Abstract. 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 im·olved in pure compre~sion 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. HoweYer, it is predicted that QR becomes less than QL at some period shorter than 50 sec, the crossover period being diagnostic of the thickness of the 'Q crust' or lithosphere.

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 P, 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.

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.

The most important and interesting source of elastic radiation in geophysics is an earthquake, or tectonic source, because the radiation field from such an energy source provides information on the largely unknown stress field within the earth. The actual mechanisms or processes of material failure undoubtedly can be described parametrically by the radiation field in terms of rupture velocity, rupture geometry, and the initial and residual stress within the region of failure. Accurate estimates of stress and the parameters of failure are therefore of particular significance in any description of the physical state of the material and would not be unrelated to the larger‐scale dynamical processes taking place within the earth. A number of methods and theories are presently used in estimating some of these parameters. The present study is intended to extend the dynamical theory of tectonic sources in order to provide a more complete description of earthquakes in terms of these basic parameters of rupture, including prestress. No assumptions are made concerning the nature of equivalent forces at the source or of their time dependence. The theory predicts the spatial and temporal form of the radiation field in terms of the initial prestress field and the basic rupture parameters. These predictions follow from the recognition that an earthquake is a relaxation source and that such a phenomenon is described analytically as an initial‐value problem. Consequently, such a source satisfies the conservation of energy and linear and angular momentum conditions required for a spontaneous source. The radiation field is produced by the continuous reduction of stored potential strain energy in the elastic medium surrounding a growing rupture zone, where it is assumed that the rupture, or at least a part of the total rupture zone, has a well‐defined boundary at a given time to which boundary conditions are applicable. The compatibility of this geometrically sharp, time‐varying boundary condition with probable failure processes in the earth is examined and judged to be good. Analytical expressions for the radiation field from an arbitrary source of elastic radiation are given, and within the framework of this formulation the properties of a spontaneous tectonic source are contrasted with ‘applied force’ sources and their special properties, as well as with some of the field observations of earthquake radiation fields. These considerations demonstrate the need for a more general and complete description of tectonic sources in order to explain all the observations and, more fundamentally, to deduce more precisely the nature of the physical processes of failure in the earth. It is concluded that a relaxation theory will provide the flexibility required to describe the characteristics of the observed radiation field and will also provide estimates of rupture parameters bearing on the processes of failure and the state of the material. A complete development of the dynamical relaxation theory for tectonic sources, including considerati...

Stress wave radiation from underground explosions has been observed to contain an anomalous shear wave contribution which is most likely of tectonic origin. In this paper the theoretical radiation field to be expected from an explosion in a prestressed medium is given under the assumption that no secondary low symmetry faulting on a large scale occurs and that the total tectonic component of the field is due to stress relaxation around the roughly spherical fracture zone created by the explosive shock wave. Evidence for the occurrence of this simple kind of tectonic source is considered, and it is concluded that this model is appropriate in many, if not most, instances involving underground explosions. Expressions for the spectrum of the radiation field and its spatial radiation pattern are given in terms of multipole expansions for the components of the rotation potential and the dilatation potential. Several possible rupture formation models are treated. All models show that the tectonic radiation is of simple quadrupole form, as has been observed. The energy radiated due to stress relaxation is considered in detail, and it is also shown that, in terms of the energy released, a dislocation source can be used as an equivalent for the stress relaxation effects.The theoretical energy partition between compressional and shear waves for the tectonic field is in the ratio of (approximately) 1 to 10, so that tectonic stress release does not affect the direct compressional body wave particulariy, but gives rise to totally anomalous SH polarized waves (e.g. Love waves) and affects Rayleigh type surface waves significantly, as is also observed. The theory can be applied to obtain estimates of source dimensions and the orientation and magnitude of the initial prestress field in the region of the explosion. In addition, application of this particular form of the general tectonic source theory to deep earthquakes and volcanic earthquakes also appears to be reasonable in view of the probable high symmetry of the failure or phase transition regions for such events.