UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Free air laser strain meter (a feasibility study of a new method of observing tectonic motions) Cannon, Wayne 1970

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1971_A1 C35.pdf [ 7.05MB ]
Metadata
JSON: 831-1.0053452.json
JSON-LD: 831-1.0053452-ld.json
RDF/XML (Pretty): 831-1.0053452-rdf.xml
RDF/JSON: 831-1.0053452-rdf.json
Turtle: 831-1.0053452-turtle.txt
N-Triples: 831-1.0053452-rdf-ntriples.txt
Original Record: 831-1.0053452-source.json
Full Text
831-1.0053452-fulltext.txt
Citation
831-1.0053452.ris

Full Text

THE FREE AIR LASER STRAIN METER (A F e a s i b i l i t y Study of a New Method of Observing Tectonic Motions) by WAYNE HARRY CANNON B.Sc, University of B r i t i s h Columbia, 1963 M.Sc, University of B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of GEOPHYSICS We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1970 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 t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e 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 a n d S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f G E O P H Y S I C S The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a A p r i l 2 0 , 1 9 7 1 11 ABSTRACT T h e o r e t i c a l i n v e s t i g a t i o n i n t o the f e a s i b i l i t y o f making l a s e r s t r a i n meter measurements over l a r g e d i s t a n c e s through the u n c o n t r o l l e d atmosphere i n d i c a t e t h a t such o b s e r v a -t i o n s are l i k e l y p o s s i b l e w i t h p r e s e n t day technology. The o b s e r v i n g d e v i c e , h e r e - i n c a l l e d a f r e e a i r l a s e r s t r a i n meter, would use l a s e r output at three f r e q u e n c i e s , a p p r o p r i a t e l y spaced i n the spectrum, coupled w i t h the d i s p e r s i o n o f the atmosphere to separate " g e o m e t r i c a l " f l u c t u a t i o n s from " r e -f r a c t i v e " f l u c t u a t i o n s o f the o p t i c a l path l e n g t h between the end m i r r o r s o f the s t r a i n meter. The g e o m e t r i c a l f l u c t u a t i o n s can be averaged to r e v e a l t e c t o n i c changes i n d i s t a n c e be-tween the end m i r r o r s . T h e o r e t i c a l e x p r e s s i o n s f o r a v e r a g i n g times, con-f i d e n c e l i m i t s , s t r a i n s e n s i t i v i t y , a c c u r a c y o f o b s e r v a t i o n s , f r i n g e count r a t e , f r i n g e v i s i b i l i t y , a p e r t u r e s i z e and l a s e r power are d e r i v e d i n terms o f the r e l e v a n t p h y s i c a l , geo-p h y s i c a l , and atmospheric parameters. Free a i r l a s e r s t r a i n meters, l i k e c o n v e n t i o n a l l a s e r s t r a i n meters, appear to be capable o f measuring e a r t h s t r a i n over d i s t a n c e s o f s e v e r a l Kms. up to the l i m i t o f frequency s t a b i l i t y o f the l a s e r i n the presence o f moderate atmospheric t u r b u l e n c e . In the f r e e a i r l a s e r s t r a i n meter the l a s e r f r e -quency s t a b i l i t y and not atmospheric e f f e c t s i s the u l t i m a t e f a c t o r l i m i t i n g s t r a i n s e n s i t i v i t y . i i i Over d i s t a n c e s o f s e v e r a l k i l o m e t e r s f r e e a i r l a s e r s t r a i n meters are t h e o r e t i c a l l y able to make s t r a i n measurements more s e n s i t i v e l y , ( ~ -v* 10 X L - 10 ^ ) , and more r a p i d l y , ( o b s e r v a t i o n i n t e r v a l A T ^ 10 minutes - 1 hour ) , than any oth e r d e v i c e , known or proposed, o p e r a t i n g through the u n c o n t r o l l e d atmosphere. The s u c c e s s f u l o p e r a t i o n o f f r e e a i r l a s e r s t r a i n meters would r e p r e s e n t an improvement over p r e s e n t 3 4 methods by a f a c t o r o f 10 - 10 i n s e n s i t i v i t y of o b s e r v a t i o n . The r e a l i z a t i o n o f f r e e a i r l a s e r s t r a i n meters would p r o v i d e g e o p h y s i c i s t s i n the f i e l d s o f geodesy and t e c t o n i c s w i t h a t o o l o f unusual c a p a b i l i t i e s and as such would be r e c o g n i z e d as a major advance. i v TABLE OF CONTENTS Page ABSTRACT i i L IST OF FIGURES v i i ACKNOWLEDGEMENTS v i i i CHAPTER I PREFACE 1 CHAPTER I I INTRODUCTION A. GEODETIC MEASUREMENTS OF TECTONIC MOTIONS 1 . The Geod imete r 4 2 . A c c u r a c y o f Measurement 5 3 . R e s u l t s 7 B. LASER STRAIN METER MEASUREMENTS OF TECTONIC MOTION 1 . The Lase r S t r a i n M e t e r 9 2 . A c c u r a c y o f Measurement 13 3. R e s u l t s 14 C. OUTDOOR OPTICAL INTERFEROMETRY 1 . Y. V a i s a l a 16 2 . K. E. E r i c k s o n 18 CHAPTER I I I FREE AIR LASER STRAIN METERS A. CONCEPTION 22 B. THEORY 1 . S e p a r a t i o n o f " G e o m e t r i c a l " and " R e f r a c t i v e " O p t i c a l Pa th L e n g t h F l u c t u a t i o n s 25 2 . S t a t i s t i c a l A n a l y s i s o f F ree A i r Lase r S t r a i n M e t e r O b s e r v a t i o n s t o Revea l T e c t o n i c M o t i o n s 38 3. S t a t i s t i c a l A n a l y s i s o f F l u c t u a t i o n s i n Ray-Path o f a Laser Beam Pr o p a g a t i n g Through a Tu r b u l e n t Atmosphere a. The Problem o f Random F l i g h t s b. C a l c u l a t i o n o f P r o b a b i l i t y D e n s i t y F u n c t i o n f o r e ( t ) c. C a l c u l a t i o n o f P r o b a b i l i t y D e n s i t y F u n c t i o n f o r 6S ( A t ) 4. Free A i r Laser S t r a i n Meter O b s e r v a t i o n s 5. Accuracy o f Obs e r v a t i o n s a. E r r o r Propagation A n a l y s i s b. Systematic E r r o r s \ c. Random E r r o r s d. Accuracy o f F r i n g e Counting C. FRINGE OBSERVATION 1. Degenerating E f f e c t s o f the Atmosphere on a Laser Beam i 2. Fraunhofer D i f f r a c t i o n f o r a Double Rectangul Aperture 3. F r i n g e V i s i b i l i t y 4. F r i n g e Counting Rate D. POWER REQUIREMENTS CHAPTER IV CONCLUSIONS A. SUMMARY OF RESULTS B. APPLICATIONS OF FREE AIR LASER STRAIN METERS 1. T e c t o n i c Research 2. Earthquake P r e d i c t i o n 3. E n g i n e e r i n g V X Page APPENDIX A - l THE GAS LASER 1. O p e r a t i n g C h a r a c t e r i s t i c s 144 2. F r e q u e n c y S t a b i l i t y 149 APPENDIX A-2 THE REFRACTIVE INDEX OF THE ATMOSPHERE 1. F o r m u l a f o r R e f r a c t i v e I n d e x 154 2. S t a t i s t i c a l P r o p e r t i e s o f A t m o s p h e r i c 158 R e f r a c t i v e I n d e x APPENDIX A-3 MATHEMATICAL APPENDIX 164 APPENDIX A -4 GEOPHYSICAL AND ATMOSPHERIC PARAMETERS 171 APPENDIX A-5 L I S T OF SYMBOLS 177 BIBLIOGRAPHY 180 LIST OF FIGURES F i g u r e Page II--1 Schematic o f C o n v e n t i o n a l Laser S t r a i n Meter 11 II' -2 The V a i s a l a Comparator 17 II--3 The i n t e r f e r o m e t e r o f K. E. E r i c k s o n 20 III--1 Schematic of a Free A i r Laser S t r a i n Meter System Zk I I I -2 P a r a m e t e r i z a t i o n o f the Atmospheric Ray Paths 26 I I I -3 A n a l y s i s o f Free A i r Laser S t r a i n Meter Ob s e r v a t i o n s kZ I I I -4 Angular Parameters o f Atmospheric Ray-Path 51 I I I -5 S t r a i g h t Line Approximation to the Atmospheric Ray-Path 59 I I I -6 Fraunhofer D i f f r a c t i o n from a Double Rectangular Aperture 97 I I I -7 I n t e n s i t y P a t t e r n , I ( x y ) , from Fraunhofer D i f -f r a c t i o n Through a Double Rectangular Aperture 101 I I I -8 Coherence Between Apertures #1 and #2 105 IV -1 Example o f Observed Earthquake S t r a i n P e r c u r s o r 1*K> A - l -1 Frequency D i s t r i b u t i o n o f Laser Output V+7 v i i i ACKNOWLEDGEMENTS I would l i k e to f o r m a l l y acknowledge the encourage-ment and counsel o f Dr. M. W. Ovenden and Dr. D. E. Smylie whose support d u r i n g t h i s r e s e a r c h i s g r e a t l y a p p r e c i a t e d . In a d d i t i o n I would l i k e to thank my c o l l e a g u e , Mr. 0. Jensen, f o r the c o n t r i b u t i o n o f many h e l p f u l c r i t i c i s m s and s u g g e s t i o n s . I am a l s o g r a t e f u l to the Department o f Geophysics, U.B.C. under Dr. R. D. R u s s e l l f o r f i n a n c i a l support d u r i n g t h i s r e s e a r c h . ^Sincere thanks i s owed to Miss J . Kalmakoff f o r a very able and accurate job of t y p i n g . 1 CHAPTER I PREFACE Throughout h i s t o r y advances i n man's a b i l i t y to a c c u r a t e l y measure d i s t a n c e s both atomic and g a l a c t i c have r e s u l t e d i n profound changes i n h i s world view. T h i s i s i l l u s t r a t e d by c o n t r a s t i n g the a n c i e n t Greeks w i t h t h a t o f any o f t h e i r contemporaries. While other Europeans h e l d t h a t the w o r l d was f l a t w i t h "edges" which were to be avoided the Greeks o f the time h e l d the world to be a r i g i d sphere sus-pended i n the space o f the heavens. T h i s enlightenment on b e h a l f o f the Greeks i s owed to Eratosthenes who measured the r a d i u s o f the globe i n about 240 B.C. [ 1 ] . I t i s no a c c i d e n t t h a t E r a t o s t h e n e s accomplished t h i s w h ile head o f the L i b r a r y of A l e x a n d r i a , Egypt, f o r i t was from the E g y p t i a n s , who had developed i t to resurvey the a r a b l e lands o f the N i l e V a l l e y a f t e r i t s annual f l o o d , t h a t the Greeks l e a r n e d the s c i e n c e o f geodesy. Modern g e o d e t i c measurements can e s t a b l i s h the c o o r d i n a t e s o f any l o c a t i o n on the ear t h ' s s u r f a c e to w i t h i n a c i r c l e a few meters i n diameter [ 2 ] . T h i s advance i n accuracy o f measurement r e v e a l s many un d u l a t i o n s i n the shape of the geoi d and has changed man's view of the p l a n e t ' s i n t e r i o r from s t a t i c to dynamic. 2 W h i l e i t i s c o n c e d e d t h a t on t h e b a s i s o f g e o l o g i c e v i d e n c e l a r g e t e c t o n i c m o t i o n s h a v e o c c u r r e d i n t h e e a r t h ' s c r u s t p r o d u c i n g c o r r e s p o n d i n g l y l a r g e c h a n g e s o f g e o d e t i c c o o r d i n a t e s , t i m e d e p e n d e n c e i s n o t a c o n s i d e r a t i o n i n m o s t g e o d e t i c s u r v e y s . T e c t o n i c c h a n g e s i n g e o d e t i c c o o r d i n a t e s , e x c e p t i n c a s e s o f c a t a s t r o p h e , o c c u r a t r a t e s s m a l l e n o u g h t o h a v e b e e n o f n o c o n c e r n t o c l a s s i c a l g e o d e s y . H o w e v e r w i t h t h e d e v e l o p m e n t o f t h e l a s e r s t r a i n m e t e r a n d l o n g b a s e l i n e r a d i o i n t e r f e r o m e t r y t h e o b s e r v a t i o n o f t e c t o n i c m o t i o n s i n r e a l t i m e b o t h l o c a l l y a n d g l o b a l l y i s p o s s i b l e [ 3 ] . T h e s e a d v a n c e s i n man's a b i l i t y t o m e a s u r e d i s t a n c e w i l l o n c e a g a i n a l t e r h i s w o r l d v i e w a n d r e n d e r g e o d e s y a t i m e d e p e n d e n t d i s c i p l i n e . A s p a r t o f t h i s g e n e r a l t r e n d t h i s t h e s i s i s p r e -s e n t e d as a p r o p o s a l t o c o m b i n e t h e p a r t i c u l a r a d v a n t a g e s o f g e o d i m e t e r s , i . e . , o p e r a t i n g a b i l i t y o v e r l o n g d i s t a n c e s t h r o u g h t h e u n c o n t r o l l e d a t m o s p h e r e , w i t h t h o s e o f t h e l a s e r s t r a i n m e t e r , i . e . , h i g h s t r a i n s e n s i t i v i t y r e s u l t i n g f r o m t h e d i r e c t i n t e r f e r e n c e o f p h o t o n s , t o r e a l i z e a " f r e e a i r l a s e r s t r a i n m e t e r " t o b e a p p l i e d t o g e o d e t i c a n d t e c t o n i c s t u d i e s . To i n t r o d u c e t h i s t h e s i s a n d p r e s e n t b a c k g r o u n d i n f o r m a t i o n p e r t i n e n t t o i t s u n d e r s t a n d i n g i t i s n e c e s s a r y t o b r i e f l y d i s c u s s t h e two m e t h o d s o f o b s e r v i n g t e c t o n i c m o t i o n s , t h e g e o d e t i c s u r v e y m e t h o d a n d t h e l a s e r s t r a i n m e t e r m e t h o d ; a n d i n a d d i t i o n d i s c u s s two n o t a b l e a c h i e v e m e n t s i n t h e f i e l d o f o u t d o o r o p t i c a l i n t e r -f e r o m e t r y . The p r e s e n t a t i o n o f t h e t h e s i s h a s b e e n a somewhat d i f f i c u l t t a s k as i t d r a w s f r o m s e v e r a l d i v e r s e a r e a s o f p h y s i c s . I h a v e c h o s e n t o s i m p l i f y t h e p r e s e n t a t i o n b y i n c l u d i n g s e v e r a l c a l c u l a t i o n s a n d s u p p o r t i n g a r g u m e n t s i n a s e r i e s o f a p p e n d i c e s t o w h i c h t h e r e a d e r c a n r e f e r i f s o moved b y c u r i o s i t y o r d o u b t . 4 CHAPTER I I INTRODUCTION A. GEODETIC MEASUREMENTS OF TECTONIC MOTIONS 1. The G e o d i m e t e r The g e o d i m e t e r [4] i s an e l e c t r o - o p t i c a l d i s t a n c e m e a s u r i n g d e v i c e w h i c h u s e s t h e t r a n s i t t i m e o f a l i g h t s i g n a l b e t w e e n s o u r c e a n d r e c e i v e r t o m e a s u r e d i s t a n c e . The g e o d i -m e t e r e m i t s a c o l l i m a t e d l i g h t beam whose i n t e n s i t y i s modu-l a t e d a t r a d i o f r e q u e n c i e s b y means o f a K e r r c e l l l i n k e d t o a c r y s t a l o s c i l l a t o r . The l i g h t beam i s d i r e c t e d t o a d i s t a n t r e t r o - r e f l e c t i n g d e v i c e a n d t h e r e t u r n e d m o d u l a t e d l i g h t beam i s p h o t o - d e t e c t e d b y a r e c e i v e r on t h e g e o d i m e t e r . The t o t a l p h a s e d i f f e r e n c e b e t w e e n t h e e m i t t e d a n d r e c e i v e d m o d u l a t e d l i g h t s i g n a l s i s p r o p o r t i o n a l t o t h e r o u n d t r i p d i s t a n c e b e t w e e n g e o d i m e t e r a n d r e f l e c t o r . T h i s p h a s e d i f -f e r e n c e i s m e a s u r e d e l e c t r o n i c a l l y t o w i t h i n a m u l t i p l e o f 2TT a n d t h u s t h e e m i t t e r - r e c e i v e r d i s t a n c e i s d e t e r m i n e d t o w i t h i n an i n t e g r a l number o f w a v e l e n g t h s o f t h e m o d u l a t e d beam. R e p e a t i n g t h e m e a s u r e m e n t a t s e v e r a l m o d u l a t i o n f r e -q u e n c i e s r e d u c e s t h e a m b i g u i t y i n t h e o r d e r number o f i n t e r -f e r e n c e b e t w e e n t h e e m i t t e d a n d r e c e i v e d s i g n a l a n d h e n c e r e d u c e s t h e a m b i g u i t y i n t h e e m i t t e r - r e c e i v e r d i s t a n c e . I n g e n e r a l , o b s e r v a t i o n s a t t h r e e c l o s e l y s p a c e d m o d u l a t i o n f r e -q u e n c i e s a l o n g w i t h k n o w l e d g e o f t h e a p p r o x i m a t e e m i t t e r -r e c e i v e r d i s t a n c e o b t a i n e d f r o m maps, a e r i a l p h o t o g r a p h s , o r 5 rougher surveys are s u f f i c i e n t to allow a p r e c i s e measure of the r e q u i r e d d i s t a n c e . Geodimeters can work e f f e c t i v e l y up to d i s t a n c e s o f 20-50 km. 2. Accuracy o f Measurement The l a r g e s t s i n g l e source of e r r o r i n a l l t e r r e s t r i a l e l e c t r o n i c d i s t a n c e measurement i s the u n c e r t a i n t y i n the mean s i g n a l p r o p a g a t i o n v e l o c i t y due to an u n c e r t a i n t y i n the mean group r e f r a c t i v e index value o f the atmosphere along the propa-g a t i o n p a t h . The mean r e f r a c t i v e index value along the beam i s o f t e n e s t i m a t e d from measurements o f atmospheric p r e s s u r e , tempera-tu r e and humidity taken at one or more p o i n t s along the l i g h t p a t h . Under f a v o u r a b l e m e t e o r o l o g i c a l c o n d i t i o n s d i s t a n c e measurement r e p r o d u c i b l e to one or two p a r t s i n 10 6 can be made. T h i s c o r -responds to an e r r o r o f ±1-2 cm. i n a 10 km. d i s t a n c e . More commonly, however, survey c l o s u r e e r r o r s o f s e v e r a l p a r t s i n 10 6 are encountered [ 5 ] . A more s o p h i s t i c a t e d and p o t e n t i a l l y more acc u r a t e method o f measuring the mean r e f r a c t i v e index along the l i g h t path i s through the o b s e r v a t i o n o f the r e l a t i v e atmospheric d i s p e r s i o n o f two or three d i f f e r e n t c o l o r geodimeter beams. T h i s method r e f l e c t s a true s p a t i a l average of the r e f r a c t i v e index along the l i g h t path and hence i s i n p r i n c i p l e s u p e r i o r to the method o f spot sampling of atmospheric p r e s s u r e , tempera-tu r e and humidity. 6 In t h e c a s e o f d r y a i r t h e m a g n i t u d e o f t h e r e l a -t i v e d i s p e r s i o n b e t w e e n g e o d i m e t e r beams o f two d i f f e r e n t c o l o r s i s p r o p o r t i o n a l t o t h e mean a i r d e n s i t y a n d h e n c e t h e mean r e f r a c t i v e i n d e x a l o n g t h e l i g h t p a t h . T h e o b s e r v a t i o n o f t h i s d i s p e r s i o n g i v e s s u f f i c i e n t i n f o r m a t i o n t o c a l c u l a t e t h e s p a t i a l a v e r a g e d e n s i t y f o r d r y a i r a n d h e n c e t h e d e s i r e d mean r e f r a c -t i v e i n d e x . I n p r a c t i c e h o w e v e r , t h e a t m o s p h e r e c o n t a i n s v a r i a b l e amounts o f w a t e r v a p o u r w i t h a d i s p e r s i o n d i f f e r e n t f r o m t h a t o f a i r [ A - 2 ] , T h e t w o - c o l o r d i s p e r s i o n m e t h o d , by n e g l e c t i n g t h e \\ p r e s e n c e o f w a t e r v a p o u r , i s s u b j e c t t o e r r o r s i n t h e e s t i m a t e d \ v a l u e o f mean r e f r a c t i v e i n d e x o f up t o s e v e r a l p a r t s i n 1 0 6 d e p e n d i n g on a t m o s p h e r i c c o n d i t i o n s [ 7 ] . T h i s s h o r t c o m i n g o f i t h e t w o - c o l o r d i s p e r s i o n m e t h o d c a n be r e d u c e d b y s p o t s a m p l i n g t h e h u m i d i t y a t p o i n t s a l o n g t h e l i g h t p a t h . T he t w o - c o l o u r d i s p e r s i o n m e t h o d s u p p l e m e n t e d w i t h an i n d e p e n d e n t m e a s u r e -j ment o f t h e mean h u m i d i t y a l o n g t h e l i g h t beam c a n i n p r i n c i p l e p r o v i d e an e s t i m a t e o f t h e mean r e f r a c t i v e i n d e x a l o n g t h e r a y p a t h a c c u r a t e t o one p a r t i n 1 0 7 [ 7 ] . Th e a d d i t i o n o f a t h i r d c o l o r w i l l r e n d e r t h e d i s -p e r s i o n m e t h o d s e n s i t i v e t o s p a t i a l a v e r a g e s a l o n g t h e beam o f p r e s s u r e , t e m p e r a t u r e , a n d h u m i d i t y ; t h u s a l l o w i n g an e s t i -m a te o f t h e mean r e f r a c t i v e i n d e x a c c u r a t e t o 2-3 p a r t s i n 1 0 8 [ 7 ] . T h e t h r e e - c o l o r d i s p e r s i o n m e t h o d i g n o r e s v a r i a t i o n s i n C 0 2 c o n t e n t o f t h e a t m o s p h e r e , t h e n e x t m o s t s i g n i f i c a n t a t m o s -p h e r i c v a r i a b l e a f t e r w a t e r v a p o u r . H o w e v e r t h e v a r i a t i o n s i n 7 C 0 2 c o n t e n t e x p e c t e d t o be e n c o u n t e r e d i n t h e u n c o n t r o l l e d a t m o s p h e r e a r e s o s m a l l as t o c a u s e r e f r a c t i v e i n d e x v a r i a -t i o n s o f o n l y one p a r t i n 1 0 9 [ 2 1 ] . A f u r t h e r l i m i t a t i o n on t h e a c c u r a c y o f g e o d i m e t e r l e n g t h m e a s u r e m e n t i s t h e u n c e r t a i n t y i n t h e l i g h t beam modu-l a t i o n f r e q u e n c y . T h e f r e q u e n c y s t a b i l i t y o f g e o d i m e t e r o s c i l -l a t o r s i s a p p r o x i m a t e l y one p a r t i n 1 0 7 a n d s o g e o d i m e t e r m e a s u r e m e n t s t o h i g h e r a c c u r a c i e s a r e n o t p o s s i b l e w i t h o u t a c a l i b r a t i o n o f t h e i n s t r u m e n t a g a i n s t some more a c c u r a t e o s c i l -l a t o r . S i n c e t h e u s e o f g e o d i m e t e r s t o o b s e r v e t e c t o n i c m o t i o n s r e q u i r e s r e p e a t e d m e a s u r e m e n t s o f a g i v e n d i s t a n c e s e p a r a t e d b y a n i n t e r v a l o f s e v e r a l m o n t h s , o s c i l l a t o r s t a b i l i t y i s an i m p o r t a n t c o n s i d e r a t i o n i n s u c h o b s e r v a t i o n s . 3. R e s u l t s One o f t h e m o s t t h o r o u g h a n d e x t e n s i v e p r o g r a m s o f m e a s u r i n g t e c t o n i c m o t i o n s by t h e g e o d e t i c s u r v e y m e t h o d h a s b e e n c a r r i e d o u t c o n t i n u o u s l y s i n c e 1959 i n C a l i f o r n i a b y t h e S t a t e o f C a l i f o r n i a D e p a r t m e n t o f W a t e r R e s o u r c e s [ 8 ] , I n a p r o g r a m i n t e n d e d t o a i d t h e d e s i g n a n d l a y o u t o f t h e a q u a d u c t s f o r t h e two b i l l i o n d o l l a r C a l i f o r n i a S t a t e W a t e r P r o j e c t a n e t w o r k o f o v e r 3000 km. o f g e o d i m e t e r l i n e s h a s b e e n l a i d o u t i n a c r i s s - c r o s s f a s h i o n o v e r t h e S a n A n d r e a s a n d r e l a t e d f a u l t s y s t e m s . G e o d e t i c l e n g t h d e t e r m i n a t i o n s o f t h e s e l i n e s a r e 8 r e p e a t e d a t i n t e r v a l s o f t h e o r d e r o f m o n t h s . A t m o s p h e r i c r e f r a c t i v e i n d e x c o r r e c t i o n s t o t h e g e o d i m e t e r m e a s u r e m e n t s a r e made f r o m d a t a o b t a i n e d b y s p o t s a m p l i n g o f t h e a t m o s p h e r i c p r e s s u r e t e m p e r a t u r e a n d h u m i d i t y a t p o i n t s a l o n g t h e g e o d i -m e t e r l i n e . T h e a c c u r a c y o f m e a s u r e m e n t as i n d i c a t e d b y c l o s u r e e r r o r s i s t y p i c a l l y one p a r t i n 10 6, a l t h o u g h t h i s f i g u r e c a n v a r y somewhat a c c o r d i n g t o o b s e r v i n g c o n d i t i o n s . Movement on t h e C a l i f o r n i a f a u l t s y s t e m i s f o u n d t o v a r y , i n a r a t h e r c o m p l e x m a n n e r f r o m l o c a t i o n t o l o c a t i o n , f r o m z e r o t o o v e r 4 c m . / y e a r . C h a n g e s i n t h e r a t e o f f a u l t movement are f o u n d t o be r e l a t e d t o t h e o c c u r r e n c e o f e a r t h -q u a k e s w i t h i n t h e r e g i o n s . A n o m a l o u s l e n g t h c h a n g e s o f g e o -d i m e t e r l i n e s g e n e r a l l y f o r e s h a d o w l o c a l e a r t h q u a k e s . T h e r e s u l t o f t h e C a l i f o r n i a m e a s u r e m e n t s s t r o n g l y i n d i c a t e t h a t l a r g e movements a n d h i g h r a t e s o f movement p r e c e d e an e a r t h -q u a k e b y h o u r s o r d a y s a n d t h a t c o n t i n u o u s o r n e a r c o n t i n u o u s m o n i t o r i n g o f s u s p e c t e d e a r t h q u a k e s i t e s c o u l d be u s e d t o f o r e c a s t e a r t h q u a k e s a n d i s s u e s h o r t - r a n g e w a r n i n g s . The d e v e l o p m e n t o f t w o - c o l o r d i s p e r s i o n l a s e r g e o -d i m e t e r s y s t e m s h a s r e a c h e d t h e f i e l d t e s t i n g s t a g e [9] a n d a l t h o u g h a c o m p l e t e t h r e e - c o l o r d i s p e r s i o n s y s t e m h a s n o t y e t o p e r a t e d t h e e l e m e n t s o f s u c h a s y s t e m h a v e b e e n u s e d s e p a r a t e l y [7]. S u c h d e v i c e s c a n be e x p e c t e d t o b e c a p a b l e o f a u t o m a t i c a n d c o n t i n u o u s s t r a i n m o n i t o r i n g w i t h s t r a i n s e n s i t i v i t i e s o f 10"7 - 10"8 i n t h e n e a r f u t u r e . 9 B. LASER STRAIN METER MEASUREMENTS OF TECTONIC MOTIONS 1. The Laser S t r a i n Meter La s e r s t r a i n meters g e n e r a l l y c o n s i s t o f a unimodal frequency s t a b i l i z e d l a s e r ; the output of which i s d i r e c t e d i n t o an i n t e r f e r o m e t e r whose r e f l e c t i n g elements are f i x e d to the s o l i d e a r t h . The i n t e r f e r o m e t e r i s coupled to a readout system which monitors the i n t e r f e r e n c e f r i n g e p a t t e r n . A change i n the l e n g t h o f the i n t e r f e r o m e t e r caused by e a r t h s t r a i n i s accompanied by a change i n the order number o f the i n t e r f e r e n c e ; and i s measured d i r e c t l y i n terms o f the wave-l e n g t h o f the l a s e r output by c o u n t i n g f r i n g e s . Automatic and continuous m o n i t o r i n g o f the i n t e r f e r e n c e f r i n g e s r e s u l t s i n a r e c o r d o f e a r t h s t r a i n as a f u n c t i o n o f time. L a s e r s t r a i n meters are c o n s t r u c t e d u s i n g both the Mi c h e l s o n and Fabrey-Perot type i n t e r f e r o m e t e r s . One method o f u s i n g the M i c h e l s o n scheme i n v o l v e s the c o n s t r u c t i o n o f an equal-arm i n t e r f e r o m e t e r i n which the m i r r o r c o n s t i t u t i n g one arm i s not coupled to the e a r t h but h e l d r i g i d w h i l e the m i r r o r c o n s t i t u t i n g the oth e r arm i s f i x e d to the s o l i d e a r t h and i s f r e e to respond to e a r t h s t r a i n . Any r e s u l t i n g i n e q u a l i t i e s i n the len g t h s o f the arms o f the i n t e r f e r o m e t e r produce changes i n the f r i n g e p a t t e r n from which changes i n ambient s t r a i n c o n d i t i o n s are i n f e r r e d . 1 0 T h e F a b r e y - P e r o t i n t e r f e r o m e t e r i s m o s t o f t e n u s e d i n t h e c o n s t r u c t i o n o f l a s e r s t r a i n m e t e r s . I n one i l l u s t r a -t i o n o f t h e F a b r e y - P e r o t s c h e m e , F i g . I I - l , b o t h m i r r o r s a r e f i x e d t o t h e s o l i d e a r t h a n d a n e v a c u a t e d p i p e i n t e r p o s e d b e t -ween them l e a v i n g o n l y s m a l l a i r g a p s a t e a c h e n d . I n t h i s scheme t h e l a s e r beam t r a v e r s e s an e v a c u a t e d p a t h b e t w e e n t h e m i r r o r s . T h i s p r e v e n t s c h a n g e s i n t h e o p t i c a l p a t h l e n g t h b e t -ween t h e e n d m i r r o r s c a u s e d b y p r e s s u r e , t e m p e r a t u r e , a n d h u m i d i t y f l u c t u a t i o n s f r o m b e i n g o b s e r v e d as " a p p a r e n t " s t r a i n , i n d i s t i n g u i s h a b l e f r o m " r e a l " s t r a i n b y s u c h an i n s t r u m e n t . T h e a i r g a p s b e t w e e n t h e e n d m i r r o r s a n d t h e t r a n s p a r e n t e n d s o f t h e e v a c u a t e d p i p e a r e k e p t as s m a l l as p o s s i b l e f o r t h e same r e a s o n . M o s t l a s e r s t r a i n m e t e r s u s e a r e t r o - r e f l e c t i n g d e v i c e ; o f t e n a p l a n e m i r r o r m o u n t e d a t t h e f o c u s o f a c o n v e r g i n l e n s ; a s t h e e n d e l e m e n t o f t h e F a b r e y - P e r o t r e s o n a t o r . T h i s i n s u r e s t h a t t h e beam w i l l a l w a y s be r e t u r n e d t o i t s p r o p e r p o s i t i o n on t h e f r o n t e l e m e n t o f t h e F a b r e y - P e r o t e t a l o n i n s p i t e o f m i s a l i g n m e n t s w h i c h o c c u r due t o t e c t o n i c a c t i v i t y . When t h e l a s e r i s c o u p l e d t o t h e i n t e r f e r o m e t e r i t i s d e s i r a b l e t o r e d u c e t h e l i g h t r e f l e c t e d f r o m t h e o p t i c a l s u r f a c e s o f t h e i n t e r f e r o m e t e r b a c k i n t o t h e l a s e r t o a minimum. A l l o w i n g r e f l e c t e d l i g h t t o r e - e n t e r t h e l a s e r c a n r e s u l t i n t h e s t i m u l a t i o n o f u n w a n t e d modes o f o s c i l l a t i o n a n d must t h e r e f o r e be s u p p r e s s e d . T h i s i s g e n e r a l l y done i n l a s e r s t r a i n m e t e r s b y d e l i b e r a t e s l i g h t m i s a l i g n m e n t o f t h e i n t e r -f e r o m e t e r s u r f a c e s as w e l l as b y t h e u s e o f q u a r t e r - w a v e BEAM SPLITTER EVACUATED PIPE RETROREFLECTOR DETECTOR FIG. I I - l SCHEMATIC OF CONVENTIONAL LASER STRAIN METER 12 c o a t i n g s . Q u a r t e r - w a v e c o a t i n g s a r e e f f e c t i v e i n s u p p r e s s i n g r e f l e c t i o n s f r o m t h e o p t i c a l s u r f a c e s b e c a u s e o f t h e h i g h m o n o c h r o m a t i c i t y o f t h e l a s e r o u t p u t . The f r i n g e c o u n t e r u s u a l l y e m p l o y s two p h o t o m u l -t i p l i e r t u b e s f o c u s s e d on o p p o s i t e s i d e s o f a f r i n g e . As t h e f r i n g e s move i n r e s p o n s e t o e a r t h s t r a i n t h e o u t p u t o f t h e p h o t o m u l t i p l i e r b e c o m e s u n b a l a n c e d , t h e m a g n i t u d e a n d s i g n o f t h e i m b a l a n c e b e i n g a m e a s u r e o f t h e f r i n g e d i s p l a c e m e n t . A means i s p r o v i d e d f o r t h e f r i n g e c o u n t e r t o move r a p i d l y t o a s u c c e s s i v e f r i n g e when t h e f r i n g e d i s p l a c e m e n t r e a c h e s a g i v e n l i m i t , u s u a l l y one f r i n g e w i d t h . F r i n g e c o u n t e r s c a p a b l e o f c o u n t i n g t o one h u n d r e d t h p a r t o f a f r i n g e a n d a s r a p i d l y as a m e g a c y c l e h a v e b e e n b u i l t . Two F a b r e y - P e r o t i n t e r f e r o m e t e r s a t r i g h t a n g l e s p r o v i d e a s e t o f r e f e r e n c e a x e s r e l a t i v e t o w h i c h c o m p o n e n t s exl a n d e 2 2 o f t h e s t r a i n t e n s o r e... c a n b e m e a s u r e d . T h e o f f d i a g o n a l c o m p o n e n t s e 1 2 a n d e 2 1 c a n b e m e a s u r e d b y i n t r o d u c i n g a t h i r d s t r a i n m e t e r o r i e n t e d a t 4 5 ° t o t h e f i r s t t w o . L a s e r s t r a i n m e t e r s e x h i b i t some n o t a b l e a n d a d v a n -t a g e o u s p r o p e r t i e s n o t commonly f o u n d i n m e a s u r i n g d e v i c e s . T h e l a s e r s t r a i n m e t e r r e s p o n d s l i n e a r l y t o e a r t h s t r a i n o v e r a n " i n f i n i t e " r a n g e o f a m p l i t u d e . T h e y a l s o e x h i b i t a f l a t 1 3 amplitude response with zero phase s h i f t over a l l f r e q u e n c i e s of t e c t o n i c a c t i v i t y from s e c u l a r s t r a i n ( D . C . ) up to the h i g h e s t f r e q u e n c i e s observable by the readout system. 2. Accuracy o f Measurement S t r a i n , b e i n g d e f i n e d o n l y a t a p o i n t , cannot be s t r i c t l y observed by any extended d e v i c e . The l a s e r s t r a i n meter measures the average s t r a i n over an extended r e g i o n comparable i n dimensions to the l e n g t h o f the s t r a i n meter arms. Although s h o r t e n i n g the arms of the s t r a i n meter f a c i -l i t a t e s a t r u e r measure of p o i n t s t r a i n i t a l s o r e s u l t s i n a lower s t r a i n s e n s i t i v i t y . In a d d i t i o n the q u a n t i t y u s u a l l y of g e o p h y s i c a l i n t e r e s t i s the ambient s t r a i n f i e l d which i s i measured b e s t by a s t r a i n meter of c o n s i d e r a b l e e x t e n s i o n . For these r e a s o n s j l a s e r s t r a i n meters range i n l e n g t h from a few meters to 1 km. i j The accuracy o f measurement o f l a s e r s t r a i n meters i s l i m i t e d mainly by the s t a b i l i t y o f the l a s e r frequency S V ( T ) [ A - l , eqn. A - l - 4 ] . I f D i s the l e n g t h between the m i r r o r s o f a Fabrey-Perot l a s e r s t r a i n meter and A D ( T ) i s the change i n the l e n g t h D i n an i n t e r v a l of time T ; then the accuracy w i t h which the s t r a i n , A ^ T ^ , i s determined cannot exceed S v ( t ) = ^.V^T) . Thus l a s e r s t r a i n meters p r e s e n t l y have s h o r t p e r i o d s t r a i n s e n s i t i v i t i e s as h i g h as one p a r t i n 1 0 1 2 [14] and long term s t r a i n s e n s i t i v i t i e s of one p a r t i n 1 0 1 0 [15]. 14 3. R e s u l t s L a s e r s t r a i n m e t e r s a r e a b l e t o o b s e r v e m o s t a l l o f t h e s e i s m i c a c t i v i t y n o r m a l l y r e c o r d e d b y s t a n d a r d s e i s m i c e q u i p m e n t s u c h as e a r t h q u a k e e v e n t s a n d m i c r o s e i s m s . H o w e v e r b e c a u s e o f t h e i r f l a t r e s p o n s e t o t e c t o n i c a c t i v i t y down t o z e r o f r e q u e n c y ( s e c u l a r s t r a i n ) t h e y h a v e b e e n m o s t u s e f u l i n s t u d y i n g l o n g t e r m , l o w f r e q u e n c y t e c t o n i c m o t i o n s . I n t h i s r e g a r d t h e y a r e u n r i v a l l e d a n d c o n s e q u e n t l y h a v e r e v e a l e d a v a r i e t y o f t e c t o n i c p h enomena, some o f w h i c h a r e n o t o b s e r v a b l e by o t h e r d e v i c e s . L a s e r s t r a i n m e t e r s w h i c h a r e s e t up s t r a d d l i n g known f a u l t z o n e s a r e a b l e t o o b s e r v e movement on t h e f a u l t i n r e a l t i m e . T h e s e m e a s u r e m e n t s r e v e a l t h a t c r e e p on t h e f a u l t i s e p i s o d i c , a p p e a r i n g p e r i o d i c a l l y w i t h a b a c k - a n d - f o r t h m o t i o n s u p e r - i m p o s e d on i t [ 1 4 ] . L a s e r s t r a i n m e t e r s a r e u s e d t o o b s e r v e t i d e s i n t h e s o l i d e a r t h c a u s e d b y l u n a r g r a v i t a t i o n a l a t t r a c t i o n . E a r t h t i d e s a r e o b s e r v e d t o h a v e a 12 h o u r p e r i o d w i t h a s t r a i n a m p l i t u d e o f a b o u t 5 p a r t s i n 1 0 8 [ 1 6 ] w h i c h c o r r e s p o n d s r o u g h l y t o a v e r t i c a l a m p l i t u d e a t t h e e q u a t o r o f t h e e a r t h o f 30 cm. The a m p l i t u d e o f e a r t h t i d e s i s o b s e r v e d b y l a s e r s t r a i n m e t e r s t o be a b o u t t e n t i m e s l a r g e r t h a n n o r m a l i n t h e v i c i n i t y o f a l a r g e f a u l t c o n f i r m i n g t h e o r e t i c a l p r e d i c t i o n s o f t h e b u i l d up o f s t r a i n a r o u n d f a u l t s i n t h e e a r t h ' s c r u s t . L a s e r s t r a i n m e t e r s h a v e a l s o b e e n u s e d t o o b s e r v e f u n d a m e n t a l modes o f o s c i l l a t i o n o f t h e e a r t h f o l l o w i n g l a r g e e a r t h q u a k e s . 15 T e c t o n i c p r o c e s s e s w h i c h a r e a t t r i b u t e d t o l o c a l m o u n t a i n b u i l d i n g p r o c e s s e s a r e a l s o o b s e r v e d by l a s e r s t r a i n m e t e r s l o c a t e d i n C o r d i l l e r a r e g i o n s [ 1 6 ] . T h e s e p h e n o m e n a a p p e a r as m i c r o - t r e m o r s a c c o m p a n i e d b y u n r e c o v e r e d s t e p c h a n g e s i n t h e a m b i e n t s t r a i n o f a few p a r t s i n 1 0 9 . S u c c e s s i v e s t r a i n s t e p s u s u a l l y h a v e t h e same s i g n w h i c h i s i n d i c a t i v e o f some c u m u l a t i v e g e o l o g i c p r o c e s s . One o f t h e m o s t i n t e r e s t i n g p henomena r e v e a l e d b y l a s e r s t r a i n m e t e r s a r e r e s i d u a l u n r e c o v e r e d s t r a i n s o f t h e o r d e r o f p a r t s i n 1 0 8 o b s e r v e d a t d i s t a n c e s o f t h o u s a n d s o f km. f r o m e a r t h q u a k e e p i c e n t r e s w h i c h a p p e a r t o b e d i r e c t l y a s s o c i a t e d w i t h t h e s e i s m i c e v e n t [ 1 7 ] , R e c e n t e v i d e n c e s u g -g e s t s t h a t a r a p i d i m p o s i t i o n o f s t r a i n a t t e l e s e i s m i c d i s -t a n c e s may p r e c e d e e a r t h q u a k e s b y p e r i o d s r a n g i n g f r o m s e v e r a l m i n u t e s t o s e v e r a l h o u r s . I n t h i s r e g a r d V a l i a n d B o s t r o m [15] r e p o r t o b s e r v i n g o n a l a s e r s t r a i n m e t e r l o c a t e d i n W a s h i n g t o n S t a t e U.S.A. an a p p r o x i m a t e l y l i n e a r a c c u m u l a t i o n o f s t r a i n a m o u n t i n g t o 49 p a r t s i n 1 0 9 o v e r a p e r i o d o f 30 m i n u t e s p r e c e d i n g a C e n t r a l A m e r i c a n e a r t h q u a k e o f m a g n i t u d e 5.7. E x t e n d i n g l a s e r s t r a i n m e t e r s , as t h e y a r e c u r r e n t l y c o n c e i v e d , t o l e n g t h s o f s e v e r a l km. w o u l d f a c i l i t a t e a m e a s u r e o f t h e r e l a t i v e t e c t o n i c m o t i o n o f a d j a c e n t c r u s t a l b l o c k s o f u n p r e c e d e n t e d a c c u r a c y ; e x c e e d i n g t h e a c c u r a c y o f t h e g e o d e t i c m e a s u r e m e n t s b y a f a c t o r o f 1 0 3 - 1 0 s d e p e n d i n g on t h e l e n g t h 16 of time over which the o b s e r v a t i o n i s made. T h i s i s not ach i e v e d i n p r a c t i c e simply because of the t e c h n i c a l d i f f i c u l t i e s and c o s t i n v o l v e d i n c o n s t r u c t i n g evacuated systems s e v e r a l km. i n l e n g t h . The encumberance of the evacuated p i p e extending the f u l l d i s t a n c e between the p o i n t s over which r e l a t i v e d i s -placements are to be measured has l i m i t e d l a s e r s t r a i n meters to lengths o f 1 km. or l e s s . C. OUTDOOR OPTICAL INTERFEROMETRY ' \ The o b s e r v a t i o n o f i n t e r f e r e n c e between two l i g h t beams which have t r a v e r s e d an a p p r e c i a b l e d i s t a n c e along separate paths through the u n c o n t r o l l e d atmosphere ( f r e e - a i r ) i . i s d i f f i c u l t because o f atmospheric t u r b u l e n c e . F l u c t u a t i o n s 1 i n atmospheric c o n d i t i o n s are accompanied by c o r r e s p o n d i n g i; f l u c t u a t i o n s i n r e f r a c t i v e index. The atmospheric r e f r a c t i v e index f l u c t u a t i o n s produce s e v e r a l d e g e n e r a t i n g e f f e c t s i n the l i g h t beams which reduce i n t e r f e r e n c e e f f e c t s to l e v e l s which are o f t e n below observable l i m i t s . In s p i t e o f the d i f f i c u l t i e s i n h e r e n t i n f r e e - a i r i n t e r f e r o m e t r y at o p t i c a l wavelengths, f r e e - a i r i n t e r f e r o m e t r y has been a p p l i e d by s e v e r a l workers to problems i n geodesy and meteorology. The work of Y. V a i s a l a and K. E. E r i c k s o n i s n o t a b l e i n t h i s r e g a r d . 1. Y. V a i s a l a In.1929 , Y. V a i s a l a [18] developed a l i g h t i n t e r -f e rence comparator capable o f measuring g e o d e t i c b a s e l i n e s up to lengths o f 0.864 km. through the u n c o n t r o l l e d atmosphere 17 t o a c c u r a c i e s w h i c h a r e c l a i m e d t o be one p a r t i n 1 0 7 . T h e c o m p a r a t o r i s shown s c h e m a t i c a l l y i n F i g . 11-2. L i g h t f r o m a w h i t e i n c a n d e s c e n t s o u r c e L i s c o l l i m a t e d b y a l e n s C a n d d i r e c t e d i n t o two a x i a l l y a l i g n e d F a b r e y - P e r o t r e s o n a t o r s d e f i n e d b y p a i r s o f h a l f s i l v e r e d m i r r o r s M1M2 a n d M 1 M 3 . W h i t e l i g h t i n t e r f e r e n c e f r i n g e s a r e o b s e r v e d i n t h e t e l e s c o p e T when t h e two i n t e r f e r i n g beams h a v e e q u a l o p t i c a l p a t h l e n g t h s . T h i s o c c u r s when t h e d i s t a n c e M1M3 i s an i n t e g r a l m u l t i p l e o f t h e d i s t a n c e MjM2 T h e d i s t a n c e M i M 2 i s e s t a b l i s h e d i n t h e f i e l d b y i n s e r t i n g a q u a r t z g a uge b l o c k 1 m e t e r i n l e n g t h b e t w e e n t h e m i r r o r s . A w h i t e l i g h t s o u r c e i s u s e d f o r a c c u r a c y s i n c e w h i t e l i g h t f r i n g e s d i s a p p e a r i f t h e o p t i c a l p a t h d i f f e r e n c e b e t w e e n t h e beams i s b u t a few w a v e l e n g t h s w h e r e a s a m o n o c h r o m a t i c s o u r c e w i l l p r o d u c e i n t e r f e r e n c e f r i n g e s b e t w e e n beams whose p a t h 18 lengths d i f f e r by many m i l l i m e t e r s . The use o f the V a i s a l a comparator i d e a l l y i n v o l v e s l e v e l t e r r a i n with the l i g h t beam about 1 meter above the ground. Temperature readings are taken s i m u l t a n e o u s l y w i t h the f r i n g e o b s e r v a t i o n s at s e v e r a l p o i n t s along the l i g h t path. The markers between which the comparator measures d i s t a n c e are u s u a l l y s e t i n l a r g e concrete b l o c k s , b u r i e d underground f o r s t a b i l i t y . The comparator can be used i n any weather f o r d i s t a n c e s of up to 0.1 km. but n i g h t - t i m e o b s e r v a t i o n s are needed f o r long b a s e l i n e work. The V a i s a l a comparator has been used to e s t a b l i s h s e v e r a l s t a n d a r d g e o d e t i c b a s e l i n e s around the w o r l d : A r g e n t i n a 0.480 km.; the Netherlands 0.576 km.; Germany 0.864 km, and F i n l a n d 0.864 km. 2. K. E. E r i c k s o n E r i c k s o n (19) has used fees-air i n t e r f e r o m e t r y to i n -v e s t i g a t e the accuracy w i t h which o p t i c a l l e n g t h comparison can be made without c l o s e c o n t r o l or sampling of the atmos-phere. E r i c k s o n ' s t h e o r e t i c a l p r o p o s a l i n v o l v e s the obser-v a t i o n o f channel spectrum i n t e r f e r e n c e between white l i g h t beams which have t r a v e r s e d d i f f e r e n t paths through f r e e - a i r i n the comparator. E r i c k s o n ' s theory i n d i c a t e s the p o s s i b i l i t y o f e s t a b l i s h i n g the o r d e r numbers o f i n t e r f e r e n c e f o r f r i n g e s o f the spectrum by counting the number of f r i n g e s i n the channel 19 s p e c t r u m s e p a r a t i n g t h r e e s t a n d a r d w a v e l e n g t h s . K n o w i n g t h e o r d e r n umbers o f i n t e r f e r e n c e f o r e a c h o f t h e s e t h r e e p o i n t s i n t h e c h a n n e l s p e c t r u m a n d t h e r e f r a c t i v i t y f o r m u l a f o r t h e a t m o s p h e r e he i s t h e n a b l e t o c a l c u l a t e t h e c o n t r i b u t i o n t o t h e i n t e r f e r e n c e o r d e r a r i s i n g o u t o f p u r e l y g e o m e t r i c a l p a t h d i f f e r e n c e s b e t w e e n t h e beams. I t was h o p e d t h a t t h i s m e t h o d a p p l i e d t o a d e v i c e s u c h as a V a i s a l a c o m p a r a t o r w o u l d f a c i -l i t a t e a c c u r a t e d e t e r m i n a t i o n s o f g e o d e t i c b a s e l i n e s by o p t i c a l means n o t r e q u i r i n g a n y m e a s u r e m e n t o f a t m o s p h e r i c p a r a m e t e r s a l o n g t h e l i g h t . p a t h . E r i c k s o n ' s t h e o r y r e q u i r e s t h e r e l a t i v e r e f r a c t i v i t y o f t h e a t m o s p h e r e t o be i n d e p e n d e n t o f a t m o s p h e r i c c o n d i t i o n s . T h i s c o n d i t i o n d o e s n o t h o l d f o r v a r i a b l e a m o u n t s o f w a t e r v a p o u r i n t h e a t m o s p h e r e . As a c o n s e q u e n c e o f t h i s E r i c k s o n ' s i m e t h o d i n v o l v i n g t h r e e w a v e l e n g t h s d o e s n o t f a c i l i t a t e f r e e -a i r l e n g t h c o m p a r i s o n s much more a c c u r a t e t h a n t h o s e a c h i e v e d b y t h e V a i s a l a c o m p a r a t o r , one p a r t i n 1 0 7 , a n d r e q u i r e s a m e a s u r e o f a t m o s p h e r i c w a t e r v a p o u r c o n t e n t a l o n g t h e l i g h t p a t h s . H o w e v e r , i t s h o u l d be n o t e d t h a t E r i c k s o n ' s m e t h o d e x t e n d e d t o f o u r w a v e l e n g t h s c a n c o p e w i t h a t m o s p h e r i c v a r i a -t i o n s i n w a t e r v a p o u r c o n t e n t . T h i s i n v o l v e s d e t e r m i n i n g t h e o r d e r o f i n t e r f e r e n c e a t a f o u r t h p o i n t i n t h e c h a n n e l s p e c t r u m a n d e x t e n d i n g t h e s p e c t r a l r a n g e f a r i n t o t h e u l t r a -v i o l e t . 20 To demonstrate the p r a c t i c a b i l i t y o f h i s theory E r i c k s o n c o n s t r u c t e d a f r e e - a i r i n t e r f e r o m e t e r , shown sche-m a t i c a l l y i n F i g . 11-3. F I G . 11-3 T H E I N T E R F E R O M E T E R O F K . E . E R I C K S O N R a d i a t i o n from the white l i g h t source i s d i v i d e d by the beam s p l i t t e r and each f r a c t i o n o f the beam i s sent i n o p p o s i t e d i r e c t i o n s around an outdoor path of t o t a l l e n g t h 0.115 km. The channel spectrum i n t e r f e r e n c e p a t t e r n i s recorded p h o t o g r a p h i c a l l y by a camera. The beams were w i t h i n 1.5 meters of the ground over t h e i r e n t i r e path where atmos-p h e r i c c o n d i t i o n s are normally l e a s t f a v o u r a b l e . Under con-d i t i o n s too poor to o b t a i n a channel spectrum the f r i n g e s remained v i s i b l e but were unsteady and warped. Under good c o n d i t i o n s the f r i n g e s were v i s i b l e c o n t i n u o u s l y and l o n g , unbroken records of f r i n g e motion were o b t a i n e d . The ap e r t u r e 21 o f E r i c k s o n ' s i n t e r f e r o m e t e r was o f t h e o r d e r o f 3 cm. a c r o s s . I t was n o t e d t h a t u n d e r v e r y u n f a v o u r a b l e c o n d i t i o n s t h e s m a l l s c a l e a t m o s p h e r i c r e f r a c t i v e i n d e x i n h o m o g e n e i t i e s w e r e s u f f i -c i e n t l y i n t e n s e t o d e s t r o y p h a s e c o h e r e n c e a c r o s s t h e a p e r t u r e c a u s i n g t h e i n t e r f e r e n c e f r i n g e s t o r a p i d l y a p p e a r a n d d i s a p p e a r . T h i s e f f e c t was o b s e r v a b l e i n p h o t o g r a p h s o f t h e f r i n g e s t a k e n a t 64 f r a m e s p e r s e c o n d . E r i c k s o n c o n c l u d e s t h a t g o o d c h a n n e l s p e c t r u m f r i n g e s c o u l d be o b t a i n e d o v e r d i s t a n c e s o f 0.5 km. o r m o r e . 22 CHAPTER I I I FREE AIR LASER STRAIN METERS A. CONCEPTION The f r e e a i r l a s e r s t r a i n meter i s a c o n c e p t i o n f o r extending l a s e r s t r a i n meter measurements to d i s t a n c e s o f s e v e r a l k i l o m e t e r s or more through the u n c o n t r o l l e d atmosphere. There are two p r i n c i p l e impediments to the r e a l i z a t i o n o f f r e e a i r l a s e r s t r a i n meters. The f i r s t i s the d i f f i c u l t y o f o b s e r v i n g i n t e r f e r e n c e between o p t i c a l s i g n a l s which have t r a v e r s e d an a p p r e c i a b l e d i s t a n c e along separate paths o f a t u r b u l e n t atmos-phere. The second i s the d i f f i c u l t y o f i n t e r p r e t i n g any i n t e r -f e rence f r i n g e s observed i n terms of t e c t o n i c d i s p l a c e m e n t s . The d i f f i c u l t y o f i n t e r p r e t a t i o n can be overcome by the use o f t h r e e l a s e r s of d i f f e r e n t f r e q u e n c i e s . The d i s p e r s i o n o f the atmosphere at o p t i c a l f r e q u e n c i e s w i l l allow the s e p a r a t i o n o f the f l u c t u a t i o n s i n o p t i c a l path l e n g t h bet-ween the end m i r r o r s i n t o " r e f r a c t i v e " and "geometric" c o n t r i -b u t i o n s . T e c t o n i c a c t i v i t y can be r e v e a l e d by f i l t e r i n g the data and o b s e r v i n g any trends i n the s e r i e s of path l e n g t h f l u c t u a t i o n s . I t i s proposed to overcome the d i f f i c u l t y o f observ-i n g i n t e r f e r e n c e f r i n g e s by the use of a "double r e c t a n g l e " i n t e r f e r o m e t e r and r e s t r i c t i n g the area o f acceptance o f the r e c t a n g u l a r a p e r t u r e to a "coherence p a t c h " s i z e . L i g h t from 23 l a s e r s o u r c e s i s no more immune t o t h e d e g e n e r a t i n g e f f e c t s o f a t m o s p h e r i c t u r b u l e n c e t h a n o p t i c a l r a d i a t i o n f r o m a ny o t h e r s o u r c e , h o w e v e r , l a s e r s o f f e r t h e a d v a n t a g e s o f l a r g e s p a t i a l a n d t e m p o r a l c o h e r e n c e a t t h e o u t p u t a n d h i g h i n t e n s i t y b o t h o f w h i c h w i l l b e u s e f u l i n m i n i m i z i n g o b s e r v a t i o n d i f f i -c u l t i e s . One p h y s i c a l a r r a n g e m e n t f o r a f r e e a i r l a s e r s t r a i n m e t e r , w h i c h makes no c l a i m t o b e i n g o p t i m a l , i s shown i n F i g . I I I - l . I t v c o n s i s t s o f t h r e e f r e q u e n c y s t a b i l i z e d l a s e r s Li , L 2 a n d L 3 w i t h f r e q u e n c i e s V i , v 2 , V 3 r e s p e c t i v e l y whose o u t p u t s a r e c o m b i n e d b y m i r r o r s Mi , M 2 M 3 a n d d i r e c t e d b y m i r r o r Mi» t o a r e t r o - r e f l e c t i n g m i r r o r M 6 a t a d i s t a n c e D . A p o r t i o n o f t h e c o m b i n e d o r i g i n a l o u t p u t o f t h e t h r e e l a s e r s i s t r a n s m i t t e d b y Mi» a n d d i r e c t e d i n t o a p r i s m p i !. . I n a s i m i l a r m a n n e r a p o r t i o n o f t h e com-b i n e d r e t u r n beam i s d i r e c t e d b y m i r r o r M 5 l o c a t e d a t 0 i n t o a p r i s m p 2 . T h e p r i s m s s e p a r a t e t h e w a v e f r o n t s i n t o t h r e e o p t i c a l c h a n n e l s w h i c h b r i n g t hem t o i n t e r f e r e i n p a i r s a t r e c t a n g u l a r d i f f r a c t i n g a p e r t u r e s I i , I 2 a n d I 3 a r e f r i n g e c o u n t e r s C i , C 2 a n d C 3 . T h e f r i n g e c o u n t s i n a s e r i e s o f c o n t i n u o u s c o n s e c u t i v e i n t e r v a l s a r e t h e raw d a t a f r o m w h i c h t e c t o n i c c h a n g e s i n t h e l e n g t h D a r e c a l c u l a t e d b y t h e r e a d o u t s y s t e m . L2 v M i ^ M -M 2 P i I C 2 R E A D O U T S Y S T E M M i R E T U R N B E A M T O RETROREFLEC7 P2 C 3 F I G . 111-1 S C H E M A T I C O F A F R E E A I R L A S E R S T R A I N M E T E R S Y S T E M 2 5 B. THEORY 1. S e p a r a t i o n o f "G e o m e t r i c a l " and " R e f r a c t i v e " O p t i c a l  Path Length F l u c t u a t i o n s C o n s i d e r c a r t e s i a n c o o r d i n a t e axes, F i g . I I I - 2 , w i t h the o r i g i n at ' 0 , F i g . I I I - l , o r i e n t e d so t h a t the x - a x i s p o i n t s i n the d i r e c t i o n o f the r e t r o - r e f l e c t i n g m i r r o r Mg a t a d i s t a n c e D(t) . The p a r a m e t r i c curves C^ ( Z , t^) and C^ ( 4 » f c 2 ^ r e p r e s e n t the g e o m e t r i c a l atmospheric paths, to the r e t r o - r e f l e c t o r and back, a t times t ^ and t2 r e s -p e c t i v e l y , o f a s i n g l e ray o f wavelength i = 1 , 2 , 3 of the combined output beam. dR^ ( I ,t) C, ( I , t) = ( B - l - 1 ) 'A i v * ' w d A where R^ ( I . t) = X A ( i , t) i + Y A ( i , t) * + Z A ( i , t ) k i i i i i s the p o s i t i o n v e c t o r o f an o r i e n t e d element dc^ o f the _^  i ray - p a t h C-^  (i ,t) and I i s a parameter, 0 s JL <. D(t) , i A 1< A p r o p o r t i o n a l t o d i s t a n c e a l o n g the x - a x i s ; i , j , k are u n i t v e c t o r s a l o n g the x , y and z axes r e s p e c t i v e l y . F i g . I I I - 2 shows the atmospheric ray-paths taken by s i n g l e rays of the combined l a s e r beam a t times t ^ and t 2 r e s p e c t i v e l y . The i n d i v i d u a l rays o f each wavelength i = 1 , 2 , 3 are shown as t a k i n g separate b ut c l o s e l y spaced ray-paths a t times t ^ and t 2 due to atmospheric d i s p e r s i o n . The "mean r a y - p a t h " C( i,t) i s a l s o i n d i c a t e d y PARAMETERIZATION OF THE ATMOSPHERIC RAY-PATHS 2 7 i n each c a s e . The a r c l e n g t h of t h e c u r v e s C*A (i , t ) i s i S A U ) g i v e n b y i D j t ) 1/2 L e t s A (t) = j [ c , ( i.t) • c A ( 4,t)l a* i J *" i i r - _ n 1/2 s A ( A , t ) = L c ( i ,t) • c A ( I ,t)J i i i be a p a r a m e t e r i z e d d i f f e r e n t i a l a r c l e n g t h segment o f the g e o m e t r i c a l curve C*A ( i , t) r e p r e s e n t i n g the r a y - p a t h i between the end m i r r o r s of the s t r a i n meter f o r r a d i a t i o n o f wavelength at time t . Then: D(t) S A (t) = I S A ( i ,t) d i (B-l - 2 ) 1 I 1 At every p o i n t R*-^  ( A /1) along the ray-path the i atmospheric r e f r a c t i v e index f o r r a d i a t i o n o f wavelength has the v a l u e n A (R*A ( I ,t) ) . The o p t i c a l path l e n g t h f o r i i a ray o f wavelength t r a v e l l i n g from .0 ( m i r r o r ) to. the r e t r o - r e f l e c t o r ( m i r r o r ) and back i s L-^  (t) D,(t) L A (t) = 2 ] n A (R A ( I ,t)) S A ( SL ,t) d i i i o (B-l - 3 ) 2 8 Here i t has been assumed t h a t the r e t r o - r e f l e c t o r simply r e t u r n s the ray back a l o n g i t s incoming path. T h i s assumption i s a good one f o r purposes of t h i s a n a l y s i s s i n c e the r e f r a c t i v e index c o n d i t i o n s are v i r t u a l l y , u n c h a n g e d over time i n t e r v a l s comparable t o the t r a n s i t time o f a photon between m i r r o r s and . I t s h o u l d be emphasized, however, t h a t t h i s assumption i s not nec e s s a r y f o r the de v e l o p - ment o f t h i s a n a l y s i s and c o u l d be d i s p e n s e d w i t h , r e s u l t i n g i n a s l i g h t l y more d i f f i c u l t mathematical treatment but no s i g n i f i c a n t d i f f e r e n c e i n the end r e s u l t s . The net change i n o p t i c a l path l e n g t h f o r r a d i a t i o n o f wavelength A . between times t , and t ~ i s 5L, ( t , t 5 ) 1 D ( t 0 ) ( B - l - 4 ) o Now the r e f r a c t i v e index n, (R, ) i s g i v e n by 2 9 where P (R*A ( JL ,t)) , T (R^ ( i , t ) ) , and p ( R A ( I ,t) ) i i i are the p a r t i a l p r e s s u r e o f "dry a i r " , temperature, and p a r t i a l p r e s s u r e o f water vapour a t p o s i t i o n R A ^ along the ray-path of wavelength A^ i = 1 , 2 , 3 r e s p e c t i v e l y [A-2, eqn. A-2-6 ] . S u b s t i t u t i n g the formula f o r r e f r a c t i v e index i n t o eqn. B-l-4 r e s u l t s i n eqn. B-l-5 B \ ( t l t 2 ) = 2 D ( t 2 ) J S A . ( l> Dj(ti ) t 2 ) d i - S, ( i , t . ) d l A. 1 +2f (A ±) I , F [ P ( R A ( A f t 2 J ) , T (R A ( ^ ' t 2 ) ) ] S A ( A , t 2 ) d A J i 1 1 i D ( t l ) 1 +2g(A.) P [ P ( % . ( j t , t l ' ' ' T ( ^ A . ( X ' t l ) } ] S A U f t ^ d . f c T2) \ G ^ p ( R A ( I , t 2 ) ) , T ( R X < ( j t , t 2 ) ) ] s A < ( A , t 2 ) d J l - ^ G | ^ P ( R A (^,t L ) l T U A (i,t 1)) ] s A \ u , t L )dA i = 1 , 2 , 3 (B-l-5) 30 Eqn. B-l-5 can be w r i t t e n 6 L A 1 ( t l t 2 ) = U A 1 ( t l t 2 ) + f ( H ) V ^ 1 ( t l t 2 ) + g ( A l ) W A 1 ( t l t 2 ) 5 L A 0 ( t l t 2 > = V^lV + f^2 )K^tlt2) + 5 ^ 2 ) W A 9 ( t l t 2 ) (B-l-6) 6 L A 3 ^ l t 2 ) = ^ i V + f ( A 3 ) V A 3 ( t l t 2 ) + * <A3' W A 3 1*1*2) where U A ( t l t 2 ) = 2 l V A . ( t l t 2 ) = 2l D ( t 2 ) D ( t ] L ) \ s A _ ( ^ t 2 ) d ^ - j s A _ ( 4 , t l o o D ( t 2 ) )d I (A, t 2 ) )»T(R A ( A , t 2 ) ) ] S A ( A , t 2 ) d A ? ( t l ) j F^P (R^ ( JL, t ± ) ) , T (R A ( i, t±)) J ( X, t x )d H WA ( t l t 2 ) = 2 l t 2 ) d i I G [ P ( R a . ( ^ t 2 ) ),T (R A U,t 2) ) ] s A (A, (B-l-7) D ( t l ) G[p (R A {l.t-J ) , T ( R A < ( 4 , ^ ) )Js A U , t L ) d 4 7 3 1 Suppose f o r the moment the s m a l l d i f f e r e n c e s between the i n d i v i d u a l rays forming the t r i p l e t o f rays a t time t 1 and t 2 , 0^ (. I .t^) and ( i , t 2 ) , can be i i ig n o r e d and the t r i p l e t o f rays i n eqn. B-l- 5 can be r e p l a c e d by a "mean ray- p a t h " , C ( 4 ,tj) and C ( 4 , t 2 ) , a t times t ^ and t 2 r e s p e c t i v e l y . c d . t ) = where A t) I j R( X.t) = 3 l^^.^'^) * + [ t i Y \ { i ' T h i s d e f i n i t i o n o f the "mean r a y - p a t h " i s such t h a t the mean ray i s g e o m e t r i c a l l y c e n t e r e d i n the t r i p l e t o f r a y s . S u b s t i t u t i n g the mean ray-path f o r the i n d i v i d u a l ray-paths i n each case i n eqn. B-l-7 y i e l d s eqn. B-l-8 5 1^ ( t x t 2 ) = U ( t 1 t 2 ) + f ( A 1 ) V ( t 1 t 2 ) + g ( A 1 ) W ( t 1 t 2 ) 51^ ( t L t 2 ) = U ( t 1 t 2 ) + f ( A 2 ) V ( t 1 t 2 ) + g ( A 2 ) W ( t 1 t 2 ) 5 L A ( t 1 t 2 ) = U ( t 1 t 2 ) + f ( ? , 3 ) V ( t 1 t 2 ) + g ( A 3 ) W ( t 1 t 2 ) (B-l-8) 32 where U ( t 1 t 2 ) = 2 J 3 su,t2)dj& - | s( a ,tj_)d A D ( t 2 ) I F [ p ( R ( j t , t 2 ) ) , T ( R ( i , t 2 ) ) ] s ( j l , t 2 ) d J t D,(t x) F [ p ( R ( J e , t 1 ) ) , T ( R ( A , t 1 ) j J s U . t ^ d Z ( B - l - 9 ) W(t 1t 2 ) = 2 J G ^ p(R(jJ,t 2)),T(R(^t 2 ) ) J s ( j e,t 2)d I ol w I' D ( t 2 ) - j c[p(R( i , t 1 ) ) . T ( R ( j t , t 1 ) ) ] s ( A , t 1 ) d i and o f course 1/2 S( = [ C( l,t) ' C(l. t) ] D(t) S ( t ) = s ( n ) d 3 3 Eqns. B-l-8 can be expressed i n m a t r i x form ( N f 6 L A 1 ( t l t 2 ) 1 f ( A X ) g ( A x ) u ( t 1 t 2 ) = 1 f ( A 2 ) g ( A 2 ) v ( t l t 2 ) 5 L A 1 F ( A 3 ) g ( A 3 ) w ( t ] L t 2 ) j V. J (B-l-10) The inhomogeneous terms i n eqns. B-l-10 , 61^ ( t ^ t 2 ) , i = 1 , 2 , 3 , a r e e q u a l t o the net i change i n o p t i c a l path l e n g t h i n the i n t e r v a l A t = t 2 - t ^ of a s i n g l e ray o f wavelength A ^ t r a v e r s i n g the d i s t a n c e between the end m i r r o r s o f the f r e e a i r l a s e r s t r a i n meter. The inhomogeneous terms i n eqns. B-l-10 are p r o p o r t i o n a l t o the n et f r i n g e counts observed. I f A . ^ , A 2 , and A 3 are the wavelengths " i n vacuo" o f the r a d i a t i o n from l a s e r s , L2 ' a n < ^ L 3 r e s p e c t i v e l y and i f ^ t i t 2 ^ ' C-v ( t , t 0 ) and C, ( t , t 0 ) are the net f r i n g e counts i n 2 3 the i n t e r v a l ^ t = t 2 - t ^ observed i n the i n t e r f e r e n c e p a t t e r n s formed by d i f f r a c t i n g a p e r t u r e s I i ' I2 ' a n ( ^ I 3 r e s p e c t i v e l y , then 34 ^ A ^ L V = A l C l ( t l T 2 ) 5 L A , ( t l T 2 ) = A2 Z (B-l-11) 5 L A 3 ( t l T 2 ) = A3 W2' The m a t r i x o f c o e f f i c i e n t s i n eqns. B-l-10 are known from the r e f r a c t i v e index formula f o r the atmosphere. The unknowns Vit^t^) and W ( t 1 t 2 ) are a s s o c i a t e d w i t h changes i n the o p t i c a l path l e n g t h o f the ray i n the i n t e r v a l At = t 2 - t ^ which are mainly r e -f r a c t i v e i n o r i g i n . Non-zero v a l u e s o f V ( t ^ t 2 ) and W ( t ^ t 2 ) a r i s e mainly from changes i n the atmospheric p a r a -meters o f p r e s s u r e , temperature and hu m i d i t y a l o n g the ray pa t h . The unknown U ( t ^ t 2 ) , however, i s e q u a l t o the change i n g e o m e t r i c a l l e n g t h o f the mean ray - p a t h between  the end m i r r o r s o f the s t r a i n meter i n the i n t e r v a l A t = t 2 - t ^ , as can be seen from i t s d e f i n i o n i n eqns. B - l - 9 . I t s h o u l d perhaps be emphasized t h a t the s u b s t i -—• t u t i o n o f the mean ray - p a t h C( I ,t) f o r the t r i p l e t o f ray paths ( Z , t) i n eqns. B - l - 7 does not m a n i f e s t i i t s e l f as a p h y s i c a l approximation i n the measurement p r o c e s s . I t merely d e f i n e s m a t h e m a t i c a l l y a "mean r a y " i n terms o f a t r i p l e t o f r a y s . The "mean ray", so d e f i n e d , i s no more or 35 l e s s o f an a b s t r a c t i o n and hence no more or l e s s " r e a l " than any o f the members o f the ray t r i p l e t . Thus i t f o l l o w s t h a t the mean ray (or more to the p o i n t , f l u c t u a t i o n s i n the mean ray) i s as adequate as any f o r use i n making p h y s i c a l o b s e r -v a t i o n s p r o v i d e d a means o f measuring i t i s a v a i l a b l e . m e t r i c a l l e n g t h o f the mean ray i s p r o v i d e d by a s o l u t i o n o f the eqns. B-l-10 and the e x i s t e n c e o f t h i s measure i s guaran-teed by the n o n - s i n g u l a r nature o f the mat r i x o f c o e f f i c i e n t s . I t can be shown t h a t U f t ^ ^ ) ^ s made up o f c o n t r i b u t i o n s from two u n r e l a t e d e f f e c t s . From eqn. B - l - 9 A measure o f the net f l u c t u a t i o n s i n the geo-U f t ^ ) = 2 (B-l-12) which can be w r i t t e n U{txt2) = 6 S ( t 1 t 2 ) + 2 5 D ( t x t 2 ) (B-l-13) where 6 S ( t 1 t 2 ) o and (B-l-15) 36 At t h i s p o i n t i n the a n a l y s i s the n e c e s s a r y approximation w i l l be made t h a t T h i s i s tantamount t o assuming t h a t the mean ray-path never d e v i a t e s s t r o n g l y from the s t r a i g h t l i n e path between the end m i r r o r s and t h a t the a r c l e n g t h o f the mean ray- p a t h between subsequent p o s i t i o n s o f the r e t r o - r e f l e c t o r (which d i f f e r due t o t e c t o n i c a c t i v i t y ) can be approximated by the s t r a i g h t l i n e d i s t a n c e . That t h i s i s a good approximation i s borne out e x p e r i m e n t a l l y by o b s e r v a t i o n s o f s t e l l a r images i n a s t r o -n omical t e l e s c o p e s i n which the r.m.s. v a l u e o f the f l u c t u a -t i o n s i n the angle o f a r r i v a l o f the ray i s about l " - 2 " o f ar c or r o u g h l y 10~"5 r a d . [ 25, P. 225 1. T h e o r e t i c a l p r e d i c t i o n s by Chernov [ 2 6 , P.17] f o r r.m.s. f l u c t u a -t i o n s i n the angle o f a r r i v a l o f a l i g h t r a y a f t e r h o r i z o n t a l t r a n s m i s s i o n through the e a r t h ' s atmosphere o f a d i s t a n c e o f 10 km. y i e l d s a somewhat l a r g e r v a l u e o f "the -3 o r d e r o f 10 r a d . However the e r r o r i n the val u e o f the net f l u c t u a t i o n i n g e o m e t r i c a l l e n g t h o f the mean ray- p a t h i n t r o d u c e d by t h i s assumption v a r i e s as the c o s i n e o f the angle o f a r r i v a l o f the ray a t the r e t r o - r e f l e c t o r and i s l i k e l y t o be much l e s s than one p a r t i n 10 . (B-1-16) In eqn. B-l-13, 5 S ( t , t _ ) i s a random con-37 t r i b u t i o n t o the net f l u c t u a t i o n i n g e o m e t r i c a l path l e n g t h of the mean ray which v a r i e s r a p i d l y w i t h time r e s u l t i n g from d i f f e r e n c e s i n the c h o i c e o f atmospheric ray-paths taken be-tween the end m i r r o r s M,- and M g a t times and t 2 In eqn. B-l-13, 2 o D ( t ^ t 2 ) i s a c o n t r i b u t i o n t o the net f l u c t u a t i o n i n the t o t a l g e o m e t r i c a l path l e n g t h o f the mean ray a r i s i n g out of t e c t o n i c changes i n d i s t a n c e D between the end m i r r o r s M 5 and M & . As such 2 6 D ( t 1 t 2 ) c o n t a i n s c o n t r i b u t i o n s from the e n t i r e spectrum o f t e c t o n i c motions from h i g h frequency s e i s m i c a c t i v i t y on down to e a r t h \\ t i d e s and s e c u l a r c r u s t a l d e f o r m a t i o n s . A l l o f these c o n t r x -\ b u t i o n s to 2 6 D ( t ^ t 2 ) , w i t h the e x c e p t i o n o f s e c u l a r c r u s t a l d e formation, by d e f i n i t i o n ; go t o zer o when averaged over s u f f i c i e n t l y l o n g time i n t e r v a l s . I t i s i d e a l l y d e s i r a b l e t o " f i l t e r " a s e r i e s o f continuous c o n s e c u t i v e o b s e r v a t i o n s o f U ( t ^ t ^ _ ^ ) i = 1 , 2 , 3 . . . n U ( t i t i _ 1 ) = 5 S ( t i t i _ 1 ) + 2 6 D ( t . t . _ 1 ) to remove the e f f e c t s o f the atmospheric c o n t r i b u t i o n 5 S ( T ) r e v e a l i n g the t e c t o n i c changes i n d i s t a n c e ° D ( T ) . The next s e c t i o n w i l l d e a l w i t h the s t a t i s t i c a l a n a l y s i s o f f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s t o r e v e a l t e c t o n i c motions. 38 2. S t a t i s t i c a l A n a l y s i s of Free A i r Laser S t r a i n Meter  Observations to Reveal T e c t o n i c Motions The data from f r e e a i r l a s e r s t r a i n meter observa-t i o n s c o n s i s t s o f continuous c o n s e c u t i v e sampled v a l u e s of the f l u c t u a t i o n s of the g e o m e t r i c a l l e n g t h o f the mean ray between the end m i r r o r s o f the s t r a i n meter, U ( t ^ , t ^ _ ^ ) i = l , 2 , 3 . . . n. There are undoubtedly many ways t o p r o c e s s t h i s data i n order to f i l t e r out the atmospheric c o n t r i b u t i o n s t o the f l u c t u a t i o n s i n g e o m e t r i c a l r a y - p a t h l e n g t h . The f i l t e r i n g c o u l d be done o p t i m a l l y , i n the Weiner sense, w i t h knowledge o f the power s p e c t r a o f both the atmospheric f l u c t u a t i o n s i n g e o m e t r i c a l ray-path l e n g t h and o f e a r t h s t r a i n s . The observed power spectrum o f e a r t h s t r a i n s from p e r i o d s of 2.35 hours to 4 sec. has been p u b l i s h e d by V a l i [ 3 9 ] but to the author's knowledge, no o b s e r v a t i o n s of the power spectrum o f g e o m e t r i c a l f l u c t u a -t i o n s i n atmospheric ray path l e n g t h have been made. Another a l t e r n a t i v e i s a l e a s t squares f i t of a curve t o the e n t i r e s e t o f d a t a . The method of data a n a l y s i s p r e s e n t e d i n t h i s s e c t i o n i s a r a t h e r simple one and w h i l e making no c l a i m t o be optimum, serves to i l l u s t r a t e the r e l a t i o n s h i p between the c o n f i d e n c e l e v e l of the o b s e r v a t i o n s o f t e c t o n i c motion and the r e l e v a n t p h y s i c a l , g e o g r a p h i c a l and atmospheric parameters. Le t e(t) be the e x t r a d i s t a n c e , over'the 39 s t r a i g h t l i n e d i s t a n c e , t h a t a photon t r a v e l s when t r a v e r s i n g the mean ray- p a t h from m i r r o r t o m i r r o r Mg and back a t time t . e ( t ) = 2 [ S ( t ) - D(t) J (B-2-1) e (t) i s a random v a r i a b l e w i t h p r o b a b i l i t y d e n s i t y f u n c t i o n P , . (R ) > o <. B £ 0 0 • ^he expected v a l u e (ensemble e \ t; average) o f e (t) i s g i v e n by 00 = / e p e ( t ) o e ( t ) = / p P ^ t ) ( 0 ) P The random v a r i a b l e 6 S ( t ^ t 2 ) i n eqn. B-l-13 i s r e l a t e d t o e ( t ) by 6 S ( A t) = e ( t 2 ) - e ( t 1 ) (B-2-2) where A t = t 2 - t x The random v a r i a b l e 6 S ( A t) has p r o b a b i l i t y d e n s i t y f u n c t i o n g ( p ) , - °° £ 8 £ 0 0 . The expected v a l u e o f 6 S ( i t) i s g i v e n by 40 5S( At) = J 6 P 5 s ( B )dB 5 S ( At) = e(t) - e(t) 5S( A t) = 0 (B-2-3) Let*'\ | U(A t i ) j - = | 5S(A t^) + 25D(A t,,) } i = l , 2 , 3 . . . n b e a sequence o f continuous c o n s e c u t i v e sampled o b s e r v a t i o n s o f the f l u c t u a t i o n s i n g e o m e t r i c a l l e n g t h o f the mean ray path U( A. t) i n the i n t e r v a l s A t . = t . - t . i , . D e f i n e X ( T ) to be the net f l u c -x I i - l r' i' t u a t i o n i n g e o m e t r i c a l ray-path l e n g t h i n the i n t e r v a l T = T*At- > O £ r < n . Thus X= J X ( T J = L U ( A t.) r i=i x X ( T R ) = V 5 S ( A t±) + 2 T 5D( A t.) i = l i " In t h i s c o n t e x t " t " i s an a b s o l u t e time v a r i a b l e and "T " i s a r e l a t i v e time v a r i a b l e , T = 0 a t the b e g i n n i n g of the f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s . 41 I t w i l l be assumed t h a t the v a l u e s X ( T ) r = 1 , 2 , 3 ... n r e p r e s e n t sampled v a l u e s , a t i n t e r v a l s A t o f a continuous, d i f f e r e n t i a b l e f u n c t i o n X ( T )• T h i s i s tantamount t o assuming t h a t changes i n geometric l e n g t h o f the mean ray-path caused by both atmospheric f l u c t u a t i o n s and t e c t o n i c motions are continuous and "smooth". A h y p o t h e t i c a l p l o t o f X ( T R ) v s . T i s shown i n F i g . I I I - 3 . I t c o n s i s t s o f broad band t e c t o n i c c o n t r i b u t i o n s 2 5 D ( At^) to the f l u c t u a t i o n s i n geometric ray-path l e n g t h "contaminated" by l a r g e amplitude atmospheric "noise", w i t h mean va l u e z e r o . L e t 25D^ be the average v a l u e o f X ( T ^) over the f i r s t i n sample p o i n t s T, T 0 ••• T / where m i s an L z m even i n t e g e r o f f i x e d v a l u e , c o r r e s p o n d i n g t o an i n t e r v a l o f l e n g t h AT c e n t r e d about T M 2 AT = mAt (B-2-4) In g e n e r a l , i f the e n t i r e l e n g t h o f o b s e r v a t i o n s o f X ( T ) r e p r e s e n t e d by the sampled v a l u e s X ( T ^ ) / i = 1 , 2 , 3 ... n , i s d i v i d e d up i n t o q i n t e r v a l s each o f l e n g t h AT = mA t (where m i s an even i n t e g e r o f f i x e d v a l u e and n = mq ) then; 2 5 k = l , 2 , 3 . . . q i s the average th v a l u e o f X ( T ^ ) over the m sample p o i n t s o f the k i n t e r v a l c o r r e s p o n d i n g to times i = m(k-l) + 1 , m(k-l) + 2 ... m(k-l) + m , c e n t r e d about Tm(k-l/2) 43 A curve f i t t e d t o a p l o t o f 2 5 Dj, v s . T m(k-l/2) as shown i n F i g . I I I - 3 , w i l l r e v e a l a c t u a l t e c t o n i c changes i n the s t r a i g h t l i n e d i s t a n c e D between the end m i r r o r s o f the s t r a i n meter p r o v i d i n g YI * 2 5D. k+1 6D, k = 1 , 2 , 3 where y , r e p r e s e n t e d i n F i g . I I I - 3 by the e r r o r b a r s , i s the expected d i f f e r e n c e or " e r r o r " between the sample mean of m samples o f 6 S ( A t) and the p o p u l a t i o n mean 5 S ( A t) = 0. The t e c t o n i c changes i n l e n g t h between the end m i r r o r s o f the s t r a i n meter w i l l be observed w i t h a " s i g n a l - t o - n o i s e r a t i o " o f s or b e t t e r i f 2 ( 6 D, - 5D, 1 I | „ L k+1 k j , IYI * ~  A s > 1 ( B - 2 - 5 ) The r e s i d u a l o f the averaged atmospheric f l u c t u a t i o n i n g e o m e t r i c a l r a y - p a t h l e n g t h , y , i s a m o n o t o n i c a l l y de-c r e a s i n g f u n c t i o n o f the a v e r a g i n g i n t e r v a l &T = m A t I t i s n e c e s s a r y to s e l e c t an a v e r a g i n g i n t e r v a l A T l o n g enough t h a t |y| i s l e s s , by a f a c t o r s , the s i g n a l -n o i s e r a t i o , than the expected d i f f e r e n c e i n the d i s t a n c e be-tween the end m i r r o r s o f the s t r a i n meter i n the same i n t e r v a l . Thus the av e r a g i n g i n t e r v a l i s c l e a r l y dependent i n g e n e r a l on 44 the s t a t i s t i c s o f the atmospheric f l u c t u a t i o n s i n g e o m e t r i c a l ray-path l e n g t h as w e l l as the spectrum o f t e c t o n i c a c t i v i t y . The f o l l o w i n g o b s e r v a t i o n should be noted r e g a r d i n g knowledge o f X ( T ) as a f f o r d e d by f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s . F i r s t l y , the sampling o f X ( T ) a t i n t e r -v a l s A T w i l l a l i a s f l u c t u a t i o n s i n the g e o m e t r i c a l l e n g t h o f the mean ray whose f r e q u e n c i e s exceed 2^7 T h i s f o l d i n g o f the spectrum about the Ny q u i s t frequency w i l l a l i a s f l u c t u a t i o n s w i t h f r e q u e n c i e s 1 / A T , 2 / A T , 3 / A T . . . back t o d.c. T e c t o n i c c o n t r i b u t i o n s 2 5 D ( T ) to f l u c t u a t i o n s i n the g e o m e t r i c a l ray-path l e n g t h a t these f r e q u e n c i e s are a s s u r e d l y o f v e r y s m a l l amplitude, however no such assurance can be g i v e n r e g a r d i n g the atmospheric c o n t r i b u t i o n s 5 S ( T ) . However the c e n t r a l l i m i t theorem s t a t e s t h a t y (the mean o f 6 S ( T ) averaged over an i n t e r v a l A T = m A t) i s a random v a r i a b l e , g a u s s i a n d i s t r i b u t e d about the p o p u l a t i o n mean, lb s2 o S ( T ) = 0 , w i t h st a n d a r d d e v i a t i o n g i v e n by 2 U — — r e g a r d l e s s o f the h i g h frequency s p e c t r a l d i s t r i b u t i o n o f 5 S ( T ) p r o v i d i n g the sampled v a l u e s o f 5 S ( T ) are i n d e -pendent. Independent samples o f the random v a r i a b l e 5 S ( T ) can be assured i f the sampling i n t e r v a l A T i s g r e a t e r than the p e r i o d o f the lowest frequency components i n the spectrum o f 5 S ( T ) Secondly, the r e c o n s t r u c t i o n o f the curve r e v e a l i n g t e c t o n i c changes i n l e n g t h 2 5 D ( T ) from the "sampled" v a l u e s 2 6 a t i n t e r v a l s o f A T can be accomplished by the use of many s o p h i s t i c a t e d "holds". For purposes o f t h i s a n a l y s i s 45 i t w i l l be assumed t h a t a simple " p o l y g o n a l h o l d " i s used. T h i s i s e q u i v a l e n t t o a p i e c e - w i s e s t r a i g h t l i n e a p p r o x i -mation t o 2 5 D ( T ) . L e t 2 f o i l , . - BD, 1 . , „ _ 2h = L k+1 k j k = 1 , 2 , 3...q k AT Then h^. i s the l i n e a r r a t e of change o f the d i s t a n c e D between the end m i r r o r s o f the s t r a i n meter i n the i n t e r v a l AT = T m(k+l/2) T m ( k - l / 2 ) d i s t a n c e D , 5 D^ +^ - 5 D^ . , A T = Tm(k+l/2) * T m ( k - l / 2 ) a t a s i g n a l - n o i s e r a t i o s " t o n i c p r o c e s s e s i f An observed change i n the i n the i n t e r v a l w i l l be c a l l e d " s i g n i f i c a n t and d u l y a t t r i b u t e d t o t e c -2 | h k | A T k, = 1 , 2 , 3 q (B-2-6) otherwise the o b s e r v a t i o n w i l l be c a l l e d " i n s i g n i f i c a n t a t a s i g n a l - n o i s e r a t i o s " and i t w i l l be assumed t h a t no " s i g n i f i c a n t " t e c t o n i c change i n the d i s t a n c e D has o c c u r r e d i n t h a t i n t e r v a l . By the c e n t r a l l i m i t theorem the p r o b a b i l i t y d e n s i t y f u n c t i o n f o r y the r e s i d u a l o f the average o f m samples o f the atmospheric f l u c t u a t i o n s i n ray-path l e n g t h 5 S , i s g a u s s i a n about ze r o w i t h s t a n d a r d d e v i a t i o n 46 5 S m P ( 0 ) = Y 1/2 2 m exp - 3 2 2 2ir 5S 0 5S^ (B-2-7) where 8s i s the variance of &S 6S 2 - j " 2 -00 r P g s o )<ap Let Tj ( | § | ) >' 0 ^ t) ( <; 1 , be the 2 h A T p r o b a b i l i t y that Jy| £ ~ afte r averaging over m samples of 6 S 2 ! h k l A T In t h i s sense T| ( | § | ) i s the "confidence l e v e l " of the observation of the tectonic change i n distance. T| (|?|) i s the p r o b a b i l i t y that an observation of tectonic motion which has been c a l l e d " s i g n i f i c a n t " (as i t represents a change i n the distance D , 6 D ] C + 1 - B D ^ f<jr w h i c h , , g 2l h kU T ) 4 7 i s i n f a c t "genuinely s i g n i f i c a n t " . The l a r g e r the c h o i c e of I £ I the l e s s l i k e l y i s the p o s s i b i l i t y o f the modulus o f the average o f m samples o f 5 S exceeding |y| the l a r g e r the c o n f i d e n c e i n the o b s e r v a t i o n . The f o l l o w i n g change o f v a r i a b l e Y (B-2-8) 6 S and i n eqn B-2-7 d e f i n e s a p r o b a b i l i t y d e n s i t y f u n c t i o n f o r Y' which i s g a u s s i a n w i t h parameters zero and one. P , ( B ) = -p= exp (B-2-9) L e t T|'U'| m 5 S samples o f 5 S V (|S'|> = 2 be the p r o b a b i l i t y t h a t a f t e r a v e r a g i n g over m l h k k r s m 6 S ' J o P , (B)d8 Y (B-2-10) 4 8 C l e a r l y 1)' ( | § ' | ) = T| ( | ? | ) - ^ u s TI' ( | 5 1 | ) i s t h e " c o n f i d e n c e l e v e l " o f t h e l a s e r s t r a i n m e t e r o b s e r v a t i o n and | 5 ' | i s t h e ' c o n f i d e n c e p a r a m e t e r . The c o n f i d e n c e l e v e l i s r e l a t e d b y d e f i n i t i o n t o t h e c o n f i d e n c e p a r a m e t e r by T|(U' |) = e r f -Ul (B-2-11) I t f o l l o w s from eqns B-2-9 and B-2-10 , [ A-3-4] , t h a t 2 s V m v r 6 S 2 (B-2-12) I t f u r t h e r f o l l o w s from eqns. B-2-11 and B-2-12 t h a t V 2 | h k U r ^ m 6 S ' (B-2-13) where | g 1 | i s the v a l u e o f the c o n f i d e n c e parameter a p p r o p r i a t e t o the r e q u i r e d c o n f i d e n c e l e v e l o f o b s e r v a t i o n . Eqns B-2-12 and B-2-13 r e l a t e the c o n f i d e n c e l e v e l o f a f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n t o the l e n g t h o f the o b s e r v i n g i n t e r v a l , the modulus o f the mean r a t e o f change o f d i s t a n c e between the end m i r r o r s o f the s t r a i n meter and the r.m.s. f l u c t u a t i o n s i n g e o m e t r i c a l ray-path l e n g t h . The dependence of the c o n f i d e n c e l e v e l o f the 4 9 t e c t o n i c o b s e r v a t i o n s on the mean square f l u c t u a t i o n s i n geo-m e t r i c a l ray-path l e n g t h QSZ r e q u i r e s a t h e o r e t i c a l e x p r e s s i o n f o r 5 S^ , the d e r i v a t i o n o f which i s the s u b j e c t f o r the next s e c t i o n . T h i s i s an i n t e r e s t i n g problem i n i t s own r i g h t w i t h many p e r p l e x i n g p h y s i c a l and p h i l o s o p h i c a l a s p e c t s . How-ever f o r those who are pre p a r e d t o ac c e p t the f i n a l mathematical r e s u l t o f s e c t i o n 3, eqn. B-3-14, t h i s s e c t i o n can be c o n s i d e r e d a d i g r e s s i o n from the s u b j e c t o f f r e e a i r l a s e r s t r a i n meters. Such rea d e r s need o n l y f a m i l i a r i z e themselves w i t h the d e f i n i t i o n of the symbols i n v o l v e d i n eqn. B-3-14 and proceed t o s e c t i o n 4. 50 3. S t a t i s t i c a l A n a l y s i s o f F l u c t u a t i o n s i n Ray-Path o f a  Laser Beam Pr o p a g a t i n g Through a T u r b u l e n t Atmosphere a) The Problem o f Random F l i g h t s Eqn. B-2-2 shows t h a t the random v a r i a b l e 5S( A t) i s simply r e l a t e d t o the random v a r i a b l e e (t) 5S( At) = e ( t 2 ) - e ( t x ) (B-2-2) and hence the s t a t i s t i c s o f 5S( A t ) are determined by the s t a t i s t i c s o f e (t) . Now e (t) , eqn. B-2-1, can be expressed as j e (t) = 2 j Tl - cosa( I , t) J s( i , t ) d i (B-3-1) T - - i 1 / 2 where S ( i , t) = J C ( i , t ) • C( j j,t) J i s a random v a r i a b l e , the p a r a m e t e r i z e d d i f f e r e n t i a l a r c l e n g t h segment o f the ray a t time t as a f u n c t i o n o f x and a ( i tt) i s the t o t a l 3-dimensional angle between the x - a x i s and the ray-path a t time t as a f u n c t i o n o f & . The p r o b a b i l i t y d e n s i t y f u n c t i o n Pg g ( 3 ) i s determined by the p r o b a b i l i t y d e n s i t y f u n c t i o n P (3 ) where T ( t ) i s g i v e n by eqn. B-3-1. F I G . I l l - 1 * A N G U L A R P A R A M E T E R S O F A T M O S P H E R I C R A Y - P A T H 5 2 I t has been p o i n t e d out by Wang and Uhlenbeck [33] tha t the d e r i v a t i o n of the p r o b a b i l i t y d e n s i t y f u n c t i o n f o r a random v a r i a b l e z ( t ) z ( t ) - K U , t ) y U ) d * where K(£,t) i s a giv e n k e r n e l and y ( i l ) a random v a r i a b l e w i t h known p r o b a b i l i t y d e n s i t y f u n c t i o n other than g a u s s i a n i s an unsolved problem i n the theory o f Brownian motion. I t i s c l e a r from eqn. B-3-1 t h a t the problem o f the d e r i v a t i o n o f the p r o b a b i l i t y d e n s i t y f u n c t i o n f o r e ( t ) i s a member of t h i s c l a s s . However a s o l u t i o n f o r e ( t ) i s o b t a i n a b l e i f the atmospheric r a y - p a t h can be approximated by a s e r i e s o f s t r a i g h t l i n e s . When c a s t i n t h i s a p p r o x i -mating form the problem i s e a s i l y s o l u b l e by methods of ana-l y s i s developed by t h e o r e t i c i a n s working on problems o f Brownian Motion and i n p a r t i c u l a r "the problem o f random f l i g h t s " . The problem o f random f l i g h t s i s to c a l c u l a t e the p r o b a b i l i t y t h a t a p a r t i c l e , i n i t i a l l y at the o r i g i n , w i l l occupy a g i v e n volume element of space a f t e r a f i n i t e number of a r b i t r a r y d isplacements whose p r o b a b i l i t y d e n s i t y f u n c t i o n s are known. The s o l u t i o n to t h i s problem i s due o r i g i n a l l y to A. A. Markov and has been g e n e r a l i z e d to n-dimension by Chandrasekhar [ 2 7 ] . 53 A p r o b l e m p e r t i n e n t t o t h e f e a s i b i l i t y o f f r e e a i r l a s e r s t r a i n m e t e r s i s t o c a l c u l a t e t h e p r o b a b i l i t y d e n s i t y f u n c t i o n f o r t h e t o t a l p a t h l e n g t h t r a v e l l e d b y a p a r t i c l e e n g a g e d i n random f l i g h t a f t e r t h e p a r t i c l e h a s a c h i e v e d a g i v e n s t r a i g h t l i n e t o t a l d i s p l a c e m e n t f r o m i t s i n i t i a l p o s i -t i o n . A l t h o u g h t h i s p r o b l e m i s r e l a t e d t o t h e p r o b l e m o f r a n d o m f l i g h t s i t a p p e a r s t h a t n e i t h e r M a r k o v n o r C h a n d r a s e k h a r i n v e s t i g a t e d i t . To b e g i n t h e p r o b l e m o f c a l c u l a t i n g t h e d i s t r i b u -t i o n o f a t m o s p h e r i c r a y - p a t h l e n g t h s o f a s i n g l e r a y o f a l a s e r beam i t i s n e c e s s a r y t o assume t h a t t h e a t m o s p h e r i c r e f r a c t i v e i n d e x f l u c t u a t i o n s c a n be d i v i d e d i n t o two c l a s s e s d e p e n d i n g on t h e i r s c a l e . One c l a s s , c h a r a c t e r i z e d b y s c a l e s o f d i m e n s i o n SLQ j ' o r l a r g e r , c a u s e s d e f l e c t i o n o f t h e l a s e r beam as a w h o l e w h i l e p r e s e r v i n g t h e s t r u c t u r e o f t h e wave f r o n t ; w h e r e a s t h e o t h e r c l a s s , c h a r a c t e r i z e d b y s c a l e s o f d i m e n s i o n l e s s t h a n l Q , c a u s e s i n t e r n a l d e g r a d a t i o n o f t h e l a s e r beam s u c h as " c r u m b l i n g o f t h e wave f r o n t " a n d f l u c t u a -t i o n s i n i n t e n s i t y a n d p h a s e c o h e r e n c e a c r o s s t h e beam w h i l e p r e s e r v i n g t h e beam's d i r e c t i o n o f t r a v e l . T h i s a s s u m p t i o n i m m e d i a t e l y p e r m i t s one t o a p p r o x i m a t e t h e r a y - p a t h o v e r a s t r a i g h t l i n e t o t a l d i s t a n c e D b y a s e r i e s o f s t r a i g h t l i n e s e g m e n t s o f l e n g t h I , where ZQ << D 54 O v e r d i s t a n c e s o f t h e o r d e r I o r l e s s i t i s o a s s u m e d t h a t a s i n g l e r a y may u n d e r g o many r a p i d f l u c t u a t i o n s i n d i r e c t i o n a b o u t i t s i n i t i a l d i r e c t i o n b u t t h a t i t s mean d i r e c t i o n i s c o n s t a n t ; w h e r e a s o v e r d i s t a n c e s l a r g e r t h a n J l Q i t i s a s s u m e d t h a t a s i n g l e r a y may e x p e r i e n c e many l a r g e f l u c t u a t i o n s i n d i r e c t i o n w h i c h a d d a p p r e c i a b l y t o i t s t o t a l p a t h l e n g t h . T h e a s s u m p t i o n t h a t t h e r a y - p a t h c a n be c h a r a c t e r i z e d b y a l e n g t h % s u c h t h a t f o r d i s t a n c e s l a r g e r t h a n J l Q t h e f l u c t u a t i o n s i n d i r e c t i o n o f t h e r a y a r e e s s e n t i a l w h e r e a s f o r d i s t a n c e s s m a l l e r t h a n Jl t h e y a r e t r i v i a l a n d c a n b e i g n o r e d i s b o t h d r a s t i c a n d c u r i o u s . S u p p o r t f o r m a k i n g s u c h a n a s s u m p t i o n comes f r o m t h e t h e o r y o f B r o w n i a n m o t i o n w h e r e an e x a c t l y a n a l o g o u s a s s u m p t i o n i s made c o n c e r n i n g v e l o c i t i e s ( r a t h e r t h a n d i s p l a c e m e n t s ) o f B r o w n i a n p a r t i c l e s . I t i s f u n d a m e n t a l t o t h e t h e o r y o f B r o w n i a n m o t i o n [ 2 7 ] t o assume t h a t t h e r e e x i s t s a t i m e s c a l e A t s u c h t h a t f o r i n t e r v a l s o f t i m e t < A t t h e v e l o c i t y u o f a B r o w n i a n p a r t i c l e i s e s s e n t i a l l y c o n s t a n t w h e r e a s i t s a c c e l e r a t i o n ^ j r c a n s u f f e r many a n d p o s s i b l y l a r g e f l u c t u a t i o n s ; w h e r e a s f o r i n t e r v a l s o f t i m e t > A t t h e v e l o c i t y u o f a B r o w n i a n p a r t i c l e c a n v a r y a p p r e c i a b l y . T h i s a s s u m p t i o n r e s t r i c t s t h e v a l i d i t y o f t h e t h e o r e t i c a l p r e d i c t i o n s t o t i m e s T >> A t T h e j u s t i f i c a t i o n f o r t h i s a s s u m p t i o n i n t h e c a s e o f B r o w n i a n m o t i o n s comes s o l e l y f r o m t h e s u c c e s s o f t h e t h e o r e t i c a l p r e d i c -t i o n s . b) c a l c u l a t i o n o f t h e P r o b a b i l i t y D e n s i t y F u n c t i o n f o r  € ( t ) We s h a l l b e g i n b y a s s u m i n g g a u s s i a n p r o b a b i l i t y d e n s i t y f u n c t i o n s f o r t h e a z i m u t h 6 and e l e v a t i o n 0 o f t h e u n i t t a n g e n t v e c t o r C t o t h e r a y - p a t h C ( - * , t ) , F i g . I I I - 4 P e ( 6 ) = P0<P> = —„i 2rr 9' yfjv 02 exp exp 29' 20' - GO 8 ^ 00 - 0 0 ^ p CO Beckmann (22) h a s p o i n t e d o u t t h a t r e g a r d l e s s o f t h e d e t a i l e d d i s t r i b u t i o n o f t h e m i c r o s c a l e f l u c t u a t i o n s i n r e f r a c t i v e i n d e x , t h e f l u c t u a t i o n s i n d i r e c t i o n o f t h e r a y w i l l be g a u s s i a n d i s -t r i b u t e d b y t h e C e n t r a l L i m i t Theorem. C h e r n o v (26, P. 22) a s s u m i n g a g a u s s i a n 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 r e f r a c t i v e i n d e x f l u c t u a t i o n s on a m i c r o -s c a l e ( r ) = exp : 2 A h a s a l s o d e r i v e d t h e a b ove p r o b a b i l i t y d e n s i t y f u n c t i o n s P f t ( B) and P^(P ) a n ^ 9 shown t h a t f o r i s o t r o p i c a t m o s p h e r e t u r b u l e n c e e7" = 0' where xQ i s t h e " i n n e r s c a l e " o f t u r b u l e n c e , ~2 t h e mean s q u a r e r e f r a c t i v e i n d e x f l u c t u a t i o n and t h e f o r m u l a e i s v a l i d f o r p r o p a g a t i o n d i s t a n c e s X >> r Q , m e a s u r e d a l o n g t h e r a y - p a t h , y F I G . 111-4 A N G U L A R P A R A M E T E R S O F A T M O S P H E R I C R A Y - P A T H 57 The t o t a l angle a between the ray and the " s t r a i g h t l i n e path" or the extension of the i n i t i a l d i r e c t i o n of propagation, represented by the x -ax is i n F i g . I I I -4 , i s given by cos a = cos 9 cos 0 I f a , 9 , and 0 are a l l small angles then 1 - 1/2 a 2 = (1-1/2 9 2) (1-1/20 2) 2 2 2 a' = 9 Z + 0^ 2 2 The p r o b a b i l i t y densi ty funct ions for 9 and 0 can be r e a d i l y c a l c u l a t e d (A-3-1) V ( P > = /. 1 9 exp J - - J = - J 0'$ B $ oo / 2 f f p 2 92 P ,2 ( B ) = r- • ==- exp [ -£-] [ 1^] oo 2 and the p r o b a b i l i t y densi ty funct ion for a , Pw 2 (Y ) < where "0" symbolizes c o n v o l u t i o n . PQ(2 ( p) c a n ke r e a d i l y c a l c u l a t e d (A-3-2) to be P « 2 ( P ' = "^F" S X P ~ I ^  I 0 B oo [*] (B->2) where "2 = 4 / i g p,2 C o n s i d e r the atmospheric ray-path C ( ^ , t ) between \\ m i r r o r s M,- and\ M,- and back o f a f r e e a i r l a s e r s t r a i n meter 5 1 6 to be approximated by s t r a i g h t l i n e segments o f l e n g t h &,Q and l e t a, be the t o t a l angle between the s t r a i g h t l i n e path, K j . th r e p r e s e n t e d by the x - a x i s , and the k segment of the r a y - p a t h k = 1 , 2 , 3 . . . n , F i g . I I I - 5 . The e x t r a d i s t a n c e , over the s t r a i g h t l i n e d i s t a n c e , i" t h t r a v e l l e d by the ray on the k segment o f i t s path d u r i n g i t s t o - a n d - f r o journey between the end m i r r o r s i s ^ k i = 2 J_ O " K J k = l , 2 , 3 . . . . n A € K - 2 [ ^ 0 " 4 c o s a k ] I f a-^ i s a s m a l l angle 2 cos a^. = 1 - 1/2 y S T R A I G H T L I N E A P P R O X I M A T I O N T O T H E A T M O S P H E R I C R A Y - P A T H 60 2 A€ = a k k = l , 2 , 3 . . . n . k The t o t a l e x t r a path l e n g t h , over the s t r a i g h t l i n e d i s t a n c e s , t r a v e l l e d by the ray i s e - t 4 t A, 4 k=l •• k k=l The problem o f c a l c u l a t i n g the p r o b a b i l i t y d i s -t r i b u t i o n f o r € , the e x t r a path l e n g t h , as opposed to c a l c u l a t i n g i t s mean square v a l u e i s c o m p l i c a t e d by the f a c t t h a t i n t e g r a l s (or summations) taken a l o n g the curved ray-path cannot be r e p l a c e d by i n t e g r a l s (or summations) al o n g the s t r a i g h t l i n e path as i s f r e q u e n t l y done i n problems o f t h i s s o r t . Thus w i t h the curved ray-path approximated by s t r a i g h t l i n e segments o f l e n g t h -A ; the d e f l e c t i o n s i n azimuth and e l e v a t i o n , 9^ and 0^ , r e f e r to the angles tl"l between the p r e s e n t d i r e c t i o n o f the k l i n e segment and the e x t e n s i o n o f the k-1 l i n e segment. The t o t a l d e f l e c t i o n i n azimuth and e l e v a t i o n , @^ and , r e l a t i v e t o the e x t e n s i o n o f the i n i t i a l d i r e c t i o n o f p r o p a g a t i o n (or the x - a x i s ) i s g i v e n & k = e L + e 2 + e 3 + . . . + e k 61 ^ k = ^ 1 + ^2 + ^3 + ^ k S i n c e each a n g l e 6^. and 0^. i s a g a u s s i a n random v a r i a b l e w i t h v a r i a n c e and 0 2 ; i t f o l l o w s t h a t and are g a u s s i a n random v a r i a b l e s w i t h v a r i a n c e s g i v e n by = k e 2 " 02" As b e f o r e the s m a l l angle approximation h o l d s and the t o t a l th, angle between the k segment of the ra y - p a t h and the s t r a i g h t l i n e path, a, , i s r e l a t e d t o the t o t a l a z imuthal and e l e v a t i o n angles a n ( ^ $ k ^ 2 Thus a ^ has a p r o b a b i l i t y d e n s i t y f u n c t i o n Pa2 <P> " -r . k °k where '4 = © 2 - * ( 9 2 W 2 ) 0 and k i s the number o f d e f l e c t i o n s . I f the t u r b u l e n c e i s assumed to be i s o t r o p i c , then 62 and 2 a e 2 = 0 2 o r o 2 Given the p r o b a b i l x t y d e n s i t y f u n c t i o n f o r (X ^ , P 2( B) / i t f o l l o w s immediately t h a t the p r o b a b i l i t y d e n s i t y f u n c t i o n f o r = o "k ' p A € k ^ P ) ' •*-s 9 " i - v e n by I t i s c l e a r t h a t s i n c e t h a t the p r o b a b i l i t y d e n s i t y f u n c t i o n f o r € w i l l be g i v e n by the c o n v o l u t i o n o f a s e r i e s of n inhomogeneous p r o b a b i l i t y d e n s i t y f u n c t i o n s P A f ( P ) ' e a c h d i f f e r i n g by the parameter a P € ( Y ' = P A € ( P } 8 P A € ( M » 8 P f i g (P ) 1 2 n T h i s i s an e x c e e d i n g l y awkward s o l u t i o n and can o n l y be g i v e n i n terms o f a s e r i e s summation, the l e n g t h o f the sum v a r y i n g w i t h the d i s t a n c e D . We s h a l l i n s t e a d d e r i v e an approximate s o l u t i o n and j u s t i f y i t by the f o l l o w i n g arguments. The s o l u t i o n t o the c o r r e s p o n d i n g homogeneous X 2 o a k exp (B-3-3) 63 problem i s simple and was d e r i v e d by Markov and Chandrasekhar. I t should be noted t h a t the above problem i s "quasi-homogeneous" i n the sense t h a t a l l members o f the s e t o f n p r o b a b i l i t y d e n s i t y f u n c t i o n s are o f the same mathematical form - an e x p o n e n t i a l - and d i f f e r o n l y among themselves by d i f f e r e n t v a l u e s o f the parameter a 2. • I f c o n s e c u t i v e f l u c t u a t i o n s i n the ray d i r e c t i o n can be c o n s i d e r e d u n c o r r e l a t e d ( t h i s does not imply independence - f o r they are dependent) then the v a r i a n c e o f the p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n f o r € i s the sum o f the v a r i a n c e s f o r the random v a r i a b l e s ^ which make up i t s sum. Thus we s h a l l o b t a i n an approximate p r o b a b i l i t y d e n s i t y f u n c t i o n f o r € ( t ) by c o n v o l v i n g n homogeneous e x p o n e n t i a l s t o g e t h e r t o o b t a i n the mathematical form o f p € ( t ) ( 8 ) and then s c a l e i t a p p r o p r i a t e l y by i n s i s t i n g t h a t the v a r i a n c e o f p € ( t ) ^  P ^ ^Y equal t o the sum o f the v a r i a n c e s o f A g ^ . k = 1 , 2 , 3 . . . n. The c o n v o l u t i o n o f a homogeneous s e t o f n ex-p o n e n t i a l p r o b a b i l i t y d e n s i t y f u n c t i o n s can be e a s i l y done u s i n g the F a l t u n g Theorem. I f 1 a e x P i s a member o f the s e t and oe (B-3-4) o C M as i t s Laplace t r a n s f o r m then A £ a s + i a (See Appendix A-3-3) By the F a l t u n g Theorem the Laplace t r a n s f o r m o f P f ( B ) i s Q € (s) where Qc (s ) = ^ (Br3-5) T h i s i s a t a b u l a t e d Laplace t r a n s f o r m and has i n v e r s i o n ( B ) g i v e n by 1 n - l a" ( n - l ) l exp The f r e e parameter "a" s h a l l be f i x e d by the c o n d i t i o n on the v a r i a n c e o f P^ ( p ) . The v a r i a n c e o f (8 ) i s g i v e n by the i n t e g r a l K be 2 c k o e kJ2 a 2 exp o o 6 *» o o Jl dp which can be shown (29, P. 317) to be 6 5 A £ 2 = 2 (k A ? )2 o o T h e r e f o r e the sum o f the v a r i a n c e s i s g i v e n by K k=l k=l k=l which f o r l a r g e n can be approximated by n -^ „ 3 . 2,2 ^ k = i " ( 4 a o } The v a r i a n c e o f (B ) i s g i v e n by the i n t e g r a l J°° n+1 — 1 — a n ( n - l ) I exp d p which can be shown (29, P. 317) to be £ 2 = a 2 n(n+l) which f o r l a r g e n can be approximated by 72 2 2 £ = a n so We s h a l l i n s i s t t h a t ^ 2 = ^rr A £ ? k k 2 2 n 3 . rt 2 2  a n = X < K ao > T h e r e f o r e a = which g i v e s a p r o b a b i l i t y d e n s i t y f u n c t i o n f o r € ( t ) t ) < P > - (r:) n / 2-where n -1 (4 <£) V D s exp [-ft] * ~ 2 4 V j r u 2 4 a ^ = r (B-3-6) and s i n c e the ray never d e v i a t e s s t r o n g l y from the s t r a i g h t l i n e path n = •?- (B-3-7) ^ S e t t i n g /7 (B-3-8) the p r o b a b i l i t y d e n s i t y f u n c t i o n f o r € (t) can be w r i t t e n p € ( t ) < P > = Ci^ 1 exp o ^ B ^ oo (B-3-9) 67 c ) C a l c u l a t i o n o f t h e P r o b a b i l i t y D e n s i t y F u n c t i o n  f o r S S ( A t ) T h e c h a n g e i n g e o m e t r i c a l r a y - p a t h l e n g t h 6 S ( A t ) i s a r a n d o m v a r i a b l e g i v e n b y 6 S ( A t ) = e ( t 2 ) - eTti) a n d h a s p r o b a b i l i t y d e n s i t y f u n c t i o n ^ 5 5 ^ ^ , - « - y - » f w h e r e P 6 S ^ ) - P E 2 ( B ) 0 P £ i (-3) P 6 S ( ^ ) = j P e 2 ( 3 ) P £ I ( B - Y ) d 3 (B-3-10) - CO H o w e v e r s i n c e P ^ C Y ) i s n o t d e f i n e d f o r n e g a t i v e v a l u e s o f t h e a r g u m e n t e q n . B-3-10 m u s t b e c o n s i d e r e d s e p a r a t e l y f o r y > 0 a n d y < 0 CO 1. Y > 0 P 6 S ( Y ) = J P e 2 ( B ) P E i ( e - Y ) d S (B-3-11) Y 2. Y < 0 P 6 S ( Y ) = j P E 2 ( B ) P £ I ( B - Y ) d 3 o S i n c e e ( t i ) a n d e ( t 2 ) a r e i d e n t i c a l l y d i s t r i -b u t e d r a n d o m v a r i a b l e s i t i s o b v i o u s t h a t ^ $ 5 ^ ^ * s a n e v e n f u n c t i o n o f y s o c o n s i d e r i n g o n l y t h e c a s e y > 0 a n d 68 s u b s t i t u t i n g eqn. B-3-7 i n t o eqn. B-3-11 n (n-1)! I e n " 1 ( B - Y ) n " 1 e x p [ - q $ ] e x p [ - q ( 3 - Y ) ] d 3 n (n-1) e x p [ q Y ] | [ 3 ( 3 - Y ) ] n " 1 e x p [ - 2 q $ ] d B T h i s i s a t a b u l a t e d i n t e g r a l [29] and where T(n) i s the gamma f u n c t i o n and K n _ i ^ l S the m o d i f i e d B e s s e l f u n c t i o n o f order n-h . Since p g s ^ * s e v e n f o l l o w s t h a t (B-3-12) < < The v a r i a n c e 6S 2 i s g i v e n by 6S' • I Y2 p 6 S ( Y ) d Y 69 where P g s ^ * s g i v e n by eqn. B-3-12 6S 2 = 2 | Y 2 P 6 S (Y)dy o l^'^—z [ l T| n4 K n . ( q | T | ) d T T h i s i s a l s o a t a b u l a t e d i n t e g r a l [29] and 6 S 7 = (B-3-13) q 2 S u b s t i t u t i n g eqns. B-3-6, B-3-7, and B-3-8 i n t o eqn. B-3-13 gi v e s 6S2" = 327r(y2)2ft»D» (B-3-14) 3 r o 70 4 . Free A i r Laser S t r a i n Meter Observations S u b s t i t u t i n g eqns. B - 3 - 1 4 and B - 2 - 4 i n t o eqn. B - 2 - 1 3 y i e l d s ' ' s JJI2 JLQD V 327T At Eqn. B - 4 - 1 i s a r e l a t i o n s h i p between the o b s e r v a t i o n time " AT " r e q u i r e d to observe, w i t h a s p e c i f i e d c o n f i d e n c e l e v e l "7| " and s i g n a l - n o i s e r a t i o "s" and i n the presence o f mean square atmospheric r e f r a c t i v e index f l u c t u a t i o n s " n " , a t e c t o n i c change i n d i s t a n c e assumed to be proceedxng a t a l i n e a r r a t e "h^" a n d t l i e d i s t a n c e "D" over which the o b s e r v a t i o n i s t o be made. The q u a n t i t i e s " r 0 " > " ^ 0 " > a n c * "A t " are e s s e n t i a l l y atmospheric c o n s t a n t s equal n u m e r i c a l l y to the " i n n e r s c a l e o f t u r b u l e n c e " , the "outer s c a l e o f t u r b u -l e n c e " , and the " c o r r e l a t i o n time f o r atmospheric r a y - p a t h s " , r e s p e c t i v e l y . The magnitude o f the c o n f i d e n c e parameter, " J§'|" / a p p r o p r i a t e f o r the s p e c i f i e d c o n f i d e n c e l e v e l "7^" i s g i v e n by eqn. B - 2 - 1 1 . S o l v i n g eqn. B - 4 - 1 e x p l i c i t l y f o r AT y i e l d s - [• 3 2 x _ ^ t I l / 3 [a ?*oD|S'|-| 2 / 3 (B-4-2) I 3 J [ 2 M r° 71 Choosing D = 5 km ( 5 x l 0 5 cm.) r = 1 cm. o .3 = 10 J cm. |h^| = 1 cm./year (3.17x10 ^ cm/sec) 1 0 - 1 4 .< ^ l O " 1 2 At = 2 X 1 0 " 1 sec s = 2 (For a d i s c u s s i o n c o n c e r n i n g the a p p r o p r i a t e magnitude o f the above p h y s i c a l q u a n t i t i e s - see /Appendix A-4) Then 55 \t\2/3 sec ^ A f £ 1.25 x 10 3 | j ' | 2 / 3 sec. (B-4-3) depending on the v a l u e of the mean square r e f r a c t i v e index f l u c t u a t i o n s . E v a l u a t i n g eqn. B-4-3 f o r a 99 p e r c e n t c o n f i d e n c e l e v e l o f o b s e r v a t i o n r e q u i r e s the c o n f i d e n c e parameter |'!§»| to e q u a l 2.58. At a 99 p e r c e n t c o n f i d e n c e l e v e l o f o b s e r v a t i o n i t f o l l o w s t h a t 104 sec ^  A T ^ 2.35 x 10 3 s e c . (B-4-4) Thus eqn. B-4-4 i n d i c a t e s t h a t under c o n d i t i o n s o f 2 -14 weak atmospheric t u r b u l e n c e , p. ^ 10 , a f r e e a i r l a s e r s t r a i n meter has a p r o b a b i l i t y o f 0.99 o f o b s e r v i n g , over a d i s t a n c e o f 5 km. and w i t h a s i g n a l - n o i s e r a t i o o f 2:1, a t e c t o n i c change i n d i s t a n c e p r o c e e d i n g l i n e a r l y a t a r a t e o f 72 1 cm/year i n 104 seconds (1.73 minutes) o f o b s e r v i n g time. I f D i s changing a t the r a t e o f 1 cm/year the ne t change i n D i n 104 seconds i s 5D = 3.29 x 10 ^ cm. The r e s u l t i n g s t r a i n s e n s i t i v i t y o f the f r e e a i r l a s e r s t r a i n meter ::under such c o n d i t i o n s would be 5 D = 3.29X10"6  D 5 x l 0 5 4 ^ = 6.6 x I O " 1 2 < B- 4" 5> T h i s s t r a i n s e n s i t i v i t y exceeds the l i m i t o f p r e s e n t l y a t t a i n a b l e l a s e r f r e q u e n c y s t a b i l i t i e s f o r such i n t e r v a l s (12). Under mo d e r a t e - t o - i n t e n s e atmospheric t u r b u l e n c e , 2 —12 |j. ^ 1 0 , the f r e e a i r l a s e r s t r a i n meter can make the same 3 o b s e r v a t i o n a t a 99 p e r c e n t c o n f i d e n c e l e v e l i n 2.35 x 10 seconds (39 m i n u t e s ) . A t a r a t e o f 1 cm/year the d i s p l a c e m e n t 3 -5 o c c u r r i n g xn 2.35 x 10 seconds xs 6D = 7.45 x 10 cm. The r e s u l t i n g s t r a i n s e n s i t i v i t y o f the f r e e a i r l a s e r s t r a i n meter under such c o n d i t i o n s would be 6 D 7 . 4 5 x l 0 ~ 5 D 5x10 5 5 D = 1.5 x I O " 1 0 (B-4-6) D T h i s s t r a i n s e n s i t i v i t y corresponds t o p r e s e n t l y a t t a i n a b l e 73 l a s e r frequency s t a b i l i t i e s f o r such i n t e r v a l s (12). Viewed i n another manner these r e s u l t s i n d i c a t e t h a t a f r e e a i r l a s e r s t r a i n meter c o u l d make "sampled o b s e r v a -t i o n s " o f t e c t o n i c changes i n d i s t a n c e p r o c e e d i n g a t r a t e s o f 1 cm/year or b e t t e r over a d i s t a n c e o f 5 km. w i t h samples measurements o c c u r r i n g every 1.73 minutes i n the presence o f weak atmospheric t u r b u l e n c e and every 39 minutes i n the presence o f m oderate-to-intense atmospheric t u r b u l e n c e . In the f r e e a i r l a s e r s t r a i n meter t h e r e are s e v e r a l f a c t o r s l i m i t i n g the d i s t a n c e over which o b s e r v a t i o n s can be made. Among these a r e : atmospheric d e g e n e r a t i o n o f phase c o -herence and subsequent l o s s o f f r i n g e v i s i b i l i t y , i n c r e a s e o f atmospheric modulation o f the l a s e r frequency and subsequent j i t t e r i n the f r i n g e p a t t e r n , and the frequency s t a b i l i t y o f the l a s e r i t s e l f . 1 I t w i l l be seen i n Chapter I I I , S e c t i o n C t h a t g i v e n s u f f i c i e n t l a s e r power and s u f f i c i e n t l y f a s t f r i n g e c o u n t e r s the l i m i t i n g e f f e c t s o f the atmosphere can be made a r b i t r a r i l y s m a l l , however, the l a s e r frequency s t a b i l i t y p l a c e s an i n t r i n s i c l i m i t on the system. I f the l a s e r frequency s t a b i l i t y as a f u n c t i o n o f o p e r a t i n g time T i s — - — then a "coherence time" 6 t ( T ) can be d e f i n e d as A nec e s s a r y o p e r a t i n g c o n d i t i o n f o r a f r e e a i r l a s e r s t r a i n meter i s t h a t 74 2D « c 6 t ( T ) (B-4-8) 6 t ( T ) decreases m o n o t o n i c a l l y w i t h i n c r e a s i n g T and a l l e l s e b e i n g e q u a l , A T i n c r e a s e s m o n o t o n i c a l l y w i t h D Thus an a r b i t r a r y i n c r e a s e i n the o b s e r v i n g d i s t a n c e D , even i f p e r m i t t e d by other c o n s t r a i n t s , would l e a d e v e n t u a l l y to a v i o l a t i o n of c o n d i t i o n B-4-8. Thus the l i m i t i n g c o n d i t i o n on o b s e r v i n g d i s t a n c e imposed by the coherence l e n g t h of the l a s e r i s c 2D « 6 y (AT ) 327T A t " [ 1 / 3 f s 2ZQ DlS'l A T [ 32TT i t ] 2 lhk! r o 2/3 5. Accuracy o f O b s e r v a t i o n a. E r r o r P r o p a g a t i o n A n a l y s i s The major source o f e r r o r i n f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s i s due to u n c e r t a i n t i e s i n the elements o f the m a t r i x of c o e f f i c i e n t s c oupled w i t h o b s e r v a t i o n a l e r r o r s i n the inhomogeneous terms i n eqn. B - l - 1 0 . I n a c c u r a c i e s i n these q u a n t i t i e s w i l l r e s u l t i n i n c o r r e c t v a l u e s f o r the s o l u t i o n s of U ( A T ) , the f l u c t u a t i o n s i n g e o m e t r i c a l r a y -p a t h l e n g t h . The " o b s e r v a t i o n a l eqns." - eqns. B - l - 1 0 75 *\ l g ( A x ) u A2 l f ( A 2 ) g ( A 2 ) V 5 L A , l f ( A 3 ) g ( A 3 ) w o V J r k * can be w r i t t e n 6 L A X + <*1 6 L A •+ q 2 6 L A . + *3 1 f C A ^ + € 1 2 g ^ ) + € 1 3 1 f ( A 2 ) + e 2 2 g ( A 2 ) + € 2 3 1 f ( A 3 ) + 6 3 2 g ( A 3 ) + € 3 3 u + p. V + p. w + P 3 where the "barred" q u a n t i t i e s r e f e r to the exa c t v a l u e s and q , £ , .and p are t h e i r a b s o l u t e e r r o r s . The o b s e r v a t i o n a l eqn. B-l-10 can be w r i t t e n L = C X where L = * \ + q i 5 L A 2 + 2^ 76 u + P i x V + p . w + P 3 1 f ( A 0 ) + ^ 2 g ( A ( } ) + € 1 f ( A 3 ) + ^ g ( A , + 23 33 The "exact" eqn. can be w r i t t e n C X where " 5 L , 111 77 if v W" l K X j ) g ( x 1 ) IL rtx2) g(x2) 1 ±Tx3) I(x3) Now X = X + p L = L + £ C = E C where 78 q2 V 3 / E I + v 1 * v l l v12 '21 '31 32 13! '22 "23 1 + v 33 and i t i s assumed t h a t v ^ « 1 By s u b s t i t u t i o n i t f o l l o w s t h a t the o b s e r v a t i o n a l eqns X = C" 1 L can be w r i t t e n *X * £ = (E £ ) ~ 1 (L + q) X + p_ - G E (L + a^  79 b u t E -1 I -so *X + p_ = £" 1 ( I - v ) (L + q) and d r o p p i n g terms of the o r d e r o f v 3, X + £ = C L + G ( q - v L ) T h e r e f o r e one can i d e n t i f y £ " r ( q - v L ) The column v e c t o r p_ r e p r e s e n t i n g a b s o l u t e e r r o r s i n the unknown U , V , and W can be decomposed i n t o s y s t e m a t i c and random e r r o r s ^ ^ s y s t + ^ r a n d 80 where (B-5-1) rand (B-5-2) b. Systematic E r r o r s The s y s t e m a t i c e r r o r s i n the m a t r i x of c o e f f i c i e n t s m a n i f e s t themselves as a " s c a l i n g " by a c o n s t a n t f a c t o r of the observed r a t e of t e c t o n i c motion and cannot be e l i m i n a t e d ex-c e p t by more a c c u r a t e knowledge o f the d i s p e r s i o n formula f o r the atmosphere. An e s t i m a t e of the magnitude of t h i s s c a l i n g can be made based on p r e s e n t knowledge of the d i s p e r s i o n formula f o r the atmosphere. The o n l y s y s t e m a t i c e r r o r s a f f e c t i n g the measurement are the e r r o r s i n U , the f l u c t u a t i o n i n g e o m e t r i c a l r a y - p a t h l e n g t h , g i v e n by p. From eqn. B-5-1 i s y s t V j k \ (B-5-3) 81 T h i s c a l c u l a t i o n r e q u i r e s v a l u e s f o r the elements of the ma t r i x v The elements of the m a t r i x ]/ are r e l a t e d the e r r o r s i n the m a t r i x o f c o e f f i c i e n t s by d e f i n i t i o n C = ( I_ + v_ ) *C which gives r i s e t o nine e q u a t i o n s i n nine unknowns. 1 + V l l + V12 + V13 = 1 V 2 1 + 1 + V22 + V23 = 1 V12 v 32 V 3 1 + V32 + 1 + V33 [ T ( A 2 . ) - f ^ ) ] '+ v 1 3 [ f ( A 3 ) - f ^ ) ] - v 2 l [ f - ( A 2 ) - T ( A l ) ] + v 2 3 | j U 3 ) - T ( A 2 ) ] [ f ( A 2 ) - f ( A 2 ) ] - v 3 1 [ f ( A 3 ) - I ' ( A 1 ) ] €22 13 v 1 2 [ * r ( A 2 ) - g - ( A X ) ] + V L 3 [ ? ( A 3 ) - * ( 7 ^ ) ] - V 2 i U ( ^ 2 ) - + v 2 3 ( g ( A 3 ) - g ( A 2 ) } = € 2 3 - V 3 2 [ ? ( A 3 ) - g ( A 2 ) ] - V 3 l [ g ( * 3 , - g ( ^ ) ] = Solving the f i r s t sat of three equations f o r V l t ,yix, , reduces 82 them to a set of six equations. v12 A l + v13 A 2 v12 61 + v13 5 2 \ 2 * 3 -v 21 A l + V23 ( A 2 " A l } = % 22 21 6 1 + v 2 3 ( 6 2 " 5 1 } ~ % 23 v32 ( A2 " A l } " V31 A 2 " %: -v 32 ( 6 2 • 6 1 ) - v 3 1 6 2 = \ . where A l = f ( A 2 ) f (A x) I A2 = f ( A 3 ) 5 1 = g ( A 2 ) g ( A 3 ) g ( A 1 ) . g ( ^ ) The s o l u t i o n s to the nine e q u a t i o n s are 6 1 3 ( A 2 - A 1 } " ^ . 2 ( 5 2 - 6 l ) 83 ^L2 5 2 " €13 A 2 v12 A x 6 2 ~ A 2 6 X ^3 A l ~ 612 6 1 v V 13 A x 6 2 " A 2 6 X €22 ( 6 2 " 5 1 ) " e 2 3 ( A 2 " A l ) 21 A 1 ( 6 2 - 6 1 ) - 6 1 ( A 2 - A 1 ) *22 52 €2 3 A2 V22 A i ( 5 2 _ 5 1 ) - 6 1 ( A 2 - A 1 ) v 6 2 3 A l - ^2 6 1 23 " A 1 ( 5 2 _ 5 1 ) " 6 1 ( A 2 " V ^ 3 < S 2 - A l ) - €32< 62- 6l>  3 1 = 6 2 ( A 2 - A 1 ) - A2< 62 - 61> ^32 ^2 - ^33 A 2 V32 = ~ 6 2 ( A 2 - A 1 ) - A 2 ( 6 2 - 5 ; L ) v €32 6 1 - 633 A l 33 " 6 2 ( A 2 - A l } - A 2 ( 6 2 - 6 1 ) Choosing l a s e r wavelengths A^ = 10 |i , A^ -= 0.5 = 0.3 n , i t f o l l o w s from the d i s p e r s i o n formula £ox t h e atmosphere (A-2 eqns. A-2-7 and A-2-8) t h a t A 1 = 1..JBJD4 :x 10" A 2 = 5.384 x 10~ 6 , 6 = 2.265 x I O - 6 , &2 = 6.291 x 1.0" The r e f r a c t i v e index formula c l a i m s an accur a c y o f one p a r t ±n 9 10 (20) i n which case €. . are of the o r d e r o f 1 x 10 I D randomly d i s t r i b u t e d about z e r o . Thus from the above s o l u t i o n s i t f o l l o w s t h a t 84 V i j W 2 x 1 0 3 (B-5-4) and are a l s o randomly d i s t r i b u t e d about z e r o . From eqn. B-5-3 i t f o l l o w s t h a t the s y s t e m a t i c e r r o r i n U ( A t ) i s P l s y s t P l s y s t = < v 1 3 _ 6 L A i + 6 L A 2 + \ 3 * ^ ) + ( v 2 1 8 L A i + v 2 2 6 1 ^ + ^ 3 6 ^ ) + ( V 3 1 6 L A X + V32 \ + V33S3> (B-5-5) However the exact s o l u t i o n f o r U(A t) i s g i v e n by UC«)= C - J 6 L A i + 6 L ^ + C - J - 8 L A 3 Thus the e x a c t v a l u e o f the f l u c t u a t i o n i n g e o m e t r i c a l ray-path l e n g t h U ( A t ) gets s c a l e d by the s y s t e m a t i c e r r o r s t o a v a l u e o f U(A t) + p^ SySt ' T * i e s c a l : L n 9 ' f a c t o r i s g i v e n by the r a t i o U( A t) + p, x *1 s y s t U ( A t ) 85 The s c a l i n g f a c t o r can be e s t i m a t e d by assuming t h a t to a f i r s t o r d e r ( n e g l e c t i n g d i s p e r s i o n ) the f l u c t u a t i o n s i n o p t i c a l path l e n g t h are a l l e q u a l f o r wavelengths A ^ , A 2 and A ^ Thus §L, = 6L, = 6T\ A l A2 A 3 and P l s y s t = 5 L A ( v l l + V12 + v13 } + ^12 ' 6 L A 2 ( V 2 1 + V22 + V23 > + " l 3 6 L A 3 ( V31 + V32 + V33 ) I f i t i s assumed, on the b a s i s o f eqn. B-5-4, t h a t the are drawn from a g a u s s i a n d i s t r i b u t i o n about z e r o w i t h s t a n d a r d - 3 d e v i a t i o n o f the o r d e r o f 2 x 10 then p 1 s y s t can be e s t i m a t e d t o be P l s y s t = ^ x l O " * 3 £ i i 5 L A i ^ 1 2 6 l A 2 + " I 3 6 l A 3 ] and the s c a l i n g f a c t o r i s g i v e n by 86 U( A t ) + p, . 1 ±2 / 3 x l O 3  ^1 s y s t _ _ U ( A t) ~ 1 U( A t) + p, S Y S t = 1.0 ± 0.00346 Sin c e t h i s s c a l i n g f a c t o r i s p r e s e n t i n each d e t e r -m i n a t i o n o f U(A t) i t w i l l a l s o be p r e s e n t i n t h e i r mean and so w i l l be r e f l e c t e d i n the observed t e c t o n i c motions. In o t h e r words, p r e s e n t day knowledge of the atmospheric d i s p e r s i o n would r e s u l t i n s y s t e m a t i c e r r o r s i n the observed t e c -•tonic motions o f the order o f -0.3% c. Random E r r o r s The random e r r o r s i n f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s are the r e s u l t o f i n a c c u r a c i e s i n the f r i n g e count. Keeping the random e r r o r s below a maximum a c c e p t a b l e l e v e l w i l l p l a c e l i m i t a t i o n s on the accuracy t o which f r i n g e s must be counted. From eqn. B-5-2 c " 1 ^ i rand i j The o n l y random e r r o r s a f f e c t i n g the measurement are the e r r o r s i n U , the f l u c t u a t i o n i n g e o m e t r i c a l r a y - p a t h l e n g t h , g i v e n 87 by 2-p l rand ~ j = l C i j q j I f f r i n g e s are counted on each channel t o ± £ p a r t s o f a f r i n g e , then and '1 rand = t€l~lW * « 2 ~ U *2 ± A 3 < B - 5 " 6 ) The worst case o c c u r s when a l l £ have the same s i g n . Thus P l rand ^ * (fl!" \ + ~ l l A 2 + ^ 1 3 A 3 ] I f wavelengths A^ = 1 0 (j, , A 2 = 0 . 5 p. , A 3 = 0 . 3 p. are chosen P l rand « « * 3 [ 3 3 - 3 " l l + L ' 6 6 ^ 1 2 + ^ 1 3 ~ ] ( B - 5 - 7 ) NOW = c o £ - p i J  i j d e t - [ c i j l and from the d e f i n i t i o n o f C\. i t f o l l o w s t h a t 88 " * - l c , : = 1 + g ( A 1 ) f A 1 - A 2 ] + f ( A 1 ) [ s ^ s j 11 A l 6 2 " A 2 6 - l g ( A x ) A 2 - f ( A 1 ) 6 2  C12 = A ! 5 2 " A 2 5 i f(\) \ - g ( A 1 ) A L r " = A l 6 2 " A2 5 1 and u s i n g the above wavelengths and d i s p e r s i o n formula f o r the atmosphere (A-2 eqns A-2-7 and A-2-8) i t f o l l o w s t h a t i " ^ ) = 6.487 x 10" 5 f(\) = 7.749 x 10~ 5 g ( A 2 ) = 6.714 x 10" 5 f < A 2 ) " 7.930 x 1 0 - 5 g"(A3) = 7.180 x 10~ 5 ^ ^ 3 ) = 8.288 x 10~ 5 and so C " l ^ -93.7 C 1 2 = 163.5 — —68.8 S u b s t i t u t i n g these f i g u r e s i n t o eqn. B-5-7 g i v e s P l rand « 8 ' 7 4 x 1 0 ~ 2 * c m « 89 d. Accuracy o f F r i n g e Counting The proposed method o f o b s e r v a t i o n o f the f r e e a i r l a s e r s t r a i n meter r e l i e s on a v e r a g i n g a s e r i e s o f continuous c o n s e c u t i v e o b s e r v a t i o n s o f the net f l u c t u a t i o n s i n the geo-m e t r i c a l r a y - p a t h l e n g t h t o reduce the atmospheric e f f e c t s (which have a long term mean o f zero) t o " i n s i g n i f i c a n t " l e v e l s . The q u a n t i t y "y " r e p r e s e n t e d by the e r r o r b a r s i n F i g . I I I - 3 i s the expected d i f f e r e n c e or " e r r o r " between the sample mean o f M samples o f the atmospheric f l u c t u a t i o n i n g e o m e t r i c a l r a y -path l e n g t h , 6S( A t ) , and i t s p o p u l a t i o n mean, 8S( A t ) = 0 . The e x a c t v a l u e o f 6S i s contaminated by the e f f e c t s o f bot h s y s t e m a t i c and random e r r o r s 5 S o b s ~ 5 S e x a c t + p l s y s t + p l rand The s y s t e m a t i c e r r o r s are i n h e r e n t i n the o b s e r v i n g method and can be combined w i t h 6 S . t o g i v e 6S . ex a c t ^ atmos 6S , = 6S . + p, , obs atmos r l rand 2 The q u a n t i t y 6S appearing i n eqn. B-2-13 should 2 i d e a l l y be 6 S , b u t w i l l i n the course o f f r e e a i r l a s e r atmos 2 s t r a i n meter o b s e r v a t i o n s be s u b s t i t u t e d f o r by 5 s o b s 6 s 2 = 6s 2 + p 2 , obs atmos r l rand 90 In o r d e r t o p e r m i t t h i s i t i s nec e s s a r y t h a t 2 2 p l rand « 6 S a t m o s . Pl rand = < ± ' C l l A l ± *2 ^  \ ± € 3 ^ 3 I f the f r i n g e o b s e r v a t i o n e r r o r s are assumed t o be u n c o r r e l a t e d i n each channel w i t h a mean v a l u e of ze r o then P l r a n d = = U A l > 2 + « 2 ^ V* + 1*3*11 V* U s i n g the above v a l u e s f o r A^ , A 2 , and A^ P l r a n d " [ ( 3 3 " 3 ' S'A )2 + ( 1 - 6 6 ° U > 2 + < ^ 3 > 2 ] v*»_l ^—1 1 and u s i n g the above v a l u e s f o r , C^ 2 a n d C^^ p l rand = 8 • 8 3 x 1 0 3 & cm 2 .. (B-5-8) 2 The v a l u e of 5 S a t m o s ^ s 9^- v e n ky eqn. B-3-14 to be 91 T a k i n g * o 2 _ 32TT (n 2 ) 2 4 2 p 2 O b . = x atmos - 2 3 r o 2 - i n " 1 2 \i = 1 0 o 103 cm D = 5 x 10 5 cm r = 1 cm (For a d i s c u s s i o n c o n c e r n i n g the a p p r o p r i a t e magnitude of these q u a n t i t i e s - see Appendix A-4) I f i t i s r e q u i r e d t h a t f. i ' 2 2 p l rand ^ 6 Satmos then 8.83 x 10" 3 € 2 « 8.39 x I O - 6 € 2 << 10" 3 or 6 » 10 92 Thus i t i s seen t h a t c o u n t i n g f r i n g e s to an accuracy of ±10" 2 p a r t s of a f r i n g e i s s u f f i c i e n t to f a c i l i t a t e f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s over a d i s t a n c e of 5 km i n the presence of moderate atmospheric t u r b u l e n c e . Present methods of f r i n g e o b s e r v a t i o n i n c o n v e n t i o n a l l a s e r s t r a i n meters are accurate to s m a l l f r a c t i o n s of a f r i n g e . The 1 km. long l a s e r s t r a i n meter i n the Cascade Tunnel of the Stevens Pass i n the s t a t e of Washington, U.S.A. counts f r i n g e s to an accuracy of ± 1 0 " 2 p a r t s of a f r i n g e (39). A small l a s e r s t r a i n meter has been designed by S a l i s b u r y and Gangi (42) which i s capable of measuring s t r a i n s of one p a r t i n 1 0 1 0 over a d i s -tance of 1 meter. T h i s s t r a i n meter operates w i t h an 0.6328y He - Ne l a s e r and such measurements correspond to f r i n g e obser-v a t i o n s of ±3.06 x 10"k p a r t s of a f r i n g e . Moreover i t i s c l a i m e d (39) t h a t the f l u c t u a t i o n s i n the number of photons i n the l a s e r output due to i n h e r e n t n o i s e c h a r a c t e r i s t i c s of the l a s e r are so low t h a t measurements ac c u r a t e to ±10" 5 p a r t s o f a f r i n g e are p r a c t i c a l . 93 C. FRINGE OBSERVATION 1. D e g e n e r a t i n g E f f e c t s o f t h e A t m o s p h e r e o n a L a s e r Beam The d e g e n e r a t i n g e f f e c t s o f a t u r b u l e n t a t m o s p h e r e on a l a s e r beam p r o p a g a t i n g t h r o u g h i t c a n be r o u g h l y d i v i d e d i n t o two c l a s s e s ; " w h o l e beam" e f f e c t s a n d " i n t e r n a l " e f f e c t s , d e p e n d i n g o n w h e t h e r t h e e f f e c t i s p r i m a r i l y c a u s e d b y t u r b u l -e n c e s c a l e s l a r g e r o r s m a l l e r t h a n ZQ . "Whole beam e f f e c t s " i n c l u d e : ( 1 ) D e v i a t i o n s o f d i r e c t i o n o f t h e e n t i r e beam -" q u i v e r i n g " . ( 2 ) L a t e r a l d i s p l a c e m e n t s o f t h e e n t i r e beam -" s p o t d a n c i n g " . ( 3 ) F l u c t u a t i o n s i n c r o s s - s e c t i o n a l a r e a o f t h e e n t i r e beam - " b r e a t h i n g " . T h e w h o l e beam e f f e c t s a r e t h o s e w h i c h c o n t r i b u t e t o t h e e x t r a r a y - p a t h l e n g t h o v e r t h e s t r a i g h t l i n e d i s t a n c e , e ( t ) , a n d h e n c e t o 6 S ( A t ) . W h o l e beam e f f e c t s p r o d u c e f l u c t u a t i o n s i n t h e t r a n s i t t i m e o f t h e beam a n d i n t h e c a s e o f l a s e r o u t p u t m a n i f e s t t h e m s e l v e s a s a f r e q u e n c y m o d u l a t i o n . S i n c e a n e l e c t r o m a g n e t i c d i s t u r b a n c e c a n be d e s c r i b e d a s a c o m p l e x s c a l a r f u n c t i o n o f p o s i t i o n d e f i n e d b y a n a m p l i -t u d e a n d a p h a s e ( i f p o l a r i z a t i o n e f f e c t s a r e i g n o r e d ) i t f o l l o w s t h a t t h e r e m a i n i n g " i n t e r n a l " d e g e n e r a t i n g a t m o s p h e r i c e f f e c t s 94 c a n be d i v i d e d i n t o two s u b - c l a s s e s - " a m p l i t u d e e f f e c t s " a n d " p h a s e e f f e c t s " . " A m p l i t u d e e f f e c t s " c a n b e d e s c r i b e d a s : (4) F l u c t u a t i o n s i n i n t e n s i t y o f v a r i o u s r e g i o n s w i t h i n t h e beam - " b o i l i n g " . " P h a s e e f f e c t s " c a n b e d e s c r i b e d a s : (5) F l u c t u a t i o n s i n t r a n s i t t i m e b e t w e e n i n d i v i d u a l r a y s o f t h e beam p r o d u c i n g " c r u m b l i n g o f t h e wave f r o n t " o r " p h a s e f l u c t u a t i o n s " o v e r t h e beam's c r o s s s e c t i o n . »\ F o r t h e o p e r a t i o n o f a f r e e a i r l a s e r s t r a i n m e t e r o n l y e f f e c t s \ (4) a n d (5) a r e p o t e n t i a l l y t r o u b l e s o m e . i S i n c e t h e a t m o s p h e r i c r e f r a c t i v e i n d e x , e v e n u n d e r t h e m o s t t u r b u l e n t c o n d i t i o n s , i s a c o n t i n u o u s f u n c t i o n o f p o s i t i o n , i t f o l l o w s t h a t t h e f r o n t s o f c o n s t a n t p h a s e , w h i c h c a n b e a s s u m e d t o b e p l a n e a t t h e e x i t p u p i l o f t h e l a s e r , a l w a y s r e m a i n c o n n e c t e d s u r f a c e s i n s p a c e as t h e beam p r o p a -g a t e s t h r o u g h t h e a t m o s p h e r e . T h u s a l a s e r beam p r o p a g a t i n g t h r o u g h a t u r b u l e n t a t m o s p h e r e c a n b e d e s c r i b e d a s b e i n g " l o c a l l y c o h e r e n t " . T h i s i s i n t e n d e d t o d e s c r i b e t h e f a c t t h a t s u r r o u n d i n g a n y g i v e n p o i n t on a f r o n t o f c o n s t a n t p h a s e t h e r e e x i s t s a " c o h e r e n c e p a t c h " o v e r w h i c h t h e p h a s e f r o n t c a n b e a p p r o x i m a t e d , t o a n y d e s i r e d a c c u r a c y , b y a p l a n e . T h e d i m e n s i o n s o f t h e c o h e r e n c e p a t c h d e c r e a s e m o n o t o n i c a l l y w i t h i n c r e a s i n g s p e c i f i e d d e g r e e o f c o h e r e n c e a n d w i t h i n -c r e a s i n g d i s t a n c e o f p r o p a g a t i o n . I f t h e p o r t i o n o f t h e i n c o m i n g w a v e f r o n t i n c i d e n t o n t h e d i f f r a c t i n g a p e r t u r e s i n 95 the i n t e r f e r o m e t e r s I i , I 2 and I 3 , F i g . 111-1, has dimensions comparable to a r e g i o n of h i g h coherence, then i n t e r f e r e n c e f r i n g e s w i t h a c o r r e s p o n d i n g l y h i g h v i s i b i l i t y w i l l r e s u l t . Thus by s u f f i c i e n t l y r e s t r i c t i n g the d i f f r a c t i n g a p e r t u r e and c o r r e s p o n d i n g l y i n c r e a s i n g the power output of the l a s e r a f r e e a i r l a s e r s t r a i n meter can be made to opera i n a c o n d i t i o n unperturbed by the d e g e n e r a t i n g e f f e c t s of atmospheric t u r b u l e n c e . 96 2. Fraunhofer D i f f r a c t i o n from a Double Rectangular  Aperture The i n t e r f e r o m e t e r s I i , I2 and I 3 o f a f r e e a i r l a s e r s t r a i n meter, F i g . 111 -1, w i l l c o n s i s t o f double r e c t a n g u l a r d i f f r a c t i n g a p e r t u r e s . The o p t i c a l system w i l l be arranged so t h a t l i g h t t r a n s m i t t e d by m i r r o r Mi» i s i n c i -dent on r e c t a n g u l a r a p e r t u r e #1 and l i g h t from the r e t r o -r e f l e c t o r o f f m i r r o r M s i s i n c i d e n t on r e c t a n g u l a r a p e r t u r e #2. I t i s assumed t h a t the a p e r t u r e dimensions are equal and s m a l l enough to ensure reasonable coherence a c r o s s the a p e r t u r e . I t i s f u r t h e r assumed t h a t the a i r y d i s c o f the o p t i c a l channel p r e c e d i n g the d i f f r a c t i n g a p e r t u r e i s l a r g e enough to ensure the e x i s t e n c e o f a r e g i o n o f the wavefront f o r i n t e r f e r e n c e which i s unperturbed by d i f f r a c t i o n of the o p t i c a l channel. The l i g h t i n c i d e n t on a p e r t u r e #1 w i t h angular frequency u>i and the l i g h t i n c i d e n t on a p e r t u r e #2, having t r a v e r s e d the t u r b u l e n t atmosphere, w i l l be p a r t i a l l y coherent and have instantaneous angular frequency W2 , ii)2 / u i . The instantaneous frequency W2 v a r i e s randomly, w i t h mean value wi , due to the e f f e c t i v e frequency modula-t i o n imposed on the l a s e r beam by the t u r b u l e n t atmosphere. Con s i d e r two r e c t a n g u l a r d i f f r a c t i n g a p e r t u r e s #1 and #2, F i g . 111-6, i n the 5n-plane and assume t h a t the o p t i c a l d i s t u r b a n c e can be r e p r e s e n t e d by a continuous s i n g l e v a l u e d 98 c o m p l e x f u n c t i o n o f p o s i t i o n , t h e s q u a r e o f t h e m o d u l u s o f w h i c h i s p r o p o r t i o n a l e v e r y w h e r e t o t h e i n t e n s i t y o f t h e o p t i c a l d i s t u r b a n c e a t t h a t p o i n t . T h e t o t a l d i s t u r b a n c e a t a p o i n t x y i n t h e x y - p l a n e i s g i v e n b y t h e sum o f t h e c o n t r i b u t i o n s , a d d e d a c c o r d i n g t o p h a s e , f r o m t h e s o u r c e e l e m e n t s d £ d n i n t h e £ n - p l a n e . I f t h e o p t i c a l d i s t u r b a n c e i n t h e £ n - p l a n e i s r e p r e s e n t e d b y <KSn) = A(£n)exp-i3(Sn;t)dCdn w h e r e A ( £ n ) i s t h e a m p l i t u d e p e r u n i t a r e a a n d g(^n;t) i s t h e t o t a l p h a s e o v e r t h e £ n - p l a n e T h e n t h e s o u r c e e l e m e n t d £ d n a t £n , i n t h e £n.-plane, p r o d u c e s a d i s t u r b a n c e a t x y , i n t h e x y - p l a n e , g i v e n b y dB = £ i | H i e x p - i ( B ( 5 n ; t ) - <j>)d£dn o w h e r e 4> i s a p h a s e a n g l e r e s u l t i n g f r o m t h e t r a n s i t t i m e f r o m t h e p o i n t £n t o t h e p o i n t x y <j) = 2ir — X 99 T h e t o t a l d i s t u r b a n c e a t x y i s g i v e n b y B ( x y ) = Sn p l a n e M ^ l ! e x p - i [ 3 ( 5 n ; t ) 2irr ]d£dn ( C - 2 - 1 ) From F i g . I I 1 - 6 r 2 = r 2 + (x - 5 ) 2 + ( y - n) ( C - 2 - 2 ) a n d R2 = r 2 + x 2 + y 2 ( C - 2 - 3 ) t h e r e f o r e f r o m e q n s . C-2-2 a n d C-2-3 •2 = R2 + 5 2 + n 2 - 2 ( x ? + yn) ( C - 2 - 4 ) c o m p l e t i n g t h e s q u a r e i n e q n . C-2-4 .2 _ (R 2 + e • n 2 ) ^ 2 xc + yn (R 2 + + n 2) (xg + y n ) 2 R2 + £ 2 + n 2 ( C - 2 - 5 ) 99a F r a u n h o f e r d i f f r a c t i o n h o l d s when o r wnen ( C - 2 - 6 ) x^ + V" « 1 R 2 + £ 2 + n 2 s o r = R - x K + y n R ( C - 2 - 7 ) I f A ( 5 n ) a n d B ( £ n ; t ) a r e a s s u m e d t o be c o n s t a n t o v e r t h e C n - p l a n e e q n . C-2-1 c a n be w r i t t e n B ( x y ) = R c - a 2irr> dn d£ e x p - i ( w i t - — ) + c b J M - c a d£ e x p - i ( w 2 t - ^-^0 ( C - 2 - 8 ) w h e r e wi , w 2 a r e t h e e f f e c t i v e a n g u l a r f r e q u e n c i e s o f t h e l a s e r o u t p u t o b s e r v e d o v e r a p e r t u r e s #1 a n d #2 r e s p e c t i v e l y . Make a c h a n g e o f v a r i a b l e s i n t h e f i r s t i n t e g r a l V « - K r ' = r ( n O 9 9 b Eqn. C-2-8 then becomes B ( x y ) = r A o R a -c exp-i(wit - — J + exp-i(w2t - — j — ) B ( x y ) = 1°^  R c b j d n jdc -c a c o s ( w i t - ) + c o s ( w 2 t - ) -i(sin(wit - ) + sm(w2t - J) Using the formulae f o r the a d d i t i o n o f s i n e s and c o s i n e s 2r A B ( x y ) = —|-c b f M -c a d£ c o s ( f i i t - j- ( r ' + r ) ) c o s ( f t 2 t - £( r» i s i n (ftxt - J ( r ' + r ) ) c o s ( f i 2 t - (r» (C-2-9) •where 2 " fi2 = (C-2-10) 9 9 c From eqn. C-2-7 i t f o l l o w s t h a t •• + r = 2(R - g l ) i _ r - 2|1 (C-2-11) S u b s t i t u t i n g eqns. C-2-11 i n t o eqn. C-2-9 g i v e s B(xy) = i A c b -c a cos (n xt + ^ - ( ^ - R ) ) c o s ( f i 2 t - £L 2TT X £ . I s i n («it + I^ CP- " R ) ) c o s ( f t 2 t - 2TT X £ \ X~ R~3 (C-2-12) The i n t e g r a t i o n o f eqn. C-2-12 i s s t r a i g h t f o r w a r d and g i v e s c o s C 2 ^ - n xt) + i s i n C 2 - ^ - n at) (C-2-13) The i n t e n s i t y o f the l i g h t i n xy-plane i s giv e n by I(xy) = B*B 99d I(xy) = 1 6 c 2 ( b - a ) r A o R 2 . ? T T X s i n j ^ ( b - a ) ( x>-a)) s m 2 2iryc XR f 2 i T y c l 2 I J cos' - fi (C-2-14) S i n c e the l i g h t i n c i d e n t on the two a p e r t u r e s i s on l y p a r t i a l l y coherent the r e s u l t a n t i n t e n s i t y p a t t e r n I (xy). must be m o d i f i e d t o a l l o w f o r the p a r t i a l coherence. Eqn. C-2-14 can be w r i t t e n I(xy) = 16c 2 (b-a) 1 - s i n 2 | ^ - ( b + a ) - fl2t The e f f e c t o f the p a r t i a l l y coherent r a d i a t i o n i s t o reduce the v i s i b i l i t y o f the i n t e r f e r e n c e f r i n g e s t o the v a l u e V , 0 < V < 1 , [ 3 4 ] . Thus I(xy) = 16c 2 (b-a) m2 1 - V s i n : J | (b--a) - fi2t (C-2-15) 100 A sk e t c h o f the i n t e n s i t y p a t t e r n I(xy) f o r f r i n g e v i s i b i l i t y V ^ 0.4 i s shown i n F i g . 111-7 . From formula C-2-15 i t i s c l e a r t h a t the f r i n g e s are d i s p l a c e d i n the p o s i t i v e or nega-t i v e x - d i r e c t i o n at a r a t e p r o p o r t i o n a l t o , and depending on the s i g n o f , 2Sl2 o - m ' 2 I f the o p t i c a l path l e n g t h L i s changing at a r a t e _ dL ^. . . . . r p . i . . . dT 1 dL o f g^- , the t r a n s i t time T i s changing at a r a t e = dT I f the v e l o c i t y o f l i g h t i s assumed to be c o n s t a n t , a l i n e a r r a t e o f change of the t r a n s i t time o f the l i g h t s i g n a l imposes a c o n s t a n t d.c. s h i f t on the e f f e c t i v e frequency o f the l i g h t dT s i g n a l o f an amount Aw = wi T h e r e f o r e A u i dL Aw = wi - o ) 2 = 7ft and o > o)i dL " 2 ~ TE ar The net f r i n g e count f o r the wavelength X i n an i n t e r v a l t i t 2 i s gi v e n by i Y ? f 2 d L . ( t ) C X ^ t 2 > = 2 T J 2^dt " 4Tc- j - d T - d t t i t i 101 y ——• ')(] A 11 V 0 . 4 F I G . I I I-7 I N T E N S I T Y P A T T E R N , I ( x y ) , F R O M F R A U N H O F E R D I F F R A C T I O N T H R O U G H A D O U B L E R E C T A N G U L A R A P E R T U R E F R I N G E V I S I B I L I T Y V ~ 0 . 4 102 C x ( t i t 2 ) = \ ( L x ( t 2 ) - L x ( t i ) ) X C x ( t ! t 2 ) = 6 L x ( t i t 2 ) (C-2-16) E q n . C - 2 - 1 6 e v a l u a t e d f o r Xi , A 2 a n d X 3 i s s e e n t o b e i d e n t i c a l w i t h e q n s . B - l - 1 1 . 103 3. F r i n g e V i s i b i l i t y The coherence l e n g t h ( i . e . the maximum d i f f e r e n c e i n path l e n g t h t h a t can be i n t r o d u c e d between two l i g h t s i g n a l s and s t i l l produce observable i n t e r f e r e n c e e f f e c t s ) o f l a s e r output frequency s t a b i l i z e d to °» ~ i s equal to ^coh ^ o~v A l i m i t a t i o n on the d i s t a n c e D over which a f r e e a i r l a s e r s t r a i n meter can operate i s simply coh For a l a s e r w i t h a frequency ^10 1 ** Hz. and a s h o r t term f r e -quency s t a b i l i t y o f one p a r t i n 1 0 1 1 the coherence l e n g t h i s of the order of 300 km. Since a s h o r t term frequency s t a b i l i t y S V ( T ) o f one p a r t i n 1 0 1 1 i s not d i f f i c u l t to achieve over i n t e r v a l s T of the order o f 10" 3 sec (the t r a n s i t time o f a photon between the end m i r r o r s o f a f r e e a i r l a s e r s t r a i n meter) [ A - l ] , i t appears t h a t the frequency s t a b i l i t y o f l a s e r s i s ample f o r the r e a l i z a t i o n o f f r e e a i r l a s e r s t r a i n meters of l e n g t h s up to 100 km. However the d e g e n e r a t i o n of phase coherence w i t h i n the beam due to atmospheric t u r b u l e n c e and the 104 s u b s e q u e n t d e c r e a s e o f f r i n g e v i s i b i l i t y p r o v e s t o be a more s e v e r e l i m i t a t i o n o n o p e r a t i n g d i s t a n c e t h a n c o h e r e n c e l e n g t h . The c o m p l e x c o h e r e n c e f u n c t i o n y f x ) b e t w e e n P.1P2 t h e o p t i c a l d i s t u r b a n c e a t p o i n t s p i = ( 5 i n i ) i n a p e r t u r e #1 and p 2 = (£2*12) i n a p e r t u r e #2, F i g . I I I - 8 , i s d e f i n e d [ 3 5 , c h a p t . 10] as y ( T ) = EJ E l ( C - 2 - 1 7 ) P l P 2 [•_, (t)<j>* (t)<J> (t)<j>* 0 0 ] * ., Px P i P2 P2 w h e r e t h e b a r d e n o t e s a t i m e a v e r a g e a n d w h e r e 4> p i = A ( t ; i n i ) e x p - i B (S im ; t ) d £ d n <!>_. = A ( £ 2 n 2 ) e x p - i 6(52n 2 ;t)dcdn P2 ( C - 2 - 1 8 ) a r e t h e o p t i c a l d i s t u r b a n c e s o v e r t h e e l e m e n t o f a r e a d£dn s u r r o u n d i n g p o i n t s p i a n d P2 r e s p e c t i v e l y . S u b s t i t u t i n g e q n s . C-2-18 i n t o e q n s . C-2-17 y i e l d s Y„ n CO = e x p - i [B (?mi ; t + T ) - 3C52H2 ;t)]d^dn P1P2 I f r e l a t i v e t i m e d e l a y x b e t w e e n t h e i n t e r f e r i n g l i g h t s i g n a l s i s s m a l l c o m p a r e d t o t h e c o h e r e n c e t i m e 1/Sv , 105 1 APERTURE #1 APERTURE #2 • •IVs ( M O FIG. I 11-8 COHERENCE BETWEEN APERTURES ttl AND %2 106 w h e r e 6v i s t h e s p e c t r a l w i d t h o f t h e " m o n o c h r o m a t i c " r a d i a -t i o n , t h e n Y ( T ) = Y ( o ) P1P2 J TPiP2 J S i n c e f r e e a i r l a s e r s t r a i n m e t e r s m u s t a l w a y s o p e r a t e u n d e r t h e s e c o n d i t i o n s , i t f o l l o w s t h a t Y „ = e x p - i [ 8 ( £ x m ; t ) - B ( C 2 n 2 ; t ) ] d 5 d T 1 ( C - 2 - 1 9 ) P1P2 F o r p u r p o s e s o f c a l c u l a t i n g t h e f r i n g e v i s i b i l i t y l e t 3 ( £ i n i ; t ) = wt ( C - 2 - 2 0 ) e ( ? 2 T i 2 ; t ) = w t + e(5n;t) w h e r e u i s t h e mean f r e q u e n c y o f t h e two a p e r t u r e s a n d 9(?2n2*,t) i s a r a n d o m p h a s e a n g l e a t t h e p o i n t p 2 = (£2*12) m e a s u r e d r e l a t i v e t o t h e mean p h a s e a n g l e a c r o s s t h e a p e r t u r e . 9 ( ? 2 n 2;t) r e s u l t s f r o m t h e d e g e n e r a t i n g e f f e c t s o f a t m o s p h e r i c t u r b u l e n c e o n t h e beam. From e q n s . C-2-19 a n d C-2-20 Y „ _„ = e x p - i 9 U2TI2 ; t ) d £ d n P1P2 107 A mean c o m p l e x c o h e r e n c e f u n c t i o n , <y> , f o r t h e r a d i a t i o n i n c i d e n t o n t h e two a p e r t u r e s c a n be c a l c u l a t e d b y an i n t e g r a l o v e r t h e s o u r c e a r e a S o f t h e £ n - p l a n e • i II < Y > = xr JJ t c o s 0 ( ? 2 n 2 ; t ) - i s i n e ( 5 2 n 2 ; t ) ] d ^ d n p l a n e w h e r e <•> i n d i c a t e an a v e r a g e o v e r t h e £ n - p l a n e . S i n c e 9 ( £ 2 T l 2 ; t ) i s n o t d e f i n e d o u t s i d e o f a p e r t u r e #2 c b <Y> = 2 c ( b - a ) | d n | d ? ^ c o s e ^ 2 n 2 ; t ) - i s i n 9 ( C 2 n 2 ; t ] •c a B e c a u s e t h e p h a s e f l u c t u a t i o n s a c r o s s a p e r t u r e #2 a r e r a n d o m w i t h mean v a l u e z e r o i t f o l l o w s t h a t c 5 J d n j s i n 9 ( £ 2 n 2 ; t ) d 5 = 0 -c a an d _ 5  V = I<Y>1 = 2c(b-a) | d n j c o s 9 ( 5 2 n 2 ; t ) d ? - c a Now e ( 5 2 n 2 ; t ) = 8 ( £ 2 n 2 ; t ) - u t 108 w h e r e B ( S 2 n 2 ; t ) i s t h e i n s t a n t a n e o u s t o t a l p h a s e a n g l e a t t h e p o i n t p 2 a n d art = 8 Q i s t h e mean p h a s e a n g l e a c r o s s a p e r t u r e #2. H e n c e c b •c a (C - 2 - 2 1 ) I n t e r c h a n g i n g t h e o r d e r o f i n t e g r a t i o n a n d t i m e a v e r a g i n g c b V = 1 -- c a I f i t i s a s s u m e d t h a t t h e d i m e n s i o n s o f t h e d i f f r a c t i n g a p e r t u r e a r e s m a l l e n o u g h t o a l l o w o n l y s m a l l p h a s e d e v i a t i o n s a c r o s s t h e a p e r t u r e t h e n c b V - 1 2 c ( b - a ) j d n j d S i (e-B 0: c a I f t h e f r i n g e v i s i b i l i t y i s r e q u i r e d t o be l a r g e r t h a n a g i v e n minimum, V Q , t h e n c b } d n } d 5 ( 3 - 3 c ) 2 ^ 2 ( i . V o ) •c a < ( B - 3 0 ) 2 > - 2 ( 1 - V 0 D ( C - 2 - 2 2 ) 109 I f t h e a t m o s p h e r i c p h a s e f l u c t u a t i o n s o f t h e l a s e r o u t p u t a r e h o mogeneous a n d i s o t r o p i c t h e n <(6-6 Q) 2> i n e q n . C-2-22 i s r e c o g n i s e d as t h e p h a s e s t r u c t u r e f u n c t i o n D g g ( p ) > [A-2] <7Fe07r> = DBB(P> w h e r e p = [ - £ 2 ) z + ( n Q - n 2 ) ] 2 • T h u s f o r a f r i n g e v i s i b i l i t y o f V o r b e t t e r ' o D g 3 (p) - 2(1-V Q ) ( C - 2 - 2 3 ) E q n . C-2-23 i m p o s e s a r e s t r i c t i o n on t h e a r e a o f t h e i n c o m i n g w a v e f r o n t , a n d h e n c e on t h e d i m e n s i o n s o f t h e r e c t a n -g u l a r d i f f r a c t i n g a p e r t u r e s , a c c e p t a b l e f o r u s e i n a f r e e a i r l a s e r s t r a i n m e t e r . The a r e a o f t h e i n c o m i n g w a v e f r o n t a c c e p t a b l e f o r i n t e r f e r e n c e i n a f r e e a i r l a s e r s t r a i n m e t e r , t o p r o d u c e f r i n g e s o f v i s i b i l i t y V Q o r g r e a t e r , m u s t n o t h a v e l i n e a r d i m e n s i o n s e x c e e d i n g p Q , w h e r e f o r p - p Q e q n . C-2-23 i s s a t i s f i e d . T h e n a t u r e o f t h e p h a s e s t r u c t u r e f u n c t i o n f o r l a s e r r a d i a t i o n p r o p a g a t i n g t h r o u g h a t u r b u l e n t a t m o s p h e r e h a s b e e n i n v e s t i g a t e d b y Beckmann [22] a n d F r i e d a n d C l o u d [ 3 2 ] . F r i e d a n d C l o u d h a v e c a r r i e d o u t a v e r y t h o r o u g h a n a l y s i s s t a r t i n g w i t h an 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 t m o s p h e r i c r e f r a c t i v e i n d e x f l u c t u a t i o n s C , . ( r ) whose f o r m i s p r e d i c t e d b y t h e 110 K o l m o g o r o f f t h e o r y o f t u r b u l e n c e t o be 2 C ( r ) = 1 - I ( r / R ) J ( C - 2 - 2 4 ) w h e r e R Q i s t h e " o u t e r s c a l e " o f t u r b u l e n c e , t h e y d e v e l o p e d a t h e o r e t i c a l p h a s e s t r u c t u r e f u n c t i o n f o r h o r i z o n t a l p r o p a -g a t i o n o f l a s e r o u t p u t D R R ( p ) w h e r e k = - y i s t h e wave number o f t h e l a s e r r a d i a t i o n a n d D i s t h e d i s t a n c e o f h o r i z o n t a l p r o p a g a t i o n , y 2 i s t h e mean s q u a r e r e f r a c t i v e i n d e x f l u c t u a t i o n s . B eckmann b e g i n s w i t h a g a u s s i a n 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 r e f r a c t i v e i n d e x f l u c t u a t i o n s w h e r e R i s t h e " c o r r e l a t i o n d i s t a n c e " a n d p r o c e e d s t o d e r i v e a t h e o r e t i c a l p h a s e s t r u c t u r e f u n c t i o n f o r h o r i z o n t a l p r o p a g a t i o f l a s e r o u t p u t D f t f t ( p ) _2 5 = 5 . 8 2 y 2 k 2 D ( C - 2 - 2 5 ) C ( r ) = e x p [ - ^ ] ( C - 2 - 2 6 ) D Q C ( P ) = 4/TT y 2 k 2 D R 1 - e x p - P ( C - 2 - 2 7 ) U l A comparison of the t h e o r e t i c a l predictions of F r i e d and Cloud with those of Beckmann requires a r e l a t i o n -ship between the " c o r r e l a t i o n distance" R , eqn. C-2-26, and the outer scale of turbulence R Q , eqn. C-2-24. When attempting such a comparison a good deal of confusion of d e f i n i t i o n s i s encountered. This d i f f i c u l t y has been pointed out by Strohbehn (24). According to conventional d e f i n i t i o n s (23), (24) , (31) the parameter R i n Beckmann*s formula, eqn. C-2-26, corresponds to the "inner scale" of turbulence r Q . For the atmosphere r /^» 1 cm (19), (25), (32) . However both beckmann and Chernov ref e r to the parameter R as the "cor-r e l a t i o n distance" and suggest much large values for i t . Chernov (26) sets R = 60 cm , however t h i s value, based on experimental evidence presented by Liebermann (26, Pp. 5, 8) seems appropriate for sea water and not for the atmosphere. Beckmann claims a value of R appropriate for the atmosphere 3 i s R = 6 x 10 cm but apparently o f f e r s no j u s t i f i c a t i o n for the use of so large a number. Thus i t i s c l e a r that whatever i s meant by the parameter R i n Beckmann's expression for the phase structure function D ( p ) , eqn. C-2-27 , i t i s not the PP inner scale turbulence, r Q 112 P o s s i b l e i n s i g h t i n t o t h e n a t u r e o f t h e c o n f u s i o n i s g a i n e d b y c o n s i d e r i n g t h e w o r k o f F r i e d a n d C l o u d . T h e i r t h e o r e t i c a l i n v e s t i g a t i o n s l e d them t o t h e c o n c l u s i o n t h a t t h e g e n e r a l p h a s e s t r u c t u r e f u n c t i o n f o r h o r i z o n t a l t r a n s m i s s i o n d e p e n d s i n a f a i r l y c o m p l i c a t e d way on b o t h t h e i n n e r a n d o u t e r s c a l e s o f t u r b u l e n c e , r Q a n d R Q r e s p e c t i v e l y [ 3 2 ] . H o w e v e r s u b s e q u e n t i n v e s t i g a t i o n [40] showed t h a t t h e d e p e n d e n c e on t h e o u t e r s c a l e o f t u r b u l e n c e R was m o s t d o m i n a n t a n d t h a t o t h e p h a s e s t r u c t u r e f u n c t i o n c o u l d be a p p r o x i m a t e d e v e r y w h e r e t o a h i g h d e g r e e o f a c c u r a c y b y e q n . C - 2 - 25. T h u s F r i e d a n d C l o u d a r e a b l e t o show t h a t , t o a g o o d a p p r o x i m a t i o n , t h e o n l y a t m o s p h e r i c v a r i a b l e s n e e d e d t o d e s c r i b e t h e p h a s e s t r u c -t u r e f u n c t i o n a r e t h e mean s q u a r e r e f r a c t i v e i n d e x f l u c t u a -t i o n s j l 2 * , a n d t h e o u t e r s c a l e o f t u r b u l e n c e , R Q I f one a s s u m e s t h i s r e s u l t t o be g e n e r a l l y t r u e ( a n d Beckmann e x p r e s s e s t h a t h o p e ) t h e n s i n c e B e c k m a n n 1 s p h a s e s t r u c t u r e f u n c t i o n c o n t a i n s o n l y two a t m o s p h e r i c v a r i a b l e s , y 2" a n d R , i t i s c l e a r t h a t R must be i n t i m a t e l y r e l a t e d t o t h e o u t e r s c a l e o f t u r b u l e n c e R_ . I f t h e a u t o c o r r e l a -o t i o n f u n c t i o n o f F r i e d a n d C l o u d , b a s e d on t h e K o l m o g o r o f f t h e o r y o f t u r b u l e n c e , i s a c c e p t e d a l o n g w i t h t h e i r e m p i r i c a l e s t i m a t e s f o r R t h e n R c a n be s i m p l y r e l a t e d t o R „ b y o o J d e f i n i n g R t o be t h e d i s t a n c e o v e r w h i c h 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 F r i e d a n d C l o u d d e c r e a s e s t o 1/e o f i t s v a l u e a t z e r o . 113 I f t h e " c o r r e l a t i o n d i s t a n c e " o f a t m o s p h e r i c r e f r a c -t i v e i n d e x f l u c t u a t i o n s R i s d e f i n e d a s t h e d i s t a n c e o v e r w h i c h 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 d r o p s t o 1/e t h e n a com-p a r i s o n o f Beckmann's r e s u l t s w i t h t h o s e o f F r i e d a n d C l o u d c a n be made b y m e a s u r i n g R i n u n i t s o f R Q . F r o m t h e d e f i n i t i o n o f R a n d e q n s . C-2-24 a n d C-2-26 i t f o l l o w s t h a t 3 R = [ 2 ( l - l / e ) ] 2 " R R = 1.42 R. ( C - 2 - 2 8 ) a n d B eckmann's e q n . C-2-27 b e c o m e s D g 3 ( p ) » 5.68 / ? y 2 D R Q e x p 2.03R ( C - 2 - 2 9 ) U s i n g F r i e d a n d C l o u d ' s r e s u l t s i t f o l l o w s f r o m e q n s C-2-23 a n d C-2-25 t h a t 1 5 5.82 P 2 k 2 D R Q r pJ - 2 ( l - V o ) w h i c h i m p l i e s < P " P. 2 d - V o ) R o 2 5 5.82 u 2 k 2 D 3 ( C - 2 - 3 0 ) 114 U s i n g Beckmann's r e s u l t s i t f o l l o w s f r o m e q n s . C-2-23 a n d C-2-29 t h a t 5.68 / ? y 2 k 2 D R 1 - e x p 2.03R - 2 ( l - V o ) w h i c h i m p l i e s P - P 0 = 2.03 R l o g o e 2 ( 1 - V n ) 5.68 / ? y 2 k 2 D R, ( C - 2 - 3 1 ) C l e a r l y , p Q , t h e l a r g e s t p e r m i s s i b l e d i m e n s i o n s f o r t h e d i f f r a c t i n g a p e r t u r e s , d e p e n d s o n t h e wave number k a n d t h e mean s q u a r e a t m o s p h e r i c r e f r a c t i v e i n d e x f l u c t u a t i o n s y"2 , d e c r e a s i n g m o n o t o n i c a l l y w i t h i n c r e a s i n g v a l u e s o f b o t h . T a b l e I I I - 1 shows t h e r e s u l t i n g v a l u e s f o r p Q when e q n s . C-2-30 a n d C-2-31 a r e e v a l u a t e d f o r D = 5 x 1 0 s cm (5 km.), R Q = 1 0 3 cm a n d f r i n g e v i s i b i l i t y V Q = 0.5 115 F r i e d $ C l o u d , E q n . C-2-30 y 2 = 1 0 " 1 2 y 2 = I O " 1 0 Beckmann, E q n . C-2-31 y 2 = 1 0 " 1 2 y 2 = 1 0 " 1 0 A i = 1 0 u X 2 = 0 . 5 y X 3 = 0.3y p =0.920 cm p =0.0583 cm o 0 p =0.0249 cm p =0.00158 cm 0 0 p =0.0136 cm p =0.000860 cm 0 0 p =3.19 cm p =0.319 cm o 0 p Q = 0 . 1 5 9 cm p Q = 0 . 0 1 5 9 cm p Q = 0 . 0 9 6 0 cm p Q = 0 . 0 0 9 6 0 cm T A B L E I I I - l I n t h e i r a n a l y s i s o £ t h e e f f e c t s o f a t u r b u l e n t a t m o s -p h e r e on t h e p r o p a g a t i o n o f l a s e r o u t p u t , F r i e d a n d C l o u d u s e d a m o d e l a t m o s p h e r e whose mean s q u a r e f l u c t u a t i o n i n r e f r a c t i v e i n d e x w e r e much s m a l l e r , i n t h e o p t i c a l r e g i o n o f t h e s p e c t r u m , t h a n t h o s e i n d i c a t e d i n t a b l e I I I - l . T h e y p r e f e r r e d 1 0 " 1 * < y 2 < 1 0 " 1 2 , w h e r e a s Beckmann's a n a l y s i s u s e d 1 0 " 1 2 < y 2 < 1 0 " 1 0 . Beckmann h i m s e l f p o i n t s o u t t h a t y 2 ^ 1 0 " 1 0 i s " i m p r o b a b l y h i g h " a n d t h a t y 2 ~ 1 0 " 1 2 i s " a more l i k e l y v a l u e " f o r t h e mean s q u a r e a t m o s p h e r i c r e f r a c t i v e i n d e x f l u c t u a -t i o n . T h e o n l y " o v e r l a p " i n t h e r a n g e o f mean s q u a r e r e f r a c t i v e i n d e x f l u c t u a t i o n s t h a t e a c h a u t h o r f e e l s i s a p p l i c a b l e t o h i s r e s p e c t i v e t h e o r y i s t h e v a l u e y 2 ^ 1 0 " 1 2 . The f o l l o w i n g 116 v a l u e s o f p a r e c o n s i d e r e d t o be r e p r e s e n t a t i v e o f t h e o mean o f t h e two t h e o r i e s f o r f r i n g e v i s i b i l i t y 0.5 a n d y~2 % 1 0 " 1 2 / m o d e r a t e - t o - i n t e n s e t u r b u l e n c e . w a v e l e n g t h d i f f r a c t i n g a p e r t u r e d i m e n s i o n s X i = 10 u p o ^ 1 cm X 2 = 0.5 y p Q 1 0 " 1 cm X 3 = 0.3 y p Q -v 5 x 1 0 " 2 cm 117 4. F r i n g e C o u n t i n g R a t e The f l u c t u a t i o n s i n t r a n s i t t i m e o f t h e w h o l e beam o f t h e l a s e r o u t p u t a p p e a r a t t h e r e c e i v e r a s a m o d u l a t i o n u on t h e o p e r a t i n g f r e q u e n c y w o f t h e l a s e r . I f 8 i s t h e t o t a l mean p h a s e a v e r a g e d a c r o s s t h e a p e r t u r e o f t h e r a d i a t i o n i m p i n g i n g on a p e r t u r e #2 t h e n B - B Q + * (C- 4 - 1 ) \ w' = to + u (C - 4 - 2 ) dS d B o where w ' = 3 T "*"s t * i e i n s t a n t a n e o u s frequency, u = i s the mean frequency and u = ^ i s the modulation frequency, u i s a random v a r i a b l e w i t h mean value z e r o . F r o m e q n . C-2-10 a n d C-2-14 i t i s o b v i o u s t h a t t h e i n s t a n t a n e o u s f r i n g e c o u n t i n g r a t e i s s i m p l y to' - w = u . Th u s t h e f r i n g e c o u n t i n g d e v i c e o f a f r e e a i r l a s e r s t r a i n m e t e r m u s t be a b l e t o r e s p o n d a t r a t e s o f u o r h i g h e r . Beckmann [ 2 2 ] h a s t h e o r e t i c a l l y i n v e s t i g a t e d t h e p r o b l e m o f p a r a s i t i c f r e q u e n c y m o d u l a t i o n o f l a s e r o u t p u t i n d u c e d b y p a s s a g e o f t h e beam t h r o u g h a t u r b u l e n t a t m o s p h e r e S t a r t i n g w i t h 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 f o r r e f r a c t i v e i n d e x g i v e n b y e q n . C-2-26 a n d a s s u m i n g t h a t t h e f i e l d o f 118 a t m o s p h e r i c r e f r a c t i v e i n h o m o g e n e i t i e s i s " f r o z e n i n " , t h a t i s n e g l e c t i n g t u r b u l e n t d i f f u s i o n c o m p a r e d t o w i n d as a means o f t r a n s p o r t i n g r e f r a c t i v e i n h o m o g e n e i t i e s , Beckmann d i v i d e s t h e v e c t o r p r o b l e m o f w i n d i n d u c e d f r e q u e n c y m o d u l a t i o n i n t o i t s l o n g i t u d i n a l a n d t r a n s v e r s e c o m p o n e n t s . B eckmann f i n d s t h e f r e q u e n c y m o d u l a t i o n c a u s e d b y up o r downwind c o m p o n e n t s o f d r i f t t o be i n s i g n i f i c a n t c o m p a r e d t o c r o s s w i n d d r i f t , a r e s u l t w h i c h c o u l d be a n t i c i p a t e d i n -t u i t i v e l y . B e c k m a n n ' s c h o i c e o f 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 r e f r a c t i v e i n d e x f l u c t u a t i o n s l e a d s t o a p a r a s i t i c m o d u l a t i o n u w h i c h i s g a u s s i a n w i t h mean v a l u e z e r o a n d v a r i a n c e u 2 g i v e n b y -2 • A vT2 k 2 D ,2 2 t u z c r o s s w i n d = ^ \lR ^ v + v z ' ( C - 4 - 3 ) u 2 u p w i n d = 2.84 y z k R Q V x ( C - 4 - 4 ) w h e r e v = v i + v j + v k i s t h e w i n d v e l o c i t y . T h e s t a n d a r d d e v i a t i o n s o f t h e f r e q u e n c y m o d u l a t i o n s a r e a c r o s s w i n d = 4 ^ y 2 k 2 D ( y2 + y2 1.42R. ( C - 4 - 5 ) a u p w i n d = [2.84 y 2 k ^ v ^ (C-4-6) 119 From these r e s u l t s i t can be seen t h a t the l a s e r frequency modulation due t o c r o s s w i n d e f f e c t s i s f a r g r e a t e r than t h a t due to upwind e f f e c t s - a r e s u l t one would a n t i c i p a t e i n t u i t i v e l y . From eqn. C-4-5 i t can be seen t h a t f o r a wind v e l o c i t y o f 10 m/sec. , moderate t o i n t e n s e atmospheric 2 -12 t u r b u l e n c e , i . e . JJ. ^ 10 , D />> 5 km and R Q ^ 10 m t h a t the s t a n d a r d d e v i a t i o n i n the f r i n g e count r a t e i s o f the 4 o r d e r o f 2 x 10 f r i n g e s per second. Thus a f r i n g e o b s e r v a -t i o n system w i t h a band width of 55 KHz would be a b l e t o make f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s 99% o f the time under such c o n d i t i o n s . F r i n g e c o u n t e r s c u r r e n t l y o p e r a t i n g on c o n v e n t i o n a l l a s e r s t r a i n meters have band-widths o f up t o 1 MHz and would f a c i l i t a t e f r e e a i r l a s e r s t r a i n meter o b s e r v a -t i o n s under the most d i s t u r b e d atmospheric c o n d i t i o n s i m a g i n a b l e . 120 D. POWER REQUIREMENTS The amplitude and phase o f l a s e r r a d i a t i o n are complimentary v a r i a b l e s i n the quantum mechanical sense t h a t the o b s e r v a t i o n o f one i n t r o d u c e s an u n c e r t a i n t y i n t o the va l u e of the o t h e r . The p a r t i c u l a r u n c e r t a i n t y r e l a t i o n s h i p enjoyed by these v a r i a b l e s [36] i s ANAp ^ 1 (D-l-1) where N i s the number o f photons a s s o c i a t e d w i t h the l a s e r output a v a i l a b l e f o r use i n the o b s e r v a t i o n and p i s the t o t a l phase. A f r e e a i r l a s e r s t r a i n meter f r i n g e counter capable o f c o u n t i n g a t a r a t e o f 10^ f r i n g e s per second w i t h an accuracy o f ±10 p a r t s o f a f r i n g e i s e f f e c t i v e l y making an -2 o b s e r v a t i o n o f (3 w i t h an u n c e r t a i n t y AP ~ 2ir x 10 —6 i n an i n t e r v a l A T ~ 10 sec. T h i s o b s e r v a t i o n by eqn. D-l-1, i n t r o d u c e s an u n c e r t a i n t y i n N o f the o r d e r AN ~ 15.9 S i n c e o b s e r v i n g d e v i c e s i n g e n e r a l cannot operate anywhere near the l i m i t s o f o b s e r v a t i o n a l a c c u r a c y imposed on them by quantum mechanics, the o b s e r v a t i o n o f p t o an accuracy o f ~AP w i l l r e q u i r e N photons where N » AN . In g e n e r a l N = a A N 121 The c o n s t a n t a i s u s u a l l y l a r g e and depends on the s e n s i t i v i t y and n o i s e c h a r a c t e r i s t i c s o f the o b s e r v i n g d e v i c e . An estimate of the number o f photons power requirements f o r f r e e a i r l a s e r s t r a i n meters can be e a s i l y o b t a i n e d from c o n s i d e r a t i o n o f the i n f o r m a t i o n c a p a c i t y o f r a d i a t i o n d e t e c t o r s . _2 Counting f r i n g e s to an accuracy o f ±10 p a r t o f a f r i n g e i n a cumulative f a s h i o n , as r e q u i r e d by the o p e r a t i o n o f a f r e e a i r l a s e r s t r a i n meter, demands, i n e f f e c t , t h a t the f r i n g e counter be capable o f d i s c e r n i n g 50 d i s t i n c t l e v e l s of i n t e n s i t y on each f r i n g e . I f a l l l e v e l s o f i n t e n s i t y on the f r i n g e are assumed "a p r i o r i " t o be e q u a l l y probable, the i n f o r m a t i o n o b t a i n e d by the observer upon d i s c o v e r i n g h i s o b s e r v i n g d e v i c e to be "focussed" on a g i v e n l e v e l i s 5.65 b i t s In a d d i t i o n , t o f a c i l i t a t e cumulative f r i n g e c o u n t i n g , the o b s e r v i n g d e v i c e must be able t o e x t r a c t a s i g n (+ ve or - ve) f o r each observed l e v e l to i n d i c a t e whether the f r i n g e motion which brought the o b s e r v i n g d e v i c e t o t h a t p o s i t i o n on the f r i n g e was to the l e f t or t o the r i g h t . T h i s o b s e r v a t i o n r e q u i r e s an a d d i t i o n a l one b i t per l e v e l . In t h i s manner the o b s e r v a t i o n of each d i s t i n c t l e v e l o f the f r i n g e i n t e n s i t y f u r n i s h e s 6.65 b i t s of i n f o r m a t i o n to the o b s e r v e r . Thus cumulative f r i n g e c o u n t i n g a t a r a t e o f 10^ 1 2 2 , -2 f r i n g e s per second t o an a c c u r a c y o f 110 p a r t o f a f r i n g e Q r e q u i r e s an i n f o r m a t i o n channel c a p a c i t y o f 3.33 x 10 b i t s per second. Jones [371 has i n v e s t i g a t e d the i n f o r m a t i o n e f f i c i e n c y o f v a r i o u s r a d i a t i o n d e t e c t o r s and has found them to v a r y w i d e l y i n e f f i c i e n c y from 10 1 b i t s per photon f o r -3 the 1P21 p h o t o m u l t i p l i e r tube, 7.2 x 10 b i t s per -5 photon f o r the human eye, to 2.0 x 10 b i t s per photon f o r the 6849 image o r t h i c o n tube. The i n f o r m a t i o n e f f i c i e n c y of the r a d i a t i o n d e t e c t o r s employed i n l a s e r s t r a i n meters should be i n the upper end o f t h i s range of v a l u e s s i n c e the o b s e r v i n g d e v i c e s are a t l i b e r t y t o e x p l o i t p h o t o a m p l i f y i n g schemes such as those used i n p h o t o m u l t i p l i e r tubes. A sug-g e s t e d v a l u e f o r the i n f o r m a t i o n e f f i c i e n c y o f the photo--2 d e t e c t o r s o f l a s e r s t r a i n meters i s t h e r e f o r e 10 b i t s per photon. [ Although Jones' a n a l y s i s s t r i c t l y o n l y a p p l i e s to r a d i a t i o n from n o n - l a s e r sources the use o f h i s f i g u r e s i n t h i s t h e s i s i s a prudent and c o n s e r v a t i v e c h o i c e s i n c e the use o f l a s e r sources w i t h t h e i r v e r y low i n h e r e n t n o i s e c h a r a c -t e r i s t i c s can o n l y be expected t o improve these f i g u r e s . ] _2 Thus cumulative f r i n g e c o u n t i n g to the n e a r e s t 10 p a r t o f a f r i n g e a t a r a t e o f 10^ f r i n g e s per second r e q u i r e s a f l u x o f 3.33 x 10"^ photons per second i n c i d e n t en the r a d i a t i o n d e t e c t o r s . The geometry o f the d i f f r a c t i n g a p e r t u r e s and the placement o f the r a d i a t i o n d e t e c t o r s o f a f r e e a i r l a s e r 123 s t r a i n meter c o u l d p r o b a b l y be arranged so t h a t the r a d i a t i o n d e t e c t o r s i n t e r c e p t 0.1% o f the t o t a l number o f photons e n t e r i n g the a p e r t u r e s . S i n c e h a l f o f these photons must e n t e r through a p e r t u r e #2 i t f o l l o w s t h a t the o p e r a t i o n o f a f r e e a i r l a s e r 1 3 s t r a i n meter r e q u i r e s a f l u x of 1 . 6 7 x 10 photons per second i n c i d e n t on the second a p e r t u r e from the d i s t a n t r e t r o r e f l e c t o r . The area o f the second a p e r t u r e i s o f the order o f 2 p 0 ( ^ ) t h e r e f o r e the photon f l u x a t the second a p e r t u r e must be o f the order o f 1 ( A ) ~ 1 ' 6 l K 1 q 1 3 Photons ( D - 1 - 2 ) p ^ ( A ) cm sec I f i t i s assumed t h a t the l a s e r r a d i a t i o n d i v e r g e s a t an angle 9 ( A ) , then a f t e r p r o p a g a t i n g a d i s t a n c e d i t i s spread over an area A ( A ) A ( A) = J 9 ( A ) 2 d 2 cm 2 ( D-l - 3 ) An e s t i m a t e f o r the angle o f d i v e r g e n c e 9 ( A ) can be o b t a i n e d by assuming the beam d i v e r g e s a t i t s d i f -f r a c t i o n l i m i t i n which case 1 2 4 9(A) = (D-l-4) o where A i s the wavelength o f the r a d i a t i o n i n q u e s t i o n and d Q i s the diameter of the e x i t p u p i l of the l a s e r . I f i s assumed t h a t the diameter o f the e x i t p u p i l of the l a s e r i s d = 1 cm then e v a l u a t i o n 9(A) f o r A., = 10 n , A 9 = 0.5 u , A 3 = 0.3|j i t f o l l o w s t h a t 1.22 x 10" 3 rad 6.1 x I O - 5 rad -5 3.66 x 10 rad e(A 2) = e(A ) = I t has been p o i n t e d out by Bender [ 43 ] t h a t atmospheric s c a t t e r i n g p l a c e s an upper l i m i t on the degree o f c o l l i m a t i o n a c h i e v a b l e i n a l a s e r beam o f r o u g h l y -5 10 rad . However g i v e n t h a t the d i f f r a c t i o n l i m i t o f the -5 l a s e r system exceeds 10 rad i t appears p r a c t i c a b l e to d e s i g n an o p t i c a l system capable of a c h i e v i n g c o l l i m a t i o n v e r y near the d i f f r a c t i o n l i m i t . Laser t h e o d o l i t e s used i n c o n s t r u c t i o n are a d v e r t i z e d which have achieved an a zimuthal -5 c o l l i m a t i o n o f 6 x 10 rad a t 0.6328 [i . T h i s even exceeds the " d i f f r a c t i o n l i m i t " as c a l c u l a t e d by the above 125 formula however i t i s achieved a t the expense of r a t h e r poor e l e v a t i o n c o l l i m a t i o n . For these reasons i t seems reasonable to r e t a i n the above c a l c u l a t e d v a l u e s as r e p -r e s e n t a t i v e o f the angle of d i v e r g e n c e o f the beams. I f E i s the power output o f the l a s e r i n watts, then the photon f l u x of the beam a t a d i s t a n c e D from the e x i t p u p i l i s _ # \ E , n 7 photons ,-. , c x = A (A) h v X 1 0 2 ( D - 1 " 5 ) v ' cm sec -27 where h i s Planck's c o n s t a n t h = 6.63 x 10 e r g - s e c . and v = c / A i s the o p e r a t i n g frequency o f the l a s e r . L e t E Q ( ^ ) ke the power output r e q u i r e d ( i n the absence o f atmospheric a t t e n u a t i o n and f l u c t u a t i o n s ) to supply the n e c e s s a r y f l u x 1 (A) (given by eqn. D-l-2) , a t the second d i f f r a c t i n g a p e r t u r e o f a f r e e a i r l a s e r s t r a i n meter. Thus from eqns. D-l-2 and D-l-5 E M = 1 . 6 7 A ( A ) h v x 1 ( ) 6 w a t t s "o<*> 126 S u b s t i t u t i n g f o r A ( A ) and f o r v E ( A ) = 1-677T 8 - ( A ) T>< he x 1(j6 w a t t s P 2 ( A ) (D-l-6) I f D = 5 x 10 5 cm (5 km) c = 3 x 10 cm/sec \ 1 = 10 3 c m ( 1 0 ^ 9 ( A 1 ) = 1 . 2 2 x l 0 ~ 3 r a d P Q ( A 1 ) = 1 cm A = 5xl0" 5cm(0.5^) 9 ( A 2 ) = 6.1x10 5 r a d P 0 ( ^ 2 ) = 1 0 l c m A 3 = 3x10 5 c m ( 0 . 3 u ) 9 ( A 3 ) = 3.66x10 5 rad P Q ( A 3 ) = 5x10 2cm Then E o * A l ^ = 3 9 0 x 1 0 ~ 3 w a t t s E Q ( A 2 ) = 1.94 watts (D-l-7) E Q ( A 3 ) = 4.66 watts In e s t i m a t i n g the above power requirements no a l l o w -127 ance has been made f o r atmospheric a b s o r p t i o n or f l u c t u a t i o n s i n i n t e n s i t y due to atmospheric t u r b u l e n c e . Atmosphere ab-s o r p t i o n a t these f r e q u e n c i e s i s l e s s than 1% per thousand f e e t [38] . Thus atmospheric a t t e n u a t i o n o f the power o f the l a s e r output as a f u n c t i o n o f d i s t a n c e D can be ex-p r e s s e d as E ( A ) = E Q ( A ) exp - a Q -o (D-l-8) where D 3.1 x 10 cm (1000 f t ) and a 0.01 I n t e r n a l f l u c t u a t i o n s i n i n t e n s i t y o f the beam, however, r e q u i r e a c o n s i d e r a b l e i n c r e a s e i n power over E Q ( A ) to be overcome. Beckmann [22] has shown t h a t the l o g a r i t h m o f the r e l a t i v e amplitude o f the beam, A / A Q » i s a random v a r i a b l e g a u s s i a n d i s t r i b u t e d about zero w i t h s t a n d a r d d e v i a t i o n . 1/2 l o g A / A , 64I/T la 2 D 3 3R 3 (D-l-9) S i n c e the i n t e n s i t y I ^ A and l o g A / A Q = 2 l o g l / l Q i t f o l l o w s t h a t the r e l a t i v e i n t e n s i t y of the beam i s l o g -no r m a l l y d i s t r i b u t e d about zero w i t h standard d e v i a t i o n 128 1/2 l o g I /I = 2 64 1 L 3R~ ( D - l - 1 0 ) From these r e s u l t s i t f o l l o w s t h a t f o r random f l u c t u a t i o n s i n the i n t e n s i t y I 1/2 ( D - l - 1 1 ) f o r 9 9 . 5 % o f the time. Combining eqns. D - l - 6 , D - l - 8 , and D - l - 1 1 i t f o l l o w s t h a t f o r a 9 9 . 5 % c o n f i d e n c e l e v e l the r e q u i r e d power i s E ( A ) = 1-677T 9 2 ( A ) D 2 h c x 1 Q 6 A P ( A ) exp 5 . 1 5 6 4 V wu ] 3 R ' watt ( D - l - 1 2 ) 1 2 9 5 Thus i f D = 5 x 10 cm 7 = i o " 1 2 R = 10 3 cm o a = 0.01 4 D = 3.1 x 10 cm o -2 7 h = 6.63 x 10 e r g s - s e c c = 3 x 10 cm/sec * x = 10u 9 ^ ) = 1.22 x 10 3 r a d p o ^ A l ^ = 1 C I t l A 2 = 0.5^ ©(^2^ = 6 ' 1 x ! 0 ~ 5 r a d P 0 ^ A 2 ) = 1 0 ~ l c m A 3 = 0 . 3 n e ( A 3 ) = 3 - 6 6 x 1 0 " 5 r a d p o ^ A 3 ^ = 5 x l 0 _ 2 c m I t f o l l o w s t h a t the r e q u i r e d l a s e r powers a r e : E ( A 1 ) = 1.19 watts E ( A 2 ) = 5.92 watts E ( A 3 ) = 14.2 watts 130 CHAPTER IV CONCLUSIONS A. SUMMARY OF RESULTS Th e r e s e a r c h p r e s e n t e d i n t h i s t h e s i s i n d i c a t e s t h a t l a s e r s t r a i n m e t e r s c a p a b l e o f m a k i n g s a m p l e d m e a s u r e m e n t s o f e a r t h s t r a i n , a c c u r a t e t o t h e l i m i t o f t h e f r e q u e n c y s t a b i l i t y o f t h e l a s e r , w h i l e o p e r a t i n g t h r o u g h u n c o n t r o l l e d a t m o s p h e r e o v e r d i s t a n c e s o f s e v e r a l k i l o m e t e r s a r e i n d e e d f e a s i b l e w i t h e x i s t i n g t e c h n o l o g y . T h e o b s e r v i n g d e v i c e , h e r e - i n c a l l e d a " f r e e a i r l a s e r s t r a i n m e t e r " , w o u l d c o n s i s t o f t h r e e f r e q u e n c y s t a b i l i z e d l a s e r s , a p p r o p r i a t e l y s p a c e d i n t h e s p e c t r u m , whose o u t p u t s a r e c o m b i n e d i n t o a s i n g l e beam a n d d i r e c t e d t h r o u g h t h e a t m o s p h e r e t o a d i s t a n t r e t r o r e f l e c t o r . T h e c o m b i n e d r e t u r n beam w o u l d be s e p a r a t e d a c c o r d i n g t o w a v e l e n g t h a n d p o r t i o n s t h e s i z e o f a " c o h e r e n c e r e g i o n " w o u l d be d i r e c t e d i n t o t h r e e d o u b l e r e c t a n g u l a r i n t e r f e r o m e t e r s w h e r e i t i n t e r f e r e s w i t h a p o r t i o n o f t h e beam d i r e c t f r o m t h e r e s p e c t i v e l a s e r . A t e a c h i n t e r f e r o m e t e r a means o f c u m u l a t i v e f r i n g e c o u n t i n g i s p r o v i d e d . T h e c u m u l a t i v e f r i n g e c o u n t s i n a n i n t e r v a l A t a r e u s e d t o c o m p u t e t h e n e t f l u c t u a t i o n i n g e o m e t r i c a l r a y - p a t h l e n g t h i n t h e i n t e r v a l . A s e r i e s o f c o n t i n u o u s c o n s e c u t i v e o b s e r v a t i o n s o f t h e f l u c t u a t i o n s i n g e o m e t r i c a l r a y - p a t h l e n g t h c a n be a v e r a g e d o v e r an i n t e r v a l A T t o re m o v e t h e a t m o s p h e r i c f l u c -t u a t i o n s a n d r e v e a l a n y t e c t o n i c c h a n g e s i n l e n g t h t h a t h a v e t a k e n p l a c e b e t w e e n t h e e n d m i r r o r s o f t h e l a s e r s t r a i n m e t e r . 131 The d i s p e r s i v e nature of the atmosphere to e l e c t r o -magnetic r a d i a t i o n o f o p t i c a l f r e q u e n c i e s f a c i l i t a t e s a mathe-m a t i c a l s e p a r a t i o n o f the " g e o m e t r i c a l " and " r e f r a c t i v e " c o n t r i -b u t i o n s to the f l u c t u a t i o n s i n o p t i c a l path l e n g t h . In f a c t i f U ( t ^ t 2 ) i s the g e o m e t r i c a l f l u c t u a t i o n i n r a y - p a t h l e n g t h A t = t 2 - t x and V ( t ] [ t 2 ) i n the i n t e r v a l s o - c a l l e d r e f r a c t i v e c o n t r i b u t i o n s then f W ( t 1 t 2 ) are where A. output and (t^t,,) f > f \ \ ( t l t 2 ) 1 f (*]_) u ( t l t 2 ) \ ( t l t 2 ) = 1 f ( A 2 ) g ( A 2 ) v ( t l t 2 ) \ ( t ^ ) 1 f ( A 3 ) g ( A 3 ) w ( t l t 2 ) > \ = A x C 1 ( t 1 t 2 ) \ = A 2 C 2 ( t l t 2 ) \ = A 3 C 3 ( t A 3 are the wavelengths " i n vacuo " o f the C-, ( t , t n ) are the r e s -A 3 1 2 p e c t i v e c u m u l a t i v e f r i n g e counts i n the i n t e r v a l A t . The elements o f the m a t r i x o f c o e f f i c i e n t s are d e r i v e d from the formula f o r atmospheric r e f r a c t i v e index n = 1 + f ( A ) F ( P , T ) + f ( A ) g ( p , T ) The a c c u r a c y o f p r e s e n t day knowledge of the r e f r a c t i v e index 132 formula and contemporary cumulative f r i n g e c o u n t i n g a b i l i t i e s f a c i l i t a t e measurement o f t e c t o n i c motions to an a c c u r a c y of r o u g h l y ±1 % or b e t t e r . The r e l a t i o n s h i p between the c o n f i d e n c e l e v e l o f the f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s and the a v e r a g i n g time f o r the atmospheric g e o m e t r i c a l ray-path f l u c t u a t i o n s was d e r i v e d S o l v e d e x p l i c i t l y f o r A X t h i s g i v e s A t = sec. These r e s u l t s i n d i c a t e t h a t a t e c t o n i c motion o f | h ^ | = 1 cm/year between r e f l e c t o r s a d i s t a n c e D = 5 km a p a r t can be d e t e c t e d w i t h a 99% c o n f i d e n c e l e v e l ( |£M = 2.58 ) and a s i g n a l - n o i s e r a t i o s = 2 i n the presence o f moderate-~2 -12 t o - i n t e n s e atmospheric t u r b u l e n c e J A ~ 10 i n about A T = 39 minutes o f o b s e r v a t i o n time. 133 Free a i r l a s e r s t r a i n meters appear t o be t h e o r e t i c a l l y c a pable o f measuring changes i n the r e g i o n a l ambient s t r a i n 1 2 s t a t e o f p a r t s i n 1 0 ' i n the presence of weak atmospheric i n t e n s e atmospheric t u r b u l e n c e . In each case the f r e e a i r l a s e r s t r a i n meter appears t o be t h e o r e t i c a l l y capable o f measuring e a r t h s t r a i n up to the l i m i t o f the l a s e r frequency s t a b i l i t y f o r the c o r r e s p o n d i n g o b s e r v a t i o n i n t e r v a l . The i n t e n s i t y d i s t r i b u t i o n I(xy) from the double r e c t a n g u l a r F raunhofer d i f f r a c t i o n p a t t e r n c a s t by l a s e r r a d i a t i o n o f f requency and wavelength A p a s s i n g through a p e r t u r e s o f dimensions 2 c , b - a , i s t u r b u l e n c e and p a r t s i n 1 0 1 0 i n the presence o f moderate-to Kxy) . 2 7TX , , » 2 27TVC s i n AR 1-V s i n 134 w h e r e °,2 = ——^ » u = wi - w 2 i s a n e f f e c t i v e f r e q u e n c y m o d u l a t i o n o f t h e r a d i a t i o n i n c i d e n t o n t h e s e c o n d a p e r t u r e a s a r e s u l t o f i t s j o u r n e y t h r o u g h a t u r b u l e n t a t m o s p h e r e . V i s t h e f r i n g e v i s i b i l i t y . T he f r i n g e c o u n t r a t e °.2 i s p r o p o r t i o n a l t o t h e r a t e o f c h a n g e o f o p t i c a l p a t h l e n g t h L ft? = — d L c eft" a n d t h u s c u m u l a t i v e f r i n g e c o u n t i n g f o r an i n t e r v a l A t a c t u a l l y m e a s u r e s t h e n e t c h a n g e i n o p t i c a l p a t h l e n g t h i n u n i t s o f X T h e e f f e c t i v e b a n d w i d t h r e q u i r e d o f a f r i n g e c o u n t e r f o r a f r e e a i r l a s e r s t r a i n m e t e r t o h a v e a 99% o p e r a t i n g c a p a b i l i t y i s " b a n d w i d t h = 2 ' 5 7 4 / ? y2 k 2 D 1.42 R - I 1/2 ( v 2 • v 2 ) Hz, w h e r e k = 2TT v z a r e c o m p o n e n t s o f w i n d v e l o c i t y t r a n s v e r s e t o t h e s t r a i n m e t e r a x i s , R Q i s t h e o u t e r s c a l e o f t u r b u l e n c e a n d y 2 i s t h e mean s q u a r e r e f r a c t i v e i n d e x f l u c t u a t i o n . T h u s i t i s s e e n t h a t a f r i n g e c o u n t e r w i t h a m e g a c y c l e b a n d w i d t h i s a d e q u a t e f o r f r e e a i r l a s e r s t r a i n m e t e r o b s e r v a -t i o n s i n t h e p r e s e n c e o f i n t e n s e a t m o s p h e r i c t u r b u l e n c e a n d h i g h 135 winds over d i s t a n c e s o f s e v e r a l k i l o m e t e r s . To i n s u r e h i g h f r i n g e v i s i b i l i t y i t i s nece s s a r y t o r e s t r i c t the d i f f r a c t i n g a p e r t u r e t o s m a l l dimensions p < p where i s the dimension o f a coherence p a t c h . I f V0 i< the minimum a c c e p t a b l e f r i n g e v i s i b i l i t y then two t h e o r e t i c a l e x p r e s s i o n s f o r Po = 5.82 v. 2 k 2 D have been d e r i v e d . |3/5 cm. p = -2.03 R In r o o 2 ( l - V o ) 5.68 7r H-2 k 2 DR J o 1/2 cm. the d i f f e r e n c e i n these two formulae b e i n g the c h o i c e o f 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 r e f r a c t i v e index v a r i a t i o n s . For a minimum f r i n g e v i s i b i l i t y o f V Q = 0.5 and o b s e r v a t i o n s made over 5 km i n the presence o f moderate-to-i n t e n s e atmospheric t u r b u l e n c e the f o l l o w i n g v a l u e s o f pQ were c a l c u l a t e d 7\1 = 10[i A 2 = 0.5(j. A 3 = 0.3M. p (A ^ ) /v 1 cm p Q (A 2) 10 cm _2 p Q ( A 3 ) r~ 5 x 10 cm 1 3 6 The r e s t r i c t i o n o n a p e r t u r e s i z e i m p l i e s a minimum r e s t r i c t i o n o n r e q u i r e d l a s e r p o w e r . I n f o r m a t i o n e f f i c i e n c i e s o f p r e s e n t d a y p h o t o d e t e c t o r s a r e o f t h e o r d e r o f 1 0 " 2 b i t s / p h o t o n . C u m u l a t i v e f r i n g e c o u n t i n g a t r a t e s o f a m e g a c y c l e w i t h a c c u r a c i e s o f ± 1 0 ' f r i n g e .8 i m p l i e s a c h a n n e l c a p a c i t y o f 3 . 3 3 x 1 0 b i t s / s e c o n d a n d r e q u i r e s a f l u x o f r o u g h l y >13 1 . 6 7 x 1 0 p h o t o n s / s e c i n t o t h e s e c o n d d i f f r a c t i n g a p e r t u r e . A t m o s p h e r i c a t t e n u a t i o n o f l a s e r p o w e r c a n b e e x p r e s s e d a s ctD, E(X) = EQ(X) e x p - 2 ( j p w h e r e a ^  0.01 a n d D „ ^ 3.1 x 101* cm . o I n t e r n a l f l u c t u a t i o n s i n i n t e n s i t y I o f t h e l a s e r beam a b o u t i t s mean i n t e n s i t y I Q due t o a t m o s p h e r i c t u r b u l e n c e a r e s u c h t h a t I - I Q e x p - 5.15 64/TT U2 D 3 3 R Q f o r 9 9 . 5 % o f t h e o p e r a t i n g t i m e . C o m b i n i n g t h e e f f e c t s o f a t m o s p h e r i c a t t e n u a t i o n a n d i n t e n s i t y f l u c t u a t i o n s i t f o l l o w s t h a t s u f f i c i e n t o p e r a t i n g p o w e r f o r f r e e a i r l a s e r s t r a i n m e t e r o b s e r v a t i o n s i s p r o v i d e d 137 by a l a s e r whose output i s E(A) = 1.67* 92 ( A ) D 2hc x 1 Q 6 g x p A P^(A) 5.15 6 4 ^ u D 2^3 3 R~ 1/2 2aD D watts where 9(A) i s the angle o f spread o f the l a s e r beam, h i s P l anck's c o n s t a n t and c i s the v e l o c i t y o f l i g h t . For o b s e r v a t i o n s over D = 5 km w i t h d i f f r a c t i o n l i m i t e d l a s e r beams and i n the presence o f i n t e n s e atmospheric t u r b u l e n c e the f o l l o w i n g l a s e r powers are r e q u i r e d A L = 1CV E ( A 1 ) ~ 1.19 watts A 2 = 0 .5u E ( A 2 ) ~ 5.92 watts A 3 = 0.3|i E ( A 3 ) ~ 14.2 watts 1 3 8 B. A P P L I C A T I O N S OF FREE A I R LASER STRAIN METERS 1. T e c t o n i c R e s e a r c h The s u c c e s s f u l o p e r a t i o n o f f r e e a i r l a s e r s t r a i n m e t e r s w o u l d p r o v i d e g e o p h y s i c i s t s i n b o t h t h e f i e l d o f t e c -t o n i c s a n d t h e f i e l d o f g e o d e s y w i t h a t o o l o f u n u s u a l c a p a -b i l i t i e s a n d as s u c h w o u l d be r e c o g n i z e d as a m a j o r a d v a n c e . As a r e s e a r c h t o o l i n t e c t o n i c s , f r e e a i r l a s e r s t r a i n m e t e r s a r e a b l e t o d i r e c t l y o b s e r v e many l o n g t e r m t e c t o n i c p r o c e s s e s w i t h u n p r e c e d e n t e d a c c u r a c y a n d e a s e . S u c h o b s e r v a -t i o n s a l o n g w i t h t h e g r o s s g e o l o g i c a l f e a t u r e o f t h e e a r t h a r e among t h e e s s e n t i a l e l e m e n t s t o be a c c o u n t e d f o r b y t h e o r i e s o f e a r t h q u a k e m e c h a n i s m , m o u n t a i n b u i l d i n g , o c e a n f l o o r s p r e a d -i n g a n d c o n t i n e n t a l d r i f t . T h e o b s e r v a t i o n a n d m e a s u r e m e n t o f s u c h p h e n o m e n a b y f r e e a i r l a s e r s t r a i n m e t e r s c o u l d p l a y a n i m p o r t a n t r o l e i n t h e d e v e l o p m e n t o f s u c h t h e o r i e s . F r e e a i r l a s e r s t r a i n m e t e r o b s e r v a t i o n s c o u l d b e o f c r u c i a l i m p o r t a n c e i n r e s o l v i n g many p r o b l e m s i n t e c t o n i c s . M e a s u r e m e n t s made a c r o s s t h e r i f t v a l l e y s o f t h e w o r l d p a r t i -c u l a r l y i n E a s t A f r i c a , t h e M i d d l e E a s t a n d I c e l a n d a s w e l l as b e t w e e n i s l a n d s s t r a d d l i n g an o c e a n i c r i d g e w o u l d y i e l d i n f o r m a t i o n o f some c o n s i d e r a b l e i m p o r t a n c e t o t h e d e v e l o p m e n t o f c u r r e n t t h e o r i e s o f t e c t o n i c s . 1 3 9 2. E a r t h q u a k e P r e d i c t i o n A m a j o r g o a l o f e a r t h q u a k e e n g i n e e r i n g r e s e a r c h i s t h e p r e d i c t i o n o f e a r t h q u a k e s , h o u r s , d a y s o r e v e n w e e k s i n a d v a n c e . A m a j o r p a r a m e t e r u s e d i n e v a l u a t i n g e a r t h q u a k e p r e c u r s o r s i s t h e c o m b i n e d s t a t e o f r e g i o n a l s t r a i n . E x p e r i e n c e i n C a l i f o r n i a [ 8 ] w i t h g e o d i m e t e r d i s t a n c e m e a s u r e m e n t b e t w e e n p o i n t s s t r a d d l i n g known f a u l t s show "an e x t r e m e l y h i g h r a t e o f movement p r i o r t o a n d d u r i n g t h e e a r t h q u a k e w i t h some r e l a x a -t i o n f o l l o w i n g , F i g . I V - 1 . T h i s s u g g e s t s t h a t l a r g e movements a n d h i g h r a t e s o f movement a few h o u r s o r m i n u t e s p r i o r t o a n e a r t h q u a k e may p r o v i d e a n e f f e c t i v e c r i t e r i o n f o r s h o r t - r a n g e e a r t h q u a k e w a r n i n g s . C o n t i n u o u s o r n e a r c o n t i n u o u s m o n i t o r i n g o f s u s p e c t e d s i t e s w o u l d b e n e c e s s a r y t o p r o v i d e t h i s k i n d o f f o r e c a s t " . C u r r e n t e f f o r t s t o u s e s t r a i n p r e c u r s o r s t o s t u d y e a r t h q u a k e a n d f a u l t i n g m e c h a n i s m s i n v o l v e s t h e u s e o f g e o d i -m e t e r s t o a c c u r a t e l y m e a s u r e t h e d i s t a n c e b e t w e e n two p o i n t s s e t i n t h e g r o u n d o n r e p e a t e d o c c a s i o n s , o f t e n s e p a r a t e d b y m o n t h s , a n d a c c o u n t f o r a n y c h a n g e s i n l e n g t h o b s e r v e d i n t e r m s o f t e c t o n i c a c t i v i t y . T h e a c c u r a c y o f g e o d i m e t e r d i s t a n c e m e a s u r e m e n t i s b e t w e e n one p a r t i n 1 0 6 a n d one p a r t i n 1 0 7 a n d a s s u c h i s a n o r d e r o f m a g n i t u d e l e s s t h a n t h e p r e c i s i o n r e q u i r e d t o i n v e s t i g a t e , i n d e t a i l , t h e e a r t h q u a k e p r e c u r s o r p a r a m e t e r o f e a r t h s t r a i n [ 9 ] . I n a d d i t i o n g e o d i m e t e r s l a c k t h e l o n g t e r m s t a b i l i t y n e e d e d t o c o n d u c t c o n t i n u o u s m e a s u r e -m e n t s o f e a r t h s t r a i n b u i l d u p . 1 4 0 9 pm T I M E (PST) ON J A N U A R Y 19, I960 F I G . I V-1 E X A M P L E O F O B S E R V E D E A R T H Q U A K E S T R A I N P R E C U R S O R 1 4 1 R e s e a r c h e r s i n t h e U n i t e d S t a t e s o f A m e r i c a h a v e d e v e l o p e d t h e G e o d e t i c L a s e r S u r v e y S y s t e m (GLASS) c a p a b l e o f m a k i n g d i s t a n c e m e a s u r e m e n t s a c c u r a t e t o s e v e r a l m i l l i m e t e r s o v e r d i s t a n c e s o f up t o 40 km. T h i s c o r r e s p o n d s t o a n a b i l i t y t o m e a s u r e s t r a i n p r e c u r s o r s o f a b o u t 5 p a r t s i n 1 0 8 . T h e r e s e a r c h p r e s e n t e d i n t h i s t h e s i s i n d i c a t e s t h a t f r e e a i r l a s e r s t r a i n m e t e r s c o u l d make s t r a i n m e a s u r e m e n t s a f a c t o r o f 1 0 2 t o IO1* more a c c u r a t e l y t h a n GLASS o v e r a d i s t a n c e o f 5 km. P r o v i d i n g l a s e r p o w e r was a v a i l a b l e l a r g e r o b s e r v i n g d i s t a n c e s w o u l d r e s u l t i n p r o p o r t i o n a t e l y h i g h e r s t r a i n s e n s i -t i v i t i e s . T he f r e e a i r l a s e r s t r a i n m e t e r d e r i v e s i t s a d v a n t a g e i n s e n s i t i v i t y f r o m t h e f a c t t h a t i t a) i n t e r f e r e s p h o t o n s a n d n o t a m o d u l a t e d l i g h t s i g n a l b) d i r e c t l y o b s e r v e s t h e c h a n g e i n d i s t a n c e 6D w i t h o u t m e a s u r i n g t h e d i s t a n c e D . I n s o d o i n g i t r e t a i n s t h e same a b s o l u t e s e n s i -t i v i t y t o c h a n g e s i n d i s t a n c e o v e r i t s e n t i r e r a n g e o f o p e r a t i n g a b i l i t i e s . T h e a b i l i t y o f a f r e e a i r l a s e r s t r a i n m e t e r t o make c o n s e c u t i v e o b s e r v a t i o n s o f c h a n g e s i n t h e a m b i e n t r e g i o n a l s t r a i n a t i n t e r v a l s o f a few m i n u t e s makes i t i d e a l l y s u i t e d t o t h e p u r p o s e o f e a r t h q u a k e p r e d i c t i o n . I n a d d i t i o n t h e p o t e n t i a l l y h i g h s t r a i n s e n s i t i v i t y , 1 0 " 1 0 - 1 0 " 1 2 , o f t h e 142 f r e e a i r l a s e r s t r a i n m e t e r h o l d s h o p e f o r p r e d i c t i o n o f t e l e -s e i s m i c e v e n t s . S t r a i n p r e c u r s o r s o f t e l e s e i s m i c e v e n t s h a v e b e e n o b s e r v e d up t o h a l f a n h o u r b e f o r e a t e l e s e i s m i c e v e n t [ 1 5 ] . T h e o r e t i c a l p r e d i c t i o n s f o r t h e m a g n i t u d e o f s u c h s t r a i n s be a p p r o x i m a t e l y i n t h e r a n g e 1 0 " 9 - 1 0 " 1 2 , [ 1 7 ] , S m y l i e [ 4 1 ] h a s r e c e n t l y c a l c u l a t e d , f o r a " r e a l e a r t h " m o d e l , t h e t h e o r e t i c a l d i s p l a c e m e n t f i e l d s r e s u l t i n g f r o m f a u l t i n g w i t h i n t h e e a r t h . K n o w l e d g e o f s u c h d i s p l a c e m e n t f i e l d s c o u p l e d w i t h a w o r l d - w i d e o r c o n t i n e n t a l w i d e n e t w o r k o f l a s e r s t r a i n m e t e r s may l e a d t o a n e f f e c t i v e s y s t e m o f e a r t h q u a k e p r e d i c t i o n . As a r e s u l t one c a n r e a s o n a b l y l o o k f o r w a r d t o t h e d a y when " e a r t h q u a k e w a r n i n g s " ( s i m i l a r t o h u r r i c a n e w a r n i n g s ) w i l l be i s s u e d r e s u l t i n g i n t h e s a v i n g o f t h o u s a n d s o f l i v e s e a c h y e a r . 1 4 3 3. E n g i n e e r i n g C o n t e m p o r a r y t r e n d s t o w a r d l a r g e r man made s t r u c t u r e s r a i s e s more c o m p l e x p r o b l e m s r e g a r d i n g s t r u c t u r a l s t a b i l i t y a n d i n t e g r i t y o f dams, b r i d g e s , a n d b u i l d i n g s . Dams, f o r e x a m p l e , a r e s u b j e c t e d t o l a r g e i s o s t a t i c r e a d j u s t m e n t s . S u c h m o t i o n s o f t e n i n v o l v e d i s p l a c e m e n t s o f i n c h e s o r f e e t a n d w i l l f o r c e d e f o r m a t i o n w i t h i n t h e s t r u c t u r e i t s e l f . S t a n d a r d e n g i n e e r i n g m e t h o d s o f s u r v e y i n g t o e s t a b l i s h s u c h m o t i o n s a r e c l e a r l y l e s s s e n s i t i v e a n d c o n v e n i e n t t h a n t h e u s e o f a f r e e a i r l a s e r s t r a i n m e t e r . A r e t r o r e f l e c t o r m o u n t e d o n t h e s t r u c t u r e (dam, b r i d g e , o r b u i l d i n g ) w i l l f a c i l i t a t e r e m o t e s e n s i n g o f t h e d e f o r m a t i o n . I n t h i s way q u i c k a n s w e r s c a n be o b t a i n e d t o s u c h q u e s t i o n s a s i n t e r n a l d e f o r m a t i o n s o f t h e s t r u c -t u r e , d i s p l a c e m e n t s i n t h e s u r r o u n d i n g l a n d m a s s , a n d t h e i n t e r -p l a y b e t w e e n t h e two. 144 APPENDIX A - l A - l . THE GAS LASER 1. O p e r a t i n g C h a r a c t e r i s t i c s T h e p r i n c i p l e s o f o p e r a t i o n o f g a s l a s e r s a r e d i s -c u s s e d e x t e n s i v e l y b y v a r i o u s a u t h o r s [10] i n c o n t e m p o r a r y t e x t b o o k s a n d j o u r n a l s o f o p t i c s . F o r t h i s r e a s o n o n l y t h o s e c h a r a c t e r i s t i c s o f g a s l a s e r s p e r t i n e n t t o t h e o p e r a t i o n o f l a s e r s t r a i n m e t e r s w i l l be i n t r o d u c e d i n t h e f o l l o w i n g d i s -c u s s i o n . T h e g a s l a s e r o s c i l l a t o r i s j u s t one o f s e v e r a l t y p e s o f l a s e r o s c i l l a t o r s c a p a b l e o f g e n e r a t i n g c o h e r e n t e l e c t r o -m a g n e t i c r a d i a t i o n a t o p t i c a l w a v e l e n g t h s . B a s i c a l l y , a g a s l a s e r c o n s i s t s o f a p l a s m a t u b e f i l l e d w i t h a m i x t u r e o f g a s e s a n d e x c i t e d b y a n e l e c t r i c a l d i s c h a r g e . T h e g a s m i x t u r e i s s e l e c t e d t o s u s t a i n a p o p u l a t i o n i n v e r s i o n among t h e a t oms o f t h e g a s p o p u l a t i n g t h e e x c i t e d a t o m i c o r m o l e c u l a r e n e r g y l e v e l s . U n d e r t h e s e c o n d i t i o n s a p h o t o n e m i t t e d b y t h e s p o n -t a n e o u s d e c a y o f a n atom i n t h e h i g h l y p o p u l a t e d h i g h e r e n e r g y s t a t e h a s a l a r g e r p r o b a b i l i t y o f s t i m u l a t i n g t h e e m i s s i o n o f a n o t h e r s u c h p h o t o n t h a n o f b e i n g a b s o r b e d b y a n a t o m i n a l o w e r e n e r g y s t a t e . The r e s u l t i s a c a s c a d i n g , a v a l a n c h e - l i k e e m i s s i o n o f s t i m u l a t e d p h o t o n s , w h i c h r a p i d l y b e c o m e s h i g h l y d i r e c t i o n a l , a l i g n e d p a r a l l e l t o t h e p l a s m a t u b e , l a s t i n g a s l o n g as t h e p o p u l a t i o n i n v e r s i o n i s m a i n t a i n e d . T h e e n d s o f t h e p l a s m a t u b e a r e b r e w s t e r w i n d o w s p r o d u c i n g a n o u t p u t o f l i n e a r l y p o l a r i z e d r a d i a t i o n . 1 4 5 The c o n v e r s i o n o f a " l a s e r " , w h i c h a c t s a s a n o p t i c a l a m p l i f i e r , t o a " l a s e r o s c i l l a t o r " w i t h d e f i n i t e modes o f o s c i l l a t i o n r e q u i r e s f e e d b a c k . T h i s i s p r o v i d e d s i m p l y b y p l a c i n g m i r r o r s a t e a c h e n d o f t h e p l a s m a t u b e c r e a t i n g a n o p t i c a l r e s o n a n t c a v i t y . O u t p u t f r o m t h e l a s e r o s c i l l a t o r i s o b t a i n e d b y m a k i n g one m i r r o r p a r t i a l l y r e f l e c t i n g . F o r p u r p o s e s o f t h i s t h e s i s t h e d i s t i n c t i o n b e t w e e n a l a s e r a n d a l a s e r o s c i l l a t o r w i l l b e d r o p p e d a n d t h e w o r d " l a s e r " w i l l b e u s e d t o r e f e r t o a " g a s l a s e r o s c i l l a t o r " . v» '\ \ The r a d i a t i o n f i e l d i n a l a s e r c a v i t y i s s u s t a i n e d i n d i s c r e t e modes d e f i n e d b y t h e c a v i t y d i m e n s i o n s a n d l a b e l l e d TEM ; i n a s i m i l a r m a n n e r t o t h e modes i n a m i c r o w a v e mnq r e s o n a t o r . T h e n u m b e r s m , n a n d q d e f i n e t h e number o f modes i n t h e s t a n d i n g e l e c t r o m a g n e t i c wave i n t h e x , y a n d z-d i r e c t i o n s r e s p e c t i v e l y ; w h e r e z i s t a k e n t o be p a r a l l e l t o t h e a x i s o f t h e p l a s m a t u b e . T h e c o n f i g u r a t i o n s o f t h e e l e c t r o -m a g n e t i c f i e l d i n x - y d i r e c t i o n s a r e c a l l e d s p a t i a l modes w i t h m a n d n b e i n g t y p i c a l l y i n t h e r a n g e 0 t o 3. T h e c o n f i g u r a -t i o n s o f t h e e l e c t r o m a g n e t i c f i e l d i n t h e z - d i r e c t i o n a r e c a l l e d t e m p o r a l modes w i t h q b e i n g ^ 1 0 6 . T h e d i s p a r i t y i n t h e mag-n i t u d e o f t h e s u b s c r i p t s m , n a n d q h a s r e s u l t e d i n a n o t a t i o n c o n v e n t i o n f o r d e s c r i b i n g l a s e r modes o n l y i n t e r m s o f t h e s p a t i a l s u b s c r i p t s ; TEM F o r t e m p o r a l mode r e s o n a n c e t h e l e n g t h L b e t w e e n t h e e n d m i r r o r s o f t h e l a s e r c a v i t y m u s t b e an i n t e g r a l number o f h a l f w a v e l e n g t h s X o f t h e l a s e r r a d i a t i o n 2L q cm. (A-l-1) T h i s c o r r e s p o n d s t o a f r e q u e n c y o f l a s e r o s c i l l a t i o n v ( A - l - 2 ) w h e r e c i s t h e v e l o c i t y o f l i g h t i n t h e l a s e r medium. T h e f r e q u e n c y s e p a r a t i o n b e t w e e n t e m p o r a l modes c o r r e s p o n d i n g t o c o n s e c u t i v e v a l u e s o f q i s A v T a s c a n b e a c c o m m o d a t e d u n d e r t h e d o p p l e r b r o a d e n e d g a i n p r o f i l e w i l l b e o p e r a t i n g s i m u l t a n e o u s l y i n a l a s e r , F i g . A - l - 1 . S i n c e e a c h t e m p o r a l mode ( b e i n g a t a s l i g h t l y d i f f e r e n t f r e -q u e n c y f r o m i t s n e i g h b o u r ) w i l l p r o d u c e a n i n d e p e n d e n t s e t o f i n t e r f e r e n c e f r i n g e s i t i s d e s i r a b l e f o r p u r p o s e s o f l a s e r i n t e r f e r o m e t r y t o e l i m i n a t e a l l b u t one t e m p o r a l mode. T h i s Av ( A - l - 3 ) I n g e n e r a l a s many t e m p o r a l modes o f o s c i l l a t i o n 1 4 7 l-s Av v c F I G . A - l - 1 F R E Q U E N C Y D I S T R I B U T I O N O F L A S E R O U T P U T 1 4 8 i s u s u a l l y a c c o m p l i s h e d b y b o t h s h o r t e n i n g t h e l e n g t h o f t h e l a s e r c a v i t y a n d c o n s e q u e n t l y i n c r e a s i n g Av^ u n t i l o n l y one mode a p p e a r s ; a n d r a i s i n g t h e l o s s l e v e l o f t h e p l a s m a t u b e u n t i l a l l modes b u t t h e one n e a r v c , t h e c e n t r e f r e q u e n c y o f t h e g a i n p r o f i l e , e x h i b i t g a i n o f l e s s t h a n o n e . When t h i s i s a c h i e v e d t h e l a s e r i s s a i d t o o p e r a t e i n a u n i m o d a l f a s h i o n . U n i m o d a l l a s e r s w i t h v e r y l a r g e p o w e r o u t p u t s a r e d i f f i c u l t t o c o n s t r u c t , h o w e v e r , u n i m o d a l T E M q o l a s e r s w i t h p o w e r o u t p u t up t o 1 0 w a t t s a r e a v a i l a b l e c o m m e r c i a l l y . '\ T h e s p a t i a l modes o f t h e l a s e r o s c i l l a t i o n d e t e r m i n e t h e i n t e n s i t y d i s t r i b u t i o n a c r o s s t h e o u t p u t beam. T he s p a t i a l modes p r e s e n t i n a l a s e r o s c i l l a t i o n a r e c o n t r o l l e d b y t h e s h a p e a n d a l i g n m e n t o f t h e o p t i c a l r e s o n a t o r a s w e l l a s t h e n a t u r e o f t h e e d g e l o s s e s p r e s e n t i n t h e p l a s m a t u b e . F o r p u r p o s e s o f l a s e r i n t e r f e r o m e t r y t h e T E M q q mode i s d e s i r a b l e . T h i s mode d o e s n o t c o n t a i n a n y p h a s e s h i f t s i n t h e e l e c t r o m a g -n e t i c f i e l d a c r o s s t h e o u t p u t beam. T h e TEM ^  mode a l s o r oo o f f e r s t h e a d v a n t a g e o f s m a l l e s t s p o t s i z e a n d l o w e s t a n g u l a r d i v e r g e n c e o f t h e o u t p u t beam f o r a g i v e n s i z e o f e x i t p u p i l . T h e o u t p u t beam o f a u n i m o d a l l a s e r o p e r a t i n g i n t h e T E M Q O mode h a s a g a u s s i a n i n t e n s i t y d i s t r i b u t i o n w i t h p l a n e o r s p h e r i c a l f r o n t s o f c o n s t a n t p h a s e . 1 4 9 2. F r e q u e n c y S t a b i l i t y T h e s t a b i l i t y o f t h e f r e q u e n c y o f o s c i l l a t i o n o f l a s e r o u t p u t i s a m a t t e r o f p r i m e i m p o r t a n c e i n l a s e r i n t e r -f e r o m e t r y s i n c e f r e q u e n c y s t a b i l i t y i s s y n o n y m o u s w i t h wave-l e n g t h s t a b i l i t y . T h e f r e q u e n c y s t a b i l i t y o f a l a s e r S V ( T ) i s g i v e n b y S v ( T ) - ( A - l - 4 ) w h e r e 6 V ( T ) i s t h e r a n g e o f f r e q u e n c y o v e r w h i c h t h e o u t -p u t o f a l a s e r v a r i e s i n t i m e T when o p e r a t i n g a t a f r e q u e n c y v . T h e f r e q u e n c y s t a b i l i t y o f a l a s e r i s c h a r a c t e r i z e d b y a l o n g - t e r m a n d a s h o r t - t e r m s t a b i l i t y d e p e n d i n g o n w h e t h e r T i s g r e a t e r o r l e s s t h a n some c h a r a c t e r i s t i c t i m e , u s u a l l y o f t h e o r d e r o f h o u r s 1 A q u a n t i t y c l o s e l y r e l a t e d t o f r e q u e n c y s t a b i l i t y a n d o f e q u a i i m p o r t a n c e i n l a s e r i n t e r f e r o m e t r y i s f r e q u e n c y r e s e t t a b i l i t y , R v , w h i c h d e s c r i b e s t h e a c c u r a c y w i t h w h i c h t h e f r e q u e n c y V q c a n b e r e p r o d u c e d a f t e r t h e l a s e r i s p e r t u r b e d a n d r e a d j u s t e d . v - v R = °- ( A - l - 5 ) v v w h e r e v - v Q i s t h e a v e r a g e d e v i a t i o n o f a number o f s e t t i n g s f r o m t h e d e s i r e d f r e q u e n c y v Q 1 5 0 T h e u l t i m a t e s h o r t t e r m f r e q u e n c y s t a b i l i t y o f a l a s e r i s d e t e r m i n e d b y t h e n a t u r a l s p e c t r a l w i d t h , 6v , o f t h e a t o m i c t r a n s i t i o n o c c u r r i n g i n t h e l a s e r . T h i s l i n e w i d t h h a s b e e n c a l c u l a t e d b y S c h a w l o w a n d Townes [11] t o b e Sv = 8* h v ( A v c ) 2 H z . ( A - l - 6 ) W w h e r e A v c i s t h e s p e c t r a l w i d t h a t h a l f i n t e n s i t y o f t h e d o p p l e r b r o a d e n e d s p e c t r a l l i n e o f t h e l a s e r t r a n s i t i o n , W i s t h e o u t p u t p o w e r o f t h e l a s e r a n d h i s P l a n c k ' s c o n s t a n t . T h i s e q u a t i o n p r e d i c t s a f r e q u e n c y s p r e a d o f a b o u t 10" 2 H z . o r a maximum f r e q u e n c y s t a b i l i t y S ( x ) „ ^ I O 1 6 - 1 0 " 1 7 n 1 J v max f o r t y p i c a l o p t i c a l l a s e r s [ 12]. H o w e v e r t h e f r e q u e n c y s t a -b i l i t y a c h i e v e d i n p r a c t i c e i s l i m i t e d b y t h e Q o f t h e o p t i c a l c a v i t y t o v a l u e s s e v e r a l o r d e r s o f m a g n i t u d e l e s s t h a n t h e t h e o r e t i c a l maximum. S i n c e t h e p r a c t i c a l l i m i t o n l a s e r f r e q u e n c y s t a b i l i t y i s t h e Q o f t h e o p t i c a l r e s o n a t o r , l a s e r f r e q u e n c y s t a b i l i t y d e p e n d s a l m o s t e n t i r e l y on t h e s t a b i l i t y o f t h e c a v i t y d i m e n -s i o n s . T h e p r i n c i p l e c a u s e s o f l a s e r f r e q u e n c y f l u c t u a t i o n a r e c h a n g e s i n a m b i e n t p r e s s u r e a n d t e m p e r a t u r e a n d d e t u n i n g due t o e x t e r n a l m e c h a n i c a l a n d a c o u s t i c a l v i b r a t i o n s . P a s s i v e m e t h o d s o f f r e q u e n c y s t a b i l i z a t i o n s u c h as m e c h a n i c a l a n d t h e r m a l i s o l a t i o n a n d t h e u s e o f i n v a r s p a c e r s b e t w e e n t h e 1 5 1 m i r r o r s a r e i n a d e q u a t e f o r a c c u r a t e l a s e r i n t e r f e r o m e t r y a n d f o r s u c h p u r p o s e s a c t i v e m e t h o d s o f f r e q u e n c y s t a b i l i z a t i o n - a r e u s e d . A c t i v e f r e q u e n c y s t a b i l i z a t i o n s y s t e m s a l l e m p l o y a c a v i t y l e n g t h c o n t r o l e l e m e n t ( u s u a l l y a p i e z o e l e c t r i c c r y s t a l ) o n w h i c h one o f t h e c a v i t y m i r r o r s i s m o u n t e d . Th e c o n t r o l e l e m e n t i s d r i v e n b y a s e r v o s i g n a l whose m a g n i t u d e a n d p o l a r i t y c o r r e s p o n d t o t h e m a g n i t u d e a n d s i g n o f t h e d e v i a t i o n o f t h e l a s e r f r e q u e n c y f r o m t h e d e s i r e d f r e q u e n c y . T h e v a r i o u s a c t i v e f r e q u e n c y s t a b i l i z i n g s y s t e m s d i f f e r one f r o m t h e o t h e r i n t h e t e c h n i q u e b y w h i c h t h e e r r o r s i g n a l i s d e r i v e d . I n g e n e r a l , s c h e m e s f o r s t a b i l i z i n g l a s e r s f a l l i n t o two c l a s s e s ; t h o s e i n w h i c h t h e e r r o r s i g n a l i s d e r i v e d f r o m t h e l a s e r i t s e l f ( s e l f - s t a b i l i z i n g ) a n d t h o s e i n w h i c h t h e e r r o r s i g n a l i s d e r i v e d f r o m an e x t e r n a l r e f e r e n c e c e l l ( e x t e r n a l l y s t a b i l i z e d ) . I n t h e s e l f - s t a b i l i z i n g scheme an a . c . v o l t a g e i s a p p l i e d t o t h e p i e z o e l e c t r i c c r y s t a l w h i c h , b y v a r y i n g t h e c a v i t y d i m e n s i o n s , f r e q u e n c y m o d u l a t e s t h e l a s e r . F r e q u e n c y m o d u l a t i o n i s a c c o m p a n i e d by an a m p l i t u d e m o d u l a t i o n o f t h e beam i n t e n s i t y b y v i r t u e o f t h e s h a p e o f t h e g a i n p r o f i l e . T h e beam a m p l i t u d e m o d u l a t i o n i s i n p h a s e w i t h t h e a p p l i e d v o l t a g e f o r f r e q u e n c i e s v > v c a n d o p p o s e d i n p h a s e f o r f r e q u e n c i e s v < v . T h e o u t p u t beam i n t e n s i t y i s m o n i t o r e d 1 5 2 b y a p h o t o d i o d e a n d p h a s e d e t e c t e d p r o d u c i n g a n e r r o r s i g n a l w h i c h when a p p l i e d t o t h e c o n t r o l e l e m e n t w i l l k e e p t h e l a s e r f r e q u e n c y v n e a r v c , t h e c e n t r e f r e q u e n c y o f t h e g a i n p r o f i l e . The s e n s i t i v i t y o f t h i s m e t h o d o f f r e q u e n c y s t a b i l i -z a t i o n s u f f e r s f r o m t h e d i s a d v a n t a g e o f b e i n g d e p e n d e n t on t h e s h a p e o f t h e g a i n p r o f i l e n e a r v c , w h i c h c a n be b a d l y f l a t t e n e d b y t h e Lamb d i p r e s u l t i n g i n a l o w s e n s i t i v i t y t o f r e q u e n c y d r i f t s . N e v e r t h e l e s s t h i s f r e q u e n c y s t a b i l i z a t i o n s cheme i s a b l e t o s t a b i l i z e l a s e r o s c i l l a t i o n s t o two p a r t s i n 1 0 9 f o r many d a y s [ 1 3] a n d a l l o w a r e s e t t a b i l i t y o f s e v e r a l p a r t s i n 1 0 9 [ 1 2 ] . One o f t h e m o s t s u c c e s s f u l e x t e r n a l s t a b i l i z a t i o n s c h e m e s i n v o l v e s r e s o n a n t a b s o r p t i o n o f p a r t o f t h e l i n e a r l y p o l a r i z e d l a s e r o u t p u t i n an a b s o r b i n g c e l l ; t h e a b s o r p t i o n p r o f i l e o f w h i c h h a s b e e n Zeeman s p l i t b y t h e a p p l i c a t i o n o f an a x i a l m a g n e t i c f i e l d [ 1 3 ] . T h e a b s o r p t i o n c e l l medium b e c o m e s d i c h r o i c ; p r e f e r e n t i a l l y a b s o r b i n g r i g h t h a n d c i r -c u l a r l y p o l a r i z e d l i g h t f o r f r e q u e n c i e s v > v c a n d p r e -f e r e n t i a l l y a b s o r b i n g l e f t h a n d c i r c u l a r l y p o l a r i z e d l i g h t f o r f r e q u e n c i e s v < v c • A n e r r o r s i g n a l i s d e r i v e d b y a l t e r n a t e l y s a m p l i n g t h e LHCP a n d RHCP o u t p u t f r o m t h e a b s o r b e r u s i n g a KDP e l e c t r o o p t i c s w i t c h d r i v e n b y a s q u a r e wave g e n e r a t o r a n d r e q u i r i n g t h a t b o t h o u t p u t s be o f e q u a l 1 5 3 i n t e n s i t y . T h i s e r r o r s i g n a l a p p l i e d t o t h e l e n g t h c o n t r o l e l e m e n t k e e p s t h e l a s e r f r e q u e n c y v n e a r v c . T h i s s y s t e m i s t h e o r e t i c a l l y c a p a b l e o f l o n g t e r m f r e q u e n c y s t a b i l i t i e s o f two p a r t s i n 1 0 1 2 a n d h a s a c h i e v e d f r e q u e n c y s t a b i l i z a t i o n o f f o u r p a r t s i n 1 0 1 0 f o r e i g h t m i n u t e s [ 1 3 ] . V a l i [ 3 9 ] h a s p o i n t e d o u t t h a t t h e u t i l i z a t i o n o f r e s o n a n t a b s o r p t i o n i n a n a r r o w m o l e c u l a r r o t a t i o n - v i b r a t i o n l i n e o f m e t h a n e a t 3.39u h a s b e e n u s e d t o f r e q u e n c y s t a b i l i z e a l a s e r r e s u l t i n g i n a r e s e t t a b i l i t y o f b e t t e r t h a n one p a r t i n 1 0 1 1 a n d p o s s i b l e f r e q u e n c y s t a b i l i t y o f one p a r t i n 1 0 1 2 . O t h e r m e t h o d s i n v o l v i n g t h e u s e o f t h e d i s p e r s i v e p r o p e r t i e s o f t h e i n v e r t e d p o p u l a t i o n t o c o n s t r u c t a f r e q u e n c y d i s c r i m i n a t o r a r e u s e d t o s t a b i l i z e l a s e r s . T h e s e m e t h o d s h a v e r e s u l t e d i n l a s e r f r e q u e n c y s t a b i l i t i e s o f one p a r t i n 1 0 1 0 f o r a p e r i o d o f e i g h t h o u r s d u r i n g w h i c h f o r p e r i o d s o f a m i n u t e o r s o t h e l a s e r was f r e q u e n c y s t a b i l i z e d t o one p a r t i n 1 0 1 2 . T h e v e r y s h o r t t e r m (x *\» 1 - 1 0 " 3 s e c . ) f r e q u e n c y s t a b i l i t y o f l a s e r s i s i n v e s t i g a t e d b y h e t e r o d y n i n g two s i m i l a r f r e q u e n c y s t a b i l i z e d l a s e r s t o g e t h e r s i n c e n o d i r e c t means o f c o u n t i n g a t o p t i c a l , f r e q u e n c i e s e x i s t s . S u c h e x p e r i m e n t s r e v e a l b e a t f r e q u e n c i e s o f 1-20 H z . o v e r p e r i o d s r a n g i n g f r o m a few t e n s o f m i l l i s e c o n d s t o a few s e c o n d s i n d i c a t i n g t h a t f o r p e r i o d s o f t h i s o r d e r l a s e r f r e q u e n c i e s c a n be s t a b i l i z e d t o one p a r t i n 1 0 1 3 [ 1 2 ] . 154 APPENDIX A-2 A - 2 . THE R E F R A C T I V E INDEX OF THE ATMOSPHERE 1. F o r m u l a f o r R e f r a c t i v e I n d e x T h e q u a n t u m t h e o r y o f a t o m i c a n d m o l e c u l a r p o l a r i z a -b i l i t y c o u p l e d w i t h t h e c l a s s i c a l t h e o r y o f e l e c t r o m a g n e t i s m y i e l d t h e L o r e n t z - L o r e n z r e l a t i o n w h i c h g i v e s t h e r e f r a c t i v e i n d e x n o f a d i e l e c t r i c i n t e r m s o f i t s s p e c i f i c r e f r a c t i v i t y R ( X ) a n d i t s d e n s i t y p i nI " 1 = R ( X ) p ( A - 2 - 1 ) n A + 2 T h e s p e c i f i c r e f r a c t i v i t y i s c o n s i d e r e d t o be i n v a r i a n t u n d e r c h a n g e s i n d e n s i t y f o r a g i v e n d i e l e c t r i c a n d t o b e a f u n c t i o n o n l y o f t h e w a v e l e n g t h o f t h e i n c i d e n t r a d i a t i o n X . I f t h e d i e l e c t r i c i s a h o m o g e n e o u s , s i n g l e c o m p o n e n t f l u i d t h e n 4TT N O A ( X )  R(X) = « -2-JJ ( A - 2 - 2 ) w h e r e N Q i s A v o g a d r o s n u m b e r , M i s t h e m o l e c u l a r w e i g h t o f t h e m a t e r i a l a n d a ( X ) i s i t s m o l e c u l a r p o l a r i z a b i l i t y . I f t h e d i e l e c t r i c c o n s i s t s o f s e v e r a l c o m p o n e n t s t h e n 4"1-1 - I R i ( X ) P i ( A - 2 - 3 ) n x • 2 i 155 w h e r e R^(A) i s t h e s p e c i f i c r e f r a c t i v i t y o f t h e i * * 1 c o m p o n e n t t h a n d i s t h e p a r t i a l d e n s i t y o f t h e i c o m p o n e n t . A l t h o u g h t h e e a r t h ' s a t m o s p h e r e c o n t a i n s many g a s e o u s c o m p o n e n t s t h e L o r e n t z - L o r e n z r e l a t i o n i s c a p a b l e o f g i v i n g t h e a t m o s p h e r i c r e f r a c t i v e i n d e x t o a c c u r a c i e s o f p a r t s i n 1 0 1 0 i f t h e s u m m a t i o n i n e q n . 3-3 i s t e r m i n a t e d a t i = 3 , [20] . T h u s < - 1 — r • R ^ P i + R 2(X)p 2 + RaCJOp, (A-2-4) " X + 2 w h e r e Ri , R 2 a n d R 3 a r e t h e s p e c i f i c r e f r a c t i v i t i e s o f C 0 2 - f r e e " a i r " , w a t e r v a p o u r , a n d c a r b o n d i o x i d e r e s p e c t i v e l y a n d p i , p 2 a n d p 3 a r e t h e i r r e s p e c t i v e p a r t i a l d e n s i t i e s * I f t h e amount o f C 0 2 p r e s e n t i n t h e a t m o s p h e r e i s a s s u m e d t o be c o n s t a n t a t 0.035 m o l a r % t h e n t h e a t m o s p h e r i c r e f r a c t i v e i n d e x c a n be e x p r e s s e d a s 2 - i = R l ( X ) p - + R 2 ( X ) P 2 (A-2-5) nx + 2 w h e r e R|(A) i s t h e s p e c i f i c r e f r a c t i v i t y o f d r y " a i r " a n d * " A i r " i s d e f i n e d a s a m i x t u r e o f g a s e s c o n t a i n i n g 78.09 m o l a r % N 2 , 20.95 m o l a r % 0 2 , 0.93 m o l a r % A r , 0.03 m o l a r % C 0 2 w h e r e a s " a t m o s p h e r e " r e f e r s t o t h a t m i x t u r e o f g a s e s s u r r o u n d i n g t h e p l a n e t e a r t h . T h e s p e c i f i c r e f r a c t i v i t i e s o f N 2 , 0 2 a n d A r a r e s o s i m i l a r t h e y a r e a l l i n c l u d e d i n t h e f i r s t t e r m , R i ( X ) . 1 5 6 p[ i s t h e c o r r e s p o n d i n g p a r t i a l d e n s i t y . T h e a s s u m p t i o n o f c o n s t a n t CO2 c o n t e n t i n t h e f r e e l y c i r c u l a t i n g a t m o s p h e r e i s s u p p o r t e d b y e x p e r i m e n t a l e v i d e n c e . The v a r i a b i l i t y o f CO2 c o n t e n t i n a t m o s p h e r i c s a m p l e s h a s b e e n f o u n d t o be l e s s t h a n a few t h o u s a n d t h s o f one p e r c e n t f r o m i t s n o r m a l v a l u e o f 0.035 m o l a r % [ 2 1 ] . > An e m p i r i c a l f o r m u l a f o r a t m o s p h e r i c r e f r a c t i v e i n d e x c o r r e s p o n d i n g t o e q n . 3-5 i s g i v e n b y Owens [20] a s nx = 1 + »\ nx = 1 + f ( X ) F ( P , T ) + g ( X ) G ( p , T ) ( A - 2 - 6 ) w h e r e t h e d i s p e r s i o n f a c t o r s f ( X ) a n d g ( X ) a r e g i v e n b y f ( X ) = oini A J. 683939.7 . 4547.3 2371.4 + + 130 - 1/X 2 38.9 - 1/X 2 x I O "8 ( A - 2 - 7 ) g ( X ) = 6487 31 + 5 8 - 0 5 8 . 0-71150 + 0.08851 x 1 0 " 8 ( A - 2 - 8 ) a n d t h e d e n s i t y f a c t o r s F ( P , T ) a n d G ( p , T ) a r e g i v e n b y F ( P , T ) = £ 1 + P ( 5 7 . 9 0 x 1 0 " 8 - 9 - 5 2 5 £ x 1 0 - k 0.25844i ( A - 2 - 9 ) 157 G ( p , T ) = E 1 + p 1 + 3.7 x 10 2.37321 x I O " 3 + 2-23566 710.792 + 7.75141 x IO 1*) ( A - 2 - 1 0 ) a n d X = w a v e l e n g t h o f l i g h t i n m i c r o n s T = t e m p e r a t u r e o f a t m o s p h e r e i n °K P = p a r t i a l p r e s s u r e o f d r y a i r i n m i l l i b a r s p = p a r t i a l p r e s s u r e o f w a t e r v a p o u r i n m i l l i b a r s T h e d e v i a t i o n s f r o m u n i t y o f t h e a t m o s p h e r i c r e f r a c -t i v e i n d e x i s r e p r e s e n t e d b y u a n d i s m e a s u r e d i n u - u n i t s . One u - u n i t i s a r e f r a c t i v e i n d e x d e v i a t i o n o f 1 0 " 6 . F o r m u l a 3-6 g i v e s t h e a b s o l u t e v a l u e o f t h e r e f r a c t i v e i n d e x a c c u r a t e t o one p a r t i n 1 0 9 o v e r t h e r a n g e o o 2302 A < X < 20,586 A 0 < P < 4000 mb. 0 < p < 100 mb. 2 4 0 ° K < T < 3 3 0 ° K 1 5 8 2. S t a t i s t i c a l P r o p e r t i e s o f t h e A t m o s p h e r i c R e f r a c t i v e  I n d e x The e a r t h ' s a t m o s p h e r e i s i n a s t a t e o f t u r b u l e n t m o t i o n . T h e v a l u e s o f w i n d v e l o c i t y as w e l l as t h e a t m o s p h e r i c p r e s s u r e , t e m p e r a t u r e , a n d h u m i d i t y u n d e r g o i r r e g u l a r f l u c t u a t i o n s i n s p a c e a n d t i m e w h i c h c a u s e r a n d o m s p a t i a l a n d t e m p o r a l f l u c t u a -t i o n s i n t h e a t m o s p h e r i c r e f r a c t i v e i n d e x n . T h e a t m o s p h e r i c r e f r a c t i v e i n d e x n c a n be c o n s i d e r e d t o be a r a n d o m f i e l d d e p e n d e n t on t h e t h r e e c a r t e s i a n s p a t i a l c o o r d i n a t e s x , y , z a n d t i m e T . I f R = x i + y j + z k i s t h e p o s i t i o n v e c t o r r e l a t i v e t o c a r t e s i a n a x e s ox , oy , a n d o z w i t h u n i t v e c t o r s i , j , a n d k r e s p e c t i v e l y t h e n f r o m e q n . A - 2 - 6 n ( R , x ) = 1 + u ( R , T ) ->• A c o m p l e t e s t a t i s t i c a l d e s c r i p t i o n o f n ( R , x ) i s g i v e n b y t h e s e t o f j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n s B ^ y ( R i ) y ( R 2 ) . . .y ( R ^ ; T X , T 2 t i ) i = 1 , 2 , 3 w h e r e 3 j _ ( y ( R i ) y ( R 2 ) .. . y ( R ^ ) ; T i , x 2 . . . . x ^ ) i s d e f i n e d a s t h e p r o b a b i l i t y t h a t a l l yCR^) b e i n t h e r a n g e y t o y + dp a t t i m e s x^ r e s p e c t i v e l y . A c o m p l e t e s t a t i s t i c a l d e s c r i p t i o n o f m o s t r a n d o m p r o c e s s e s t h u s i n v o l v e s an i n f i n i t e s e t o f s u c h f u n c t i o n s . I n p r a c t i c e a n i n c o m p l e t e b u t c o m p u t a t i o n a l l y u s e f u l d e s c r i p t i o n o f a r a n d o m p r o c e s s r e s u l t s i f " i " i s a l l o w e d t o assume v a l u e s i = 1 , 2 ; c o r r e s p o n d i n g t o 3 i ( y ( R i ) ; x x ) a n d 62 (y ( R i ) y (R2) ; T I X 2 ) 159 Moments o f v a r i o u s o r d e r s c o r r e s p o n d i n g t o d e n s i t y -f u n c t i o n s 3 i a n d £2 a r e d e f i n e d u n ( R \ T l ) - j y n ( R T ) 3 1 ( y ( R 1 ) ; T 1 ) d y u n ( R i T 1 ) y m ( R 2 T 2 ) = j J y n ( R ! T i ) y m ( R 2 T 2 ) 3 2 ( y ( R i ) y ( R 2 ) ; T 1 r 2 ) d y x d y 2 o o A g a i n f r o m a p r a c t i c a l s t a n d p o i n t t h e m o s t i m p o r t a n t o f t h e s e a r e t h e l o w e s t o r d e r moments 00 ( A - 2 - 1 1 ) u(Ri T X ) y ( R 2 T 2 ) = j J y ( R \ T i ) y ( R 2 T 2 ) 3 2 (y (Ri ) y (R 2) ;T 1 T 2 ) d y i d y 2 o o ( A - 2 - 1 2 ) w h i c h a r e t h e mean v a l u e y a n d t h e a u t o c o v a r i a n c e y 2 a s a f u n c t i o n o f s p a c e a n d t i m e r e s p e c t i v e l y . I f t h e a t m o s p h e r i c r e f r a c t i v e i n d e x f l u c t u a t i o n s a r e c o n s i d e r e d t o be ho m o g e n e o u s a n d i s o t r o p i c t h e s t a t i s t i c a l d e s c r i p t i o n o f the m i s i n v a r i a n t w i t h r e s p e c t t o t r a n s l a t i o n s a n d r o t a t i o n s o f t h e c o o r d i n a t e a x e s . I f t h e a t m o s p h e r i c 160 r e f r a c t i v e i n d e x f l u c t u a t i o n s a r e c o n s i d e r e d t o be s t a t i o n a r y t h e i r s t a t i s t i c a l d e s c r i p t i o n i s i n v a r i a n t w i t h r e s p e c t t o t r a n s l a t i o n s i n t i m e . T h e a s s u m p t i o n o f i s o t r o p y p e r m i t s t h e v e c t o r R a p p e a r i n g i n e q n s . A-2-11 a n d A-2-12 t o be r e p l a c e d b y i t s m o d u l u s R oo y ( R i T i ) • | u ( R , T ) 8 1 ( u ( R i ) ; T 1 ) d u ( A - 2 - 1 3 ) y ( R i T O y ( R 2 x 2 ) = j J y ( R i x x ) y ( R 2 x 2 ) $ 2 ( y ( R i ) y ( R 2 ) ; x 1 x 2 ) d y x d y 2 o o ( A - 2 - 1 4 ) The a s s u m p t i o n s o f h o m o g e n i e t y a n d s t a t i o n a r i t y p e r m i t o ne t o a r b i t r a r i l y s e t R x = 0 x x = 0 a n d t o d e f i n e v a r i a b l e s r = R 2 - R i a n d t = x 2 - X i . T h u s y = J u ( R i T i ) B , ( n ( R i ) ; T , ) d y ( A - 2 - 1 5 ) o a n d y ( R , x 1 ) y ( R 2 x 2 ) = B ( R 2 - R x ) , ( x 2 - T , ) ) y ( R i T i ) y ( R 2 x 2 ) = B ( r , t ) ( A - 2 - 1 6 ) 1 6 1 T h e a u t o c o v a r i a n c e n o r m a l i z e d b y d i v i s i o n b y t h e v a r i a n c e o f t h e r e f r a c t i v e i n d e x f l u c t u a t i o n s iT2" g i v e s 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 c ( r , t ) c ( r , t ) = ~ B ( r , t ) . ( A - 2 - 1 7 ) . y 2 T h e a s s u m p t i o n s o f h o m o g e n i e t y , i s o t r o p y a n d s t a t i o n a -r i t y o f a t m o s p h e r i c r e f r a c t i v e i n d e x f l u c t u a t i o n s r e q u i r e s some s c r u t i n y . C l e a r l y t h e r e a l a t m o s p h e r e c o n f o r m s t o n o n e o f t h e s e i d e a l d e s c r i p t i o n s . T h e a t m o s p h e r e i s c h a r a c t e r i z e d s p a t i a l l y b y l a r g e v e r t i c a l a n d h o r i z o n t a l g r a d i e n t s i n p r e s s u r e , t e m p e r a -t u r e , a n d h u m i d i t y ; a n d t e m p o r a l l y b y s e a s o n a l a n d d a i l y v a r i a -t i o n s i n t h e s e same p a r a m e t e r s . H o w e v e r f o r c a l c u l a t i n g t h e e f f e c t o f a t m o s p h e r i c t u r b u l e n c e on t h e p r o p a g a t i o n o f l a s e r o u t p u t , p a r t i c u l a r l y t h e e f f e c t on t h e p h a s e c o h e r e n c e , t h e a s s u m p t i o n s a r e j u s t i f i e d . K o l m o g o r o v h a s p o i n t e d o u t a n d e x p e r i m e n t h a s v e r i f i e d [ 2 3 , p . 79] t h a t t h e s m a l l s c a l e c o m p o n e n t s o f t u r b u l e n c e t e n d t o w a r d h o m o g e n i e t y a n d i s o t r o p y . T h e a s s u m p t i o n s o f h o m o g e n i e t y a n d i s o t r o p y a r e j u s t i f i e d s i n c e i t i s t h e s m a l l s c a l e t u r b u l e n c e w h i c h w i l l b e t h e p r i m a r y c a u s e o f d e g e n e r a t i o n o f p h a s e c o h e r e n c e i n t h e l a s e r s i g n a l [ 2 4 ] . T h e l a r g e r s c a l e s o f t u r b u l e n c e w i l l t e n d t o d e f l e c t t h e beam a s a w h o l e w h i l e p r e s e r v i n g t h e s t r u c -t u r e o f t h e p h a s e f r o n t s . W h i l e t h e s t a t i s t i c a l d e s c r i p t i o n o f a t m o s p h e r i c t u r b u l e n c e i s a d m i t t e d l y t i m e d e p e n d e n t a n d h e n c e 1 6 2 non-stationary on a time scale of days, the assumption of s t a t i o n a r i t y i s j u s t i f i e d on the short time scales involved in the i n v e s t i g a t i o n of phase coherence i n the laser s i g n a l . The t r a n s i t time of a photon along a free a i r laser s t r a i n meter of length 3 km i s of the order of 10" 5 sec and the s t a t i s t i c s of atmospheric r e f r a c t i v e index fl u c t u a t i o n s are very nearly stationary over such short time i n t e r v a l s . Assuming the autocorrelation function for atmospheric r e f r a c t i v e index fluctuations i s a single parameter function of r , C ^ ( r ) , i t remains only to assign a s p e c i f i c dependence on r and to define the parameter i n physical terms. While i t has been shown experimentally [24] that there does not e x i s t a r e f r a c t i v e index autocorrelation function v a l i d for a l l atmospheric conditions several t h e o r e t i c a l predictions concern-ing the nature of the r e f r a c t i v e index autocorrelation function have been made. Chernov [20, p. 29] presents a simple proof that any r e f r a c t i v e index autocorrelation function C ^ ( r ) must exhibi t the property that d C (r) _ J U L "cTF = 0 (A-2-18) r=0 This i s a necessary consequence of the fact that the atmospheric r e f r a c t i v e index i s a continuous and d i f f e r e n t i a b l e function of p o s i t i o n . The Kolmogoroff theory of turbulence predicts a 1 6 3 2 7 r e f r a c t i v e i n d e x a u t o c o r r e l a t i o n f u n c t i o n [ 3 2 ] C ( r ) = 1 - b f r / R ) uu o w h e r e b i s a c o n s t a n t a n d R^ i s t h e o u t e r s c a l e o f t u r b u l e n c e . o W h i l e t h i s a u t o c o r r e l a t i o n f u n c t i o n a n d o t h e r s s u c h a s cu u ( r ) = e x p - ( r / R 0 ) v i o l a t e t h e c o n d i t i o n r e q u i r e d a t r=0 t h e y a r e u s e d b y many a u t h o r s who c o n s i d e r them t o be a d e q u a t e r e p r e s e n t a t i o n s o f a t m o s p h e r i c r e f r a c t i v e i n d e x f l u c t u a t i o n s . T a t a r s k i [ 2 5 ] h a s shown t h e a t m o s p h e r i c p a r a m e t e r s o f p r e s s u r e , t e m p e r a t u r e a n d h u m i d i t y b e h a v e a p p r o x i m a t e l y i n w h i c h c a s e 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 s o f t h e a t m o s p h e r i c r e f r a c t i v e i n d e x a n d w i n d v e l o c i t y a r e t h e same [24] e x c e p t f o r a c o n s t a n t o f p r o p o r t i o n a l i t y . H o w e v e r , 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 t h e v e l o c i t y d i s t r i b u t i o n i n a h o m o g e n e o u s t u r b u l e n t medium t e n d s t o b e g a u s s i a n [ 2 3 , p . 94] a n d s o f o r t h i s r e a s o n a s w e l l a s m a t h e m a t i c a l s i m p l i c i t y some a u t h o r s assume t h e a u t o c o r r e l a t i o n f o r a t m o s p h e r i c r e f r a c t i v e i n d e x f l u c t u a t i o n s t o be g a u s s i a n C y y ( r ) = e x p - ( r 2 / r p 1 6 4 APPENDIX A-3 A-3. MATHEMATICAL APPENDIX A-3-1 Given P E ( B ) -/2TT6 : exp 2e : < n < co - R - oo to c a l c u l a t e P Q 2 ( Y ) 0 - y - oo I f p{a - 8 - b} = | P Q(6)dB 0 < a < b then p{a - 6 2 - b} = p{/a" - 6 - /F> + p ( - / F - 9 - - / a } p{a - 6 2 - b> = | P e ( 8 ) d B + f P Q(B)dB -/a -/a" I make a change o f v a r i a b l e s : i n the f i r s t i n t e g r a l l e t B = y'* i n the second i n t e g r a l l e t B = -y2 V p{a - e 2 - b> = | P f l ( / Y ) d Y f P f l(-/Y) _e 2/Y i 6 2/Y dy p{a - 0 2 - b} = | B P B ( / Y ) d Y P 6(-/Y) a 2 / Y i 2 / Y 165 • P e 2 ( 3 ) d B = | 2/y d Y P e 2 ( 3 ) = P e ( / B ) «• P 0 ( - / 3 ) 2/F 0 - 8 - ~ P Q 2 ( 8 ) = exp /2TT8 26 2Q2 0 - $ - co 29 ; 166 A-3-2 G i v e n P e 2 ( 8 ) = e x p 0 - 8 - » / 2 T T 8 2 6 2 o 2 26 ; P ^ 2 ( 8 ) - e x p / 2 T T B 2 6 2 Q 2 8_1_ 2 6 T 0 - 8 - -a 2 = 6 2 + <J>2 t o c a l c u l a t e P a 2 ( ? ) Now a 2 " = 9 2 + cj>2 a n d i f t h e a t m o s p h e r i c t u r b u l e n c e 2* i s i s o t r o p i c 8 Z = <}> ~ J~ ' S u b s t i t u t i n g a n d r e s u l t s i n 8 2 = v i n P e 2 ( 3 ) 8 2 = n i n P < j ) 2 ( B ) P Q 2 ( v ) = / T T O 2 V exp p<j>2 ^  = —zr exp / i r a 2 n a Now P 0 . ( 0 = P 6 2 ( v ) 0 P + « ( n ) 167 w h e r e " ® " s y m b o l i z e d c o n v o l u t i o n . p««<« I P e 2 ( X ) P + 2 ( 5 - X)dX P 0 2 ( C ) = } P e a ( X ) P ^ i ( C - X)dX s i n c e P 0 2 ( v ) = 0 f o r v < 0 P^Cn) = 0 f o r n < 0 TTOT £ /xu-x) IT OT J e x p dX a dX t i e i n t e g r a l c a n b e e a s i l y shown t o e q u a l ir b y c o m p l e t i n g t h e s q u a r e 5 « . j dX A ( C - X ) A 2/4 - ( X - C / 2 ) 2 1 6 8 169 A-3-3 I f o la' o The L a p l a c e T r a n s f o r m o f p A £ ( 3 ) ^ s g i v e n b y "I-Q i e C s ) = | e " s B P A e ( 6 ) d 6 o o e x p -3 r i " s + — l a 2 1 o J O e x p • - 8 f 1 1 S + A - T o « s + l a ' O s + Jl a' o 170 A-3-4 I f x i s a r a n d o m v a r i a b l e , g a u s s i a n d i s t r i b u t e d w i t h p a r a m e t e r s 0 a n d 1 , t h e n x h a s p r o b a b i l i t y d e n s i t y f u n c t i o n P Y ( B ) = — e x p B2 T h e p r o b a b i l i t y t h a t - a - x - a i s g i v e n b y p ( - a - x - a) = 2 j P x ( B ) d B p ( - a - x - a) / I ? J a i — i e x p B2 1~ dB By d e f i n i t i o n e r f z -/iF | e x p [ - B 2 ] d B b y t h e c h a n g e o f v a r i a b l e s B' = B//2" i t i s c l e a r t h a t p ( - a - x - a) = e r f [ 2 _ ] / I 1 7 1 APPENDIX A-4 A-4 . GEOPHYSICAL AND ATMOSPHERIC PARAMETERS The m a g n i t u d e s a s s i g n e d t h e v a r i o u s g e o p h y s i c a l a n d a t m o s p h e r i c p a r a m e t e r s u s e d i n t h i s t h e s i s a r e s u b j e c t , i n some i n s t a n c e s , t o c o n s i d e r a b l e d e b a t e . The f o l l o w i n g d i s c u s s i o n i s p r e s e n t e d a s an a t t e m p t t o j u s t i f y t h e v a l u e s c h o s e n f o r compu-t a t i o n i n t h i s t h e s i s . 1) "D" - T h e d i s t a n c e D o v e r w h i c h t h e t e c t o n i c o b s e r v a t i o n s a r e t o be made was a r b i t r a r i l y s e t a t 5 km. D = 5 km . was c h o s e n i n t h e h o p e t h a t i t was l a r g e e n o u g h t o e n s u r e o b s e r -v a t i o n s o f r e g i o n a l , a s o p p o s e d t o l o c a l , t e c t o n i c m o v e m e n t s . D = 5 km. i s l a r g e e n o u g h i n m o s t c a s e s f o r o b s e r v i n g s t a t i o n s s t r a d d l i n g a f a u l t z o n e t o be s e t b a c k f r o m t h e a r e a o f a c t i v e f a u l t i n g a n d t h u s e a c h be s i t u a t e d o n r e l a t i v e l y l a r g e a n d s t a b l e g e o l o g i c p l a t f o r m s . 2) ' j h j j ' 1 - T h e c h o i c e o f |h^|= 1 c m / y e a r was made o n t h e b a s i s o f e v i d e n c e [8] t h a t t h i s f i g u r e i s t y p i c a l o f f a u l t m o t i o n on m a j o r f a u l t s y s t e m s . I t i s a l s o t y p i c a l o f p r o j e c t e d c o n t i n e n t a l d r i f t v e l o c i t i e s . 3) " s " - T h e c h o i c e o f a s i g n a l - t o - n o i s e r a t i o S = 2 i s e n t i r e l y a r b i t r a r y . 4) " A t " - The a p p r o p r i a t e m a g n i t u d e f o r A t c a n be d e s c r i b e d a s t h e s m a l l e s t t i m e i n t e r v a l p o s s i b l e ( t o e n s u r e r a p i d d a t a a c q u i s i t i o n ) o v e r w h i c h t h e a t m o s p h e r i c r a y - p a t h s c a n b e c o n s i d e r e d u n c o r r e l a t e d . An e s t i m a t e o f A t c a n be o b t a i n e d f r o m t h e f r e q u e n c y o f s c i n t i l l a t i o n o f l i g h t s o u r c e s o b s e r v e d through an u n c o n t r o l l e d atmosphere. The power s p e c t r a of such f l u c t u a t i o n s i n l i g h t i n t e n s i t y have been determined experimen-t a l l y under a v a r i e t y of atmospheric c o n d i t i o n s . These power s p e c t r a a l l show a broad peak e x t e n d i n g from 5 Hz - 250 Hz (25, p. 220), (30). S i n c e the temporal f l u c t u a t i o n s i n images viewed through a t u r b u l e n t atmosphere shown no a p p r e c i a b l e f o u r i e r components w i t h p e r i o d s l o n g e r than 2 x 10 x sec t h i s v a l u e was chosen f o r the magnitude o f A t . The magnitude of A t i s not v e r y c r i t i c a l i n d e t e r m i n i n g the o b s e r v a t i o n i n t e r v a l A T s i n c e &X A t 1 / 3 ... 5) Turbulence s c a l e s - A good d e a l o f c o n f u s i o n p e r -s i s t s i n the l i t e r a t u r e r e g a r d i n g " s c a l e s of t u r b u l e n c e " . To b e g i n w i t h , the " i n n e r (outer) s c a l e o f t u r b u l e n c e " i s u s u a l l y v a g u e l y d e f i n e d (31) as a measure of, but not n e c e s s a r i l y e q u a l t o , the dimensions o f the s m a l l e s t ( l a r g e s t ) inhomogenieties p r e s e n t i n the v e l o c i t y f i e l d . There i s a g e n e r a l l y accepted p r a c t i c e o f a s s o c i a t i n g the i n n e r and o u t e r s c a l e s o f t u r b u l e n c e w i t h the s o - c a l l e d " m i c r o - s c a l e " and " i n t e g r a l s c a l e " d e r i v e d m a t h e m a t i c a l l y from the 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 C ( r ) f o r the f i e l d . The micro s c a l e , A , i s the i n t e r c e p t on the r - a x i s o f a p a r a b o l a f i t t e d t o the a u t o c o r r e l a t i o n f u n c t i o n a t the o r i g i n Physically, the micro scale must e x i s t for reasons of continuity but many a n a l y t i c a l autocorrelation functions such as the ex-ponential do not have a "micro scale". The " i n t e g r a l scale", L Q , i s defined as the value of r which i s numerically equal to the i n t e g r a l of the auto-c o r r e l a t i o n function from zero to i n f i n i t y . CD L Q = J C(r) dr o P h y s i c a l l y , the i n t e g r a l scale i s expected to e x i s t since c o r r e l a t i o n functions approach zero for large values of the argument. The guassian autocorrelation function i s an a n a l y t i c a l autocorrelation function widely used i n t h e o r e t i c a l work because of i t s mathematical manageability C(r) = exp 2 r R2 The parameter R i s usually referred to as the c o r r e l a t i o n d i s -tance - the value of r at which the autocorrelation function has dropped to 1/e of i t s value at the o r i g i n . Unfortunately the gaussian autocorrelation function i s blessed with the curious mathematical property that i t s micro scale i s larger than i t s  i n t e g r a l scale and the association between the inner and outer scales of turbulence and the micro- and i n t e g r a l scales res-p e c t i v e l y cannot be made when gaussian autocorrelation functions are used. 174 6. " r " - S i n c e the s m a l l e s t s c a l e o f t u r b u l e n c e p r e s e n t o i n the atmosphere has been determined by radar s c a t t e r i n g and other methods to v a r y from s e v e r a l m i l l i m e t e r s (2 5) to s e v e r a l c e n t i m e t e r s (32) a v a l u e o f r Q = 1 cm was chosen f o r ease of computation. The magnitude o f r Q i s not a c r i t i c a l f a c t o r i n -2/3 d e t e r m i n i n g the o b s e r v a t i o n time A «C s i n c e A t * r Q 7. "Si "/ "R " - The magnitude of the parameter JL , o o 3 o the s t r a i g h t l i n e segments used to approximate the curved ray-path, i s d i f f i c u l t t o e s t i m a t e . The mathematical a n a l y s i s p r e s e n t e d i n t h i s t h e s i s i s v a l i d under the assumption r Q « -4 «. D S i n c e r v>~ 1 cm and D ^ 10 cm a m a t h e m a t i c a l l y c o n v e n i e n t o c h o i c e f o r i s 10 3 cm or 10 meters. However th e r e are o p h y s i c a l reasons f o r t h i s c h o i c e of *^ as w e l l . I t would o b v i o u s l y be p h y s i c a l l y r e a s o n a b l e t o equate X w i t h the o u t e r s c a l e o f t u r b u l e n c e , R . In most p h y s i c a l o o problems, e.g. f l u i d flow through a p i p e , the o u t e r s c a l e o f t u r b u l e n c e i s d e f i n e d by obvious p h y s i c a l c o n s t r a i n t s , e . g . the i n n e r diameter o f the p i p e . The atmosphere i s a t u r b u l e n t f l u i d c o n f i n e d by o n l y one w a l l , the s u r f a c e o f the e a r t h ; thus one expects the s i z e o f the l a r g e s t u n i t s o f t u r b u l e n c e t o depend on a l t i t u d e , h F r i e d and C l o u d (32) p r e s e n t a graph o f e x p e r i m e n t a l l y determined v a l u e s of R Q v s . h They c l a i m an e m p i r i c a l f i t t o the d a t a i s g i v e n by R = V b h h i s i n meters b = 4 meters Thus a s s i q n i n g = R = 1 0 meters i s c o n s i s t e n t 3 3 o o w i t h a l a s e r beam p r o p a g a t i n g p a r a l l e l to the ground a t an average h e i g h t o f 2 5 meters. T h i s i s p h y s i c a l l y reasonable f o r a f r e e a i r l a s e r s t r a i n meter making o b s e r v a t i o n s over a d i s t a n c e o f 5 km. The c a l c u l a t e d o b s e r v i n g i n t e r v a l A *C o n l y depends L 2/3 ° n o ' _ _ 2 8. " u- " - The mean square f l u c t u a t i o n s i n atmos-p h e r i c r e f r a c t i v e index as a f u n c t i o n o f h e i g h t h (meters) above sea l e v e l based on microwave r e f r a c t o m e t e r o b s e r v a t i o n s i s g i v e n e m p i r i c a l l y by F r i e d and Cloud (32) as 2 -12 | i = 1.2 x 10 exp h_ h h = 1600 m o Because o f the e f f e c t s o f water vapour, p r e s e n t a t microwave f r e q u e n c i e s , but g r e a t l y reduced a t o p t i c a l f r e q u e n c i e s , F r i e d and C l o u d p r e f e r an a l t e r n a t i v e formula 2 r _ , f t-14 p. = 6.7 x 10 exp h_ h h = 3200 m o which, they c l a i m , agrees more c l o s e l y w i t h v a l u e s f o r JJ, d e r i v e d a s t r o n o m i c a l l y . 2 Beckmann (22) chooses l a r g e r v a l u e s f o r y and —12 2 —10 uses 10 < p. < 10 . Beckmann p o i n t s out, however, 2* -10 t h a t u- = 10 i s an "improbably h i g h v a l u e " and 2* -12 c o n s i d e r s p = 10 to be "a more l i k e l y v a l u e " . C o n s o r t i n i e t a l (44) have r e c e n t l y p u b l i s h e d d a t a d e r i v e d from f l u c t u a t i o n s i n l a s e r beams t r a n s m i t t e d over a h o r i z o n t a l path which r e v e a l measured mean square r e f r a c t i v e -14 -13 index f l u c t u a t i o n s between 7 x 10 and 2 x 10 Thus f o r purposes o f t h i s t h e s i s 10 < u. < 10 ~~2 -14 Weak atmospheric t u r b u l e n c e corresponds t o u. = 10 and moderat e - t o - i n t e n s e atmospheric t u r b u l e n c e corresponds t o ~~2 -12 177 LIST OF SYMBOLS P p a r t i a l p r e s s u r e o f d r y a i r [ra.m. Hg] T atmospheric temperature [°K] p p a r t i a l p r e s s u r e o f water vapour [m.m. Hg] 2 2 2 1 / 2 x,y,z c a r t e s i a n s p a t i a l c o o r d i n a t e s , r = ( x + y + z ) A A A i , j , k u n i t v e c t o r s a l o n g axes ox , oy , oz t a b s o l u t e time T r e l a t i v e time measured from the b e g i n n i n g o f the f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s D d i s t a n c e over which s t r a i n measurements are to be made through the u n c o n t r o l l e d atmosphere h^ l i n e a r approximation t o the r a t e o f change of d i s t a n c e between the ends o f the s t r a i n meter i n the k^h av e r a g i n g i n t e r v a l A t sampling i n t e r v a l o f the f l u c t u a t i o n s i n atmospheric ray-path l e n g t h , e q u a l t o the c o r r e l a t i o n time f o r l i g h t rays p a s s i n g through the u n c o n t r o l l e d atmosphere A T a v e r a g i n g i n t e r v a l f o r s u p p r e s s i o n o f atmospheric f l u c t u a t i o n s i n ray-path t o r e v e a l t e c t o n i c changes i n d i s t a n c e s s i g n a l n o i s e r a t i o o f the f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s | 5*| c o n f i d e n c e parameter o f f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s , 0 s s. °° 7] (I e'l ) c o n f i d e n c e l e v e l o f f r e e a i r l a s e r s t r a i n meter o b s e r v a t i o n s , 7)(|t|) = e r f ( / / 2 ) , 0 ^ TK|§'|) <; 1 1^ 2 mean square r e f r a c t i v e index f l u c t u a t i o n r i n n e r s c a l e o f atmospheric t u r b u l e n c e o R Q o u t e r s c a l e o f atmospheric t u r b u l e n c e S ( t ) a r c l e n g t h o f atmospheric ray-path a t time t 178 SL s t r a i g h t l i n e s e g m e n t , a s e r i e s o f w h i c h a r e u s e d t o a p p r o x i m a t e t h e a t m o s p h e r i c r a y - p a t h . SL i s e q u a l i n m a g n i t u d e t o t h e o u t e r s c a l e o f t u r b u l e n c e U ( A t ) t h e n e t f l u c t u a t i o n i n g e o m e t r i c r a y - p a t h l e n g t h i n t h e i n t e r v a l A t X ( T ) t h e f l u c t u a t i o n i n g e o m e t r i c r a y - p a t h l e n g t h a s a f u n c t i o n o f t i m e x I c u r v e p a r a m e t e r f o r m a t h e m a t i c a l r e p r e s e n t a t i o n o f a t m o s p h e r i c r a y - p a t h . I i s p r o p o r t i o n a l t o d i s t a n c e t r a v e l l e d p a r a l l e l t o t h e x - a x i s . e ( t ) t h e e x t r a d i s t a n c e , o v e r t h e s t r a i g h t l i n e d i s t a n c e , t r a v e l l e d b y a p h o t o n when t r a v e r s i n g t h e r a y - p a t h b e t w e e n t h e e n d m i r r o r s o f a f r e e a i r l a s e r s t r a i n m e t e r , "\ 6S f l u c t u a t i o n i n g e o m e t r i c a l r a y - p a t h l e n g t h a ( J l ) t o t a l a n g l e b e t w e e n t h e d i r e c t i o n o f t h e r a y a n d t h e x - a x i s X w a v e l e n g t h i n v a c c u o o f l a s e r r a d i a t i o n k = wavenumber o f l a s e r r a d i a t i o n v f r e q u e n c y o f l a s e r r a d i a t i o n a) = 2TTV a n g u l a r f r e q u e n c y o f l a s e r r a d i a t i o n £,n c o o r d i n a t e s i n t h e p l a n e o f t h e r e c t a n g u l a r d i f f r a c t i n g a p e r t u r e s x , y c o o r d i n a t e s i n t h e p l a n e o f t h e f r i n g e o b s e r v a t i o n V f r i n g e v i s i b i l i t y 8 t o t a l p h a s e o f l a s e r r a d i a t i o n p s p a t i a l v a r i a b l e i n a p l a n e n o r m a l t o t h e d i r e c t i o n o f p r o p a g a t i o n o f t h e l a s e r beam D g g ( p ) p h a s e s t r u c t u r e f u n c t i o n C ( r ) 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 r e f r a c t i v e i n d e x f l u c t u a t i o n s y y v 1 V minimum a c c e p t a b l e f r i n g e v i s i b i l i t y o p Q maximum a l l o w a b l e d i m e n s i o n s f o r d i f f r a c t i n g a p e r t u r e s 179 v = v i + v j + v k w i n d v e l o c i t y h P l a n c k ' s c o n s t a n t c v e l o c i t y o f l i g h t 0 ( A ) a n g l e o f d i v e r g e n c e o f l a s e r beam D Q 1% p o w e r a t t e n u a t i o n d i s t a n c e f o r l a s e r beam p r o p a -g a t i n g t h r o u g h t h e a t m o s p h e r e . D *v» 3.1 x 10* cm. 180 BIBLIOGRAPHY WELLS, H. G.; A S h o r t H i s t o r y o f t h e W o r l d . T he M a c M i l l a n Co. , New Y o r k , 1 9 2 2 . JACOBS, J . A., R U S S E L L , R. D., WILSON, J . T.; P h y s i c s a n d G e o l o g y , McGraw H i l l I n c . , T o r o n t o , 1 9 5 9 . GOLD, T.; R a d i o M e t h o d f o r t h e P r e c i s e M e a s u r e m e n t o f t h e R o t a t i o n P e r i o d o f t h e E a r t h . S c i e n c e , v . 1 5 7 , 302-304, J u l y 2 1 , 1 9 6 7 . BERGSTRAND, E . ; T h e G e o d i m e t e r S y s t e m - A S h o r t D i s c u s s i o n o f i t s P r i n c i p a l F u n c t i o n a n d F u t u r e D e v e l o p m e n t . J . G e o p h y s . R e s . , v . 6 5 , No. 2, 4 0 4 - 4 0 9 , F e b r u a r y 1 9 6 0 . OWENS, J . C . ; A R e v i e w o f A t m o s p h e r i c D i s p e r s i o n M e a s u r e m e n t s i n G e o d e s y a n d M e t e o r o l o g y . P a p e r p r e s e n t e d a t t h e C o n f e r e n c e o n " R e f r a c t i o n E f f e c t s i n G e o d e s y a n d E l e c t r o n i c D i s t a n c e M e a s u r e m e n t . " U n i v e r s i t y o f New S o u t h W a l e s , S y d n e y , A u s t r a l i a , November 5-8, 1 9 6 8 . BENDER, P. L . a n d OWENS, J . C ; C o r r e c t i o n o f O p t i c a l D i s t a n c e M e a s u r e m e n t s f o r t h e F l u c t u a t i n g A t m o s p h e r i c I n d e x o f R e f r a c t i o n . J . G e o p h y s . R e s . , v . 7 0 , No. 1 0 , 2 4 6 1 - 2 4 6 2 , May 1 9 6 5 . THOMPSON, M. C ; S p a c e A v e r a g e s o f A i r a n d W a t e r V a p o u r D e n s i t i e s b y D i s p e r s i o n f o r R e f r a c t i v e C o r r e c t i o n o f E l e c t r o - M a g n e t i c Range M e a s u r e m e n t s . J . G e o p h y s . R e s . , v . 7 3 , No. 10, 3 0 9 7 - 3 1 0 2 , May 1 9 6 8 . HOFMANN, R. B.; G e o d i m e t e r F a u l t Movement I n v e s t i g a t i o n s i n C a l i f o r n i a . P u b l i c a t i o n o f t h e D e p a r t m e n t o f W a t e r R e s o u r c e s , S t a t e o f C a l i f o r n i a , U.S.A., B u l l e t i n No. 1 1 6 - 6 . FOWLER, R. A.; E a r t h q u a k e P r e d i c t i o n f r o m L a s e r S u r v e y i n g . N.A.S.A. R e p o r t S P - 5 0 4 2 , S c i e n t i f i c a n d T e c h n i c a l I n f o r m a t i o n D i v i s i o n , O f f i c e o f T e c h n o l o g y U t i l i z a -t i o n , N a t i o n a l A e r o n a u t i c a l a n d S p a c e A d m i n i s t r a t i o n , W a s h i n g t o n , D . C , 1 9 6 8 . BLOOM, A. L . ; Gas L a s e r s . J o h n W i l e y a n d S o n s , I n c . , New Y o r k , 1 9 6 8 . SCHAWLOW, A. L . a n d TOWNES, C. H.; I n f r a r e d a n d O p t i c a l M a s e r s . P h y s . R e v . , v . 1 1 2 , No. 6, 1 9 4 0 - 1 9 4 9 , D e c e m b e r 1 5 , 1 9 5 8 . 181 12. BIRNBAUM, G.; F r e q u e n c y S t a b i l i z a t i o n o f Gas L a s e r s . P r o c . I . E . E . E . , v . 5 5 , No. 6, 1 0 1 5 - 1 0 2 5 , J u n e 1967. 13. WHITE, A. D,; F r e q u e n c y S t a b i l i z a t i o n o f Gas L a s e r s . I . E . E . E . , J . Quantum E l e c t r o n i c s , v . QE-1, No. 8, 3 4 9 - 3 5 7 , November 1965. 14. V A L I , V.; M e a s u r i n g E a r t h S t r a i n s b y L a s e r . S c i e n t i f i c A m e r i c a n , v . 2 2 1 , No. 6, 8 9 - 9 5 , D e c e m b e r 196 9 . 15. V A L I , V.; S t r a i n s R e c o r d e d o n a H i g h M a g n i f i c a t i o n I n t e r -f e r o m e t r i c S e i s m o g r a p h . N a t u r e , v . 220, 1 0 1 8 - 1 0 2 0 , D e c e m b e r 7, 1 9 6 8 . 16. V A L I , V., KROGSTAD, R. S., MOSS, R. W.; O b s e r v a t i o n o f E a r t h T i d e s U s i n g a L a s e r I n t e r f e r o m e t e r . J . A p p l i e d P h y s . , v . 3 7 , No. 2, 5 8 0 - 5 8 2 , F e b r u a r y 1 9 6 6 . 17. PRESS, F . ; D i s p l a c e m e n t s , S t r a i n s , a n d T i l t s a t T e l e -s e i s m i c D i s t a n c e s . J . G e o p h y s . R e s . , v . 7 0 , No. 10, May 1 5 , 1965. 18. H 0 N K A S L 0 , M T . ; M e a s u r e m e n t o f S t a n d a r d B a s e l i n e s w i t h t h e V a i s a l a L i g h t I n t e r f e r e n c e C o m p a r a t o r . J . G e o p h y s . R e s . , v . 6 5 , No. 2, 4 5 7 - 4 6 0 , F e b r u a r y 1 9 6 0 . 19. ERICKSON, K. E . ; L o n g - P a t h I n t e r f e r o m e t r y t h r o u g h a n U n c o n t r o l l e d A t m o s p h e r e . J . O p t . S o c . A m e r i c a , v . 5 2 , No. 7, 7 8 1 - 7 8 8 , J u l y 1 9 6 2 . 20. OWENS, J . C.; O p t i c a l R e f r a c t i v e I n d e x o f A i r : D e p e n d e n c e o n P r e s s u r e , T e m p e r a t u r e , a n d C o m p o s i t i o n . A p p l i e d O p t i c s , v . 6, No. 1, 5 5 - 5 8 , J a n u a r y 1 9 6 7 . 21. T I L T O N , L . W.; S t a n d a r d C o n d i t i o n s f o r P r e c i s e P r i s m R e f r a c t o m e t r y . J . R e s . N a t . B u r e a u o f S t a n d a r d s , v . 14, 3 9 3 - 4 1 8 , A p r i l 1 9 3 5 . 22. BECKMANN, P.; S i g n a l D e g e n e r a t i o n i n L a s e r Beams P r o p a g a t e d t h r o u g h a T u r b u l e n t A t m o s p h e r e . J . R e s . N a t . B u r e a u o f S t a n d a r d s , v . 69-A, No. 4, 6 2 9 - 6 4 0 , A p r i l 1 9 6 5 . 23. LUMLEY, J . L . a n d PANOFSKY, H. A.; The S t r u c t u r e o f A t m o s -p h e r i c T u r b u l e n c e . J o h n W i l e y § S o n s , I n c . , New Y o r k , 1964. 24. STROHBEHN, J . W.; The F e a s i b i l i t y o f L a s e r E x p e r i m e n t s f o r M e a s u r i n g A t m o s p h e r i c T u r b u l e n c e P a r a m e t e r . J . G e o p h y s . R e s . , v . 7 1 , No. 24, 5 7 9 3 - 5 8 0 8 , D e c e m b e r 1 5 , 1 9 6 6 . 182 25. T A T A R S K I , V. I . ; Wave P r o p a g a t i o n i n a T u r b u l e n t M edium. T r a n s l a t e d b y R. A. S i l v e r m a n , D o v e r B o o k s I n c . , New Y o r k , 1 9 6 1 . 26. CHERNOV, L . A.; Wave P r o p a g a t i o n i n a Random Medium. T r a n s l a t e d b y R. A. S i l v e r m a n , D o v e r B o o k s I n c . , New Y o r k , 1960. 27. CHANDRASEKHAR, S.; S t o c h a s t i c P r o b l e m s i n P h y s i c s a n d A s t r o n o m y . R e v i e w s o f M o d e r n P h y s i c s , v . 1 5 , No. 1, 1-89, J a n u a r y 1 9 4 3 . 28. H a n d b o o k o f M a t h e m a t i c a l F u n c t i o n s . M. A b r a m o w i t z a n d I . S t e g u n E d . , D o v e r P u b l i c a t i o n s I n c . , New Y o r k , 1 9 6 5 . 29. GRADSHTEYN, I . S. a n d RYZHIK, I . M.; T a b l e s o f I n t e g r a l s S e r i e s a n d P r o d u c t s . T r a n s l a t e d b y A. J e f f r e y , A c a d e m i c P r e s s , New Y o r k , 1965. 30. D E I T Z , P. H.; O p t i c a l M e t h o d f o r A n a l y s i s o f A t m o s p h e r i c E f f e c t s o n L a s e r Beams. P r o c . Symposium o n M o d e r n O p t i c s , P o l y t e c h n i c I n s t i t u t e o f B r o o k l y n , 7 5 7 - 7 7 4 , M a r c h 1967. 31. ROUSE, H.; A d v a n c e d M e c h a n i c s o f F l u i d s . P. 2 8 2 , J o h n W i l e y a n d S o n s I n c . , 1 9 5 9 . 32. F R I E D , D. L . a n d CLOUD, J . D.; A t m o s p h e r i c T u r b u l e n c e a n d I t s E f f e c t o n L a s e r C o m m u n i c a t i o n S y s t e m s . 2nd R e p o r t , E l e c t r o - o p t i c a l L a b o r a t o r y T e c h n i c a l Memo-ra n d u m #91, N o r t h A m e r i c a n A v i a t i o n I n c . , S p a c e a n d I n f o r m a t i o n S y s t e m s D i v i s i o n , J u n e 1 5 , 1 9 6 4 . 33. WANG, M. C., a n d UHLENBECK, G. E . ; On t h e T h e o r y o f t h e B r o w n i a n M o t i o n . R e v . M o d e r n P h y s . , v . 1 7 , N o s . 2 a n d 3, 3 2 3 - 3 4 2 , A p r i l - J u l y 1 9 4 5 . 34. LONGHURST, R. S.; G e o m e t r i c a l a n d P h y s i c a l O p t i c s . 2nd E d . , p . 104, L o n g m a n s , L o n d o n I n c . , 1 9 6 7 . 35. BORN, M. a n d WOLFE, E . ; P r i n c i p l e s o f O p t i c s . C h a p t . 10, 3 r d R ev. e d . , P e r g a m o n P r e s s , New Y o r k , 1 9 6 5 . 36. DE B R OGLIE, L . ; New P e r s p e c t i v e s i n P h y s i c s . C h a p t . 1, B a s i c B o o k s I n c . , New Y o r k , 1 9 6 2 . 37. JONES, R. C ; I n f o r m a t i o n C a p a c i t y o f R a d i a t i o n D e t e c t o r s . J . Op. S o c . A m e r i c a , v . 52, No. 1 1 , 1 1 9 3 - 1 2 0 0 , November 1962 . 183 38. ASHFORD, R. D.; p e r s o n a l communication. 39. VALI, V.; Some E a r t h S t r a i n Measurements w i t h Laser I n -ferometer. Earthquake Displacement F i e l d s and the R o t a t i o n of the E a r t h , 2 06-216, D. R e i d e l Pub. Co., H o l l a n d , 1970. 40. CLOUD, J . D. and FRIED, D. L.; The S i g n i f i c a n c e of the Inner S c a l e o f Turbulence t o the Phase S t r u c t u r e F u n c t i o n . E l e c t r o - o p t i c a l L a b o r a t o r y T e c h n i c a l Memorandum #120, North American A v i a t i o n , Inc., Space and I n f o r m a t i o n Systems D i v i s i o n , J u l y 1964. 41. SMYLIE, D. E.; p e r s o n a l communication. 42. GANGI, A. F.; E r r o r A n a l y s i s o f a Laser S t r a i n Meter. Earthquake Displacement F i e l d s and the R o t a t i o n o f the E a r t h , 217-229, D. R i e d e l Pub. Co., H o l l a n d , 1970. 43. BENDER, P. L.; Laser Measurements o f Long D i s t a n c e s . Proc. J.E.E.E., v. 55, No. 6, 1039-1045, June 1967. 44. CONSORTINI, A., RONCHI, L., STEFANUTTI, L.; I n v e s t i g a t i o n o f Atmospheric Turbulence by Narrow Laser Beams. A p p l i e d O p t i c s , v. 9, No. 11, 2543 - 2547, November 1970. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0053452/manifest

Comment

Related Items