3.2.1. The Fourier Expansion

According to the Fourier expansion [26], the phase corresponding to each frequency component can be obtained as

$$f\_{fourir}(\mathbf{x}) = \frac{a\_0}{2} + \sum\_{k=1}^{\infty} a\_k \cos(k\lambda \mathbf{x}) + \sum\_{k=1}^{\infty} b\_k \sin(k\lambda \mathbf{x}) = \frac{a\_0}{2} + \sum\_{k=1}^{\infty} c\_k \sin(k\lambda \mathbf{x} + \phi\_k) \tag{1}$$

where *a*0, *a*1, *a*2, ... , *ak* are the cosine coefficients of the Fourier factor and *b*1, *b*2, ... , *bk* are the sine coefficients of the Fourier factor. And *x* is a discrete time variable, as to the temperature and strain data in Lieshihe bridge, *x* = 1/1440 × [1, 2, 3,. . . ,1440], owing to the sampling frequency is 1/1 min. λ is the reference frequency of the raw data, and φ*k* is the phase of *k*-th order. Moreover, the amplitude of each order data signal is

$$c\_k = \sqrt{a\_k^2 + b\_k^2} \tag{2}$$

The phase of each order of the signal is

$$\phi\_k = \arctan(a\_k/b\_k), k = 1, 2, 3, \dots \tag{3}$$

3.2.2. Determining the Minimum Order of the Fourier Series Expansion P

The expansion order is determined by trial-by-level trials until the residuals root mean square error (RMSE) meets the requirements. The S order RMSE(S) between the Fourier expansion value *ffourier*

and the temperature data *ftemp* can be calculated by *RMSE*(*S*) = *Si*=<sup>1</sup>(*ff ourier*−*f*temp) *S* , where *S* is the expansion order. The minimum expansion order *M* of temperature is determined by *RMSE*(*m*) < 0.2. Then, the expansion order *N* of strain data is also obtained in the same way. At last, the larger value of *M* and *N* is taken as *P* as the final expanded order.

The value of *RMSE* is related to the absolute value of the data value, the degree of dispersion, and so on. As a result, *RMSE* has no certain criterion for different kinds of data. The criterion of *RMSE* is determined by the correlation coefficient of fitted data and raw data in this paper.

#### *3.3. Calculating the Phase Di*ff*erence*

The phase difference Δφ*i* between the separated structural response data *fsr*,*tem* and the temperature data *ftemp* are solved at the same frequency. The specific steps are as follows.

#### 3.3.1. Obtain the Phase of Temperature and Strain Respectively

Calculate the phase φ*sr*,*<sup>k</sup>* of the structural response data *fsr*,*tem* and the phase φ*temp*,*<sup>k</sup>* of the temperature data *ftemp* according to the Fourier series approximation expression [27]. The Fourier series approximation expression for the structural response data *fsr*, *f ourier* and temperature data *ftemp*, *f ourier* are

$$f\_{sr, fourir}(\mathbf{x}) = \frac{a\_{sr,0}}{2} + \sum\_{k=1}^{P} c\_{sr,k} \sin(k\lambda\_{sr}\mathbf{x} + \phi\_{sr,k}) \tag{4}$$

$$f\_{temp, fourir}(\mathbf{x}) = \frac{a\_{temp, 0}}{2} + \sum\_{k=1}^{p} c\_{temp, k} \sin(k\lambda\_{temp}\mathbf{x} + \phi\_{temp, k})\tag{5}$$

where λsr and <sup>λ</sup>*temp* represent the reference frequency of strain data and temperature data, respectively. They can be computed by 2π/L. L is the length of the normalized cycle, which is closely related to the baseline period of the raw data and can be calculated automatically by the MATLAB Fourier series fitting program. Lambda varies with data of different days. It is mainly dependent on the shape feature of the data. As the shape of data in different single days is roughly similar, so the value of Lambda is approximate 5π/2.

#### 3.3.2. Calculate the Phase Difference

The phase difference Δφ*i* can be calculated according to the following formula:

$$
\Delta \phi\_i = \phi\_{temp, i} - \phi\_{sr, i}, i = 1, 2, 3, \dots, P \tag{6}
$$

where φ*temp*,*<sup>i</sup>* is the *i-*th order temperature data phase and φ*sr*,*<sup>i</sup>* is the *i*-th order structure response data phase.

#### *3.4. Determining the Total Phase Di*ff*erence and Lag Time*

Through a mass data research, the total phase difference can be obtained by the weighted summation of phase differences in each order. Moreover, the weight is proportional to the square of the frequency amplitude of each order.

$$\mathbf{w}\_{j}(\mathbf{x}\_{1}, \mathbf{x}\_{2}, \cdots, \mathbf{x}\_{P}) = c\_{\text{temp}, j}^{2} / \sum\_{j=1}^{P} c\_{\text{temp}, j}^{2} \tag{7}$$

where <sup>w</sup>*j* is the phase weight of the *j*-th order.

The delay effect on the correlation is eliminated or reduced by translating phase difference ϕ, which can be calculated by the following equation:

$$\wp = \sum\_{i=1}^{P} w\_i \times \Delta \phi\_i \tag{8}$$

where ϕ is the total delay phase to be eliminated.

Since the temperature data changes in cycles of days and the overall trend is a half-sine function, the lag time can be determined from the ratio of the lag phase difference to the half cycle of the sine function:

$$T\_{\rm lag} = \varphi \times 1440 / \lambda\_{\rm sr} \tag{9}$$

where *<sup>T</sup>*lag is the lag time in minutes.
