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The Hilbert-Hankel Transform and its Application to Shallow Water Ocean Acoustics PDF

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TK7 855 .141 .R43 .1K eRWR~~Sor,eo, -Q .I S. 5SlRr 51 .. -7 Lir 1/CI 3MAR 16 i988 (LeVAR,.S The Hilbert-Hankel Transform and its Application to Shallow Water Ocean Acoustics RLE Technical Report No. 513 January 1986 Michael S. Wengrovitz Research Laboratory of Electronics Massachusetts Institute of Technology Cambridge, MA 02139 USA This work has been supported in part by the Advanced Research Projects Agency monitored by ONR under Contract No. N00014-81 -K-0742 and in part by the National Science Foundation under Grant ECS84-07285. Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Research Laboratory of Electronics Room 36-615 Cambridge, MA 02139 The Hilbert-Hankel Transform and its Application to Shallow Water Ocean Acoustics Michael S. Wengrovitz Technical Report No. 513 - January 1986 This work has been supported in part by the Advanced Research Projects Agency monitored by ONR under Contract No. N00014-81- K-0742 and in part by the National Science Foundation under Grant ECS-8407285. UNCLASSIFED :URIYT CLASS3IPCATION OF TNIS P401 I I REPORT DOCUMENTATION PAGE I N,- iPOTSI NSCUIITY CLASIFICATION .~~~~~l ~~~~~l~s~P~T~&hTI1c- --VV A.....q.....K. ING --- I....... 2& SaCUJRITY CAUICATIO.AUTIAITYV 3. OISTRISUTTONAVAI.A^ILJTY Ol ARPRT Approved for public release; distribution D.o CLo.IPICTINhiOOWNGRAOIN4 SCISOUL unlimited & PERFORMING ORGANIZATION RPORwT PUMISER(S S. MONITORING OGANiZATIN REP<RT NUMOIIR(S & NAMO OP PIIFORMING ORGANIIZATION OIFICS SYMOL 7& NAME OP MONITORING ORGINIZATION Research Laboratory of Elec arodnemw Office of Naval Research .Vassachusetts Institute of Te hnology Mathematical and Information Scien. Div. w. AOOE.U (Ctiy. SA ai ZZP Codes . AOORISa (CUy. $Mw - ZZP Cde 77 Massachusetts Avenue 800 North Quincy Street [ Cambridge. MA 02139 Arlington, Virginia 22217 h MEU OP PUNOING/ISPNSOIqNG l OFIC31 SYMOL. .PROCUREMNT INSTlIUMItNT IONTIPICATION AMII.ISR ORGANIZ10? ((LIt dsbM Advanced Research Projects gency N00014-81 - K-0742 II AOOREs 1Cty. SaM ido ZIP Cde) 1. SOtiNC OP PUNOiNG 460 1400 Wilson Boulevard PROGRAM PiROJcr" ?ASK WORK UNIT .rlington, Virginia 22217 ELEMENT O. NO. NO. No. ,, tt.r,.i.ie~ w .mfgcz~mm#aiThHei lbe_r t-Ha~n~k~~e~ ~~~~~049-506 049-506 ransform a it Applic.tinn tn... 2. PESONAL AUTMOR(S) I SCAA ?" Michael S. Wengrovitz TrY Pt OF IPOnRrT a 131 ?oIME OVERSTO 14 . OAT1 OP RIPORT (P.. .. D.s 15. PAGi CUNT Technical PROM *_o January 1986 482 I L. SUPPILMNTAY NOTATION l ? .CO A COOlS I& SUGACT IAS 1CMa ne M *4M Wd Mf4t Mi NMn "E4LO41 GROUP SUE R. . q 11. AmTnRACT tConadm on ffr if rm end imb by MoeS Nluberl In the shallow water acoustics problem, a time-harmonic source is placed in the ocean and a hydrophone records the acoustic pressure field as a function of range from the source. In this thesis, new techniques related to the synthetic generation, acquisition, and inversion of this data are developed. |I A hybrid method for accurate shallow water synthetic data generation is presented. The method is based on computing the continuum portion of the field using the Hankel transform and computing the trapped portion analytically. In the related problem of extracting the reflection coefficient, it is shown that the inversion can be highly sensitive to errors in the Green's function estimate. This sensitivity can be I eliminated by positioning the source and receiver above the invariant critical depth (cont.) 20. OISTRIUTIONIAVAILAILIJTY OP ASTRACT 21. ATRACT CURITY CASSICAT NjCLAWIP1EO1/ULM61lMT10 SAAG AS NP?. =o-r1c usP CI Unclassified 22Al NAyrMa l MOP .-IR HGSPaOlNl SIS LS INOIVIOUA. 22. TILEP"ONE UMIER 22c OFPICI SYMIAOL df nbd eAI4 Code) Ky Rnorvc (617) 253-2569 SqCUJri CL.A.MPICATIO oP m"i PAG.. 19. Abstract continued of the waveguide. The theory of a new transform, referred to as the Hilbert-Hankel transform, is developed. Its consistency with the Hankel transform leads to an approximate real- part/imaginary-part sufficiency condition for acoustic fields. An efficient reconstruc- tion method for obtaining the complex-valued acoustic field from a single quadrature component is developed and applied to synthetic and experimental data. The Hilbert- Hankel transform is a unilateral version of the Hankel transform and its application to this problem is based on the outgoing nature of the acoustic field. The theory of this transform and its one-dimensional counterpart can be applied to a wide class of problems. The Hilbert-Hankel Transform and its Application to Shallow Water Ocean Acoustics by Michael S. Wengrovitz Submitted in partial fulfillment of the requirements for the degree of Doctor of Science at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution. January 30, 1986 Abstract In the shallow water acoustics problem, a time-harmonic source is placed in the ocean and a hydrophone records the acoustic pressure field as a function of range from the source. In this thesis, new techniques related to the synthetic generation, acquisition, and inversion of this data are developed. A hybrid method for accurate shallow water synthetic data generation is presented. The method is based on computing the continuum portion of the field using the Hankel transform and computing the trapped portion analytically. In the related problem of extracting the reflection coefficient, it is shown that the inversion can be highly sensitive to errors in the Green's function estimate. This sensitivity can be eliminated by positioning the source and receiver above the invariant critical depth of the waveguide. The theory of a new transform, referred to as the Hilbert-Hankel transform, is developed. Its consistency with the Hankel transform leads to an approximate real- part/imaginary-part sufficiency condition for acoustic fields. An efficient reconstruc- tion method for obtaining the complex-valued acoustic field from a single quadrature component is developed and applied to synthetic and experimental data. The Hilbert- Hankel transform is a unilateral version of the Hankel transform and its application to this problem is based on the outgoing nature of the acoustic field. The theory of this transform and its one-dimensional counterpart can be applied to a wide class of problems. Thesis Supervisors: Alan V. Oppenheim, Professor of Electrical Engineering, Massachusetts Institute of Technology. George V. Frisk, Associate Scientist, Woods Hole Oceanographic Institution. 1 Acknowledgements I wish to thank my thesis supervisors, Professor Alan Oppenheim and Dr. George Frisk, for their guidance, encouragement and support of this work. Their insights, intuition, and uncompromising standards have contributed greatly to my intellectual and personal growth. They have truly been super supervisors. I am also grateful to Professor Arthur Baggeroer for serving as a thesis reader and to Dr. Robert Spindel for serving as the chairman of my thesis defense. I thank all the members of the MIT Digital Signal Processing Group and the WHOI Department of Ocean Engineering for many interesting technical discussions and for making this research so enjoyable. In particular, discussions with Evangelos Milios, Meir Feder, Webster Dove, David Izraelevitz, Thrasyvoulos Pappas, Avideh Zakhor, and Patrick Van Hove at MIT and Jim Miller, Subramanian Rajan, Jim Lynch, Jim Doutt, Chris Dunn, and Arthur Newhall at WHOI have been useful. I am particularly grateful to Doug Mook, now at Sanders Associates, for stimulating technical discussions in the early stages of this work. I would also like to thank Andy Kurkjian at Schlumberger-Doll Research, and Dave Stickler at the Courant Institute for their useful comments and advice over the years. I also thank Giovanni Aliberti for making the computer cooperate and Becky Johnson for her help in preparing the figures in this text. I gratefully acknowledge the financial support of the Fannie and John Hertz Foun- dation throughout my stay at MIT. In addition, I thank the Woods Hole Oceano- graphic Institution for their support during my summers at Woods Hole. A special thanks goes to my wife, Debbie. Her constant understanding, advice, encouragement, and above all, patience greatly contributed to this work. To Steven, my son, also goes a special thanks for your patience with me. 2

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Jan 30, 1986 Hankel transform and computing the trapped portion analytically. In the 22c OFPICI SYMIAOL yra M.- Hall dnbd f. eAI4. Code). Ky Rnorvc.
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