TY - JOUR A1 - Molkenthin, Christian A1 - Scherbaum, Frank A1 - Griewank, Andreas A1 - Kühn, Nicolas A1 - Stafford, Peter J. A1 - Leovey, Hernan T1 - Sensitivity of Probabilistic Seismic Hazard Obtained by Algorithmic Differentiation: A Feasibility Study JF - Bulletin of the Seismological Society of America N2 - 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. Y1 - 2015 U6 - https://doi.org/10.1785/0120140294 SN - 0037-1106 SN - 1943-3573 VL - 105 IS - 3 SP - 1810 EP - 1822 PB - Seismological Society of America CY - Albany ER - TY - THES A1 - Molkenthin, Christian T1 - Sensitivity analysis in seismic Hazard assessment using algorithmic differentiation Y1 - 2016 ER - TY - JOUR A1 - Molkenthin, Christian A1 - Donner, Christian A1 - Reich, Sebastian A1 - Zöller, Gert A1 - Hainzl, Sebastian A1 - Holschneider, Matthias A1 - Opper, Manfred T1 - GP-ETAS: semiparametric Bayesian inference for the spatio-temporal epidemic type aftershock sequence model JF - Statistics and Computing N2 - The spatio-temporal epidemic type aftershock sequence (ETAS) model is widely used to describe the self-exciting nature of earthquake occurrences. While traditional inference methods provide only point estimates of the model parameters, we aim at a fully Bayesian treatment of model inference, allowing naturally to incorporate prior knowledge and uncertainty quantification of the resulting estimates. Therefore, we introduce a highly flexible, non-parametric representation for the spatially varying ETAS background intensity through a Gaussian process (GP) prior. Combined with classical triggering functions this results in a new model formulation, namely the GP-ETAS model. We enable tractable and efficient Gibbs sampling by deriving an augmented form of the GP-ETAS inference problem. This novel sampling approach allows us to assess the posterior model variables conditioned on observed earthquake catalogues, i.e., the spatial background intensity and the parameters of the triggering function. Empirical results on two synthetic data sets indicate that GP-ETAS outperforms standard models and thus demonstrate the predictive power for observed earthquake catalogues including uncertainty quantification for the estimated parameters. Finally, a case study for the l'Aquila region, Italy, with the devastating event on 6 April 2009, is presented. KW - Self-exciting point process KW - Hawkes process KW - Spatio-temporal ETAS model KW - Bayesian inference KW - Sampling KW - Earthquake modeling KW - Gaussian process KW - Data augmentation Y1 - 2022 U6 - https://doi.org/10.1007/s11222-022-10085-3 SN - 0960-3174 SN - 1573-1375 VL - 32 IS - 2 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Molkenthin, Christian A1 - Scherbaum, Frank A1 - Griewank, Andreas A1 - Leovey, Hernan A1 - Kucherenko, Sergei A1 - Cotton, Fabrice Pierre T1 - Derivative-Based Global Sensitivity Analysis: Upper Bounding of Sensitivities in Seismic-Hazard Assessment Using Automatic Differentiation JF - Bulletin of the Seismological Society of America N2 - 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. Y1 - 2017 U6 - https://doi.org/10.1785/0120160185 SN - 0037-1106 SN - 1943-3573 VL - 107 SP - 984 EP - 1004 PB - Seismological Society of America CY - Albany ER - TY - JOUR A1 - Gianniotis, Nikolaos A1 - Schnoerr, Christoph A1 - Molkenthin, Christian A1 - Bora, Sanjay Singh T1 - Approximate variational inference based on a finite sample of Gaussian latent variables JF - Pattern Analysis & Applications N2 - Variational methods are employed in situations where exact Bayesian inference becomes intractable due to the difficulty in performing certain integrals. Typically, variational methods postulate a tractable posterior and formulate a lower bound on the desired integral to be approximated, e.g. marginal likelihood. The lower bound is then optimised with respect to its free parameters, the so-called variational parameters. However, this is not always possible as for certain integrals it is very challenging (or tedious) to come up with a suitable lower bound. Here, we propose a simple scheme that overcomes some of the awkward cases where the usual variational treatment becomes difficult. The scheme relies on a rewriting of the lower bound on the model log-likelihood. We demonstrate the proposed scheme on a number of synthetic and real examples, as well as on a real geophysical model for which the standard variational approaches are inapplicable. KW - Bayesian inference KW - Posterior estimation KW - Expectation maximisation Y1 - 2016 U6 - https://doi.org/10.1007/s10044-015-0496-9 SN - 1433-7541 SN - 1433-755X VL - 19 SP - 475 EP - 485 PB - Springer CY - New York ER - TY - JOUR A1 - Molkenthin, Christian A1 - Scherbaum, Frank A1 - Griewank, Andreas A1 - Kuehn, Nicolas A1 - Stafford, Peter T1 - A Study of the sensitivity of response spectral amplitudes on seismological parameters using algorithmic differentiation JF - Bulletin of the Seismological Society of America N2 - 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. Y1 - 2014 U6 - https://doi.org/10.1785/0120140022 SN - 0037-1106 SN - 1943-3573 VL - 104 IS - 5 SP - 2240 EP - 2252 PB - Seismological Society of America CY - Albany ER - TY - JOUR A1 - Lontsi, Agostiny Marrios A1 - Garcia-Jerez, Antonio A1 - Camilo Molina-Villegas, Juan A1 - Jose Sanchez-Sesma, Francisco A1 - Molkenthin, Christian A1 - Ohrnberger, Matthias A1 - Krüger, Frank A1 - Wang, Rongjiang A1 - Fah, Donat T1 - A generalized theory for full microtremor horizontal-to-vertical [H/V(z,f)] spectral ratio interpretation in offshore and onshore environments JF - Geophysical journal international N2 - Advances in the field of seismic interferometry have provided a basic theoretical interpretation to the full spectrum of the microtremor horizontal-to-vertical spectral ratio [H/V(f)]. The interpretation has been applied to ambient seismic noise data recorded both at the surface and at depth. The new algorithm, based on the diffuse wavefield assumption, has been used in inversion schemes to estimate seismic wave velocity profiles that are useful input information for engineering and exploration seismology both for earthquake hazard estimation and to characterize surficial sediments. However, until now, the developed algorithms are only suitable for on land environments with no offshore consideration. Here, the microtremor H/V(z, f) modelling is extended for applications to marine sedimentary environments for a 1-D layered medium. The layer propagator matrix formulation is used for the computation of the required Green’s functions. Therefore, in the presence of a water layer on top, the propagator matrix for the uppermost layer is defined to account for the properties of the water column. As an application example we analyse eight simple canonical layered earth models. Frequencies ranging from 0.2 to 50 Hz are considered as they cover a broad wavelength interval and aid in practice to investigate subsurface structures in the depth range from a few meters to a few hundreds of meters. Results show a marginal variation of 8 per cent at most for the fundamental frequency when a water layer is present. The water layer leads to variations in H/V peak amplitude of up to 50 per cent atop the solid layers. KW - Numerical modelling KW - Earthquake hazards KW - Seismic interferometry KW - Site effects KW - Theoretical seismology KW - Wave propagation Y1 - 2019 U6 - https://doi.org/10.1093/gji/ggz223 SN - 0956-540X SN - 1365-246X VL - 218 IS - 2 SP - 1276 EP - 1297 PB - Oxford Univ. Press CY - Oxford ER -