sin n0 are very small. Using more than two sets of observations ameliorates the problem, but cannot 

 eliminate it completely. 



A Constrained Estimate 



The minimum-bias estimator having the least mean-square error of Ref. 1 is actually the zero-bias 

 estimator of expression 18. Unfortunately, expression 18 may be negative for some values of d. One 

 way to relieve the implausibility of the resulting solution is to add an arbitrary function of the form of 

 Eq. (19), but wath coefficients bg, bi2, etc., chosen to minimize the negative excursions of the estimator. 

 This is done in Ref. 1. Wagstaff, in Ref. 5, rightly criticizes the remedy as ineffectual. The resulting 

 solution "looks" nicer, but it is no better cui approximation to the true distribution in several examples. 

 There are anjilogous instances in signal detection where invoking similar constraints results in a real im- 

 provement, but Ref. 1 fails to show any reason why improvement should result m the present case, and 

 Ref. 5 shows examples where it does not. 



OTHER USEFUL ESTIMATORS 



References 6 £md 7 describe two other ancdytical methods of resolving ambiguities in noise fields 

 which are not ejisily described in terms of the minimization of some error functions. These are, respec- 

 tively, the hot-spot method and the BAR method. Both of these are defined by algorithms rather than 

 by minimization processes, and both of them are nonlinear. Inasmuch as they jure nonlinear, analyses in 

 terms of Fourier components of the spatial distribution are not appropriate. 



Hot Spot Analysis 



To a first approximation, the hot-spot method matches the observations as well as possible with 

 plane waves arriving from a small number of points or directions (called hot-spots, from which the 

 algorithm gets its name). The fit with the observations is improved by having more and more hot-spots 

 of progressively lower intensities. Within the limits of consistency of multiple observations, the fit can 

 be made arbitrarily good, for any field whatsoever can be represented by a continuum of plane waves. 



The hot-spot solution suffers from two potential disadvantages. First, the criterion of fit is to 

 match the measured values, and as indicated, it may be possible to match the measurements very closely 

 with 8in estimator which is very far from the true field. Second, the solution is sometimes used to 

 justify the assumption that noise comes from a small number of hot-spots. Such an argument is circular 

 and must be validated with additional observations. 



The hot-spot method was originally designed for use under circumstances where observations were 

 ' made with an array at two or more geographic locations, with the possibility of a different cirray heading 

 at each location. This is different from the situation postulated in this paper, where all observations are 

 made at the same location but possibly different array heading angles. When different positions axe 

 available for observation, the hot-spot method applied to a large number of observations should give not 

 only an unambiguous resolution of direction, but an unambiguous distribution of noise sources in two 

 dimensions. However, that is beyond the scope of this discussion. 



5 

 R. A. Wagstaff, A discussion and comparison of five methods utilizing the Towed Array to Assess the Azimuthal Direc- 

 tionality of Ambient Noist Naval Underseas Center Report TP-374, (Jan. 1974) 



J. P. Holland, T. F. Cannan, Jr., J. J. Hanrahan, and J. B. Bairstow, Preliminary Results of Azimuth Directionality 

 Measurements of Ambient Noise Ob.'served During EASTLANT II, 6-14 August 1972 NUSC Technical Report 4431 



(27 Sept. 1972) ■■ 



J. P. Holland, T. F. Cannan, Jr., J. B. Bairstow, and F. E. Rembetski, EASTLANT II Ambient Noise Study, Vols. 1 

 and 2 NUSC Technical Reports 4487-1 and 4487-2, (1 Jan. 1973) and (13 Aug. 1973) 



901 



