Uniform Half-space Research Articles

Quasi-static strains and tilts due to faulting in a viscoelastic half-space

abstract Recently derived quasi-static surface displacements resulting from a finite rectangular vertical strike-slip fault in a viscoelastic half-space are used to derive the surface strains and tilts for both Voigt and Maxwell viscoelastic models. Contour maps are obtained for various strain and tilt components. The variation with time and epicentral distance is studied in some representative cases. Detailed numerical calculations reveal significant differences between the viscoelastic and the elastic results in the case of a vertical strike-slip fault. This contrasts with the results for a vertical dip-slip fault in a uniform half-space, where the surface strains and tilts for the viscoelastic and the elastic models are identical.

Deep seismic crustal studies in Manitoba

abstract The results of 84 deep seismic sounding recordings scattered over an area of 250,000 km 2 (average spacing, 25 km) with each recording yielding up to 3 depth points for each interface mapped, are compiled on maps giving crustal thickness and depth to the transition between an upper and a lower crustal layer. Ranges are from a few kilometers up to about 700 km. A velocity model is derived as follows: upper crustal layer, α = 6.05 km/sec, β, = 3.46 km/sec; lower crustal layer, α = 6.85 km/sec, β, = 4.00 km/sec (with the strong possibility that these values are averages, with a small vertical velocity gradient in the lower layer); uppermost part of upper mantle, α = 7.92 km/sec, β, = 4.60 km/sec. A sub-Mohorovicic event, showing triplication, is found. It may represent a boundary in a layered medium, or a second-order discontinuity in a medium with a vertical gradient. If it is the former, a velocity transition to a velocity in the range 8.3 to 8.4 km/sec is indicated at an average depth of 50 km. Lateral velocity variation was found to be small. A method of calculating velocities from wide-angle reflections for a 3-layer medium overlying a uniform half-space is given. The derived deep crustal structure yields important geological information, showing downward extension of geological features visible at the surface. These include the boundary between the Churchill and Superior geological provinces of the Canadian shield, and a trough-like thickening of the upper crustal layer (accompanied by a thinning of the total crustal section) which has been traced 350 km from the English River area of Ontario, across the south basin of Lake Winnipeg, Manitoba. It lies beneath a geological unit known as the English River gneissic belt. In addition, faulted boundaries of crustal blocks are shown in the deep seismic structure. The derived seismic structure shows correlation with regional gravity and magnetic data.

The Electromagnetic Perturbation Fields of Conductivity Anomalies within the Earth

Summary The electromagnetic perturbation fields of anomalous two-dimensional conductivity structures are considered. It is assumed that the total field in a conducting half-space with inhomogeneities consists of a normal component and an anomalous or perturbation component. The equations for the perturbation component are solved by an extension to the general numerical method developed by Jones and Price, Jones and Pascoe, and Pascoe and Jones, which solves for the total field distributions. A model of a conducting half-space consisting of quarter spaces of different conductivity and also a model in which a conductivity anomaly is embedded in an otherwise uniform half-space are investigated. The distributions of the perturbation fields are shown.

Seismic Wave Scattering by Obstacles on Interfaces

Summary A unified treatment of the scattering of seismic waves by small obstacles is presented. The method is based on an integral equation formulation and the use of first-order perturbation theory. This approach is used to consider elastic wave scattering by interface irregularities and obstacles at the interface of two elastic media, whose elastic properties are different from those of the bounding media. The results obtained by this method are shown to reduce to those of previous authors when applied to their more specialized problems. It is well known that in regions of irregular topography much noise is present in both reflection and refraction seismic records; this is attributed to scattering by surface and sub-surface topography and inclusions with elastic properties which differ from those of the surrounding matrix. Also much of the signal-related noise at seismic recording stations appears to be due to scattering by obstacles at or below the surface in the vicinity of the station. Gilbert & Knopoff (1960) considered the scattering produced by a surface irregularity on a uniform half space. Their treatment was subsequently extended by Hudson (1967) to deal with a three-dimensional surface obstacle whose elastic properties differ from those of the half space. This method is suitable for small obstacles whose bounding surfaces have only small slopes. Recently Aki & Larner (1970) and Slavin & Wolf (1970) have considered larger interface irregularities by constructing solutions in which the boundary condition mismatch is minimized by a least squares technique. In this paper we will consider only the first order perturbations for small obstacles, andpresentaunified treatment of the scattering of elastic wavesby interface irregularities and obstacles at the interface of two elastic media, whose elastic properties differ from those of the bounding media. The method is based on an integral equation approach and has the advantage that it is an extension of the normal methods for layered media. The results obtained reduce to those of Gilbert & Knopoff and of Hudson when applied to their more specialized problems. 2. Basic equations for an irregular interface We consider isotropic media, and for simplicity only two-dimensional problems where all stresses and displacements are independent of the coordinate y. A harmonic time dependence of the form exp (- iwt) will be assumed throughout, though it will not be explicitly represented. Away from the neighbourhood of irregular interfaces we shall consider media

The depth location by induced polarization of a target equivalent to a simple induced dipole in a semi-infinite homogeneous medium

An electric field set up in the earth by a dipole transmitter causes an isolated metallic body to exhibit induced polarization effects (the induced electric field). Assuming that the induced electric field is proportional to the applied field at any point, it is shown that in the simple case of a small spheroidal body in a uniform half-space, the position of a target body, for maximum response to be registered at the receiver, is at the intersection of two lines drawn at an angle of 59° 26′ of depression from the transmitter and the receiver. This angle is different from the 45° currently used, but agrees with the results of field experience, which shows that on drilling to small targets in uniform ground, the depths are greater than indicated by the induced polarization method.

The Relative Excitation of Seismic Surface and Body Waves by Point Sources

Summary Experimental evidence shows that earthquakes can be differentiated from explosions on teleseismic records by their relatively greater amplitude of Rayleigh waves compared with the amplitude of the direct P-wave; in effect the ratio of magnitude estimates mb/Ms, is greater for explosions than for earthquakes. In this paper we have tried to assess the contribution of source mechanism alone to this effect. We have used point dilatation and double-couple sources to represent explosions and earthquakes and derived the amplitudes of the waves generated from such sources in a uniform half space. The results show that source mechanism is an important factor governing the value of mb/Ms.

Calculation of surface motions of a layered half-space

Abstract A new method is presented for computing the transient response of a set of horizontally stratified, linearly elastic layers overlying a uniform half-space and excited by vertically incident, transient plane waves. In addition, a simple approximate method of satisfactory accuracy is developed that reduces the computing time required. Calculated responses are compared with motions recorded under Union Bay in Seattle to evaluate the agreement between recorded and calculated motions.

Weight Minimization Of Dynamically Loaded 3-DFoundation On Layered Medium

Minimum-weight design of a machine foundation on an inhomogeneous soil is considered. The soil model under the base of block corresponds to a layer with linearly varying properties overlying a uniform half-space. Furthermore, the block may be surrounded by a backfill. The optimal designs of dynamically loaded rectangular machine foundation, including coupled sliding-rocking vibrations, are found for a number of realistic inhomogeneous supporting media. The results obtained in the case of the layered soil are compared with those for a uniform half-space. It illustrates the problem of adequate modelling of the nature of the soil profile providing an insight into the action of the soilfoundation-machine system from the point of view of performance and safety.

A dislocation model for the Fairview Peak, Nevada, earthquake

abstract Maruyama's equations have been used to calculate both vertical and horizontal surface displacements for a Volterra dislocation in a uniform half space. The shape of the dislocation surface is assumed to be a plane rectangle. A least-squares procedure has been used to determine the dislocation surface which best fits the geodetic data for the Fairview Peak, Nevada, earthquake of December 16, 1954. The best model strikes N9°E, has a total length of 50 km, a width of 8. ± 2. km, a dip of 57° ± 6°, and a fault displacement of 2.3 ± 0.4 m normal dip slip and 2.9 ± 0.4 m dextral strike slip. The width of the model is probably somewhat less than the true width of the fault, but the other parameters appear to be quite reasonable. The N9°E strike of the model is thought to be more realistic than the N4°W to N11°W strikes indicated by nodal-plane solutions. It appears that the Dixie Valley earthquake of December 16, 1954 was associated with normal slip on a fault plane parallel to, but offset from, the fault plane of the Fairview Peak earthquake.

I. P. 法における異常部の影響

Generally the resistivity curves are affected by the topography and the near-surface conditions, butthe frequency effect curves are not so much affected.The two-dimensional conducting square which lies imbedded in the uniform half-space is shown in Fig. 4, and the resistivity curves and the frequency effect curves are shown in Figs. 5, 6 and 7 respectively.Local anomalous conditions are shown in Fig. 8 which is the case of the topography and are shown in Fig. 11 which is the case of near-surface conditions, and the resistivity curves are shown in Figs. 9, 10 and 12, 13 respectively. In these cases, the resistivity curves are considerably complicated.The two-dimensional conducting squares which lie imbedded under topographic conditions are shown in Figs. 14 and 18. The resistivity curves and the frequency effect curves are shown in Figs. 15, 16; 17 and 19, 20; 21 respectively. In these cases, the resistivity curves can not point out real anomalies but the frequency effect curves can point out real anomalies fairly well, comparing with the curves in Fig. 7.

Body wave radiation patterns from elementary sources within a half space

abstract The known expressions for the polar radiation patterns due to a horizontal or a vertical force, applied at a point within a uniform half space, are used to obtain the body wave radiation patterns from several other elementary seismic sources. Polar radiation patterns from seven elementary line sources, i.e., horizontal and vertical double forces without moment, horizontal and vertical single couples, center of dilatation, center of ratation, and double couple without moment, are first derived. Similar point sources in the three-dimensional space are also considered and the corresponding polar as well as azimuthal radiation patterns are obtained for P, SV, and SH waves. The results obtained include the effect of finite depth of the source below the free surface. Some of the results of Burridge et al for double-couple type seismic sources near a free surface are reproduced in a simple manner. For the elementary point sources considered here, the azimuthal radiation patterns for a uniform half-space are found to be identical with those for an infinite homogeneous medium. However the polar radiation patterns appear to be profoundly affected by the proximity of the free surface.

Probing sea-bottom sediments with microseismic noise

Simultaneous measurements of the fluctuating pressure just above the sea bottom and of the vertical velocity of the sea bottom can provide information on the character and thickness of sediments, since the magnitude of pressure variation on the ocean bottom, generated by a passing Rayleigh wave is a function of the frequency and phase velocity of the Rayleigh wave. Such Rayleigh wave fluctuations are naturally present as a result of world-wide microseismic activity and are generally prominent over an octave of frequency centered at v = 0.15 cps. The simplest case to consider is that of a uniform ocean (density ρ1, depth H, sound velocity V1) overlying an infinite uniform half-space (phase velocity C). A pertinent ratio involving pressure p and vertical velocity v, then [Bradnet, 1962], is The calculation can be generalized to any number of layers by using Haskell [1953] matrices. The first Rayleigh mode results of such calculations, for a number of models, are shown in Figure 1. Second and third Rayleigh mode curves are presented in Figure 2 and Figure 3. The layer parameters are tabulated at the end of this letter. Figure I shows that the pressure-velocity ratio varies over a wide range and promises to be a sensitive indicator of bottom structure, as long as the microseismic energy is predominately in the first Rayleigh mode.

Higher Order Born Approximations for the Inversion of Love Wave Dispersion

Summary The procedure for the inversion of Love wave dispersion has been extended to the case of an arbitrary number of homogeneous layers overlying a uniform half-space. With this process the iterative inversion procedure can proceed as often as required for convergence subject to the limitation of the Born approximation.

Experimental investigation of PL modes in a single layer

Abstract A two dimensional seismic model has been set up to simulate the problem of elastic wave propagation in a single layer overlying a uniform half space. Both the source and the receiver are mounted on the free surface of the layer. Seismograms are presented as a funciton of range. In addition to the Rayleigh and shear modes, PL modes are observed. Experimentally determined phase and group velocities compare fairly well with theoretical curves. The decay factor for PL is maximum at the arrival time of P waves in the half-space. There is also a secondary maximum at the arrival time of P waves in the layer. Although the decay of PL is small, phase equalization of PL does not yield the initial pulse shape because the mode embraces an insufficient frequency band to permit good resolution.

The Motion of the Surface of a Uniform Elastic Half-Space Produced by a Buried Torque-Pulse

In this investigation we determine the motion of the surface of a uniform elastic half-space produced by the application of a torque-pulse at a point below the surface. The axis of the torque is taken to be parallel to the surface and its time variation is assumed to be represented by the Heaviside unit function H(t). Our results are compared with those of Pinney (1954), who treated the same problem by a different method. The principal feature of physical interest which we found is that, for ranges where the direct S wave is preceded by a diffracted SP wave, the displacement at the surface starts with an infinite amplitude at the time of arrival of the SP wave, and that this is followed by an even stronger infinity at the time of arrival of the S wave. Also, for small ranges, for which there is no SP wave, the displacement starts with a sharp pulse in the form of a Dirac delta function. None of these features was brought out by Pinney's curves. The results are shown in Figures 2, 3 and 4.

Motion of the Surface of a Uniform Elastic Half-Space Produced by a Buried Pulse

An exact solution is obtained for the motion of the surface of a uniform elastic half-space due to the application at a depth H below the surface of a concentrated vertical force. The time-variation of the applied force is assumed to be represented by the Heaviside unit function. The solution for the horizontal and vertical components of displacement cast in the form of single integrals over a fixed range, and these have been evaluated on the electronic computer of the Weizmann Institute (WEIZAC). The assumed source emits both S waves and P waves. Beyond a distance r1 from the epicenter, which is equal to H/√ in the case λ = μ, the original S wave is converted on reaching the surface into a diffracted SP wave traveling along the surface. At large ranges, the SP phase is more pronounced than the P phase. The S phase is marked by a finite jump for r<r1, and by a logarithmic infinity for r>r1. The coefficient of the logarithmic term is zero both at r=r1 and at large ranges, having a sharp maximum at r=1.004r1. There is no Rayleigh wave at r<r1. At large ranges (r/H≫1) the solution, as a function of the reduced time τ = ct/R, approaches the form of the solution for the surface pulse.

The seismic pulse, an example of wave propagation in a doubly refracting medium

An exact and closed solution is given for the motion produced on the surface of a uniform elastic half-space by the sudden application of a concentrated pressure-pulse at the surface. The time variation of the applied stress is taken as the Heaviside unit function, and its concentration at the origin is such that the integral of the force over the surface is finite. This problem gives an instructive illustration of wave propagation in a doubly refracting medium, since both shear waves and compressional waves are excited, and they travel with different speeds. There is, in addition, the Rayleigh surface wave. For a medium in which the elastic constants <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\mu</tex> are equal, the vertical component of displacement <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{o}</tex> at the surface is given by: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{o} = 0</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau &lt; \frac{1}{\sqrt{3}}</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{o} = -\frac{Z}{\pi\mu r}\{\frac{3}{16}-\frac{\sqrt{3}}{32\sqrt{\tau^{2}}-{\frac{1}{4}}} - \frac{\sqrt{5+3\sqrt{3}}}{32\sqrt{\frac{3}{4}+\frac{\sqrt{3}}{4}-\tau^{2}} + \frac{\sqrt{3\sqrt{3}-5}}{32\sqrt{\tau^{2}+\frac{\sqrt{3}}{4}-\frac{3}{4}}}\}</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\frac{1}{\sqrt{3}}&lt; \tau &lt; 1</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{o}= - \frac{Z}{\pi\mu r\{\frac{3}{8} - \frac{\sqrt{5 + 3\sqrt{3}}{16\sqrt{\frac{3}{4} + \frac{\sqrt{3}}{4} - \tau^{2}}\}, 1 &lt; \tau &lt; \frac{1}{2}\sqrt{3+\sqrt{3}}</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{o} = - \frac{Z}{\pi\mu r}\frac{3}{8}, \tau &gt; \frac{1}{2}\sqrt{3 + \sqrt{3}}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\tau = (ct/r)</tex> , <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> -shear wave velocity, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-Z</tex> is the surface integral of the applied stress. The horizontal component of displacement is obtained similarly in terms of elliptic functions. A discussion is given of the various features of the waves. It is pointed out that in the case of a buried source, an observer on the surface will, under certain circumstances, receive a wave which travels to the surface as an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</tex> wave along the ray of total reflection, and from there along the surface as a diffracted <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</tex> wave. An exact expression is given for this diffracted wave. The question of the suitability of automatic computing machines for the solution of pulse propagation problems is also discussed.