Locating Events Using Time Reversal and Deconvolution: Experimental Application and Analysis

J Nondestruct Eval (2015) 34:2 DOI 10.1007/s10921-015-0276-x

Johannes Douma · Ernst Niederleithinger · Roel Snieder

Received: 28 March 2014 / Accepted: 13 January 2015 © Springer Science+Business Media New York 2015

Abstract Time reversal techniques are used in ocean 1 Introduction acoustics, medical imaging, seismology, and non-destructive evaluation to backpropagate recorded signals to the source Several methods are used to evaluate acoustic signals gen- of origin. We demonstrate experimentally a technique which erated by events in media such as water, rocks, metals or improves the temporal focus achieved at the source location concrete. Most of them are summarized as acoustic emis- by utilizing deconvolution. One experiment consists of prop- sion methods (AE), which are used to localize and character- agating a signal from a transducer within a concrete block ize the point of origin of the events. Sophisticated methods to a single receiver on the surface, and then applying time have been developed in seismology to localize and charac- reversal or deconvolution to focus the energy back at the terize earthquakes. Time reversal (TR) has been a topic of source location. Another two experiments are run to study the much research in acoustics due to its robust nature and abil- robust nature of deconvolution by investigating the effect of ity to compress the measured scattered waveforms back at changing the stabilization constant used in the deconvolution the source point in both space and time [1–4]. This has led to and the impact multiple sources have upon deconvolutions’ TR being applied in a wide variety of fields such as medicine, focusing abilities. The results show that we are able to gen- communication, ocean acoustics, seismology or nondestruc- erate an improved temporal focus at the source transducer tive evaluation. However, continued work is being done to using deconvolution while maintaining the robust nature of improve TR’s ability to focus energy. Some newly devel- time reversal. Additionally, deconvolution’s costs are neg- oped techniques use an array of input transducers, measure ligible due to it being a preprocessing step to the recorded the wave field with an array near the desired focal spot, and data. The technique can be applied for detailed investigation then optimize the spatial and temporal focusing [5–13]. Other of the source mechanisms (e.g. cracks) but also for monitor- methods use an array of input transducers and optimize the ing purposes. temporal focusing at an output transducer [14–18]. In this paper, we present evaluation experiments to com- Keywords Time reversal · Deconvolution · Concrete · pare conventional time reversal to an improved variant which Acoustic emission uses deconvolution (DC). We explore the application of DC, which is a primitive though robust version of the inverse filter [5,9], to calculate the optimal signal for backpropaga- · J. Douma R. Snieder tion. The ultrasound experiments are performed on a concrete Center for Wave Phenomena, Colorado School of Mines, Golden, CO, USA block which has sources embedded within. Instead of using e-mail: [email protected] a large array of receivers, the experiments use only a sin- R. Snieder gle receiver to record the scattered waveforms. TR or DC is e-mail: [email protected] then applied to the measured scattered waveforms to calcu- B late the TR and DC signal. The calculated signals are then E. Niederleithinger ( ) backpropagated from a transducer on the surface of the block BAM Federal Institute for Materials Research and Testing, Berlin, Germany into the medium and recorded at the original source location e-mail: [email protected] transducer. By this experiment, we are able to explore and 123 2 Page 2 of 9 J Nondestruct Eval (2015) 34:2 compare the capabilities of TR and DC to focus the mea- upheld during an experiment. This led us to explore the appli- sured waveforms at a point in both space and time. We show cation of deconvolution. that DC significantly improves the temporal focus compared We can rewrite Eq. 1 in a more generalized form (using a with TR. We also run two different experiments to study the convolution notation, rather than the integral form) as robust nature of deconvolution by investigating the effect of ( ) = ( ) ( ) ≈ δ( ), changing the stabilization constant used in the deconvolution F t g t R t t (2) and the impact multiple sources have upon deconvolutions’ where denotes convolution, F(t) is the focal signal or focusing abilities. source reconstruction, R(t) = GAB(t)S(t) is the recorded We have shown previously that deconvolution improves signal measured at the receiver location B from the initial temporal focusing [19]. It was then shown by Douma source propagation where S(t) represent the source as a func- et al. [20] and Douma and Snieder [21] that improved tem- tion of time, and g(t) is the signal necessary to be back propa- poral focusing leads to improved spatial focusing as well gated for focusing. We are able to go from Eqs. 1 to 2 because for both the acoustic and elastic case. Douma et al. [20], we only investigate signals between the two points A and B, Douma and Snieder [21] and Ulrich et al. [19] only used a and remove the Green function notation to indicate we do single source within their experiments and numerical stud- not have infinite bandwidth. Thus, we remove some of the ies. However, within the earth or concrete, multiple fractures unrealistic conditions that are required for Eq. 1 to hold. For may be generated within the time window being recorded. a TR process, the signal for backpropagation is purely the Therefore, the goal of this paper is evaluate the robust nature time reversed recorded signal: g(t) = R(−t). Our goal is to and focusing capabilities of deconvolution when multiplied calculate the optimal signal g(t) such that the focal signal source wave fields are generated within a concrete block. Due F(t) approximately equals a Dirac delta function δ(t). to deconvolution allowing one to focus an arbitrary source Deconvolution equates to inverse filtering by transforming function a source location [19], one could potentially use to the frequency domain, thus Eq. (2) becomes deconvolution to improve the characterization of the earth or concrete through virtual sources [23–27]. F(ω) = g(ω)G(ω)S(ω) ≈ 1. (3) Equation (3)isusedtosolveforg(ω), 1 G(ω)∗ S(ω)∗ 2 Deconvolution Theory g(ω) = = , (4) G(ω)S(ω) |G(ω)S(ω)|2 Time reversal (TR) is a process used to compress the mea- where ∗ denotes complex conjugation. Equation 4 is, how- sured scattered waveforms at a point in both space and time ever, unrealistic for experimental use in the event that there is to ideally a Dirac delta function δ(t). It uses the recorded a limited bandwidth, significant background noise, or more impulse response which can be represented by a Green func- specifically, if R(ω) = 0 at any frequency. To avoid the asso- tion GAB that accounts for the wave propagation between two ciated singularity, we add a constant to the denominator of points A and B. TR then simply reverses the signal in time the last term of Eq. 4 to ensure that we never divide by 0, and propagates it back from the receiver location into the hence Eq. 4 becomes, same medium. By doing so, one expects the energy to focus G(ω)∗ S(ω)∗ at the source location. The TR process can be represented by g(ω) = , 2 (5) the following equation, |G(ω)S(ω)| +

∞ where is a constant related to the original received signal GAB(τ)GAB(τ − t)dτ = δ(t), (1) as −∞ = γ mean |G(ω)S(ω)|2 . (6) where reciprocity has been used to replace the Green’s function GBA with GAB. According to Eq. 1, the TR process, The quantity γ , which is sometimes referred to as the which is equivalent to the autocorrelation of GAB(t), should waterlevel parameter [22], is a constant chosen to optimally ideally lead to equal a delta function. In practice, however, reduce the effect of noise introduced through the DC pro- one cannot truly recreate a Dirac delta function focus due cedure. This quantity may equal any positive number where to one or more conditions, necessary to satisfy Eq. 1, not deconvolution could fail due to noise. Here we use γ = 0.9 upholding. In order for it to work perfectly, one must record for all experiments. The value 0.9 was chosen based on opti- for infinite time, Green’s functions are assumed to have a mizing the focus energy in a process similar to that developed flat, infinite bandwidth, the medium is not attenuative, and by Clayton and Wiggins [22]. Equation 5 gives the solution one must have full coverage of the wavefield at a surface for g(ω). One only has to inverse Fourier transform this result surrounding the points A & B. These requirements are not to retrieve the “optimal” DC signal in the time domain to be 123 J Nondestruct Eval (2015) 34:2 Page 3 of 9 2 backpropagated such that one gets a approximate Dirac delta function focus.

3 Experimental Set Up

A laboratory experiment was created and run in the Civil Engineering lab of Colorado School of Mines. A 30 × 30 × 37 cm3 concrete block was cast from 72kg Quickcrete mix (No. 1101, max aggregate grain size <4mm), with about 60liters water and 5kg of additional gravel (5–15mm grain size). The concrete used has a nominal compressive strength of 27.9MPa after curing for 28days in accordance to ASTM C39/ASTM 387. The block contains only minimal reinforcement as shown in Fig. 1a. Three ultrasonic piezo transducers (type Acsys SO807, center frequency 60kHz, labeled ‘ES’ in Fig. 1) were attached to the reinforcement in order to cast them within the concrete block. These sources (transducers) are visible in Fig. 1a. These sources were oriented differ- ently in order to generate more complex waveforms and to study the effect of varying source orientation. Source 3 was oriented perpendicular to the other two sources. Broadband Acsys sensors type 1803 (center frequency about 100 kHz, labeled ‘PT’ in Fig. 1) were used as external transducers. They are piezo-based and feature a spring-loaded 2mm diam- eter ceramic tip for contact to the concrete. These transducers are most sensitive in the direction perpendicular to the surface it is attached to. The transmitted signal is generated by a custom made rectangular signal generator/amplifier (BAM US in Fig. 1). It is triggered by a TTL impulse which is issued by our data acquisition device (National Instruments model 6366). The recorded signals at the external sensor are first high pass filtered at a frequency of 1kHz and amplified by Fig. 1 Figures and diagrams indicating experimental set up and work- a Stanford Research low noise preamplifier (SR 566) before flows used during acquisition and backpropagation. Note, the diagrams being digitized and recorded. This was necessary to remove indicating the workflows are meant to show the tools used during acquisition and backpropagation and do not accurately describe the interior low frequency noise present in our data. The workflow and set of the concrete for every experiment in this paper. a Source set up and up used for acquiring the data is shown in Fig. 1b. Addition- reinforcement to be placed within the concrete block, b acquisition ally, in order to reduce noise, we have used time averaging workflow, c backpropagation workflow by stacking over 144 runs. For backpropagation, the setup is reversed. The BAM US device is removed. The transmitter signal generator identified by electromagnetic crosstalk between transmitter is replaced by the digital/analog converter integrated in and receiver cables, generating a small but easy to recognize the data acquisition device, sending the computed, time impulse in the receiver data. The laboratory contained other reversed/deconvolved waveforms to the external sensor. The noise due to multiple experiments being run simultaneously. embedded sensor is used as receiver, again using the pream- Due to a high noise lab environment and a lack of a power plifier before AD conversion and recording. This reversed set amplifier for the backpropagated transmitter signal, we up is shown in Fig. 1c. We have used a sampling frequency of apply an additional 2kHz high-pass Butterworth filter on all 2MHz and 20,000 samples per trace (10ms recording time). data. A 4,000 sample (2ms) pre-trigger interval was set. The rea- In order to test the stability of deconvolution for different son a longer pre-trigger time was used is because it was shown values of the regularization parameter γ , we run the exact by Ulrich et al. [19] that with a longer pre-trigger time, we same experiment as for a single source described above and were able to improve the focus for deconvolution. Amplitude shown in Fig. 1b. Once the signal was recorded, we applied resolution is 16 bit. True zero time of the transmitter could be DC multiple times with different gamma values in order to 123 2 Page 4 of 9 J Nondestruct Eval (2015) 34:2 generate the different signals to be backpropagated. We then Time Reversal ran the same workflow as shown in Fig. 1c for each DC 1 signal separate, recording the focused wavefield at the source 0 location each time. For multiple sources, we executed the workflow described Amplitude −1 6 6.5 7 7.5 8 8.5 9 9.5 10 above and shown in Fig. 1b three times (once for every Time(ms) source). This was necessary because we did not have the equipment capabilities in this laboratory to generate a source Deconvolution 1 function at all three source locations at different onset times. We recorded these three generated wavefields separately and 0 then superimposed them. Due to the experiment being run separately three times, each recorded signal was normalized Amplitude −1 6 6.5 7 7.5 8 8.5 9 9.5 10 independently. This caused our recorded signals for all three Time(ms) sources to vary between amplitudes of −1 and 1. Thus, it destroyed the relative amplitude variation one would expect Fig. 2 Normalized temporal focus measured at the embedded source location using (top panel) time reversal, and (bottom panel) deconvo- for three different sources at different locations and orien- lution for a single source and single receiver set up tations. Once superimposed, TR or DC was applied and we carried out the same back propagation workflow as shown in Fig. 1c. During the backpropagation, one restores the The DC and TR signals were then backpropagated from relative amplitudes in the focus achieved because of reci- the transducer on the surface of the block into the same procity. medium and recorded at the original source location transducer. Figure 2 show the refocused waves recorded at the source location where Fig. 2a is the temporal focus achieved using TR while Fig. 2b represents the temporal 4 Data Analysis focus achieved using the DC calculated signal. The temporal focus achieved using TR has significant side-lobes away 4.1 Single Source Experiment from the time of focus; the temporal focus achieved using DC has suppressed most of these side-lobes and was able to pro- The purpose of this experiment is to study the capabilities duce a better focus. In order to quantify this improvement, we of TR and DC to focus the measured waveforms at a point calculate the amount of energy in a 0.02 ms window around in time. The experiment began with propagating a defined the time of focus compared to the total energy of the signal. 60kHz source function from the embedded source towards The temporal focus achieved using TR only had 41 % of the the external receiver. The receiver’s direction of measure- total energy within this window while DC’s temporal focus ment was perpendicular to the direction of source emission. had 80% of the total energy within this window. Thus, DC Once our wave field was recorded at the single receiver, TR is able to generate a significantly better temporal focus than or DC was applied to calculate the back propagating signals. TR. Our source function used wasn’t a Dirac delta function For a single source, deconvolution ideally achieves an but deconvolution still improved the focus significantly as it improved temporal focus. This is due to there being a single improved the reconstruction of our source function. Once we term in the denominator as shown in Eq. 5, which leads to had shown that deconvolution was able to improve the focus the following, at a point in time, we continued our experimental studies to G(ω)G∗(ω)S∗(ω) 1 investigate the robust nature of deconvolution. g(ω)G(ω) ≈ ≈ , (7) |G(ω)S(ω)|2 + S(ω) 4.2 Regularizing the Deconvolution where G(ω) represents the Green’s function describing the propagation between source and receiver, S(ω) represent the The purpose of this experiment was to study the robust nature source function in the frequency domain, and g(ω) the signal of DC by changing the location of the receiver and investigat- we are trying to solve for with deconvolution. Equation 7 ing the effect of regularizing the deconvolution through the should approach 1/S(ω) as approaches 0. Therefore, the parameter used in Eq. 5. This experiment started the same focus achieved using deconvolution approaches an optimal way as our previous single source experiment. We first prop- reconstruction of the inverse of the source function and not agated a defined 60 kHz source function from the embedded necessarily a Dirac delta function. When the source function source towards the external receiver. For this experiment, is a delta function, S(ω) is constant, this leads to a delta the receiver’s direction of measurement was parallel to the function at the focal point. direction of source emission. The recorded signal was then 123 J Nondestruct Eval (2015) 34:2 Page 5 of 9 2

Fig. 3 Normalized temporal Gamma=.001 Gamma=.01 Gamma=.1 Gamma=.9 focus measured at the embedded 1 1 1 1 source location using different values for γ in the deconvolution (shown in red) 0 0 0 0 and time reversal (shown in Amplitude blue) (Color ﬁgure online) Amplitude Amplitude Amplitude −1 −1 −1 −1 6 8 10 6 8 10 6 8 10 6 8 10 Time(ms) Time(ms) Time(ms) Time(ms)

Gamma=1 Gamma=10 Gamma=100 TR 1 1 1 1

0 0 0 0 Amplitude Amplitude Amplitude Amplitude −1 −1 −1 −1 6 8 10 6 8 10 6 8 10 6 8 10 Time(ms) Time(ms) Time(ms) Time(ms) deconvolved using various values of γ which was the con- 0.8 stant scalar number used to characterize the regularization 0.7 term defined in Eq. 6. These calculated deconvolved signals were propagated back into the medium from the receiver 0.6 location and recorded at the source transducer. 0.5 Theoretically, we would expect the focused wavefield to 0.4 contain significant amount of noise at low γ values. As γ 0.3

increases, the temporal focus is expected to improve because Temporal Focus we reduce the effect of noise and force our signal to generate 0.2 a better approximate Dirac delta function focus. However, 0.1 γ −2 −1 0 1 2 if becomes too large, one approaches the temporal focus −3 10 10 10 10 10 10 achieved using TR. This can be seen as follows: For time Gamma reversal, g(t) = R(−t), therefore, g(ω) = R∗(ω).Ifγ is large, becomes large in the sense that >>|R(ω)|2, and Fig. 4 The temporal focus, defined as the amount of energy in a 0.02ms window around the time of focus compared to the total energy of the Eq. 5 reduces to signal, as function of γ . High temporal focus indicates most of the energy is compressed at the time of focus 1 1 g(ω) ≈ G(ω)∗ S(ω)∗ = R∗(ω), (8) γ = . γ which implies that our deconvolved signal is just a scaled be 0 9. However, even for different values, one still version of the time reversed signal. achieves some form of a temporal focus as shown in Fig. 3. Figure 3 shows the normalized focused wavefield at the source location for TR and DC for different values of γ .For 4.3 Multi Source Experiment this experiment, we quantified the temporal focus the same way as the previous experiment with the identical window The purpose of this experiment was to study the effect of mul- size of 0.02ms used. The optimal DC’s temporal focus was tiple sources when using deconvolution. We began by emit- 79% (for a gamma value of 0.9) while TR had a temporal ting the same 60kHz source function from different source focus of 47 %. We would not expect to see the exact same transducers within the concrete block at different onset times. temporal focusing numbers as in our single source experi- The experiment was repeated three times to record each ment because we changed the direction of displacement we source wavefield separately which normalized the recorded record and the distance between the source and receiver. signals independently. The employed normalization caused Figure 4 highlights the effect of gamma by showing the our signals for all three sources to vary between ampli- temporal focus as a function of γ .Ifγ becomes small, the tudes of −1 and 1. The three recorded wavefields due to temporal focus achieved decreases. However, as γ becomes the three sources were then superimposed before TR or DC large, the temporal focus approaches TR’s temporal focus was applied. Figure 5 shows the superimposed wavefield. of 47%. The experiment showed that the optimal value to Note the complexity of the wavefield due to scattering within 123 2 Page 6 of 9 J Nondestruct Eval (2015) 34:2

Recorded Data If there are three sources, the recorded signal in the fre- 1 quency domain is given by

R(ω) = G1 S1 + G2 S2 + G3 S3, (10) 0.5 where R(ω) is the recorded signal in the frequency domain, 0 and the subscripts indicate the source transducer used. The

Amplitude inverse signal obtained by deconvolution is given by −0.5 1 Dt (ω) = G1 S1 + G2 S2 + G3 S3 −1 ( + + )∗ = 1 G1 S1 G2 S2 G3 S3 . 0 1 2 3 4 5 6 7 8 9 10 ∗ (G1 S1 +G2 S2 +G3 S3) (G1 S1 +G2 S2 +G3 S3) Time(ms) (11) Fig. 5 Recorded scattered waveforms at the receiver location due to three source waveﬁelds being emitted at different times We simplify the above solution and add the regulation term = γ mean(|R(ω)|2) to get, Time Reversal ∗ (G1 S1 + G2 S2 + G3 S3) 1 D (ω) = , t 2 2 2 |G1 S1| +|G2 S2| +|G3 S3| + Crosstalk + 0 (12)

Amplitude = ∗ ∗ + ∗ ∗ + ∗ ∗ + −1 where Crosstalk G1 S1G2 S2 G1 S1G3 S3 G2 S2G1 S1 0 1 2 3 4 5 6 7 8 9 10 ∗ ∗+ ∗ ∗+ ∗ ∗ (ω) G2 S2G3 S3 G3 S3G1 S1 G3 S3G2 S2 , and Dt represents Time(ms) the deconvolved signal when the recorded signal contains Deconvolution three source wavefields. 1 If we recorded each sources’ wavefield separate and applied deconvolution first before the superposition of the 0 wavefields, we would get: Amplitude −1 (G S )∗ (G S )∗ (G S )∗ 0 1 2 3 4 5 6 7 8 9 10 D (ω) = 1 1 + 2 2 + 3 3 , s 2 2 2 Time(ms) |G1 S1| + 1 |G2 S2| + 2 |G3 S3| + 3 (13) Fig. 6 Back propagation signals calculated using Time reversal (top panel) and deconvolution (bottom panel). These signals are backprop- where Ds(ω) represents the deconvolved signal when decon- agated into the medium for the multi-source experiment volution is applied before superposition. One might expect that for a real scenario, where multiple sources are present, the concrete sample. Due to the complicated nature of the deconvolution would break down due to the influence of recorded signal, one can assume that the cross-correlation of crosstalk. Because the recorded wavefields generated by each each source wave field is negligible: source are extremely complex, as shown in Fig. 5,terms such as G G∗ are small (Eq. 9). Therefore, the influence ∗ ≈ = 1 2 Gi G j 0fori j (9) of the crosstalk terms is minimal and we can assume it van- where G for i = 1, 2, 3 is the Green’s function characteriz- ishes. This provides us with the following solution that relates i (ω) (ω) ing the source wavefield for sources 1, 2, or 3. Equation 9 is Ds to Dt , crucial in explaining why deconvolution is stable for multiple (G S )∗ (G S )∗ (G S )∗ D (ω) = 1 1 + 2 2 + 3 3 s 2 2 2 sources. |G1 S1| + 1 |G2 S2| + 2 |G3 S3| + 3 We apply time reversal and deconvolution to the super- (G S + G S + G S )∗ ≈ 1 1 2 2 3 3 = D (ω), imposed signal consisting of the three source wavefields to 3 2 2 2 3 t |G1 S1| +|G2 S2| +|G3 S3| + generate our TR and DC signals shown in Fig. 6. Deconvo- (14) lution’s signal differs from time reversal signal in its acausal nature, due to our pre-trigger time, where DC adds infor- where we assume |G1 S1|≈|G2 S2|≈|G3 S3|. mation past 8ms while TR has zero amplitude after 8ms. Figure 7 shows that the approximation (14) holds. Fig- Additionally, the three different source wavefields are still ure 7 demonstrates that after normalizing Ds(t) and Dt (t), clearly visible in the DC signal. Below, we demonstrate why one can note that there does not seem to be a obvious dif- deconvolution is able to focus the wavefield due to multiple ference between Ds(t) and Dt (t). Therefore, one may con- source at each source location. clude Thus, the crosstalk term may be ignored and deconvo- 123 J Nondestruct Eval (2015) 34:2 Page 7 of 9 2

D (t) t then used the transducers within the concrete block as our 1 receivers. Figure 8 shows the focused waveﬁelds at each of the three sources for TR, shown in the top panels, versus DC, 0 shown in the bottom panels. For sources 1 and 2, deconvo- Amplitude −1 lution compresses the sidelobes substantially better than TR. 0 1 2 3 4 5 6 7 8 9 10 However, for source 3, deconvolution does not signiﬁcantly Time(ms) improve the focus compared to time reversal. D (t) s The orientation of our sources is an important factor. 1 Sources 1 and 2 were oriented perpendicular to source 3, and as a result the recorded waves excited by source 3 are stronger 0 than those excited by sources 1 and 2. However, as previously

Amplitude −1 stated, we ran each source wavefield propagation separately 0 1 2 3 4 5 6 7 8 9 10 which normalized the recorded signals independently caus- Time(ms) ing the amplitudes of each recorded source wavefield to vary − Fig. 7 Comparison of deconvolution signal calculation. Top panel between amplitudes of 1 and 1. Thus, our superimposed shows the DC signal after applying deconvolution to the superposi- signal shown in Fig. 5 does not show a higher amplitude for ( ) tion of the three recorded wavefields Dt t . Bottom panel shows the the source 3 wavefield. However, when we back propagate DC signal after applying deconvolution to the three recorded signals our TR and DC wavefield, due to reciprocity, the source 3 before adding them to each other D (t). All signals are normalized s wavefield focus will have a higher amplitude. This causes the crosstalk terms to be negligible for the source 3 focus because, lution is stable and able to focus the wavefield due to mul- | (ω)| >> | (ω)| | (ω)| >> | (ω)|. tiple sources at each source location. We were able to do G3 G1 and G3 G2 (15) this calculation because we recorded each source wavefield Under these conditions, Eq. 12 reduces to, separately. (G S )∗ One does not need to have priori knowledge of the source D (ω) ≈ 3 3 , (16) t | |2 + signal in order to apply deconvolution and detect the sources. G3 S3 One can essentially modify the focus to be any arbitrary which is what we had before. source function as shown in Ulrich et al. [19]. We have just Figure 8 show that Eq. 15 holds because, for source 3, assumed the source function to be a Dirac delta function for the focus has significantly higher relative amplitude than the these experiments. crosstalk terms. For sources 1 and 2, the maximum amplitude In order to keep the experiment realistic, we back prop- of the crosstalk is closer to the maximum amplitude of its agated the DC signal which was calculated after the super- focus. position of the three separate wavefields. Figure 6 shows In conclusion, for multiple sources, deconvolution is able the signals calculated using TR and DC which are propa- to focus the energy at the source location at the correct time. gated back into the medium from the receiver location. We It is arguable whether time reversal or deconvolution is better

Fig. 8 Temporal focus Source 1: TR Source 2: TR Source 3: TR measured at the three embedded 1 1 1 source location using (top panels) time Reversal’s signal, and (bottom panels) 0 0 0 deconvolution’s Dt (t) signal Amplitude Amplitude and back propagating it from the Amplitude transducer on the surface of the −1 −1 −1 concrete sample 0 5 10 0 5 10 0 5 10 Time(ms) Time(ms) Time(ms)

Source 1: DC Source 2: DC Source 3: DC 1 1 1

0 0 0 Amplitude Amplitude Amplitude −1 −1 −1 0 5 10 0 5 10 0 5 10 Time(ms) Time(ms) Time(ms)

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In: Proceedings of the Eighth International Symposium of Wireless Personal Communications Conference, Aalborg, Dan- We have introduced in an experimental study a simple mark, pp. VI-124–VI-129 (2005) though robust method for determining the optimal signal for 15. Qiu, R.C., Zhou, C., Guo, N., Zhang, J.Q.: Time reversal with MISO for ultrawideband communications: experimental results. backpropagation such that one gets an improved temporal IEEE Antennas Wirel. Propag. Lett. 5, 1–5 (2006) focus at the source location. Deconvolution was shown to 16. Blomgren, P., Kyritsi, P., Kim, A., Papanicolaou, G.: Spatial focus- have an optimal regularization parameter, γ , for improved ing and intersymbol interference on multiple-input-single-output temporal focusing. If one increases γ , the temporal focus time reversal communication systems. IEEE J. Ocean. Eng. 33, 341–355 (2008) approaches that of TR; if one decreases γ , one increases 17. Zhou, C., Guo, N., Qiu, R.C.: Experimental results on multiple- the effect of noise and the temporal focus decreases dra- input single-output (MISO) time reversal for UWB systems in matically. However, Fig. 4 shows that one still attains a an office environment. In: MILCOM’06 Proceedings of the 2006 temporal focus even for different values of γ . Additionally, IEEE Conference on Military Communications, pp. 1299–1304. IEEE Press, Piscataway, NJ (2006) deconvolution does not break down when there are multi- 18. Zhou, C., Qiu, R.C.: Spatial focusing of time-reversed UWB elec- ple source wavefields being propagated. This is due to the tromagnetic waves in a hallway environment. In: Proceedings of influence of the crosstalk terms being minimal for the com- the Thirty Eighth Symposium on System Theory, pp. 318–322 plicated waveforms generated by scattering in the concrete. (2006) 19. Ulrich, T.J., Douma, J., Anderson, B.E., Snieder, R.: Improving Thus, deconvolution has a robust nature comparable to that spatio-temporal focusing and source reconstruction through decon- of time reversal while having the potential to dramatically volution. Wave Motion (2014). doi:10.1016/j.wavemoti.2014.10. improve the focus. In conclusion, the simple and robust 001 nature of deconvolution allows it to be implemented as a 20. Douma, J., Snieder, R., Fish, A., Sava, P.: Locating a microseismic event using deconvolution. In: Proceedings of the 83rd preprocessing step in order to improve focusing at the source Annual International Meeting, Society of Exploration Geophysi- location. cists (2013)

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