Scaling relations for earthquake source parameters and magnitudes

A data set of 41 moderate and large earthquakes has been used to derive scaling rules for kinematic fault parameters. If effective stress and static stress drop are equal, then fault rise time, τ, and fault area, S, are related by τ = 16S^(1/2)/(7π^(3/2)β), where β is shear velocity. Fault length (parallel to strike) and width (parallel to dip) are empirically related by L=2W. Scatter for both scaling rules is about a factor of two. These scaling laws combine to give width and rise time in terms of fault length. Length is then used as the sole free parameter in a Haskell type fault model to derive scaling laws relating seismic moment to M_S (20-sec surface-wave magnitude), M_S to S and m_b (1-sec body-wave magnitude) to M_S. Observed data agree well with the predicted scaling relation. The “source spectrum” depends on both azimuth and apparent velocity of the phase or mode, so there is a different “source spectrum” for each mode, rather than a single spectrum for all modes. Furthermore, fault width (i.e., the two dimensionality of faults) must not be neglected. Inclusion of width leads to different average source spectra for surface waves and body waves. These spectra in turn imply that m_b and M_S reach maximum values regardless of further increases in L and seismic moment. The m_b versus M_S relation from this study differs significantly from the Gutenberg-Richter (G-R) relation, because the G-R equation was derived for body waves with a predominant period of about 5 sec and thus does not apply to modern 1-sec m_b determinations. Previous investigators who assumed that the G-R relation was derived from 1-sec data were in error. Finally, averaging reported rupture velocities yields the relation v_R = 0.72β

Time-domain observation and synthesis of split spheroidal and torsional free oscillations of the 1960 chilean earthquake: Preliminary results

The rotationally and elliptically split normal modes of the earth are observed for the 1960 Chilean earthquake by analysis in the time domain. One hundred and fifty hours of the Isabella, California, strain record are narrow band filtered about the central frequency of each split multiplet to isolate the complex wave form resulting from the interference of the different singlets. We compute synthetic seismograms using our previous theoretical results, which show the dependence of the amplitude and phase of the singlets on source location, depth, mechanism, and the position of the receiver. By comparing these synthetics to the filtered record, we conclusively demonstrate the splitting of modes whose splitting had not been definitely resolved: torsional modes (_0T_3, _0T_4) and spheroidal modes (_0S_4, _0S_5). The splitting of _0S_2 and _0S_3 is reconfirmed. We obtain good agreement between the synthetics and the filtered data for a source mechanism (previously determined from long-period surface waves) of thrust motion on a shallow dipping fault

Normal modes of a laterally heterogeneous body: A one-dimensional example

Various methods, including first- and second-order perturbation theory and variational methods have been proposed for calculating the normal modes of a laterally heterogeneous earth. In this paper, we test all three of these methods for a simple one-dimensional example for which the exact solution is available: an initially homogeneous “string” in which the density and stiffness are increased in one half and decreased in the other by equal amounts. It is found that first-order perturbation theory (as commonly applied in seismology) yields only the eigenvalues and eigenfunctions for a string with the average elastic properties; second-order perturbation theory is worse, because the eigenfunction is assumed to be the original eigenfunction plus small correction terms, but actually may be almost completely different. The variational method (Rayleigh-Ritz), using the unperturbed modes as trial functions, succeeds in giving correct eigenvalues and eigenfunctions even for modes of high-order number. For the example problem only the variational solution correctly yields the transient solution for excitation by a point force, including correct amplitudes for waves reflected by and transmitted through the discontinuity. Our result suggests but does not demonstrate, that the variational method may be the most appropriate method for finding the normal modes of a laterally heterogeneous earth model, particularly if the transient solution is desired

Attenuation measurements of split normal modes for the 1960 Chilean and 1964 Alaskan earthquakes

Measurements of attenuation for the Earth's longest period modes can be significantly biased by the effects of frequency splitting. Using our previously developed methods of time domain synthesis of split normal modes, we measure Q without such a bias. We also conduct numerical experiments to confirm the errors in Q measurements which result from neglecting the effects of splitting. In contrast to frequency domain this time domain technique allows us to reject data below the ambient noise level for each mode. The Q's of the longest period spheroidal (_0S_(2–0)S_5) and torsional (_0T_(3–0)T_4) modes are determined using long (500 hr) records from the Chilean and Alaskan earthquakes

Split free oscillation amplitudes for the 1960 Chilean and 1964 Alaskan earthquakes

Splitting of the Earth's normal modes was observed for both the 1960 Chilean and 1964 Alaskan earthquakes. The strong peaks in the observed spectrum of each split multiplet correspond to singlets with much higher amplitudes than the others. Using theoretical results we have derived elsewhere (Stein and Geller, 1977a), we are able to predict this pattern. We show that the source mechanisms inferred for these earthquakes from surface waves are consistent with the observed pattern of relative spectral amplitudes of the split modes. However other mechanisms, such as a slow isotropic volume change, are also consistent with the split-mode amplitudes and are excluded only by additional data

Shear-wave velocity at the base of the mantle from profiles of diffracted SH waves

Profiles of SH waves diffracted around the core (Sd) for three deep events at stations across North America and the Atlantic (Δ = 92° to 152°) are used to determine the properties of the lower mantle in the vicinity of the core-mantle boundary (CMB). The S-wave velocity above the CMB is found to be β_c = 7.22 ± 0.1 km/sec, in agreement with gross earth models, but higher than previously reported values from direct measurements of Sd. The frequency imdependence of the Sd ray parameter argues strongly against the possibility of a low-velocity zone immediately above the core mantle boundary. We compute synthetic seismograms for Sd by summing normal modes. A comparison of the present data with a synthetic profile for earth model 1066A gives excellent agreement at periods greater than 45 seconds. Synthetics for other models are used to substantially constrain the possibility of significant rigidity of the uppermost layer of the core

The ecological validity of the self-explanation effect: the deleterious effect of music on self-explanations

One of the most powerful ways to boost learning is to require students to self-explain—to generate written or verbal explanations of their study material as they are studying. Although self-explaining is known to enhance learning across a wide range of ages and study materials, this empirical work has focused almost exclusively on optimal study conditions. Here we explore if selfexplaining is similarly effective in the presence of background music, a distraction students commonly elect to incorporate into their study routines. In the first study, 32 university students were asked to learn about neuronal action potentials while we varied both self-explaining and the presence of loud background music. Results indicated self-explaining enhanced learning during silent study but actually impaired learning while listening to loud background music. To determine a threshold for this interaction, a second experiment was conducted (N=64) in which the music variable was manipulated at 4 levels: silent, quiet, moderate, and loud. We found increasing music volume impaired learning overall, and that this effect was particularly pronounced when students were instructed to self-explain. Overall, self-explaining is a powerful but potentially brittle learning technique, one which may not mesh well with common study habits