Source: Rogers (1981).

Source: Rogers (1981).

Colossus employs several concepts: (1) refractive cycle, or skip distance; (2) deflection of energy into the bottom at high angles by scattering from the sea surface; and (3) a simplified Rayleigh two-fluid model of the bottom for sand or mud sediments. With a few free parameters (including water depth), about 100,000 measurements spanning the frequency range from 100 Hz to 10 kHz were fitted within stated error bounds.

Figure 5.10 Composite illustration of the eight TL curves resulting from the coefficients presented in Table 5.2 (Rogers, 1981).

Figure 5.10 Composite illustration of the eight TL curves resulting from the coefficients presented in Table 5.2 (Rogers, 1981).

Along with the concepts of skip distance and bottom loss, the Colossus model used the AMOS results (Marsh and Schulkin, 1955) for a deep-water isothermal surface duct, but with an average thermocline appropriate for shallow water. It also used measurements of actual transmission losses in shallow water off the east coast of the United States as a function of frequency, categorized by bottom type (sand or mud) and by season. Two other mechanisms characteristic of shallow-water processes were also included in the Colossus model. The first mechanism was a "near-field anomaly" correction in the direct radiation zone that included the gain due to multiple bottom and surface bounces. The second was an energy-conservation rule that was used to establish the effective shallow-water attenuation coefficient (at), which included the additional loss due to the coupling of energy from the wind-roughened sea surface to the bottom.

The bilinear gradient used in the Colossus model is composed of two constant, linear segments drawn toward the surface and toward the bottom from the depth of maximum sound speed (or temperature). This profile corresponds to the bilinear gradient with a surface duct (Category II-A) that was illustrated previously in Table 5.1. By placing the layer depth (maximum sound speed) at the sea surface, a linear negative-gradient profile (Category I-B) is obtained. Alternatively, placing the layer depth at the sea floor produces a linear positive-gradient sound-speed profile (Category I-A). As noted in Table 5.1, profiles from Categories I-A, I-B and II-A collectively comprised about 77 percent of all sound-speed profiles encountered in shallow-water regions.

In Colossus, the sound ray cycles have one upward radius of curvature (positive sound-speed gradient) for surface bounces and one downward radius of curvature (negative gradient) for bottom bounces. Based on the depth of the surface layer and the water depth, a single effective skip distance (H) is formulated. Multiples of this effective skip distance are used to define a zone of direct ray paths (20 log^ R, where R is the range), a zone of mode stripping (15 log^ R), and a zone of single-mode control (10 log^ R). The mode-stripping process was found to be complete at a range equal to 8H, where H is the skip distance.

In the Marsh-Schulkin (or Colossus) model, TL is a function of sea state (or wave height), bottom type, water depth, frequency and the depth of the positive-gradient layer. The skip distance (H) is used as a reference to define regions where wavefront spreading follows square, three-halves and firstpower laws as a function of range (R). Accordingly, three equations were developed to provide for the gradual transition from spherical spreading in the near field to cylindrical spreading in the far field:

TL = 20log10 R + aR + 60 - Kl (5.8) Intermediate range (H < R < 8H)

TL = 15 log10 R + aR + at[(R/H) - 1] + 5 log10 H + 60 - Kl (5.9) Long range (R > 8H)

TL = 10 log10 R + aR + at[(R/H) - 1] + 10 log10 H + 64.5 - Kl

where H = [(L + D)/3]1/2 is the skip distance (km), L the mixed layer depth (m), D the water depth (m), R the range (km), a the absorption coefficient (dBkm-1), at the effective shallow-water attenuation coefficient (dB per bounce) and Kl the near-field anomaly (dB).

The absorption coefficient (a) can be estimated from Figure 3.14. The effective shallow-water attenuation coefficient (at) and the near-field anomaly (Kl) are functions of frequency, sea state and bottom composition. Values of at range from about 1 to 8 dB per bounce. Typical values of Kl range from about 1 to 7 dB. Complete tables of at and Kl are contained in the paper by Marsh and Schulkin (1962b).

According to Schulkin and Mercer (1985), the model's chief criticisms have been that it could not be adjusted for arbitrary negative sound-speed gradients (it uses the same constant gradient in all cases), and that it uses empirical bottom loss values. Consequently, the Colossus model has been extended to accommodate arbitrary gradients in both negative and bilinear sound-speed profiles. These extensions use new general expressions for the skip distance, the near-field anomaly, and the reflection coefficients. The extended model (Schulkin and Mercer, 1985) and the model by Rogers (1981) were found to give about the same predictions when the same inputs were used.

5.4 Arctic models

5.4.1 Arctic environmental models

Basic descriptors of the Arctic marine environment that require specialized algorithms for use in propagation models are limited to absorption and surface (under-ice) scattering (Etter, 1987c; Ramsdale and Posey, 1987). The generation of other parameters, such as sound speed and bottom scattering, appear to be adequately supported by existing algorithms that are valid over a wide range of oceanic conditions.

Absorption is regionally dependent mainly due to the pH dependence of the boric acid relaxation. In the Arctic, the pH range is roughly 8.0-8.3 (versus 7.7-8.3 for nominal sea water), but the greatest variability occurs much closer to the sea surface than in other ocean areas. Absorption formulae appropriate for use in the Arctic regions were presented by Mellen et al. (1987c).

Existing under-ice scattering loss models appropriate for inclusion in mathematical models of acoustic propagation were reviewed and evaluated by Eller (1985). Chapter 3 described physical models of under-ice roughness. Many under-ice scattering loss models can be incorporated directly into existing propagation models that were constructed using a modular architecture.

5.4.2 Arctic propagation models

Developments in acoustic propagation modeling for the Arctic Ocean have been very limited. Much of the past effort on characterizing propagation in the Arctic has been devoted to gathering acoustical data and developing empirical models based on that data. While these models tend to be site and season specific with little generality, they do provide basic information on the frequency and range dependence of acoustic propagation in ice-covered regions.

In general, there are four factors peculiar to the Arctic environment that complicate the modeling of acoustic propagation: (1) the ice keels present a rapidly varying surface; (2) the reflection, transmission and scattering properties at the water-ice interface are not well known; (3) the measurement of under-ice contours is difficult; and (4) the diffraction of sound around ice obstacles may be important. In the Arctic half-channel, ray theory may provide a useful predictive method, whereas its utility in temperate-water surface ducts may actually be quite limited. The utility of ray theory in the Arctic derives from the fact that the Arctic half-channel is between one and two orders of magnitude greater in gradient (and much greater in depth) than the temperate-water surface ducts. The strong positive thermocline and halocline produce an exceptionally strong positive sound-speed gradient in the subsurface layer. This markedly shortens the ray-loop length, causing many surface reflections for rays with small grazing angles (Mobile Sonar Technology, 1983, unpublished manuscript).

The open ocean region in the Arctic environment also poses potential problems as far as existing propagation models are concerned. Surface reflection losses, which may not be significant in deep-ocean propagation, become more important in the upward-refracting environment of the Arctic since multiple surface reflections now play a dominant role. Moreover, multiple ducts are prevalent in this region and many propagation models do not adequately treat them.

Two basic modeling approaches are currently being pursued: the application of ice-scattering coefficients to existing numerical models of acoustic propagation and the development of empirical models. These two approaches are discussed in more detail below.

5.4.3 Numerical models

Numerical models of underwater acoustic propagation specifically designed for ice-covered regions are limited. Since the Arctic half-channel acts as a low-pass filter (discriminating against higher-frequency components), and since bottom interaction is not as important as in other ocean regions (because of the upward-refracting sound-speed profile), most modeling applications in the Arctic have employed either normal mode (Gordon and Bucker, 1984) or fast-field (Kutschale, 1973, 1984) approaches. These models are considered most appropriate for prediction of low-frequency (<350Hz) propagation at long ranges (>25nm). Other modeling techniques (e.g. ray theory and parabolic equation) have also proved suitable for calculation of transmission losses once the required ice-scattering algorithms were incorporated (e.g. Chin-Bing and Murphy, 1987). For example, Stotts et al. (1994) developed a ray-theoretical propagation model called ICERAY that is valid for under-ice environments, and at least one implementation of the PE technique has been modified to include the effects of ice scattering (refer back to Table 4.1).

Shot signals in the Arctic channel propagate via low-order normal modes, a result of the constructive interference of RSR rays traveling in the upper few hundred meters of the water column. Due to scattering at the rough boundaries of the ice, only low frequencies (typically less than 40 Hz) can propagate to long distances in the Arctic channel. If the channel is sufficiently deep along the entire propagation path, as over an abyssal plain, then the received wave trains display an impulsive character corresponding to arrivals of deep-penetrating RSR rays while the latter part of the wave trains retain the nearly sinusoidal character typical of low-order modes traveling in the upper layers of the water column (Kutschale, 1984). Using the normal-mode theory of Pekeris (1948), the measured frequency dispersion of low-order modes has often been used to infer acoustic properties of sediments in shallow-water waveguides. Kutschale and Lee (1983) and Kutschale (1984) utilized a similar approach to infer acoustic properties of bottom sediments in the Arctic based on the dispersion of high-order normal modes. A normal mode model was used to interpret the frequency dispersion of bottom-interacting wave trains. The MSPFFP model was used to derive a geoacoustic model that predicted synthetic waveforms matching the dispersion profiles of the bottom-interacting signals. More information on MSPFFP can be found in Table 4.1.

5.4.4 Empirical models

Empirical models are inherently limited by the databases from which they were derived. Attempts to fit results from a large data set with simplistic curves generally imply large errors in the model results. Comparisons of model results with data are clearly quite limited by a lack of comprehensive data sets. A recent intercomparison of available empirical techniques applicable to the Arctic revealed large unresolved discrepancies in the manner in which under-ice scattering losses were computed (Deavenport and DiNaploi, 1982). Two of the better-known empirical models are described below. Marsh-Mellen model

The Marsh-Mellen Arctic transmission loss model (Marsh and Mellen, 1963; Mellen and Marsh, 1965) is based on observations made during the summers of 1958 and 1959 between Arctic drift stations separated by 800-1,200 km. The measured arrivals were found to consist of a dispersive, quasi-sinusoidal wave train in the 10-100 Hz frequency range. A halfchannel model in which the higher frequencies were attenuated by under-ice scattering explained these features. Using these and other experimental data, long-range, low-frequency (<400 Hz) TL data in the Arctic were fitted with an equation of the form

where r0 is the skip distance for the limiting ray, Ns is the number of surface reflections, R is the range in meters (R = —0Ns), and as is the loss per bounce. The wind-generated ocean wave spectrum (Marsh, 1963) was taken to approximate the ice roughness.

Propagation II: mathematical models (Part Two) 175 Buck model

The Buck Arctic transmission loss model (Buck, 1981) consists of a short-range (10-100 nm) and a long-range (100-1,000 nm) model for low-frequency (<100 Hz) transmission loss in that part of the Arctic Ocean deeper than 1,000 m. The crossover range at 100 nm is where higher-order, deeper cycling modes begin to dominate the acoustic propagation. These preliminary empirical models represent linear regression fits to winter data collected in the Beaufort Sea in 1970, in the Fram Strait in 1977, and in an intermediate area in 1979 for a source depth of 244 m and a receiver depth of 30 m:

Short range (10-100 nm)

TL = 62.4 + 10 log10 R + 0.032 f + 0.065 R + 0.0011 fR (5.12)

Long range (100-1,000 nm)

TL = 68.5 +10log10 R + 0.07f-0.0015sR + 0.000487fsR (5.13)

where f is the frequency (Hz), R the range (nm) and s the standard deviation ice depth (m), also referred to as the under-ice roughness parameter. A chart of the standard deviation ice depth (s) suitable for use in Equation (5.13) is presented in Figure 5.11 (from Buck, 1985).

Figure 5.11 Estimates of under-ice roughness in standard deviation (m) about the mean ice depth. These estimates were derived, in part, from nuclear submarine sonar data analyzed by L.A. LeSchack (Buck 1985).

There is some doubt regarding the applicability of Equations (5.12) and (5.13) to summer conditions and to other source-receiver depth combinations. Therefore, these equations should be used with caution. To obtain coarse estimates, refer to Figure 3.29 where average curves of Arctic TL versus frequency have been derived from measured data (Buck, 1968).

5.5 Data support requirements

The development of numerical models requires data with which to support model initialization and model evaluation. Initialization of propagation models requires various descriptors of the ocean environment including the water column, the sea surface and the sea floor. Wave-theoretical models tend to be more demanding of bottom sediment information than do the ray-theoretical models. Evaluation of propagation models requires TL measurements that are keyed to descriptions of the prevailing ocean environment. Such coordinated measurements are necessary to ensure that the model is initialized to the same environment for which the TL data are valid (e.g. Hanna, 1976).

Data support requirements are further complicated by the fact that longrange TL can rarely be considered to occur in a truly range-independent environment. With few exceptions, changes in sound speed or water depth (among other parameters) can be expected not only as a function of range but also as a function of azimuth (or bearing). Consequently, data management in support of model development and operation can be formidable, and is often a limiting factor in the proper employment and evaluation of numerical models.

When working with a variety of propagation models (either different models within one category or models from different categories, as illustrated previously in Figure 4.1), it becomes obvious very quickly (sometimes painfully so) that each model not only requires somewhat different parameters but sometimes requires varying format specifications for any given parameter. In response to this situation, two related developments have occurred. First, gridded databases (i.e. those organized by latitude and longitude) have been established that contain all the required ocean environmental parameters in a standardized format with automated update and retrieval mechanisms. Second, model-operating systems (see Chapter 10) have been created to automatically interface the standard databases with the various models resident in the operating system, and to accommodate the format specifications peculiar to each of the models.

A graphic example of a retrieval from a gridded ocean database is presented in Figure 5.12. This presentation of range-dependent sound speed and bathymetry data is suitable for use in many existing propagation models. Chapter 10 presents a summary of available oceanographic and acoustic databases.

Environment: Indian Ocean Latitude: 10° 0.0' S Month: May Longitude: 80° 0.0' E Bearing: 0.0° Reference sound speed: 1,515 ms-1

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