Range (km)

Figure 5.8 Downslope propagation over a constant 5° slope. Results were generated using a PE propagation model (Jensen and Schmidt, 1984).

5.3.4 Empirical models

Two noteworthy empirical algorithms have been developed for use in predicting transmission loss in shallow water, both of which provide depth-averaged estimates for range-independent ocean environments. One model (Rogers, 1981) was derived from theoretical (physics-based) considerations. The second model (Marsh and Schulkin, 1962b; Schulkin and Mercer, 1985), also known as Colossus, was derived from field measurements obtained from a limited number of geographic areas.

5.3.4.1 Rogers model

Rogers (1981) found that virtually all shallow-water TL curves could be described by an equation of the form

where R is the range, and A, B and C are coefficients.

For the case of a negative sound speed gradient (i.e. sound speed decreases with increasing depth), Rogers obtained the following equation:

where R is the range (m), H the water depth (m), P the bottom loss (dBrad-1), 0L the limiting angle (rad) and aw the absorption coefficient of sea water.

The term 15log10 R represents the spreading loss for the mode-stripping regions. Thereafter, the spreading loss corresponds to cylindrical spreading (10log10 R). The limiting angle (Ql) is the larger of Qg or 0c, where Qg is the maximum grazing angle for a skip distance (i.e. the maximum-RBR ray) and Qc is the effective plane-wave angle corresponding to the lowest propagating mode:

where g is the magnitude of the negative sound-speed gradient (s-1), cw the maximum (sea surface) sound speed (ms-1), and f the frequency (Hz).

The bottom loss (P) was derived from the theoretical expression for the Rayleigh reflection coefficient for a two-fluid lossy interface, for small grazing angles. For most cases of interest (small values of Ql), the bottom loss can be approximated as

where Nq = cw/cs, Mq = ps/pw, pw is the density of sea water, ps the sediment density and Ks the sediment attenuation coefficient (dB m" -1 kHz-1).

For example, at a fixed frequency of 200 Hz, Rogers (1981) considered eight different sound-speed profiles for which sound speed decreased monotonically with depth (Figure 5.9). Using Equation (5.3), the coefficients A, B and C were determined for a number of test cases between 5 and 100 km. The maximum deviation between Equation (5.3) and the actual transmission loss curves generated by a normal mode model was also reported (Table 5.2). The sediment properties were based on Hamilton (1980). The intent was to demonstrate the importance of the sound-speed profile (versus the average gradient in the duct) in determining depth-averaged transmission loss. The eight TL curves generated using the coefficients presented in Table 5.2 are plotted in Figure 5.10 to demonstrate the spread in values. At a range of 100 km, the minimum and maximum TL values differ by more than 20 dB even though the overall gradients of all eight sound-speed profiles were the same. These results demonstrate the importance of the shape of the soundspeed profile, and not just the overall gradient, in determining transmission loss in a shallow-water waveguide.

The model by Marsh and Schulkin (1962b), also referred to as Colossus, is an empirical model for predicting TL in shallow water (e.g. Podeszwa, 1969).

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