Array modeling of active sonar clutter

Transcription

1 1 Array modeling of active sonar clutter Douglas A. Abraham CausaSci LLC P.O. Box 589 Arlington, VA 5 Published in IEEE Journal of Oceanic Engineering Vol. 33, no., pp , April 8 Abstract Active sonar systems operating in shallow water environments often deal with excessive false alarms, generically referred to as clutter, that are more numerous than expected for Rayleigh-distributed reverberation. The clutter probability density function, and therefore the probability of false alarm P fa, depends on the scattering sources, propagation conditions, sonar system and signal processing. Clutter statistics are often approximated by the K distribution where the shape parameter α provides an inverse relationship to P fa with decreases in α representing an increase in P fa. In this paper, the effect of sonar array processing on clutter statistics is evaluated by first modeling clutter at the hydrophone level and then analyzing the resulting K-distribution shape parameter after conventional beamforming. When the transmit waveform bandwidth is narrowband with respect to the array processing and propagation consists primarily of low-angle paths, α was found to be separable into the product of a cluttersource scattering effect, an array-processing effect, and a coupled transmit-waveform and propagation effect. The array effect was found to coarsely follow the array beamwidth, although precise evaluation is straightforward given the array beampattern. As might be expected, array design or processing that tends to increase the beamwidth was found to increase α. Uniform shading was seen to provide a practical, though not exact, lower bound on α for common array shading functions. For circular arrays with unaliased spatial sampling, an asymptotic beampattern was found to provide a very accurate approximation to α. These results should be useful in predicting the performance of sonar systems in clutter dominated areas, in the design of arrays and array processing, the inversion of clutter data for clutter-source scattering parameters, or to improve signal processing algorithms aiming to reduce or control clutter-related false alarms. Index Terms active sonar, clutter, array processing, beamforming, K distribution I. INTRODUCTION AND BACKGROUND False alarms in active sonar systems have traditionally been assumed to arise from reverberation comprising a multitude of individual scatterers producing a matched filter envelope following the Rayleigh probability density function PDF. However, with increasing transmit waveform bandwidth and array size, the number of independent scatterers within a sonar resolution cell can be small enough to void the central Manuscript received June 9, 7; revised February, 8; accepted February 1, 8. First published October 1, 8; current version published October 31, 8. This work was supported by the U.S. Office of Naval Research under Contract N14-7-C-9. Associate Editor: K. E. Wage. limit theorem argument that leads to the Rayleigh-distributed envelope. Empirical statistical analysis of reverberation and clutter data has identified several dependencies of the envelope PDF on the environment and sonar system including bottom backscattering [1], [], [3], [4], [5], [6], multipath [7], [8], transmit waveform bandwidth [9], [8], [1], and array beamwidth [11], [1], [13], [14]. These data analyses predominantly entail fitting measured data to a variety of PDF models e.g., Weibull, log-normal, Rayleigh mixtures, and the K distribution, although [1], [1], [5], [14], [9] additionally derive models to explain the observed dependence. In this paper, a model is developed for clutter measured at an array of hydrophones and used to characterize and analyze the modeled statistics of clutter after conventional beamforming. The aforementioned real data analysis and intuition relating the sonar resolution cell size to clutter statistics indicates that, as the array beamwidth narrows, the clutter envelope PDF tail becomes heavier, resulting in an increase in the probability of false alarm P fa. The model proposed by McDaniel [1] articulates this dependence by relating the half-power array beamwidth to the degrees of freedom of one component of a multiplicative clutter model. A similar assumption has been made in [14] where the shape parameter of the K distribution [15] is taken as proportional to the array beamwidth. These models essentially require an assumption that the beampattern is a rectangular function in angle with width equal to the prescribed beamwidth. This type of assumption has been shown to yield inaccurate prediction of the effect of the transmitwaveform autocorrelation function ACF on clutter statistics [16], potentially challenging the accuracy of modeling clutter statistics on array beamwidth. As such, the primary goals of this paper are to build intuition and provide more accuracy in predicting how array processing affects clutter statistics. The basis assumed in this paper for representing clutter statistics after matched filtering and beamforming is the K distribution [15]. This choice is supported by empirical studies showing that a wide range of sonar reverberation and clutter e.g., see [6], [17], [18], [14], [7] is accurately described by the K distribution. Additionally, from a theoretical modeling perspective, the K distribution can arise from scattering from certain types of rough surfaces [19], [], [1, Ch. 7] or from a finite number of exponentially sized discrete scatterers or patches [14]. Coupling this with the results of [16] where it was shown that the K distribution provides a reasonable

2 approximation to clutter statistics affected by propagation using a limit-distribution argument argues for approximating the beamformed, matched filtered clutter time series PDF by a K distribution with parameters found by matching the first and second moments of the matched filter intensity. This paper is organized as follows. In Sect. II the model derived in [16] to account for the effect of multipath propagation on clutter statistics is extended to represent clutter received by an array of hydrophones. The statistics after conventional beamforming are then characterized by a K distribution with parameters obtained through the aforementioned limitingdistribution/moment-matching approximation. Simplifications for isotropic scattering conditions and narrowband transmit waveforms are derived in Sect. III. The remaining sections are devoted to evaluating the effect of array processing on clutter statistics for different array types with a focus on the shape parameter of the K distribution owing to its relationship to the sonar P fa. Equi-spaced line arrays are considered in Sect. IV along with the effects of array shading and coupled multipath and array processing effects. The effect of directional sensors or composite arrays on clutter statistics is evaluated in Sect. V using a line array of triplet hydrophones or vector sensors as examples. Finally, circular arrays with beamforming in the same plane as the array are considered in Sect. VI. II. ARRAY MODELING OF CLUTTER For a single omnidirectional receiver placed at depth z = d and at the origin in the horizontal x, y plane and a collocated, omnidirectional source, suppose that the clutter response is divided into that arriving from infinitesimally small angular sectors of width dφ emanating from the origin in x, y and from scattering in a given plane e.g., the bottom z = d b or the surface z = as illustrated in Figs. 1 and. For environments that may be characterized by a linear relationship between the scattering sources and the signal received at the single sensor, the response from the sector centered at azimuthal angle φ after matched-filtering and basebanding may be described as x φ τ = h τ rṽ φ r dr 1 where r is horizontal range, ṽ φ r represents the scattering of the clutter sources and h τ r is an impulse response representing both the effect of matched filtering and acoustic propagation for delay time τ. A tilde denotes the basebanded or complex-envelope version of a signal with the exception of α and λ definined later in this section where it indicates approximation. From the development of [16], h τ r may be described as h τ r = a i Rs γ i r i r where R s τ is the basebanded ACF of the transmit waveform, a i, r i, θ g,i are amplitude, horizontal range, and grazing angle associated with the ith of the p multipath contributing to clutter from the source described by ṽ φ r at delay time τ, and γ i = cos θ g,i /c w where c w is the speed of sound. As seen in Fig. 1, these are the paths originating at the parts of the bottom such that they all arrive at time τ. Note that the characterization of assumes an isospeed environment and appropriately accounting for any non-zero Doppler in the matched filtering; however, extensions to more general situations should follow readily. Fig. 1. Schematic of the environment. The multipath used to describe clutter statistics represent the contributions from various parts of the bottom or other clutter source that arrive at the same delay time. Hybrid paths are not shown. Development of a model for clutter as received on an array of hydrophones is most easily done in the frequency domain, therefore the dependence of on delay time τ must be articulated to allow evaluation of the Fourier transform of 1. For an isospeed environment, this may be done by first defining the horizontal range r = c wτ cos θ g 3 where θ g is in the range of the grazing angles of the paths contributing to reverberation at time τ e.g., the average grazing angle or the one associated with the dominant path. Equation may then be rewritten as h τ r = a i Rs ν i τ τ i γ i r 4 where ν i = γi γ cos θ g, γ = cos θ g /c w, and τ i are delay offsets defined as τ i = γ i r r i 5 Delay offsets are used in order to describe the dependence of h τ r on delay time in such a way that eqs. 1 and may be approximated as time-invariant in the vicinity of τ = r/ c w cos θ g which enables evaluation of the Fourier transform of 1. If the multipath characterization a i, r i, θ g,i changes slowly with τ over the combined extent of the ACF width approximately 1/W where W is the waveform bandwidth and maximum delay offset, then in analyzing the clutter for a specific delay time τ these parameters may be assumed to be constant leading to time invariance. This is a reasonable assumption for moderate to long delays in a shallow water environment where the paths contributing to clutter have shallow grazing angles. In [16], eqs. 1 and were derived for scattering from the bottom; however, with the appropriate choice of a i, r i, θ g,i the relationship is equally valid for scattering from other boundary reverberation sources

3 3 such as a layer of bubbles near the ocean surface or from a school of fish with minimal vertical extent. Scattering fields with large vertical extent should include an additional integral in 1 over depth. The Fourier transform of 1 can now be derived under the assumption that a i, τ i, ν i, and γ i are constant and that ṽ φ r = for r <, X φ ω = = = Ṽφ e jωτ x φ τ dτ S ω/ν i a i ν i e jωτi/νi ωγ/ cos θ g p ] e jωγir/νi ṽ φ r dr [ S ω/ν i a i ν i e jωτi/νi 6 where S ω is the basebanded version of the transmit waveform spectrum i.e., the spectrum is shifted down so the center of the band is at zero Hertz and then low-pass filtered. Extending 6 to a vector formulation representing the signal received on an array of n hydrophones in the farfield of the clutter scattering results in X φ ω = Ṽφ ωγ/ cos θ g S ω/ν i a i ν i e jωτi/νi φ, θ a,i, ω where θ a,i is the arrival angle in elevation of the ith path at the receiving array and d φ, θ, ω is the array steering vector [, Sect..] pointing in direction φ, θ with the phase center of the array at the origin in x, y. The array steering vector imparts the phase adjustment corresponding to the time delay of the wave propagating from the origin to the location of each element. This assumes that variations in the multipath characterization are small over the extent of the array. Note that the array data, steering, and beamforming vectors in this paper have dimension n-by-1 and are denoted by bolded variables. Finally, superimposing the response from all angular sectors results in the total array response in the form of the Fourier transform of the array data vector Xω = = π π X φ ω dφ Ṽ φ ωγ/ cos θ g S ω/ν i a i ν i e jωτi/νi d φ, θ a,i, ω dφ. For large delay times in a shallow water environment, the grazing angles will be quite low resulting in the approximations ν i 1, cos θ g 1, θ a,i, and { π } Xω S ω Ṽ φ ωγ a i e jωτi d φ,, ω dφ where, in general, the multipath amplitudes and delays will vary with azimuth. From this frequency-domain characterization of the data received on an array, it can be seen that the effect of the transmit waveform is fully separable while the bottom scattering, array sensing, and multipath are separable within an angular sector. A. Conventional beamforming While 8 represents the clutter as received on an array of hydrophones and will be useful for purposes such as hydrophone-level clutter simulation, determining the effect of beamforming on clutter statistics is of primary interest as it more directly dictates the sonar s false alarm performance. An expectation supported by real data analysis [11], [1], [14] is that clutter for arrays with narrow beamwidths is heavier tailed than that for systems with broader beams owing to the smaller number of scatterers contributing to the beam time-series. In particular, it has been assumed that the shape parameter of the K distribution is proportional to the beamwidth of the sonar [14]. This is akin to the rectangular approximation to the transmit waveform ACF used in [16] which was seen to be accurate, with a correction factor, for large time-bandwidth product transmit waveforms. The array model of the previous section enables the following analysis of the statistics of the beamformer output which will be seen to support the expectation derived from real data analysis and provide greater accuracy than the simple rectangular beampattern approximation. Applying a frequency-domain conventional beamformer pointing in direction ψ = ψ a, ψ e azimuth and elevation 1 angle as illustrated in Fig. with weight vector w ψ, ω to the array data results the frequency-domain beam output Ỹ ψ ω = w H ψ, ω Xω. 1 Uniformly shading the array results in setting the beamforming weight vector to the array steering vector, w ψ, ω = d ψ a, ψ e, ω, while non-uniform shading is represented by multiplying the steering vector by a diagonal matrix of weights, w ψ, ω = diag {w 1, w,..., w n } d ψ a, ψ e, ω. 11 In many situations, including the examples of this paper, beamforming is only done in the horizontal i.e., ψ e = owing to either a lack of vertical directionality in the sonar receiver or an interest in longer ranges. The clutter component of the beam output is obtained by inserting 8 into 1 and then may be simplified to the form Ỹ ψ ω = π Ṽ φ γω cos θ g Cψ,φ Ω dφ 1 1 In beamforming, the elevation angle represents the angle above horizontal at which the beam is pointed. It is similar in concept to the departure angle of a ray in acoustic propagation and, in the case of an isospeed, range-independent environment, would also be related to the grazing angle of a path at the bottom.

4 4 Fig.. Schematic of a line-array receiver place horizontally in the water column. The azimuthal beam-steering angle is measured from broadside to the array i.e., the y axis while the elevation angle is measured from horizontal. An elemental azimuthal scattering sector is shown with width dφ in the topdown view shown in part a. where Ω = ω/ cos θ g and C ψ,φ Ω = S γω/γ i a iγ e jωτiγ/γi γ i w H ψ, Ω cos θ g d φ, θa,i, Ω cos θ g. 13 The beam output time series can be described by applying an inverse Fourier transform to Ỹψω and replacing Ṽφ γω in 1 with the Fourier transform definition, accounting for the scale γ, ỹ ψ τ = = 1 π π π where c ψ,φ t is the inverse Fourier transform of 13 and it has again been exploited that ṽ φ r = for r <. It is convenient to view the function c ψ,φ t as a spatio-temporal template with which the bottom is probed by the sonar system accounting for the spreading induced by multipath. As in [16], it is assumed that ṽ φ r is a K-distributed random process with scale parameter λ r, φ and shape parameter α r, φ dr dφ = 1 β r, φ r dr dφ 15 where β r, φ is the number of scatterers per square meter. The factor of dividing β r, φ arises from assuming a finite number of discrete scatterers with an exponentially distributed size [14]. As derived in [16] through a limit-distribution argument, the PDF of the envelope of ỹ ψ τ is assumed to be closely approximated by the K distribution owing to its composition as the integral of a weighted K-distributed random process. Extending the results of [16, eqs. 33 and 34], which provide the shape and scale parameters for the K-distribution approximation to the PDF of the envelope of x φ τ as in 1, to account for the integration over φ found in 14 results in the following forms for the shape α and scale λ parameters of the K-distribution approximation to the statistics of ỹ ψ τ, [ π α r, φ λ r, φ M r, φ dr dφ] and α = where λ = = 1 π α r, φ λ r, φ M r, φ dr dφ [ π β r, φ λ r, φ M r, φ r dr dφ] π π π β r, φ λ r, φ M r, φ r dr dφ 16 β r, φ λ r, φ M r, φ r dr dφ β r, φ λ r, φ M r, φ r dr dφ 17 M r, φ = cψ,φ τ cos θ g γr. 18 As eqs. 16 and 17 can be derived from moment equations see [9], they are useful in characterizing the statistics of ỹ ψ τ even if the clutter source scattering or resulting beamformed, matched filter envelope data are not precisely K distributed; large values of α represent a nearly Rayleigh distributed envelope while small values of α represent heavier tailed clutter. A. Isotropic scattering III. SIMPLIFICATIONS Assuming that the clutter source scattering is isotropic i.e., spatially stationary or at least slowly varying over the spatial extent of the multipath spreading and some angular extent containing the main beam and perhaps first sidelobes of the e jωγr ωτ dω array processing, it is reasonable to approximate the scatterer Cψ,φ Ω cos ṽ φ r dr dφ θ g spatial density β r, φ and individual scatterer power λ r, φ c ψ,φ τ cos θ g γr as constant in the vicinity of r, ψ in eqs. 16 and 17. ṽ φ r dr dφ 14 Further assuming that τ cos θ g is significantly greater than the temporal extent of c ψ,φ t, which is nominally 1/W plus the spreading induced by multipath and the time required for propagation across the array, results in α β r, ψ and cw τ λ λ r, ψ c w { π π [ π c ψ,φ t dt dφ] π c ψ,φ t 4 dt dφ 19 c ψ,φ t 4 } dt dφ c ψ,φ t. dt dφ

5 5 Note that the term encompassed in braces in 19 will have units of meters while that in is nondimensional. Equations 19 and characterize the higher order moments of clutter after matched filtering and beamforming in terms of simple descriptions of the scattering of a clutter source β r, φ and λ r, φ, the environment-induced multipath, transmit waveform, receiving array, and array processing with these latter four entering into the formation of the spatio-temporal template function c ψ,φ t as described in the Fourier domain in 13. The remainder of the paper focuses on simplifying and evaluating α as it characterizes how the array processing affects the tail of the clutter PDF, which directly impacts the probability of false alarm and is therefore of primary interest. B. Narrowband transmit waveforms When the array and transmit waveform are such that a narrowband assumption can be made [3, Sect. 8.3], [, pg. 34], the steering vector may be fixed to that for the center frequency of the transmit waveform ω c and assumed constant over frequency which simplifies C ψ,φ Ω in 13 to C ψ,φ Ω = where S γω/γ i a iγ γ i e jωτiγ/γi b ψ, φ, θ a,i 1 b ψ, φ, θ = w H ψ, ω c d φ, θ, ω c is the array beam pattern note that this is an unsquared beampattern. This results in the spatio-temporal template function having the form of a sum of weighted and delayed transmit-waveform ACFs, c ψ,φ t = a i b ψ, φ, θ a,i R s tγ i /γ τ i. 3 At short range relative to water depth, variations in the grazing and arrival angles over the different paths can significantly impact the clutter statistics. An example illustrating the use of 3 and the coupling of the effects of propagation and array processing on clutter statistics for an equi-spaced line array may be found in Sect. IV-C. At long ranges in shallow water environments, when the grazing and arrival angles for the different paths can be assumed small i.e., γ i /γ = cos θ g,i / cos θ g 1 and θ a,i, 3 simplifies further to c ψ,φ t b ψ, φ a i Rs t τ i 4 where b ψ, φ = b ψ, φ,. This approximation may also be appropriate for shorter ranges when the receiving array has low sensitivity to changes in arrival angle. As noted before, and worth repeating, the multipath parameters a i, τ i, θ g,i, θ a,i are assumed to be fixed with respect to the variable t in 3 and 4. Inserting 4 into 19 results in the beam-output shape parameter α β r, ψ cw τ cw [ π b ψ, φ dφ where = β r, ψ p a R i s t τ i dt p a R i s t τ i 4 dt [ ] cw Ks W 5 π b ψ, φ 4 dφ cw τ K a n [ π b ψ, φ dφ K a n = π b ψ, φ 4 dφ ] 6 represents the effect of array processing and may be viewed as an effective array beamwidth and [ p a R ] i s t τ i dt K s W = p a R i s t τ i 4 7 dt represents the effects of both the transmit waveform and propagation. When there is only one path or if the transmitwaveform ACF is too coarse i.e., the waveform bandwidth is too small to resolve the paths, 7 will simplify to [ R ] s t dt K s W K s W = R s t 4 8 dt which is identical to [16, eq. 37]. Note that K a n, Ks W, and K s W are scale invariant; that is, a multiplicative scale applied to the beampattern, ACF, or multipath amplitudes will not alter their value. The form of 5 illustrates that the effects of matched filtering and beamforming on the clutter statistics are separable when evaluated through the equivalent K-distribution shape parameter, confirming intuition arising from the linearity of the two operations. However, the effect of the transmit waveform and propagation remain entwined. Intuition indicating an approximate dependence of α on the area of the sonar resolution cell may be substantiated by rewriting 5 as α = β r, ψ [ ] cw Ks W [ rk a n] cos θ g 9 which is the product between the scatterer spatial density normalized by to account for exponentially sized discrete scatterers and the effective area of the sonar resolution cell. The latter is formed as the product of the cross-range extent the quantity in the first set of brackets formed by the horizontal range times an effective array beamwidth and an effective range extent the quantity in the second set of brackets accounting for multipath propagation, grazing angle, and the transmit waveform. ]

6 6 When the transmit-waveform ACF and the array beampattern are approximated by rectangle functions with, respectively, width 1/W and φ BW and propagation consists of a single path, the shape parameter simplifies to α = β r, ψ c w r φ BW 4W cos θ g. 3 As shown in [16] for the transmit waveform ACF, the rectangle-function approximation is not alway accurate. However, it confirms intuition that the effect of array processing on clutter statistics is dominated by the array beamwidth and reinforces interpreting K a n as an effective beamwidth of the array with respect to clutter statistics. Explicitly, the definition of a narrow-band waveform relative to array processing exploited in this paper is that the array beampattern bψ, φ, θ be constant over the band of the transmit waveform. Clearly this is never precisely true, leading to confusion over when a transmit waveform may be considered narrowband. As noted in [4] the definition should not be rooted in the assumptions, but on the ensuing analysis. As such, answering this question requires the evaluation of α using the broadband definitions i.e., 19 using the inverse Fourier transform of 13 to determine when 5 is an adequate approximation. This task is beyond the scope of the present paper, but the topic of future research. In the following sections, the effect of array processing on clutter statistics under the narrowband assumption K a n, is evaluated for a variety of array configurations and array processing situations. The independent variable n is chosen here for the array effect K a n as it is most often the number of sensors that dictates the size of an array and therefore the array beamwidth, although it should be noted that there are other dependencies. IV. EQUI-SPACED LINE ARRAYS The effect on clutter statistics from beamforming a line array is considered in this section. Except where noted otherwise, it is assumed that the array is placed horizontally in a shallow water environment as illustrated in Fig., is comprised of n sensors spaced every half-wavelength, and that beamforming is performed in the horizontal i.e., ψ e = and ψ a ψ. In Sect. IV-A an approximation to K a n is derived when the beamformer has uniform shading and compared with various beamwidth-based approximations. The effect of nonuniform shading is examined in Sect. IV-B and the coupling of the effect of multipath and array processing is illustrated with an example in Sect. IV-C. A. Large-n approximation Consider an equi-spaced line array of n hydrophones with inter-sensor spacing d. From [, eq..131], the unsquared beampattern with a uniform weighting on the array is b ψ, φ = 1 sin πnd λ n sin 31 πd λ where λ = cw f is the wavelength and = sin φ sin ψ 3 with the azimuthal angles being defined as zero at broadside to the array see Fig.. Describing the beampattern as a ratio of sinc functions = sin µ µ affords the following large-n approximation b ψ, φ = sinc πnd λ πnd sinc sinc. 33 πd λ λ which, when substituted into the integrals in 6 results in the need for the integral π b ψ, φ k dφ = π/ π/ = λ πndλ 1 1 sin ψ πnd πndλ 1 1+sin ψ 4λ πnd cos ψ b ψ, φ k dφ 1 k sin µ µ dµ sin ψ + λµ πnd k sin µ dµ, 34 µ for k = and 4 where the simplification exploits the transformation µ = πndλ 1 sin φ sin ψ, assumes n is large and ψ π/, with an expectation of greater accuracy when ψ is near zero. Noting that sin x/x dx = π/ and sin x/x 4 dx = π/3 [5, pgs. 431 and 463], the largen approximation to the array processing effect may be derived as K a n 3λ nd cos ψ 35 which is 3 divided by the effective length of the array measured in wavelengths. Figure 3 shows the large-n approximation to K a n in comparison with the exact value obtained via numerical integration for an equi-spaced line array with half-wavelength inter-sensor spacing. Note that although K a n has units of radians, the figures will show the value in degrees. The approximation is quite good even for arrays as small as 16 elements where the error is less than 5%. Approximating the beampattern b ψ, φ by a rectangular function in bearing with beamwidth φ bw ψ, n results in K a n = φ bw ψ, n, which emphasizes that the units of K a n are radians. However, as can be seen in Fig. 3, neither the 3-dB nor 6-dB beamwidths provide an accurate approximation to the array processing effect for a uniformly-weighted, equi-spaced line array. As discussed in Sect. IV-B, this is most likely caused by the relatively high sidelobes produced by the rectangular shading. The beamwidths shown in Fig. 3 are computed numerically as the total angle over which the beampattern is within 3 or 6 db of the peak value note that since bψ, φ is the unsquared beampattern, the conversion to db takes the form 1 log 1 bψ, φ. The large-n and rectangular-beampattern approximations are evaluated as a function of steering direction in Fig. 4 for n = 64 where the large-n approximation is seen to be quite good out to 6 where the error is just under 5%. Near endfire e.g., ψ > 8, the 6-dB beamwidth provides the best approximation with error less than 5%. For smaller values of n, the bearing at which the large-n approximation deteriorates decreases e.g., for n = 3 the error is less than 5% for

7 7 bearings 5 and the region near endfire where the 6-dB beamwidth provides the best approximation increases e.g., for n = 3, it is approximately ψ > 75. For example, a Hann window cosine squared applied to an equi-spaced line array results in the expected decrease in K a n with n as seen in Fig. 5 where it is compared with the 6-dB beamwidth. In this example and the following one, the zero-valued endpoints of the Hann window are discarded yielding the shape of an order n+ Hann window. In contrast to the results of a uniform array weighting shown in Figs. 3 and 4, the 6-dB beamwidth as computed from the Hannshaded array beampattern is a reasonably accurate predictor of K a n with error less than 5% for large enough n and steering directions near broadside. This appears to be a result of the lower sidelobes produced by the Hann window 31.4 db down as opposed to the uniform weighting for which the sidelobes are 18.4 db higher. Evaluation of other windows with relatively high sidelobes e.g., the Tukey window and low sidelobes e.g., the Hamming window confirms this correlation. The lower sidelobes essentially reduce the error involved in approximating the beampattern as a rectangular function in azimuth with width equal to the 6-dB beamwidth. Fig. 3. Effect of array processing on α, K an, for a uniformly shaded line array of n hydrophones spaced every half wavelength steered to broadside ψ =. Neither the 3- or 6-dB beamwidths are as accurate as the large-n approximation. Fig. 5. Effect of array processing on α for a Hann weighted conventional beamformer for an array of n hydrophones spaced every half wavelength steered to various angles. The 6-dB beamwidth is seen to provide a good approximation, especially for large n and near broadside to the array. Fig. 4. Effect of array processing on α as a function of steering angle ψ for a uniformly shaded line array of n = 64 hydrophones spaced every half wavelength. Steering away from broadside in this example can cause up to an eightfold increase in α, which could significantly reduce P fa, though at the expense of a commensurate reduction in directivity. The large-n approximation degrades as ψ departs broadside and fails near endfire while the 6-dB beamwidth improves near endfire. B. Effect of array shading In practice, arrays are typically shaded with a window function to reduce the effects of sidelobes. Though it may be feasible to analytically evaluate 6 for certain windows, it is straightforward and simple to evaluate it numerically for any window function. This is especially true for equi-spaced line arrays where the beampattern can be efficiently evaluated using a fast Fourier transform. In general, there is a trade-off between sidelobe height and mainlobe width in the design of window functions [6], which might lead one to suspect that any use of a window that widens the mainlobe would result in an increase in K a n compared with an unweighted aperture. Such an increase implies a lightening of the clutter PDF tails compared with the uniformly weighted array. To investigate this possibility, the array effect for window functions formed from discrete prolate spheroidal sequences DPSS [, eq. 3.35] is evaluated. The DPSS windows essentially focus the beampattern within a certain region about the main response axis of the beamformer. When this spatial bandwidth is zero, the window simplifies to uniform weighting. Increasing the spatial bandwidth monotonically increases the null-to-null beamwidth and monotonically decreases the maximum sidelobe level. Despite this, as seen in Fig. 6, the array effect can result in a small decrease from that for a uniform window when the focusing window is small,

8 8 nominally within the null-to-null beamwidth. As a side note, the DPSS beamwidth φ bw shown in Fig. 6 is related to ψ in [, eq. 3.35] by φ bw = arcsin ψ λ/πd. Fig. 6. Change in the equivalent shape parameter between discrete prolate spheroidal sequence DPSS beamforming weights and a uniformly weighted array as a function of DPSS beamwidth. The null-to-null beamwidth of the uniformly weighted array is shown as an approximate demarcation point above which the DPSS weighting produces lighter clutter PDF tails than the uniformly weighted array. The equivalent shape parameter ratio is also shown for several common window functions which are all seen to result in lighter clutter tails than the uniform weighting. Clearly, the unweighted equi-spaced line array does not provide a lower bound on the effect of array processing on clutter statistics. However, for most practical windows it is reasonable to use it as a lower bound. This is evidenced in Fig. 6 for the Hann order n +, Hamming, triangular and Tukey windows by noting where on the DPSS window function curves the value of the equivalent shape parameter ratio for these windows lies. It may also be seen by comparing the ψ = line in Fig. 5 for the Hann window to the results for uniform shading seen in Fig. 3. The Hann-shaded array results in K a n larger than the asymptotic approximation of 35 for all n and larger than the exact value for all n > 3. The discrepancy at n = 3 arises from the Hann order n + window essentially not having any sidelobes. When n = there is no difference as the shadings are equivalent. Thus, approximations to the array beamwidth that depend on the array shading e.g., as described in [, Sect. 3.1] will be useful in incorporating the effect of array processing on clutter statistics into sonar performance prediction tools such as tactical decision aids where many evaluations might be required and extreme precision is not necessary. In many such situations, the large-n approximation of 35 should prove adequate arguing that the uniform weighting provides an approximate lower bound for other window functions. However, for situations where computational burden is not an issue or precision is paramount, numerical evaluation of 6 is preferred. C. Multipath example The combined effect on α of the environment-induced multipath, matched filtering, and array sensing and processing may be evaluated through eqs. 3 and 19. In this section an example is used to highlight the potential coupling of the array-processing effect and the effect of multipath. The example is not intended to represent a precise evaluation of multipath conditions but merely to illustrate the potential coupling of these two effects. Consider a horizontal line array where the array s conical beams cause multipath with non-zero elevation angles to arrive at an azimuthal angle closer to broadside than their angle of origin. This phenomenon is expected to increase α for certain steering directions owing to the high-elevation-angle multipath contribution of azimuthal angles toward endfire. Similarly, a reduction in α might be expected when steering close to endfire where the high-angle paths don t enter into the beam. The effect is expected to lessen as the steering angle nears the array broadside where all multipath enter the same beam or as range increases where the elevation angles contributing to clutter become shallow. To illustrate this effect and evaluate how significant it might be, consider an isospeed, range-independent environment with isotropic scattering, a co-located source and receiver placed in the middle of the water column, and restrict multipath to the direct path and the upward-going path with only one surface reflection i.e., source-surface-bottom clutter-surface-receiver. Using the rectangular approximation to the transmit waveform ACF R s τ = rect W τ, and evaluating 3 at t/w results in c ψ,φ t/w = a b ψ, φ, θ rectt +a 1 b ψ, φ, θ 1 rect t τ 1 36 where a i, θ i are the amplitude and grazing angle which is also the elevation angle of the path at the receiver of the direct path i = and the surface reflected path i = 1. Through a change of variables in 19, it can be shown that integrating c ψ,φ t/w as opposed to c ψ,φ t results in a factor of W in the denominator of α and allows characterizing the delay of the surface path with respect to the direct path τ 1 = W τ 1 in 36 in terms of the bandwidth-to-center-frequency-ratio, water depth in terms of wavelengths d b /λ c, and horizontal range in terms of water depths r d = r/d b, τ 1 = W τ 1 = W f c d b λ c r d 4 r d + 1/4 [ ] r d r d 4 4 r d W f c d b λ c 37 Evaluating α requires the integral c ψ,φ t/w k dt = [a b ψ, φ, θ ] k min 1, τ 1 + [a b ψ, φ, θ + a 1 b ψ, φ, θ 1 ] k 1 τ 1 U 1 τ 1 + [a 1 b ψ, φ, θ 1 ] k [1 + τ 1 max 1, τ 1 ] 38 for k = and 4 where the step function Ux = 1 for x and is otherwise zero. Thus, the horizontal range, water depth and bandwidth-tocenter-frequency ratio dictate the normalized path delay with

9 9 the latter two proportionately increasing the path separation and the range proportionately decreasing it. When the normalized delay is greater than one, the two paths are separable and act to increase α for the beams in which they contribute simultaneously as dictated by the array beampattern and the grazing angles for the two paths, which are only dependent on the range to water depth ratio, θ = arcsin 1 4r d and 3 θ 1 = arcsin. 4 4r d + 1 Clearly, at longer ranges the grazing angles approach zero so both paths contribute to the same beam for all steering angles. However, at shorter ranges and steering angles off broadside to the array, the different paths will contribute to different beams. To illustrate how this impacts α, consider a line array steered to azimuthal direction ψ. As shown in Fig. 4, α is expected to increase as ψ moves from broadside to endfire in the absence of multipath. To isolate the impact of multipath coupling with the array sensing, α for the multipath environment is normalized by the shape parameter for the direct path environment as a function of steering angle. Depending on the path overlap as dictated by τ 1, these curves may still be greater than one at broadside, so the above ratio is further normalized by its value at broadside. This latter normalization is a constant with respect to the steering direction, essentially removing the gross effect of the multipath on the shape parameter and allowing comparison of different ranges on the same plot. The net effect, which is the impact on the clutter statistics of multipath being separated into different azimuthal beams by array processing as a function of steering angle, is shown in Fig. 7 for an array of n = 64 sensors spaced every half wavelength with W/f c =.1, d b /λ c = 5 and a variety of ranges. For this example, the path amplitudes were assumed to only be affected by the backscattering e.g., a i = sin θ i so the surface reflection is considered lossless. Owing to the ratios in 19, it is only necessary to articulate the relative path amplitude. As seen in Fig. 7 for this particular example, there is a potentially significant effect at shorter ranges where α can be approximately 3% larger or smaller than if the paths contributed to the same azimuthal beam. As might be inferred from eqs. 38 4, the effect is dominated by the range altering the multipath arrival angle, so larger arrays with narrower beamwidths will result in greater deviation. Fortunately, the effect decreases with range to the point where it is likely not measurable e.g., at ranges somewhat greater than 5 water depths in this example. V. DIRECTIONAL SENSORS When an array is comprised of individual sensors having their own directivity, the composite beampattern is simply the product of the two individual beampatterns [, pg. 75]. Thus, Fig. 7. Horizontal line array example illustrating the coupling between the array effect and the effect of multipath on α. High elevation-angle multipath contributing to azimuthal angles closer to broadside from their angle of origin cause both increases and decreases in α as a function of the steering angle. K a n from 6 may be described as [ π b ψ, φ b 1 ψ, φ dφ K a n = π b ψ, φ b 1 ψ, φ 4 dφ ] 41 where b ψ, φ is the beampattern of an individual sensor and b 1 ψ, φ is the beampattern formed from an array of such sensors were they omnidirectional. As an example, consider a line array of directional components such as vector sensors [7], [8], [9] or hydrophone triplet sensors [3], [31], [3], [33]. In sonar array processing, line arrays of such sensors allow resolution of the left/right ambiguity inherent in line arrays of omni-directional sensors. The simplest way to accomplish this is with a two-stage beamformer where in the first stage each individual triplet or vector sensor is beamformed broadside to the line array, resulting in a cardioid beampattern as shown in Fig. 8a for a single triplet sensor. In the second stage, the directional-sensor outputs are combined using a standard line-array beamformer i.e., as if the sensors were omni-directional and steered to azimuthal angle ψ. The advantage of this technique is that the first beamforming stage need only be performed for the left and the right sides of the array, not for every different value of ψ. As such, this technique will be referred to as the fixed-firststage FFS beamformer. When ψ is near zero, the null in the cardioid pattern of the individual directional sensors rejects most of the energy coming from the ambiguous beam located at π ψ. However, as ψ tends toward endfire to the line array, the rejection accomplished by the fixed cardioid pattern Fig. 8a becomes less effective. An alternative to this method involves steering the individual directional sensors to ψ in the first beamforming stage while placing a null in the direction of the ambiguous beam of the line array π ψ, which results in the sensor-level beampattern shown in Fig. 8b for a triplet sensor. The improved rejection

10 1 ψ, the beampattern for a single triplet sensor can be shown to be Fig. 8. Azimuthal beampatterns of a single triplet sensor a steered to broadside to the array of triplet sensors and b steered to the array steering direction ψ with a null in the ambiguous beam π ψ. Regardless of where the individual triplets are steered, the line array is steered to ψ with confounding energy coming from the ambiguous beam at π ψ. b ψ, φ = w H ψd φ = 1 [ ] sin κ cos φ 1 + cos κ cos φ cos κ cos ψ + sin κ cos ψ + cos κ cos ψ 44 where κ = 3πd λ /. When κ is small, which is typically the case but still depends on frequency [33], 44 may be approximated as b ψ, φ 1 [ 1 + cos φ ] 45 cos ψ which is seen to satisfy the distortionless response constraint of b ψ, ψ = 1 and provide ambiguous-beam nulling with b ψ, π ψ =. of the ambiguous beam for ψ may outweigh the increased computational requirements of this variable-first-stage VFS beamformer. With a distortionless response constraint in the steering direction and an assumption of a loud interferer, the beamforming weight vector for the null steerer in the first stage e.g., following [, Sect ] of this latter method is I d Id H I d ψ w ψ = d H ψ d H I d I I d Id H I d H I d I d ψ 4 where I is the identity matrix, d ψ is the steering vector of the components of one directional sensor pointing to angle ψ, and d I is the steering vector pointing in the direction of the desired null i.e., d I = d π ψ. There are other techniques for combining the individual sensor data as discussed in [3], [3], [33] for triplet arrays and [9] for vector sensors. However, the weight vector shown in 4 provides easily evaluated beampatterns for both triplet arrays and vector sensors and is therefore selected to examine the effects of a directional sensor on clutter statistics. A. Line array of triplet sensors Consider a triplet sensor where the three hydrophones lie equally spaced on a circle of diameter d λ in units of wavelengths placed perpendicular to the axis of the line array as shown in Fig. 9. If the topmost sensor is assumed to be in the vertical plane of the line array, the steering vector for this sensor may be obtained from [3, eq. ] as [ ] T d φ = 1 e j 3 πd λ cos φ e j 3 πd λ cos φ 43 where it is useful to note that d λ = r/λ = kr/π where r is the radius of the triplet array, λ is the wavelength and k = πf/c is the wavenumber. Using the beamforming weight vector in 4 with the steering vector in 43 and d I = d π Note that in exploiting the steering vector definition of [3] the azimuthal angle θ in [3] is converted from referencing endfire to broadside by setting θ = π/ φ and the steering angle in elevation φ in [3] is taken as π/. Fig. 9. Schematic description of a line array of triplet sensors. In this paper the topmost hydrophone of each triplet sensor is assumed to be in the vertical plane of the array i.e., in the x, z plane where y =. The triplet beampattern in 45 may be used in 41 along with 31 for b 1 ψ, φ to examine the effect of nulling the ambiguous beam of the line array. When the nulling is effective, a reduction of the array processing effect K a n or the equivalent shape parameter by a factor of two compared with that obtained by a line array of omni-directional sensors is expected because half as many scatterers should contribute to the clutter at any given time. The FFS beamformer may be evaluated by using b, φ from 45 and results in the expected halving of the shape parameter when steered near the array broadside as seen in Fig. 1 for n = 64 triplet sensors placed in a line array with λ/ spacing. However, a significant departure from halving is observed as the line array is steered toward endfire. The improved ambiguous-beam rejection of the VFS beamformer is seen to provide consistent results much closer to array endfire. This implies that the FFS beamformer should not be used when estimating clutter source parameters e.g., geoacoustic inversion without carefully accounting for the effect of array processing as a function of ψ. Altering the number of sensors in the line array affects how far from broadside the departure from halving occurs. As might be expected, larger arrays result in a greater region of validity for the halving approximation. B. Line array of vector sensors A similar effect to that seen for the triplet sensors is expected for vector sensors which independently measure the

11 11 Fig. 1. Effect of using a line array of triplet or vector sensors with the individual directional sensors steered either to broadside or to null backscattering from the ambiguous beam of the line array. As expected, the equivalent shape parameter is approximately half that obtained with a line array of omnidirectional sensors with deterioration as the steering direction departs from broadside. particle velocity in three dimensions and the acoustic pressure at a given point. A line array of such sensors beamformed in the horizontal results in a steering vector [7] d φ = [ 1 cos φ sin φ ] T 46 where the first component is the acoustic pressure followed by the two velocity components in the horizontal plane after appropriately scaling by the characteristic impedance ρc [34, pg. ]. The vertical velocity component is discarded owing to the zero weight it receives when beamforming in the horizontal. Following the development of the previous section, the beampattern for the weight vector defined by 4 is b ψ, φ = 1 [ ] cos φ 1 + sin φ sin ψ + cos ψ 1 + sin 47 ψ which satisfies the distortionless response and ambiguousbeam nulling constraints. Noting that when ψ =, the triplet- and vector-sensor beampatterns of 45 and 47 are identical, applying the FFS beamformer to an array of vector sensors will produce the same results as for an array of triplet sensors. As seen in Fig. 1, the VFS beamformer produces very similar, though not identical, results to that of a line array of triplet sensors. Although the vector sensor array departs slightly more from direct proportionality to the omni-directional sensor result than the triplet sensor array, it is more ably predicted by the large-n approximation of 35 i.e., by dividing 35 by. This latter result is not shown graphically owing to its similarity to the curves found in Fig. 1. VI. CIRCULAR AND CYLINDRICAL ARRAYS Consider an array comprising n hydrophones at equally spaced intervals on a circle of radius r in the horizontal plane. Such an array might be found in a sonar buoy where it would be used to form beams azimuthally at zero elevation angle i.e., in the same plane as the array. This circular array differs from the triplet array considered in Sect. V-A where the beamforming was performed in the plane perpendicular to the triplet array. The array beampattern for the horizontal circular array for a plane wave coming from azimuth φ with the beamformer steered to azimuth ψ may be obtained from [, pg. 83] with θ = θ = π/ and φ = ψ, b n ψ, φ = 1 n 1 e jπdλ[cosφ πi/n cosψ πi/n], 48 n i= where the function has been normalized to be one along the main response axis, b n ψ, ψ = 1, and d λ is the array diameter in wavelengths, d λ = r/λ. Though not computationally difficult to evaluate, 48 may be approximated when n is large simply by phrasing b n ψ, φ as a Riemann sum and taking the limit as n, resulting in b ψ, φ = J πd λ 1 cos ψ φ 49 where J x is the zeroth order Bessel function. The effect of array processing on α for a circular array is shown in Fig. 11 as a function of d λ for several values of n and for the limiting beampattern of 49. The independent variable in this example is d λ for the array effect i.e., K a d λ rather than n as it will be seen that d λ plays the dominant role in dictating the effect on α. The point at which the array is just sampling without spatial aliasing, when n = πd λ + 1 [, pg. 8] or, equivalently, d λ = n 1 / π, is noted on the figure by an asterisk for each value of n. As long as n is slightly greater than this value or d λ slightly less, the asymptotic beampattern obtained from letting n produces a very accurate approximation to K a d λ. That is, when n is large enough for unaliased spatial sampling, the effect of array processing on α only depends on the array diameter d λ and not on the number of sensors. When d λ exceeds this value i.e., to the right of the asterisks in Fig. 11, the effects of spatial aliasing and grating lobes are observable as a sharp increase in K a d λ followed by highly varying values. The 6-dB beamwidth for the asymptotic beampattern is shown for comparison; however, as with the uniform shading on the equally-spaced line array, the high sidelobes the first of which is only 7.9 db down cause a poor approximation. The dashed line in Fig. 11 shows the relationship K a n = dλ.7, 5 which, though derived empirically, is a reasonable approximation to the array effect using the large-n approximation to the beampattern and illustrates that the effect of beamforming on α for a circular array is inversely related to the diameter of the array, with inverse proportionality between one over the diameter and one over the square root of the diameter. A cylindrical array of sensors may be viewed as a line array of circular subarrays and, as described in Sect. V, the resulting beampattern may therefore be described as the product of the beampatterns of the two component array forms. Thus, for a