Measurement of autoproduct fields in a Lloyd's mirror environment

Whenever an acoustic field is analyzed to obtain information about its source or the environment through which it has passed, the analysis nearly always seeks to extract such information from within the acoustic field's bandwidth. However, confinement of information extraction to frequencies within the bandwidth (i.e., to in-band frequencies) may not be a restriction in general. As shown recently (Worthmann and Dowling, 2017), nonzero-bandwidth acoustic fields may carry additional information at frequencies above and below the original field's bandwidth (i.e., at out-of-band frequencies). This out-of-band information may be extracted by forming the frequency-difference and frequency-sum autoproducts, which are quadratic products of frequency-domain complex field amplitudes at a pair of in-band frequencies. The frequency-difference and frequency-sum autoproducts can, when averaged through the signal bandwidth, mimic genuine acoustic fields at frequencies below and above the signal bandwidth, respectively. In the recent study, autoproducts were investigated theoretically in a Lloyd's mirror environment. In this study, those predictions are tested by measuring autoproduct fields in a laboratory water tank and comparing those fields to theoretical autoproduct fields and theoretical out-of-band fields. The results presented here show that experimentally measured autoproducts can correlate strongly with genuine out-of-band fields at more than 80% of the possible difference frequencies and at nearly 60% of the possible sum frequencies, percentages that compare favorably with theory.

The autoproducts share features with other better-known concepts. The shifting to lower and higher frequencies is also an attribute of the parametric array (Westervelt, 1963); however, the out-of-band frequencies generated by the parametric array exist due to nonlinearities in the propagation medium, whereas autoproducts do not physically exist, and are created through an intentional mathematical nonlinearity. The construction of autoproducts as a product of complex field amplitudes at a pair of frequencies is reminiscent of the mutual coherence function in statistical optics (Beran and Parrent, 1964; Born and Wolf, 1999) and the formative concepts of Δk-radar (Weissman, 1973; Popstefanija et al., 1993). Furthermore, single Fourier transforms of the frequency-difference autoproduct lead to the Wigner-Ville transform and the ambiguity function (Cohen, 1989).

Prior experimental studies have used autoproducts in the source-localization techniques of beamforming (Abadi et al., 2012; Douglass et al., 2017) and matched-field processing (Worthmann et al., 2015, 2017) within the realm of underwater acoustics. However, these techniques do not utilize the autoproduct fields themselves but rather cross-spectral density matrices, allowing the localization schemes to be independent of the source waveform. In contrast, this study measures autoproduct fields directly and requires a known source waveform to verify the claims in Worthmann and Dowling (2017). Moreover, although this study uses water as the propagation medium, the techniques outlined here need not be confined strictly to underwater acoustics, and could also apply to other acoustic-wave propagation regimes such as diagnostic ultrasound or room acoustics.

The results in Worthmann and Dowling (2017) predict that autoproduct fields will correlate strongly with genuine out-of-band acoustic fields when the in-band acoustic field is well described by elementary ray acoustics provided that two conditions are satisfied: (i) the gradient of the acoustic pressure is dominated by phase variations and not amplitude variations and (ii) multipath arrivals have sufficient temporal separation so that the products of the arrival-time differences with the signal bandwidth are much greater than unity. Simulations in a homogeneous acoustic half-space bounded by a flat, pressure-release surface (a Lloyd's mirror environment) confirmed these conditions. Regions very close to the source gave poor spatial cross-correlations between the autoproducts and genuine out-of-band acoustic fields due to a violation of condition (i), while regions near the reflecting surface gave poor cross-correlations due to a violation of condition (ii). Away from the source and surface, near-perfect cross-correlations were found. However, these predictions were made using a set of assumptions that may or may not apply in reality, e.g., that the acoustic environment is homogeneous with a perfectly known sound speed and perfectly known geometry. Thus, for this paper, experimental measurements were collected to quantitatively support or refute the achievability of the theory in practice.

The remainder of this paper is divided into four sections. Section II provides an overview of the relevant theory and defines the theoretical fields and comparison metrics used throughout the rest of the paper. Section III details the experimental method, including efforts to characterize the source and the acoustic environment. Section IV presents the measurements of autoproduct fields and the comparisons to theoretical fields. Section V provides a summary of this work and the conclusions drawn from it.

This section provides an overview of relevant autoproduct theory and specifies how the theory and measurements may be compared. Definitions for autoproduct quantities are provided first, including normalization and cross-correlation procedures. Theoretical acoustic and autoproduct fields for a Lloyd's mirror environment are then presented and discussed.

In this paper, $P(r,ω)$ represents the frequency-domain complex pressure field measured at position $r$ and angular frequency $ω$ due to an omnidirectional point source at position $rs$ that broadcasted a waveform with source spectrum $S(ω)$⁠. The frequency-difference autoproduct (⁠$APΔ$⁠) and frequency-sum autoproduct (⁠$APΣ$⁠), constructed from $P(r,ω)$⁠, are defined as

where the asterisk denotes a complex conjugate, and $ω±≡ω±Δω/2$ defines a pair of frequencies with difference frequency $Δω=ω+−ω−$ and sum frequency $Σω=ω++ω−=2ω$⁠. Bandwidth-averaged autoproducts, denoted by angle brackets $〈 〉$⁠, are defined as In this formulation, $ΩL≤ω≤ΩH$ defines the signal bandwidth, $ΩC=(ΩL+ΩH)/2$ defines the in-band center frequency, and $ΩBWΔ=ΩH−ΩL−Δω$ and $ΩBWΣ=min(2ΩH−Σω,Σω−2ΩL)$ define the signal bandwidth available for averaging $APΔ$ and $APΣ$⁠, respectively. By defining the limits on the integrals of Eqs. (2a) and (2b) in this manner, the integrands contain only acoustic-field amplitudes within the signal bandwidth, and encompass all possible in-band frequency pairs that have a difference of $Δω$ or a sum of $Σω$⁠. Note that $ΩBWΔ$ and $ΩBWΣ$ take on the maximum value of $ΩH−ΩL$ at $Δω=0$ and $Σω=2ΩC$⁠, respectively. Moreover, although the difference- and sum-frequency bandwidths (⁠$0≤Δω≤ΩH−ΩL$ and $2ΩL≤Σω≤2ΩH$⁠, respectively) may include some frequencies within the signal bandwidth, $Δω$ and $Σω$ are still termed out-of-band in this paper for simplicity. Finally, note that the definitions of the bandwidth-averaged autoproducts in Eqs. (2a) and (2b) include the source spectrum.

To compare autoproduct fields with out-of-band acoustic fields, a normalization must be performed since autoproducts and ordinary fields have different units (pressure-squared vs pressure). To normalize an acoustic or autoproduct field within a chosen region $V$⁠, the following procedure is applied, where $( )norm$ denotes the normalized version of a field quantity $( )$⁠:

After normalization, the figure of merit considered here for how closely two fields align is the complex spatial cross-correlation coefficient, $χ$⁠, defined by

where the “$Δ$” and “$Σ$” subscripts denote quantities associated with the frequency-difference and frequency-sum autoproducts, respectively, and $GΔ,Σ$ denotes the genuine out-of-band acoustic field that $⟨APΔ,Σ⟩$ should mimic. $GΔ,Σ$ is the Green's function (or impulse response) for the same acoustic environment evaluated at the out-of-band frequency $Δω$ or $Σω$⁠, respectively; however, it may satisfy different boundary conditions than the original in-band Green's function, $G(r,ω)≡P(r,ω)/S(ω)$ (see Worthmann and Dowling, 2017). The complex cross-correlation coefficient $χ$ is guaranteed to fall within the complex unit circle, and a perfect match of fields in phase and normalized magnitude within $V$ corresponds to $χ=1+0i$⁠, while a perfect lack of correlation corresponds to $χ=0$⁠. In addition, the expression $2(1−Re[χΔ,Σ])$ is equivalent to the L₂-norm of the difference between $⟨APΔ,Σ⟩ norm$ and $GΔ,Σnorm$⁠. In prior work (Worthmann and Dowling, 2017), only the magnitude of $χ$ was reported. [Furthermore, an error exists in Eq. (6) of that paper; the absolute value bars should be outside the integral, not inside.] However, the full complex value of $χ$ is considered here since a constant relative phase factor (such as an overall sign difference) between $⟨APΔ,Σ⟩$ and $GΔ,Σ$ can be considered to be a discrepancy between the fields.

To apply the preceding theory to the experimental Lloyd's mirror environment, start with a cylindrical coordinate system, where $r$ denotes the radial distance (range) from the source, $z$ denotes the axial distance from the surface (depth), and the source is at depth $z=d$⁠. In this environment, the theoretical in-band Green's function is

where $r1,2=r2+(z∓d)2$ are the direct and reflected path lengths, respectively, between the field point $(r,z)$ and source $(0,d)$⁠, $τ1,2=r1,2/c$ are the corresponding arrival times, and $c$ is the speed of sound for the propagation medium. The second term in Eq. (5), corresponding to the reflected path, includes the pressure-release surface reflection coefficient of −1.

Inserting the complex field from Eq. (5) into Eqs. (1) and (2) yields the following bandwidth-averaged autoproducts:

where $Δτ=τ2−τ1$ is the difference in arrival times, and $sinc(x)=sin (x)/x$⁠. The first two terms in Eqs. (6a) and (6b) arise from same-path products (direct-direct and reflected-reflected) and are the self terms. The last term in Eqs. (6a) and (6b) arises from different-path products (direct-reflected and reflected-direct, simplified to a single term) and is the cross term.

The self terms in Eqs. (6a) and (6b) are similar, but not identical, to the field in Eq. (5) evaluated at $Δω$ or $Σω$⁠; thus, they can potentially correlate strongly with an out-of-band acoustic field. Compared to Eq. (5), each self term has an additional factor of $r1$ or $r2$ in the denominator, so a strong correlation is obtained when $1/r1,2$ does not vary significantly in the region of interest (see Worthmann and Dowling, 2017), i.e., when $r1,2$ is much larger than $V$ in Eqs. (3) and (4). Additionally, the reflected-path (second) self term in both Eqs. (6a) and (6b) carries a positive sign, while the reflected-path term in Eq. (5) carries a negative sign. This sign difference arises because the quadratic nonlinearity rectifies the −1 reflection coefficient. Consequently, the autoproduct fields in Eq. (6) appear to mimic genuine out-of-band acoustic fields in a Lloyd's mirror environment having a rigid, rather than pressure-release, surface boundary condition. Further discussion of this modified reflection coefficient can be found in Worthmann and Dowling (2017).

The cross term in Eqs. (6a) and (6b) is not present in Eq. (5), and thus permits or inhibits the autoproducts' potential mimicry of genuine fields, depending on its magnitude compared to the self terms. The magnitude is controlled by a sinc function, which arises from the bandwidth averaging. In regions where $ΩBWΔ,ΣΔτ≫2π$ (i.e., away from the surface), the sinc factor suppresses the cross term and permits potentially strong cross-correlations between autoproducts and out-of-band fields. However, near the surface, an “interference layer” exists where the cross term is insufficiently suppressed, leading to poor cross-correlations. The nominal interference layer thickness, $h$⁠, is taken to be the region where $ΩBWΔ,ΣΔτ≤2π$⁠. For a given range, source depth, signal bandwidth, and difference- or sum-frequency selection, $h$ is given implicitly by

In regions where $r≫|hΔ,Σ±d|$ (i.e., where propagation directions are close to horizontal), this relationship simplifies to $hΔ,Σ∼πrc/ΩBWΔ,Σd$⁠. Thus, longer ranges thicken the interference layer, while greater bandwidth averaging and source depth shrink the interference layer, and these trends also hold when the small-angle approximation does not apply.

To measure bandwidth-averaged autoproduct fields and compare them to theory, the acoustic field, source waveform, and environmental parameters (geometry and sound speed) must be known. This section describes how these quantities were measured or characterized for a Lloyd's mirror laboratory environment.

In this study, an acoustic field was generated and measured in a 107-cm-diameter cylindrical plastic water tank, filled to a depth of 90 cm with fresh water. A sound source and single receiver were used to broadcast sound from a fixed location and record the pressure waveform at a variable location, respectively. Source and receiver locations were chosen to ensure that arrival times via the direct and surface-reflected paths were temporally well separated from arrival times via tank-bottom and tank-side-wall reflected paths. Sound traveling on these additional paths was removed by time gating the measured signals, allowing the finite water tank to imitate a semi-infinite half-space.

Figure 1 provides a schematic of the experimental setup in panel (a) and a corresponding photograph in panel (b). The source, an ITC-1042 transducer (International Transducer Corp., Santa Barbara, CA), was positioned at depth $d=$ 200 mm, and broadcasted a nominally 50-μs Gaussian-enveloped sinusoidal pulse with a nominal center frequency of 70 kHz. The receiver, a Reson TC4013 transducer (Teledyne Reson A/S, Slangerup, Denmark), was mounted via thin stainless-steel tubing to a 0.01-mm-resolution height gauge, and was placed at depths between 0 and 100 mm in increments of 1 mm, and at depths between 100 and 400 mm in increments of 5 mm. The finer spacing near the surface was chosen to carefully measure the autoproducts in regions where the cross terms may be poorly suppressed (see Sec. II B). Additionally, tank-spanning bars allowed the receiver assembly to translate in range, which allowed data collection from three different source-to-receiver ranges: $r=$ 175, 325, and 475 mm. The orientation of the transducers shown in Fig. 1(b) was chosen to best achieve broadcast and receiving directional uniformity. Additionally, for each range and depth pair, the source broadcasted three identical pulses, providing an opportunity to check the repeatability of the measurements. The signal-to-noise ratio observed in the experiment was approximately 40 dB. A total of 1449 recordings (161 depths, 3 ranges, and 3 trials) serve as the experimental field measurements used in the remainder of the paper.

To calculate the bandwidth-averaged autoproducts defined in Eq. (2), the source waveform and signal bandwidth must be known. While the nominal source waveform applied to the broadcast transducer could potentially provide this information, the imperfect frequency response of the transducer pair lead to noticeable distortion of the intended waveform. To compensate, a measured source waveform was determined from the time-domain acoustic-field measurements.

To determine the measured source waveform, direct-path waveforms were first extracted from recordings in which the arrival times were well separated (defined here as an arrival-time difference of 150 μs or more, where 50 μs before and after the nominal pulse duration were included to provide robustness). This requirement allowed 291 of the 1449 recordings to be used, which came primarily from the deeper sampling depths. These recordings were then scaled up by a factor of $r1$ (to counteract spherical spreading losses), shifted in time to maximize their temporal cross-correlations with the nominal source waveform, and coherently averaged. Figure 2 shows the nominal, extracted, and coherent-average waveforms in the time domain in panel (a) and frequency domain in panel (b). In each panel, the nominal waveform (dashed black curve) is given a maximum amplitude of unity, the coherent average (solid black curve) is scaled to contain the same total signal energy as the nominal waveform, and the 291 individual signal samples (semitransparent red curves) are scaled by the same factor as the coherent average.

Figure 2 reveals imperfections in the broadcast and receiving transducers' responses. In the time domain, the recorded waveforms were temporally stretched to a duration of approximately 75 μs, consistent across all 291 signal samples. In the frequency domain, a shift in spectral peak from 70 to approximately 80 kHz is observed, consistent with the resonant frequency of the broadcast transducer, nominally 79 kHz. Additionally, Fig. 2(b) shows an unexpected dip in direct-path waveform amplitudes near 63 kHz. A corresponding plot made using reflected-path waveforms (not shown) does not contain this dip. Thus, the origin of this dip is most likely frequency-dependent directionality of the broadcast and/or receiving transducers. Moreover, after correcting for spherical spreading losses, the coherent-average waveform for the reflected path had 1.3 dB more signal energy than for the direct path, providing further evidence for imperfectly uniform transducer directionality. This observed variation is consistent with the broadcast and receiving transducers' uniformity specifications, nominally ±1–2 dB.

Based on this source-characterization effort, the experimentally determined $S(ω)$ used to construct the measured fields is the coherent average of the direct-path waveforms, shown with the solid black curve in Fig. 2(b). Also, the experimentally determined bandwidth (defined here to be the range of frequencies that contain 99.9% of the recorded signal energy, rounded to the nearest 5 kHz) is $ΩL/2π$ = 40 kHz to $ΩH/2π$ = 110 kHz, with a measured center frequency of $ΩC/2π$ = 75 kHz.

Evaluation of the theoretical acoustic and autoproduct fields requires values for environmental parameters—specifically, the ranges and depths of the source and receiver, and the sound speed. Although the nominal experimental values could be used, inaccuracies of order 1% could significantly influence comparisons between measured and theoretical fields, since the values of $ωτ1,2$ in Eq. (5) are of order 10² for the bandwidth (40 to 110 kHz) and source-to-receiver ranges used in the experiment. Accurate environmental parameters are especially important at the higher frequencies of interest in this study, which can exceed 200 kHz for the frequency-sum autoproduct. Therefore, to mitigate the effects of experimental uncertainties and ensure the fairest comparisons of autoproduct measurements to theory, four experimental parameters were determined a posteriori by optimizing the cross-correlation of measured and theoretical in-band acoustic fields.

The four optimized parameters were a source-depth offset, receiver-depth offset, receiver-range offset, and sound-speed offset from the nominal experimental values. The source-depth offset (⁠$δd$⁠) was a vertical offset to the nominally 200-mm depth of the geometric center of the broadcast transducer. Throughout the experiment, receiver depth relative to an uncertain water-surface zero point was determined via the height gauge. The receiver-depth offset (⁠$δz$⁠) was a vertical offset to all measured receiver depths to account for uncertainty in the zero point, and the location of the receiver's acoustic center. Similarly, changes in source-to-receiver range were well known, but the spatial location of zero range was not because of the finite sizes of the transducers; the receiver-range offset (⁠$δr$⁠) accounted for this range uncertainty. Finally, the sound-speed offset (⁠$δc$⁠) allowed refinement of the nominal experimental sound speed of 1451.2 m/s, determined from a measured water temperature [see Eq. (5.6.8) in Kinsler et al., 2000].

The parameter optimization was performed by maximizing the overall correlation of measured in-band fields with theoretical in-band fields at all depths, trials, ranges, and in-band frequencies. The maximized quantity was the real part of the bandwidth-averaged cross-correlation between the measured and theoretical in-band fields, $Re[〈χIB〉]$⁠. Here, $χIB$⁠, and its bandwidth average $〈χIB〉$⁠, were determined from

where $δu=[δd, δz, δr,δc]$ is the vector of parameters to be optimized, $D$ = 400 mm is the deepest nominal depth of the measurements, and $Nt$ = 3 and $Nr$ = 3 are the number of trials and ranges, respectively. In Eq. (8a), $Pnm(z,ω)$ is the measured complex field at the nth range and mth trial, taken to occur at the nominal receiver depth incremented by $δz$⁠. $Gn(z,ω,δu)$ is the theoretical in-band Green's function for the nth range, calculated from Eq. (5) with the nominal source depth, receiver range, and sound speed incremented by $δd$⁠, $δr$⁠, and $δc$⁠, respectively. Here, and in the remainder of the paper, the normalization defined in Eq. (3) takes $V$ to be a vertical line segment spanning 400 mm of depth, and normalization is performed separately for each range and trial. In Eq. (8a), the integral gives the cross-correlation between the measured and theoretical in-band field at one range, trial, and frequency; the two summations average the cross-correlation over the three trials and three ranges. In Eq. (8b), the integral averages over all in-band frequencies to produce $〈χIB〉$⁠. Then, $Re[〈χIB〉]$ is maximized by varying $δu$⁠. Without optimization (i.e., $δu=[0,0,0,0]$⁠), $Re[〈χIB〉]=0.906$⁠. When a nonlinear optimization was performed, the following parameter corrections were found: $δd=+1.62 mm$⁠, $δz=−2.13 mm$⁠, $δr=−0.59 mm$⁠, and $δc=−0.76 m/s$⁠, yielding $Re[〈χIB〉]=0.980$⁠. All four fitted parameters are within their known or estimated ranges of uncertainty. In the remainder of this paper, unless otherwise specified, the optimized values of the source depth, receiver depth, receiver range, and sound speed are used.

Using the measurements described in Sec. III, experimental in-band fields and autoproduct fields may be computed and then compared to theoretical fields. This section presents these results visually with field plots and numerically with the cross-correlation coefficient $χ$⁠.

The in-band Green's functions—the constituents of the autoproduct fields—may be determined from the measured data as $Pnm(z,ω)/S(ω)$⁠, and from theory using Eq. (5). Figure 3 shows plots of the real parts of the measured and theoretical normalized Green's functions at different ranges (⁠$r$⁠) and frequencies (⁠$f=ω/2π$⁠). Panels (a), (b), and (c) of Fig. 3 are evaluated at the center frequency, 75 kHz, and at $r$ = 175, 325, and 475 mm, respectively. These three panels give a cross-correlation coefficient of $χIB=0.98−0.02i$⁠. Panels (d) and (e) are evaluated at the middle range of 325 mm, and at the lower and upper limits of the signal bandwidth, 40 and 110 kHz, respectively. In each plot, the horizontal axis is the depth (0 mm ≤ $z$ ≤ 400 mm), the vertical axis varies from −4 to +4, the normalized theoretical Green's function is the solid black curve, and the normalized measured Green's function is plotted with red ×'s for the three trials. As the plots show, the spatial sampling is five times finer for $z$ ≤ 100 mm, and the repeatability between trials is excellent. Overall, the measured data match the theoretical results well—as they should when the in-band field is well described by Eq. (5)—since the environmental parameters were optimized to maximize the cross-correlation between the measured and theoretical in-band fields.

From the in-band Green's functions, the bandwidth-averaged autoproducts, $⟨APΔ,Σ⟩$⁠, may be calculated using Eq. (2) and normalized via Eq. (3). Once normalized, the measured autoproduct fields may then be compared to the theoretical autoproduct fields from Eq. (6), and to theoretical out-of-band acoustic fields from Eq. (5) with the modified surface boundary condition discussed in Sec. II B. In Fig. 4, the normalized real parts of five $⟨APΔ⟩$ fields and five $⟨APΣ⟩$ fields are plotted vs depth on the horizontal axis. The red ×'s are the field measurements for the three trials, the solid black curves are the theoretical autoproduct fields, and the blue dotted curves are the theoretical out-of-band acoustic fields at the difference or sum frequencies. Additionally, the vertical dashed lines show the depth of the interference layer [see Eq. (7)], below which autoproduct fields are expected to correlate strongly with corresponding out-of-band fields. Panels (a), (b), and (c) of Fig. 4 are evaluated at a difference frequency (⁠$Δf=Δω/2π$⁠) of 30 kHz, and at $r$ = 175, 325, and 475 mm, respectively. Together, these three panels give $χΔ=0.96−0.03i$⁠, calculated as described below. Panels (d) and (e) show results for the 325-mm range at difference frequencies of 5 and 60 kHz, respectively. Panels (f), (g), and (h) are evaluated at the sum frequency (⁠$Σf=Σω/2π$⁠) of 150 kHz (twice the center frequency), and at $r$ = 175, 325, and 475 mm, respectively. These three panels give $χΣ=0.95+0.00i$⁠. Panels (i) and (j) are evaluated at the middle range of 325 mm, and at sum frequencies of 115 and 185 kHz, respectively, which are halfway between twice the center frequency and twice the lowest and highest in-band frequencies.

In all ten panels of Fig. 4, the measured autoproducts (red ×'s) closely follow the theoretical autoproducts (solid black curves), except for very shallow depths where the receiving transducer was not sufficiently submerged. Furthermore, trial repeatability is very good, and the measured autoproducts closely follow their associated theoretical out-of-band fields for depths below the interference layer (⁠$z>hΔ,Σ$⁠).

To provide a comparison of experimental and theoretical results at all possible frequencies, Fig. 5 shows the real parts of the cross-correlations between experimental fields and genuine acoustic fields as a function of below-band (⁠$Δω)$⁠, in-band (⁠$ω)$⁠, or above-band (⁠$Σω)$ frequency. At each frequency, the cross-correlation encompasses all depths, ranges, and trials. The in-band cross-correlation $χIB$⁠, discussed in Sec. III C, is given in Eq. (8a). The below- and above-band cross-correlations, $χΔ$ and $χΣ$⁠, respectively, are given analogously by

where $〈APΔ,Σ〉nm$ is the measured bandwidth-averaged autoproduct at the nth range and mth trial, and $Gn$ uses the modified (+1) surface reflection coefficient, unlike in Eq. (8a).

The black (central) curves in Fig. 5 show cross-correlations for the in-band field as a function of in-band frequency. The solid black curve is $Re[χIB]$⁠, and compares the in-band field measurements to the theoretical in-band fields. The average of this curve over the signal bandwidth is 0.980, as described in Sec. III C. The black dashed line represents the maximum possible cross-correlation, which for the in-band field is simply unity. The thin black dotted curve is $Re[χIB]$ without optimization (i.e., $δu$ = [0,0,0,0]), and provides a measure of the robustness (or sensitivity) to experimental uncertainties. The cross-correlation between the measured and theoretical data (solid black curve) exhibits a noticeable dip around 63 kHz, dropping by approximately 0.03. This drop in cross-correlation is attributed to the directionalities of the broadcast and receiving transducers (see Sec. III B). The steady drop in the cross-correlation for the unoptimized in-band results with increasing frequency (decreasing wavelength) is understandable since the cross-correlation is sensitive to path-length variations whenever they are a nonnegligible fraction of the acoustic wavelength.

The blue (red) curves on the left (right) of Fig. 5 show the cross-correlations between the frequency-difference (-sum) autoproducts and the corresponding theoretical acoustic fields at the difference (sum) frequencies. The frequency-difference autoproduct is plotted from 0 to 70 kHz, the full difference-frequency bandwidth, and the frequency-sum autoproduct is plotted from 80 to 220 kHz, the full sum-frequency bandwidth. The solid curves show $Re[χΔ,Σ]$ from Eq. (9), and specify how well the measured autoproduct fields match their corresponding genuine out-of-band fields. The dashed curves show the cross-correlation between theoretical autoproduct fields and theoretical out-of-band fields, and are an upper bound for the performance of the measured data shown with the solid curves. The inability of the measured autoproducts to achieve the upper bound can again be linked to the transducers' directionalities. The drop in cross-correlation at high $Δω$ and at low and high $Σω$ can be attributed to the larger thickness of the interference layer relative to the total size of the normalization region $V$⁠, or to the insufficient signal bandwidth available for suppressing the cross terms via averaging, $ΩBWΔ,Σ$⁠. Similarly, the highest cross-correlations occur at frequencies where $ΩBWΔ,Σ$ is a maximum and the interference layers are thinnest (⁠$Δω=0$ and $Σω=2ΩC$⁠). Figure 5 shows that the real part of the cross-correlation between measured autoproduct fields and theoretical out-of-band fields exceeds 0.90 for difference frequencies between 0 and 60 kHz, and for sum frequencies between 110 and 190 kHz, which correspond to 86% and 58% of the difference- and sum-frequency bandwidths, respectively. Theory predicts these percentages to be 89% and 77%, respectively.

In Fig. 5, the blue (red) dotted curve shows the cross-correlation between measured frequency-difference (-sum) autoproduct fields and their corresponding theoretical out-of-band fields when the nominal experimental parameters without optimization are used. Thus, the sensitivity of the autoproducts to imperfect environmental parameters is easily seen. As expected for low frequencies, the blue dotted curve is not significantly lower than the results with optimized parameters (solid blue curve), implying that the frequency-difference autoproduct is robust to environmental uncertainties. The red dotted curve, however, falls well below the results with optimized parameters (red solid curve), showing the expected sensitivity to environmental uncertainties at high frequencies. This sensitivity motivated the in-band optimization described in Sec. III C. In both cases, the divergence between the results with optimized and unoptimized parameters generally grows with increasing difference or sum frequency. Again, this fact is consistent with fixed distance-and-sound-speed errors but increasing frequency. The imaginary parts of the cross-correlation $χ$ were omitted from Fig. 5 for clarity and because they are typically small compared to the real parts.

The purpose of this study was to determine whether the theoretical claims made in prior work (Worthmann and Dowling, 2017) about the ability of autoproducts to mimic genuine out-of-band acoustic fields could be realized experimentally in a Lloyd's mirror environment. Measurements of a 40- to 110-kHz acoustic field in a laboratory water tank were used to calculate bandwidth-averaged autoproducts. After characterizing the source waveform and acoustic environment, these autoproducts were found to correlate very well with genuine out-of-band acoustic fields (⁠$Re[χΔ,Σ]≥0.90$⁠) for difference frequencies between 0 and 60 kHz, and for sum frequencies between 110 and 190 kHz. The correspondence of measured and theoretical fields was found to be excellent. The observed minor discrepancies likely arise primarily from the (uncompensated) directionalities of the broadcast and receiving transducer pair.

This research effort supports the following three conclusions. First, frequency-difference and frequency-sum autoproduct fields constructed from acoustic-field measurements in a Lloyd's mirror environment can mimic out-of-band acoustic fields at the difference and sum frequencies in an equivalent environment with a modified surface reflection coefficient. This conclusion supports the unconventional claim made in prior work (Worthmann and Dowling, 2017) that acoustic fields with nonzero bandwidth may provide acoustic-field information at frequencies outside this bandwidth. This conclusion is drawn from Figs. 4 and 5, which show high spatial correlations between measured autoproducts and out-of-band fields for the same environment with the modified reflection coefficient. In addition, the theoretically predicted deviations between autoproducts and genuine acoustic fields within the near-surface interference layer were observed experimentally, providing further support for the predictions and limitations stated in prior work.

Second, the experimental effort needed to successfully measure the autoproducts does not appear to exceed that of successfully measuring the in-band Green's function. Efforts in this study to characterize the source waveform and determine accurate environmental parameters using in-band analysis serve to directly improve in-band results, and indirectly improve autoproduct results. In other acoustic environments or experimental setups, additional or reduced efforts of this sort may be required. However, no experimental difficulties unique to measuring autoproducts have been uncovered by this study.

Third, the frequency-difference autoproduct is less sensitive to mild mismatch between the actual and measured distances and sound speed than the frequency-sum autoproduct. However, both display a general trend of increasing mismatch sensitivity with increasing difference or sum frequency. This sensitivity trend is consistent with the expectations for genuine sound fields at increasing acoustic frequencies, and thus supports the first conclusion as well.

This research was supported by the Office of Naval Research under Award No. N00014-16-1-2975 and the National Science Foundation Grant Fund No. DGE 1256260.