A physics-based method for beamforming signals measured on a horizontal array is developed with an application to underwater active sonar systems. The proposed striation-based beamformer coherently combines the pressure from each element in the array at different frequencies, and these frequencies are selected based on a striation hypothesis. The linear frequency shift and corresponding phase term introduced in the array weight vector accounts for multipath-induced fading, producing beam output with increased signal gain. The method is demonstrated using data collected on an array towed in the North Atlantic. The combination of the striation-based beamformer with the waveguide invariant concept to improve tracker performance is discussed.

## I. Introduction

Multipath propagation is a ubiquitous feature of underwater acoustic propagation, with constructive and destructive interference between the paths often producing richly structured intensity patterns. A familiar near-field example is the Lloyd’s mirror pattern. An example at more distant ranges is the plot of intensity versus range and frequency generated by Weston and Stevens using data collected on a bottom-mounted receiver as a towed source passed.^{1} The plot shows clear local minima at the source’s closest point of approach to the receiver. Weston’s work influenced early Russian studies of interference phenomena,^{2} most famously the work by Chuprov.^{3} Chuprov distilled the main features of interference patterns into a single scalar parameter that he called the waveguide invariant. Figure 1 shows passive sonar results analogous to those originally derived by Weston and Chuprov in both the time-frequency domain [the boat signatures in Fig. 1(a)] and in the range-frequency domain [simulated transmission loss (TL)^{4} in Fig. 1(b)].

Modern signal processing algorithms for passive sonar have increasingly recognized the importance of interference phenomena. Using vertical arrays, the waveguide invariant has been used to create a virtual receiver^{5} and to shift the focal range in a time-reversal experiment.^{6} Coherent processing of data collected on a bottom-mounted horizontal array was used to estimate both the waveguide invariant and the rate-to-range ratio of a moving source.^{7} A frequency-shift compensation method to increase the spatial coherence of horizontal arrays was demonstrated.^{8} Passive range estimation using a moving point receiver was evaluated using data collected in the Mediterranean.^{9} A transform method was developed to recompress a signal that has propagated through a multimode acoustic waveguide.^{10}

There are a number of complications in trying to incorporate interference phenomena into active sonar signal processing. In active sonar, a source transmits a signal that interacts with a target of interest with the resulting scattered field then measured at a receiver that may or may not be near the original source. Although recent work has demonstrated the presence of striations in active sonar data,^{11} the two-way nature of the propagation problem (source-to-target and target-to-receiver) and possibly frequency-dependent nature of the scattering problem might make the observed interference pattern more complicated than those produced in passive sonar.^{12} Additionally, active sonar systems often have lengthy pulse repetition intervals (PRIs), implying much coarser time resolution than shown in Fig. 1(a).

The impact of coarser time sampling is that data from conventional active sonar systems cannot typically be processed to yield a plot like Fig. 1(a) with a recognizable interference structure. For example, with a PRI of 1 min, only a single row of Fig. 1(a) would be captured in an active return, with no information on how this structure evolves over time.

Finally, an active sonar system might have an array aperture such as a towed horizontal array where output is calculated at multiple bearing angles using conventional beamforming. Conventional narrowband beamforming applied at a single frequency essentially takes a horizontal slice through an image like Fig. 1(b) and combines the complex pressure coherently across the array. Destructive interference between the paths can create deep fades over portions of the array thereby decreasing the effective aperture, decreasing gain, and increasing vulnerability to noise. More importantly, depending on the array aperture and the multipath spread across the array, the beamformed output may no longer contain intensity variations indicative of the interference phenomenon.^{13}

In the present work, a technique called striation-based beamforming for active sonar is developed. Rather than process at a single frequency across the array, the output from multiple phones is summed across a phone-dependent frequency whose value is calculated based on a striation hypothesis. The frequency shift is selected based on the trajectory of high intensity bands (striations) such as those observed in images such as Fig. 1(b). In effect, one beamforms along the striations insuring a strong signal across the entire array and increasing gain. The method is demonstrated using actual towed array data collected in deep water environments at ranges less than a deep-water convergence zone.

## II. Striation-based beamforming

A horizontal line array (HLA) is an interesting case to consider for interference processing because the observation space (i.e., element space) provides direct exposure to the structure of the interference. Consider an *N*-element HLA with element spacing *d.* Assume the array is towed at a constant depth and that array tilt can be ignored. Let *r _{n}* be the one-way range from the target to the

*n*th element of the array. If the target is in the far-field of the array

where *R* is the range to the first (reference) element and Δ*r _{n}* is the differential range. The target’s bearing relative to endfire is

*ϕ*. The approximate form of Eq. (1) neglects the quadratic term in

_{t}*n*caused by wavefront curvature and other higher order terms.

Evaluating Eq. (1) across the array for *n* = 0, 1, 2, … (*N* − 1) produces a series of range points that populate the horizontal axis in Fig. 1(b). Given that an HLA is likely to have half wavelength element spacing, the density of these points is sufficient to sample the interference structure. Furthermore, the frequency content at each phone produces the vertical dimension. Thus, a single measurement along an HLA can be used to produce observations of the structure as shown in Fig. 1(b).

A conventional plane wave beamformer uses weights with a phase relationship based on Eq. (1), with an output

where *k* = 2*πf*/*c* is the wavenumber, *p _{n}* is the pressure measured on the

*n*th array element,

*υ*is the corresponding conventional array weight, and

_{n}*P*is evaluated at a particular frequency, say

_{c}*f*=

*f*

_{0}. The calculation might be repeated for many

*f*

_{0}and combined coherently with a matched filter or possibly averaged for a broadband output. If there is good signal-to-noise and an accurate value for

*c*

_{0}is assumed, the output should be a maximum when the look direction

*ϕ*matches the target’s bearing,

*ϕ*. Note that the processor assumes that the phase depends linearly on the sensor index

_{t}*n*both through the approximate form of the differential range and the beamformer weights. If the active return was truly a plane wave, the intensity would be uniform across the array. This will not be true in the ocean waveguide, however, as constructive and destructive interference will create an intensity pattern analogous to those in Fig. 1.

Striation-based beamforming is an alternative to conventional processing. Rather than process each element at the same frequency, an offset is introduced across the array. Let Δ_{f} be the average frequency shift between any two elements in the array when evaluated across a line of constant intensity. Then the frequency for a line of constant intensity as a function of array element index *n* is

Note that Eq. (4) describes a family of sampled lines in the two-dimensional space defined by frequency and sensor index. In many situations the lines of constant intensity can be described in terms of the waveguide invariant. For a known (scalar) value of *β*, the frequency shift in Eq. (4) would then be given by Δ_{f} = *βf*_{0}/*R* Δ*r _{n}*. However, in the more general case Δ

_{f}is obtained by observation (i.e., such as the lines in Fig. 1) or by testing a sequence of possible values.

If the value of *f*_{0} is selected to be on a high-intensity striation, the signal will be uniformly strong across the entire array. Conversely, the signal will be weak across the array if *f*_{0} is selected in a trough (intensity minimum). The magnitude of striation-based beamformer output will consequently oscillate as a function of *f*_{0} for a specified look direction. While the frequencies yielding output maxima are naturally important, the pattern of peaks and troughs is itself of value as will be discussed later.

It is tempting to simply evaluate Eqs. (2) and (3) at *f*_{0} and examine the output. The result would be unsatisfactory for the present purpose, however, as it can be shown that the beamformer would not focus at the correct bearing of the target.^{14} An additional phase correction is required for active sonar. The remainder of this section shows why the additional phase correction is needed and sketches a signal processing strategy for its attainment.

For an HLA operating in the ocean waveguide, one must often use a normal mode expansion for the measured acoustic pressure.^{15} To show why the phase correction is needed, however, a simple planewave model for the return pulse will suffice. Assuming an omnidirectional source co-located with the HLA, the frequency-shifted pressure

where *k _{n}* = 2

*πf*/

_{n}*c*

_{0}is the wavenumber at frequency, and

*f*is given by Eq. (4). Without loss of generality the transmitter is assumed collocated with the first array element. The factor 2

_{n}*R*+ Δ

*r*is the two-way range from the transmitter to the target and back to the

_{n}*n*th array element. The complex

*S*(

*f*) lumps together the effects of the transmitted waveform and the target’s possibly frequency-dependent scattering function.

_{n}The part of Eq. (5) independent of the differential range can be represented in terms of a Taylor’s series expansion of the phase about *f*_{0}. Retaining terms through first order gives

where *ψ _{m}* is the coefficient for the

*m*th term in the expansion. Using this and then evaluating Eq. (5) at

*f*gives

_{n}where *A*(*f*_{0}) contains all the zeroth order terms.

Consider the terms in Eq. (7). The terms included in *A*(*f*_{0}) are independent of *n* and thus do not affect the bearing *ϕ* where a striation-based beamformer would focus. The term *υ _{n}*(

*ϕ*,

_{t}*f*) has a phase linear in

_{n}*n*and is compensated with conventional beamforming with weights in Eq. (3). The term exp(

*iψ*

_{1}

*n*Δ

_{f}) introduces an additional linear phase shift across the array and across a frequency band defined by Eq. (4). The weighing in the striation-based beamformer must compensate for this additional phase shift or it will not focus at the correct target bearing.

Based on the above, the striation-based beamformer is proposed as

with weight vector components *w _{n}* chosen to match the phase in Eq. (7)

Note the above striation beamforming weight is a combination of spatial processing [conventional beamforming using the weight given in Eq. (3)] and a spatial-spectral combination using the compensation coefficient $\psi \u03021$. This factor $\psi \u03021$ compensates for the phase introduced by the waveform (in other words, provides a matched filter) but additionally it adjusts for the phase introduced by the slope of the striation. If this slope is known (for example, provided by a waveguide invariance prediction) and the waveform and target range are known, all terms in Eq. (6) are known and $\psi \u03021$ can be calculated for a desired target bearing. In practice, errors in the estimate for the target range or pulse distortions may introduce a sufficient error that iterating provides a better result. The iteration proceeds by first calculating the conventional beamformer output with Eqs. (2) and (3), then using this bearing angle to determine the value of $\psi \u03021$ to provide optimal focus of the striation output (i.e., determining the slope of the striations).

Finally, it should be noted that the above processing incorporates a coherent combination across frequency, and thus is similar to the more usual matched filter processing used in active sonar pulse compression. However, unlike matched filtering, a frequency-dependent output [i.e., *P _{s}*(

*f*)] is produced that preserves the intensity variations. This output is produced with a combined spectral-spatial filter $\psi \u03021$ which preserves the striations structure and provides a beam focus at the correct target bearing—neither of which occurs with standard matched filtering followed by conventional spatial beamforming.

^{13}

## III. Experimental results

Striation-based beamforming is tested using active-sonar data collected in the North Atlantic. A surface vessel towed both a transmitter and a 128-element HLA receiver. The transmitter sent hyperbolic frequency modulated (HFM) pulses that were in the mid-frequency (1 to 10 kHz) band. The response was measured on and then rebroadcast from a distant echo-repeater that mimicked a point-target. The PRI between pulses was on the order of 1 min. The experiment was performed in deep water and the one-way range *R* to the echo-repeater was on the order of 20 km in range.

Each pulse was processed first by taking an 8 s window centered around the echo repeater return. A short-time Fourier transform was calculated (without matched filtering) yielding *p _{n}*(

*f*) for each of the

*N*= 128 phones. Note that this processing isolates the echo repeater returns in time to produce a “target spectrogram” that contains spectral content from only the target, i.e., clutter from other range bins is excluded. The procedure was repeated for each pulse.

Figure 2 shows target spectrograms where $|pn(f)|2$ is plotted for two different pulses. The vertical axis is phone number *n* that can be translated into either distance along the array using the known element spacing *d* or range *r _{n}* using Eq. (1). Striations for this active-sonar data are clearly evident for these two pulses as they are for each of the 37 pulses in the data set. The striation patterns are analogous to the striations observed for the passive-sonar simulation shown in Fig. 1(b).

Note that in Eq. (8) the output for frequency *f*_{0} is produced by coherently combing data across phones and across frequencies; the phase of the weight vector is matched to this selection. Conceptually, the output is formed from data along a line in the phone-frequency spectrum with a slope chosen so the line falls along the striations. For the HLA data shown in Fig. 2, the phone-level signal-to-noise ratio was high enough to directly determine the value of Δ_{f} (i.e., the slope of the striations was measured from the target spectrogram). The value of the compensation coefficient $\psi \u03021$ was then determined by estimating the phase slope at a single phone across the frequency band, so no iteration was necessary.

Figure 3 shows beamforming results for a single pulse. Conventional and striation-based beamforming results are in the left and center panels, respectively. The echo-repeater’s bearing is identified as *ϕ _{t}* = 166.5°. The right panel plots the output of both beamformers as a function of processing frequency

*f*

_{0}when evaluated at the look direction

*ϕ*=

*ϕ*. The striation-based result oscillates about the conventional result exhibiting a pattern of peaks and troughs. When

_{t}*f*

_{0}is selected to be on a high-intensity striation, the striation-based beamformer offers several decibels of gain. The interference pattern is thus analogous to that of a TL curve, with particular frequencies where constructive interference occurs and those with destructive interference. In this analogy, the conventional output is similar to a TL curve averaged over several ranges (i.e., ranges across the array).

While the increased gain offered by striation-based beamforming is useful, the method’s primary benefit may arise when it is incorporated into solutions of a combined detection/tracking problem. The pattern of peaks and troughs in Fig. 3(c) is for a particular active sonar pulse. For subsequent pulses, the pattern will be similar but shifted slightly. Previous work^{16} had proposed an Invariance-based Extended Kalman Filter (IEKF), which improved tracker detection performance and clutter rejection by incorporating the time-dependent intensity peaks. The striation patterns obtained by striation beamforming could be incorporated into the IEKF, or a similar formulation, to enhance tracker performance.

To summarize, a method of striation-based beamforming for active sonar has been developed. Rather than process at a single frequency as is done with a conventional beamformer, the frequency is shifted across the array. The result is a uniformly strong signal producing increased processing gain and resolution. The result has been demonstrated using actual towed array data collected in the North Atlantic.

## Acknowledgments

This work was supported by the Office of Naval Research.