CWP-713

Optimization of time reversal focusing throughdeconvolution

T.J. Ulrich1, Johannes Douma2, Brian E. Anderson1, and Roel Snieder21 Geophysics Group EES-17, MS D443, Los Alamos National Laboratory, Los Alamos, New Mexico 875452 Center for Wave Phenomena, Colorado School of Mines, Golden, Colorado 80401

ABSTRACTIn this study a technique is demonstrated to optimize the ability of time re-versal to both spatially and temporally focus, or compress, elastic wave energy.This method allows for optimization of time reversal focusing for any numberof channels by calculating the required time reversed signal to be rebroadcastfrom the time reversal mirror element using only the original recorded scatteredsignal from that element. The speed, simplicity and robust nature of this pro-cess for any number of time reversal mirror elements, including only a singlechannel, provide the ability to enhance applications currently using time re-versal for tasks such as communications, nondestructive evaluation, lithotripsyand imaging, among others. It is also demonstrated theoretically and numer-ically that good temporal focusing implies that the radius in the sphericallysymmetric part of the spatial focus is small.

Key words: time reversal, signal processing, communications

1 INTRODUCTION

Time Reversal (TR) in acoustics has been the focusof much research (Parvulescu, 1961; Fink, 1997; Lar-mat et al., 2010). This has led to many applicationsin a wide variety of fields, such as medicine, communi-cations, nondestructive evaluation (NDE), seismology,etc. In many of these applications, the ability to use TRdepends upon its ability to precisely compress the mea-sured scattered waveforms to a point in both time andspace, ideally a delta function, δ(t), which is commonlyreferred to as TR focusing. The desire to enhance theTR process to focus wave energy has lead researchersto develop a variety of ways to accomplish this. Sometechniques use arrays of input transducers, measure thewave field with an array near the desired focal spot, andthen optimize the spatial and temporal focusing (Tan-ter et al., 2000, 2001; Montaldo et al., 2004; Vignonet al., 2006; Gallot et al., 2011; Bertaix et al., 2004;Roux and Fink, 2000; Aubry et al., 2001; Jonsson et al.,2004). Other methods use an array of input transduc-ers and optimize the temporal focusing at an outputtransducer (Daniels and Heath, 2005; Qiu et al., 2006;Blomgren et al., 2008; Zhou et al., 2006; Zhou and Qiu,2006). Some of these techniques require, or at least ben-efit greatly from, large arrays, while others may enhancetemporal focusing at the expense of spatial localization.

We explore a simple approach and use deconvolu-tion, a primitive, though robust, version of the inversefilter (Tanter et al., 2000; Gallot et al., 2011), to givean input signal that gives a delta function δ(t) as anoutput at the focal point, and show laboratory experi-ments with ultrasound that the temporal focusing thusobtained is superior to that obtained by using time re-versal. By scanning the sample near the focal point, wefind that this improved temporal focusing is accompa-nied by a better spatial focusing. We then show theoreti-cally that improved temporal focusing implies enhancedspatial focusing for the spherical average of the focus.The method described in this paper enhances both spa-tial and temporal focusing for any number of array ele-ments, including a single channel. In contrast to previ-ous applications of the inverse filter (Tanter et al., 2000,2001; Gallot et al., 2011), we also explore the the effec-tiveness of using deconvolution for small time reversalmirrors (TRM), i.e., using 1-8 channels, and in the caseof limited available bandwidth.

2 DECONVOLUTION AS AN INVERSEFILTER

The time reversal process relies on the assumption thatfrom a recorded impulse response, or Green function

126 T.J. Ulrich, Johannes Douma, Brian E. Anderson, and Roel Snieder

GAB between two points A and B, a simple reversal intime of that signal and rebroadcast of it back into thesame propagating medium, from A to B or vice versa,will focus an impulsive function δ(t) at the appropriatepoint in space (A or B); or

∫ ∞−∞

GAB(τ)GAB(τ − t)dτ ?= δ(t) . (1)

where reciprocity has been used to replace the Greenfunction GBA with GAB . Expression (1) states that theautocorrelation of GAB(t) is ideally equal to a deltafunction. In practice, actual TR experiments cannotrecreate a true delta function focus as one or more condi-tions necessary to satisfy eq. (1) are typically violated.For example, due to either time, physical memory, orattenuation constraints, it is not possible to record foran infinitely long period of time. Also, the Green func-tions here are assumed to contain an infinite bandwidth,which is routinely not the case in experiments. For thisreason we, as others before us, asked the question: whatwould be required to most closely approach an impulse-like TR focus given typical experimental limitations?

Going back to eq. (1), we can rewrite this (using aconvolution notation, rather than the integral form) as

f(t) = g(t) ? r(t) ≈ δ(t) , (2)

where f(t) is the acoustic wave signal measured at thesource, r(t) is the signal send from the source, and g(t) isthe signal that is back propagated to the source. Here werestrict ourselves to looking only at signals between thetwo points A and B and thus have removed them fromthe notation. We also change from a Green functionnotation so as not to imply that we necessarily have aninfinite bandwidth available. From eq. (2), we aim toapproximate the focused signal f(t) to a delta functionby assuming that the recorded impulse response r(t)contains information enough to produce an optimizedsignal g(t) to rebroadcast in our TR focusing procedure.

Deconvolution equates to inverse filtering by trans-forming to the frequency domain, thus eq. (2) becomes

f(ω) = g(ω)× r(ω) ≈ 1 , (3)

which can easily be manipulated to provide g(ω) sim-ply from measuring r(ω) (or rather from the FFT ofr(t)). As such, we can now calculate the optimized signalwhich should produce the most impulse-like TR focusfrom

g(ω) =1

r(ω)=

r∗(ω)

|r(ω)|2 . (4)

This expression gives a mathematical expression forg(ω), but this may not be practical for experimentaluse in the event that there is a limited bandwidth, ormore precisely if r = 0 at any frequency. To avoid this

0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0

TR O

ptimi

zed C

oda

T i m e ( m s )

0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0- 1 . 0- 0 . 50 . 00 . 51 . 0

TR St

anda

rd Co

da

T i m e ( m s )

Figure 1. Time reversed signals used for back propagation,

top (blue): original response r(−t) due to a 5µs 200kHz tone

burst; bottom (red): optimized signal g(t) from the deconvo-lution procedure described herein.

we simply add a constant to the denominator to ensurethat we never divide by 0, hence we replace eq. 4 by

g(ω) =r∗(ω)

|r(ω)|2 + ε, (5)

where ε is a constant related to the original receivedsignal as

ε = γ ×mean(|r(ω)|2) . (6)

We used an arbitrary value of γ = 1 for the experimentsdescribed in the next section.

As the above derivation has been performed in thefrequency domain, it is necessary to transform back tothe time domain to recover the optimized signal, g(t)to be used in the time reversal experiments. It is worthnoting that this procedure produces a g(t) that can beused directly for rebroadcasting, i.e., there is no needto perform a TR operation g(t) → g(−t) to obtain afocused signal. While this completes the deconvolutionoptimization procedure used in this paper, it is onlya portion of the inverse filter procedure as defined byTanter et al. (2000), and utilized most recently by Gallotet al. (2011). Performing the full inverse filter procedurerequires the singular value decomposition (SVD) of thematrix of filtered waveforms. For reasons that will beaddressed in the following section this step is neithernecessary nor appropriate for this study.

3 EXPERIMENTAL VALIDATION

The purpose of this study is to optimize TR in typicalexperimental situations, specifically those where onlya few channels are used with reverberant signals in aclosed cavity. With this in mind, it is prudent to testthe methodology experimentally, as presented above. Todo this, we used a sample made of aluminum, approx-imately 1000cm3 volume though irregularly shaped. A

Optimization of time reversal focusing through deconvolution 127

source transducer (2 cm PZT disk) was attached to thesample using epoxy. This defines, and fixes, our sourceposition A. The receiver, a non-contact laser vibrome-ter (PSV 301, OFV 5000 controller, Polytec Inc.) waspositioned at a point B on the surface. As we cannotperform a classical TR experiment with a laser vibrom-eter, since it cannot act as both receiver and source, weperformed reciprocal TR experiments, i.e., where theoriginal source and back propagation signals are emit-ted from the same location/transducer, thus focusingthe wave energy to the original receiver location. Thisuse of TR in a reciprocal sense is now a widely usedform of TR, and indeed is the method originally used byParvelescu (Parvulescu, 1961) in the first known demon-stration of TR in acoustics (also known as phase conju-gation). It is due to the use of reciprocal TR that SVDis unnecessary. The SVD step of the inverse filter pro-cedure is applicable to the classical TR procedure asits purpose is to remove correlated noise that may existsimultaneously in all of the received signals. ReciprocalTR requires that each received signal r(t) is obtained in-dividually, i.e., not simultaneously. For this reason thereis no expected correlation of noise from one measurmentof r(t) to the next.

The original impulse, a 5µs shaped toneburst of200kHz, was broadcast from the source transducer us-ing a 12-bit arbitrary waveform generator with a conver-sion rate of 10MHz. The response to this impulse wasrecorded by the laser vibrometer (5mm/s/V, 250kHzbandwidth) using a 14-bit digitizer also sampling at10MHz. The generator and digitizer were synced suchthat the source impulse was centered in a 3.2768ms win-dow, thus the first portion of the received signal r(t)contains no causal signal. This also means that the TRfocus will occur at 1.6384ms during the rebroadcast.

In fig. 1, both the reversed original response r(−t)and the result of the optimization procedure g(t) can beseen. Each of these signals were used independently toachieve a TR focus. These focused signals, correspond-ing to the back propagation of these signals, can befound in fig. 2 (a). There are a few key points to benoted in these two figures. First, note the acausal sig-nal, i.e., signal appearing after t = 1.64ms, in the op-timized signal. This portion of the signal is necessaryfor the reconstruction of a symmetric focal signal. Ar-bitrarily setting g(t > 1.64ms) = 0 does not affect theimpulse-like focus at 1.6384ms in fig. 2 (a). However, itdoes destroy the symmetry that is seen in fig. 2 (a) forthe optimized focus.

In fig. 2, the two important features of note are: i)the temporal compression , and ii) maximum achiev-able amplitudes. The normalization of the focal signals(and the original source signal) shown in fig. 2 (a) pro-vide a clear demonstration of the enhanced temporalcompression. This can be quantified by calculating thetotal energy near the focal time (50 samples centeredat 1.6384ms, coresponding to the duration of the origi-

0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 00 . 00 . 40 . 81 . 21 . 62 . 0 ( b )

Veloc

ity (m

m/s)

T i m e ( m s )

( a )

1 . 5 8 1 . 6 0 1 . 6 2 1 . 6 4 1 . 6 6 1 . 6 8 1 . 7 0- 0 . 10 . 00 . 10 . 20 . 3

S o u r c e O p t i m i z e d S t a n d a r d

Norm

alized

Amplit

ude

T i m e ( m s )

Figure 2. Focused signals resulting from the back propaga-tion of the TR signals shown in fig. 1, compared with the

original source function. (a) illustrates the structure of the

signals near the focal time, each of which have been nor-malized to the total signal magnitude to accurately compare

the temporal compression of the signal. (b) are envelopes of

the focused signals for the entire duration of the signal. En-velopes are presented in absolute velocity to illustrate the

difference in maximum achieved focal amplitude.

nal source) and normalizing by the total energy in thesignal. Performing this metric for these signals, we findthat the original source has 100% of the energy in this50 sample window, while the standard and optimized fo-cused signals contain ∼ 15% and 70% of the total energyrespectively. Fig. 2 (b) shows further evidence in thelack of energy in the optimized signal (red) at all timesaway from the focal window, whereas the standard TRfocal signal exhibits temporal side lobes symmetricallyaround the focal time. The second point to be notedfrom these figures can be seen in fig. 2 (b). The maxi-mum achieved focal amplitude for the optimized processis approximately one half of that achieved by the stan-dard TR process, a fact that we observe in all tests ofthis technique for all numbers of channels 1-8.

While figs. 1 and 2 effectively illustrate the opti-mization of temporal compression, albeit at the expenseof the amplitude, they do not in any way address thespatial extent of the focus. To explore this, the samelaser vibrometer was used in a scanning mode to mea-sure the wavefield in a 5 × 5cm2 region around the focuspoint (i.e., location B) with a resolution of 2mm. Theimages in fig. 3 show the wavefield measured at the timeof focus (t = 1.6384ms) for both the standard and opti-mized TR process. The standard TR process in fig. 3 (a)shows the typical focal spot of approximately λ/2 with afringe, or ring, appearing at approximately 2λ from thefocal point. Using the optimized signal g(t) from fig. 1,on the other hand, appears to have a smaller spatialextent, see fig. 3 (b), but most notably lacks any struc-ture away from the central focus. Normalizing the twoimages and performing a simple subtraction (standard -optimized) verifies the more peaked nature of the opti-

128 T.J. Ulrich, Johannes Douma, Brian E. Anderson, and Roel Snieder

position (mm)

posit

ion

(mm

)Standard TR Focus

0 10 20 30 40 500

10

20

30

40

50

mm

/s

2

4

6

8

10

position (mm)

posit

ion

(mm

)

Optimized TR Focus

0 10 20 30 40 500

10

20

30

40

50

mm

/s

1

2

3

4

5

position (mm)

posit

ion

(mm

)

Normalized Difference

0 10 20 30 40 500

10

20

30

40

50

arb.

0.03

0.02

0.01

0

0.01

0.02

0.03

Figure 3. Images of the spatial extent of the focus. (a) thestandard TR focus, corresponding to the blue signals in figs. 1and 2; (b) the optimized focus, corresponding to the red sig-

nals in figs. 1 and 2; and (c) difference of normalized images

(a)-(b).

mized focus (central blue region) and visually enhancesthe ring structure present in the standard focus (fig. 3(c)).

Both the enhanced temporal and spatial compres-sion can be quantified by examining the amount of en-ergy in the focal region, defined as 10 µs (the full widthof the source pulse) centered around 1.6384 ms, and thediffraction limit (λ/2) as a spatial definition. In bothcases, spatial and temporal, the deconvolution method

Figure 4. (a) Ratio of energy in the focal time (5µs cen-

tered at 1.6384 ms) to the total energy of in the focal signals

displayed as a function of the number of channels, i.e., TRMelements, used. (b) The ratio of energy in the focal region

(as defined by the diffraction limit λ/2) to the total energy

in the scan region (x-direction linear scan, 0-70 mm, 1mmspatial resolution, focal point centered at 35 mm) displayed

as a function of the number of channels, i.e., TRM elements

used.

enhances the energy compression over the standard TRprocedure. When performing these same tests for vari-able number of channels (1-8) the results are consis-tent showing the maximal compression is achieved atapproximately 4-5 TRM elements (for either deconvolu-tion or standard TR), after which time total amplitudeincreases linearly with number of channels (not shown)but further enhancement to focusing is minimal (fig. 4).To accomplish this study as a function of TRM ele-ments, the same experimental procedure as previouslydescribed was used for up to 8 individual transducersaffixed at 8 unique locations on the aluminum sample.For 1 channel, the values in Fig. 4 result from an aver-age of the temporal and spatial metrics across the re-sults from the 8 individual channels. The results for Nchannels come from summing the data of N channelsto produce the two metrics and then averaging thesemetric values across various combinations of N of the 8channels. The results for 8 channels is a summation ofall 8 individual data to produce one value each for thetemporal and spatial metrics.

In conclusion, the main objective for the use of de-

Optimization of time reversal focusing through deconvolution 129

convolution was to improve the temporal focusing. Weshowed experimentally that simple deconvolution canachieve this, albeit at the expense of the amplitude. Wethen discovered and showed experimentally that, notonly did the method improve the temporal focus, thespatial focus was also improved. In the next section, wepresent one explanation for both improved spatial andtemporal focusing.

4 IMPROVED TEMPORAL FOCUSINGLEADS TO IMPROVED SPATIALFOCUSING

In this section, we will show that better temporal fo-cusing implies better spatial focusing for the sphericalaverage of the focus. We begin by first considering thewave field near its focal spot at r = 0 and consider themedium to be locally homogeneous in that region. Thesolution of the Helmholtz equation in a homogeneousmedium can be written as

p(r, θ, ϕ, ω) =

∞∑l=0

m=l∑m=−l

almjl(kr)Ylm(θ, ϕ) , (7)

see table 8.2 of ref. Arfken and Weber (2001). In thisexpression jl denotes the spherical Bessel function, Ylm

the spherical harmonics, and k = ω/c. According to ex-pressions (11.147) and (11.148) of ref. Arfken and Weber(2001), jl(0) = 0 for l ≥ 1 and j0(0) = 1. This meansthat at the focal point r = 0 only the terms l = m = 0contribute. Using that Y00 = 1/

√4π (table 12.3 of ref.

Arfken and Weber (2001)), this means that at the focalpoint

p(r = 0, θ, ϕ, ω) =a00√4π

. (8)

The properties of the wave field at the focal point thusonly depend on the coefficient a00. Since the l = m = 0component of the spherical harmonics expansion givesthe spherically symmetric component of the wave field,the properties of the wave field at the focal point canonly bear a relation to the spherically symmetric com-ponent of the wave field. The properties of temporalfocusing can thus only be related to the spherically sym-metric component of the spatial focusing.

Because of this, we restrict ourselves to the spher-ically symmetric component (l = m = 0) of the wavefield at the focal point. Using that j0(kr) = sin(kr)/kr,the spherically symmetric component of expression (7)is given by

p(r, ω) = p0e−ikr − eikr

r, (9)

with p0 = −a00/(2ik√

4π). The coefficient p0 dependson frequency. Using the Fourier convention f(t) =

∫p0(ω)e−iωtdω, and using that k = ω/c, expression (9)

corresponds, in the time domain, to

p(r, t) =f(t+ r/c)− f(r − r/c)

r. (10)

In this expression, f(t + r/c) denotes the wave that isincident on the focus, and f(t− r/c) the outgoing waveonce it has passed through the focus. The field at thefocus follows by Taylor expanding f(t± r/c) in r/c andtaking the limit r → 0, this gives

p(r = 0, t) =2

cf ′(t) . (11)

In this expression and the following, the prime denotesthe time derivative. Expression (11) states that thewave field at the focus is the time derivative of theincoming wave field.

Equation (11) gives the temporal properties of thefocus. In order to get the spatial properties, we considerthe wave field near the focal point at time t = 0. Set-ting t = 0 in expression (10), and making a third orderTaylor expansion of f(±r/c) gives

p(r, t = 0) =2

cf ′(0) +

r2

3c3f ′′′(0) . (12)

Using equation (11) to eliminate f , gives

p(r, t = 0) = p(r = 0, t = 0) +1

6c2p′′(r = 0, t = 0)r2 .

(13)This is a parabolic approximation for the wave field nearthe focus, with coefficients related to the wave field andthe focus and its second time derivative. The radius Rof the focal spot can be estimated by setting the lefthand side of equation (13) equal to zero, this gives

R =√

6c2

√−p(r = 0, t = 0)

p′′(r = 0, t = 0). (14)

A good temporal focusing means that the temporalcurvature of the wave field at the focal point is strong,this means that −p′′/p is large, and that the radius Ris small. Good temporal focusing thus implies that theradius in the spherically symmetric part of the spatialfocus is small.

5 CONCLUSION

Here we have introduced a simple method for determin-ing the optimal signal to be used in a TR experimentin order to maximally compress the focus in both timeand space. This method has been experimentally veri-fied, albeit without fully varying all possible parameters.A more detailed study is currently underway to quan-tify the effects due to variations in available bandwidth,

130 T.J. Ulrich, Johannes Douma, Brian E. Anderson, and Roel Snieder

and material properties (e.g., homogeneous vs. hetero-geneous media, range of attenuations, wave speeds, etc.)

While temporal and spatial focusing have been en-hanced with this method, there is a cost in the max-imum achieved amplitude. This fact, coupled with theeffect of the acausal portion of g(t) may lead one to thinksimplistically about g(t) as containing a signal to focusthe energy at the appropriate time and place, and an-other signal that, simultaneously, actively cancels noiseat all other times and locations. Thus the optimized fo-cus is cleaner and more impulse-like, but not as strong,as some of the energy being transmitted is being exertedin suppressing sidelobes, spatially and temporally.

We have also shown, theoretically and numerically,that if one has good temporal focusing, the radius in thespherically symmetric part of the spatial focus is small.This concept can turn out to be useful in future researchbecause it allows one to work on improving the temporalfocusing in order to improve the spatial focusing.

ACKNOWLEDGMENTS

Work supported by Los Alamos National LaboratoryInstitutional Support (LDRD).

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