### Site layout and soil characteristics

In this section, a numerical example is used to validate the proposed conditional simulation algorithm for non-stationary SVGMs on non-uniform sites. Figure 1 shows the site layout, where the ground motions at sites 1, 4 and 7 are measured while the ground motions at sites 2, 3, 5 and 6 are to be simulated.

The EPSD functions of all sites adopt the non-stationary Kanai-Tajimi spectrum with different parameters, which can be expressed as (Deodatis 1996),

$$Sleft(omega, tright)={left|A(t)right|}^2overline{S}left(omega right)$$

(13)

$$A(t)={a}_1texp left(-{a}_2tright)$$

(14)

$$overline{S}left(omega right)={overline{S}}_0cdot frac{1+4{xi}_g^2{left(frac{omega }{omega_g}right)}^2}{{left[1-{left(frac{omega }{omega_g}right)}^2right]}^2+4{xi}_g^2{left(frac{omega }{omega_g}right)}^2}cdot frac{{left(frac{omega }{omega_f}right)}^2}{{left[1-{left(frac{omega }{omega_f}right)}^2right]}^2+4{xi}_f^2{left(frac{omega }{omega_f}right)}^2}$$

(15)

where *A*(*t*) is the modulation function with parameters *a*_{1} = 0.906 and *a*_{2} = 1/3; (overline{S}left(omega right)) is the stationary Kanai-Tajimi spectrum; the spectrum parameters ({overline{S}}_0), *ω*_{g}, *ξ*_{g}, *ω*_{f}, and *ξ*_{f} denote the PSD, domain frequency of the foundation soil, damping ratio of the foundation soil, central frequency of the filter, and damping radio of the filter, respectively. These spectrum parameters depend on different soil conditions. In this study, sites 1 and 7 are assumed to be rock or stiff soil whereas site 4 is assumed to be deep cohesionless soil. The spectrum parameters of other sites are assumed to change linearly with the distances from sites 1, 4 and 7. The spectrum parameters of all sites are listed in Table 1.

In addition, the coherencies of the sites are modelled by using the Harichandran-Vanmarcke coherency model (Harichandran and Vanmarcke 1986), which is expressed as,

$${gamma}^{pq}left(omega right)=Aexp left[-frac{2{d}_{pq}}{alpha theta left(omega right)}left(1-A+alpha Aright)right]+left(1-Aright)exp left[-frac{2{d}_{pq}}{theta left(omega right)}left(1-A+alpha Aright)right]$$

(16)

where *θ*(*ω*) = *k*{1 + [*ω*/*ω*_{0}]^{b}}^{−1/2}, *ω*_{0} = 2*πf*_{0}, *A* = 0.736, *α* = 0.147, *k* = 5210, *f*_{0} = 1.09, and *b* = 2.78; *d*_{pq} is the distance between any two sites *p* and *q* (*p* = 1,2,…,7; *q* = 1,2,…,7).

### Conditional simulation

Based on the EPSD functions of all sites and the coherency model shown in Eq. (16), the time histories are first simulated using the unconditional simulation algorithm proposed by Deodatis (1996), which are adopted as the measured (reference) time histories for later comparison analysis. The duration *T* and the time step Δ*t* of the simulated time histories are 10.24 s and 0.01 s, respectively. The unconditionally simulated time histories at sites 1, 4, and 7 are also used as the recorded data in the proposed conditional simulation algorithm. Rather than assuming that all EPSD functions are known and identical, as done in the previous studies, different EPSD functions at sites 1, 4, and 7 are adopted, and the EPSD functions of the other unmeasured sites are estimated based on the IDW interpolation.

Based on the proposed conditional simulation algorithm, the simulated time histories at all sites, which are compatible with the recorded ones at sites 1, 4, and 7, can be obtained. Considering the stochastic characteristic of the simulated samples (Hu et al. 2017), a total of 10,000 samples are generated to obtain a more accurate estimation of EPSD functions and correlation functions. Figure 2 shows only one representative sample. The blue dot lines in the figure denoted as “Simulated” are the conditionally simulated time histories, while the red solid lines denoted as “Target” are the actual time histories obtained from the unconditional simulation algorithm. The simulated time histories are similar to the target ones at sites 2, 3, 5, and 6, indicating that the proposed conditional simulation algorithm can properly generate the time histories of the non-stationary SVGMs.

The correlation functions *R*^{pq}(*t*_{1}, *t*_{2}) (*p* = 1,2,…,7; *q* = 1,2,…,7) of the simulated non-stationary SVGMs can be approximately represented by the ensemble average of the 10,000 sample sets. As shown in Fig. 3, the simulated auto-correlation functions agree well with the target obtained from Eq. (4). It verifies that the proposed conditional simulation algorithm can offer an unbiased estimation of the time-variant auto-correlation functions. Fig. 4 also shows the cross-correlation functions between site 2 and sites 1, 4, and 7 individually. The simulated cross-correlation functions also match the target ones well, indicating that the simulated time history at site 2 is simultaneously compatible with the records at sites 1, 4, and 7. Also, the peak values of the cross-correlation functions reduce with the increasing distance at the same time instant, i.e., *peak*(*R*^{12}) > *peak*(*R*^{24}) > *peak*(*R*^{27}) at *t*_{2} = 3 s. The cross-correlation functions between site 3 and sites 1, 4, and 7 show a similar trend. This trend is generally consistent with the expectation.

The EPSD functions at sites 2, 3, 5, and 6 can be estimated based on the simulated time histories by using the Priestley estimation method (Priestley 1965). Figure 5 shows the comparison between the target EPSD functions and the estimated EPSD functions at sites 2 and 3. Here, the estimated EPSD functions are the ensemble average obtained from 100 sample sets of the simulated time histories. Figure 6 shows the slices of the EPSD functions at sites 2 and 3, which correspond to *t* = 3 s in Fig. 5. In Fig. 6, “One sample” denotes the raw estimation from one representative simulated time histories, “Estimated” denotes the ensemble average from 100 sample sets of the simulated time histories, and “Target” denotes the target one. Figures 5 and 6 show that the simulated time histories at the unmeasured sites are consistent with their target EPSD functions.

Based on the estimated EPSD functions, the estimated coherence ({hat{gamma}}^{pq}left(omega right)) between sites *p* and *q* can be calculated as follows,

$${hat{gamma}}^{pq}left(omega right)=frac{1}{T}{int}_0^Tfrac{{hat{S}}^{pq}left(omega, tright)}{sqrt{{hat{S}}^{pp}left(omega, tright){hat{S}}^{qq}left(omega, tright)}} dt$$

(17)

where ({hat{S}}^{pq}left(omega, tright)), ({hat{S}}^{pp}left(omega, tright)) and ({hat{S}}^{qq}left(omega, tright)) denote the estimated cross- and auto-EPSD functions; and *T* is the time duration.

The unconditional simulation algorithm proposed by Deodatis (1996) exhibited a large error in the imaginary part of the cross-EPSD functions. Thus, only the real part of the cross-EPSD functions is used in Eq. (17). The estimated coherences between site 2 and other sites are shown in Fig. 7. A very good agreement with the target ones can be observed.

Moreover, the acceleration response spectra at unmeasured sites 2, 3, 5, and 6 are estimated based on the average acceleration response spectra obtained from 100 sample sets of the simulated time histories under the damping ratio *ξ* = 5%. The estimated acceleration response spectra are compared with the target ones, which are obtained from the measured time histories generated by the unconditional simulation algorithm. As shown in Fig. 8, the estimated acceleration response spectra are overall closed to those of the target ones, especially when the structural fundamental period *T*_{s} is larger than 0.3 s, which corresponds to the period of most long-span bridges. Slight deviations can be observed when the structural fundamental period *T*_{s} is around 0.2 s.

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