Ground Motion Occurrence Rates for Scenario Spectra

It is common practice for probabilistic seismic hazard analysis (PSHA) to use the uniform hazard spectra (UHS) to describe the ground motion. A short-coming of UHS is that it represents an envelope of many different earthquakes that control the hazard at different spectral periods. As an alternative, the UHS can be broken into a suite was developed by Baker and Cornell (2006), in which conditional mean spectra (CMS) are developed that represent realistic earthquake scenarios given a specified spectral acceleration at a single period. Using only the CMS as scenarios is too restrictive and does not provide enough scenarios to reproduce the hazard. The concept of the CMS is expanded to produce three scenario spectra for each CMS. The three scenarios represent the mean (the CMS) and two lower fractiles of the conditional spectra. Using these three scenario spectra for three difference spectral periods (0.2, 0.5, and 2.0 sec) and four different return periods (250, 500, 1000, and 2500 years) results in 36 scenario spectra. Rates for these 36 representative scenarios can be derived that approximate the hazard curves at all three spectral periods simultaneously. The advantage of the new approach over the CMS approach is that it provides rates of occurrence of the scenario spectra in addition to providing realistic scenario spectra. These scenario spectra with their associated rates of occurrence can be used in seismic risk calculations for estimating the probability of structural performance

Empirical estimation of high-frequency ground motion on hard rock

Site effects for hard-rock sites are typically computed using analytical models for the effect of κ0, the high-frequency attenuation parameter. New datasets that are richer in hard-rock recordings allow us to evaluate the scaling for hard-rock sites (e.g., VS30 > 1500 m=s). The high-frequency response spectra residuals are weakly correlated with κ0, in contrast to the strong scaling with κ0 in the analytical models. This may be due to site-specific shallow resonance patterns masking part of the effect of attenuation due to damping. An empirical model is developed for the combined VS30 and κ0 scaling for hard-rock sites relative to a reference site condition of 760 m=s (i.e., correction factors that should be used for going from soft rock to hard rock, taking into account the net effect of VS and κ0). This empirical model shows high-frequency amplification that is more similar to the analytical prediction corresponding to a hard-rock κ0 of 0.020 s rather than the typical value of 0.006 s, which is commonly used for hard-rock sites in the central–eastern United States. Compared to the current analytical approach, this leads to a reduction of high-frequency (>20 Hz) scaling of about a factor of 2

Taxonomy of κ: A review of definitions and estimation approaches targeted to applications

In a way perhaps not dissimilar to stress drop (Atkinson and Beresnev, 1997), the high-frequency attenuation parameter κ (kappa), introduced by Anderson and Hough (1984), is one of the most used yet least understood or agreed-upon parameters in engineering seismology. It describes the deviation at high frequencies between observed Fourier amplitude spectra calculated from seismograms and an ω−2 source model, such as the Brune (1970) model. Almost 30 years after its introduction, κ is used by seismologists and engineers alike and constitutes an important input parameter for several applications. Perhaps because of its importance, it is estimated, physically explained, and used in many different ways. This note aims to illustrate the multiple approaches to its estimation, and to suggest that, in order to reduce ambiguities, the parameter should always be given a notation consistent with its measurement and application to help avoid inconsistencies in its application of κ scaling to ground-motion models. Hanks (1982) observed that above a given frequency the acceleration spectrum decays sharply. He termed this frequency fmax (e.g., Fig. 1a) and attributed it mainly to local site conditions. Soon after, Anderson and Hough (1984) introduced an alternative parameter to model this decay, which is the one most commonly used today: κ. They measured κ directly from the high‐frequency part of the acceleration Fourier amplitude spectrum of a record. Above a certain frequency (which they named fe but we will call here f1), the overall shape of the spectrum generally decays exponentially with frequency; the decay constant is most easily measured by finding a linear approximation to the spectrum plotted in log–linear space. The slope of the linear approximation is −πκ (e.g., Fig. 1b). In this note we use the notation κr for individual observations of κ, for example, the κ value corresponding to the slope of a particular record; this record may be at any epicentral distance Re≥0. Anderson and Hough (1984) also observed that κr at individual stations increases with distance and concluded that it includes components related both to the local geology of the top few km of crust beneath the station and to the regional structure. They then suggested that the site component of κ (denoted κ0) could be computed by extrapolating the κr values to zero epicentral distance, thus correcting for the regional effect of anelastic Q. In this note, we discuss the use of κ0 in various engineering seismology applications today and why interest in this parameter has been revived. We briefly discuss its possible physical interpretations, and detail the known approaches to estimate κ0 from seismic records. We group these approaches into families according to basic features, such as the range of frequencies over which κ0 is computed and the trade‐off with path effects. We then discuss the alternative option for estimating κ0 when site‐specific records are not available, based on empirical correlations with VS30. We collect previously published correlations and demonstrate the scatter observed across different studies. Finally, we make suggestions as to how κ0 estimation can be made in a more consistent way with the applications that use it, and how existing correlations can be made more consistent to improve both the inference of κ0 in the absence of site‐specific data and the physical understanding of κ0

A Seismologically Consistent Surface Rupture Length Model for Unbounded and Width-Limited Event

A new surface-rupture-length ($SRL$) relationship as a function of magnitude ($\mathbf{M}$), fault thickness, and fault dip angle is presented in this paper. The objective of this study is to model the change in scaling between unbounded and width-limited ruptures. This is achieved through the use of seismological-theory-based relationships for the average displacement scaling and the aid of dynamic fault rupture simulations to constrain the rupture width scaling. The empirical dataset used in the development of this relationship is composed of $123$ events ranging from $\mathbf{M}~5$ to $8.1$ and $SRL~1.1$ to $432~km$. The dynamic rupture simulations dataset includes $554$ events ranging from $\mathbf{M}~4.9$ to $8.2$ and $SRL~1$ to $655~km$. For the average displacement ($\bar{D}$) scaling, three simple models and two composite models were evaluated. The simple average displacement models were: a square root of the rupture area ($\sqrt{A}$), a down-dip width ($W$), and a rupture length ($L$) proportional model. The two composite models followed a $\sqrt{A}$ scaling for unbounded ruptures and transitioned to $W$ and $L$ scaling for width-limited events, respectively. The empirical data favors a $\bar{D} \sim \sqrt{A}$ scaling for both unbounded and width-limited ruptures. The proposed model exhibits better predictive performance compared to linear $\log(SLR)\sim\mathbf{M}$ type models, especially in the large magnitude range, which is dominated by width-limited events. A comparison with existing $SRL$ models shows consistent scaling at different magnitude ranges that is believed to be the result of the different magnitude ranges in the empirical dataset of the published relationships.Comment: 21 pages, 11 figure

A methodology for the estimation of kappa (κ) for large datasets. Example application to rock sites in the NGA-East database

This report reviews four of the main approaches (two band-limited and two broadband) currently used for estimating the site κ0: the acceleration slope (AS) above the corner frequency, the displacement slope (DS) below the corner frequency, the broadband (BB) fit of the spectrum, and the response spectral shape (RESP) template. Using these four methods, estimates of κ0 for rock sites in Central Eastern North America (CENA) in the shallow crustal dataset from NGAEast are computed for distances less than 100 km. Using all of the data within 100 km, the mean κ0 values are 8 msec for the AS approach and 27 msec for the DS approach. These mean values include negative κ estimates for some sites. If the negative κ values are removed, then the mean values are 25 msec and 42 msec, respectively. Stacking all spectra together led to mean κ0 values of 7 and 29 msec, respectively. Overall, the DS approach yields 2–3 times higher values than the AS, which agrees with previous observations, but the uncertainty of the estimates in each case is large. The AS approach seems consistent for magnitudes down to M3 but not below. There is large within-station variability of κ that may be related to differences in distance, Q, complexity along the path, or particular source characteristics, such as higher or lower stress drop. The station-to-station differences may be due to site-related factors. Because most sites have been assigned Vs30 = 2000 m/sec, it is not possible to correlate variations in κ0 with rock stiffness. Based on the available profile, the individual spectra are corrected for crustal amplification and only affect results below 15 Hz. Since the AS and DS approaches are applied over different frequency ranges, we find that only the DS results are sensitive to the amplification correction. More detailed knowledge of individual near-surface profiles may have effects on AS results, too. Although κ is considered to be caused solely by damping in the shallow crust, measurement techniques often cannot separate the effects of damping and amplification, and yield the net effect of both phenomena. The two broadband approaches, BB and RESP, yield similar results. The mean κ0_BB is 5±0.5 msec across all NEHRP class A sites. The κ0_RESP for the two events examined is 5 and 6 msec. From literature, the average value of κ0 in CENA is 6 ± 2 msec. This typical value is similar to the broadband estimates of this study and to the mean κAS when all available recordings are used along with all flags. When only recordings with down-going FAS slope are selected from the dataset, the mean value of κAS increases by a factor of 2–3. To evaluate the scaling of high-frequency ground motion with κ, we analyze residuals from ground motion prediction equations (GMPEs) versus κ estimates. Using the κ values from the AS approach, the average trend of the ln(PSA) residuals for hard-rock data do not show the expected strong dependence on κ, but when using κ values from the DS approach, there is a stronger correlation of the residuals, i.e., a κ that is more consistent with the commonly used analytically based scaling. The κDS estimates may better reflect the damping in the shallow crust, while the κAS estimates may reflect a net effect of damping and amplification that has not been decoupled. The κDS estimates are higher than the κAS estimates, so the expected effect on the high-frequency ground motion is smaller than that expected for the κAS estimates. An empirical hard-rock site factor model is developed that represents the combined Vs-κ0 site factor relative to a 760 m/sec reference-site condition. At low frequencies ( 10 Hz), the residuals do not show the strong increase in the site factors as seen in the analytical model results. A second hard-rock dataset from British Columbia, Canada, is also used. This BC hard-rock residuals show an increase in the 15–50 Hz range that is consistent with the analytical κ0 scaling for a hard-rock κ0 of about 0.015 sec. The variability of the PSA residuals is also used to evaluate the κ0 scaling for hard-rock sites from analytical modeling. The scatter in existing κ0 values found in literature is disproportionately large compared to the observed variability in high-frequency ground motions. We compared the predicted ground-motion variability based on analytical modeling to the observed variability in our residuals. While the hard-rock sites are more variable at high frequencies due to the additional κ0 variability, this additional variability is much less than the variability predicted by the analytical modeling using the variability from κ0-Vs30 correlations. This is consistent with weaker κ0 scaling compared to that predicted by the analytical modelling seen in the mean residuals

Squeezing Kappa (κ) out of the transportable array: A strategy for using bandlimited data in regions of sparse seismicity

The κ parameter (Anderson and Hough, 1984), and namely its site-specific component (κ0), is important for predicting and simulating high-frequency ground motion. We develop a framework for estimating κ0 and addressing uncertainties under the challenging conditions often imposed in practice: 1. Low seismicity (limited, poor-quality, distant records); 2. Limited-bandwidth data from the Transportable Array (maximum usable frequency 16 Hz); 3. Low magnitudes (ML1.2-3.4) and large uncertainty in stress drop (corner frequency). We cannot resolve stress drop within the bandwidth, so we propose an approach that only requires upper and lower bounds on its regional values to estimate κ0. To address uncertainties, we combine three measurement approaches (acceleration spectrum slope, AS; displacement spectrum slope, DS; broadband spectral fit, BB). We also examine the effect of crustal amplification, and find that neglecting it can affect κ0 by up to 35%. DS estimates greatly exceed AS estimates. We propose a reason behind this bias, related to the residual effect of the corner frequency on κAS and κDS. For our region, we estimate a frequency-independent mean S-wave Q of 900±300 at 9-16 Hz, and an ensemble mean κ0 over all sites of 0.033±0.014 s. This value is similar to the native κ0 of the NGA-West2 ground motion prediction equations, indicating that these do not need to be adjusted for κ0 for use in Southern Arizona. We find that stress drop values in this region may be higher compared to estimates of previous studies, possibly due to trade-offs between stress drop and κ0. For this dataset, the within-approach uncertainty is much larger than the between-approach uncertainty, and it cannot be reduced if the data quality is not improved. The challenges discussed here will be relevant in studies of κ for other regions with band-limited data, e.g., any region where data come primarily from the TA

Selection of ground motion prediction equations for the global earthquake model

Ground motion prediction equations (GMPEs) relate ground motion intensity measures to variables describing earthquake source, path, and site effects. From many available GMPEs, we select those models recommended for use in seismic hazard assessments in the Global Earthquake Model. We present a GMPE selection procedure that evaluates multidimensional ground motion trends (e.g., with respect to magnitude, distance, and structural period), examines functional forms, and evaluates published quantitative tests of GMPE performance against independent data. Our recommendations include: four models, based principally on simulations, for stable continental regions; three empirical models for interface and in-slab subduction zone events; and three empirical models for active shallow crustal regions. To approximately incorporate epistemic uncertainties, the selection process accounts for alternate representations of key GMPE attributes, such as the rate of distance attenuation, which are defensible from available data. Recommended models for each domain will change over time as additional GMPEs are developed

Understanding the physics of kappa (κ): Insights from a downhole array

At high frequencies, the acceleration spectral amplitude decreases rapidly; this has been modelled with the spectral decay factor κ. Its site component, κ0, is used widely today in ground motion prediction and simulation, and numerous approaches have been proposed to compute it. In this study, we estimate κ for the EUROSEISTEST valley, a geologically complex and seismically active region with a permanent strong motion array consisting of 14 surface and 6 downhole stations. Site conditions range from soft sediments to hard rock. First, we use the classical approach to separate local and regional attenuation and measure κ0. Second, we take advantage of the existing knowledge of the geological profile and material properties to examine the correlation of κ0 with different site characterization parameters. κ0 correlates well with Vs30, as expected, indicating a strong effect from the geological structure in the upper 30 m. But it correlates equally well with the resonant frequency and depth-to-bedrock of the stations, which indicates strong effects from the entire sedimentary column, down to 400 m. Third, we use our results to improve our physical understanding of κ0. We propose a conceptual model of κ0 with Vs, comprising two new notions. On the one hand, and contrary to existing correlations, we observe that κ0 stabilizes for high Vs values. This may indicate the existence of regional values for hard rock κ0. If so, we propose that borehole measurements (almost never used up to now for κ0) may be useful in determining these values. On the other hand, we find that material damping, as expressed through travel times, may not suffice to account for the total κ0 measured at the surface. We propose that, apart from material damping, additional site attenuation may be caused by scattering from small-scale variability in the profile. If this is so, then geotechnical damping measurements may not suffice to infer the overall crustal attenuation under a site; but starting with a regional value (possibly from a borehole) and adding damping, we might define a lower bound for site-specific κ0. More precise estimates would necessitate seismological site instrumentation

Understanding single-station ground motion variability and uncertainty (sigma) – Lessons learnt from EUROSEISTEST

Accelerometric data from the well-studied valley EUROSEISTEST are used to investigate ground motion uncertainty and variability. We define a simple local ground motion prediction equation (GMPE) and investigate changes in standard deviation (σ) and its components, the between-event variability (τ) and within-event variability (φ). Improving seismological metadata significantly reduces τ (30-50%), which in turn reduces the total σ. Improving site information reduces the systematic site-to-site variability, φS2S (20-30%), in turn reducing φ, and ultimately, σ. Our values of standard deviations are lower than global values from literature, and closer to path-specific than site-specific values. However, our data have insufficient azimuthal coverage for single-path analysis. Certain stations have higher ground-motion variability, possibly due to topography, basin edge or downgoing wave effects. Sensitivity checks show that 3 recordings per event is a sufficient data selection criterion, however, one of the dataset’s advantages is the large number of recordings per station (9-90) that yields good site term estimates. We examine uncertainty components binning our data with magnitude from 0.01 to 2 s; at smaller magnitudes, τ decreases and φSS increases, possibly due to κ and source-site trade-offs Finally, we investigate the alternative approach of computing φSS using existing GMPEs instead of creating an ad hoc local GMPE. This is important where data are insufficient to create one, or when site-specific PSHA is performed. We show that global GMPEs may still capture φSS, provided that: 1. the magnitude scaling errors are accommodated by the event terms; 2. there are no distance scaling errors (use of a regionally applicable model). Site terms (φS2S) computed by different global GMPEs (using different site-proxies) vary significantly, especially for hard-rock sites. This indicates that GMPEs may be poorly constrained where they are sometimes most needed, i.e. for hard rock

Seismic risk assessment for developing countries : Pakistan as a case study

Modern Earthquake Risk Assessment (ERA) methods usually require seismo-tectonic information for Probabilistic Seismic Hazard Assessment (PSHA) that may not be readily available in developing countries. To bypass this drawback, this paper presents a practical event-based PSHA method that uses instrumental seismicity, available historical seismicity, as well as limited information on geology and tectonic setting. Historical seismicity is integrated with instrumental seismicity to determine the long-term hazard. The tectonic setting is included by assigning seismic source zones associated with known major faults. Monte Carlo simulations are used to generate earthquake catalogues with randomized key hazard parameters. A case study region in Pakistan is selected to demonstrate the effectiveness of the method. The results indicate that the proposed method produces seismic hazard maps consistent with previous studies, thus being suitable for generating such maps in regions where limited data are available. The PSHA procedure is developed as an integral part of an ERA framework named EQRAM. The framework is also used to determine seismic risk in terms of annual losses for the study region