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discussion of this method together with several examples of its application. Tolstoy and Clay (1966: 33-6) discussed solutions in waveguides.

When the sea surface is rough, the vertical motion of the surface modulates the amplitude of the incident wave and superposes its own spectrum as upper and lower sidebands on the spectrum of the incident sound. Moreover, when there is a surface current, the horizontal motion will appear in the scattered sound and cause a Doppler-shifted and Doppler-smeared spectrum.

3.4.3 Turbidity and bubbles

3.4.3.1 Open ocean

The presence of bubble layers near the sea surface further complicates the reflection and scattering of sound as a result of the change in sound speed, the resonant characteristics of bubbles and the scattering by bubbly layers (e.g. Leighton, 1994).

Hall (1989) developed a comprehensive model of wind-generated bubbles in the ocean. The effects on the transmission of short pulses in the frequency range 1.25-40 kHz were also examined. For long-range propagation, Hall concluded that the decrease in the near-surface sound speed due to bubbles does not significantly affect the intensity of the surface-reflected rays.

3.4.3.2 Coastal ocean

Coastal waters are often characterized by suspensions of solid mineral particles that are agitated by waves, currents or river outflows, in addition to microbubbles that are generated at the sea surface by wind and wave action or at the sea floor by biochemical processes (Richards and Leighton, 2001a,b). Suspended solid particles and microbubbles jointly modify the complex acoustic wavenumber, thus influencing the acoustic properties of the medium and thereby affecting the performance of acoustic sensors operating in such turbid and bubbly environments.

Consequently, the acoustic attenuation coefficient (Section 3.6) in shallow coastal waters is of interest to designers and operators of Doppler-current profiles, sidescan-surveying sonars and naval mine-hunting sonars operating in the frequency range from tens of kilohertz to several hundred kilohertz and possibly up to 1 MHz. At these frequencies, attenuation due to suspended particulate matter is an important contribution to the total attenuation coefficient (Richards, 1998). Typical suspensions contain particles in the size range 1-100 ^m, where a variety of shapes and concentrations from 0.1 to 4kgm-3 are possible (Brown et al., 1998). Microbubbles with radii in the range 10-60 ^m will be resonant in the frequency interval 50-300 kHz. Preliminary calculations using viscous-damping theory suggest that particulate concentrations on the order of 0.1 kg m-3 may be important, even possibly reducing the detection range of sonars by a factor of two relative to clear water at a frequency of 100 kHz. Observations have shown that concentrations of this level, or greater, often occur in coastal waters and have been detected several tens of kilometers offshore of the Amazon river, in the Yellow Sea and in the East China Sea offshore of the Yangtze and Yellow rivers (Richards et al., 1996).

The presence of microbubbles increases acoustic attenuation through the effects of thermal and viscous absorption and scattering. Unlike particles, however, the resonant scattering of bubbles can be important - the scattering cross-section of a bubble near resonance can be much larger than its geometric cross-section. Moreover, bubbles cause the compressibility of the medium to be complex, thereby resulting in dispersion. The effect of bubbles on the phase speed should be used to modify the sound-speed profile when computing ray paths in bubbly layers. A numerical procedure was developed by Norton et al. (1998) to parameterize bubble clouds in terms of an effective complex index of refraction for use in high-fidelity models of forward propagation.

The effective attenuation coefficient in turbid and bubbly environments can be expressed as (Richards and Leighton, 2001a):

a = aw + ap + ab and ap = av + as where a is the total volume attenuation coefficient of sea water containing suspended particles and microbubbles, aw the physico-chemical absorption by clear sea water (see Section 3.6), ap the plane-wave attenuation coefficient due to a suspension of solid particles (neglecting thermal absorption) and ab the attenuation coefficient for a bubbly liquid. Furthermore, ap is composed of two terms: av is the attenuation coefficient associated with the visco-inertial absorption by suspended particles and as the attenuation coefficient associated with scattering by suspended particles.

3.4.4 Ice interaction

Acoustic interaction with an ice canopy is governed by the shape of the under-ice surface and by the compressional wavespeeds (typically 1,3003,900 ms-1) and shear wavespeeds (typically, 1,400-1,900 ms-1) (see Untersteiner, 1966; Medwin et al., 1988).

McCammon and McDaniel (1985) examined the reflectivity of ice due to the absorption of shear and compressional waves. They found that shear wave attenuation is the most important loss mechanism from 20° to 60° incidence for smooth ice at low frequencies (<2kHz).

In Arctic regions, the presence of a positive-gradient sound-speed profile and a rough under-ice surface (with a distribution of large keels) may lead to significant out-of-plane scattering. The acoustic impacts of this scattering are twofold. First, significant beam widening may result from the multiple interactions with the randomly rough under-ice surface. Second, the presence of ice keels in the vicinity of the receiver leads to multiple source images or beam-steering errors arising from interactions of the acoustic signal with the facets of the local under-ice surface.

Because of the overwhelming effect of ice on the propagation of sound in the Arctic, the magnitude of the excess attenuation observed under the ice should be determined by the statistics of the under-ice surface. Available ice-ridge models can be used to generate such statistics. These models can be categorized according to two classes: discrete models and continuous statistical models. These two classes of ice-ridge models are briefly described below.

Discrete ice-ridge models prescribe a representative ridge shape, or an ensemble of ridge shapes, to calculate the statistics of the surface from the discrete statistics of the known ice structure. Continuous statistical ice-ridge models treat the under-ice surface as a stochastic process. This process is then analyzed using the techniques of time-series analysis in which the under-ice surface can be characterized by its autocorrelation function. Continuous statistical models can give a more complete description of the under-ice roughness than can the discrete models; however, they are limited in application to those surfaces that can be completely specified by a Gaussian depth distribution.

The model developed by Diachok (1976) will be described since it is considered to be representative of the class of discrete ice-ridge models known to exist and because of its intuitive appeal. The discrete models are also more robust (i.e. require less knowledge of the under-ice surface) than the continuous statistical models. Furthermore, Diachok's model has been incorporated into existing propagation models with some success.

According to Diachok's model, sea ice may be described as consisting of floating plates, or floes, about 3 m thick, occasionally interrupted by ridges, which are rubble piles formed by collisions and shear interactions between adjacent floes. Ridge dimensions vary widely, but are nominally about 1 m high, 4 m deep and 12 m wide, with the ridge lengths generally being much greater than the depths or widths. A representative average spacing between ridges (the spacing is random) is about 100 m. Ice-ridge orientation is commonly assumed to be directionally isotropic, although limited empirical data suggest that, at least locally, there may be a preferred orientation. The physical model of reflection developed by Twersky (1957) was used.

A comparison between measured contours and simple geometrical shapes suggests that ridge keel contours may reasonably be represented by a halfellipse (as in Figure 3.10) and that ridge sail contours may be described using a Gaussian distribution function. The relative dimensions of this geometrical model are indicated in Figure 3.10. The exact solution of under-ice scattering off a flat surface with a single semi-elliptical cylindrical boss of infinite extent was developed by Rubenstein and Greene (1991).

Figure 3.10 Geometrical model of sea-ice ridges (Diachok, 1976).

LePage and Schmidt (1994) extended the applicability of perturbation theory to under-ice scattering at low frequencies (10-100 Hz) by including the scattering of incident acoustic energy into elastic modes, which then propagate through the ice. Kapoor and Schmidt (1997) developed a canonical model in which the under-ice scattering surface was represented as an infinite elastic plate with protuberances.

3.4.5 Measurements

Three basic experimental techniques have been employed to measure forward reflection losses at the sea surface:

1 comparing the amplitude or energy of pulses returned from the surface with that of the direct arrival;

2 using the Lloyd mirror effect and observing the depth of the minima as the frequency is varied; and

3 measuring the attenuation in the surface duct.

Based on a compilation of results in the literature by Urick (1982: chapter 10), it appears that surface losses are less than 1 dB (per bounce) at frequencies below 1 kHz, and rise to about 3 dB (per bounce) at frequencies above 25 kHz.

3.5 Sea-floor boundary

The sea floor affects underwater sound by providing a mechanism for:

1 forward scattering and reflection loss (but is complicated by refraction in the bottom);

2 interference and frequency effects;

3 attenuation by sediments;

4 noise generation at lower frequencies due to seismic activity; and

5 backscattering and bottom reverberation.

Items (1)-(3) will be discussed below. Item (4) will be discussed in Chapter 6 and item (5) in Chapter 8. Urick (1982: chapter 11) provided a comprehensive summary of sound reflection and scattering by the sea floor. The single most important physical property that determines the acoustic characteristics of sediments is their porosity.

The return of sound from the sea floor is more complex than from the sea surface for several reasons: (1) the bottom is more variable in composition; (2) the bottom is often stratified (layered) with density and sound speeds (both shear and compressional) varying gradually or abruptly with depth; (3) bottom characteristics (composition and roughness) can vary over relatively short horizontal distances; and (4) sound can propagate through a sedimentary layer and either be reflected back into the water by sub-bottom layers or be refracted back by the large sound-speed gradients in the sediments.

These mechanisms can be incorporated into mathematical models through the specification of appropriate "boundary conditions." The complexity of these boundary conditions will depend upon the level of known detail concerning the composition and structure of the sea floor, and also to some degree on the sophistication of the mathematical model being used.

The specification of boundary conditions at the sea floor has assumed greater importance due to increased interest in the modeling of sound propagation in shallow-water areas. Such propagation, by definition, is characterized by repeated interactions with the bottom boundary. Acoustic interactions with highly variable sea-floor topographies and bottom compositions often necessitate the inclusion of both compressional- and shear-wave effects, particularly at lower frequencies. A fluid, by definition, cannot support shear stresses. Therefore, in modeling acoustic propagation in an ideal (boundless) fluid layer, only compressional-wave effects need be considered. As an approximation, saturated sediments are sometimes modeled as a fluid layer in which the sound speed is slightly higher than that of the overlying water column. The basement, however, can support both compressional and shear waves, and rigorous modeling of acoustic waves that interact with and propagate through such media must consider both types of wave effects. As an approximation, shear-wave effects are sometimes included in the form of modified attenuation coefficients.

3.5.1 Forward scattering and reflection loss 3.5.1.1 Acoustic interaction with the sea floor

Westwood and Vidmar (1987) summarized pertinent developments in the modeling of acoustic interaction with the sea floor. It is convenient to partition the discussion according to low-frequency and high-frequency bottom interaction. The transition between low and high frequencies is imprecise but can be considered to occur near 200 Hz.

At low frequencies and low grazing angles, acoustic interaction with the sea floor in deep ocean basins is simple and well understood. The relatively long acoustic wavelengths are insensitive to details of small-scale layering in the sediments. Moreover, for low grazing angles, there is little interaction with the potentially rough substrate interface. Accordingly, the sea floor can be accurately approximated as a horizontally stratified and depth-dependent fluid medium. The major acoustical processes affecting interaction with the sea floor are: (1) reflection and transmission of energy at the water-sediment interface; (2) refraction of energy by the positive sound-speed gradient in the sediments; and (3) attenuation within the sediments. Modeling of this interaction is further enhanced by the availability of established methods for estimating the geoacoustic profile (i.e. sound speed, density and attenuation as functions of depth) of deep-sea sediments, given the sediment type and physiographic province.

In contrast, bottom interaction at high frequencies is not well understood. The relatively short wavelengths are more sensitive to the small-scale sediment layering. These layers are reported to have an important effect on the magnitude and phase of the plane-wave reflection coefficient. Stochastic techniques with which to analyze the effects of the near-surface sediment layering are being developed, but they do not yet incorporate potentially important acoustical processes such as refraction and shear-wave generation. Modeling at high frequencies is further frustrated by the high spatial variability of sediment layering.

The concept of "hidden depths" (Williams, 1976) states that the deep ocean sediment structure well below the ray turning point has no acoustical effect. This concept is important because it focuses attention on those low-frequency processes occurring in the upper regions of the sediments (see Knobles and Vidmar, 1986).

3.5.1.2 Boundary conditions and modeling

The ideal forward reflection loss of sound incident on a plane boundary separating two fluids characterized only by sound speed and density was originally developed by Rayleigh (1945: Vol. II, 78). This model is commonly referred to as Rayleigh's law. In the simplest model incorporating absorption, the bottom can be taken to be a homogeneous absorptive fluid with a plane interface characterized by its density, sound speed and attenuation coefficient. In the case of sedimentary materials, all three of these parameters are affected by the porosity of the sediments.

In underwater acoustics, a common idealized model for the interaction of a point-source field with the sea floor is the so-called Sommerfeld model (after A.N. Sommerfeld). This model consists of an isospeed half-space water column overlying an isospeed half-space bottom. The bottom has a higher sound speed than the water. Thus, a critical angle exists in the plane-wave reflection coefficient. For large grazing angles, energy is partially reflected and partially transmitted at the water-bottom interface. For small grazing angles, energy is totally reflected back into the water column. Energy incident near the critical angle produces a complex phenomenon known as the lateral, or head, wave (Chin-Bing et al., 1982, 1986; Westwood, 1989a; see also the discussions by Clay and Medwin, 1977: 262-3; Frisk, 1994: 32). This boundary condition is referred to as an "impedance (or Cauchy) boundary."

Another commonly assumed boundary condition for the sea floor is the homogeneous Neumann bottom-boundary condition. Here, the derivative of the pressure normal to the boundary vanishes (Frisk, 1994: 32-3). There is no phase shift in the reflected wave. For harmonic time dependence and constant density, this condition is also termed a "rigid boundary."

Hall and Watson (1967) developed an empirical bottom reflection loss expression based largely on the results of Acoustic Meteorological and Oceanographic Survey (AMOS) (Marsh and Schulkin, 1955). Ainslie (1999) demonstrated that much of the complexity of bottom interaction could be represented in simple equations for the reflection coefficient when expressed in the form of a geometric series. Such simplifications can be useful in modeling acoustic propagation in shallow water where repeated interactions with the seabed are expected. Moreover, Ainslie et al. (1998b) presented benchmarks for bottom reflection loss versus angle at 1.5, 15 and 150 Hz for four different bottom types, each comprising a layered fluid sediment (representing sand or mud) overlying a uniform solid substrate (representing limestone or basalt). These benchmarks provide ground-truth reference solutions against which the accuracy of other models can be assessed. The benchmarks are calculated using exact analytical solutions where available (primary benchmarks) or they are calculated using a numerical model (secondary benchmark). While the secondary benchmarks are approximate, they provide useful diagnostic information. Robins (1991) developed a FORTRAN program called PARSIFAL to compute plane-wave reflection coefficients from a sediment layer modeled as an inhomogeneous fluid overlying a uniform substrate.

Tindle and Zhang (1992) demonstrated that the acoustic-reflection coefficient for a homogeneous fluid overlying a homogeneous solid with a low shear speed could be approximated by replacing the solid with a fluid having different parameters. Zhang and Tindle (1995) subsequently simplified these expressions by approximating the acoustic-reflection coefficients of solid layers with a fluid described by suitably chosen (proxy) parameters.

Westwood and Vidmar (1987) developed a ray-theoretical approach called CAPARAY for simulating the propagation of broadband signals interacting with a layered ocean bottom. CAPARAY can simulate a time series at a receiver due to an arbitrary source waveform by constructing a frequency domain transfer function from the eigenray characteristics.

76 Propagation I: observations and physical models 3.5.1.3 Geoacoustic models

Geoacoustic models of the sea floor more properly account for the propagation of sound in sediments (Anderson and Hampton, 1980a,b; Hamilton, 1980). As summarized by Holland and Brunson (1988), geoacoustic models of marine sediments can be formulated in one of three ways: (1) by empirically relating geoacoustic and geophysical properties of the sediments (e.g. Hamilton, 1980); (2) by using the Biot-Stoll model to relate sediment geoacoustic properties to geophysical properties on the basis of physical principles (Biot, 1956a,b; Stoll, 1974,1980,1989); and (3) by using an inversion technique to generate sediment geophysical parameters from bottom loss measurements (e.g. Hovem et al., 1991 (see especially section 3, Modelling and inversion techniques); McCammon, 1991; Rajan, 1992; Dosso et al., 1993; Hovem, 1993; Frisk, 1994).

The Biot-Stoll model (Biot, 1956a,b; Stoll, 1974, 1980, 1989) provides a comprehensive description of the acoustic response of linear, porous materials containing a compressible pore fluid. The model predicts two types of compressional waves and one shear wave. Recent applications in underwater acoustics with references to the key historical literature were provided by Beebe et al. (1982) and by Holland and Brunson (1988). Routine operational employment of this model is complicated by the input of more than a dozen geophysical parameters, some of which are difficult to obtain even in laboratory environments.

McCammon (1988) described the development of a geoacoustic approach to bottom interaction, called the thin layer model. This model, which is based on an inversion technique, contains a thin surficial layer, a fluid sediment layer and a reflecting sub-bottom half-space. There are 10 input parameters to this model: sediment density, thickness, sound-speed gradient and curvature, attenuation and attenuation gradient, thin-layer density and thickness, basement reflectivity and water-sediment velocity ratio (Figure 3.11). The model generates bottom-loss curves as a function of grazing angle over the frequency range 50-1,500 Hz. The model makes several assumptions: it relies upon the "hidden depths" concept of Williams (1976), the sediments are isotropic, the roughness of the sediment and basement interfaces and multiple scattering within the layers are neglected and shear wave propagation is ignored.

Sample outputs from this thin layer model are presented in Figure 3.12. A ratio (cs/cw) > 1 (where cs is the sound speed in the upper sediment and cw is the sound speed at the base of the water column) predicts a critical angle 0c = cos-1 (cw/cs), below which most of the incident energy is reflected; that is, the bottom loss is nearly zero. By comparison, a ratio (cs/cw) < 1 would refract the incident energy into the sediments and result in greater losses at small angles.

A qualitative comparison of bottom loss versus grazing angle for (cs/c w) > 1 and (cs/cw) < 1 is presented in Figure 3.13. It has been

Basement Basement loss

Figure 3.11 Thin layer model for sediment reflected and refracted paths.

(McCammon, 1988; J. Geophys. Res., 93, 2363-9; published by the American Geophysical Union.)

Figure 3.12 Variation of bottom loss (dB) as a function of grazing angle for a frequency of 1,000 Hz. Curves are presented for various values of the ratio of the upper sediment sound speed (cs) to the sound speed at the base of the water column (cw). (McCammon, 1988; J. Geophys. Res., 93, 2363-9; published by the American Geophysical Union.)

Figure 3.12 Variation of bottom loss (dB) as a function of grazing angle for a frequency of 1,000 Hz. Curves are presented for various values of the ratio of the upper sediment sound speed (cs) to the sound speed at the base of the water column (cw). (McCammon, 1988; J. Geophys. Res., 93, 2363-9; published by the American Geophysical Union.)

demonstrated that high-porosity sediments (e.g. mud and silt) have sound speeds less than that of the overlying water (Urick, 1983: 138-9; Apel, 1987: 386). Qualitatively, then, the comparison presented in Figure 3.13 contrasts the effects of high-porosity [(cs/cw) < 1] and low-porosity sediments

The bottom loss upgrade (BLUG) model, which was a modular upgrade designed for incorporation into existing propagation models to treat bottom

Figure 3.13 Qualitative illustration of bottom loss versus grazing angle for low-porosity and high-porosity bottoms.

loss, was based on a geoacoustic (or inverse) approach. The low-frequency bottom loss (LFBL) model has subsequently replaced the BLUG model.

3.5.2 Interference and frequency effects

Stratification and attendant scattering within the bottom produce pulse distortion, as does reflection at grazing angles less than the critical angle. Zabal et al. (1986) developed a simple geometric-acoustic model to predict frequency and angle spreads as well as coherence losses to sonar systems. The sea floor was modeled by homogeneous and isotropic slope statistics. The facets are planar and reflect specularly, thus giving rise to the name "broken mirror" model.

3.5.3 Attenuation by sediments

Sound that propagates within sediment layers is subject to the effects of attenuation. A variety of sediment attenuation units are commonly used in the underwater acoustic and marine seismology communities. The relationships among these units can become very confusing when attempting to enter values into propagation models.

Mathematically, acoustic attenuation (a ) is expressed in the exponential form as e-ax using the units of nepers (Np) per unit distance for a. The acoustic attenuation can be converted to the units of decibels per meter

Propagation I: observations and physical models 79 using the relationship:

Some propagation models require that the attenuation be specified in units of decibels per wavelength (X):

Another term commonly encountered in underwater acoustic modeling is the attenuation coefficient (k), which is based on the concept that attenuation (a) and frequency (f) are related by a power law:

a(dBm-1) = kfn where f is measured in kilohertz and n is typically assumed to be unity. Over the frequency range of interest to underwater acousticians, attenuation is approximately linearly proportional to frequency.

A compilation of sediment attenuation measurements made by Hamilton (1980) over a wide frequency range showed that the attenuation in natural, saturated sediments is approximately equal to 0.25f (dBm ) when f is in kilohertz. There was a tendency for the more dense sediments (such as sand) to have a higher attenuation than the less dense, higher-porosity sediments (such as mud). The attenuation in sediments is several orders of magnitude higher than in pure water.

In the Arctic, acoustical parameters of the sea floor and sub-bottom are poorly known. Difficulties in obtaining direct core samples to great depths limit the database from which to extract the parameters needed to determine many of the major acoustical processes in bottom interaction. Estimating these geoacoustical parameters based on data from contiguous areas may not be meaningful since the basic processes of sedimentation at work under the pack ice are unique to that environment. Sedimentation rates are very low, being dominated by material carried by the ice rather than by material of biologic origin, as is the case in more temperate areas. The ice pack may also carry large boulders of glacial origin and deposit them in the Arctic Ocean. The low sedimentation rate leaves the boulders exposed as potential scatterers for acoustic energy over a wide range of frequencies.

3.5.4 Measurements

The standard method for measuring bottom loss is to use pings or explosive pulses and to compare the amplitude, intensity or energy density (integrated intensity) of the bottom pulse with that of the observed or computed pulse traveling via a direct path.

Bottom loss data typically show a loss increasing with angle at low angles, followed by a nearly constant loss extending over a wide range of higher angles (refer back to Figure 3.13). High-porosity bottoms (having a sound speed less than that of the overlying water) tend to have a maximum loss at an angle between 10° and 20° where an angle of intromission (i.e. no reflection, but complete transmission into the bottom) would be expected to occur in the absence of attenuation in the bottom. When narrowband pulses are used, measured losses are often irregular and variable, showing peaks and troughs due to the interference effects of layering in the bottom. Measured data rarely show a sharp critical angle (as would be inferred from the Rayleigh reflection model) because of the existence of attenuation in the bottom (Urick, 1983: chapter 5).

3.6 Attenuation and absorption in sea water

Sound losses in the ocean can be categorized according to spreading loss and attenuation loss. Spreading loss includes spherical and cylindrical spreading losses in addition to focusing effects. Attenuation loss includes losses due to absorption, leakage out of ducts, scattering and diffraction. Urick (1982: chapter 5) summarized the relevant literature pertaining to this subject.

Absorption describes those effects in the ocean in which a portion of the sound intensity is lost through conversion to heat. Field measurements of the absorption coefficient (a), typically expressed in units of decibels per kilometer, span the frequency range 20 Hz to 60 kHz. In practice, absorption loss (in dB) is computed as the product of a and range (r) using self-consistent units for range.

The dependence of a on frequency is complicated, reflecting the effects of different processes or mechanisms operating over different frequency ranges. The equation developed by Thorp (1967) is probably the best known and is valid at frequencies below 50 kHz:

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