The general topic of this paper is the passive reconstruction of an acoustic transfer function from an unknown, generally nonstationary excitation. As recently shown in a study of building response to ground shaking, the paper demonstrates that, for a linear system subjected to an unknown excitation, the deconvolution operation between two receptions leads to the Green's function between the two reception points that is independent of the excitation. This is in contrast to the commonly used cross-correlation operation for passive reconstruction of the Green's function, where the result is always filtered by the source energy spectrum (unless it is opportunely normalized in a manner that makes it equivalent to a deconvolution). This concept is then applied to high-speed ultrasonic inspection of rails by passively reconstructing the rail's transfer function from the excitations naturally caused by the rolling wheels of a moving train. A first-generation prototype based on this idea was engineered using noncontact air-coupled sensors, mounted underneath a test railcar, and field tested at speeds up to 80 mph at the Transportation Technology Center (TTC), Pueblo, CO. This is the first demonstration of passive inspection of rails from train wheel excitations and, to the authors' knowledge, the first attempt ever made to ultrasonically inspect the rail at speeds above ∼30 mph (that is the maximum speed of common rail ultrasonic inspection vehicles). Once fully developed, this novel concept could enable regular trains to perform the inspections without any traffic disruption and with great redundancy.

## Introduction

The passive reconstruction of the Green's function (or, generally, a transfer function) between two points of a medium without an active controlled excitation has been of interest in many scientific fields. The most commonly used operation for this purpose is a cross-correlation between two receivers that has been shown theoretically to lead, for example, to the coherent Green's function of a medium subjected to diffuse random fields. The time-averaged cross-correlation essentially builds up the coherent portion of the traveling waves between the two receivers by a constructive averaging process. This process has been shown to work in both closed and open systems [17]. Applications of passive reconstruction of the Green's function by cross-correlations of random fields has been demonstrated in ultrasonics including guided waves [1,816], seismology [1721], underwater acoustics [7,22,23], dynamic characterization of highway bridges excited by street traffic [24,25], of hydrofoils excited by flow-induced vibrations [26], and of buildings excited by ground shaking [27].

It is also known that the cross-correlation operator filters the reconstructed impulse response between the two receiving points by the source energy spectrum [4]. Snieder [5] explicitly remarks how the cross-correlation operator may not give the correct frequency dependence of the Green's function without correction for the energy spectrum. This problem was circumvented by Snieder and Safak [27] by replacing the cross-correlation operation with a deconvolution operation in a wave reverberant system (the Millikan Library, Pasadena, CA) with wave radiation losses into the ground, and subjected to ground shaking as the excitation. By deconvolving the acceleration responses between different floors of the building, the authors were able to reconstruct the interstory transfer function without the effects of the ground excitation and without the effects of the building's coupling to the ground.

The elimination of the excitation source in passive transfer function reconstruction is clearly necessary in those cases where the excitation source is unknown and generally unstable, e.g., nonstationary and/or random. This is the case, for example, of the acoustic excitation of a rail track by the rolling wheels of a moving train. Structural inspection of rails for internal defect detection is a task that is typically carried out by ultrasonic tests performed by piezoelectric transducers hosted in fluid-filled wheels [28], known by the rail inspection community as rolling search units (RSUs). The typical inspection is performed today by specialized vehicles (hi-railers) that travel at relatively low speeds (∼30 mph maximum) compared to regular train traffic (∼60 mph and above).

For several years, the authors have conducted research to improve the RSU approach for ultrasonic rail inspections by introducing noncontact excitation and sensing of the ultrasonic waves in the rail steel [2934]. The various systems developed previously, however, still relied on the traditional active–passive approach requiring the controlled excitation source. The present paper introduces a radically new approach of ultrasonic rail inspection. This new approach consists of passively reconstructing the impulse response of the rail at two points by deconvolving the responses of noncontact air-coupled receivers under excitations naturally caused in the rail by the rolling wheels of a moving train. Therefore, the train wheels replace the controlled ultrasonic excitation of the rail by a transducer or the like. Passive reconstruction of the rail transfer function from acoustic noise was recently discussed using the cross-correlation operation in laboratory tests involving contact piezoelectric transducers [35]. For the present objective of enabling the passive rail inspection by using the wheel excitations (that are nonstationary and random), the need to eliminate the effect of the source leads to the use of the deconvolution operation instead. A first-generation prototype based on this passive inspection idea was field tested at the Transportation Technology Center (TTC), at speeds of up to 80 mph, therefore, at much higher speeds than what possible with conventional RSU inspection units.

The paper first presents a theoretical treatment of cross-correlation versus deconvolution as it applies to an open dynamic system excited by an unknown source. It then describes the passive-only, noncontact inspection prototype that was designed and built around this idea. It finally reports representative results from the first passive-only rail inspection tests conducted at TTC at speeds between 25 mph and 80 mph.

## Theoretical Considerations: Cross-CorrelationVersus Deconvolution

Let us consider the case schematically shown in Fig. 1, showing a rail track dynamically excited by a rolling wheel W, and whose acoustic response is measured by two receivers at locations A and B. Let us also assume that both receivers are only sensitive to waves propagating unidirectionally from left to right in Fig. 1 (this is the case of air-coupled acoustic sensors oriented at a specific Snell's law angle). The primary objective of this exercise is to isolate the transfer function of the test structure (the rail in the present application) between location A and location B, GAB(ω), without the effect of the source excitation (the rolling wheel in the present application). If GAB(ω) can be isolated, discontinuities in the rail (such as defects) can be detected as a change in the transfer function, similarly to a pitch-catch active–passive ultrasonic guided-wave approach [3034]. This scenario does not change whether the wave propagation problem is one-dimensional (1D) or three-dimensional (3D), since the Green's function concept applies to both cases.

The problem is more easily formulated in the frequency domain. Assuming linearity, the response measured at A from the random wheel excitation, VA(ω), is given by the convolution (product in the frequency domain) between the wheel excitation spectrum W(ω), the transfer function (Green's function) of the rail between the wheel and location A, WA(ω), and the frequency response of the receiving sensor, A(ω)
$VA (ω)=W(ω)⋅WA(ω)⋅A(ω) response at A$
(1)

where the symbol $⋅$ means product.

The response measured at B will be similar to the above expression, with the addition of the transfer function (Green's function) of the rail between location A and location B, GAB(ω), and considering (in general) a different frequency response for the receiving sensor, B(ω)
$VB(ω)=W(ω)⋅WA(ω)⋅GAB(ω)⋅B(ω) response at B$
(2)

Option 1: Cross-correlation operation

The frequency-domain cross-correlation between the responses at A and at B, XcorrAB(ω), yields the following result:
$XcorrAB(ω)=VA*(ω)⋅VB(ω)=W*(ω)⋅WA*(ω)⋅A*(ω)⋅W(ω)⋅WA(ω)⋅GAB(ω)⋅B(ω)=|W(ω)|2⋅|WA(ω)|2⋅A*(ω)⋅B(ω)⋅GAB(ω)$
(3)

where the asterisk (*) means complex conjugate, and $| |$ indicates modulus. The terms $| |2$ are auto-correlations, since $Autocorr(ω)=F*(ω)⋅F(ω)=|F(ω) |2$, which physically correspond to the energy spectrum of the function.

If the two receivers have the same response A(ω) = B(ω) = R(ω), that is a reasonable assumption if the same receiver types are used, Eq. (3) simplifies to
$XcorrAB(ω)=|W(ω)|2⋅|WA(ω)|2⋅|R(ω)|2⋅GAB(ω)$
(4)

Equations (3) and (4) contain the desired impulse response of the rail between A and B, $GAB(ω)$, but “scaled” or “colored” by the energy spectra of (a) the wheel excitation, (b) the transfer function of the rail between the wheel excitation and receiver A, and (c) the receiver responses. In a test configuration where the inspection probe is moved along the rail, spectra (a) and (b) are generally expected to change, whereas spectra (c) are invariant (meaning that the response spectra of the receivers do not change during a test). Since the inspection relies on tracking changes to the rail transfer function alone, $GAB(ω)$, the effects of (a) and (b) need to be eliminated. That is why the cross-correlation operator is not a good metric for a passive inspection approach that relies on a varying excitation.

Option 2: Normalized cross-correlation operation

Here, we show that the autocorrelation of the responses from either A or B can be used as a suitable normalization factor to isolate the desired impulse response $GAB(ω)$. For example, normalizing the cross-correlation of Eq. (3) by the autocorrelation of receiver A yields
$NormXcorrAB(ω)=XcorrAB(ω)AutocorrA(ω)=|W(ω)|2⋅|WA(ω)|2⋅A*(ω)⋅B(ω)⋅GAB(ω)|W(ω)|2⋅|WA(ω)|2⋅|A(ω)|2=B(ω)A(ω)GAB(ω)$
(5)
resulting in the desired transfer function $GAB(ω)$, only scaled by the receivers' responses A(ω) and B(ω). Since the receivers do not change during the inspection, the ratio B(ω)/A(ω) is just a constant scale factor that cannot cause false positive detections. If needed, B(ω)/A(ω) could also be calibrated once for the particular receivers used, and eliminated altogether from the test output. If the two receivers have the same response A(ω) = B(ω) = R(ω), Eq. (5) further simplifies to
$NormXcorrAB(ω)=XcorrAB(ω)AutocorrA(ω)=GAB(ω)$
(6)

which is the perfect reconstruction of the impulse response of the test object.

The cross-correlation of Eq. (3) can be also normalized by the autocorrelation of receiver B. In this case
$NormXcorrAB(ω)=XcorrAB(ω)AutocorrB(ω)=|W(ω)|2⋅|WA(ω)|2⋅A*(ω)⋅B(ω)⋅GAB(ω)|W(ω)|2⋅|WA(ω)|2⋅|GAB(ω)|2⋅ |B(ω)|2=A*(ω)B*(ω)GAB(ω)|GAB(ω)|2$
(7)

resulting in the impulse response, $GAB(ω)$, only scaled by its own energy spectrum $|GAB(ω)|2$ and by the receivers' (conjugate) responses $A*(ω)$ and $B*(ω)$. Therefore, Eq. (7) is also a suitable metric for a passive inspection.

In summary, the cross-correlation between receiver A and receiver B, once normalized by the autocorrelation of either receiver (but not both simultaneously), provides a metric that is able to isolate changes in the test object's impulse response $GAB(ω)$ in a passive-only manner and without the influence of the variable excitation. Changes in $GAB(ω)$ can be then directly related to the presence of an internal flaw between A and B without any spurious effects that may cause false detections.

Option 3: Deconvolution operation

It is here demonstrated that the deconvolution operation yields results that are formally equivalent to the normalized cross-correlation discussed earlier. Deconvolving receiver A from receiver B yields
$DeconvBA(ω)=VB(ω)VA(ω)=W(ω)⋅WA(ω)⋅GAB(ω)⋅B(ω)W(ω)⋅WA(ω)⋅A(ω)=B(ω)A(ω)GAB(ω)$
(8)
That is the test object's impulse response only scaled by the receiver responses (that do not change during a test). This result is identical to the normalized cross-correlation of Eq. (5). If the receivers have the same response, Eq. (8) further simplifies to
$DeconvBA(ω)=VB(ω)VA(ω)=GAB(ω)$
(9)

i.e., the “ideal” impulse response reconstruction.

$DeconvAB(ω)=VA(ω)VB(ω)=W(ω)⋅WA(ω)⋅A(ω)W(ω)⋅WA(ω)⋅GAB(ω)⋅B(ω)=A(ω)B(ω)1GAB(ω)=A(ω)B(ω)GAB*(ω)|GAB(ω)|2$
(10)
that is exactly the conjugate version of the normalized cross-correlation of Eq. (7), i.e., its time-reversed version. If the receivers have the same response, Eq. (10) simplifies to
$DeconvAB(ω)=VA(ω)VB(ω)=GAB*(ω)|GAB(ω)|2$
(11)

i.e., the desired impulse response, only time-reversed and scaled by its own energy spectrum.

In summary, the deconvolution is a suitable operation for the inspection because it is able to isolate changes in the test object's impulse response $GAB(ω)$ in a passive-only manner without the influence of changing excitation spectrum and other spurious factors.

The time-domain impulse response function can be then retrieved from the frequency domain through an inverse Fourier transform
$GAB(t)=12π ∫−∞∞GAB(ω) eiωt dω$
(12)

Physically, the ideal $GAB(t)$ corresponds to the response of the test object (rail) at location B from an impulse excitation at A. It will therefore contain both standing waves (at low frequency values) and traveling waves (at high frequency values). In practice, however, the passively reconstructed version of this Green's function will retrieve usable data only within the frequency spectrum of the excitation and that of the receivers' response. As long as this transfer function is stable along a test run, it will enable robust defect detection. In other words, only the reconstruction of a stable transfer function between two points of the structure (not necessarily the retrieval of the ideal Green's function) is the sufficient condition for successful passive structural inspection. A similar argument of passive reconstruction of an “imperfect” Green's function has been, for example, presented in studies of structural monitoring of a highway bridge by cross-correlation of directional traffic-induced acoustic sources [25].

The derivation presented in this section assumed a single excitation source. If multiple sources exist, their contribution will superimpose (assuming linearity) and their effect will still be eliminated by the deconvolution operation because of the common contribution to the two receivers.

## Averaging to Increase the Signal-to-Noise Ratio (SNR) of the Passive Reconstruction

The impulse response $GAB$ emerges from the constructive interference of wave modes continuously excited by the random (wheel) excitation and propagating in the rail along the line connecting the two receivers. Furthermore, the rail structure at hand is practically a one-dimensional waveguide, where the random wave fields travel along one direction that is also the direction of alignment of the receiver pairs. This is a desirable feature, since studies of cross-correlations of multidirectional diffuse acoustic fields [5,22,36] have demonstrated that the transfer function is mostly reconstructed by wave fields aligned with the receivers.

The constructive interference process under continuous excitation clearly benefits from signal averaging in time. Following a known result in cross-correlation of diffuse fields [5,22,25,36], the rate of convergence (SNR) of this kind of process can be written as
$SNR ∝T Δfe−αD$
(13)

where T is the length of the recording time window, $Δf$ is the source bandwidth, α is the linear attenuation coefficient in the test material (in dB/m), and D is the distance between the two receivers.

Experimentally, the SNR can be determined by the peak amplitude of the reconstructed impulse response in the time domain, $GAB(t)$, divided by the standard deviation of the total reconstructed signal taken away from the expected arrival time (“noise”)
$SNR=Max(GAB(t))Std(GAB(t))$
(14)

The directivity beam consideration of Refs. [22,36] do not apply to the rate of convergence in the case at hand since all sources are in the “end-fire” direction of the receiving array. Similarly, geometrical spreading effects included in Ref. [22] can be neglected in a unidirectional waveguide. Of primary importance from Eq. (13) is the fact that long recording time windows (besides large bandwidths) help with the emergence of the passively reconstructed transfer function.

An additional requirement in the application at hand, where both excitation and reception are moving along the test piece (in-motion scanning test), is the stationarity of the reconstructed transfer function that can only be ideally guaranteed for a fixed position of the test object (rail). Therefore, a compromise must be found between the long recording times required by the averaging process and the stationarity (or spatial localization) of the transfer function reconstruction. The test speed clearly affects this compromise, since higher speeds will have to result in shorter recording times to maintain sufficient spatial localization. The topic of the recording time window will be further discussed with the experimental results presented in Sec. 4.3.

## Application to High-Speed Inspection of Rails

Sections 4.14.5 describe a system developed on the idea of passive reconstruction of the impulse response from deconvolution operations applied to high-speed inspection of rails. When a train is in motion, the rotating wheels produce continuous dynamic excitations of the rail, in a way that is sometimes treated as an array of point sources at the rail/wheel interface [37]. Such excitation is also randomized and made generally nonstationary by the unevenness and irregularities of the wheel and rail surfaces [38], as well as acceleration and braking actions. The challenge is therefore to extract a transfer function of the rail that is not affected by the variability of this type of excitation.

### Passive-Only Rail Inspection Prototype.

The signal acquisition from the air-coupled receivers was accomplished by a National Instruments (NI) PXI unit running LabVIEW real-time to guarantee deterministic processing. The data unit also recorded a tachometer transistor-transistor logic pulse, that marked the spatial position of the test car with a resolution as small as 1 in (2.54 cm), and a high-speed camera (SONY ICX-424 with a 6 mm C-mount lens) with appropriate illumination (30 k Lumens LED flood light) that recorded videos of the rail during each run. The purpose of the camera was to verify the presence of visible discontinuities in the rail, i.e., joints and welds, when the prototype detected an anomaly.

### Test Lay-Out and Procedure.

Test runs were made at both TTC's railroad test track (RTT) that allowed maximum test speeds of 80 mph, and at TTC's RDTF that allowed maximum test speeds of 25 mph. A locomotive was used to tow the DOTX-216 car instrumented with the passive-only prototype. The tested lengths depended on test speed and were about 18,000 ft for the RTT and about 2000 ft for the RDTF at the highest speeds.

The runs at the RTT were conducted between markers R42 and R25 (see Fig. 3(a)). As shown in the figure, this test zone featured a tangent track in the middle, with curved tracks at the beginning and at the end. As many as three joints and seven welds were identified in the RTT test zone through visual survey and the high-speed camera (a snapshot of a joint from the camera is shown in Fig. 3(a)). Runs at the RTT test track were conducted at the speeds of 10–80 mph (max speed allowed) in 10 mph increments. This was the first time, to the authors' knowledge, that an ultrasonic inspection system was tested in motion on a rail at speeds larger than ∼30 mph.

In addition to the high-speed tests at the RTT, some test runs were conducted at the RDTF at speeds of 25 mph (max speed allowed). Specifically, these runs were conducted at the Technology Development Section of the RDTF facility (Fig. 3(b)), a mostly curved track with several known defects. The authors were quite familiar with the RDTF facility, since it was the test site for the earlier active–passive research prototypes [3034]. This was the first time that ultrasonic defect detection in the RDTF track was attempted in a passive-only manner.

### Data Processing and Proof-of-Concept Transfer Function Reconstruction.

The data processing steps implemented in the prototype are schematized in Fig. 4.

For each run, recordings from the pairs of air-coupled receivers at locations A and B under the continuous wheel excitation were first amplitude clipped to within the average ±3 standard deviations. This is a common step also taken in cross-correlation of diffuse fields to prevent isolated spikes to contribute disproportionally to the passive reconstruction of the impulse response. The signals were then processed by the deconvolution operation of Eqs. (8) and (9). The time-domain transfer function between A and B, GAB(t), was finally obtained by averaging over four sensor pairs the frequency-domain GAB(ω) after bandpass filtering and inverse Fourier transform of Eq. (12).

Figure 5(a) shows a representative recording from one of the receivers in the RTT track at 60 mph. This trace shows a high signal strength variability expected from the random wheel/rail contact conditions (surface unevenness, acceleration, braking, etc.). The result of the deconvolution between two receivers is shown in Fig. 5(b) for two different snapshots of the raw recording that correspond, respectively, to a “quiet” zone and to a “loud” zone. The loud zone is caused by rail unevenness, flanging of the wheels or other conditions that exacerbate the wheel-rail contact. The deconvolved signal of Fig. 5(b) shows the clear arrival of a wave mode at about 140 μs, that is the expected travel time of the leaky surface wave in the rail between the two receiver locations. As discussed in Sec. 2, the emergence of this wave arrival results from the constructive interference of the multiple wheel-generated waves that creates the coherent wavefront. The two plots in Fig. 5(b) also show that the reconstructed wavefront arrival at 140 μs is very similar for both the quiet zone and the loud zone. This confirms the ability to reconstruct a stable coherent arrival independently of the instantaneous strength of the wheel excitation source.

Figure 5(c) compares the wave arrival reconstructed in a pristine section of the rail to one reconstructed near a joint location in a loud zone of the same RTT track. The 140 μs arrival for the joint location shows a substantial drop in amplitude as a result of the wave scattering from the discontinuity. The finite energy detected at the joint could be the result of a closed joint (leaking some wave energy across) or to the fact that some of the sensor pairs laid to one side of the joint at this specific recording time window. Since internal rail flaws also act as wave scatterers, the potential exists for this passive-only approach to be used for rail flaw detection.

It was discussed in Sec. 3 how the rate of convergence of the passively reconstructed transfer function (or equivalently its SNR) is expected to increase with increasing recording time lengths according to Eq. (13). It was also discussed how, for the application at hand involving a scanning inspection, an additional constraint is the spatial localization of the transfer function that is being reconstructed. This second constraint imposes an upper limit to the length of the recording time window that also depends on test speed, with higher speeds requiring shorter time windows for spatial localization. Accordingly, a study was conducted to determine the time window length that resulted in a good compromise between SNR of the prototype's reconstructed transfer function and spatial localization in the rail. Figure 6 plots the SNR of the reconstructed transfer function at the 140 μs arrival, calculated from Eq. (14), for various recording time lengths and for three runs on the RTT track at the speeds of 30 mph, 50 mph, and 80 mph. The three curves end at different times since shorter time windows are required for higher speeds as discussed earlier. The first observation from the plots in Fig. 6 is the confirmation that the SNR generally increases with increasing recording time. However, the rate of increase is seen to drop for the longest recording times considered, due to the loss of spatial localization in the rail and consequent nonstationarity of the reconstructed transfer function. The figure also shows that the SNR generally decreases with increasing test speed, as a result of the increased standard deviation of the incoherent portion of the wheel-generated excitation. The time window durations that correspond to a spatial localization in the rail to within 8 in (20.3 cm) are marked by stars for each of the three speeds. These points correspond to a SNR of ∼12 at 30 mph, ∼9 at 50 mph, and ∼4.5 at 80 mph. These values were chosen as the final recording time windows to provide an acceptable compromise between achievable SNR and spatial localization. This choice also effectively meant that the prototype “averaged” the transfer function of the rail over an 8 in (20.3 cm) “gage length.”

The final signal processing step that was implemented in the prototype was an outlier analysis to compute a statistically robust metric (herein referred to as “damage index”) related to the strength of the reconstructed transfer function. A similar statistical analysis was implemented in the previous active–passive versions of the rail inspection systems developed by the authors [3034], and it does not need to be described here in detail. In summary, the damage index (DI) was calculated as the Mahalanobis squared distance [45,46] defined in a multivariate sense
$DI=(x−x¯)T⋅Cov−1⋅(x−x¯)$
(15)

where $x$ is the feature vector extracted from the passively reconstructed transfer function, $x¯$ is the mean vector of the baseline distribution, $Cov$ is the covariance matrix of the baseline distribution, and T represents transposed. The feature vector contained the following three metrics: maximum amplitude, root mean square, and variance of the reconstructed transfer function. The statistical computation of the DI normalizes the data by the normal (baseline) data variability that occurs during a run. As such, compared to a simple deterministic metric, the DI of Eq. (15) dramatically increases the probability of detection and decreases the probability of false alarms of this kind of inspection, as amply shown in the aforementioned prior works. Moreover, the baseline distribution of the reconstructed signal features in the prototype was collected adaptively at each position along the rail, and considering the preceding 350 ft (107 m) of rail. Finally, an “exclusive” [45] version of the baseline was adopted, whereby extreme values of the DI (i.e., values larger than mean + twice the standard deviation) were removed from the baseline computation. This removal ensured that only pristine portions of rail were considered in the baseline computation.

### Results From Transportation Technology Center's Railroad Test Track.

The runs at the RTT were conducted between markers R42 and R25 at the speeds of 10–80 mph in 10 mph increments. Figures 7 and 8 show representative results of the DI computed from the multivariate outlier analysis of the passively reconstructed transfer function for five runs at different test speeds, according to the process described in Sec. 4.3. The distance covered by each run depended on the intended target speed, with longer distances needed to reach the higher speeds at “steady-state.” The distances covered for each of the runs are schematized in the top drawings of Figs. 7 and 8. The peaks in the DI traces labeled as “joint #” and “weld #” were confirmed by either information provided by TTC staff or by images collected by the video camera at those specific locations.

Figure 7 reports the results from a 30 mph run and from a 50 mph run. At 30 mph, Fig. 7(a), the plot shows a remarkably clear detection of two joints (joint 2 and joint 3) and two welds (weld 4 and weld 5), with flat noise floor with low risk of false positive detections in the clean part of the rail. It can also be noticed that the trace does not appear degraded when moving from the tangent portion of the track to the curved portion (after position ∼4500 ft in Fig. 7(a)). The result for 50 mph, Fig. 7(b), shows a similarly clear trace, confirming the detection of joint 2 and weld 4 of the 30 mph run, and showing additional true detections (weld 1, weld 2, and weld 3) in the additional distance covered. The fact that weld 3 was detected at 50 mph but not at 30 mph suggests that the lower-speed may be generally less desirable than the higher-speed. This is an interesting result possibly associated with the fact that slower rotational speeds of the wheels have difficulty generating higher acoustic frequencies that were monitored by the first version of the prototype. Additional discussion on the role of test speed is provided in Sec. 5. The few small peaks that are also visible, but not marked by a label (e.g., the peak at position ∼4400 ft in Fig. 7(b)) could be caused by an unknown discontinuity in the track (e.g., a weld or internal defect) or by a false positive detection. Further investigations (e.g., an independently conducted detailed survey of the track such as a conventional ultrasonic inspection with RSU rolling probes) would be needed to determine the true nature of these “unaccounted” peaks. Overall, Fig. 7 shows good promise for the possibility to (a) extract a stable transfer function of the rail in a passive-only manner by exploiting the natural wheel excitations and (b) process the passively reconstructed transfer function to detect rail discontinuities in a statistically robust manner. This is especially interesting since no previous attempt exists, to the authors' knowledge, to ultrasonically inspect a rail at speeds higher than ∼30 mph.

Figure 8 reports the results of a 60 mph run, a 70 mph run, and an 80 mph run. The 60 mph run, Fig. 8(a), continues to show a very clear detection of true discontinuities (joint 2, joint 3, weld 2, and weld 3). The clean portions of the rail continue to show an almost flat noise floor and therefore small risk of false positive detections. As discussed earlier, the few peaks seen beyond the labeled discontinuities could be due to undetected true discontinuities or to false detections, with further investigations needed to clarify this question. The 70 mph trace in Fig. 8(b) confirms all true detections of the 60 mph run, with additional true detections seen in the additional portion of rail covered (joint 1, weld 7). The noise floor at 70 mph is slightly raised compared to the lower speeds, likely a result of the increased mechanical vibrations (e.g., sensor misalignment) caused by the very high-speed. For example, railroad contractors assisting with the field tests indicated that the accelerations expected at the car axle or below the car primary suspension at high speeds can be as high as 30 g (root-mean-square) in the vertical direction (during sustained operation) and 100 g (root-mean-square) in all directions (during shocks). Clearly, these operational conditions should be considered extremely severe for a “typical” operation of an air-coupled ultrasonic receiver. Nevertheless, remarkably, all of the three confirmed joints and the seven confirmed welds in the test track are actually detected at the 70 mph speed. A similar result is provided by the 80 mph trace in Fig. 8(c), where all of the confirmed discontinuities remain detected with the exception of weld 5. This is a quite remarkable result considering, again, the potential difficulties associated with attempting to operate an inspection system mounted to the axle of a train car running at 80 mph.

### Results From Transportation Technology Center's Rail Defect Test Facility.

While the main focus of the TTC field tests was to test the stability of the transfer function's passive reconstruction at sustained speeds, the instrumented DOTX-216 car was moved to the Technology Development Zone of the RDTF track for a preliminary test on defect detection potential. The RDTF contains various known rail flaws.

Figure 9 shows traces from a run conducted at the maximum allowed speed of 25 mph, zoomed-in for three different zones of the track, all consisting of a curved lay-out. Zone 1 in Fig. 9(a), from left to right, shows the clear detections of a joint, a “crushed head” defect and a weld. Zone 2 in Fig. 9(b) shows the detections of a “shelling” defect, a “transverse defect” simulated by a saw cut and extending for 20% of the rail head area (HA), a joint, a “detail fracture” defect extending for 7% of the rail HA, and a second joint. The peak at position ∼5116 ft prior to the first joint could be due to an unmapped rail condition or it could be a false positive. Zone 3 in Fig. 9(c) shows the detection of a detailed fracture defect extending for 20% of the rail HA, a joint, a 20% HA “transverse fissure” defect, and another joint. The peak at position ∼5242 ft is not accounted for. The fact that the joint peaks are generally wider than the other defect peaks is the result of the substantial wave reflection/scattering experienced across a joint that is sensed at multiple positions of the sensor head relative to the rail.

Overall, the results in Fig. 9 show the potential for detecting relevant rail flaws by the passive approach. The fact that 25 mph was the maximum speed permitted on the RDTF track did not allow to study the defect detection performance at sustained speeds. Since the approach relies on wheel generated noise as the acoustic excitation of the rail, both signal strength and signal frequency bandwidth are expected to increase with increasing wheel rotational speed. It is therefore possible that the speed of 25 mph will require the monitoring of lower frequencies for optimum rail inspection performance, and additional field tests are being planned to confirm this hypothesis.

## Discussion and Conclusions

The general research topic addressed in this paper is the passive-only extraction of an acoustic transfer function between two receiving points of a linear system subjected to a continuous dynamic excitation that is arbitrary and generally nonstationary. The specific application of this topic considered here is the ultrasonic inspection of railroad tracks by exploiting the natural excitations provided by the rotating wheels of a running train. This implementation would enable to perform rail inspections at speeds that are unthinkable by current RSU ultrasonic search units that can conventionally test rails at a maximum speed of ∼30 mph. Moreover, the ability to inspect the rail at regular traffic speed would enable individual trains to perform the testing without any traffic disruptions while guarantying an unprecedented level of inspection redundancy (since multiple trains run several times on the same track). This increased redundancy would lead to increased probability of defect detection and reduced probability of false alarms.

Several researchers have well established the passive reconstruction of a transfer function (Green's function) of a system subjected to random acoustic fields from time-averaged cross-correlations of measurements at two receiving points. However, cross-correlation (or a normalized version of it where the autocorrelations of both receivers are used as the normalization factor) results in a Green's function that is affected by the energy spectrum of the excitation. This problem would make it impossible to inspect a system subjected to a highly variable excitation (such as the nonstationary wheel excitation of a rail). Utilizing the convolution theorem, the paper shows that a deconvolution operator (or, equivalently, a cross-correlation operator normalized by only one of the receivers but not both) is theoretically able to isolate the Green's function passively without the influence of the excitation source spectrum. While the theoretical development provided in the paper examines an open system with directional receivers (which applies to the case study of wheel excitations of a rail probed by air-coupled ultrasonic receivers), the deconvolution operation was previously shown to provide this important result also in reverberating systems with reflecting boundaries [27]. It is also important to highlight that a sufficient condition for an effective structural inspection is not to reconstruct the “ideal” Green's function of the test object, but rather to reconstruct a stable transfer function of the object, i.e., one that is invariant during the scanning of a defect-free region of the object.

The length of the recording time window for the deconvolution operation is an important factor in the application at hand. Since the receivers are moving relative to the test object (rail), a compromise must be identified between the long recording times necessary for the emergence of the transfer function by the constructive interference of the random excitations, and the short times required for the spatial localization of the reconstructed transfer function. The paper identifies a suitable time duration that is able to provide a satisfactory signal-to-noise ratio of the reconstructed transfer function at varying test speeds.

A first-generation prototype for the passive-only inspection of rails was designed, constructed and mounted on the DOTX-216 test car for field tests at the TTC. The prototype utilizes pairs of capacitive air-coupled ultrasonic receivers for truly noncontact probing of the rail. An outlier analysis statistical processing applied to the reconstructed transfer function helped to increase the detection of true rail discontinuities and decrease the detection of false alarms. The prototype was tested on the RTT track of TTC at speeds as high as 80 mph, where no ultrasonic inspection system for rails has ever been tested—to the authors' knowledge. The results show that the potential indeed exists to reconstruct stable transfer functions of the rail at these speeds, despite the variability of the wheel excitations, and consequently detect rail discontinuities such as joints and welds. For example, the run at 70 mph detected all of the three known joints and the seven known welds. The run at 80 mph also successfully detected all known discontinuities with the exception of one missed weld.

Limited tests were also conducted on the RDTF defect farm of TTC, where much lower speeds were allowed. Preliminary analysis of a run at 25 mph on the RDTF indicated the potential for the system to detect rail defects such as crushed heads, detail fractures and transverse fissures. Clearly, for a true assessment of defect detection performance the trade-off between probability of detection and probability of false alarms will have to be quantitatively determined. One way to do this is to generate receiver operating characteristic curves on a well mapped rail track. This step will be part of a second field test that is being planned for the second-generation prototype with a focus on defect detection performance.

The important role of test speed on the inspection performance of the passive system has not been fully characterized. Since the technique proposed relies on wheel generated excitations, both strength and bandwidth of the reconstructed transfer function are expected to increase with increasing wheel rotational speed. This consideration would suggest that different frequency bands may have to be optimally tracked in a way that is adaptive to the current test speed. Several design options to implement this bandwidth adaptability are being considered for the second-generation prototype.

Another relevant factor is the lay-out of the track, specifically a tangent track versus a curved track. For example, on a curved track the wheels will flange differently for different speeds depending on whether the train is above or below the “balance” speed for that curve. Different wheel flanging conditions will clearly change the strength and bandwidth of the excitation source. While the proposed signal processing is theoretically able to eliminate the source from the transfer function reconstruction, its signal-to-noise ratio (or rate of emergence) will likely be affected by the source bandwidth. The second-generation prototype is being designed to further increase the opportunities for signal averaging by exploiting multiple time snapshots for the same receiver pair. Increasing the averaged samples will improve the rate of convergence of the reconstructed signal and thereby further stabilize the performance in curves regardless of the train speed.

An additional aspect that is being improved in the second-generation prototype is the “co-location” of the separate receiver pairs that are averaged together. The physical dimensions of the receivers utilized in the first-generation system resulted in a spatial offset that was not optimal to ensure spatial localization of the transfer function.

A final open question is the role of rail lubrication on the passive inspection idea. The tests reported in the paper did not provide an opportunity to examine this factor (no lubrication was applied in addition to the lubrication that may have been already present on the rail at the time of the tests). Clearly, different lubrication conditions will change the wheel-rail contact. The presence of water on the rail could have a similar effect. Additional studies will be needed to shine light on these questions.

## Acknowledgment

Several individuals and organizations were instrumental for the preparation and the conduction of the field test at the Transportation Technology Center. They include: FRA Program Manager Dr. Robert Wilson and FRA Office of Research, Development and Technology Division Chief Gary Carr for technical advice and project oversight; Eric Sherrock and Jeff Meunier of ENSCO for field test support; FRA on-site representatives Jay Baillargeon and Luis Maal for field test observation; TTCI Program Manager Dr. Dingqing Li for field test logistics and support.

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