Subbottom profilers are sonars used to image shallow (<100 m) subsurface structure beneath the seafloor (or the bottom of rivers or lakes). In contrast to simple echosounders that use acoustic energy reflected off the bottom to measure the depth, subbottom profilers provide a record of acoustic energy reflected by layers beneath the seafloor. The frequencies used to achieve penetration range from 1 kHz to 20 kHz, with lower frequencies providing greater penetration. Conventional subbottom profilers (Figure 1) use single frequency (CW) pulses. Starting in the 1960's, 3.5 kHz CW subbottom profilers became standard equipment on oceanographic research vessels. The ability of CW subbottom profilers to resolve discrete reflecting layers in the subsurface is limited by the duration of the transmit pulse. Shorter pulses allow the sonar to image finer structure. However, the signal available for achieving penetration scales with the pulse duration because longer pulses equal more power in the water. Thus, the operation of these instruments has always involved a tradeoff between penetration and resolution of the subsurface layering.
Figure 1. Conventional subbottom profiler record from a hull mounted 3.5 kHz sonar (Heinrich, 1986). The horizontal lines are spaced 100 msec apart, corresponding to a depth difference of 75 meters. The seafloor is approximately 4500 m deep.
In recent years, a new generation of subbottom profilers has emerged which use swept-frequency, or chirp, transmit pulses (Figures 2-3). In contrast to CW profilers, the time resolution of chirp sonars can be much smaller than the pulse length, Higher resolution is achieved using match filtering (cross correlation) of the raw data with the source pulse. Because the resolution does not depend linearly on the transmit duration (and in fact improves with increasing duration), sonars can generate long transmit pulses to achieve high signal levels and greater penetration.
Figure 2. Chirp subbottom sample from a hull mounted system operating in 1330 m depth . A 3-7 kHz sweep was used to collect these data.
Figure 3. Chirp subbottom profile collected at 1100 m depth by an autonomous underwater vehicle (AUV) flying 50 m above the seafloor. The chirp sweep is 2-10 kHz, resulting in a potential vertical resolution of about 10 cm. These data, collected by the Monterey Bay Aquarium Research Institute, were processed using the MB-System software package (Caress and Chayes, 1996; Caress and Chayes, 2003-2009; Schmidt et al., 2004)
This document presents a very basic view of the signal processing concepts underlying modern chirp subbottom profilers. The various commercially available systems all incorporate some degree of additional sophistication in the source signal construction and match filtering scheme, but the relationships between the source frequency sweep, the source duration, and the potential resolution are common to all chirp sonars.
A swept-frequency or chirp signal has a time-varying frequency. In the simplest and most common case, the frequency varies linearly in time between a starting value and an ending value over a defined duration. Such a signal can be expressed as:
x(t) = A cos( 2 pi ( (f2 - f1) t**2 / (2 d) + f1 t + P) )
where f1 is the starting frequency in Hz, f2 is the ending frequency, d is the duration in seconds, P is the starting phase, and A is the amplitude.
Figure 4. Chirp signal sweeping from 2 kHz to 10 kHz over a 20 millisecond duration. Top: Untapered signal. Bottom: Chirp signal tapered with a fourth root sine function.
Figure 5. Comparison of power spectra of chirp signals. Top: power spectrum of untapered chirp. Bottom: power spectrum of tapered chirp. The reduction in the power at the low and high frequencies due to the tapering is evident.
Figure 4 displays a chirp signal generated using a frequency sweep from 1 - 6 kHz, a duration of 20 msec or 0.02 seconds, a starting phase of zero, and an amplitude of 1. The upper plot shows the unmodified chirp; the lower plot shows the the chirp signal tapered by multiplication with a fourth root sine function. Figure 5 presents power spectra of the untapered and tapered chirp signals. As expected, the tapering reduces the power at the low and high frequency ends of the peak.
Figure 6. Comparison of Klauder Wavelets from untapered (top) and tapered (bottom) chirp signals. The Klauder Wavelets are the autocorrelation of the chirp source, and represent the best possible resolution of an impulse response. The advantage of tapering the source is clear, as the untapered chirp produces a Klauder Wavelet that is far more oscillatory.
Figure 7. Comparison of power spectra of the Klauder Wavelets. Top: power spectrum of untapered chirp. Bottom: power spectrum of tapered chirp. As expected, the power spectra of the Klauder Wavelets are similar to those of the chirp signals.
A chirp sonar works by using a long, variable frequency source signal, such as those in Figures 4 and 5, and then cross correlating that source with the data time series containing arrivals that have been reflected back towards the sonar. This match filtered, or correlate, time series will contain peaks where the source function correlates with arriving signals in the data. For any given source, the best possible correlation can be estimated by cross correlating the source with itself, simulating the situation where the data contains an exact, unmodified version of the source. This autocorrelation of the source chirp is called the Klauder Wavelet. Figure 6 displays the Klauder Wavelets for the chirp sources of Figure 4. These wavelets have the largest peak centered at zero time (the arrival time), but also show smaller peaks on either side of the main peak, with oscilliatory tails stretching off both before and after the peack. The Klauder Wavelet for the untapered chirp (top) is clearly more oscillatory than that for the tapered chirp (bottom). Thus, applying tapering to the source improves the potential time resolution of the sonar. In this case, if one measures the width of the tapered chirp wavelet by the location of the second zero crossings, arrivals can be resolved in time to +/- 0.2 millisecond. Figure 7 displays the power spectra of the tapered and untapered chirp Klauder Wavelets. The wavelets contain the same frequency structure as the original chirp signals, but are quite compressed in time.
It is important to remember that the Klauder Wavelets and the time series resulting from the match filtering are correlates, and no longer represent physical observed quantities such as hydrophone amplitude. In particular, the Klauder Wavelets are acausal because they contain nonzero structure for times before zero. Real, causal signals cannot have structure arriving before zero time. A simple consequence is that arrival times of reflected signals should be picked at correlation peaks rather than at leading edges when working with match filtered chirp sonar data.
Because of the Klauder Wavelet structure, real subbottom profiler correlate data is usually quite complex. For instance, finely layered sediments will result in many closely spaced arrivals in the raw data. Following successful match filtering, the correlate signal will mix together the positive and negative peaks of the Klauder Wavelets, making a time series that is difficult to interpret. This observation is demonstrated with simple synthetic data in Figures 8-13. Figures 8, 10, and 12 show the construction of synthetic data by convolving a chirp pulse with an impulse model simulating a simple subsurface structure with both positive and negative responses corresponding to abrupt increases and decreases in sound speed (seismic velocity). In all three cases noise is added to the data as well. Figures 9, 11, and 13 show the results of match filtering these noisy synthetic data with the appropriate chirp pulse function. The upper plots show the raw correlate, and demonstrate that both positive and negative correlation peaks are observed in the vicinity of each arrival, making the determination of event polarity and the picking of event arrival times very problematic.
A commonly used approach to solve this interpretational difficulty is to construct an envelope function that delineates arrivals less ambiguously. The envelope is calculated by constructing a complex time series in which the original correlate forms the real part, and the Hilbert transform of the correlate is the imaginary part, and then taking the magnitude of the combined complex time series. The result, as seen in the lower plots of Figures 9, 11, and 13, is a positive-only function that has a single peak for each arrival, regardless of the polarity. The great majority of marine subbottom profiler plots in the literature are plots of the envelope function [Henkart, 2006], including those in Figures 2 and 3.
Figure 8. Construction of a 2 second long synthetic time series to test chirp processing. Top: An impulse function with both positive and negative impulses of different amplitudes. In the context of subbottom profilers, these impulses represent acoustic impedence contrasts within the sediments that produce positive or negative polarity reflections of acoustic energy. Middle: convolution of the tapered chirp source with the impulse function. Bottom: Addition of random (white) noise to the convolved signal. The noise has an amplitude greater than some of the impulses and less than others.
Figure 9. Results of chirp processing. Top: Correlate time series resulting from the cross correlation of the tapered chirp source (Fig. 4) with the synthetic noisy data (Fig. 8). Peaks in the correlate correspond to the initial impulse locations, but have both positive and negative spikes. Bottom: Envelope function calculated by taking the magnitude of a complex time series composed of the correlate as the real part and the Hilbert transform of the correlate as the imaginary part. The envelope is a positive only, lower frequency time series that nicely resolves all of the original impulses with about the correct relative amplitudes.
Figure 10. A 200 milliseond section of the synthetic time series. Top: impulse function with closely spaced positive and negative impulses of different amplitudes.Middle: Convolution of the tapered chirp source with the impulse function. Bottom: Addition of random (white) noise to the convolved signal. The first pair of signals is higher amplitude than the noise, and the second pair is hidden by the noise.
Figure 11. Results of chirp processing shown for a 200 milliseond section of the synthetic time series. Top: Correlate time series resulting from the cross correlation of the tapered chirp source (Fig. 4) with the synthetic noisy data (Fig. 10). Peaks in the correlate correspond to the initial impulse locations, but have both positive and negative spikes. Bottom: Envelope function calculated by taking the magnitude of a complex time series composed of the correlate as the real part and the Hilbert transform of the correlate as the imaginary part. The envelope is a positive only, lower frequency time series that resolves all of the original impulses with about the correct relative amplitudes.
Figure 12. A 30 milliseond section of the synthetic time series bet. Top: Impulse function with one closely spaced pair of positive and negative impulses of different amplitudes. Middle: Convolution of the tapered chirp source with the impulse function. Bottom: Addition of random (white) noise to the convolved signal.
Figure 13. Results of chirp processing shown for a 30 milliseond section of the synthetic time series. Top: Correlate time series resulting from the cross correlation of the tapered chirp source (Fig. 4) with the synthetic noisy data (Fig. 12). Peaks in the correlate correspond to the initial impulse locations, but have both positive and negative spikes. The short time interval viewed in this plot allows the Klauder wavelet structure of the arrivals to be visible in the correlate. Bottom: Envelope function calculated by taking the magnitude of a complex time series composed of the correlate as the real part and the Hilbert transform of the correlate as the imaginary part. The envelope is a positive only, lower frequency time series that resolves all of the original impulses with about the correct relative amplitudes.
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