## 2 Materials and Methods
### Excitation Modelling
#### 2.1.3 The Proposed Spatiotemporal Mining-Induced Kinematic Excitation (SMIKE) Model  
The main objective of this work was to propose a new model of kinematic excitation for multiple-support structures located in areas of mining activity and exposed to mining-triggered seismicity. The proposed model of nonuniformity of ground motion takes into account three distinct reasons for spatial variability, namely:  
* The loss of coherency resulting from scattering of waves in the heterogeneous ground and their superpositioning;
* The wave-passage effect resulting from the difference in the arrival times of waves at different supports;
* The site-response effect that takes into account the local soil conditions. The loss of coherency of mining-induced seismic waves was obtained by applying the random field generator based on the spatial correlation function (see Section 2.1.2) to produce the time histories of accelerations on consecutive structure supports based on the originally recorded shock. The proposed SMIKE model has two parameters: the space scale parameter \(\alpha\) and the apparent wave velocity in the ground \(v\). The random field generator requires the assumption of space scale parameter \(\alpha\). To determine this parameter (depending on local geological conditions), it was assumed that the empirical regression functions proposed for the local mining regions (see Section 2.1.4), which describe the reduction in the maximal amplitude with the distance from the epicenter, also define the loss in wave coherency. Hence, the space scale parameter can be determined by fitting the spatial correlation curve (see Equation (6)) into the regression function defined for a given mining activity region. The regression functions for different mining areas across the world are delivered by seismologists on the basis of continuous monitoring [32; 33; 34]. The deterministic approach was applied to account for temporal wave variability (wave passage effect). The records obtained for the consecutive supports were shifted in time by the value needed for the wave front, moving with the experimentally determined apparent wave velocity, to travel the distance between supports. This strategy needs the experimentally obtained apparent wave velocity \(v\) and avoids the necessity of identifying the time correlation parameter necessary to define the spatiotemporal correlation function (see Equation (12)). In the literature, this parameter is assumed rather freely and ranges from 1 to 200 [29; 30; 31]. 1.4 Empirical Regression Relationship of Amplitude Decay with a Distance from the Epicenter for Space Scale Parameter Estimation  
The space scale parameter \(\alpha\) is one of the main parameters of the proposed SMIKE model. Since the patterns for seismic wave propagation for mining quakes depend highly on local geological conditions, the physico-mechanical properties of the soil, and the terrain type, this parameter should be estimated for particular areas of mining activity. In Poland, regions of mining-triggered seismicity are continuously monitored, and the parameters of tremors, such as accelerations or velocities of the ground, are recorded continuously [35; 36; 37; 38; 39]. Investigations related to the reconstruction of ground vibration acceleration fields on the basis of point recordings are the subject of seismologists' research in mining-triggered areas. The registrations of accelerations or velocities at seismic stations are used to create regression relationships that enable the prediction of the level of ground vibrations at different locations in mining regions. One of the most seismically active regions of Poland is the Upper Silesian Coal Basin (USCB), in which the threat of mining seismicity and the intensity of tremors have recently increased [40]. In this region, the Upper Silesian Regional Seismological Network, installed in 1974, records all seismic events with local magnitude.