A General Bayesian Solution to the Bridge Weigh-in-Motion Equations

Bridge weigh-in-motion (B-WIM) utilizes bridges as weighing scales to estimate the axle weights of passing vehicles. Conventional B-WIM relies on a static assumption, obtaining axle weights via least-squares fitting between measured and predicted responses. However, the system of equations may be ill-conditioned, resulting in significant errors in the estimated individual axle weights. To address this limitation, this paper proposes a novel Bayesian B-WIM approach, using the maximum posterior probability to determine axle weights. The posterior probability incorporates prior knowledge about the likely axle weights and the likelihood of the bridge response. These are characterized by the mean and covariance of several factors at both system calibration level, represented by influence line ordinates, and at individual vehicle level, represented by the axle weights. If the bridge response is treated as a sequence of independent variables with identical standard deviations for each time scan, the Bayesian solution reduces to the classical Moses B-WIM algorithm. If each axle weight is independent and identically distributed, the Bayesian formulation reduces to the Regularized B-WIM algorithm. The proposed approach is evaluated using both simulated and field test data. Results are compared against the Moses and Regularization algorithms. For all calculations, the Bayesian approach achieves the highest accuracy, with particularly notable improvements in the identification of individual axle weights. In field tests, the mean absolute errors of the first axle weights are reduced by 21.7% compared with the Moses algorithm and 14.0% compared with the Regularization algorithm. The proposed method can effectively mitigate the effects of fluctuations in axle weights, bridge influence lines, and measurement noise on weighing accuracy.