Radiation Pattern Of Rayleigh Waves Research Articles

Hypothesis tests on Rayleigh wave radiation pattern shapes: a theoretical assessment of idealized source screening

SUMMARYShallow seismic sources excite Rayleigh wave ground motion with azimuthally dependent radiation patterns. We place binary hypothesis tests on theoretical models of such radiation patterns to screen cylindrically symmetric sources (like explosions) from non-symmetric sources (like non-vertical dip-slip or non-VDS faults). These models for data include sources with several unknown parameters, contaminated by Gaussian noise and embedded in a layered half-space. The generalized maximum likelihood ratio tests that we derive from these data models produce screening statistics and decision rules that depend on measured, noisy ground motion at discrete sensor locations. We explicitly quantify how the screening power of these statistics increase with the size of any dip-slip and strike-slip components of the source, relative to noise (faulting signal strength) and how they vary with network geometry. As applications of our theory, we apply these tests to (1) find optimal sensor locations that maximize the probability of screening non-circular radiation patterns and (2) invert for the largest non-VDS faulting signal that could be mistakenly attributed to an explosion with damage, at a particular attribution probability. Finally, we quantify how certain errors that are sourced by opening cracks increase screening rate errors. While such theoretical solutions are ideal and require future validation, they remain important in underground explosion monitoring scenarios because they provide fundamental physical limits on the discrimination power of tests that screen explosive from non-VDS faulting sources.

A Pseudo-3D Model for Electromagnetic Acoustic Transducers (EMATs)

Previous methods for modelling Rayleigh waves produced by a meander-line-coil electromagnetic acoustic transducer (EMAT) consisted mostly of two-dimensional (2D) simulations that focussed on the vertical plane of the material. This paper presents a pseudo-three-dimensional (3D) model that extends the simulation space to both vertical and horizontal planes. For the vertical plane, we combines analytical and finite-difference time-domain (FDTD) methods to model Rayleigh waves’ propagation within an aluminium plate and their scattering behaviours by cracks. For the horizontal surface plane, we employ an analytical method to investigate the radiation pattern of Rayleigh waves at various depths. The experimental results suggest that the models and the modelling techniques are valid.

A Novel Variable-Length Meander-Line-Coil EMAT for Side Lobe Suppression

Typically, meander-line-coil electromagnetic acoustic transducers (EMATs) with wires of a constant length are used to generate Rayleigh waves. However, significant side lobes are observed in this design. In this paper, a novel meander-line-coil with variable-length wires is proposed to suppress side lobes of the Rayleigh waves’ beam. This novel EMAT is simulated by using a wholly analytical method, which involves the coupling of an analytical EM model and an analytical US model to generate the radiation pattern of Rayleigh waves. The Rayleigh waves’ beam directivity of this novel variable-length meander-line-coil (VLMLC) is quantitatively analyzed by the half power beamwidth and the sidelobe level; the effect of the length step on the Rayleigh waves’ beam directivity is discussed. A 50-mm VLMLC with a length step of 8 mm is fabricated and experimentally analyzed; experimental results showed that novel VLMLCs are capable of suppressing side lobes of the Rayleigh waves’ beam, which is not achievable by the conventional meander-line-coil. To the best of our knowledge, there has been no report so far of the VLMLCs EMAT.

Dichotomy in mode propagation of coseismic ionospheric disturbance: Observations from 11 April 2012 Indian Ocean earthquake

AbstractThe ionosphere response to the great intraplate Indian Ocean earthquake of 11 April 2012 (Mw 8.6) and its largest aftershock (Mw 8.2) is analyzed using GPS‐aided total electron content (TEC) measurements. Data from the dense GPS networks, SuGAr (Sumatran GPS Array) and the permanent Andaman‐Nicobar array, formed the near‐field observations at distances 250–1200 km from the epicenter. Stations such as IISC, DGAR, and few others provided measurements over 2000 km from the epicenter. The coseismic ionospheric disturbances (CIDs) with a propagation velocity of 930–1262 m/s, equals the speeds of the shock acoustic waves, arrive within 10–18 min after the earthquake occurrence. The observed phenomenon of CID splitting into two modes, north and south of the epicenter, is akin to the well‐documented effects of anisotropy on wave propagation. Closer to the epicenter, to its south, the propagation velocity of CID is ~1 km/s, and farther southeast of the network the velocity reduces to 500–600 m/s. In contrast, toward Andaman in the north, the CID propagation velocity increases to 2–3.5 km/s. The zenith angle of the line of sight between the GPS receiver and satellite appears to influence the amplitude of the TEC fluctuations. The anomalous azimuthal variation of the Rayleigh wave radiation pattern best explains the observed N‐S asymmetry of CID.

Location and mechanism of very long period tremor during the 2008 eruption of Okmok Volcano from interstation arrival times

We describe continuous, very long period (VLP) tremor that occurred during the 2008 eruption of Okmok Volcano, Alaska. Due to its low frequency content in band from the 0.2–0.4 Hz, the wave field of the VLP tremor is relatively free of path effects. From continuous recordings of the VLP tremor on 2 three‐component broadband and 3 single‐component short‐period instruments, we devise a method to locate the epicenter of the tremor based on interstation arrival times computed with cross correlation. We find the epicenter since the vertical and radial components of the VLP tremor wave field are dominated by Rayleigh waves and the time shifts are related to lateral propagation. Over the 4 h period studied, this procedure yields a location NNW of Cone D, close to the new cone built over the course of the eruption. Similar analysis using the transverse horizontal components from the 2 three‐component broadband instruments yields strong constraints on the source mechanism of the VLP tremor. We observe an anomalous interstation arrival time due to the existence of a nodal plane in the Love wave radiation pattern. The orientation of a compensated linear vector dipole (CLVD) source estimated from the transverse components closely aligns with the regional maximum horizontal stress direction. The depth of the CLVD source is constrained by matching the vertical components to the Rayleigh wave radiation pattern at all five stations. We find the VLP tremor source depth to be 2 km BSL, positioned between the magma chamber at Okmok (>3 km BSL) and the surface.

Specific features of surface wave radiation by a shallow source

The uncertainty of the seismic moment tensor inversion from records of long period surface waves radiated by a shallow source is considered. The special cases of sources of pure thrust, normal, and strike-slip faults are examined in detail. Results of numerical modeling of Love and Rayleigh wave radiation patterns and frequency dependences of the radiation intensity are presented for these types of sources. It is shown that, in the general case, the focal mechanism and seismic moment of a shallow double couple can be uniquely determined from long period surface waves only if one of the nodal planes is subhorizontal. The threshold (maximum) value of the dip angle of this plane is determined by the source depth, the spectral range of observed surface waves, and the model structure in the vicinity of the source.

Seismic source mechanisms for quarry blasts: modelling observed Rayleigh and Love wave radiation patterns from a Texas quarry

SUMMARY A theoretical understanding of the mechanisms by which quarry blasts excite seismic waves is useful in understanding how quarry blast discriminants may be transported from one region to another. An experiment in Texas with well-placed seismic stations and a cooperative blasting engineer has shed light on some of the physical mechanisms of seismic excitation at short periods (0.1‐3 Hz). Azimuthal radiation patterns of the 0.2‐3 Hz Rayleigh and Love waves are diagnostic of two proposed mechanisms for non-isotropic radiation from quarry blasts. Observations show that the Love and Rayleigh wave radiation patterns depend upon the orientation of the quarry benches. Two possible mechanisms for non-isotropic radiation are (1) the lateral throw of spalled material and (2) the presence of the topographic bench in the quarry. The spall of material can be modelled by vertical and horizontal forces applied to the free surface with time functions proportional to the derivative of the momentum of the spalled material. We use wavenumber integration synthetics to model the explosion plus spall represented by seismic moment tensor sources plus point forces. The resulting synthetics demonstrate that the magnitude of the SH (Love) compared with the SV (fundamental Rayleigh or Rg) in the short period band (0.5‐3 Hz) may be explained by the spall mechanism. Nearly all of the available mass must participate in the spall with an average velocity of 2‐5 m s −1 to provide sufficient impulse to generate the observed Love waves. Love wave radiation patterns from such a mechanism are consistent with the spall mechanism. We modelled the effects of the topographic bench using 3-D linear finite-difference calculations to compute progressive elastic wavefields from explosion sources behind the quarry bench. These 3-D calculations show SH radiation patterns consistent with observations while the SV radiation patterns are not consistent with observations. We find that the radiation patterns from the explosion behind the 3-D bench cannot be modelled by a modified moment tensor. The 3-D effects of the bench are more complicated than the representation by a moment tensor with a single reduced horizontal couple. The 3-D finite-difference synthetics exhibit strong azimuthal asymmetry and polarity reversals in the outgoing P-SV waves (P, S and Rg) radiated behind the bench for V p/V s ratios between 2 and 3. Both mechanisms may contribute to the non-isotropic radiation patterns but the spall mechanism is the simplest physical mechanism that explains the bulk of the observations. Adjustments to the time functions for the horizontal force, the vertical force and the explosion source may further refine the remaining differences between prediction and the observations.

Examination of slow and late moment release of the 1988 Spitak and 1991 Racha, Caucasus, earthquakes

SUMMARY Two large shallow earthquakes occurred in the Caucasus, part of the collisional belt caused by the northward movement of the Arabian plate with respect to the Eurasian plate: the 1988 December 7 Spitak (Armenian) (M,6.7) and 1991 April 29 Racha (M,6.9) earthquakes. In the present study, slow and late moment releases of the earthquakes, which were previously proposed based on the body-wave analyses, are examined using long-period surface waves. Results from spectral inversions using fundamental-mode surface waves show that the source durations of both earthquakes are about 20 s. Although additional investigations utilizing the long-period surface waves, which are optimized to detect slow and late moment release, were employed, no evidence of slow and late moment release was found for either earthquake. For the Racha earthquake, the obtained source duration is consistent with the result from body-wave analyses. Within the resolution of the body-wave data, the simple P and SH waveforms are well modelled by a source time history model with a duration of 20 s. However, for the Spitak earthquake, the source duration obtained from the surface waves is about 30 s shorter than the source model suggested in the body-wave analysis. This discrepancy is attributed to a slow and late thrust-fault slip in the body-wave source model, which was proposed in order to interpret the late arrivals of P waves as the result of source complexity. Numerical experiments predict that Rayleigh-wave radiation is sensitive to the radiation from the proposed slow and late slip. A comparison between the observed and predicted Rayleigh-wave radiation patterns, however, provides no indication of the surface-wave radiation from the slow and late slip. It is unlikely that the discrepancy in the source model is only due to errors in modelling the propagation of long-period surface waves, because for the Racha earthquake the source models obtained from body and surface waves are consistent. The present results thus indicate that the late arrivals of P waves are probably caused by something other than source complexity, for example complex local structure near the earthquake source in the continental collisional belt. For the Spitak earthquake, a large non-double-couple component is observed in the moment tensor solution obtained at long periods. If the non-double-couple component is due to source complexity, it should be produced within the 20 s rupture process.

Source parameters of the 1989 Loma Prieta Earthquake determined from long‐period Rayleigh waves

The source parameters of the Loma Prieta earthquake are determined using long‐period Rayleigh waves recorded by USGS/ERIS, IDA/IRIS, and GEOSCOPE stations. The source mechanism is well‐constrained by the Rayleigh wave radiation pattern, with a dip = 70 (±5)°, strike = 130 (±5)°, rake = 135 (±5)°, and moment = 3.4 (±0.5) × 1019 Nm (Mw = 7.0). This mechanism is generally consistent with independent body wave determinations. The most stable long‐period waves, with periods from 200 to 275 s, indicate that the source process has a centroid time of about 10 s, somewhat longer than that indicated by body waves (about 5–6 s). This discrepancy cannot be uniquely attributed to source effects because of uncertainties in the propagation corrections. The importance of using surface waves with short propagation paths for analysis of moderate size earthquakes such as the Loma Prieta event is demonstrated by the unreasonably long source durations inferred from R3 arrivals.

Spatial variation of ground motion determined from accelerograms recorded on a highway bridge

AbstractA set of time-synchronized strong-motion accelerograms, obtained on the San Juan Bautista 156/101 Separation Bridge in California during the 6 August 1979 Coyote Lake earthquake (ML = 5.9), are used to study the spatial variation of ground motion at the bridge site, including traveling wave effects and the influence of multiple-support excitation. Analysis of the ground motion recorded at the base of two of the bridge supports (32.6 m apart) revealed the presence of a differential support excitation having a period of ≈ 3 sec, much longer than any structural periods of the bridge. This signal also appeared as a noticeable long-period component in the superstructure displacements. Analysis of the vertical and radial components of the 3-sec ground motion indicated that ground displacements were retrograde for the duration of strong shaking, with several cycles exhibiting elliptical particle motions. These findings suggest that long-period differential support motions were induced by phase delays in a Rayleigh wave traveling across the bridge site. Further support to this premise is given by the location of the bridge site near a maxima of the Rayleigh wave radiation pattern for the Coyote Lake earthquake (based on published focal mechanism data). Traveling wave effects were also detected for compressional body waves by a correlation analysis which indicated a time delay of ≈ 7 msec between P-wave arrivals at two of the bridge supports.

Source process and tectonic implications of the great 1975 North Atlantic earthquake

Summary. The Atlantic segment of the Africa-Europe plate boundary has usually been interpreted as a transform boundary on the basis of the bathymetric expression of the Gloria fault and dextral strike-slip first-motion mechanisms aligned along the Azores-Gibraltar line of seismicity. The 1975 May 26 earthquake (Ms=7.9) was assumed to fit into this framework because it occurred in the general area of this line and has a similar first-motion focal mechanism (strike=288, dip=72, slip angle=184). However, several anomalies cast doubt on this picture: the event is abnormally large for an oceanic transform event; a sizeable tsunami was excited; the aftershock area is unusually small for such a large event; and most significantly, the epicentre is 200 km south of the presumed plate boundary. The Rayleigh wave radiation pattern indicates a change in focal mechanism to one with a significant dip-slip component. The short duration of the source time history (20 s, as deconvolved from long-period P-waves), the lack of directivity in the Rayleigh waves, and the small one-day aftershock area suggest a fault length less than 80 km. One nodal plane of the earthquake is approximately aligned with the trace of an ancient fracture zone. We have compared the Pasadena 1-90 record of the 1975 earthquake to that of the 1941 North Atlantic strike-slip earthquake (200km to the NNW) and confirmed the large size of the 1941 event (M=8.2). The non-colinear relationship of the 1975 and 1941 events suggests that there is no well-defined plate boundary between the Azores and Gibraltar. This interpretation is supported by the intraplate nature of both the 1975 event and the large 1969 thrust event 650 km to the east. This study also implies that the largest oceanic strike-slip earthquakes occur in old lithosphere in a transitional tectonic regime.

Source geometry and mechanism of 1978 Tabas, Iran, earthquake from well located aftershocks

Aftershocks of the September 16, 1978 Tabas earthquake located from close-in observations made during a four-week fielding of temporary stations have been analyzed for the purpose of delineating detailed source geometry of the 1978 earthquake. Spatial distribution of aftershocks and their composite focal mechanism suggest that the geometry of faulting is far from planar. Aftershocks define two prominent alignment. The southern alignment strikes E-W to WNW-ESE, whereas the northern alignment strikes in a N-S to NNW-SSE direction with an abrupt change of nearly 55–60 degrees near 33.4°N latitude. Both field observations of surface faulting pattern and systematic variation of principal directions of stress axes computed from aftershock focal mechanisms are consistent with the upthrusting and imbrication of a wedge shaped crustal block with the wedge angle of about 120 degrees. Both geological and seismological evidence suggest that the deformed zone is truncated at the southern edge by preexisting E-W fault structures. New observations may provide a partial answer to the unexplained farfield asymmetry of the long period Rayleigh wave radiation pattern recently observed for the mainshock across IDA network.

A method for the near-source anisotropy by the pair-event inversion of Rayleigh-wave radiation patterns

Summary. A novel method is proposed for retrieving the 3-D orientation of axes of symmetry of near-source anisotropy by a non-linear inversion of observed radiation patterns of seismic displacement spectra of Rayleigh waves. If faulting is generated within an anisotropic source region, body force equivalents for the faulting are in general not a double couple but the sum of three orthogonal dipole forces (Kosevich; Kawasaki & Tanimoto). As a result of the third dipole force, radiation patterns of Rayleigh waves are deformed, the deformation amounting to several per cent of those for an isotropic source medium. The non-linear inversion is carried out to find the optimum fault plane solutions giving the minimum square residual between observed and theoretical radiation patterns in some period range. In order to remove effects of heterogeneity along propagation paths, a pair-event scheme is involved in the inversion, which denotes taking spectral amplitude ratios and differential phases of the seismic displacement spectra of the pair-events having close hypocentres and different fault plane solutions. The uniqueness of the fault plane solutions of the non-linear inversion is afforded a proof by the Monte-Carlo experiment. The non-linear inversion is repeated for some possible types of symmetry of the near-source orthotropic anisotropy due to the preferred orientation of olivine crystals as mantle materials. Square residuals thus obtained are compared with each other to see which orientation gives the minimum. The method is applied to pair-events which occurred in the anomalous mantle beneath the Mid-Atlantic Ridge. This leads to a discovery that one type of symmetry of the preferred orientations with a-, b- and c-axes aligned vertical, parallel to and perpendicular to the trend (N11E) of the ridge axis, respectively, is most likely existing in the anomalous mantle.

Source characterization of two Reykjanes Ridge earthquakes: Surface waves and moment tensors; P waveforms and nonorthogonal nodal planes

Well‐constrained fault plane solutions from P wave first motions for mid‐ocean ridge normal faulting earthquakes usually require nonorthogonal nodal planes. Local structural effects and/or departures from a double‐couple source mechanism have been invoked to explain this phenomenon. In order to obtain an independent determination of the source mechanisms for the April 24, 1970, and April 3, 1972, events on the southern Reykjanes Ridge, we invert the Rayleigh wave radiation pattern to obtain the source moment tensor. The moment tensor formulation should be particularly well suited to this problem because it is not restricted a priori to a double‐couple source mechanism. A potential drawback of the technique, however, is the requirement that phase velocities along the earthquake‐station paths be known very accurately in order to obtain the source phase from the observed phase, and an objective of this study was to determine whether a regionalized phase velocity model compiled from published dispersion curves is adequate. The results of the moment tensor inversion for both events indicate shallow normal faulting with the tension axis approximately horizontal and perpendicular to the local strike of the ridge. Apparent departures from a pure double‐couple source seem to result from errors in the data and the poor resolution of the Mxz and Myz components of the moment tensor for shallow sources. After performing the inversion under a series of increasingly more stringent constraints we conclude that the data for both events are compatible with pure double‐couple sources with moments of 4.8 and 7.5 × 1024 dyn cm, respectively. We then show that interference between P, pP, and sP due to shallowness of the source can account for the observed nonorthogonality and match the observed P waveforms for the April 3, 1972, event with theoretical seismograms calculated for a shear fault whose orientation is consistent with the surface wave solution. The best fit to the data is obtained for a long, narrow fault (13 km by 3 km), with rupture initiating near the seafloor. The moment indicated by the P waves is 7.5 × 1024 dyn cm. These source parameters give an average displacement of about 60 cm and a stress drop of 30–60 bars, taking into account various uncertainties. Although we might expect attentuation to be high in the mid‐ocean ridge environment, the average attenuation required to fit the teleseismic data is not higher than normal (t* = 1 s). The P waves from the April 24, 1970, earthquake were too small to be suitable for quantitative modeling by synthetic seismograms but are qualitatively consistent with a shallow fault model similar to that for the larger event. We conclude that the faulting process described by these two earthquake mechanisms is directly related to the formation of rift valley topography.

The focal mechanisms and depths of small earthquakes as determined from Rayleigh-wave radiation patterns

abstract Many earthquakes of body-wave magnitude less than about 5.5 are located in key positions for defining tectonic processes at plate boundaries and in the interiors of plates. Due to the sparsity of observed body-wave data from these events, their focal mechanisms and depths cannot be well determined from body-wave data alone. Earthquakes with mb as low as 4.5 radiate surface-wave trains which can be observed at teleseismic distances. In this study, Rayleigh waves observed at WWSSN stations have been inverted to obtain focal mechanisms and depths of two small earthquakes (mb = 4.7 and mb = 5.1) in the Caribbean plate region. An event pair inversion patterned after Weidner and Aki (1973) is contrasted with a single event inversion which makes use of the seismic moment tensor formalism (Mendiguren, 1977; Gilbert, 1970) to linearize the relationship between observed signal and model. The event pair approach requires master events whose focal parameters are well known and which are located near the smaller events. The single event approach, on the other hand, does not require master events, but is limited to amplitude data only. Phase information can be utilized by considering a pair of events located near each other and recovering the difference in phase for the event pair. For the events studied, the single event analysis yields the same solution as the event pair analysis. In general, amplitude is more sensitive to mechanism than phase. However, noise in amplitude increases more rapidly than noise in phase as the size of the event is reduced.

Seismic moment of great deep shocks

The long-period Rayleigh waves were investigated for the largest four deep shocks in 1963–1973 to determine the seismic moment by the same technique as used for shallow earthquakes. The results could be used for a quantitative comparison of source parameters between shallow and deep events. Three of the four shocks occurred beneath the South American continent (the Colombia earthquake, 1970; the western Brazil earthquake, 1963; the Peru—Bolivia border earthquake, 1963) and the other beneath the Japan Sea (1973). The focal depths are 653, 576, 593 and 575 km, respectively. The largest value of seismic moment was obtained as 2.1 · 10 28 dyncm for the Colombia earthquake. This value is still about forty times smaller than that for the great Alaskan earthquake. A slight inconsistency was found between the first-motion diagram and the Rayleigh wave radiation pattern for the Colombia earthquake and the Peru—Bolivia border earthquake.

Focal mechanism of the Iwate-Oki earthquake of June 12, 1968.

The focal mechanism of the Iwate-Oki earthquake (Ms=7.2) of June 12, 1968, is studied on the basis of P-wave first motion and Love and Rayleigh wave radiation patterns. This earthquake represents a low-angle thrust fault with a left-lateral strike-slip component. The direction of slip vector is N55°W, which is parallel to that of the neighboring Tokachi-Oki earthquake (Ms=7.9). The source parameters determined are: dip angle=30° and dip direction=N119°W for the fault plane; dip angle=76° and dip direction= N125°E for the auxiliary plane; seismic moment=5.1×1026 dyne·cm; fault dimension=80×30km2; slip dislocation=48cm; stress drop=12bar; strain drop=0.27×10-4; released strain energy=0.70×1022erg. This earthquake was accompanied with small tsunamis. The sea bottom crustal deformation calculated from the seismic fault parameters fits the tsunami data very well. The generation mechanism of tsunami of the Iwate-Oki earthquake resembles that of the Tokachi-Oki earthquake of 1968.

The Bengough, Saskatchewan, Earthquake of July 26, 1972

A minor earthquake occurred on July 26, 1972 near Bengough, Saskatchewan, at 49°21′ N ± 5′, 104°56′ W ± 4′ with origin time 03h58m19s ± 1s GMT. The magnitude was mb 3.7 ± 0.2 and the focal depth less than 10 km. The radius of perceptibility was 30 km and the maximum intensity was IV on the Modified Mercalli scale. There was no reported damage. The large ratio of Love- to Rayleigh-wave amplitudes indicated that the event was not due to the alleged sudden collapse of a solution cavity in the salt formations. The Love- and Rayleigh-wave radiation patterns are consistent with a vertical strike-slip fault striking approximately N 30 °E. Three other earthquakes are known to have occurred in the general area of the Williston Basin—the largest in 1909 was probably about mb[Formula: see text].

Mechanisms of earthquakes and plate motions in the east Pacific

Mechanisms of earthquakes occurring on plate boundaries have been used to determine the relative motions of lithospheric plates in the east Pacific. By using both P-wave first motions and the radiation pattern of Rayleigh waves in the focal mechanisms determinations, the uncertainty in strike is held to about ± 10°. On the Chile fracture zone and the fracture zone at 4°S on the East Pacific Rise, the direction of motion is not parallel to the trend of epicenters. This supports the hypothesis that these fracture zones are composed of series of en-echelon transform faults. Two regions were found on the East Pacific Rise, in which the slip directions probably are due to local tectonic activity, rather than motion between the Pacific and Nazca plates.

Use of reciprocity theorem for obtaining Rayleigh wave radiation patterns

abstract The reciprocity theorem is used to obtain Rayleigh wave radiation patterns from sources on the surface of or within an elastic semi-infinite medium. Nine elementary line sources first considered are: horizontal and vertical forces, horizontal and vertical double forces without moment, horizontal and vertical single couples, center of dilatation (two dimensional case), center of rotation, and double couple without moment. The results are extended to the three dimensional case of similar point sources in a homogeneous half space. Haskell's results for the radiation patterns of Rayleigh waves from a fault of arbitrary dip and direction of motion are reproduced in a much simpler manner. Numerical results on the effect of the depth of these sources on the Rayleigh wave amplitudes are shown for a solid having Poisson's ratio of 0.25.