0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 Range (nm)

Figure 5.12 Example of a range-dependent ocean environment extracted from a gridded database appropriate for use in many propagation models.

0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000.0 Range (nm)

Figure 5.12 Example of a range-dependent ocean environment extracted from a gridded database appropriate for use in many propagation models.

One of the newest automated ocean data products is the generalized digital environmental model (GDEM), developed by Dr Tom Davis at the US Naval Oceanographic Office (Wells and Wargelin, 1985). GDEM uses a digitized database of major oceanic features together with their climatic location to display temperature, salinity and sound-speed profiles on a SO7 latitude and longitude grid. A simple parabolic fit is used to describe the deep portion of the sound-speed profile. The more variable near-surface fields of temperature, salinity and sound speed are constructed using orthogonal polynomials between the surface and a depth of 800 m. GDEM has been rigorously compared with reliable climatologies (Teague et al., 1990). Refer to Chapter 10 (specifically Table 10.5) for further details regarding GDEM. The specification of sound-speed profiles for initialization of propagation models is very important, and the following sections will elaborate on this topic.

In preparation for a typical modeling run, a sound-speed profile is reconstructed (or synthesized) from a tabulation of sound-speed values at discrete

Sound-speed profile curve-fitting techniques c (z) ^__c (z) -

Segmented constant gradient z

Segmented constant gradient

• Discontinuous first derivative leads to false caustics in ray tracing techniques

• Line segments in which 1/c2 varies linearly with depth permits mode functions to be expressed in terms of Airy functions in normal mode techniques.

Curvilinear or continuous gradient

■ Quadratic equations - fit data points to within specified tolerances

1 Cubic splines - do not result in closed-form solutions for ray path equation in ray tracing techniques; continuous first and second derivatives.

■ Conic sections and hyperbolic cosines - result in closed-form solutions for ray path equation using elliptical and hyperbolic ray paths in ray tracing techniques.

■ Exponential forms - facilitate matrizant solution in fast field theory techniques.

Figure 5.13 Summary of sound-speed profile curve-fitting techniques.

depths. The best functional representation of the synthesized sound-speed profile varies with the type of modeling technique employed. There are basically two different approaches in use, each with its own advantages and disadvantages, as summarized in Figure 5.13.

The sound-speed profile can be constructed by connecting the discrete points with straight-line segments. Because of discontinuities in the first derivative of the resulting function, transmission losses calculated on the basis of ray-tracing techniques are undefined at certain ranges (see Pedersen, 1961). These regions are referred to as false caustics since they are false regions of infinite intensity.

In some propagation models, the sound-speed profile is fitted with segments in which the inverse of the sound-speed squared (1/c2) varies linearly with depth. This often permits a more efficient mathematical solution in which the mode functions are expressed in terms of Airy functions.

5.5.1.2 Curvilinear or continuous gradient

Curvilinear profile approximations that preserve the continuity of slope as well as sound speed have been developed by Pedersen and Gordon (1967).

The sound-speed profile can also be fitted with quadratic equations within specified tolerances (Weinberg, 1969, 1971).

Cubic splines can be used to approximate the sound-speed profile for application to ray-theoretical techniques (Solomon et al., 1968; Moler and Solomon, 1970). However, this method does not result in closed-form solutions for the rays. Approximating profiles with conic sections and hyperbolic cosines results in closed-form solutions for the ray-path equations when using elliptical and hyperbolic ray paths (Flanagan et al., 1974).

Exponential forms are sometimes used in those modeling techniques employing fast-field theory to facilitate a matrizant solution. To simplify the mathematical treatment of long-range sound propagation in the ocean, some investigators (Munk, 1974; Flatté, 1979) have introduced what is termed a "canonical" model of the sound-speed profile. This model has an exponential form that is valid in the vicinity of the deep sound channel axis:

where c(z) is the sound speed as a function of depth, C1 the sound speed at channel axis (z1 ), n the dimensionless distance beneath channel axis, B the scale depth, s the perturbation coefficient and ya the fractional sound-speed gradient for adiabatic ocean.

Munk (1974) used the following typical values: C1 = 1,492 ms-1, B = 1.3 km, zi = 1.3 km, ya = 1.14 x 10-2km-1 and s = 7.4 x 10-3.

In problems involving acoustic propagation over great distances (greater than 30 nm or 56 km), the fact that the reference plane is situated on a spherical surface can no longer be ignored. The curvature of Earth's surface is roughly equivalent to a linear sound-speed profile. A linear surface duct is closely analogous to a whispering gallery.

First-order spherical Earth curvature corrections are usually applied to the sound-speed profile before any curve-fitting techniques are employed. These corrections are typically of the form (Watson, 1958; Hoffman, 1976)

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