Faculty of Science and Engineering Department of Imaging and Applied Physics Non-Gaussian and non-homogeneous Poisson models of snapping shrimp noise Matthew W. Legg This thesis is presented for the Degree of Doctor of Philosophy of Curtin University of Technology February 2010 Declaration To the best of my knowledge and belief this thesis contains no material previously published by any other person except where due acknowledgment has been made. This thesis contains no material which has been accepted for the award of any other degree or diploma in any university. ||||||||||{ ||||||||||{ Signature Date Abstract The problem of sonar detection and underwater communication in the presence of impulsive snapping shrimp noise is considered. Non-Gaussian amplitude and non- homogeneous Poisson temporal statistical models of shrimp noise are investigated from the perspective of a single hydrophone immersed in shallow waters. New statistical models of the noise are devised and used to both challenge the superiority of existing models, and to provide alternative insights into the underlying physical processes. A heuristic amplitude statistical model of snapping shrimp noise is derived from first principles and compared with the Symmetric-α-stable model. The models are shown to have similar variability through the body of the amplitude probability density functions of real shrimp noise, however the new model is shown to have a superior fit to the extreme tails. Narrow-band detection using locally optimum detectors derived from these models show that the Symmetric-α-stable detector retains it's superiority, despite providing a poorer overall fit to the amplitude probability density functions. The results also confirm the superiority of the Symmetric-α-stable detector for detection of narrow- band signals in shrimp noise from Australian waters. The temporal nature of snapping from a field of shrimp is investigated by considering the snapping as a point process in time. Point process analysis techniques are drawn from the fields of optics, neuro-physics, molecular biology, finance and computer sci- ence, and applied to the problem of snapping shrimp noise. It is concluded that the snapping is not consistent with a homogeneous Poisson process and that correlations iii exist in the point process on three different time scales. The cause of short time correla- tions is identified as surface reflected replicas, and models of medium time correlations are investigated. It is shown that a Cox-Ingersoll-Ross driven doubly-stochastic Poisson model is able to describe the medium time correlations observed from the counting pro- cess, but a kth-order interval analysis reveals that there is more information contained within the snapping than can be described by the model. Analysis of shrimp snap times over a full day provides evidence of correlation between snap events on long time scales. Simulation of ocean noise is conducted to illustrate the use of such temporal models, and implications for their use in detection algorithms are discussed. Acknowledgements I would first and foremost like to thank my wife Tracy and my children Sarah and Jacob for all of the evenings and weekends that they allowed me to use for this study. Special thanks go to my official supervisors Dr Alec Duncan, Prof. Tony Zaknich and Dr Mike Greening for their guidance and support, and for simultaneously allowing me the time to explore in shady corners and for not allowing me to be lost forever in complete darkness. Special thanks also go to my unofficial supervisors Dr Derek Bertilone and Dr Dave Matthews, who bore the brunt of my immediate questions and continually encouraged me to get the thesis completed. Thanks go to Dr Dave Matthews, Dr Rod MacLeod, Damien Killeen, Paul Moses and Mark Savage for their assistance with field work, and to Dr Doug Cato, Dr Rob McCauley, Dr Brian Ferguson and Jane Cleary for providing (respectively) data from the Spencer Gulf, Feather Reef and Sydney Harbour. Special thanks go to Dr Dave Matthews for providing Seal Island, Seal Cave, Exmouth Jetty and Pyrmont data. I would also like to thank the many people who helped with reviewing, proofreading and grammatical editing of this thesis. Special thanks go to Dr Derek Bertilone for an in-depth review of the mathematical derivation in this thesis. Also to my sister Caz for her continued tutoring and guidance in grammar and writing style. I would like to thank the examiners for the time and effort they have expended to ensure this thesis is of a high standard. v Support and resources were provided by the Department of Imaging and Applied Physics and the Centre for Marine Science and Technology (CMST) at Curtin Uni- versity, and the Defence Science and Technology Organisation (DSTO). I extend my gratitude to the people who have helped with administration within these two organi- sations. Contents 1 Introduction1 1.1 Thesis aims..................................2 1.2 Research methodology............................3 1.3 Supporting field work............................4 1.4 Thesis format and outline..........................6 2 Snapping shrimp noise8 2.1 Snapping shrimp...............................9 2.2 Acoustic characteristics of individual shrimp snaps............ 10 2.3 Snapping shrimp and ambient noise.................... 11 2.4 Shrimp noise analyses and applications................... 18 2.5 Summary of snapping shrimp noise..................... 21 CONTENTS vii 3 Amplitude models 23 3.1 Models of snapping shrimp noise...................... 24 3.1.1 The Gaussian distribution...................... 24 3.1.2 The Gaussian-Gaussian mixture distribution........... 26 3.1.3 The α-stable distribution...................... 29 3.2 A heuristic model of snapping shrimp noise................ 34 3.2.1 Derivation of the density function................. 36 3.2.2 Derivation of the distribution function............... 45 3.3 Ambient noise: a Gaussian-Garnele mixture model............ 46 3.3.1 Parameter estimation for the Gaussian-Garnele distribution... 47 3.4 Comparison of SαS and Gaussian-Garnele models........................ 50 3.5 Frequency dependence of the pressure amplitude pdf........... 61 3.6 Experimental investigation of pdf asymmetry in real shrimp noise.... 65 3.7 Summary of amplitude models....................... 72 4 Temporal models 74 4.1 Random point processes........................... 75 CONTENTS viii 4.2 Homogeneous Poisson model........................ 77 4.3 Event detection................................ 78 4.3.1 Pre-detection filtering........................ 79 4.3.2 Threshold detection with dead-time................ 81 4.3.3 Consequences of threshold detection................ 83 4.4 Inter-snap interval histogram analysis................... 83 4.4.1 The uniform conditional test.................... 87 4.5 Kth order interval analysis.......................... 89 4.6 Intensity function analysis.......................... 99 4.7 Fano-factor analysis............................. 106 4.8 The cause of short time effects....................... 124 4.9 Modelling medium and long time effects.................. 125 4.9.1 Doubly-stochastic Poisson models................. 126 4.9.2 The Ornstein-Uhlenbeck model................... 128 4.9.3 The Cox-Ingersoll-Ross model.................... 132 4.9.4 Asymptote for infinite counting time................ 140 4.9.5 A 24 hour time-series........................ 140 CONTENTS ix 4.10 Summary of temporal models........................ 144 5 Application to simulation and detection 148 5.1 Simulated ocean ambient noise....................... 148 5.2 Detection in snapping shrimp noise..................... 156 5.3 Summary of applications........................... 160 6 Discussion 161 6.1 Discussion of amplitude probability density function models....... 162 6.2 Emerging knowledge of the temporal nature of snapping shrimp noise. 164 6.3 Implications for sonar detection and communications........... 167 7 Conclusions 169 7.1 Thesis conclusions.............................. 169 7.2 Recommendations for further work..................... 171 A Field Measurements 173 A.1 AWharf (AW)................................. 173 A.2 Busselton Jetty (BUWO).......................... 176 A.3 Cockburn Sound measurements....................... 178 CONTENTS x A.3.1 10 March 2008 (CS-A,CS-D).................... 178 A.3.2 1 December 2005 (CS-B)...................... 181 A.3.3 25 November 2005 (CS-C)...................... 184 A.4 Nornalup-Walpole Estuary (WP)...................... 187 A.5 24 hour recording at the AWharf (AW24)................. 189 B Field data sets 193 B.1 Seal Island.................................. 193 B.1.1 Seal Island Cave (SEAL-A and SEAL-B)............. 193 B.1.2 Seal Island (SEAL-C)........................ 196 B.2 Feather Reef (FR).............................. 196 B.3 Spencer Gulf (SG).............................. 199 B.4 Sydney Harbour (SYD-A and SYD-B)................... 201 C Logarithmic smoothing algorithm 203 List of Figures 2.1 Top and side views of a snapping shrimp.................. 13 2.2 Sketches of shrimp claw closure....................... 14 2.3 Detail of an individual shrimp snap..................... 15 2.4 Shallow water ambient noise spectrum................... 16 2.5 Shrimp noise spectra at various locations................. 17 2.6 An empirical probability density function of real shrimp noise compared with a Gaussian fit.............................. 18 3.1 Probability density function of real shrimp noise and a Gaussian fit... 27 3.2 Probability density function