This additional manual quality control reduced the. The moment tensor is linearly related to the potency tensor through the fourthorder elastic moduli tensor e. The stress tensor and the moment tensor are very similar ideas. Moment tensor components can be dealt with centroidbased clustering.
Im going to write this equation in matrix form, and well play with it a little bit before we see where the inertia tensor. Moment tensor moment tensors are well known in other areas of mechanics where they are usually called spontaneous inelastic stresses or transformational stresses following a nomen clature introduced by eshelby. The moment tensor solution is typically displayed graphically using a socalled beachball diagram. Momenttensor summation to derive the active crustal. The moment of inertia tensor engineering libretexts. Seismic moment tensors are able to describe whether the failure is shearing or tearing i. It is in theory possible to solve a full moment tensor from inversion of a few seismograms, using normalmode data, surface waves or body waves.
Defining the scalar moment of a seismic source with a. The pattern of energy radiated during an earthquake with a single direction of motion on a single fault plane may be modelled as a double couple, which is described mathematically as a special case of a second order tensor similar to those for stress and strain known as the moment tensor. From extensive studies of moment tensor sources, it appears that many seismic sources are very well represented by an almost pure doublecouple model with m 1. Determination and uncertainty of moment tensors for. Bayesian inference and interpretation of centroid moment tensors of. Convert moment tensor components to strike, dip and rake moment tensor construction is shown by using the moment components in northeastdown coordinate system convention and conversion to strike, dip and rake. Any moment rate tensor can be expressed as mo 2mtomo, where mt is the total scalar moment and ivi is the source mechanism, a symmetric tensor of unit euclidean norm describing the. The definition of the moment of inertia for an area involves an integral of the form. Moment tensor solutions the moment tensor solution is typically displayed graphically using a socalled beachball diagram. An atomic unit of electric dipole moment is a0e, where a0 is the radius of the first bohr orbit for hydrogen and e is the magnitude of the electronic charge. Quadrupole and higher moments there are two common definitions for the quadrupole moment of a charge distribution. Rapid fullwave centroid moment tensor cmt inversion in. Pdf deriving source time function and moment tensor from. Astronomy research centre arc at the university of victoria.
Furthermore, our definitions of m t 0, m i and m d are in accord with the parameterization of the moment tensor into a deviatoric part represented by t which lies in. A positive outcome constitutes the first time that a significant body of observational evidence supports the hypothesis of a volumetric component due to source medium damage by. Get the scalar moment of the moment tensor frobenius norm based the scalar moment is calculated based on the euclidean frobenius norm silver and jordan, 1982. Source mechanism and dc moment tensor inversion the seismicity data are from both surface and downhole monitoring networks figure 1. Mar 25, 2011 the following is an analysis of moment tensor inversion results for nevada test site nts explosions in which an interpretation of the results is put to a test. Using the definitions of the normal and slip vectors in terms of fault strike, dip and slip directions, it is straight forward to write the moment tensor for any fault. I and j are used as symbols for denoting moment of inertia. However, a brief discussion of internal forces in slender members will be provided in section 9. The elements of the moment inertia can be written as a 3. The moment of inertia describes the angular acceleration produced by an applied torque. On the resolution of the isotropic component in moment tensor.
Express letter parametrization of general seismic potency and. Express letter parametrization of general seismic potency. Moment tensors th e seismic moment tensor smt is a symmetric, rank 2 adam baig and ted urbancic, engineering seismology group canada tensor, which therefore has six independent components. The definition of general moment tensors is given by the volume integral of moment tensor density m i j x over a source region v s. A firstorder tensor is simply another name for a vector u. Encyclopedia of physical science and technology third edition, 2003. Tensor project have estimated the moment tensor solution for earthquake m 5. To parametrize the source type from the moment tensor, we note that the tensor has three real eigenvalues. As an example, we present the application of som to moment tensors calculated for.
The first and least common in chemistry equates the quadrupole moment tensor with the second moment tensor a defined above. Types moment tensor focalmechanism with moment tensor parameters. A moment tensor m defines the strength of a seismic source in terms of its seismic moment, usually denoted by the scalar quantity m. Our cmt inversion algorithm does not require manual intervention, when combined with. Here we infer centroid moment tensors cmts for 11 events with m.
Im going to write this equation in matrix form, and well play with it a little bit before we see where the inertia tensor comes from. Mar 12, 2014 where repeated indices are summed over, and iij, the moment of inertia7, is given by iij m h x2. Model waveform depends only on the six elements of the moment tensor. While we have seen that the computational molecules from chapter 1 can be written as tensor products, not all computational molecules can be written as tensor products. Physically, this is a representation of a shear dislocation source without any volume changes. Moment tensor inversion of induced seismicity under. Much as a stress tensor describes the state of stress at a particular point, a moment tensor describes the deformation at the source location that generates seismic waves. Physical interpretation of moment tensor and the energetics.
Force is a vector it has a magnitude specified in newtons, or lbf, or whatever, and a direction. The moments of inertia for the entire area are obtained by integration. The forces described by the moment tensor are not the actual forces acting at the source because the moment tensor. Tensors you cant walk across a room without using a tensor the pressure tensor.
The moment tensor description is not restricted to earthquake sources, but covers also other types of seismic sources such as explosions. The sourcetime function is often assumed to have a simple form, a single pulse appearing in all elements of the moment tensor. Seismic moment tensors smts are represented by beachballs. One method of graphically representing the mechanism as described by the moment tensor is. The moment tensor represents equivalent body forces of a seismic source. The moment tensor characterizes the event magnitude, fracture type e. Most of the commonly used properties of moment functions and the mathematical framework for the derivation of basic theorems and results are discussed in detail. Moment tensor inversion of induced seismicity under different. The second definition and the one we will use almost exclusively defines it as 22 11 3 222 nn. Centroid moment tensor an overview sciencedirect topics. We first construct a greens function library, and calculate the synthetic seismograms for a point moment tensor source using the discrete wavenumber integration method bouchon, 1981.
There are several definitions, but it is normally stated in the traceless form i. Rapid fullwave centroid moment tensor oxford academic journals. Parametrization of general seismic potency and moment. Compute the moments of inertia of the bounding rectangle and halfcircle with respect to the x axis. The most commonly used specification of seismic sources is the moment tensor defined as the integral of the stress glut over the source volume e. A moment tensor is a representation of the source of a seismic event. Diagonal elements represent force couples that describe volumetric changes.
Moment tensor decompositions revisited springerlink. The general definition of the moment of inertia matrix is. The quadrupole moment tensor q is a ranktwo tensor3. The quantity 2 miri is the second moment of the i th mass with respect to or about the axis, and the sum 2. Defining the scalar moment of a seismic source with a general. The moment tensor description is not restricted to earthquake sources, but covers. The moment of inertia of the shaded area is obtained by subtracting the moment of inertia of the halfcircle from the moment of inertia of the. You cant align the wheels on your car without using a tensor the inertia tensor. This catalog contains moment tensor solutions for over 10,000 large earthquakes since 1977. Matrices that satisfy this restriction are called hermitian for such matrices, the principal moments can always be found, and they are always real see proof in text iij iij this mathematics will come up again in quantum mechanics principal moments. It depends on the shape and mass distribution of the body. V s m i j x d v gilbert 1971 chose stress drop difference between static stresses before and after an event as the moment tensor density, while backus and mulcahy 1976a chose stress glut difference between model stress and actual physical stress.
The first part discusses the fundamental concepts behind important moments such as geometric moments, complex moments, legendre moments, zernike moments, and moment tensors. Nov 20, 2019 the definition of general moment tensors is given by the volume integral of moment tensor density m i j x over a source region v s. Oct 16, 2014 the moment tensor describes equivalent body forces acting at a seismic point source burridge and knopoff 1964 and is a basic quantity evaluated for earthquakes on all scales from acoustic emissions to large devastating earthquakes. As previously mentioned, most tectonic seismicity is caused. The eigenvalue represents the moment of the dipole. Determine the moment of inertia of the shaded area with respect to the x axis. Name psmeca plot focal mechanisms on maps synopsis psmeca filesjparametersrwesteastsouthnorth btickinfo cpen ddepmindepmax efill gfill. It is the output of the function iafter the independent variable. Topic moment tensor inversion and moment tensor interpretation. Moment tensor decomposition is a method for deriving the isotropic iso, doublecouple dc, and compensated linear vector dipole clvd components from a seismic moment tensor. The above equation is solved using linear least squares for a given source depth.
It is a basic quantity evaluated for earthquakes on all scales from acoustic emissions to large devastating earthquakes see entries longperiod momenttensor inversion. We denote a point source moment tensor by m, a moment tensor density by m. In this distribution only the deviatoric seismic moment tensor is solved for, and the inversion yields the m ij which is decomposed into the scalar seismic moment, a doublecouple moment tensor and a compensated linear vector dipole moment tensor. Mg0 is the scalar moment of the nth focal mechanism, and f is a function of the strike, dip, and rake of this focal mechanism aki and richards, 1980. Moment functions in image analysis theory and applications.
Th ese eigenvalues determine the proportion of isotropic, compensated linear vector dipole, and doublecouple contributions to the full moment tensor. Overview of momenttensor inversion of microseismic events. The concept of moment of inertia was introduced by leonhard euler. The harvard centroid moment tensor catalog department of. The tensor of inertia gives us an idea about how the mass is distributed in a rigid body. Towards seismic moment tensor inversion for source mechanism. Parametrization of general seismic potency and moment tensors. The resistance that is shown by the object to change its rotation is called moment of inertia.
A great effort for calculating moment tensors for deeper sources has been made by several authors in recent years. The decomposition of moment tensors into isotropic iso, doublecouple dc and compensated linear vector dipole clvd components is a tool for classifying and physically interpreting seismic. Faultplane solutions from moment tensor inversion for. We determined the moment tensor of a subset of the induced earthquakes using both dc and full moment tensor models. Properties beachballsource the original author of this focal mechanism networkauthor code. We also acknowledge previous national science foundation support under grant numbers 1246120, 1525057, and 14739.
For an analogy, retreat to the case of a real valued function. Another key factor in recovering seismic moment tensor is the determination of the time variation of the source, st. The scalar moment returned by this function differs from the standard decomposition based definition of the scalar moment for nondoublecouple moment tensors. Equation 2 shows that the tensor, f 0, is a moment normalized tensor, that is, a moment weighted average of the f tensors. Th e smt is responsible for the observed radiation pattern from the event. The scalar moment is calculated based on the euclidean frobenius norm silver and jordan, 1982. The inertia tensor is both real and symmetric in particular, it satisfies. The determination of the composite moment tensor is based on a joint inversion for one common moment tensor using amplitudes of p andor s waves or using full waveforms observed at a limited number of stations but for multiple earthquakes. The theory so far is standard, but the special point is that the kernels will be.
The quadrupole moment tensor has thus 9 components, but because of transposition symmetry and zerotrace property, in this form only 5 of these are independent for a discrete system of point charges or masses. Determination of the moment of inertia mitsubishi colt 2009 pdf of an. The moment of area of an object about any axis parallel to the centroidal axis is the sum of mi about its centroidal axis and the prodcut of area with the square of distance of from the reference axis. Feasibility of moment tensor inversion from a single.
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