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A two-component Arctic ambient noise model Greening, Michael Victor 1994

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A TWO-COMPONENTARCTIC AMBIENT NOISE MODELbyMICHAEL VICTOR GREENINGB.Sc., University of Victoria, 1983M.Sc., University of Victoria, 1987A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of OceanographyWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1994©Michael Victor Greening, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of C) (2 ! A PH ‘The University of British ColumbiaVancouver, CanadaDate 25, 19C/DE-6 (2/88)Supervisors: Dr.P.Zakarauskas and Dr.P.H .LeBlondAbstractShort term Arctic ambient noise spectra over the frequency band 2- 200Hz are presented along with a two component noise model capable of reproducingthese spectra. The model is based on the measured source spectrum and the spatial,temporal and source level distributions of both active pressure ridging and thermalice cracking. Modeled ambient noise levels are determined by summing the inputenergy of the distributions of ice cracking and pressure ridging events and removingthe propagation loss.Measurements were obtained on a 22-element vertical array along with a 7-elementhorizontal array deployed beneath the Arctic pack ice in 420 meters of water. Over900 thermal ice-cracking events were detected in approximately 2 hours of data collected over several days during April 1988. The source directivity for events beyond40 wavelengths range was found to be accurately represented by a dipole with anapproximate 3 dB increase above the dipole directivity pattern near 60° - 65° causedby the leaked longitudinal plate wave. A technique for measuring the bottom reflectivity function by correcting the bottom reflection of a thermal ice crack for themeasured directivity is presented. The spatial distribution of thermal ice-crackingevents is consistent with a uniform distribution. Source levels were measured from110 to 180 dB//pPa2/Hz at 1 m with the distribution of all events approximating alinearly decreasing function on a log-dB scale of the number of events versus source11level.Near the end of the data collection period, measurements from a nearby activepressure ridge were obtained. Evidence is presented that the infrasonic peak observednear 10 Hz in Arctic ambient noise spectra may result from a frequency dependentpropagation loss acting on the source spectrum of pressure ridging.Both modeled and measured ambient noise spectra show that ice cracking maydominate the spring-time ambient noise to frequencies as low 40 Hz. Below 40 Hz, theambient noise is dominated by a single or few active pressure ridges at ranges of tensof kilometers. Above 40 Hz, the ambient noise is dominated by a large distribution ofthermal ice-cracking events with over 50% of the total noise level produced by eventswithin 6 km range and over 80% produced by events within 30 km range.111A Two- ComponentArctic Ambient Noise ModelTable of ContentsABSTRACT iiTABLE OF CONTENTS ivLIST OF TABLES viiLIST OF FIGURES viiiACKNOWLEDGEMENTS xviiCHAPTER 1 INTRODUCTION 11.1 Objective 11.2 Approach and Contents 2CHAPTER 2 REVIEW OF SOURCE AND PROPAGATION 6MODELS2.1 Plate Waves 72.2 Dilatational Point Source 142.3 Horizontal and Vertical Point Force 162.4 Propagation Model 21ivTable of ContentsCHAPTER 3 DATA COLLECTION 343.1 Environment and Instrumentation 343.2 Source Detection and Description 36CHAPTER 4 ARCTIC AMBIENT NOISE MEASUREMENTS .. 58CHAPTER 5 PRESSURE RIDGING 695.1 Spatial Distribution 695.2 Source Spectrum 705.3 Propagation Loss 715.4 Modeled Pressure Ridge Noise 75CHAPTER 6 THERMAL ICE CRACKING 926.1 Spatial Distribution 936.2 Source Directivity 946.3 Source Spectrum 105.6.4 Source Level Distribution 1066.5 Modeled Thermal Ice-Cracking Noise 115VTable of ContentsCHAPTER 7 MODELED AMBIENT NOISE 148CHAPTER 8 SUMMARY 154REFERENCES 157APPENDIX A SEABED REFLECTIVITY FUNCTION 164viList of Tables3-I Cross correlation of the number of detected ice-cracking events per 42minute with environmental conditions.4-I Mean and variance of measured environmental conditions and the 63number of detected ice-cracking events per minute for the fourclasses of detected power spectra.4-TI Cross correlation between the measured ambient noise levels at 10 64Hz and 200 Hz with the measured environmental conditions and thenumber of detected ice-cracking events per minute.5-I Parameters of ice and bottom layers used for propagation loss mod- 78eling in Safari.6-I Mean, variance and skewness of the distribution of the value of m 119in the sinm8 model of source directivity as a function of range andfrequency.viiList of Figures2-1 Longitudinal (symmetric) and Flexural (antisymmetric) modes in a free 27plate.2-2 Dispersion curves for first three symmetric and antisymmetric modes in 28a free plate for a Poisson’s ratio of 0.34.2-3 Coordinate system used for a pair of horizontal point forces F applied at 29the inner surface of a cylindrical cavity with radius a and height h.2-4 Vertical directivity predicted from a pair of horizontal point forces applied 30at the surface of 6 m thick ice for frequencies of 50, 100, 150 and 200 Hz.2-5 Vertical directivity predicted from a vertical point force applied at the 31surface of 6 m thick ice for frequencies of 50, 100, 150 and 200 Hz.2-6 Vertical directivity predicted from both horizontal and vertical point 32forces applied at the surface of 3 m thick ice for frequencies of 50, 100,150 and 200 Hz.vii’2-7 Vertical directivity predicted from both horizontal and vertical point 33forces applied at the surface of 9 m thick ice for frequencies of 50, 100,150 and 200 Hz.3-1 a) Air temperature, b) solar radiation, c) barometric pressure, 43d) wind speed, e) wind direction, and f) current speed for severaldays during time of experiment.3-2 Measured sound speed profile along with the profile used for analysis 49such as range finding, propagation loss estimation and vertical sourceangle estimation.3-3 Array configuration. A linear equispaced vertical array with a 12 m 50hydrophone spacing and the first hydrophone 18 m below sea level isused along with a low redundancy L shaped horizontal array with 7hydrophones at 102 m depth.3-4 Time series of received acoustic pressure for an ice-cracking event at 51480 m horizontal range.3-5 Time series of received acoustic pressure for an ice-cracking event at 523150 m horizontal range.3-6 Number of detected ice-cracking events per minute for each of the 53two-minute ambient noise recordings.3-7 Time series of an event (or events) with distinct modal properties such 54as changing amplitude and phase with depth.ix3-8 Time series of an ice-cracking event at 400 m horizontal range which 55shows a leaked longitudinal plate wave arrival preceding the directarrival.3-9 Time series of the received acoustic pressure of three hydrophones in 56the vertical array for two minutes during a nearby (approximately 2km range) active ridge building event.3-10 Time series of the received acoustic pressure of three hydrophones in 57the vertical array for one minute during a time of active thermal icecracking but several days prior to the ridge building event shown inFig. 3-9.4-1 Average ambient noise spectra and standard deviation for four classes 65of measured noise. a) Class 1. Strong infrasonic peak at 8 Hz withlittle or no thermal ice cracking. b) Class 2. Strong infrasonic peakat 8 Hz with some thermal ice cracking. c) Class 3. Weaker andbroader infrasonic peak at 4 Hz with some thermal ice cracking. d)Class 4. Very intense thermal ice cracking with no noticeable infrasonicpeak.5-1 Received spectral level of a typical pulse within the ridge building 79event.5-2 First 3 modes at 100 Hz in water 420-rn deep with a sound speed profile 80as shown in Fig. 3-2. Note that the modes are trapped near the surfaceresulting in propagation loss that is dependent on spreading and iceinteraction.x5-3 First 3 modes at 20 Hz in water 420-rn deep with a sound speed profile 81as shown in Fig. 3-2. For this low a frequency, the modes are nolonger trapped near the surface (as in Fig. 5-2) resulting in bottominteraction which in turn increases the propagation loss.5-4 Propagation loss determined using the wave model SAFARI for a 82source at 0.2-rn depth in 7-rn thick ice for water 420-rn deep and depthaveraged over all hydrophones in the vertical array.5-5 Propagation loss determined using the normal modes model KRAK- 83ENC with Burke-Twersky scattering for water 420-rn deep and depthaveraged over all hydrophones in the vertical array. An average of 11.5keels per kilometer with keel depths of 5.3 m and keel half widths of11.9 rn is used.5-6 Received spectra of 3 ice-cracking events at different ranges when all 84multiple arrivals including direct path, bottom reflection, under-icereflection, etc. are used.5-7 Resulting ambient noise spectra produced by distant ridge building 85events using the environmental conditions given in Table 5-I to determine propagation loss.5-8 Resulting ambient noise spectra produced by a ridge building event at 8690 km range for sources at varying depths in the ice.5-9 Resulting ambient noise spectra produced by a ridge building event at 8790 km range for different levels of ice absorption.xi5-10 Resulting ambient noise spectra produced by a ridge building event at 8890 km range for different shear wave characteristics in the bottom. a)Ambient noise for different bottom shear wave speeds. b) Ambientnoise for different bottom shear wave absorptions.5-11 Resulting ambient noise spectra produced by distant ridge building 90events using the normal modes model KRAKENC with Burke-Twerskyscattering.5-12 Propagation loss as a function of frequency and total water depth for 91a source at 200 km range.6-1 Spatial distribution of all detected events about the vertical array. 1206-2 Comparison of the vertical directivity of a pair of horizontal point 121forces with the sintm8 model.6-3 Comparison of the vertical directivity of a vertical point force with the 122sinmO model.6-4 Vertical directivity of an ice-cracking event at 450 m range which fits 123the sintm 8 model very well.6-5 Vertical directivity of an ice-cracking event at 300 m range which shows 124a small fluctuating pattern about the best fit to the model sintm8.xli6-6 Vertical directivity of an ice-cracking event at 500 m range which does 125not fit the sinm6 model.6-7 Normalized distribution of the directivity index m for all events within 1262000 m range and with at least a 3 dB signal to noise ratio.6-8 Normalized distribution of the directivity index m for one octave fre- 127quency bands centered at 48 Hz, 96 Hz and 145 Hz. Distributions ofm are given for all events less than approximately 40 wavelengths in6-8a and for all events greater than this range but less than 2000 m in6-8b.6-9 Vertical directivity of an ice-cracking event at 300 m range showing 129little or no dependence on frequency.6-10 Vertical directivity of an ice-cracking event at 450 m range showing 130strong dependence on frequency.6-11 Vertical directivity of an ice-cracking event at 1150 m range showing 131dependence on frequency.6-12 Vertical directivity of an ice-cracking event at 140 m range showing an 132excess in pressure level of 1 dB - 2 dB above the sinmO model at asource angle of 60° - 65°.6-13 Synthetically generated time series for an ice-cracking event at 500 m 133range and 5.5 m depth in 6 m thick ice.xiii6-14 Vertical directivity of a synthetically produced ice crack at 500 m range 134and 5.5 m depth in 6 m thick ice with a compressional absorption inthe ice of 0.2 dB/wavelength.6-15 Vertical directivity of a synthetically produced ice crack at 500 m range 135and 0.25 m depth in 6 m thick ice with a compressional absorption inthe ice of 0.2 dB/wavelength.6-16 Vertical directivity of a synthetically produced ice crack at 500 m range 136and 0.25 m depth in 6 m thick ice with a compressional absorption inthe ice of 2.0 dB/wavelength.6-17 Vertical directivity of a synthetically produced ice crack at 2000 m 137range and 0.25 m depth in 6 m thick ice with a compressional absorption in the ice of 2.0 dB/wavelength.6-18 Received power spectrum of the hydrophone at 150 m depth as a func- 138tion of time for the event shown in figure 3-8.6-19 Source spectral level of a typical ice-cracking event. 1396-20 Detection threshold function indicating the probability of detecting an 140event of a given received level.6-21 Distribution of received levels of all detected events. 1416-22 Median number of events per square kilometer per minute per 1 142dB//1.iPa/Hz at 1 m source level interval versus source level.xiv6-23 The measured and modeled normalized distributions of received levels 143for all detected events over the range interval 0 - 1 km.6-24 The measured and modeled normalized distributions of source ranges 144for all detected events.6-25 Modeled thermal ice cracking noise. 1456-26 Required range to model 80% (within 1 dB) of the total noise produced 146from all thermal ice cracking events out to a range of 200 km.6-27 Required range to model 50% (within 3 dB) of the total noise produced 147from all thermal ice cracking events out to a range of 200 km.7-1 Comparison of two component noise model to average ambient noise 150and standard deviation of real noise, a) Modeled noise for ridge oflevel +3 dB at 70 km range with thermal ice cracking at -8.2 dB andlocal ice cracking at -3 dB; versus class 1 real data from Fig. 4-la.b) Modeled noise for ridge of level +3 dB at 70 km range with thermalice cracking at +0.4 dB and local ice cracking at +0 dB; versus class 2real data from Fig. 4-lb. c) Modeled noise for ridge of level -5 dB at40 km range with thermal ice cracking at -3.0 dB and local ice crackingat +0 dB; versus class 3 real data from Fig. 4-ic. d) Modeled noisefor ridge of level -5 dB at 40 km range with thermal ice cracking at+5.3 dB and local ice cracking at +4 dB; versus class 4 real data fromFig. 4-id.xvA-i Polar plot of the pressure level for a single event arriving at the array 173along both the direct arrival and bottom reflected paths for the octaveband centered at 96 Hz and corrected for spherical spreading loss.A-2 Scatter plot of the pressure level ratio between the bottom-reflected 174arrivals and that extrapolated from the direct arrival, using a sinmOfunction corrected for spherical spreading losses, as a function of grazing angle at the bottom.A-3 Sound pressure ratio between the measured direct arrivals above 3Q0 175and that extrapolated from angles below 300, using a sin6 directivityfunction corrected for the spherical spreading losses, as a function ofsource angle, for all 30 events that fit the criteria of pg. 159.A-4 Averaged and plus and minus one-standard-deviation pressure level 176ratios between the bottom-reflected arrivals, and that extrapolatedfrom the direct arrival, using a sin6 directivity function correctedfor the spherical spreading losses, as a function of grazing angle at thebottom.A-5 Sound pressure ratio between the bottom-reflected arrivals and that 177extrapolated from the direct arrival, using a sintme directivity functioncorrected for the spherical spreading losses, as a function of grazingangle at the bottom, for a few individual events.xviAcknowledgementsI would like to thank all the members of my supervisory committee for theirassistance in the research and preparation of this thesis. I would like to especiallythank Pierre Zakarauskas for his guidance in this work and for being a friend andgolfing partner.The financial support provided by Jasco Research Ltd., Datavision ComputingServices Ltd. and especially Defence Research Establishment Pacific is gratefullyacknowledged. The announced closure of Defence Research Establishment Pacific atthe end of this thesis came as a surprise to many and it will be sorely missed amongthe scientific community.Finally, my deepest thanks and love go to my spouse, Cynthia Lane, who alwaysbelieved in me and supported me and this work in every way. As everything in ourlives, this thesis is a result of our work together.xviiChapter 1Introduction1.1 ObjectiveThe study of ocean acoustics is valuable in many diverse fields such as fishing,oil exploration, weather prediction, navigation and ship detection. In the open ocean,ambient noise is generated mainly by wind, precipitation and distant shipping. Thus,knowledge of the ambient noise can be used directly in measuring weather or indirectlyin designing sonar systems which can detect signals in a noisy background. In the icecovered Arctic, ambient noise is generated mainly by thermal ice cracking, ridgingor grinding of ice, wind blown snow, impacting waves or melting icebergs. Thus,ambient noise measurements in the Arctic can be used directly for determining iceproperties such as breakup or indirectly by aiding in the design of sonar systems. Dueto the increase in mineral exploration and military use of the Arctic, it is apparentthat the study of ice generated noise is very important.It is generally assumed that the underwater ambient noise below 1000 Hz in theice-covered Arctic is produced by the summation of many discrete sources, which areusually attributed to either thermal ice cracking or active ridge building.14 However, most previous studies of underwater Arctic acoustic noise have examined eitherindividual source mechanisms or the ambient noise levels separately. This thesis develops a two component noise model, incorporating both thermal ice cracking andactive pressure ridging as source mechanisms, which is capable of reproducing the1low frequency ambient noise spectra.The model is based on the measured source spectrum, directivity and the spatial,temporal and source level distributions of thermal ice cracking along with measurements and estimates of these parameters for active pressure ridging. Modeled ambientnoise levels are determined by summing the input energy of the distributions of icecracking and pressure ridging events and removing the propagation loss.The model also examines the short term variability of the ambient noise and relatesthis to environmental conditions. Finally, the model reveals the relative importanceof each source term over the frequency band examined and the spatial and strengthdistributions of individual events which sum to give the ambient noise. This lastresult is an important factor in many sonar applications as it reveals whether theambient noise is produced at close or far range.1.2 Approach and ContentsIn concept, the approach of the two component ambient noise model is relatively simple. The ambient noise spectrum over the frequency band 2 - 200 Hz isassumed to be produced by thermal ice cracking and active pressure ridging. Thus, bysumming the acoustic field generated by the distribution of these events, the ambientnoise can be reproduced and the relative contributions of each noise source at the receiver can be determined. In practice however, this procedure becomes very difficult.A source and propagation model must first be combined which is capable of reproducing the measured field of a single event at any range. One requires knowledge of2the source characteristics along with accurate estimates of the impulse response of theice and the associated propagation loss due to refraction, absorption and boundaryinteraction. The fields produced by all events which occur over a given time interval must then be summed to produce the resulting ambient noise. This summationrequires knowledge of the distribution of events in space, time and strength.Chapter two introduces the theoretical source and propagation models used torepresent an event occurring in a floating ice sheet. The source model used is a purelydilatational point source (monopole) in the ice. Although a vertical or horizontalpoint force may be more representative of a thermal ice crack, the dilatational pointsource was found to be an adequate representation of an ice-cracking event for the lowfrequencies examined in this thesis. The propagation model used was a full wavefieldtechnique which solves the four dimensional partial differential wave equation byapplying a series of integral transforms to reduce the problem into a series of ordinarydifferential equations separated in depth. This model was chosen because it takes fullaccount of elastic properties in the ice and bottom and because it is applicable to bothclose and far range. Another advantage of the source and propagation models chosenis the ease of combining these models together. The use of this combined source andpropagation model will be justified in chapters five, six, and seven by comparing themodeled fields with measurements.Chapter three describes the experimental setup and the identification and description of individual thermal ice-cracking and pressure ridging events. A set of 69two-minute samples of ambient noise were collected on both vertical and horizontal arrays suspended beneath the ice. Individual transient events were detected bythreshold clipping and by looking for unlikely groupings of local peaks. It was found3that the acoustic mode dominated the energy received in the water from ice cracks,with only occasional contributions due to the leaked longitudinal plate wave.Ambient noise measurements are shown in chapter four. These measurementsare divided into four distinct classes by examining the power spectra. Each classis compared to both environmental conditions and the measured rate of thermal icecracking. These classes are shown in chapters five, six and seven to depend on theoverall level or rate of thermal ice cracking and the distance to the dominant activepressure ridge.Chapter five describes active pressure ridging and its role on the ambient noise.The spectrum of a single ridging event is shown along with an estimate of the spatialdistribution of active pressure ridges. The large separation between active pressureridges prompted an investigation of the received power from a single event at varyingrange. It was found that the received power from a single event could accuratelyreproduce the infrasonic peak in ambient noise spectra.Chapter six describes thermal ice-cracking events and their role on the ambientnoise. The source spectrum and directivity of individual events is shown and compared to measurements by other authors and to results from the theoretical sourcemodels (dilatational point source, vertical point force and horizontal point force) introduced in chapter two. The source spatial, temporal and strength distributionsare calculated from over 900 detected events and these are used to model ambientnoise levels produced by thermal ice cracking. Appendix A shows how the bottomreflectivity function can be calculated by extending the directivity measured from thedirect arrival of a thermal ice-cracking event to the bottom reflected arrival.4Chapter seven shows how the summation of the large distribution of thermal icecracking events along with a single or few active pressure ridging events at varyingranges can reproduce the ambient noise measurements.Finally, chapter eight summarizes the results and conclusions of this thesis alongwith several interesting features of thermal ice cracking and active pressure ridgingobserved along the way.5Chapter 2Review of Sourceand Propagation ModelsThis chapter gives a brief review of the theoretical source and propagationmodels used to represent an event occurring in a floating ice sheet. This chapteris included to give the reader an understanding of the mathematics involved and topredict the type of waves expected in the water from a source in the ice. The purposeof this thesis is not to develop a theoretical source or propagation model which willexplain the physical principles involved in Arctic noise generation or propagation butis rather to measure the characteristics of individual noise events and combine themwith a propagation model to reproduce the Arctic ambient noise. In doing so, existingsource and propagation models developed by other authors are used. For the sourcemodel, we are interested mainly in what type of field the model produces in the waterand if this is a realistic representation of the measured fields created by real events.For the propagation model, we are interested in how well the model represents realityand reproduces measured effects along with how easily it may be used for differentsites. Results of the theoretical models are compared to measurements.62.1 Plate WavesWe start with a model presented by Press and Ewing5 which gives solutionsfor plane waves propagating in a floating ice sheet. Although this model does notexamine waves radiated into the water in the near field of a source, it serves as auseful starting point for determining the types of waves which may exist in the ice orradiate into the water in the far field of a source imbedded in the ice. For purposes ofthis work, the far field of the source is meant as being at a range distant enough thatthe radiated waves in the ice may be represented by plane waves. This problem wasoriginally examined by Press and Ewing to determine if elastic waves transmitted inthe ice could be used to find the thickness and mechanical strength of the ice and forposition fixing and long range signaling.The model consists of an infinite ice plate of thickness 2ff floating on an infinitehalf-space of water below with the atmosphere represented as a vacuum above. Weare considering plane waves so Cartesian co-ordinates are used with the x-axis atthe mid-plane of the ice and parallel to the direction of propagation and the positivez-axis vertically downward.A solution is sought which satisfies both the wave equations in the ice and wateralong with the boundary conditions at the ice/vacuum and ice/water interfaces.7The governing wave equations in the ice and water are:1 2i72j 1 &(P18t =2 1at2 = 0,2 1624,2= 0, (2.1)v2 atwhere 1 represents the ice, 2 represents the water, is the bulk compressional wavespeed in the ice, f3 is the bulk shear wave speed in the ice, v2 is the speed of soundin water and 4, and are the scalar and vector displacement potentials defined by:i=Vq5+Vx&. (2.2)Note that 4, represents the irrotational component of the displacement field (compressional waves) while & represents the incompressible component of the displacementfield (shear waves). Inviscid fluids do not support shear waves and thus 2 = 0.Then for a system with the x-axis parallel to the direction of propagation, ô/6y =0, and equation 2.2 becomes:— 64,U—ax azaV — 6z üx84,w = (2.3)The result is two uncoupled problems with the particle displacements in the xand z directions depending only on 4, and while the particle displacement in they direction depends only on 4,,, and The particle displacements in the x and zdirections produce the dilatational, or P wave and the vertically polarized shear, orSV wave while the particle displacement in the y direction produces the horizontally8polarized, or SH wave. Note that the particle displacement in the y direction containsonly a shear component and theoretically should not generate waves in the water. Forthis reason, the SH wave will not be examined further. It should be noted thoughthat current research has found that strong SH waves generated by rubbing ice platesproduce a pure tone signal near 800 Hz which can be measured in the water.6 It hasalso been found that sources in the water near the under-ice surface produce strongSH waves in the ice plate.7 A problem is that the coupling mechanism from SHwaves in the ice plate to energy in the water is unknown and ignoring the SH wavesmay effect the estimates of propagation loss used later. Solutions for the P and SVwaves which satisfy the wave equations 2.1 can easily be shown to be of the form:qi = [A sinh(z) + B cosh(Ez)] exp[i(kx —[C sinh(iz) + D cosh(ijz)j exp[i(kx — wt)j,c”2 = E exp(—z) exp[i(kx — wt)], (2.4)where= k2 (1 —c2/c),= k2 (1= k2(1—c/v). (2.5)The constants A, B, C, D and E can be found by substituting into the boundaryconditions. The boundary conditions are the conservation of normal particle displacement and of normal and shear stresses across the boundaries. For the vacuuminterface, the particle displacements at the interface of the non-vacuum are unconstrained but the stresses must vanish; while for water, only the shear stress must9vanish. Thus, the boundary conditions reduce to:(T)1 = 0 at z=—H(T)1 = 0 at z=—H(T2)1 = (T)2 at z = H(T)1 = 0 at z—H= at z = H (2.6)where the stress tensor T13 is defined as T3 = CijkiSki , with Cijkl being the elasticstiffness constants and Ski being the strain defined as 8k1 = (ôuk/8ai + üul/8xk)Assuming the ice to be an isotropic solid, the stress tensor can be reduced to T, =2,us + AskkS, where t and A are Lame’s constants. For a fluid, the stress tensorreduces further to T3 = AskkS. Lame’s constants are related to the density p andthe compressional and shear speeds as = (Ai-f-2i)/p1, = m/pi, V2 = A2/p.Thus, the boundary conditions can be written as:A V2q Ui +°8j’) = 0 at z = —Hxôz + 8x2—) = 0 at z —H,\ VqiH-2 ( + ) = A V2q at z = H+—) = 0 at z = H8& 8b1 842at z=H (2.7)8z ôx 8zThen by substituting q, and q into the boundary conditions, we get 5equations with the 5 unknowns A, B, C, D and E. A solution for all 5 unknownsexists when the determinant of the coefficients vanishes.10Press and Ewing5 found a solution to these equations by using a Poisson’s ratio of0.25 for the ice. This ratio is equivalent to using ) = i’i since Poisson’s ratio is givenas o = \/2 + i’)• Measurements show that Poisson’s ratio is approximately 0.32 forice (a = 3500 m/sec, 3 = 1800 m/sec) and thus, the solutions by Press and Ewingare not exact but may be used for determining the wavetypes which may exist in thefloating ice system described. Stein8 also solved the above equations numerically on acomputer. The solutions for these equations will be discussed below but the reader isdirected to the above papers for an in-depth study on the calculation of the solutions.First, examine the waves in a free plate with a vacuum above and below. Thesolutions for the F and SV waves in a free plate are called Lamb waves and break downinto symmetric and antisymmetric modes where the particle displacements u and ware symmetric or antisymmetric about the midplane of the plate. The symmetricmodes have u with the same sign about the midplane and w with opposite signabout the midplane while the antisymmetric modes have u with opposite sign andw with the same sign. These particle displacements cause the boundaries of theplate to periodically dilate and contract for symmetric modes or periodically flexfor antisymmetric modes as shown in Fig. 2-1. Thus, the symmetric modes arealso known as longitudinal or dilatational modes while the antisymmetric modes arealso known as flexural modes. Although the symmetric modes are sometimes calledlongitudinal modes, it should be noted that both the longitudinal and flexural modesare produced from combinations of both longitudinal (F) and vertically polarizedshear (SV) waves.Some of the characteristics of the symmetric and antisymmetric plate modes canbe found by examining a dispersion curve for these waves as shown in Fig. 2-2 (see11Ref. 9). For a plate of thickness 2H, there are only two possible traveling wavesin the plate when kH —* 0. These are called the zeroth order symmetric (s0) andantisymmetric (a0) modes and correspond to the real roots of the governing equations.There are also an infinite number of purely imaginary roots to the equations butthese represent motions which grow or decay exponentially along the plate and arenot traveling waves. For small kH, the lowest order symmetric (or longitudinal)mode has a phase speed equal to the plate longitudinal speed (ci O.9a) and isnondispersive. As kH increases, the mode becomes dispersive as the phase velocitydecreases. As kH continues to increase, the mode becomes a surface Rayleigh wavepropagating along the upper and lower surfaces with a phase velocity equal to thatof the Rayleigh wave (r 0.9/3). For small kH, the lowest order antisymmetric(or fiexural) mode is highly dispersive with a phase velocity which starts at zerofor an infinitely thin plate and increases with increasing kH. At very high kH, theantisymmetric mode also becomes a surface Rayleigh wave propagating along theupper and lower surfaces of the plate.As kH increases, new traveling waves appear corresponding to the first, second andhigher order symmetric (Si, 2, s3, ...) and antisymmetric (ai, a2, a3, ...) modes.At the critical values of kH, when an even or odd number of longitudinal P ortransverse SV half waves fit into the thickness of the plate, a new mode producedby an uncoupled purely longitudinal or transverse standing wave in z is formed. Justabove the critical value of kH for a given mode, both P and SV waves couple togetherto form a standing wave in z. These P and SV waves travel along the plate as coupledwaves by propagating at different angles to match phase speeds. As kH continues toincrease, the P wave eventually travels in a direction parallel to the axis of the plate.12A further increase in kH transforms the P wave into inhomogeneous surface waveswhich decay exponentially from the boundaries. Thus, in the limit as kH —* oc, onlyan SV wave is left which propagates with a phase speed equal to that of the bulktransverse wave speed.When a floating plate is considered with a fluid halfspace below, the modes discussed above are no longer purely symmetric or antisymmetric due to the differentboundary conditions on the upper and lower surfaces of the plate. For small kH, thelowest order symmetric or longitudinal mode travels with a phase speed of approximately 90% of the bulk dilatational wave speed. This is greater than the speed ofsound in water and thus, this wave radiates into the water at an angle of cosO v/c1.This mode is often referred to as the “leaky” plate wave due to this radiative loss intothe water. For large kH, the symmetric mode becomes a Rayleigh wave propagating along the air/ice interface. For small kH, the lowest order antisymmetric modeis usually called the flexural mode and is highly dispersive as for a free plate. Forlarge kH, the antisymmetric mode turns into a Scholte wave propagating along theice/water interface with a phase speed slightly less than the speed of sound in water.Because of the low phase speed, the first antisymmetric mode does not radiate intothe water but instead decays exponentially away from the interface. As for the freeplate, the higher order symmetric and antisymmetric modes are introduced as kHincreases but these modes are highly attenuated due to leakage into the water and donot propagate far beyond the source which generated these modes.To determine the arrival times of these modes, the group velocities must be known.13The group velocities are given by:ciw d(ck) deCg= =c+k. (2.8)Thus, at the limits as kH —* oo for all modes, or kH —* 0 for s (for whichdc/dk = 0), the group speeds of all modes are the same as the phase speeds. Forintermediate values of kH, the group speed of the zeroth order symmetric mode staysnear the Rayleigh wave speed (approximately 90% of the bulk transverse wave speed)while the group speed of the zeroth order antisymmetric mode rapidly increases tothe Scholte wave speed (somewhat less than the sound speed in water). Higher ordermodes are introduced with group speeds near zero which quickly converge about thebulk transverse wave speed as kH increases.2.2 Dilatational Point SourceThe above presentation examined plane waves in the ice and their radiationinto the water. It gave an indication of the type of waves which travel to a hydrophonemainly along a path in the ice plate and radiate into the water near the hydrophone.Stein8 also examined the waves produced in the near field of an acoustic monopolewhich travel to a hydrophone mainly along a path in the water. Stein started withthe same model as Press and Ewing except that he added a purely dilatational pointsource in the ice. Introduction of a point source in the ice requires a change tocylindrical coordinates and the governing wave equations in the ice and water become:142 1 82 8(r)V 4’ — = s(t)—8(z — zn),V2 1 O2 —V2:2 2 = : (2.9)V2 ôtwhere the right hand side of the first equation represents a forcing function at ranger = 0 and depth z = z0.Again, the boundary conditions are the conservation of normal particle displacement and of normal and shear stresses across the boundaries. Solutions to the forcedwave equations with these boundary conditions contain all the solutions of the previous unforced system along with a wave radiated into the water in the near field ofthe source. Stein calls this solution the acoustic mode since it travels from source toreceiver nearly entirely as an acoustic wave in the water. This mode is believed to becaused by unequal flexing of the ice plate in the vicinity of the source. The acousticmode for a monopole source in the ice was shown by Stein to radiate into the wateras:wcc sine, (2.10)v2Rwhere R is the propagation path length from the source to the receiver and 6 is thedeclination angle of the ray from horizontal. Results shown in chapters three andsix indicate that the energy observed in the water below the ice is dominated by theacoustic mode with only occasional small contributions from the plate waves.152.3 Horizontal and Vertical Point ForcesStein’s model of a dilatational point source (monopole) in the ice correctlyaccounts for the large energy contained in the acoustic mode under the ice. However,a point source is a crude representation of an ice crack and may not reproduce thevertical directivity (energy radiated as a function of declination angle) of a crackcorrectly. Xie’° and Xie and Farmer’1 have examined the radiation patterns producedby a single vertical point force and a pair of horizontal point forces applied at thesurface of the ice.Because the acoustic mode is assumed to be generated by flexing of the ice platein the vicinity of the source, Xie and Farmer derived the radiation patterns of thevertical and horizontal point forces by starting with plate vibration theory insteadof platewave propagation theory. They calculate the sound field under the ice as anintegral form of the product of the impulse response of the ice and the forcing functionof the crack. An outline of the derivation of these sound fields is given below while acomplete derivation is available in Xie1° and Xie and Farmer’1.The impulse response of the ice is found by considering an incident plane wave ofamplitude P02 on the upper surface of the ice and determining the resulting reflected(For) and transmitted (F2) waves as:= exp [ik0 (x sinO0 + z cos60)J,For = A exp [ik0 (x sin&0 — z cos0)j,= B exp [ik2 (x sin92 + (z — H) cos92)j, (2.11)where o represents the air, 2 represents the water, H is the ice plate thickness and A16and B are the reflection and transmission coefficients of the ice plate.Then, by separating the motion of the plate into symmetric (s) and antisymmetric(a) modes, the symmetric oscillations of the plate can be represented as the sum ofthe pressures acting on the plate while the antisymmetric oscillations are caused bythe differences. Thus, using Snell’s law of k0 sin60 = k2 sin , get:P5 =-‘oi z=O + For zO + ‘2t z=H= (1+A+B) exp(ik2xsin8, (2.12)Pa = F0 z=O + For z=O — P2t Iz=H= (1 + A — B) exp (ik2xsin2). (2.13)Then the vertical component of the plate vibration velocity is the sum of thevelocities of the symmetric and antisymmetric modes at the upper surface of the iceand the difference of the velocities at the lower surface of the ice. Thus,V z0 (Pa/Za) + (P8/Z), (2.14)V zH = (Fa/Za) — (P5/Z8), (2.15)where Z3 and Za are the impedances of the symmetric and antisymmetric modes inthe plate.17But the vertical components of the plate vibration velocity can also be expressedas:1 (8FO,____vz=o I +ZWp0 \ ÔZ z=O 0z z=U= 1—A (ik0 cos0) exp (ik2x sin82), (2.16)iwpo1 (8P2tVz-H = jjP2 Z zH=——(ik2 cosO2) exp (ik2x sin82). (2.17)iwp2By combining the above equations, the transmission coefficient can be solved as:B—2 z2 (Z8— Za) (2 18)— ZsZa+4ZZ+(Z+)( s Za)where Z0) = (p0c/cosO), and z0(2 =(p2c/cosO).The impedances can be found for a thin plate (H << )) by examining the symmetric and antisymmetric mode equations:8U5 2 ÜUS H 1 2_ _2 ‘\____— CsOx2 =—- (o — 1) + C5 Ox2 (2.19)M’ +E1H3___= Pa, (2.20)where U is the vertical displacement of the plate, P is the applied pressure, c8 is thelongitudinal plate wave speed c5 = JE1/p , M = p1H and o and E1 are thePoisson’s ratio and elastic modulus of the plate which are related to the Poisson’sratio and elastic modulus of the ice as o = o/(1—) and E1 = E/(1 — o2). Thenby assuming that U and P are harmonic of the form U = U° exp[i (kix sinG1 — wt)jand P = P° exp[i (kix sinG1 — wt)j, the symmetric and antisymmetric impedances18can be found from Z = P/iwU as:2iE1 [1 — (c/c2)sin2&j (2.21)wH [1 — — (c5/2) sinZa = —iwiVi [1— (cf/c2)4 sin82] (2.22)with (4 = (w2E1H3/12M)Now the sound field produced under the ice by a point force applied at the surfaceof the ice will be solved using the principle of reciprocity. In other words, place apoint source at some location M(r2,z2) under the ice (see Fig. 2-3) and find thescattered field P8 at the crack location. Then, the sound field at M caused by a crackin the ice can be expressed in terms of the scattered field P8 and the forcing functionf of the crack in an integral form.For a point source under the ice at location M(r2,z2), and with strength Q, theresulting radiation field measured below the ice is:iwp2QPg =— 47rRexp (ik2R) (2.23)where R is the distance from the point (?2, z2). The scattered sound field P8 measuredat some location N(r, z) above the ice is then found by multiplying each path fromM(r2,z2) to N(r, z) by the transmission coefficient B21 of the ice and integrating overall paths. Note that B21 is the transmission coefficient from the water to the air andis thus related to the transmission coefficient B already found by interchanging Zb0)and Zr). Following the above procedure, get:z, r2,z2)= wp2kQf H1) (k2 — r sin&)B21(6 e2z2c5O_k00s00)sin8 d8(2.24)19where ff) is the zero order Hankel function of the first kind and I’ denotes the pathof integration.Then assuming the crack initially forms as a cylindrical cavity of radius a andheight h at the upper surface of the ice, the radiation field measured at M(r2,z2)below the ice is given as:—1 jj 8P5,z,r2 f(,z)addz (2.25)iwp0Q s —8rwhere a cylindrical coordinate system (r, z, q) is used, f(5, z) is the forcing function ofthe crack, and the integration is performed over the entire inner surface of the cylindrical cavity. Note that although a cylindrical cavity is assumed here, this methodcan be used to construct solutions for cracks of other shapes.For a pair of horizontal point forces, the forcing function is given as: f(q, z) ë,. =(F/a) [S(ç) 6(z) + 6( — ir) S(z)] er where r is a unit vector along the radial directionand F is the magnitude of the two point forces. The field at M(r2,z2) is:P(r2,z) iFp2ksine2Bi(&)e2R. (2.26)Pocc sin9B21(0). (2.27)For a vertical point force, the forcing function is given as: f(q, r) ë,. =(F/i’) 6(çb) 6(r) e where is a unit vector along the z direction and F is themagnitude of the vertical point force. The field at M(r2,z2) is:P(r2,z) 1cosBi(6)eik2R. (2.28)z(2)cos6 B21(&). (2.29)20The directivity patterns produced by both a pair of horizontal point forces anda vertical point force are shown in Figs. 2-4 and 2-5 respectively. Figs. 2-6 and 2-7show the resulting directivity patterns for the horizontal or vertical point forces foreither thinner or thicker ice. Comparisons of both measured and modeled directivityare shown in detail in chapter six. They show that for the frequencies examined andfor grazing angles below the critical angle of the bottom, the acoustic monopole, thevertical point force and the pair of horizontal point forces all give a reasonable approximation to the vertical directivity of real ice-cracking events. Based on these results,the acoustic monopole was chosen as a source model for an individual event due to itsease of implementation. This model can then be combined with the power spectrumof an individual event along with the spatial, temporal and strength distributions ofevents to reproduce the ambient noise.2.4 Propagation ModelAcoustic wave propagation deals with finding solutions to the wave equation.These solutions may be very complicated in nature due to the variability of theenvironment (changing sound speed, irregular bottom stratification, rough ice cover,etc.) and closed form solutions exist for simplified cases only. Thus, models of wavepropagation in the Arctic vary from purely analytic to numeric methods. Analyticmodels use appropriate approximations to the boundary or initial conditions thuspermitting algebraic solutions to a reduced wave equation. Numerical models allowaccurate specification of boundary and initial conditions and give direct numericalsolutions to the wave equation.21The most common types of propagation models are normal mode, parabolic equation, ray theory, finite element or finite differences, and full wavefield techniques. Thenormal mode model provides an exact solution to the wave equation for long rangepropagation by determining the modes of propagation in a bounded fluid. However,it is not appropriate for grazing angles greater than that of the bottom critical anglefor which the field cannot be broken into discrete modes, and it usually ignores shearwaves. The parabolic equation model is similar to the normal mode model in that itcalculates modes of propagation in a fluid. It is also restricted to grazing angles lessthan critical and generally does not include shear waves. Ray theory works by tracingrays from the source to the receiver and coherently summing the pressure field of allrays at the receiver. Ray theory is appropriate mainly for close range and usuallydoes not accurately model long range effects such as caustics or shadow zones. Finiteelement or finite differences models are purely numerical techniques which break upthe differential wave equations into piecewise polynomial functions. These techniquesare very general and can approximate the boundary and initial conditions to anydesired accuracy. However, these techniques are extremely computationally intensiveand difficult to implement.The propagation model left is the full wavefield technique, also known as a fastfield technique or a Green’s function technique. This type of propagation model solvesthe wave equation for horizontally stratified media by applying a series of integraltransforms to the wave equation. This procedure reduces the four dimensional partialdifferential equation (three space coordinates and one time) into a series of ordinarydifferential equations separated in depth only. These equations are then solved withineach layer in terms of unknown amplitudes which are determined by matching the22boundary conditions between layers. For our purposes, the advantages of the fullwavefield technique over previously mentioned techniques are that it is a fully elasticmodel capable of including shear waves and that it is applicable to both close and farrange.The Kuperman-Schmidt full wavefield technique, SAFARI12”3(Seismo-AcousticFast field Algorithm for Range Independent environments) was used for the propagation model part of our noise model. This technique seeks solution to the partialdifferential wave equation in cylindrical coordinates:(V2— c2(r’z) ) (r,z,t) =F8(r,z,t) (2.30)where F8(r, z, t) is the forcing term and I1 refers to both the scalar 4 and vector &wave potentials with c(r, z) representing the bulk compressional or transverse wavespeeds respectively. Note that the azimuthal dependence is removed by using anazimuthally independent source placed on the z axis.The first step to solving the partial differential wave equation is to remove thetime dependence by using the Fourier transform:1f(w) =— J F(t)e”tdt, (2.31)27r-x,to get (V2 + k(r,z)) Ti(r,z,w) =f8(r,z,w), (2.32)where km(r, z) = w/c(r, z) is the wavenumber of the media. Note that capital [i isstill used after the Fourier transform to avoid confusion with the vector wave potentialb. Then, for a range independent environment with the source on the vertical axis,equation 2.32 becomes:(V2 + k(z)) (r,z,w) = f8(z,w) 8(r). (2.33)23The next step is to remove the range dependence by using the Hankel transform:(k)= f g(r)J0(kr) r dr, (2.34)to get (- - (k - k(z))) (k,z,w) = (2.35)where k is the horizontal wavenumber (related to the phase speed of the wave in thegiven layer). Thus, we have an ordinary differential equation in depth only.Solutions are then found for each angular frequency w as:‘(k, z) = 1,(k, z) + A(k) 1P(k, z) + A(k) z) (2.36)where the argument w is dropped but implied, and (k, z) and k, z) are solutionsto the homogeneous wave equation, ‘(k, z) is a particular solution to the inhomogeneous wave equation, and A(k) and A+(k) are arbitrary coefficients determinedfrom the boundary conditions.The complete solution (r, z, t) to the partial differential wave equation is thenfound by performing the inverse Hankel and Fourier transforms,1(r,z,w)=(2.37)(r,z,t)= f (r,z,w)edw. (2.38)The trick to the full wavefield technique is to restrict the depth dependent soundspeed and the forcing term to cases where the solutions ‘I’(k,z), 4’ik,z), and+(k, z) are known analytically and the unknown coefficients A(k) and A+(k) canthen be solved numerically by matching the boundary conditions. In practice, c(z)is usually restricted to be either a constant over a given depth interval or is allowedto change with depth according to a specified formula for which an analytic solution24to the wave equation is known. In SAFARI, elastic solid layers are restricted tohaving constant sound speeds over specified depth intervals. This restriction leads tosolutions which are simple exponential functions of the form:ilJ+(k,z) =‘1r(k,z) = e° (2.39)where a2(k) = — (w/c(z))2. Fluids however do not generally have constant soundspeeds with depth or changes in sound speed in large discrete steps (although thisbehavior is allowed in SAFARI). From the definition of the bulk compressional soundspeed in a fluid, c (\ + 2p)/p, SAFARI assumes that p and p are constant butallows )i to vary inversly with depth so that, c = 1/(az + b). This assumption leadsto solutions which are Airy functions of the form:q(k, z) = Ai{(pw2a)2/3(k2 — pw2(az + b))J,qY(k,z) = Bi[(pw2a)13 (k2 — pw2(az + b))]. (2.40)Note that is used here instead of ‘1 because fluids do not support shear waves.In general, the boundary conditions between two elastic solids are the conservation of normal and tangential particle displacements and the conservation of normaland si-Lear stresses across the boundaries. For any interface not involving two elasticsolids, slip may occur and the horizontal particle displacements no longer need tobe conserved. For any interface involving a vacuum, the particle displacements atthe interface of the non-vacuum are not constrained. For a solid/fluid interface, theshear stress must vanish while for a solid/vacuum interface, both the normal andshear stresses must vanish. Finally, for a fluid/fluid interface, only normal particle displacements and stresses are involved and these must be conserved across the25boundary while for a fluid/vacuum interface, only the normal stress is involved andit must vanish.Finally, for our particular problem, an acoustic monopole source in the ice is usedas the forcing function (as described in the source model) and we start with the partialdifferential wave equations:(V2- c2(r’z) ) (r,z,t) = s(t) S(z - zn),(V2- c(r,z) 2) (r,z,t) = 0. (2.41)For this case, the solutions to the homogeneous depth separated problems are firstdetermined as exponential or Airy functions depending on the layer type and how thesound speeds are defined within the layer. The particular solution to the nonhomo—geneous depth separated wave equation is then given as:= (2.42)The full solution to the depth separated wave equations is then determined by fittingto the boundary conditions. Finally, the range, depth and time dependent fields aredetermined by numerically integrating the inverse Hankel and Fourier transforms ofthe full solution from the depth separated wave equations. For further details on themathematical implementation of this propagation model, the user is directed to theSAFARI user’s guide.1326>—-H >—--:—rI t — E— 1’ > ‘ —> ( I — — —‘ ‘ ( I —- I/ ———— I ‘ IFigure 2-1. Longitudinal (symmetric) and Flexural (antisymmetric) modesin a free plate. Arrows indicate particle displacements.2754c-3ci)a)ciC/)ci)C’)(‘310 10kHFigure 2-2. Dispersion curves for first three symmetric and antisymmetricmodes in a free plate for a Poisson’s ratio of 0.34. Ct and k are the bulktransverse phase speed and wavenumber respectively. (Reproduced fromI.A.Viktorov9)2 4 6 828Figure 2-3. Coordinate system used for a pair of horizontal point forcesF applied at the inner surface of a cylindrical cavity with radius a andheight h. (Reproduced from Y.Xie’°)rN( r, z )C0 F F13C2M(r2 , z2)z29Hor i. zonta ILflro-DIwLwwUiLWU-ICFigure 2-4. Vertical directivity predicted from a pair of horizontal pointforces applied at the surface of 6 m thick ice for frequencies of 50. 100,150 and 200 Hz.50‘Do‘so0D30Hor—izonta ILI)-DI\-JQiLDU)U)UiLW0-ICFigure 2-5. Vertical directivity predicted from a vertical point force applied at the surface of 6 m thick ice for frequencies of 50, 100, 150 and200 Hz.50100‘ScDc31Hor—zcnta 1LO0\-,wLU)U)ciiLwu-I0Figure 2-6. Vertical directivity predicted from both horizontal (solid) andvertical (dashed) point forces applied at the surface of 3 m thick ice forfrequencies of 50, 100, 150 and 200 Hz.60Hz100 Hz150 Hzaoo Hz32Hor-izonta 1QJ0\-IUiLDU)U)UiLUFigure 2-7. Vertical directivity predicted from both horizontal (solid) andvertical (dashed) point forces applied at the surface of 9 m thick ice forfrequencies of 50, 100, 150 and 200 Hz.33Chapter 3Data Collection3.1 Environment and InstrumentationThe data analyzed in this thesis were collected on the pack ice off the northerncoast of Ellesmere Island over several days during April 1988. The ice cover remainedrelatively stable but was very rough, consisting of a mixture of both new and multiyearice. The ice thickness varied from approximately 2 m on re-frozen leads to 7 m onfloes and to larger values on ridges. An ice thickness of 7 m is much larger thanthe 2 m to 4 m thick ice commonly found in the central Arctic but is very commonfor the Lincoln Sea area north of Ellesmere Island.’4 Snow cover was as thick as 60cm, although in the rough pack ice many areas remain exposed. Measurements ofair temperature, solar radiation, barometric pressure, and wind speed and directionwere obtained every 15 minutes throughout the duration of the experiment and areshown in Fig. 3-1. Air temperatures and winds were measured from the top of a 10 mtower while solar radiation was measured at 1 m height away from the shadow of anystructures. Currents were measured at 25 m depth during each recording of ambientnoise and are also shown in Fig. 3-1.The data were collected over an area of continental shelf in 420 meters of waterwith a sound speed profile shown in Fig. 3-2. A seismic survey showed the bottomto be acoustically flat to beyond 1000 Hz and depth measurements within a fewkilometers around the array indicated a shallow slope of less than 10.34To aid in modeling propagation ioss, measurements of the seabed and ice properties were also obtained. A bottom grab sample15 and an acoustic seismic refractionsurvey’6 indicated that the bottom consisted of a layer of sand approximately 15thick with a density of 1.8 g/cm3 and a compressional wave speed of approximately1800 rn/sec. Below the upper 15 rn thick layer was a second bottom layer with anestimated density of 1.9 g/cm3 and a compressional wave speed of 2000 rn/sec. Thesebottom characteristics were supported by measurements of the bottom reflectivity using ice cracking noise as a source (described further in Appendix A). Realistic shearwave speeds of 300 rn/sec and 500 rn/sec were used for the first and second bottomlayers respectively and compressional and shear wave absorptions of 0.5 dB/.) and0.25 dB/ were chosen17 to complete the bottom properties required for later analysis. For the ice, compressional and shear wave speeds of 2800 m/sec and 1750 rn/secalong with compressional and shear wave absorptions of 2.0 dB/.\ and 3.0 dB/) respectively were used based on previous measurements obtained from smoother andthinner ice.18 Measurements of the plate wave speed at the experimental site wereobtained by examining the energy radiated into the water and are consistent with thevalues of cornpressional and shear wave speeds used.A set of 69 two-minute samples of ambient noise were collected at the timesindicated in Fig. 3-1. The data were collected on a vertical array of 24 hydrophonesand a 7-hydrophone horizontal array as shown in Fig. 3-3. The vertical array wasequispaceci with a hydrophone separation of 12 m and the top hydrophone was 18rn below sea level. However, the second and third hydrophones (at 30-rn and 42-rndepth) were not working and hence could not be used in any of the analysis. TheL-shaped horizontal array was at 102 m depth and used a low redundancy spacing35with a total separation of 12.5 m in one direction and 200 m in the other. A samplingrate of 516 Hz was used, and the data were filtered using a 150-Hz low-pass filter witha high-frequency roll-off of 45 dB per octave.After collecting the two-minute noise samples mentioned above, a pressure ridgebuilt itself approximately 2 km from the experimental site. Continuous ambient noisemeasurements were recorded for several hours during the time of this ridge buildingevent.3.2 Source Detection and DescriptionAs stated in the introduction, the Arctic ambient noise is believed to beproduced by the summation of individual transient noise sources produced by the ice.The detection of a transient signal in a noisy background is a common but difficultproblem in signal processing, and the many techniques that are used depend on thesource and background characteristics. Two simple techniques were used here.The first technique was to scan a suitable length (generally 2 minutes) of datafrom a single hydrophone to determine the average pressure peak height, and thenrescan the same data while recording the position of all peaks greater than someuser-supplied multiple of the average peak height. Although this technique is notvery robust, it was useful for finding the transient sources that had a high signal-to-noise ratio. Fig. 3-4 shows the unprocessed output of the vertical array for an eventat 480-rn range. This figure clearly shows multipath arrivals corresponding to a directpath, a bottom reflection, and a bottom and under-ice reflection. These arrivals are36all produced from the acoustic mode (i.e. energy entering the water in the vicinityof the source). Neither the longitudinal plate wave nor the flexural plate wave weredetected for this event. The range of the source was determined using a ray tracingmodel which calculates the eigenrays (the acoustic rays joining a source to a target)for a range independent, vertically stratified fluid with a flat bottom and surface. Therange independence and flat bottom hypothesis were confirmed by a seismic surveyand depth measurements, as described previously.Close inspection of the transient in Fig. 3-4 shows that an ice-cracking eventmay contain several closely spaced high amplitude peaks. These may correspond toa single arrival path such as the direct arrival for the 480-rn range source shown, orto multipath arrivals overlapping in time for sources at longer range as shown in Fig.3-5. This observation led to the development of a second detection technique’9’2°thatseeks sections of data where high amplitude peaks are clustered closely together. It isbased on the probability of finding x values out of a subsample of length y within thetop fraction N of the entire sample z and is given by a standard statistical formulaas:yF(x,y,N) = i!(y— i)!NZ (1— N). (3.1)By scanning the full length of data for one channel, a distribution of peak heights canbe obtained which is then used on a second scan of the data to find all subsectionsof data of length y with x peaks within the top fraction N of the peak heights. Byadjusting x, i and N to detect all transients in a section of data which has beenexamined visually, these values can then be used to detect transient events in all thedata.Over 900 ice-cracking events were detected using the above procedure. Ice37cracking events were sought only in the 69 two-minute samples of noise collectedprior to the time when the pressure ridging event occurred. The number of eventsdetected per minute for each data sample is shown in Fig. 3-6. The strong correlation between the number of detected events per minute and a decreasing temperature(shown in Table 3-I) suggests that most of these events are likely to be thermalice-cracking events.21’2 This hypothesis is further supported by the observation thatapproximately 83% of the data files containing more than the median number ofevents detected per minute occurred from 7 PM to 4 AM local time while 91% ofthe data files with fewer events occurred from 4 AM to 4 PM. Although thermal icecracking contains a broad peak in its power spectrum near 300 - 500 Hz (see Refs.1,2,10,11,21-23), energy is still expected at the lower frequencies measured in thisthesis. This is described in more detail in section 6.3.Several possible explanations are given for the events which occur when the airtemperature is rising. The first possibility is that these events are still thermal ice-cracking events and that the snow cover creates a thermal delay between the airand ice temperatures. Thus, the ice temperature may still be decreasing as theair temperature starts to rise. A thicker snow cover would allow for a greater lagbetween air and ice temperatures and at the same time insulate the ice from theextreme air temperature changes. Thus, areas with snow cover would exhibit fewerthermal ice-cracking events than areas without snow cover but these events couldoccur during times when the air temperature may be increasing. Other possibleexplanations are that these are not thermal ice-cracking events but are caused bysome other environmental condition such as wind, current or tides which builds upstress in the ice until the ice fractures. Dyer24 suggests that wind or current shear38acting on the hummocks of old pressure ridge sites could twist the hummock causingfractures in the ice plate surrounding the ridge. A comparison of the characteristicsof the events which occur during times of cooling with those of the events which occurduring times of warming was not performed here. However, a difference in the powerspectra between events occurring during warming and cooling periods was found byZakarauskas et.al.2°who speculate that different source mechanisms are involved forthese conditions.Visual confirmation of each of the detected events showed that the false alarmrate for the above detection technique was less than 10%. Visual confirmation wasperformed using the entire vertical array, and a false alarm is defined as an event forwhich no distinct arrival paths can be seen across the array. Note that although atransient ice-cracking event usually exhibits multiple arrivals across the entire verticalarray, oniy the direct path may be visible for an event which is close enough that thebottom grazing angle is larger than the bottom critical angle. Also, for very distantevents, the direct path and possibly even low order reflections may disappear due toboth the source directivity and the effects of the upward refracting sound speed profile.The probable cause of a false alarm is the fluctuations in the background noise levelover the length of the two-minute sample. Visual examination of the entire length ofthree data files obtained during quiet, moderate and noisy background levels showedthat over 90% of the total observed ice-cracking events were detected.Several of the events detected using the first detection technique were markedlydifferent from that shown in Figs. 3-4 or 3-5. These events have no discernibleindependent arrivals such as direct path, bottom or under-ice reflections. They alsoextend over longer periods of time than the ice-cracking events previously discussed,39and have characteristics of modal propagation such as nulls and phase changes withchanging depth as shown in Fig. 3-7. Although the exact range of these events couldnot be determined because of the lack of individual arrivals, their modal propertiesindicate that they were very distant events. These events were found to have amaximum power near 10 Hz with a bandwidth of approximately 20 Hz and are latershown to be likely caused by active pressure ridging. Because these events are verydifferent from the other transient events detected, they were not included in thedatabase of thermal ice-cracking events, and all the data were high-pass filtered at30 Hz before the second detection method was used. Subsequent analysis used theoriginal unfiltered data.Finally, an arrival due to leakage of the longitudinal plate wave in the ice wasseen on approximately 20% of the transients with source ranges less than 1 km andon a very small number at greater ranges. This leaked plate wave appeared as aprecursor to the acoustic mode, as shown in Fig. 3-8, and always contained only asmall fraction of the total energy of the acoustic mode. The absence of this arrival formost sources appears, as will be shown later, to be due to absorption and scatteringby discontinuities in the ice. For those events at greater range that exhibit a leakedplate wave, comparison of the arrival times of the plate wave and the acoustic modeshow that the plate wave did not originate at the source but was instead producedby energy from the acoustic mode re-entering the ice. The flexural plate wave, whichdecays exponentially into the water was not seen on any of the data. Therefore, thetotal energy observed in the water was produced mainly by the acoustic mode (energyentering the water in the vicinity of the source) with occasional small contributionsdue to plate waves propagating in the ice. This result agrees with observations from40other authors.8”°”25A few days after the ambient noise data were collected, a pressure ridge builtitself approximately 2 km from the experimental site. The location and time of thisactive pressure ridge are known because of some experimental equipment which wasat the site during the building process. Continuous ambient noise measurements wererecorded for several hours during the time of this ridge building event. Fig. 3-9 showsthe unprocessed output of three hydrophones in the vertical array for two minutesof the ridge building event. This figure is representative of approximately thirty tosixty minutes of the ridge building event showing two or three large pulses everyminute with each pulse lasting on the order of five to tens of seconds. This is quitedifferent from one of the two-minute data samples shown in Fig. 3-10 which wascollected several days prior to the ridge building event and shows typical thermal icecracking events with short bursts of energy lasting a maximum of approximately onesecond (individual thermal ice-cracking events typically last 0.05-0.1 sec with multiplearrivals extending the observed signal to a maximum of approximately 1 sec as seenin Figs. 3-4 and 3-5).41Table 3-I. Cross correlation of the number of detected ice-cracking events per minutewith environmental conditions. Temperature and solar radiation changes were measured over one hour prior to the ambient noise measurements.Environmental CorrelationParameter CoefficientCurrent Speed 0.19Wind Speed -0.12Temperature -0.22Temperature Change -0.59Solar Radiation -0.35Solar Rad Change -0.33Barometric Pressure -0.0242-10-15a)a)4-a)H-30-35Figure 3-la. Air temperature for several days during time of experiment. Crosses indicate times when two-minute ambient noise sampleswere recorded.10 11 12 13 14 15 16April43400300EI200t31ct50C’)1000Figure 3-lb. Solar radiation for several days during time of experiment. Crosses indicate times when two-minute ambient noise sampleswere recorded.10 11 12 13 14 15 16April4430.530.4C)zC30.330.2E0Ct530.130.0Figure 3-ic. Barometric pressure for several days during time of experiment. Crosses indicate times when two-minute ambient noise sampleswere recorded.10 11 12 13 14 15 16April4512100(-I)0C’)20Figure 3-id. Wind speed for several days during time of experiment.Crosses indicate times when two-minute ambient noise samples wererecorded.10 11 12 13 14 15 16April46350300a)D-D2000.4-0(2). 150C: 100500Figure 3-le. Wind direction for several days during time of experiment. Crosses indicate times when two-minute ambient noise sampleswere recorded.10 11 12 13 14 15 16April4710 I I I I I+8 * +0ci--H- ++ + +• +• +÷+a- +C,)+ ++( • -H +• +2• +0 I I I i I I I10 11 12 13 14 15 16AprilFigure 3-if. Current speed for several days during time of experiment.Crosses indicate times when two-minute ambient noise samples wererecorded.4801435 1440 1445 1450 1455 1460Velocity (m/sec)Figure 3-2. Measured sound speed profile along with the profile used foranalysis such as range finding, propagation loss estimation and verticalsource angle estimation.100200300400500143049Figure 3-3. Array configuration. A linear equispaced vertical array witha 12 m hydrophone spacing and the first hydrophone 18 m below sea levelis used along with a low redundancy L shaped horizontal array with 7hydrophones at 102 m depth.50E-CQi(00CFigure 3-4. Time series of received acoustic pressure for an ice-crackingevent at 480 m horizontal range. The output of all functioning hydrophones in the vertical array is shown and clearly indicates the direct arrival (D), the bottom reflection (B), and the bottom and surfacereflection (BS).0 0.2TImB (s6c)0.6 0.8 151CII [BSB BSBS 8SBSB0.6 0.8Tima (sac)Figure 3-5. Time series of received acoustic pressure for an ice-crackingevent at 3150 m horizontal range. The output of the hydrophones in thevertical array is shown and indicates the direct arrival (D), the bottomreflection (B), the bottom and surface reflection (BS) and subsequentreflections from the bottom and surface (BSB, BSBS, BSBSB). Note that• the reflections are compressed in time when compared to a closer rangeevent as shown in Fig. 3-4.18sq7810212615017q1980 0.2 1523025j2015>LU01002z50Figure 3-6. Number of detected ice-cracking events per minute for eachof the two-minute ambient noise recordings.April53vJVVW-vI-’ -‘\fVV/WWJVWVWVWvI ‘WVWV’JW’iWW WVVJVVV%.JVVW‘\rvv\AJVWvWwv i “ v— v v v ‘JV,’AJ’Af\jVVV’-’VV’zzz=26cn 27029q______________1.5 2 2.8 3Time (s6c)Figure 3-7. Time series of an event (or events) with distinct modal properties such as changing amplitude and phase with depth. An examplecan be seen at approximately 0.9 sec. It has a null at 186 m depth and a1800 phase shift between the received signal above and below this depth.1021261601791982223.654L D 8 BSI-’E-c0Ui(UUFigure 3-8. Time series of an ice-cracking event at 400 m horizontal rangewhich shows a leaked longitudinal plate wave arrival (L) preceding thedirect arrival (D), followed by a bottom reflection (B) and a bottom andsurface reflection (BS).0 0.6TimB (s6c)0.6 15518Figure 3-9. Time series of the received acoustic pressure of three hydrophones in the vertical array for two minutes during a nearby (approximately 2 km range) active ridge building event.S\-,C 160-p0UiID10U0(JihiL I. 1 1 Jih .LiJI,L.m4w,U0 20 °rO 60Time (sec)80 100 12056182: iso-pw100aFigure 3-10. Time series of the received acoustic pressure of three hydrophones in the vertical array for one minute during a time of activethermal ice cracking but several days prior to the ridge building eventshown in Fig. 3-9.3DTi.ma (sec)57Chapter 4Arctic Ambient NoiseMeasurementsThe 69 two-minute samples of ambient noise recordings were separated intofour distinct classes by examining their power spectra. These are shown in Figs. 4-lato 4-ld. Class one shows a peak with a level of approximately 86 dB//1tPa2/Hz at8 Hz and a fall-off to 48 dB//Pa2/Hz at 200 Hz. Class two shows the same peaknear 8 Hz but the fall-off at higher frequency has two stages with a rapid decrease inlevel out to 35 Hz and a slower decrease beyond 35 Hz to a level of 57 dB//Pa2/Hzat 200 Hz. Class three shows a weaker and broader peak at 4 Hz with a fall-off athigher frequency to 58 dB/fliPa2/Hz at 200 Hz. Class four shows a continual butslow decrease in noise level with increasing frequency from a level of 85 dB//Pa2/Hzat 2 Hz to 66 dB/fltPa2/Hz at 200 Hz. The minimum number of data sets used inany class was 14.The four classes of ambient noise spectra are compared to environmental conditions and the number of detected ice cracking events occurring per minute in Table 4-I.The most striking feature of the comparison of ambient noise spectra with environmental parameters is the increase in noise levels above 35 Hz as both the temperaturechange becomes more negative and the number of detected ice cracking events perminute increases. This observation is consistent with previous measurements of thermal ice cracking and temperature change, although thermal ice cracking is usually58associated with higher frequencies from 300 - 500 Hz (see Refs. 1,2,10,11,21-23). Itcan also be seen that the noise levels above 35 Hz increase as the average solar radiation decreases and the solar radiation change becomes more negative; however,the large variance in these measurements makes any correlations suspect. The othernoticeable feature of the comparison of ambient noise spectra with environmentalparameters is the high wind speeds and barometric pressures associated with thestrong peak at 8 Hz as seen in Figs. 4-la and 4-lb. This result agrees with previousmeasurements made by Makris and Dyer.26The correlations between the ambient noise levels at 10 Hz and 200 Hz withthe environmental conditions are shown in Table 4-IT. There are good correlationsbetween the ambient noise level at 200 Hz and both the number of detected eventsand a negative temperature change. A moderate correlation is also shown betweenthe ambient noise level at 10 Hz and the barometric pressure. Again, this result isconsistent with the correlation study performed by Makris and Dyer.26 Surprisingly,the correlation between the noise level at 10 Hz and the wind speed was very low.Our measurements and those of other authors indicate that the ambient noise at200 Hz is dominated by thermal ice cracking. The source mechanism for the infrasonicpeak however, is not yet positively identified.Buck27 showed that low frequency noise correlates well with wind stress and thusmight be due to ice movement or pressure ridging. He also examined the correlationof noise at 1200 Hz with that at various lower frequencies and found that correlationdropped as frequency decreased. Due to the large propagation loss at high frequencies,he suggested that the 1200-Hz noise must be local, and that the low frequency noise59is more dependent on distant ice movement.Although they did not study ambient noise, Thorndike and Colony28 showed thatapproximately 50% of the long term (several months) average ice motion was dueto the geostrophic wind with the rest due to ocean circulation. Over shorter timescales, they found that more than 70% of the variance of the ice motion could beexplained by fluctuations in the geostrophic wind. They showed theoretically thatice motion should depend not only on wind and current shearing stresses acting onthe ice, but also on the normal stresses of the sea surface tilt and on the Corioliseffect. Makris and Dyer26 later examined the cross-correlation of low frequency noisewith the above forces along with two composite stresses forming the horizontal loadon a vertical section of the ice sheet and the stress moment acting about the centralhorizontal plane of the ice sheet. They found the highest cross-correlations wereobtained using the stress moment which is dominated by opposing current and windshearing stresses.The high cross-correlations shown for both low frequency ambient noise and icemotion with the wind and current shearing stresses applied to the ice has led to thegeneral belief that low frequency ambient noise in the Arctic is caused by large scaleice motion and the resulting pressure ridging. Several models of pressure ridging haveappeared in the literature,3’429each explaining some feature of pressure ridging or itsrelation to ambient noise. Parmerter and Coon29 developed a model of pressure ridgeformation consisting of two ice sheets moving together to close a lead. As the icesheets move together, the rubble from the lead thrusts over or under a sheet, bendingthe sheet until the stresses fracture the ice into large blocks which then continue tobe over or under thrust. They were able to reproduce measured profiles of pressure60ridges and showed that wind forcing contained enough energy to produce ridging.However, they did not examine the spectral shape or level of noise that would beproduced. Pritchard3 developed a model which converted measurements of ice driftinto pressure and shear ridging events. He showed that the modeled noise resultingfrom pressure ridging was more highly correlated with the measured noise than wasthe modeled noise produced by shear ridging. Using only pressure ridging as a noisesource, he was able to simulate 46% of the variance of the noise at 10 Hz and 32 Hzmeasured over 120 days in the Beaufort Sea and up to 80% of the variance of the noiseover shorter time periods of 20 days. Buck and Wilson1 measured the source levelper unit length of a pressure ridging event which occurred near their campsite andused it to predict the average ridge separation required to reproduce the backgroundnoise level. Although they had no direct measurements of the average active ridgeseparation to verify their work, they predicted an average separation of 37 km whichcompared favorably with pan sizes of 10 - 100 km measured by SEASAT SAR. Notehowever that Buck and Wilson obtained their measurements in April while the pansizes were measured in October and also that they assumed the noise generated byall pressure ridges to be identical to that measured for this single ridge.Dyer2’suggests that although pressure ridge building is indeed very noisy, the lowfrequency noise may not be dominated by this type of ice fracturing event due to thelow percentage of ice which is undergoing active ridge building. He also states that aplausible source mechanism must produce energy with the correct spectral shape (i.e.it must have a peak in the spectrum near 10 Hz), which has not yet been shown forpressure ridging events. Dyer proposes a possible source mechanism involving windor current shear acting on old pressure ridge sites which need not be active. The61wind or current pushes the hummock of the old ridge thereby transferring stress tothe thinner, weaker ice sheet. This stress results in both circumferential and radialcracks emanating from the ridge site. Further wind or current shear can then twistthe ridge vertically, bending the ice sheet between several radial cracks. Finally, thecracks will slip allowing the ice sheet to vibrate back to its stable position. Dyershows that there is enough energy input in the ice to produce such a fracture andthat the resulting radiation will peak near 11 Hz.In chapter five, I show that a source mechanism with a broad spectral peak near 10Hz is not required to reproduce the infrasonic peak in the ambient noise and presentevidence that this peak is due to the effects of frequency-dependent propagation lossacting on the spectral shape of distant ridge building events. Chapter six will show theeffects of thermal ice cracking on the ambient noise while chapter seven will indicateover which frequency band pressure ridging and thermal ice cracking dominate bycombining each source type to reproduce the measured ambient noise spectra.62Table 4-I. Mean and variance of measured environmental conditions and the numberof detected ice-cracking events per minute for the four classes of detected power spectra shown in Figs. 4-ia to 4-id. Temperature, barometric pressure and solar radiationchanges were measured over one hour prior to the ambient noise measurements.Environmental Class 1 Class 2 Class 3 Class 4Parameter Fig 4-ia Fig 4-lb Fig 4-ic Fig 4-idCurrent Speed 3.0 + 1.9 4.6 + i.8 4.5 ± 0.6 4.4 ± 1.3( cm/sec)Wind Speed 7.8 + 1.5 7.2 + 2.0 3.3 + 2.0 4.6 + 1.1(m/sec)Temperature -21.3 + i.4 -21.8 + 2.3 -18.3 + 2.i -21.6 + 2.3(°C)Temp Change 0.35 + 0.i6 -0.26 + 0.16 -0.i8 ± 0.i5 -0.45 + 0.30(°C/hour)Solar Radiation 154 + 70 123 + 82 i03 + 73 65 + 23(W/m2)Solar Rad Change 24 + 34 -2 + 30 5 + 25 -i8 + 14(W/m2hour)Barometric Pressure 30.40 +0.07 30.35 + 0.12 30.15 ±0.ii 30.22 ±0.11(in-Hg)Bar Pres Change 0.00 ± 0.01 0.00 ± 0.01 0.01 + 0.03 -0.02 ± 0.01(in-Hg/hour)Detected Events 1.0 + 1.0 7.2 ± 5.6 3.3 + 1.3 22.2 + 5.2( events/mm)63Table 4-IT. Cross correlation between the measured ambient noise levels at 10 Hzand 200 Hz with the measured environmental conditions and the number of detectedice-cracking events per minute. Temperature, barometric pressure and solar radiationchanges were measured over one hour prior to the ambient noise measurements.Environmental Correlation CoefficientParameter 10 Hz 200 HzCurrent Speed -0.06 0.14Wind Speed 0.18 -0.11Temperature -0.01 0.01Temp Change 0.20 -0.72Solar Radiation 0.28 -0.23Solar Rad Change 0.20 -0.33Barometric Pressure 0.52 -0.07Bar Pres Change -0.19 -0.05Detected Events -0.18 0.7664Nzc’iI100908070605040Frequency (Hz)Figure 4-la. Average ambient noise spectra (solid) and standard deviation (dashed) for Class 1 type ambient noise spectra indicating stronginfrasonic peak at 8 Hz with little or no thermal ice cracking.10 10065NIc’J0-oci)>ci)-Jci)C,)0zFrequency (Hz)I I I I I I II I I I III///—/\\100908070605040Figure 4-lb. Average ambient noise spectra (solid) and standard deviation (dashed) for Class 2 type ambient noise spectra indicating stronginfrasonic peak at 8 Hz with some thermal ice cracking.jII I I I I I I III10 10066Nzc’J0-4--Da)>a)1a)(1)0z100908070605040Frequency (Hz)Figure 4-ic. Average ambient noise spectra (solid) and standard deviation (dashed) for Class 3 type ambient noise spectra indicating weakerand broader infrasonic peak at 4 Hz with some thermal ice cracking.10 10067Nzc’J0-D>ci)_1ci)Cl)0zFrequency (Hz)I I I I III—I I I I III\. /‘‘\100908070605040—V JI liii I I I II10 - 1001Figure 4-id. Average ambient noise spectra (solid) and standard deviation (dashed) for Class 4 type ambient noise spectra indicating veryintense thermal ice cracking with no noticeable infrasonic peak.68Chapter 5Pressure RidgingActive pressure ridges occur when two ice sheets are pushed together, usuallyby wind or current, forcing one sheet over the other. This bends the sheets until theresulting stresses fracture the ice into large blocks which pile up forming long rowswith a triangular shaped vertical cross section. The full effects of active pressureridging on the ambient noise spectra from 2 - 200 Hz will be examined here. It willbe shown how the source spectral level of an active pressure ridge, when correctedfor the propagation loss of a distant event at 40 km or more, produces the infrasonicpeak found in the Arctic ambient noise spectra.A simple method of determining the role of active pressure ridges on the ambientnoise would be to sum the received energy from all occurring active ridges at a giventime. This approach requires knowledge of the spatial, temporal and source leveldistributions of active pressure ridges, about which very little is known. It will beshown that the measured ambient noise below 40 Hz can be accurately reproducedby a single active pressure ridge at a range of tens of kilometers.5.1 Spatial DistributionSonar measurements of the under surface of the ice performed by the TJSSNautilus30 give an average of 6 keels/km across the entire Arctic basin. Laser altime69try measurements of the upper surface of the ice show reasonable agreement with anaverage ridge separation of approximately 100 m (see Ref 31). However, these measurements show the sum of both active and old, inactive ridges. Members of DefenceResearch Establishment Pacific have been conducting experiments in the Arctic packice over spans of one or more months each year since 1986. The occurrence of an active pressure ridge at or near an experimental site is very rare suggesting that activepressure ridges are only a small fraction of the ridges observed using sonar or laseraltimetry. Buck and Wilson4 estimate an average active pressure ridge separation of37 km to account for the ambient noise levels and compare this separation to pansizes of 10 - 100 km measured by SEASAT SAR. Pans are defined as groups of icefloes which move together. Because of the large spatial separation of active pressureridges, the component of ambient noise caused by pressure ridging may be dominatedby a single or few events occurring at some large distance from the array. Thus,examining a single pressure ridging event as a function of range may provide insightinto the role of pressure ridging on the ambient noise. This is what is done here. Itrequires only knowledge of the source spectral level and the propagation loss.5.2 Source SpectrumThe received spectrum of each 5 - 10 sec pulse within the ridge building eventof Fig. 3-9 had a spectral shape and overall levels within 5 dB of that shown in Fig.5-1. The source spectral level estimated from these received levels is found to bemonotonously decreasing and agrees with the source level per unit length of an activepressure ridging event measured by Buck and Wilson.4The spectral shape we observed70persisted for at least 1.5 hours with a continuous rise in level of approximately 15 dBover the first thirty minutes followed by a relatively consistent ambient noise levelover the next sixty minutes with the exception of two or three large pulses occurringevery minute with durations of five to tens of seconds (as shown in Fig. 3-9) andlevels 10 - 15 dB above the ambient level. The absolute levels shown in Fig. 5-1correspond to one of the pulses observed after the initial thirty minute rise. Detailson the entire duration of the active pressure ridge are not available but the eventrecorded by Buck and Wilson lasted approximately two days.4An obvious feature of the received spectral shape of the ridge building event isthe increase in level with decreasing frequency. This observation is in contrast withthe spectral shape of the ambient noise which exhibits a broad peak near 10 - 20 Hz.Thus, either a different source mechanism is dominant over this frequency band orthe propagation loss must increase dramatically below 20 Hz.5.3 Propagation LossPropagation loss in ice covered Arctic water behaves as a band-pass filterwith the minimum propagation loss occurring in the octave 15- 30 Hz (see Ref. 32).For high frequencies, the upward refracting sound speed profile traps energy in thesurface channel as shown in Fig. 5-2. This trapping results in propagation loss thatis dependent on spreading and ice interaction and thus increases with increasing frequency. At low frequency, modal leakage from the surface channel results in bottominteraction (as shown in Fig. 5-3) which increases with decreasing frequency andgreatly increases the propagation loss. The increase in propagation loss at low fre71quencies was shown by Mime33 who found that the received energy in 2500-rn waterin the Arctic over the frequency band 12- 24 Hz decreased faster with increasingrange than over the band 24- 48 Hz. This minimum in the propagation loss has beenmeasured by several authors for the open ocean also.3436Modeling propagation loss in the Arctic is a very difficult problem because of therough ice cover. To date, no complete model is available which allows both a fullyelastic ice layer and incorporates a realistic approximation of the under-ice roughness.Development of such a model would be a thesis in itself and will not be performed inthis work. Instead, results from two different propagation models will be given.The first propagation loss model used was the wave model SAFARI which includesrefraction effects from the sound speed profile (Fig. 3-2) and the effects of absorptionand shear waves in both the ice and bottom (ice and bottom characteristics areoutlined in Table 5-I). SAFARI implements absorption by allowing km in Eq. 2.32to be complex of the form = km(1 — iS). This results in plane harmonic waveshaving the form:F(r,t) A exp[i(wt— km)]= A exp(—Skmr) exp[i(wt— km)] (5.1)which is a wave decaying exponentially in range. The parameter 5 is defined byspecifying the attenuation coefficient ‘y in dB per wavelength. For linearly frequencydependent attenuation, the attenuation coefficient is given as:= —20 log[F(r + )i,t)/F(r,t)]= —20 log [exp(—Skm))j= 4O7rSlog e. (5.2)72The result of the propagation loss determined by SAFARI, shown in Fig. 5-4,does indeed indicate that at longer ranges the propagation loss has a minimum atapproximately 50 Hz and greatly increases below 10 Hz. Combining this pattern withthe spectrum of ridge building could easily result in a peak near 10 Hz as found inthe ambient noise.Although this model allows a fully elastic ice layer, it does not model the under-iceroughness correctly. It instead uses the Kirchholf approximation which assumes thatat any point, the surface can be treated as being locally flat. This allows the reflectedfield at any point to be determined using:UrRUi (5.3)(54)8n 8nwhere U and U are the incident and reflected fields, n is the normal to the surfaceat the scattering point in question, and R is the reflection coefficient obtained byassuming infinite plane waves and an infinite plane interface. The Kirchhoff scatteringrequires small roughness with small slope and a radius of curvature of the roughsurface much less than the incident wavelength. Both measurements3’and modeling29show that ridge keels form in a roughly elliptical shape with keel depths as large as10 times the ice thickness and ridge widths often as small as 3 times the keel depth.Thus, the Kirchhoff approximation is a very poor estimate of the under-ice roughness,especially at the shallow grazing angles involved with long range propagation, andwill lead to an underestimation of the propagation loss.The second propagation loss model used was the normal modes model KRAKENC.This model also includes refraction effects from the sound speed profile and allows73fully elastic bottom layers. The implementation of this model as used for this workdid not incorporate an elastic ice layer though and instead used a scattering modeldeveloped by Burke and Twersky37 to estimate the under-ice reflection coefficient as afunction of frequency and grazing angle. This scattering model determines reflectivityfrom a uniform distribution of protuberances on a perfectly reflecting boundary. Themodel considers plane waves incident on a one dimensional distribution of semielliptic, infinitely long cylinders. A reflection coefficient for the rough surface is thenconstructed by combining the effects of the scattering for each individual protuberance. Reflection coefficients are then estimated for a given average keel depth, keelwidth and number of keels per kilometer. The propagation loss estimated from thismodel also indicates a minimum near 50 Hz and a large increase below 10 Hz as shownin Fig 5-5. Combining this pattern with the spectrum of ridge building could alsoeasily result in a peak near 10 Hz as found in the ambient noise. Estimates of theaverage keel depth, keel width and number of keels per kilometer were obtained fromDiachok.3’This model provides an accurate estimate of the under-ice propagation losswhen both the source and receiver are in the water but does not include the effects ofhaving the source in the ice. A more detailed comparison of the two propagation lossmodels and the effects of changing various environmental parameters will be given inthe next section.Evidence of the band-pass filtering effect of propagation at our experimental siteis obtained by examining the spectral levels of the thermal ice cracking events asa function of source range. When the spectral levels of these events are calculatedusing all the bottom and under-ice reflections, a relatively flat spectrum results forclose range events but a broad peak appears in the spectrum which narrows slightly74and shifts towards lower frequency as the source range increases. This behavior isshown in Fig. 5-6 which indicates that the events are being band-pass filtered withthe very low frequencies (below 20 Hz) being attenuated at a faster rate than thehigher frequencies (above 40 Hz) in agreement with the predictions shown in Figs.5-4 and 5-5.5.4 Modeled Pressure Ridge NoiseWhen the received spectral level of the pressure ridging event at 2 km rangeis corrected for propagation ioss to larger ranges using the SAFARI wave model, theresulting spectra exhibit broad peaks at 10- 30 Hz as shown in Fig. 5-7. This figureassumes a source depth of 5.5 m while the true source depth is unknown. Fig. 5-8shows that the modeled peak at 10 - 30 Hz persists for varying source depths in the ice.Buck and Wilson1 found the acoustic source of an active pressure ridge they examinedto be in the keel of the ridge at a depth below the thickness of the surrounding iceplates. To keep the modeled source within the ice, a source near the bottom of theice at 5.5 m depth is used for all further modeling. Although the source depth haslittle effect on the frequency at which the infrasonic peak occurs, Fig. 5-8 does showthat it affects the spectral levels by as much as 6 dB with the lowest source levelsoccurring for a source at mid depth. These observations agree with results found byStein.8The modeled peaks at 10- 30 Hz shown in Figs. 5-7 and 5-8 resemble the infrasonicpeak found in many real data ambient noise spectra with the exception that themodeled ambient noise level remains relatively flat until 30 - 60 Hz while the real75ambient noise level often begins decreasing at 10 - 20 Hz (Figs. 4-la and 4-ib). Themodeled results also show that the peak narrows as the range of the event increasesand suggest that the infrasonic peak could be explained if active pressure ridges wereseparated by several hundred kilometers. This idea disagrees with measurementsof pan sizes and an alternative explanation for the narrow peak is examined in thepropagation loss model.As mentioned in the previous section, the wave model SAFARI uses the Kirchhoffapproximation of small roughness with small slope to estimate the interface roughness.This is a poor approximation to the under-ice roughness and leads to an underestimation of the propagation loss. Livingston and Diachok38 estimated the under-icereflectivity at approximately 18 Hz and 24 Hz along several tracks using matchedfield processing and found the scattering loss to be approximately 10 times as largeas the Kirchhoff’ predictions.In an effort to more accurately reproduce the propagation loss caused by under-icescattering when using the SAFARI wave model, the absorption loss in the ice wasallowed to increase. Although this is an entirely different loss mechanism, it has thedesired characteristics of increasing the loss with increasing frequency and with eachinteraction with the ice. The resulting spectrum of a ridging event at 90 km awayfor varying ice absorptions is shown in Fig. 5-9. This figure clearly shows that as theice absorption increases, the modeled infrasonic peak narrows and becomes a betterapproximation of the peak observed in real data in Figs. 4-la and 4-lb.Because of the uncertainty in bottom shear characteristics, the effects of smallchanges in the shear wave speed and absorptions in the bottom layers were examined.76These effects are shown in Figs. 5-l0a and 5-lob. Fig. 5-lOa shows that as the bottomshear wave speed increases, the infrasonic peak in the ambient noise spectrum becomesmore pronounced and shifts slightly towards higher frequency. Fig. 5-lob shows thatthe infrasonic peak also becomes more pronounced as the shear wave absorptionincreases.The received spectral level of the pressure ridging event at 2 km range correctedfor propagation ioss to larger ranges using the KRAKENC normal modes model withBurke-Twersky ice scattering is shown in Fig. 5-11. This figure shows a broad peakbetween 8 - 20 Hz depending on source range. This result agrees well with the infrasonic peak found in the ambient noise spectra. This model appears to accuratelyreproduce the shape of the ambient noise spectra without the need to artificially increase the absorption levels in the ice as required by the SAFARI model with Kirchhoff scattering. However, this model simply estimates reflection coefficients from theunder-ice scattering caused by ridge keels and does not include an actual ice layer.Thus, it does not include the effects of having the source in the ice. Because theactive pressure ridges examined here and the thermal ice cracks examined in the nextchapter all occur in the ice, this model will not be used further.All the modeling up to this point has assumed a water depth of 420 m as found atthe experimental site. It is interesting to examine the propagation loss as a functionof bottom depth as shown in Fig. 5-12. This figure shows that at very far range,the propagation loss minimum is near 30 Hz for water 500 m deep and increases toapproximately 50 lIz for water 3000 m deep. It also shows that the increased lossat frequencies above and below the frequency of minimum propagation loss is muchhigher for shallow water than deep water. This last effect becomes more pronounced77as the source range increases.Because of the deficiencies of the propagation loss models available for the Arctic,some caution should be used when drawing conclusions from their usage. Estimatingthe range of an active pressure ridge strictly by using the ambient noise spectrum islikely to be inaccurate because of the uncertainty in the propagation loss and possibledifferences in the spectral shape of active pressure ridges. It is shown though that forour data, both the spectral shape and level of the ambient noise below 40 Hz can beaccurately modeled by active pressure ridges at ranges of tens of kilometers.Table 5-I. Parameters of ice and bottom layers used for propagation loss modelingin Safari. C,, and C5 are the sound speeds for the compressional and shear waveswhile K,, and K5 and the compressional and shear wave attenuations.Ice Bottom Sub-BottomThickness (m) 6 15 -Density (g/cm3) 0.9 1.8 1.9C,, (m/sec) 2800 1800 2000C5 (m/sec) 1750 300 500K,, (dB/)) 2.0 0.5 0.5K5 (dB/)i) 3.0 0.25 0.25RMS roughness (m)upper 1.0 - -lower 4.0 - -78NI0zLci)>ci-Ja)>ciC-)ci)1201151101051009590Frequency (Hz)Figure 54. Received spectral level of a typical pulse within the ridgebuilding event.1 10 100790400500Figure 5-2. First 3 modes at 100 Hz in water 420-rn deep with a soundspeed profile as shown in Fig. 3-2. Note that the modes are trapped nearthe surface resulting in propagation loss that is dependent on spreadingand ice interaction.100200300600800100200400500600Figure 5-3. First 3 modes at 20 Hz in water 420-rn deep with a soundspeed profile as shown in Fig. 3-2. For this low a frequency, the modesare no longer trapped near the surface (as in Fig. 5-2) resulting in bottominteraction which in turn increases the propagation loss.81N>C.)Cci)0ci)U-100160Range (km)Figure 5-4. Propagation loss in dB determined using the wave model SAFARI for a source at 0.2-rn depth in 7-rn thick ice for water 420-rn deepand depth averaged over all hvdrophones in the vertical array.0 10 20 30 40 5082N>C-)Ca)0a)U-100101100Range (km)Figure 5-5. Propagation loss in dB determined using the normal modesmodel KRAKENC with Burke-Twersky scattering for water 420-rn deepand depth averaged over all hydrophones in the vertical array. An averageof 11.5 keels per kilometer with keel depths of 5.3 m and keel half widthsof 11.9 m is used.0 20 40 60 808390 I I I26Cm•—— 330Cm•-- 1100Cm60 -—. \“ -• \..>0ci) /cl:i:zn//40 ••• I I I I10 100Frequency (Hz)Figure 5-6. Received spectra of 3 ice-cracking events at different rangeswhen all multiple arrivals including direct path, bottom reflection, underice reflection, etc. are used. Approximately 20 ice-cracking events at eachrange were examined. Although absolute levels varied between individualevents, spectral shapes were consistent for any given range.84Nc’a13a)>a)-J-Da)a)0Figure 5-7. Resulting ambient noise spectra produced by distant ridgebuilding events using the environmental conditions given in Table 5-I todetermine propagation loss.\\ — —— % S..—”\S.-—/N/\-S100908070605040I ••‘ •/.\/,••/•/ ,/ . \\/ \ \.1•1•I./\\ \.\\\30km60km90km120 km150 km\\. \\‘ .‘.\\\1 10 100\Frequency (Hz)85Nc’JcE5Ucr-o>-Jci)00100908070605040Frequency (Hz)Figure 5-8. Resulting ambient noise spectra produced by a ridge buildingevent at 90 km range for sources at varying depths in the ice.1 10 10086NzcJ0-o>-J0100908070605040Frequency (Hz)Figure 5-9. Resulting ambient noise spectra produced by a ridge buildingevent at 90 km range for different levels of ice absorption. Levels of absorption for the compressional wave in the ice are given in dB/A with shearwave absorption 1.5 times the compressional wave absorption.1 10 10087NC”Ct50-Dci)>ci)-J-Dci)ci)0250, 400300, 450350, 500400, 55003, 04 (rn/see)I\‘\J/100908070605040/.,// //////1 10 100Frequency (Hz)Figure 5-lOa. Resulting ambient noise spectra produced by a ridge buildingevent at 90 km range for different bottom shear wave speeds. Speeds aregiven in m/sec for bottom (C3) and sub-bottom (C4) layers respectively.Compressional and shear wave absorptions in the ice of 10 and 15 dB/.\are used respectively.8810090NcN80- 70>a)-J-c60a)-c05040Frequency (Hz)Figure 5-lOb. Resulting ambient noise spectra produced by a ridge buildingevent at 90 km range for different shear wave absorptions in the bottom.Both bottom layers use the same level of absorption given in dB/). Cornpressional and shear wave absorptions in the ice of 10 and 15 dB/A areused respectively.1 10 10089Nc’.JaG)>-J0,,— —\100908070605040\/ ,• .-.‘7,,/ \ \./\\N\30km60km90km120 km150 km/ \\‘.\.‘1 10 100Frequency (Hz)Figure 5-11. Resulting ambient noise spectra produced by distant ridgebuilding events using the normal modes model KRAKENC with BurkeTwersky scattering. A source depth of 5 m is used along with an averageof 11.5 keels per kilometer with keel depths of 5.3 m and keel half widthsof 11.9 m.901N>C.)ci)ci)U-1500 2000 2500 3000Bottom Depth (m)Figure 5-12. Propagation loss in dB as a function of frequency and totalwater depth for a source at 200 km range.500 100091Chapter 6Thermal Ice CrackingThermal ice cracks are small ice fracturing events which occur near the surfaceof the ice as it contracts during times of atmospheric cooling. By using the spatial,temporal and source level distributions of thermal ice cracking, the average energy perunit area entering the water due to thermal ice cracking can be determined. This levelin turn is used to determine the average energy received at a hydrophone suspendedbelow the ice by summing the energy input over all locations about the hydrophoneand subtracting the associated propagation loss for each location.In order to develop the thermal ice cracking component of the ambient noise model,the spatial, temporal and source level distributions along with the source spectrumand directivity of thermal ice cracking must be known. I derive the temporal andsource level distributions of thermal ice cracking from the received levels and ranges ofindividual events along with the propagation loss and probability of detecting an eventat a given range and source level. The spatial distribution of events is measured andcompared to results by other authors. The directivity of individual events is measuredand compared to the source models introduced in chapter two.926.1 Spatial DistributionIn determining the spatial distribution, the range of an individual source isfound using a vertically stratified ray tracing model with the sound speed profileshown in Fig. 3-2. This model calculates the eigenrays for all propagation paths(direct arrival, bottom reflection and multiple reflections) to all hydrophones on thevertical array. Ranges were determined to the nearest 10 rn out to 2 km and tothe nearest 50 m beyond 2 km. The bearing of a source was determined using onlythe direct arrival path on the horizontal array and assuming straight ray propagation.Tests at several angles showed a maximum error of approximately 100 between straightray and refracted ray propagation for a source at 100-rn range. The error was reducedto less than 10 for a source at 500-rn range. The bearing of each event was determinedto the nearest increment of 5°.A scatter plot of all the detected events, shown in Fig. 6-1, reveals that mostdetected events are within 5 km of the vertical array. This grouping occurs becausethe increased propagation loss associated with increasing range makes an event of agiven source level harder to detect with increasing range. When propagation loss isconsidered, the lack of preferred source locations or directions supports the idea of alarge scale spatially isotropic noise field and agrees with previous results for areas ofArctic pack ice.39 This isotropy gives some justification for the proposed method ofdetermining an average input energy per unit area. Note that two forms of short termfluctuations of the average input energy can occur. The first is a strength fluctuationin the overall input level applied to all locations as the rate of ice cracking changes.The second is a random statistical fluctuation in the spatial distribution which may93cause local areas of weak or intense ice cracking. These random statistical fluctuationshave more effect on the ambient noise when occurring at close range due to the smallernumber of events and lower propagation loss associated with close range. Thus,statistical fluctuations will be applied only within 1 km of the hydrophone. Thesetwo types of fluctuations could account for some of the differences between classes ofreal ambient noise spectra measured in chapter four and are included in the model.6.2 Source DirectivityAn important characteristic in modeling the ambient noise produced by individual events is the vertical directivity of these events, which is a measure of therelative power radiated from a source as a function of vertical angle. Due to thehigh propagation loss associated with near vertical rays, long range propagation isrestricted to angles near horizontal. Therefore, the source directivity at these shallowangles is critical in modeling the ambient noise.In the open ocean, a major source of noise is entrapped bubbles from rain orbreaking waves. A spherical bubble which expands and contracts in the water is amonopole source radiating equal energy in all directions. When the bubble depth issmall compared to the wavelength of the energy radiated, interference of the bubblewith its image source above the surface results in dipole radiation. Acoustic sourcesin the ice are more complicated, with the measured radiation pattern depending onthe shape of the source, the elastic properties of the ice and the complicated interactions with the rough ice surface and ice bottom. Therefore, there is no obviousradiation pattern for a source in the ice, and several theoretical models have been94proposed.8”°”4042 The models of Stein8, Xie’° and Xie and Farmer’1 have beenshown in chapter two. Stein8 predicts a dipole radiation by using a point source inthe ice. Xie1° and Xie and Farmer1’ predict an angularly dispersive directivity byusing plate vibration theory to describe the motion of the ice plate and the resultingwaves produced from either a vertical point force or a pair of horizontal point forcesat the surface of the ice. Langley’° predicts a higher order multipole radiation patternby using an extended source in the ice. Kim4’ uses a combined mathematical andnumerical model to find the radiation pattern of typical sources used in geophysicalstudies such as strike-slip, dip-slip and tensile cracks. Dyer42 states that the ice fracturing process of a crack is important above 200 Hz while the deformation unloadingor plate vibration of the crack is important below 200 Hz. He also states that the lowfrequency plate vibration should have a dipolar radiation while the higher frequencyfracture process will have an octopolar radiation.Observation of the number of detectable ice-cracking events with range,’3’1theratio of energy in the direct arrival versus bottom reflection,23 and the ratio of energyin a lead pressure ridge to that of a floe pressure ridge4 have shown that ice crackingevents often radiate more energy vertically than horizontally. These papers all assumea dipole directivity to determine other characteristics of the source or propagationparameters. However, the theoretical models mentioned above do not all agree that adipole directivity is the appropriate form to use. Also, the only published data showing direct measurements of the vertical directivity (at more than two angles) knownto the author are those given by Stein.8’25 Stein shows the directivity of two eventsas measured on a 24-element horizontal array with a 1-km aperture. The low signalto-noise ratio along with the limited span of vertical source angles available from a95horizontal array resulted in an inconclusive measurement which neither supportednor contradicted a dipole directivity model.This section presents direct measurements of the vertical directivity of 160 ice-cracking events and compares them to results obtained from the wave model SAFARIand to the directivity predicted by the source models of Stein, Xie and Farmer. Usingthe 22-element vertical array, the received signals spanned source angles from 10 to80° below horizontal. Only those events within 2 km of the array and with a minimumsignal-to-noise ratio of 3 dB were used.The vertical directivity of an individual ice-cracking event was obtained in thefollowing manner. An event is first identified and isolated as described in chapterthree. The received pressure level in the direct arrival is then measured for eachhydrophone. Pressure levels are averaged over one octave bands centered at 48, 96and 145 Hz. These levels must then be corrected for background noise, sphericalspreading loss and the focusing of rays due to changes in the refractive index. Thenoise is removed by subtracting the pressure level contained in an equal length ofdata immediately preceding the direct arrival. Measurements of ambient noise pressure levels showed a maximum change of only 0.5 dB across the vertical array for thefrequencies examined. Because of this small change, the noise level was sampled atonly one hydrophone and assumed constant along the array. The effects of sphericalspreading and ray focusing are determined by using SAFARI to calculate the propagation loss from a source at the surface to the receivers in the vertical array usingthe sound speed profile shown in Fig. 3-2. Reflections from the surface, bottom andunder-ice are eliminated from the propagation loss by removing the ice and usinginfinite half spaces of isovelocity water above the source and below 420-rn depth with96sound speeds equal to those of the surface and bottom water respectively. Absorptionin the water can be ignored as it is insignificant (approximately 10 dB/km at 100Hz) for these short ranges and low frequencies.45 A ray tracing model can then beused to give the vertical source angle to each hydrophone for a given range, and thus,measurements of radiated pressure level as a function of angle are obtained.For a given ice-cracking event, the measured sound pressure level Pd(O) in thedirect arrival at each hydrophone (corrected for spreading loss, focusing and background noise) is used to calculate the vertical directivity pattern of the source. Wechose to parametrize the directivity of the source by a model of the form:46’7P(8) = P0 sinm8 (6.1)where P0 is the source pressure level (in dB//1iPa/Hz at lm) and P(8) is the modeledradiated pressure level in the direct arrival at source angle 8, normalized back to 1m. However, it was found to provide a useful parametrization of the directivity atshallow angles which controls long range propagation. The continuous parameters mand P0 may be determined by minimizing the squared difference (Pd — P)2.Figs. 6-2 and 6-3 show the best fit of the above model to the horizontal andvertical point forces used as a source model by Xie’° and Xie and Farmer1’for anglesless than 45°. They show very close agreement for frequencies of 150 Hz or less.Although the fit is not accurate at 200 Hz, it is still reasonable for angles less than30°. It is shown in Appendix A that the bottom critical angle is approximately 40°and thus, our model of 5111m8 provides a reasonable estimate to nearly all the energypredicted from distant horizontal or vertical point forces.Fig. 6-4 shows the measured directivity of an ice-cracking event as a function97of source angle along with the best fit directivity pattern calculated using the abovemodel. This figure shows that the model fits these data very well. Of the 160ice-cracking events examined, 95 (approximately 60%) were found to fit the modelwith an average deviation of less than 0.75 dB per hydrophone. A further 42 events(approximately 26%) were found to fit the model with an average deviation of 0.75- 1.5 dB. The deviations in the first case tended to be random while those in thesecond case generally followed some small fluctuating pattern as shown in Fig. 6-5.The 95% confidence limits (assuming a time scale of approximately 0.15 sec for thedirect arrival) in the pressure levels are 1.2 dB at 150 Hz, 1.5 dB at 100 Hz and 2.2 dBat 50 Hz. Thus, these deviations from the fit could be caused by the estimation errorin determining the pressure levels from the short time samples. Although our simplemodel does not account for these fluctuations, it provides a useful first estimate ofthe directivity of these events and therefore was used to model all the above events.The remaining 23 events (approximately 14%) were found to have complex directivitypatterns often exhibiting a peak in the directivity at source angles ranging from 100to 350 as shown by the example in Fig. 6-6. The directivity of these events could notbe approximated by our model and they were not used in any further processing. Acomparison of the directivity of these events with horizontal or vertical point forces(Fig. 2-7) shows that this type of directivity pattern may result from a source inthicker ice. It will also be shown, while examining the directivity of modeled eventsusing SAFARI, that it is still possible to reproduce such a directivity pattern using amonopole source.Although the model of the form sintm 8 provided an acceptable fit to the directivityof approximately 86% of the 160 transient events analyzed, the directivity index m98of the model was found to vary between individual events. The distribution of m forall 137 ice-cracking events which fit the model is shown in Fig. 6-7. This distributionhas a mean of 0.87, a variance of 0.20 and a positive skew. This distribution alsoshows two peaks for values of m of approximately 0.6 and 1.0.If the distribution of m is examined for different horizontal source ranges as shownin Fig. 6-8, it is seen that the lower values of m occur mainly for sources at ranges ofless than approximately 40 wavelengths (1300 m at 48 Hz, 650 m at 96 Hz and 430m at 145 Hz) while for further range events, the values of m are narrowly distributedabout 1.0. Values of the mean, variance and skewness of these distributions areshown in Table 6-I. The distribution of m was examined for different sets of horizontalsource range but the clearest separation between events with m narrowly distributedabout 1.0 and those with m broadly distributed about a smaller value occurred at40 wavelengths. The lower values of m for sources at shorter range may be causedby nearfield effects which are not included in the transmission loss. Therefore itappears that the vertical directivity of an ice-cracking event at long range may beapproximated as a dipole and thus the source may be approximated as a monopole.Note that the models of Langley,4°Kim,4’ and Xie and Xie and Farmer” may alsoall approximate a dipole directivity pattern when observed at long range and lowfrequency.It was also found that not only did the order m of the model depend on range, butalso the goodness of fit of this model to the measured source directivity. For the 96Hz band, of the 65 events with an average deviation above 0.75 dB, only 2 were at arange of 60 wavelengths or greater. Also, all of the 23 events with complex directivitypatterns were at ranges of 20 - 40 wavelengths. Therefore, the model sin6 appears99to be a good fit to the directivity of an ice crack at long range while deviations fromthis model are concentrated at shorter ranges. These results will also be comparedto modeled directivity produced by SAFARI.Although a dipole directivity pattern (monopole source) was found to accuratelyrepresent most of the detected events, an attempt was made to positively identifyone of the source models. One method was to examine the directivity for differentfrequencies simultaneously. Stein’s model of a monopole source gives purely dipoledirectivity for all frequencies while Xie and Farmer’s model of horizontal or verticalpoint forces predicts a directivity pattern dependent on frequency. Most events werefound to either radiate equal energy at all frequencies or have no noticeable patternsdistinguishing different frequencies as shown in Fig. 6-9. This result is consistent witha monopole source in the ice but may also be produced by horizontal or vertical pointforces occurring in thinner ice as shown in Fig. 2-6. Approximately 15% of the eventswithin 400 m range were found to have distinctly different radiation patterns for 50and 100 Hz than for 150 and 200 Hz as shown in Fig. 6-10. At the lower frequencies,these events were found to fit our directivity model with values of m between 0.6 to0.8. The higher frequencies did not fit our model but were instead found to peak at15° to 300. This peak in the directivity at 15° to 30° is consistent with the radiationpredicted by horizontal or vertical point forces for events occurring in thicker ice asshown in Fig. 2-7. Also, approximately 10% of the events beyond 1000 m range werefound to have different radiation patterns for different frequencies as shown in Fig.6-11. These events were all found to contain more energy at 100 and 150 Hz than at50 or 200 Hz.The second method used to try to distinguish the source model was to exam100me events occurring at close enough range that several hydrophones received energyat source angles greater than 700 below horizontal. The pair of horizontal pointforces predicts a sharp decrease in energy for these large source angles while boththe monopole source and vertical point force do not. Only seven events satisfied theabove condition with four events showing a decrease in energy above 70° while theremaining three events did not. This inconsistency in the level of energy above 70°could be explained by cracks occurring on vertical faces of upthrust ice but may alsobe a result of near field effects not considered in any of the source models. Unfortunately, a definite source mechanism could not be identified from the data but theplate vibration theory used by Xie’° and Xie and Farmer11 appears to be able toaccount for all observed effects if areas of different ice thicknesses are present.Finally, although the sinm8 model was an accurate fit to the directivity of 86%of the events identified, an excess in the sound pressure levels of 1- 3 dB above the fitwas found centered about a 60° - 65° source angle as shown in Fig. 6-12. This increasewas found on 9 out of 13 or 70% of the events that had source angles this high, whilein total, fluctuations of a few dB were found on only 30% of the 137 events used.Better statistics on this increase in energy may be obtained by examining the bottomreflected arrivals for which more events occur with a 60° source angle. These arrivalsshowed an increase in pressure level of 3 dB above the model fit (after correctionfor bottom reflectivity) with an approximately Gaussian distribution centered at 60°with a 5° standard deviation as shown in Appendix A. This appendix also shows howthermal ice-cracking events can be used to measure the bottom reflectivity function.The increase in pressure levels of both the direct arrival and the bottom reflectionat 60° may be explained as the leaked plate wave which coincides with the acoustic101wave at this angle and therefore cannot be separated in time from the acoustic waveas it could for the event in Fig. 3-8. Using plate vibration theory, Xie’° and Xie andFarmer” also predict an increase in energy in the water at this angle for a thin plateas the plate vibration impedance vanishes. This increase is clearly shown in Figs. 2-4to 2-7. Thus, this increase should be included in the directivity of the source and aproposed model of the vertical directivity is:2P(8) = P0 [sinm8 + C exp L) j] (6.2)where P(8) is the modeled pressure level in the direct arrival at source angle 8, P0 isthe pressure level of the source, m is the directivity index, C is a constant giving therelative contributions of the acoustic mode and the leaked longitudinal plate wave,and 8L and w are the critical angle and beamwidth of the leaked longitudinal platewave respectively. For our data, C 1.0, 8, 60° and w 5°, while m 1.0 forhorizontal source ranges greater than 40 wavelengths but decreases to approximately0.7 for shorter range events. Note that the contribution to the directivity due to theleaked longitudinal plate wave will have little effect on the long range propagation ifthe bottom critical angle is less than 60° above horizontal. Also, although the leakedplate wave contributes to the acoustic mode, it was noted earlier that as the rangeincreases the leaked plate wave itself is rarely observed. The modeling discussed nextsuggests that the absence of this wave could be due to scattering and absorption in theice. Finally, it should be remembered that this proposed model of source directivitycomes from a statistical average over m, where the distribution about m at long rangeis fairly narrow but broadens for shorter ranges.To compare the source model with measurements of vertical directivity, a simplethree-layer horizontally stratified model which approximated the experimental site102was used as the environmental input for SAFARI. It consisted of a vacuum halfspaceabove, followed by a layer of ice and a water halfspace below. Ice thicknesses of both3 m and 6 m were used with compressional and shear speeds of 2700 rn/sec and 1750rn/sec respectively. Although an average ice thickness of 3 rn for the central Arctichas been measured in August,3°our measurements of ice thickness obtained duringlate spring at multiple locations and over several years have usually shown thicknessesof 6 m or more for level undeformed ice and agrees with other measurements made inthe Lincoln Sea area.14 The change in ice thickness had little effect on the modeledsource directivity and only the results for 6 rn thick ice are shown. An isovelocitywater layer with a sound speed of 1450 rn/sec was chosen to minimize the number oflayers used, thereby reducing artifacts introduced at boundaries. Also, since only thedirect path from source to receivers was used for measuring the directivity, an infinitehalf space of water was chosen to prevent interference between the direct path andbottom reflections. Although the water layer in the model does not resemble that atthe site, the differences will alter only the spreading losses and source angles of thereceived signal, both of which can be determined and corrected easily.The ice crack itself was modeled as a monopole pulse source at varying depths inthe ice. The pulse form used was:f(t) sin(2 f t) — sin(4 f t) 0 <t <fr’ (6.3)where f is the center frequency or peak frequency of the signal (100 Hz for ourdata). This pulse form was chosen as it approximates the frequency response seenby our system. Fig. 6-13 shows the modeled response for a source in the ice at arange of 500 m, a depth in the ice of 5.5 m and for a value of absorption in the icewhich approximates smooth ice.18 Although thermal ice cracking is expected to occur103near the ice surface where thermally applied stress is greatest, a deep source is usedhere for illustrative purposes and the effects of source depth will be examined. Notethat both the leaked plate wave and direct path acoustic mode are clearly visibleacross the entire array while the fiexural wave is visible on the shallowest receiveronly. The direct path acoustic mode also shows interference from multiple arrivals(indicated by the separation of a single pulse near the surface into two pulses fordeeper receivers). These multiple arrivals could be caused by the unequal fiexuralvibrations of the ice plate near the source as described by Stein8 and result in highlynon-dipolar simulated radiation patterns as shown in Fig. 6-14. This figure is to becompared with the measured directivity shown in Fig. 6-6.The directivity pattern of the acoustic mode of a source in the ice as modeledby SAFARI was found to depend on source depth, source range and the level ofabsorption in the ice. As the source range or level of absorption in the ice increased,interference from the multiple arrivals decreased and the directivity pattern shiftedfrom the non-dipolar pattern shown in Fig. 6-14 to a more dipolar radiation pattern.Dependence on depth was more complex with the directivity pattern most closelyapproximating a dipole for a source near mid-depth in the ice but becoming morecomplicated as the source approached the ice surface and even further complicated asthe source approached the ice bottom. Fig. 6-15 shows the resulting directivity whena shallow source depth of only 0.25 m is used. Fig. 6-16 shows the directivity fora shallow source when the level of absorption in the ice is increased to approximaterough ice18 while Fig. 6-17 shows the directivity when the source range is increased to2000 m. In general, the modeled directivity always approximated a dipole for distantevents while for shorter ranges, it approximated a dipole for events near mid-depth104in the ice with high levels of ice absorption but became significantly non-dipolar asthe depth changed or the level of ice absorption decreased. This behavior comparesfavorably with measured results which also always approximated a dipole at longrange but sometimes had complex directivity patterns at short range. The mixtureof dipolar and non-dipolar source directivity measurements at short range can easilybe caused by differences in source depth or by local areas of low or high absorptionin the ice. Finally, high absorption in the ice also eliminates the leaked plate wave inthe simulation, which could explain why this arrival is not always observed.6.3 Source SpectrumAlthough ambient noise caused by thermal ice cracking is shown to contain abroad peak in its spectrum near 300 - 500 Hz (see Refs. 1,2,10,11,21-23), very littlehas been published on the spectrum of individual events. Xie and Farmer1’show thereceived spectra of individual ice-cracking events for frequencies above 100 Hz andfind a peak near 300 - 600 Hz with a rapid decrease in energy at higher frequenciesbut a slow decay with an almost flat spectrum for the lower frequencies. Zakarauskaset.al.2° have also published spectra of individual events, including that of a thermalice crack. That spectrum was fairly flat from 25 - 400 Hz, with a negative slope above400 Hz. Dyer42 shows a frequency-time analysis of an ice cracking event from 30 to 590Hz and shows that individual components or peaks within the event have band widthsand time lengths of approximately 100 Hz and 10 msec respectively. He also showsthat individual components tend to progress in time from higher to lower frequency.This progression from high to low frequency can be seen in the direct arrival for the105bottom hydrophones of Fig. 3-4. When the spectra of individual pulses from a singleice-cracking event are averaged together, Dyer’s results would also indicate a fairlyflat spectrum from 30 - 400 Hz.Unfortunately, the sampling rate used to collect data for this thesis does not allowas fine a frequency-time resolution as that used by Dyer and the spectra of individualpulses within a single crack are averaged together. A frequency-time analysis of thehydrophone at 150 m depth for Fig. 3-8 is shown in Fig. 6-18. This figure shows thatthe leaked plate wave arrives at approximately 0.26 seconds with energy concentratedbelow 120 Hz. It is followed by the direct arrival at 0.35 seconds, the bottom reflectionat 0.60 seconds and the bottom, under-ice reflection at 0.80 seconds. For the data usedin this thesis, the received spectra are averaged over the entire length of the directarrival. It was then found that nearly all the ice-cracking events had a relativelyflat spectrum over the frequency band 2 - 200 Hz, as shown in Fig. 6-19. Althoughaveraging the spectra of individual pulses from a single ice-cracking event does notallow identification of the mechanism involved for each pulse, the emphasis of thisthesis is to determine if the ambient noise can be modeled by the individual eventsand not to determine all the mechanisms involved in a single ice crack.6.4 Source Level DistributionIn measuring the source level distribution of underwater noise in the Arctic,several papers have related the ambient noise level to environmental correlates;2’2618however, only one report known to the author contains information on the numberof individual events detected or the distribution of strengths of these events.1’Unfor106tunately, that thesis reports only on those events which were detected and does notattempt to correct the measurements to derive the number of events which actuallyoccurred. Thus, the measurements are dependent on both site and array characteristics such as propagation loss and the detection threshold. The measurements inthis thesis have been corrected to produce distributions of all the events which actually occur, not just those which are detected, and thus should be site and systemindependent.Two techniques were used to determine the source intensity level of individualevents. The first technique was used only for events within 2 km of the vertical arrayand for which the direct arrival could be isolated from other arrival paths. For eachhydrophone in the vertical array, the measured sound pressure level Fd() in the directarrival is first averaged over a one-octave band centered at 96 Hz and corrected forspreading loss and ambient noise. Spreading loss includes both spherical spreadingalong the ray path and focusing of rays due to changes in the refractive index withdepth. This loss is determined using SAFARI with the sound speed profile shown inFig. 3-2. Ambient noise is determined by measuring the pressure level in an equallength of data immediately preceding the direct arrival. The true source pressurelevel can then be determined by correcting for the source directivity. As shown inthe section on source directivity, it can be estimated by using a model of the formF(O) = P0 sinme where P0 is the true source pressure level, P(6) is the modeledsource pressure level in the direct arrival at source angle 0, and m is the directivityindex (m = 1 corresponds to a dipole). The source angle S at each hydrophone isgiven by the ray tracing model used to determine range. The continuous parametersm and P0 may be determined by minimizing the squared difference (Pd(S) — P(S))2.107P0 is then squared to give the source intensity level. These strength measurementscan be used to fine tune or verify the second source level estimation technique.The second method of determining the source level of an individual event canbe used for events at any range and requires only a single hydrophone, although lessvariability in the results was obtained by averaging over all hydrophones in the verticalarray. The received intensity level from all arrivals including the direct path, bottomreflection and any multiple reflections is summed, averaged over a one octave bandcentered at 96 Hz, and corrected for ambient noise. The true source level can then beobtained by correcting for the propagation loss from the source to the hydrophone.Propagation loss was estimated using the wave model SAFARI with the sound speedprofile shown in Fig. 3-2 and the ice and bottom characteristics outlined in Table5-I. In an attempt to correct for the underestimation of loss caused by the Kirchhoffapproximation of the under-ice surface roughness, the shear and compressional waveabsorptions in the ice were allowed to vary as outlined in chapter five. The sourcelevels calculated by this technique were then compared to the levels determined bythe first method to estimate the correct levels of absorption to use. It was found thata compressional wave absorption of between 1.0 and 10.0 dB/) with a shear waveabsorption 1.5 times as large generally gave source levels within 1 - 2 dB of the levelsdetermined using the first method. A wide range of values of absorption fit the modeldue to the small ice interaction involved for short range events. A more accurateestimate of absorption is obtained later when events at further range are used forwhich the propagation loss depends more strongly on ice interaction.Due to the increased signal to noise ratio which results from using all arrivalpaths, and the ability to determine source levels for events at any range, the sec108ond estimation technique was used to determine the source level of all the detectedevents. Although the exact absorption rates required to model both the absorptionand scattering in the ice are not known, they can be estimated more accurately byexamining the distributions of the number of detected events and their source levelsas a function of range. The next paragraphs will explain the determination of thetrue source level distribution.The observed distribution of source levels differs from the true distribution becausethe further away an event occurs, the louder it must be to be detected. Hence, theobserved distribution of source levels is biased toward stronger events. This biascan be avoided by using only the events which occur very close to the array. Thisprecedure however would demand the analysis of very long records to accumulate thedesired statistics. The method I have chosen is to use all events which are detectedand correct them for the increasing source level required to detect an event as thesource range increases. Consider first only the events which are detected over thesmall range interval r — Sr to r + Sr. Then for a spatially uniform distributionof events, the number of events which will be detected over this range interval as afunction of source level SL is given by:Dr(SL) = 2irr8r n E(SL) T(SL — FLr) (6.4)where E(SL) is the true source level distribution of all occurring events which hasbeen normalized so that E(SL) = 1 where SLmin and SLmar are theminimum and maximum detected source levels; n is the number of events/area/timeoccurring over the entire interval of observed source levels; PLr is the propagation109loss at range r and T(SL — PLr) is a detection threshold function representing theprobability of detecting an event with a received level RL = SL — PLr. Although thismethod is sufficient to estimate the true source level distribution, better statistics areobtained by summing over all ranges at which events are detected. Thus, the observeddistribution of source levels for all detected events is given as:rmaxD(SL) = D7.(SL)rmax= 27r8r Ii E(SL) r T(SL— PLr). (6.5)Then by measuring the total number of events detected at each source level D(SL)over some time interval, the number of events occurring per unit area per time intervalat each source level is given as:-D(SL)E(SL)= 28r max r T(SL— FLr) (6.6)The remaining tasks are to estimate the detection threshold function T(RL), andto check the accuracy of the ice absorption terms used in the propagation loss. Thedetection threshold function should allow detection of all events with a received level(source level minus propagation loss) greater than 2 - 3 dB above the ambient noiselevel, while events below this level have some finite but decreasing probability of beingdetected, as shown schematically in Fig. 6-20. By examining the distribution of received levels of all events, as shown in Fig. 6-2 1, one could assume that all events withreceived levels above 75 dB//1tPa2/Hz are detected while there is an exponentially decreasing probability of detecting an event below this level which falls to approximately11010% at 60 dB//1iPa2/Hz. However, because this fall-off below 75 dB//pPa2/Hz maybe due in part or even wholly to the true strength distribution, both the level atwhich the detection probability begins to fall and the rate of exponential fall are leftas variables which must be determined by some other means.Two methods are available for ensuring the consistency of the model and inputvariables. The first technique is to use the true source level distribution calculatedfrom Eq. 6.6 to predict the observed source level distribution of detected eventsover a finite but large interval of ranges Lr — r2 — ri and compare the result to themeasured distribution. Because the measured source levels depend on the propagationloss used, it is easier to compare the distribution of levels measured at the receiver.This distribution can be determined by a variation of Eq. 6.5 as:Dr(RL) = 28r n T(RL) r E(RL + PLr). (6.7)r =r 1This test was performed over the range intervals of 0 - 1 km, 1 - 3 km, 3 - ‘7 km and7 - 15 km.The second technique for ensuring the consistency of the model is to use the truesource level distribution calculated from Eq. 6.6 to predict the range distributionof detected events D(r), and compare to the measured range distribution. Thisdistribution can be determined by summing Eq. 6.4 from the minimum to maximumdetected source levels in the following manner:SLmaxD(r) = 27rr6r m E(SL) T(SL — PLr). (6.8)SL=SLmin111Thus, by using a given level of ice absorption to determine propagation loss, thenumber of events detected at each source level D(SL) over some time period is firstmeasured. This number is then combined with a given detection threshold functionT(RL) to estimate the true source level distribution n E(SL) using Eq. 6.6. The truesource level distribution is then used to estimate the observed source level distributionof detected events over a given range interval using Eq. 6.7 and the observed rangedistribution of detected events using Eq. 6.8. Finally, by varying the ice absorptionsand the detection threshold function, the error between the measured and calculateddistributions of Eqs. 6.7 and 6.8 can be minimized resulting in the best-fit true sourcelevel distribution. The absolute level of nE(SL) can be further confirmed by summingEq. 6.5 over all detected source levels or by summing Eq. 6.8 over all ranges andcomparing the results to the total number of events detected.The median number of events detected over the survey for all source levels was3.1 events per minute. This number converts to a median source level distribution asshown in Fig. 6-22. This figure shows that a median of four events occur per squarekilometer per minute over a 1 dB band of source levels centered at 110 dB//Pa2/Hzat 1 m. It also shows that more events occur at the low power end of the distributionand the number of events decreases with increasing source level. The decrease inoccurring events with increasing source level is proportional to SL with a 0.08for source levels below 160 dB//iPa2/Hz at 1 m and a 0.12 for source levelsabove. The mean and maximum number of events detected per minute are 6.6 and31.0 respectively. Thus, the mean and maximum source level distributions can beobtained by multiplying the number of events shown in Fig. 6-22 by approximately2.1 and 10.0 respectively.112The shape of the source level distribution curve of Fig. 6-22 (a linearly decreasingfunction on a log-dB scale of the number of events versus source level with a changeto a more steeply decreasing slope at higher source levels) agrees with the measured distribution of earthquake magnitudes reported by Gutenberg and Richter.49This relation is often an indication of a self similarity feature of the quantity beingmeasured.5°The general idea is that large earthquakes are composed of many smallerearthquakes which in turn are composed of even smaller earthquakes and that thesmallest “building block” earthquakes and the resulting large earthquakes have manysimilar characteristics (are scale invariant). Aki5’ proposed that the magnitude distribution curve of earthquakes may be related to their self similar (or fractal) natureand this relation was later shown to be so by Rundle.52 The self similarity structureof earthquakes was also shown in their spatial53 and temporal54’5 distributions.Several authors5658 have developed simple slip-stick mechanical models whichare capable of reproducing the spatial, temporal and strength distributions of earthquakes. These models are usually a system of blocks hung by springs from a stationaryplatform. The blocks are then coupled together by other springs and rest on a frictional moving surface. Consider first a single block hung by a spring from a stationaryplatform and resting on a moving surface. The block sticks to the moving surface andmoves with it until the spring force exceeds the frictional coefficient between the blockand the surface. At this point, the block slips along the surface back towards its originbeneath the spring. Now consider many blocks hung from the stationary platform ina two dimensional grid and connected by horizontal springs. If the horizontal springconstant is much smaller than the vertical spring constant, when one block slips, itdoes not effect the neighboring blocks. This situation corresponds to that in which113small earthquakes occur frequently when the crust is sufficiently fractured. Whenthe horizontal spring constant is greater than the vertical spring constant, the slip ofone block propagates to many surrounding blocks corresponding to the situation ofa large earthquake. A slip-stick mechanism has also been shown to be important inthe propagation of thermal ice cracks23’59 and this mechanism may explain why themagnitude distributions of earthquakes and thermal ice cracks are similar.Although the distribution of source levels for thermal ice cracks was obtainedby examining source levels at 96 Hz only, the relatively flat power spectrum of icecracking shown in Fig. 6-19 suggests that this relation is valid for the entire frequencyrange of 2 - 200 Hz. Also, it should be noted that no attempt was made to distinguishbetween times of high and low ambient noise levels, so the distribution of Fig. 6-22represents an averaged distribution for the entire observation period.The self-consistency of the model is shown by the comparisons of the measuredand calculated distributions for Eqs. 6.7 and 6.8 shown in Figs. 6-23 and 6-24respectively. Only the range interval of 0 - 1 km is shown in Fig. 6-23; the remainingrange intervals of 1 - 3 km, 3 - 7 km and 7 - 15 km have similar accuracies. Thebest-fit ice absorptions determined by the model were 8.0 dB/) for the compressionalwave and 12.0 dB/.) for the shear wave. These absorption rates are high compared tothe ice absorptions previously measured for smooth ice18 but may be realistic for thisfairly rough ice, especially when including the effects of under-ice scattering. Also,these values of absorption agree favorably with those required in the pressure ridgemodel and the resulting source levels agree within 1 - 2 dB//iPa/Hz of the sourcelevels determined using the direct arrival only.1146.5 Modeled Thermal Ice Cracking NoiseUsing the temporal and source level distributions given in Fig. 6-22 alongwith the fact that the events are spatially uniform, the mean energy input into theice per square kilometer as a function of source power can be determined as:P2(P) = St N(P) P (6.9)where N(P) is the mean number of events occurring at source power P per squarekilometer per minute, and St is the mean time duration in minutes of an event. Fromour data, St was found to be approximately 0.1 seconds. The total average energyentering the ice per square kilometer is then obtained by summing over all sourcepowers as:PmaxP2= St > N(P) P. (6.10)P=PrninWhen summing over only the observed source powers (Pmin = 0.1 Pa2, Pmax = 106Pa2), the average energy input is 116.6 dB/km2//pPa at 1 m. When extended 50dB above and below the observed powers (from 106 - 1011 Pa2) using the SL fit(SL = 10 Log P with P given in iiPa2), the average energy input increased by only1.1 dB. Because the true minimum and maximum source powers are unknown, theaverage energy input of 116.6 dB/km2/iPa at 1 m is used and the small increaseobtained by extending the range of source powers will be ignored.Using the average energy input per square kilometer calculated above, the average115energy received at a hydrophone from a given source location is simply the averageinput minus the propagation loss associated with that location. The average receivedenergy is then summed over all possible source locations to determine the ambientnoise produced by thermal ice cracking. Our model steps out in range increments of100 m to a total range of 200 km and determines the input energy for each rangeincrement based on the area of the annulus ir(rmax+rmjn)(rmax—rm;n). Propagationloss is then averaged from the midrange of the annulus to 21 equispaced receiversfrom 100- 300 m depth using the Kuperman-Schmidt propagation model SAFARI.An arbitrary maximum range of 200 km was chosen based on the propagation modelused but later results will show that much shorter maximum ranges could have beenused.Finally, the short term fluctuations in the strength distribution and the randomstatistical fluctuations in the spatial distribution of thermal ice cracking must bemodeled. The strength distribution of Fig. 6-22 represents an average distributionover the entire observation period. Insufficient data exist to produce separate distributions for intense or quiet times of ice cracking and these times are assumed to resultin a strength distribution curve which is simply shifted up or down respectively. Thisshift can be estimated for real data by comparing the number of events detected perminute for a data sample with the average number of events detected per minute forall the data used to produce the strength distribution curve. For all 69 data files usedin the strength distribution curve, the average number of detected events per minutewas 6.6. Thus, for Fig. 4-id, with an average of 22.2 events detected per minute, thestrength distribution curve is assumed to be shifted up by a factor of 3.4. This shiftin turn results in an increase in the average input energy per square kilometer by a116factor of 10 Log(3.4) = 5.3dB.The random statistical fluctuations in the spatial distribution have a large effecton the ambient noise levels when occurring at small range. These effects are includedby varying the average input noise level within 1 km of the hydrophone. Ignoring shortterm strength fluctuations, the modeled ambient noise due to thermal ice crackingis shown in Fig. 6-25 for the input noise level within 1 km of the array varyingfrom -20 to +20 dB above the average input noise level. When short term strengthfluctuations are included, the curves in Fig. 6-25 are simply shifted up or down by theappropriate factor. Thus, to model the real data in Fig. 4-id, the curves of Fig. 6-25are shifted up by 5.3 dB as determined in the previous paragraph. Ice and bottomcharacteristics used to produce Fig. 6-25 are given in Table 5-I with the exceptionthat compressional and shear wave absorptions in the ice were five times higher in anattempt to better compensate for the effects of under-ice roughness. For frequenciesabove 40 Hz, Fig. 6-25 shows the general characteristics of the real data shown inFigs. 4-la to 4-id.A comparison between real and modeled data above 40 Hz shows the validity ofallowing the close range input noise to vary. Fig. 4-la, which is approximated by alower close range than average modeled input noise level, represents data files withvery few detected thermal ice cracking events. Of those events detected, only twoevents were within 1 km of the array and the received level of both these events wasless than the ambient noise level at the time (events with a SNR < 0 can still bedetected because of the array gain of the vertical array). At the other extreme, Fig.4-id, which is approximated by a higher close range input noise level, represents datafiles with many detected thermal ice cracking events. These data files were each found117to contain several events within 1 km of the array with received levels at least 15 dBhigher than the background ambient noise level. No other data files contained suchevents.Finally, the thermal ice cracking model outlined above is capable of determiningthe relative contribution of close versus far range events in producing the ambientnoise. Fig. 6-26 shows the required range to model 80% (within 1 dB) of the totalnoise energy produced from all thermal ice cracking events out to a range of 200 km.This result is independent of the short term strength fluctuations but does depend onthe short term spatial fluctuations. For frequencies near 10 Hz, if the local thermalice cracking level is low (within 1 km), events beyond 100 km must be considered tomodel the thermal ice cracking noise. However at this frequency, unless local thermalice cracking levels are very high, the ambient noise is dominated by pressure ridgingand thermal ice cracking need not be considered. For frequencies above 40 Hz, eventswithin 30 km range suffice to model within 1 dB of the total ambient noise. Fig.6-27 shows that for frequencies above 40 Hz, only the events within 6 km range arerequired to model 50% (within 3 dB) of the total ambient noise.118Table 6-I. Mean, variance and skewness of the distribution of the value of m in thesinm6 model of source directivity as a function of range and frequency.Range (m) Frequency (Hz) Mean Variance Skew0-2000 96.7 0.87 0.21 0.681300-2000 48.4 0.95 0.14 -0.10650-2000 96.7 1.04 0.15 0.54430-2000 145.1 1.05 0.18 0.470-1300 48.4 0.68 0.24 0.880-650 96.7 0.68 0.23 1.580-430 145.1 0.70 0.44 1.7311915105ci) Qc3)c-510-15-15 -10 -5 0 5 10 15Range (km)Figure 6-1. Spatial distribution of all detected events about the verticalarray which is centered at (0,0) km range. The number of events detecteddecreases with increasing range due to propagation loss but the lack ofpreferred locations or directions suggests a spatially uniform distributionof occurring events.1201.• ••.• ••I I I I I I I I I I I I I I I I— •••• ••• • ••••••I• I•• I•• •I I •• •I.• I— I.• II• I • II II.• I II•I••IIII••• •i• .1 ••I•I III•••••• I I • I I• I II II•I •.I IIII•II.I•II II I I I I I I I I t I i i I I I I I I i iHor-izor,ta 1-DI\-,wLDU)U]UiLWQ-IFigure 6-2. Comparison of the vertical directivity of a pair of horizontalpoint forces with the sinm8 model for angles less than 45°. The solid lineshows the directivity of the horizontal point forces while the dashed lineshows the best fit of the sinm8 model with the values of m = 0.89, 0.90,1.05 and 1.54 for 50, 100, 150 and 200 Hz respectively.0121Hor-izcrta 1Figure 6-3. Comparison of the vertical directivity of a vertical point forcewith the sinm6 model for angles less than 450 The solid line shows thedirectivity of the vertical point force while the dashed line shows the bestfit of the sintm8 model with the values of m = 1.00, 1.00, 1.14 and 1.62for 50, 100, 150 and 200 Hz respectively.U)UiLDLI)(1)UILu-I050Hz100 Hz150 Hz200 Hz122wrc-UIwC3U)U)wLL1)U-I0Figure 6-4. Vertical directivity of an ice-cracking event at 450 m range.The output is shown on a polar scale with each cross representing themeasured pressure level in the direct arrival (after correction for propagation loss) for a different hydrophone, or source angle. The dashed linecorresponds to the best fit to a model of the form sintmO with the valueof m = 0.76.123Hor-izonta 1m13QiCDU)U)QJL0Figure 6-5. Vertical directivity of an ice-cracking event at 300 m rangewhich shows a small fluctuating pattern about the best fit to the modelsinO. The value of m = 1.00.124HorizontalI-’-UQiCD1J3wwC0Figure 6-6. Vertical directivity of an ice-cracking event at 500 m rangewhich does not fit the sintmO model. The best fit with a value of m0.58 is shown.125C0-Q(1)>C’)Ca)0.0000.00.2000.1500.1000.0500.5 1.0 1.5 2.0 2.5 3.0mFigure 6-7. Normalized distribution of the directivity index m (binwidth= 0.1) for all events within 2000 m range and with at least a 3 dBsignal to noise ratio. Pressure levels were determined by averaging overa one octave band centered at 96 Hz.126C0-4-D-Q-4-(I)>-4-C’)Cci)0.0000.00.2000.1500.1000.0500.5 1.0 1.5 2.0 2.5 3.0mFigure 6-8a. Normalized distribution of the directivity index m (binwidth= 0.1) for one octave frequency bands centered at 48 Hz, 96 Hz and145 Hz. Distributions of m are given for all events less than approximately40 wavelengths.127C0•1D-o-eu)a>%4-C,)Ca)a0.0000.00.2000.1500.1000.0500.5 1.0 1.5 2.0 2.5 3.0mFigure 6-8b. Normalized distribution of the directivity index i-n (binwidth= 0.1) for one octave frequency bands centered at 48 Hz, 96 Hzand 145 Hz. Distributions of m are given for all events greater than 40wavelengths but less than 2000 m.128LI)-a’\-,QiLU)(I)QiLU)00Figure 6-9. Vertical directivity of an ice-cracking event at 300 m rangeshowing little or no dependence on frequency.SD100‘SD200129m-DcvLDU)U)cvLUFigure 6-10. Vertical directivity of an ice—cracking event at 450 m rangeshowing strong dependence on frequency.130Hor—jzor,taLfl.1-U,wLDU)InIiiLWQ_I0Figure 6-11. Vertical directivity of an ice-cracking event at 1150 m rangeshowing some dependence on frequency.60Hz100 Hz160 Hz200 Hz131Horizonta ILflI-” Q-UtQiC(I)(1)QiLu,u-ICFigure 6-12. Vertical directivity of an ice-cracking event at 140 m rangeshowing an excess in pressure level of 1 dB - 2 dB above the sinm8 modelat a source angle of 600 - 65°. The value of m = 0.56.132L D P_____________I30_---sq79 11’________1)’10221261S0______________198222____________________________270___0. 1 0.2 0.3 0.q 0.6 0.6Tima (sec)Figure 6-13. Synthetically generated time series for an ice-cracking eventat 500 m range and 5.5 m depth in 6 m thick ice. Both the leaked longitudinal plate wave (L) and direct arrival acoustic mode (D) are clearly seenacross the entire array (at all depths) while the low frequency flexuralwave (F) is seen following the direct arrival (at 0.35- 0.6 sec) only forthe 18 m depth.133Hor-izorita 1m-oUiCDU)U)UiC0Figure 6-14. Vertical directivity of a synthetically produced ice crack at500 m range and 5.5 a depth in 6 m thick ice. The ice crack is modelledas a monopole source and the compressional absorption in the ice is 0.2dB/wavelength. The best fit to the sintm8 model has a value of m =0.71.134-UQiL3U)U)wL0Figure 6-15. Vertical directivity of a synthetically produced ice crackat 500 m range and 0.25 m depth in 6 m thick ice. The compressionalabsorption in the ice is 0.2 dB/wavelength and the value of m 0.88.135Hor i. zonta IU)‘-4‘‘0-UIwLU)U)QJLci’CFigure 6-16. Vertical directivity of a synthetically produced ice crackat 500 m range and 0.25 m depth in 6 m thick ice. The compressionalabsorption in the ice is 2.0 dB/wavelength and the value of m = 0.99.136Hor—izorta ILI)‘-4r\o-a1wLDw(nwLU-ICFigure 6-17. Vertical directivity of a synthetically produced ice crack at2000 m range and 0.25 m depth in 6 m thick ice. The compressionalabsorption in the ice is 2.0 dB/wavelength and the value of m = 1.10.The receivers were at a depth 4 times larger than in figures 6-13 to 6-16so the same source angles would be used as for those figures.137210—0.0___ABOVE 110—-- 105-110__100-10595-10090- 9585- 9080- 851 75- 8070- 75_ _65 701 60- 65BELOW 60Figure 6-18. Received power spectrum of the hydrophone at 150 m depthas a function of time for the event shown in figure 3-8.N>0Cci)110U100.2 0.4 0.6 0.8Time (sec)1.0138140 I130120>-JG)C.) 1100C/)100 I I I liii I I liii10 100Frequency (Hz)Figure 6-19. Source spectral level of a typical ice-cracking event.139>.‘_4Q-a’-1nj-Q0CC0-IUwUiUNL+SNRRace i ved LevelFigure 6-20. Detection threshold function indicating the probability ofdetecting an event of a given received level. An event with a received levelof approximately 2 dB - 3 dB (SNR) above the noise level (NL) shouldalways be detected, with a decreasing probability of detection for eventswith received power below this level. The rate of decrease is determinedby the model for calculating the source level distribution.140ci)cDci)4-•’cI)C>0)•—>III ()0.4-C543210Received Level (dB//iPa2/Hz)Figure 6-2 1. Distribution of received levels of all detected events. Receivedlevel is the source level minus the propagation loss from the source toreceiver.40 60 80 100 120141I I I f I I I I I I ( I I I I I>CQ)c—J-4->NWICt5z’a-110-110-2ioloio200Source Level (dB//tPa2/Hz at 1 m)Figure 6-22. Median number of events per square kilometer per minuteper 1 dB//tPa/Hz at 1 m source level interval versus source level.100I I I I I120 140 160I I I1801426504.)C>>III 0.4-C10Figure 6-23. The measured and modeled normalized distributions of received levels for all detected events over the range interval 0 - 1 km. Themodeled distribution is calculated from Eq. 6.7 using the source level distribution determined previously and shown in Fig. 6-22.40 60 80 100 120Received Level (dB//iPa2/Hz)143(l)W•1C C>0—CCC0•’- -cDa)Qa3.02.52.01.51.00.50.0Range (km)Figure 6-24. The measured and modeled normalized distributions of sourceranges for all detected events. The modeled distribution is calculated fromEq. 6.8 using the source level distribution determined previously and shownin Fig. 6-22.0 2 4 6 8 10 12 1414410090Nzc’Ja. 80j706005040Frequency (Hz)Figure 6-25. Modeled thermal ice cracking noise. The close range inputnoise (within 1 km of the array) is allowed to vary from -20 dB (lowercurve) to +20 dB (upper curve) above the average input noise level witha 5 dB interval between individual curves. Compressional and shear waveabsorptions in the ice of 10.0 and 15.0 dB/A respectively were used.1 10 10014514012010080c,)C6040200Frequency (Hz)Figure 6-26. Required range to model 80% (within 1 dB) of the totalnoise produced from all thermal ice cracking events out to a range of 200km. Compressional and shear wave absorptions in the ice of 10.0 and 15.0dB/) respectively were used with the close range input noise (within 1km) varying from -20 dB (upper curve) to +20 dB (lower curve) above theaverage input noise level with a 5 dB interval between individual curves.1 10 100146605040-‘‘30c,)20100Frequency (Hz)Figure 6-27. Required range to model 50% (within 3 dB) of the totalnoise produced from all thermal ice cracking events out to a range of 200km. Compressional and shear wave absorptions in the ice of 10.0 and 15.0dB/A respectively were used with the close range input noise (within 1km) varying from -20 dB (upper curve) to +20 dB (lower curve) above theaverage input noise level with a 5 dB interval between individual curves.1 10 100147Chapter 7Modeled Ambient NoiseA comparison of the two component noise model with the four classes ofmeasured real ambient noise spectra is shown in Figs. 7-la to 7-ld. The sound speedprofile shown in Fig. 3-2 along with the measured and estimated bottom and iceparameters from Table 5-I were used to determine propagation loss for the modelwith the exception that the compressional and shear wave absorptions used in theice were five times higher (10.0 and 15.0 dB/\ respectively) in an effort to bettercompensate for the effects of under-ice roughness (as outlined in chapters 5.4 and6.4). These figures show that the measured ambient noise spectra can be reproducedby a single active pressure ridging event, along with a distribution of thermal icecracking events. For frequencies below 40 Hz, the ambient noise spectrum may bedetermined by the range and level of the strongest received active pressure ridge.For frequencies above 40 Hz, the ambient noise spectrum is determined by thermalice cracking, with overall levels and spectral shape dependent on the intensity of icecracking and the relative strength of local to average events. For purposes of ourmodel, local events are considered to be within 1 km of the hydrophone.It was also noted that all of the data files used in classes 1 and 2 for real dataoccurred during a 66 hour span within the middle of the experiment while all exceptone of the data files used in classes 3 and 4 occurred before or after this time. Thisseparation suggests that active pressure ridging was occurring at approximately 40km range during the entire 100 hours of ambient noise measurements and that for a14866 hour span of time in the middle of the experiment, a much stronger active pressureridge built itself at a range of approximately 70 km. Although the stronger and moredistant active pressure ridge appears to be associated with times of weaker thermalice cracking, no correlation is assumed due to the short time sample and the factthat the two source mechanisms are controlled by different environmental conditions.Also, although the frequency of the infrasonic peak in the ambient noise spectrumis related to the range of the strongest received active pressure ridge, estimating itsrange is likely to be inaccurate because of possible differences in active pressure ridgespectra and the inadequacies of current propagation loss modeling for the Arctic.All the events as shown in Fig. 3-7 also occurred only during the middle 66 hoursat which time a strong active pressure ridge occurred at a range of approximately70 km. For frequencies below 40 Hz, the spectra of these events were found to havesimilar shapes with higher overall levels as the ambient noise spectra at these times.This observation suggests that these events originate from the active pressure ridge at70 km range and may correspond to the several second long bursts of intense energy(see Fig. 3-9) found in active pressure ridges.Finally, it can be noticed from Figs. 7-la to 7-ic that the modeled ambient noisespectra have negative slope below 3 Hz while the measured spectra have a positiveslope. The reason for this discrepancy is unknown but may be due to the assumedspectral shape of active pressure ridging which was based on measurements from asingle event using equipment which was not very reliable below 4 Hz.14910090NC\ict50:±-o>-J60Cl)0z5040Frequency (Hz)Figure 7-la. Comparison of two component noise model (solid) to average ambient noise (dash) and standard deviation (dash-dot) of real noise.Modeled ridge levels are relative to that shown in Fig. 5-1. Thermal icecracking levels are relative to the average input energy determined fromFig. 6-22 with local levels applied above this but only for ranges within Ikm of the hydrophone. Modeled noise for ridge of level +3 dB at 70 kmrange with thermal ice cracking at -8.2 dB and local ice cracking at -3 dB;versus class 1 real data from Fig. 4-la.1 10 10015010090Nc”J0a)>a)-JG) 60U)0z5040Figure 7-lb. Comparison of two component noise model (solid) to average ambient noise (dash) and standard deviation (dash-dot) of real noise.Modeled ridge levels are relative to that shown in Fig. 5-1. Thermal icecracking levels are relative to the average input energy determined fromFig. 6-22 with local levels applied above this but only for ranges within 1km of the hydrophone. Modeled noise for ridge of level +3 dB at 70 kmrange with thermal ice cracking at +0.4 dB and local ice cracking at +0dB; versus class 2 real data from Fig. 4-lb.1 10 100Frequency (Hz)15110090Nc’.jC”0cJ)>-J60Cl)0z5040Frequency (Hz)Figure 7-ic. Comparison of two component noise model (solid) to average ambient noise (dash) and standard deviation (dash-dot) of real noise.Modeled ridge levels are relative to that shown in Fig. 5-i. Thermal icecracking levels are relative to the average input energy determined fromFig. 6-22 with local levels applied above this but only for ranges within1 km of the hydrophone. Modeled noise for ridge of level -5 dB at 40 kmrange with thermal ice cracking at -3.0 dB and local ice cracking at +0 dB;versus class 3 real data from Fig. 4-ic.1 10 10015210090NcJUQ)>-J60Cl)0z5040iFrequency (Hz)Figure 7-id. Comparison of two component noise model (solid) to average ambient noise (dash) and standard deviation (dash-dot) of real noise.Modeled ridge levels are relative to that shown in Fig. 5-1. Thermal icecracking levels are relative to the average input energy determined fromFig. 6-22 with local levels applied above this but only for ranges within1 km of the hydrophone. Modeled noise for ridge of level -5 dB at 40 kmrange with thermal ice cracking at +5.3 dB and local ice cracking at +4dB; versus class 4 real data from Fig. 4-id.10 100153Chapter 8SummaryThis thesis has shown that the spring-time Arctic ambient noise spectra measured in the pack-ice over the frequency band 2 - 200 Hz can be modeled by a combination of active pressure ridging and thermal ice cracking. A single or few activepressure ridges at ranges of tens of kilometers produces the low frequency end of theambient noise spectra up to approximately 40 Hz while a distribution of thermal ice-cracking events produces the higher frequency end. Over 50% of the ambient noiseproduced by thermal ice cracking is generated by events occurring within 6 km ofthe hydrophone while over 80% is generated by events occurring within 30 km of thehydrophone.In developing the two-component noise model of low frequency ambient noise,several characteristics of both thermal ice cracking and active pressure ridging weredetermined. These are outlined in the following paragraphs.It was found that the energy measured in the water from events occurring in theice was dominated by the acoustic source with contributions from leaked plate wavesfalling off rapidly away from the source due to scattering and absorption in the ice. Itwas also found that the measured vertical source pressure directivity of individual icecracking events approximated a dipole radiation pattern for events at ranges beyond40 wavelengths. For shorter ranges, the radiation pattern often became either lessdirectional or more complex. The directivity is also expected to become more complex154at higher frequencies. These results could be reproduced using a simple monopolesource in the ice but are also consistent with either a pair of horizontal point forcesor a vertical point force in the ice. Superimposed on this radiation pattern is a 3 dBincrease in pressure level near 600. Using plate wave theory, this increase in pressurelevel may be explained as the leaked longitudinal plate wave coinciding with theacoustic mode at these source angles. Using plate vibration theory for a thin plate,this increase in pressure level may be explained by the plate vibration impedancevanishing at this angle.The spatial distribution of thermal ice-cracking events measured in the roughArctic pack ice was found to be consistent with a large-scale spatially isotropic distribution of noise sources. The distribution of source intensity levels measured from110 to 180 dB//Pa2/Hz at 1 m decreases with increasing source level as 10with a 0.08 for source levels below 160 dB//iPa2/Hz at 1 m and a 0.12 forsource levels above. The shape of this source level distribution curve agrees with themeasured distribution of earthquake magnitudes and this result is believed to be dueto the self similarity nature of the slip-stick process involved in both earthquake andice crack propagation. A median of four events per square kilometer per minute havesource levels within a 1 dB band centered at 110 dB//Pa2/Hz at 1 m. The meanand maximum number of events occurring at any source level are approximately 2.1and 10.0 times the median.Evidence was also presented in this thesis that the infrasonic peak found in theArctic ambient noise is due to the band-pass filtering effect of the propagation losson the noise generated by ridge building. A ridge building event was shown to have apower spectrum which increases with decreasing frequency. However, the propagation155loss was shown to have a minimum near 30 Hz and increase dramatically below 20 Hzdue to leakage of energy out of the near surface sound speed channel. Combining thesource spectrum and propagation loss produces a spectrum at longer ranges (beyond40 km) which has a peak near 10 Hz. This explanation is supported by the observedpower spectra of ice cracking events as a function of range and by the occurrence ofseveral distant events in the data which have time scales and power spectra consistentwith both ridge building and ambient noise.Finally, it was shown that it is possible to use local thermal ice-cracking events tomeasure the reflectivity of the seabed as a function of reflection angle. This methodcircumvents the difficulties and expense of introducing artificial sound sources throughthe thick Arctic pack ice.156References1. A.R.Milne and J.H.Ganton, “Ambient noise under Arctic sea ice,” J.Acoust.Soc.Am., 36, 855-863 (1964).2. A.R.Milne, “Thermal tension cracking in sea ice: A source of underice noise,”J.Geophys.Res., 77, 2177-2192 (1972).3. R.S.Pritchard, “Arctic Ocean background noise caused by ridging of sea ice,”J.Acoust.Soc.Am., 75, 419-427 (1984).4. B.M.Buck and J.H.Wilson, “Nearfield noise measurements from an Arctic pressure ridge,” J.Acoust.Soc.Am., 80, 256-264 (1986).5. F.Press and M.Ewing, “Propagation of elastic waves in a floating ice sheet,”Trans.Am.Geophy.Union., 32, 673-678, (1951).6. Y.Xie and D.M.Farmer, “The sound of ice break-up and floe interaction,”J.Acoust.Soc.Am., 91, 1423-1428, (1992).7. B.E.Miller and H.Schmidt, “Observation and inversion of seismo-acoustic wavesin a complex arctic ice environment,” J.Acoust.Soc.Am., 89, 1668-1679, (1991).8. P.J.Stein, “Acoustic monopole in a floating ice plate,” Doctoral thesis, Department of Ocean Engineering, MIT., Cambridge, Woods Hole OceanographicInstitution, Woods Hole, Massachusetts, (1986).9. I.A.Viktorov, Rayleigh and Lamb Waves, Plenum Press, New York, 1967.10. Y.Xie, “An acoustic study of the properties and behavior of sea ice,” Doctoral157thesis, Department of Oceanography, University of British Columbia, Vancouver, British Columbia, Canada, (1991).11. Y.Xie and D.M.Farmer, “Acoustical radiation from thermally stressed sea ice,”J.Acoust.Soc.Am., 89, 2215-2231, (1991).12. W.A.Kuperman and H.Schmidt, “Rough surface elastic wave scattering in ahorizontally stratified ocean,” J.Acoust.Soc.Am., 79, 1767-1777, (1986).13. H.Schmidt, “SAFARI, Seismo-Acoustic Fast field Algorithm for Range Independent environments, User’s guide,” Rep. SR-113, SACLANT Undersea ResearchCenter, San Bartolomeo, Italy, 1988.14. R.H.Bourke and A.S.Mcbaren, “Contour mapping of Arctic Basin ice draft androughness parameters,” J.Geophys.Res., 97, 17,715-17,728, (1992).15. L.A.Mayer and J.Marsters, “Measurements of geophysical properties of Arcticsediment cores,” DREP Contractors Report 89-19, Defence Research Establishment Pacific, Victoria, B.C., Canada, (1989).16. J.P.Todoeschuck, J.M.Ozard and J.M.Thorleifson, “Refraction and reflectionexperiments with a vertical array of hydrophones in the Lincoln Sea,” EOS, 69,1320 (1988).17. D.M.F.Chapman, “Surface-generated noise in shallow water: a model,” Proc.I.O.A., 9 (4), 1-11, (1987).18. G.H.Brooke and J.M.Ozard, “In-situ measurement of elastic properties of seaice,” in Underwater Acoustic Data Processing, Kluwer Academic Publishers,113-118, 1989.15819. P.Zakarauskas, C.J.Parfitt and J.M.Thorleifson, “Statistiques de bruits ambiants transitoires dans 1’Arctique,” in Proceedings of the First French Conference on Acoustics, Lyon, France, April 1990, Edited by P.Filippi andM.Zakharia, (Les editions de Physique, Les Ulis Cedex, France),pp.733-736.20. P.Zakarauskas, C.J.Parfitt and J.M.Thorleifson, “Automatic extraction of springtime Arctic ambient noise transients,” J.Acoust.Soc.Am., 90, 470-474 (1991).21. J.H.Ganton and A.R.Milne, “Temperature-and wind-dependent ambient noiseunder midwinter pack ice,” J.Acoust.Soc.Am., 38, 406-411 (1965).22. A.R.Milne, J.H.Ganton and D.J.McMillin, “Ambient noise under sea ice andfurther measurements of wind and temperature dependence,” J.Acoust.Soc.Am., 41, 525-528 (1967).23. D.M.Farmer and Y.Xie, “The sound generated by propagating cracks in seaice,” J.Acoust.Soc.Am., 85, 1489-1500 (1989).24. I.Dyer, “Speculations of the origin of low frequency Arctic Ocean noise,” in SeaSurface Sound, ed: B.R.Kerman, Kluwer Academic Publishers, 1988.25. P.J.Stein, “Interpretation of a few ice event transients,” J.Acoust.Soc.Am., 83,617-622 (1988).26. N.C.Makris and I.Dyer, “Environmental correlates of pack ice noise,”J.Acoust.Soc.Am., 79, 1434-1440 (1986).27. B.M.Buck, “Arctic acoustic transmission loss and ambient noise,” in ArcticDrifting Stations, ed: J.E.Sater, The Arctic Institute of North America, 1968.15928. A.S.Thorndike and R.Colony, “Sea ice motion in response to geostrophic winds,”J.Geophys.Res., 87, 5845-5852 (1982).29. R.R.Parmerter and M.D.Coon, “Model of pressure ridge formation in sea ice,”J.Geophys.Res., 77, 6565-6575, (1972).30. A.S.Mcbaren, “Analysis of the under-ice topography in the Arctic Basin asrecorded by the USS Nautilus during August 1958,” J. Arctic Institute of NorthAmerica, 41, 117-126, (1988).31. O.I.Diachok, “Effects of sea-ice ridges on sound propagation in the Arctic Ocean,”J.Acoust.Soc.Am., 59, 1110-1120, (1976).32. R.J.Tjrick, Principles of Underwater Sound, McGraw-Hill Book Company, NewYork, 1983.33. A.R.Milne, “Sound propagation and ambient noise under sea ice,” in Underwater Acoustics, Vol 2, ed: V.M.Albers, Plenum Press, New York, 1967.34. M.J.Sheehy and R.Halley, “Measurement of the attenuation of low-frequencyunderwater sound,” J.Acoust.Soc.Am., 29, 464-469 (1957).35. R.J.Urick, “Low-frequency sound attenuation in the deep ocean,” J.Acoust.Soc.Am., 35, 1413-1422 (1963).36. A.C.Kibblewhite and R.N.Denham, “Long-range sound propagation in the SouthTasman Sea,” J.Acoust.Soc.Am., 41, 401-411 (1966).37. J.E.Burke and V.Twersky, “Scattering and reflection by elliptically striated surfaces,” J.Acoust.Soc.Am., 40, 883-895, (1966).16038. E.Livingston and O.Diachok, “Estimation of average under-ice reflection amplitudes and phases using matched-field processing,” J.Acoust.Soc.Am., 86, 1909-1919, (1989).39. R.J.Nielsen et.al., “TRISTEN/FRAM IV Arctic ambient noise measurements”,NTJSC Technical Document 7133, Naval Underwater Systems Center, New London, Connecticut, 1984.40. A.J.Langley, “Acoustic emission from the Arctic ice sheet,” J.Acoust.Soc.Am.,85, 692-701 (1989).41. J.S.Kim, “Radiation from directional seismic sources in laterally stratified mediawith application to Arctic ice cracking noise,” Doctoral thesis, Department ofOcean Engineering, MIT., Cambridge, Massachusetts, 1989.42. I.Dyer, “Source mechanisms for Arctic Ocean ambient noise,” in Sea SurfaceSound (2), Natural Physical Sources of Underwater Sound, ed: B.R.Kerman,Kluwer Academic Publishers, 1993.43. A.R.Milne, “Statistical description of noise under shore-fast sea ice in winter,”J.Acoust.Soc.Am., 39, 1174-1182 (1966).44. M.Townsend-Manning, “Analysis of central Arctic noise events,” Master’s Thesis, Department of Ocean Engineering, MIT., Cambridge, Massachusetts, 1987.45. L.M.Brekhovskikh and Y.Lysanov, Fundamentals of Ocean Acoustics, Springerseries in electrophysics, New York, 1982.46. R.M.Hamson, “The theoretical responses of vertical and horizontal line arrays towind-induced noise in shallow water,” J.Acoust.Soc.Am., 78, 1702-1712, (1985).16147. P.Zakarauskas and J.M.Thorleifson, “Directionality of ice cracking events,”J.Acoust.Soc.Am., 89, 722-734, (1991).48. C.R.Greene and B.M.Buck, “The influence of atmospheric pressure gradient onunder-ice ambient noise,” J.Underwater Acoust., 28, 529-538, (1978).49. B.Gutenberg and C.F.Richter, Seismicity of the earth arid associated phenomena, Princeton University Press, Princeton, New Jersey, 1954.50. B.B.Mandelbrot, The Fractal Geometry of Nature, W.H.Freeman, New York,1982.51. K.Aki, “A probabilistic synthesis of precursory phenomena,” in Earthquake Prediction: An International Review, eds: D.W.Simpson and P.G.Richards, American Geophysical Union, Washington, D.C., 1981.52. J.B.Rundle, “Derivation of the complete Gutenburg-Richter magnitude-frequency relations using the principle of scale-invariance,” J.Geophys.Res., 94,12,337-12,342 (1989).53. Y.Y.Kagan and L.Knopoff, “The spatial distribution of earthquakes: The two-point correlation function,” Geophys.J.R.Astron.Soc., 62, 303-320 (1980).54. Y.Y.Kagan and L.Knopoff, “Stochastic synthesis of earthquake catalogs,”J.Geophys.Res., 86, 2853-2862 (1981).55. Y.Ogata, “Statistical models for earthquake occurrences and residual analysisfor point process,” J.Am.Stat.Assoc., 83, 9-27 (1988).56. J.M.Carlson and J.S.Langer, “Mechanical model of an earthquake fault,”Phys.Rev.A, 40, 6470-6484 (1989).16257. S.R.Brown, C.H.Scholz and J.B.Rundle, “A simplified spring-block model ofearthquakes,” Geophys.Res.Lett., 18, 215-218 (1991).58. M.Matsuzaki and H.Takayasu, “Fractal features of the earthquake phenomenonand a simple mechanical model,” J.Geophys.Res., 96, 19,925-19,931 (1991).59. Y.P.Doronin and D.E.Kheisin, Sea Ice, Amerind Publishing Co., New Delhi,India, 1977.60. M.J.Buckingham and S.A.S.Jones, “A new shallow-ocean technique for determining the critical angle of the seabed from the vertical directionality of theambient noise in the water column,” J.Acoust.Soc.Am., 81, 938-946 (1987).163Appendix ASeabed Reflectivity FunctionDetailed knowledge of the seabed properties is often needed in order to predict propagation characteristics in the ocean. Amongst the most useful parametersare the compressional and shear sound speeds. These two parameters are sufficient tocharacterize the reflection as a function of angle if the bottom is flat and no layeringis present. Layering or roughness of the bottom may modify the reflection functionin a frequency-dependent manner. The techniques traditionally used to extract theseabed properties are: laboratory measurement of physical parameters of coring orgrab samples, and in-situ acoustic reflection or refraction experiments. Analysis ofsamples provides unambiguous and precise values of physical properties of the substrata. Reflection and refraction measurements provide information about the acoustic effects of layering and roughness. Sources for reflection experiments are explosivecharges, although Buckingham and Jones6° have developed a technique in which theinter-sensor coherence of the ambient noise in shallow water is used to determine thecritical angle of the seabed. Unfortunately, the environmental model Buckinghamand Jones used does not include shear, and the difference in loss between the regionsbelow and above critical angle must be large for the technique to be used successfully.The concept of making use of the ambient noise as an acoustic source is appealing,as it eliminates the time-consuming task of deploying sources. This is a problemespecially in the Arctic environment, where one must cut through the thick (3 - 7m) Arctic pack ice in order to drop the charges. The harsh weather conditions in164the Arctic often makes it difficult to operate driffing equipment. Thus, a techniquewhich requires only one hole in the ice for deployment of the receiving array may helpincrease the possible number of sites surveyed.In this Appendix, a technique is described for measuring the reflection coefficientof the Arctic seabed as a function of angle with a single vertical array of hydrophones.Noise from naturally occurring jce cracking was used as the acoustic source. A complication arises compared to the use of charges as sources, since ice cracking noise hassome directivity. We parametrize the source directivity for every crack. The reliability of the method depends on the goodness of fit of the parametrization employedto the reflectivity function. That parametrization was found in chapter six to beadequate.The range of the source is first determined by comparing the difference in arrivaltimes across the array with those predicted by a ray-based model using both the directarrival and the multiple reflections from the seabed and under-ice surface. Then thesource directivity is determined using the direct path arrival only. The expected pressure level at angles corresponding to the bottom path is extrapolated using the sourcedirectivity index thus calculated. It is then straightforward to compute the bottomreflection coefficient as a function of grazing angle by using various combinations ofsource locations and hydrophones from different depths.For a given ice-cracking event, the vertical directivity of the pressure level of theevent is first parametrized as:P(&) = p jnm9 (A.l)as shown in chapter five. P0 is the source pressure level (in dB//1iPa/Hz at im) and165P(8) is the modeled radiated pressure level in the direct arrival at source angle 8,normalized back to 1 m. The continuous parameters m and P0 are found for each eventby minimizing the squared difference (Pd — P)2 where Pd is the measured pressurelevel received in the direct arrival. Over 80% of the 160 transients used to determinesource directivity were found to closely fit the above parametrization. Those eventsthat were found to fit the model are used to extract the reflectivity function fromthe seabed. The reflectivity function is found by comparing the pressure levels Prmeasured in the bottom-reflected arrivals at each hydrophone with those predictedby extrapolating from Eq. A.i using the parameters m and P0 extracted from thedirect arrivals (Fig. A-i). The difference between the two is assumed to be theloss due to reflection at the seabed. The arrival at each hydrophone corresponds to adifferent bottom scattering angle which is calculated using the ray propagation modelmentioned above. Thus, the reflectivity can be plotted against the bottom grazingangle as shown in Fig. A-2.The validity of the extrapolation of the directivity measured at low angles to higherangles was tested using direct arrivals from events close enough to span both low andhigh angles at the array. The test was performed the following way. All events thatmet the following conditions were selected from the database: 1 - Having a minimumof 3 dB S/N; 2 - Having at least five points arriving below 300; 3 - Having at leastone point arriving above 430• Thirty events were found to fit these criteria. Only thepoints arriving at angles below 30° were used for the fitting of the parameters m andP0. The pressure level at the other hydrophones was estimated by extrapolation ofEq. A.1. The difference between the measured and the extrapolated pressure levelsis shown in Fig. A-3. The difference is seen to cluster around 0 dB, except for a few166individual events. This result is in contrast with the ratio of Fig. A-2 for reflectedarrivals, which exhibits a definite fall off above 400.Fig. A-3 tells us that results from individual events are not reliable, but if a largenumber of events are used, with much overlap in their angle coverage, noise and errorsin fit are averaged out.The bottom reflectivity function measured for all events, using the method described, is shown in Fig. A-2 as a scatter diagram. Despite the large scatter in thereflectivity measurement, one can discern that the critical angle, defined as the angle at which the reflectivity begins to drop, occurs somewhere between 35° and 40°grazing angle. In order to make the extracted directivity function more explicit, themeasured values are averaged using a window 1.5° wide from 30° to 45° and 3° wideelsewhere. The reflectivity function thus averaged is displayed in Fig. A-4 along withthe reflectivity calculated using a single bottom layer with three different compressional sound speeds. The critical angle is now seen to occur near 350 and correspondsto a bottom compressional sound speed of approximately 1800 m/sec with minimumand maximum compressional speeds of approximately 1750 m/sec and 1900 m/secrespectively. A bottom shear speed of 300 m/sec with compressional and shear waveabsorptions of 0.5 dB/.\ and 0.25 dB/) were used. Increasing the absorption in thebottom produces a smoother and less abrupt drop in the reflectivity as the criticalangle is approached. The large uncertainty in the measurements however makes estimating the bottom absorptions very difficult and levels up to six times larger thanthose specified above were found to fit within the uncertainty. Increasing the shearspeed results in a quicker drop in reflectivity past critical angle and a very largerange of shear speeds was found to fit within the uncertainty. The reasons for the167large uncertainty in the measurements are discussed later.The estimated compressional sound speed in the bottom of 1800 m/sec with aminimum of 1750 m/sec and maximum of 1900 m/sec can be compared with the yelocity measurements of 1794 and 1683 rn/sec obtained from two bottom grab samplesand a measurement of 1980 + 18 rn/sec obtained from a seismic refraction survey.The value extracted in the present study compares rather well with those obtainedfrom the bottom grab samples. These grab samples were collected directly below themidpoint of the horizontal array. It does not agree very well with that obtained fromthe seismic refraction survey. The reason for this is likely due to the lower frequencyused in the refraction survey. The seismic refraction survey used a 60 lb charge detonated at 341 m depth resulting in a peak bubble pulse frequency of approximately21 Hz. The arrival times of the head wave from a shot at a range of 2686 m was thenmeasured at 8 of the hydrophones in the vertical array using a sampling rate of 2064samples per second. The lower frequency of the seismic refraction survey results ina deeper propagation path, and higher sound speeds, than those measured using theice-cracking events as sources.A rise in the reflectivity function around 60° is also very noticeable. This riseis not actually due to an increased reflectivity at this angle, but is an artifact dueto the use of an incomplete source directivity function (Eq. A.1) which does notinclude the leaky plate wave radiating around 60° from the ice at the site. Thisincrease in level was also seen in the directivity measurements of thermal ice crackingobtained in chapter six. If the source’s pressure levels at high angles are higher thanthose predicted by the model source function of Eq. A.1, the difference between themeasured and predicted sound pressure levels will be greater than it should, and will168show up in the reflectivity function. The extra radiated pressure level due to the platewave may now be quantified. It corresponds to a pressure level of approximately 3dB above the sintm 6 model with an approximately Gaussian distribution centered on600 and a standard deviation of approximately 50 The radiation angle 6L of a leakyplate wave into the water below corresponds to the critical angle between the twomedia. A grazing angle of 60° at the water-seabed interface corresponds to a sourceangle of 60.5°. The leaky ice plate wave speed is thus found to be 2914 m/sec.The value for the leaky plate wave speed can be compared to a direct measurementmade at the site using a hammer and geophones. The speed thus obtained was 3050 +100 rn/sec. The error interval for the value obtained from the present study overlapswith that obtained directly using the hammer and geophones.Several factors contribute to errors in both the grazing angle and the reflectivityvalue. Since the duration of each arrival is of finite length, it happens in some casesthat the direct and bottom-bounce partially overlap. A boundary is nonethelessplaced between the two arrivals, and therefore some of the direct arrival’s pressurelevels will be assigned to the bottom-bounce arrival. The pressure levels for the directarrival are underestimated, and that for the bottom-bounce overestimated. This errortends to happen for only a few channels toward the bottom of the array for distantevents, where the inter-arrival delay is at a minimum.The main source of scatter in Fig. A-2 is due to misfit of the value of the directivityindex m. A slight misfit im the value of m at the low angles corresponding to directarrivals leads to a rather large difference in the extrapolated pressure levels at thelarger angles corresponding to bottom-bounce arrivals. This error in extrapolation169of the expected bottom-bounce pressure levels leads to a systematic error in thereflectivity function for one given event. To make this result evident, the reflectivityfunction for a few events for the arrival at each hydrophone is plotted in Fig. A-5,where the arrivals corresponding to one single event are joined together by a solid line.One may now notice that although there can be a large difference in the reflectivityvalue corresponding to different events arriving at the same angle, the scatter aroundthe mean for a single event is much smaller. The scatter from the multiple eventsresults in a large uncertainty in the reflectivity at a given angle making the estimationof bottom properties other than the compressional sound speed very difficult.The residual scatter within a single event is thought to correspond to the irreducible error in measurement due to the relatively low signal-to-noise ratio, of between 3 and 10 dB and the short time length of the direct or bottom reflected arrival.This scatter becomes larger beyond the bottom critical angle for which signal to noiseratios are even lower. Moreover, since the same estimate of the noise level is usedfor all channels, small fluctuations of the ambient noise level with depth or inequalities in absolute calibration of each channel would lead to further random inequalitiesbetween channels.The estimate of arrival angles depends entirely on estimating time delays betweendifferent channels for a given arrival. If the time delays can be measured to anaccuracy of one sampling point for a given channel, then the accumulated error acrossthe array leads to an error in angle measurements of the order of 2.3° correspondingto an error in the bottom compressional sound speed of 45 rn/sec.The study of source directivity through the measurement of the pressure levels170in the direct arrivals yielded very little data on the relative strength and width ofthe leaky plate wave contribution. This lack of data is because very few eventsoccur close enough to the array for the direct arrival to occur at 600 or more. Betterstatistics are available through the present study, since it corresponds to a much largernumber of arrivals: because bottom-bounce events with a source angle of around 60°arriving at the array lay on a much larger circle around the array, there is a greatersurface area providing suitable cracking events. In this study, the leaky plate wavecontribution shows up as an artifact in the reflectivity function, which would nothappen if its contribution was included in the source model instead. Inclusion ofthe plate wave would be desirable, since one wants the reflectivity function derivedthrough this method to represent the true reflectivity function of the seabed. A newsource directivity model, as proposed in chapter six is the sum of the sinmO modelcontribution and the leaky plate wave contribution represented by a Normal function,P(8) = P0 [sinmO + c exp [(6 6L)2]J (A.2)where C is a constant giving the relative contributions of the acoustic mode and theleaked longitudinal plate wave, and O. and w are the critical angle and beamwidthof the leaked longitudinal plate wave respectively. For our data, C 1.0, 60°,and w 5°. It is not known whether the radiated pressure level and radiating angleof the plate wave will be the same in other ice cover conditions. On the encouragingside, Brooke and Ozard18 reported that the leaky plate wave speed was almost thesame ( 2950 m/sec) in 2.5 m thick new ice, as it was at the present much thickerice.It has been shown in this Appendix that it is possible to use local ice crackingevents and a vertical array of hydrophones to measure the reflectivity of the seabed as171a function of reflection angle. Although a large uncertainty exists in the reflectivityat a given angle, the critical angle at the seabed interface is easily extracted fromthe reflectivity function, and can be used to determine the sediment sound speed.The steps involved in using this method are: detect and isolate ice cracking events,determine the range of each event using a ray tracing model, separate the direct arrivaland the bottom-bounce arrival from each other and higher order arrivals, subtract thenoise contained in an interval of time before the events, fit the pressure level in thedirect arrival at each hydrophone to the source directivity model in order to extractthe parameter m, and thus extrapolate the pressure level to that expected at anglescorresponding to bottom-bounce paths. The difference between the expected and themeasured bottom-bounce pressure level is assumed to be due to reflection loss at theseabed. The method is fairly time-consuming, and it may take several days for atrained operator to extract the full directivity function at one site. The amount ofrecording needed depends entirely on the frequency of suitable ice cracking events atthe time of recording. During some periods, no ice cracking is taking place at all,and the ambient noise is totally stationary. At some other times, one event occursevery few seconds, and a few minutes of data may be all that is needed to make thereflectivity measurements.172Horazonta 1-oQiLJU)U)wL0Figure A-i. Polar plot of the pressure level for a single event arrivingat the array along two paths for the octave band centered at 96 Hz,corrected for spherical spreading loss. The diamonds indicate the pressure level measured at each hydrophone from the direct arrivals, andthe crosses that of the bottom-reflected arrivals. The drop in pressurelevel for grazing angles greater than about 400 corresponds to the passagethrough the critical angle.173Figure A-2. Scatter plot of the pressure level ratio between the bottom-reflected arrivals and that extrapolated from the direct arrival, using asinmO function corrected for spherical spreading losses, as a function ofgrazing angle at the bottom. Each cross indicates the pressure level ratioat one hydrophone for one event.qsRn31 (da9)174Figure A-3. Sound pressure ratio between the measured direct arrivalsabove 300 and that extrapolated from angles below 30°, using a 5111m6 directivity function corrected for the spherical spreading losses, as a functionof source angle, for all 30 events that fit the criteria of pg. 166. The soundpressure ratios as calculated for a given event for different hydrophones arelinked up with a solid line.3D 36 qs SO 66 60An9la (da9)17520. -2>-8-10Angle (deg)Figure A-4. Averaged plus and minus one-standard-deviation (solid lines)pressure level ratios between the bottom-reflected arrivals, and that extrapolated from the direct arrival, using a sintm8 directivity function correctedfor the spherical spreading losses, as a function of grazing angle at the bottom. Dashed lines show the bottom reflectivity calculated using bottomcompressional sound speeds of 1750 m/sec (lower), 1800 rn/sec (middle)and 1900 rn/sec (upper).0 20 40 60 80176-a>‘>Ii0)0)Figure A-5. Sound pressure ratio between the bottom-reflected arrivalsand that extrapolated from the direct arrival, using a sintmO directivityfunction corrected for the spherical spreading losses, as a function ofgrazing angle at the bottom, for a few individual events. The soundpressure ratios as calculated for a given event for different hydrophonesare linked up with a solid line.An9la (da9)177

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