An application of normal mode theory to the retrieval of structural parameters and source mechanisms from seismic spectra

1975 ◽  
Vol 278 (1280) ◽  
pp. 187-269 ◽  
Keyword(s):  
Moment Tensor ◽  
The Other ◽  
Free Oscillations ◽  
Stress Release ◽  
Modal Data ◽  
Mechanical Structure ◽  
The Earth ◽  
Principal Axes ◽  
Source Mechanisms ◽  
The Moment

A cyclic process of refining models of the mechanical structure of the Earth and models of the mechanism of one or more earthquakes is developed. The theory of the elastic-gravitational free oscillations of the Earth is used to derive procedures for resolving nearly degenerate multiplets of normal modes. We show that a global network of seismographs (W.W.S.S.N.) permits resolution for angular orders l ≤ 76 and for frequencies a) w ≤ 0.090 s -1 . The peak or centre frequency of each nearly degenerate multiplet is interpreted to be a gross Earth datum. Together, the data are used to refine models of the mechanical structure of the Earth. The theory of free oscillations is used further to derive procedures for retrieving the mechanism, or moment tensor, of an earthquake point source. We show that a globa network of seismographs permits retrieval for frequencies 0.0125 s-1 ≤ w ≤ 0.0825 s-1 . We show that refined models of structure and mechanism lead to improved resolution and retrieval, and that an array of sources further complements the resolution of multiplets. We present a ‘standardized dataset’ of 1064 distinct, observed eigenfrequencies ol the elastic-gravitational free oscillations of the Earth. These gross-Earth data are compiled from 1461 modes reported in five studies: 2 modes reported by Derr (1969), 159 modes observed by Brune & Gilbert (1974), 240 modes observed by Mendiguren ( 1973), 248 modes observed by Dziewonski & Gilbert (1972,1973) and 812 modes reported here. It is our opinion that the establishment of a standardized dataset should precede the establishment of a standardized model of the Earth. Two new Earth models are presented that are compatible with the modal data. One is derived from model 508 (Gilbert & Dziewonski 1973) and the other from model B1 (Jordan & Anderson 1974). In the outer core and in the lower mantle, below a depth of about 950 km, the differences between the two models are negligibly small. In the inner core there are minor differences and in the upper mantle there are major differences in detail. The two models and the modal data are compatible with traditional ray data, provided that appropriate baseline corrections are made to the latter. The source mechanisms, or moment tensors, of two deep earthquakes, Colombia (1970 July 31) and Peru-Bolivia (1963 August 15), have been retrieved from the seismic spectra. In both cases the moment tensor possesses a compressive (implosive) isotropic part. There is good evidence that isotropic stress release begins gradually, over 80s before the origin time derived from the onset of short-period P and S waves. During the process of stress release the principal axes of the moment rate tensor migrate. The axis of compression is relatively stable, the compressive stress rate is dominant, and the other two axes rotate about the axis of compression. We speculate that earthquakes, occurring deep within descending lithospheric plates, are not sudden shearing movements alone but do exhibit compressive changes in volume such as would be associated with a phase change. We further speculate that compressive changes in volume may occur without sudden shearing movements, that there may be ' silent earthquakes’.

Using Seismic Source Parameters to Model Frequency-Dependent Surface-Wave Radiation Patterns

2020 ◽  
Vol 91 (2A) ◽  
pp. 992-1002 ◽  
Author(s):  
Boris Rösler ◽  
Suzan van der Lee
Keyword(s):  
Surface Wave ◽  
Moment Tensor ◽  
Source Parameters ◽  
Love Waves ◽  
Wave Radiation ◽  
Radiation Patterns ◽  
Frequency Dependent ◽  
The Earth ◽  
Rayleigh And Love Waves ◽  
Source Mechanisms

Abstract The excitation of surface waves depends on the frequency-dependent eigenfunctions of the Earth, which are determined numerically. As a consequence, radiation patterns of Rayleigh and Love waves cannot be calculated analytically and vary with source depth and with frequency. Owing to the importance of surface-wave amplitudes for inversions of source processes as well as studies of the elastic and anelastic structure of the Earth, assessing surface-wave radiation patterns for different source mechanisms is desirable. A data product developed in collaboration with the Incorporated Research Institutions for Seismology (IRIS) Consortium provides visualizations of the radiation patterns for Rayleigh and Love waves for all possible source mechanisms. Radiation patterns for known earthquakes are based on the moment tensors reported by the Global Centroid Moment Tensor project. These source mechanisms can be modified or moment tensor components can be chosen by the user to assess their effect on Rayleigh- and Love-wave radiation patterns.

A Discussion on the measurement and interpretation of changes of strain in the Earth - Derivation of source parameters from low-frequency spectra

1973 ◽  
Vol 274 (1239) ◽  
pp. 369-371 ◽  
Keyword(s):  
Focal Mechanism ◽  
Moment Tensor ◽  
Source Parameters ◽  
Low Frequency ◽  
Frequency Spectra ◽  
Isotropic Part ◽  
The Earth ◽  
Double Couple ◽  
The Moment ◽  
Deviatoric Part

By recording several components of tilt, strain and acceleration at one location, one can determine the focal mechanism, or moment tensor, of an earthquake. Alternatively, recordings made at several locations can be used. The moment tensor can be decomposed into its isotropic part and its deviatoric part. When the eigerrvalues of the deviator are in the sequence (— 1, 0, 1) the equivalent double couple can be found.

La riflessione impossibile e il rispecchiamento nel mondo. Dall’esperienza infantile alla surréflexion

2020 ◽  
Vol 22 ◽  
pp. 135-151
Author(s):  
Prisca Amoroso ◽  
Keyword(s):  
The Body ◽  
The Other ◽  
The Sun ◽  
The Earth ◽  
Way Of Knowing ◽  
The Moment

This essay builds on two questions: the relation of the child with the other and the child’s way of knowing, in which the resistance of the unreflected is not yet problematized. Through a reconstruction of Merleau-Ponty’s critique of Piaget’s idea of the child’s linear intellectual progression toward reflexive abstraction, I highlight the moment of unreflection by taking up the notion of ultra-thing, which Merleau-Ponty borrows from Henry Wallon. These ultra-things are entities with which the child entertains a vague relation and which always remain at the horizon of her perception without yet being possessed in a representation or grasped in a concept. They include, for example, the sun, the sky, the Earth, the body as an object, existence before the birth of the child – uninhabitable dimensions or, to the contrary, ones that are necessarily inhabited. The concept of ultra-thing has not been sufficiently explored in Merleau-Pontian studies and its importance remains underappreciated. This essay thus formulates a hypothesis about the relation between ultra-things and hyper-reflection.

Oxford Calculators

2018 ◽  
Author(s):  
Edith Dudley Sylla
Keyword(s):  
Fourteenth Century ◽  
Minimum Time ◽  
The Other ◽  
Heavy Body ◽  
The Earth ◽  
Maximum Time ◽  
Night And Day ◽  
The Moment

‘Oxford Calculators’ is a modern label for a group of thinkers at Oxford in the mid-fourteenth century, whose approach to problems was noticed in the immediately succeeding centuries because of their tendency to solve by ‘calculations’ all sorts of problems previously addressed by other methods. If for example the question was, what must a monk do to obey the precept of his abbot to pray night and day, a ‘calculator’ might immediately rephrase the question to ask whether there is a minimum time spent in prayer that would be sufficient to fulfil the abbot’s precept, or a maximum time spent that would be insufficient to fulfil the precept. Or, if grace was supposed to be both what enables a Christian to act meritoriously and a reward for having so acted, then a calculator might ask whether the degree of grace correlated with a meritorious act occurs at the moment of the meritorious act, before the act when the decision to act is being made, or after the act when the reward of increased grace is given. If a body was hot at one end but cold at the other, then a calculator might ask not whether it is to be labelled hot or cold, but how hot it is as a whole. Finally, if it was asked whether a heavy body acts as a whole or as the sum of its parts, then a calculator might take the case of a long thin rod falling through a tunnel pierced through the centre of the earth and attempt to calculate how the rod’s velocity would decrease as parts of the rod passed the center of the cosmos, if it acted as the sum of its parts. Of these four questions, the last two were asked by Richard Swineshead, a mid-fourteenth century fellow of Merton College, Oxford, whose Liber calculationum (Book of Calculations) led to his being given the name ‘Calculator’. By association with Richard Swineshead, other Oxford masters including Thomas Bradwardine, Richard Kilvington, William Heytesbury, Roger Swineshead and John Dumbleton have been labelled the ‘Oxford Calculators’. Their work contains a distinctive combination of logical and quantitative techniques, which results from the fact that it was often utilized in disputations on sophismata (de sophismatibus). This same group of thinkers, with emphasis on their mathematical rather than logical work, has been called the ‘Merton School’, because many but not all of the Calculators were associated with Merton College, Oxford. Besides calculatory works, the same authors wrote works in which calculatory techniques are not so prominent, including commentaries on Aristotle, mathematical compendia and commentaries on Peter Lombard’s Sentences.

Can plasticity explain microseismic source mechanisms?

2021 ◽  
Author(s):  
Viktoriya Yarushina ◽  
Alexander Minakov
Keyword(s):  
Seismic Moment ◽  
Field Experiments ◽  
Moment Tensor ◽  
Failure Process ◽  
Source Mechanism ◽  
Published Data ◽  
Seismic Moment Tensor ◽  
Special Cases ◽  
Source Mechanisms ◽  
The Moment

<p>The microseismic events can often be characterized by a complex non-double couple source mechanism. Recent laboratory studies recording the acoustic emission during rock deformation help connecting the components of the seismic moment tensor with the failure process. In this complementary contribution, we offer a mathematical model which can clarify these connections. We derive the seismic moment tensor based on classical continuum mechanics and plasticity theory. The moment tensor density can be represented by the product of elastic stiffness tensor and the plastic strain tensor. This representation of seismic sources has several useful properties: i) it accounts for incipient faulting as a microseismicity source mechanism, ii) it does not require a pre-defined fracture geometry, iii) it accounts for both shear and volumetric source mechanisms, iv) it is valid for general heterogeneous and anisotropic rocks, and v) it is consistent with elasto-plastic geomechanical simulators. We illustrate the new approach using 2D numerical examples of seismicity associated with cylindrical openings, analogous to wellbore, tunnel or fluid-rich conduit, and provide a simple analytic expression of the moment density tensor. We compare our simulation results with previously published data from laboratory and field experiments.  We consider three special cases corresponding to "dry" isotropic rocks, "dry" transversely isotropic rocks and "wet" isotropic rocks. The model highlights theoretical links between stress state, geomechanical parameters and conventional representations of the moment tensor such as Hudson source type parameters. </p>

Uncertainties in Seismic Moment Tensors Inferred from Differences between Global Catalogs

2021 ◽  
Author(s):  
Boris Rösler ◽  
Seth Stein ◽  
Bruce D. Spencer
Keyword(s):  
Moment Tensor ◽  
Centroid Moment Tensor ◽  
Moment Tensors ◽  
Double Couple ◽  
Source Mechanisms ◽  
Order Of Magnitude ◽  
The Difference ◽  
The Moment ◽  
Source Geometry ◽  
Scalar Moment

Abstract Catalogs of moment tensors form the foundation for a wide variety of seismological studies. However, assessing uncertainties in the moment tensors and the quantities derived from them is difficult. To gain insight, we compare 5000 moment tensors in the U.S. Geological Survey (USGS) and the Global Centroid Moment Tensor (Global CMT) Project catalogs for November 2015–December 2020 and use the differences to illustrate the uncertainties. The differences are typically an order of magnitude larger than the reported errors, suggesting that the errors substantially underestimate the uncertainty. The catalogs are generally consistent, with intriguing differences. Global CMT generally reports larger scalar moments than USGS, with the difference decreasing with magnitude. This difference is larger than and of the opposite sign from what is expected due to the different definitions of the scalar moment. Instead, the differences are intrinsic to the tensors, presumably in part due to different phases used in the inversions. The differences in double-couple components of source mechanisms and the fault angles derived from them decrease with magnitude. Non-double-couple (NDC) components decrease somewhat with magnitude. These components are moderately correlated between catalogs, with correlations stronger for larger earthquakes. Hence, small earthquakes often show large NDC components, but many have large uncertainties and are likely to be artifacts of the inversion. Conversely, larger earthquakes are less likely to have large NDC components, but these components are typically robust between catalogs. If so, these can indicate either true deviation from a double couple or source complexity. The differences between catalogs in scalar moment, source geometry, or NDC fraction of individual earthquakes are essentially uncorrelated, suggesting that the differences reflect the inversion rather than the source process. Despite the differences in moment tensors, the location and depth of the centroids are consistent between catalogs. Our results apply to earthquakes after 2012, before which many moment tensors were common to both catalogs.

MAGNETOSPHERIC TRANSMISSION FUNCTION TO SEPARATE NEAR EARTH PRIMARY AND SECONDARY COSMIC RAYS

2005 ◽  
Vol 20 (29) ◽  
pp. 6678-6680 ◽  
Author(s):  
P. BOBIK ◽  
M. BOSCHINI ◽  
M. GERVASI ◽  
D. GRANDI ◽  
E. MICELOTTA ◽  
...  
Keyword(s):  
Cosmic Rays ◽  
Field Model ◽  
Primary Particle ◽  
Charged Particles ◽  
Transmission Function ◽  
The Other ◽  
Solar Cycles ◽  
The Earth ◽  
The Moment

The main features of charged particles accessing the Earth magnetosphere have been studied by tracing their trajectories. The reconstruction code has allowed us to perform two simulations of Cosmic Rays (CRs) accessing the AMS detector, one for the 1998 data, and the other for the 2005 (at the moment, the IGRF data are available up to that year). The parameters of the external field model for 2005 have been estimated from the solar conditions in 1982 and 1984, two solar cycles before. The CRs have been assumed to be isotropically impinging on the AMS detector, flying at 400 km altitude with energies reproducing the AMS-01 observed spectrum. The computation of allowed and forbidden primary particle trajectories has enabled us the estimate of the Transmission Function in both periods. A comparison with the overall (primary and secondary) AMS-01 data provides by subtraction the determination of the secondary spectrum.

Note on Selection of Coordinate Systems for Geodynamics

1975 ◽  
Vol 26 ◽  
pp. 395-407
Author(s):  
S. Henriksen
Keyword(s):  
Sea Level ◽  
The Other ◽  
Coordinate Systems ◽  
Mean Sea Level ◽  
The Earth ◽  
The One ◽  
Static Systems ◽  
Selection Of

The first question to be answered, in seeking coordinate systems for geodynamics, is: what is geodynamics? The answer is, of course, that geodynamics is that part of geophysics which is concerned with movements of the Earth, as opposed to geostatics which is the physics of the stationary Earth. But as far as we know, there is no stationary Earth – epur sic monere. So geodynamics is actually coextensive with geophysics, and coordinate systems suitable for the one should be suitable for the other. At the present time, there are not many coordinate systems, if any, that can be identified with a static Earth. Certainly the only coordinate of aeronomic (atmospheric) interest is the height, and this is usually either as geodynamic height or as pressure. In oceanology, the most important coordinate is depth, and this, like heights in the atmosphere, is expressed as metric depth from mean sea level, as geodynamic depth, or as pressure. Only for the earth do we find “static” systems in use, ana even here there is real question as to whether the systems are dynamic or static. So it would seem that our answer to the question, of what kind, of coordinate systems are we seeking, must be that we are looking for the same systems as are used in geophysics, and these systems are dynamic in nature already – that is, their definition involvestime.

The Final ‘Thank You’

Derrida Today ◽  
2010 ◽  
Vol 3 (1) ◽  
pp. 21-36
Author(s):  
Grant Farred
Keyword(s):  
South Africa ◽  
South African ◽  
Jacques Derrida ◽  
Political Transition ◽  
The Other ◽  
World Cup ◽  
The Core ◽  
Nelson Mandela ◽  
Black President ◽  
The Moment

‘The Final “Thank You”’ uses the work of Jacques Derrida and Friedrich Nietzsche to think the occasion of the 1995 rugby World Cup, hosted by the newly democratic South Africa. This paper deploys Nietzsche's Zarathustra to critique how a figure such as Nelson Mandela is understood as a ‘Superman’ or an ‘Overhuman’ in the moment of political transition. The philosophical focus of the paper, however, turns on the ‘thank yous’ exchanged by the white South African rugby captain, François Pienaar, and the black president at the event of the Springbok victory. It is the value, and the proximity and negation, of the ‘thank yous’ – the relation of one to the other – that constitutes the core of the article. 1

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