# Study on the Influence of a Soft Soil Interlayer on Spatially Varying Ground Motions
## 1 Introduction  
The spatial variation of ground motions is moving researchers and engineers' focus toward the seismic response characteristics of large-scale or extended structures [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11] and engineering site [12; 13; 14] under spatially varying earthquake ground motions (SVEGMs). However, the empirical analysis of SVEGMs requires the availability of spatially dense strong motion arrays, which are indeed rare even on a worldwide scale. Therefore, artificial simulation of SVEGMs is now widely employed and many simulation methods have been developed [15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25].  
In addition, spatial coherency function measures the similarity of the motions at two different locations [26] and indicates the degree to which the data recorded at the two stations are related. A lot of coherency models or relevant studies [27; 28; 29; 30; 31; 32] have been reported to facilitate the generation of SVEGMs based on theoretical derivation or measured spatial ground motions at dense seismographic arrays, in which specific site conditions (e.g., topography and soft soil interlayer) were not considered. Therefore, a lot of studies were performed on the influence of site conditions on the spatial coherency [26; 33; 34; 35; 36; 37].  
However, in general, these studies were on the basis that the soil property was linear and/or the shear modulus of soil layers were regularly increasing as depth, and underlying soft soil interlayer or non-linear soil behavior was not taken into account.  
On one hand, besides the topography [38; 39; 40], the existence of soft interlayer can also change the propagation path of wave from bedrock to surface in comparison with corresponding regularly and horizontally stratified site and can significantly amplify or attenuate the ground motion [41; 42; 43]. On the other hand, it has long been recognized that, during strong ground motions, non-linear soil behavior tends to reduce site amplification at high frequencies due to the increase of damping and also tends to shift resonance frequencies of site toward lower frequencies due to the degradation of shear modulus [44; 45; 46; 47; 48; 49; 50]. In the decay stage of ground motion, the shear modulus and damping of soil may recover and thus the amplification and filtering effects may recover as well [51; 52]. Taking all abovementioned into consideration, the nonlinear soil behavior during earthquake may considerably change the amplification and filtering effect of soil on ground motion and thus the soft interlayer that may reflect highly non-linearity might possess non-negligible influence on the spatial coherency between SVEGMs.  
Therefore, a site with local soft soil interlayer was assumed first, and two numerical examples were set in the present paper to: (i) Study the influence regularity of the properties of soft soil interlayer itself on the lagged spatial coherency; and (ii) demonstrate the difference of spatial coherency characteristics between linear elastic soil and non-linear soil cases. In example 1 (linear soil case), by combining the 1D wave propagation theory with classical spectral representation method (SRM), numerous quasi-stationary SVEGMs in horizontal out-of-plane, horizontal in-plane, and vertical in-plane directions were simulated for the target site, where varying shear wave velocity, buried depth, and thickness of the soft interlayer were considered. In example 2 (non-linear soil case), fully non-stationary SVEGMs were generated in those three directions, into which the spectral non-stationarity of ground motion caused by non-linear soil behavior was introduced. Besides the shear wave velocity, buried depth, and thickness of the soft interlayer, varying PGA of input motion was also considered in this example. Then mean lagged spatial coherency between SVEGMs under each circumstance above mentioned was estimated, and a series of comparisons were conducted.