Noise Source Localization

Unlike the whale species that produce clicks, propeller-driven vehicles produce contin- uous noise. In a shallow water waveguide that supports strong multipath characteris- tics, a whale click will show up in the received waveform as a series of impulses having offsets corresponding to the travel time of each multipath ray. For a noise-producing source, such as a boat, looking at the waveform is useless for discerning anything

about the multipath structure. Noise, by definition, is a sequence of random num- bers, or samples. In other words, the value at any given time is a random variable. If the source produces a random sequence of samples, then the received waveform will be the sum of delayed copies of this sequence. To be precise, the delayed sequence may not exactly match the original sequence due to propagation effects, but the de- layed sequence will be “coherent” with respect to the source. In other words, there is a nonzero degree of predictability between them. Even though the received waveform is the sum of coherent versions of the same sequence, it still appears as a random sequence of samples.

The correlation operation is a very useful tool for analyzing this kind of sequence. Auto-correlation is applied to a single channel, whereas cross-correlation is applied to two different channels. The auto-correlation can thus be thought of as a spe- cialization of the cross-correlation in which the same channel is used both times. Cross-correlation is applied not to absolute points in time, but rather to relative spans of time. In the case of boat noise for which a direct arrival and a sequence of multipath arrivals are being recorded, the cross-correlation operation can expose the coherence between each pair of arrivals. This operates by “compressing” the broad- band noise into a broadband impulse, hence the name “pulse compression.” This will be described in greater detail in the next chapter. However, it is important to point out that this is the underlying mechanism that is used extensively for localization of noise-type sources. Pulse compression, as it is used here, is formally based on the generalized correlation algorithm from Carter and Knapp (1976). Knapp and Carter

(1976) showed that the Cram´er-Rao lower bound on time delay estimation is reached by the generalized cross correlation algorithm.

The advantages of using the correlations of multipath versus only the correlation of direct arrivals have been studied extensively over the past several decades. These investigations have mostly been formulated by deriving the optimal theoretical accu- racy of source location using the Cram´er-Rao lower bound (CRLB) as a metric. The CRLB is the lowest achievable variance of an unbiased estimator, and works by con- sidering how much information about the unknown parameters (i.e. source location) is contained in the measured data (i.e. recorded source noise including multipath). These bounds hold under certain conditions such as no relative target motion, stochas- tic Gaussian signals, and additive uncorrelated Gaussian noise (Rendas and Moura,

1991). It also assumes that all parameters (i.e. depth of seabed reflections, sound speeds of the medium) are fixed and known a priori.

There has been a multitude of work in this area. Target localization using multipath obtained from an autocorrelator has been studied by Ianniello (1986). For multiple hydrophones, the CRLB of the target’s range and depth based on pulse compression of radiated noise from the target was given byFriedlander(1988). Two paths (one direct and one reflected) from each of two vertically aligned receivers to a submerged source were considered. Bandwidth and signal-to-noise ratio (SNR) determined the CRLB for target location. A similar study was conducted by Rendas and Moura (1991).

Abel and Lashkari (1987) investigated using multipath information from multiple hydrophones in different orientations. Yuan and Salt (1993) and Yuan et al. (2000) used simulations to try to characterize the maximum possible efficiency from pre- extracted time delay measurements between vertical line array elements. Lee et al.

(2002) used bearing angles and time-difference of arrival between two multipaths on a bottom-mounted array to perform underwater source localization. Cram´er-Rao analysis has also been applied to matched field processing by Hursky et al. (2004).

In 1968, Van Trees(2001a) expanded the CRLB to include non-deterministic param- eters, in a formulation that is often referred to as the Bayesian CRLB. The term “Bayesian” is used to indicate that prior information about the parameters are be- ing accounted for. This was the basis for the study by Hamilton and Schultheiss

(1993) that examined the performance impact of having imperfect knowledge of the bathymetry by treating the depth of the seabed reflections as normally-distributed random variables. Later studies have expanded on this by applying the CRLB or Bayesian CRLB to gauge the performance limits of trackers that reduce location es- timation variance by modeling the Markov chain nature of measurements observed from moving targets (Dauwels,2005, Lehmann and Williamson, 2007).

Refraction is a difficult phenomenon to model with the CRLB analysis due to the need to obtain partial derivatives of the cross spectrum with respect to parameters that define the refraction (i.e. sound speed profile slope). Franchi and Jacobson (1972) and Rendas and Moura (1990)are two studies that have examined this issue.