Approximate extraction of late-time returns via morphological component analysis

Underwater remote sensing using active sonar is typically performed by ensonifying the seafloor and processing the echoes to characterize the response from the objects and the environment. Synthetic aperture sonar (sas) processing is one of the primary methods used to generate imagery of the scattering intensity of the ensonified scene and utilizes acoustic scattering phenomena that is akin to geometric optics. As such, the image formation algorithm only accounts for the early-time response of an object and tightly couples the arrival time of acoustic energy with its spatial location. However, the overall response from an acoustically interrogated scene, especially the objects, supports additional responses including elastic scattering as well as structural resonances. This late-time energy does not conform to the image formation model, is improperly associated with pixels during the image reconstruction process, and appears in the image as smearing or blurring (Plotnick and Marston, 2016), see the top right subplot of Fig. 4. Additional artifacts arise due to the fact that the late-time signal structure has been spectrally modified by the acoustic coupling, structural vibration, and re-radiation back to the receiver. Differences in the signal structure could be exploited via custom signal processing approaches, if those structures can be separated from each other.

Various methods of separating early-time and late-time returns exist, from simple time gating to subtracting the response of a rigid object from an elastic one with an identical geometry, to more advanced recursive algorithms (Azimi-Sadjadi , 1998; Azimi-Sadjadi , 1995; Hall and Marston, 2022; Hall , 2016; Jia , 2017; Morse and Marston, 2002; Morse , 1998). A fundamental challenge with time-gating is that the early-time and late-time responses from a field of scatterers will overlap in time, preventing a clean separation of components. Subtracting the responses of objects with the same geometry but different material properties is an important tool but limited to analytic or laboratory settings.

In this paper, we model the measured data as a superposition of these multiple components plus white noise at an unknown level. We use a recent convex optimization framework called morphological component analysis (MCA) (Selesnick, 2014; Starck , 2004) to specify distinguishing properties of the components we wish to extract. MCA identifies each component of an additive mixture by its ability to be sparsely represented by a unique linear operator, such as a frame or a dictionary. MCA and L₁-regularization based techniques have been applied to signal separation problems in other fields (Donoho and Kutyniok, 2009; Nguyen , 2022; Parekh , 2015; Reddy and Rao, 2019; Starck , 2005). Here, we present two sparse representation frameworks for discriminating acoustic phenomena and compare their performance.

Separation of the early-time and late-time returns from a non-homogeneous field of scatterers is particularly challenging due to the diversity of acoustic effects (Pareige , 1989). While high-Q elastic responses such as whispering gallery modes produce long-duration ringing, low-Q modes such as surface wavepackets produce short-duration ringing, which can either arrive coincidentally with the geometrically scattered return or later in time (Kargl and Marston, 1989). Although we are motivated by the separation of the early-time and late-time responses, the paper will focus on the related problem of separating the short-duration and long-duration components of a time series. In this context, a short duration component is any signal component which has a short time duration, e.g., smaller than the pulse bandwidth reciprocal, regardless of the physical source. This will include both the initial geometrically scattered return from the object as well as any late-time wavepackets resulting from surface coupling. Long duration components will generally include long tailed exponential decays caused by high-Q resonance modes. The reason we focus on short-duration/long-duration separation versus early-time/late-time is that we are motivated by the application to sonar imaging, where time series may feature multiple superimposed returns with start times that are not known a priori. As such none of the separation techniques presented here will rely on time gating.

Section II describes the MCA framework and the two sparsification transforms featured in this paper. In Sec. III we demonstrate the approach by separating the short-duration and long-duration components of an analytic time series produced by Stanton's elastic cylinder model (Stanton, 1988). Next in Sec. IV we discuss the use of MCA separated time series in sas image formation. Section V applies MCA techniques to experimental data collected using an in-air circular synthetic aperture sonar (AirSAS) (Blanford , 2019) and demonstrates short-duration/long-duration separation on AirSAS imagery. Finally, Sec. VI concludes with a discussion of MCA as applied to acoustic time series.

In MCA, our ability to identify a component via its sparse representation hinges on the aptness and mutual exclusivity of each linear operator. In other words, each transform $A i$ should admit sparse representation of its corresponding component signal $y i$⁠, but should be inefficient in representing the other components. We wish to decompose y into a sum of short-duration components $y 1$ and long-duration components $y 2$ (as proxies for the early-time and late-time returns as discussed in Sec. I), and thus the problem at hand is to design $A 1$ and $A 2$ to describe those respective phenomena. We focus on over-complete tight-frame operators $A i$ (Han , 2007) which, by definition, satisfy $A i A i * = p i I$ for $p i > 0$⁠. The subsequent sections discuss specific, promising selections of $A i$ for our application.

At convergence, the choice of μ does not impact the solution for Algorithm 1, but it impacts the convergence rate. In the subsequent experiments μ is set to $λ / p$ where p is the 99th percentile of the initial coefficient magnitudes. This causes the first soft threshold of the algorithm to zero-out 99% of the coefficients.

A particularly simple, yet effective, form of MCA is to let the first representation be the identity, $A 1 = I$⁠, and the second be the unitary discrete Fourier transform, $A 2 = F$⁠. We refer to this as FFT MCA and in this case the solution to Eqs. (3) or (4) splits a signal into two components, with the former sparse in the time domain and the latter sparse in the frequency domain. A consequence of Fourier duality is the $y 1$ component tends to be made up of broadband, short-duration elements while $y 2$ tends to be made up of long-duration elements with a narrower spectrum.

The goal when applying ESP frames to MCA is to find two sets of envelopes $e l i$⁠, where the superscript indicates the component index and the subscript the envelope index, such that the signal components $y i$ are sparsely represented by one set of frame vectors but not the other. In the ideal case, $y i$ is actually equal to a frame vector for $A i$⁠. The specific choice of envelope is often informed by the physics associated with the signal in question. In this paper, we wish to separate the long-duration high-Q signal components from the short-duration acoustic response of an elastic object. As such, we will use decaying exponentials as envelopes for $A 2$ since exponentially decaying sinusoids are an excellent signal model for long-duration ringdown (Hambric, 2006). For $A 1$ we will use extremely short rectangular windows since they flexibly capture short-duration signals.

As an aside, if $A 1$ is generated using a single one-hot vector as an envelope, while $A 2$ is generated using a single constant function as an envelope, then the resulting representations are extremely similar to the representations used in FFT MCA. This mode of ESP MCA effectively generalizes FFT MCA, albeit not in strict mathematical terms.

We are generally interested in separating out the short-duration returns of an elastic object from the long-duration ones. From a physical perspective, the short-duration returns include the initial return of the ping reflecting off the rigid geometry of the object, as well as low-Q elastic effects and additional short-duration late-time phenomena. The long-duration returns primarily include the high-Q resonance modes of the object. From a signal analysis perspective, however, the specific form of what constitutes a short-duration or long-duration component is ultimately defined by the MCA representations. For FFT MCA the short-duration returns are represented using one-hot vectors, since $A 1 = I$⁠, while the long-duration returns are represented using sinusoids, since $A 2 = F$⁠. ESP MCA will use a frame built from short rectangular windows to capture the short-duration components (representing them as very short windowed sinusoids) and a frame built from exponentially decaying envelopes to capture the long-duration components (representing them as exponentially decaying sinusoids). The very generic windows used for the ESP representation were chosen for their ability to flexibly represent short-duration and long-duration returns from elastic objects. Utilizing higher fidelity ray theoretic characterizations (España , 2014; Gipson and Marston, 1999; Williams and Marston, 1986) of acoustic returns from elastic objects to derive ESP envelopes is an ongoing topic of research. There are a wide variety of generic, tunable wavelets (e.g., Gabor, TQWT), shearlets, and other frame-based representations that are compatible with the MCA framework. Application of these tools to acoustics is an active area of study (Donoho and Kutyniok, 2009; Meng , 2020; Reddy and Rao, 2019).

A more recent approach is to use machine learning techniques to generate data-driven dictionaries (Olshausen and Field, 1996; Zhang , 2015) However, it is still an open research question how to learn dictionaries that can be efficiently inverted in SALSA-like algorithms (Cisse , 2017; Hwang , 2019). Jointly learning multiple dictionaries for the purpose of MCA further complicates the problem and is also an open research question (Cowen , 2019; Deligiannis , 2017; Guo , 2021; Peyré , 2007, 2010). In the context of this paper, another downfall of learned dictionaries is that they may not be interpretable, and hence after the learning program may not be able to be fine-tuned to specific phenomena. Hence our choice of the ESP frame, which utilizes data but remains analytic and can be tuned.

In this section we will demonstrate the MCA approaches presented in Sec. II on an analytic acoustic signal produced by the Stanton elastic cylinder model (Stanton, 1988, Sec. I B). Stanton's model was chosen because it is a relatively simple model which still supports a wide variety of elastic effects, including short-duration/low-Q wave packets as well as long-duration/high-Q structural resonances. In addition Stanton's model has been shown to agree with results from finite-element methods in the case of broadside returns from 40 mm long, 20 mm diameter copper cylinders (Gunderson , 2017), which is relevant given the application in Sec. V. Notably Stanton's model does not encapsulate all of the relevant physics, particularly off broadside. Ensonification of a cylinder at oblique angles produces waves, such as meridional or helical waves, which travel along the axis of the cylinder. These waves can interact with the ends of the cylinder to produce a significant elastic response (Blonigen and Marston, 2000; Gipson and Marston, 1999; Plotnick , 2014) not accounted for by Stanton's approximation. Considering the generic nature of the MCA atoms, however, the Stanton approximation will be sufficient to demonstrate the overall MCA approach.

The Stanton model parameters used in this paper were chosen to represent backscatter from a solid aluminum cylinder in water with a diameter of 15.25 cm and a length of 30.5 cm. The receiver is located 2 m from the cylinder with a centered broadside orientation, and the signal is sampled at f_s = 300 kHz. Stanton's model provides the frequency representation of the cylinder's broadside impulse response. In order to minimize effects caused by the finite nature of the impulse, we apply a Butterworth filter of order 3 and threshold 0.25 to the synthesized time series. This helps to remove spectral discontinuities and produces a more natural impulse. The resulting time series, and corresponding spectrum, are shown in Fig. 1. The deep nulls at 15, 23, and 30 kHz are caused by low-Q surface wave elastic effects, which can be precisely characterized in the context of ray theory (La Follett , 2011; Marston , 2010). These effects are short duration and may add constructively or destructively to the geometric scattering response. The intention is for these effects to be included in the short-duration component. The sharper, shallower nulls at 18, 28, 39, and 42 kHz correspond to high-Q geometric resonance modes. These are long duration signals and are one of our primary interests for the long-duration component.

We begin by applying FFT MCA to the Stanton signal using BP with 1000 iterations and $λ = 0.1 λ max$⁠. After performing the separation, we get the results shown in Fig. 2. There we have plotted the original time series, the short-duration component, and the long-duration component in the early time, in the late time, and the frequency domain. The results are quite good. FFT MCA correctly separates the loud initial response into the short-duration component while the signal tail is entirely separated into the long-duration component. Quantitatively, the short-duration component has an early-time error of 4.44% while the long-duration component has a late-time error 3.70%. The behavior of the spectrum is particularly interesting. The bulk of the spectral power for the impulse response is separated into the short-duration signal, including the wide nulls caused by the low-Q elastic responses. The sharp short high-Q nulls however have been turned into distinct spikes in the spectrum of the long duration component. This can aid in sonar signal processing and has been demonstrated previously (Hall , 2019; Marston , 2010).

The results of ESP MCA utilizing 1000 iterations of BP with $λ = 0.1 λ max$ are shown in Fig. 3. For this time series, ESP MCA closely mirrors FFT MCA. We see that the short-duration component captures almost all of the initial response, and most of the spectral power, with an early-time error of 4.04%. The long-duration component on the other hand captures the entire late-time response, with a late-time error of 3.67%, and as a result has some clear peaks at the high-Q null locations. Comparing the performance of both methods we see that FFT MCAand ESP MCA perform about the same in terms of relative error, which is confirmed by a visual inspection of the separated components.

sas image reconstruction fundamentally depends on the coherence between time series and is sensitive to small errors in phase (Carrara , 1995; Cook and Brown, 2009). MCA does not explicitly preserve this coherence, and hence its separated components' time series are not guaranteed to produce high-quality images. However, we consistently observe high-quality image reconstructions from outputs of MCA (e.g., Figs. 5 and 6). We have several hypotheses as to why this works.

The primary driver for high-quality image reconstruction from MCA outputs is that we have designed component y₁ to contain the energy that is compatible with image reconstruction, namely, short-duration signals. Short duration signals are consistent with the fundamental assumption of sas image formation, i.e., that the scene is made up of point scatterers. Other components, such as long-duration returns or late-arriving short-duration returns, are not expected to constructively/destructively combine. It follows that if a collection of received time series $y ( j )$ is separated into short-duration components $y 1 ( j )$ and long-duration components $y 2 ( j )$ then $y 1 ( j )$ contains the specular returns and will produce a “correct” image of the underlying scene. Assuming the separation is done using BP we will have $y 2 ( j ) = y ( j ) − y 1 ( j )$⁠, so that the image produced by $y 2 ( j )$ will consist of everything in the original image that is not in the short-duration component image. Notably we do not claim that $y 2 ( j )$ reconstructs “correctly” since its components do not satisfy the underlying assumptions of the reconstruction process.

While ideal separation produces a short-duration component that respects image formation, in practice the time series separation will not be perfect. This is due to the fact that we are using SALSA to produce an approximate separation and because the complexity of the time series and generic nature of our atoms imply that even a fully converged regularization will not completely isolate the desired components. As such it is important to understand how failed or partial separation interacts with the image formation process.

In this section we will apply the MCA techniques presented in Sec. II to experimentally generated AirSAS time series (Blanford , 2019). Experimental AirSAS data were collected on two targets: an 8-in. long, 2-in. diameter copper pipe with 0.032-in. thick walls, and an 8-in. long, 2-in. diameter air-filled, hollow copper cylinder with 0.032-in. thick walls and end caps. The targets were centered on a turntable and rotated in 1° increments relative to a transducer array consisting of loudspeaker tweeter (Peerless OX20SC02-04) and a microphone (GRAS 46AM). The tweeter transmits a 1 ms 30–10 kHz linear frequency modulated (LFM) chirp and the microphone receives the signals backscattered from the target. Motion, timing, signal generation and capture is controlled from a National Instruments data acquisition platform. The recorded signals are match filtered with the transmitted waveform. For this paper we only utilize the 3–8 ms portion of each time series.

We apply FFT MCA and ESP MCA to the resulting dataset for the 0.032-in. hollow copper cylinder object in Sec. V A and the more complicated dataset collected from a 0.032-in. copper pipe in Sec. V B.

Despite the fact that the AirSAS data is much more complex than the analytic signal from Sec. III, with multiple returns arriving at different times, for this section we will continue use the performance metrics from Eq. (6). However, for the AirSAS cylinder data we use an early time-interval I₁ from 4 to 6 ms and a late-time interval I₂ from 6 to 8 ms. Since each AirSAS object scan includes 360 different time series, we will report the relative error averaged over all aspect angles, which introduces some variation into the error. While we will still view these metrics as a measure of separation performance, the fact that we expect there to be late-time short-duration energy (particularly in Sec. V B) means that even in the case of perfect separation we do not expect either m₁ or m₂ to be zero. More broadly these metrics provide only a rough indication of overall performance and are only possible in laboratory settings. In situ estimation of separation performance is an open question.

The first dataset we will consider are the AirSAS time series collected from the 0.032-in. hollow copper cylinder. In some respects, these time series are similar to the Stanton model used in Sec. III, since at most aspect angles there is a single bright initial return potentially followed by a long-duration low power component. Notably cylindrical objects support a wider class of non-rigid phenomenon than Stanton's model. Various representations of the hollow copper cylinder experimental data are shown in Fig. 4. The top left subplot is a logarithmically scaled color plot of the time series magnitude. The bottom left subplot shows the associated normalized target strength and is the spectra of each time series normalized across all aspect angles. The top right subplot is a logarithmically scaled color plot of the polar format algorithm (PFA) (Doerry, 2012) generated image magnitude. The bottom right subplot shows the object's k-space representation, which is the magnitude of the two-dimensional Fourier transform of the complex PFA image. The long-duration signal is clearly present in the time series representation, in bands from –10° to 90° and 180° to 280°. This late time energy is also apparent in the PFA image. Not readily apparent in either of the spectral representations is a faint signature corresponding to this late-time energy.

As discussed in Sec. IV, MCA is broadly compatible with the signal processing and image reconstruction algorithms used with the AirSAS data. For this section, we will apply MCA to the match filtered AirSAS time series individually, splitting each into short-duration and long-duration components. We then apply PFA to the separated time series to reconstruct a pair of images, one corresponding to the short-duration components and the other to the long-duration components. We produce normalized target strength representations corresponding to the short-duration and long-duration components as well.

To begin, we will apply FFT MCA using 1000 iterations of BP with $λ = 0.1 λ max$ to the 0.032-in. hollow copper cylinder as described above. Since we are utilizing BP, the separated time series, as well as the corresponding PFA images, will add up exactly to the original dataset from Fig. 4. After image formation, the PFA images associated with the separated short-duration and long-duration components are shown in Fig. 5, along with their normalized target strength representations, on a pairwise common color scale. The separation looks fairly clean in the time domain image, with the extended ringing response from the cylinder principally in the long-duration image while the brighter geometric scattering response appears in the short-duration image. There does appear to be some bleed through of the object into the long-duration image. It should be noted that the experimental data contains significant multipath returns, particularly those caused by reflections off of the turntable (Williams , 2010). For FFT MCA, these multipath returns appear in the long-duration component. The average short-duration early-time error is 44.9% while the average long-duration late-time error is 23.7%. One particularly interesting set of features are the hyperbolic signatures present at 20° and 210° in the long-duration normalized target strength plot, since these signatures were masked by the much brighter short-duration response in the bottom-left plot of Fig. 4.

For ESP MCA we will use the same process as above, applying BP to individual time series and producing a pair of PFA images associated with each component. We continue to use rectangular windows and decaying exponentials as our envelopes, with the same window lengths and time constants as Eq. (7). Heuristic experimentation showed these parameters produced reasonable results, although we will see that specific performance characteristics can be attained by using shorter windows and larger time constants. Using BP with 1000 iterations and $λ = 0.1 λ max$ produces the PFA images shown in Fig. 6. The separation is quite effective with the bright initial scattering almost completely contained within the short-duration response with an average early-time short-duration error of 19.0%. Furthermore, ESP MCA separation appears to put more of the turntable multipath in the short-duration component. The long-duration component has most, but not all, of the late-time response and has an average error of 49.0%. Interestingly, there is some late-time energy in the short-duration image at the acoustic coupling angles that was not present in the FFT MCA. This energy does appear to take the form of late arriving wave packets and one theory is that these discrete returns are late arriving pulses associated with surface waves propagating on the cylinder. A ray theory based analysis may be able to determine if this is indeed the case, but is outside the scope of the paper. If the late time returns are the result of surface wave packets, then it is more appropriate for them to be part of the short-duration image. The fact that they are not present in the long-duration image negatively impacts the long-duration late-time relative error, which is consistent with our goal of separating the signal into early-time and late-time components.

Overall, compared to the FFT MCA separation we have sacrificed accuracy in the long-duration late-time for increased accuracy in the short-duration early-time. As we will demonstrate in Sec. V B the ESP frame approach is flexible so this separation could be further tuned by changing the envelope parameters. (Recall that using a one-hot and constant envelope will largely reproduce the FFT result.) Moreover, there is evidence that not all of the short-duration components are early-time, which impacts the reliability of the early-time/late-time error metrics. Visually the separation appears best with the ESP MCA approach as the hyperbolic late-time features are clearer and at a higher relative power.

The second data set we will consider are the AirSAS time series collected from the 0.032-in. copper pipe. This dataset is significantly more complex than the 0.032-in. hollow copper cylinder dataset, with obvious short-duration late-time energy present in the time-series. This will exercise the MCA approach to signal separation by demonstrating separation of a long duration signal superimposed on a sequence of repeated short duration signals, but also highlights the weakness of our performance metric. Figure 7 shows logarithmically scaled color plots of the 0.032-in. copper pipe time series data, the associated normalized target strength, as well as the PFA image and its associated k-space representation. The aforementioned late-time short-duration energy is present in discrete “rings” around the object from –10° to 90° and 180° to 280°. These rings are caused by multipath effects involving the interior of the pipe. A related analysis of this coupling to internal pipe modes is given in España (2014, Appendix B). We wish to understand how our MCA tools respond to this energy.

Since FFT MCA is signal agnostic we apply it to the 0.032-in. copper pipe just as in Sec. V A. Using 1000 iterations of FFT MCA BP with $λ = 0.1 λ max$ we produce the separated PFA images shown in Fig. 8. It is immediately apparent that most of the late-time energy, including the shorter duration late-time “rings,” are contained in the long duration image. As a result, the average late-time, error for the long-duration component is a relatively low 17.7%, at the cost of a higher 64.1% error for the early-time short-duration component. It is slightly unexpected that these apparently short duration signals can be more sparsely represented in the frequency domain; however, analysis of the spectrum shows that while this late time energy is apparently time limited it is nevertheless not particularly broad band. This is due to the fact that these late arriving wave packets are shaped reflections of the pulse used to ensonify the object.

Next, we will separate the copper pipe time series using the same ESP MCA parameters as Sec. V A. The short and long-duration PFA images resulting from 1000 iterations of BP with $λ = 0.1 λ max$ are shown in Fig. 9. We see that unlike the FFT MCA much of the power of the late-time short-duration wavepackets has been correctly placed in the short-duration PFA image. The separation is a bit muddled overall, although there is clear distributed late-time energy in the long-duration plot over the expected range of angles. For ESP MCA, the long-duration error is worse than the FFT case, with an average late-time error of 62.5%, but the average short-duration early-time error is a better 27.4%. Also, notable is that the normalized target strength plots in Figs. 8 and 9 emphasize different features. The “V” shaped signatures between 10 and 20 kHz around 130° and 300° seem to move from the short-duration component to the long-duration component.

In this case we get something that looks more like the FFT separation, with most of the late-time power (including the short-duration multipath components) in the long-duration image. There is more late-time energy in the short-duration image for this alternative ESP MCA than FFT MCA and the average short-duration early-time error is larger at 74.6%. The long-duration late-time average error 15.9%, an improvement over FFT MCA.

In summary the performance metrics for the copper pipe, the alternative ESP MCA produces the best early time image and ESP MCA produces the best late time image. However, the overall performance of each method depends on whether the motivating need for MCA signal separation would benefit from late-time short-duration energy grouped with the initial scattering return or with the more diffuse late-time return.

This section demonstrates some of the complexities of applying MCA to experimental data. Performance of MCA representations will depend on the signals in question and determining the correct separation parameters is not always obvious. In the case of ESP MCA, adjusting the envelope parameters can be done in a more principled fashion than adjusting the λ_i-parameters. In either case, finding a way to allow the MCA representations to be data informed, rather than completely model driven, is a potential topic for further research.

More broadly we have demonstrated the ability to separate sas imagery into distinct morphological components using both FFT MCA and ESP MCA. The FFT MCA approach is generally superior at producing separations with lower late-time error while ESP MCA has lower early-time error. Importantly both MCA methods were designed to separate short-duration and long-duration components, rather than early-time/late-time components and will associate the late-time short-duration energy in the AirSAS data with geometric scattering. An open question is if differences in the morphology between early-time and short-duration late-time energy could be used to further the overall goal of early-time/late-time separation.

Motivated by the problem of separating the early-time and late-time returns from the acoustic response of an elastic object, we have presented a pair of MCA techniques which can successfully separate the short-duration and long-duration components of a time-series without the need for a reference signal or time gates. This partly isolates the late-time returns, with the geometric scattering response generally present in the short-duration component and high-Q resonances present in the long-duration components. Successful separation was achieved on both analytic model data as well as experimentally collected in-air sonar time series.

The FFT MCA approach is signal agnostic and does a reasonable job of signal separation, performing very similarly to ESP MCA in the Stanton model case. FFT MCA also has the benefit of being extremely fast but is rigid and cannot be tuned to fit a particular signal outside of the traditional λ_i parameters. The ESP MCA application had the best metric scores in analytic time series analysis. ESP MCA has a flexible signal model which can be tuned to support a wide variety of signals as demonstrated in Sec. V B. It does take orders of magnitude longer to run than FFT MCA, however.

In practice we found that MCA can turn sharp nulls in the spectrum of a time series, which are easily filled in by Gaussian noise, into peaks in the spectrum of the long-duration component, which are easier to identify in the presence of noise. This is most obvious in the Stanton model impulse response (see Figs. 2 and 3) but can also be seen in experimentally collected AirSAS data with the hyperbolic signatures that appear in the long-duration normalized target strength plots for the 0.032-in. hollow copper cylinder dataset (see Fig. 6) but are obscured by the stronger short-duration geometric response in the unseparated data. The ability to utilize features derived from separated components is an ongoing topic of research.

Additionally, Secs. IV and V demonstrate compatibility of MCA with synthetic aperture sonar image reconstruction. The ESP MCA and FFT MCA techniques produced short-duration and long-duration images which split the initial loud response of thin-walled cylindrical objects from the diffuse energy produced by long-duration ringing. This was done without time gating and in the presence of both experimental noise and overlapping returns with varying start times. Notably we have presented examples where long-duration late-time energy is separated from superimposed short-duration late-time energy (see Fig. 9). While the performance metric results were less clear cut for the experimental data, ESP MCA was capable of performance similar to FFT MCA while being significantly more flexible in its signal model. While both MCA approaches were designed to separate short-duration/long-duration components, the ultimate goal is to separate early-time/late-time components in order to preserve the assumptions of the image formation model. It is an open question if the spectral characteristics of early-time versus short-duration late-time responses could be used in an MCA context.

Moving forward there are quite a few potential applications of MCA to acoustic signals. The filtering process could be used in the formation of sas imagery to either reduce late-arriving energy, allowing for a sharper representation of the object, or to highlight late-time ringing energy, identifying objects with elastic behaviors from those without. Additionally, the spectral peaks resulting from high-Q modes in the long-duration component could be used as features for classification. The approach itself could be refined by utilizing signal-dependent representations, particularly those informed by ray theoretic characterizations of specific elastic effects. More broadly, MCA could be used to at least partially separate any components of a signal which feature sparse representation, such as overlapping acoustic returns from two pings with significantly different spectra, and as such FFT MCA and ESP MCA provide flexible tools for tackling a fundamental acoustics challenge.

This work was sponsored in part by the Department of the Navy, Office of Naval Research under ONR Award Nos. N00014-18-1–2820, N00014-19-1–2221, and N00014-22-1–2620.

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It is an immediate corollary that if $| | e l | | = ( N L ) − 1 / 2$ for all l then the vectors ${ a k , l , m }$ form a Parseval frame. Notably, the fact that ${ a l , k , m }$ is a tight frame can also be derived by viewing it as a multi-window STFT. More importantly, the conditions on the envelopes are minimal: even a set of unrelated envelopes will admit a frame under this procedure.

An important corollary to the above is that under certain limited conditions the SALSA operators E_i preserve superposition.

Corollary 1. Suppose $w = ∑ k = 1 K S n k y$ and that $d i = A i E i w$ can be decomposed into k disjoint sets of coefficients $d i , k$ such that $A i * d i , k = S n k y$. Then $E i w = ∑ k = 1 K S n k E i y$⁠.

The conditions in the previous corollary might occur for one of several reasons relevant to sas time series:

• If $d i = 0$ then $d i , k = 0$ is a trivial decomposition.

• If the supports of the envelopes for $A i$ are all much narrower than $n k − n l$ for all $k , l = 1 , … , K$⁠, then we can let $d i , k$ be the restriction of $d i$ to those time shifts in the support of $S n k y$⁠.