Convoluted Reciprocity and Other Methods for Vehicle Speed Estimation in Bridge Weigh-in-Motion Systems

The initial and simplest idea to identify the vehicle’s speed was to study the peaks on the measured signals. The time delay between peaks in signals at separate locations, together with the actual distance between them, provides an estimate of the event’s speed (Liljencrantz et al. 2007), an idea that has also been termed peak-to-peak (OBrien and Žnidarič 2001). However, this was quickly adopted only as a reasonable first guess. Kalhori et al. (2017) clearly showed, with laboratory and field tests, that the peak-to-peak approach did not provide acceptable speed estimates. Others have tried to improve the idea by automating peak detection using wavelets (Chatterjee et al. 2006; Kalin et al. 2006). The peak-to-peak approach was reported to be adequate for BWIM installations on orthotropic bridges of 6–10 m span lengths (OBrien and Žnidarič 2001). However, this is not the case for longer bridges or bridges with thick slabs (Kalhori et al. 2017).

Arguably, the most common method to estimate speed is calculating the cross-correlation (or simply correlation) between signals measured at two different sections in the longitudinal direction (Yu et al. 2016). The speed is estimated by the distance between sensors divided over the time lag that maximizes the correlation. For this method to work perfectly, both signals should be time-shifted copies. In BWIM installations, this would require that the influence lines at the two measured locations should be identical in shape. In practice, this symmetry is rarely achievable. However, if the influence lines feature very sharp peaks that enable individual axle detection, then the speed estimation using correlation has been found to give sufficiently accurate results (Hajializadeh et al. 2020). Also, the experimental campaign by Zolghadri et al. (2016) to estimate speeds for different bridges and vehicle types concluded that the correlation method is good enough, reporting average errors of 9% approximately. Therefore, it is a widely implemented solution for speed estimation, being a recent example of the study by MacLeod and Arjomandi (2022). However, when the asymmetry of the influence lines is too large, the correlation method provides wrong speed estimates. This is known, and correction strategies have been suggested. Kalin et al. (2006) proposed a correction based on qualified maxima, reporting satisfactory results even for simply supported bridges. Liljencrantz et al. (2007) suggested using a factor to be obtained during the system calibration to correct the results, which was then implemented by Liljencrantz and Karoumi (2009). Both correction strategies will be discussed further in the “Correlation” section.

More recently, a new possibility has emerged that consists of the combination of different signals from different sections in the longitudinal direction. He et al. (2017) and Chen et al. (2018, 2019) showed that it is possible to combine three measured strain responses from various locations to obtain a signal that is related to a shorter and sharper influence line. The use of two of those signals was used then to estimate speed via a variation of the peak-to-peak method. He et al. (2017) acknowledged that the correlation method could be used with this pair of signals but was not explored further. Later, these ideas were theoretically framed into the point load approximation (PLA) (Cantero 2021), which showed that the combination of measurements, using the factors from a finite difference scheme, produces signals that approximate the vehicle loading function. For appropriate sensor layouts, it is possible then to obtain signal pairs that are similar in shape but time-shifted. This property has never been used to estimate the speed of passing vehicles in combination with the correlation method.

For completeness, there exist other possibilities to estimate speed based on alternative ideas or sensor technology. For instance, using shear strain signals has been proven to provide accurate speed estimates (Kalhori et al. 2017; Bao et al. 2016) via the peak-to-peak method in combination with signal processing. Liljencrantz et al. (2007) suggested matching the IL length in the time domain to the recorded signal. However, for this method, the axle configuration of the passing vehicle must be known before, something that the other methods do not require. Algohi et al. (2018) processed the acoustic emissions of trucks passing over the expansion joints of the bridge to infer the speed of the vehicle. Speed estimates can also be obtained with synchronized camera recordings via image processing (Ojio et al. 2016) and computer vision (Jian et al. 2019). Another possibility, reported by Kim et al. (2022), is instrumenting the supports to measure the reaction forces to obtain the entry and exit times of individual vehicles. Sekiya et al. (2018) and Mustafa et al. (2021) used accelerometers at different sections of the bridge to identify the speed from peaks in the signals. Also, if sufficient calibration events exist, it is even possible to train a machine learning model to identify the speed of the vehicle (Zhou et al. 2021).

This study focused on the speed estimation problem for the most common BWIM installation, which utilizes sensors on the bridge only and measures strain at multiple locations. Therefore, alternative methods involving different sensor technology, sensors on the pavement, or cameras were not considered. Furthermore, the analysis in the following concentrated on physics-based methods rather than data-based (machine learning) solutions. Also, it was assumed that no prior information about the vehicle crossing event is known, except that it traverses the bridge at a constant speed.

The review in the “Review of Existing Speed Estimation Methods” section has shown that, despite a range of solutions to estimate the speed of passing vehicles, there are difficulties and limitations. There is a need to find a universally valid method for speed estimation. Therefore, the goal was to introduce a generic method that could be applied to any bridge typology. This paper proposed a new concept, called convoluted reciprocity (CR), that enables the definition of a theoretically consistent method to determine the speed. In addition, the study validated the use of PLA signals together with the correlation method. This was done theoretically, numerically, and validated with results from laboratory experiments. The study compared the performances of established and new methods.

The following section first presents the numerical model to simulate bridge responses under traffic loading and introduces the new theoretical finding called CR. The “Speed Estimation Methods” section reports a numerical investigation of the different speed estimation strategies, completed with extended numerical studies, to compare their performances in the subsequent section. Finally, the paper validates the results experimentally in the “Experimental Validation” section.

Validated open-source software VBI-2D (D. Cantero, “VBI-2D - Vehicle-bridge interaction simulation and validation framework for Matlab,” submitted.) was used in this study to simulate bridge responses loaded by passing vehicles. This tool numerically solves the coupled vehicle–bridge interaction (VBI) problem for finite-element models of bridges defined with beam elements that are traversed by multibody vehicles. Various levels of vehicle model complexity can be simulated, supported by the complementary tool VEqMon2D (Cantero 2022). In particular, in this study, two-axle vehicle [Fig. 1(a)] and five-axle [Fig. 1(b)] articulated truck models were used. The bridge was represented as a simply supported beam [Fig. 1(c)] of length L with constant section properties. The numerical model was used to simulate the bending moments, which are proportional to the strain signals that would be measured in a BWIM installation. Notation S_X is adopted for the signals, indicating that, for example, S₂₅ is the bending moment (or strain) measured with a sensor located at 25% of L (the bridge span).

Table 1 shows the model properties used in this study to describe concrete bridges of different span lengths, which are based on the values reported by Li (2006) for an elastic modulus of 35 × 10⁹ N/m². On the other hand, the mechanical properties of the vehicles have been taken exactly as provided by Cantero and González (2015) for the two-axle vehicle and by OBrien et al. (2010) for the five-axle truck. In addition, the simulation includes road profile irregularities of Class A, according to the international standard ISO-8608 (ISO 1995). Because bridge responses under normal operational traffic conditions are expected to behave within the linear elastic range, which is an underlying assumption in BWIM installations, the linear numerical model used in this study is a valid proxy of the problem.

The quasi-static response of a bridge due to a passing vehicle can be calculated using the influence line. More precisely, the response at Location A is the convolution of the influence line at A by the loading function (Frøseth et al. 2017), as shown in the following equation:where P(t) = loading function, which is made of a series of impulse functions of magnitudes P_j, one for each axle j, that is time-shifted according to the axle distances and the speed of the vehicle v; and I_A = influence line of the measured load effect at the Location A of the bridge.

Now, consider the influence line I_B at another Location, B. The convolution of Eq. (1) by I_B is

Using the definition in Eq. (1) and considering the commutative property of the convolution operator, Eq. (2) can be rewritten as follows:

This relation in Eq. (3) has been termed CR. The expression shows that the convolution of the response at Location A by the influence line at Location B is the same as the convolution of the response at B by the influence line at A. This is a simple yet powerful theoretical result that can be utilized for speed estimation. By definition, an influence line is described in the space domain. However, the influence lines, I_A(x) and I_B(x), can be transformed into the time domain by knowing (or assuming) the vehicle’s traversing speed v. The v value that satisfies Eq. (3) corresponds to the speed of the vehicle. This constitutes the basis of the proposed speed estimation method, which is illustrated in the “Speed Estimation Methods” section

Moreover, Eq. (3) can be extended to a more generic case. Multiplying it by the loading function P_C(t), for a vehicle of known speed (called here calibration vehicle), gives the following equation:

Again, using the definition in Eq. (1), one can derive the following relationship:where S_X,C(t) = measured response at location X due to the passage of the calibration vehicle. Eq. (5) indicates that the convoluted reciprocity relationship also exists between two Generic bridge locations A and B for responses for two independent vehicle crossing events.

Defining CR_AB as the convolution of the response at A due to a vehicle by the response at B due to a second vehicle, Eq. (5) can be rewritten as follows:

This relationship [Eq. (6)] is valid for the responses of any two vehicles regardless of their axle configuration. Therefore, for the passage of a theoretical single-axle unit load vehicle and an unknown truck, the relationship gives Eq. (3) again.

Similar to previously with the influence lines, the signals for the (calibration) vehicle passage at known speed v_C can be readily transformed into the space domain, S_A,C(x) and S_B,C(x). To estimate the speed of other vehicles, one can find the new value of v that satisfies Eq. (5). The speed estimation procedure based on calibration vehicle signals is called CRV here, whereas the procedure based on influence lines is referred to as CRI. Both approaches will be investigated separately because their results are somewhat different, and they have different practical implications.

Note that the aforementioned formulation was derived in terms of functions with continuous time (or space) as the argument. However, in practice, the signals and influence lines consist of arrays of values sampled at instances in time. Nevertheless, the aforementioned results are also valid for discrete representations of the quantities involved. Also, note that the previous derivation has considered only quasi-static effects. Real measured bridge responses due to passing vehicles also include dynamic components, which introduce perturbations and inaccuracies when using CR for speed estimation. Nevertheless, this simple idea provides a theoretical method to estimate vehicle speeds. Therefore, these ideas will be numerically and empirically validated in the following sections.

A method to estimate the speed of a vehicle crossing event is by studying the time difference between peaks in the signals. One possibility is to use the times for maximum (or minimum) values of signals measured at separate locations. This idea applied to the quasi-static signals of the running example [Fig. 2(a)] gives a speed estimate of 180 m/s (800% error), which is not a valid result. However, this simple idea has worked for some ideal cases (OBrien and Žnidarič 2001) and has been used as an initial guess for more refined speed estimation methods.

Alternatively, it is possible to work with the time differences between peaks that correspond to the same axle of the vehicle. For instance, this is illustrated for the peaks corresponding to the first axle in the quasi-static responses in Fig. 2(a). Knowing the distance between sensors, this idea provides a perfect speed estimation (0% error). However, when the signals include disturbances from the dynamic response, as shown in Fig. 2(b), the estimated speed is 19.15 m/s (−4.25% error). Therefore, this method might provide appropriate estimates, but it relies on the identification and matching of the peaks corresponding to vehicle axles.

The most popular and established method to estimate speed is studying the correlation between two signals. The time lag that gives the highest correlation is used as an indication of the vehicle’s travel time from one sensor to the next. The underlying assumption for this method is that both signals match in shape when time-shifted. However, for BWIM systems, it is generally not possible to find two locations on the bridge that give the same signal for a passing vehicle. Nevertheless, it is often used because it provides appropriate speed estimates, particularly when the sensors capture local responses of the structure and the signals have very distinct peaks (Kalin et al. 2006). For the running example, the correlation between the quasi-static signals in Fig. 2(a) is shown in Fig. 3. The time lag corresponding to the maximum correlation is used to estimate the speed, which in this case is 21.18 m/s (5.88% error). Fig. 3 also shows the lag value to obtain the correct speed of the event. This example shows that the correlation method does not work perfectly, even in the case of quasi-static signals.

The discrepancies in speed estimates using the correlation method are well-known. Some correction strategies have been proposed to improve its accuracy and facilitate its use in a wider set of structural configurations. One such correction strategy relies on the definition of a calibration factor C_Veh, which was reported in Liljencrantz et al. (2007) and has also been used in Zolghadri et al. (2016). This calibration factor is obtained by comparing the known speed of a passing vehicle to the result obtained with the correlation method. However, it can be shown that this calibration factor is specific for each vehicle since it depends on the number of axles, the spacing between them, and their relative load. Fig. 4 shows the calibration factor for a range of different axle spacings and axle weight distributions. The factor was calculated for the two-axle vehicle using ideal quasi-static responses of the bridge. Therefore, this correction strategy could be used only to estimate the speed of vehicles identical to the calibration vehicle, thus requiring different factors for different vehicles. This is not a practical strategy for road bridges due to the heterogeneity of the vehicles in traffic. However, this might be a feasible approach for railway bridges, where generally there exist only a limited number of types of locomotives. For completeness, the performance of the correlation method corrected with C_Veh for the running example is reported in Table 2.

Another correction strategy is suggested in Kalin et al. (2006) based on identifying a few qualified maxima (usually 2 or 3) in both signals. These maxima are points in the signals with certain requirements to ensure that they correspond to significant peaks and are not just simply local extremes. The correction strategy calculates the time differences for each pair of maxima and proceeds with the time difference that is closest to the time lag obtained by simple correlation. Indeed, this correction strategy improves the correlation method for quasi-static signals obtaining perfect speed estimation (0% error). When including dynamic components in the signals, the strategy identifies four points as qualified maxima, as shown in Fig. 5. These maxima occur at slightly different instances in time (compared to the quasi-static case), affecting the accuracy of the final speed estimation, as reported in Table 2.

Table 2 summarizes the speed estimates for the running example using the correlation method and eventual correction strategy. For the ideal case (quasi-static response), the method works perfectly only when a correction strategy is applied. Even though the use of the calibration factor seems simple and reliable, it only works for the same vehicle configuration, rendering it impractical for road bridges. The correction based on qualified maxima is more general but requires peak identification in the signals, which is not necessarily always possible. On the other hand, when the dynamic responses are considered, the accuracies degrade even when including correction strategies. Inevitably, when introducing the disturbances in the ideal signals, the speed estimation errors deviate.

Another possibility for speed estimation is combining multiple responses into signals that are better suited for the correlation method. This can be achieved with the PLA presented in Cantero (2021), which factors and adds the responses using a finite difference scheme. With this procedure, it is possible to obtain signals with the same shape at different locations along the beam when using equally separated sensors. The results are time-shifted symmetric signals that are ideal inputs for speed estimation by correlation.

To illustrate this method on the running example, it is necessary to use additional sensor readings to those indicated in Fig. 1(c). As suggested in Cantero (2021), one possibility is to define virtual sensors located at the supports, which are known to give zero-valued signals. These five sensors (three real and two virtual) are equally spaced between each other and permit the fabrication of time-shifted versions of the same signal via PLA. The three nonzero signals for the running example are shown in Fig. 6(a), including dynamic effects. The first PLA signal can be obtained by combining the responses at 25% and 50% of the span together with the virtual sensor signal at the left support (0% of L). Similarly, the second PLA signal is the combination of S₅₀ and S₇₅ together with the virtual zero response at the right support (S₁₀₀). The resulting PLA signals are shown in Fig. 6(b). As can be seen, even when including the dynamic effects, both PLA signals have remarkably similar shapes. The correlation method based on these PLA signals gives a speed estimate of 19.78 m/s (−1.10% error). Not shown here, but the same exercise using quasi-static signals renders a perfect speed estimation (0% error).

The theoretical relationship derived in the “Methods” section, called CR, can be used to obtain speed estimates based on influence lines or responses from calibration vehicles. In this section, the speed estimation is illustrated only using CRI, but a similar procedure and results are obtained using CRV.

The speed estimation method is described schematically in Fig. 7 for the case of the running example. First, the procedure requires information from two sensors, namely, their influence lines and the responses due to the passing vehicle of unknown speed. Then, the method rescales the influence lines assuming different speeds and evaluates the validity of the convoluted reciprocity equality [Eq. (6)]. For any given speed, the discrepancy between the right-hand side and the left-hand side of the equation is evaluated with the norm of their difference. The value that minimizes that difference is the estimated speed of the crossing vehicle.

The illustrative example shown in Fig. 7 used the ideal quasi-static bridge responses and resulted in perfect speed estimation (20 m/s). However, when including the dynamic components of the bridge response in the signals, the convoluted reciprocity procedure provides a speed estimate of 20.41 m/s (2.06% error). For reference, Fig. 8 shows the norms of the convoluted reciprocity for the range of speeds considered, observing the small discrepancy due to the disturbances introduced by the dynamic components in the signals.

The procedure illustrated in Figs. 7 and 8 found the minimum norm from a parametric study on a range of speeds. However, this procedure can also be implemented as a numerical minimization problem. It was found that the objective function, namely the norm of the difference between convoluted reciprocities, is a convex function that features a single well-defined minimum. Adopting established optimization procedures is generally more computationally efficient and results in the same speed estimate (except for very small numerical discrepancies related to the discretization size and the adopted tolerances in the minimization procedure). In the remainder of this study, the vehicle speed via CR is obtained using MATLAB’s fminsearch command to find the minimum, which uses the simplex search method (MATLAB 2022).

As described in Fig. 7, the speed estimation via the convoluted reciprocity used the norm as the metric to evaluate the similarity between both sides in Eq. (6). However, there exist different possibilities to define this norm. The generalized expression of p-norm (norm of order p) for a vector r is shown in Eq. (7), a definition that gives either the sum in absolute value (p = 1), the Euclidean norm (p = 2), or the maximum (p = ∞), among the possibilities:

A priori, it was not known what norm order would work best for the proposed speed estimation procedure. Therefore, this was evaluated numerically, studying the same problem for a selection of different norm orders. A Monte Carlo simulation with 4,000 events was used to identify what is the best metric to use as an indicator in the speed estimation procedure using the convoluted reciprocity. For each simulated event, the CR procedure was repeated for four different norms (1-norm, 2-norm, 3-norm, and ∞-norm). Fig. 9 shows the number of times (in relative terms) that each norm was identified as the best-performing metric. The results indicate that generally 1-norm seems to be the norm that most frequently best-identified speed. Therefore, the 1-norm was adopted as the default norm to evaluate the difference in the convoluted reciprocity expression.

The “Speed Estimation Methods” section presents presents different speed estimation methods based on one single example in detail. To assess the actual performance of each of them, they must be tested under a range of structural configurations and loading scenarios. Therefore, a new Monte Carlo analysis was performed with 20,000 events to compare the performances of several speed estimation methods. In particular, the study evaluated the following methods:

Fig. 10 shows the mean error and corresponding standard deviation for each of the four methods, where the results are divided into bridge spans and vehicle types. In general, the results show that the correlation (with correction) does perform well enough for certain spans and vehicle types. However, its mean error can be significant (up to 9% error) in some cases. In addition, the correlation error displays large variability in performance, reflected in a large standard deviation. In comparison, the methods based on PLA and CR give consistently small mean errors and standard deviations. The results clearly indicate that PLA and CR approaches outperform the standard speed estimation method based on the correlation of two signals. To confirm these results and evaluate further the difference between methods, the study was continued under laboratory conditions with real measured signals and discussed in the next section.

A vehicle crossing a bridge was experimentally reproduced in the laboratory using a scaled setup of a beam traversed by a model vehicle. Fig. 11 shows the four-axle model vehicle used in the experiment, which consisted of the main body and trailer with axle weights and spacing as provided in Table 3. The vehicle was moved across the access tracks and beam by a pulley system, which provided constant speed while crossing the beam. The vehicle was moved back and forth across the beam 12 times, which resulted in 24 crossing events. This loading sequence was performed for two different constant speeds, namely, at V1 = 0.551 and V2 = 1.048 m/s.

The beam, also shown in Fig. 11, was a 5.6-m-long steel beam in a simply supported configuration spanning 5.4 m. The properties of the beam are listed in Table 4. The 300-mm-wide Steel I-beam was positioned on the side, so the edges of the model bridge were the flanges, and the web constituted the deck on which the model vehicle was moving on. The vehicle was moving over steel rails with an imposed roughness corresponding to a Scaled class A ISO-8608 (ISO 1995) profile. The beam was instrumented with strain gauges located at 25% (= L/4), 50% (= L/2), and 75% (= 3L/4)of L, which resulted in signals S₂₅, S₅₀, and S₇₅, respectively, adopting the same configuration and notation as in Fig. 1. The sampling frequency was 2,048 Hz using the NI9237 data acquisition system.

Fig. 12(a) shows a sample of the measured strain signals during a single-vehicle model passage traveling at V2 speed. These signals have been processed to remove noise with a zero-phase first-order Butterworth lowpass filter with a 1 Hz cutoff frequency. With the known information about the vehicle (Table 3) and traversing speed, the influence lines at each sensor location were obtained via the matrix method (OBrien et al. 2006). The final influence lines considered in this study [shown in Fig. 12(b)] are taken as the average result extracted from several vehicle passages.

The three novel methods presented in this document were used to estimate vehicle speeds using the signals obtained from the laboratory experiments. First, the PLA method utilized the three recorded beam responses and assumed the existence of two virtual sensors at the supports to obtain signals of similar shape to subsequently estimate the speed via correlation. Also, both possibilities of the convoluted reciprocity approach were investigated separately. The CRI approach relied on the average influence lines presented in Fig. 12(b), whereas CRV employed the bridge responses for the first event with Speed V1. The estimated speeds for each method and event are shown in Fig. 13, where they can be compared against the correct value of the model vehicle. All three methods have certain variability in results but predict the speed with similar accuracy. There is no obvious trend or difference between methods.

To evaluate the speed estimation performance of each method, the results in Fig. 13 are summarized in statistical terms in Table 5. For completeness, this table also includes the results obtained using the correlation method with and without correction strategy. The results clearly show that for the novel speed estimation methods, the mean errors are one order of magnitude smaller than those based on correlation. As expected, the plain correlation method offered quite poor results. This was significantly improved when including the correction strategy based on qualified maxima. Nevertheless, on average, the estimated speeds showed more than 10% error and a significant dispersion in results. On the other hand, the results indicate that the PLA method offered the best performance, obtaining the smallest mean and standard deviation among all the methods. Then, the second-best method is CRI, closely followed by CRV, featuring average errors less than 2% and similar standard deviations.