Preface 1. Basic tools 2. Elasticity and Hooke's law 3. Seismic wave propagation 4. Effective media 5. Granular media 6. Fluid effects on wave propagation 7. Empirical relations 8. Flow and diffusion 9. Electrical properties Appendices.

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The Rock Physics Handbook: Tools for Seismic Analysis of Porous Media

Responding to the latest developments in rock physics research, this popular reference book has been thoroughly updated while retaining its comprehensive coverage of the fundamental theory, concepts, and laboratory results. It brings together the vast literature from the field to address the relationships between geophysical observations and the underlying physical properties of Earth materials - including water, hydrocarbons, gases, minerals, rocks, ice, magma and methane hydrates. This third edition includes expanded coverage of topics such as effective medium models, viscoelasticity, attenuation, anisotropy, electrical-elastic cross relations, and highlights applications in unconventional reservoirs. Appendices have been enhanced with new materials and properties, while worked examples (supplemented by online datasets and MATLAB® codes) enable readers to implement the workflows and models in practice. This significantly revised edition will continue to be the go-to reference for students and researchers interested in rock physics, near-surface geophysics, seismology, and professionals in the oil and gas industries.

The Rock Physics Handbook

Preface 1. Introduction 2. The elastodynamics and its simple solutions 3. Seismic rays and travel times 4. Dynamic ray tracing. Paraxial ray methods 5. Ray amplitudes 6. Ray synthetic seismograms Appendix. Fourier transform, Hilbert transform and analytical signals References Index.

Seismic Ray Theory

Our understanding of the dynamics of plate motions is based almost entirely upon modeling of present-day plate motions A fuller understanding, how- ever, can be derived from consideration of the history of plate motions Here we investigate the kinematics of the last 120 Myr of plate motions and the dynamics of Cenozoic motions, paying special attention to changes in the character of plate motions and plate-driving forces We analyze the partitioning of the observed surface velocity field into toroidal (transform/spin) and poloidal (spreading/subduction) motions The present-day field is not equipartitioned in poloidal and toroidal compo- nents; toroidal motions account for only one third of the total The toroidal/poloidal ratio has changed substan- tially in the last 120 Myr with poloidal motion decreasing significantly after 43 Ma while toroidal motion remains essentially constant; this result is not explained by changes in plate geometry alone We develop a self- consistent model of plate motions by (1) constructing a straightforward model of mantle density heterogeneity based largely upon subduction history and then (2) cal- culating the induced plate motions for each stage of the Cenozoic The "slab" heterogeneity model compares rather well with seismic heterogeneity models, especially away from the thermochemical boundary layers near the surface and core-mantle boundary The slab model pre- dicts the observed geoid extremely well, although com- parison between predicted and observed dynamic topog- raphy is ambiguous The midmantle heterogeneities that explain much of the observed seismic heterogeneity and geoid are derived largely from late Mesozoic and early Cenozoic subduction, when subduction rates were much higher than they are at present The plate motion model itself successfully predicts Cenozoic plate motions (glob- al correlations of 07-09) for mantle viscosity structures that are consistent with a variety of geophysical studies We conclude that the main plate-driving forces come from subducted slabs (! 90%), with forces due to litho- spheric effects (eg, oceanic plate thickening) providing a very minor component (" 10%) For whole mantle convection, most of the slab buoyancy forces are derived from lower mantle slabs Unfortunately, we cannot re- produce the toroidal/poloidal partitioning ratios ob- served for the Cenozoic, nor do our models explain apparently sudden plate motion changes that define stage boundaries The most conspicuous failure is our inability to reproduce the westward jerk of the Pacific plate at 43 Ma implied by the great bend in the Hawai- ian-Emperor seamount chain Our model permits an interesting test of the hypothesis that the collision of India with Asia may have caused the Hawaiian-Emperor bend However, we find that this collision has no effect on the motion of the Pacific plate, implying that impor- tant plate boundary effects are missing in our models Future progress in understanding global plate motions requires (1) more complete plate reconstruction infor- mation, including, especially, uncertainty estimates for past plate boundaries, (2) better treatment of plate boundary fault mechanics in plate motion models, (3) application of numerical convection models, constrained by global plate motion histories, to replace ad hoc man- tle heterogeneity models, (4) better calibration of these heterogeneity models with seismic heterogeneity con- straints, and (5) more comprehensive comparison of global plate/mantle dynamics models with geologic data, especially indicators of intraplate stress and strain, and constraints on dynamic topography derived from the stratigraphic record of sea level change

https://courses.washington.edu/ess502/Lithgow-BertelloniRichards1998RG.pdf

The dynamics of cenozoic and mesozoic plate motions

The purpose of this paper is twofold: from one side we provide a general survey to the viscoelastic models constructed via fractional calculus and from the other side we intend to analyze the basic fractional models as far as their creep, relaxation and viscosity properties are considered. The basic models are those that generalize via derivatives of fractional order the classical mechanical models characterized by two, three and four parameters, that we refer to as Kelvin–Voigt, Maxwell, Zener, anti–Zener and Burgers. For each fractional model we provide plots of the creep compliance, relaxation modulus and effective viscosity in non dimensional form in terms of a suitable time scale for different values of the order of fractional derivative. We also discuss the role of the order of fractional derivative in modifying the properties of the classical models.

Creep, relaxation and viscosity properties for basic fractional models in rheology

SUMMARY A ray-theoretical relation is established between the autocorrelation function of the slowness fluctuations of a random medium and the autocorrelation function of the traveltime fluctuations on a profile perpendicular to the general propagation direction of an originally plane wave. Although this relation can be inverted exactly, it is preferable for applications to use the results of a forward calculation for a modified exponential autocorrelation function which represents slowness fluctuations with zero mean. The essential parameters of this autocorrelation function, standard deviation E and correlation distance a, follow by simple relations from the maximum and the zero crossing of the corresponding autocorrelation function of the traveltime fluctuations. The traveltime analysis of 2-D finite-difference seismograms shows that E and a can be reconstructed successfully, if the wavelength-tocorrelation-distance ratio is 0.5 or less. Otherwise, E is underestimated and a overestimated; however, both effects can be compensated for. The average traveltime, as determined from the finite-difference seismograms, is slightly, but systematically shorter than the traveltime according to the average slowness, i.e. the wave prefers fast paths through the medium. This is in agreement with results of Wielandt (1987) for a spherical low-velocity inclusion in a full-space and with results of Soviet authors, summarized by Petersen (1990). The velocity shift is proportional to E*, it has dispersion similar to the dispersion related to anelasticity, and it increases with the pathlength of the wave.

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Seismic-wave traveltimes in random media

Accepted 1989 February 1. Received 1989 January 28; in original form 1988 September 26 SUMMARY paper investigates four generalized Maxwebl bodies (GMBs) and the classical Maxwell body (CMB) with respect to their potential to explain postglacial uplift. The GMBs are the Lomnitz body, defined by a logarithmic creep law, the Q power-law body, defined by a power-law dependence of the quality factor on frequency, the Caputo body, defined by a special stress-strain relation with memory, and the Burgers body. The Burgers body is a 4-parameter model, the other GMBs have 3 and the CMB has only 2 parameters. Theoretical uplift curves are calculated for simple space-time models of the Canadian and Fennoscandian deglaciations, and hedgehog inversion yields the successful model parameters. The GMBs explain the data better than the CMB. The parameters of the Burgers body are only rather weakly constrained because of trade-offs among them. From the success of the GMBs we conclude that long time-scale transient creep is a possible rheological behaviour of mantle rocks. Viscosity extrapolation from the time constant of postglacial uplift, about 10 ka, to the time constant of mantle convection, roughly 100 Ma, is made with the effective viscosity and yields approximately ld2 Pas (Q power-law body), loz3 Pas (Caputo body) and loz5 Pas (Lomnitz body). Hence, postglacial uplift does not strongly constrain the long time-scale mantle viscosity. It provides only a lower bound, about loz1 Pas; this value follows from data interpretation with the CMB model in which steady-state creep is present alone. Rayleigh number estimates for whole-mantle convection are clearly supercritical for the CMB, the Q power-law body and the Caputo body, but only about critical or even subcritical for the Lomnitz body. Both postglacial uplift and mantle convection is satisfactorily explained by the CMB, the Q power-law body and the Caputo body.

Rheological models and interpretation of postglacial uplift

SUMMARY Seismic waves in a random medium (with standard deviation E and correlation distance a of the relative slowness fluctuations) prefer fast paths, and therefore the apparent velocity of wave propagation is larger than the velocity which corresponds to the volume average of slowness. This velocity shift can be determined by ray perturbation theory (Snieder & Sambridge 1992), by the Huygens method (Podvin & Lecomte 1991) and by wave theory (Muller, Roth & Korn 1992). We apply all three methods to plane-wave propagation through a 2-D acoustic medium with Gaussian or exponential autocorrelation function of the slowness fluctuations. Ray perturbation theory gives numerical and analytical results, but has path-length (L) limitations. The Huygens method, which also gives the ray-theoretical velocity shift, can be used for Lla ratios of seismological interest. Wave theory shows that the velocity shift also depends on the wavelength A and that for Ala less than about 0.1 the velocity shift agrees with the result of the Huygens method. For Ala = 1 the wave-theoretical (i.e. true) shift is lower than the Huygens-method shift by a factor of 0.25 to 0.5. Simple formulae for the E dependence of the Huygens-method shift at long path lengths (Lla 2 80) are given, and a correction factor is derived which approximately transforms plane-wave 2-D into spherical-wave 3-D velocity shifts; the latter correspond to 3-D two-point ray tracing. For short-period seismic waves, propagating to teleseismic distances, mantle heterogeneity with E = 1 per cent and a = 100 km produces a velocity shift of about 0.2 per cent. Shifts of this order can explain the difference in earth models, derived from free oscillations on the one hand and from short-period body waves on the other. A velocity shift (or velocity dispersion) due t o anelasticity would be

Velocity shift in random media

Summary. We investigate a recent suggestion of Lay & Helmberger that there is a world-wide discontinuity in 5’-wave velocity, about 250-300 km above the core-mantle boundary, with an increase of 2.5-3 per cent from above to below. Their model predicts pronounced pulse-form changes of long-period SH body waves in the epicentral distance range 95-120”, a range which has not been studied by them. Similar pulse-form changes are expected in P-wave seismograms, under the reasonable assumption of a similar increase in P velocity. Nine long-period seismogram sections of SH- and P-waves from deepfocus earthquakes below Fiji, West Brazil, the Sea of Japan, the Sea of Okhotsk and the Banda Sea are presented with practically no indication of the complications expected. Also, the decay constants of pdiff, as predicted by the new model, differ strongly from observed values. Thus, there is clear evidence against this model. The pulse-form peculiarities between mantle S and ScS in transverse-component seismograms at distances from 70” to 80°, which Lay & Helmberger interpreted as reflections from the discontinuity of their model, must have a different cause.

Evidence against a discontinuity at the top of D

SUMMARY

Plane-wave ray tracing has been performed through 2-D random media with Gaussian and exponential autocorrelation functions of the slowness perturbations. The standard deviation is ɛ, correlation length a and propagation distance L; computations have been performed for ɛ= 1 and 3 per cent and for L/a= 2.5 to 60. Up to L/a= 5 or 10, regular focusing and defocusing of the rays is observed. Then, irregular behaviour develops, apparently without foci, but with increasing deviations q from the straight rays of a homogeneous medium. Two neighbouring rays at the top of a rectangular cross-section usually arrive far from each other at the bottom, and rays can even turn. The standard deviation σ of q, normalized by L, is approximately σ/L=ɛ(L/a)1/2.



From the traveltimes of the rays at the bottom of the cross-section, the first arrivals were determined and compared with the first arrivals according to the Huygens method of Podvin & Lecomte (1991). Both ray tracing and the Huygens method are high-frequency methods, but their traveltimes do not always agree. The reason is that the Huygens method gives first arrivals including diffractions around the slow parts of the structure, whereas ray tracing, in the sense of initial-value ray tracing, gives only transmitted rays. As a consequence, traveltimes calculated by ray tracing are systematically late with respect to the Huygens-method traveltimes. The difference becomes significant for L/a greater than 10 to 20, it increases with ɛ, and it is more pronounced for exponential than for Gaussian media. For instance, the velocity shift (with respect to the inverse of the volume average of the slowness) for an exponential medium, L/a= 60 and ɛ= 3 per cent, is –0.5 per cent according to ray tracing, whereas it is + 1.3 per cent according to the Huygens method. For ɛ= 1 per cent the corresponding numbers are 0 and +0.3 per cent, i.e. the difference between ray tracing and the Huygens method is still significant.



A conclusion from our calculations is that, even in the mildly laterally heterogeneous earth models of seismic tomography, ray tracing may give traveltimes that are significantly biased, i.e. overestimated. Another result is that, in random media, ray-tracing first-arrival traveltimes often have much stronger short-scale spatial variations than Huygens-method traveltimes which include wavefront healing due to diffractions. Both effects, traveltime bias and short-scale variations, limit the applicability of ray tracing already for weak random media.

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G. Müller

Papers

Seismic-wave traveltimes in random media

Rheological models and interpretation of postglacial uplift

Velocity shift in random media

Evidence against a discontinuity at the top of D

Ray tracing in random media