Revealing hidden information with quadratic products of acoustic field amplitudes

Acoustic waves are omnipresent in modern life and are well described by the linearized equations of fluid dynamics. Once generated, linear acoustic waves carry and collect information about their source and the environment through which they propagate, and this information may be retrieved by analyzing recordings of these waves. Because of this, acoustics is the primary means for imaging and remote sensing in otherwise opaque environments, such as the Earth's oceans and crust and the interior of the human body. For these information-retrieval tasks, acoustic fields are nearly always interrogated within their recorded frequency range or bandwidth. However, this frequency-range restriction is not general; acoustic fields may also carry hidden information at frequencies outside their bandwidth that can be revealed by analyzing the quadratic products of a pair of complex frequency-domain field amplitudes having different frequencies. The two unique quadratic field products, known as the frequency-difference and frequency-sum autoproducts, can be utilized for remote sensing at the difference and sum of the two constituent frequencies, respectively, even if these difference and sum frequencies lie outside the recorded field's bandwidth. Despite some fundamental limitations, forming and analyzing the autoproducts enables a variety of acoustic remote sensing applications that were long thought to be impossible. In particular, analysis of the frequency-difference autoproduct allows the detrimental effects of sparse-array recordings, random scattering, and other unknown source-to-receiver propagation effects to be suppressed when the recorded acoustic field has sufficient bandwidth. Examples and applications from laboratory and ocean propagation experiments are provided that involve frequencies from a few Hertz to more than 100 kHz and propagation distances from tens of centimeters to more than 100 km.

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Figure 1

Time- and frequency-domain representations of acoustic field waveforms. Acoustic recordings are made in the time domain (left) but are often transferred via a Fourier transform to the frequency domain (right) for analysis. The frequency range where the field has nonnegligible complex amplitude is the field's bandwidth. In-band and out-of-band frequencies fall within and outside the field's bandwidth, respectively, with out-of-band frequencies occurring both below and above the field's in-band frequency range.

Figure 2

Schematic and coordinate system for a two-path laboratory water tank experiment intended to demonstrate that out-of-band field information is hidden within in-band signal recordings. A nominally omni-directional acoustic source located at $\left(r,z\right)=(0,d=200\mathrm{mm})$ broadcasted a 40–110 kHz pulse. The receiver was placed at 161 locations on a vertical line segment from $z=0$ (the water surface) to a depth of 400 mm at fixed horizontal distance of $r=325\mathrm{mm}$.

Figure 3

Comparisons of measured and theoretical fields as a function of depth for the experimental setup shown in Fig. 2. In all panels, the amplitudes are normalized using Eq. (5), with $V$ as the vertical line segment from $\mathrm{depth}=0$ to $\mathrm{depth}=400\mathrm{mm}$, and the measurements are shown with red “×”s. The measured field and theoretical in-band field (black curve) from Eq. (7) at 75 kHz are shown in (a). The measured autoproducts are shown with theoretical autoproducts (black curves), and theoretical out-of-band fields from Eq. (7) with a plus sign between terms (blue-dotted curves) for below-band frequencies of $\mathrm{\Delta }\omega /2\pi =5\mathrm{kHz}$ in (b) and 30 kHz in (c), and for above band frequencies of $\mathrm{\Sigma }\omega /2\pi =150\mathrm{kHz}$ in (d) and 185 kHz in (e). The dashed vertical lines show the depth below which autoproducts should match out-of-band fields in the Fig. 2 environment. All five panels have the same horizontal axes and vertical axes and show excellent matches between the various curves and measured data.

Figure 4

A picture looking through the water surface of the water-tank experimental setup. The source is the black sphere in the lower left quadrant of the picture at the end of the bronze pole. The receivers appear in a diagonal row near the top of the picture. The 18 distributed white spheres (air-filled ping-pong balls) are scatterers that complicate the source-to-array propagation and thereby hinder in-band source localization from the array-recorded signals alone.

Figure 5

Measured (a) and ideal (b) time-domain acoustic fields for the experimental setup shown in Fig. 4 for broadcast of a 200-μs Gaussian-windowed pulse with an in-band frequency range from 150 to 200 kHz. The measured field in (a) includes scattering while the ideal field in (b) does not.

Figure 6

Source localization correlation outputs from the recordings shown in Fig. 5. The source location is marked by a white circle outlined in black. The correlation peak location is marked by a white triangle outlined in black. The receiving array is at $x=0$ and $\left|y\right|\le 28\mathrm{cm}$. Panel (a) shows in-band conventional results for $B\left({\mathbf{r}}_t\right)$ from Eq. (9). Panel (b) shows below-band frequency-difference results for $B_{\mathrm{\Delta }}({\mathbf{r}}_t,\mathrm{\Delta }\omega )$ using Eq. (10) with $\mathrm{\Delta }\omega /2\pi =20\mathrm{kHz}$. The below-band result is successful (the markers overlap) while the in-band technique is not.

Figure 7

Generic ocean experimental geometry showing a vertical array at $r=0$, a sound source at range $r=R$ and depth $z=Z$, both in water of depth $D$. Sample depth-dependent sound speed profiles for the shallow and deep ocean experiments are shown at the right.

Figure 8

Measured (a) and computed (b) time-domain impulse responses obtained from matched filtering for the shallow-ocean experiment. The source signal utilized here was a 100-ms-duration frequency sweep from 11.2 to 32.8 kHz. The measured impulse responses show more arrivals and the effects of scattering when compared to the computed ones.

Figure 9

Source localization correlation outputs from the recordings shown in Fig. 8. The source location is marked by the white circle. The receiving array at $r=0$ is shown by black dots. Panel (a) provides in-band conventional results for $B\left({\mathbf{r}}_t\right)$ from Eq. (9). Panel (b) provides below-band frequency-difference results for $B_{\mathrm{\Delta }}({\mathbf{r}}_t,\mathrm{\Delta }\omega )$ using Eq. (10) incoherently averaged from $\mathrm{\Delta }\omega /2\pi =50$ to 500 Hz. The in-band technique fails, but below-band result is modestly successful. The 3-km periodicity in (b) is an unavoidable part of this acoustic environment.

Figure 10

Measured (a) and computed (b) time-domain impulse responses (aka time fronts) obtained from matched filtering for the deep-ocean experiment. The source signal utilized here was a 135-s-duration frequency sweep from 200 to 300 Hz. The orange stripe at the top of (b) is noise from an experimental artifact. The boundary interacting time fronts are the diagonal features that extend from the top to the bottom of both panels. The louder refraction-guided arrivals appear as zigzags in the upper two-thirds of each panel. The measured time fronts in (a) show less regularity and different timing than the computed time fronts in (b).

Figure 11

Two-pulse ensemble-averaged source localization scatter plots from the first 101 (correctly measured) experimental pulses from the deep-ocean experiment in a 300-km-range by 6-km-depth search plane. Blue squares mark the locations of correlation maxima. The circle marks the location of the source. The red ellipse defines the localization-success boundary. Its dimensions are ±5% of the overall search range and depth. Panel (a) shows in-band conventional results for $B\left({\mathbf{r}}_t\right)$ from Eq. (9). Panel (b) shows below-band frequency-difference results for $B_{\mathrm{\Delta }}({\mathbf{r}}_t,\mathrm{\Delta }\omega )$ using Eq. (10) incoherently averaged across $\mathrm{\Delta }\omega /2\pi =3.75$, 4.00, and 4.25 Hz with a phase correction applied to $w$ to account for caustics along the 130 km propagation path. The in-band scheme is successful 1 time out of a 100. The below-band scheme is successful 92 times out of 100. In (b), the inset plot shows the cloud of successful below-band localizations.