@article{LandwehrKuehnSchefferetal.2016,
  author    = {Landwehr, Niels and Kuehn, Nicolas M. and Scheffer, Tobias and Abrahamson, Norman A.},
  title     = {A Nonergodic Ground-Motion Model for California with Spatially Varying Coefficients},
  series = {Bulletin of the Seismological Society of America},
  volume    = {106},
  journal   = {Bulletin of the Seismological Society of America},
  publisher = {Seismological Society of America},
  address   = {Albany},
  issn      = {0037-1106},
  doi       = {10.1785/0120160118},
  pages     = {2574 -- 2583},
  year      = {2016},
  abstract  = {Traditional probabilistic seismic-hazard analysis as well as the estimation of ground-motion models (GMMs) is based on the ergodic assumption, which means that the distribution of ground motions over time at a given site is the same as their spatial distribution over all sites for the same magnitude, distance, and site condition. With a large increase in the number of recorded ground-motion data, there are now repeated observations at given sites and from multiple earthquakes in small regions, so that assumption can be relaxed. We use a novel approach to develop a nonergodic GMM, which is cast as a varying-coefficient model (VCM). In this model, the coefficients are allowed to vary by geographical location, which makes it possible to incorporate effects of spatially varying source, path, and site conditions. Hence, a separate set of coefficients is estimated for each source and site coordinate in the data set. The coefficients are constrained to be similar for spatially nearby locations. This is achieved by placing a Gaussian process prior on the coefficients. The amount of correlation is determined by the data. The spatial correlation structure of the model allows one to extrapolate the varying coefficients to a new location and trace the corresponding uncertainties. The approach is illustrated with the Next Generation Attenuation-West2 data set, using only Californian records. The VCM outperforms a traditionally estimated GMM in terms of generalization error and leads to a reduction in the aleatory standard deviation by similar to 40\%, which has important implications for seismic-hazard calculations. The scaling of the model with respect to its predictor variables such as magnitude and distance is physically plausible. The epistemic uncertainty associated with the predicted ground motions is small in places where events or stations are close and large where data are sparse.},
  language  = {en}
}
@article{KuehnScherbaum2016,
  author    = {Kuehn, Nicolas M. and Scherbaum, Frank},
  title     = {A partially non-ergodic ground-motion prediction equation for Europe and the Middle East},
  series = {Bulletin of earthquake engineering : official publication of the European Association for Earthquake Engineering},
  volume    = {14},
  journal   = {Bulletin of earthquake engineering : official publication of the European Association for Earthquake Engineering},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {1570-761X},
  doi       = {10.1007/s10518-016-9911-x},
  pages     = {2629 -- 2642},
  year      = {2016},
  abstract  = {A partially non-ergodic ground-motion prediction equation is estimated for Europe and the Middle East. Therefore, a hierarchical model is presented that accounts for regional differences. For this purpose, the scaling of ground-motion intensity measures is assumed to be similar, but not identical in different regions. This is achieved by assuming a hierarchical model, where some coefficients are treated as random variables which are sampled from an underlying global distribution. The coefficients are estimated by Bayesian inference. This allows one to estimate the epistemic uncertainty in the coefficients, and consequently in model predictions, in a rigorous way. The model is estimated based on peak ground acceleration data from nine different European/Middle Eastern regions. There are large differences in the amount of earthquakes and records in the different regions. However, due to the hierarchical nature of the model, regions with only few data points borrow strength from other regions with more data. This makes it possible to estimate a separate set of coefficients for all regions. Different regionalized models are compared, for which different coefficients are assumed to be regionally dependent. Results show that regionalizing the coefficients for magnitude and distance scaling leads to better performance of the models. The models for all regions are physically sound, even if only very few earthquakes comprise one region.},
  language  = {en}
}