n g e r r o r was due mainly to i n t e r p o l a t i n g e r r o r s . For LP F i l t e r t h i s was due mainly to 1 34 the t r a n s i t i o n bandwidth. Furthermore, a p o s s i b l e source of e r r o r i s i n s t r u m e n t a l measurement e r r o r i n o b t a i n i n g z ( t ) , which has not been i n c l u d e d here. T h i s can be e a s i l y modelled as an a d d i t i o n a l v a r i a b l e i n v k , white Gaussian in nature. 7 . 3 Suggestions f o r F u r t h e r Research By now i t i s obvious that the r e s t o r a t i o n problem i n v o l v e s many f a c t o r s , a l l of which r e q u i r e a c c u r a t e m o d e l l i n g . In t h i s t h e s i s the most r e l e v a n t f a c t o r s were i d e n t i f i e d and i s o l a t e d , and the r e s e a r c h were c a r r i e d out based on some p r e s c r i b e d assumptions on these parameters. Thus, f o r example, to l i m i t mode of motions to only heave and p i t c h , and to v a l i d a t e the measurement formula, the assumption of deep, f u l l y developed sea was made. Within the scope of t h i s t h e s i s , the tasks set out e a r l i e r has been accomplished, namely, more a c c u r a t e m o d e l l i n g of the measurement formula and i n v e s t i g a t i o n i n t o e f f e c t s of v a r y i n g system parameters, and comparison of d i f f e r e n t f i l t e r i n g a l g o r i t h m s . However, there are c l e a r l y room f o r f u r t h e r r e s e a r c h , which i n c l u d e s : Higher order m o d e l l i n g of S n ( c o ) , and comparing the performance of the r e s u l t i n g high-order Kalman F i l t e r to the other f i l t e r s . - F u r t h e r work on m o d e l l i n g of the s e a f l o o r i s l i k e l y to a r r i v e at a more a c c u r a t e , but more complex s p a t i a l spectrum. T h i s may r e q u i r e a d i f f e r e n t form of the Kalman F i l t e r 135 I n v e s t i g a t i o n of n o n - s t a t i o n a r y c o n d i t i o n s , which may be due to v a r y i n g u, L 2 , i>, e t c . Besides c a l l i n g f o r new techniques, these c o n d i t i o n s may a l s o a l t e r the nature of the measurement formula i t s e l f . For example, v a r y i n g wind speed probably i m p l i e s a d e v e l o p i n g storm, and t h e r f o r e r e q u i r e s a two-dimensional spectrum. V a r y i n g L 2 may r e s u l t i n v a r y i n g T c ; consequently the measurement formula becomes n o n l i n e a r . Some p o s s i b l e f i l t e r i n g a l g o r i t h m s have been proposed : the adaptive ARMA F i l t e r , Extended Kalman F i l t e r and S e l f - T u n i n g F i l t e r . Experimental works are needed to compare t h e i r performances. - Slamming has been mentioned as a d i s t i n c t l y n o n - s t a t i o n a r y d i s t u r b a n c e , and occurs commonly with s m a l l e r s h i p s . Again, t h i s c a l l s f o r d i f f e r e n t f i l t e r i n g s t r a t e g y . Ship a n a l y s i s assumes l i n e a r s h i p response, but t h i s may not be true f o r shallow water [56]. In such case, higher order spectrum i s r e q u i r e d , as w e l l as a r e d e f i n i t i o n of the measurement formula. Thus, f u r t h e r reaearch would be d i r e c t e d toward a broadening set of m o d e l l i n g parameters, as w e l l as i n v e s t i g a t i o n i n t o the e f f e c t s of n o n - s t a t i o n a r i t y and the a p p r o p r i a t e f i l t e r i n g a l g o r i t h m s to deal with these problems. 136 BIBLIOGRAPHY [1] A k a l , T . , " A c o u s t i c a l C h a r a c t e r i s t i c s of the S e a f l o o r " , i n Physics of Sound in Marine Sediments (ed. L. Hampton), Plenum Press, NY, 1974,pp.447-480. [2] Bartram,J.F. and R.R.Ramseyer,"Fifth Generation D i g i t a l Sonar S i g n a l Processing",IEEE J.Oceanic Engineering OE-2, No.4, October 1977, pp.337-343. [3] Belanger,P.B.,"Estimation of Noise Covariance M a t r i c e s f o r a L i n e a r Time-Varying S t o c h a s t i c Process", Automatica 10 1974, pp.267-275. [4] Berkson,J.M. and J . E . M a t t h e w s , " S t a t i s t i c a l C h a r a c t e r i s t i c s of S e a f l o o r Roughness", IEEE J.Oceanic Engineering OE-9,No.1, January 1984, pp.48-51. [5] Bryson, A.E. and L . J . He n r i c k s o n , " E s t i m a t i o n u s i n g sampled data c o n t a i n i n g S e q u e n t i a l l y C o r r e l a t e d Noise", /. Space cr af t and Rockets 5, No. 6,June 1968, pp.662-665. [6] Chamberlain,S.G. and J.C. G a l l i , " A Model f o r Numerical S i m u l a t i o n of Nonstationary Sonar Reverberation using L i n e a r S p e c t r a l P r e d i c t i o n " , IEEE J.Oceanic Engineering OE-8, No.1, January 1983, pp.21-36. [7] Chapman,N.R., and I.Barrodale,"Deconvolution of Marine Seismic Data using the 1,-norm", Geophys. J.Roy.Aslr on.Soc. ,72, 1983, pp.93-100. [8] Chen, Y.H.,"Ship V i b r a t i o n s i n Random Seas", /. Ship Research 24 , No.3, September 1980, pp.156-169. [9] Chin,L.,"Advances i n Computational E f f i c i e n c i e s of L i n e a r F i l t e r i n g " , i n Advances in the Techniques and Technology of the Appl i cat i on of Nonlinear F i l t e r s and Kalman F i l t e r s (ed. C.T.Leondes), NATO AGARDograph No.256, 1982, pp.4-1 to 4-29. [10] C i o f f i , J . M . , a n d T . K a i l a t h , " F a s t , R e c u r s i v e - L e a s t - S q u a r e s T r a n s v e r s a l F i l t e r f o r Adaptive F i l t e r " , I E E E Trans. Acousti cs, Speech, and Signal Processing ASSP-32, No.2, A p r i l 1984, pp.304-337. 1 37 [11] Clay,C.S. rand W.K.Leong,"Acoustical Estimates of the Topology and Roughness Spectrum of the S e a f l o o r southwest of I b e r i a n P e n i n s u l a " , i n Physics of Sound in Marine Sediments (ed. L.Hampton), Plenum Press, New York, 1974, pp.373-446. [12] DeSouza,C.,and J.M.Fernandez,"Convergence P r o p e r t i e s of an Optimal F i l t e r Design with C o r r e l a t e d Output Noise", IEEE Trans. Automatic Control AC-30, No.5, May 1985, pp.487-488. [13] El-Hawary,F.,"Compensation of Source Heave by use of a Kalman F i l t e r " , IEEE J.Oceanic Engi neer i ng O E - 7 , No.2, A p r i l 1982, pp.89-96. [14] Friedlander,B.,"System I d e n t i f i c a t i o n Techniques f o r Adaptive S i g n a l P r o c e s s i n g " , I E E E Trans. Acoustics, Speech, and Signal Processi ng A S S P - 3 0 , No.2, A p r i l 1982, pp.240-246. [15] F r i e d l a n d e r , B . , " A Recursive Maximum L i k e l i h o o d A l g o r i t h m f o r ARMA L i n e Enhancement", IEEE Trans. Acoustics, Speech, and Signal Processing ASSP-30,No.4,August 1982,pp.651-657. [16] Gelb,A. , A p p l i ed Optimal Estimation,MIT Press, Cambridge, 1974. [17] Goodwin,G.C.,and K.S.Sin, Adaptive Filtering, Prediction and Control, P r e n t i c e - H a l l , E n g l e w o o d C l i f f s , 1984. [18] Godbole,S.S.,"Kalman F i l t e r i n g with no A P r i o r i I nformation about Noise - White Noise Case : I d e n t i f i c a t i o n of C o v a r i a n c e s " , IEEE Trans. Automatic Control A C - 1 9 ,October 1974, pp.561-563. [19] Guthrie,K.M.,"A Wave Theory of SOFAR S i g n a l Shape", J.Acoustic Soc.Am. 56, No.3, September 1974, pp.827-836. [20] Gylys,V.B.,"Design of Real-Time E s t i m a t i o n A l g o r i t h m f o r Implementation i n M i c r o p r o c e s s o r and D i s t r i b u t e d P r o c e s s i n g Systems", i n Advances in the Techniques and Technology of the Application of Nonlinear Filters and Kalman Filters (ed. C.T.Leondes), NATO AGARDograph No.256, 1982, pp.5-1 to 5-34. 138 [21] Halamandaris,H. and D.Ozdes,"A Kalman F i l t e r Augmented Marine N a v i g a t i o n System", i n Theory and Applications of Kalman Filtering (ed. C.T.Leondes), NATO AGARDograph No.139, 1970, pp.517-537. [22] Hampton,R.T.L.,"Stochastic A l g o r i t h m f o r S e l f - A d a p t i v e F i l t e r i n g and P r e d i c t i o n " , NASA Semi-Annual Report 03-002-006, p a r t A, January 1971. [23] Hertz,D.,"A Fast D i g i t a l Method of e s t i m a t i n g the A u t o c o r r e l a t i o n of a Gaussian S t a t i o n a r y Process", I E E E Trans. Acoustics, Speech, and Signal Processing ASSP-30,No.2,April 1982, p.329. [24] Kaila t h , T . , " T h e Innovation Approach to D e t e c t i o n and E s t i m a t i o n Theory", Proc. IEEE 58 ,May 1970, pp.680-695. [25] Kitagawa,G., and W.Gersch,"A Smoothness P r i o r s Time-varying AR C o e f f i c i e n t Modeling of N o n s t a t i o n a r y Covariance Time S e r i e s I E E E Trans. Automatic Control AC-30,No.1, Jaunary 1985, pp.48-56. [26] Lewis,E.V.,"The Motion of Ships i n Waves", i n Principles of Naval Architecture (ed. J.P.Comstock),SNAME, New York, 1967, pp.607-717. [27] L j u n g , L . , " A n a l y s i s of General Recursive P r e d i c t i o n E r r o r I d e n t i f i c a t i o n A l g o r i t h m " , Automatica 17, No.1,1981,pp.89-99. [28] Maybeck,P.S. ,St ochasti c Models, Estimation and Control, v o l s . I , I I , Academic Press, New York, 1979/82. [29] Mehra,R.K.,"On the I d e n t i f i c a t i o n of V a r i a n c e s and Adaptive Kalman F i l t e r i n g " , IEEE Trans. Automatic Control AC-15, No.2, A p r i l 1970,pp.175-184. [30] Mehra,R.K.,"Approaches to Adaptive F i l t e r i n g " , I E E E Trans. Automatic Control AC-17, October 1972, pp.693-698. [31] Mendel,J.M.,"Computational Requirements f o r a D i s c r e t e Kalman F i l t e r " , IEEE Trans. Automatic Control AC-16, No. 6, December 1972, pp.748-758. 139 [32] M i t c h e l l , R . L . , a n d D.A.McPherson,"Generating Nonstationary Random Sequences",IEEE Trans. Aerospace and E l e c t r o n i c Systems AES-17 , No.4, J u l y 1981, pp.553-560. [33] M o i r , T . J . , and M.J.Grimble,"Optimal S e l f - T u n i n g F i l t e r i n g , P r e d i c t i o n , and Smoothing f o r D i s c r e t e M u l t i v a r i a b l e Processes", IEEE Trans. Automatic Control AC-22,No.2, February 1984, pp.128-137. [34] Monden,Y.,T.Komatsu,S.Arimoto, " S t a t i s t i c a l Design of Nearly Linear-Phase ARMA F i l t e r s " , IEEE Trans. Acoust i cs, Speech, and Signal Processi ng ASSP-32, No.5, October 1984, pp.1097-1100. [35] M u l l i s , C . T . , and R.A.Roberts,"The Use of Second Order Information i n the Approximation of Discrete-Time L i n e a r Systems",IEEE Trans. Acoustics, Speech, and Signal Processi ng ASSP-24,No.3, June 1976, pp.226-238. [36] Nelson,L.W., and E.Stear, "The Simultaneous On-Line E s t i m a t i o n of Parameters and S t a t e s i n L i n e a r Systems", IEEE Trans. Automatic Control AC-21, February 1976, pp.94-98. [37] Newman,J.N., Marine Hydrodynamics , MIT Press, Cambridge, 1978. [38] Ogata,K.,St ate Space Analysis of Control Systems, P r e n t i c e - H a l l , Englewood C l i f f s , 1967. [39] Osborne,A.R.,"The S i m u l a t i o n and Measurement of Random Ocean Wave S t a t i s t i c s " , in Proc. International School of Physics : Topics in Ocean Physics (ed. A.R.Osborne, P . M . R i z z o l i ) , N o r t h - h o l l a n d , Amsterdam, 1982. [40] Panuska,V.,"A New Form of the Extended Kalman F i l t e r f o r Parameter E s t i m a t i o n i n L i n e a r System with C o r r e l a t e d Noise", IEEE Trans. Automatic Control AC-25,No.2, A p r i l 1980, pp.229-235. [41] P h i l l i p s , O . M . , T h e Dynamics of the Upper Ocean , Cambridge U n i v e r s i t y Press, 1977. [42] P i e r s o n , W.J.,and L.Moskowitz,"A proposed S p e c t r a l Form f o r F u l l y Developed Wind Seas based on the S i m i l a r i t y 140 Theory of S . A . K i t a i g o r o d s k i i " , J. Geophysical Research, 69,No.24, December 1964, pp.5181-5190. [43] Prado,G.,"Optimal E s t i m a t i o n of Ship's A t t i t u d e s and A t t i t u d e Rates", IEEE J.Oceanic Engineering OE-4,No.2,April 1979, pp.52-59. [44] Price,W.G.,and R.E.D.B i shop probabilistic Theory of Ship Dynami cs , Chapman and H a l l , London, 1974. [45] Rabiner,L.R.,R.W.Schafer,and D.Dlugos, " C o r r e l a t i o n Method f o r Power Spectrum E s t i m a t i o n " , i n Programs for Digital Signal Processi ng , IEEE Press, New York, 1979, pp.2.2-1 to 2.2-14. [46] Rao,D.B.,and S.Y.Kung, "Adaptive Notch F i l t e r i n g f o r the R e t r i e v a l of S i n u s o i d s i n Noise", IEEE Trans. Acoust i cs, Speech, and Signal Processing ASSP-32,No.4,August 1984, pp.791-802. [47] Salvesen,N.,E.O.Tuck,and O . F a l t i n s e n , "Ship Motions and Sea Loads", Trans. SNAME 78, 1970, pp.250-279. [48] Scharf,L.L.,and J . C . L u b y , " S t a t i s t i c a l Design of Au t o g r e s s i v e Moving Average D i g i t a l F i l t e r s " , IEEE Trans. Acoustics, Speech, and Signal Processing ASSP-27 ,No.3, June 1979, pp.240-247. [49] Sidar,M.M.,and B.F.Doolin,"On the F e a s i b i l i t y of Real-Time P r e d i c t i o n of A i r c r a f t C a r r i e r Motion at Sea", IEEE Trans. Automatic Control AC-28 , No.3, March 1983, pp.350-355. [50] S6derstrom,T.,L.Ljung,and I.Gustavsson,"A T h e o r e t i c a l A n a l y s i s of Recursive I d e n t i f i c a t i o n Methods", Automatica 14, 1978, pp.231-244. [51] Triantafyllou,M.S.,M.Bodson,and M.Athans,"Real Time E s t i m a t i o n of Ship Motions using F i l t e r i n g Techniques", IEEE J.Oceanic Engineering OE-8 , No. 1 , January 1983, pp.9-20. [52] Ulrych,T.J.,and M.Ooe, "A u t o r e g r e s s i v e and Mixed A u t o r e g r e s s i v e Moving Average Models and S p e c t r a " , i n Nonlinear Methods of Spectral Analysis (ed. S.S. Haykin), S p r i n g e r - V e r l a g , Berlin,- 1979, pp.73-125. 141 [53] van S c h o o n e v e l d , C . , " D i g i t a l Logarithmic N o r m a l i z a t i o n of Sonar S i g n a l s : S e r i a l P r o c e s s i n g " , i n Signal Processing ( e d . J . W . R . G r i f f i t h s , P . L . S t o c k l i n , and C.van Schooneveld), Academic P r e s s , London/New York, 1973, pp.671-688. [54] Weiss,I.M.,and T.W.Devries,"Ship Motion Measurement F i l t e r Design", IEEE J.Oceanic Engi neeri ng OE-2,No.4,October 1977, pp.325-330. [55] Widrow,B.,J.R.Glover,Jr.,et a l , " A d a p t i v e Noise C a n c e l l i n g : P r i n c i p l e s and A p p l i c a t i o n s " , Proc.IEEE 63,No.12, December 1975, pp.1692-1716. [56] Yamanouchi,Y.,"Nonlinear Response of Ships on the Sea", i n Stochastic Problems in Dynamics (ed. B,L.Clarkson), Pitman, London, 1971. [57] Zohar,S."Fortran Subroutines f o r the S o l u t i o n of T o e p l i t z Sets of L i n e a r Equations", IEEE Trans. Acoustics, Speech, and Signal Proces si ng ASSP-27,No.6,December 1979, pp.656-658. 142 APPENDIX A : Sea and Heave Spe c t r a RESPONSE- AMPLITUDE OPERATOR L a r q e S h i p o ~ o f (Hz) RESPONSE AMPLITUDE OPERATOR S P E C T R U M 0 Smal l S h i p  gure A 1 . 1 F i g u r e A 1 . 2 HEAVE S P E C T R U M M E A S U R E M E N T E R R O R S P E C T R U M Wind Speed D = 8 0= 10 A = 12 V = 14 0.2 0.3 0.4 (a) f(Hz) HEAVE S P E C T R U M (b) '(Hz) M E A S U R E M E N T E R R O R S P E C T R U M Encounter Angle 0 = 0 0 = 40 A = 80 V = 120 (c) 0.2 0.3 Encounter Angle • = 0 O = 40 A = 80 V = 120 02 0.3 U.4 f ( H z ) it* i t * Figure A 1 . 3 Large Ship o C 9 *Ar O HEAVE SPECTRUM Wind Speed • = 8 O = 10 A = 12 V = 14 MEASUREMENT ERROR SPECTRUM (a) (b) f(H Z) 0.4 HEAVE SPECTRUM Encounter Angle 0 = 0 0 = 40 A = 80 V = 120 0.3 0.4 MEASUREMENT ERROR SPECTRUM (d) Encounter Angle • = 0 0 = 40 A = 80 V = 120 f(Hz) 0.3 0.4 cn F i g u r e A1 . 4 Small Ship 1 46 APPENDIX B : Kalman Algorithms B.1 Optimal F i l t e r i n g f o r Model S 2 The measurement v e c t o r £ k i n s i g n a l model S 2 i s given by *k= z k+1~* z k (B1.1) Ac c o r d i n g to (3.31b), £ k a l s o equals H'x k+v k'. Thus, £ k - H ' x k - v k ' = 0 (B1.2) A d j o i n i n g (B1.1) t o (3.31a), x k + 1=*x k+rw k + D U k-H'x k-v k' ] =(*-DH')x k+D^ k+Tw k-Dv k' (B1.3) where D i s a matrix to be determined. (B1.3) can be v i s u a l i z e d as an a l t e r n a t i v e form of the s t a t e e q u a t i o n , with (B1.2) the measurement e q u a t i o n . The process d r i v i n g n o i s e f o r t h i s model i s rw k-Dv k', which i s c o r r e l a t e d with v k ' with c o v a r i a n c e E [(Tw ( k)-Dv'(1))v'(l ) T ] = ( r S-DR')6 ( k-l) To convert the s t a t e model to one with u n c o r r e l a t e d d r i v i n g n o i s e s , choose D t o be D=rSR'~ 1 (B1.4) From c o n v e n t i o n a l Kalman F i l t e r e quations, the update equation i s giv e n by x k | k = x k | k - 1 + K k U k - H ' x k | k _ 1 ) (B1.5) where K k, the Kalman Gain, i s g i v e n by K k=P kH' TR'- 1=M kH , T(H ,M kH , T+R') _ 1 (B1.6) The propaga t i o n equation i s given by xk+1 | k = * x k | k + D U k - H ' x k | k ] (B1.7) P k i s the c o v a r i a n c e matrix of the e r r o r i n e s t i m a t i n g x k , i . e . , P k = E [ ( x k | k - x k ) ( x k | k - x k ) T ] 1 47 = E [ e k e k T ] But e k = x k j k _ 1 - x k + x k | k - x k | k _ 1 =xk| k-1 _ x k + R kUir H , *k|k-1 > = x k|k-1- x k +Kk( v 'k- H '< sk|k-1-x k> > = ( I - K k H ) ( x k | k . 1 - x k ) + K k v ' k H e n c e , P k = E [ e k e k T ] = ( I - K k H ' ) M k ( I - K k H ' ) T + K k R ' K k T where M k + 1 = E [ ( x k | k _ , - x k ) ( x k | k _ , - x k ) T ] =(*-DH*)Pk(*-DH')T+rQrT-DR'DT (B1.9) (B1.10) (B1 .1 1 ) The update and-propagation equations may be combined to gi v e x k + 1 | k = * < S k | k _ 1 + K k r k ) + D [ ^ k ~ H ' ( x k | k _ 1 + K k r k ) ] =*x k| k_ 1+(*K k-DH'K k)r k+D[^ k-H'x k| k. 1] =* xk|k-1 +(*K k+D-DH'K k)r k =4>x k| k_i+(D+(*-DH')K k)r k =* xkjk-1 +Gk rk where G k=D+(*-DH')K k i s the p r e d i c t i o n g a i n . (B1.12) (B1.10) through (B1.12) g i v e s the measurement-differencing Kalman F i l t e r equations,as quoted i n the set of equations(3.33) For a second-order system, c e r t a i n s i m p l i f i c a t i o n s can be made. * and T are both 2x2 m a t r i c e s . Let * and T be r= 011 ^12 021 <>22 t , 0 0 t 2 (B1.13) (B1.14) Since t h e r e i s only a s i n g l e measurement, £, v' R' and * are a l l s c a l a r s . H and * are giv e n by H=(0 c) (B1.15) 148 *=(exp(-vA/L 2)) Let R = ( r 1 ) , and Q be d i a g o n a l , given by (Bi.16) Q= Q1 0 0 q 2 (B1.17) Then, the parameters r e q u i r e d i n the f i l t e r equations are given by H'=H<t>-*H=(hn h l 2 ) (B1.18) with hi i =c<i>21 h , 2 = c ( 0 2 2 - e x p ( - v A / L 2 ) ) R'=R+ H r Q ( H D T=r 1+q 2c 2t 2 2 (B1.19) S=Q(HDT= 0 q 2 c t 2 D = T S R '-1 = q 2 c t 2 2 / R ' and the i n n o v a t i o n sequence r=£-H'x={-h,,x,-h, 2x 2. (B1.20) (B1.21 ) (B1.22) 1 49 B.2 Innovation C o r r e l a t i o n A l g o r i t h m D e f i n e p r e d i c t i o n e r r o r e k as e k = x k _ x k | k - 1 (B2.1) with c o v a r i a n c e M k. Def i n e i n n o v a t i o n sequence r k as rk=*k-H' xk|k-1=H'e k+v k' (B2.2) Using (B1.12) ek+1 = xk+1~ xk+1|k =* xk + r wk-* xk|k-1-Gk(H'e k+v k') =(*-G kH')e k+rw k-G kv k' (B2.3) Taking k steps back from i , e i = [ * - G k H ' ] k e i _ k - I [ * - G k H ' ] D " 1 G k V i _ j ' + I [ * - G k H ' ] D ~ 1 r w i _ j (B2.4) Let C k be the c o r r e l a t i o n of i n n o v a t i o n r k , C k = E [ r i r i _ k T ] = E [ ( H ' e i + V i ' ) ( H ' e i _ k + v i _ k ' ) T ] = H , E [ e i e i _ k T ] H ' T + H ' E [ e i v i _ k ' T ] (B2.5) But E [ e i e i _ k T ] = ( * - G k H ' ) k M k and E [ e i v i _ k , T ] = ( * - G k H ' ) k _ 1 ( r s - G k R ' ) Hence, C k=H'(*-G kH') kM kH' T+ H'(*-G kH') k " 1 (rs-G kR') =H'(*-G kH*) k _ 1(*-G kH f)M kH , T+ H'(*-G kH') k _ 1(rS-G kR') -H*(*-G kH') k" 1[*M kH , T-G kH'M kH , T +rS-G kR'] =H'(*-G kH') k _ 1[*M kH , T +rS-G k(H ,M kH' T+R')] =H'(*-G kH') k~ 1[*M kH , T+rS-G kC 0] (B2.6) where C 0=H'M kH' T+R'=E[ r i r i _ k T ] (B2.7) (From (B2.2)) Note t h a t as a consequence of i n n o v a t i o n p r o p e r t y , optimal G k should make C k -*> 0, Vk# 0. T h i s o p t i m a l G k i s give n by G k=C 0 _ 1(*M kH , T +rS ) (B2.8) 150 Pr o o f : From (B1.12), G k=D+($-DH')K k =D(I-H'K k)+*K k =D (I -H' MkH' T C o - 1 ) +* M kH' T ^ - 1 =D(C 0" 1(C D-H'M kH' T))+*M kH' TC 0 _ 1 =rSR'" 1(C 0" 1R*)+*M kH' TC 0~ 1 =C 0 _ 1(*M kH' T+rs) If f i l t e r i s suboptimal, then C k#0 i n (B2.6). (B2.6) may then be w r i t t e n e x p l i c i t l y i n terms of the e m p i r i c a l C k : C i i c i N H'(#-G kH' H'(*>-GkH' ) i 2 _ 1 * H ' ( * - G k H ' ) l N " 1 * (M kH' T+*" 1rS-#" 1G kC 0) Defi n e A=MH'T+*"1TS Then, A=A# 11 Ci2 piN_ From (B2.7) R =C 0-H'M kH' T =C 0-H*A+H ' *" 1rS From (B2.3) M = E [ e k + 1 e k + 1 T ] =(*-G kH')M k(*-G kH- ) T +rQr T+G kR fG k T-rsG k T-(rsG k T)T =$M k* T-G kH'M k* T-*M kH' TG k T +G kH'M kH' TG k T+G kR'G k T-G k(TS)T -rs'G k T +rQr T =*Mk*T+rQrT+£2 where 0=G kC oG k T-G kA T* T-*AG k T (B2 . 9 ) (B2.10) (B2.1 1 ) (B2.12) (B2.13) 151 S u b s t i t u t i n g M back i n t o (B2.11), M = * 2 M k ( * 2 ) T + * n * T + n + * r Q r T * T + r Q r T Repeating n times and s e p a r a t i n g terms i n v o l v i n g Q on L.H.S., "£$ J rGTT ( * J ) T=M k -*kMk (*k ) T - I * J 0 ( * 3 ) T k=1 ,.. , n P r e m u l t i p l y both s i d e s by H' and p o s t m u l t i p l y by (<i>~k)TH'T, we have I H ' * 3 r Q r T ( * 3 - k ) T H . T = H'M k(*- k) TH' T-H'<I> kM kH' T -EH' * 3 n ( * 3 ~ k ) TH' T (B2.14) From (B2.9), MkH'T=A-4>~1 TS H'M k=(A -*~ 1 .rS) T S u b s t i t u t e i n (B2.14) Z H ' * 3 r Q r T ( $ 3 - k ) T H ' T + S TrT(*~k-1 ) T H ' T _ H ^ k - 1 r s 7=0 = A T ( * - k ) T H ' T - H ' * k A - l H ' * J n ( * J - k ) T H ' T k=1 ,.. ,n (B2.15) In the case of n=2, two equations may be w r i t t e n : For k=1, H ' r Q ( H , * " 1 r ) T + H r Q T ( H , * " 2 r ) T - H T Q ( H D T = A T ( * _ 1 ) T H T - H ' * A - H ' n * - 1 H ' T (B2.16) ( H T ) Q ( ( H ' * - 1 r ) T - ( H r ) T ) + ( H r ) Q ( H ' * " 2 r ) T = ( A T - H ' n ) ( H ' * ~ 1 ) T - H ' * A (hut, h l 2 t 2 ) qi o >1~ 0 q 2 F 2 + (0 c t 2 ) q, 0 0 q 2 F3 F 4 = b i (B2.17) hut] qi Pi + h l 2 t 2 q 2 F 2 + c t 2 q 2 F 4 =b. For k=2, H ' r Q r T ( * - 2 ) T H ' T + H ' * r Q r T ( * - 1 ) T H ' T + H r Q T r T ( # - 3 ) T - H ' * r Q ( H D T = A T ( # - 2 ) T H T - H ' * 2 A - H ' n ( * - 2 ) T H ' T - H ' * n ( * - 1 ) T H ' T (B2.18) 1 52 (hut} h 1 2 t 2 ) Ql O" > 5 0 q 2 . F 6 (0 c t 2 ) Q1 0 0 q 2 11 12 + ( F 7 F 8 ) - < F7 F 8 ) <3i 0 Ql 0 0 q 2 q 2 0 ]=b c t -F 9 F 1 0 2 h i l t , q , F 5 + h l 2 t 2 q 2 F 6 + F 7 F g q i + F 8 F l 0 q 2 + c t 2 q 2 F 1 2 + c t 2 q 2 F 8 = b 2 ( B 2 . 1 9 ) T h u s t h e L . H . S . o f t h e e q u a t i o n s a r e l i n e a r i n (q^ , q 2 ) T , and t h e R . H . S . d e p e n d s on A a n d Q o n l y . C o m b i n i n g ( B 2 . 1 7 ) and ( B 2 . 1 9 ) , a n d w r i t i n g i n m a t r i x f o r m , h 1 1fc1 F 1 h l l t i F5 + F 7 F 9 h l 2 t 2 F 2 + c t 2 F 4 h l 2 t 2 F 6 + F 8 F 1 0 + c t 2 ( F 1 2 + F 8 ) <3l Q2 o r , H e n c e , <31 .52. P31" 52 = 0 ( B 2 . 2 0 ) ( B 2 . 2 1 ) A l s o , f r o m ( B 2 . 7 ) E [ r i r i - K T ] = H ' M k H , T + R ' S t a t i s t i c a l h y p o t h e s i s t e s t i n g i n d i c a t e s t h a t a good c h o i c e o f L i k e l i h o o d F u n c t i o n f o r d e t e c t i n g e s t i m a t i o n f a i l u r e w o u l d be i n t h e f o r m o f sum o f n a t u r a l l o g s o f c o n d i t i o n a l d e n s i t i e s f o r c o m p o n e n t s o f t h e i n n o v a t i o n s L ( t k ) d L * 7 n f r ^ i ) | r ( t / - 1 ) , . . , r ( t z ) ( P i | P i - i , . . , P i > ( B 2 . 2 2 ) S u p p o s e t h e i n n o v a t i o n s e q u e n c e c o n s i s t s o f i n d e p e n d e n t , z e r o - m e a n G a u s s i a n random v a r i a b l e s , t h e c o n d i t i o n a l d e n s i t y c a n be d e f i n e d a s 1 53 f < Pi I Pi-1 Pl> = " k 2 ( t i ) -i • • i 2 ( t i ) (B2.23) where 0)c 2(tj.) * s the estimate of the v a r i a n c e of in n o v a t i o n Using t h i s , and s u b s t i t u t i n g (B2.23) i n t o (B2.22), one ob t a i n s the form of L i k e l i h o o d F u n c t i o n given i n (3.46). Thus, i f r k 2 ( t i ) becomes c o n s i s t e n t l y l a r g e r than that p r e d i c t e d by a k 2 ( t i ) over the most recent N samples, L ( t k ) w i l l become more and more ne g a t i v e , i n d i c a t i n g divergence of f i l t e r . On the other hand, c o n s i s t e n t l y i n c r e a s i n g L ( t k ) i n d i c a t e s improvement, s i n c e r k 2 ( t i ) must be g e t t i n g s m a l l e r . given no f a i l u r e has occurred, and i s equal to H'MkH'T+R*. 1 54 APPENDIX C : Design A l g o r i t h m of L i n e a r Phase ARMA P r e - F i l t e r Let S n(cj) be the measurement noise spectrum. F i r s t , o b t a i n the corresponding c o v a r i a n c e sequence {c k} by DFT. Let H j ( z ) be the AR F i l t e r of order n a , whose p e r i o d i c spectrum approximates Sn(a>) : K K i . e . , design H, (z) = — ^ — = 1 - I a i z 1 G ( z ) such that r ^ e c j ^ , k=0 , ± 1 , . . . ,±n. ( C . 1 ) ( C . 2 ) The f i l t e r c o e f f i c i e n t s are obtained by s o l v i n g the normal equations Ra=c (C.3) where R= c 0 Ci .... c n a_ 1 C, CG .... C n a _2 cna - 1 c=(C,,C2,.. a=(a,, a 2 , . C ) T na *na> T and K2=Cn-L a\C\ w 1*1 A A (C.3a) (C.3b) (C.3c) (C.3d) The matrix R i s T o e p l i t z and the equation (C.3a) may be so l v e d e f f i c i e n t l y by v a r i o u s f a s t a l g o r i t h m s , such as the Levinson-Durbin a l g o r i t h m , or the Zohar a l g o r i t h m [ 5 7 ] . A F o r t r a n s u b r o u t i n e implementing the l a t t e r i s i n c l u d e d i n Appendix F. The u n i t impulse response h k of H(z) may be generated r e c u r s i v e l y from a k as 155 h u = 0 n a «6k+Z a^h^-^ k < 0 k > 0 (C . 4 ) The f i l t e r G(z) i s the whitening f i l t e r . However, s i n c e n a i s u s u a l l y very l a r g e , i t w i l l be i n e f f i c i e n t to implement G(z) d i r e c t l y . T h e r e f o r e , a much lower order ARMA(m,n) f i l t e r i s designed to approximate AR(n a) f i l t e r . Let H 2 ( z ) be the t r a n s f e r f u n c t i o n of the time-delayed ARMA f i l t e r , and l e t N0=m+n be the o v e r a l l order : H 2 ( z ) = Z b k z Aero * - k Ia-;z~3 j'i J - T B(z) A(z) The c o e f f i c i e n t s {aj}, {b k} are obt a i n e d by mini m i z i n g e m n=(1/2ir)J^|A(e3^) H(e3«) - e 3 ™ B(e>>) | 2dw which leads to the f o l l o w i n g normal equations (C.5) (C.6) ( 1 , a 1 , . . . , a m ; b 0 , b i , . . . , b n ) Rm hT HT xn+1 ( C 7 ) m and H r denotes the (m+l)x(m+1) and (m+l)x(n+l) where R,T o e p l i t z m a t r i c e s Rnr-•m •m H T = h T . . . h T + m m LT+n-m (C.8) Equation (C.7) may be s o l v e d r e c u r s i v e l y by, f o r example, the M u l l i s - R o b e r t s A l g o r i t h m [35] f o r r=N/2-n and v a r i o u s combinations of (m,n) su b j e c t to the c o n s t r a i n t m+n=NQ. In each case, the corresponding e m n i s c a l c u l a t e d . Then, the one with the lowest e m n i s s e l e c t e d . 1 56 APPENDIX D : S i m u l a t i o n Algorithm, Data and A n a l y s i s The random phase method i s a form of s i m u l a t i o n i n the frequency domain. T h i s method was proposed by T.Borgman and developed by A.R.Osborne[39]. The idea behind the random phase method i s a c t u a l l y q u i t e simple. Let £(t) be the random s e r i e s to be simulated, with spectrum S£(f). Then, given a s u f f i c i e n t l y l a r g e N, £(t) may be approximated by s i m u l a t i o n of N F o u r i e r s e r i e s having random phase 4>n £(t)=£ C n c o s ( c j n t - * n ) (D.I) where 4>n i s u n i f o r m l y d i s t r i b u t e d i n (0 ,2 i r ) cj n=27rf n E q u i v a l e n t l y , {(t)=E | ( a n c o s a j n t + b n s i n o n t ) (D.2) The simulated £(t) w i l l have a spectrum S£(f) i f C n = / 2 S ( f n ) M n (D.3) where Af n=fjj/N f f l , the upper l i m i t of frequency, i s the Nyquist frequency. f N = l / 2 A t , where At i s the p e r i o d of the time r e s o l u t i o n s e r i e s . Thus, the time At depends on the maximum frequency chosen; c o n v e r s e l y , the frequency r e s o l u t i o n A f n depends on the time l i m i t (0,T) of the time s e r i e s . T h e i r r e l a t i o n s are summarised as A t = l / 2 f N (D.4) Af n=1/T (D.5) For l a r g e T, N i s l a r g e . T h e r e f o r e summations i n (D .D are e x t e n s i v e . To a v o i d t h i s , we choose to s e l e c t unequal S ( £ ) 157 0 - i r- f N F i g u r e D.I E q u a l A r e a P a r t i t i o n o f S p e c t r u m S ( f ) f r e q u e n c y i n t e r v a l A f n o f e q u a l a r e a p a r t i t i o n s u n d e r t h e power s p e c t r u m . T h i s way, N c a n be much s m a l l e r ( F i g . D . 1 ) . L e t M 0 be t h e t o t a l a r e a u n d e r S £ ( f ) f r o m 0 t o f N . M 0 = J ^ S U f ) d f (D.6) The e q u a l a r e a p a r t i t i o n A 0 i s t h e n g i v e n by A 0 = M 0 / N (D .7 ) S t a r t i n g f r o m f c = 0 , e a c h f r e q u e n c y b i n w i t h M n i s f o u n d i t e r a t i v e l y u s i n g N e w t o n - R a p h s o n m e t h o d : r*'S*(f ) d f - A 0 SUt) (D.8) f n + 1 = f n + A f n (D.9) The p r o c e d u r e i s r e p e a t e d ( f o r N t i m e s ) u n t i l f N i s r e a c h e d . C n i n ( D . 3 ) i s now a c o n s t a n t ( e q u a l s / 2 A 0 ) , a n d may be moved o u t s i d e t h e summat i on i n (D .1 ) ^ t m ^ C k J , c o s ^ n t n T ^ n ) 0 < " * N, 0 £ m < M, T=MAt (D .10 ) E x p a n d t h e c o s i n e t e r m i n (D .10 ) a s c o s ( w n t m ) c o s 0 n + s i n ( w n t m ) s i r 0 n (D .11 ) L e t a n = w n A . t = 2 i r f n A t (D .12) A n = c o s t > n (D .13 ) 158 Input S(f),M.N.fN.e Compute At.M0.A0 Ini f o = ° tialize n = f D,n = 1 Iteration of fj using (D.8) Generate 0n, 0 « n £ N on (0,2w) n=0, m = 0 Ck=/2\j, I„=0. 0 «J m < M Calculate A n,B a. o B using (D.I2) to (D.I4) \ m +1 = C k[ A ncos(ma n) + Bnsin(m a n ] + \ m m = m + 1 n = n + 1 Yes Stop Figure D.2 Simulation Algorithm 159 Bn=sin</>n (D.14) Then, fmn'^^ni^n" C k [ A n c o s ( m a n ) +B nsin(ma n)] (D.15) Note that £(t) i s obtained by f i r s t summing over a l l m ( i . e . , a c r o s s t i m e ) , then over a l l n ( i . e . , a c r o s s f r e q u e n c y ) . The s i m u l a t i o n procedure i s summarised i n Fig.D.2 The simulated time s e r i e s must be s u b j e c t e d to some v e r i f i c a t i o n s . T h i s i n c l u d e s v e r i f i c a t i o n of s p e c t r a l content, such as comparison of s p e c t r a , and v e r i f i c a t i o n of Gaussian nature of the time s e r i e s . The l a t t e r has been the b a s i c assumption i n both heave and s e a f l o o r p r o f i l e . A convenient way i s to compute a histogram for the simulated r e c o r d s , and c a l c u l a t e c e r t a i n s t a t i s t i c a l parameters, such as mean, standard d e v i a t i o n , and skewness. De f i n e the c e n t r a l moment m r c of a histogram (x^, y^,i=1,..,k), about the mean x by m r c =1 y i ( x i - x ) r / Z y i (D.16) The parameters are d e f i n e d as mean=m1c=M (D.17) standard deviation=m2 C=o (D.18) skewness=m3 C/a 3=s (D.19) kurtosis=m 4 c/a 4=k (D.20) The standard d e v i a t i o n i s r e l a t e d to the s i g n i f i c a n t wave he i g h t H s by H s=4o (D.21) 160 For a zero-mean Gaussian process, the time s e r i e s n ormalized to a has the p r o p e r t i e s u=0, o=1, s=0 and k=3. For sea waves, as p o i n t e d i n S e c t i o n 2.1, n o n l i n e a r behaviour may l e a d t o more peaked c r e s t s , r e s u l t i n g i n k u r t o s i s l a r g e r than 3. In t h i s t h e s i s , the random phase method i s used to simulate both measurement noi s e and s e a f l o o r time s e r i e s . Corresponding to system sampling time At i s chosen to be 0.5s. Thus, ftf=1 H z r which i s s u f f i c i e n t f o r both s e r i e s . A t o t a l of 2400 p o i n t s are generated i n a l l c a s e s ; t h i s corresponds to T=1200s. For power spectrum e s t i m a t i o n , the c o r r e l a t i o n method due to Rabiner, Schafer and Dlugos[45] i s used. Hamming window and FFT s i z e of 1024 are adopted. A 60-bin histogram of the normalized time s e r i e s i s c a l c u l a t e d i n each case, along with the f i r s t t h ree s t a t i s t i c a l parameters, f o r v e r i f i c a t i o n purposes. The program SIMUL i n Appendix F c o n t a i n s the complete software package to a c h i e v e a l l of the above o b j e c t i v e s . F i g u r e s D.3 through D.7 show the r e s u l t s of s i m u l a t i o n of s e a f l o o r p r o f i l e s having c o r r e l a t i o n d i s t a n c e s L2=50m, 100m, 150m, 300m, and 600m. In each case, (a) i s a p l o t of the p r o f i l e , (b) t r a c e s the estimated versus the given spectrum, and (c) i s the histogram of the normalized e l e v a t i o n . Samples of the r e s u l t s f o r measurement e r r o r s e r i e s £ f o r l a r g e and sma l l s h i p s are shown i n F i g u r e s D.8 through D.13. 161 The f o l l o w i n g t a b l e s summarise the s t a t i s t i c a l parameters of the s i m u l a t i o n . Table D.1 S e a f l o o r P r o f i l e S i m u l a t i o n L 2 (m) 0 s 50 -0.0364 0.9236 -0.0001 1 00 -0.0110 0.8372 -0.0002 150 0.0409 0.8882 -0.0001 300 0.2880 0.9283 0.0004 600 0.0678 0.3959 -0.0001 Table D.2 Measurement E r r o r S i m u l a t i o n -- Large Ship u (ms~ 1) u 0 s 9 -0.1335 1 .1202 -1.0616 10 -0.0420 1.0205 -1 .2172 1 1 -0.0725 0.9471 -0.6124 12 0.0974 1.0090 0.6585 1 3 0.0988 1 . 147 0.5007 1 5 -0.1257 0.9721 -0.2691 1 7 0.1075 0.9911 0.1861 Table D.3 Measurement E r r o r S i m u l a t i o n • - Small Ship u (ms~ 1) u 0 s 6 -0.0793 1.0024 -1 .8705 8 0.0220 1 .0081 0.4048 10 -0.0803 1.0032 -0.6131 12 -0.1325 1.0109 -0.4832 13 -0.0947 0.9728 -0.3474 15 -0.1045 0.9801 -0.1784 17 -0.1370 1.0006 -0.2833 162 In a l l cases, the r e s u l t s showed c l o s e s p e c t r a l approximations, p a r t i c u l a r l y f o r measurement e r r o r s i m u l a t i o n s . For s e a f l o o r p r o f i l e s , there were o b v i o u s l y some d i f f i c u l t y f o r the s i m u l a t i o n a l g o r i t h m to generate enough higher frequency data and there were some e x c e s s i v e low-frequency d a t a . In the case of L2=600m the r e v e r s e was t r u e . T h i s may account f o r the low o, and may be due to small data l e n g t h generated. On the other hand, there were higher d e v i a t i o n s i n skewness i n measurement e r r o r s i m u l a t i o n s . T h i s was expected, s i n c e measurement e r r o r i s an o s c i l l a t o r y p r o c e s s , i t i s more s e n s i t i v e to skewing towards p o s i t i v e or negative e l e v a t i o n s . For a l l p r a c t i c a l purposes, we accept these generated data on account of t h e i r c l o s e agreement with the given s p e c t r a l c h a r a c t e r i s t i c s . S e a f l o o r P r o f i l e o= 1 0 . 0 I = 50.0 7 -0.0364 o= 0.9236 s= -0.0001 - ? 0 -2 .0 -1-0 6.0 10 2.0 3.0 Normal ized h F i g u r e D.3 o 7 J O o o u= -0.0110 o= 0.8372 s= -0.0002 n ° CP O . ^ % On °CP UJUUUUUU I I ' 1 1 'X 3 0 -3 .0 -2 .0 - 1 0 0.0 10 2 0 o.u Normalized h Figure D.4 S e a f l o o r P r o f i l e o = 10.0 \= 150.0 0.0 1 200 .0 400 .0 600 .0 t ( s e c ) i 800.0 1000.0 1200 o 6 I o = s = 0.0409 0.8882 -0.0001 O GOD ° _ O ^ c P C D C D O O o i pmrjp i ' -3.0 - 2 . 0 -1.0 0.0 V0 2.0 3.0 N o r m a l i z e d h Figure D.5 S e a f l o o r P r o f i l e o= 10.0 I = 300.0 *o!o ' 2 0 O 0 ' 4oarJ ' eoao ' eoao 1000.0 1260.0 \ (sec) sh ( f ) § r x -f (Hz) to <o -OH a = s = o 0 o 0 octo O Oo - 3 . 0 - 2 . 0 -1.0 0.0 Normal ized h 1.0 0.2880 0.9283 0.0004 °o UUJLUJULJJDa 2.0 3.0 Figure D.6 o o LL — 0 . 0 6 7 8 0 = 0 . 3 9 5 9 s = - 0 . 0 0 0 1 o %°o c, CD o O o o o -mmirjLiJGrjOD i 1 i ' -yo-2.0 - i o 0 0 1 0 2.0 3 N o r m a l i z e d h F i g u r e D.7 c ; 7 S e r i e s L a r g e S h i p u = 11.0 6= 0 . 0 H = 2 . 5 4 0 1 0.0 50.0 100.0 t ( s e c ) 150.0 200 s, 7 (f) o co I "a O I I t> O O -3.0 1 I 1 - 2 . 0 / / = - 0 . 0 7 2 5 0 = 0 . 9 4 7 1 s = - 0 . 6 1 2 4 o QD OOTID -1.0 0.0 N o r m a l i z e d ( z T—' 1.0 2.0 3.0 F i g u r e D.B £ 7 S e r i e s Lorge Ship u- 13 .0 e= o.o H = 3 . 4 9 6 1 « f ( H z ) oo © I ID O I u= 0 . 0 9 8 8 o= 1 .1470 s= 0 . 5 0 0 7 CP cP cP O OJ I T fOO i ' i ' ' 3.0 - 2 . 0 - 1 . 0 0.0 1.0 2.0 3.0 Normal ized ( Figure D.9 S e r i e s Lorge Ship u= 17.0 6= 0.0 H = 4 . 6 3 3 5 50.0 I 100.0 t (sec) 150.0 O I u= 0.1075 0= 0.9911 s= 0.1861 OP**** CO u o O 0 00 o CD CD O a -3.0 -2.0 -V0 0.0 Nor ma Ii zed 1.0 2.0 F i g u r e D.10 0.0 c ; 7 S e r i e s 50.0 100.0 t (sec) 150.0 200.0 s;7 (f) 00 6 I N l O I I li= -0.1325 o= 1.0109 s= -0.4832 CD OP, o w o o oCfcP o p op o 0 o a -3.0 -1—^  -2.0 T --1.0 0.0 Normal ized £. 1.0 - r — * 2.0 3.0 F i g u r e D.11 S e r 100.0 t ( s e c ) 150.0 200 sJ7 (0 oo o -I O I c = s = -0.1045 0.9801 -0.1784 o o ° ° CD O O ° ° o q o o -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 Normal ized £ 2 F i g u r e D .12 £ 7 S e r i e s 100.0 t (sec) 150.0 200.0 s „ (0 o d-oo d-l C l O i °o o o o H= -0.1370 o= 1.0006 s= -0 .2833 Oo o o o -3.0 -2.0 -1.0 0.0 1.0 Normal ized ( 2.0 3.0 Figure D.13 174 APPENDIX E:S i m u l a t i o n R e s u l t s T h i s S e c t i o n groups the performance curves of v a r i o u s f i l t e r s p e r t a i n i n g to d i s c u s s i o n s i n S e c t i o n s 5.3 - 5.5. 0.00 Normal ized rms e r r o r oos o.o O B 020 1 w C n n 3E •—- co c II n — >-t co >-i 3 O I -n — c r < NJ fO II UI ON O O 0 i-h 3 ?! a» a i -— 3 C 0J II 3 CO Tl g 01 l - 1 i rr —m r 3 II — I O o cn 3 3 0) co 3* 73 81 o . 81 0.23 • ' • 0 30 In 8" b 0.00 Normalized rms e r r o r oos OB OS 0.20 i . . . . i . . — ^ 0 25 030 a 0.00 Normal ized rms e r r o r 0.09 o n 0 0 0 M Oi 81 0.23 o 03 C n v—.. 9 (i UI g -PJ • 8. O cu 2 ^ CO c II — >1 —• 1 3 * 3 O b 01 1 I 1 - o - c S r < o b fO II VI CTi o o O M i 3 Pi — Q) cr — 3 C Q» o II 3 b 3 0J I-1 1 rr o -—ft) 5 - M b f 3 N) l Q II » CO 1 o b o r 3 0) M. » • 03 crt Q * 2 to 3* w-TJ Normal ized rms e r r o r ooo oos o n 0.S 0.20 023 5Z.I g So *0 5.0 Botch No 0.0 0.0 200.0 4000 eoo.o t (sec) 8000 noo.o uoo.o Normo l i zed C o r r e l a t i o n C o e f f i c i e n t of r r 11 1 (. 11111 • 1111. r T • 100 200 300 40 0 I og U 500 60.0 F i g u r e E1.2 RMS E r r o r Curves of Adaptive Kalman F i l t e r i N=80, NBATCH=400, Large Ship, u=l3ms~ 1, L2=600m Estimated 01 l i i i i i i 10 2.0 3.0 «-0 i& tO 7.0 Batch No F i g u r e E l . Estimated R — i — 6.0 —I— 7.0 to 2.0 —1— 3.0 Batch No - i — 5.0 8.0 Likelihood Function -e- -B-1 0 s o B o t c h No cont i nued 4.0 50 Botch No 400.0 eoo.o « (sec) BO0 0 WOO.O 1200.0 Norma l i zed C o r r e l a t i o n C o e f f i c i e n t of r, 30.0 400 l o g k 500 60.0 CO F i g u r e E1.3 RMS E r r o r Curves of Adaptive Kalman F i l t e r i N=80, NBATCH=400, Large Ship, u=13ms _ 1, L2=150m Estimated 01 Batch No Estimated 02 Botch No Figure El. 3 Estimated R Likelihood Function 3 3: 34 4 0 5 0 B a t c h No c o n t i n u e d 6 « o •» E If k. o z 8 d 8! 0.0 o o *- o j d-1.0 200.0 400.0 600.0 t (sec) 800.0 WOO.O 1200.0 P e r c e n t a g e Count ^ L im i t -o- -B-i 1 2.0 3.0 4.0 Botch No -r—*• 5.0 I • 6 0 7.0 8.0 3 0.0 # d 3 i 00 200.0 400.0 600.0 t ( s e c ) i • 8000 ! ' noo.o 1200.0 N o r m a l i z e d C o r r e l a t i o n C o e f f i c i e n t of r f c I. | | | H " IIM I 1 WO 200 30.0 400 log l< 30 0 CO o F i g u r e E1.4 RMS E r r o r Curves of Adaptive Kalman F i l t e r i N=80 F NBATCH=400, Large Ship, u=13ms , L2-bUm Estimated QI Batch No Estimated Q2 3-, 3 3 P F i g u r e E l . 4 3n 3 3 3 Estimated R - S -? 3. i 3 t.o 2.0 I 30 *0 i.O Batch No 60 h o : P 3-a «-Likelihood Function 20 4 0 SO B o t c h No cont inued C (fl M cn Z 3C II CO CO O RJ - 1-1 z o 13) 1 > -3 O n c X n II < o ui o - o c 0» > i i Qj LQ OJ (fl TJ rr CO :r < !-•• fD TJ - :* cu c i - 1 II 3 - CJ CO 3 3 Ul **1 I >-• II cn o 3 oa o.o Percentage 20.o j a o •o.o 50.0 sao 7Q;o a a o a a o j o a o 0.00 N-1 o CO j o o Z O OI -10 -0.8 -0.8 -0 .* - 0 2 0.0 0.2 0> 3A a 0.8 o" b co -1 o CD a >o co o o c w 3* 81 0.8 10 Normalized rms error 0.05 a.« °.° ° ; 2 0 . . " i 2 3 0J0. IS -100.0-75.0 -50.0 -25.0 25.o sao 73.0 m o o -1 3 a N CD C L o o (fl Q o o o fD CD 8 | 381 Estimated Q1 Estimated 02 Si I n n rp n 4 0 3.0 B a t c h No F i g u r e E1 Est mated R -€J 1.0 JO 3.0 -1 r— 4.0 5.0 Botch No S1 3 3. Likelihood Function Si r a., 4.0 3 0 B a t c h No cont inued s? d o 8 00 200.0 »000 800.0 t (sec) 800.0 woo.o '200.0 200.0 400.0 600.0 t (sec) 8000 1000.0 1200 n E s t i m a t e d R L ike l ihood Funct ion or o o * -e- -B- -B-3 0 4 0 so p n t c h N o 6.0 8.0 CD F i g u r e E1.6 RMS E r r o r Curves of Adaptive Kalman F i l t e r i n g - P a r t i a l Adaptation. i N=80, NBATCH=400 Large Ship, u=13ms ' rL 2=50m o 1. o o> d 10 E c o o z -Jo — .—.. i—i—>—•—•—•—i—•—•—•—•— 0.0 o o si-o D ~c ° 0) d o_ * 200.0 400.0 600.0 t ( sec ) 800.0 1000.0 1200.0 P e r c e n t a g e C o u n t ^ L i m i t -o 2.0 3.0 4.0 5.0 B o t c h No 6.0 7.0 8 0 0.0 -9 200.0 400.0 600.0 t ( s e c ) 800.0 1000.0 1200.0 Norma l i zed C o r r e l a t i o n C o e f f i c i e n t of r ( I. M i l l r 11111 i ' M i l 20 0 «0.0 I a<q k CD cn Figure E1.7 RMS E r r o r Curves of Adaptive Kalman F i l t e r i n g - F u l l Adaptation, iv,=80, NBATCH = 800 Large Ship, u=13ms ',L2=50m E s t i m a t e d Q2 o si-o • s-o 8-O 9 o Botch No F i g u r e E l E s t i m a t e d R r o 3. o 74- 1 1 1 — — i r -(.0 2.0 3.0 *0 SO 6.0 B o t c h No L ike l ihood Funct ion a: o i a • <0 5 0 B o t c h No cont i nued 1200.0 4.0 5.0 B o t c h No Figure E1 8 RMS E r r o r Curves of Adaptive Kalman F i l t e r i n g - F u l l Adaptation, i M = 8 0 f NBATCH=800 Small Ship, u=13ms T,L 2=l50m Estimated Q1 4.0 5.0 B o t c h No — i — 7 0 Estimated 02 <crS -e-2 0 4.0 50 B o t c h No F i g u r e E l Estimated R B o t c h No L ike l ihood Functi a:, d -e-4.0 5 0 B a t c h No c o n t i n u e d 189 F i g u r e E2.1 Performance Curves of RLSI F i l t e r as a F u n c t i o n of Q - Large Ship, u=9ms"1 (a)L=2 (b)L=11 190 E2.1 Performance Curves of RLSI F i l t e r as a F u n c t i o n of Q - s m a l l Ship, u=6ms_1 (a)L=2 (b)L=11 191 (a) 50.0 60.0 Seaf loo r L2 o = 50m o = 80m A = 10Qm + = 150m x = 300m © = 600m (b) 60.0 .2 Performance Curves of RLSI F i l t e r as a F u n c t i o n of L - Large Ship, u=9ms"1 (a)Q=0.82 (b)Q=0.85 1 92 (a) 40.0 50.0 60.0 S e a f l o o r L2 D = 50m o = 80m A = 100m * = 150m x = 300m o = 600m 10.0 20.0 30.0 L 40.0 50.0 60.0 (b) 2 Performance Curves of RLSI F i l t e r as a F u n c t i o n of L - Small Ship, u=6ms~1 (a)Q=0.78 (b)Q=0.8 1 in 61 CM (a) 0.02 0.03 0.04 0.05 — I 0.06 Fc 0.07 0.08 0.09 0.K) 6 1 ^6 -1 Seaf loor L2 • = 50m o = 100m * = 150m • = 300m x = 600m (b) 0.02 0.03 0.04 0.05 0.06 Fc 0.07 0.08 0.09 —1 0.10 F i g u r e E3.1 Performance Curves of LP F i l t e r as a F u n c t i o n of C u t o f f Frequency F~ (a)Large Ship, u=12ms _ 1 (a)Small Ship, u=12ms _ 1 1"1 ' I ' 1 ' I ' — I — ' — I — ' — I— • — 1— • — I — > — I— I — I— I — I— . — I— , — I 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 f i l ter Length (a ) b o Seof loor L2 D = 50m o = 100m * = 150m • = 300m x s= 600m (b) i - i — • — i — ' — i — • — i — 1 — i — • — i — • — i — 1 — i — • — i — < — i — 1 — i — • — i — ' — i — • — i 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0 120.0 130.0 n i te r Length F i g u r e E 3 . 2 P e r f o r m a n c e C u r v e s o f L P F i l t e r a s a F u n c t i o n o f F i l t e r L e n g t h ( a ) L a r g e S h i p , u ^ n m s " 1 , F c = 0 .05 Hz ( a ) S m a l l S h i p , u=l7ms ] , F c = 0 . 0 5 Hz 195 APPENDIX F: Programs The major r o u t i n e s used i n the software f o r s i m u l a t i o n and experiments are i n c l u d e d here. There are 5 p a r t s :-1. SIMUL - c o n t a i n i n g the s i m u l a t i o n program to generate the s e a f l o o r p r o f i l e sequence and the measurement e r r o r sequence. C o n s i s t s of r o u t i n e s SIM, CORR and HIST. 2. RLSI - the f i l t e r i n g r o u t i n e implementing the RLSI F i l t e r . 3. KFCOR - the f i l t e r i n g r o u t i n e implementing the MeasurementDifferencing Kalman F i l t e r . 4. ADPKF - the f i l t e r i n g r o u t i n e implementing the Adaptive Kalman F i l t e r . A l l i nput, output and other f i l e - k e e p i n g procedures have been omitted. The v a r i a b l e s and l i b r a r y r o u t i n e s used i n the programs are d e f i n e d i n the heading. 196 c  c * SIMUL.D * C C INTEGRATED SIMULATION. DATA GENERATION AND SPECTRAL C ESTIMATION PROGRAM BY RANDOM PHASE METHOD C C INPUT PARAMETERS : C 1. SIMTYP. NPTS - TYPE OF SIMULATION!1=MEASUREMENT SERIES. C 2=SEAFLOOR);NO. OF POINTS TO BE GENERATED. C 2. SIGMA. V. L2 - SPECTRAL AMPLITUDE, SHIP SPEED. CORRELATION DISTANCE C U. V, DEGREE. DELAY .SIZE - WIND AND SHIP SPEED, ENCOUNTER ANGLE C ROUNDTRIP TIME DELAY. SIZE OF SHIP C 3. FMIN, FMAX, NBINS, STEP, INIT - LOWER AND UPPER LIMIT OF C FREQUENCY OF SPECTRUM IN SIMULATION, NO. OF BINS. ITERATION STEP C STEP SIZE. INITIAL FREQUENCY C 4. SEED - SEED TO GENERATE SET OF RANDOM NUMBERS C SE. SFLR - EXTERNAL FUNCTIONS DEFINING THE MEASUREMENT ERROR C AND SEAFLOOR SPECTRA C OUTPUT:-C H - SIMULATED SERIES C XA -ESTIMATED POWER SPECTRUM C XFR X COORDINATES OF XA C S -IDEAL POWER SPECTRUM (SE OF SFLR) C FREQ -X COORDINATES OF S C C -c c REALM INIT. L2 REAL*8 MO, FMIN.FMAX INTEGER SIMTYP.SIZE EXTERNAL SE,SFLR COMMON /BLK1/ SIZE.U.V,DEGREE.DELAY,SIGMA.L2,RANGE COMMON /BLK2/ SEED,NBINS DIMENSION XA(1024),XFR(1024).FREQ(600),S(60O).COUNT(200) C DIMENSION XC0RDC20O), TIME(2500).H(2500) C READ(5.902) SIMTYP. NPTS 902 F0RMAT(2I5) 5 CONTINUE C GOTO(10.5).SIMTYP C C SEAFLOOR SIMULATION C 5 READ(5,904)SIGMA.V,L2 904 FORMAT(3F8.2) GO TO 20 C C MEASUREMENT ERROR SIMULATION C 10 READ(6.919)U.V,DEGREE.DELAY, SIZE 919 F0RMAT(4F7.2.I2) READ(5,907)FMIN,FMAX,NBINS.STEP.INIT 907 F0RMAT(2F10.8,I5.2F10.8) C C SIMULATION C 20 CALL SIM(FMIN,FMAX.STEP,INIT,SIMTYP,NPTS.MO.TIME,HS . H) C C SPECTRAL ANALYSIS OF SIMULATED TIME SERIES C CALL CORR(SIMTYP,NPTS,MO,XFR,XA,FREQ,S.NHF1.YMAX.H) C C HISTOGRAM ANALYSIS C CALL HI ST(NPTS.MO,XCORD.COUNT.NBIN,MEAN,STO,SKEW.H) STOP END 1 97 c  c C SUBROUTINE : SIM C C SUBROUTINE TO SIMULATE TIME SERIES FOR 240O M (1200 SEC) C AS SPECIFIED BY THE SPECTRUM SE (MEASUREMENT ERROR) OR C SFLR (SEAFLOOR) C H(M) IS THE GENEATED TIME SERIES C DEFINITIONS OF VARIABLES :-C N1 - TOTAL NO. OF FREQUENCY BINS C MO - TOTAL AREA UNDER POWER SPECTRUM C AO - AREA OF 1 FREQUENCY BIN C FLOW - LOWER LIMIT OF FREQUENCY BIN AT CURRENT ITERATION C F1 -ESTIMATED UPPER LIMIT OF FREQ BIN BEFORE ITERATION C FNEXT -ESTIMATED UPPER LIMIT OF FREO BIN AFTER ITERATION C H - OUTPUT (SIMULATED) SERIES C C - -- — -c SUBROUTINE SIM(FMIN.FMAX.STEP.INIT.SIMTYP.M.MO.TIME . HS . H) C DIMENSION FREQ(201),PHI(201),TIME(2500),H(2500) REAL*8 MO,AO.AREA.FI,FLOW.FNEXT.FREQ.CK REAL*8 ANGLE,ARG.XI.EPSILN.PHI,VAR,TEMP REALM L2.INIT INTEGER SIMTYP,SIZE EXTERNAL SE, SFLR COMMON /BLK 1/ SIZE.U,V.DEGREE.DELAY,SIGMA.L2,RANGE COMMON /BLK2/ SEED,NBINS C 4 RANGE =8.O*ATAN(1.0) N1=NBINS+1 DELT=0.5/1.0 IF(SIMTYP .EQ. 0) GO TO 5 MO=DQUANK(SFLR,FMIN,FMAX,0.OOOOOO1,TOL,ERROR) GO TO 6 5 MO=DOUANK(SE,FMIN,FMAX,0.000O0O1,TOL,ERROR) 6 AO=MO/NBINS REMA =M0 FREQ(1)=0.0 FI=INIT FLOW=FMIN HS=4.O*USQRT(M0) C ITERATIONS TO EVALUATE FREQUENCY BINS C DO 30 I=2,N1 10 GOTO(8,7),SIMTYP 7 AREA =DQUANK(SFLR,FLOW,FI ,0.0000001,TOL.ERROR) FNEXT=FI-((AREA-AO)/SFLR(FI)) GO TO 9 8 AREA = DQUANK(SE,FLOW,FI,O.OOOOOO1,TOL,ERROR) FNEXT=FI-((AREA-AO)/SE(FI)) 9 EPSILN = DABS(FNEXT-FI) IF (EPSILN .LT. 0.000001 ) GO TO 20 FI•FNE XT GO TO 10 20 FLOW=DABS(FNEXT) FREQ(I)"DABS(FNEXT) FI"FNEXT+STEP 30 CONTINUE C C INPUT OF RANDOM SEED AND GENERATE RANDOM PHASE C READ(5.909)SEED 909 FORMAT(F10.4) X"RAND(SEED ) DO 100 K=1,N1 X-FRAND(O.O) PHI(K)=X*RANGE 100 CONTINUE C C INITIALIZE H ARRAY C DO 110 M*1,2401 H(M)=0.0 110 CONTINUE c C SUMMATIONS TO OBTAIN H C CK=DSORT(2.0*AO) DO 130 d=1,N1 ANGLE = RANGE *FREQ( J)*DELT VAR=PHI(J) A=DCOS(VAR) B=DSIN(VAR) DO 120 M=1,2401 ARG=(M-1)*ANGLE H(M)=H(M)+CK*(A*DCOS(ARG)+B*DSIN(ARG)) 120 CONTINUE 130 CONTINUE C DO 160 1=1,2401 TIME(I)=0.5*FL0AT(I-1) 160 CONTINUE RETURN END C C c  C SUBROUTINE : CORR C SUBROUTINE TO ESTIMATE THE POWER SPECTRUM OF A GIVEN C TIME SERIES USING A MODIFIED VERSION OF THE CORRELATION C METHOD OF RABINER, SCHAFER. AND DLUGOS ( PROGRAMS FOR C DIGITAL SIGNAL PROCESSING, IEEE PRESS, PP.2.2.11 -C 2.2.14, 1979) . C C - VARIABLE DEFINITIONS: C SIMTYP TYPE OF SERIES 0=MEASUREMENT ERROR C 1=SEAFL00R C N NUMBER OF SAMPLES IN SERIES C M SECTION SIZE (MUST BE A POWER OF 2) C 2 =< M =< 1024 C FS SAMPLING FREQUENCY IN HZ C IWIN WINDOW TYPE 1=HANNING C 2'HAMMING C L NUMBER OF CORRELATION POINTS(LAGS) C USED IN SPECTRAL ESTIMATE C 2 =< L •< M/2+1«513 C NFFT FFT SIZE USED TO COMPUTE SPECTRAL C ESTIMATE FROM WINDOWED CORRELATION C FUNCTION (MUST BE A POWER OF 2) C 2*L-1 =< NFFT «< MAXM C OUTPUT :-C XFR, FREQ -DISCRETE FREQUENCY SERIES C XA -MAGNITUDE OF ESTIMATED POWER SPECTRUM C S -IDEAL SPECTRUM (SE OR SFLR) C C - -C SUBROUTINE CORR(SIMTYP,N,MO,XFR,XA,FREQ,S,NHF1.YMAX,H) DIMENSION XA(1024), XFR(513).FREQ(600),S(GOO) ,H(2500) REAL MM.C.KK.L2.M0 INTEGER M, SIMTYP,ORIG.LOGPL,SS,SIZE COMPLEX X(1024), Z(513). XMN, XI, YI COMMON/BLK1/ SIZE.U,V.DEGREE,DELAY.SIGMA,L2,RANGE COMMON /BLK2/ SEED,NBINS EXTERNAL SE,SFLR DATA FS/2.0/ C C READ IN INPUT PARAMETERS C READ(5.997) M,IWIN.L.NFFT 997 F0RMAT(4I6) C C NSECT IS THE TOTAL NUMBER OF ANALYSIS SECTIONS C LSHFT IS THE SHIFT BETWEEN ADJACENT ANALYSIS SAMPLES C LSHFT = M/2 MHLF1 « LSHFT + 1 NSECT = (FLOAT(N ) + FLOAT(LSHFT)-1.)/FLOAT(LSHFT) 1 9 9 c C LOOP TO ACCUMULATE CORRELATIONS C 8 0 SS = 1 NRDY » M NRDX = LSHFT DO 9 0 1=1,MHLF1 Z ( I ) - ( 0 . . 0 . ) 9 0 CONTINUE DO 190 K=1,NSECT NSECT1 = NSECT - 1 I F ( K . L T . N S E C T 1 ) GO TO 110 NRDY = N - ( K - 1 ) * L S H F T IF (K.EQ.NSECT) NRDX = NRDY IF (NRDY.EQ.M) GO TO 110 NRDY1 = NRDY + 1 DO 100 I=NRDY1,M X ( I ) = ( 0 . 0 . ) 100 CONTINUE C C READ NRDY SAMPLES FROM X GENERATOR STARTING AT SAMPLE SS C 110 CALL GETX(XA, NRDY, SS.H) DO 120 I"1.NRDY X ( I ) = C M P L X ( X A ( I ) , X A ( I ) ) 120 CONTINUE 180 NRDX1 • NRDX + 1 DO 170 I=NRDX1,M X ( I ) = CMPLX(0. ,AIMAG(X(I ) ) ) 170 CONTINUE C C CORRELATE X AND Y SECTIONS C DO EVEN-ODD SEPARATION AND ACCUMULATE CONJG(X)*Y C CA L L F F T ( X . M, 0 ) DO 180 1=2,LSHFT J «= M + 2 - I XI •= ( X ( I ) + C 0 N J G ( X ( J ) ) ) * . 5 YI » ( X ( J ) - C O N J G ( X ( I ) ) ) * . 5 YI = C M P L X ( A I M A G ( Y I ) , R E A L ( Y I ) ) 2(1) «= Z ( I ) + C O N J G ( X I ) » Y I 180 CONTINUE XI « X ( 1 ) Z ( 1 ) - Z ( 1 ) + C M P L X ( R E A L ( X I ) * A I M A G ( X I ) , 0 . ) XI • X(MHLF1) Z(MHLF 1 ) = Z ( M H L F 1 ) + C M P L X ( R E A L ( X I ) * A I M A G ( X I ) , 0 ) SS - SS • LSHFT 190 CONTINUE C C INVERSE DFT TO GIVE CORRELATION C DO 2 00 1-2.LSHFT J = M + 2 - I X ( I ) - 2(1) X ( J ) « C O N J G ( Z ( I ) ) 2CO CONTINUE X ( 1 ) •= Z ( 1 ) X(MHLF1 ) = Z(MHLF1) CAL L F F T ( X , M, 1) FN - F L O A T ( N ) DO 2 1 0 I=1.MHLF1 X A ( I ) » R E A L ( X ( I ) ) / F N 2 1 0 CONTINUE C C C WINDOW CORRELATION USING L VALUES TO GIVE SPECTRAL ESTIMATE C READ I N WINDOW TYPE AND WINDOW LENGTH C C PI • 4.0*ATAN(1.0) DO 2 3 0 I=2,L IF (IWIN.EO.1) GO TO 215 X A ( I ) - X A ( I ) * ( 0 . 5 4 + 0 . 4 6 » C 0 S ( P I * F L 0 A T ( I - 1 ) / F L 0 A T ( L - 1 ) ) ) GO TO 220 215 X A ( 1 ; « A A ( I ) * ( 0 . 5 * ( 1 - C O S ( 2 * P I ' F L O A T ( I - 1 ) / F L O A T ( L - 1 ) ) ) ) 220 d « NFFT + 2 - 1 X A ( J ) = X A ( I ) 230 CONTINUE NLAST = NFFT + 1 - L L1 » L + 1 DO 240 I"L1,NLAST XA(I) » 0. 240 CONTINUE DO 250 1=1,NFFT X( I ) «= CMPLX(XA(I) .0. ) 250 CONTINUE C C FFT ON WINDOWED CORRELATION C FUNCTION TO OBTAIN POWER SPECTRUM C CALL FFT(X. NFFT, 0) XFS - FS/FLOAT(NFFT) NHF = NFFT/2 NHF1 = NHF + 1 DO 260 1 = 1,NHF1 XFR(I)«FL0AT(I-1)*XFS XA(I)=CABS(X(I)) IF(SIMTYP .EQ. 0 .AND. I .LT. 10)XA(I)=0.0 260 CONTINUE C C READ IN MIN AND MAX VALUE OF SPECTRUM C READ(5,1001)YMIN,YMAX 1001 FORMAT(2F8.1) C IF (SIMTYP .EQ. 0) GO TO 330 C DO 320 1=1,400 ARG = (I-1)*0.001 S(I ) =SFLR(ARG) FREQ(I)=ARG 320 CONTINUE GO TO 345 330 DO 340 1=1,400 ARG=(I-1)»0.001 FREQ(I)=ARG S( I) = SE(ARG) 340 CONTINUE 345 RETURN END C c  C SUBROUTINE: GETX C OBTAIN NRD SAMPLES OF GENERATED SERIES STARTING AT SS c  c SUBROUTINE GETX(X, NRD, SS.H) DIMENSION X(1024) ,H(2500) COMMON /BLK2/ SEED,NBINS INTEGER NRD.SS C C X = ARRAY OF SIZE NRD TO HOLD GENERATOR OUTPUT DATA C NRD • NUMBER OF SAMPLES TO BE CREATED C SS • STARTING SAMPLE OF GENERATOR OUTPUT C DO 10 1=1,NRD K«(SS-1) +<I-1) X(I)=H(K) 10 CONTINUE RETURN END C C- -C SUBROUTINE : FFT C FFT PROGRAM USING DECIMATION IM TIME ALGORITHM c  c SUBROUTINE FFT(X , N, INV) C C X « 2**M COMPLEX ARRAY THAT INITIALLY CONTAINS INPUT C AND ON RETURN CONTAINS TRANSFORM C N • 2**M POINTS C INV «• O. DIRECT TRANSFORM C INV * 1. INVERSE TRANSFORM C COMPLEX X(1), U, W. T. CMPLX M = AL0G(FL0AT(N))/AL0G(2.) + .1 NV2 «= N/2 NM1 » N - 1 J - 1 DO 40 1=1,NM1 IF (I.GE.d) GO TO 10 T - X(J) X(d) ' X(I) X(I) • T 10 K • NV2 20 IF (K.GE.J) GO TO 30 d - d - K K - K/2 GO TO 20 30 d = d + K 40 CONTINUE PI = 4.0*ATAN(1.0) DO 70 L«1,M LE = 2«»L LEI = LE/2 U "= ( 1.0.0.0) W •= CMPLX(C0S(PI/FL0AT(LE1)),-SIN(PI/FL0AT(LE1))) IF (INV.NE.O) W = CONdG(W) DO 60 d=1,LE 1 DO 50 I=d,N,LE IP » I • LE1 T « X(IP)*U X(IP) = X(I) - T X(I) = X(I) + T 50 CONTINUE U = U*W 60 CONTINUE 70 CONTINUE IF (INV.EQ.O) RETURN DO 80 I-I.N X(I) - X(I )/CMPLX(FLOAT(N),0. ) 80 CONTINUE RETURN END C c  C SUBROUTINE : HIST C SUBROUTINE TO GENERATE HISTOGRAM OF TIME SERIES C AND CALCULATE STATISTICAL PARAMETERS OF THE HISTOGRAM C INPUT VARIABLES :-C XMIN -MINIMUM NORMALIZED MAGNITUDE C XMAX -MAXIMUM NORMALIZED MAGNITUDE C DX -BIN WIDTH C NBIN -TOTAL NO. OF BINS C C -c SUBROUTINE HI ST(N.MO.XCORD,COUNT,NBIN,MEAN,STD,SKEW,H) DIMENSION C0UNT(200). XCORD(200).BIN(20O),H(2500).X(250O) INTEGER SIZE, BIN REAL*4 LOWLIM, MEAN.STD,SKEW.KURT,MC1,MC2,MC3.MC4 REAL'S MO,ROOT COMMON /BLK1/ SIZE,U,V,DEGREE,DELAY,SIGMA,L2.RANGE COMMON /BLK2/ SEED.NBINS DATA FI,MC1,MC2,MC3/4*0.0/ C C NORMALIZE H C ROOT=DSQRT(MO) SIG=SNGL(ROOT) DO 5 I=1,N X(I)=H(I)/SIG 5 CONTINUE c C READ IN INPUT PARAMETERS C READ(5,8)XMIN,XMAX,DX,NBIN 8 F0RMATOF8.4.I4) C C USE SHELL SORT ROUTINE TO ARRANGE C NORMALIZED H IN ASCENDING ORDER C CALL SORT(X.N) C DO 20 1=1.NBIN BIN(I)=0.0 20 CONTINUE C J=1 LOWLIM=XMIN UPLIM=XMIN+DX C 21 IF(X(J) LT. LOWLIM)GO TO 22 GO TO 23 22 d = J-M GO TO 21 C C ACUMULATE COUNT FOR EACH BIN C 23 DO 30 1=1.NBIN 25 IF (X(J) .GT. UPLIM) GO TO 28 BIN(I)=BIN(I)+1 d = J+1 IF(J .GT. N) GO TO 31 GO TO 25 2B UPLIM=UPLIM+DX LOWLIM=LOWLIM+DX 30 CONTINUE C C XCORD(I) IS THE MEDIAN POINT OF BIN(I) C 31 XCORD(1)=XMIN+DX/2.0 DO 40 1=2.NBIN XCORD(I ) = XCORD(1-1) + (DX) 40 CONTINUE C DO 45 1=1.NBIN COUNT(I)=FLOAT(BIN(I))/FLOAT(N) IF(COUNT(I) .EO. 0.0) COUNT(I) =1.E-4 COUNT(I)=AL0G10(C0UNT(I)) 45 CONTINUE C C C CALCULATE STATISTICAL PARAMETERS OF HISTOGRAM C DO 60 1=1.NBIN MC1=MC1+(BIN(I)*XCORD(I)) FI»FI+BIN(I ) MC2=MC2+(BIN(I)*(XCORD(I )**2)) MC3=MC3+(BIN(I)*(XCORD(I )**3) ) 60 CONTINUE MEAN=M"</FI STD=MC2/FI SKEW=MC3/(FI*(SIG*3)) C RETURN END 203 c c  C SUBROUTINE : SORT C SHELL SORT ROUTINE c  SUBROUTINE SORT(A.N) REAL A(2500) C M=N 1 M=M/2 IF(M .LE. O) GO TO 6 K = N-M 2 I=J 3 IF(A(I) .GT. A(I+M)) GO TO 5 4 J=J+1 IF(J .GT. K) GO TO 1 GO TO 2 5 TEMP=A(I) A(I)=A(I+M) A(I+M)=TEMP I«I-M IF ( I .LT. 1) GO TO 4 GO TO 3 6 RETURN END c  C SUBROUTINE : RLSI C INPLEMENT RLSI FILTER C INPUT : H0.H1.L.Q (FROM CALLING ROUTINE) C INPUT SEREIS :-C H - SEAFLOOR SERIES C ZI MEASUREMENT SERIES C OUTPUT SERIES :-C ESTH - ESTIMATED SEAFLOOR SERIES C ESTRMS - ACCUMULATED RMS ERROR c  SUBROUTINE RLS(NPTS.NMAX.L,0.HO,H1) C REAL MO.M1,M2 COMMON /BLK2/ H(2500). ZI(2500),ESTH(2500),ESTRMS( 2500) DATA E3.MO,M2,SUM/0.0.1.0.0.0,0.0/ C NMAX=NPTS-L L1=L+1 20 DO 30 I-1.NPTS NL=I-L H0X«H0+H1 H1X=H1 MO=MO+0**I SUM=SUM+(FLOAT(I)*(0**I)) M1=-SUM M2»M2+((FLOAT(I))**2)*(0**I) ALPHA=M2/((MO*M2)-(M1**2)) BETA=-M1/((MO*M2)-(M1**2)) HO=HOX+ALPHA*(ZI(I)-HOX) H1=H1X+BETA*(ZI(I)-HOX) IF(I .LT . L1) GO TO 30 ESTH(NL)=H0-L*H1 E3=E3+(ESTH(NL)-H(NL))**2 ESTRMS(NL)=S0RT(E3/FL0AT(NL)) 30 CONTINUE RETURN END c  C SUBROUTINE : KFCOR C IMPLEMENT MEASUREMENT DIFFERENCING KALMAN FILTER. C - DEFINITIONS OF VARIABLES C M - A APRIORI COVARIANCE C P - A POSTERIORI COVARIANCE C PO -INITIAL VALUE OF P C 01, 02 - ELEMENTS OF 0 COVARIANCE MATRIX C R1, R2 - ELEMENTS OF R COVARIANCE MATRIX C G - KALMAN GAIN MATRIX C X - STATE VECTOR C NPTS - TOTAL NO OF DATA POINTS C CALLING ROUTINE MUST SUPPLY THE FOLLOWING PARAMETERS :-C PHI - STATE TRANSITION MATRIX C T - SYSTEM DISTURBANCE DISTRIBUTION MATRIX C C SCALING FACTOR IN APPROXIMATING TRANSFER FUNCTION C L2 - SEAFLOOR CORRELATION DISTANCE C PO.O.R C INPU. sEREIS :-C H - SEAFLOOR SERIES C ZI - MEASUREMENT SERIES C OUTPUT SERIES :-C ESTH - ESTIMATED SEAFLOOR SERIES C ESTRMS - ACCUMULATED RMS ERROR C ESTXI - ESTIMATED NOISE SERIES C c  C SUBROUTINE KFCOR(NX I,H,ZI.PHI,T.DEC,C,PO,0,R,ESTH,ESTXI.ESTRMS,L2,NPTS) C DIMENSION H(2401),ZK2401).ESTXI(2401),ESTRMS(2401),ESTH(2401) DIMENSION PHK2.2) ,T(2) COMMON /BLK2/ D(2,2),HR(2,2),HT(2,2),RR(2,2),PHIDH(2,2),TOT( 2,2). DRD(2,2),G(2,2),P(2.2),M(2.4),X(2).RSEO(2401),PHIDHT(2.2) REAL L2.M DATA M( 1,2),M(2, 1 ) ,X( 1 ) ,X(2)/4*0.0/ C C INITIALIZATION OF ERROR COVARIANCES C M( 1 , 1 )=P0 M(2.2)=PO C R1=R R2 = R 01=0 02=0 C DO 38 1=1,2 DO 39 J=1 ,2 G(I,J)=0.0 39 CONTINUE 38 CONTINUE C C CALCULATE PARAMETERS OF SYSTEM MATRICES C CALL PAR(NXI.PHI,T,R1,R2,Q1,02.DEC,C,L2) C C BEGIN PROCESSING C E»0.0 C DO 50 I=2,NPTS CALL UPDATE(PHI,ZI,I.DEC.C) ESTXI(I)=C*X(2) ESTH(I)=ZI(I)-ESTXI(I) E=E+(ESTH(I)-H(I))**2 ESTRMS(I ) = (SQRT(E/FL0AT(I-1)))/10.0 CALL PROP 50 CONTINUE C C RETURN END 205 c  C SUBROUTINE : PAR C CALCULATE PARAMETERS OF SYSTEM MATRICES REQUIRED C IN KALMAN EOUATIONS FOR N=2 c c SUBROUTINE PAR(NXI.PHI.T.R1.R2,Q1,Q2.DEC,C,L2 ) C DIMENSION PHK2.2). S(2.2).T(2) COMMON /BLK2/ D(2,2).HR(2.2),HT(2,2).RR(2.2),PHIDH(2,2),TQT(2,2), DRD(2.2),G(2.2),P(2.2),M(2.4),X(2).RSE0(24O1),PHIDHT( 2 . 2) REAL L2 DATA HR(2,1),HR(2.2),RR(1.2),RR(2,1)/4*0.0/ DATA bl i . 1 ) ,S( 1 .2) .5(2, 1 )/3*0.0/ DATA D( 1 . 1 ) ,D(1.2),D(2.1)/3*0.0/ DATA TQT(1.2),TQT(2,1),DRD(1,1),DRD(1.2),DRD(2.1)/5*0.0/ C ARG=-1.0/L2 DEC=EXP(ARG) HR( 1 ,1)=C*PHI(2.1) HR( 1 ,2)=C*(PHI(2,2)-DEC) C CALL TRANSP(HR,HT,2,2) VAR1=Q2*((C*T(2))**2) RR(1,1)=R1+VAR1 RR(2,2)=R2 S(2,1 )=Q2*C*T(2) C DENOM=1./(R2*(R1+VAR1)) VAR2=R2*Q2*C*(T(2),*2) D(2,1)=VAR2*DEN0M C PHIDH(1,1)=PHI(1,1) PHIDH(1.2)=PHI( 1.2) PHIDH(2,1) = PHI(2.1)-(HR(1,1)*VAR2 *DENOM) PHIDH(2.2) = PHI(2.2)-(HR( 1,2)*VAR2 *DENOM) C CALL TRANSP(PHIDH.PHIDHT,2,2) C TQT( 1 , 1 )=Q1»T( 1 )**2 TQT(2,2)=Q2*T(2)'*2 DRD(2.2)*-(R1+Q2*(C**2)*(T(2)**2))*(VAR2**2 )*(DENOM**2) C RETURN END c  C SUBROUTINE : UPDATE C UPDATE OF STATE VECTOR C - DEFINITIONS OF VARIABLES C ZI NOISY MEASUREMENTS C PSI DERIVED MEASUREMENT c  SUBROUTINE UPDATE(PHI.ZI,I.DEC.C) C DIMENSION ZK2401 ) . PHM2.2). MH( 2 . 2 ) , HMH( 2 , 2 ) , B ( 2 . 2 ) , BT ( 2 , 2 ) DIMENSION IPERM(10),DlFF(2).PK(2,2),DPK(2.2) COMMON /BLK2/ D(2,2),HR(2,2).HT(2.2),RR(2,2),PHIDH(2,2),TQT(2,2), DRD(2,2),G(2,2),P(2,2),M(2.4),X(2),RSEQ(2401),PHIDHT(2,2) REAL M.MH C C UPDATE OF GAIN G C CALL MULT(M,HT,MH,2,2) CALL MULT(HR,MH,HMH,2,2) CALL ADD(HMH,RR,B,2,2 ) CALL FINV(2.2,B,IPERM.2.BT,DET.JEXP.COND) CALL MULT(MH,BT,G,2,2) C TAKE IN NEXT MEASUREMENT AND FORM PSI VECTOR C 11=1-1 PSI=ZI(I)-DEC*ZI(I1) RSEO(I 1 )«PSI-HR(1.1)*X(1)-HR(1,2)*X(2) C C UPDATE OF X VECTOR C CALL MULT(PHIDH.G.PK,2,2) CALL ADD(D.PK.DPK,2.2) XN1-PHK1,1)*X(1) + PHI(1 ,2)*X(2)+DPK(1, 1 )*RSEO(I 1) XN2 = PHI(2.1)*X(1) + PHI(2,2)*X(2)+DPK(2,1)*RSEQ( I 1) X(1)«XN1 X(2)=XN2 C 20 RETURN END C C SUBROUTINE : PROP C PROPAGATION EQUATIONS c  SUBROUTINE PROP C DIMENSION GH(2.2),GHT(2,2) ,MH(2.2),B1(2,2),RG(2.2),GRG(2,2) DIMENSION GT(2.2).B2(2.2 ) ,B3(2.2 ) .B4(2,2) COMMON /BLK2/ 0(2,2),HR(2,2),HT(2,2),RR(2,2),PHIDH(2,2),TQT(2,2), DRD(2.2),G(2,2) ,P(2.2) ,M(2,4),X(2),RSEQ(2401).PHIDHT(2,2) REAL M.MH LOGICAL FORM C C UPDATE OF P MATRIX C CALL MULT(G.HR,GH,2.2) GH(1. 1 )=1.-GH(1,1) GH(2.2)=1.-GH(2,2) GH(1,2) = -GH( 1,2) GH(2.1)=-GH(2,1) CALL TRANSP(GH,GHT,2,2) CALL MULT(M.GHT,MH,2,2) CALL MULT(GH.MH,B1.2,2) C CALL TRANSP(G,GT,2,2) CALL MULT(RR,GT,RG,2,2) CALL MULT(G,RG,GRG.2,2) CALL ADD(B1,GRG,P,2,2 ) C C UPDATE OF M MATRIX C CALL MULT(P,PHIDHT.B2,2,2) CALL MULT(PHIDH,B2.B3,2.2) CALL ADD(B3,TQT,B4,2,2) CALL ADD(B4,DRD.M,2.2) C 20 RETURN END c  C SUBROUTINE : ADPKF C IMPLEMENT ADAPTIVE KALMAN FILTER USING THE INNOVATION C CORRELATION ALGORITHM C CALLING ROUTINE MUST SUPPLY THE FOLLOWING PARAMETERS :-C PHI - STATE TRANSITION MATRIX C T - SYSTEM DlS URBANCE DISTRIBUTION MATRIX C C - SCALING FACTOR IN APPROXIMATING TRANSFER FUNCTION C L2 - SEAFLOOR CORRELATION DISTANCE C PO.QO.RO C OO, RO - INITIAL VALUES OF THE ERROR COVARIANCE MATRICES C NSS - INITIAL BATCH SIZE C NSAMPL - SAMPLE SIZE OF INNOVATION CORRELATION C NBATCH - TOTAL NO. OF BATCH C KBATCH - CURRENT BATCH NO. C IN - SIZE OF THE SET OF INNOVATION CORRELATION C MODE - MODE OF ADAPTATION C (T=FULL. F"PARTIAL ) C INPUT SEREIS :-C H - SEAFLOOR SERIES C ZI - MEASUREMENT SERIES C OUTPUT SERIES :-C ESTH - ESTIMATED SEAFLOOR SERIES C ESTRMS - ACCUMULATED RMS ERROR C ESTXI - ESTIMATED NOISE SERIES C LF - LIKELIHOOD FUNCTION C DEFINITIONS OF VARIABLES :-C QE, RE - CURRENT ESTIMATES OF COVARIANCE MATRICES C RHO - CORRELATION COEFFICIENTS C c  c SUBROUTINE FILTER(NX I.H.ZI,PHI,T,DEC,C,PO,OO,RO,ESTH,ESTXI.ESTRMS, L2,NPTS,NSS.NBATCH.IN,NSAMPL.OE.RE.RHO,LIMIT,MODE.LF.MXLIM,COUNT) C DIMENSION H(2401 ) ,ZI (2401 ) .ESTXI (2401 ) . E STRMS ( 2401 ) ,r.STH(2401 ) DIMENSION PHI(2,2).T(2),LF(10).COUNT(10) DIMENSION Q(2),0E(10,2),RE(lO).RHO(GO) COMMON /BLK2/ B,D(2,1).HR(1,2).HT(2,1),RR,PHIDH(2,2),TOT(2,2), DRD(2.2),G(2.2),P(2.2),M(2.4),X(2).RSEO(2401).PHIDHT(2.2) C0MM0N/BLK3/ HTAUT(2,1),HP(1,2),HPIT(2,1),HPI2T(2,1),HP2(1.2). PHINV(2.2),PHIT(2.2).PIT(1,2).FI(2.2) REAL L2.LIMIT.LF.MXLIM LOGICAL MODE REAL*8 C.PHI.T.OE.RE.O.R.DO.DR REAL*8 B.D.HR,HT.RR,PHIDH,TOT,DRD.G,GK,P,M,X,PHIDHT,RSEO REAL*8 HTAUT.HP.HPIT,HPI2T,HP2.PHINV,PHIT,PIT,FI DATA M(1.2),M(2.1),X(1 ) ,X(2),G(1.1),G(2.1)/6*0 0/ C C INITIALIZATION C M(1.1)=DBLE(PO) M(2,2)=DBLE(PO) D0=DBLE(0O) DR=DBLE(RO) C C CALCULATE PARAMETERS C CALL PAR(NXI.PHI.T.DEC,C,L2.MODE) CALL GETPAR(C,DO.DO.DR,PHI,T) C C BEGIN PROCESSING FOR FIRST BATCH C CON«FLOAT(NSAMPL) LIMIT=MXLIM*1.96/(SORT(CON)) E-0.0 RSEO(1)»0.0 R1=R 00 50 1=2,NSS K1-I-1 CALL UPDATE(PHI.ZI.I.DEC.C) 30 ESTXI(I)=C*X(2) ESTH(I)=ZI(I)-ESTXI(I) E=E+(ESTH(I)-H(I))**2 ESTRMS(I)=(SQRT(E/FLOAT(I-1)))/10.0 CALL PROP 50 CONTINUE c C COMMENCE ADAPTATION C KBATCH*1 NI-NSS+1 OE(1,1)=DBLE(0O) OE(1,2)=DBLE(QO) RE(1)=DBLE(RO) C C ADAPTATION TO ESTIMATE NEXT OE.RE C 100 CALL ESTOR(KBATCH,NSS.NSAMPL,K1.IN.PHI.00.0,R,MODE) C C AVERAGE OE,RE BY STOCHASTIC APPROXIMATION C K2=KBATCH+1 0E(K2,1)=0E(KBATCH,1) + (Q(1)-OE(KBATCH.1))/FL0AT(K2) QE(K2.2)=QE(KBATCH,2)+(Q(2)-QE(KBATCH,2))/FL0AT(K2) RE(K2)=RE(KBATCH)•(R-RE(KBATCH))/FLOAT(K2) C C RECALCULATE Q.R RELATED PARAMETERS C CALL GETPAR(C.0E(K2,1).OE(K2.2),RE(K2),PHI,T) NMAX=NI+NSAMPL-1 C C FILTERING FOR NEXT BATCH C DO 60 I»NI,NMAX K1=I-1 CALL UPDATE(PHI,ZI,I.DEC,C) ESTXI(I )<=C*X(2) ESTH(I) = ZI(I)-ESTXI(I ) E = E+(ESTH(I)-H(I))* *2 ESTRMS(I ) = (SORT(E/FLOAT(1-1)))/10.0 CALL PROP 60 CONTINUE C C TEST FOR OPTIMALITY C CALL TEST(KBATCH,NSS,NSAMPL.K1.CKOUNT.RHO.LIMIT.ELF) LF(KBATCH)=ELF COUNT(KBATCH)=CK0UNT 65 KBATCH=KBATCH+1 IF(KBATCH .EQ. NBATCH )G0 TO 120 NI=I+1 GO TO 10O 120 CALL TEST(KBATCH,NSS,NSAMPL,K1,CKOUNT.RHO.LIMIT.ELF) LF(KBATCH ) = ELF COUNT ( KBATCH ) -=CKOUNT C 150 RETURN END C C SUBROUTINE : PAR C CALCULATE PARAMTERS OF SYSTEM MATRICES c  C SUBROUTINE PAR(NXI,PHI,T.DEC.C.L2.MODE) C DIMENSION T(2) . PHK2.2). IPERM( 10) DIMENSION PHINV2(2.2),PHINV3(2,2),PHISQ(2.2),HTAU(1,2),HPI(1,2),HPI2(1 . 2) DIMENSION PT(1,2).PITT(2,1 ).P2T( 1 ,2).P2TT(2. 1 ) ,P3TT(2.1) DIMENSION F1(2),F2(2),F3(2),F4(2),F(2.2) COMMON /BLK2/ B,D(2,1),HR(1,2),HT(2.1).RR,PHIDH(2,2),TQT(2,2). DRD(2,2).G(2,2),P(2,2).M(2,4),X(2).RSEQ(2401),PHIDHT(2,2) C0MM0N/BLK3/ HTAUT(2,1).HP(1.2),HPIT(2,1),HPI2T(2,1),HP2( 1.2). PHINV(2.2),PHIT(2,2),PIT(1.2),FI(2,2) LOGICAL MODE REAL'S B . D , HR , HT , RR , PHIDH, TQT , DRD, G , GK , P , M, X , PHi'DHT , RSEQ REAL'S HTAUT.HP.HPIT.HPI2T.HP2.PHINV.PHIT.PIT,FI REAL'S C.T.PHI,PHINV2,PHINV3.PHISQ,HTAU.HP I.HP12.PT.PITT,P2T REAL'S F1.F2.F3.F4.F.DET.COND.P2TT.P3TT 209 REAL L2 C ARG=-1.0/L2 DEC-EXP(ARG) HR(1, 1 )*C*PHI(2, 1) HR( 1 ,2)-C*(PHI(2.2)-DEC) C HT(1,1)«HR(1,1) HT(2, 1)=HR(1.2) C CALL INV(2,2.PHI.I PERM,2,PHINV,DET,JEXP,COND) CALL DGMULT(PHI.PHI,PHI SO,2.2.2,2.2,2) CALL DGMULT(PHINV,PHINV.PHINV2,2.2.2.2.2.2) CALL DGMULT(PHINV2,PHINV,PHINV3.2,2.2.2.2,2) CALL DGTRAN(PHI,PHIT,2,2.2,2) C HTAU( 1 . 1)*HR(1, 1 )*T( 1 ) HTAU(1,2)=HR(1,2 )*T(2 ) HTAUT( 1 . 1)«HTAU(1,1) HTAUT(2,1 )«HTAU(1,2) C C CALL DGMULT(HR.PHI,HP.1,2.2.1.2. 1 ) CALL DGMULT(HR,PHINV.HPI.1.2.2.1,2,1) C PIT(1,1)=HPI(1 . 1)*T( 1) PIT(1,2)=HPI(1,2)*T(2) IF(.NOT. MODE)G0 TO 50 HPIT(1,1)=HPI( 1,1) HPIT(2,1)=HPI(1,2) CALL DGMULT(HR.PHI SO,HP2,1.2.2.1,2,1) CALL DGMULT(HR,PHINV2,HPI2,1.2.2.1.2,1) HPI2T(1,1)=HPI2( 1.1) HPI2T(2.1)*HPI2(1.2) C PT(1,1)=HP(1. 1 )*T( 1 ) PT(1,2)=HP(1.2)*T(2) PITT(1.1 )-PIT( 1,1) PITT(2,1)=PIT(1,2) P2T(1,1)=HPI2(1.1)*T( 1 ) P2T(1,2)»HPI2(1,2)*T(2) P2TT(1. 1 )=P2T( 1,1) P2TT(2. 1)«P2T(1,2) P3TT(1,1)=(HR(1.1)*PHINV3(1,1)+HR(1.2)*PHINV3(2,1))*T(1) P3TT(2.1)-(HR(1,1)*PHINV3(1,2)+HR(1,2)*PHINV3(2.2))*T(2) C F1(1 )«PITT( 1 , 1)+P2TT(1.1)-HTAUT(1,1) F1(2)«=PITT(2, 1 )+P2TT(2. 1 )-HTAUT(2, 1 ) F2( 1 )-P2TT(1, 1 )+P3TT(1,1) F2(2)-P2TT(2,1)+P3TT(2.1) F3(1 )=PT(1,1) F3(2)=PT(1.2) P4(1 )=PITT( 1 . 1)-HTAUT(1,1) F4(2)=PITT(2.1)-HTAUT(2,1) C F(1.1)-HTAU(1,1)*F1(1) F(1,2)=HTAU(1 ,2)*F1(2) F(2,1)-F3(1)*F4(1)+HTAU(1,1)*(F2(1)) F(2,2)=F3(2)*F4(2)+HTAU(1,2)*(F2(2)) CALL INV(2,2,F,IPERM,2,FI.DET,JEXP,COND) C 50 RETURN END c  C SUBROUTINE : GETPAR C UPDATE OF PARAMETERS RELATED TO 0 AND R DURING ADAPTATION c  SUBROUTINE GETPAR(C,01.02,R,PHI.T) C DIMENSION T(2),PHI(2.2) COMMON /BLK2/ B , D( 2 . 1 ) , HR( 1 . 2 ) ,HT ( 2 . 1 ) , RR , PHIDH( 2 , 2 ) . TOT( 2 . ) , DRD(2.2).G(2.2),P(2,2).M(2.4),X(2).RSE0(24O1),PHIDHT( 2 . 2) C REAL'S T.PHI.01.02,R.DENOM,VAR,DRD REAL*8 B,D.HR,HT,RR,PHIDH,TOT.DRD,G,GK,P,M,X,PHIDHT,RSEO DATA TQT(1.29.T0T82.1).D(1.0/3*0.0/ DATA DRD(1,1),DRD(1,2),DRD(2.0/3-0.0/ 210 TQT(1.1)=Q1*T(1)"2 TQT(2,2)=Q2*T(2)"2 RR=R+Q2*(C*T(2))**2 CENOM=1./RR VAR=Q2*C*(T(2)**2) D(2, 1 )=VAR'DENOM C PHIDH(1. 1 )=PHI(1.1) PHIDH(1.2)=PHI(1.2) PHIDH(2.1)«PHI(2.1)-(HR(1.1)»VAR*DEN0M) PHIDH(2.2)=PHI(2.2)-(HR(1,2)*VAR*DENOM) C CALL DGTRAN(PHIDH,PHIDHT,2.2,2.2) DRD(2.2)=-RR*(D(2.1 )"2) C RETURN END c  C SUBROUTINE : ESTQR C ESTIMATION OF 0 AND R C - DEFINITIONS OF VARIABLES C CK INNOVATION CORRELATIONS C CVEC - SET OF INNOVATION CORRELATIONS C C - --SUBROUTINE ESTQR(KBATCH.NSS.NSAMPL.K1.IN,PHI,OO,0,R,MODE) C DIMENSION PHI(2,2),CVEC(100).A(100,2),AT(2.100),AINV(2,100). LAMDA(2,1).OMEGA(2.2),S(2) DIMENSION PGH(2,2).TEMP(2,2),TEMP2(2,2).AA(2.2),AAINV(2,2). IPERM(10).AC(2).PG(2,1).PG(2) DIMENSION GCG(2,2).GLP(2.2).PLG(2.2).GL(2.2).VARK1,2),VAR2(2,1).VAR3(2.1) C COMMON /BLK2/ B,D(2,1).HR(1,2),HT(2.1).RR.PHIDH(2.2),TOT ( 2.2), DRD(2.2),G(2.2),P(2.2),M(2,4),X(2),RSEQ(2401),PHIDHT(2, 2) C0MM0N/BLK3/ HTAUT(2,1),HP(1.2),HPIT(2,1),HPI2T(2.1),HP2(1,2), PHINV(2,2).PHIT(2,2).PIT(1,2).FI(2.2) REAL'S B,D,HR,HT,RR,PHIDH,TOT,DRD,G,GK,P,M,X,PHIDHT,RSEO REAL *8 HTAUT,HP,HPIT.HP12T.HP2,PHINV,PHIT.PIT.FI REAL*8 C.PHI,CK,C0.CVEC,PGH,TEMP,TEMP2.A,AA,AAINV,AINV,AC.PG REAL'S PGC,LAMDA,GL.GLP.PLG.GCG,OMEGA,VAR1.VAR2,VAR3,VAR4 REAL'S VAR5,VAR6,VAR7,B2,B1.DET,COND.0,S,R LOGICAL MODE C C CALCULATE THE CK'5 AND FORM CVEC C C IN1=IN+1 IF(KBATCH .EQ. 1)G0 TO 5 KSS-K1-NSAMPL+2 GO TO 6 5 KSS«K1-NSS+2 6 CALL GETCK(NSAMPL,KSS,1,CK) CO=CK C DO 10 1=2.IN1 CALL GETCK(NSAMPL,KSS,I,CK) 11=1-1 CVEC(I 1)«CK 10 CONTINUE C C FORM A AND AINV C PGH(1,1)-PHI(1,1)-GK(1.1)*HR( 1.1) PGH(1.2)=PHI(1,2)-GK(1,1)*HR(1,2) PGH(2.1)=PHI(2.1)-GK(2.1)*HR( 1,1) PGH(2,2)=PHI(2,2)-GK(2,1)*HR(1,2) C CALL DGC0PY(PGH.TEMP,2.2,2.2) C A(1.1)-HP(1,1) A( 1 ,2)=HP(1.2) 21 1 DO 20 1=2. IN IF(I .EQ. 2)G0 TO 25 CALL DGMULT(TEMP.PGH.TEMP2.2,2.2.2.2.2) CALL DGC0PY(TEMP2.TEMP.2,2.2,2) 25 A(I.1)=HR(1.1)*(TEMP(1.1 )*PHI(1,1)+TEMP(1.2)*PHI(2,1)) +HR(1,2)*(TEMP(2,1)*PHI(1.1)+TEMP(2.2)*PHI(2,1)) A( I ,2)=HR(1.1)*(TEMP(1.1)*PHI(1,2) + TEMP(1,2)*PHI(2.2)) +HR(1,2)*(TEMP(2,1)*PHI(1.2)+TEMP(2.2)*PHI(2.2)) 20 CONTINUE C CALL DGPR0D(A,A,AA,2,IN,2,50,50,2,1.0.1) CALL INV(2,2.AA,I PERM.2.AAINV.DET,JEXP.COND) CALL DGPR0D(AAINV.A.AINV,2.2,IN,2.50,2,0.1,10) C C FORM LAMDA C CALL DGMATV(AINV.CVEC.AC,2.IN,2) CALL DGMULT(PHINV,GK,PG,2,2,1,2,2,2) PGC(1)=PG(1.1 )*C0 PGC(2)=PG(2,1)*C0 LAMDA(1.1)=AC(1)+PGC(1) LAMDA(2,1)=AC(2)+PGC(2) IF(.NOT. MODE)G0 TO 50 C c  C FULL ADAPTATION C FORM OMEGA C CALL DGPROD(GK,GK,GCG.2.1.2.2.2,2.0.1.1) DO 30 1=1,2 DO 40 d=1,2 GCG(I,d)=C0*GCG(I,J) 40 CONTINUE 30 CONTINUE C CALL DGPROD(GK,LAMDA,GL,2,1.2,2,2,2,0.1.10) CALL DGPROD(GL,PHI,GLP,2,2,2,2,2,2.0,1.10) CALL DGTRAN(GLP.PLG.2,2,2.2) CALL DGSUB(GCG,GLP,TEMP,2.2,2,2.2) CALL DGSUB(TEMP,PLG.OMEGA,2,2,2.2,2) C C C CALCULATE B1.B2 AND 0 C CALL DGMULT(HR,OMEGA,VAR1.1,2.2,1,2,1) B1 = (LAMDA( 1. 1)-VAR1( 1 , 1))*HPIT( 1. 1)*(LAMDA(2. 1)-VAR1( 1,2))*HPIT(2. 1)-HP( 1, 1 ) * LAMDA( 1,1) -HP(1,2)*LAMDA(2,1) CALL DGMULT(0MEGA,HPI2T,VAR2,2,2,1,2.2,2) CALL DGMULT(OMEGA,HP IT,VAR3.2.2, 1,2,2,2) VAR4 = LAMDA(1, 1)»HPI2T( 1, 1 ) + LAMDA(2, 1)*HPI2T(2. 1) VAR5=LAMDA(1.1)*HP2(1,1)+LAMDA(2,1)*HP2(1,2) VAR6=HR(1,1)*VAR2(1,1)+HR(1,2)*VAR2(2.1) VAR7=HP(1.1)»VAR3(1.1)+HP(1,2)*VAR3(2.1) C B2=VAR4-VAR5-VAR6-VAR7 C 0( 1 ) = FI( 1 , 1 )»B1 + FI( 1 ,2)*B2 0(2) = FI(2, 1)*B1 + FI(2,2)*B2 C GO TO 60 C c  C C CALCULATE S AND R C 50 0(1)=00 0(2)=00 C 60 S(1)=0(1)*HTAUT(1,1) S(2)=0(2)*HTAUT(2.1) C R=C0-(HR(1.1)*LAMDA(1,1)+HR(1,2)*LAMDA(2.1))+(PIT(1,1)*S(1)+PIT(1,2)»S(2)) C lOO RETURN END c  C SUBROUTINE : GETCK C FORM THE INNOVATION CORRELATION C SUBROUTINE GETCK(NSAMPL.KSS,K,CK) C COMMON /BLK2/ B,D(2,1 ) ,HR(1,2),HT(2,1),RR,PHIDH(2,2),TOT(2,2), DRD(2,2),G(2,2),P(2,2),M(2,4),X(2),RSEO(2401),PHIDHT(2,2) REAL'S B,D,HR,HT,RR,PHIDH,TOT.DRD.G.GK,P,M.X,PHIDHT,RSEO REAL'S PROD,CK,FD,SUM C SUM=0.0 INIT-KSS+K-1 NEND=KSS+NSAMPL-1 DO 10 I-INIT.NEND IK=I-K+1 PROD=RSEQ(I)*RSEO(IK) SUM=SUM+PROD 10 CONTINUE FN"FLOAT(NSAMPL) FD'DBLE(FN) CK=SUM/FD C RETURN END C --C SUBROUTINE : TEST C TEST FOR OPTIMAL ITY BY COUNTING OFF-LIMIT CORRELATION C COEFFICIENTS AND CALCULATING THE LIKELIHOOD FUNCTION c  SUBROUTINE TEST(KBATCH,NSS,NSAMPL,K1,PERCNT,RHO,LIMIT,LF) C DIMENSION CVEC(GO),RH0(6O) COMMON /BLK2/ B,0(2 , 1),HR(1,2),HT(2.1),RR,PHIDH(2,2),TOT( 2,2) . DRD(2,2),G(2,2).P(2,2).M(2,4),X(2),RSEO(2401),PHIDHT(2,2) REAL LIMIT,LF REAL*8 CO,CVECCK,VALUE,RATIO INTEGER COUNT REAL'S B,0,HR.HT,RR,PHIDH,TOT,DRD.G,GK,P,M,X,PHIDHT.RSEO C KSS=K1-NSAMPL+2 DO 10 I-1.60 CALL "FTCK(NSAMPL,KSS,I,CK) CVEC(I)=CK lO CONTINUE C COUNT»0 C0=DABS(CVEC(1)) C C FORM CORRELATION COEFFICIENTS AND COUNT OFF-LIMIT VALUES C DO 20 1=1,60 RATIO=CVEC(I)/C0 RHO(I)=SNGL(RATIO) VALUE=ABS(RHO(I)) IF(VALUE .GT. LIMIT ) COUNT"COUNT* 1 20 CONTINUE PERCNT=100.0*(FLOAT(COUNT)/60.0) C C CALCULATION OF LIKELIHOOD FUNCTION C VAR«1.0/B VAR1«ABS(VAR) LF=-(B/NSAMPL)*CO-AL0G(VAR1) C C RETURN END 

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