Measurements of the correlation of the frequency-difference autoproduct with acoustic and predicted-autoproduct fields in the deep ocean^a) Free

Array-signal-processing techniques are commonly used to study acoustic fields that have propagated through the ocean, as well as to infer information about acoustic sources and the ocean environment. In general, an array-signal-processing technique is scientifically useful if it allows a user to infer measurable, stable, and identifiable information from acoustic recordings. In this context, acoustic information is measurable if a signal can be recorded and distinguished from corrupting influences such as acoustic or electronic noise. The inferred information is stable if measurements taken near one another in space or time are predictably related, and is identifiable if it reliably provides quantitative physical indications about, or parameter values for, the acoustic source(s) or phenomena in the acoustic environment. One commonly used technique in this field is frequency-domain cross correlation between a measured signal and a predicted Green's function for the medium (Urick, 1976). In a known acoustic environment, this technique can produce measurable, stable, and identifiable information about sound source(s). However, for pressure fields in uncertain environments (Tolstoy, 1989) or ocean environments that are altered by internal waves (Garrett and Munk, 1971; Flatte et al., 1983), this technique is difficult to apply because predicting the Green's function with the requisite accuracy in phase may not be possible. For a fixed source-receiver range, successfully cross correlating a measured and predicted field in an uncertain environment increases (decreases) in difficulty with increasing (decreasing) frequency. The research reported here demonstrates that the bandwidth-averaged frequency-difference autoproduct, a quantity derived from measured pressure fields with non-zero bandwidth, can be used for cross correlation analysis because—when the difference frequency is chosen appropriately—it makes this process stable for pressure fields that have propagated over hundreds of kilometers in an uncertain, deep-ocean environment.

The bandwidth-averaged frequency-difference autoproduct (Lipa et al., 2018) is a quantity that has been used to shift measured field information downward to appear at frequencies below the bandwidth of the recorded field. The autoproduct is calculated from measured (in-band) field information via a quadratic product of complex field amplitudes at two different frequencies. The frequency-difference autoproduct can be thought of as a surrogate field; it is not a genuine acoustic field. Nevertheless, it can mimic a genuine field at a frequency that is the difference of its two constituent frequencies, but, more importantly, the frequency-difference autoproduct is a measurable quantity that can produce stable information for cross correlation with a pressure field at the difference frequency in a multipath environment without refraction. The purpose of this manuscript is to document the extent to which the frequency-difference autoproduct retains these cross correlation properties with respect to predicted genuine and surrogate (autoproduct) fields that propagated over long distances through a time-varying deep-ocean environment. In particular, this manuscript presents the results of a study that extends the prior Lloyd's mirror cross correlation analysis (Lipa et al., 2018) to acoustic fields propagating in the deep ocean using mode-code simulations (Porter, 1992) and measurements collected during the PhilSea10 experiment (Worcester et al., 2013). This manuscript demonstrates that frequency-differencing techniques can be used to produce measurable field-derived information that complements time-front analysis (Colosi et al., 2019) and leads to stable cross correlations between measured and calculated fields even at long propagation distances in an uncertain deep-ocean environment, based on a relatively simple understanding of the acoustic environment. This stability does not exist for frequency-domain cross correlation between measured and predicted in-band pressure fields. Propagation simulations using the nominal PhilSea10 environment show that cross correlation between the frequency-difference autoproduct and a genuine low-frequency acoustic field is not stable with respect to source-array range, whereas cross correlation between a simulated and a calculated frequency-difference autoproduct is stable with respect to changing source-array ranges. Furthermore, propagation measurements from the PhilSea10 experiment show that the frequency-difference autoproduct correlates consistently to a computed autoproduct field across a wide difference-frequency bandwidth.

One preferred array-signal-processing method for deducing physical information from acoustic recordings is time-front analysis (Munk et al., 1995). This method typically works by measuring an acoustic field—with some bandwidth and a known source signal waveform that is removed using factor-inverse filtering (Birdsall and Metzger, 1986)—that has propagated through in the deep ocean and then using this information to construct the measured time-domain Green's function (response) of the environment in the bandwidth of the broadcast signal. This time-domain Green's function carries identifiable environmental information and is measurable. In many acoustic environments (Worcester et al., 2013), it is found to be stable over distances of hundreds of kilometers and over time periods of years. Last, it is found to carry identifiable information about the acoustic sound speed profile (Colosi et al., 2013), internal wave fluctuations (Colosi et al., 2019), and spice (Dzieciuch, 2014; Worcester et al., 2006). Correlation analysis of the frequency-difference autoproduct differs from time-front methods because the correlation analysis is conducted in the frequency domain and could possibly provide identifiable information that is complementary to the data available from time-front analysis. Here, this technique is considered to be complementary to time-front analysis in that these methods can be employed using the same measuring device in the same experiment, and they may be useful in identifying information beyond that deduced from time-front analysis.

Unstable or unreliable cross correlation between measured and predicted fields is, in general, problematic for array-signal-processing techniques and can have negative implications for beamforming (Morgan and Smith, 1990), and matched field processing (Baggeroer et al., 1993). For acoustic signals nominally at or above 10 Hz propagating over long distances (50 km or greater) in the deep ocean, a lack of complete environmental knowledge can lead to low correlation between measured and predicted acoustic fields (Tolstoy, 1989). Spatial cross correlation between measurements and predictions arises from a spatially coherent and predictable phase structure in the measurements. Such spatial coherence has been used to quantify the limitations of beamforming (Cox, 1973; Finette and Oba, 2003). Spatial coherence lengths in the ocean have been studied with both horizontal and vertical transducer arrays (Wan et al., 2009), in the presence of challenging ocean-environment phenomena like internal waves (Duda et al., 2012; Lunkov and Petnikov, 2014) and uncertain channel depths (Carey, 1998), and over long propagation distances (Andrew et al., 2005) to quantify, and possibly understand, the influence of various environmental characteristics. In general, spatial coherence, which leads to spatial cross correlation of field predictions and measurements, has been shown to increase with decreasing signal frequency, decreasing propagation distance, and decreasing environmental complexity (Flatte, 1979). Frequency differencing attempts to promote array spatial coherence over long propagation distances, even with environmental complexities that are not known or controlled, by decreasing the frequency of the field information.

The frequency-difference autoproduct is a mathematical construct with some similarity to other, more familiar, signal-processing concepts, such as heterodyning and bilinear time-frequency analysis. In microwave circuit design, where generating and analyzing signals in changing bandwidths can be challenging, heterodyning is the process of shifting the center frequency of a recorded signal by multiplying it by a standard, typically sinusoidal, reference signal (Dunn, 2012). The frequency-difference autoproduct also seeks to shift the frequency of a recorded signal, but it does so, not by multiplying the recorded signal by a standard signal, but rather by multiplying the recorded signal by itself at a different frequency. This formulation, then, is like that of $Δk$-radar and -sonar (Popstefanija et al., 1993; Sarabandi, 1997; Silva et al., 2008) but has been developed independently and coherently exploits several features that are unique to the frequency-differencing process.

Prior experimental studies have used array measurements of the frequency-difference autoproduct for beamforming (Abadi et al., 2012; Douglass et al., 2017; Douglass and Dowling, 2019) and remote source localization (Worthmann et al., 2015; Geroski and Dowling, 2019). The beamforming studies have either used water tank measurements, where the source-array range is less than 1 m, or have used measurements taken in the shallow ocean, here defined as less than 200 m water-column depth, using sources placed 1–3 km from the array. These prior studies, especially Douglass et al. (2017), also demonstrate the added robustness that can be achieved using frequency-difference beamforming over conventional beamforming. However, within these techniques, measured complex autoproduct fields are reduced to cross-spectral density matrices that allow the algorithm's results to be independent of the waveform broadcast by the source. This study, because it assumes knowledge of the waveform broadcast by the source, utilizes the signal-bandwidth-averaged complex frequency-difference autoproduct field itself, as in Worthmann and Dowling (2017) and Lipa et al. (2018). Thus, the known source waveform is removed from the measurements so that the final cross correlation results are indicative of deep-ocean acoustic propagation characteristics alone. Furthermore, while this study analyzes fields recorded in the deep ocean, these results are expected to readily generalize to any refracting environment.

One major difference between the frequency-difference autoproduct and a genuine acoustic-pressure field at the difference frequency arises when sound interacts with a reflecting boundary (Worthmann and Dowling, 2017). For a surface with a complex reflection coefficient, $R$⁠, the apparent reflection coefficient for the autoproduct is $R2$⁠, a positive-definite real number, because of the intentional quadratic nonlinearity. A second major difference occurs within refracting environments. Here, sound waves traveling on ray paths that pass through one or more caustics collect a 90° phase shift (Jensen et al., 2011) for each caustic. These phase shifts also arise when the acoustic field is calculated as a sum of normal modes (Zhuravlev, 1989; D'Spain et al., 2002), and caustics are referred to as dislocations in the acoustic field. However, the frequency-difference autoproduct lacks caustic phase shifts (Worthmann and Dowling, 2020). At long ranges in the deep ocean, caustic phase shifts affect a significant portion of a measured acoustic field in a non-uniform manner. Therefore, high spatial cross correlation of an autoproduct to an acoustic-pressure field at the difference frequency is not expected at frequencies and ranges that are high enough for caustics (field dislocations) to form. Thus, autoproduct-to-field cross correlation is expected to decline with increasing frequency independent of any other environmental factors. This study explores this expected cross correlation degradation in the deep ocean in Sec. IV and demonstrates that this degradation can be overcome by correlating the frequency-difference autoproduct to a calculated frequency-difference autoproduct rather than a pressure field at the difference frequency. In particular, high cross correlations between measurements and predictions are possible if the frequency-difference autoproduct is matched to a calculated autoproduct field and the difference frequency is low enough.

The remainder of this manuscript is divided into five sections. Section II provides an overview of the relevant theory, and defines the fields analyzed in this study and the cross correlation coefficients used herein. Section III summarizes the PhilSea10 experiment, including the relevant array and signal management details, and includes an analysis of the observed environmental fluctuations. Section IV presents simulation results from an environment and geometry that mimic that of the PhilSea10 experiment and illustrate the limitations associated with correlating the frequency-difference autoproduct to a pressure field at the difference frequency. Section V provides spatial-correlation results for the autoproduct fields determined from six sources used in the PhilSea10 experiment compared to predicted acoustic and calculated-autoproduct fields. Section VI provides a summary of the results of this study and provides the conclusions drawn from it.

This section provides an overview of the relevant autoproduct theory, including the definition of the frequency-difference autoproduct and its signal-bandwidth average, the reason for the absence of caustic phase shifts within the autoproduct field, and the specific formulas used to compare measured and predicted fields.

The frequency-difference autoproduct is constructed from the complex frequency-domain pressure field $Pr→,ω$ at position $r→$ as a quadratic product involving two measured frequencies:

where the asterisk denotes complex conjugation, $ω$ is the in-band frequency, $Δω$ is the difference frequency, and together $ω$ and $Δω$ must be consistent with $ΩL≤ω±Δω/2≤ΩH$⁠, where $ΩL$ and $ΩH$ are the acoustic field's lower and upper frequency limits, respectively. In this study, the autoproduct is referred to as a surrogate field because it does not satisfy the Helmholtz equation in a general medium. In contrast, pressure fields are referred to as genuine field quantities because they do satisfy the Helmholtz equation. This definition of the autoproduct includes the source-broadcast signal spectrum $S(ω)$ and unintended multipath cross terms. For the analysis presented here, the signal dependence in Eq. (1) is removed by division, and cross terms are suppressed by signal-bandwidth averaging to form the bandwidth-averaged autoproduct:

where $ΩBW=ΩH−ΩL−Δω$ is the signal bandwidth available for averaging. For the measurements used in this study, the signal waveform is removed through factor inverse filtering (Birdsall and Metzger, 1986), described in Sec. III, rather than division. This is done because the phase of a source waveform removes the coherence that the bandwidth average aims to promote. Note that the autoproduct (1) and bandwidth-averaged autoproduct (2) are, in general, complex quantities, since the pressure amplitudes used in computing them are at two different frequencies. In general, for a sufficiently broadband signal, $Δω$ may be chosen between $ΩL$ and $ΩH$⁠. However, $Δω$ is referred to herein as being below-band or out-of-band for simplicity since $Δω<ΩL$ throughout this manuscript.

All calculated fields for this study were produced with the modal sum code kraken (Porter, 1992). However, the need for a caustic phase correction when correlating a predicted field to a measured autoproduct is most easily explained assuming the genuine complex pressure field, $Pr→,ω$⁠, in a refracting environment is well described by a sum of ray-path contributions:

where $r→$ is the receiver location, $r→s$ is the source location, N is the number of eigenrays between the source and receiver, $An$ is the complex amplitude of a given eigenray, $τn(r→,r→s)$ is the travel time from source to receiver along the nth ray, and $mn$ is the Keller, Maslov, Arnold, and Hörmander (KMAH) index of the nth ray (Kratsov and Orlov, 1999). The KMAH index counts the number of caustics that each eigenray traverses.

Using the ray approximation specified in Eq. (3), the bandwidth-averaged autoproduct field, $APΔr→,Δω$⁠, can be determined analytically from Eq. (2) for a given set of ray-path amplitudes and arrival times, as shown in Eq. (4). Here, $APΔr→,Δω$ is separated into two sums of ray-path pairs, one summation that arises from multiplying ray-path contributions from the same path (hereafter referred to as a self-terms) and a double summation that results from multiplying ray-path contributions from different paths (referred to as cross terms).

Here, $ΩC$ is the center frequency of the measured signal, and sinc represents the unnormalized cardinal sine function. In Eq. (4), the self-term sum looks like a ray-path summation at the difference frequency without any caustic phase shifts. While the KMAH index does appear in the cross term summation, all these terms are suppressed in magnitude by the bandwidth averaging processing when $ΩBWτn′−τn≫π$ and so are ideally zero. Thus, the bandwidth-averaged autoproduct is largely free of caustic phase shifts when the cross term sum in Eq. (4) is suppressed. This lack of caustic phase shifts in the autoproduct is expected to be problematic when correlating it to a genuine acoustic field, which includes such phase shifts.

In general, two fields can be spatially correlated along an array of receivers in the frequency domain by taking a complex inner product of the signals at each array element location. The correlation coefficient definition used here was chosen to match previous studies in this area (Lipa et al., 2018) while accounting for the discrete nature of a receiving array with $J$ elements:

where $F$ is a field measured along the array that is correlated with a calculated or predicted field $w$⁠. The resulting correlation coefficient, $χ$⁠, is a complex quantity in general. Because of the normalizations, the magnitude of $χ$ will fall between 0 and 1, with increasing magnitude indicating an increasingly better match between $F$ and $w$⁠. The phase of $χ$ represents a weighted-average phase difference between the measured (⁠$F$⁠) and predicted (⁠$w$⁠) fields.

Three versions of the complex correlation defined by Eq. (5) are considered in this paper. The first, $χP,P¯ω$⁠, is an in-band pressure-field cross correlation where $Fr→,ω$ is the recorded complex pressure-field $Pr→,ω$⁠, and $wr→,ω$ is a mode-code predicted complex pressure field from a simplified range-independent ocean environment intended to nominally match the actual ocean environment. The effects of the complex phase differences between these fields are shown in Sec. IV. The second correlation, $χAP,P¯Δω$⁠, is a below-band measured autoproduct to pressure-field correlation where $F$ is $APΔr→,Δω$ and $wr→,Δω$ is a mode-code predicted complex pressure field at the same location in the same nominal environment with adjusted boundary conditions as described below. The third correlation, $χAP,AP¯Δω$⁠, is a below-band measured autoproduct to calculated autoproduct correlation where $F$ is $APΔr→,Δω$ and $w$ is $wΔr→,Δω,$ a computation of the bandwidth-averaged autoproduct using kraken:

As described in Worthmann and Dowling (2017) and Lipa et al. (2018), the autoproduct feels rectified boundary conditions, so the nominal environment's boundary conditions must be adjusted when computing $wr→,Δω$ from Eq. (6). Here, the modes at the difference frequency $Δω$ were calculated with +1 and +0.5 reflection coefficients on the sea surface and floor, respectively. The seafloor reflection coefficient was chosen to nominally match the attenuation of sea-surface and -floor interacting sound found in the PhilSea10 experiments. The surrogate field specified by Eq. (6) is a predicted autoproduct where caustic phase contributions are essentially absent and the cross term sum is suppressed by the bandwidth average, as discussed in Sec. II B.

Acoustic and autoproduct field measurements utilized in this study come from the deep-ocean environment of the 2010–2011 Philippine Sea Experiment (PhilSea10) (Worcester et al., 2013). The experimental geometry and a sound speed profile measured near the distributed vertical line array (DVLA) are shown in Fig. 1 [reprinted from Geroski and Dowling (2019) for completeness]. The sound speed minimum was at a depth of 960 m, and the water-column depth was 6.37 km. The DVLA was comprised of 149 individual elements with variable hydrophone spacing (20, 40, or 60 m) and was deployed between the depths of 180 and 5380 m. The data used in this study came from six sources positioned near the sound speed minimum of the water column at ranges from 129 to 450 km from the DVLA in a variety of compass directions. Each source transmitted a linear-frequency-modulated (LFM) chirp with mildly differing bandwidth and duration of 135 s. Relative locations and signal-bandwidth information for these sources are given in Table I. Acoustic-pressure data were recorded from each hydrophone at a rate of 1953.125 Hz. Measurements used in this study were collected over approximately a year from 25 April 2010 to 25 March 2011.

The acoustic recordings were demodulated and processed via factor-inverse filtering (Birdsall and Metzger, 1986) to compress the chirp modulation achieving a nominal boost in the signal-to-noise ratio of 41 dB. The factor-inverse filter matches the source spectrum but only removes the modulation without flattening the amplitude of the spectrum. This differs from matched filtering and preserves the inherent time resolution of the full source bandwidth for a small cost in SNR. For the purpose of dividing the source waveform out of the bandwidth-averaged autoproduct, as in Eq. (2), the spectral amplitude of the broadcasts was assumed to be constant. Although the source levels were not constant across the transmitting bandwidths [see Worcester et al. (2013)], the consequences of ignoring this variation are found to be minimal because of the normalizations in Eq. (5) and because complex field correlations are more sensitive to the relative phases of the fields than their relative amplitude.

Figure 2 shows an example of acoustic pressure time fronts, factor-inverse-filtered signal amplitude (in color) vs time and depth, for the farthest PhilSea10 source, T3, positioned approximately 450 km from the array. Figure 2(a) shows measurements from an individual pulse (number 3 of 356) for the 225 Hz ≤ $f$ ≤ 325 Hz signal bandwidth and is generally representative of all the measured data used in this study. Figure 2(b) shows equivalent simulations for the same bandwidth in a range-independent environment having the depth and sound speed profile measured near the DVLA (see Fig. 1). As noted above, the ocean floor's reflection coefficient was set to +0.5 within these simulations. Figure 2(c) shows simulated acoustic-field time fronts in the same environment for a below-band signal having frequency content in the nominal difference-frequency bandwidth, 0.0625 ≤ $f$ ≤ 15 Hz, of this study. To visually suppress the Gibbs phenomena that appear with a sharp low-pass filter, this field was calculated using frequencies up to 50 Hz, with a smooth high-order filter that suppresses frequencies above 15 Hz. In addition, the ocean surface reflection coefficient was set to +1 for the results in Fig. 2(c) because autoproduct fields respond to pressure-release surfaces as if they were hard surfaces (Worthmann and Dowling, 2017; Lipa et al., 2018), and this ocean surface boundary condition change allows the predicted below-band acoustic pressure time fronts to better match equivalent autoproduct results.

Analysis of time fronts is generally conducted to remotely study ocean processes because time fronts are measurable, stable, and identifiable after propagating long distances through a time-varying deep-ocean environment. Frequency-domain cross correlation between a measured and predicted high-frequency pressure field is generally not stable or found to carry any identifiable information. This correlation loss occurs because, although the general appearance of the time fronts in Figs. 2(a) and 2(b) is reassuringly similar, there is substantial waveform mismatch. For example, if the simulation results are shifted in time to best-match the measured time fronts, the minimum root mean square (rms) timing mismatch between the measured and simulated time fronts is 74 ms, which is more than 20 periods at 275 Hz, the signal's center frequency. Thus, at the in-band frequencies, the measured and simulated arrival patterns are almost entirely uncorrelated. Figure 2(c) shows the effects of the bandwidth change from 225 Hz ≤ $f$ ≤ 325 Hz down to 0.0625 Hz ≤ $f$ ≤ 15 Hz and is provided for comparisons to the first two panels of Fig. 2 and both panels of Fig. 3. In the lower bandwidth, the noted rms time mismatch is potentially much less important, since 74 ms is only 1.1 periods at 15 Hz, or 0.074 periods at 1 Hz. Therefore, frequency-domain cross correlation should be more stable in this below-band frequency range. Thus, it stands to reason that correlations of the frequency-difference autoproduct could possibly be used as a supplement to the process of forming time fronts, if below-band frequency-domain cross correlation is measurable, stable, and identifiable for fields that have propagated over long distances in the ocean.

The first step in establishing this possibility is to show that the bandwidth-averaged frequency-difference autoproduct is measurable. Figure 3 shows simulated and measured frequency-difference autoproduct time fronts for 0.0625 ≤ $Δf$ ≤ 15 Hz. Figure 3(a) shows measured autoproduct time fronts constructed from the pressure-field time fronts shown in Fig. 2(a). Figure 3(b) shows equivalent autoproduct time fronts constructed from the simulated pressure field time fronts shown in Fig. 2(b) for the same bandwidth. The noteworthy feature of both panels of Fig. 3 is their similarity to each other and to Fig. 2(c), the primary difference being the extended horizontal glints that appear near the intersection of time fronts and are most prominent at the turning-point depths in both panels of Fig. 3. These glints are remnants of the cross term sum that are not fully suppressed by the signal-bandwidth average when the time difference between apparent ray-path arrivals is small [see Eq. (4)]. For both Figs. 3(a) and 3(b), the same filtering process that is used to visually suppress the Gibbs phenomenon in Fig. 2(c) is applied to the measured and computed autoproduct fields.

This visual correspondence between Figs. 2(c), 3(a), and 3(b) suggests that the time-amplitude character of a genuine acoustic field with 0.0625 Hz ≤ $f$ ≤ 15 Hz matches that of measured and simulated autoproducts with 0.0625 ≤ $Δf$ ≤ 15 Hz. In addition, this correspondence is potentially the most important because it implies that a worthwhile prediction of the frequency-difference autoproduct can be obtained from Eq. (6) without knowing the original signal bandwidth (225 Hz ≤ $f$ ≤ 325 Hz), since Eq. (6) depends on $Δω=2πΔf$ and not on the in-band frequency $ω=2πf$⁠. Furthermore, the visual correspondence of Fig. 2(c) and either panel of Fig. 3 suggests, but does not prove, that the frequency-difference autoproduct has the capacity to reveal below-band information from a 275-Hz-center-frequency signal with 100 Hz of bandwidth that has propagated through the deep ocean.

As shown in Fig. 4, the visual robustness of the autoproduct time fronts persists when other signal broadcasts are considered. Figure 4(a) shows measured acoustic travel-time variations for the in-band field time fronts (225 to 325 Hz) near the sound channel axis from 376 source transmissions for the farthest PhilSea10 source, T3, 450 km from the array. The vertical white lines are separated by two standard deviations of the measured travel-time variations at this depth. Transmission number, timing, and broadcast information for each of the six PhilSea10 sources is provided in Table II. The third column of this table lists the overall rms timing variations (⁠$Δτ$⁠) across the entire receiving array, and all of these values are in the tens of milliseconds. Thus, $fΔτ≫1$ for the entire in-band frequency range, so measured-to-predicted pressure-field correlations are likely to be unstable. However, the story may be different for the frequency-difference autoproduct. Figure 4(b) shows equivalent measured acoustic travel-time variations for frequency-difference autoproduct time fronts (0.0625 to 15 Hz) constructed from the in-band acoustic-field data shown in Fig. 4(b). Although the same rms timing variations are apparent, they represent less than 0.2 difference-frequency wave periods up to $Δf∼$ 3 Hz at all source-array ranges. Thus, frequency-domain cross correlations between measured and predicted autoproducts may be stable at difference frequencies of a few Hz and lower. The stability of these cross correlations is further explored in Sec. V.

Figures 2(c), 3, and 4(b) suggest that the time-amplitude character of a measured autoproduct can be matched to that of a simulated below-band field or a simulated autoproduct. However, cross correlations of such fields might still be low unless there is also good phase matching between the two fields being correlated. Thus, for completeness, the two parts of Fig. 5 show the relative magnitude and phase character of a measured autoproduct (black), a simulated autoproduct (red), and a simulated below-band field (blue) at identical difference and below-band frequencies of 1.0 Hz as observed on, and predicted for, the PhilSea10 array for a source-array range of 450 km. The simulated and measured autoproduct fields are computed using Eq. (2), and the simulated below-band field is directly computed using kraken. In this case, the nature of the measured and simulated autoproduct field magnitude (linear, no dB) and phase is comparable. In Fig. 5(a), the jagged amplitude curves of the two autoproducts are imperfectly matched but show the same general character, while the pressure field displays smoother magnitude variation in depth. This imperfect amplitude matching could lead to lowered cross correlation values, as will be shown in Sec. V. In Fig. 5(b), the pressure field shows little phase change over the array; however, the phases of the two autoproducts show more variation and track each other fairly well down to a depth of approximately 3 km, where the measured autoproduct amplitude is small. This suggests that the 1.0-Hz cross correlation between the measured and simulated autoproducts will be higher than the 1.0-Hz cross correlation between the measured autoproduct and the below-band pressure field.

Using the nominal range-independent PhilSea10 environment shown in Fig. 1, transmissions from each of the six sources to the receiving array were simulated using the field calculation routine available within the kraken normal mode program to produce fields and autoproducts. To mimic the dynamic-ocean time-front mismatch shown in Fig. 4 within the normal mode program, simulated receiver locations were randomly shifted in range by a zero-mean Gaussian-distributed distance. The standard deviation of this distance, $Δr$⁠, approximately corresponds to a rms propagation-time fluctuation of $Δτ=Δr/c$⁠. These random range shifts were added independently to each individual receiver for 200 Monte Carlo trials for each PhilSea10 source. Here, increasing $Δr$ corresponds to an increasing level of mismatch. The equivalent predicted fields, $w$ in Eq. (5), were computed using the same field calculation procedure without any shifts in receiver locations.

The results of this comparison study appear in Fig. 6, which shows the coherent average of 200 Monte Carlo trials of cross correlation between mismatched fields and predicted fields for a given amount of rms time-front fluctuation, $Δτ$⁠, at the center frequency of each source in the PhilSea10 environment (see Table II) or at a difference frequency $Δf=1 Hz$⁠. The figure shows the magnitude of these coherent averages plotted vs $Δτ$ for $10−5 s≤Δτ≤1 s$⁠. Before describing the figure, it is worth noting that the difference between the coherent and incoherent ensemble average of the 200 cross correlation values is small (within 2 dB) where the average cross correlation is high. However, when the average cross correlation is low, this difference is more noticeable, with the incoherent average being higher. This difference would not be noticeable in Fig. 6, as these low cross correlation values are below the lower extent of the y axis.

The specific results shown in Fig. 6 are for in-band pressure fields correlated to in-band pressure fields (⁠$χP,P¯$⁠, in red), autoproducts [see Eq. (2)] correlated to pressure fields at the difference frequency (⁠$χAP,P¯$⁠, in black), and autoproducts correlated to autoproducts (⁠$χAP,AP¯$⁠, in blue). All simulated fields with mismatch $[F$ in Eq. (5)] are generated with a nominal, deep ocean geoacoustic bottom boundary condition, and predicted fields [$w$ in Eq. (5)] are generated using a reflection coefficient $R=0.5$⁠. The different PhilSea10 sources appear as lines of the same color in both panels. In Fig. 6(a), the autoproduct-to-autoproduct cross correlation, $χAP,AP¯$⁠, and the in-band correlation, $χP,P¯$⁠, are highest at small $Δτ$ values. However, compared to the in-band field results, the cross correlations involving the autoproduct both persist to $Δτ$ values that are 2–3 orders of magnitude higher. Despite this advantage, the cross correlation between the autoproduct and pressure field at the difference frequency fails to approach unity, even as $Δτ→0$⁠, because of deep-ocean caustic phase shifts in the pressure fields that are not in the autoproducts. Moreover, $χAP,P¯$ is inconsistent at different ranges in the ocean, indicating that the autoproduct and pressure field experience different waveguide effects when propagating over long distances in the deep ocean. Finally, it can be observed that one source (⁠$r=395$ km) maintains unity $χP,P¯$ for more mismatch than the other sources. This comes from the fact that this source transmitted at a lower frequency (⁠$fc=172.5$ Hz). Figure 6(b) displays the same information as Fig. 6(a), with the horizontal axis non-dimensionalized by multiplying $Δτ$ by the field or autoproduct center frequency. With this change, the $χP,P¯$ and $χAP,AP¯$ results collapse to a single curve. Thus, Fig. 6(b) clearly shows that genuine acoustic fields and autoproducts are equally robust to time-front mismatch that is small compared to the temporal period of their center frequency. However, because the autoproducts are governed by a user-selectable below-band difference frequency, they can be constructed to be robust to larger rms time-front mismatch.

To better explore the inconsistent cross correlation values observed in Fig. 6(b) for $χAP,P¯$⁠, this simulation study was extended to the experimental situation, where $Δτ$ is fixed for each PhilSea10 source at its observed value (see Table II) and the difference frequency is varied. Here, $Δf$ values were chosen within the 0.0625–15 Hz bandwidth in such a way that nearly equal values of $ΔfΔτ$ were generated for each source. In general, time-front mismatch is expected to cause a signal to lose coherence across a receiving array when $Δτ$ is larger than a temporal period (Skolnik, 1980). When this quantity, $ΔfΔτ$⁠, approaches unity from below, the measured-to-predicted field cross correlation along a receiving array is expected to degrade, a trend that is followed in Fig. 7 for $χAP,P¯$ (blue dots) and $χAP,AP¯$ (black dashed-dotted curve). The red dashed-dotted curves are vertically shifted versions of the black dashed-dotted curves. Here, it is also seen that the simulated autoproduct with mismatch correlates better with the predicted autoproduct than with a genuine field at the difference frequency. This effect mirrors that observed in Fig. 6, where the cross correlations differed in range. Furthermore, the blue points are scattered owing to the changes in cross correlations at different ranges; this scatter and the separation of the black curves in Fig. 6 arise for the same reasons.

There are two reasons for this superior autoproduct-to-autoproduct correlation shown in Fig. 7. The first reason is illustrated in Fig. 7(a), where $χAP,P¯$ and $χAP,AP¯$ are plotted vs $ΔfΔτ$⁠. Here, the angle brackets indicate an incoherent ensemble average across the six PhilSea10 sources. At long ranges in a refracting multipath environment, uncompensated caustic phase shifts within a genuine field cause it to decorrelate from the equivalent autoproduct, which lacks such phase shifts. Thus, $χAP,P¯$ reliably falls below $χAP,AP¯$⁠. Interestingly, this correlation loss is independent of $ΔfΔτ$ too. The second reason is illustrated in Fig. 7(b), where $χAP,P¯$ and $χAP,AP¯$ are plotted vs $ΔfΔτ$ and the angle brackets have the same meaning as in Fig. 7(a). Here, $χAP,P¯$ is even lower than $χAP,P¯$ in Fig. 7(a) because $χAP,P¯$ does not have a consistent phase for each source. However, $χAP,AP¯$ and $χAP,AP¯$ are essentially identical in Fig. 7, and this indicates that $χAP,AP¯$ does have a consistent phase across all six source-array ranges. Furthermore, the difference between the average $χAP,P¯$ results in Figs. 7(a) and 7(b) is approximately a factor of $6$ (7.8 dB), the expected loss factor for adding six randomly distributed unity-amplitude phasors, and this indicates that the autoproduct and the low-frequency field have a randomly distributed correlation phase, arising from caustic-induced mismatch. Given this information, it seems that frequency-domain cross correlation between a measured autoproduct and pressure field is unlikely to be stable in an experimental environment. However, this simulation study suggests that this frequency-domain cross correlation between a measured and a predicted autoproduct may be stable in an experimental environment.

Given the superior correlation between the simulated frequency-difference autoproduct with mismatch and the predicted autoproduct, the three correlations were evaluated with measured propagation data to assess the stability of this process. For fair and uniform comparisons, 16 s time records from individual transmissions were examined for all sources, and the time fronts for each transmission (like the example shown in Fig. 2) were nominally centered within this time record. The time windows used for each source are listed in Table II. For completeness, every successful transmission was employed for this study and includes 224 individual signal samples for the nearest source (T6, 129 km downrange from the receiving array) and more than 350 samples for all other sources. The exact number of transmissions considered for each source is listed in Table II, as well as the date and time stamp for the first and final recordings considered. Last, many of the recordings from the shallowest ten receivers are dominated by array strumming noise. Although not present for all measurements from all sources, these receivers were omitted from this analysis in the interest of providing fair comparisons between all data-derived results in the study.

All recordings used in this portion of the study were cross correlated to the predicted fields defined in Sec. II. Probability density functions (PDFs) for all cross correlation measurements taken are shown in Fig. 8 at the center frequencies of the transmissions [Fig. 8(a)] and at a difference frequency of 0.5 Hz [Fig. 8(b)]. Given the time-front fluctuations observed during this experiment (due in part to the energetic eddy field observed in the Philippine Sea), the cross correlations between the measured and predicted pressure fields shown in Fig. 8(a) for the broadcast signal center frequencies are predictably abysmal and resemble the cross correlations between noise and a predicted field. However, the cross correlation results between the measured frequency-difference autoproduct [computed using Eq. (2)] and its corresponding predicted autoproduct shown in Fig. 8(b) for $Δf$ = 0.5 Hz are much higher and indicate the frequency-difference autoproduct has a predictable phase structure along the receiving array, even when there are challenging levels of ocean-induced time-front fluctuation. The highest measured correlation at this difference frequency was $χ=0.93$ (source at $rs=379 km$⁠), and all sources had at least one transmission where $χ>0.9$⁠. The correlations between the autoproduct and a genuine acoustic-pressure field are not shown, because they are not expected to be stable in this operating environment, as discussed in Sec. IV.

Given the results shown in Fig. 8, the difference frequency was swept from the minimum possible (⁠$Δf=0.0625 Hz$⁠, set by the signal sample time duration) up to $Δf=15 Hz$⁠, which is high enough so that even the correlations from the best correlating fields are suppressed by measured-to-predicted field mismatch. The results of this parameter study are shown in Fig. 9, with the results fitted to the simulated results in Fig. 7(b) (dark dashed curve) overlaid with the measurements. For all six sources, the non-dimensional quantity at which correlation is predicted by simulation to be lost is $ΔfΔτ≈0.3$⁠, and this is seen to be relatively accurate for measured correlations as well. The actual decorrelation point for each source exists in the range $0.30≤ΔfΔτ≤0.40$ (⁠$−0.53≤log10ΔfΔτ≤−0.4$⁠), and the simulation curve with loss falls nicely within the measurements as well. On average, 2.5 dB of loss in cross correlation is observed at low difference frequencies in measurement vs simulation and is likely caused by time-averaged mismatch between the sound speed profile used in the kraken calculations (it was measured near the PhilSea10 receiving array and assumed range-independent) and the actual range-dependent sound speed profiles existing in the ocean at the time of the measurements. Overall, Figs. 6, 7, and 9 indicate that acoustic fields and autoproducts decorrelate similarly based on the appropriate frequency time-fluctuation product, $fΔτ$ or $ΔfΔτ$⁠.

The purpose of this paper is to present results that extend the below-band laboratory water tank correlation findings in Lipa et al. (2018) to measured autoproduct fields constructed from recorded sound that propagated long distances through the deep ocean and to demonstrate that the autoproduct field is a predictable, measurable, and stable phenomenon in the deep ocean. For this purpose, measurements taken during the PhilSea10 experiment, as well as simulations of this experimental environment and geometry, were used to determine correlation results. For all six PhilSea10 sources considered, the frequency-difference autoproduct was found to correlate well, $χ≈0.8$⁠, to an approximate autoproduct at low difference frequencies (⁠$ΔfΔτ<0.2$⁠, where $Δτ$ is the root mean square propagation-time fluctuation). However, this cross correlation decayed to the noise level as the difference frequency was increased from several Hz to 15 Hz. The correspondence between correlation results obtained from simulations and measurements was found to be excellent, with the remaining discrepancies likely arising from features of the experiments, such as range and compass-heading dependence of the sound speed profile, that were neglected (for simplicity) within the simulations and differing or insufficient amounts of bandwidth averaging of the frequency-difference autoproduct.

The research reported here supports the following three conclusions. First, that the bandwidth-averaged frequency-difference autoproduct [as defined by Eq. (2)] constitutes a measurable quantity derived from broadband sound that propagated through the deep ocean. While the autoproduct requires a signal with some bandwidth, an analysis of the necessary amount of bandwidth is not presented. Second, that cross correlation between a measured and a predicted bandwidth-averaged frequency-difference autoproduct is a stable process at low difference frequencies in the deep ocean even for pressure fields in the hundreds of Hz, which have propagated over hundreds of kilometers. Third, although it is possible to correlate a frequency-difference autoproduct with an out-of-band pressure field at the difference frequency, this process is not stable for varied difference frequencies and source ranges. This lack of stability arises because the frequency-difference autoproduct does not include the caustic phase shifts or field dislocations that are commonly observed in refracted sound fields in the deep ocean. Thus, the frequency-difference autoproduct at best imperfectly correlates with an appropriate genuine acoustic field arising under the same conditions.

Given that the frequency-difference autoproduct is measurable and stable, it is possible that it also carries identifiable physical information. For example, this study shows that the difference frequency that causes autoproduct decorrelation (⁠$Δfd$⁠) is directly related to rms fluctuations in time fronts (⁠$Δτ ∼ 0.3/Δfd$⁠). Therefore, the frequency-difference autoproduct may offer remote sensing possibilities and insights into the character of the ocean environment that complement or confirm in-band acoustic-field analysis. Here, it should also be noted that the construction and examination of the frequency-difference autoproduct may take place entirely in post-processing using conventional in-band recordings. Therefore, there is nothing preventing both in- and out-of-band field possibilities from being analyzed for a given experimental dataset.

And finally, the primary results presented here are an extension of previous studies of the frequency-difference autoproducts to the much larger scale, more complicated, refracting environment of the deep ocean. This study demonstrates the capability of the frequency-difference autoproduct to mimic a predicted autoproduct computed from nominal environmental information. In particular, measured frequency-difference autoproducts show phase coherence with such a predicted (and readily calculated) autoproduct field [see Eq. (6)] over long ranges and over long periods of time in an environment with challenging levels of fluctuation. Furthermore, the current results conclusively demonstrate that the frequency-difference autoproduct maintains this phase coherence over much greater distances than does the recorded in-band field. The results of this study may be useful in other fields where guided waves are common, such as seismology, structural acoustics, or possibly even radar.

The authors would like to acknowledge useful discussions with Dr. Brian Worthmann and the support of the Office of Naval Research through Award No. N00014-16-1–2975. The authors would also like to acknowledge the sponsorship of the Raytheon Advanced Studies Scholarship for continuing graduate study.