## 1 Introduction  
The overall use of numerical modeling techniques based on the finite element (FE) method, as well as experimental techniques for operational modal analysis (OMA), made the updating of numerical models based on modal parameters quite widespread [1]. These model-updating methodologies are widely used for: (i) developing highly accurate numerical models [2, 3, 4, 5]; (ii) modeling structures under operational conditions with unknown levels of degradation and/or geometrical/mechanical parameters with very high levels of uncertainty [7, 8]; (iii) monitoring the evolution of the structural behavior during retrofit operations [9]; and (iv) identifying structural damage [10, 11], among others. In most situations, these methodologies are based on the minimization of an objective function, composed by the residuals between numerical and experimental modal parameters, through the iterative variation of sensitive parameters of the numerical model [4]. During the optimization process, particularly due to variations on the numerical parameters' values, several changes in the order of the numerical mode shapes are frequently registered. To guarantee that these modifications do not affect the search for the optimal solution, it must be ensured that, in all iterations throughout the optimization process, the residuals of the objective function are calculated between an experimental mode and its correspondent numerical counterpart. Thus, the convergence of the optimization problem fundamentally depends on an efficient, automatic and stable mode-pairing technique to perform the correct assignment between numerical and experimental modes [12]. Most of the mode-pairing techniques are based on metrics for evaluating the correlation between two vectors (in this case, modal ordinate vectors), or between Frequency Response Functions (FRFs) [13; 14]. Such metrics, applied to experimental and numerical results, allow to quantify the degree of correlation between the modes, and, consequently, to assign each experimental mode to its numerical counterpart [12]. The metrics based on FRFs are less widespread in the field of mode pairing compared to those based on modal ordinate vectors, mainly due to the difficulties associated with the estimation of the experimental FRFs in large-scale structures. Nevertheless, the Frequency Domain Assurance Criterion (FDAC), proposed by [15], can be used for mode pairing with FRFs. The FDAC computes the correlation between two FRFs for different frequency shifts, and provides, for each frequency, a scalar between 0 and 1, with 0 meaning no correspondence and 1 meaning full correspondence. Among the modal-ordinate-vector-based criteria, one of the first to be applied is known as the Modal Scale Factor (MSF), proposed by [16]. The MSF is defined as the inner product of the two modal ordinate vectors to be compared, scaled by the inner product of either one of these vectors, making it significantly dependent on the normalization of the modes [17]. This dependence on the normalization factor sometimes causes a problem in practical applications, since it is often not possible to apply the same normalization to experimental and numerical modes, especially when using OMA techniques to gather experimental data. A very widespread criterion that is independent from normalization is the Modal Assurance Criterion (MAC), proposed by [16]. The MAC is defined as the square of the inner product of the two vectors, scaled by the product of the two inner products of the two vectors by themselves, resulting in a real scalar between 0 and 1 [18].