It is often represented with beach balls just like the focal mechanism or fault plane solution. Moment tensor solutions the moment tensor solution is typically displayed graphically using a socalled beachball diagram. The inertia tensor about some origin located at position relative to the center of mass is easily found to be. This is known as the parallel axis theorem 24 examples of moment. The forces described by the moment tensor are not the actual forces acting at the source because the moment tensor. In this video i will calculate the 9 components of a simple example mass of m1 is place on the yaxis 0,1,0 of an inertia tensor matrix. In addition, based on analysis of synthetic seismograms we develop and evaluate a workflow for inverting source mechanisms moment tensors. The line id in each indicates the type of the moment tensor solution 1 and 1 indicate the surfacewave frequencydomain solutions, 2 and 2 indicate the completewaveform timedomain solutions, and 3 and 3 indicate a nearfield waveform inversion.
The moment of inertia mi of a plane area about an axis normal to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and passing through the given axis. For those of you who think i did some witchcraft with. Im going to write this equation in matrix form, and well play with it a little bit before we see where the inertia tensor. To parametrize the source type from the moment tensor, we note that the tensor has three real eigenvalues. A moment is an expression involving the product of a distance and a physical quantity. Lea january 2020 1 the magnetic moment tensor our goal here is to develop a set of moments to describe the magnetic. Tensor which depends on two directions one normal to the area and other that of the force on it. Although the moment tensor has some limitations, it is very useful and widely used. The moment tensor is linearly related to the potency tensor through the fourthorder elastic moduli tensor e. The source parameters of the iranian earthquakes were improved by the advent of synthetic seismogram techniques during the 1970s by the harvard and usgs centroid moment tensor cmt.
Full moment tensor and dc moment tensor for 10 events monitored by downhole station network. Microseismic microseismic microseismic moment tensors. Lecture notes on vector and tensor algebra and analysis. Principal axes of inertia weve spent the last few lectures deriving the general expressions for l and trot in terms of the inertia tensor both expressions would be a great deal simpler if the inertia tensor was diagonal. I is relative to the centroid of the crosssection area more generally moment of inertia is defined as i zz. Force is a vector it has a magnitude specified in newtons, or lbf, or whatever, and a direction. It depends on the bodys mass distribution and the axis chosen, with larger moments. Seismic moment tensors smts are represented by beachballs. In both of these examples, the functions involved were vector valued functions of vector variables. We show that the solid angle subtended by the microseismic array, viewed from the source location, is a key consideration for stability of the inversion. In the integrals, where is the mass density, and a volume element. Moment of inertia tensor definition of moment of inertia. Higher order multipole moments are represented by higherorder tensors.
An equivalent definition of a tensor uses the representations of the general linear group. Moment tensor solutions come online rapidly, for example, on the web page of gfz. Moment definition is a minute portion or point of time. A seismic moment tensor is a mathematical representation of the moments generated by a seismic event. 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. Conclusion based on the synthetic simulation and the study of induced seismic events from an oilgas field, we can state that reliable moment tensor inversion for small events requires. Two vectors, u and v can also be combined via an inner product to form a new scalar thus u v example. Moment tensor decompositions revisited springerlink. This additional manual quality control reduced the. The moment tensor represents equivalent body forces of a seismic source.
The matrix of the values is known as the moment of inertia tensor. Jost and herrmann 1989 a students guide to moment tensors. Insights from moment tensor inversion farshid forouhideh and david w. There is an action of the general linear group on the set of all ordered bases of an ndimensional vector space. The experiment is conducted by recording the time taken for the respective point of mass to being rotated by a fixed load until the. Determination of event locations and magnitudes leads to estimations of the geometry of the fracture zone and certain dynamics of the fracturing. On the basis of theoretical consideration to the equation of motion in continuum mechanics, we clarified the physical meaning of the backusmulcahy moment tensor and derived a fundamental equation that the moment tensor of a seismic event is mathematically equivalent to the volume integral of. According to the rules of matrix multiplication the above equation means. Moment of inertia illinois institute of technology. I and j are used as symbols for denoting moment of inertia. Abstract we compare several published definitions of the scalar moment m o, a measure. We will close by highlighting available tensor software libraries and by presenting the most prominent open research questions in the tensor.
This is known as the parallel axis theorem 24 examples of moment of inertia. Moment tensor this is a mathematic description of the seismic source. The moment of inertia, i, of a beam depends on the geometrical properties of the crosssection area a of a beam. The inner product of force and velocity gives the scalar power being delivered into or being taken out of a system. Express letter parametrization of general seismic potency. 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. The tensor depends of the source strength and fault orientation. The resistance that is shown by the object to change its rotation is called moment of inertia. The moment of inertia about an arbitrary axis through the center of mass, in the direction of the unit vector is.
Dividing the sum of the area moments by the total area we. Inertia tensor in different coordinate systems so far ive called the numbers iij a tensor, but have provided no definition of what a tensor is to provide one, consider what happens to the iij if one rotates the coordinate system well start with the simpler case of observing how a vector transforms under such a rotation. Event locations and magnitudes give limited insight into the processes that control the growth and dynamics of hydraulic fractures. Nov 04, 2018 today i show you how to take the definition of angular momentum of a rigid body, and use it to derive the components of the inertia tensor. Acoustic emission shear crack moment tensor tensile crack moment tensor. The moment tensor inversion can be performed on the base of a linear system in the form d gm, where d represents waveform data, m is the model vector in this case, the six independent moment tensor components, and g is the greens functions, describing the response of the chosen velocity model at the sensor location, upon basic excitations at the source location. The moment tensor solution is typically displayed graphically using a socalled beachball diagram. A path to understanding frac growth m icroseismic monitoring is a valuable tool in understanding the effi cacy of hydraulic fracture treatments. Reason why the moment tensor is symmnetric, and why forces always come in oppositely oriented couples. 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.
Lecture notes on vector and tensor algebra and analysis ilyal. Moment tensor inversion of induced seismicity under different. Oct 16, 2014 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 sources. This is a measure of earthquake size and energy release where. The concept of moment of inertia was introduced by leonhard euler. Moment tensors and lowfrequency approxima spatial and temporal components of ff1, the motions ment tensor of first degree.
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. Moment tensor solutions a useful tool for seismotectonics. Using these parameters with an earth model and wave propagation code, synthetic seismograms are made that fit the observations. In its inertial properties, the body behaves like a circular cylinder.
Note that each component of the moment of inertia tensor can be written as either a sum over separate mass elements, or as an integral over infinitesimal mass elements. May 12, 2018 in this video i will calculate the 9 components of a simple example mass of m1 is place on the yaxis 0,1,0 of an inertia tensor matrix. Consider a body which is on the whole electrically neutral, but in which there is a separation of charge such that there is more positive charge at one end and more negative charge at the other. The general expression of a symmetric moment tensor is represented as dufumier and rivera 1997 where. In this experiment, the moment of inertia of flywheel is being studied by variating the point of mass of flywheel. Determination and uncertainty of moment tensors for. Area properties of various geometrical shapes compiled by jack a.
Moment tensor is a basic concept of source representation in seismology, established in the 1970s. The moment tensor is a square matrix composed of nine elements. Overview of momenttensor inversion of microseismic events. Note that t is not a tensor density in the mathematical sense. Solutions are provided in form of pdf files showing moment tensor. When the axes are such that the tensor of inertia is diagonal, then these axes are called the principal axes of inertia. Home for beachball patterns moment tensor source typesthe fundamental lune of the unit sphere. Defining the scalar moment of a seismic source with a. The seismic moment tensor mcan be decomposed using eigenvalues and an. By expert choice, one could define which is most likely the actual fault plane, and model an extended rupture along this plane.
One method of graphically representing the mechanism as described by the moment tensor is. Faultplane solutions from moment tensor inversion for. The book focuses on focal mechanisms and moment tensor solutions. Each point on the lune represents a beachball pattern. Then there will be a series of twoline entries, one for each moment tensor solution. Tensors have their applications to riemannian geometry, mechanics, elasticity, theory of relativity, electromagnetic theory and many other disciplines of science and engineering. Since an increasing quantity and quality of seismic data allow inverting for accurate moment tensors and interpreting details of the source process, an efficient and physically reasonable. They have the further property that they are linear functions, i.
Seismic moment tensors are able to describe whether the failure is shearing or tearing i. However, a brief discussion of internal forces in slender members will be provided in section 9. Moment of inertia tensor synonyms, moment of inertia tensor pronunciation, moment of inertia tensor translation, english dictionary definition of moment of inertia tensor. The moment of inertia describes the angular acceleration produced by an applied torque. In this paper, the most common moment tensor decompositions are revisited, new equivalent formulas of the decompositions are derived, suitable norms of the moment tensors are discussed and the properties of commonly used sourcetype plots are analysed. Manuel berberian, in developments in earth surface processes, 2014. 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.
The variation of the induced dipole moment with the direction of the applied electric field is only one example, the one we will use for our example of a tensor. If the moment tensor is defined in a ned system the relation to these parameters is e. It depends on the shape and mass distribution of the body. Centroid moment tensor an overview sciencedirect topics. 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. The most common type of the moment tensor is the doublecouple dc source which represents the force equivalent of shear faulting on a planar fault. The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of ricci. It is the output of the function iafter the independent variable. Thus, even if the number of elementary earthquake events were distributed according to the powerlaw with the exponent. The polar moment of inertia is related to an axis which is basically perpendicular to the plane of an area. Our understanding of reservoir behaviour can be enhanced by considering the seismic moment tensor representations of these events, which serve as a direct snapshot of the instantaneous deformation of the surrounding rock by the seismicity. These define the orientations of two possible fault planes and the slip directions on them. A mathematical representation of the movement on a fault during an earthquake, comprising of nine generalized couples, or nine sets of two vectors. Tensors for beginners albert tarantola september 15, 2004 1 tensor notations the velocity of the wind at the top of eiffels tower, at a given moment, can be represented by a vector v with components, in some local, given, basis, vi i 1,2,3.
Dahm t 1996 relative moment tensor inversion based on ray theory. Because there are no magnetic monopoles, the dominant contribution to b at a great distance from a current. Difference in earthquake mechanisms between first motion. Pdf moment tensor decompositions revisited researchgate. For some models of the source, the moment tensor can define the possible fault planes from earthquake data. Th ese eigenvalues determine the proportion of isotropic, compensated linear vector dipole, and doublecouple contributions to the full moment tensor. If all of the area is assumed to comprise infinitely small areas da then the polar moment of inertia is the sum of all of these areas x r2. For an analogy, retreat to the case of a real valued function. An introduction to tensors for students of physics and.
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