A complete suite of closed analytical expressions is presented for the surface displacements, strains, and tilts due to inclined shear and tensile faults in a half-space for both point and finite rectangular sources. These expressions are particularly compact and free from field singular points which are inherent in the previously stated expressions of certain cases. The expressions derived here represent powerful tools not only for the analysis of static field changes associated with earthquake occurrence but also for the modeling of deformation fields arising from fluid-driven crack sources.

/pdf/surface-deformation-due-to-shear-and-tensile-faults-in-a-2vckanzx6g.pdf

Surface deformation due to shear and tensile faults in a half-space

A complete set of closed analytical expressions is presented in a unified manner for the internal displacements and strains due to shear and tensile faults in a half-space for both point and finite rectangular sources. These expressions are particularly compact and systematically composed of terms representing deformations in an infinite medium, a term related to surface deformation and that is multiplied by the depth of observation point. Several practical suggestions to avoid mathematical singularities and computational instabilities are also presented. The expressions derived here represent powerful tools both for the observational and theoretical analyses of static field changes associated with earthquake and volcanic phenomena.

http://arrowsmith510.asu.edu/TheLectures/Lecture18/Okada_1992.pdf

Internal deformation due to shear and tensile faults in a half-space

The conventional magnitude scale M suffers saturation when the rupture dimension of the earthquake exceeds the wavelength of the seismic waves used for the magnitude determination (usually 5–50 km). This saturation leads to an inaccurate estimate of energy released in great earthquakes. To circumvent this problem the strain energy drop W (difference in strain energy before and after an earthquake) in great earthquakes is estimated from the seismic moment M_0. If the stress drop Δσ is complete, W = W_0 = (Δσ/2μ)M_0 ∼ M_0/(2×10^4), where μ is the rigidity; if it is partial, W_0 gives the minimum estimate of the strain energy drop. Furthermore, if Orowan's condition, i.e., that frictional stress equal final stress, is met, W_0 represents the seismic wave energy. A new magnitude scale M_w is defined in terms of W_0 through the standard energy-magnitude relation log W_0 = 1.5M_w + 11.8. M_w is as large as 9.5 for the 1960 Chilean earthquake and connects smoothly to M_s (surface wave magnitude) for earthquakes with a rupture dimension of about 100 km or less. The M_w scale does not suffer saturation and is a more adequate magnitude scale for great earthquakes. The seismic energy release curve defined by W_0 is entirely different from that previously estimated from Ms. During the 15-year period from 1950 to 1965 the annual average of W_0 is more than 1 order of magnitude larger than that during the periods from 1920 to 1950 and from 1965 to 1976. The temporal variation of the amplitude of the Chandler wobble correlates very well with the variation of W_0, with a slight indication of the former preceding the latter. In contrast, the number N of moderate to large earthquakes increased very sharply as the Chandler wobble amplitude increased but decreased very sharply during the period from 1945 to 1965, when W_0 was largest. One possible explanation for these correlations is that the increase in the wobble amplitude triggers worldwide seismic activity and accelerates plate motion which eventually leads to great decoupling earthquakes. This decoupling causes the decline of moderate to large earthquake activity. Changes in the rotation rate of the earth may be an important element in this mechanism.

/pdf/the-energy-release-in-great-earthquakes-27rg7m8mky.pdf

The energy release in great earthquakes

A Viking Lander 1 image was modeled as mixtures of reflectance spectra of palagonite dust, gray andesitelike rock, and a coarse rocklike soil. The rocks are covered to varying degrees by dust but otherwise appear unweathered. Rocklike soil occurs as lag deposits in deflation zones around stones and on top of a drift and as a layer in a trench dug by the lander. This soil probably is derived from the rocks by wind abrasion and/or spallation. Dust is the major component of the soil and covers most of the surface. The dust is unrelated spectrally to the rock but is equivalent to the global-scale dust observed telescopically. A new method was developed to model a multispectral image as mixtures of end-member spectra and to compare image spectra directly with laboratory reference spectra. The method for the first time uses shade and secondary illumination effects as spectral end-members; thus the effects of topography and illumination on all scales can be isolated or removed. The image was calibrated absolutely from the laboratory spectra, in close agreement with direct calibrations. The method has broad applications to interpreting multispectral images, including satellite images.

Spectral mixture modeling - A new analysis of rock and soil types at the Viking Lander 1 site. [on Mars]

Preface 1. Origin and history of the Solar System 2. Composition of the Earth 3. Radioactivity, isotopes and dating 4. Isotopic clues to the age and origin of the Solar System 5. Evidence of the Earth's evolutionary history 6. Rotation, figure of the Earth and gravity 7. Precession, wobble and rotational irregularities 8. Tides and the evolution of the lunar orbit 9. The satellite geoid, isostasy and post-glacial rebound 10. Elastic and inelastic properties 11. Deformation of the crust: rock mechanics 12. Tectonics 13. Convective and tectonic stresses 14. Kinematics of the earthquake process 15. Earthquake dynamics 16. Seismic wave propagation 17. Seismological determination of Earth structure 18. Finite strain and high pressure equations of state 19. Thermal properties 20. The surface heat flux 21. The global energy budget 22. Thermodynamics of convection 23. Thermal history 24. The geomagnetic field 25. Rock magnetism and paleomagnetism 26. Alternative energy sources and natural climate variations: some geophysical background Appendix A. General reference data Appendix B. Orbital dynamics (Kepler's laws) Appendix C. Spherical harmonic functions Appendix D. Relationships between elastic moduli of an isotropic solid Appendix E. Thermodynamic parameters and derivative properties Appendix F. An Earth model: mechanical properties Appendix G. A thermal model of the Earth Appendix H. Radioactive isotopes Appendix I. A geological time scale 2004 Appendix J. Problems References Index.

Physics of the Earth

The S transform, which is introduced in the present correspondence, is an extension of the ideas of the continuous wavelet transform (CWT) and is based on a moving and scalable localizing Gaussian window. It is shown to have some desirable characteristics that are absent in the continuous wavelet transform. The S transform is unique in that it provides frequency-dependent resolution while maintaining a direct relationship with the Fourier spectrum. These advantages of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis, whereas the localizing scalable Gaussian window dilates and translates.

Localization of the complex spectrum: the S transform

The S transform, an extension to the ideas of the Gabor transform and the Wavelet transform, is based on a moving and scalable localising Gaussian window and is shown here to have characteristics that are superior to either of the transforms. The S transform is fully convertible both forward and inverse from the time domain to the 2-D frequency translation (time) domain and to the familiar Fourier frequency domain. Parallel to the translation (time) axis, the S transform collapses as the Fourier transform. The amplitude frequency-time spectrum and the phase frequency-time spectrum are both useful in defining local spectral characteristics. The superior properties of the S transform are due to the fact that the modulating sinusoids are fixed with respect to the time axis while the localising scalable Gaussian window dilates and translates. As a result, the phase spectrum is absolute in the sense that it is always referred to the origin of the time axis, the fixed reference point. The real and imaginary spectrum can be localised independently with a resolution in time corresponding to the period of the basis functions in question. Changes in the absolute phase ofa constituent frequency can be followed along the time axis and useful information can be extracted. An analysis of a sum of two oppositely progressing chirp signals provides a spectacular example of the power of the S transform. Other examples of the applications of the Stransform to synthetic as well as real data are provided.

Localisation of the complex spectrum : The S transform

Closed analytical expressions for the displacement fields of inclined, finite strike-slip and dip-slip faults are given. They may be readily used in the numerical computation of displacements, and, by differentiation, strain and stress fields may be derived. The expressions are valid both at the surface and at depth.

The displacement fields of inclined faults

The S-transform is an invertible time-frequency spectral localization technique which combines elements of wavelet transforms and short-time Fourier transforms. In previous usage, the frequency dependence of the analyzing window of the S-transform has been through horizontal and vertical dilations of a basic functional form, usually a Gaussian. In this paper, we present a generalized S-transform in which two prescribed functions of frequency control the scale and the shape of the analyzing window, and apply it to determining P-wave arrival time in a noisy seismogram. The S-transform is also used as a time-frequency filter; this helps in determining the sign of the P arrival.

The S‐transform with windows of arbitrary and varying shape

An image is a function, f(x, y) , of the independent space variables x and y . The global Fourier spectrum of the image is a complex function F(k x , k y ) of the wave numbers k x and k y . The global spectrum may be viewed as a construct of the spectra of an arbitrary number of segments of f(x, y) , leading to the concept of a local spectrum at every point of f(x, y) . The two-dimensional S transform is introduced here as a method of computation of the local spectrum at every point of an image. In addition to the variables x and y , the 2-D S transform retains the variables k x and k y , being a complex function of four variables. Visualisation of a function of four variables is difficult. We skirt around this by removing one degree of freedom, through examination of ‘slices’. Each slice of the 2-D S transform would then be a complex function of three variables, with separate amplitude and phase components. By ranging through judiciously chosen slice locations the entire S transform can be examined. Images with strictly periodic patterns are best analysed with a global Fourier spectrum. On the other hand, the 2-D S transform would be more useful in spectral characterisation of aperiodic or random patterns.

Lalu Mansinha

Papers

Localization of the complex spectrum: the S transform

Localisation of the complex spectrum : The S transform

The displacement fields of inclined faults

The S‐transform with windows of arbitrary and varying shape

Pattern analysis with two-dimensional spectral localisation: Applications of two-dimensional S transforms