A Study of the Sensitivity of Response Spectral Amplitudes on Seismological Parameters Using Algorithmic Differentiation

Abstract

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.

Sensitivity of Probabilistic Seismic Hazard Obtained by Algorithmic Differentiation: A Feasibility Study

“…The partial derivatives (equation 2) of the output of a complex model with respect to its inputs can be efficiently and accurately obtained via AD (Griewank et al, 2012): for example, in the case of a general circulation model for an ocean (Marotzke and Giering, 1999). An additional, more contextually relevant, example of applying this approach is given by Molkenthin et al (2014), in which sensitivities for predicted ground motions computed using the Stochastic Method SIMulation (SMSIM; Boore, 2003) technique are obtained. AD is a chain-rule-based method and has the advantage that it can, in principle, be applied directly to existing computer code of a model, irrespective of the model complexity.…”

“…Table 4 Sensitivities of the Annual Rate of Exceedance A computer implementation of the stochastic method for ground-motion simulation is SMSIM (Boore, 2003). AD has been successfully implemented for SMSIM based on an enhanced version of its computer source code in order to obtain sensitivities of response spectra (Molkenthin et al, 2014). For detailed information regarding AD of SMSIM the reader is referred to Molkenthin et al (2014).…”

“…AD has been successfully implemented for SMSIM based on an enhanced version of its computer source code in order to obtain sensitivities of response spectra (Molkenthin et al, 2014). For detailed information regarding AD of SMSIM the reader is referred to Molkenthin et al (2014).…”

“…This extremely powerful and flexible approach has been extensively used for performing sensitivity analyses of complex models in other domains such as oceanography (Marotzke and Giering, 1999) and aerodynamics (Gauger et al, 2008). Two of the first applications using AD in seismology are from Sambridge et al (2007) and Molkenthin et al (2014). A major advantage of AD is that it can be applied directly on the computer source code representation of a model.…”

“…We derive relative sensitivity coefficients for different input parameters (seismicity parameters as well as those related to the ground-motion model). This study builds on the work of Ordaz (2004), which provided closed-form solutions for simple cases of PSHA, and the study of Molkenthin et al (2014), which successfully implemented AD for a widely employed stochastic groundmotion simulation model in order to obtain sensitivities of response spectra using AD.…”