The Anelasticity of the Earth

The attenuation of seismic waves is one manifestation of the earth's anelasticity and is not unrelated to the response of the earth to stresses of longer duration. The well-known difficulties involved in the extraction of meaningful amplitude information from body waves have prevented an accurate determination of attenuation of seismic energy versus depth. Most of these difficulties are not present in surface wave and free oscillation measurements, but there are complexities of interpretation. A method is developed for the analysis of the amplitudes of dispersed wave trains and free oscillations which yields the anelasticity (Q) as a function of depth in the earth just as the frequency spectrum yields the elasticity-density structure. The advantages and limitations of the method are essentially identical to those of the dispersion method. The amplitude decay versus period for toroidal oscillations and Love waves was computed for a variety of hypothetical Q distributions in the earth. Those models which satisfy the available attenuation measurements have a broad, highly attenuating zone in the upper mantle and a high-Q lower mantle. The range of Q for shear waves in these models is from about 80 in the upper mantle to about 2000 in the lower mantle. A rapid increase in Q beginning at about 400 km seems to be a required feature. This is probably the most direct evidence for inhomogeneity, possibly a phase change, beginning at this depth. The details of this transition zone must await more accurate data on surface wave attenuation. The high Q of the lower mantle seems to imply temperatures substantially below the melting point, and it probably precludes extensive lower mantle convection. There is no need to invoke a frequency-dependent Q in order to satisfy available body and surface wave data in the period range 10 seconds to 30 minutes, although a Q that is frequency dependent cannot be ruled out

Dynamics in prestressed media with moving phase boundaries: a continuum theory of failure in solids

Spontaneous failure in a solid medium is described as a localized transition of the material from one physical state to another, characterized in part by contrasting rheological properties and density. Such a process is viewed as a local disordering of the relatively ordered structure of the solid due to any variety of causes, such as massive microfracturing or shear melting, and can be confined to a very thin zone, but nevertheless of finite volume such that a volumetric transition energy can be defined. This leads to the description of failure as a generalized phase transition in a prestressed continuum, with instability and transition zone growth being driven by the energy contributions from the relaxation of stress in the surrounding medium. Direct application of mass, momentum and energy conservation to such a generalized phase transition leads to ‘jump’ conditions specified on the growing boundary surface of the transition zone, that relate the rupture growth to discontinuous changes in the dynamic field variables across the failure zone boundary. These field discontinuities are, in turn, related to the localized changes in physical properties induced by failure. Dynamical conditions for rapid spontaneous failure growth in a stressed medium are investigated in some detail, and we find that the failure boundary growth can be simply expressed in terms of energy ‘failure condition’ and a dynamic growth condition specifying the rupture velocity. These results imply that the integral energy change associated with earthquakes is in the range 10^4−10^6 erg/g. Further the failure growth rate is shown to be expressible in terms of the rheological properties of the material before and after failure. For shear melting resulting in a low viscosity fluid, for example, the rupture velocity will be near the shear velocity of the original material. A general Green's function solution for the radiation due to stress relaxation in the medium surrounding the growing failure zone is given and provides the basis for detailed computations of the strain or displacement field changes due to spontaneous failure processes. In particular, it is shown that the jump conditions for the growing transition zone boundary appear naturally as surface integral terms over the boundary. Since these boundary conditions contain the failure rate explicitly, then these terms include effects that have not been represented in previous integral representations of the radiation field resulting from failure. Further, we show that the formal Green's integral representation for the dynamical wave field can be used with known, simple Green's functions to generate approximate solutions for complex failure processes occurring in media with inhomogeneous material properties and prestress

Variational Markov Chain Monte Carlo for Bayesian smoothing of non-linear niffusions

In this paper we develop set of novel Markov chain Monte Carlo algorithms for Bayesian smoothing of partially observed non-linear diffusion processes. The sampling algorithms developed herein use a deterministic approximation to the posterior distribution over paths as the proposal distribution for a mixture of an independence and a random walk sampler. The approximating distribution is sampled by simulating an optimized time-dependent linear diffusion process derived from the recently developed variational Gaussian process approximation method. Flexible blocking strategies are introduced to further improve mixing, and thus the efficiency, of the sampling algorithms. The algorithms are tested on two diffusion processes: one with double-well potential drift and another with SINE drift. The new algorithm's accuracy and efficiency is compared with state-of-the-art hybrid Monte Carlo based path sampling. It is shown that in practical, finite sample, applications the algorithm is accurate except in the presence of large observation errors and low observation densities, which lead to a multi-modal structure in the posterior distribution over paths. More importantly, the variational approximation assisted sampling algorithm outperforms hybrid Monte Carlo in terms of computational efficiency, except when the diffusion process is densely observed with small errors in which case both algorithms are equally efficient

Fine structure of the upper mantle

The spectral amplitudes and travel times of seismic body waves are used to determine mantle velocity structures appropriate to distinct structural provinces within the western continental United States. In addition to basic amplitude and time data, travel-time delays and Pn velocity data from other studies are used as constraints in the systematic inversion of the data for mantle structure. The regional structures for the upper mantle determined in this manner show collectively rather sharp zones of transition (high velocity gradients) near 150, 400, 650 km and possibly near 1000 km. Comparatively, the regional structures indicate strong lateral variations in the upper mantle structure down to 150 km and possibly as deep as 200 km. The structures appropriate to the Rocky Mountain and Colorado plateau physiographic provinces show low-velocity zones capped by high-velocity lid zones, with variability in both the lid and the low-velocity zone properties from province to province and within these provinces to a much lesser degree. The mantle properties obtained for the Basin and Range contrast sharply with the plateau and mountain structures, with the lid zone being very thin or absent and abnormally low velocities extending from, or very near, the base of a thin crust to 150 km. The velocity determinations are coupled with estimates of the variation of the intrinsic dissipation function (Q) as a function of depth and frequency. These results show a pronounced low-Q zone corresponding to the average low-velocity zone depth range for the velocity models. The data suggest a frequency-dependent Q, with Q increasing with frequency. In total the results of the study strongly suggest phase transitions in the mantle, including a partially melted region corresponding to the low-velocity zone, the latter being highly variable in its properties over the region studied and strongly correlated with tectonic activity

Detection, analysis, and interpretation of teleseismic signals: 1. Compressional Phases from the Salmon Event

The travel times and amplitude spectrums of first- and later-arrival P phases from the Salmon event are computed on the basis of polarization filter outputs. The interpretation of the P wave radiation field is made in terms of crust and mantle structure using the first- and later-arrival P phases and their amplitude spectrums. The observed seismic field corresponds with that expected from a symmetric, purely compressive source. The essential features of the observed travel times and amplitudes are explained in terms of regional mantle structures. These structures provide first-order fits to the observed data and are characterized by low-velocity zones which terminate with rapid and continuous increases in velocity near depths of 130 km. The velocity structures also show a strong velocity gradient near 330 km. The regional models differ most strongly in the relative extent and magnitude of the velocity decrease in the Iow-velocity zone

A Comparative Study of the Elastic Wave Radiation from Earthquakes and Underground Explosions

A detailed analysis of the surface wave radiation from two underground explosions (BILBY and SHOAL) and an earthquake near Fallon, Nevada, whose epicentre is only 60 km from SHOAL, indicates that: (1) at long periods the surface wave radiation from the earthquake can be explained by a pure quadrupole (double couple) source, but at higher frequencies the radiation pattern contains asymmetries which suggest effects due to rupture propagation; these would require higher-order multipole terms in the source equivalent representation; (2) the surface waves from the explosions can be explained by superimposed monopole and quadrupole sources, with no indication of higher-order multipole terms, at least in the period range comparable to that in which the earthquake signal was recorded; (3) the principal conclusion of this study is that the anomalous radiation from the explosions is probably due to stress relaxation around the shock-generated shatter zone and not due to earthquake triggering. Comparative analysis of SHOAL and FALLON shows that: (1) the ratio of the Love wave amplitude generated by the earthquake to the Love wave amplitude from the explosion increases with period, which implies a larger source dimension for FALLON; (2) the normalized spectral ratio of Love wave amplitude to Rayleigh wave amplitude, considered as a function of period, is nearly constant and close to unity for the explosions, but larger for the earthquake by a factor of two or three, and increasing with period. These differences might be useful in distinguishing earthquakes from explosions (at least in the magnitude range of the events used in this study, m_b 4.4 and above), as well as for estimating source parameters, such as stress, which are of fundamental geophysical interest

Attenuation of Seismic Energy in the Upper Mantle

The amplitude attenuation and phase dispersion for Love and Rayleigh waves in the period range 50 to 300 sec is determined from two earthquakes by digital techniques. A distribution of Q, or anelasticity, is determined for the upper mantle which satisfies the amplitude decay data for Love and Rayleigh waves and which is consistent with available body wave data. An argument is made for a longitudinal wave Q of about 2.4 to 2.6 times the Q for shear waves. This implies that very small losses are involved in pure compression compared to the losses in shear. This is an argument against the importance of certain mechanisms, such as thermoelastic losses, in the mantle. The Q for shear waves in the upper 400 km of the mantle seems to vary from about 50 to about 150. The Q for mantle Rayleigh waves is greater than the Q for mantle Love waves, both theoretically and experimentally. However, it is predicted that Q_R becomes less than Q_L at some period shorter than 50 sec, the crossover period being diagnostic of the thickness of the 'Q crust' or lithosphere

Theoretical Rayleigh and Love Waves from an Explosion in Prestressed Source Regions

Expressions and synthetics for Rayleigh and Love waves generated by theoretical tectonic release models are presented. The multipole formulas are given in terms of the strengths and time functions of the source potentials. This form of the Rayleigh and Love wave expressions is convenient for separating the contribution to the Rayleigh wave due to the compressional and shear-wave source radiation and the contribution of the upgoing and downgoing source radiation for both Rayleigh and Love waves. Because of the ease of using different compression and shear-wave source time functions, these formulas are especially suited for sources for which second- and higher-order moment tensors are needed to describe the source, such as the initial value cavity release problem. A frequently used model of tectonic release is a double couple superimposed on an explosion. Eventually, we will compare synthetics of this and more realistic models in order to determine for what dimensions of the tectonic release model this assumption is valid and whether the Rayleigh wave is most sensitive to the compressional or shear-wave source history. The pure shear cavity release model is a double couple with separate P- and S-wave source histories. The time scales are proportional to the source region's dimension and differ by their respective body-wave velocities. Thus, a convenient way to model the effect of differing shot point velocities and source dimensions is to run a suite of double-couple source history calculations for the P- and SV-wave sources separately and then sum the different combinations. One of the more interesting results from this analysis is that the well-known effect of vanishing Rayleigh-wave amplitude as a vertical or horizontal dip-slip double-couple model approaches the free surface is due to the destructive interference between the P- and SV-wave generated Rayleigh waves. The individual Rayleigh-wave amplitudes, unlike the SH-generated Love waves, are comparable in size to those from other double-couple orientations. This has important implications to the modeling of Rayleigh waves from shallow dipslip fault models. Also, the P-wave radiation from double-couple sources is a more efficient generator of Rayleigh waves than the associated SV wave or the P wave from explosions. The latter is probably due to the vertical radiation pattern or amplitude variation over the wave front. This effect should be similar to that of the interaction of wave-front curvature with the free surface