## Mathematical models

7.1 Background

Chapter 6 discussed aspects of ambient-noise measurements in the ocean. In particular, Figure 6.2 portrayed the average deep-sea ambient-noise spectra as originating from shipping traffic at low frequencies (~5Hz to ~200Hz) and from surface weather at high frequencies (~200Hz to ~50kHz). In the range 50-500 Hz, both mechanisms (shipping and weather) contribute to the observed noise levels. The significance of these dual mechanisms has influenced the course of noise model developments.

Mathematical models of noise in the ocean can be segregated into two categories: ambient-noise models and beam-noise statistics models. Ambient-noise models predict the mean levels sensed by an acoustical receiver when the noise sources include surface weather, biologics and such commercial activities as shipping and oil drilling. Beam-noise statistics models are more specialized in that they predict the properties of low-frequency shipping noise for application to large-aperture, narrow-beam passive sonar systems. The latter models use either analytic (deductive) or simulation (inductive) techniques to generate statistical descriptions of the beam noise. In this context, beam noise is defined as the convolution of the receiver beam pattern with the sum of the intensities from the various noise sources. The analytic models calculate statistical properties directly from the components (e.g. source level, propagation loss) while the simulation models use Monte Carlo techniques.

### 7.2 Theoretical basis for noise modeling

Mathematical models of noise in the ocean predict both the level and directionality (vertical and horizontal) of noise as a function of frequency, depth, geographic location and time of year. Both categories of noise models (ambient-noise and beam-noise statistics) consist of two components: a transmission loss (TL) component and a noise level and directionality component. In principle, the TL can be computed internal to the noise model or it can be input externally from other (stand-alone) model predictions or from field measurements.

Ambient-noise models treat noise sources as variable densities distributed over large areas. This approximates the generation of wind noise and distant shipping noise. Consequently, the TL calculations in ambient-noise models can be range-averaged (as opposed to point-to-point). This greatly relaxes the accuracy to which TL must be known. Alternatively, beam-noise statistics models treat noise sources (individual ships) as discrete sources, and thus require point-to-point representations of TL.

Empirical regression formulae can sometimes satisfy low-fidelity modeling requirements. The loss of fidelity stems principally from a lack of directionality information (vertical and horizontal) and also from a lack of temporal and spatial resolutions. Moreover, the use of regression formulae to estimate the ambient-noise levels (but not directionality) presumes that the noise levels can be considered independently of the sonar system characteristics. Rigorous noise modeling convolves the system beam pattern (i.e. receiver response) with the calculated noise field levels and directionalities (Wagstaff, 1982).

Sadowski et al. (1984) reviewed regression formulae appropriate for the estimation of average ambient-noise spectra below 100 kHz, including noise sources arising from ocean turbulence, shipping traffic, surface weather (both wind and rain) and molecular agitation. Wagstaff (1973) also presented regression formulae that were incorporated into an ambient-noise model that was valid over the frequency range 10-500 Hz. Ross (1976: chapter 8) reviewed regression formulae appropriate for shipping noise levels. For a brief history of such work, see Ross (1993). Bj0rn0 (1998) summarized the general characteristics of ambient-noise in littoral waters.

Using radiated-noise measurements collected from 272 ships over the period 1986-92, Wales and Heitmeyer (2002) updated the classical merchant ship radiated-noise regression formulae utilized by Ross (1976). These classical regression formulae postulated that the source spectrum for an individual ship was proportional to a baseline spectrum whose constant of proportionality was determined by power-law exponents for the ship speed (sixth power) and ship length (second power). The reanalysis by Wales and Heitmeyer (2002) over the frequency range 30-1,200 Hz now represented the individual ship spectra by a modified rational spectrum. At high frequencies (400-1,200 Hz), most of the individual spectra showed a simple power-law dependence on frequency with exponents concentrated around a mean value of about 2. At low frequencies (30-400 Hz), many of the source spectra exhibited a more complex dependence on frequency with greater spectral variability across the ensemble.

Bradley and Bradley (1984) developed an elaborate empirical model called the geophysics ambient-noise model. This model provided estimates of seasonal deep-water ambient-noise levels and azimuthal directionalities over the frequency range 25Hz-15kHz. This model was based on a comprehensive empirical database of shipping and wind noise, but not noise due to biologics or industrial activity. The model further assumed a nominal receiver depth of 100 m, thus ignoring any depth dependence. Hamson (1997) reviewed techniques for modeling shipping and wind noise over the frequency range 50-3,000 Hz, concentrating mainly on work performed after 1980. Noise level, horizontal and vertical directionalities, and the noise responses of arrays were used to characterize the ambient-noise field. Harrison (1996) used a simple ray approach to approximate the full-wave treatment of noise levels and coherence in range-independent ocean environments. Alvarez et al. (2001) used an approach based on genetic algorithms to study the physical characteristics of measured underwater ambient noise in the frequency range 10-2,000 Hz. The resulting predictability of the recorded signals was attributed, in part, to the contributions of shipping noise.

### 7.3 Ambient-noise models

A simple model of ambient noise in the ocean would consist of an infinite layer of uniform water with a plane surface over which the sources of noise (shipping and weather) were uniformly distributed. In this model, the ambient-noise level would be independent of depth (Urick, 1983: chapter 7). Of course, a more realistic model would include volume absorption in addition to the effects of refraction and boundary reflections over long-range paths (Talham, 1964).

To compute the low-frequency component of noise due to distant shipping, three inputs are required: (1) the density of shipping as a function of azimuth and range from the receiver; (2) the source level of the radiated noise for each generic type of merchant ship; and (3) the TL as a function of range between the near-surface sources (ships) and the depth of the receiver. Then, contributions from successive range rings centered about the receiver can be summed to obtain the level of shipping noise as a function of azimuth at the receiver. The theory behind this kind of modeling, in addition to some observational data on the density of shipping traffic in the North Atlantic, has been described by Dyer (1973), among others.

The high-frequency component of noise due to surface weather is usually computed on the assumption that it is locally generated and isotropic. Thus, only the weather conditions (sea state or wind speed) prevailing in the immediate vicinity of the receiver need be considered, in addition to any localized rain shower or biologic activity.

Modeling the vertical directionality of deep-water ambient-noise can be approached by means of the simple model mentioned earlier. Consider a bottomless, uniform ocean, without refraction or attenuation, having a surface covered with a dense, uniform distribution of noise sources. Furthermore, let each unit area of the surface radiate with an intensity I(9) at a distance of 1 m. Then, at point P (the receiver) in Figure 7.1(a) the incremental intensity dl produced by a small circular annulus of area dA at horizontal range r is (Urick, 1983: chapter 7):

JsZ.