K. Aki and P.G. Richards, Quantitative Seismology - Theory and Methods (W. H. Freeman, San Francisco, 1980), Vol. 1, 2.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....source distribution may be replaced by a suitable collection of multipoles, cf. 13, 8] The resulting series, multipole expansion, is redundant and strongly related to other expansions of the acoustic field. Seismic sources and ice cracks have been described using moments in the literature [1]. However, modeling of the ocean acoustic sources has mostly been limited to a monopole for matched field processing (one exception being surface generated noise [11, 7] For near field problems, it is to be expected that this is not a good approximation when the region of acoustic excitation ....

K. Aki and P.G. Richards. Quantitative Seismology: Theory and Methods, vol. 1 & 2. W.H. Freeman & Co., San Francisco, (1980).

....to the complexity of the underlying wave propagation phenomena. Based on the traditional uncoupled analyses, ratio between the material stiffness and mass density is determined from travel times, while the damping characteristics are inferred from the spatial amplitude decay of seismic signals (Aki and Richards 1980). In cases where the full record of the seismic source is also available, site characterization can be effected more completely in terms of the experimental frequency response functions which relate the applied load to the surface motion in the frequency domain. The field measurement of such ....

Aki, K. and Richards, P.G. (1980). "Quantitative Seismology: Theory and Methods", Vol. 2, W.H. Freeman and Co., San Francisco.

....local axis (s) along the fault. b) Approximate a curved fault with a series of small planar fault elements ( Deltas) In this case, off plane stress expression for a flat source is required. Formulation It usually begins with a representation theorem, equation (3. 2) in Aki and Richards (1980)[1], for example. The displacement field u i ( x; t) in a (x 1 ; x 2 ; x 3 ) coordinate system is written by a spatio temporal convolution over the fault Sigma u i ( x; t) Z 1 Gamma1 d Z Sigma Deltau j ( c jkpq k ( G ip ( x Gamma ; t Gamma ) q d Sigma (1) where ....

Aki, K., and P. G. Richards, 1980, Quantitative Seismology: Theory and Methods, Freeman and Co., San Francisco.

....2 0 p = 2 f( ffi(x)ffi 0 (z Gamma z s ) Consider the standard point source problem Lg j 4g ( 2 fl 2 0 g = Gammaffi (x)ffi(z Gamma z s ) 51 associated with the reduced wave equation. The solution, the free space Green s function, can be written in terms of the Weyl integral [1] as g = i = 8 2 ) Z Z e i OE= OE z d: Here the phase OE is given by OE = Delta x q fl 2 0 Gamma 2 jz Gamma z s j with x = x 1 ; x 2 ) 1 ; 2 ) and Im[ q fl 2 0 Gamma 2 ] 0. Hence, L[ Gamma 2 f z g] 2 f( ffi(x)ffi 0 (z Gamma z s ) Thus we ....

K. Aki and P. G. Richards. Quantitative Seismology-Theory and Methods, volume 1. W. Freeman, 1980.

....to the complexity of the underlying wave propagation phenomena. Based on the traditional uncoupled analyses, ratio between the material stiffness and mass density is determined from travel times, while the damping characteristics are inferred from the spatial amplitude decay of seismic signals (Aki and Richards 1980). In cases where the full record of the seismic source is also available, site characterization can be effected more completely in terms of the experimental frequency response functions which relate the applied load to the surface motion in the frequency domain. The field measurement of such ....

Aki, K. and Richards, P.G. (1980). "Quantitative Seismology: Theory and Methods", Vol. 2, W.H. Freeman and Co., San Francisco.

....point x and r = x # y # z ) p 1 #p 2 #p 3 )betheslowness vector. Then, p i is related to the normal velocityofthewavefrontby the relation p i = n i v # i =1# 2# 3# where n i is the direction cosines of the wavefront normal and v is the wavefront normal velocity (i.e. phase velocity)# see [1]. From the above equation, we obtain jrK j 2 = 1 v 2 K # K = P# SV# SH: 6.1) Here the subscript K is introduced to indicate different VTI traveltimes and VTI phase velocities. The VTI phase velocities are known as follows (see [33] v 2 P ( ff 2 0 h 1 sin 2 D ( i # v ....

K. Aki and P. G. Richards, Quantitative seismology - Theory and methods, W. H. Freeman & Co., 1980.

....two component shot record show that the Pand SV waves can be decomposed completely and the subsurface image for P P waves is consistent with P SV waves. 2. Basic equations for decomposition In a general 3 D isotropic medium, elastic waves propagation is described by the linearized wave equation (Aki and Richards, 1980) ae 2 u i t 2 = x i r 2 u i aeX i (1) where u i = u 1 ; u 2 ; u 3 ) T denote the displacement component, X i is the external body force component, ae is the density of the medium, and are lam e parameters, and is the divergence of the displacement or dilation, ....

Aki, K., and P.G. Richards, 1980, Quantitative Seismology: Theory and Methods, Vol. 1 and 2, W.H. Freeman, New York.

....subset of geophysics, primarily within solid Earth geophysics. 1 General Geophysical Texts General texts on geophysics include Fowler [76] Garland [77] Jeffreys [95] and Stacey [188] Bolt has written a number of very readable introductions to seismology [26, 27, 28, 30, 31] Aki and Richards [3] treat theoretical seismology at a fairly advanced level; Bullen and Bolt [40] and Lay and Wallace [107] are more accessible. Merrill and McElhinny [117] is a very readable introduction to geomagnetism and paleomagnetism; Backus et al. 15] is more theoretical. Blakely [23] discusses elements of ....

K. Aki and P.G. Richards. Quantitative Seismology: Theory and Methods. W.H. Freeman, New York, 1980.

....at a smooth interface between two regions with smoothly varying parameters. The amplitude of the scattered waves is determined essentially by the reflection coefficients. It is well known how to calculate these for two constant coefficient media and a plane interface (see e.g. Aki and Richards [1], chapter 5) In the case of smoothly varying media they determine the scattering in the limit of high frequency, see Taylor [22] for a treatment of reflection and transmission of waves using microlocal analysis. For the acoustic case see also Hansen [11] Mathematically the reflection and ....

Keiiti Aki and Paul G. Richards. Quantitative seismology: theory and methods, volume 1. Freeman, San Francisco, 1980.

....r = x ; y ; z ) p 1 ; p 2 ; p 3 ) be the slowness vector. Then, p i is related to the normal velocity of the wavefront by the relation p i = n i v ; i = 1; 2; 3; where n i is the direction cosines of the wavefront normal and v is the wavefront normal velocity (i.e. phase velocity) see [1]. From the above equation, we obtain jrK j 2 = 1 v 2 K ; K = P; SV; SH: 6.1) Here the subscript K is introduced to indicate different VTI traveltimes and VTI phase velocities. The VTI phase velocities are known as follows (see [33] v 2 P ( ff 2 0 h 1 sin 2 D ( i ....

K. Aki and P. G. Richards, Quantitative seismology - Theory and methods, W. H. Freeman & Co., 1980.

....information about the elastic parameters. If the wave is attenuated during propagation, through viscous loss of energy, the estimates of the elastic parameters will be incorrect. Reflection coefficients are also affected by viscous effects in the media which also can lead to incorrect estimates, [1, 33]. 1.4.3 Solutions to the viscoelastic wave equation There exist closed form solutions to the elastic wave equation for both transient and continuous sources in three dimensional homogeneous media [1] Frequency domain solutions are as easy to find for viscoelastic as for elastic materials but ....

....are also affected by viscous effects in the media which also can lead to incorrect estimates, 1, 33] 1.4. 3 Solutions to the viscoelastic wave equation There exist closed form solutions to the elastic wave equation for both transient and continuous sources in three dimensional homogeneous media [1]. Frequency domain solutions are as easy to find for viscoelastic as for elastic materials but transient time domain solutions are more difficult to find. There exist analytic solutions for viscoelastic materials with transient sources, 28] 77] The solutions show a main slower wave, which is ....

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K. Aki and P. G. Richards. Quantitative seismology - Theory and methods. W. H. Freeman & Co., 1980.

....the local power spectral density does indeed contain information about the large scale structure of the medium. In fact, just by looking at the surfaces, we can get a good idea of what the mean sound speed profile actually is. An interesting case, which is close to what happens in the earth [79], is a sound speed profile that FREQUENCY CONTENT OF RANDOMLY SCATTERED SIGNALS 72 increases linearly with depth from speed 1 at the surface to speed 2, say. Then we have turning points at all depths. The surface plots of Fig. 5.5 exhibits what we expect: a curved line of arrival and then a ....

....SCATTERED SIGNALS 83 eliminate u to obtain Deltap c 0 2 p = 3=2 f ( ffi 0 (z Gamma z s ) where c 0 is a constant. Let us first consider the usual point source problem DeltaOE c 2 OE = Gammaffi (x)ffi(y)ffi(z) Then its solution is given by the Weyl integral [79] OE = i 8 2 Z Z e i S S z d; where S = Delta x 1 c 2 Gamma 2 1=2 jzj with x = x; y) 1 ; 2 ) and Im Gamma 1=c 2 Gamma 2 Delta 1=2 0. Thus the solution for the time transformed pressure p will be p = Gamma 3=2 f( z i 8 2 Z Z ....

K. AKI and P.G. RICHARDS, Quantitative Seismology-Theory and Methods, Vol. 1, W. Freeman, 1980.

....combination of background parameters and slowness gives FP (z; p) The temporal frequency variable is denoted ; z 0 is the source depth, and q P is the p wave attenuation factor. Similar expressions define the geometric optics amplitudes A S and A d . For a reference to these calculations see [AKI and RICHARDS, 1980]. Inversion Algorithms We used two inversion algorithms in our work. The first, output least squares (OLS) inversion, has been described many times in the geophysical literature (see [TARANTOLA, 1987] In this algorithm, model parameters are adjusted to minimize the mean square error between ....

AKI, K. and RICHARDS, P. (1980). Quantitative Seismology: Theory and Methods. Freeman, San Francisco.

....The only alternative we are aware of that may achieve the same result as the multiresolution method is to use the fact that, in the case of ODE s, there is a preferred direction. For this preferred direction, an exact solution may be obtained by using propagator matrices for each layer (see e.g. [1] for a description of this technique) Once such a solution is computed, one can easily obtain its projection on all scales. The multiresolution homogenization method simply finds an equation on each scale which has as its solution the projection of the exact solution to that scale. Thus, ....

K. Aki and P. Richards. Quantitative Seismology: Theory and Methods. W.H. Freeman, San Francisco, 1980.

.... 1970; Gilbert 1972; Herman 1980; Natterer 1986] acoustical and optical tomography [Schomberg 1978; Kak 1984; Natterer 1986] radio astronomy [Bracewell and Riddle 1967] and seismic tomography [Nolet 1987] In geophysical applications to the whole earth [Aki, Christoffersson, and Husebye 1976; Aki and Richards 1980] or to smaller scale reconstruction problems such as borehole to borehole scanning with either seismic or electromagnetic probes [Bois, LaPorte, Lavergne, and Thomas 1972; Lager and Lytle 1977] the assumption of straight ray paths, although very common, is often a poor approximation [Dines and ....

....found the optimum slowness s b = fls in the given direction, we next attempt to improve the model by finding another direction in the slowness vector space that gives a still better fit to the traveltime data. As many others have done, we first compute a damped least squares [Marquardt 1963; Aki and Richards 1980; Nolet 1987; Bording, Gersztenkorn, Lines, Scales, and Treitel 1987; Scales, Gersztenkorn, and Treitel 1988] solution s . Next we note that both of the points found so far are guaranteed to lie in the nonfeasible part of the vector space at least one and generally about half of the ray paths ....

K. Aki and P. G. Richards (1980), Quantitative Seismology --- Theory and Methods, Vol. II, Freeman, San Francisco, Section 12.3.

....and Delta 0 denotes the dual space. Moreover we assume that ff k 6= 0 and k are distinct for 1 k N . In this system, the N point sources are assumed to start the vibration. For instance, this kind of point sources can be related with models in seismology (e.g. Aki and Richards [1]) Here we are engaged with the determination of point dislocation sources from boundary measurements: Let = t) and T 0, Gamma ae Omega be given. Then determine N 2 N , ff k 6= 0, 2 R, k 2 Omega Gamma 1 k N , from the normal derivative u (x; t) x 2 Gamma, 0 t T . Our ....

Aki., K. and Richards, P.G., Quantitative Seismology Theory and Methods, Volume I, Freeman, New York, 1980.

.... vertical boreholes in oil field applications [Rector, 1995] We could also consider the problem of inverting for wave slowness when the absolute traveltimes and locations of the sources must also be inferred from the data, as is normally the case in earthquake seismology for whole Earth structure [Aki and Richards, 1980]. 3.1 Wave slowness models When a sound wave or seismic wave is launched into a medium, it takes time for the influence of the wave to progress from a point close to the source to a more distant point. The time taken by the wave to travel from one point of interest to the next is called the ....

....be discussed later in this paper. 4 Feasibility constraints for density distribution in the Earth A rather different class of problems involves analysis of free oscillations of the Earth in order to deduce its density structure [MacDonald and Ness, 1961; Gilbert, 1971; Jordan and Anderson, 1974; Aki and Richards, 1980; Gilbert, 1980; Ben Menahem and Singh, 1981; Lapwood and Usami, 1981; Morelli and Dziewonski, 1987; Snieder, 1993] We concentrate on toroidal modes since they are independent of gravitational effects. Let B and S be certain functionals of the exciting eigenfunctions. Then, for each normal mode ....

K. Aki and P. G. Richards, Quantitative Seismology: Theory and Methods, Vol. I, Freeman, New York, 1980, Chapter 8, pp. 337--381.

....plane wave, and let # # denote the angle of incidence relative to the vertical axis. The direction of propagation in the lower layer is determined by Snell s law of refraction: the waves in the two layers must have the same apparent velocity , which is the reciprocal of the ray parameter [2]. The apparent velocity is the velocity of a point where a wavefront intersects a horizontal line or equivalently the velocity seen by an observer who restricts attention to a horizontal line. The apparent velocities in the two layers are c cos # and c # cos # # and hence we have c cos # = ....

K. Aki and P. G. Richards, Quantitative seismology: Theory and methods, vol. I, II, Freeman, San Francisco, 1980.

....be: E ae 1 0 (1 ) 2 oe Gamma1 Gamma 2 = ae = ae ; z 0 : 5. 27) Material parameter values in the region z 0, i.e. 0 ; ae, were chosen so that, in the absence of random fluctuations, the half space would correspond to the first layer of the Gutenberg Earth model [18]. This layer has a compressional wave velocity of 6.14 km. sec. a shear wave velocity of 3.55 km. sec. and a density of 2.74 gm. cm 3 . Then, adopting the length, velocity, and density scales mentioned in section 2, we obtained the following scaled, nondimensionalized material parameter ....

K. Aki and P. Richards, Quantitative Seismology - Theory and Methods, vol. I, W. Freeman, San Francisco (1980).

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K. Aki and P.G. Richards, Quantitative Seismology - Theory and Methods (W. H. Freeman, San Francisco, 1980), Vol. 1, 2.

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K. Aki and P. G. Richards, Quantitative Seismology: Theory and Methods, vol. II (Freeman, 10 New York, 1980).

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Aki, K., and Richards, P.G., 1980, Quantitative seismology: theory and methods, vol. 2: W. H. Freeman.

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Aki, K., Richards, P.G. (1980) Quantitative Seismology: Theory and Methods. Volumes I, II. W.H.Freeman and Company, San Francisco.

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