Nonlinear Inversion of Acoustic Scalar and Vector ... - Semantic Scholar

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 37, NO. 4, OCTOBER 2012

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Nonlinear Inversion of Acoustic Scalar and Vector Field Transfer Functions Steven E. Crocker, Member, IEEE, James H. Miller, Senior Member, IEEE, Gopu R. Potty, Member, IEEE, John C. Osler, and Paul C. Hines

Abstract—A study to investigate the use of the acoustic vector field, separately or in combination with the scalar field, to invert for geoacoustic properties of the seafloor was conducted. The analysis was performed in the context of the 2004 Sediment Acoustics Experiment (SAX04) conducted in the Northern Gulf of Mexico (GOM) where a small number of acoustic vector sensors were deployed in close proximity to the seafloor. The acoustic vector sensors were located both above and beneath the seafloor interface where they measured the acoustic pressure and the acoustic particle acceleration. A variety of acoustic waveforms were transmitted into the seafloor at normal incidence. Motion data provided by the buried vector sensors were affected by a suspension response that was sensitive to the mass properties of the sensor, the sediment density, and shear wave speed. The suspension response for the buried vector sensors included a resonance within the analysis band of 0.4–2.0 kHz. The response was sufficiently sensitive to the local geoacoustic properties, that it was integrated into the inverse methods developed for this study. Inversions of real and synthetic data sets showed that information about sediment shear wave speed was carried by the suspension response of the buried sensors, as opposed to being contained inherently within the vector acoustic field. Index Terms—Acoustic vector sensor, geoacoustic inversion, optimization methods, seismoacoustics, underwater acoustic propagation.

I. INTRODUCTION

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STIMATION of seabed sediment properties by inversion of acoustic field data has been the subject of considerable attention. A substantial body of literature [1]–[3] exists on inversion of acoustic scalar field data (e.g., acoustic pressure) for

Manuscript received October 18, 2011; revised April 24, 2012; accepted June 26, 2012. Date of publication August 24, 2012; date of current version October 09, 2012. This work was supported by the Naval Undersea Warfare Center Division In-House Laboratory Independent Research (ILIR) program, by the U.S. Office of Naval Research (ONR) under Grants N000141010WX20956 and N000140310883, and by the Defence Research and Development Canada—Atlantic (DRDC Atlantic) under Grant 30ak01. Associate Editor: N. Ross Chapman. S. E. Crocker is with the Sensors and Sonar Systems Department, Naval Undersea Warfare Center, Newport, RI 02841-1708 USA (e-mail: [email protected]). J. H. Miller is with the Department of Ocean Engineering, University of Rhode Island, Narragansett, RI 02882 USA and also with the NATO Undersea Research Centre, La Spezia, Italy. G. R. Potty is with the University of Rhode Island, Narragansett, RI 02882 USA. J. C. Osler is with the NATO Undersea Research Centre, La Spezia 19126, Italy. P. C. Hines is with the Defense Research and Development Canada—Atlantic (DRDC Atlantic), Dartmouth, NS B2Y3Z7 Canada. Digital Object Identifier 10.1109/JOE.2012.2206852

various properties of the seabed. Most of this work has been centered on the inversion of one or more acoustic quantities derived from scalar field measurements. Inversion of the acoustic vector field data has received less attention. The vector field, defined here as the acoustic particle displacement (and its temporal derivatives), has the potential to convey information about the medium through which the wave field propagates that may not be available using the scalar field alone. Relatively few investigations on the use of the acoustic vector field in geoacoustic inversions have been published. Peng and Li [4] used a small vertical array of acoustic vector sensors to record the transmission loss for broad band signals in a shallowwater wave guide. They successfully inverted the acoustic scalar field in combination with the vertical component of vector field for the water depth, sediment compression wave speed, and attenuation. Information in the vector field was shown to reduce the variance in water depth and sediment compression wave speed estimates, compared to inversion of scalar field data alone. Santos et al. [5] inverted complex representations of the acoustic scalar and vector field for sediment compression wave speed and density. A sensitivity study using synthetic data showed that the method was sensitive to the target parameters. Experimental results appear to have been consistent with the sensitivity study, however a comparison with results obtained by inversion of scalar field data alone was not reported. Koch [6] inverted acoustic field data from a vector sensor line array that was horizontally moored to the seafloor in 85 m of water. The output of a plane wave beamformer used to track a moving broadband source was inverted for geoacoustic parameters including sediment compression wave speed, gradient, attenuation, and density. Separate inversions of scalar and vector field data provided consistent sets of parameter estimates. In this study, inversion of vector field data provided no performance advantage relative to inversion of scalar field data. Osler et al. [7] performed a series of measurements as part of the 2004 Sediment Acoustics Experiment (SAX04), in which four acoustic vector sensors were deployed in close proximity to the seafloor; two suspended above the seabed, and two buried within it. While not among their original objectives, the resultant data set provided a convenient opportunity to pose new inverse problems that exploited the unique features of the experimental arrangement. In particular, their SAX04 data set facilitated the development of new methods [8] to invert acoustic field data provided by an array of acoustic vector sensors that spanned the seafloor interface. The objective of this study was to test the postulate that the vector field contains information that could be used to invert

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for geoacoustic properties of the seafloor, and that this same information was not available in the scalar field alone. It is a simple matter to show that this postulate is correct for the simplest case of measurements relating to a plane propagating wave. Consider an experiment in which two hydrophones, separated by a known distance, observe the passage of a plane propagating wave. Countless students of acoustics have inverted the equation for propagation where , , and represent the distance traversed, phase speed, and elapsed time to yield an estimate of the phase speed based on the known separation and measurement of the time taken to traverse that separation. If the experimental apparatus were also to include a measurement of the acoustic vector field, we could then estimate the density of the medium using the characteristic impedance where , , and are the (scalar) acoustic pressure, the (vector) acoustic particle velocity, and density, respectively. There is also the potential for vector field data provided by an instrument to contain information exceeding that carried by the vector field in the absence of the instrument. It will be shown that the motion of an instrument, in this case an inertial acoustic vector sensor, may not equal that of the material in which the instrument is in direct physical communication. The instrument motion may be influenced by a frequency-dependent transfer function between itself and the propagating medium. Since the vector field data reflect the motion of the instrument case, not necessarily that of the propagating medium, the suspension response may carry useful information about the environment in which the sensor is embedded. Section II discusses the acoustic field measurements including a description of the test site, experiment geometry, and acoustic field data. Frequency-dependent transfer functions between the incident acoustic vector field (e.g., acoustic particle velocity) and the resultant motion of the acoustic vector sensor are derived for a sensor immersed in an inviscid fluid and for a sensor embedded in an elastic solid. Section III presents the inversion method that was developed for this effort. The technique inverts acoustic transfer functions between sensor pairs to include accounting for the corruption of the acoustic vector field data provided by the buried vector sensors. The correction is integrated into the inverse method such that useful information about the shear wave speed at the buried sensor locations is returned. Uncertainty estimates based on analysis of an objective function derived from the Euclidean norm of the weighted error vector are described. Results for the inversion of synthetic and experimental data are presented in Section IV. II. ACOUSTIC FIELD MEASUREMENTS A. Test Site Description SAX04 was conducted in 16.7 m of water located 1 km offshore of Fort Walton Beach, FL, as illustrated in Fig. 1. Experimental work began in early September 2004 and continued through the middle of November. This study was based on a subset of the SAX04 data and was collected on October 29, 2004. Among the primary considerations for selection of the test site was the need for the seafloor to have a relatively high

Fig. 1. Location of SAX04. Coastline data were extracted from the Global Self-consistent, Hierarchical, High-resolution Shoreline Database (GSHHS) [9]. Acoustic data were collected aboard the R/V Seward Johnson at location 30 23.232 N, 86 38.706 W.

critical angle (e.g., 25 –30 ) to support the study of acoustic penetration and scattering. Characteristics of the desired test site included benign bathymetry and a homogeneous sediment half-space with a sediment-to-sound-speed ratio greater than about 1.1, characteristics satisfied by the relatively flat and sandy bottom off Fort Walton Beach. Hurricane Ivan made landfall 100 km west of the test site on September 16, six weeks before the data used for this study were collected. The storm came ashore as a category 3 hurricane with sustained winds of 105 kt [10]. Ivan produced significant wave heights of 12 m at the experiment site resulting in the destruction and loss of bottom-mounted equipment [11]. Ivan also created major changes to the distribution of sediment along the near-shore shelf, to include the test site [12]. Following passage of the storm, water retreating from the estimated 3.0–4.5-m storm surge transported a significant amount of mud, likely derived from the lagoon north of Santa Rosa Island onto the shelf. A nearly continuous drape of mud was deposited at the test site and surrounding shelf waters. Subsequent wind–wave activity mobilized and mixed the sediments, ultimately forming lens-shaped mud inclusions buried within the recently disturbed seabed [13]. During July–August 2005, nine months after the passage of Hurricane Ivan, the SAX04 test site and surrounding waters were surveyed by R/V G.K. Gilbert. The survey included the collection of numerous grab samples and cores, four of which (e.g., 4A–4D) were collected in the immediate vicinity of the test site. Core 4A showed no evidence of layering, while core 4B contained evidence of a muddy layer at a depth of 4 m. Fig. 2 illustrates data for cores 4C and 4D where clear evidence for muddy layers at 140 and 100 cm was found [14]. Evidence of mud flasers was also found in several grab samples and cores surrounding the SAX04 test site, including one (e.g., core 25) located 11 km to the west, that had a prominent low-impedance layer at a depth of 65 cm [12]. While the experiment site was

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Fig. 3. Experiment geometry and geoacoustic parameterization. Fig. 2. Sediment composition measured in cores collected at the SAX04 test site aboard R/V G.K. Gilbert. Core 4C was collected at location 30 23.237 N, 86 38.709 W. Core 4D was collected at location 30 23.230 N, 86 38.705 W.

selected for its moderately well-sorted, medium quartz sand, the presence of mud layers and lenses created a more heterogeneous environment that included small-scale features with the potential to influence the acoustic propagation. Less clear is whether these features were created by Hurricane Ivan or were the result of prior storm activity. B. Experiment Geometry Fig. 3 illustrates the experimental arrangement used to collect the data on which this work was based. The experiment was conducted in a water depth of 16.7 m. An acoustic projector was suspended above the seafloor at a height above bottom of 8.4 m. Directly beneath the projector were located four Wilcoxon Research TV-001 [15] acoustic vector sensors arranged in a vertical line that spanned the seafloor. Two sensors (e.g., VS5 and VS6) were suspended above the bottom at heights of 10 and 25 cm. Two sensors were buried in the sediments (e.g., VS1 and VS2). The intended burial depths were 50 and 100 cm, although there was uncertainty in the actual burial depths as reported in [7]. The vector sensors were all oriented to collect the vertical component of the acoustic particle acceleration on the same sensor axis (e.g., -axis). All data were collected at normal incidence, thus the observable vector field was represented by data from a single accelerometer in each acoustic vector sensor. C. Acoustic Field Data Acoustic field data used to invert for estimates of the geoacoustic properties of the seabed consisted of gated, continuouswave transmissions with 100-ms pulse width. Data selected for use were a subset of those collected for the SAX04 sound-speed dispersion experiments. While data were collected for transmit frequencies ranging from 200 Hz to 18 kHz, data for the inversion were limited to 400–2000 Hz. Data collected at lower frequencies were not used due to insufficient acoustic power output by the SX100 projector. Data at higher frequencies were not

used due to an undocumented resonance in the acoustic vector sensor at 4.5 kHz. Thus, the data used for these inversions included acoustic waveforms with transmit frequencies of 400, 600, 800, 1000, 1200, 1600, and 2000 Hz. Acoustic data were collected using a Nicolet Liberty data acquisition system. The system acquired 32 simultaneously sampled data channels with 16-b resolution. The sample rate was 40 kHz. The input range of the data channels was 5.4613 V, set to prevent clipping of signals received from the acoustic projector operated at a nominal source level of 175 dB re 1 Pa at 1 m. The source-to-receiver ranges were less than 10 m for all sensors. Initial processing of the SAX04 data included bandpass filtering, application of the sensor calibration coefficients, and extraction of time-aligned waveforms. Alignment of the individual waveforms onto a common time base was desired to facilitate the use of complex signal representations in the inversion process. Had the signals not been aligned to a common time base, then the phase of the individual waveforms would have varied due to the process used to apply gates to the data as opposed to being influenced only by the experimental conditions, including the environment. Filtered, calibrated, and aligned time series for a buried vector sensor (e.g., VS1) are illustrated in Fig. 4. The figure shows the first 30 ms of all 28 waveforms that were transmitted at 800 Hz. All 28 instances of the trigger signal that were used to drive the acoustic projector are illustrated in Fig. 4(a). The acoustic pressure and vertical particle acceleration measured with a buried acoustic vector sensor are shown in Fig. 4(b) and (c). All data were passed through a third-order Butterworth filter with bandwidth proportional to the transmit frequency. Specifically, the pass bands for the filters were one octave wide with the center frequency equal to the transmit frequency. D. Acoustic Vector Sensors 1) Sensor Description: Inertial vector sensors typically combine one scalar measurement with one or more vector measurements performed at a collocated point in space. The vector measurements are usually provided by piezoelectric

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TABLE I WILCOXON RESEARCH TV-001 SPECIFICATIONS [15]

Fig. 4. Acoustic waveform data measured by buried vector sensor (e.g., VS1). The transmit frequency was 800 Hz. The pulse repetition frequency was 2.0 Hz. A total of 28 waveforms were transmitted. (a) Trigger signal used to drive the acoustic projector. (b) Acoustic pressure . (c) Vertical particle acceleration .

accelerometers or moving coil geophones [16] arranged to measure three orthogonal components of the acoustic vector field. Acoustic vector sensors employing accelerometers have benefited from recent advances in single crystal piezoelectric materials [17] that have facilitated the design and manufacture of accelerometers with high sensitivity in miniaturized packages relative to previous sensors based on piezoelectric ceramic materials. Much of the work to develop and employ acoustic vector sensors [18]–[20] has been directed toward applications unrelated to environmental characterization. Characteristics of the TV-001 vector sensors used for this study are provided in Table I. In-water calibrations of these vector sensors revealed the presence of an undocumented resonance at 4.5 kHz [7]. As a result, the upper limit of the analysis band was established at 2.0 kHz to preclude data contamination at frequencies near resonance. 2) Acoustic Vector Sensor Dynamic Response Model: Acoustic vector sensors are designed for operation in fluids, typically seawater. Many acoustic vector sensors of the inertial class, including the TV-001, measure the acoustic vector field with an arrangement of accelerometers housed in an evacuated water tight sensor case [21]–[24]. Data provided by an inertial vector sensor reflect the motion of the sensor package, not necessarily that of the surrounding medium. Thus, requirements for the successful design and use of acoustic vector sensors are more stringent than those for hydrophones due to the dynamics of the vector sensing mechanism.

Fig. 5. Acoustic vector sensor dynamic model. Model parameters include the , mass displaced by the sensor , added (or inertial) mass sensor mass , sensor velocity , and incident acoustic particle due to sensor motion . velocity

Practical applications using vector sensors invariably involve a mount, or similar physical constraint, that is applied to the sensor case. This has the effect of modifying the response by incorporating a suspension response into the sensor velocity transfer function. Design requirements for an effective vector sensor mount include: 1) a natural frequency well outside the intended range of acoustic sensing; 2) fix the average position and orientation of the sensor body; 3) permit movement of the sensor body in response to the acoustic field; 4) isolate the sensor from structure-born noise; and 5) not distort the response of the sensor in either magnitude or phase [25]. Fig. 5 illustrates the dynamic model of a rigid sensor body embedded in a viscoelastic material that supports both shear and compression waves. The embedding material moves with velocity when set into motion by a seismoacoustic wave field. The sensor body moves with velocity in response to the motion of the propagating medium. The mass of the sensor is and the added mass due to motion of the sensor body is . The mass of material displaced by the sensor body is . The steady-state equation of motion (1), derived for an accelerated frame of reference in which the propagating medium is at rest, was developed [7] as an extension of the analysis for the dynamic response of an ocean-bottom seismometer [26]. In this noninertial reference frame, the sensor velocity is and its acceleration is . The temporal derivative of momentum, including the contribution from the added mass,

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is . The forcing function acting on the sensor body is the oscillatory term due to the net weight of the sensor in the pseudogravitational field opposed by the “buoyancy” of the sensor due to the displaced embedding material. A restoring force due to the resistance and stiffness of the mechanical impedance between the sensor body and the propagating medium is (1) Rearrangement of (1) provides the sensor-to-medium-velocity ratio for a rigid sensor body, embedded in a linear viscoelastic material, oscillating with a wavelength that is sufficiently long that the medium velocity can be approximated as spatially uniform near the sensor and time harmonic. The mass and stiffness of the sensor leads were assumed to be negligible. The transfer function between the motion of the viscoelastic medium and the motion of the embedded acoustic vector sensor becomes (2) for a Oestreicher [27] derived the mechanical impedance rigid spherical body, oscillating in a compressible viscoelastic material. The mechanical impedance, including contributions for the added mass, resistance, and stiffness, was provided as (3) as a function of the sphere radius , shear wave number , compression wave number , and medium density

Fig. 6. Added mass and resistance for an acoustic vector sensor immersed in an inviscid, compressible fluid. Terms are provided in nondimensional form as indicated in each panel. Values applicable to the vector sensors suspended above the seafloor during SAX04 are indicated with markers.

(3) The shear and compression wave numbers are related to their respective material properties by (4) and (5), where and are the coefficients of shear elasticity and shear viscosity. Likewise, and are the coefficients of volume elasticity and volume viscosity (4) (5) 3) Acoustic Vector Sensor in an Inviscid Fluid: The special of a small rigid sphercase for the mechanical impedance ical body oscillating in an inviscid, compressible fluid is given when the shear elasticity, shear viscosity, and volume viscosity vanish (e.g., ). In this case, the mechanical impedance reduces to (6)

Under these circumstances, the mechanical impedance results from the inertia of the added mass and a radiation resistance associated with the acoustic field. A useful form of (6) is (7) where terms for the added mass given by

and resistance

are

(8) (9) and is the surface area of the sphere. The compression wave speed is . Nondimensional forms of the added mass and resistance are presented in Fig. 6. When the sensor body is small relative to a wavelength (e.g., ), inspection of (8) and (9) shows that the added mass is well approximated as half the displaced mass, the resisis proportional to , and the impedance is contance trolled by the added mass [e.g., ]. Under these

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conditions, the velocity transfer function of (2) reduces to the classic case [28] where the sensor-to-fluid-velocity ratio is given by (10) where and are the densities of the sensor and fluid, respectively. Consistent with the results of (2), the sensor velocity is equal to the fluid particle velocity when the sensor is neutrally buoyant (e.g., ). The velocity of an acoustic vector sensor immersed in an inviscid, compressible fluid equals that of the incident acoustic vector field only when the sensor is small relative to a wavelength (e.g., ) and the sensor displaces its own mass in the fluid (e.g., neutrally buoyant). Acoustic vector sensors designed for use in seawater, including those used for this study, usually satisfy both of these conditions. Thus, the velocity transfer function for the vector sensors suspended above the seafloor was taken to be unity [e.g., ]. 4) Acoustic Vector Sensor in an Elastic Solid: The mechanof a small, rigid spherical body embedded ical impedance in marine sediment was similarly reduced to simplified form by approximating the sediment as an incompressible, elastic medium. Setting , the impedance of the buried sensor becomes (11) An alternate form for the mechanical impedance of (11) separates the individual contributions as (12) where terms for the added mass are given by ness

, resistance

Fig. 7. Impedance and approximation error for buried vector sensor. In regions where values are less than zero, the negative is annotated to facilitate display.

, and stiff(13) (14) (15)

and is the shear wave speed in the sediment. As before, the surface area of the spherical body is . The added mass is equal to half the mass of sediment displaced by the sensor. In the case where the sediment is further approximated as lossless (e.g., ), the individual terms are all independent of frequency. When viscous losses are included, frequency-dependent attenuations are represented in the resistance and stiffas a complex shear wave speed ness (16) Fig. 7 provides a comparison of the full and simplified expressions for the mechanical impedance of a buried sensor using parameters that are representative of the experimental conditions. The resistance and reactance (e.g., the real and

imaginary parts of the impedance) are plotted as functions of frequency. Sediment properties were compiled from test site measurements with density (2.04 g/cm ), compression wave speed (1680 m/s), compression wave attenuation (1.0 dB/m/kHz), shear wave speed (120 m/s), and shear wave attenuation (30 dB/m/kHz) as reported for SAX99 [29] and SAX04 [7]. The approximation error for this case is also illustrated where the specific inversion frequencies are indicated with markers. As shown in the figure, the difference between the full and simplified impedance was less than 1% throughout the measurement band, with the error tending to increase with frequency. The full and simplified expressions for the impedance were also compared as a function of sediment shear wave speed, as illustrated by Fig. 8. The comparison was performed at the upper end of the analysis band due to the generally increasing error at the higher frequencies as the finite volume elasticity of the sediment became more important. Fig. 8(a) shows good agreement between the full and simplified impedance for a wide range of sediment shear wave speeds. Fig. 8(b) provides a more detailed view of the approximation error. While the error in the resistance increased somewhat for decreasing shear wave speed,

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Fig. 9. Buried sensor velocity transfer function . Individual curves representing shear wave speeds of 25–150 m/s are indicated in the figure.

Fig. 8. Shear-speed-dependent impedance and approximation error at 2000 Hz . In regions where values are less than zero, the negative is annotated red to facilitate display.

both the resistance and reactance errors were less than 2% for all probable values of this parameter. The simplified expression for the mechanical impedance of a buried sensor (12) is in a convenient form for use in the velocity transfer function of (2). Recall that accounted only for the resistance and stiffness of the mechanical impedance between the sensor body and the surrounding medium. In an elastic bottom where , this term becomes (17) and were provided as (14) and (15). Thus, the where velocity transfer function for the buried vector sensor becomes

(18)

As was the case for a vector sensor immersed in an inviscid fluid, inspection of (18) shows that the velocity of the sensor equals that of the medium in which it is embedded when the sensor displaces its own mass of embedding material, regardless of the resistance and stiffness terms for the impedance. Analysis of the velocity transfer function (18) for the buried sensor indicates the presence of a suspension resonance within the processing band. As illustrated in Fig. 9, the resonant frequency was sensitive to the shear wave speed where it increased from 320 to 1910 Hz for shear wave speeds of 25 and 150 m/s, respectively. Thus, data provided by the vector channels of the buried acoustic vector sensors were quite likely to have been affected by a frequency-dependent transfer function between the acoustic vector field and the data representing that field. In principle, data provided by the vector channels of the buried sensors could be corrected by compensating for the frequency-dependent transfer function associated with the suspension response. However, implementation of the correction would require a priori information about the sediment shear wave speed and density, parameters that were to have been estimated by the inversion. As a result, the inversion method needed to account for unknown sediment parameters that contributed to a frequency-dependent transfer function located between the acoustic vector field and data representing that field.

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III. INVERSION METHOD Methods to invert the complex acoustic transfer functions between pairs of sensors were developed. Central to these methods was the definition of an objective function that was normalized to facilitate sensitivity comparisons among data vectors of differing lengths and magnitudes. A sensitivity analysis was performed to verify behavior of the objective function when operating on the different acoustic quantities for which it was derived. Analysis was performed using synthetic data generated by the Ocean Acoustic and Seismic Exploration Syntheses (OASES) seismoacoustic model [30]. A. Inversion Data Complex acoustic transfer functions were a natural choice to support the inversion process since they carry information about the physical properties of the paths over which the wave had traversed and they were readily computed. A set of overlapping paths that spanned the space sampled by the acoustic vector sensors was chosen as the basis for the inversion process. Fig. 10 illustrates the acoustic paths for which transfer functions were computed. The location of each vector sensor is indicated, as are the end points of each path over which each transfer function was computed. Transfer function subscripts identify the end points of each acoustic path. Specific implementations operated on acoustic scalar, acoustic vector, and combined scalar–vector field transfer functions. The acoustic scalar field results from the convolution of a source pressure with the impulse response function for the acoustic channel between the source and receiver positions, as shown in (19), where “ ” is the convolution operator. The frequency-domain representation is provided by the Fourier transform of the convolution integral. Likewise, the acoustic vector field at position resulting from the scalar field at position is provided as (20). All acoustic field calculations were performed using OASES–OASP, a software module within OASES for the calculation of broad band transfer functions

(19) (20) The acoustic scalar and vector field transfer functions [e.g., and ] between the locations occupied by sensor and sensor are given by (21) and (22) where the respective positions of the sensors are and (21) (22) The forward model used to support the inversion was required to predict not only the acoustic field variables, but also to consider any distortion to measurements of the acoustic particle velocity resulting from the sensor suspension response. This re-

Fig. 10. Acoustic transfer function paths. Relative locations for acoustic vector sensors VS5, VS6, VS1, VS2 are annotated. Acoustic paths over which the where transfer functions were computed are indicated. For example, and is the acoustic path extending between VS5 and VS2.

quired replacement of the acoustic particle velocity with the velocity of the sensor case in the forward model. Thus, between any two sensors and , the acoustic vector field transfer function observed for the path between them becomes (23) (24) where is the velocity transfer function for the suspension response of the th sensor. The velocity transfer function for a vector sensor that was buried in the seabed was with and equal to the density and shear wave speed at the location of the th sensor. When the th vector sensor was suspended above the seafloor, the velocity transfer function was . 1) Data Reduction: Acoustic transfer functions among the six overlapping paths illustrated in Fig. 10 were computed for all transmit frequencies. In addition, the standard deviations were computed for use in the weight vector (26) included as part of the objective function (25). These data were used as the basis of the inversion process. Transfer functions among the four acoustic paths that traversed the seafloor interface are illustrated in Figs. 11 and 12. In the simplest case of a plane propagating wave, the transfer functions for the scalar and vector acoustic fields are identical, where the acoustic pressure and particle velocity are everywhere related by the characteristic impedance of the medium. While the circumstances for a spherically divergent wave are slightly more complicated, they are important only for source-to-receiver ranges on the order of a wavelength or less. Thus, for the purposes of this discussion, we consider the incident wave field to be approximately planar. Features of the environment, reflecting boundaries in particular, change the nature of the field from a purely propagating wave, to one with standing wave characteristics. Thus, information about the nature of reflecting surfaces becomes embedded in the acoustic transfer functions with the result that the scalar and vector acoustic transfer functions are no longer identical.

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Fig. 11. Complex acoustic transfer functions for paths and between vector sensors suspended above the seafloor (e.g., VS5 and VS6) and vector sensor VS1 buried at a nominal depth of 50 cm. Error bars represent one standard deviation.

Consider also the effect of sediment shear elasticity in the case of a buried vector sensor where motion of the sensor case may not equal that of the surrounding sediment. In this case, a frequency-dependent suspension response that is sensitive to both the sediment shear wave speed and the density contrast between the sensor and the sediment was shown to exist. The effect of this response will also be reflected in the vector acoustic transfer function such that information about the sediment can be exploited by an inversion process. B. Objective Function Definition: An objective function was defined to facilitate both the global search strategy and to estimate the associated uncertainties in the parameter estimates. Among the attributes of the desired objective function were that it was: 1) based on a known functional form with a history of good performance in similar inverse problems; 2) normalized in a way that simplified interpretation of results; and 3) weighted individual measurements by the inverse of their respective variances. Objective functions based on the Euclidean norm have been shown [31] to provide good performance when the error vector , the data , and forward model predictions are complex, with the vector of model parameters.

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Fig. 12. Complex acoustic transfer functions for paths and between vector sensors suspended above the seafloor (e.g., VS5 and VS6) and vector sensor VS2 buried at a nominal depth of 100 cm. Error bars represent one standard deviation.

The objective function included weights that were inversely proportional to the data variances and normalized by the magnitude of the data vector to simplify interpretation of inversion results (25) with the Hermitian transpose. The weight vector fined as

was de-

(26)

was the number of observations, was the stanwhere dard deviation of the th observation, and was the transpose operator. Thus, the inverse problem was reduced to a directed search for the set of model parameters that minimized the difference between the acoustic measurements and the prediction of a physically realistic forward model in a least squares sense, as measured by the objective function . 1) Sensitivity: The sensitivity of the objective function (25) to the estimated geoacoustic parameters was evaluated using a

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TABLE II ENVIRONMENT PARAMETERIZATION FOR SYNTHETIC DATA

set of synthetic observations computed for parameter values reported for the SAX04 experiment site [7]. The marginal sensitivity of each parameter was then assessed by calculating the value of the objective function across the range of plausible values for each parameter while holding all other parameters constant at their reported value. The bottom parameterization included the reported low-impedance reflector as summarized in Table II. The analysis was performed for five distinct cases. In the first case, it was assumed that only the scalar field was measured using typical hydrophones. In the second case, it was assumed that only the vector field was measured by an ideal (e.g., distortion free) sensor. The third case included both scalar and vector field data, again assuming noise- and distortion-free measurements. In the fourth case, it was assumed that only the vector field was measured, and that the measurements were distorted by the suspension response of a sensor with the same mass properties as the TV-001 acoustic vector sensor (e.g., neutrally buoyant in seawater). The fifth case included both the scalar and vector fields as observed by a sensor with the mass properties of the TV-001 acoustic vector sensor. Results of the analysis showed that the objective function was sensitive to the geoacoustic parameters of interest. Objective function values ranged from a low of zero when all parameters were correctly estimated (relative to the synthetic data set) to highs ranging from 0.2 to 0.5 when estimation errors were inserted into the calculation. The greatest sensitivities were realized at depths sampled by the buried sensors, with somewhat reduced sensitivity to the geoacoustic properties at depths exceeding 1 m. Objective function sensitivity varied only weakly among the five sensor configurations outlined above with two exceptions. As illustrated in Fig. 13, the objective function was most sensitive to overestimates in the compression wave speed when inverting scalar transfer functions alone. The sensitivity study also identified the means by which information about shear wave speed in the sediment was communicated. The sensitivity study showed that the objective function responded to the shear wave speed only when a buried vector sensor was in direct physical contact with the sediment at the depth of the estimate. As shown in Fig. 13, the objective function was insensitive to the sediment shear wave speed when the vector field measurements were modeled as distortion free, as

Fig. 13. Sensitivity study result for surface sediment layer. Legend entries annotated as buoyant correspond to objective function values computed assuming an acoustic vector sensor with the same mass properties as the Wilcoxon Research TV-001. Legend entries annotated as neutral correspond to objective function values where the sensor was assumed to displace its own mass of sediment, thus providing a distortion-free (e.g., ideal) measurement of the acoustic particle motion. Note that the objective function was insensitive to the shear wave speed when the sensor was assumed to provide a distortion-free measurement of the acoustic particle motion.

would be the case for a sensor that displaced its own mass of sediment. Thus, information about the shear wave speed in the sediment appears to have been associated with the frequency-dependent sensor suspension response (18), as opposed to being carried inherently within the acoustic vector field itself. Sensitivity of the objective function to sensor burial depth errors was also examined. It was found that depth errors on the order of five centimeters resulted in objective function values of about 0.2, demonstrating the sensitivity of the inversion method to uncertainties in the sensor locations. This sensitivity was due to the property of complex acoustic transfer functions, where the phase measurements were susceptible to corruption by uncertainty in the path length between sensors. C. Uncertainty Analysis A complete solution to an inverse problem includes not only the parameter estimates, but also some measure of the uncertainties in those estimates. Thus, assessment of the value represented by the information in the acoustic vector field considered both the parameter estimates and their associated uncertainties. This was accomplished through implementation of an empirical

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method to characterize the uncertainties in the geoacoustic parameter estimates for different approaches to the inversion of acoustic scalar and vector field data. Parameter uncertainty in nonlinear inverse problems is typically expressed as the a posteriori probability distribution [32]. This study adopted a maximum-likelihood approach [33] where the a priori P and a posteriori G probability distributions are related through a likelihood function as indicated by (27) The 1-D marginal a posteriori probability density function for the th parameter is provided by integrating the -dimensional probability density with respect to all parameters for and to yield

(28) Evaluation of (28) may be accomplished by several means including grid search, Monte Carlo, and importance sampling. Grid search is a computationally impractical approach when the number of parameters in the model vector is larger than about four. Monte Carlo methods sample the distributions randomly as a means to reduce the computational work to estimate the integrals, but may spend a significant fraction of the computational effort in regions that contribute little to the value of the integral. Importance sampling attempts to exploit some knowledge about the integrands to develop nonuniform sampling distributions that concentrate in areas that contribute most to the integral. Nonlinear optimization approaches such as genetic algorithms and simulated annealing employ such importance sampling. These approaches use a generating distribution for selecting the next model vector. However, this distribution is unknown and evolves over the course of the optimization. When performing a nonlinear inversion, one generates a large number of observations of candidate solutions from the total model parameter space . The objective function values associated with these model runs can be used to approximate the above integrals. The a posteriori probability for the th model vector using the observations is provided as (29)

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where is the estimated parameter vector for the optimum value of the objective function and is a constant that is particular to each optimization. A common value for is the average of the 50 best objective functions obtained during the optimization minus the best value of the objective function [33] (32)

D. Differential Evolution Differential evolution [34] is a particular evolutionary algorithm proposed to solve optimization problems in continuous search spaces. It is a population-based stochastic heuristic characterized by simplicity, effectiveness, and robustness. While it shares some similarities with more traditional evolutionary algorithms, it does not use binary encoding nor does it use a probability density function to self-adapt its parameters as an evolution strategy. Instead, differential evolution performs mutation based on the distribution of the solutions in the current population. In this way, search directions and possible step sizes depend on the locations of the individuals selected to calculate the mutation vectors. A system of nomenclature to identify the variants of the differential evolution algorithm has been widely adopted. The classic algorithm variant is identified as DE/rand/1/bin where “DE” refers to differential evolution, “rand” indicates that individuals are selected at random to compute the mutation values, “1” is the number of member pairs used to compute a difference vector, and “bin” specifies that binomial recombination is used. Algorithm variants differ in their mutation and recombination operators. The mutation operators vary in the number of member pairs used to develop mutations, and in the relative influence given the best member of the population in the formation of the trial vectors. Recombination operators vary in the way traits are selected for incorporation into the trial vectors. Several studies have sought to characterize the performance of the algorithm variants [35]–[37], including self-adaptive algorithms [38], [39], for a variety of benchmark problems. While some of these approaches have demonstrated improved performance in selected benchmark cases, none has yet demonstrated the kind of robust, reliable performance on the wide variety of problems as has been demonstrated by the classic DE/rand/1/bin algorithm that was used for inversions performed during this study. IV. RESULTS

in the model vector, the marginal For the th parameter probability distribution for obtaining the particular value can be estimated using (30) Since the likelihood function is usually related to the objective function through an exponential, an estimate of the empirical likelihood function is (31)

A. Inversion of Synthetic Data The primary objective for inversion of a synthetic data set was to verify the performance of the differential evolution algorithm when operating on acoustic data for which the associated geoacoustic parameters were exactly known. Since it was known that a solution existed for which the objective function was exactly zero, failure to converge under these conditions would have indicated that the proposed inverse methods were fundamentally flawed. The algorithm developed to support these inversions was provided with limited authority to determine the number of layers used in the geoacoustic parameterization. The parameterization

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Fig. 14. Results for inversion of synthetic acoustic transfer functions. The burial depths of vector sensors 1 and 2 were modeled as exactly known.

could contain a maximum of three finite layers overlying a homogeneous sediment half-space. At its simplest, the environmental description could be reduced to just the sediment halfspace, without any finite layers. A second objective for the synthetic inversion was to assess the variances of the parameter estimates under ideal conditions. Thus, the synthetic inversions provided insight into the performance of the inversion method when operating under the best case scenario. In total, six inversions were performed using synthetic acoustic transfer functions. Scalar acoustic, vector acoustic, and combined scalar–vector acoustic transfer functions were inverted. In addition, inversions with and without sensor burial depth errors were performed. In all cases, synthetic data were generated using buried sensor depths of 61 and 98 cm, as estimated by a regularized inversion of a more comprehensive acoustic data set as reported in [7]. In the first instance, these inversions were performed with the forward model operating with the true burial sensor depths of 61 and 98 cm. The inversions were repeated using the same synthetic data set. However, in this instance, the forward model was operated with the intended sensor burial depths of 50 and 100 cm, representing the case where the sensor burial depths were not exactly known. Results for the inversions of synthetic acoustic transfer functions are summarized in Figs. 14 and 15. Geoacoustic property estimates were generally good when the sensor burial depths were treated as having been exactly known, as shown by Fig. 14. Inspection of the figure shows that all three inversion

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Fig. 15. Results for inversion of synthetic acoustic transfer functions. The burial depths of vector sensors 1 and 2 were modeled as not exactly known.

methods correctly parameterized the geoacoustic environment as two discrete layers overlying a sediment half-space. Recall that this was accomplished by providing the inverse algorithm with the authority to determine the number of finite layers used to parameterize the environment. In this way, the number of layers included in the geoacoustic parameterization was reduced to the minimum required to describe the data. The inversion was provided with the authority to reduce the complexity to a simple sediment half-space. The maximum complexity allowed by the inversion included three discrete layers situated above the half-space. The inversion based only on synthetic vector acoustic transfer functions provided generally better estimates for sediment density and compression wave speed. Property estimates in the thin, low-impedance layer were less definitive, where only the inversion of combined scalar–vector acoustic transfer functions correctly identified the decrease in compression wave speed. However, when the characteristic impedance was considered, the inversion of vector acoustic transfer functions again provided the better estimate. While not immediately obvious from a casual inspection of Fig. 14, inversion of (synthetic) vector acoustic transfer functions provided the best estimate for shear wave speed. As shown in Table III, the shear wave speed estimates of 143 m/s and 117 m/s compared favorably with the simulated values of 120 and 100 m/s in the first two sediment layers. While inversion of scalar acoustic transfer functions returned estimates

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TABLE III INVERSION RESULTS FOR SYNTHETIC ACOUSTIC TRANSFER FUNCTIONS. VECTOR SENSOR BURIAL DEPTHS WERE MODELED AS EXACTLY KNOWN

TABLE IV INVERSION RESULTS FOR SYNTHETIC ACOUSTIC TRANSFER FUNCTIONS. VECTOR SENSOR BURIAL DEPTHS WERE MODELED AS NOT EXACTLY KNOWN

of 98 m/s and 104 m/s that more closely approximate that simulated values, consideration of the standard deviations shows that the inversion did not converge on a particular value for this parameter. Instead, the shear wave speeds were approximately uniformly distributed throughout the allowed range of 0–200 m/s. Thus, inversions of scalar acoustic transfer functions were insensitive to shear wave speed, confirming results of the sensitivity study as depicted in Fig. 13. Results for inversion of simulated data when the sensor burial depths employed by the forward model contained errors are illustrated in Fig. 15 and Table IV. None of the inversion methods returned accurate estimates for the sediment density and compression wave speed individually. However, estimates of the characteristic impedance were reasonably good due to offsetting errors in density and compression wave speed. In addition, the

inversions that included vector acoustic transfer functions successfully detected the presence of the low-impedance boundary. Thus, it appears that estimation of the characteristic impedance was more robust than estimates for the density and compression wave speed individually. B. Inversion of SAX04 Field Data Acoustic field data from SAX04 were inverted for the geoacoustic properties of the bottom. Inversions were performed using scalar field transfer functions, vector field transfer functions, and both. As was done for the inversion of synthetic data, two different experiment configurations were assumed. In the first (e.g., inversion A), the buried sensor depths were taken to be 61 and 98 cm as was estimated by a regularized inversion of acoustic field data [7]. A second inversion (e.g., inversion B)

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TABLE V INVERSION A OF SAX04 ACOUSTIC TRANSFER FUNCTIONS. THE INVERSION WAS PERFORMED WITH BURIED SENSOR DEPTHS OF 61 AND 98 cm

Fig. 16. Results for inversion of complex acoustic transfer functions measured during SAX04. Inversion A assumed buried sensor depths of 61 and 98 cm below the seafloor as reported [7].

was performed for which the buried sensors were assumed to have been at the intended burial depths of 50 and 100 cm. Inversion results where the forward model was operated with sensor burial depths of 61 and 98 cm (e.g., inversion A) are sum-

marized in Fig. 16 and Table V. As Fig. 16 shows, geoacoustic property estimates for depths less than about 0.5 m were highly variable and improbable. The best estimates for the respective parameter bounds indicated in the figure were based on results published previously for this test site [7]. The range includes the geoacoustic properties estimated for a thin, low-impedance layer that was hypothesized to exist at depth of less than 1 m. Inversion results where the forward model was operated with sensor burial depths of 50 and 100 cm (e.g., inversion B) are summarized in Fig. 17 and Table VI. While the geoacoustic parameter estimates in the upper meter of sediment were less chaotic than inversion A, the density estimates were improbable. In this case, the inversion based on combined scalar–vector acoustic transfer functions provided an estimate for the characteristic impedance that most closely conformed to expectation. It would be tempting to also note that this inversion result suggested the presence of a thin, dissimilar layer at a depth that was consistent with previous analysis [7]. However, without the benefit of a sediment core taken directly through this acoustic path, it is not possible to state with confidence that this feature corresponds to a discrete sediment layer. A noteworthy feature of these inversions was the shear wave speed estimates. Two distinct and consistent sets of estimates were provided, one at each sensor depth. The shear wave speed at the depth of the shallow sensor (e.g., VS1) was estimated by inversions A and B as 34 and 35 m/s, respectively. Estimates at the deeper sensor (e.g., VS2) were 74 and 69 m/s. The estimated values are generally consistent with regressions reported by Hamilton [40] for compression wave and shear wave speed in sandy sediments. The shear wave speed returned by Hamilton’s regression was 52 m/s. However, these estimates were not consistent with values based on SAX99 measurements [41], where the shear wave speed was reported as 120 m/s. At least two

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TABLE VI INVERSION B OF SAX04 ACOUSTIC TRANSFER FUNCTIONS. THE INVERSION WAS PERFORMED WITH BURIED SENSOR DEPTHS OF 50 AND 100 cm

may have influenced the environment local to the sensors cannot be ruled out. V. CONCLUSION

Fig. 17. Results for inversion of complex acoustic transfer functions measured during SAX04. Inversion B assumed sensors were buried at intended depths of 50 and 100 cm below the seafloor [7].

factors may account for the difference. First, disturbance of the seabed by the passage of Hurricane Ivan shortly before the experiment may have disrupted the bottom such that shear speeds were reduced. Also, the possibility that the act of sensor burial

The primary objective of this work was to test the postulate that the acoustic vector field contains information that could be used to improve estimates of the geoacoustic properties of the seafloor. This question was not addressed in the context of a traditional geoacoustic inverse problem where acoustic data are typically collected over path lengths on the order of kilometers. Instead, data collected for this experiment sampled less than one wavelength of the seismoacoustic field using sensors deployed in close proximity to the seafloor, on both sides of the interface. Thus, a new inverse problem approach was required to test the postulate that information in the acoustic vector field could measurably improve a given parameter estimate. The inverse method developed for this work operated on complex representations of the acoustic field. The complex acoustic transfer functions between various pairs of acoustic vector sensors were inverted for the local geoacoustic properties. Inversion of acoustic transfer functions was adopted to facilitate comparison of inversion results using only scalar field data, only vector field data, and when both were included. An objective function was defined to guide a directed search for the set of parameter estimates that resulted in the lowest mismatch between the observed data and the prediction of a forward model. An evolutionary algorithm was used to perform the directed search. Inversion of a synthetic data set showed that acoustic vector field data alone, or when combined with scalar field data, could improve geoacoustic parameter estimates in this particular experimental setting. However, inversion of synthetic data also showed that the method was sensitive to uncertainty in the

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acoustic path lengths among the sensors, the buried sensors in particular. This sensitivity to path length errors was associated with the use of complex acoustic transfer functions where the phase figured prominently in the acoustic path description. Thus, sensor burial depth errors on the order of 5 cm were shown to significantly degrade the inversion performance for a synthetic data set that was free of all other sources of noise and uncertainty. Analysis of synthetic data also showed that information about the shear wave speed in the sediment was not carried by the acoustic vector field in this experimental setting. On the contrary, information about the shear wave speed was shown to reside in the frequency-dependent suspension responses of the buried sensors. These inversion methods exploited the corruption in the vector channel data that was caused by the dynamic response of the elastic suspension between the sensor and the medium in which it was embedded. In the absence of this corrupting suspension response, the vector field transfer functions were insensitive to the elasticity of the sediment. Unfortunately, inversion of the experimental data did not confirm the conclusions based on the analysis of synthetic data. As reported previously, the actual sensor burial depths were uncertain [7]. Analysis of the synthetic data set predicted that these inversion methods were quite sensitive to acoustic path length errors. Sensor burial depth errors on the order of 5 cm were shown to introduce significant errors in the inversion results. Thus, good performance of the methods described here may require more precise knowledge of the acoustic sensor locations than was realized during SAX04.

[6] R. A. Koch, “Proof of principle for inversion of vector sensor array data,” J. Acoust. Soc. Amer., vol. 128, no. 2, pp. 590–599, Aug. 2010. [7] J. C. Osler, D. M. F. Chapman, P. C. Hines, G. P. Dooley, and A. P. Lyons, “Measurement and modeling of seabed particle motion using buried vector sensors,” IEEE J. Ocean. Eng., vol. 35, no. 3, pp. 516–537, Jul. 2010. [8] S. E. Crocker, “Geoacoustic inversion using the vector field,” Ph.D. dissertation, Dept. Ocean Eng., Univ. Rhode Island, Kingston, RI, 2011. [9] National Geophysical Data Center, “A global self-consistent, hierarchical, high-resolution shoreline database,” 2010 [Online]. Available: http://www.ngdc.noaa.gov/mgg/shorelines/gshhs.html [10] S. R. Stewart, “Tropical cyclone report, Hurricane Ivan,” National Hurricane Center, 2005 [Online]. Available: http://www.nhc.noaa. gov/2004ivan.shtml [11] R. T. Guza and W. O’Reilly, “Attenuation of ocean waves by ripples on the seafloor,” Scripps Institution of Oceanography, La Jolla, CA, Tech. Rep., 2007. [12] W. C. Vaughan, K. B. Briggs, J.-W. Kim, T. S. Bianchi, and R. W. Smith, “Storm-generated sediment distribution along the northwest Florida inner continental shelf,” IEEE J. Ocean. Eng., vol. 34, no. 4, pp. 495–515, Oct. 2009. [13] K. B. Briggs, A. H. Reed, D. R. Jackson, and D. Tang, “Fine-scale volume heterogeneity in a mixed sand/mud sediment off Fort Walton Beach, FL,” IEEE J. Ocean. Eng., vol. 35, no. 3, pp. 471–487, Jul. 2010. [14] P. C. Hines, J. C. Osler, J. G. E. Scrutton, and L. J. S. Halloran, “Time-fo-flight measurements of acoustic wave speed in a sandy sediment at 0.6–20 kHz,” IEEE J. Ocean. Eng., vol. 35, no. 3, pp. 502–515, Jul. 2010. [15] The TV-001 Miniature Vector Sensor. Gaithersburg, MD: Wilcoxon Research, Inc., 2003. [16] C. H. Sherman and J. L. Bulter, Transducers and Arrays for Underwater Sound. New York: Springer-Verlag, 2007, ch. 4, p. 185. [17] J. C. Shipps and K. Deng, “A miniature vector sensor for line array applications,” in Proc. OCEANS Conf., 2003, pp. 2367–2370. [18] A. Nehorai and E. Paldi, “Acoustic vector-sensor array processing,” IEEE Trans. Signal Process., vol. 42, no. 9, pp. 2481–2491, Sep. 1994. [19] B. A. Cray and A. H. Nuttall, “Directivity factors for linear arrays of velocity sensors,” J. Acoust. Soc. Amer., vol. 110, no. 1, pp. 324–331, Jul. 2001. [20] K. B. Smith and A. V. van Leijen, “Steering vector sensor array elements with linear cardiods and nonlinear hippioids,” J. Acoust. Soc. Amer., vol. 122, no. 1, pp. 370–377, Jul. 2007. [21] B. A. Cray, “Acoustic vector sensor,” U.S. Patent 6370084, 2002. [22] K. K. Deng, “Underwater acoustic vector sensor using transverse-response free, shear mode, PMN-PT crystal,” U.S. Patent 7066026, 2006. [23] T. B. Gabrielson, D. L. Gardner, and S. L. Garrett, “A simple neutrally buoyant sensor for direct measurement of particle velocity and intensity in water,” J. Acoust. Soc. Amer., vol. 97, no. 4, pp. 2227–2237, Apr. 1995. [24] K. Kim, T. B. Gabrielson, and G. C. Lauchle, “Development of an accelerometer-based underwater acoustic intensity sensor,” J. Acoust. Soc. Amer., vol. 116, no. 6, pp. 3184–3392, Dec. 2004. [25] J. A. McConnell, “Analysis of a compliantly suspended acoustic velocity sensor,” J. Acoust. Soc. Amer., vol. 113, no. 3, pp. 1395–1405, Mar. 2003. [26] J. C. Osler and D. M. F. Chapman, “Quantifying the interaction of an ocean bottom seismometer with the seabed,” J. Geophys. Res., vol. 103, no. B5, pp. 9879–9894, May 1998. [27] H. L. Oestreicher, “Field and impedance of an oscillating sphere in a viscoelastic medium with an application to biophysics,” J. Acoust. Soc. Amer., vol. 23, no. 6, pp. 707–714, Nov. 1951. [28] E. Skudrzyk, The Foundations of Acoustics. New York: SpringerVerlag, 1971, ch. 18, pp. 361–363. [29] M. J. Buckingham, “Compressional and shear wave properties of marine sediments: Comparisons between theory and data,” J. Acoust. Soc. Amer., vol. 117, no. 1, pp. 137–152, Jan. 2004. [30] H. Schmidt, “OASES Version 3.1 User Guide and Reference Manual,” Massachuesetts Inst. Technol., Cambridge, MA, 2006 [Online]. Available: http://acoustics.mit.edu/faculty/henrik/oases.html [31] C. F. Mecklenbräuker and P. Gerstoft, “Objective functions for ocean acoustic inversion derived by likelihood methods,” J. Comput. Acoust., vol. 8, no. 2, pp. 259–270, Jun. 2000.

ACKNOWLEDGMENT The authors would like to thank the team of divers from the Applied Physics Laboratory, University of Washington, Seattle, and the officers and crew of the R/V Seward Johnson for their support. M. O’Connor, J. Scrutton, M. Mackenzie, D. Caldwell, and I. Haya designed and operated the experimental kit. J. Smith provided expert advice regarding diving operations. D. M. F. Chapman provided insights regarding soil–sensor interaction. The authors would also like to thank the reviewers for their insights and useful suggestions for the improvement of this manuscript. REFERENCES [1] G. J. Heard, D. Hannay, and S. Carr, “Genetic algorithm inversion of the 1997 Geoacoustic Inversion Workshop test case data,” J. Comput. Acoust., vol. 6, no. 1, pp. 61–71, Mar. 1998. [2] M. R. Fallat and S. E. Dosso, “Geoacoustic inversion via local, global, and hybrid algorithms,” J. Acoust. Soc. Amer., vol. 105, no. 6, pp. 3219–3230, Jun. 1999. [3] G. R. Potty, J. H. Miller, P. H. Dahl, and C. J. Lazauski, “Geoacoustic inversion results from the ASIAEX East China Sea experiment,” IEEE J. Ocean. Eng., vol. 29, no. 4, pp. 1000–1010, Oct. 2005. [4] P. Han-Shu and L. Feng-Hua, “Geoacoustic inversion based on a vector hydrophone array,” Chin. Phys. Lett., vol. 24, no. 7, pp. 1977–1980, Jul. 2007. [5] P. Santos, O. Rodriguez, P. Felisberto, and S. Jesus, “Geoacoustic matched-field inversion using a vertical vector sensor array,” in Proc. Underwater Acoustic Meas., Technol. Results, Heraklion, Crete, Greece, Jun. 2009, pp. 29–34.

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[32] P. Gerstoft, “Inversion of seismoacoustic data using genetic algorithms and a posteriori probability distributions,” J. Acoust. Soc. Amer., vol. 95, no. 2, pp. 770–782, Feb. 1994. [33] P. Gerstoft and C. F. Mecklenbräuker, “Ocean acoustic inversion with estimation of a posteriori probability distributions,” J. Acoust. Soc. Amer., vol. 104, no. 2, pp. 808–819, Aug. 1998. [34] R. M. Storn and K. V. Price, “Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim., vol. 11, no. 4, pp. 341–359, Dec. 1997. [35] G. Jeyakumar and C. S. Velayuthan, “An empirical comparison of differential evolution variants on different classes of unconstrained global optimization problems,” in Proc. World Congr. Nature Biologically Inspired Comput., 2009, pp. 866–871. [36] E. Mezura-Montes, J. Velázquez-Reyes, and C. A. Coello, “A comparative study of differential evolution variants for global optimization,” in Proc. 8th Annu. Conf. Genetic Evol. Comput., 2006, pp. 485–492. [37] D. Zaharie, “A comparative analysis of crossover variants in differential evolution,” in Proc. Int. Multiconf. Comput. Sci. Inf. Technol., 2007, pp. 171–181. [38] K. Zielinski, X. Wang, and R. Laur, “Comparison of adaptive approaches for differential evolution,” in Parallel Problem Solving From Nature, ser. Lecture Notes in Computer Science, G. Rudolph, T. Jansen, S. Lucas, C. Poloni, and N. Beume, Eds. Heidelberg, Germany: Springer-Verlag, 2008, vol. 5199, pp. 641–650. [39] G. Jeyakumar and C. S. Velayuthan, “A comparative performance analysis of differential evolution and dynamic differential evolution variants,” in Proc. World Congr. Nature Biologically Inspired Comput., 2008, pp. 463–468. [40] E. L. Hamilton, “ and Poisson’s ratios in marine sediments,” J. Acoust. Soc. Amer., vol. 66, no. 4, pp. 1093–1101, Oct. 1979. [41] M. D. Richardson, K. B. Briggs, L. D. Bibee, P. A. Jumars, W. B. Sawyer, D. B. Albert, R. H. Bennett, T. K. Berger, M. J. Buckingham, N. P. Chotiros, P. H. Dahl, N. T. Dewett, P. Pleischer, R. Flood, C. F. Greenlaw, D. V. Holliday, M. H. Hulbert, M. P. Hutnak, P. D. Jackson, J. S. Jaffee, H. P. Johnson, D. L. Lavoie, A. P. Lyons, C. S. Martens, D. E. McGehee, K. D. Moore, T. H. Orsi, J. N. Piper, R. I. Ray, A. H. Reed, R. F. L. Self, J. L. Schmidt, S. G. Schock, F. Simonet, R. D. Stoll, D. Tang, D. E. Thistle, E. I. Thoros, D. J. Walter, and R. A. Wheatcroft, “Overview of SAX99: Environmental considerations,” IEEE J. Ocean. Eng., vol. 26, no. 1, pp. 26–53, Jan. 2001.

Steven E. Crocker (M’09) was born in Boston, MA. He studied geology at the University of Massachusetts, Lowell, where he received the B.S. degree in environmental science in 1984. While on active duty with the U.S. Navy, he attended the Naval Postgraduate School, Monterey, CA, earning the M.S. degree in engineering acoustics (with distinction) in 1991. In 2011, he received the Ph.D. degree in ocean engineering from the University of Rhode Island, Narragansett. He served as a Commissioned Officer of the U.S. Navy from 1985 until 1994. Upon leaving military service, he joined the research staff at Caterpillar’s Peoria Proving Ground, where he investigated noise control technologies for cooling air systems in mobile machine applications until his return to the coast in 1997. From 1997 to 1999, he provided consulting services for J&A Enterprises, a small engineering firm in Marblehead, MA, specializing in commercial marine acoustics. He was employed with Raytheon Integrated Defense Systems, Portsmouth, RI, from 1999 until 2002, where he led the development of an acoustic array for a maritime surveillance system. In 2002, he joined the Naval Undersea Warfare Center Division Newport, Newport, RI, where his research interests include advanced acoustic sensor technology, acoustical measurement techniques, and acoustical oceanography. Dr. Crocker is a member of the Acoustical Society of America and the IEEE Oceanic Engineering Society.

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James H. Miller (S’77–M’81–SM’06) received the B.S. degree in electrical engineering from Worcester Polytechnic Institute, Worcester, MA, in 1979, the M.S. degree in electrical engineering from Stanford University, Stanford, CA, in 1981, and the Doctor of Science degree in oceanographic engineering from Massachusetts Institute of Technology/Woods Hole Oceanographic Institution, Cambridge/Woods Hole, MA, in 1987. He was on the faculty of the Department of Electrical and Computer Engineering, Naval Postgraduate School (NPS), from 1987 to 1995. Since 1995, he has been on the faculty of the Department of Ocean Engineering, University of Rhode Island (URI), Narragansett, where he holds the rank of a Professor. In 2011, he joined the NATO Undersea Research Centre, La Spezia, Italy. He is a founder of FarSounder, Inc., Providence, RI, a startup company developing forward-looking sonar for vessels, underwater vehicles, and divers. During 2001–2003, he was a member of National Academy of Sciences Panel on Noise in the Ocean. He serves on the National Marine Fisheries Service Panel on Acoustic Criteria for Marine Mammals. He also served on the Marine Mammal Commission Subcommittee on the Impacts of Acoustics on Marine Mammals. He has more than 100 publications in the area of acoustical oceanography, signal processing, and marine bioacoustics. Dr. Miller was elected Fellow of the Acoustical Society of America in 2003. He has served as an Associate Editor for Underwater Sound for the Journal of the Acoustical Society of America, responsible for scattering, inverse methods, and fish acoustics. He is a member of Sigma Xi, Tau Beta Pi, Eta Kappa Nu, the Acoustical Society of America, and the Marine Technology Society. In 1993, he received the NPS Menneken Faculty Award for Excellence in Scientific Research. In 1999, he received the URI Marshall Award for Faculty Excellence in Engineering. Gopu R. Potty (M’09) was born in Trivandrum, India. He received the graduate degree in civil engineering from the University of Kerala, Trivandrum, India, in 1985, the M.S. degree in ocean engineering from the Indian Institute of Technology—Madras, Chennai, India, in 1987, and the Ph.D. degree in ocean engineering from the University of Rhode Island, Narragansett, in 2000. He was a Research Associate at the Indian Institute of Technology from 1987 to 1988. From 1988 to 1995, he was with the Department of Ship Technology, Cochin University of Science and Technology, Cochin, India. Since 2000, he has been with the University of Rhode Island and currently he is an Assistant Professor (Research) there. His research interests include nonlinear sediment inversion, time–frequency analysis techniques, and marine bioacoustics. Dr. Potty is a member of the Acoustical Society of America and the IEEE Oceanic Engineering Society. John C. Osler was born in Montreal, QC, Canada. He studied geophysics, receiving the B.Sc. degree with honors from McGill University, Montreal, QC, Canada, in 1986, and geological oceanography, receiving the Ph.D. degree from Dalhousie University, Halifax, NS, Canada, in 1993. In 1993, he joined the Defence Research Establishment Atlantic (now Defence Research & Development Canada—Atlantic), Dartmouth, NS, Canada, and spent one year as a term Defence Scientist and two years as a Visiting Fellow. He was a Scientist at the NATO Undersea Research Centre, La Spezia, Italy, from 1996 to 1999. He returned to DRDC Atlantic as a Defence Scientist in 1999 where he led the Maritime Environmental Awareness Group from 2004 until 2011. In 2011, he returned to the NATO Undersea Research Centre. His research interests include seabed-interacting acoustics and techniques for the rapid environmental assessment of oceanographic and seabed conditions. Dr. Osler is a Fellow of the Acoustical Society of America, and a member of the Canadian Acoustical Association, the American Geophysical Union, and the Canadian Geophysical Union. In 1994 and 1995, he was awarded the Edgar and Millicent Shaw Postdoctoral Prize by the Canadian Acoustical Association. In 1997, he received the Canadian Acoustic Association Director’s Award for best paper by a professional over 30.

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Paul C. Hines was born and raised in Glace Bay, Cape Breton. He received the B.Sc. (honors) degree in engineering—physics from Dalhousie University, Halifax, NS, Canada, in 1981 and the Ph.D. degree in physics from the University of Bath, Bath, U.K. in 1988. His research on acoustic scattering from ocean boundaries earned him the Chesterman Medal from the University for “Outstanding Research in Physics.” He joined the Defence Research Establishment Atlantic [now Defence Research & Development Canada—Atlantic (DRDC Atlantic)], Dartmouth, NS, Canada, where he researched towed array self-noise. Upon returning to DRDC Atlantic in 1989,

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 37, NO. 4, OCTOBER 2012

he joined Acoustic Countermeasures group to work on acoustic scattering and time spreading. From 1996 until 2003, he led several research groups that focused on experimentation and modeling to support sonar research. He is a seasoned experimentalist and has been chief scientist for several collaborative international research trials. Since 2003, he has managed projects in Rapid Environmental Acoustics and Classification for underwater acoustics and is currently Principal Scientist in the Underwater Sensing section at DRDC. He is an Adjunct Professor at the Department of Graduate Studies, Dalhousie University. His present research interests include acoustic scattering, vector sensor processing, and the application of aural perception in humans, to target classification sonar. Dr. Hines is a Fellow of the Acoustical Society of America.