The effects of refraction and caustics on autoproducts

In linear acoustics, a broadband source creates an acoustic field containing those same frequencies—herein, termed the in-band field. Interestingly, quadratic products of the complex amplitudes of the in-band field (autoproducts) have recently been shown to mimic an out-of-band field—a field with frequency content either below or above the original bandwidth (Worthmann and Dowling, 2017). The properties of the autoproducts have been found to improve a variety of signal processing techniques, such as beamforming with sparse arrays (Abadi et al., 2012; Douglass et al., 2017; Douglass and Dowling, 2019), matched field processing in uncertain environments (Worthmann et al., 2015, 2017; Geroski and Dowling, 2019), and passive cavitation imaging for higher resolution (Abadi et al., 2017).

The extent of the autoproducts' field mimicry of genuine out-of-band acoustic fields has not been extensively studied. However, the available theoretical (Worthmann and Dowling, 2017) and experimental (Lipa et al., 2018) studies have found that for acoustic fields adequately described by the ray approximation, the autoproducts' mimicry of out-of-band acoustic fields has two primary limitations. First, the amplitude of the rays must be sufficiently slowly varying in space relative to the wavelength. Second, in multipath environments, the minimum difference in ray arrival times should be larger than the inverse of the bandwidth available for averaging. Under these two conditions, autoproducts have been shown to have strong cross correlations with out-of-band acoustic fields.

Not all acoustic fields adequately satisfy the ray approximation. The effects of diffraction, in particular, are not captured in the most elementary ray approximation. The effects of diffraction on the autoproducts are interesting to study because the inherent frequency dependence must somehow limit the extent to which the autoproducts and their bandwidth averages mimic out-of-band fields. Two separate studies of the effects of diffraction on the autoproducts have been completed. The first study by Worthmann and Dowling (2020), a companion to this one, addresses the autoproducts' ability to mimic out-of-band fields in the presence of barriers where only diffracted sound appears in ray-acoustic shadow zones. The results from this first study indicate that diffraction does not significantly impact the ability of autoproducts to successfully mimic out-of-band fields except at shadow-zone edges. The second study, this one, addresses the autoproducts' ability to mimic out-of-band fields in environments with strong refraction where caustics are formed, and only diffracted sound appears outside the caustic. Near and beyond caustics, some detrimental effects on the autoproducts' mimicry of out-of-band fields are expected at a level comparable to the barrier diffraction results. However, even more interesting are the cumulative effects that caustics may have on the acoustic waves that have traveled through them, particularly when such waves are no longer near caustics.

Caustics occur when refraction causes neighboring ray paths to cross, a situation that is common in the ocean. The locus of these ray-path crossing points, the caustic, typically forms a one-dimensional curve in a two-dimensional range-depth plane. On one side of this caustic curve, at least two rays reach every point, while on the other side, there may be no rays. Within elementary ray acoustics (i.e., the solution to the eikonal and transport equations only), the field amplitude evaluated on a caustic is predicted to be infinite (because a ray's cross-sectional area drops to zero), while just beyond the caustic, the field amplitude is predicted to be identically zero (due to the absence of rays; Chapman, 2004; Jensen et al., 2011). These field features are, of course, unphysical as diffraction would enforce continuous, finite field amplitudes around the caustic, but it does so with a frequency dependence, which may or may not be compatible with what is needed for autoproducts to mimic out-of-band fields. Strictly speaking, there are many types of caustics, including fold-type, cusp-type, swallowtail-type, butterfly-type, among others (Kravtsov and Orlov, 1993). For the purposes of this study, only the simplest type—fold-type caustics—are considered, although some generalizations are given to accommodate these higher-order caustics.

To study the effects of caustics on autoproducts, simple environments that exhibit caustic phenomena are desired. This preference for simplicity is twofold. First, the effects of caustics can be studied separately from other wave propagation effects, such as reflections. And, second, exact solutions to the Helmholtz equation can be used both for computational speed and suppression of grid-resolution and numerical-solution artifacts. For this study, a range-independent waveguide with a depth-dependent sound speed profile is chosen, and Helmholtz-equation solutions in the form of modal sums are utilized. The chosen waveguides reside in unbounded environments with continuous sound speed profiles and do not include reflecting boundaries, which may direct sound into the ray-acoustics shadow zone beyond caustics. Furthermore, to avoid unphysical effects, the chosen sound speeds should not diverge to infinity or approach zero anywhere. Finally, for mathematical simplicity, waveguides that are symmetric in depth (⁠$z$⁠) are considered because the field from sources placed on the axis of symmetry (⁠$z=0$⁠) only require even-symmetry modes.

To satisfy these requirements, refractive index, $n(z)≡c∞/c(z)$⁠, profiles of the following form are considered:

where $n0$ is the maximum refractive index in the medium (strictly greater than unity and occurs at $z=0$⁠), $L$ is the half-height of the inhomogeneous portion of the waveguide, which is symmetric in depth about $z=0$⁠, $c∞$ is the sound speed at depths $|z|≥L$⁠, and $f$ is a function to be chosen. Its properties are $f(0)=0, f(1)=1$⁠, $f$ is continuous, and $f$ varies monotonically between $|z|/L=0$ and $|z|/L=1$⁠. Furthermore, $f$ should correspond to a sound speed profile that creates caustics and have a closed form solution to the depth-separated Helmholtz equation,

where $k∞=ω/c∞$ is the wavenumber in the homogeneous portion of the environment medium (⁠$|z|>L$⁠), and $ψ$ and $kr$ are the mode shapes and corresponding eigenvalues, respectively. Subject to all of these constraints, the authors have found two possible choices for $f(x)$⁠, namely, $x$ and $x2$⁠, referred to hereafter as the $n2$-linear and $n2$-quadratic sound speed profiles, respectively. Figure 1 is a plot of the sound speed versus depth for $n2$-quadratic (solid line) and $n2$-linear (dotted line) profiles, using $c∞, n0,$ and $L$ of 1500 m/s, 1500/1450, and 100 m, respectively.

Use of modal decomposition in an unbounded environment can create some mathematical difficulties. Chief among these is the presence of so-called leaky modes or modes which correspond to sound that propagates in both the inhomogeneous and homogeneous regions (Jensen et al., 2011). This is in contrast to trapped modes, which correspond to sound that propagates in the inhomogeneous region but decay exponentially in the homogeneous region, and evanescent modes, which decay exponentially in both regions. Acoustic fields in realistic ocean-waveguide environments may also include branch-cut contributions from a discontinuous sound speed profile (e.g., at the water–sediment interface in the shallow ocean), although the sound-speed continuity requirement specified here does not require such a branch cut (Bartberger, 1977; Stickler, 1975). Evanescent modes can typically be safely neglected several wavelengths from the source. Trapped modes are essential for accurately determining the field in the inhomogeneous region. Leaky modes are important in the homogeneous region, but the physics of interest here occurs within the inhomogeneous region, where leaky modes may still be important. Determination of trapped modes is typically straightforward, but leaky modes can present many difficulties (Labianca, 1973; Tindle et al., 1976; Tindle, 1979; Stickler and Ammicht, 1980; Buckingham and Giddens, 2006). For the two environments considered here, leaky modes are neglected in the $n2$-quadratic profile and evaluated approximately in the $n2$-linear profile. Omission of the leaky modes does not locally affect the satisfaction of the wave equation—however, without the leaky modes, the point source that launches the field is no longer perfectly omnidirectional. Overall, the trapped modes should provide the diffraction behavior sought in this study, but including the leaky modes allows for more confident quantitative results. Other numerical wave propagation schemes can also be used (e.g., parabolic equation solvers like RAM; Collins, 1996), although their satisfaction of the wave equation may no longer be mathematically exact and may be computationally expensive at high frequencies.

In Sec. II, autoproducts and their associated cross correlations are defined. In Sec. III, the acoustic field in the $n2$-quadratic profile is found, autoproducts are computed, and cross correlations between autoproducts and out-of-band fields are presented alongside a discussion of the results. In Sec. IV, analogous results are provided for the $n2$-linear profile. In Sec. V, the results from this study are summarized, and the conclusions that can be drawn from it are provided.

Define $P(r,ω)$ to be the solution to the inhomogeneous Helmholtz equation,

where $rs$ is the location of the source. Notably, the definition of $P(r,ω)$ here is synonymous with a Green's function for a single-frequency source. In terms of measured data, the source waveform is assumed to be known so that an impulse response can be estimated from the data.

Bandwidth-averaged frequency-difference (subscript $Δ$⁠) and frequency-sum (subscript $Σ$⁠) autoproducts are defined as

where $Σω=ΩL+ΩH$⁠, and the bandwidth of the original acoustic field is contained within $ΩL≤ω≤ΩH$⁠. The expectation is that these bandwidth averaged frequency-difference and frequency-sum autoproducts may mimic acoustic fields at the difference frequency (⁠$Δω$⁠) and sum frequency (⁠$Σω$⁠), respectively, even when $Δω<ΩL$ or $Σω>ΩH$⁠. A direct equivalence between autoproducts and genuine out-of-band acoustic fields is not possible because they have different units (pressure-squared versus pressure). To correct this, a normalization scheme is defined where autoproducts and out-of-band fields are divided by their root-mean-square amplitudes computed within a certain normalization region, $V$⁠. Then, after normalization, a cross correlation metric can be defined which quantifies how closely autoproducts mimic out-of-band fields over the normalization region. This is given by

where $PΔ,Σ(r)$ refers to a genuine acoustic field at the difference or sum frequency, $Δω$ or $Σω$⁠, respectively. Here, the normalization regions are defined to be rectangular regions in range and depth. Notably, $dV$ is evaluated as $drdz$⁠, not $rdrdz$⁠. And finally, an autoproduct is nominally considered to mimic an acoustic field at the difference or sum frequency when $|χΔ,Σ|≥0.90.$

In this section, a depth-symmetric and range-independent waveguide with refractive index given by Eq. (1) with $f(x)=x2$ is detailed. In this environment, trapped modes take on the form of confluent hypergeometric functions (Abramowitz and Stegun, 1965), also known as Kummer M functions, which are matched at the $|z|=L$ boundary with exponentially decaying functions. Here, only trapped modes (⁠$k∞≤kr≤n0k∞$⁠) are considered, primarily due to the prohibitively large computational expense associated with the complex root finding algorithms that would be necessary to include leaky modes, too. Evaluating confluent hypergeometric functions with complex parameters is especially expensive because a symbolic calculation is required instead of a simple floating-point calculation (Nardin et al., 1992).

For this study, the following parameters are chosen: $c∞=1500$ m/s, $n0=1500/1450$⁠, and $L=100$ m. These acoustic fields are studied over depths from 0 to $±$100 m and over ranges from 0 to 5 km. Figure 2(a) indicates the trajectory of the rays as calculated from a source at $(r,z)=(0,0)$⁠, and Fig. 2(b) indicates the transmission loss (amplitude) from the modal sum calculation on a decibel (logarithmic) color scale for a 5 kHz source at the origin.

Figure 2(a) shows that some of the rays propagating at high angles (relative to horizontal) escape from the domain into the homogeneous region (not pictured), where they continue on straight-line path trajectories to $|z|→∞$⁠; these rays roughly correspond to the leaky modes because they indicate acoustic energy is lost (leaked out) from the inhomogeneous part of the waveguide. Other rays with shallower launch angles are able to return to $z=0$ and continue in a cyclic manner up and down the waveguide. One of the salient features in Fig. 2 is the apparent ray cycle distance of slightly more than 1 km. In this environment, the shallow angle rays seem to converge to the same range—however, this is only approximately true as can be seen at the third or fourth convergence zone, where the ray intersections begin to smear out in range. The caustics that are formed can be found just before each of the convergence zones, where adjacent rays get very close together and, while not immediately obvious from the ray diagram, actually cross each other. It is more obvious in the amplitude plot in Fig. 2(b), where the acoustic field has a much higher amplitude along the caustic. Notably, because this field was calculated with a modal decomposition, the field at the caustic does not have an infinite amplitude as predicted by ray theory, nor does the nearby shadow zone contain amplitudes of identically zero.

Other interesting features in Fig. 2(b) are the interference patterns apparent at depths near $|z|=L$⁠, which can be seen as a sort of checkerboard pattern. Additionally, there appears to be no sound propagating at high angles (see the deep blue region near $r=0$⁠). These behaviors are artifacts of not including the leaky modes—the acoustic field for a true point source would not have this checkerboard pattern and would have sound propagating at high angles into the homogeneous medium. Despite these artifacts, the qualitative results described here are expected to be accurate despite the omission of the computationally challenging leaky modes because the retained trapped modes do include diffraction.

In Fig. 3(a), the magnitude of a genuine acoustic field at the difference frequency $Δω/2π=500$ Hz is shown above Fig. 3(b), where the magnitude of the corresponding bandwidth-averaged frequency-difference autoproduct $⟨APΔ⟩$ is given. Here, the bandwidth of the in-band field is $ΩL/2π$= 1 kHz to $ΩH/2π$ = 5 kHz. The color scale for Fig. 3(a) spans 25 dB, while for Fig. 3(b), it spans double that—the doubled decibel scale is a result of the fact that the autoproducts are formed from a multiplication of two in-band fields and, thus, have effectively twice the transmission loss as a single in-band field.

It is evident in Fig. 3 that the autoproduct amplitudes do not match the out-of-band field amplitudes particularly well. Part of that is due to the checkerboard patterns in Fig. 3(a), which are related to the absence of the leaky modes—the checkerboard pattern in Fig. 3(b) is suppressed by the bandwidth averaging defined in Eq. (4). The autoproduct field amplitude increase near caustics in Fig. 3(b) is narrow in size compared to the acoustic field amplitude increase near caustics in Fig. 3(a); their width in Fig. 3(b) is on the order of the wavelength at the in-band frequencies, not the out-of-band frequencies. The spatial extent of the acoustic field amplitude increase near the convergence zones in Fig. 3(b) is similarly smaller compared to their analogous extents in Fig. 3(a). However, these plots only indicate magnitude, not phase. At 500 Hz, phase varies too quickly to be conveniently displayed in a figure. Nevertheless, phase matching is crucial for high cross correlation values, whereas amplitude mismatch is not so severely detrimental.

The cross correlation between the autoproduct and out-of-band fields shown in Fig. 3 is shown in Fig. 4 with the magnitude and phase plotted separately. In Fig. 4(a), the magnitude of $χΔ$ is plotted on a linear color scale with strong cross correlations near unity corresponding to white and poor cross correlations near zero corresponding to black. In Fig. 4(b), the phase angle of $χΔ$ is plotted with cyan corresponding to a phase angle of zero. The normalization regions here are rectangles that extend 50 m in range and 5 m in depth.

Figure 4(a) shows that the magnitude of the cross correlations in the well-ensonified regions (i.e., where rays are an adequate description of the local acoustic field) are very nearly unity. This magnitude plot also suggests mediocre cross correlation results on the edge of the well-ensonified region and in the shadow zones. These modest cross correlations are primarily a result of the amplitude variations near the caustic and are not phase variations. In the shadow zone, there are some regions with poor cross correlations; however, due to the lack of leaky modes, the acoustic fields simulated in the shadow zones are not necessarily trustworthy. Furthermore, fields deep in the shadow zone would likely suffer from poor signal-to-noise ratios in practical applications, therefore, any autoproduct discrepancies in the shadow zone would likely be dominated by noise-related discrepancies.

In Fig. 4(b), the phases of the cross correlations show a more interesting result. Whereas studies in barrier diffraction effects have shown $χΔ$ phases generally well under 90° (Worthmann and Dowling, 2020), these plots show the full $±$180° variation. More interestingly, between convergence zones, the well-ensonified region seems to show a constant phase shift. As the sound passes through each convergence zone, there appears to be a net +90° phase shift in the cross correlation. This behavior is due to the caustic phase shift.

In elementary ray acoustics, infinite field amplitudes develop at caustics. However, if an additional term is carried out in the ray acoustics expansion, the so-called WKB (Wentzel-Kramers-Brillouin) approximation, this amplitude can be shown to be finite, but an additional phase factor of $−i$ develops for all acoustic waves passing through the caustic (Jensen et al., 2011; Ludwig, 1966). For environments with more complicated types of caustics (i.e., cusp-type), this caustic phase can become $−1$⁠, or for particularly more exotic types of caustics, this phase shift can even be $+i$ (Kravtsov and Orlov, 1993). A generalization of this phase shift for all caustics types is the KMAH index (Cerveny, 1977; Chapman, 2004; Kravtsov and Orlov, 1993), which is an integer describing the effective number of simple (i.e., fold-type) caustics a particular ray has passed through; the net phase shift that acoustic fields described by that ray inherit, then, is $(−i)n$⁠, where $n$ is the cumulative KMAH index of that ray.

To understand the effect of this caustic phase shift on the phase of $χΔ$⁠, it is convenient to think of a ray passing through a caustic as comparable to a ray reflecting off a boundary where the acoustic field experiences an effective reflection coefficient of $R$⁠, where $R=−i$ for simple caustics. The frequency-difference autoproduct, which is computed as $P(ω+)P*(ω−)$⁠, sees an effective reflection coefficient of $|−i|2=1$⁠. In other words, the frequency-difference autoproduct does not include a caustic phase shift. Therefore, in the cross correlation definition in Eq. (6), when $APΔPΔ*$ is calculated, the net phase observed is $+i$ or +90°: exactly the same as observed in Fig. 4(b) after the rays have passed through the first caustic. After each caustic, there is an accumulation of a +90° phase shift on $χΔ$⁠. This is clearly seen in Fig. 4(b), where the phase of $χΔ$ in the regions between caustic zones varies with range as 0°, +90°, $+$ 180°, and −90°, and the edge of the next 0° region can be seen near the 5 km range.

Figures 5 and 6 are analogous to Figs. 3 and 4, respectively, except that they depict the frequency-sum autoproduct instead, where the sum frequency $Σω/2π$ is 6 kHz, averaged over a difference frequency bandwidth from zero to $(ΩH−ΩL)/2π=(5−1) kHz=4 kHz$⁠. Specifically, Fig. 5(a) shows an amplitude plot of an out-of-band field at 6 kHz, Fig. 5(b) shows an amplitude plot of the frequency-sum autoproduct with twice the dynamic range, and in Figs. 6(a) and 6(b), the magnitude and phase of $χΣ$⁠, the cross correlation between the two plots shown in Figs. 5(a) and 5(b), respectively, are shown. Note that the normalization region in these plots is also defined as rectangles that are 50 m in range and 5 m in depth.

Figures 5(a) and 5(b) are much more similar to one another than Figs. 3(a) and 3(b), particularly, in the relative sizes of the caustics and convergence zones. This is primarily because the ratio of the center frequency (3 kHz) to the difference frequency (500 Hz) is larger than the ratio of the sum frequency (6 kHz) to the center frequency (3 kHz). As a result, Fig. 6(a) shows stronger cross correlations near the caustics than Fig. 4(a) does. Additionally, it can be seen in Fig. 6(b) that the frequency-sum autoproduct exhibits similar behavior in the phase of $χΣ$ as $χΔ$ does in Fig. 4(b) except that the phase trend is reversed. Because the frequency-sum autoproduct is defined as $P(ω2)P(ω1)$⁠, the effective reflection coefficient for passing through a caustic is $(−i)2=−1$⁠. Then, calculating $χΣ$ as proportional to $APΣPΣ*$⁠, the net phase is, thus, $(−1)(−i)*=−i$⁠, or a −90° phase shift for each caustic that is passed through. This can be found in Fig. 6(b), where the phase between convergence zones for increasing range varies as 0°, −90°, ±180°, +90°, and then followed again by 0° at the far right of Fig. 6(b).

Overall, the −90° caustic phase shift causes phase shifts in $χΔ,Σ$ that are proportional to the number of caustics through which a ray has passed. However, the magnitude of the cross correlation is generally still quite high in well-ensonified regions, suggesting that autoproduct-based source localization techniques will not be impeded significantly. These findings are limited to the $n2$-quadratic profile, where the majority of the well-ensonified regions contain only one ray path. This changes for other profiles, such as the $n2$-linear profile.

In this section, the range-independent waveguide has a refractive index given by Eq. (1), where $f(x)=x$⁠. In this environment, mode shapes in the inhomogeneous region are given by Airy functions (of the first and second kind) and matched across the $|z|=L$ with either exponential decaying mode shapes in the homogeneous region for trapped modes or sinusoidal varying mode shapes in the homogeneous region for leaky modes. Both trapped and leaky modes are calculated exactly, although the eigenvalue root-finding calculation for the leaky modes is performed numerically. Details of this derivation are omitted here for brevity (see Worthmann, 2019).

As in Sec. III, the following parameters are used: $c∞=1500$ m/s, $n0=1500/1450$⁠, and $L=$ 100 m, with the plot domain varying from 0 to 5 km in range and $±$100 m in depth. Figures 7(a) and 7(b) show the ray trace and the transmission loss (amplitude) plots, respectively, for a source at $(r,z)=(0,0)$ and frequency of 5 kHz. Figure 7(b) has a dynamic range of 100 dB.

In this environment, there are still rays that propagate at high angles (which correspond to leaky modes) and rays that propagate at low angles (which correspond to trapped modes). In this environment, there are many caustics, which all stem from the origin. Additionally, there are spatial regions reached by a variety of ray numbers, including zero rays (e.g., in the shadow zones, as seen in the A regions of Fig. 7(a), one ray [e.g., near the source, and after the first caustic, as seen in the B regions of Fig. 7(a)], two rays [e.g., the roughly triangular region where rays approach and return from the first caustic, as seen in the C regions of Fig. 7(a)], and many rays [such as the rectangular region with three rays, where two are approaching and returning from the second caustic, and a third ray has exited the first caustic and is passing through the second caustic, as seen in the D regions of Fig. 7(a)].

Notably, due to the inclusion of the leaky modes, the acoustic field amplitudes at high propagation angles are nonzero and close to the source, the amplitudes decay as approximately $1/r$⁠, where $r$ is the distance from the source as anticipated for a point source before refraction has had a significant effect on the wave propagation. The shadow zones have amplitudes approximately 30 dB below the well-ensonified regions, and the structure observed in the shadow zones, including the pattern of nulls in depth, agrees with numerical codes and arise due to the inclusion of the leaky modes.

In Fig. 8(a), the magnitude of the out-of-band field is plotted at the difference frequency $Δω/2π=$ 1 kHz, and in Fig. 8(b), the magnitude of the autoproduct is plotted for the same difference frequency but bandwidth averaged between $ΩL/2π=$ 4 kHz and $ΩH/2π=$ 6 kHz. Figure 8(a) also spans 100 dB, and for reasons similar to those in Figs. 3 and 5, the autoproduct plot spans 200 dB.

In Fig. 8, just as in Fig. 3, the downshifting of frequencies from a center frequency of 5 kHz to a difference frequency of 1 kHz creates significant differences in amplitudes. Near the caustics in Fig. 8(b) but closer to the sound-channel axis, there exist striations that run parallel to the caustic, which correspond to the interference layer. Interference layer is a term that is used to describe the region where rays have a time-difference-of-arrival that is smaller than the inverse of the bandwidth available for averaging, which, here, is 1 kHz (see Worthmann and Dowling, 2017, for further discussion of interference layers). Still closer to the axis than this interference layer, there is an interference pattern comprised of a pattern of peaks that appears approximately as distorted hexagons. These hexagons are roughly the same size as in the genuine out-of-band field in Fig. 8(a), but they appear at slightly different locations. In this case, hexagons appear due to the presence of three rays traveling in different directions. The sizes of the hexagons are proportional to the wavelength at the difference frequency, and their shapes are associated with the relative angles of the three rays contributing to the acoustic field in that region.

The cross correlations between Figs. 8(a) and 8(b) can be found in Fig. 9, where the normalization regions are rectangles spanning 50 m in range and 2 m in depth. In terms of the magnitude of $χΔ$⁠, the high-angle regions near the source have strong cross correlations. Even the first shadow zone, along with subsequent shadow zones, have strong cross correlations—except at the shadow zone nulls, where poorer cross correlations exist—likely due to amplitude variations. Generally, it seems that in single-ray-path regions, there are strong cross correlations. In the single-ray-path regions, the caustic phase shift can be seen in Fig. 9(b) as well, where a cumulative +90° phase shift is observed in $χΔ$ after each caustic.

However, unlike in the $n2$-quadratic profile, this $n2$-linear profile features regions with multiple ray paths. In particular, there are regions where two rays exist—one that is approaching the caustic and another that has passed through it and is now leaving the caustic. In these environments, much poorer cross correlation magnitudes are observed. In these regions, there also exist some exceptionally poor (near zero) cross correlations, particularly, along lines that appear to travel parallel to the caustic but in the two-path regions. These arise due to interference layers, which are associated with not having sufficient bandwidth with which to suppress the cross terms of the autoproduct by averaging. This effect, combined with the amplitude variations associated with the larger downshift in frequency (from 5 kHz center frequency to 1 kHz difference frequency), creates many regions of poor cross correlations in Fig. 9(a).

Another set of autoproducts is, thus, defined such that it does not shift down in frequency as far and uses much more bandwidth for averaging. In Fig. 10(a), the magnitude of the field at a difference frequency of 5 kHz is shown alongside Fig. 10(b), where a bandwidth averaged autoproduct is given with a $Δω/2π$ equal to 5 kHz, using an in-band bandwidth of $ΩL/2π=$ 1 kHz to $ΩH/2π=$ 9 kHz. Note that in this case, the difference frequency is not outside the bandwidth of the original field—instead, the difference frequency is equal to the center frequency. Autoproducts formed in this manner are not expected to be particularly interesting for signal processing purposes because the difference frequency field is already available in-band. However, for comparison purposes here, a center-frequency-to-difference-frequency ratio of unity is convenient as is the wide bandwidth available for averaging. As a result, the amplitude variations and interference layers observed in Fig. 8 are now mitigated.

In Fig. 11, the magnitude and phase of $χΔ$ is plotted for the two fields given in Fig. 10. Here, both Figs. 11(a) and 11(b) show a smoother spatial variation in $χΔ$ than in Fig. 9 but, otherwise, show roughly the same trends regarding which regions have which cross correlation magnitude and phase. The lines of poor cross correlation magnitude that appear to follow the caustics are actually just below the true caustic in the well-ensonified region. Along the caustic, strong cross correlation magnitudes exist. Just below the caustic there are interference layers that are still not averaged away, although their spatial extent is thin, relative to the size of the normalization regions. The extent of these interference layers is effectively only one normalization region in size, unlike in Fig. 9, where the interference layers spanned many normalization regions in size. In the shadow zones, generally strong cross correlation magnitudes still exist with the exception near the nulls that are formed in the shadow zone, where poor cross correlations emerge, some dropping as low as zero.

In the single-path regions, the phase of $χΔ$ can be seen here to vary as before with a +90° phase shift being picked up after each caustic [in Fig. 11(b), see the cyan, violet, and red regions, along with the light green region at the far-right edge], and these regions all have a corresponding $|χΔ|$ near unity.

In the two-path regions, the phases of $χΔ$ are now halfway between the +90° increments. Specifically, consider the blue, magenta, orange, and bright green regions, which are associated approximately with +45°, +135°, −135°, and −45°, respectively. These regions have cross correlation magnitudes of approximately 0.7. To understand this behavior, consider an environment in which a unity amplitude plane wave reflects off a boundary with an effective reflection coefficient of $(−i)$⁠, which is chosen to mimic passage of a ray through a caustic. In this two-path, plane wave environment, the acoustic field can be written as

where $τ1,2$ are plane wave propagation times for rays before and after reflecting from the boundary. Forming a bandwidth-averaged autoproduct from this creates

The final term in Eq. (8) is the cross term, which can be neglected when $ΩBWΔ(τ2−τ1)≫2π$ (i.e., outside the interference layer). By normalizing these two fields over some normalization region [such as a region with plenty of variation in $τ1,2$⁠, such that $Δω(τ2−τ1)≥2π$], it can be shown that

Therefore, in the two-path region where rays are entering/exiting a caustic, when evaluated outside of the interference layer and when amplitude variations can be neglected, the cross correlation has a magnitude of $1/2≈0.71$ and a phase angle of +45°, just as observed in the darker blue regions of Fig. 11(b). Generalizing this result for rays entering a caustic with a KMAH index of $n$ and exiting the caustic with a KMAH index of $m$⁠, $χΔ=in(1+im−n)/2$⁠. In other words, when passing through a simple (fold-type) caustic which increments the KMAH index by 1, $χΔ$ will have a magnitude of $1/2$ and a phase angle of $n×90°+45°$⁠. For a KMAH increment of two, such as in a cusp-type caustic (Kravtsov and Orlov, 1993), $χΔ$ acquires a magnitude of zero. And for a KMAH increment of three, which might happen in some more exotic types of caustics (see Chap. 10 of Kravtsov and Orlov, 1993), $χΔ$ returns to a magnitude of $1/2$ but has a phase angle of $n×90°−45°$⁠.

Finally, in regions that contain three rays, such as in the slightly lighter blue region bordering the one-path cyan region and the two-path blue region in Fig. 11(b), the overall phase angle of $χΔ$ can be thought of as a linear combination of the phase angle expected of $χΔ$ for each ray individually. But because the two rays entering/exiting the caustic are of higher amplitude than the one ray passing through the caustic, the net phase angle of $χΔ$ in these three-path regions is very close to that of the neighboring two-path region.

Due to the symmetry and geometry of this problem, near $z=0$⁠, there are many rays propagating into and out of very shallow angle caustics. At exactly $z=0$⁠, there are technically infinitely many caustics. This situation is not particularly realistic and is primarily an artifact of having a perfectly symmetric waveguide with a perfectly $n2$-linear sound speed profile with a source located perfectly along the line of symmetry. Therefore, the almost random cross correlations observed near $|z|=0$ are not particularly insightful for more realistic environments. Additionally, frequency-sum autoproduct plots behave mostly as anticipated with the primary change being that the phase angles of $χΣ$ have the opposite sign relative to the plots of $χΔ$ shown in Figs. 9(b) and 11(b), and, therefore, for this reason, these plots are omitted.

Autoproducts have been found to be useful in a variety of applications (Abadi et al., 2012; Worthmann et al., 2015; Douglass et al., 2017, Douglass and Dowling, 2019), but their theoretical limitations have only been considered within the ray approximation of acoustics. In this study, the effect of refraction and diffraction near caustics on the ability of autoproducts to mimic out-of-band fields was assessed. This was done by considering simple, range-independent waveguide environments with two different sound speed profiles: $n2$-quadratic and $n2$-linear. Cross correlations between autoproducts and out-of-band fields were computed, and their magnitude and phase were understood in the context of ray approximations. Particularly of interest were regions of acoustic fields near caustics, shadow zones, convergence zones, and regions well-described by rays that may have passed through caustics. Below, five conclusions from this study are stated and described.

First, caustics introduce phase shifts in genuine acoustic fields that are absent from both frequency-difference and frequency-sum autoproducts. This leads to overall phase offsets between genuine-acoustic and autoproduct fields that may or may not be a problem for autoproduct-based signal processing techniques. A recent study (Geroski and Dowling, 2019) shows that compensation for these caustic phase problems is possible for autoproduct-based source localization techniques.

Second, in single-ray-path regions, generally strong cross correlation magnitudes are observed; however, there exists an overall phase shift between autoproducts and out-of-band fields. This phase shift in the cross correlation for frequency-difference (frequency-sum) changes by +90° (−90°) for each caustic that the acoustic waves pass through. But because many source localization algorithms typically only depend on the relative phase difference across the array, as long as the array is wholly contained within the same one-ray-path region (or at least approximately so), then this phase shift is not expected to be detrimental. Alternatively, if the source waveform were known, the number of caustics through which a particular ray has passed (modulo 4) could be estimated.

Third, in regions with multiple rays, particularly regions where rays entering and leaving a particular caustic coexist, cross correlations between autoproducts and out-of-band fields are degraded with the upper bound of cross correlation magnitudes in these regions being only $1/2$ instead of the ideal value of 1. This is very problematic for signal processing algorithms, such as matched field processing, which require accurate phase estimates. Therefore, it is anticipated that in environments where multiple rays coexist and have passed through different numbers of caustics the ability of a technique such as frequency-difference matched field processing would be substantially degraded. However, it is noted that mild caustic effects can still be overcome—for example, the KAM 11 environment considered in Worthmann et al. (2015, 2017) technically included rays which passed through caustics, but because several other rays were measured which did not pass through caustics (and were instead reflected), source localization, here, was not substantially degraded. Therefore, while the presence of caustics is not helpful for frequency-difference matched field processing, it may not necessarily cause it to catastrophically fail.

Fourth, regions with multiple rays were found to exhibit interference layers, the size of which are inversely proportional to the bandwidth available for averaging the autoproducts. These interference layers derived from caustics differ only in shape from the interference layers formed from flat reflecting boundaries, as described in prior work (Worthmann and Dowling, 2017). Depending on environmental parameters, rays may travel on separate paths, but can still arrive at a particular point at nearly the same time (such as what happens near caustics). If the product of the time-difference-of-arrival and the bandwidth available for averaging is not much greater than unity, then cross terms formed in the autoproducts can substantially degrade autoproduct-to-out-of-band-field cross correlations.

Fifth, beyond the difficulties the caustic phase introduces, amplitude variations exist which are related and proportional to how far from unity the ratio of the out-of-band frequency to the center frequency of the in-band field is. In other words, shifting down in frequency (via the frequency-difference autoproduct) too far can lead to amplitude differences compared to a genuine field at the downshifted frequency because the autoproduct retains amplitude features found in the in-band signal.

Overall, this study shows that caustic phase shifts substantially change the capacity for autoproducts to mimic out-of-band fields, and provide a challenge for autoproduct-based signal processing algorithms which rely on accurate phases, such as matched field processing. Nevertheless, it is possible that ad hoc phase corrections may prove effective for acoustic source localization in environments with significant refraction (for example, see Geroski and Dowling, 2019).

This research was supported by the Office of Naval Research under Award No. N00014-11-1-0047 and the National Science Foundation Grant Fund No. DGE 1256260.