The nearly coincident forms of the relations between seismic moment M_0 and the magnitudes M_L, M_S, and M_w imply a moment magnitude scale M = ⅔ log M_0 - 10.7 which is uniformly valid for 3 ≲ M_L ≲ 7, 5 ≲ M_s ≲ 7½, and M_w ≳ 7½.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JB084iB05p02348%4010.1002/%28ISSN%292169-9356.FAULT1

A moment magnitude scale

Theoretical predictions of seismic motions as a function of source strength are often expressed as frequency-domain scaling models. The observations of interest to strong-motion seismology, however, are usually in the time domain (e.g., various peak motions, including magnitude). The method of simulation presented here makes use of both domains; its essence is to filter a suite of windowed, stochastic time series so that the amplitude spectra are equal, on the average, to the specified spectra. Because of its success in predicting peak and rms accelerations (Hanks and McGuire, 1981), an ω -squared spectrum with a high-frequency cutoff ( f m), in addition to the usual whole-path anelastic attenuation, and with a constant stress parameter (Δ σ ) has been used in the applications of the simulation method. With these assumptions, the model is particularly simple: the scaling with source size depends on only one parameter—seismic moment or, equivalently, moment magnitude. Besides peak acceleration, the model gives a good fit to a number of ground motion amplitude measures derived from previous analyses of hundreds of recordings from earthquakes in western North America, ranging from a moment magnitude of 5.0 to 7.7. These measures of ground motion include peak velocity, Wood-Anderson instrument response, and response spectra. The model also fits peak velocities and peak accelerations for South African earthquakes with moment magnitudes of 0.4 to 2.4 (with f m = 400 Hz and Δ σ = 50 bars, compared to f m = 15 Hz and Δ σ = 100 bars for the western North America data). Remarkably, the model seems to fit all essential aspects of high-frequency ground motions for earthquakes over a very large magnitude range .

Although the simulation method is useful for applications requiring one or more time series, a simpler, less costly method based on various formulas from random vibration theory will often suffice for applications requiring only peak motions. Hanks and McGuire (1981) used such an approach in their prediction of peak acceleration. This paper contains a generalization of their approach; the formulas used depend on the moments (in the statistical sense) of the squared amplitude spectra, and therefore can be applied to any time series having a stochastic character, including ground acceleration, velocity, and the oscillator outputs on which response spectra and magnitude are based .

Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra

Digital simulation of random processes and its applications

We have taken advantage of the recent increase in strong-motion data at close distances to derive new attenuation relations for peak horizontal acceleration and velocity. This new analysis uses a magnitude-independent shape, based on geometrical spreading and anelastic attenuation, for the attenuation curve. An innovation in technique is introduced that decouples the determination of the distance dependence of the data from the magnitude dependence. The resulting equations are log A = − 1.02 + 0.249 M − log r − 0.00255 r + 0.26 P r = ( d 2 + 7.3 2 ) 1 / 2 5.0 ≦ M ≦ 7.7 log V = − 0.67 + 0.489 M − log r − 0.00256 r + 0.17 S + 0.22 P r = ( d 2 + 4.0 2 ) 1 / 2 5.3 ≦ M ≦ 7.4 
where A is peak horizontal acceleration in g , V is peak horizontal velocity in cm/ sec, M is moment magnitude, d is the closest distance to the surface projection of the fault rupture in km, S takes on the value of zero at rock sites and one at soil sites, and P is zero for 50 percentile values and one for 84 percentile values. 

We considered a magnitude-dependent shape, but we find no basis for it in the data; we have adopted the magnitude-independent shape because it requires fewer parameters.

Peak horizontal acceleration and velocity from strong motion records including records from the 1979 Imperial Valley, California, earthquake

Peak horizontal acceleration and velocity from strong-motion records including records from the 1979 imperial valley, California, earthquake

In this study of the dynamics of building-soil interaction, the soil is modeled by a linear elastic half-space, and the building structure by an n-degree-of-freedom oscillator. Both earthquake response and steady-state response to sinusoidal excitation are examined. By assuming that the interaction system possesses n + 2 significant resonant frequencies, the response of the system is reduced to the superposition of the responses of damped linear oscillators subjected to modified excitations. The results are valid even though the interaction systems do not possess classical normal modes. For the special cases of single-story systems and the first modes of n-story systems, simplified approximate formulas are developed for the modified natural frequency and damping ratio, and for the modified excitation. Example calculations are carried out by the approximate and more exact analysis for one-story, two-story and ten-story interaction systems.

The results show that interaction tends to decrease all resonant frequencies, but that the effects are often significant only for the fundamental mode for many n-story structures and are more pronounced for rocking than for translation. If the fixed-base structure has damping, the effects of interaction on the earthquake responses are not always conservative, and an increase or decrease in the response can occur, depending on the parameters of the system.

Dynamics of building-soil interaction

Strong shaking of structures during large earthquakes may result in some cases in partial separation of the base of the structure from the foundation. A simplified problem of this type, the dynamic response of a rocking rigid block allowed to uplift, is examined here. Two foundation models are considered: the Winkler foundation and the much simpler ‘two-spring’ foundation. It is shown that an equivalence between these two models can be established, so that one can work with the much simpler two-spring foundation. Simple solutions of the equations of motion are developed and simplified methods of analysis are proposed. In general, uplift leads to a softer vibrating system which behaves non-linearly, although the response is composed of a sequence of linear responses. As a result the apparent rocking period increases with the amount of lift-off. The corresponding apparent ratio of critical damping decreases, in general, with the amplitude of the response. Compared to the case without lift-off, the response of the system may increase or decrease because of the uplift, depending on the excitation and the parameters of the system.

Rocking of slender rigid bodies allowed to uplift

Simulated earthquake motions suitable for design calculations are generated with the aid of a digital computer. The artificial accelerograms are sections of a random process with a prescribed power spectral density, multiplied by envelope functions chosen to model the changing intensity at the beginning and end of real accelerograms.
Two each of four different types of acce lerograms have been generated to represent ground acceleration for a variety of earthquakes. Type A models the acceleration in a Magnitude 8 shock and type B the motion expected in a Magnitude 7 earthquake. The shaking expected in the epicentral area of a Magnitude 5 or 6 earthquake is modeled by type C and the motion close to the fault in a shallow Magnitude 4 or 5 shock is represented by type D.
The two records of each type were generated to serve as the two horizontal components of earthquake shaking, or as two independent samples of possible ground shaking in the same type of event. Included in the report are accelerograms and derived velocity and displacement curves as well as response spectra.

Simulated earthquake motions

This report presents a numerical method for computing response spectra from strong-motion earthquake records. The method is based on the exact solution to the governing differential equation and gives a three to four-fold saving in computing time compared to a third order Runge-Kutta method of comparable accuracy. An analysis was made of the errors introduced at various stages in the calculation of spectra so that allowable errors could be prescribed for the numerical integration. Using the proposed method of computing or other methods of comparable accuracy, example calculations show that the errors introduced by the numerical procedures are much less than the errors inherent in the digitization of the acceleration record.

Included as appendices to the report are computer programs in Fortran IV, with instructions for their use, for computing spectra, for correction of the baseline of the digitized record, and for the computation of ground velocity and displacement.

Digital calculation of response spectra from strong-motion earthquake records

The problem of determining linear models of structures from seismic response data is investigated using ideas from the theory of system identification. The approach is to determine the optimal estimates of the model parameters by minimizing a selected measure-of-fit between the responses of the structure and the model. Because earthquake records are normally available from only a small number of locations in a structure, and because of noise in the records, it is necessary in practice to estimate parameters of the dominant modes in the records, rather than the stiffness and damping matrices of the linear model. A new algorithm is developed to determine the optimal estimates of the modal parameters. After tests with simulated data, the method is applied to a multi-storey building using records from the 1971 San Fernando earthquake in California. New information is obtained concerning the properties of the lower modes of the building and the time-varying character of the equivalent linear parameters.

Paul C. Jennings

Papers

Dynamics of building-soil interaction

Rocking of slender rigid bodies allowed to uplift

Simulated earthquake motions

Digital calculation of response spectra from strong-motion earthquake records

Structural identification using linear models and earthquake records