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Although the methodological framework of probabilistic seismic hazard analysis is well established, the selection of models to predict the ground motion at the sites of interest remains a major challenge. Information theory provides a powerful theoretical framework that can guide this selection process in a consistent way. From an information- theoretic perspective, the appropriateness of models can be expressed in terms of their relative information loss (Kullback-Leibler distance) and hence in physically meaningful units (bits). In contrast to hypothesis testing, information-theoretic model selection does not require ad hoc decisions regarding significance levels nor does it require the models to be mutually exclusive and collectively exhaustive. The key ingredient, the Kullback-Leibler distance, can be estimated from the statistical expectation of log-likelihoods of observations for the models under consideration. In the present study, data-driven ground-motion model selection based on Kullback-Leibler-distance differences is illustrated for a set of simulated observations of response spectra and macroseismic intensities. Information theory allows for a unified treatment of both quantities. The application of Kullback-Leibler-distance based model selection to real data using the model generating data set for the Abrahamson and Silva (1997) ground-motion model demonstrates the superior performance of the information-theoretic perspective in comparison to earlier attempts at data- driven model selection (e.g., Scherbaum et al., 2004).
The estimation of minimum-misfit stochastic models from empirical ground-motion prediction equations
(2006)
In areas of moderate to low seismic activity there is commonly a lack of recorded strong ground motion. As a consequence, the prediction of ground motion expected for hypothetical future earthquakes is often performed by employing empirical models from other regions. In this context, Campbell's hybrid empirical approach (Campbell, 2003, 2004) provides a methodological framework to adapt ground-motion prediction equations to arbitrary target regions by using response spectral host-to-target-region-conversion filters. For this purpose, the empirical ground-motion prediction equation has to be quantified in terms of a stochastic model. The problem we address here is how to do this in a systematic way and how to assess the corresponding uncertainties. For the determination of the model parameters we use a genetic algorithm search. The stochastic model spectra were calculated by using a speed-optimized version of SMSIM (Boore, 2000). For most of the empirical ground-motion models, we obtain sets of stochastic models that match the empirical models within the full magnitude and distance ranges of their generating data sets fairly well. The overall quality of fit and the resulting model parameter sets strongly depend on the particular choice of the distance metric used for the stochastic model. We suggest the use of the hypocentral distance metric for the stochastic Simulation of strong ground motion because it provides the lowest-misfit stochastic models for most empirical equations. This is in agreement with the results of two recent studies of hypocenter locations in finite-source models which indicate that hypocenters are often located close to regions of large slip (Mai et al., 2005; Manighetti et al., 2005). Because essentially all empirical ground-motion prediction equations contain data from different geographical regions, the model parameters corresponding to the lowest-misfit stochastic models cannot necessarily be expected to represent single, physically realizable host regions but to model the generating data sets in an average way. In addition, the differences between the lowest-misfit stochastic models and the empirical ground-motion prediction equation are strongly distance, magnitude, and frequency dependent, which, according to the laws of uncertainty propagation, will increase the variance of the corresponding hybrid empirical model predictions (Scherbaum et al., 2005). As a consequence, the selection of empirical ground-motion models for host-to-target-region conversions requires considerable judgment of the ground-motion analyst
The use of ground-motion-prediction equations to estimate ground shaking has become a very popular approach for seismic-hazard assessment, especially in the framework of a logic-tree approach. Owing to the large number of existing published ground-motion models, however, the selection and ranking of appropriate models for a particular target area often pose serious practical problems. Here we show how observed around-motion records can help to guide this process in a systematic and comprehensible way. A key element in this context is a new, likelihood based, goodness-of-fit measure that has the property not only to quantify the model fit but also to measure in some degree how well the underlying statistical model assumptions are met. By design, this measure naturally scales between 0 and 1, with a value of 0.5 for a situation in which the model perfectly matches the sample distribution both in terms of mean and standard deviation. We have used it in combination with other goodness-of-fit measures to derive a simple classification scheme to quantify how well a candidate ground-rnotion-prediction equation models a particular set of observed-response spectra. This scheme is demonstrated to perform well in recognizing a number of popular ground-motion models from their rock-site- recording, subsets. This indicates its potential for aiding the assignment of logic-tree weights in a consistent and reproducible way. We have applied our scheme to the border region of France, Germany, and Switzerland where the M-w 4.8 St. Die earthquake of 22 February 2003 in eastern France recently provided a small set of observed-response spectra. These records are best modeled by the ground-motion-prediction equation of Berge-Thierry et al. (2003), which is based on the analysis of predominantly European data. The fact that the Swiss model of Bay et al. (2003) is not able to model the observed records in an acceptable way may indicate general problems arising from the use of weak-motion data for strong-motion prediction
Composite ground-motion models and logic trees: Methodology, sensitivities, and uncertainties
(2005)
Logic trees have become a popular tool in seismic hazard studies. Commonly, the models corresponding to the end branches of the complete logic tree in a probabalistic seismic hazard analysis (PSHA) are treated separately until the final calculation of the set of hazard curves. This comes at the price that information regarding sensitivities and uncertainties in the ground-motion sections of the logic tree are only obtainable after disaggregation. Furthermore, from this end-branch model perspective even the designers of the logic tree cannot directly tell what ground-motion scenarios most likely would result from their logic trees for a given earthquake at a particular distance, nor how uncertain these scenarios might be or how they would be affected by the choices of the hazard analyst. On the other hand, all this information is already implicitly present in the logic tree. Therefore, with the ground-motion perspective that we propose in the present article, we treat the ground-motion sections of a complete logic tree for seismic hazard as a single composite model representing the complete state-of-knowledge-and-belief of a particular analyst on ground motion in a particular target region. We implement this view by resampling the ground-motion models represented in the ground-motion sections of the logic tree by Monte Carlo simulation (separately for the median values and the sigma values) and then recombining the sets of simulated values in proportion to their logic-tree branch weights. The quantiles of this resampled composite model provide the hazard analyst and the decision maker with a simple, clear, and quantitative representation of the overall physical meaning of the ground-motion section of a logic tree and the accompanying epistemic uncertainty. Quantiles of the composite model also provide an easy way to analyze the sensitivities and uncertainties related to a given logic-tree model. We illustrate this for a composite ground- motion model for central Europe. Further potential fields of applications are seen wherever individual best estimates of ground motion have to be derived from a set of candidate models, for example, for hazard rnaps, sensitivity studies, or for modeling scenario earthquakes
The most recent intense earthquake swarm in the Vogtland lasted from 6 October 2008 until January 2009. Greatest magnitudes exceeded M3.5 several times in October making it the greatest swarm since 1985/86. In contrast to the swarms in 1985 and 2000, seismic moment release was concentrated near swarm onset. Focal area and temporal evolution are similar to the swarm in 2000. Work hypothysis: uprising upper-mantle fluids trigger swarm earthquakes at low stress level. To monitor the seismicity, the University of Potsdam operated a small aperture seismic array at 10 km epicentral distance between 18 October 2008 and 18 March 2009. Consisting of 12 seismic stations and 3 additional microphones, the array is capable of detecting earthquakes from larger to very low magnitudes (M<-1) as well as associated air waves. We use array techniques to determine properties of the incoming wavefield: noise, direct P and S waves, and converted phases.
In probabilistic seismic-hazard analysis, epistemic uncertainties are commonly treated within a logic-tree framework in which the branch weights express the degree of belief of an expert in a set of models. For the calculation of the distribution of hazard curves, these branch weights represent subjective probabilities. A major challenge for experts is to provide logically consistent weight estimates (in the sense of Kolmogorovs axioms), to be aware of the multitude of heuristics, and to minimize the biases which affect human judgment under uncertainty. We introduce a platform-independent, interactive program enabling us to quantify, elicit, and transfer expert knowledge into a set of subjective probabilities by applying experimental design theory, following the approach of Curtis and Wood (2004). Instead of determining the set of probabilities for all models in a single step, the computer-driven elicitation process is performed as a sequence of evaluations of relative weights for small subsets of models. From these, the probabilities for the whole model set are determined as a solution of an optimization problem. The result of this process is a set of logically consistent probabilities together with a measure of confidence determined from the amount of conflicting information which is provided by the expert during the relative weighting process. We experiment with different scenarios simulating likely expert behaviors in the context of knowledge elicitation and show the impact this has on the results. The overall aim is to provide a smart elicitation technique, and our findings serve as a guide for practical applications.
Aleatory variability in ground-motion prediction, represented by the standard deviation (sigma) of a ground-motion prediction equation, exerts a very strong influence on the results of probabilistic seismic-hazard analysis (PSHA). This is especially so at the low annual exceedance frequencies considered for nuclear facilities; in these cases, even small reductions in sigma can have a marked effect on the hazard estimates. Proper separation and quantification of aleatory variability and epistemic uncertainty can lead to defensible reductions in sigma. One such approach is the single-station sigma concept, which removes that part of sigma corresponding to repeatable site-specific effects. However, the site-to-site component must then be constrained by site-specific measurements or else modeled as epistemic uncertainty and incorporated into the modeling of site effects. The practical application of the single-station sigma concept, including the characterization of the dynamic properties of the site and the incorporation of site-response effects into the hazard calculations, is illustrated for a PSHA conducted at a rock site under consideration for the potential construction of a nuclear power plant.
The PEGASOS project was a major international seismic hazard study, one of the largest ever conducted anywhere in the world, to assess seismic hazard at four nuclear power plant sites in Switzerland. Before the report of this project has become publicly available, a paper attacking both methodology and results has appeared. Since the general scientific readership may have difficulty in assessing this attack in the absence of the report being attacked, we supply a response in the present paper. The bulk of the attack, besides some misconceived arguments about the role of uncertainties in seismic hazard analysis, is carried by some exercises that purport to be validation exercises. In practice, they are no such thing; they are merely independent sets of hazard calculations based on varying assumptions and procedures, often rather questionable, which come up with various different answers which have no particular significance. (C) 2005 Elsevier B.V. All rights reserved
Seismic-hazard assessment is of great importance within the field of engineering seismology. Nowadays, it is common practice to define future seismic demands using probabilistic seismic-hazard analysis (PSHA). Often it is neither obvious nor transparent how PSHA responds to changes in its inputs. In addition, PSHA relies on many uncertain inputs. Sensitivity analysis (SA) is concerned with the assessment and quantification of how changes in the model inputs affect the model response and how input uncertainties influence the distribution of the model response. Sensitivity studies are challenging primarily for computational reasons; hence, the development of efficient methods is of major importance. Powerful local (deterministic) methods widely used in other fields can make SA feasible, even for complex models with a large number of inputs; for example, automatic/algorithmic differentiation (AD)-based adjoint methods. Recently developed derivative-based global sensitivity measures can combine the advantages of such local SA methods with efficient sampling strategies facilitating quantitative global sensitivity analysis (GSA) for complex models. In our study, we propose and implement exactly this combination. It allows an upper bounding of the sensitivities involved in PSHA globally and, therefore, an identification of the noninfluential and the most important uncertain inputs. To the best of our knowledge, it is the first time that derivative-based GSA measures are combined with AD in practice. In addition, we show that first-order uncertainty propagation using the delta method can give satisfactory approximations of global sensitivity measures and allow a rough characterization of the model output distribution in the case of PSHA. An illustrative example is shown for the suggested derivative-based GSA of a PSHA that uses stochastic ground-motion simulations.
Probabilistic seismic-hazard analysis (PSHA) is the current tool of the trade used to estimate the future seismic demands at a site of interest. A modern PSHA represents a complex framework that combines different models with numerous inputs. It is important to understand and assess the impact of these inputs on the model output in a quantitative way. Sensitivity analysis is a valuable tool for quantifying changes of a model output as inputs are perturbed, identifying critical input parameters, and obtaining insight about the model behavior. Differential sensitivity analysis relies on calculating first-order partial derivatives of the model output with respect to its inputs; however, obtaining the derivatives of complex models can be challenging.
In this study, we show how differential sensitivity analysis of a complex framework such as PSHA can be carried out using algorithmic/automatic differentiation (AD). AD has already been successfully applied for sensitivity analyses in various domains such as oceanography and aerodynamics. First, we demonstrate the feasibility of the AD methodology by comparing AD-derived sensitivities with analytically derived sensitivities for a basic case of PSHA using a simple ground-motion prediction equation. Second, we derive sensitivities via AD for a more complex PSHA study using a stochastic simulation approach for the prediction of ground motions. The presented approach is general enough to accommodate more advanced PSHA studies of greater complexity.
Response spectra are of fundamental importance in earthquake engineering and represent a standard measure in seismic design for the assessment of structural performance. However, unlike Fourier spectral amplitudes, the relationship of response spectral amplitudes to seismological source, path, and site characteristics is not immediately obvious and might even be considered counterintuitive for high oscillator frequencies. The understanding of this relationship is nevertheless important for seismic-hazard analysis. The purpose of the present study is to comprehensively characterize the variation of response spectral amplitudes due to perturbations of the causative seismological parameters. This is done by calculating the absolute parameter sensitivities (sensitivity coefficients) defined as the partial derivatives of the model output with respect to its input parameters. To derive sensitivities, we apply algorithmic differentiation (AD). This powerful approach is extensively used for sensitivity analysis of complex models in meteorology or aerodynamics. To the best of our knowledge, AD has not been explored yet in the seismic-hazard context. Within the present study, AD was successfully implemented for a proven and extensively applied simulation program for response spectra (Stochastic Method SIMulation [SMSIM]) using the TAPENADE AD tool. We assess the effects and importance of input parameter perturbations on the shape of response spectra for different regional stochastic models in a quantitative way. Additionally, we perform sensitivity analysis regarding adjustment issues of groundmotion prediction equations.
The ellipticity of Rayleigh surface waves, which is an important parameter characterizing the propagation medium, is studied for several models with increasing complexity. While the main focus lies on theory, practical implications of the use of the horizontal to vertical component ratio (H/V-ratio) to Study the subsurface structure are considered as well. Love's approximation of the ellipticity for an incompressible layer over an incompressible half-space is critically discussed especially concerning its applicability for different impedance contrasts. The main result is an analytically exact formula of H/V for a 2-layer model of compressible media, which is a generalization of Love's formula. It turns out that for a limited range of models Love's approximation can be used also in the general case. (C) 2003 Elsevier B.V. All rights reserved
Empirical ground-motion models used in seismic hazard analysis are commonly derived by regression of observed ground motions against a chosen set of predictor variables. Commonly, the model building process is based on residual analysis and/or expert knowledge and/or opinion, while the quality of the model is assessed by the goodness-of-fit to the data. Such an approach, however, bears no immediate relation to the predictive power of the model and with increasing complexity of the models is increasingly susceptible to the danger of overfitting. Here, a different, primarily data-driven method for the development of ground-motion models is proposed that makes use of the notion of generalization error to counteract the problem of overfitting. Generalization error directly estimates the average prediction error on data not used for the model generation and, thus, is a good criterion to assess the predictive capabilities of a model. The approach taken here makes only few a priori assumptions. At first, peak ground acceleration and response spectrum values are modeled by flexible, nonphysical functions (polynomials) of the predictor variables. The inclusion of a particular predictor and the order of the polynomials are based on minimizing generalization error. The approach is illustrated for the next generation of ground-motion attenuation dataset. The resulting model is rather complex, comprising 48 parameters, but has considerably lower generalization error than functional forms commonly used in ground-motion models. The model parameters have no physical meaning, but a visual interpretation is possible and can reveal relevant characteristics of the data, for example, the Moho bounce in the distance scaling. In a second step, the regression model is approximated by an equivalent stochastic model, making it physically interpretable. The resulting resolvable stochastic model parameters are comparable to published models for western North America. In general, for large datasets generalization error minimization provides a viable method for the development of empirical ground-motion models.