## 1 Introduction
An alternative approach [23] forgoes direct measurements in favor of calculating the rotation of each structural segment from measured strain data, subsequently deriving displacement through cumulative summation. However, Bayesian methods invariably necessitate preliminary or approximate measurement of the target responses, i.e., before estimation, some true estimation results should be available, which can be a limitation given budget constraints. Thus, while Bayesian methods provide a valuable framework for data fusion, they still require a certain level of investment in advanced measurement technology or an adequate number of sensors, potentially conflicting with the budgetary restrictions often encountered in engineering projects. An alternative approach involves the use of surrogate models through machine learning techniques [24; 25; 26; 27; 28; 29; 30]. These methods sidestep the construction of finite element or state-space models, opting instead to directly simulate a surrogate model. This model captures the transfer matrix that links monitoring data with the responses to be estimated. However, similar to Bayesian-based methods, machine learning approaches typically necessitate some level of measured responses at the target location. This requirement has become a focal point of research, often referred to as \"missing data recovery\" [31; 32; 33] or \"response reconstruction\" [34; 35; 36]. Despite these research efforts, current studies have yet to tackle scenarios where no sensors are installed at specific locations. Notably, some research [37] has suggested that machine learning can address this issue. It posits that a trained transfer matrix between strain and acceleration at one cross-section could be applied to other cross-sections, thus eliminating the need for partial measurements. However, this research has only been validated using a simply supported beam, where the strain and acceleration mode shapes are identical. This results in a consistent transfer matrix across all cross-sections, which is presumed to be a critical factor in validating the proposed method. For general beam bridges with multiple spans, the mode shapes for strain and acceleration may differ, leading to variations in the transfer matrix at different cross-sections. Consequently, the effectiveness of the method as demonstrated in this significant research may be questioned when applied to more complex bridge structures. In this study, we proposed a novel finite element model-free method. Instead of a surrogate state-space model or directly trained surrogate model of transfer matrix with machine learning technologies, our method endeavors to derive surrogate mode shape functions. Through theoretical analysis of the dynamics of the simple vehicle-bridge interaction (VBI) system [38], we identified a parallel between the structural responses and the blind source separation (BSS) problem [39]. Leveraging this insight, we applied the Fast Independent Component Analysis (FastICA) algorithm [40], a prevalent technique for BSS, to compute surrogate inclination mode shapes. Further, this study employs discrete integration to deduce the surrogate displacement mode shape functions. These mode shape functions are then instrumental in formulating the surrogate inter-section transfer matrix for acceleration across various locations. A rigorous series of theoretical derivations elucidate three pivotal principles involving the identity between transfer matrices of low-frequency displacement and high-frequency acceleration, surrogate and real physical inter-section transfer matrices, and integration for the surrogate displacement mode shape function. These principles collectively validate the effectiveness of our proposed method. To confirm its reliability, we performed a series of numerical simulations and a full-scale laboratory test, demonstrating the method's practical applicability. Compared to the SSI and eigenvalue methods, our proposed method overcomes the challenge of extracting modal information from low-frequency inclination data.