A finite difference equation formulation for the equations of elasticity is presented and applied to the problem of a layered half-space with a buried point source emitting a compressional pulse. Complete theoretical seismograms for the horizontal and vertical components of displacement are obtained. The results for a specific case are compared with those found by a completely different method in order to check the validity of the finite difference methods. The agreement is excellent. The effect of different mesh sizes on the theoretical seismograms is studied next and a suitable grid system selected for the applications that follow. The development of Rayleigh waves on the surface of a half-space and the change of the Rayleigh wave with depth and pulse width are examined. The problem of a layered half-space with a high velocity bottom is considered and the refraction arrivals on the surface and on the interface are studied. The problem of interface waves on the surface separating two semiinfinite media is also examined. Interface waves are found when the physical parameters lie both inside and outside the region determined by the Stoneley equation. Finally, a series of theoretical seismograms for a layered half-space showing the variation of the surface waves as a function of depth and of the density in the lower medium is presented.

Propagation of elastic waves in layered media by finite difference methods

SUMMARY We have used both single-station and array methods to determine shallow shear velocity site profiles in the vicinity of the city of Cologne, Germany from ambient vibration records. Based on fk-analysis we assume that fundamental-mode Rayleigh waves dominate the analysed wavefield in the frequency range of 0.7‐2.2 Hz. According to this view a close relation exists between H/V spectral ratios and the ellipticity of the contributing Rayleigh waves. The inversion of the shape of H/V spectral ratios then provides quantitative information concerning the local shear wave velocity structure. However, based on tests with synthetic data believed to represent a typical situation in the Lower Rhine Embayment, dispersion curves were found to provide stronger constraints on the absolute values of the velocity‐depth model than the ellipticities. The shape of the ellipticities was found to be subject to a strong trade-off between the layer thickness and the average layer velocity. We have made use of this observation by combining the inversion schemes for dispersion curves and ellipticities such that the velocity‐depth dependence is essentially constrained by the dispersion curves while the layer thickness is constrained by the ellipticities. In order to test this method in practice, we have used array recordings of ambient vibrations from three sites where the subsurface geology is fairly well known and geotechnical information is at least partially available. In order to keep the parameter space as simple as possible we attempted to fit only a single layer over a half-space model. However, owing to earlier studies from the region, we assume a power-law depth dependence for sediment velocities. For all three sites investigated, the inversion resulted in models for which the shear wave velocity within the sediment layer both in absolute value at the surface and in depth dependence are found to be remarkably similar to the results obtained by Budny from downhole measurements. This is strong support for the interpretation of H/V spectral ratios as Rayleigh wave ellipticities. For all three sites the predicted SH-wave site amplification factors at the fundamental frequency are of the order of 5‐6 with a slightly smaller value south of Cologne.

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Determination of shallow shear wave velocity profiles in the Cologne, Germany area using ambient vibrations

Early history of soil–structure interaction

A frequency-domain parametric study using generalized consistent transmitting boundaries has been performed to evaluate the significance of topographic effects on the seismic response of steep slopes. The results show that the peak amplification of motion at the crest of a slope occurs at a normalized frequency H /λ = 0.2, where H is the slope height and λ is the wavelength of the motion. The importance of the natural site frequency is illustrated by the analysis of a stepped layer over a half-space. It was found that the natural frequency of the region behind the crest can dominate the response, relative to the topographic effect, for the conditions studied. Moreover, the effect of topography can be handled separately from the amplification due to the natural frequency of the deposit behind the crest of the slope. This concept of separating the amplification caused by topography from that caused by the natural frequency is advantageous to the development of a simplified method to estimate topographic effects.

Topographic effects on the seismic response of steep slopes

Summary
The finite element method is shown to be a powerful tool for the numerical modelling of seismic body wave propagation problems. Applications extend to both problems on a scale of interest to engineers and also to large-scale seismological problems. Solutions are sought in the time domain. Efficient programs have been written to accomplish this. The scope of numerical solutions has been greatly enhanced by the use of a previously reported scheme for exactly cancelling reflections at the boundaries of the model.

The finite difference results of Boore and the analytical results of Trifunac for the amplification due to a mountain and an alluvial valley respectively are compared with new finite element results. The new results agree well, although there are some difficulties with resonance in the alluvial valley problem. Boore's SH results have been extended to vertical P and SV incidence. A deep earthquake zone has been modelled realistically in two dimensions and earthquakes simulated at depth. It is suggested that the variation in observed amplitude across the top of the zone, due to refraction away from the slab, may be used to provide an estimate of the thickness of the slab from long-period observations of local earthquakes.

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The Application of Finite Element Analysis to Body Wave Propagation Problems

We consider a version of Lamb9s Problem in which a vertical time-dependent point force acts on the surface of a uniform half-space. The resulting surface disturbance is computed as vertical and horizontal components of displacement, particle velocity, acceleration, and strain. The goal is to provide numerical solutions appropriate to a comparison with observed wave forms produced by impacts onto granite and onto soil. Solutions for step- and delta-function sources are not physically realistic but represent limiting cases. They show a clear P arrival (larger on horizontal than vertical components) and an obscure S arrival. The Rayleigh pulse includes a singularity at the theoretical arrival time. All of the energy buildup appears on the vertical components and all of the energy decay , on the horizontal components. The effects of Poisson9s ratio upon vertical displacements for a step-function source are shown. For fixed shear velocity, an increase of Poisson9s ratio produces a P pulse which is larger, faster, and more gradually emergent, an S pulse with more clear-cut beginning, and a much narrower Rayleigh pulse. For a source-time function given by cos 2 ( πt / T ), − T /2 ≦ T /2, a × 10 reduction in pulse width at fixed pulse height yields an increase in P and Rayleigh-wave amplitudes by factors of 1, 10, and 100 for displacement, velocity and strain, and acceleration, respectively. The observed wave forms appear somewhat oscillatory, with widths proportional to the source pulse width. The Rayleigh pulse appears as emergent positive on vertical components and as sharp negative on horizontal components. We show a theoretical seismic profile for granite, with source pulse width of 10 µsec and detectors at 10, 20, 30, 40, and 50 cm. Pulse amplitude decays as r −1 for P wave and r − 1 2 for Rayleigh wave. Pulse width broadens slightly with distance but the wave form character remains essentially unchanged.

Some numerical solutions for Lamb's problem

The dispersive characteristics of a single elastic layer overlying an elastic half-space are examined in detail for the fundamental and the first and second higher modes of Rayleigh waves. Phase velocity, group velocity, and the ratio of horizontal to vertical surface displacement are computed as functions of dimensionless quantities proportional to period and wave number. The significant range for the independent variable, B 1 T/H , proves to be largely independent of the parameters of the structure. The range is 1 to 20 for the fundamental, 0.3 to cutoff for the first higher mode, and 0.2 to cutoff for the second higher mode. The most important parameter of the structure for Rayleigh wave dispersion is the shear velocity ratio. Variations in the Poisson9s ratio in the surface layer and the density contrast may produce substantial effects. Poisson9s ratio in the half-space is of least significance. The dependence on model parameters of the long-period cutoff for the higher modes is determined. Specific results are given for the following geophysical examples: continental crust, continental ice cap, sedimentary basin, alluvial overburden, and laboratory seismic models.

Dispersive characteristics of the first three Rayleigh modes for a single surface layer

Certain travel time readings in shallow engineering seismology can be attributed to a seismic wave which has propagated through the air, but in which the energy has been transmitted into the ground before reaching the detector. Waveform evidence for this conclusion is presented, based upon the effects of altering the source conditions to delay the direct wave and of embedding the geophone in sound-absorbent material. Erroneously large depths may be computed if these air wave readings are interpreted as readings produced by direct waves through the ground. Potential difficulty exists only if velocity in the near-surface material is less than 330 meters/second (1100 feet/second).

Air waves in engineering seismology

Twelve earthquakes have been documented in Minnesota in the last 120 yr. The first nine were felt, whereas the last three (all in 1979) were detected instrumentally by a six-element seismic array which has recently been put into operation. Estimated magnitudes range from 0.1 (instrumental only) to 4.8, with four of magnitude 4.3 or greater. The highest intensity values were VI to VII. Depths where obtainable are estimated at 5 to 20 km. The best documented event occurred on 9 July 1975 near Morris, Minnesota, with a magnitude of 4.6, a maximum intensity of VI, and a felt area of 82,000 km 2 covering parts of four states. The event was recorded to epicentral distances of at least 38°. The epicenters show a clear relationship to tectonic features of the state. Four epicenters lie along the newly defined Great Lakes Tectonic Zone, an east-northeast-trending belt extending across several states and into Canada. The zone separates 3,000 to 3,600 m.y. rocks of a gneissic terrane to the south from 2,700 m.y. rocks of a greenstone-granite terrane to the north. Four other events lie on known major northwest-trending faults in the greenstone-granite terrane. Two and possibly three events are associated with the western margin of the Midcontinent Rift System.

Seismic history of Minnesota and its geological significance

A six-element, short-period circular array (diameter, 27 km) has been operated in east-central Minnesota since January 1977. The goal is to monitor low-level seismicity in the Upper Great Lakes Precambrian Shield Province. Special reference is given to the Midcontinent Gravity High and to a newly recognized tectonic hinge line across central Minnesota which has been the site of several earthquakes of magnitude up to 5. Data from the array are telemented 100 km to Minneapolis for tape recording. Flexible playback arrangements permit very compressed to very expanded time scales, using common time-base display with WWVB time code. Detection capability is estimated at m b = 2.0 within 100 km and 2.5 within 300 km, although three local events of m b = 1.5, 2.6, and 3.0 were clearly recorded well above noise level. A companion paper (Mooney and Morey, 1980) sets these events in the context of historical seismicity and regional tectonics. Based upon an analysis of hundreds of events, it is shown that the array is capable of determining azimuth for near-regional events within 1° to 4° and for teleseisms within 2° to 5°. Capability for determining distance from apparent horizontal slowness is less good but in the case of local and regional events, can be improved through measurements of S minus P time and R g minus P time.

Harold M. Mooney

Papers

Some numerical solutions for Lamb's problem

Dispersive characteristics of the first three Rayleigh modes for a single surface layer

Air waves in engineering seismology

Seismic history of Minnesota and its geological significance

The minnesota seismic network