A geometric perspective on seismic moment tensors
GALLERY OF IMAGES
A geometric setting for moment tensors (pdf)Geophysical Journal International, 2012Walter Tape and Carl Tape
We describe a parametrization of moment tensors that is suitable for use in algorithms for moment tensor inversion. The parameters are conceptually natural and can be easily visualized. The ingredients of the parametrization are present in the literature; we have consolidated them into a concise statement in a geometric setting. We treat several familiar moment tensor topics in the same geometric setting as well. These topics include moment tensor decompositions, crack-plus-double-couple moment tensors, and the parameter that measures the difference between a deviatoric moment tensor and a double couple. The geometric approach clarifies concepts that are sometimes obscured by calculations.
A geometric comparison of source-type plots for moment tensors (pdf)Geophysical Journal International, 2012Walter Tape and Carl Tape
We describe a general geometric framework for thinking about source-type plots for moment tensors. We consider two fundamental examples, one where the source-type plot is on the unit sphere, and one where it is on the unit cube. The plot on the sphere is preferable to the plot on the cube: it is simpler, it embodies a more natural assumption about eigenvalue probabilities and it is more consistent with the conventional euclidian definition of scalar seismic moment. We describe the source-type plots of Hudson, Pearce and Rogers (1989) in our geometric context, and we find that they are equivalent to a plot on the cube. We therefore suggest the plot on the sphere as an alternative.
Angle between principal axis triples (pdf)Geophysical Journal International, 2012Walter Tape and Carl Tape
The principal axis angle xi0, or Kagan angle, is a measure of the difference between the orientations of two seismic moment tensors. It is the smallest angle needed to rotate the principal axes of one moment tensor to the corresponding principal axes of the other. We give a concise formula for calculating xi0, but our main goal is to illustrate the behavior of xi0 geometrically. When the first of two moment tensors is fixed, the angle xi0 between them becomes a function on the unit ball. The level surfaces of xi0 can then be depicted in the unit ball, and they give insights into xi0 that are not obvious from calculations alone. We also include a derivation of the known probability density of xi0. The density is proportional to the area of a certain surface, whose variation explains the rather peculiar shape of the probability density curve. This shape needs to be considered when analyzing distributions of empirical xi0 values. We recall an example of Willemann which shows that xi0 may not always be the most appropriate measure of separation for moment tensor orientations, and we offer an alternative measure.
The classical model for moment tensors (pdf, supplement)Geophysical Journal International, 2013Walter Tape and Carl Tape
A seismic moment tensor is a description of an earthquake source, but the description is indirect. The moment tensor describes seismic radiation rather than the actual physical process that initiates the radiation. A moment tensor ‘model’ then ties the physical process to the moment tensor. The model is not unique, and the physical process is therefore not unique. In the classical moment tensor model, an earthquake arises from slip along a planar fault, but with the slip not necessarily in the plane of the fault. The model specifies the resulting moment tensor in terms of the slip vector, the fault normal vector and the Lame elastic parameters, assuming isotropy. We review the classical model in the context of the fundamental lune. The lune is closely related to the space of moment tensors, and it provides a setting that is conceptually natural as well as pictorial. In addition to the classical model, we consider a crack plus double-couple model (CDC model) in which a moment tensor is regarded as the sum of a crack tensor and a double couple.
A uniform parameterization of moment tensors (pdf, supplement)Geophysical Journal International, 2015Walter Tape and Carl Tape
A moment tensor is a 3 x 3 symmetric matrix that expresses an earthquake source. We construct a parameterization of the five-dimensional space of all moment tensors of unit norm. The coordinates associated with the parameterization are closely related to moment tensor orientations and source types. The parameterization is uniform, in the sense that equal volumes in the coordinate domain of the parameterization correspond to equal volumes of moment tensors. Uniformly distributed points in the coordinate domain therefore give uniformly distributed moment tensors. A cartesian grid in the coordinate domain can be used to search efficiently over moment tensors. We find that uniformly distributed moment tensors have uniformly distributed orientations (eigenframes), but that their source types (eigenvalue triples) are distributed so as to favor double couples.
A confidence parameter for seismic moment tensors (pdf, supplement)Geophysical Journal International, 2016Walter Tape and Carl Tape
Given a moment tensor m inferred from seismic data for an earthquake, we define Ps(V) to be the probability that the true moment tensor for the earthquake lies in the neighborhood of m that has fractional volume V. The average value of Ps(V) is then a measure of our confidence in m. The calculation of Ps(V) requires knowing both the probability P(ω) and the fractional volume V(ω) of the set of moment tensors within a given angular radius ω of m. We explain how to construct Ps(ω) from a misfit function derived from seismic data, and we show how to calculate V(ω), which depends on the set M of moment tensors under consideration. The two most important instances of M are where M is the set of all moment tensors of fixed norm, and where M is the set of all double couples of fixed norm.
Volume in moment tensor space in terms of distance (pdf, report)Geophysical Journal International, 2017Walter Tape and Carl Tape
Suppose that we want to assess the extent to which some large collection of moment tensors is concentrated near a fixed moment tensor m. We are naturally led to consider the distribution of the distances of the moment tensors from m. This distribution, however, can only be judged in conjunction with the distribution of distances from m for randomly chosen moment tensors. In cumulative form, the latter distribution is the same as the fractional volume V(ω) of the set of all moment tensors that are within distance ω of m. This definition of V(ω) assumes that a reasonable universe M of moment tensors has been specified at the outset and that it includes the original collection as a subset. Our main goal in this article is to derive a formula for V(ω) when M is the set [Lambda]_U of all moment tensors having a specified eigenvalue triple Lambda . We find that V(ω) depends strongly on Lambda , and we illustrate the dependence by plotting the derivative curves V'(ω) for various seismologically relevant Lambdas. The exotic and unguessable shapes of these curves underscores the futility of interpreting the distribution of distances for the original moment tensors without knowing V(ω) or V'(ω). The derivation of the formula for V(ω) relies on a certain φσz coordinate system for [Lambda]_U, which we treat in detail. Our underlying motivation for the paper is the estimation of uncertainties in moment tensor inversion.
The eigenvalue lune as a window on moment tensors (pdf, supplement)Geophysical Journal International, 2019Walter Tape and Carl Tape
A moment tensor is a symmetric matrix that expresses the source for a seismic event. The fundamental lune is a certain subset of the unit sphere whose points represent the source types for all moment tensors. Familiar source types such as double couple or pure isotropic have natural locations on the lune. Although the lune consists only of source types, it serves as a low-dimensional outline of moment tensor space as a whole. For each subset B of the lune we therefore consider the associated set [B]_U of unit moment tensors that have their source types in B; we wish to get a sense of [B]_U from looking at B. We succeed in calculating both the angular diameter and the volume of [B]_U . We also calculate the angular diameter and the volume of the set [Lambda]_U of unit moment tensors that have source type Lambda, and we plot the results as contours on the lune. We show that great circle arc lengths on the lune are closely related to angles between moment tensors, and that arc length on the lune gives a natural measure of difference in source type. We show how to calculate volume elements for a variety of moment tensor coordinates. Volumes are relevant in part because we equate fractional volume with the probability that expresses randomness. We thus can find the probability that a random moment tensor have its source type in a given subset of the lune. Our results have implications for estimating moment tensor uncertainties, but we do not pursue them in this paper.
We describe a parametrization of moment tensors that is suitable for use in algorithms for moment tensor inversion. The parameters are conceptually natural and can be easily visualized. The ingredients of the parametrization are present in the literature; we have consolidated them into a concise statement in a geometric setting. We treat several familiar moment tensor topics in the same geometric setting as well. These topics include moment tensor decompositions, crack-plus-double-couple moment tensors, and the parameter that measures the difference between a deviatoric moment tensor and a double couple. The geometric approach clarifies concepts that are sometimes obscured by calculations.
A geometric comparison of source-type plots for moment tensors (pdf)Geophysical Journal International, 2012Walter Tape and Carl Tape
We describe a general geometric framework for thinking about source-type plots for moment tensors. We consider two fundamental examples, one where the source-type plot is on the unit sphere, and one where it is on the unit cube. The plot on the sphere is preferable to the plot on the cube: it is simpler, it embodies a more natural assumption about eigenvalue probabilities and it is more consistent with the conventional euclidian definition of scalar seismic moment. We describe the source-type plots of Hudson, Pearce and Rogers (1989) in our geometric context, and we find that they are equivalent to a plot on the cube. We therefore suggest the plot on the sphere as an alternative.
Angle between principal axis triples (pdf)Geophysical Journal International, 2012Walter Tape and Carl Tape
The principal axis angle xi0, or Kagan angle, is a measure of the difference between the orientations of two seismic moment tensors. It is the smallest angle needed to rotate the principal axes of one moment tensor to the corresponding principal axes of the other. We give a concise formula for calculating xi0, but our main goal is to illustrate the behavior of xi0 geometrically. When the first of two moment tensors is fixed, the angle xi0 between them becomes a function on the unit ball. The level surfaces of xi0 can then be depicted in the unit ball, and they give insights into xi0 that are not obvious from calculations alone. We also include a derivation of the known probability density of xi0. The density is proportional to the area of a certain surface, whose variation explains the rather peculiar shape of the probability density curve. This shape needs to be considered when analyzing distributions of empirical xi0 values. We recall an example of Willemann which shows that xi0 may not always be the most appropriate measure of separation for moment tensor orientations, and we offer an alternative measure.
The classical model for moment tensors (pdf, supplement)Geophysical Journal International, 2013Walter Tape and Carl Tape
A seismic moment tensor is a description of an earthquake source, but the description is indirect. The moment tensor describes seismic radiation rather than the actual physical process that initiates the radiation. A moment tensor ‘model’ then ties the physical process to the moment tensor. The model is not unique, and the physical process is therefore not unique. In the classical moment tensor model, an earthquake arises from slip along a planar fault, but with the slip not necessarily in the plane of the fault. The model specifies the resulting moment tensor in terms of the slip vector, the fault normal vector and the Lame elastic parameters, assuming isotropy. We review the classical model in the context of the fundamental lune. The lune is closely related to the space of moment tensors, and it provides a setting that is conceptually natural as well as pictorial. In addition to the classical model, we consider a crack plus double-couple model (CDC model) in which a moment tensor is regarded as the sum of a crack tensor and a double couple.
A uniform parameterization of moment tensors (pdf, supplement)Geophysical Journal International, 2015Walter Tape and Carl Tape
A moment tensor is a 3 x 3 symmetric matrix that expresses an earthquake source. We construct a parameterization of the five-dimensional space of all moment tensors of unit norm. The coordinates associated with the parameterization are closely related to moment tensor orientations and source types. The parameterization is uniform, in the sense that equal volumes in the coordinate domain of the parameterization correspond to equal volumes of moment tensors. Uniformly distributed points in the coordinate domain therefore give uniformly distributed moment tensors. A cartesian grid in the coordinate domain can be used to search efficiently over moment tensors. We find that uniformly distributed moment tensors have uniformly distributed orientations (eigenframes), but that their source types (eigenvalue triples) are distributed so as to favor double couples.
A confidence parameter for seismic moment tensors (pdf, supplement)Geophysical Journal International, 2016Walter Tape and Carl Tape
Given a moment tensor m inferred from seismic data for an earthquake, we define Ps(V) to be the probability that the true moment tensor for the earthquake lies in the neighborhood of m that has fractional volume V. The average value of Ps(V) is then a measure of our confidence in m. The calculation of Ps(V) requires knowing both the probability P(ω) and the fractional volume V(ω) of the set of moment tensors within a given angular radius ω of m. We explain how to construct Ps(ω) from a misfit function derived from seismic data, and we show how to calculate V(ω), which depends on the set M of moment tensors under consideration. The two most important instances of M are where M is the set of all moment tensors of fixed norm, and where M is the set of all double couples of fixed norm.
Volume in moment tensor space in terms of distance (pdf, report)Geophysical Journal International, 2017Walter Tape and Carl Tape
Suppose that we want to assess the extent to which some large collection of moment tensors is concentrated near a fixed moment tensor m. We are naturally led to consider the distribution of the distances of the moment tensors from m. This distribution, however, can only be judged in conjunction with the distribution of distances from m for randomly chosen moment tensors. In cumulative form, the latter distribution is the same as the fractional volume V(ω) of the set of all moment tensors that are within distance ω of m. This definition of V(ω) assumes that a reasonable universe M of moment tensors has been specified at the outset and that it includes the original collection as a subset. Our main goal in this article is to derive a formula for V(ω) when M is the set [Lambda]_U of all moment tensors having a specified eigenvalue triple Lambda . We find that V(ω) depends strongly on Lambda , and we illustrate the dependence by plotting the derivative curves V'(ω) for various seismologically relevant Lambdas. The exotic and unguessable shapes of these curves underscores the futility of interpreting the distribution of distances for the original moment tensors without knowing V(ω) or V'(ω). The derivation of the formula for V(ω) relies on a certain φσz coordinate system for [Lambda]_U, which we treat in detail. Our underlying motivation for the paper is the estimation of uncertainties in moment tensor inversion.
The eigenvalue lune as a window on moment tensors (pdf, supplement)Geophysical Journal International, 2019Walter Tape and Carl Tape
A moment tensor is a symmetric matrix that expresses the source for a seismic event. The fundamental lune is a certain subset of the unit sphere whose points represent the source types for all moment tensors. Familiar source types such as double couple or pure isotropic have natural locations on the lune. Although the lune consists only of source types, it serves as a low-dimensional outline of moment tensor space as a whole. For each subset B of the lune we therefore consider the associated set [B]_U of unit moment tensors that have their source types in B; we wish to get a sense of [B]_U from looking at B. We succeed in calculating both the angular diameter and the volume of [B]_U . We also calculate the angular diameter and the volume of the set [Lambda]_U of unit moment tensors that have source type Lambda, and we plot the results as contours on the lune. We show that great circle arc lengths on the lune are closely related to angles between moment tensors, and that arc length on the lune gives a natural measure of difference in source type. We show how to calculate volume elements for a variety of moment tensor coordinates. Volumes are relevant in part because we equate fractional volume with the probability that expresses randomness. We thus can find the probability that a random moment tensor have its source type in a given subset of the lune. Our results have implications for estimating moment tensor uncertainties, but we do not pursue them in this paper.
Miscellaneous notes (these may not display in your browser; look for the download arrow to get the pdfs)
Notes on volume change and moment tensors. These notes include excerpts from Tape and Tape (2013). We challenge the prevailing view that the isotropic component of a moment tensor is related to volume change in the source region.
Notes on the use of percentages in moment tensor decompositions. We do not use percentages ourselves.
Notes on measures of the CLVD fraction of a moment tensor (such as the epsilon parameter). We favor using lune longitude (gamma).
Notes on potency tensors. We do not use potency tensors ourselves.
Chapman-Leaney decompositions (pdf). This pdf is a set of notes we wrote regarding a 2012 GJI paper by Chapman and Leaney. Our notes were published as an online supplement of a 2014 GJI addendum paper by Chapman and Leaney. In our notes we use the lune in the context of general anisotropy to investigate the moment tensor decomposition proposed by Chapman and Leaney.
Start-up guide for plotting moment tensor source types on the eigenvalue lune
I have posted online some matlab scripts, GMT scripts, and mathematica scripts relevant for interpreting and plotting moment tensors. These files are available within the mtbeach repository on github. They can be downloaded from github as a zipped file or using git from the command line. Please email me with corrections or suggestions.
If you think you might use some of these concepts, here is a start-up guide:
Check out this pdf.
Download the mtbeach repository from github (git clone https://github.com/carltape/mtbeach.git )
Follow the setup instructions in the README file here
If you have Matlab installed, try one of the scripts to reproduce results in one of our published papers: TT2012kaganAppE.m, TT2013AppA.m, TT2015AppA.m These scripts use several functions that you may find useful.
If you have GMT installed, try out lune.pl
(Please email me if things do not work for you.)
Go here to see some example figures related to the papers above. Check out links to notes above.