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At least 289 records · Page 16Linked to original sources
USGS2014

Three-dimensional ground-motion simulations of earthquakes for the Hanford area, Washington

This report describes the results of ground-motion simulations of earthquakes using three-dimensional (3D) and one-dimensional (1D) crustal models conducted for the probabilistic seismic hazard assessment (PSHA) of the Hanford facility, Washington, under the Senior Seismic Hazard Analysis Committee (SSHAC) guidelines. The first portion of this report demonstrates that the 3D seismic velocity model for the area produces synthetic seismograms with characteristics (spectral response values, duration) that better match those of the observed recordings of local earthquakes, compared to a 1D model with horizontal layers. The second part of the report compares the response spectra of synthetics from 3D and 1D models for moment magnitude (M) 6.6–6.8 earthquakes on three nearby faults and for a dipping plane wave source meant to approximate regional S-waves from a Cascadia great earthquake. The 1D models are specific to each site used for the PSHA. The use of the 3D model produces spectral response accelerations at periods of 0.5–2.0 seconds as much as a factor of 4.5 greater than those from the 1D models for the crustal fault sources. The spectral accelerations of the 3D synthetics for the Cascadia plane-wave source are as much as a factor of 9 greater than those from the 1D models. The differences between the spectral accelerations for the 3D and 1D models are most pronounced for sites with thicker supra-basalt sediments and for stations with earthquakes on the Rattlesnake Hills fault and for the Cascadia plane-wave source.

Washington↗
USGS2025

Earthquake stress-drop values delineate spatial variations in maximum shear stress in the Japanese forearc lithosphere

Earthquake stress drop (Δσ) may increase with depth and stress in the brittle lithosphere. However, the range of uncertainty in Δσ and the lack of constraints on absolute stress make it difficult to establish whether they are correlated. Here, we investigate Δσ dependence on depth and maximum shear stress ( τ max ) based on ~11 years of seismicity in the northeastern Japanese forearc following the 2011 Tohoku-Oki megathrust earthquake. We interpret Δσ estimates computed using both individual spectra and spectral-ratio methods and find that Δσ exhibits a clear depth dependence within the seismically active upper ~60 km of the forearc lithosphere ( ~ 0.8 MPa per 10 km). We further compare Δσ values with quantitative τ max estimates from finite-element models of force balance. We find that median Δσ values increase with τ max in the brittle forearc lithosphere and that earthquake stress release is proportional to τ max . The dependence of Δσ on τ max explains the apparent depth dependence of Δσ and suggests that average Δσ values provide a relative measure of the stress at failure. In the northeastern Japanese forearc, Δσ values remained roughly constant in the decade following the Tohoku-Oki earthquake, suggesting negligible changes in failure stress in the forearc since the mainshock.

Communications Earth and Environment↗
USGS2026

Moment magnitude for small earthquakes in the Delaware basin of west Texas and southeast New Mexico, USA

The Delaware Basin region of west Texas and southeast New Mexico has become one of the most prolific regions of seismic activity in the continental United States due to widespread hydraulic fracturing and wastewater disposal injection. In response to the increased number of earthquakes in this region, rapid and accurate characterization of earthquake sources is necessary to understand the evolution of seismic activity and level of seismic hazard associated with these earthquakes. This study re-evaluates earthquake magnitudes, estimating moment magnitude (MW) for small earthquakes in the Delaware Basin using 1) moment-rate spectra derived from S-wave coda envelopes, and 2) a relative magnitude method that relies exclusively on the ratio of waveform amplitudes between highly correlated waveform pairs. The coda-envelope method produces accurate M W estimates for small earthquakes ( M 1.5 – 3) that are consistent with independent, waveform modeled moment magnitudes for events with M W > 3 . Using the relative amplitudes method to extend these M W magnitudes to many other events, we successfully provide relative moment magnitude ( M W,rel ) values for 81% of the Texas Seismological Network catalog in the Delaware Basin region, and 45% of the USGS Induced Seismicity Project’s catalog of events in southeast New Mexico. The adoption and integration of the calibrated M W,rel method with current magnitude estimation methods offers valuable insights into the relationships between local and moment magnitude and will contribute to improved characterization of widespread induced seismicity.

New Mexico, Texas↗
USGS2003

Empirical ground-motion relations for subduction-zone earthquakes and their application to Cascadia and other regions

Ground-motion relations for earthquakes that occur in subduction zones are an important input to seismic-hazard analyses in many parts of the world. In the Cascadia region (Washington, Oregon, northern California, and British Columbia), for example, there is a significant hazard from megathrust earthquakes along the subduction interface and from large events within the subducting slab. These hazards are in addition to the hazard from shallow earthquakes in the overlying crust. We have compiled a response spectra database from thousands of strong-motion recordings from events of moment magnitude ( M ) 5-8.3 occurring in subduction zones around the world, including both interface and in-slab events. The 2001 M 6.8 Nisqually and 1999 M 5.9 Satsop earthquakes are included in the database, as are many records from subduction zones in Japan (Kyoshin-Net data), Mexico (Guerrero data), and Central America. The size of the database is four times larger than that available for previous empirical regressions to determine ground-motion relations for subduction-zone earthquakes. The large dataset enables improved determination of attenuation parameters and magnitude scaling, for both interface and in-slab events. Soil response parameters are also better determined by the data.

British Columbia, California, Oregon, Washington↗
USGS2026

Efficient physics‐informed ground‐motion simulations with reduced‐order models: CyberShake implications and high‐resolution site terms for southern San Andreas fault earthquakes

Recent advances in Probabilistic Seismic Hazard Analysis (PSHA) leverage physics‐based ground‐motion simulations to estimate seismic hazard, such as the CyberShake project. However, computational costs quickly escalate when performing PSHA for numerous faults or sites and can become prohibitively expensive. To reduce computational demands, CyberShake uses reciprocity and interpolates physics‐informed corrections from simulations conducted at fewer locations, but the accuracy of these interpolations remains poorly quantified. To quantify the interpolation accuracy, we derive high‐resolution, frequency‐dependent site terms for southern California and compare them with interpolated site terms using the CyberShake approach. We accomplish this by performing a set of earthquake point‐source simulations distributed along the nonplanar fault geometry for the southern San Andreas fault (SSAF) extending from Bombay Beach to Lake Hughes. Using SeisSol, we simulate three minutes of viscoelastic seismic wave propagation for these sources and store the horizontal‐component Green’s functions for 480,000 sites. We then use a scientific machine learning approach based on interpolated proper orthogonal decomposition to construct an accurate reduced‐order model of the Green’s functions to efficiently predict effective amplitude spectra (EAS) for finite‐source rupture models of SSAF earthquakes. Using minimum curvature interpolation with tension, as used in CyberShake, we compare the interpolated site terms against our high‐resolution site terms. We identify local discrepancies with EAS differing by up to a factor of approximately three. Furthermore, we identify locations where unexpectedly high or low ground motions are missed when using the interpolated dataset for these earthquakes. We estimate that our approach may be used within CyberShake to reduce the time‐to‐solution by a factor of 336 for the entire earthquake rupture forecast. Our analysis of physics‐based site terms provides more insight into the seismic hazard due to SSAF ruptures and guides future developments by combining high‐performance computing and reduced‐order modeling techniques for PSHA.

California↗
USGS2013

Ground motions recorded in Rome during the April 2009 L’Aquila seismic sequence: site response and comparison with ground‐motion predictions based on a global dataset

The mainshock and moderate‐magnitude aftershocks of the 6 April 2009 M 6.3 L’Aquila seismic sequence, about 90 km northeast of Rome, provided the first earthquake ground‐motion recordings in the urban area of Rome. Before those recordings were obtained, the assessments of the seismic hazard in Rome were based on intensity observations and theoretical considerations. The L’Aquila recordings offer an unprecedented opportunity to calibrate the city response to central Apennine earthquakes—earthquakes that have been responsible for the largest damage to Rome in historical times. Using the data recorded in Rome in April 2009, we show that (1) published theoretical predictions of a 1 s resonance in the Tiber valley are confirmed by observations showing a significant amplitude increase in response spectra at that period, (2) the empirical soil‐transfer functions inferred from spectral ratios are satisfactorily fit through 1D models using the available geological, geophysical, and laboratory data, but local variability can be large for individual events, (3) response spectra for the motions recorded in Rome from the L’Aquila earthquakes are significantly amplified in the radial component at periods near 1 s, even at a firm site on volcanic rocks, and (4) short‐period response spectra are smaller than expected when compared to ground‐motion predictions from equations based on a global dataset, whereas the observed response spectra are higher than expected for periods near 1 s.

Rome↗
USGS2012

Spectral damping scaling factors for shallow crustal earthquakes in active tectonic regions

Ground motion prediction equations (GMPEs) for elastic response spectra, including the Next Generation Attenuation (NGA) models, are typically developed at a 5% viscous damping ratio. In reality, however, structural and non-structural systems can have damping ratios other than 5%, depending on various factors such as structural types, construction materials, level of ground motion excitations, among others. This report provides the findings of a comprehensive study to develop a new model for a Damping Scaling Factor (DSF) that can be used to adjust the 5% damped spectral ordinates predicted by a GMPE to spectral ordinates with damping ratios between 0.5 to 30%. Using the updated, 2011 version of the NGA database of ground motions recorded in worldwide shallow crustal earthquakes in active tectonic regions (i.e., the NGA-West2 database), dependencies of the DSF on variables including damping ratio, spectral period, moment magnitude, source-to-site distance, duration, and local site conditions are examined. The strong influence of duration is captured by inclusion of both magnitude and distance in the DSF model. Site conditions are found to have less significant influence on DSF and are not included in the model. The proposed model for DSF provides functional forms for the median value and the logarithmic standard deviation of DSF. This model is heteroscedastic, where the variance is a function of the damping ratio. Damping Scaling Factor models are developed for the “average” horizontal ground motion components, i.e., RotD50 and GMRotI50, as well as the vertical component of ground motion.

PEER Report↗
USGS1997

Estimating the diminution of shear-wave amplitude with distance: Application to the Los Angeles, California, urban area

The rate of decay with distance of shear-wave amplitude, computed from 20-sec S -wave spectra, is determined from TERRA scope records of small earthquakes in the greater Los Angeles area. Piecewise log-linear interpolation functions and traditional diminution functions are used to fit spectral decay to a maximum distance of 150 km. Simultaneously, isotropic source and receiver terms are determined. Separate branches of the spectral decay function are found for two categories of source depth: greater than 10 km and less than 10 km. In the hypocentral distance range of 20 to 150 km and in the frequency range of 0.5 to 8.0 Hz, an important result of the investigation is that the horizontal-component decay rate associated with deeper-crustal sources is generally greater than that associated with shallower sources and is greater than that which is estimated using more traditional models of spectral decay with distance. The same behavior generally holds for vertical-component spectra. The variation in apparent attenuation rate with source depth should affect seismic-hazard estimates associated with the rupture of blind thrust faults in the Los Angeles basin and vicinity. The results of the inversions suggest that interpolation function representations of spectral decay are sensitive to perturbations of S -wave amplitude due to crustal reflectors, such as post-critical S -wave arrivals from mid-crustal to deep-crustal velocity interfaces.

California↗
USGS1982

Estimation of response-spectral values as functions of magnitude, distance, and site conditions

We have developed empirical predictive equations for the horizontal pseudo-velocity response at 5-percent damping for 12 different periods from 0.1 to 4.0 s. Using a multiple linear-regression method similar to the one we used previously for peak horizontal acceleration and velocity, we analyzed response spectra period by period for 64 records of 12 shallow earthquakes in western North America, including the recent Coyote Lake and Imperial Valley, California, earthquakes. The resulting predictive equations show amplification of the response values at soil sites for periods greater than or equal to 0.5 s, with maximum amplification exceeding a factor of 2 at 1.5 s. For periods less than 0.5 s there is no statistically significant difference between rock sites and the soil sites represented in the data set. These results are consistent with those of several earlier studies. A particularly significant aspect of the predictive equations is that the response values at different periods are different functions of magnitude (confirming earlier results by McGuire and by Trifunac and Anderson). The slope of the least-squares straight line relating log response to moment magnitude ranges from 0.21 at a period of 0.1 s to greater than 0.5 at periods of 1 s and longer. This result indicates that the conventional practice of scaling a constant spectral shape by peak acceleration will not give accurate answers. The Newmark and Hall method of spectral scaling, using both peak acceleration and peak velocity, largely avoids this error. Comparison of our spectra with the Regulatory Guide 1.60 spectrum anchored at the same value at 0.1 s shows that the Regulatory Guide 1.60 spectrum is exceeded at soil sites for a magnitude of 7.5 at all distances for periods greater than about 0.5 s. Comparison of our spectra for soil sites with the corresponding ATC-3 curve of lateral design force coefficients for the highest seismic zone indicates that the ATC-3 curve is exceeded within about 5 km of a magnitude 6.5 earthquake and within about 20 km of a magnitude 7.5 event. The amount by which it is exceeded is largest in the period range from 0.5 to 2.0 s.

Open-File Report↗
USGS2018

Regional spectral analysis of moderate earthquakes in northeastern North America—Final Report to the Nuclear Regulatory Commission, Project V6240, Task 3

We analyze the Fourier spectra of S + Lg +surface wave groups from the horizontal and vertical components of broadband and accelerogram recordings of 120 small and moderate (2< M w <6) earthquakes recorded by Canadian and American stations sited on rock at distances from 3 to 600 kilometers. There are seven M w 4.0–4.5, six M w 4.5–5.0, and three M w ≥5 earthquakes in this event set. We test the regional spectral analysis by comparing the moment magnitudes with the moment magnitudes from the earthquake moment tensors determined by Bob Herrmann (St. Louis University) for 27 events, obtaining d M w =0.004±0.074. We determine the Lg attenuation in seven regions within northeastern North America: Charlevoix, lower St. Lawrence, Maine, Northern New York, lower Great Lakes, Ontario, and Nunavut. These attenuation estimates yield an average attenuation Q = (368±13) f (0.54±0.02) for the Appalachian region, a stronger attenuation Q = (317±16) f (0.54±0.03) for the Appalachian lowlands, and a weaker attenuation Q=(455±20) f (0.51±0.02) for Ontario and western Quebec. For events in Nunavut and northernmost Quebec, we estimate a similar attenuation for r <450 km, but a weaker attenuation Q = (773±70) f (0.27±0.06) for Lg propagation for 450< r <1700 kilometers. This far-regional attenuation allows us to analyze recordings of the 1989 Ungava and Payne Bay earthquakes obtained in Ontario and southern Quebec. We use these regional attenuations to determine the corner frequencies, stress drops, and radiated energies of the 120 earthquakes.

Open-File Report↗
USGS2017

Source spectral properties of small-to-moderate earthquakes in southern Kansas

The source spectral properties of injection-induced earthquakes give insight into their nucleation, rupture processes, and influence on ground motion. Here we apply a spectral decomposition approach to analyze P -wave spectra and estimate Brune-type stress drop for more than 2000 M L 1.5–5.2 earthquakes occurring in southern Kansas from 2014 to 2016. We find that these earthquakes are characterized by low stress drop values (median ∼0.4MPa) compared to natural seismicity in California. We observe a significant increase in stress drop as a function of depth, but the shallow depth distribution of these events is not by itself sufficient to explain their lower stress drop. Stress drop increases with magnitude from M1.5–M3.5, but this scaling trend may weaken above M4 and also depends on the assumed source model. Although we observe a nonstationary, sequence-specific temporal evolution in stress drop, we find no clear systematic relation with the activity of nearby injection wells.

Kansas↗
USGS2023

Quantifying site effects and their influence on earthquake source parameter estimations using a dense array in Oklahoma

We investigate the effects of site response on source parameter estimates using earthquakes recorded by the LArge-n Seismic Survey in Oklahoma (LASSO). While it is well known that near-surface unconsolidated sediments can cause an apparent breakdown of earthquake self-similarity, the influence of laterally varying site conditions remains unclear. We analyze site conditions across the 1825-station array on a river plain within an area of 40 km by 23 km using vertical ground motions from 14 regional earthquakes. While the source radiation pattern controls P-wave ground motions below 8 Hz, the surface geology correlates with P-wave ground motions above 8 Hz and S-wave ground motions at 2–21 Hz. Stations installed in alluvial sediments have vertical ground motions that can exceed three times the array median. We use the variation of ground motion of regional earthquakes across the array as a proxy for site effects. The corner frequencies and stress drops of local earthquakes ( M L = 0.01–3) estimated using a standard single-spectra approach show negative correlations with the site-effect proxy, while the seismic moments show positive correlations. In contrast, the spectral-ratio approach effectively shows no correlation. The overall bias is small as expected for this relatively homogeneous structure; accurate estimation of site-related biases requires at least 30 stations. Correcting for site-related biases reduces the standard deviations of the source parameters by less than 13% of the total variations. Remaining variations are partially associated with source directivity and model misfits— as small earthquakes can have complex ruptures.

Oklahoma↗
USGS2012

Implications of NGA for NEHRP site coefficients

Three proposals are provided to update tables 11.4-1 and 11.4-2 of Minimum Design Loads for Buildings and Other Structures (7-10), by the American Society of Civil Engineers (2010) (ASCE/SEI 7-10), with site coefficients implied directly by NGA (Next Generation Attenuation) ground motion prediction equations (GMPEs). Proposals include a recommendation to use straight-line interpolation to infer site coefficients at intermediate values of ̅v s (average shear velocity). Site coefficients are recommended to ensure consistency with ASCE/SEI 7-10 MCE R (Maximum Considered Earthquake) seismic-design maps and simplified site-specific design spectra procedures requiring site classes with associated tabulated site coefficients and a reference site class with unity site coefficients. Recommended site coefficients are confirmed by independent observations of average site amplification coefficients inferred with respect to an average ground condition consistent with that used for the MCE R maps. The NGA coefficients recommended for consideration are implied directly by the NGA GMPEs and do not require introduction of additional models.

Open-File Report↗
USGS2009

Comparing stochastic point-source and finite-source ground-motion simulations: SMSIM and EXSIM

Comparisons of ground motions from two widely used point-source and finite-source ground-motion simulation programs (SMSIM and EXSIM) show that the following simple modifications in EXSIM will produce agreement in the motions from a small earthquake at a large distance for the two programs: (1) base the scaling of high frequencies on the integral of the squared Fourier acceleration spectrum; (2) do not truncate the time series from each subfault; (3) use the inverse of the subfault corner frequency for the duration of motions from each subfault; and (4) use a filter function to boost spectral amplitudes at frequencies near and less than the subfault corner frequencies. In addition, for SMSIM an effective distance is defined that accounts for geometrical spreading and anelastic attenuation from various parts of a finite fault. With these modifications, the Fourier and response spectra from SMSIM and EXSIM are similar to one another, even close to a large earthquake (M 7), when the motions are averaged over a random distribution of hypocenters. The modifications to EXSIM remove most of the differences in the Fourier spectra from simulations using pulsing and static subfaults; they also essentially eliminate any dependence of the EXSIM simulations on the number of subfaults. Simulations with the revised programs suggest that the results of Atkinson and Boore (2006), computed using an average stress parameter of 140 bars and the original version of EXSIM, are consistent with the revised EXSIM with a stress parameter near 250 bars.

Bulletin of the Seismological Society of America↗
USGS2013

Vs30 and spectral response from collocated shallow, active- and passive-source Vs data at 27 sites in Puerto Rico

Shear‐wave velocity (V S ) and time‐averaged shear‐wave velocity to 30 m depth (V S30 ) are the key parameters used in seismic site response modeling and earthquake engineering design. Where V S data are limited, available data are often used to develop and refine map‐based proxy models of V S30 for predicting ground‐motion intensities. In this paper, we present shallow V S data from 27 sites in Puerto Rico. These data were acquired using a multimethod acquisition approach consisting of noninvasive, collocated, active‐source body‐wave (refraction/reflection), active‐source surface wave at nine sites, and passive‐source surface‐wave refraction microtremor (ReMi) techniques. V S ‐versus‐depth models are constructed and used to calculate spectral response plots for each site. Factors affecting method reliability are analyzed with respect to site‐specific differences in bedrock V S and spectral response. At many but not all sites, body‐ and surface‐wave methods generally determine similar depths to bedrock, and it is the difference in bedrock V S that influences site amplification. The predicted resonant frequencies for the majority of the sites are observed to be within a relatively narrow bandwidth of 1–3.5 Hz. For a first‐order comparison of peak frequency position, predictive spectral response plots from eight sites are plotted along with seismograph instrument spectra derived from the time series of the 16 May 2010 Puerto Rico earthquake. We show how a multimethod acquisition approach using collocated arrays compliments and corroborates V S results, thus adding confidence that reliable site characterization information has been obtained.

Bulletin of the Seismological Society of America↗
USGS2019

Variability in synthetic earthquake ground motions caused by source variability and errors in wave propagation models

Numerical simulations of earthquake ground motions are used both to anticipate the effects of hypothetical earthquakes by forward simulation and to infer the behaviour of the real earthquake source ruptures by the inversion of recorded ground motions. In either application it is necessary to assume some Earth structure that is necessarily inaccurate and to use a computational method that is also inaccurate for simulating the wavefield Green's functions. We refer to these two sources of error as ‘propagation inaccuracies’, which might be considered to be epistemic. We show that the variance of the Fourier spectrum of the synthetic earthquake seismograms caused by propagation inaccuracies is related to the spatial covariance on the rupture surface of errors in the computed Green's functions, which we estimate for the case of the 2009 L'Aquila, Italy, earthquake by comparing erroneous computed Green's functions with observed L'Aquila aftershock seismograms (empirical Green's functions). We further show that the variance of the synthetic seismograms caused by the rupture variability (aleatory uncertainty) is related to the spatial covariance on the rupture surface of aleatory variations in the rupture model, and we investigate the effect of correlated variations in Green's function errors and variations in rupture models. Thus, we completely characterize the variability of synthetic earthquake seismograms induced by errors in propagation and variability in the rupture behaviour. We calculate the spectra of the variance of the ground motions of the L'Aquila main shock caused by propagation inaccuracies for two specific broad-band stations, the AQU and the FIAM stations. These variances are distressingly large, being comparable or in some cases exceeding the data amplitudes, suggesting that the best-fitting L'Aquila rupture model significantly overfits the data and might be seriously in error. If these computed variances are typical, the accuracy of many other rupture models for past earthquakes may need to be reconsidered. The results of this work might be useful in seismic hazard estimation because the variability of the computed ground motion, caused both by propagation inaccuracies and variations in the rupture model, can be computed directly, not requiring laborious consideration of multiple Earth structures.

Geophysical Journal International↗
USGS2006

Analytical approach to calculation of response spectra from seismological models of ground motion

An analytical approach to calculate response spectra from seismological models of ground motion is presented. Seismological models have three major advantages over empirical models: (1) they help in an understanding of the physics of earthquake mechanisms, (2) they can be used to predict ground motions for future earthquakes and (3) they can be extrapolated to cases where there are no data available. As shown with this study, these models also present a convenient form for the calculation of response spectra, by using the methods of random vibration theory, for a given magnitude and site conditions. The first part of the paper reviews the past models for ground motion description, and introduces the available seismological models. Then, the random vibration equations for the spectral response are presented. The nonstationarity, spectral bandwidth and the correlation of the peaks are considered in the calculation of the peak response.

Earthquake Engineering and Structural Dynamics↗
USGS1994

Regional propagation characteristics and source parameters of earthquakes in northeastern North America

The vertical components of the S wave trains recorded on the Eastern Canadian Telemetered Network ( ECTN ) from 1980 through 1990 have been spectrally analyzed for source, site, and propagation characteristics. The data set comprises some 1033 recordings of 97 earthquakes whose magnitudes range from M ≈ 3 to 6. The epicentral distances range from 15 to 1000 km, with most of the data set recorded at distances from 200 to 800 km. The recorded S wave trains contain the phases S , S m S , S n , and L g and are sampled using windows that increase with distance; the acceleration spectra were analyzed from 1.0 to 10 Hz. To separate the source, site, and propagation characteristics, an inversion for the earthquake corner frequencies, low-frequency levels, and average attenuation parameters is alternated with a regression of residuals onto the set of stations and a grid of 14 distances ranging from 25 to 1000 km. The iteration between these two parts of the inversion converges in about 60 steps. The average attenuation parameters obtained from the inversion were Q = 1997 ± 10 and γ = 0.998 ± 0.003. The most pronounced variation from this average attenuation is a marked deamplification of more than a factor of 2 at 63 km and 2 Hz, which shallows with increasing frequency and increasing distance out to 200 km. The site-response spectra obtained for the ECTN stations are generally flat. The source spectral shape assumed in this inversion provides an adequate spectral model for the smaller events ( M o < 3 × 10 21 dyne-cm) in the data set, whose Brune stress drops range from 5 to 150 bars. For the five events in the data set with M o ≧ 10 23 dyne-cm, however, the source spectra obtained by regressing the residuals suggest that an ω 2 spectrum is an inadequate model for the spectral shape. In particular, the corner frequencies for most of these large events appear to be split, so that the spectra exhibit an intermediate behavior (where | ü ( ω )| is roughly proportional to ω ).

Bulletin of the Seismological Society of America↗