# Railway Bridge Condition Monitoring Using Numerically Calculated Responses from Batches of Trains
## 1 Introduction
It is mobile, economic, and efficient [19]. [PERSON] et al. [20] review drive-by, also known as indirect, methods of monitoring highway bridges. Most methods are based on the identification of dynamic characteristics of the bridge from responses measured in the vehicle, such as natural frequencies, mode shapes, and damping [21; 22; 23; 24; 25]. Some methods detect bridge damage directly from the interaction between the vehicle and bridge without using the conventional structural dynamic properties. [PERSON] and [PERSON] [26] use CE optimisation to determine the Apparent Profile (AP), where the displacements recorded by the sensors are used as the inputs. The time-shifted difference in the AP is proposed to detect bridge damage and numerical results show that it has good potential as a damage indicator. [PERSON] et al. [27] calculate APs from on-board sensors and then subtract them to calculate the 'bridge displacement profile difference' (BDPD), which is proposed as another bridge damage indicator. [PERSON] and [PERSON] [28] used a Traffic Speed Deflectometer (TSD) to detect damage in bridges. The data gathered from the TSD are post-processed and time-shifted curvature, without pitch; this is used as the damage indicator. The time-shifted curvature is derived from the displacements and is proposed as a novel damage indicator, which removes the influence of the road profile and all vehicle motions except for pitch. For railway bridges, [PERSON] et al. [29] detect damage through comparison of APs sensed by the passing vehicle. The APs are calculated using the CE optimisation method that generates a vehicle dynamic response most similar to the measured input. APs for a number of damage scenarios are inferred and compared over time to detect damage. [PERSON] et al. [30] develop a method to detect the presence of railway bridge scour using bogie acceleration measurements from a passing train. A scour indicator is defined as the difference in average Continuous Wavelet Transform coefficients between healthy and scoured batches of train crossings. The result shows that this indicator is quite effective at detecting the presence of scour and its location. Some field tests of railway bridge monitoring using onboard train measurements are introduced in the following articles. Various tests have been conducted at the Bridge Deflection Test Facility (BDTF) at the Transportation Technology Center, USA. A track deflection measurement system (TDMS) and a track geometry measurement system (TGMS) have been tested on different trains. They investigate the potential for using onboard technology to detect bridge impairment or changes in bridge behaviour [31]. [PERSON] et al. [32] present simulation and field test results of a freight car and locomotive running on a railway bridge. Simulations and tests are used to improve the onboard measurement capability of such a system and the new configuration of accelerometers is shown to provide useful results for bridge condition evaluations. Displacements are derived from the acceleration data. Then, the results from different bridge conditions are compared and relative deflections are calculated to identify the change in bridge condition. [PERSON] et al. [33] introduce a field study of drive-by bridge monitoring using acceleration measurements on an instrumented train. The dynamic responses of the train signals are used to detect the existence and location of a stiffer part of the viaduct where two spans were replaced. The results show that instrumented trains can be successfully used to monitor bridge condition and to identify the need for repair or rehabilitation. This article presents a novel method of determining AP of the track and detecting bridge damage using batches of trains. In this context, AP is deemed to include the unloaded surface profile and the deflections of the track under the applied train load. Firstly, an Inverse Newmark-\(\beta\) algorithm is introduced to calculate AP from a train measurement.