The Effect of Coherence and Noise on the Decomposition of the Time Reversal Operator David M. Fromm Charles F. Gaumond Joseph F. Lingevitch Richard Menis David C. Calvo Geoffrey F. Edelmann Elizabeth Kim Acoustics Division, Naval Research Laboratory, Washington, DC 20375-5320 Abstract – Active sonar in shallow water in shallow water is often reverberation-limited and the detectability is often limited by the presence of too many false alarms. The decomposition of the time reversal operator (DORT) is a method that potentially alleviates this problem by separating echoes from different depths in the water column. For example, DORT can separate a target in the water column from reverberation on the bottom. DORT requires a set of echoes recorded on a line array that result from a set of independent transmissions from a source array. A short derivation of DORT using the sonar equation is given. Because DORT is inherently a frequency-domain method, the time-frequency domain is derived to implement the algorithm on the data. Lastly, the similarity of DORT to adaptive beam forming is shown. In this paper, data – taken on the Atlantic shelf, east of Cape May, NJ, during Geoclutter 03 and TREX-04 experiments – is processed and shown. The data was taken with a 64 element vertical line array of source-receiver elements. The target was an echo repeater using an XF4 source from 500 to 2500 Hz or an ITC 200 source from 2500 to 3500 Hz. The data cover six 500 Hz-bands from 500 Hz to 3500 Hz. The data are processed using DORT in the time-frequency domain. The analysis produces singular values in the time-frequency domain and in the timedelay domain. It also produces singular vectors that are used with a broad-band propagation model to form backpropagation images in the range, depth, frequency or range, depth, time domain. The analysis shows that the limiting factors in this data set arise from 1) motion that causes a lack of time-invariance, 2) additive noise and 3) the independent transmission scheme. The lack of time invariance is shown to spread the echo energy into several singular indices. Additive noise is shown to contaminate the singular values and back-propagation images. The particular transmission scheme used, time division multiplexed LFMs, is shown to create large side lobes in the time domain. Alternative transmission sequences, as well as alternative source and receiver orientations, are discussed. (This work sponsored by ONR.)

I. INTRODUCTION The decomposition of the time reversal operator (DORT) is a technique based on the larger field of time reversal [1] – [4]. In the theory section, a short derivation of the procedure is given based on the sonar equation, showing the transformation from data to back-propagated images and also to various quantities suitable for A-scan display. Then a short

description of the experiment is given, followed by results. The results include maintenance of coherence, spreading of the echo energy through singular values and the spreading of the focus in back-propagated images. Alternative transmission schemes are discussed. Finally a summary and conclusions are given. II. THEORY The theory of DORT has been presented in terms of generalized time-reversal [1] and in terms of the sonar equation [4]. First a signal is transmitted by beam or element n of a set of N . This signal then propagates through a medium and scatters from one or more objects and is then received by beam or element m of a set of M . The received signal is then assembled as s !n , m , t " an array of three indices, where t is the time index. This signal is then decomposed into E epochs of duration TE using a time window w!t ", 1 # t # TE with a time shift of TS E TE

p!n , m ,t " % $ $ p!n , m ,t ( !e ' 1"TS ( t & ' 1"w!t &" . e %1 t & %1

(2.1)

Note that the window and overlap are designed to preserve the signal. Each epochal signal TE

p E !e , n , m ,t " % $ p!n , m ,t ( !e ' 1"TS ( t & ' 1"w!t &" t & %1

(2.2)

is then Fourier transformed along the time dimension PE !e , n , m , f " % F ! p E !e , n , m ,t "" and then decomposed using the singular value decomposition (SVD) along the transmit and receiver indices, namely n and m . PE !e , n , m , f " %

S

$U !e , n , s , f ")!e , s , f "V T !e , s ,m , f "

(2.3)

s %1

All of the indices are indicated, which makes the representation a bit cluttered and confusing. Note that the epoch and frequency indices are equal on the left and right hand side of the equation; the SVD is performed on each frequency component in each epoch. There are S singular components in the decomposition. Thus each epoch can be decomposed into singular components

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~ PE !e , n , s , m , f " % U !e , n , s , f ")!e , s , f "V T !e , s , m , f " so that S

p!n , m ,t " % $ ~p !n , s , m ,t "

(2.5)

s %1

where E

~p !n , s ,m ,t " % $ ~pE !e ,n , s ,m ,t "

(2.6)

e %1

and ~ ~p !e , n , s , m ,t " % F '1 P (2.7) E E !e , n , s , m , f " . Of course, attention must be paid to prevent wrap around of the inverse Fourier transform back into the time domain. The value of constructing the singular component ~ p !n , s , m , t " relies on the physical interpretation of it. In the frequency domain each singular component is the response from one resolved component [1]. With the assumption that one scatterer predominates over the frequency band, each singular component is predominantly comprised of signal from one scatterer. Each singular value can be transformed into the time domain

!

III. EXPERIMENT

(2.4)

"

The Time Reversal Experiment 04 (TREX-04) was performed on the continental shelf east of New Jersey at approximately 39º 11´ N and 72 º 48´ W. The sound speed profile was generally downward refracting with a warm water incursion at the bottom. Two ships were involved, the R/V Endeavor and the R/V Cape Henlopen. The Cape Henlopen was moored with the NRL 64-element time-reversal array (TRA) hung from the A-frame on the stern. The R/V Endeavor deployed an echo repeater consisting of either a Raytheon XF4 for frequencies below 2500 Hz or an ITC 2000 for frequencies between 2500 and 3500 Hz. The TRA was used to transmit a sequence of 250 msec, 500 Hz bandwidth linear frequency modulated (LFM) signals. Each LFM was transmitted from all the elements into beams of -5º, 0º, 5º and 10º depression angle, with positive angles oriented toward the bottom of the sound channel. At 500 Hz the predicted beams were computed using RAM PE code are shown in Fig. 1 [5]. The position of the echorepeater is at range = 300 m and depth = 40 m. Note that the 10º beam does not strongly ensonify the echo repeater.

* !e , s ,t &" % F '1 !)!e , s , f "" .

(2.8) Various quantities are derivable from * that are useful for an A-scan display. For example, the peak singular value 2 PSV p !e , s " % MAX t & 0. * !e , s , t &" -+ , the mean singular value , /

(a)

(b)

2 MSV p !e , s " % MEAN t & 0. * !e , s ,t &" -+ or integrated singular , /

(c)

2 ISV p !e , s " % SUM t & 0. * !e , s ,t &" -+1t & . These are , / related to the size of the signal at each epoch of time and in each singular index. The normalized integrated singular value NISV p !e , s " % ISV p !e , s " $ ISV p !e , s " is useful for

value

(d)

s

comparing results from DORT analysis of echoes to the results from noise. Another convenient presentation of the data is the backpropagated image of a singular component ~ ~ i !e , r , z , s , t &" % F '1 I !e , r , z , s , f " , where (2.9) ~ ~ I !e , r , z , s , f " % $ G !r , z , n , f "P !e , n , s , m , f "G !r , z , m , f "

!

"

m ,n

(2.10) and G is the Green’s function between the field point !r , z " and the beam n or m . This image has five indices. The epoch e is the total time delay after transmission and is grossly indicative of the range. The range and depth are obvious and depend on the knowledge of the Green’s function. The singular index is indicative of which scatterer is selected by DORT processing. The time t & is the timedelay difference between the echo and the Green’s functions.

Fig. 1. The four transmitted beams at 500 Hz at (a)-5º, (b) 0º, (c) 5º, and (d) 10º. No bottom reverberation was apparent in the data even though the beams were directed at the bottom. This is due to the low source level, the ambient noise level and warm water incursion covering the bottom. The signal from the echo repeater was therefore noise-limited. The responses from the echo repeater is recorded at a sampling rate of 10129 Hz and stored in files of 64 channels and 20 second duration. The data is then demodulated to the relevant 500 Hz bandwidth, matched filtered and stored as complex, time-domain data. The complex time domain data is shown as a function of phone depth versus time in Fig. 2(a) below.

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This data s !n , m ,t " are then decomposed using a rectangular window into epochs with TE % 0.250 s such that each peak is contained in one epoch. DORT processing is then performed.

(a)

A. Spreading through singular value Because DORT processing depends on maintaining coherence over the four transmissions, a sequence of four identical 0º beams are transmitted to measure the degree of coherence. The normalized integrated singular value NISV p !e , s % 1" is plotted in Fig. 4 below for data from each

(b)

Fig. 2. The transformation of the received signal from (a) element space to (b) beam space. The data are transformed from 64 elements shown in Fig. 2(a) into 17 beams shown in the lower plot, Fig. 2(b). The transformation was performed by transforming the signal into the frequency domain, applying the relevant phase shifts and retransforming into the time domain. Little signal energy outside the angular range of +/- 20º was observed. Angular spacing of 2.5º were used, generating 17 receiver beams from 64 receiver channels.

frequency bands (a), at various wind speeds (b) and various ranges (c). Two observations are noted from this analysis. First there appears to be a clear degradation of coherence with frequency in Fig. 4(a). Secondly, the value of coherence in the highest frequency band 3.0-3.5 kHz is very close to 0.59 the value of the normalized singular energy of ocean noise NISVn !e, s % 1" taken during this test. The manifestation of this degradation in coherence is shown in Figs. 5 and 6 that show back propagation images . (a)

(b)

IV. RESULTS After being transformed into beam space, each complex, time-domain signal is time shifted by multiples 0.250 s to form the four source channels. Fig. 3 shows the four source channels received in one receiver beam, with m % 1 in Fig. 3 (a) and m % 4 in Fig. 3 (c). All four peaks align at approximately 2.2 s.

(c)

Fig. 4 The normalized singular energy as functions of (a) frequency, (b) wind speed and (c) range.

(a)

(b)

(c)

(d)

Fig. 3 The four transmitted signals after matched filtering and time-shifting into four source channels.

~ The back-propagated image i !e , r , z , s ,t &" is displayed as a function of depth z and time delay t & at the epoch that contains all four components and at the range expected from GPS measurements taken on each ship. Figure 5 shows a back-propagation image of the lowest frequency data from 0.5 to 1.0 kHz. Note that the back-propagation image of the first singular value in Fig. 5(a) contains nearly all of the signal energy with no visible signal in singular value 2 through 4. Compare that with the image of data taken at the higher frequency range 2.5-3.0 kHz shown in Fig. 6 where the signal energy is spread through all four singular values. The degree of spreading seen in Fig. 6 is comparable to that shown in Fig. 7 of the back-propagated image from ocean noise in the same frequency band.

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(a)

(a)

(b)

(b)

(c)

(c)

(d)

(d)

Fig. 5 Echo back-propagated image at 0.5-1.0 kHz. with (a) s=1, (b) s=2, (c) s=3 and (d) s=4.

Fig. 6 Echo back-propagated images at 2.5-3.0 kHz. with (a) s=1, (b) s=2, (c) s=3 and (d) s=4 of noise only.

(a)

(a)

(b)

(b)

(c)

(c)

(d)

(d)

Fig. 7 Ocean-noise back propagated images at 2.5-3.0 kHz. with (a) s=1, (b) s=2, (c) s=3 and (d) s=4.

Fig. 8. Simulated-echo back-propagated images at 2.5-3.0 kHz with (a) s=1, (b) s=2, (c) s=3 and (d) s=4.

(a)

(a)

(b)

(b)

(c)

(c)

(d)

(d)

Fig. 9 Back-propagation images of the first singular values of (a) 0.5-1.0, (b) 1.0-1.5, (c) 1.5-2.0 and (d) 2.5-3.0 kHz for sea data.

Fig. 10 Back-propagation images of the first singular value of (a) 0.5-1.0, (b) 1.0-1.5, (c) 1.5-2.0 and (d) 2.5-3.0 kHz for simulated data.

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Note that the back-propagated image of noise shown in Fig. 7 contains signal energy at the extreme ends of the time axis. However, the back-propagated image of the 2.5 - 3.0 kHz data has no apparent noise at the extreme ends of the time axis. This implies that the SNR of the 2.5 - 3.0 kHz data is very high. The spreading of signal energy across the singular values in Fig. 6 is not due to noise but instead due to motion that is more important at the higher frequencies in this experiment. Data with no motion is generated using RAM PE from 2.5 to 3.0 kHz. After DORT processing and back-propagation, the results are shown in Fig. 8. Note that all of the singular energy is concentrated into the first singular value s % 1 shown in Fig. 8(a). The higher singular components in Fig. 8(b) to (d) have no energy in them at all. This implies that the spreading shown in Fig. 6 is due to motion. (a)

(b)

(c)

(d) Fig. 11. The transmission loss at 500 Hz. from the four transmitted beams at (a) -5°, (b) 0°, (c) 5°, (d) 10°. (a)

(b)

(c)

(d) Fig. 12. The transmission loss at 2500 Hz. from the four transmitted beams at (a) -5°, (b) 0°, (c) 5°, (d) 10°.

B.

Spreading of the focus Another phenomenon observed with this data set in the ~ back propagated images i !e0 , r0 , z , s % 1, t &" is the focal spreading that increases with increasing frequency. This ~ effect can be seen in Fig. 9 that shows i !e0 , r0 , z , s % 1, t &" for

(a) 0.5 - 1.0 kHz, (b) 1.0 - 1.5 kHz, (c) 1.5 - 2.0 kHz, (d) 2.5 3.5 kHz. The physical depth of the echo-repeater, measured with a depth sensor, are (a) 40 m for 0.5 - 1.0 kHz, (b) 40 m for 1.0 - 1.5 kHz, (c) 54 m for 1.5 - 2.0 kHz, and 60 m for 2.5 - 3.0 kHz. In each subplot the peak of the back-propagated signal is demarcated with green diamonds on the top and left axes. The depth is quite accurately predicted in (a), (b) and (c). At the highest frequency band, the depth is estimated to be approximately 40 m significantly different from 60 m. The focus in the lowest frequency band, both data in Fig. 9(a) and simulated in Fig. 10(a), is not well-defined. The spread of the focus is due to the fact that the echo-repeater was not well ensonified by all four beams. This is shown in Fig. 11 where the echo-repeater location is shown as a green diamond in Fig. 11(a) through (d). The echo repeater is in the center of the 0º beam but at the edge of the -5º and 10º beams. In this case the four beams do not well sample the scattering volume and therefore do not spatially isolate the echo-repeater well. Note that the quality, or appearance, of the focus in depth time-delay changes dramatically with increasing frequency. This dramatic change is not due to motion because motionfree numerical simulations produce very similar focal patterns as shown in Fig. 10 that shows numerically ~ simulated images i !e0 , r0 , z , s % 1, t &" for the corresponding frequencies, ranges and depths as in Fig. 9. Careful inspection of Figs. 9(c), 9(d), 10(c) and 10(d) reveal that there is a sharp peak surrounded by diffuse sidelobes. Figures 9(a) and 10(a) on the other hand display peaks that are more spread and are surrounded less extensively by side-lobes. This phenomenon is due to the use of the same source beams across all frequency bands. This is completely analogous to the use of using a linear array with equally spaced element over many frequency bands. At higher and higher frequency bands, grating lobes eventually appear. The use of beams with a vertical array in a shallow-water channel is more complicated; the complexities of propagation prevent precise grating lobes from appearing. Instead of grating lobe peaks appearing, a diffuse region is produced. A qualitative view of the increased coarseness of the source beams can be seen in Figs. 11 and 12 that display the transmission loss at 500 and 2500 Hz respectively. Two major differences are apparent. The most striking difference is the visibility of the grating beams in Fig. 12. There are clearly at least two sets of beams in each of the subplots; this is due to the element spacing being designed for 600 Hz. These high-angle grating beams produce echo components with significantly greater time delays than the lower-angle main beams. These longer time-delay components significantly spread the focus along the time-delay dimension. The second difference between Figs. 11 and 12 is appearance of more continuity in the spatial distribution of the field from -5° in (a) to 10° in (d). At lower frequency in Fig. 11, the progression through angle appears much more continuous than at higher frequency in Fig. 12. This coarseness of angular sampling prevents the complete

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cancellation of the back-propagated field at areas surrounding the focus. This explains the increased low-level spreading in the depth dimension. V. ALTERNATIVE TRANSMISSION SCHEMES In this experiment, a sequence of four LFM signals were transmitted. This signal set can be described as time division multiple access (TDMA) because four beams are transmitted and individually received. This set of signals are good for a proof-of-concept experiment because their separation in time permits the complete separation of the individual echoes while preserving coherence over the transmission sequence. Thus in Fig. 3(a), the decay time of the two-way propagation is clearly visible and is less than the 250 ms separation. This transmission sequence does have a major detraction if an A-scan signal, e.g. ISV p !e , s " , is plotted as in [4] then a sequence of peaks is generated along the epoch-time e axis. Thus a target echo would appear as a sequence of 2 N ( 1 peaks in the singular channel s % 1 . This phenomenon is similar to an ambiguity surface in range-Doppler estimation. Alternative transmission schemes have been discussed in [6] with the transmission of adjacent narrow band signals being termed Adaptive Instant Recording (AIR) signals. These signals have very good auto-correlation and crosscorrelation properties. However, a set of AIR signals can still produces a sequence of peaks in the time-delay domain. Continued engineering of suitable signals for multiplexing and suitable epochal windowing is needed to find signals that are optimized for specific sonar sourcereceiver geometries in noise-limited or reverberation-limited environments. VI. SUMMARY AND CONCLUSIONS In this paper, several points are made. First the maintenance of coherence is necessary to concentrate a signal component into a single singular component. The degradation of coherence in this experiment appears to be due to ship motion with less coherence at higher frequency. The signal transmission sequence took one second in this experiment. A sonar mounted on a larger ship with less heaving motion could operate at higher frequency or with a longer transmission sequence. The second point is that the response time of the environment must be taken into account so that the echoes from different source transmissions are separated from each other. This point is made also in [6] where the response time determines limitations on frequency spacing. In our TDMA transmission scheme, the response time determines the time delay of transmission sequence. The third point is that noise is decomposed into each singular component. The distribution of noise power across the components depends on the noise arrival structure as well as the dimensions of the data matrix. The fourth point is that the transmission of beams, or combinations of beams, is useful for transmitting full power of the source array [7]. The beams should be chosen to

independently well sample the scattering volume. For example, both upward and downward beams would have sampled the scattering volume in the 0.5 - 1.0 kHz case shown. Also, if back-propagation images are to be produced, the set of beams should be chosen to generate a suitable focus; if a compact focal region is desired then the beams should be close enough to permit cancellation in regions far from the focus. A fifth point is that transmitted signals with relatively long-time delays, as the grating beams in Fig. 12, can degrade the back-propagated focus. This is solved by using an array that is cut for the transmitted frequency band and transmitted angles. Finally, and more fundamentally, this paper presents a demonstration of DORT on a ship-mounted sonar. In the lower-frequency bands, the echo is concentrated into one singular component and that component is associated with a region of the ocean near to the depth and range of the echorepeater. This implies that the depth separation of scatterers in the same range resolution cell is possible, which can improve the detection of targets in a reverberation-limited environment. Acknowledgements

The authors acknowledge many fruitful and stimulating discussions with Claire Prada, Thomas Folegot, M. Fink, Alan Meyer, David Chambers and H.C. Song. This work was funded by ONR. References

[1] C. Prada, S. Manneville, D. Spoliansky and M. Fink, “Decomposition of the time reversal operator: Detection and selective focusing on two scatterers,” J. Acoust. Soc. Amer. 99, p. 2067, (1996). [2] E. Kerbat, C. Prada, D. Cassereau and M. Fink, “Imaging in the presence of grain noise using the decomposition of the time reversal operator,” J. Acoust. Soc. Amer. 113, p. 1230 (2003). [3] G. Montaldo M. Tanter and M. Fink, “Revisiting iterative time reversal processing: Application to detection of multiple targets,” J. Acoust. Soc. Amer. 115, p. 776 (2004). [4] C.F. Gaumond, D.M. Fromm, J. Lingevitch, R. Menis, G. Edelmann, D. Calvo, and E. Kim, “Application of DORT to Active Sonar,” Proceedings of Oceans 04, p. (2004). [5] M.D. Collins, “A split-step Pade solution for parabolic equation method,” J. Acoust. Soc. Amer. 93, pp. 17361742 (1993). [6] T. Folegot, J. DeRosny, C. Prada and M. Fink, “Adaptive instant record signals applied to detection with time reversal operator decomposition,” J. Acoust. Soc. Amer. 117, pp. 3757-3765 (2005). [7] J.F. Lingevitch, H.C. Song and W.A. Kuperman, “Time reversed reverberation in a waveguide,” J. Acoust. Soc. Amer. 111, pp. 2609-2614 (2002)

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