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ACOUSTIC MEASUREMENT TECHNIQUES FOR SUSPENDED SEDIMENTS AND BEDFORMS By OLEG A. MOURAENKO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2004 To my family. ACKNOWLEDGMENTS I wish to express my appreciation to my chairman, Dr. Andrew B. Kennedy, for the freedom granted to me while pursuing my research interests, and for the guidance and support. I also wish to thank the members of my graduate committee, Dr. Robert G. Dean, Dr. Robert J. Thieke, Dr. Donald N. Slinn, for their intelligence and comprehen sive supervision. My special thanks go to Dr. Renwei Mei, the external member of the committee, for his interest and involvement. I thank Dr. Peter D. Thorne for the guidance and great discussions on various aspects of acoustic measurements. I also thank Dr. Christopher R. Sherwood for providing the acoustic backscatter system and technical help. I thank my father, Alexander Mouraenko, for his patience and love. My special thanks go to the staff at the Coastal Engineering Lab, especially to Sid ney Schofield, Jim Joiner, Victor Adams, and Chuck Broward, for their help, support, and technical guidance, without which this research would not have been possible. I thank Jamie MacMahan and Justin Davis for their help and endless support, with special thanks to Brian Barr for his exceptional knowledge, great vision of life and nu merous discussions. This work was supported by the ONR Coastal Geosciences program and the Na tional Science Foundation through the National Ocean Partnership Program (NOPP). TABLE OF CONTENTS Page A C K N O W L E D G M E N T S ................................................................................................. iii LIST OF TABLES ............................................ .................................... vi LIST OF FIGURES ......................................... .................................... vii K E Y T O SY M B O L S ....................................................................................... xv A B S T R A C T ................................................................................................................... x v iii CHAPTER 1 IN T R O D U C T IO N .................................................. .. ....................................1.. .. ... 1 2 MULTIPLE REFLECTION SOLUTION FOR ACOUSTIC BACKSCATTER ........7 A acoustic B ackscatter E quation ................................... ..................... 7 Inverse Problem for Sedim ent Concentration........................................ .................. 11 M multiple L ayer A pproxim ation ................................................................................ 11 Numerical Realization of M multiple Reflections.............................. ............... 15 Determination of Sediment Size and Concentration ......................................17 Number of Paths ......................................................... .......... .. 18 Effect of M multiple Reflections .............. ............. .... ................ 19 Special Parameters for Acoustic Backscatter Equation.........................................23 Form Function and Normalized CrossSection for Elastic Spheres.................23 H ighPass M odel for Quartz Sand.................................................. ................ 24 M odified M odel for Elastic Spheres ..................................................... 25 Modified HighPass Model for Natural Sediments........................................26 Sound A ttenuation C oefficient....................................................... ................ 27 Lognormal and Normal Distributions of Sediment Sizes............................... 28 Averaged Form Factor and Normalized CrossSection Terms ........................30 Circular Transducer Directivity and Half Intensity Beam Width ....................32 N earF ield C orrection F actor ..............................................................................34 3 CALIBRATION OF THE ACOUSTIC BACKSCATTER SYSTEM....................36 C alib ratio n C h am b er ................................................................................................... 3 8 A B S T ests in the C ham ber ......................................... ........................ ................ 40 T ests in A ir and Still W ater ............................................................ ................ 40 Homogeneity of Sediment Suspension and Effect of Chamber Walls.............41 R e su lts .............. .. ........................................................................................... . 4 8 Estimation of System Function..................... ................ 48 Approxim ation by 2nd Order Polynomial...................................... ................ 49 C alibration of the A B S 1 ................................................................. ................ 50 C alibration of the A B S2 ................................... ..... ................ 53 Sensitivity Analysis For Estimation of System Function......................................55 D ependence on Sedim ent Sorting .................................................. ................ 59 D ependence on Transducer R adius ................................................ ................ 60 D term nation of P aram eter f f .................................................................................60 Estim ation of System Function for ABS1 ...................................... ................ 62 Analysis of Relationship between z and f ............. ................................... 67 "F u n P u m p T e sts ............................................................... .................. .................... 7 0 Sam ple Sedim ent Size D istribution.................................................................. 71 Prediction of Concentration and Sediment Size Distribution...........................73 P aram eters of Inversion .................................................................. ................ 74 Results of Inversion for Concentration........................................... ................ 76 Results of Inversion for Sedim ent Size ............... .................................... 81 4 MEASUREMENTS OF TARGET POSITION AND BOTTOM ECHO REMOVAL A L H O R IT H M ....................................................... .. .............................................. 8 5 A Sim plified B ottom Echo M odel ......................................................... ................ 85 Tests w ith a Steel W ire .. ................................................................................ 88 T ests w ith a W aterSedim ent Jet ........................................................... ............... 96 R esult of B ottom E cho R em oval ........................................................... ................ 98 5 CONCLUSIONS ...................................... ........ ................... 100 APPENDIX A CA LIBR A TION TE ST RE SU LTS ..........................................................................104 B EMPIRICAL MODEL FOR ACOUSTIC BACKSCATTER................................123 C ID ADVECTIONDIFFUSION MODEL ............... .............. ..................... 130 N um erical Solution ............................................................................... 132 Vertical Sedim ent Size Distribution ...... ........ .. ...... ..................... 133 D RESULTS FOR "FUNPUMP" TESTS........................................137 LIST O F REFEREN CE S .. .................................................................... ............... 145 BIOGRAPH ICAL SKETCH .................. .............................................................. 150 LIST OF TABLES Table Page 21: F irst 24 C atalan num bers......................................... ......................... ............... 19 22: Beam width for a circular piston transducer (a,= 5 mm and c= 1500 m/s). ..........34 23: Nearfield range r for a circular piston source (a,=5mm and c =1500m/s). .........35 31: Locations and concentration levels in tests for 1.08MHz transducer....................42 32: Locations and concentration levels in tests for 2.07MHz transducer....................43 33: Locations and concentration levels in tests for 4.70MHz transducer....................43 3 4 : S ed im en t sizes .......................................................................................................... 6 3 35: Estimation of parameters Ifz and ff for different sediment sizes.......................64 36: V alues of x = ka for test cases ........................................................... ................ 64 37: Concentration levels, mean and standard deviation of the mixture for "funpump" te sts ............................................................... 7 2 38: Coefficients of 2nd order polynomial for system functions................................75 41: T ests w ith 0.33m m steel w ire .............................................................. ................ 9 1 C 1: Advectiondiffusion model test parameters. .......................................133 LIST OF FIGURES Figure Page 21: Example of the scattered sound within the narrow beam....................................12 22: The multilayer approximation for a possible sound pulse traveling path. The time interval corresponds to rc/2 ............................................................... ............... 13 23: Generation of a binary tree for numerical realization of multiple reflections.......... 16 24: Difference between intensities modeled with one and three reflections for selected profiles of concentrations: a) constant profiles, e) linear profiles, bd) and fh) percent difference. ......... ..... .......................................... ...............2 1 25: Difference between intensities modeled with one and three reflections for selected profiles of concentrations: a) profiles of constant concentration with "bump", e) profiles of constant concentrations with "spike", bd) and fh) same as in F ig u re 2 4 ............................................................................................................... 2 2 26: Form function and normalized cross section for quartz sand (blue) and spheres (red): a) form function, b) normalized crosssection...........................................25 27: Modified y function for selected values of parameter fl ............... ................26 28: Circular piston directivity function, D 2 (0) ...................................... ................ 33 29: Nearfield correction factor, 1/1V(r), for a,=5mm and c=1500m/s....................35 31: C alibration cham ber side view ........................................................ ................ 39 32: RMS voltage profiles (solid lines) with scatter intervals (dashed line) from the measurements in air and still clear water: a) 1.08MHz, b) 2.07MHz, and c) 4 .7 0 M H z .............................................................................................................. 4 1 33: Calibration chambertop view. The jet system is shown with transducer positions for the sidew all proxim ity tests ........................................................... ................ 42 34: Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) for 1.08MHz transducer for sidewall proximity tests ..............................................43 35: Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) for 2.07M Hz transducer for sidewall proximity tests. .............................. ................ 44 36: Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) measured along centerline with 1.08MHz transducer.........................................45 37: Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) measured along centerline with 2.07MHz transducer.........................................45 38: Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) measured along center line with 4.70MHz transducer........................................46 39: Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) measured near the chamber wall with 1.08MHz transducer ...............................47 310: Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) measured near the chamber wall with 2.07MHz transducer ...............................47 311: Estimation of sediment size distribution used for the calibration of ABS1 by: a) sieve analysis, b) fall velocity ......................................................... ................ 50 312: Estimation of the system function for 1.08MHz transducer: a) ratio of measured and calculated voltage profiles, b) estimated system functions, and c) inverted sedim ent concentration ........................................................................ ................ 52 313: Estimation of the system function for 2.07MHz transducer: ac) same as in Figure 312. ............. ........................................................................ ....... ..... 52 314: Estimation of the system function for 4.70MHz transducer: ac) same as in F figure 312 ............. ............................................................................... ..... 53 315: Estimation of sediment size distribution for sand used in calibration of ABS2 by: a) sieve analysis, b) fall velocity ......................................................... ................ 54 316: Estimation of the system function for IMHz of ABS2: ac) same as in F figure 312 ............. ............................................................................... ..... 54 317: Values of fitted parameters /fi, a, p and q,, with corresponding errors in inverted concentrations for the 1.08MHz transducer of the ABS1.......................57 318: Same as on Figure 317 for the 2.07MHz transducer of the ABS1. ......................57 319: Same as on Figure 317 for the 4.70MHz transducer of the ABS1. ......................58 320: Estimation of system function K, for 4.70MHz transducer with corrected param eter fi : ac) sam e as in Figure 312. ....................................... ................ 59 321: Estimation of sediment size distribution for Case 4 used for calibration of ABSI by: a) sieve analysis, b) fall velocity ................................................... ................ 63 322: Estimated system functions and their means for Cases 14 with 83 =0.5 (dashed) and fitted (solid): a) 1.08, b) 2.07, and c) 4.70MHz transducers of the ABS1. .......65 323: Ratios between individual system functions for Case 14 and mean system function with 8z=0.5 (dashed) and fitted (solid): a) 1.08, b) 2.07, and c) 4.70MHz transducers of the A B S ...................................................................... ................ 66 324: Fitted parameters f#3 and corresponding parameters f, for Cases 14 ..............68 325: Values of parameters /f, and/ ,, for Cases 14 for different frequencies ............68 326: Values of j and y(ka,137) as functions of ka with approximated functions y(ka, f ) for C ases 14 (solid lines) .................................................. ................ 69 327: Sediment size distribution for Sand 1, Sand 2, and 1:1 mixture .............................72 328: Predicted with ID advectiondiffusion model profiles of concentration and sediment size distributions for maximum concentration of 0.8g/l at times 070s: size distributions at a) 30, b) 50, and c) 75cm below the free surface, d) sediment concentration profiles, e) profiles of mean sediment size in phi units, and f) profiles of standard deviations in phi units....................................................... ................ 73 329: Highpass model for backscattered intensity and its modified version used for "fun pump" test: a) form functions, b) normalized crosssections...............................76 330: Inversion by unmodified highpass model concentration profiles for "funpump" test with maximum concentration of 0.4g/l: mean profiles (thick lines) and 1.08M H z profiles (thin lines) ................................... ...................... ................ 77 331: Inversion by unmodified highpass model concentration profiles for "funpump" test with maximum concentration of 0.8g/l: mean profiles (thick lines) and 2.07M H z profiles (thin lines) ................................... ...................... ................ 78 332: Inversion by modified model concentration profiles for "funpump" test with maximum concentration of 0.8g/l: mean profiles (thick lines) and 2.07MHz profiles (th in lin e s). .............................................................................................................. 7 9 333: Comparison of concentration profiles inverted by modified highpass model with profiles calculated by the ID advectiondiffusion and empirical models (labeled as "ABn") for the "funpump" test with maximum concentration of 0.8g/l at 10 seco n d in terv als. ....................................................................................................... 8 0 334: Inverted profiles of mean sediment sizes for "funpump" test with maximum concentration of 0.8g/l by unmodified highpass model.....................................82 335: Inverted by modified highpass model profiles of mean sediment sizes for "fun pump" test with maximum concentration of 0.8g/1.............................................82 336: Comparison of inverted profiles of mean sediment sizes by modified highpass model with profiles calculated by the ID advectiondiffusion model for the "fun pump" test with maximum concentration of 0.8g/l at 10second intervals.............. 83 41: A sample of voltage profile with definition of"A"point ...................................... 87 42: Setup for the tests w ith steel w ire........................................................ ............... 89 43: Rootmeansquare voltage profiles for the tests with 0.33mm steel wire. Tests 116 with the wire location (dashed line) and the bottom location (solid line) .............92 44: Sam e as in Figure 43 for tests 1731 ................................................ ................ 93 45: Rootmeansquare voltage profiles with subtracted bottom echo profile for the tests with 0.33mm steel wire. Tests 116 with the wire location (dashed line) and the bottom location (solid line) ...................................... ....................... ................ 94 46: Sam e as in Figure 45 for tests 1731 ................................................ ................ 95 47: Rootmeansquare voltage profiles (solid) for tests with 2mm steel wire with the wire locations (dashed) and "ghost wire" reflection locations (dashdotted) from a) 1.08, b) 2.07, and c) 4.70M H z transducers. ........................................ ................ 96 48: Setup for the tests w ith the sedim ent jet............................................. ............... 97 49: Rootmeansquare voltage profiles (solid) for tests for different watersediment jet locations (dashed) from a) 1.08, b) 2.07, and c) 4.70MHz transducers ................97 410: Time series of suspended sediment concentrations estimated with a) bottom echo and b) bottom echo removed; c) free surface elevation. .....................................99 A1: Case 1: Errors in inverted concentrations from varying of parameters /fz, at, ,, and ,, for 1.08MHz transducer. ................................ 104 A2: Case 1: Errors in inverted concentrations from varying of parameters /fz, a, ,, and ,, for 2.07M Hz transducer. ...... ...... ...... ...................... 105 A3: Case 1: Errors in inverted concentrations from varying of parameters /fz, a,, ,, and o, for 4.70M Hz transducer. ...... ...... ...... ...................... 105 A4: Case 2: Errors in inverted concentrations from varying of parameters /fz, a, p, , and ,, for 1.08M H z transducer. ...... .................................. .. ...................... 106 A5: Case 2: Errors in inverted concentrations from varying of parameters /fz, a, p, , and ,, for 2.07M H z transducer. ...... .................................. .. ...................... 106 A6: Case 2: Errors in inverted concentrations from varying of parameters /fz, a, p, , and a,, for 4.70M H z transducer. ...... .................................. .. ...................... 107 A7: Case 3: Errors in inverted concentrations from varying of parameters f6z, a,, p , and a for 1.08M H z transducer. ...... .................................. .. ...................... 107 A8: Case 3: Errors in inverted concentrations from varying of parameters f6z, a,, p , and a,, for 2.07M H z transducer. ...... .................................. .. ...................... 108 A9: Case 3: Errors in inverted concentrations from varying of parameters f6z, a,, p , and ,, for 4.70M H z transducer. ...... .................................. .. ...................... 108 A10: Case 4: Errors in inverted concentrations from varying of parameters /fz, a, p, , and ,, for 1.08M H z transducer. ...... .................................. .. ...................... 109 A11: Case 4: Errors in inverted concentrations from varying of parameters /fz, a,, ,, and o, for 2.07M H z transducer. ...... .................................. .. ...................... 109 A12: Case 4: Errors in inverted concentrations from varying of parameters /fz, a, p, , and a,, for 4.70M H z transducer. ...... .................................. .. ...................... 110 A13: Case 1: Estimation of the system function for the 1.08MHz transducer initial value of parameter f6z: ac) same as in Figure 312.......................... .................... 110 A14: Case 1: Estimation of the system function for the 1.08MHz transducer with corrected parameter If : ac) same as in Figure 312. ........................................111 A15:Case 1: Estimation of the system function for the 2.07MHz transducer with initial value of parameter fz: ac) same as in Figure 312 .......................................... 111 A16: Case 1: Estimation of the system function for the 2.07MHz transducer with corrected parameter If : ac) same as in Figure 312. .............. ................112 A17: Case 1: Estimation of the system function for the 4.70MHz transducer with initial value of parameter fiz: ac) same as in Figure 312. .................... ................ 112 A18: Case 1: Estimation of the system function for the 4.70MHz transducer with corrected parameter If : ac) same as in Figure 312 .............. ................ 113 A19: Case 2: Estimation of the system function for the 1.08MHz transducer with initial value of parameter fiz: ac) same as in Figure 312. .................... ................ 113 A20: Case 2: Estimation of the system function for the 1.08MHz transducer with corrected parameter If : ac) same as in Figure 312 .............. ................ 114 A21:Case 2: Estimation of the system function for the 2.07MHz transducer with initial value of parameter fiR: ac) same as in Figure 312. .................... ................ 114 A22: Case 2: Estimation of the system function for the 2.07MHz transducer with corrected parameter If : ac) same as in Figure 312 .............. ................ 115 A23:Case 2: Estimation of the system function for the 4.70MHz transducer with initial value of parameter fiz: ac) same as in Figure 312. .................... ................ 115 A24: Case 2: Estimation of the system function for the 4.70MHz transducer with corrected parameter If : ac) same as in Figure 312 .............. ................ 116 A25:Case 3: Estimation of the system function for the 1.08MHz transducer with initial value of parameter fiz: ac) same as in Figure 312. .................... ................ 116 A26: Case 3: Estimation of the system function for the 1.08MHz transducer with corrected parameter If : ac) same as in Figure 312 .............. ................ 117 A27: Case 3: Estimation of the system function for the 2.07MHz transducer with initial value of parameter fiR: ac) same as in Figure 312. .................... ................ 117 A28: Case 3: Estimation of the system function for the 2.07MHz transducer with corrected parameter If : ac) same as in Figure 312 .............. ................ 118 A29: Case 3: Estimation of the system function for the 4.70MHz transducer with initial value of parameter fiz: ac) same as in Figure 312. .................... ................ 118 A30: Case 3: Estimation of the system function for the 4.70MHz transducer with corrected parameter Jf : ac) same as in Figure 312 .............. ................ 119 A31:Case 4: Estimation of the system function for the 1.08MHz transducer with initial value of parameter fi : ac) same as in Figure 312. ...................... ................ 119 A32: Case 4: Estimation of the system function for the 1.08MHz transducer with corrected parameter If : ac) same as in Figure 312. ............... ....... ............ 120 A33:Case 4: Estimation of the system function for the 2.07MHz transducer with initial value of parameter fi : ac) same as in Figure 312. ...................... ................ 120 A34:Case 4: Estimation of the system function for the 2.07MHz transducer with corrected parameter If : ac) same as in Figure 312. ............... ....... ............ 121 A35:Case 4: Estimation of the system function for the 4.70MHz transducer with initial value of parameter fi : ac) same as in Figure 312. ...................... ................ 121 A36: Case 4: Estimation of the system function for the 4.70MHz transducer with corrected parameter If : ac) same as in Figure 312. ............... ....... ............ 122 B1: Calibration coefficients for 1.08 MHz transducer: a) measured voltage (dots) and fitted by curves B3 and calculated coefficients b) A, c) B, and d) n...............127 B2: Calibration coefficients for 2.07MHz transducer: ad) same as in Figure B1 ....127 B3: Calibration coefficients for 4.70MHz transducer: ad) same as in Figure B1 ....128 B4: Concentration profiles converted with the empirical model from the measurements with a) 1.08, b) 2.07, and c) 4.70M Hz transducer...... ................... ............... 129 C1: Case 1zero diffusivity: sediment size distribution at distance a) 0.25, b) 0.5, and c) Im; d) mass concentration; e) approximate mean, and f) standard deviation.... 135 C2: Case 2constant diffusivity: af) same as in Figure C1. ...............................135 C3: Case 3linear diffusivity: af) same as in Figure C1............... ...................136 C4: Case 4parabolic diffusivity: af) same as in Figure C1..............................136 D1: "Funpump" test results for concentration profiles with maximum concentration of 1.2g/l: ah) sam e as for 333. ................................................................ 137 D2: "Funpump" test results for mean sediment size profiles with maximum concentration of 1.2g/l: ah) sam e as for 336....... .................... .................. 138 D3: "Funpump" test results for concentration profiles with maximum concentration of 1.6g/l: ah) sam e as for 333. ................................................................ 139 D4: "Funpump" test results for mean sediment size profiles with maximum concentration of 1.6g/l: ah) sam e as for 336....... .................... ................... 140 D5: "Funpump" test results for concentration profiles with maximum concentration of 2.0g/l: ah) sam e as for 333. ................................................................ 141 D6: "Funpump" test results for mean sediment size profiles with maximum concentration of 2.0g/l: ah) sam e as for 336....... .................... ................... 142 D7: "Funpump" test results for concentration profiles with maximum concentration of 2.4g/l: ah) sam e as for 333. ................................................................ 143 D8: "Funpump" test results for mean sediment size profiles with maximum concentration of 2.4g/l: ah) sam e as for 336....... .................... ................... 144 KEY TO SYMBOLS a [m] sediment radius [a,b] [phi] sediment size range (a, b, c) coefficients of powerlaw curve (A, B, n) parameters of empirical model at [m] transducer radius b [m] rootmeansquare width of bottom echo return B = arc, pressure correction factor c [m/s] speed of sound in water Cfl Courant number c, [m/s] speed of compressional wave in spherical particle C" n th Catalan number ct [m/s] speed of shear wave in spherical particle d [mm] sediment diameter Dt transmitter directivity function E modulus of elasticity f [MHz] sound frequency f nondimensional form function f (0, x) nondimensional form function h filter impulse response function h(1) spherical Hankel function of first kind I sound intensity J, first order Bessel function of first kind jl spherical Bessel function of first kind k [rad/m] sound wave number K, scattering parameter Kt system parameter K, system function LI complex acoustic scattering length M [kg/m3] concentration of suspended sediments MA [m] mean sediment size N number of scatters nb [1/m3] density of scatters P0 [Pa] reference sound pressure at r = ro p, coefficients of bestfit polynomial for K, PL lognormal probability density function PN normal probability density function PI. [Pa] rootmeansquared backscattered pressure P, [Pa] farfield sound pressure Px [Pa] total pressure an receiver r [m] range from transducer 91 transducer receive sensitivity ro [m] reference range rb [m] range to the bottom location Rb bottom reflection coefficient R, reflection coefficient rI. [m] nearfield range S, [m] sediment sorting, standard deviation of sediment sizes t [s] time T [C] water temperature T transmission coefficient Voltage transfer function V [m3] gated volume VO [volt] signal from mirrorreflected surface vb [volt] magnitude of bottom echo signal Vb [volt] recorded bottom echo return VeII [volt] amplitude envelope of recorded voltage Vrm [volt] recorded rootmeansquared voltage V [volt] recorded signal from suspended scatterers W [m] reference beam diameter at r =lm Wman [m] reference main lobe diameter at r =lm wS [m/s] sediment settling velocity x =ka z normalized distance a [Np/m] total sound attenuation coefficient as [Np/m] attenuation coefficient due to suspended sediments a. [Np/m] water attenuation coefficient fl form function correction parameter f3z normalized crosssection correction parameter y modified scattering correction function 70 scattering correction function Ar [m] thickness of insonified layer c relative error e2 variance of estimated concentrations e [m 2/s] sediment diffusivity sediment attenuation constant 0 [rad] latitude angle from central axis of transducer 0 [rad] half intensity beam width Oma.n [rad] width of main lobe 2 [1/m] sound wave length / [phi] mean sediment size correction factor, calculated from data p [kg/m3] water density PA acoustic density p, [kg/m3] sediment density [7i [m2] total scattering crosssection oa [phi] sediment sorting, standard deviation of sediment sizes T [s] sound pulse duration (p phi scale " normalized scattering crosssection Vf nearfield correction function T D integrated beam pattern xvii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ACOUSTIC MEASUREMENT TECHNIQUES FOR SUSPENDED SEDIMENTS AND BEDFORMS By Oleg A. Mouraenko December, 2004 Chair: Andrew B. Kennedy Major Department: Civil and Coastal Engineering The acoustic backscatter system provides a nonintrusive method for measuring of profiles of suspended sediment size and concentrations. Therefore, an inversion model is required to convert the recorded series of intensities into parameters of sediment suspen sion. The measurement precision highly depends on the choice of the inversion model. In this work the highpass backscattering model is modified to allow accurate inversion for both sediment sizes and concentrations. The modified correction function with a single parameter is introduced for the form function and normalized crosssection. The parame ters can be found from the calibration of the system in the chamber with generated ho mogeneous suspension. To account for the difference in shape and mineralogy, a set of two parameters can be determined for a particular sediment. Their use greatly improves the accuracy of mean sediment size measurements obtained with the threefrequency acoustic backscatter system. xviii The inversion model was applied to data obtained in the chamber with settling mix ture of two sediment samples. The results were compared to the predictions by the ID advectiondiffusion model. Good agreement was shown for concentrations and sediment size measurements, with the error for sediment size approximation lying within the stan dard deviation of size distribution. The accuracy of the concentration measurements was different among the transducer frequencies, and was found to depend on the sensitivity of the transducer to the particular sediment size. The sediment sizes were estimated by minimizing the variance between concentrations, determined from different frequencies, and found to depend on the accuracy of estimated concentrations. For correction of re corded voltage data with applied timevarying gain, the system functions were found to be better approximated by second order polynomials. A multiple reflection model was developed based on the multiple layer approxima tion. The numerical simulation with the initial uniform concentration profile of 3g/l shows that the effect in intensities from the multiple reflections can be up to 14%. Also, a bottom echo removal algorithm was developed in order to eliminate the effect of high intensity scattering on the concentration measurements close to the bottom. CHAPTER 1 INTRODUCTION The application of acoustics for the study of sediment transport and development of prediction models for morphological changes of coastlines and bathymetry has been widely recognized and developed during the last few decades [Thorne andHanes, 2002]. By providing remote, no intrusive methods of measurements of flow velocities, bottom topography, and profiles of suspended sediment concentration, acoustic instruments have proven a reliable way of obtaining high resolution data during field and laboratory ex periments [Hanes, 1991; Hanes et al., 2001; Hanes and Vincent, 1987; Hanes et al., 1988; Jansen, 1979; Libicki et al., 1989; Traykovski et al., 1999; Vincent and Green, 1990; Young et al., 1982]. The ultrasound at megahertz frequency range is used for the measurements of sedi ment suspension parameters within a meter range above the bottom [e.g., Smerdon et al., 1998]. The system consists of a sound transmitter and receiver. The properties of water sediment mixture along an insonified path can then be estimated from the changes in sound attenuation and from the intensity of scattered sound [Morse andlngard, 1968]. Two configurations of such systems are usually applied: bistatic, when the transmitter and receiver are spatially separated, and monostatic, when the same transducer serves as a transmitter and a receiver. The later configuration measures the backscattered intensity from the scatters in suspension sound. The acoustic backscatter system allows measurement of both suspended sediment concentration profiles and distribution of sediment sizes [e.g., Craw ford and Hay, 1993; Hay and M.Niig, 1992; Schat, 1997; Thorne andHardcastle, 1997; Thosteson andHanes, 1998]. Compared to other techniques, acoustics are a nonintrusive way of measuring sus pended sediments with a high temporal (in the order of a second) and spatial (in order of a centimeter) resolutions [Wren et al., 2000], with comparable accuracy [Hanes et al., 1988; Thorne andHardcastle, 1997; Thorne et al., 1995]. Since the acoustic backscatter system is an indirect method of measurement, an in version algorithm is required for converting measured backscattered sound intensities into parameters of suspension. The acoustic backscatter equation [e.g., Medwin and Clay, 1997, p.353] provides the basis for the development of such an algorithm. The difficulties arise from the dependence of a backscattered signal from the prop erties of scatterers, their concentration, directivity of the sound beam, and also particular electronic implementation. Semiempirical approximations of these parameters [e.g., Ta mura andHanes, 1986] can be used to convert the data into concentration profiles. Al though a better description of the parameters can be found from theoretical approxima tion of sound scattering properties of natural sediments. The acoustic properties of scatterers depend on many parameters, such as size, shape, mineralogy, distribution of those parameters in a sample, speed and direction of motion, their concentration, relative position to the sound transmitter and receiver, and others. Only a limited set of properties can be considered for the inversion. The simpli fied model for acoustic backscattering allows a user to obtain the most important proper ties for the study of sediment transport, while others can be approximated by general ex pressions. Only suspended sediment concentration, mean sediment size and sorting are considered to provide the necessary description of the watersediment suspension, while the lognormal distribution of sediment sizes in a sample is assumed. The scattering pa rameters then are approximated by two functionsthe form function and normalized crosssection. The choice of these two functions determines the accuracy and reliability of inversion model. The following works define the necessary framework for develop ment of the inversion model: [Downing et al., 1995; Faran, 1951; Hay, 1991; Hay and Mercer, 1985; Hay and Schaafsma, 1989; Neubauer etal., 1974; .iheig and Hay, 1988; Thorne and Campbell, 1992; Thorne et al., 1993a]. A multiple frequency setup needs to be used in order to estimate sediment sizes. Schaafsma [1989] used a bistatic configuration of variable frequency transducers, which covered the frequency range between 110MHz, to measure concentration and sediment sizes of a mixture of two sediment samples. He showed that concentrations could be ob tained with 20% error, while the sediment sizes can be resolved within a factor of 2. For field measurements monostatic configurations are usually used. A system consisting of a set of three or more transducers with frequencies 15MHz is able to estimate the mean sediment sizes along the insonified column [Crawford andHay, 1993; Hay and.\heig, 1992; Schat, 1997; Thosteson andHanes, 1998]. The accuracy of the estimation highly depends on correct approximation of scattering properties of the sediments, as well as on the accuracy of estimations of suspended sediment concentrations from each of the fre quencies. It has been shown that the accuracy and stability of estimations of sediment sizes can be achieved by averaging among a series of consecutive profiles, which leads to significant loss in temporal resolution [e.g., Thorne andHardcastle, 1997]. Several problems regarding measurement accuracy were addressed by the current dissertation. Their resolution improves both the accuracy and the temporal resolution of the measurements, and provides a basis for future improvements. Usually, only measurements at low suspended sediment concentrations (<10g/l) are performed. Therefore, the multiple scattering of the sound waves can be ignored [e.g., \/Vieg andHay, 1988]. Although it is true for natural sand, in the presence of stronger scatterers, like air bubbles, or at higher concentrations, as in the region near the sand bot tom, multiple scattering can become important. Here a multiple reflection model is de veloped and the effects of multiple scattering on resulted errors are shown. The acoustic properties of different natural sediments vary from sample to sample. By using the approximated parameters, such as form function and crosssection, obtained for a particular type of scatters like glass spheres or quartz sand, the inversion algorithm can lead to erroneous results if the actual sediment differs from the approximation in mineralogy and shape [Thorne and Buckingham, 2004]. Therefore, acoustical properties of the actual sediments need to be well defined before the inversion process can be done. The detailed study of each sediment sample, obtained from the experimental site, for mineralogical content, distribution of shapes, and overall acoustical performance is an even more difficult problem, which even in a case of positive solution cannot guarantee correct results, since the sediment content can change during the period of measurements. The current work describes a new scattering model, which modifies a correction function originally presented by Thorne and Buckingham [2004], with a technique to determine necessary parameters. The description of the acoustic backscatter equation, an implemen tation of the multiple layer solution, and the newly developed model are discussed in Chapter 2. The calibration of an acoustical backscatter system is performed in order to obtain system transfer function, which describes the dependence of sound intensity and recorded voltage levels. Different methods of calibration can be applied, such as by using thinwire targets [./heng andHay, 1993], calibration with a particleladen turbulent jet [Hay, 1991], or in a calibration chamber with a homogeneous suspension [Tamura andHanes, 1986; Thorne et al., 1993a]. The time varying gain (TVG) is usually applied for the amplifica tion of low level sound signals, arriving from the farther distances from the transducer. With the proper correction for the TVG, the system function should be uniform with the range and therefore, can be determined from the calibration. In fact the system functions found from the calibrations are not always uniform with the range and may depend of particular sediment sample used for the calibration [Thorne and Buckingham, 2004]. The variations in system constant can reach 25%, which result in a corresponding error in es timation of suspended sediment concentrations. By providing proper corrections to the system functions developed here, these errors can be lowered. The solution of the acoustic backscatter equation for suspended sediment concen tration can be obtained explicitly with some limitations [Lee andHanes, 1995]. Further development of the method [Thosteson andHanes, 1998] allowed us to estimate the pro files of sediment sizes. The implicit method [Thorne et al., 1993a] provides a more flexi ble way of solution, although it can become unstable; when the concentrations become large and the sediment attenuation becomes significant, even small 510% errors can re sult in unbounded profiles of concentrations from the iterative solution [Thorne et al., 1995]. If the concentration profile is approximately known, for example, from the pump sampler measurements [e.g., Thorne et al., 1993a], the correction for sediment attenua tion can be performed more accurately. In Chapter 3 calibration of the acoustic backscatter system together with a tech nique of evaluation of the system parameters and the parameters for the correction func tions is described. Also, the results of a series of laboratory tests are shown and compared to the ID advectiondiffusion model predictions for sediment size and concentration pro files. The bed echo contains important information, which can be used to improve accu racy of the acoustic backscatter system [Thorne et al., 1995]. Also, the intensity of the bed echo at low concentrations is much greater than the intensity of the sound backscat tered from suspended particles. Because the acoustic backscatter system requires averag ing and filtering of the input signal, signals from sand bottom interchange with the sig nals from suspended sediments. The algorithm for separation of these signals can provide one with the information on bed location and allow to determine concentrations of sus pended sediments in the vicinity of the bottom. The algorithm is discussed in Chapter 4. CHAPTER 2 MULTIPLE REFLECTION SOLUTION FOR ACOUSTIC BACKSCATTER Acoustic Backscatter Equation The acoustic backscatter equation is derived from the time integral of squared pres sure backscattered from the objects in an insonified volume with following assumptions [Medwin and Clay, 1997]: 1. The objects in the insonified volume are alike, which implies that they are of a similar shape, size and mineral structure, and can be substituted with a population of objects with average properties. 2. The distribution of the objects within a gated volume is random and uniform. The gated volume is a volume of fluid, insonified within the short time intervaltime gate. 3. The time gate is assumed to be large compared to the sound pulse duration. In prac tice, however, the time gate is often chosen to be equal to half of the pulse duration [e.g., Smerdon et al., 1998]. 4. The attenuation of sound within the gated volume is ignored. The farfield sound pressure from a circular piston source in isospeed medium is then equal to [Medwin and Clay, 1997] P,(t)= D (0) P tr ro ear (21) where D, (0) is the transmitter directivity function given by 265, 1P (t) is the sound pressure at the range ro along the central axis, a is the total sound attenuation coefficient given by 242, c is the speed of sound in water, and r is the range from the transducer. After the transducer emits a sound pulse, it travels through the volume of water with sus pended sediments. A portion of the sound is scattered by any particle in the direction of the receiver. The pressure at the receiver is equal to P (t) = D (0)P (t) L ear (22) r where Dr (0) is the directivity of the receiver, L, is the complex acoustic scattering length, and the factor 1/r accounts for the spherical spreading of the sound wave. The complex acoustic scattering length describes the scattering properties of the particle. It is a function of angles between incident and reflected waves relative to the particle, and the sound wave number k. For spheres it can be rewritten as [e.g., .\heig andHay, 1988] af (0,x) (23) 2 where a is the particle radius, and f (0, x) is the nondimensional form function with x = ka . In general, the form function is a complex valued function. For scatterers of simple shapes, like cylinders and spheres, it can be found analytically [Morse andIngard, 1968]. But for particles of irregular shapes, such as sand grains, approximations are used to de scribe the amplitude of backscattered form function [Craw ford and Hay, 1993; Neubauer et al., 1974; .ihelg andHay, 1988; Thorne and Buckingham, 2004; Thorne et al., 1993b]. It relates to the form function as f = f (0, x) (24) The total pressure at the receiver, P., is a sum of all backscattered pressures from individual particles. The squared pressure amplitude can be calculated as N N N PX (tr =i (t),* (t)+fii (t),(t) (25) 1=1 1=1 J=1 J1# where N is the total number of particles, and asterisk denotes complex conjugate value. The second term is the sum of the crossproducts of the pressures. In the case of a homogeneous suspension it can be shown that this term approaches zero either with in creasing the number of particles or by ensemble averaging of the total backscattered pres sure realizations [Libicki et al., 1989; Smerdon et al., 1998; Thorne et al., 1993a]. Previ ously mentioned assumptions 1 and 2 define the homogeneity of the watersediment sus pension. In practice, both methods of averaging are used to obtain the rootmeansquared (RMS) pressure profiles [e.g., Smerdon et al., 1998]. The RMS backscattered pressure for a given range r is calculated as is the integral over the gated volume of the total backscattered pressure squared, which for the gated time equal to sound pulse duration, r, can be written as [Hay, 1991] P 2 1 sinh B4r (26) 0 P0 4 bArTDr 4 B e (26) p2 2 difcs t e p where P0 = [ 0] (t) dt is the reference pressure, nb = N/V is the density of the particles in gated volume, Ar = rc/2 is the extent of the sample volume in r, and (...) implies expected value over the sediment size distribution. The term sinh B/B with B = arc is the correction factor if the attenuation over the gated volume is large [Hay, 1991; Thosteson andHanes, 1998]. The integrated beam pattern is calculated as 2)r;r/2 PD = fJ D,2 (, 0)D (), 0)sin d0dd (27) 0 0 which for the monostatic system, when transmitter and receiver are located in the same point and their directivities are the same, can be approximated as [Thorne andHardcas tie, 1997] YD 2 096 ka, > 10 (28) where a, is the transducer radius. The density of the particles can be calculated by approximation of the particle shape by spheres with radius a. Therefore, it can be written as a function of particle mass concentration, M, as nb (29) 4 (a3)ps where ps is the density of sediments. The rootmeansquare voltage, recorded by the acoustic backscatter system can be written in the form [Thorne and Hanes, 2002; Thorne andHardcastle, 1997] 1 M1/2 S= 9 = K K  e2r (210) a 1 2f2/2 Ks= Ia (21 la) K, = (K 3rc 1/2 0.96 (21 lb) S16 ) ka, K = 9T?0ro (21 1c) where 9% is the transducer receive sensitivity, T is the voltage transfer function of the system, y is the nearfield correction factor given by equation 267. K, is the system parameter, which is a constant if all other parameters of the system are fixed, or a func tion, if the time varying gain (TVG) is applied. It can be determined by the calibration with known concentrations and sediment properties [Hay, 1991; Thorne andHanes, 2002; Thorne et al., 1993a]. Inverse Problem for Sediment Concentration The parameters of the suspension such as sediment concentration and sediment size distribution parameters can be evaluated by solving the inverse to equation 210 problem [Thorne andHanes, 2002]. For the concentration, the solution can be obtained from the equation 2 M(r) = \ Vyrr e4ra (212) It cannot be solved explicitly, since the reflection parameter Ks and total attenuation co efficient a are functions of sediment size distribution and concentration. The solution by iterations is usually applied. As can be seen from the equation, the solution for the con centration can be unstable, since it has positive exponential dependence on attenuation. Multiple Layer Approximation In the derivation of the equation 26 for RMS pressure, the multiple scattering be tween the particles was ignored. Multiple scattering for the suspension of sand can be ne glected for suspended sediment concentrations of less than 10 g/1 [.\/eng and Hay, 1988]. At higher concentrations, which can be found at the ranges close to the bottom [e.g., DohmenJanssen andHanes, 2002], or in the presence of stronger scatterers in the fluid, such as a wire or bubbles, multiple reflections can affect the measurements. After the transducer emits a sound pulse, it travels within a narrow directed beam. Part of the sound wave is scattered by the suspended particles, and it may be rescattered by other particles or reflected from the transducer surface multiple times before the back scattered sound is recorded by the receiver. The receiver detects only the sound arriving from the direction within the narrow beam, which is defined by the directivity of the transducer (265). Therefore, the sound scattered backwards in the direction of the inci dent sound will most likely remain within the beam and be detected by the receiver (see Figure 21). Table 22 shows the approximate beam widths for transducers of different frequencies. D (0) 3dB Figure 21. Example of the scattered sound within the narrow beam. The RMS pressure (26) is calculated as the integral of the backscattered pressure over the time gate interval, which is again assumed to be equal to half of the pulse dura tion. During each time interval, the sound wave travels the distance Tc/2. Therefore, the whole range from the transducer surface can be represented as a set of layers with thick nesses rc/2 (see Figure 22). The center of the nth layer is located at distance r=n (213) 2 where n =0 corresponds to the transducer surface. Time 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 TRANSDUCER 0 1 J 3  leading edge 4 * trailing edge Figure 22. The multilayer approximation for a possible sound pulse traveling path. The time interval corresponds to rc/2. The sound waves can travel along many possible paths within the beam and scatter from the layers of suspended particles. In the multiple layer approximation all the paths begin at the layer 0 and pass between the layers with possible changes in direction. The mean squared pressure at the transducer at time t,, = 1r/c is equal to the sum of mean squared pressures of the sound followed all possible paths Cn 1 p2 pZn (214) wheree the C is the total number of the paths given by equation 226. Obviously, the where the Cn, is the total number of the paths given by equation 226. Obviously, the length of every path in the summation should be equal to r. Let the sound pressure corresponding to an individual path at level i be ]P. With spherical spreading and attenuation of the sound, the backscattered pressure from level j becomes R Pj = P T (215) where ; is the distance between layers, T is the transmission coefficient, and Rj is the backscattering coefficient. The backscattering coefficient defines a part of sound which is scattered from the layer of suspended particles in the direction of the incident sound wave. It is proportional to the number of scatterers within the layer and their scattering length (23). Using the same assumptions as for the acoustic backscatter equation, the coefficient can be written as Sa2f 3Mi rc R 4=)P 2 Dj (216) where the term (a f2) /4 approximates the scattering length of the scatterers in the layer j, and the term 'DrD2 Tc/2 defines the insonified volume at layer j. The transmit ting coefficient T is equal to T= e ea (217) where the attenuation coefficient is calculated using the trapezoidal rule as a, = +a ++a a + (218) 2 2 Therefore, the RMS pressure corresponding to a path of length r is equal to =P 2,ljl e 2ayr (219) Ur ^ where P,,, is initially transmitted pressure (21) with the nearfield correction factor ap plied (267). The multiplication is made over all sub paths, whose total length is Numerical Realization of Multiple Reflections The main difficulty in numerical realization is to account for all possible paths that sound waves can travel along. In the newly developed algorithm the binary tree is used to solve this problem. The generation of the binary tree is possible, because from any layer a sound wave is assumed to be traveling only along the directional beam in two direc tionsup or down. For the given number of layers, the binary tree will look like it is shown in Figure 23. The nodes of the tree correspond to the centers of the layers, and their numeration was done in a way that simplifies the generation of the binary tree. The root of the tree is located at the layer corresponding to the transducer surface. There is a set of parameters associated with every node. They include the indices of the nodes for possible up and down directions, as well as calculated reflection (216) and attenuation (242) coefficients. The same reflection and attenuation coefficients are applied to all nodes located at the same layer. After the binary tree is generated, a fast algorithm of treewalk is used [e.g., Knuth, 1997]. The walk over the binary tree starts with the node at the transducer surface, where the reflection coefficient is assumed to be 1 and the attenuation coefficient is equal the value for plain water. This implies complete reflection of the sound wave from the trans ducer surface and zero suspended sediment concentration. After the first step "down" is made (from the node "1" to "2" in Figure 23), neither reflection nor attenuation coeffi cients are known at layer 1 (node "2"); therefore they are initially assigned with the val 16 ues 1 and 0 correspondingly. The intensity of the sound at layer 1 becomes 2 I' = I, e 'Ar (220) where I, is the initial intensity of sound emitted by the transducer with the index corre sponding to the node number, and V is the nearfield correction factor calculated at r Ar. Bin numbers 0 1 2 3 4 5 6 u1 4 4 9 3 6 f2!? S5 4 27 2' CZ 4 X1 X7 X22 4 X5 2 18 5 6  :: possible direction S i possible path Figure 23. Generation of a binary tree for numerical realization of multiple reflections. A step "up" is made next (from the node "2" to "8" in Figure 23), and the calcu lated intensity of sound arriving back to layer 0 becomes I' I e Ar (221) where "primes" are used for the intensities calculated with initial values R' = 1 and a= 0. The calculated intensity I, should be equal to the measured intensity correspond ing to bin one and recorded at time r, I, This provides us with an expression from which the concentration at leyer 1 can be found, and hence, corresponding reflection and attenuation coefficients Iml, = R2e 21A, (222) '8, After the reflection and attenuation coefficients for layer 1 are determined, the in tensities along all the paths within the layers 0 and 1 can now be calculated and accumu lated according to the equation 214. The next step "down" is made (from the node "2" to "3") and the procedure is repeated until the last layer is reached. The expression to find the concentration in general form becomes I, I R e Ar (223) I' where IJ, is the sum of intensities calculated for the paths which arrive at the transducer surface at the time corresponding to bin i. The node number is found as n,= i n 1 i+1 (224) with n number of layers. Determination of Sediment Size and Concentration Three parameters need to be known at every layer to calculate reflection and at tenuation coefficients: suspended sediment concentration M, and parameters of sediment size distribution, MA and S,. If suspended sediment concentration is the only unknown parameter and mean sediment size, MA, and standard deviation, So, are known somehow (for example, from additional measurements or assumed values are used), then equation 223 can be solved via iterations [Thorne et al., 1993a]. To determine the sediment size parameters and the suspended sediment concentration more than one measurement with different sound fre quencies are needed. The techniques were developed to obtain concentration, M, and mean sediment size, MA, by using 3 transducers, while standard deviation, So, is as sumed to be known [Hay and.\heng, 1992], [Crawford and Hay, 1993], [Thosteson and Hanes, 1998]. To obtain all the parametersM, MA, and S,a system with six trans ducers can be used [Schat, 1997]. More often the 3transducer setup is used. It usually requires sorting, So, of the sediments to be known. The mean sediment size can be estimated from the minimization of the variance of sediment concentration approximated by each of the transducers c2 (MO)= (Mk M) (225) k=1 M3 where M = Mk /3. Number of Paths The number of possible paths grows very quickly with the number of layers n. For a specified n the total number of paths is equal to (n1)th Catalan number [Weisstein]. The n th Catalan number is equal to C (2n)! (226) (n+1)!n! For illustration the first 24 numbers are given in Table 21. The asymptotic form of n th Catalan number is 4" C ~4 (227) which for the number of layers n = 120 gives C120 ~ 7.58.1068. Table 21. First 24 Catalan numbers n C,, n n C,, 1 1 9 1,430 17 35,357,670 2 1 10 4,862 18 129,644,790 3 2 11 16,796 19 477,638,700 4 5 12 58,786 20 1,767,263,190 5 14 13 208,012 21 6,564,120,420 6 42 14 742,900 22 24,466,267,020 7 132 15 2,674,440 23 91,482,563,640 8 429 16 9,694,845 24 343,059,613,650 Because of the large number of the paths, the execution time becomes very long even for a small number of layers. Therefore, not all the paths can be included in calcula tion. The number of reflections which a sound wave is exposed can be chosen as a crite ria for the path to be included in the calculation. The number of reflections for each path varies between 1 and n(n 1)/2, where n is the number of layers. The reflection coeffi cient for the layer of scatterers is determined by equation 216. To approximate the order of magnitude of the coefficient, the following values can be used: r =lm, r =10lOs, c=1500m/s, T'D=0.015, p,=2650kg/m3, M=3g/1l, a=0.2mm, for which R=0.01. Two reflections result in attenuation of incident pressure on the order of 104, three106. Al though, the number of the paths can be very large, the sound waves of small amplitude will be absorbed by the suspension. Therefore, the number of reflections can be limited to 3 if the number of layers is large (n > 15). Effect of Multiple Reflections Four tests were performed to analyze the relative effect of multiple reflections on measurement of sediment concentration. The ratios of intensities calculated with one and three reflections using the model described above are plotted in Figures 24 and 25. For uniform concentration profiles (see Figure 24 ad), the effect of multiple reflections is almost uniform with distance. The difference reaches its maximum of 1214% at 3g/l, but at lg/1 it is only around 5%. For the concentration profiles, which linearly change from Og/1 to maximum values of 0.13g/1 with distance (see Figure 24 eh), the differ ence is even smalleron the order of 13%. This is probably the most common situation for field measurements, when the concentration near the transducer is much smaller than it is near the bottom. However, it is possible for a cloud of sediments to be injected by a turbulent eddy into the water column with the concentration at higher elevations to become greater than the concentration near the bottom. Figure 25 ad shows the results when uniform con centration profiles were modified by a jettype profile with maximum concentration of 2g/l, located at around 30cm distance from the transducer. The difference becomes more visible, especially at the ranges just after the jet location. Compared to uniform concen tration profiles, the intensity increases around 1.2 times. In the next test, a wiretype response was modeled with results shown in Figure 25 eh. For the profiles at 0.1 and 0.5g/l there are "ghost" peaks present, which corre spond to the second reflection of sound between the transducer surface and "wire". This situation can be modeled in the lab with a strong reflector, like steel wire (see Chapter 4). Although the second reflection is present in Figure 25 bd, there are no such "ghost" peaks in the tests with the sediment mixture jet (see Chapter 4). In the field, air bubbles can be present in the water column, and since the bubbles are strong reflectors the second reflection from them can affect the measurements at longer ranges. Concentration profiles 1.08MHz 1.08MHz 2.25MHz 2.25MHz 4.75MHz 4.75MHz 0 0.2 0.4 0.6 Distance 0.2 0.4 0.6 0.8 Distance Figure 24. Difference between intensities modeled with one and three reflections for se lected profiles of concentrations: a) constant profiles, e) linear profiles, bd) and fh) percent difference. Concentration profiles 6 4 Concentration profiles a). ia */ 2 10M 1.08MHz 2.25MHz 20 10 0 0 0.2 0.4 0.6 0.8 Distance 0 20 o 10 Concentration profiles e). 1.08MHz 2.25MHz 30 20 7 5 10 0 4.75MHz 0.2 0.4 0.6 0.8 Distance Figure 25. Difference between intensities modeled with one and three reflections for se lected profiles of concentrations: a) profiles of constant concentration with "bump", e) profiles of constant concentrations with "spike", bd) and fh) same as in Figure 24. f) A ~ZZ~f\Ezzzzr A I Special Parameters for Acoustic Backscatter Equation Form Function and Normalized CrossSection for Elastic Spheres The theoretical farfield form function for elastic spheres can be expressed as [Thorne and Campbell, 1992] f (x= (1)" (2n+1)b (228) Ix n 0 where A a12 a13 a a12 a13 b =2 a22 a23 a21 a22 a23 (229a) 0 a32 a33 0 a32 a33 4, = x,2 (P/Psji(,), nXi ( = x2 p ) (x),2 =xh (x), 12 = [2n(n+1)x ] ( )4xj (x,), 22= X (X ) 32 =2 [j( x ) xj (x ) (229b) a3 =2n(n+1) [x j: (x )j (x )i , 23 =n(n +l)j (x,), a33 = 2xj (x,)+[x2 2n(n+1)+2] j (x), where x, = x(c/c), x, = x(c/c,), c is the sound speed in the water, c, and c, are the speeds of shear and compressional waves in the spherical particle, p and p, are the fluid and particle densities, ji is the spherical Bessel function, and h~1) is the spherical Hankel function of first kind. The derivatives of the functions can be found by the recurrence re lation [Abramowitz and Stegun, 1965] S= ()+ ng (x) (230) x where g (x) is either a spherical Bessel or Hankel function. The normalized scattering cross section is defined as X= t (231) 2ra2 where <7, = L 2 is the total scattering cross section of a single particle of radius a. For the spherical particles it can be found to be S(x) = 2 (2n+1) b 2 (232) X n=0 Equations 228 and 232 involve infinite series, but only a finite number of terms can be calculated. The following approximate expression can be used to calculate the number of terms sufficient for the resulting accuracy of 0 (106) for x < 50 N106 = 1.2246x + 6.4232 (233) HighPass Model for Quartz Sand The modified highpass model for backscattered intensity [Crawford andHay, 1993; Johnson, 1977; .\heig andHay, 1988; Thorne et al., 1993b] provides simpler ex pressions for the form function and normalized scattering cross section for quartz sand fh (x)CO Kfx42 (234a) C =vexp 2exp )212 (234b) 4 K, x4 3Kx4 (235) (')C) I+ x2 4 K C4 3 K,? where v, = 0.25, x, =1.4, i7 =0.5 and V2 = 0.37, x2 =2.8, 172 =2.2, Kf =1.1, K, = 0.18, and x=ka. The functions given by equations 234 and 235 are shown in Figure 26 together with corresponding curves for quartz spheres (228 and 232). The values of the veloci ties of shear and compressional waves in quartz spheres were taken to be c, =3760m/s and c, =5980m/s. 2 1.2 a). b). 1.5 0.8 1 0.6 0.4 0.5 0.2 0 0 100 102 100 102 ka ka Figure 26. Form function and normalized cross section for quartz sand (blue) and spheres (red): a) form function, b) normalized crosssection. Modified Model for Elastic Spheres Based on the spherical model, modified expressions were developed by Thorne and Buckingham [2004] to better describe the scattering properties of natural sediments. The following empirical function was introduced 3 +0.5x+3.5 (236) x3 +3.5 and applied to the lowpass filtered spherical form function and normalized cross section in form f ( X)= (X,) f= (237) (x)= o(x,= Jz (238) The f3 parameters describe the deviation of the modified model from the spherical model due to irregularities in shapes and sizes, which occur in natural sediments. While the parameters /~. and fz are generally independent, their values are close and usually change between 1.3 and 2.2. The value of / =1.9 can be taken for both of the parameters to provide a reasonable fit with the data [Thorne and Buckingham, 2004]. It can be seen, that the function 70 (x, 8) is greater than 1 for all values of x. It implies that the values of modified form function and normalized cross section are greater than the original ones calculated by spherical model. 1.5 2  7:: :. 0.5 101 100 101 102 k.a Figure 27. Modified y function for selected values of parameter 3 . Modified HighPass Model for Natural Sediments Similarly, the highpass model can be modified to better model the properties of particular sediments. Therefore, the original function 70 (x,f8) (236) was currently modified as { 0(x,?3), /3>1 ;(xO)=Lx'+(80.5)x+3.5 (239) 1 x' +3.5 <8< The function y(x,/3) is equal to 1 for /3=0.5. The lower limit 8/ implies y(x, 3) >0 and is equal to 0.55.25/i.75 =3.85. The graphs of the function y(x,/3) for selected 83 parameters equal to 1, 0.5, 1, and 1.9 are shown in Figure 27. The modified highpass model becomes fh ( x) = y(x, /3)f ,, (240) X (x))= y(x,]3z)X, (241) Sound Attenuation Coefficient The sound attenuation coefficient defines the exponential rate of decay of sound in tensity with distance. It depends on temperature, salinity, pressure, presence of suspended particles and microbubbles, and varies with sound frequency [Richards, 1998]. The total attenuation coefficient can be written as a sum of the attenuation coefficient in clear wa ter, a,, and the attenuation due to scattering by suspended sediments, as, a = a, + a (242) The attenuation of sound in the megahertz range in seawater at small (<30m) depths is primarily a function of water temperature. The formula by Fisher and Simmons [1977] gives the approximation to within 4% c= f2 (55.92.37T+4.77.102T23.48.104T3).103 (243) where f is the sound frequency in megahertz, T is the water temperature in degrees centigrade. The sediment attenuation coefficient at a range r is given by a, =nb (244) 2 where nb is the number of particles per unit volume, <7, is the total scattering cross section of the particle. Using the equations 29 and 231 and averaging over the sediment sizes, the sediment attenuation coefficient becomes 3Ka2 a (r) = M (r) = (r)M (r) (245) 4p, a ) where 4 is the sediment attenuation constant evaluated at range r 3Ka2) ( = (246) 4p, a3 The attenuation of the sound traveling within interval [r, r2 ] due to scattering and absorption by suspended sediments can be approximated by a, = 1 fJ (r)M(r)dr (247) r2 ~ r2 r, If the sediment attenuation constant is not changing with range, then the attenuation is equal to a, = "M (248) where M is the mean sediment concentration between ranges [r', r2]. Lognormal and Normal Distributions of Sediment Sizes Sediment sizes vary from very fine (clay) to coarse (sand, gravel, shell) [e.g., .Se'ith, 1984]. Lognormal distribution can be used to approximate the actual distribution of sediment sizes. Its probability density function is given by pL S 1 (In a Ma)2 a aMSS = exp 2 (249) a S, 2S To obtain the probability density function in terms of sediment radius a in meters the following substitutions were made MA =(l+,u)ln231nl0 (250) S = = o ln2 where /u and a, are the mean and standard deviation in phi units. The phi scale is de fined as the logarithm of sediment size in millimeters. The sediment size, given in meters, can be written as (p log2 (2a103) (251) If the probability density function of a is given by equation 249, then the quantity In a is normally distributed. In terms of (p the probability density function is given by P, (p a,) =exp (252) The parameters u,, and a,, can be approximated from the sieve analysis as [Sleath, 1984] /* N1/2 // =log2 (d841d1591/2 and 07 = log92 d841 (253) d,159 where da is the diameter in millimeters for which n % of the sample by weight is finer. The raw moments of the lognormal distribution are =(a) = ap, (a M ,S)da=exp MA + (254) ,2 a2)=Ja2 pL (aMa,S)da= exp{2(Ma,+S )} (255) 0 3 =(a' =japL,(aMSa,S)da=exp 3MA,+9a (256) Averaged Form Factor and Normalized CrossSection Terms The averaged terms in the equations 211 and 246 can be calculated with the as sumption that sediment sizes obey some known distribution. Assuming the lognormal distribution (249), the following simplification can be made. The term Ka2 f2(ka) 2), where (...) denotes averaging over the sediment sizes, is equal to a2 f (ka)2 f (ka) p(a)da (257) 0 where the sediment size is distributed with the density function p (a). If p (a) is the log normal distribution, the expression above simplifies to af (ka)=a 2(ka) pL(aMa,SO)da (258a) 0 l= na=y,da= dy (258b) _a 2F ezyf(e2 1 (yM)2 f e (key) exp 2 dy (258c) 2 S 2(k 2 xp 21 y (M + 2S 2 )]2 (= (key) exp{2(M+ S ) } exp 2Sdy (258d) = J2 f(key)2 pN,(y/it,& )dy (258e) 31 =2Kf (keY)) (258f) Here (... p(x) is the expected value of the argument with the probability density function p (x). PN, (y , 6() is the normal distribution probability density function PN (Y ) )= exp (259) with J = M + 2S2 and S& = S. With backward substitution the term Ka2 f2(ka)2) becomes a2 f (ka)2 = In a = y, = dy (260a) = /2 f(ka)2 1 exp (lna da (260b) = f (ka)2pL (ap, (.) da (260c) 0 = /2 ( (ka)2) (260d) (a2) f (ka)2 ) (260e) The modified term (a2 ((ka) 2 \(ka)f) S a) PI(a (ka) ) (261a) Ka3) /3 exp{2M< + 2S } exp{ 2M +92S} f (ka)2) (261b) exp{3M0+9 2SJ} IL(a^) =e2s'i fka)p(, (261c) /A f(ka)2) =e2s fa) PL4) (261d) The expressions for the averaged normalized total scattering crosssection can be obtained following similar derivations a 2X(ka)) = a2 )(ka))P(a,a) (262) a 2X(ka)) )2 (K(ka ) P(263a) =  (263a) = e2si (a) (263b) (a) Circular Transducer Directivity and Half Intensity Beam Width The directivity of the transducer can be defined as the ratio of sound pressure at point (r, 0), where 0 is latitude angle measured from central axis of the transducer, to the axial pressure at the same range r D ( r,0) ) (264) P(r,0) Let at be the radius of transducer and k = 2r/2A be the sound wave number, where A is the sound wavelength. At great distances from the transducer (rk > 1), the farfield directivity of a circular piston source is [Medwin and Clay, 1997, p. 139; Morse and n gard, 1968, p.381] D() 2J1 (ka, sin 0) Dka sin (265) kat sin 6 where J, (x) is the first order Bessel function of first kind. The curve of the squared di rectivity function, D2 (0), is shown in Figure 28. 1 0.75 '5 0.5 . \ 0.25 I 1.6163 3.8317 6 4 2 0 2 4 6 katsinO Figure 28. Circular piston directivity function, D2 (0). The time averaged acoustic intensity, I, relates to the RMS pressure as [Medwin and Clay, 1997] P2 jI = ]m (266) PAC where PA is the acoustic density that relates to the bulk modulus of elasticity, E, as PA = E/c2. The half intensity beam width, 0, can be defined as 0 = 20 where 0 is the angle from the center axis to where the acoustic intensity is equal to the half of the axial value. For a circular piston source, the directivity function of which is given by equation 265, D2 (0) = 0.5 at ka, sin = 1.6163 Although most of the sound energy is transmitted within the cone defined by the half intensity beam width, as shown in Figure 21, the low frequency transducers can have wider main lobe and significant side lobes. Similar to the half intensity beam width, the width of the main lobe, .ma.n, can be determined from D,2 (0) = 0, which is satisfied at kat sin = 3.8317. The beam width at a given range, r, from the transducer, for example at r =1 m, can be found as W(r) = 2r sin (0/2) and Wea, (r) = 2r sin (On /2). The values of the beam widths 0 and W for frequencies 15MHz are shown in Table 22. It can be seen, that for higher sound frequencies the beam widths are narrower. Table 22. Beam width for a circular piston transducer (a,= 5 mm and c= 1500 m/s). Frequency 1 MHz 2 MHz 3 MHz 4 MHz 5 MHz ka, 21.0 41.9 62.8 83.8 104.7 0, deg 8.9 4.4 2.9 2.2 1.8 .ma.n, deg 21.1 10.5 7.0 5.2 4.2 W, m 0.15 0.08 0.05 0.04 0.03 Wma I. m 0.37 0.18 0.12 0.09 0.07 NearField Correction Factor Close to the transducer, the sound waves from different parts of the transducer sur face interact with each other, both destructively and constructively. The structure of sound pressure in the nearfield is different from the form given by 21. Regardless of the complexity, it still can be calculated accurately [Lockwood and Willette, 1973]. For the application to remote measurements of suspended sediment concentration and sediment size, the detailed information about the sound field close to the sensor is usually [Downing et al., 1995; Thorne et al., 1993a] substituted by the nearfield correction fac tor, y~(r), which is given by [Downing et al., 1995] 1+1.32z + (2.5z)32 ( z = 32 (267) 1.32z+(2.5z) where z = r/rl is the normalized distance, and r1f is the nearfield range as given by r.,= (268) The value of r1f depends on the transducer radius and the sound wavelength. For a fixed transducer size, the nearfield range increases linearly with the transducer fre quency. For illustration, the values of the nearfield range for frequencies 15 MHz are shown in Table 23. Table 23. Nearfield range r, for a circular piston source (a, =5mm and c =1500m/s). Frequency 1 MHz 2 MHz 3 MHz 4 MHz 5 MHz rf, m 0.05 0.10 0.16 0.21 0.26 0.8  56:. / 0.4 04 1 MHz  2 MHz 0.2  3 MHz / 4 MHz S 5 MHz 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Range, m Figure 29. Nearfield correction factor, 1/ly(r), for a,=5mm and c=1500m/s. The nearfield correction factor is applied to the farfield pressure in form of i/I (r). The corresponding curves for 15MHz frequency transducers are shown in Figure 29. For higher frequencies the effect of the proximity to the transducer on trans mitted sound is larger, and the nearfield region is farther. CHAPTER 3 CALIBRATION OF THE ACOUSTIC BACKSCATTER SYSTEM Calibration of the acoustic backscatter system is necessary to obtain system func tion given by equation 211 [Thorne andHanes, 2002]. For this purpose, the measure ments of profiles of backscattered sound intensity can be taken in an environment with known characteristics. Different scatterers, such as thin wire targets, glass spheres or sand, are used in the calibration in various experimental setups [e.g., Hay, 1991; Jansen, 1979; .he/ug and Hay, 1993]. Calibration in a chamber with a homogeneous suspension of sand or glass spheres is the most common technique [e.g., Ludwig andHanes, 1990; Schaafsma, 1989; Tamura and Hanes, 1986; Thorne et al., 1993a; Thosteson and Hanes, 1998]. It has several advantages over other methods. The calibration with similar scatter ers to those found in the field requires no changes in backscattering parameters such as form function or normalized crosssection. The experiments with the thin wire require modification of the acoustic backscatter equation due to cylindrical spreading of back scattered sound compared to spherical spreading from the sand grains and spheres. Also, the definition of sample volume for the wire is different from the one done for the sus pension of particles. Although a modified form of equation 210 needs to be applied [.\heig and Hay, 1993], the method with the thin wire targets provides the easiest calibra tion which requires only a simple setup (similar to shown in Figure 42 in Chapter 4). In experiments with a turbulent jet [Hay, 1991], a mixture of suspended particles and water is used. The suspension within the crosssection of the jet is not homogeneous and therefore modifications to the acoustic backscatter equation (equation 210) are needed. The turbulence provides proper mixing and makes the orientation of the particles random relative to the transducer. Freely falling particles [Jansen, 1979] can have prede fined orientation relative to the positions of transmitter and receiver, although the effect of this is not significant for particles with approximately spherical shapes [.\//,g and Hay, 1988]. For calibration and other tests, two acoustic backscatter systems were used. Both systems were developed by Centre for Environment, Fisheries and Aquaculture Science (CEFAS), Lowestoft, UK. The first system, hereafter referred as "ABS 1", has three 10mm transducers with operational frequencies of 1.08, 2.07 and 4.70MHz. The data col lection was carried out by a specially developed acquisition system [Thosteson, 1997]. The acquisition system operates at maximum rate of 88 profiles per second, and records a maximum of 4 rootmeansquare (RMS) profiles a second by averaging over 22 consecu tive raw voltage profiles. The second system, hereafter referred as "ABS2", also has three 10mm transducers, with frequencies 1.0, 2.5 and 5.0MHz. Aquatec Electronics Ltd, UK, developed the data logger, which allows obtaining raw ABS profiles at a maximum of 80 profiles per second for each frequency. The resolutions of the two systems are different. For the ABS1, the voltage is measured in a 12bit range, while the ABS2 has 16bit resolution. Spatial resolution is defined by the pulse duration. It is 10ps for the ABS1 and 13jps for the ABS2, which re sults in a bin size of approximately 7.5 and 10mm, respectively. Both systems use time varying gain (TVG). The form of the TVG was set to account for the spherical spreading, and is nominally given by a linear function of the range r, although the exact form of the TVG may deviate from linear. Calibration Chamber The recirculating chamber at the University of Florida Coastal Laboratory was used to perform the calibration of the acoustic backscatter systems. This chamber was previ ously used in a number of projects [Lee, 1994; Ludwig andHanes, 1990; Tamura and Hanes, 1986; Thosteson, 1997]. The chamber is schematically shown in Figure 31. It consists of a vertical 2m acrylic tube with inside diameter of 0.19m. A rotary pump moves a watersediment mixture through a 4cm PVC pipe upward, where it discharges into the tank through the four nozzles. The nozzles are at the same level, about 10cm be low the free surface. The jets are directed towards the center of the tank from 90degree sectors (see Figure 33), where they collide and generate a turbulent flow, mixing the suspension. The mixture flows down the chamber with a mean crosssection velocity of approximately 4cm/s. The bottom of the tank has a funnel, which leads the flow through a pipe back into the pump. The funnel prevents the sediments from accumulating at the bottom. The suction ports were closed with metal plugs, which protruded into the interior of the chamber for about 5mm. Before taking measurements, the chamber was filled with tap water and left for about 20 hours to allow air bubbles to leave the system. For additional 24 hours the sys tem was left with the pump turned on. To establish the predefined level of suspended sediment concentration, portions of sand were added into the water. Every portion was initially dried, weighed and placed for several hours in container with a small amount of water and detergent. The detergent was used to detach air bubbles from the surface of the sand grains. Excess water with deter gent was removed from the container before the sand was added into the chamber. Transducer(s) Nozzles Levels 0, 15, and 30 cm Suction ports A I Funnel Pump l 20cm. Figure 31. Calibration chamberside view. ABS Tests in the Chamber The chamber has a different environment for measurements than field experiments. Because of the laboratory setup, various artificial conditions can affect the measurements, which may lead to erroneous conclusions about estimated parameters. A series of tests were performed in the University of Florida Coastal Laboratory to investigate the effects of the chamber on calibration measurements. For every test, profiles of rootmeansquare voltage were obtained. Each profile was calculated as the average of 120 individual RMS profiles recorded at 4Hz for 30 sec onds. To obtain each individual RMS profile, the ABS was set to average 22 raw profiles. Tests were made with fewer averaged profiles to check if the setting has an effect on the measurements. The higher sampling rate theoretically can lead to the interaction between several consecutive pulses, when previous sound pulses were not fully attenuated. The tests did not reveal this possibility. Additionally, to the average profiles, the 10th and 90th percentile profiles were calculated to illustrate the scatter among the measured RMS profiles. Tests in Air and Still Water Two tests were performed to examine the lower bound for voltage profiles. For this purpose the measurements were taken with the transducers left in the air and placed in the chamber with still clear water. The resulting curves are shown in Figure 32. The curves, which correspond to the measurements in the air, take on the lowest values except for the 1.08MHz transducer, which has a peak near approximately 4cm range. None of the pro files start with zero valueson the contrary they start from the value around 50100 in relative units and decay within a few bins. Within this range there is no difference be tween profiles measured in the air and in the water. Most likely this is an effect of resid ual vibration of transducer surface after the sound pulse was emitted. The measurements in the water show that the scatter of the RMS profiles (in figures plotted with dashed lines) can be significant especially for high frequency transducers. The peaks around 9cm for 1.08 and 2.07MHz are caused by the reflection of emitted sound from the chamber walls, which will be illustrated further. 1.08MHz 2.07MHz 200 200 a). b). 150 150 / s 100 n 100 50 50  0o 0o 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 4.70MHz Range,m 150 c). 100  air water/conc. 0 g/I I \  water/conc. 0 g/1I 50 J (center line) 0 0 0.2 0.4 0.6 0.8 Range,m Figure 32. RMS voltage profiles (solid lines) with scatter intervals (dashed line) from the measurements in air and still clear water: a) 1.08MHz, b) 2.07MHz, and c) 4.70MHz. Homogeneity of Sediment Suspension and Effect of Chamber Walls The jet system and high flow rate serve to generate a homogeneous suspension. To investigate the homogeneity of the mixture in vertical and horizontal directions, samples were taken from the four openings in the side of the chamber wall (see Figure 31). The results of this investigation was previously published by Ludwig and Hanes [1990]. From their tests, it was concluded that the suspension in the chamber is homogeneous within the measurement error. Additional measurements were currently conducted to test effects of the proximity of a transducer to the jet system and sidewalls. The ABS1 system with 3 frequencies (1.08, 2.07, and 4.70MHz) was used for the tests. One transducer at a time was placed at different vertical levels relative to the nozzles (see Figure 31) and at different distances from the sidewall (see Figure 33). Nozzles R R 70mm. Suction ports Nozzles Figure 33. Calibration chambertop view. The jet system is shown with transducer po sitions for the sidewall proximity tests. For the wall proximity tests, measurements were taken at the center of the chamber corresponding to a distance of 95mm, and also at 30 and 50mm distances from the side wall. In the vertical, the transducer was placed at 3 levels: 0, 15 and 30cm. The Ocm level corresponds to the upper edge of the nozzles, therefore the jets of sandwater mixture were colliding just under the transducer surface. The profiles of RMS voltage were recorded for several different levels of sus pended sand concentration (see Tables 31, 32, and 33). Sand collected from Jackson ville Beach, FL with sieve fraction of size 0.2120.250mm was used for the test. Table 31. Locations and concentration levels in tests for 1.08MHz transducer. Distance from wall (row) 0cm 15cm 30cm and level (column) 30mm 0, 0.2, 0.5g/l 0, 0.2, 0.5g/l Og/1 50mm 0, 0.2, 0.5g/l 0, 0.2, 0.5g/l Og/1 95mm (center) 0, 0.2, 0.5, 0.8g/l 0, 0.2, 0.5, 0.8g/l 0, 0.2, 0.5, 0.8g/l 43 Table 32. Locations and concentration levels in tests for 2.07MHz transducer. Distance from wall (row) 0cm 15cm 30cm and level (column) 30mm 0.2, 0.5g/l 0.2, 0.5g/l 50mm 0.2, 0.5g/l 0.2, 0.5g/l 95mm (center) 0.2, 0.5, 0.8g/l 0.2, 0.5, 0.8g/l 0.2, 0.5, 0.8g/l Table 33. Locations and concentration levels in tests for 4.70MHz transducer. Distance from wall (row) 0cm 15cm 30cm and level (column) 95mm (center) 0.5, 0.8g/l 0.5, 0.8g/l 0.5, 0.8g/l level 0 cm / conc. 0.0 g/l level 0 cm / conc. 0.2 g/l 1.08 MHz level 15 cm / conc. 0.0 g/l level 30 cm / conc. 0.0 g/l level 15 cm / conc. 0.2 g/l level 30 cm / conc. 0.2 g/l level 0 cm / conc. 0.5 g/l level 0 cm / conc. 0.8 g/l level 15 cm / conc. 0.5 g/l 600 h). 95 mm 400 50mm 400 30mm 200 0 level 15 cm / conc. 0.8 g/l level 30 cm / conc. 0.5 g/l level 30 cm / conc. 0.8 g/l 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Range, m Range, m Range, m Figure 34. Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) for 1.08MHz transducer for sidewall proximity tests. 400 300 a 200 100 0 400 300 a 200 100 0 800 600 a 400 200 Figures 34 and 35 show the results of measurements with the 1.08 and 2.07MHz transducers for different concentrations of sand. For the 1.08MHz transducer the profiles have several peaks, which correspond to the positions of the suction port plugs. As shown earlier (Table 22), the 1.08MHz transducer has the widest directional beam. Therefore, a part of the sound can reflect from the plugs and be detected by the receiver. The profiles for 2.07MHz transducer do not have similar peaks. 2.25 MHz level 0 cm / conc. 0.2 g/l level 15 cm / conc. 0.2 g/l level 30 cm / conc. 0.2 g/l 800 800 800 a). 95 mm b)' 95 mm c)' 95 mm 600 50mm 600 50mm 600 30 mm : 30mm : : 400 400 400 : : 200 200 200 0 0 0 level 0 cm / conc. 0.5 g/l level 15 cm / conc. 0.5 g/l level 30 cm / conc. 0.5 g/l 1000 1000 1000 d). 95 mm 95 mm m. 95 mm 50 mm 50 mm 30 mm 30 mm E 500 500 500 0 0 0 level 0 cm / conc. 0.8 g/l level 15 cm / conc. 0.8 g/l level 30 cm / conc. 0.8 g/l 1500 1500 1500 g). 95mm h). 95mm i). 95mm 1000 1000 1000 500 500 500 : . 0 0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Range, m Range, m Range, m Figure 35. Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) for 2.07MHz transducer for sidewall proximity tests. The measurements along the chamber centerline at different vertical levels are shown in Figures 36, 37, and 38. The curves for the 1.08MHz have peaks at approxi mately 9 and 19cm. Comparing to the profiles at other distances from the wall (see Figure 34), these peaks appear only when the transducer is placed in the center of the chamber. The ranges at which the peaks appear correspond to multiple chamber radiuses. They can result from multiple reflection of the sound wave from side lobes between the chamber wall and transducer surface. This also demonstrates that the side lobes of the 1.08MHz transducer are quite significant. This can affect the measurements in the field as well, be cause other transducers may not detect the scatterers located within the side lobes of the 1.08MHz transducer. conc. 0.0 g/1I conc. 0.2 g/1I 400 400 a)'  Ocm cm 300  15cm 300 15 cm 30 cm 30 cm S200 30200cm 100 100 conc. 0.5 g/I conc. 0.8 g/I 600 800 9).  0 cm d). : 0 cm  15cm 600 15cm 400  30 cm 30 cm S400 4 0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Range, m Range, m Figure 36. Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) measured along centerline with 1.08MHz transducer conc. 0.2 g/1I conc. 0.5 g/1I 800 1000 a  Ocm 0cm 600 15 cm 15 cm ., : 30 cm '^, 30 cm 2 400 500 m 00 0 0.2 0.4 0.6 0.8 conc. 0.8 g/1I Range, m 1500 c). 0 cm 30 cm 500 ... 0 0 0.2 0.4 0.6 0.8 Range, m Figure 37. Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) measured along centerline with 2.07MHz transducer. The effect of proximity to the nozzles is noticeable only for the 2.07 and 4.70MHz transducers. Figures 37 and 38 show that the profiles at level Ocm are lower than at lev els 15 and 30cm. The drop in the intensity can be due to either Doppler shift of the fre quency of the backscattered sound from faster moving sand grains, which cannot be re solved by the receiver with fixed frequency, or due to inhomogeneity of suspended sedi ment concentration in the upper part of the chamber. Although separate measurements of the concentration in the jet part of the chamber were not performed, the later seems to have more effect, since the maximum speed from the jets is not very high (approximately 3m/s) and resulted Doppler shift would be about 0.2% of the transducer frequency. How ever, the measurements show that if the transducer is placed lower than the jets, the dif ference between profiles from different layers is very small. Hence, the vertical homoge neity of the concentration in the chamber is confirmed and by the current tests. conc. 0.5 g/I conc. 0.8 g/1I 1500 a 0 cm 1500 10 a). m :'n : 0 Ocm 100 .,,n, : 15cm 1000  :n', 1000 30cm ED  n. E n' 500 1 :, :.n',n 500 0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Range, m Range, m Figure 38. Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) measured along center line with 4.70MHz transducer Figures 39 and 310 show the difference between the measurements at different levels along the chamber, but now the transducers are placed at 30 and 50mm from the chamber wall. Similar to the tests when the transducers were located at the center of the chamber, the tests for 2.07MHz transducer show a slight decrease in voltage at Ocm level. 47 However, the profiles for 1.08MHz do not have peaks at ranges equal to multiples of the chamber radius. at 30 mm / conc. 0.0 g/1I a). 0 cm" S 15cm m 30 cm at 30 mm / conc. 0.2 g/I 0cm 15 cm m 'cs^ / 0 0.2 0.4 0.6 0.8 Range, m at 50 mm / conc. 0.0 g/I 150 b). 0 cm 100 15 cm  30 cm at 50 mm / conc. 0.2 g/I 300 c F : 0 cm 200o 15 cm 100 I 0 at 50 mm /conc. 0.5 g/I 600 f) o cm 400 : 15cm 200 0 0 0.2 0.4 0.6 0.8 Range, m Figure 39. Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) measured near the chamber wall with 1.08MHz transducer. at 30 mm / conc. 0.2 g/1I at 30 mm / conc. 0.5 g/1I 0 Ocm 15 cm N  v ,, 0 0.2 0.4 0.6 0.8 Range, m 800 600 2 400 200 0 1000 2 500 0 at 50 mm /conc. 0.2 g/1I b). :  Ocm \ 15 cm . : at 50 mm / conc. 0.5 g/1I 0 0.2 0.4 0.6 0.8 Range, m Figure 310. Profiles of RMS voltage (solid lines) with scattering ranges (dashed lines) measured near the chamber wall with 2.07MHz transducer. at 30 mm / conc. 0.5 g/I 800 600 2 400 200 0 1000 m 500 Results The tests show that the jet system generates homogeneous suspension in the region approximately 15cm below the nozzles. The chamber wall effects are not significant for 2.07 and 4.70MHz transducers, but for the measurements with the 1.08MHz transducer, it should not be placed in the center of the chamber to avoid multiple reflections from the side lobes of the beam. To obtain the smooth profile, the peaks corresponding to the posi tions of the suction port plugs can be filtered out by interpolating between the nearest data points. A few bins at the beginning of the profile do not contain useful information on the suspended sediment concentration, since it is contaminated by an electronic noise or residual vibration of the transducer membrane. Estimation of System Function The system functions of the acoustic backscatter, K, can be found from the cali bration by using the expression K =V Vr kat 1 1/6 2ar (31) KM1/2 0.96 3rc where K5 is calculated by equation 211, M is the level of suspended sand concentration in the chamber, Vrm, is the measured rootmeansquare profile of voltage, y is the near field correction factor (267), k and c are the sound wavenumber and the sound speed in water, a, is the transducer radius, r is the pulse duration, and a is the total attenuation coefficient (242) Since the transducer frequency is preset and changes in the speed of sound in water are relatively small, the system parameter Kt, given by equation 21 la, is usually ob tained from the calibration instead of K [Thorne andHanes, 2002]. However, the effec tive transducer radius, which determines the directivity of the sound beam and extent of the nearfield region, is usually 520% less than the geometrical radius. By using the pa rameter KA, the bestfit value for a, may be found from the calibration. The system function K, is constant if the time varying gain (TVG) is not applied, otherwise, K is a function of range r. The system function describes the properties of the system and, therefore, does not vary with sediment size or concentration. However, the functions estimated from the measurements taken for different concentrations of sediments of different size distributions will most likely be not the same due to variations in sediment shapes, mineralogy, etc., which are not accounted for in the acoustic equation 210. The modified form function and normalized crosssection function with parameters fl, and 8z, given by equations 240. and 241, may be used to account for these differ ences, as well as to better approximate the system function. Let K' be the i th estimation of the inherited system function, K, where each K' is calculated by equation 31 at the concentration level MA',eas from the measured voltage profile V, for a given sediment size distribution. The average function can be found as the mean of estimates as K = Y K (32) v V where N is the number of estimations, to provide an estimation of the system function for a particular sediment size distribution. Approximation by 2nd Order Polynomial The system function calculated by equation 32 is not necessarily a smooth func tion. Therefore, K, was fitted with a 2nd order polynomial over the range of bins nfnf < j < n (nf is the nearfield bin index and n is the total number of bins) to produce an estimation of system function K' as K = P2r2 +pr+pO (33) where p,, i =0, 1, or 2, are the coefficients of the bestfit polynomial. Calibration of the ABS1 The ABS1 was calibrated with quartz sand, collected at SISTEX'99 experiment [Vincent et al., 2001], with mean size /u =2.17 and standard deviation a7 =0.27 esti mated from the sieve analysis. From the fall velocity measurements, the mean sediment size was approximated to bel.96 in phi units. Both measurements and corresponding normal distribution curves are shown in Figure 311. The transducer radius was taken to be 90% of the geometrical radius. The modified form function and normalized cross sec tion were calculated by equations 234 and 235 with initial value of 0.5 for parameters Iz and ff 1 , 1b).  0 _[ a). b). 0 0 data fit 0.8 0.8 o 0.6 0.6 o 0o o o 0.4 0.4 j Sieve: Fall velocity: 0.2 / = 2.17 0.2 = 1.96 o =0.27 / =0.27 1 1.5 2 2.5 3 1 1.5 2 2.5 3 9 9 Figure 311. Estimation of sediment size distribution used for the calibration of ABS1 by: a) sieve analysis, b) fall velocity. The resulted system functions are shown in Figures 312314. Individual estima tions, K', are shown together with the curves for K' (marked as "fit") on plots "b" in Figures 312314. The errors on plot "b" were calculated as the average of relative root meansquare deviations from the 2nd order fit and are given by the expression 1 n 1 N K_ ( r) K r ( t ) 2 CK =nn +1 K (34) nf j = The nearfield bin index nnf for current calculations was taken to be equal to 15, but can be adjusted for any particular system. The value was chosen from the sidewall proximity tests which showed that the measurements at bins <15 are not accurate. Calculated by 210, voltage profiles were compared to the measured profiles. Their ratios are plotted on plots "a." Using 212 the inverted profiles of the suspended sediment concentration are shown on plots "c." Relative errors for each of the concentration levels between inverted and measured concentrations are also shown with the mean error, which was calculated as the average of individual relative errors: 1 N 1 .. MA, c (r, ) M'as (r) (35) N nnnf +1 =nn Me', a(r,) where the subscript "meas" denotes measured or expected concentration in the chamber and subscript "calc" denotes estimated concentration profiles calculated with 212. 2.5 2 >1.5 > 0.5 0 5 4 0 2 0 0 1 0 System coefficients '>20 10 0.2 0.4 0.6 0.8 0.2 Range, m Measured (dashed) and calculated (solid) concentrations 0.4 0.6 0.8 Range, m Conc,g/l Err,% 0.1041.0%  0.2027.1% 0.30 13.2% 0.40 9.7%  0.50 3.1% 0.70 4.2% 1.00 15.7% 1.50 28.7% 2.00 25.6% 3.00 36.2% 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Range, m Figure 312. Estimation of the system function for 1.08MHz transducer: a) ratio of meas ured and calculated voltage profiles, b) estimated system functions, and c) in verted sediment concentration. System coefficients 0.5 1.5 0) S1 o 0.5 0 0 0.2 0.4 0.6 0.8 0.2 Range, m Measured (dashed) and calculated (solid) concentrations 0.4 0.6 0.8 Range, m Conc,g/l Err,% 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Range, m Figure 313. Estimation of the system function for 2.07MHz transducer: ac) same as in Figure 312. Ratio of voltages 1.6 1.4 >1.2 0.8 0.5 0.4 0)  0.3 o 0.2 0 0 0.1 0 Ratio of voltages 250 f =:4.70MHzat = 4.50 mm b). PO = 2.170 = 0.27 200 Sf = 0.50 P = 0.50 .... :150 100 50 K =20 0 0.2 0.4 0.6 0.8 0.2 Range, m Measured (dashed) and calculated (solid) concentrations System coefficients fit Err = 4.8066% 19.7920 r2+377.1953 r+25.0449 0.4 0.6 0.8 Range, m Conc,g/l Err,% 90% 0.10 18.2% 0.20 12.7% 0.30 3.1% 0.40 26.2% 0.50 34.3% 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Range, m Figure 314. Estimation of the system function for 4.70MHz transducer: ac) same as in Figure 312. The estimated system functions show that the expected linear dependence due to the TVG is present only for the 2.07MHz transducer, although the 2nd order coefficient is still significant. The curves of voltage ratios on plot "a" for the 4.70MHz transducer (see Figure 314) are not uniform with range compared to the curves for the other two trans ducers. This may indicate erroneously estimated attenuation at each level of concentra tion. The errors corresponding to the different concentration levels are not uniform, espe cially for 1.08 and 4.70MHz. Since the estimated system functions were applied, the er rors grow for smaller and larger concentrations. Calibration of the ABS2 The calibration of the ABS2 was performed similarly to the ABS1. The sand from Jacksonville Beach, FL was used for the calibration. The estimations of sediment size 54 distribution parameters by sieve analysis and based on fall velocity are shown in Figure 315. 1  0.8 Z 0.6 S0.4 0.2 0 1.5 0.8  z^' 0 2 2.5 3 1. Figure 315. Estimation of sediment size distribution for ABS2 by: a) sieve analysis, b) fall velocity. Ratio of voltages 3 1000 a). f = 1.00MHz a = 4.50 mm b). 2.5 : : 2.41 = 0.13 800 .52 J\ Pf= 0.50 3 =:0,50 >o x 600 1 __ ...,. 400 0.5 200 0.5 0 0 0.2 0.4 0.6 0.8 1 Range, m Measured (dashed) and calculated (solid) concentra 1.5 0.2 0.4 0.6 Range, m 0 data  fit 2 2.5 3 sand used in calibration of System coefficients 0.2 0.4 tions 0.6 0.8 1 Range, m Conc,g/l Err,% 0.02 15.0% 0.04 6.7% 0.07 5.2% 0.20 3.9%  0.50 4.3% 1.00 5.0%  2.00 13.1% 0.8 1 Figure 316. Estimation of the system function for 1MHz of ABS2: ac) same as in Fig ure 312. b). Fall velocity: P = 2.28 S= 0.23 c). Mean error = 7.60% /7 n.[^  5 The resulting system functions are shown in Figure 316. Unlike the 1.08MHz transducer of the ABS 1, the curves for the system functions are very close to each other and the correction of /f parameter was not needed. The 1MHz transducers of the both systems are very similar, and one would expect to obtain similar results in calibration of both systems. However, the current tests show that the results are very different. The calibration of the 1.08MHz transducer of the ABS1 does not result in a unique system function. The application of the average of estimated system functions can lead to large errors in inverted concentrations. For these cases, a semiempirical model was developed to provide the solution for suspended sediment concentration. This model is described in Appendix B. Sensitivity Analysis For Estimation of System Function A particular estimate of system function K depends on several parameters, which can be initially approximated with some error or uncertainty. In order to investigate the sensitivity of the system function K, to changes in the parameters, such as form function, normalized crosssection, and sediment distribution, the following analysis was carried out. Using equation 31, the system function K, can be rewritten as a function of pa rameter ,fi, transducer radius at, and sediment size distribution parameters jl, and a. as K = F( lz,a,, p ) (36) The form function and normalized cross section were estimated by equations 240 and 241. The dependence of K on listed parameters is not explicit, but through the form function, normalized crosssection, beam directivity, and nearfield correction factor. Generally for calibration, these parameters need to be known and fixed. For current analysis, a subset of the listed parameters set to be free variables (i.e. by letting them vary within some intervals). This produces the new system functions, K ,,fi estimated by minimizing the error calculated by equation 34. The results of the calculations, based on the measurements for the ABS 1, are gath ered in tables in Figures 317319. For each test, some of four parameters Jfz, a,, uV , and a,, were fixed (marked with "dots"). Others were variable, for them the bestfit val ues are presented. Bounds for free parameters are shown in the last column of the table. For fixed parameters, their initial values were used. The errors of inverted concentrations together with the mean error (black line) were calculated and plotted for each test. The values of the mean errors are shown in the table in "Mean error, %" row. Values in row "Fit errors, %" were calculated by 34. The bounds of free parameters were chosen to cover a realistic, but still wide, range of values. In many cases, the minima were reached at boundaries. If unconstrained mini mization would be performed, the minimum could be reached at unrealistic values, such as Omm for transducer radius. It can be seen from the figures that not all the parameters have the same effect on calibration. The parameter fiz and mean sediment size p/ have the most significant ef fects, while transducer radius a, and sediment size sorting a, have almost no effect. Since both the mean sediment size and parameter Jfz are present in the expression for normalized crosssection, minimization on one of them can be compensated by an equivalent change in another. Because of this relationship, use of incorrect values for 8fz may lead to erroneous approximation of sediment size from the measurements. 57 45 Conc, g/I 40 0 0.10 e 0.20 35  0.30 0.40 30 0.50 0.70 25 1.00 20 ~ 1.50 W 2.00 15 0 3.00  mean 10  5  0 Test # Initial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Fit error, % 11.7 10.9 10.2 9.89 11.7 10.9 10.2 9.89 10.4 10.2 9.86 9.74 10.4 10.2 9.86 9.74 Bounds Mean error, % 20.5 18.5 16.6 16 20.5 18.5 16.6 16 17.2 16.7 15.9 15.6 17.2 16.7 15.9 15.6 P 0.50 . 2.502.502.502.502.502.502.502.50 2.50+ 2.50 at, mm 4.50 5.00 5.00 5.00 5.00 5.00 5.00 5.00 5.00 3.00 5.00 1 2.17 2.71 2.71 2.71 2.71 2.71 2.71 2.71 2.71 1.63 2.71 0 0.27 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 1.00 Figure 317. Values of fitted parameters fz, at, p and a. with corresponding errors in inverted concentrations for the 1.08MHz transducer of the ABS1. 12 T F Conc, g/1 0 0.10 10 0 0.20 0 0.30 e e e 0e e e e 0.40 e 0.50 8o 0.70 0E 1.00 20 o o ee e e e eomean Test # Initial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Fit error, % 3.57 2.84 2.65 2.65 3.57 2.84 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 2.65 Bounds Mean error, % 5.58 5.02 5.2 5.2 5.63 5.06 5.24 5.24 5.2 5.2 5.2 5.2 5.24 5.24 5.24 5.24 P3 0.50 0.770.180.040.010.770.190.040.01 2.502.50 at, mm 4.50 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 5.00 t 2.17 2.47 2.40 2.47 2.40 2.36 2.21 2.36 2.22 1.63 2.71 0 0.27 0.05 0.21 0.05 0.21 0.05 0.05 0.07 0.09 0.05 1.00 Figure 318. Same as on Figure 317 for the 2.07MHz transducer of the ABS1. 35 r T I T0 Conc, g/I 30\  0.10 30 E E o 0 e 0.20 e 0.30 25 0.40 e 0.50 o20 0 mean S15  10 0 1 1 1 1 1 1 1 1 1 1 1 1 1 Test # Initial 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Fit error, % 4.81 4.56 4.8 4.4 4.8 4.55 4.8 4.39 2.45 2.45 2.45 2.45 2.44 2.44 2.44 2.44 Bounds Mean error, % 18.9 18.5 18.9 18.3 19 18.7 19 18.4 14 14 14 14 14.2 14.2 14.2 14.2 P 0.50 . .. 1.20 1.18 1.20 1.18 1.20 1.18 1.20 1.18 2.502.50 at, mm 4.50 3.00 3.00 3.00 3.00 .. 3.00 3.00 3.00 3.00 3.00 5.00 S 2.17 2.13 2.01 2.13 2.01 2.22 2.11 2.22 2.12 1.63 2.71 G 0.27 0.05 0.05 0.05 0.05 0.19 0.19 0.19 0.20 0.05+ 1.00 Figure 319. Same as on Figure 317 for the 4.70MHz transducer of the ABS1. The biggest decrease in mean error (35)from 18.9 to 14.0%is shown in Figure 319 for the 4.70MHz transducer by varying parameter z After the corrections to sediment attenuation, not only the mean error was reduced, but also the errors for most of concentration levels. The sediment attenuation for transducer 1.08MHz was insignificant; neither p/ nor fz have any influence on errors. For the 2.07MHz transducer parameter #z was estimated to be 0.77, which results in decreasing of the originally estimated (by equation 235) normalized crosssection and therefore sediment attenuation. The corre sponding value of f8z for the 4.7MHz is 1.20, which implies that the sediment attenua tion is larger than estimated by equation 235. The system functions for the 4.70MHz transducer with corrected f3z are shown in Figure 320. The errors in inverted concentrations at low concentration levels (0.1 and 0.2g/l) were decreased more than 2 times, and also became more uniform with range. The spreading among K,it curves was significantly reduced resulting in a mean relative error of 2.4%. Ratio of voltages System coefficients 1.6 250 a). f = 4.70MHz at = 4.50 mm b). 1.4 = 2.17z = 0.27 200  > 1.2Pf = 0.50 P = 1.20 1 >"1.2 150 1 100 0.8 : 50 Err = 2.4457% 0 5 K = 198.7014. r2+405.7229 r+22.0241 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 Range, m Range, m Measured (dashed) and calculated (solid) concentrations Conc,g/l Err,% 0 .5  c). Mean error = 13.96%  0.10 6.0% ._4_  0.20 5.9% 0.4 0.30 3.9% S.. ... ......... .. 0.40 25.1% @0.2    0 0 0.1 .. ...  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Range, m Figure 320. Estimation of system function KV for 4.70MHz transducer with corrected parameter /fz: ac) same as in Figure 312. Dependence on Sediment Sorting The dependence of K on sorting is rather complicated, because of filtering the data (either theoretical or experimental) used in obtaining the form function and normal ized crosssection. From equations 261 and 263, K can be simplified to r(a2 ,1/2( K = C (r (3 .exp C2(r) = C (r)[es .exp C2(r)e2S = (37) =C3 (r) exp {SC4(r)e 2S} where S, = 7 In 2, and functions C, (r) do not depend on 0a, but on other parameters. 60 By approximating exponents to 2nd order, the following expression can be obtained K C3 (r){lC4 (r)+o2 Iln4(l+ 2C4 (r))} (38) where K now depends on a,, to second order. Therefore, small changes in a,, will have small effect on minimized relative error. This is supported by Figures 317319, where there is no consistency in calculated values for a, ; the errors sometime reach their min ima at either limits of the constraint interval for o',. Based on current analysis, it can be concluded that if the sediments are well sorted, then it becomes nearly impossible to suc cessfully approximate sorting from measurements using this minimization technique. Dependence on Transducer Radius The dependence of K, on transducer radius a, can be explicitly found and deriva tive becomes Ka = C(r) a, +y = C(r){2zVi+V,} rA Za'2 z(39) =C(r){1.010.01l(zfr)} rA where C(r) is the coefficient not depending on at. For farfield (z > )T) the relation be tween K, and at is almost linear with a very small second order term, and therefore does not significantly affect the relative error (34). However, if the nearfield region extends significantly, as in the case of 4.70MHz transducer, changes in transducer radius can be noticeable. Determination of Parameter fL For a homogeneous suspension of sediments, coefficient K, (equation 21 la) is constant with range. Therefore, any uncertainties in determination of K affect only the magnitude of the estimated system function K. Since the system function for a given transducer is unique, any deviation of the estimated system functions can be compensated by using the modified form functions (237 or 240) with the appropriate parameter ,ff. Because the coefficient K, appears as a multiplier in equation 31, the parameter flf cannot be found from measurements with a homogeneous suspension of one sediment sample. Thus measurements with different sediment samples are required. From equations 211 and 240 it can be found that there exists a constant 2 a2f 2h) Ka2 fh2)( 2 (ka, 1f ) (310) ={aY(ka,/3 ,)fh which for small standard deviation a, gives j r(ka,f/) (311) Similar to the estimation of the system functions, K', which were obtained from the measurements with particular sediments (see equations 32 and 33), let the system function K, be written in the form of a 2nd order polynomial over the range n nf < j < n K,=pr + pr+p (312) with each coefficient pk, k =0,1 or 2, found as 1 m k Pk, (313) where pk,j are the corresponding coefficients of the estimation for one sediment size sys tem function K', and m is the number of estimations. Therefore, j can also be written as K$ P2r2 +Plr+po (314) Kv p2r + pr + po which needs to be satisfied for all values or r. This may not be possible for all times, therefore the estimation of using mean quantities can be used and the following ex pression obtained 1 p p, )r 2{+(p( p)r+)(po4 p)} dr=0 (315) with the solution for j as P2 r02 +P r +P0 = 2 (316) P2 r2 +P2r + po 3 2 where r0 is some reference range. After obtaining the solution for J, parameter flf, can be found by inverting equation 311. Estimation of System Function for ABS1 To obtain the system function for the ABS1, the tests were carried out with four different sediment samples. The sediments were predominantly quartz sand. The parame ters of the size distributions were estimated from sieve analysis and are shown in Table 34. The sample from Case 3 is the same sample used for the calibration of the ABS1 and described in the previous section. Its size distribution is shown in Figure 311. The sedi ments for Case 4 were the mix of two sieve fractions0.1250.149 and 0.2970.250mm. The estimations of distribution parameters are shown in Figure 321. The system functions were calculated for all four series of tests. The results of the calculations are shown in Appendix A (see Figures A13A36). The estimated parame ters f6z and ,ff, the mean errors (calculated by equations 34 and 35), and parameter 4 are gathered into Table 35. The parameters /f, were approximated in both casesfor initial values of fz =0.5 and for bestfit values, estimated from the minimization of error 34. It was previously noted that for a wide sediment size distribution, the value of sort ing a, can have a noticeable effect on estimated quantities. For example, Figure A2 shows that if o, =0.44, then the corresponding parameter fz is equal to 1.98. But if the sorting would be estimated as o. =0.06, which is the bestfit value when a,, is taken as a free parameter, the parameter 8fz is equal to 0.19. In Table 35 the values of parameter I#z correspond to the estimated from measurements values of a (from the column for Test#8 in Figures AlA12 in Appendix A). Table 34. Sediment sizes. Case 1 Case 2 Case 3 Case 4 /, 2.67 2.74 2.17 2.16 o' 0.44 0.25 0.27 0.54 1 O 1 O a). b). 0 0 data fit 0.8 0.8 0.6 0.6 0 o o C 0 0 0 00 0 o o o t 0.4 0.4 0 0.2 Sieve: Fall velocity: 0.2 t = 2.16 0.2 = 1.99 G = 0.54 G =0.60 0 0=0 0000 1 1.5 2 2.5 3 1 1.5 2 2.5 3 9 9 Figure 321. Estimation of sediment size distribution for Case 4 used for calibration of ABS1 by: a) sieve analysis, b) fall velocity. The results in Table 35 show that for the fitted values of /, all mean errors de crease up to 50%. Case 1 for the 4.70MHz transducer is the only one, where the mean error in inverted concentrations increases slightly. Although the changes in parameters I# and /fi are also small. Table 35. Estimation of parameters /6 and fL for different sediment sizes. Case 1 Case 2 Case 3 Case 4 Initial Fitted Initial Fitted Initial Fitted Initial Fitted 1.08MHz 1# 0.5 K % 6.9 CM, % 12.3 1.04 'yf 0.90 2.07MHz 1# 0.5 K, % 7.3 M, % 11.4 1.08 'if 0.92 4.70MHz 1# 0.5 K, % 3.2 ,M % 16.8 1.16 '/f 1.10 Table 36. Values Frequency, MHz 1.08 2.07 2.5 5.3 9.5 1.04 0.91 1.98 3.6 6.8 1.06 0.85 0.33 3.1 17 1.13 1.04 of x= Case 1 0.36 0.69 0.5 3.6 6.8 1.51 2.5 0.5 2.6 5.9 1.15 1.85 0.5 5.8 21.9 1.01 0.55 for test Case 2 0.34 0.66 2.5 3.3 6.0 1.53 2.5 0.37 2.6 5.9 1.21 2.5 1.47 2.1 11.8 1.09 0.91 cases. Case 0.51 0.98 0.5 11.7 20.5 0.80 0.94 0.5 3.6 5.6 1.01 0.56 0.5 4.8 18.9 1.03 0.72 2.5 10.4 17.3 0.80 0.93 0.77 2.6 5.2 1.04 0.66 1.20 2.4 14.0 1.08 1.00 0.5 17.1 37.8 0.65 1.99 0.5 12.2 inf 0.76 0.59 0.5 6.2 42.5 0.80 0.82 2.5 14.1 26.3 0.63 2.16 2.17 8.7 17.11 0.69 0.88 2.5 4.7 18.8 0.71 1.43 3 Case 4 0.51 0.98 4.70 1.57 1.49 2.22 2.23 It was previously noticed, that even large changes in parameter 8, for the 1.08MHz transducer have small effects on estimated errorsthe fitted values for )6, ka drop to 2.5 (the lower bound), the value chosen to be the lower limit in the calculations. For the 1.08MHz frequency, the values of x = ka (see Table 36) are less than 0.51 for the Cases 14. This gives the values for the normalized crosssection of 0.01 or less (see equation 235 and Figure 26), which results in very small total attenuation (242) at low concentrations. Since, the parameter /z is found from an expression like ;I(kaJZ) h (ka) (317) where normalized crosssections is estimated from data and h is given by equation 235are small numbers, approximated from their ratio values of y can have large er rors. Therefore, the initial value for z =0.5 can be used. 1.08MHz 2.07MHz 4.70MHz 0.2 0.4 0.6 0.8 Range, m  Case 1: ta =2.67, o =0.44  Case 2: pt = 2.74, o = 0.25  Case 3: pt =2.17, o =0.27 Case 4: at = 2.16, o = 0.54  Mean 0.2 0.4 0.6 0.8 Range, m Figure 322. Estimated system functions and their means for Cases 14 with fz =0.5 (dashed) and fitted (solid): a) 1.08, b) 2.07, and c) 4.70MHz transducers of the ABS1. 50 40 30 20 10 0 250 200 150 100 50 0 66 The estimated system functions for all four cases are shown in Figure 322 together with their means calculated by equation 312. The dashed lines correspond to the func tions estimated by assuming the initial value fz =0.5. The solid lines correspond to the system functions estimated with fitted values for fz (see Table 35). The parameter j was estimated by equation 316 with ro =0.6m. Corresponding values of Jfl, were found by inverting equation 311. Both parameters are shown in Table 35. 1.08MHz 2.07MHz 1.6t . E1.2 0.8 0.6 . 4.70MHz 1.4 )' 'IV .0.8  . ...... ... .... 0.6 . 0.2 0.4 0.6 0.8 Range, m .1.2 I   Case 1: po = 2.67, o = 0.44 I .   Case 2: pt = 2.74, o = 0.25 S1 Case 3: ta = 2.17, o = 0.27 0.8 Case 4: It = 2.16, G = 0.54 0.6 0.2 0.4 0.6 0.8 Range, m Figure 323. Ratios between individual system functions for Case 14 and mean system function with f8z=0.5 (dashed) and fitted (solid): a) 1.08, b) 2.07, and c) 4.70MHz transducers of the ABS 1. Although the system function should be unique for a given system, the estimated curves have some scatter, which results from the different sediments used in each case. For a homogeneous suspension, the curves differ approximately by a constant (equation 314). Therefore, the ratios of individual estimations to the mean were calculated with the results shown in Figure 323. For the 1.08 and 2.07MHz transducers, the calculated ratios are close to constant for r >0. Im. In most cases for these two transducers, the ratios are more uniform for the system functions estimated with fitted values of /z, compared to ones estimated by using /z =0.5. This means that the corrections to the normalized cross section (equation 241) lead to a better approximation of sediment attenuation. Also it should be noted that for the 4.70MHz transducer, the ratios are less uniform with range than for other two frequencies. Analysis of Relationship between )6 and fi As can be seen from the results in Table 35, by choosing appropriate values of fz and f/, corrections to the form function and normalized crosssection can be made (equations 240 and 241), which improve the accuracy of suspended sediment concentra tions. Since the parameters Ifz and fl3. were found individually for each frequency and each sediment sample, the following questions may arise: * Can a single value 8? be found, which could be used for correction of both the form function and the normalized crosssection for every individual or for all used fre quencies? * Can a single value of fL be found which would represent a sediment sample and be unique for an applicable range of sound frequencies? * Can a single value of /z be found which would represent a sediment sample and be unique for an applicable range of sound frequencies? In Figure 324, the estimated parameters f/L are plotted versus corresponding fitted values of /z for frequencies 2.07 and 4.70MHz. It can be seen that in most cases J8f > fz, as it obtained by Thorne and Buckingham [2004]. Because of large scatter and an insufficient number of available data points, the functional relationship between Jfl3 68 and f can only be approximated with a large degree of uncertainty. Nevertheless, it can be concluded that Jfl. and I3 are most likely different even for a single frequency. 3 2 cc 0 1 2 3  2.07MHz  4.70MHz O Case 1 A Case 2 O Case 3 0 Case 4 3 2 1 0 1 2 3 Figure 324. Fitted parameters I3# and corresponding parameters f. for Cases 14. Values of fL and 8', for Cases 14 are plotted in Figure 325. It can be seen, that even with the large scatter, there is an overall trend in the parameters. 3 2 1  C1 Case 1 Case 2 Case 3 Case 4 Figure 325. Values of parameters flf and 8 for Cases 14 for different frequencies.  1.08MHz 2.07MHz 4.70MHz  A A 1 a OOf 0 0 A A 0 0 A A Figure 326 shows calculated values for j together with y(ka, fz). Dashed lines correspond values of x = ka (see Table 36). Note that values of 7(ka, f ) for the 1.08MHz transducer were not plotted since all fitted z = 2.5. Solid lines correspond to the function y(ka, 3) (equation 239) with f3 estimated as I= ( f,2o07+ f,470) (318) where f3f,207 and ff ,470 are the values of Jf3 for the 2.07 and 4.70MHz transducers. The lines fit data points for j relatively well, especially in Cases 1 and 4. Although in Case 3 the point corresponding to the 1.08MHz frequency is much lower than points for the other two frequencies and also lower than the approximated curve. Overall, equation 318 gives a good approximation for parameter f/f for a given sediment sample. 1.6  ....  Case 1 Case 2 : : \'. I 0 A: /\ case 3k ' 100 101 1.8 CC 4 k.a Figure 326. Values of and "(ka,1 3) as functions of ka with approximated functions y(ka, 3) for Cases 14 (solid lines). A similar approximation for fz will not be possible. The values of y(ka, ,8) are close to each other only for Case 4, while other three cases look random. "FunPump" Tests The homogeneous sediment suspension, generated in the chamber (Figure 31) for calibration, will most likely not be found during field or lab experiments, when the meas urements are taken above a sandy bottom, and the suspension is generated by waves and currents. The following tests were conducted in order to simulate the "real" environment, which have variable concentration and size distribution profiles of sediment suspension. A simple technique is used to generate such nonuniform profiles in the chamber. With the sediments added to the chamber, the pump system generates a homogeneous suspension throughout the water column. If at any time the pump is turned off, the sediments will settle with some terminal velocity. Since there are no other sources of sediment near the top of the chamber, a nonzero gradient in sediment concentration will be generated. Also, if there are several sediment sizes present, then the sediments with different sizes will settle at different velocities. This will also generate a nonuniform profile of sediment size distributions. The measurements were taken with the ABS1 system. Three transducers were placed at the top of the chamber at about 10cm below the free water surface pointed downwards (see Figure 31). The ABS was operating at 88 profiles per second. Every 22 profiles were averaged to obtain a single rootmeansquare (RMS) profile of voltage. The resulting RMS voltage profiles were recorded by the data logger at 4Hz. The duration of every run was three minutes, including 30 seconds of data with homogeneous suspension with the pump turned on. Then the pump was turned off, and the measurements were re corded for another two minutes, while the sediments were settling down. After that, the pump was turned back on and data were collected for another 30 seconds. Sample Sediment Size Distribution A mixture of two sediment samples was used for the tests. These are the same sedi ments as were used for the calibration (see Case 4). The first sample is the sieve fraction of size 0.1250.149mm (2.472.75 phi units) of the sand from Jacksonville, FL. It will be referred as Sand 1. The other sample is the sieve fraction of size 0.2970.250mm (1.25 1.75 phi units) will be referred as Sand 2. Assuming a uniform distribution of sediment sizes within each fraction, one can approximate the mean and standard deviation of a mixture of two sediment samples. Let the mixture contains two fractions a and a, of Sand 1 and Sand 2. Therefore, a + 2 = 1 (319) The size distributions are given by p, ((p) = ba (320) 0, [ ',b] where i corresponds to Sand 1 or Sand 2, and [a,b, b] are the size ranges. The mean and variance of sediment sizes in mixture becomes // = + a2, = a+ a2 (2 +U)_ ) 2 (321) where / and a, are the mean and standard deviation of Sand 1 and Sand 2, determined as A = ") = (322) 2 12 72 Different concentration levels were acquired by adding proper amounts of Sand 1 or Sand 2 into the chamber. The fractions of each sample are shown in Table 37 together with the resulting size distributions. Based on the fall velocity analysis, a 1:1 mixture of two samples had the mean of 1.99 and standard deviation of 0.60 (see Figure 321). Note that the values of parameters of the distribution for the 1:1 mixture in Figure 321 from the sieve analysis were different from Table 37. It is because for the calibration in Case 4 average values of the parameters were used. Table 37. Concentration levels, mean and standard deviation of the mixture for "fun pump" tests. Concentration, g/1 a, a2 P aUV, 0.4 1.00 0.00 2.61 0.08 0.8 0.50 0.50 2.06 0.57 1.2 0.67 0.33 2.24 0.53 1.6 0.50 0.50 2.06 0.57 2.0 0.60 0.40 2.17 0.55 2.4 0.50 0.50 2.06 0.57 3.5 I Sand 2: Sand 1: Sand 1 3 = 1.5 = 2.61 Sand 2 SMix 1:1 2.5 = 0.14 = 0.08 2 IL 2 0 1.5 0.5  1 1.5 2 2.5 3 3.5 Figure 327. Sediment size distribution for Sand 1, Sand 2, and 1:1 mixture. The fall velocity analysis provided estimations of size distribution functions (PDF) for both sand samples and the 1:1 mixture. The results are shown in Figure 327. Note that the size distribution for Sand 1 is wider than that approximated from the sieve analy sis. Prediction of Concentration and Sediment Size Distribution The profiles of sediment concentration and size distribution parameters at any given time after the pump was turned off can be predicted with a ID advectiondiffusion model (see Appendix C for a more detailed description of the model). The sediment dif fusivity parameter (equation C1) can be taken to be zero, because of the absence of tur bulence production when the pump is off. When the pump has just been turned off, the turbulence is not zero. Therefore, a small constant value can be used for the sediment dif fusivity to account for the residual turbulence. The sediment diffusivity was taken to be 0.0005m2/s. Other values within 50200% interval were also tested; this did not result in significant difference in predicted profiles. Size distribution at Z= 0.30 m Time, s Sediment concentrations at given times, g/1I 0.06 :10  / 2. 30 0.6 0.04 40 L 50 0.02  60 0.4  070 Size distribution at Z= 0.60 m 0.2 0.08 0.06 0  0 \Mean of sediment size distribution, i 0.04 2.8 e). U 0.02 2.6 2.2 Size distribution at Z= 0.75 m ___ 0.08 c). STD of sediment size distribution, o 0 0.6 f)0 0.04 / 0.4 02.02 0.23 0 1 1.5 2 2.5 3 3.5 0 0.2 0.4 0.6 0.8 1 Sediment size, 0 Distance from the top, m Figure 328. Predicted with ID advectiondiffusion model profiles of concentration and sediment size distributions for maximum concentration of 0.8g/l at times 0 70s: size distributions at a) 30, b) 50, and c) 75cm below the free surface, d) sediment concentration profiles, e) profiles of mean sediment size in phi units, and f) profiles of standard deviations in phi units. The suspended sediment concentration profiles and profiles of size distribution pa rameters predicted with the ID advectiondiffusion model for the initial concentration in the chamber of 0.8g/l are shown in Figure 328. All the profiles are shown from 0 to 70 seconds at 10second intervals. The initial size distribution was approximated from the fall velocity analysis (see Figure 327). The parameters of size distribution were esti mated by fitting the normal distribution curve (equation 252) to the calculated by the model distribution curves. The model results show that most of the concentration profiles have a step, which originates from the difference in terminal settling velocity of two sediment samples; the coarse sediment, Sand 2, settles faster than the finer sediment, Sand 1. A similar irregu larity can be seen in the profiles for the approximated mean sediment sizes and standard deviations. The mean sediment size at every range increases (in phi units) with time, while the sorting decreases. The difference in the sorting along the profile can be quite significant. It can be seen especially at times 10 and 20 seconds after the pump was turned off, when the STD changes from approximately 0.2 to 0.5 along a single profile. This can affect the accuracy of the inversion of the measurements taken by the ABS into concentration and sediment sizes, as already was previously discussed. Parameters of Inversion For the inversion of RMS voltage profiles, the model defined by equations 210 and 211, was used. The modified highpass model for backscattered intensity, defined by equation 240 for the form function, and by equation 241 for the normalized cross section were applied with the parameters /f, and fz determined from the calibration (see parameters for Case 4 in Table 35). The system function K was approximated by equation 312 based on all available calibration profiles (see section "Determination of Parameter Jf "). The resulting coefficients of the second order polynomial are given in Table 38. Table 38. Coefficients of 2nd order polynomial for system functions. Frequency, MHz p2 PA p0 1.08 24.65 1.162 11.62 2.07 4.118 55.37 18.56 4.70 199.9 359.6 27.18 The form function and normalized crosssection are shown in Figure 329. Both the initial unmodified highpass model curves for f ,h and X ,h (equations 234 and 235) and the modified versions f:h and %,h (equations 240 and 241) are plotted. With the provided parameters flf and 83 the modified functions have lower values than the ini tial ones. It should be pointed out that according to the plots the 1.08MHz frequency is the least sensitive to the sediment size represented by Sand 1. For this sand, the sediment attenuation, a, (equation 247), is also close to zero. Therefore in the test with 0.4g/l ini tial concentration when only Sand 1 was used (see Table 37), the inverted concentration for the 1.08MHz transducer was much lower (with an average approximately equal to 0.07g/1) than for other two transducers (see Figure 330). The mean concentration profiles were also estimated to be lower. The profiles of mean sediment sizes were found by minimization of the error be tween estimated concentration profiles for each frequency at a given range (equation 225). The standard deviation of sediment sizes needs to be known in advance. Theoreti cally, the sediment sorting can be specified at each time for every profile. But since field measurements usually lack of such information, and the sorting is estimated from the sieve or fall velocity analyses. Therefore, a single value equal to 0.54 (p was used in all calculations. 1.5 1.5 a). b).  , ',h' X h S. Sand 1: 1 \ 1 1 25 3=.2.61 G =0.08 S \.and 2: 01. F 1.50 0.5 \ 0.5 NN 1 1.5 2 2.5 3 3.5 1 1.5 2 2.5 3 3.5 Figure 329. Highpass model for backscattered intensity and its modified version used for "funpump" test: a) form functions, b) normalized crosssections. The empirical model described in Appendix B was also used to invert the ABS measurements into concentration profiles. The required parameters of the model were estimated by using the same calibration data as for system functions and parameters /f . and I6. Results of Inversion for Concentration The results of inverting the concentrations from the "funpump" test with a maxi mum concentration of 0.4g/l are shown in Figure 330. The profiles of suspended sedi ment concentration estimated from the 1.08MHz frequency transducer are shown to gether with the mean profiles. The mean profiles were calculated as an average of the es timations from the three frequencies. It was already noted in the previous section, that the resulting concentrations from 1.08MHz data are almost four times lower than expected. There may be several causes of this discrepancy. Some of them are: * Erroneous initial values for parameters of sediment size distribution, used for cali bration, which can result in biased values for parameter fi, and system functions; * Lower sensitivity of the 1.08MHz transducer to the used sediment sizes. The mean sediment size specified for the calibration was taken as an average of the sediment sizes for all used concentration levels. It was equal to 2.16 (p (see Table 34). But since a mixture of two sediment samples was used for the calibration, none of sample mean sizes was equal to it. As can be seen from Figure 329, the 1.08MHz transducer is less sensitive to the finer sample than to the coarse sediments. Since two sample sizes were not equally resolved, the use of their mean value can result in overestimation of the system function and in obtaining a biased value for f/i. By exclusion of the 1.08MHz frequency, one can improve the accuracy in estimation of the mean concentration pro files. But the decision should be made based on ability of the transducer's particular fre quency to resolve sediment sizes present in the sample. 0.4 Ini: M = 0.4 g/l, p = 2.61, % = 0.50  pump off 0.35 max off 10 sec off 20 sec off 30 sec 0.3 off 40 sec off 50 sec 0.25 off 60 sec off 70 sec 0.2 0 0.15 0.1 0.05  0 0.2 0.4 0.6 0.8 Range, m Figure 330. Inversion by unmodified highpass model concentration profiles for "fun pump" test with maximum concentration of 0.4g/l: mean profiles (thick lines) and 1.08MHz profiles (thin lines). 78 Figures 331 and 332 show inverted concentration profiles for the "funpump" test with a maximum concentration of 0.8g/l. The profiles in Figure 331 were calculated with the initial unmodified highpass model, which corresponds to the modified model with values of parameters ifl~ and fz equal to 0.5 for all frequencies. The profiles in Figure 332 were calculated with the modified model with ifl~ and fz estimated from calibra tion (see Table 35). .2 Ini: M = 0.8 g/1, P = 2.06 .5 pump off max off 10 sec 1 off 20 sec off 30 sec off 40 sec 0.8 off 50 sec 0  off 60 sec .0 0 off 70 sec 0.6 0.2 0 0 0.2 0.4 0.6 0.8 Range, m Figure 331. Inversion by unmodified highpass model concentration profiles for "fun pump" test with maximum concentration of 0.8g/l: mean profiles (thick lines) and 2.07MHz profiles (thin lines). At the initial time, when the sediment suspension in the chamber was homogeneous (profiles labeled "pump off") both models give similar results, since the system functions were calibrated by using these profiles, and the correction for fly's is not yet necessary. The high jumps in the concentration at the ranges near the transducer show that the inver sion is not accurate there for either model. After the pump was turned off and the sedi ments started to settle, the difference in the inverted profiles between two models become more evident. The unmodified model overestimates the concentration over the range r >0.5m at time 10 seconds after the pump was turned off. The profiles also have "steps" which correspond to erroneously determined sediment sizes (see discussion in the following section). lni: M = 0.8 g/1, = 2.06, = 0.57 pump off max off 10 sec 1 off 20 sec off 30 sec off 40 sec 0.8 _off 50 sec  off 60 sec 0 / 0' off 70 sec .'0 0.6 0 0.4  0.2 0 0 0 0.2 0.4 0.6 0.8 Range, m Figure 332. Inversion by modified model concentration profiles for "funpump" test with maximum concentration of 0.8g/l: mean profiles (thick lines) and 2.07MHz profiles (thin lines). The concentration profiles, both inverted using the modified model and calculated by the ID advectiondiffusion model, are shown in Figure 333. The mean profiles calcu lated with empirical model are shown as green lines. The mean concentration profiles were calculated by averaging of the profiles inverted from all three frequencies. The rela tive errors were calculated as C = () mdl ( ) (323) N M/i dl (r, ) where Mmodel is the concentration estimated by the ID advectiondiffusion model, Mac, is the inverted concentration, with the summation carried over the indices at which Model (r~) >0.05g/l, and N is the number of included indices. Errors: mean 10.9%; ABn 8.3% Errors: mean 8.4%; ABn 31.5% 4 1.5 0 sec b). 10 sec 1.08 MHz 3 2.07 MHz S1 4.70 MHz 2  Mean 5 8 0.5 / Model Error: . Mean (ABn) 0 0 Errors: mean 45.2%; ABn 17.9% Errors: mean 59.9%; ABn 15.9% 1 0.8 S0.6 0.5 0.4 o 0 0 0.5  0 0 Errors: mean 60.6%; ABn 15.0% Errors: mean 56.6%; ABn 40.2% 0.4 0.4 e). 40 s c f). 50 sec 0.3 0.3 6 0.2 02  o o.1 o. 1 Errors: mean 60.7%; ABn 49.5% Errors: mean 72.8%; ABn 39.0% 0.4 g) 60ee 0.2 g). 60 sec h). 70 sec 0.3 0.15 d 0.2 0.1 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Range, m Range, m Figure 333. Comparison of concentration profiles inverted by modified highpass model with profiles calculated by the ID advectiondiffusion and empirical models (labeled as "ABn") for the "funpump" test with maximum concentration of 0.8g/l at 10 second intervals. The sequence of plots in Figure 333 shows the dynamics of the two sediment sam ples settling. The profiles corresponding to individual frequencies (blue lines) show in creasing stratification of sediment suspension over time due to the different settling ve locity. Again, when the content of coarse sediments gets low, the concentration estimated from the 1.08MHz frequency is much lower than obtained by the other frequencies. The 4.70MHz becomes the most accurate predictor of the concentration when only fine sedi ments are present. The concentrations estimated from the 2.07MHz measurements are closest to the estimated mean concentration. The empirical model provides the best estimations for suspended sediment concen tration. Although it does not use the information on sediment size, it still can be applied for the inversion. Results of Inversion for Sediment Size The sediment size distribution parameters were also estimated. The results for the test with the maximum concentration of 0.8g/l are shown in Figures 334 and 335 for unmodified and modified models; they correspond to the results presented in Figures 331 and 332. It can be seen from Figure 334, that the unmodified model provides esti mations of mean sediment size which are closest to the expected values at very low con centrations (at time>30 seconds), but fails at higher concentrations, for example, profiles at 10 and 20 seconds. The model predicts sediment sizes which were not present in the chamber (see Figure 327), aliasing the estimations towards finer sediment sizes. The only approximation which is reasonable is at time t=0 seconds; and that profile was used for the calibration of the system, therefore the inversion is much more accurate. The estimations with the modified model are shown in Figures 335 and 336. Sev eral definite improvements can be noticed. The model now captures correctly the dynam ics of the changes in mean sediment sizes along the chamber with time. It was predicted by ID advectiondiffusion model (see Figure 328e) that the mean sediment sizes de crease with time at any horizontal level after the pump was turned off. Figure 335 shows that the inverted profiles behave similarly to those predicted by the numerical (ID) model. Also, there is a limit for estimated sediment size at approximately 2.4 (p, which is close to the mean size of Sand 1 sample, used in the test. 