A parabolic approximation for surface waves

Summary. A parabolic approximation to the equation of motion of elastic waves as a sum of surface modes and discovering a parabolic approximation be applied directly to surface waves. The approximation depends on the material properties varying slowly within a wavelength, whereas surface waves may travel in a surface wave guide whose depth is of the same order of magnitude as a wavelength. This difficulty is overcome by representing the waves as a sum of surface modes and discovering a parabolic approximation for the amplitudes as a function of position on the surface. The theory is applicable to the propagation of Love or Rayleigh waves in a structure which is vertically stratified in an arbitrary way, but varies slowly in any horizontal direction.     

Reply to comment by J. R. Murphy on 'Q for short-period P waves: is it frequency dependent?' by A. Douglas

Using network averaged spectra Murphy (1989) demonstrates that if it is assumed that the source functions of explosions at Pahute Mesa, Nevada Test Site (NTS) are as predicted by the Mueller-Murphy (M-M) source model then the average t * at around 1 Hz for P waves radiated from the test site must be about 0.75 s. With this value of t * Murphy (1989) estimates the best fitting M-M spectrum for each explosion studied, by adjusting A, t _o & c ; A & t _o being the amplitude and delay time of pP relative to P and c the wave speed for the material in which the explosion was fired. The absolute amplitudes of the theoretical spectra are obtained using a calibration factor estimated from the data. Murphy (1993) extends the analysis to explosions in granite at the Nevada, French Sahara and E. Kazakh test sites. For the French Sahara explosion t * is assumed to be 0.75 s (Murphy's estimate for NTS explosions), and for the E. Kazakh explosion a t * of 0.55 s is used. For the French Sahara and E. Kazakh explosions Murphy (1993) shows that by using the same calibration factor as for the NTS it is possible by varying A & t _o to fit the estimated average network spectra using the M-M granite source. Murphy (1993) states that the amplitudes and spectra for the largest NTS explosion in granite (PILE DRIVER) can also be predicted using the M-M model but these results are not shown.     

Scattering of horizontally-polarized shear waves by surface irregularities

Summary. The problem of the scattering of harmonic SH waves by an arbitrary surface irregularity in an otherwise semi-infinite elastic, homogeneous, isotropic two-dimensional half-space is examined in this study in order to ascertain the effect of topography on this type of seismic ground motion and to develop a useful scheme which can realistically handle arbitrary two-dimensional topography. Three geometric models are considered: a semicircular hill which is of academic interest; a mountain with a Gaussian shape which utilizes realistic dimensions and the combination of a ridge and a depression that models a region in Sylmar, California.
A singular Fredholm integral equation of the second kind for the displacement at the free surface is developed and solved numerically. In the case of the semicircular hill, horizontal ground motion can be more than twice that occurring in the case of smooth topography. The mountain simulated by a Gaussian profile experiences at its crest amplifications for certain angles of incidence and de-amplifications for other angles of incidence, as well as displacements whose amplitudes vary slowly with frequency on the side of the mountain which is in the same direction as the incident waves. The ridge-depression combination which is approximated by a sixth-order polynomial actually experienced shattered earth at its ridge crest during the San Fernando, California earthquake of 1971. This amplification is also exhibited by the results of the analysis which, predicts amplifications of over 75 per cent at the top of the ridge for waves arriving on the same side as the ridge.     

Effects of slight anisotropy on surface waves

We present a complete ray theory for the calculation of surface-wave observables from anisotropic phase-velocity maps. Starting with the surface-wave dispersion relation in an anisotropic earth model, we derive practical dynamical ray-tracing equations. These equations allow calculation of the observables phase, arrival-angle and amplitude in a ray theoretical framework. Using perturbation theory, we also obtain approximate expressions for these observables. We assess the accuracy of the first-order approximations by using both theories to make predictions on a sample anisotropic phase-velocity map. A comparison of the two methods illustrates the size and type of errors which are introduced by perturbation theory. Perturbation theory phase and arrival-angle predictions agree well with the exact calculation, but amplitude predictions are poor. Many previous studies have modelled surface-wave propagation using only isotropic structure, not allowing for anisotropy. We present hypothetical examples to simulate isotropic modelling of surface waves which pass through anisotropic material. Synthetic data sets of phase and arrival angle are produced by ray tracing with exact ray theory on anisotropic phase-velocity maps. The isotropic models obtained by inverting synthetic anisotropic phase data sets produce deceptively high variance reductions because the effects of anisotropy are mapped into short-wavelength isotropic structure. Inversion of synthetic arrival-angle data sets for isotropic models results in poor variance reductions and poor recovery of the isotropic part of the anisotropic input map. Therefore, successful anisotropic phase-velocity inversions of real data require the inclusion of both phase and arrival-angle measurements.     

Reply to comment by J. L. Mateos, O. Novaro, T. H. Seligman and J. Flores on 'Can horizontal P waves be trapped and resonate in a shallow sedimentary basin?'

Using a complete mathematical formulation, we show that the trapping of horizontal P waves in a very soft shallow alluvial layer is a minor effect. These waves do not have a stable way of propagation since in order to exist they require an incident wave and are therefore incapable of resonating in the lateral direction when confined in a basin of limited extent.     

Scattering of surface waves modelled by the integral equation method

The integral equation method is used to model the propagation of surface waves in 3-D structures. The wavefield is represented by the Fredholm integral equation, and the scattered surface waves are calculated by solving the integral equation numerically. The integration of the Green's function elements is given analytically by treating the singularity of the Hankel function at   R = 0  , based on the proper expression of the Green's function and the addition theorem of the Hankel function. No far-field and Born approximation is made. We investigate the scattering of surface waves propagating in layered reference models imbedding a heterogeneity with different density, as well as Lamé constant contrasts, both in frequency and time domains, for incident plane waves and point sources.     

Comments on'The palaeomagnetism of the Central Zone of the Lewisian foreland, north-west Scotland'by J. D. A. Piper

Summary. Piper suggested that the Lewisian has rotated 30° anticlockwise since magnetization, whereas the opposite appears more likely. The main magnetization in the Lewisian recognized by Piper and Beckmann was imposed upon cooling after the Laxfordian metamorphism at about 1750 (± 50) Ma. The palaeomagnetic pole corresponding to this magnetization is at 37.6° N, 273.2° E ( dp = 3.7°, dm = 5.2°).
In Greenland, palaeomagnetic poles similar to each other, with a mean pole at 21.6° N, 280.1° E ( K = 52, A ₉₅= 9.4°), have been determined from five widely separated regions in central West Greenland and from Angmags-salik in East Greenland. The magnetization observed in all these regions was established upon cooling after the Nagssugtoqidian metamorphism, again at about 1750 (± 50) Ma.
The Laxfordian and Nagssugtoqidian metamorphisms were equivalent. It is therefore assumed that the two palaeomagnetic poles quoted above were originally identical. Their present difference can be explained by clockwise rotation of north-west Scotland about a local rotation pole since the Lewisian became magnetized, in addition to opening of the Atlantic assuming conventional reconstructions:
(1) assuming the reconstruction of Bullard, Everett & Smith, the local rotation proposed is 39.5° (± 18.1°) about a pole of rotation at 60.3° N, 354.5° E, or
(2) assuming the reconstruction of Le Pichon, Sibuet & Francheteau, the local rotation is 28.0° (±17.7°) about a pole of rotation at 54.1° N, 354.6° E.
These proposals of local clockwise rotation of north-west Scotland accord with that of Storetvedt based on palaeomagnetic results from Devonian rocks on the north-west side of the Great Glen Fault.