Moment Tensors – A Practical Guide

Moment tensor analysis is a topic that carries a decent level of uncertainty and confusion for many people. So I’m going to lay it out as simply as I can. For this post, I’m not going to go into too many details on how moment tensors are actually calculated. But, I’m going to summarise the things I think are most important for geotechnical engineers to know for interpreting moment tensor results.

 

OK, so, what is a moment tensor?

 

A moment tensor is a representation of the source of a seismic event. The stress tensor and the moment tensor are very similar ideas. 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.

You can see the similarity between the stress and moment tensors in the figure below. The moment tensor describes the deformation at the source based on generalised force couples, arranged in a 3 x 3 matrix. Although, the matrix is symmetric so there are only six independent elements (i.e. M12 = M21). The diagonal elements (e.g. M11) are called linear vector dipoles. These are equivalent to the normal stresses in a stress tensor. The off-diagonal elements, are moments defined by force couples (moments and force couples discussed in previous blog post).

 

 

Producing a moment tensor of a seismic event requires the Green’s function. This function computes the ground displacement recorded by the seismic sensor based on a known moment tensor (the forwards problem). A moment tensor inversion is when the inverse Green’s function is used to find the source moment tensor based on sensor data.

 

 

Sure… but what’s with the beach balls?

 

It’s pretty hard to interpret a 3 x 3 matrix of numbers, so moment tensors are usually displayed as beach balls, either 2D or 3D. I will mostly discuss the 3D case; the 2D diagram is just a stereonet projection of the 3D beach ball.

The construction of a beach ball diagram is very simple. For each point on the surface of a sphere, the moment tensor describes the magnitude and direction of the first motion. If the direction of motion is inwards, towards the source, the surface is coloured white (red arrows). If the direction of motion is outwards, away from the source, the surface is coloured black (blue arrows). Where there is a border between black and white on the beach ball surface, the direction of motion is tangential (purple arrows). The direction of motion across the border is white-to-black.

The figure below shows the first ground motion on the beach ball surface, split into radial and tangential components. The lengths of the radial and tangential arrows are proportional to the strength of the P and S waves respectively. P-waves generally emanate strongest from the middle of the white and black regions. S-waves emanate strongest from the black-white borders.

 

 

The location of the pressure and tension axes can be confusing. If you look at the S-waves diagram, the tension axis is in the compressional quadrant. However, it does make more sense from the P-waves diagram. The black/white convention can also be counter-intuitive for some. ‘Black’ holes pull things inwards, the sun radiates ‘white’ light outwards, but the beach ball diagram is the opposite of that. I’m sorry I don’t know why this is the convention. Perhaps seismologists are Star Wars fans… Vader wants Luke to come to the dark side, and so this is the movement direction that he is tempted towards… that’s all I got 😊. 

 

Right, but what can I learn about the event mechanism?

 

Even with the beach ball diagram, it can still be hard to interpret the geological or physical mechanism of the event. This is why the moment tensor is often decomposed into its constituent elementary source mechanisms. To decompose the moment tensor, the matrix is rotated to zero the off-diagonal elements. This is just like finding the principal axes of a stress tensor, by zeroing the shear elements and leaving the normal stresses. So, every moment tensor can be expressed as three linear vector dipoles (orthogonal), rotated to a particular orientation. These three dipoles are referred to as the P (pressure), B (or N, neutral or null) and T (tension) principal axes.

Isotropic source

In combination, the three dipoles either result in an overall expansion or a contraction of the source volume. If the source is explosive, the largest dipole direction is the T axis and the smallest dipole is the P axis. These are reversed for an implosive source. Although, for a pure isotropic source the axis orientations have no meaning.

The isotropic component is the portion of the tensor that represents a uniform volume change. Only P-waves radiate from a purely isotropic source. A positive isotropic component is an expansion/explosion. This can be a confined blast or possibly rock bulking. A negative isotropic component is a contraction/implosion. Any pillar burst, buckling or rock ejecting into a void will likely appear as an implosion, given the path of the recorded waves around the void, all first motions will be towards the source.

Deviatoric source

When the isotropic component is removed from the moment tensor, the remainder is the deviatoric component. The deviatoric tensor results in displacement that has zero net volume change, i.e. equal movement in, equal movement out. The underlying geological process to the deviatoric component is a general dislocation of a fault. The general dislocation can be a mix of shear and normal dislocation (although still with no net volume change). To better interpret the relative proportions of shear and normal displacement, the deviatoric component can be decomposed into the DC and CLVD elemental sources.

 

 

Double Couple (DC) source

The DC source is a pure shear dislocation. It is referred to as a double couple because there are two force couples and two (alternate) fault plane orientations that equally model the expected displacement. This notion was discussed in a previous post. The shear direction on the fault is from white-to-black. You can review the orientation of the two planes in relation to your site geology. It may be the case that one of the planes makes more sense than the other or you can find the specific structure.

A pure DC source has two equal and opposite linear vector dipoles. The third dipole is zero (B or null axis). The embedded video shows the direction of first motions from a pure DC source. As mentioned already, motion is inwards for the white regions, outwards for the black regions and tangential across black-white borders. Radial movement radiates P-waves, tangential movement radiates S-waves.

 

Compensated Linear Vector Dipole (CLVD) source

The CLVD source is a normal dislocation on a plane. The normal displacement from one linear vector dipole is ‘compensated’ (hence the name) by opposing displacement from the other two linear vector dipoles so that there is no net volume change.

For a positive CLVD source, a single tensile dipole is compensated by two compressive dipoles.

 

Vice-versa for a negative CLVD source.

 

A pure CLVD source would imply a poisson’s ratio of 0.5, which is more like chewing gum or toothpaste than rock. So there is no geological example of a pure CLVD source. Although, it can make sense as a mixed source event; i.e. part isotropic, part CLVD. This event mechanism may be dominant for confined pillar crushing events. The Hudson chart indicates two key points that are a combination of isotropic and CLVD sources. A single linear vector dipole (other two dipoles are zero) decomposes to a one-third isotropic source, two-thirds CLVD. A pure tensile crack mechanism decomposes to a source 55% explosive, 45% positive CLVD.

The Hudson chart is a useful tool to visualise the moment tensor decomposition, seeing the relative proportions of the isotropic, DC and CLVD elemental sources. The vertical axis is the isotropic component, from -100% (implosion) to 100% (explosion). The horizontal axis is the deviatoric decomposition, from +100% to -100% CLVD, with 100% DC in the centre (0% isotropic, 0% CLVD). The outer border is the 0% DC line.

 

 

Final comments

 

There are many factors that can lead to uncertainty in determining the first motions of waves recorded at sensors and the final moment tensor solution. Seismic waves travelling through the rock mass divert around mining voids and go through numerous refractions, reflections and superimpositions. Noise at the sensor site can also influence the first motion analysis and the solution can also be very sensitive to poor P and S picks. Good moment tensor solutions require a sensor array that is well dispersed, covering the focal sphere in all three dimensions.

Be aware that each moment tensor solution is not going to be of equal quality, particularly small events with few sensors used. Your seismic service provider should provide you with some measure of solution accuracy to help assess this. This might be based on an assessment of the sensor configuration or a misfit analysis between the observed waveforms and the theoretical waveforms generated synthetically from the moment tensor. In general, it is better to look at trends and a convergence of evidence across multiple events rather than a single moment tensor solution. Even if you are investigating a single large event, it is probably worth reviewing the mechanisms of aftershocks and previous events in the area.

It is important not to jump blindly to the nodal plane solutions and to consider the decomposition of the moment tensor in your analysis. If the source is only 5-10% DC, the nodal planes are not very significant. The P, B, T axis are also less important for strongly isotropic sources so keep that in mind for stereonet analysis.

And one last warning about CLVD components. In tests where random noise is added to an initial noise-free, moment tensor inversion of a pure DC source, the noise serves to increase the CLVD component. So it is hard to be sure when CLVD shows up in a solution that it isn’t just noise related. In fact, seismologists often evaluate the accuracy of a moment tensor solution by how large the CLVD component is. A good solution would have a low CLVD component. This is earthquake seismology though so the range of rock mass mechanisms is less diverse than the mining environment, DC is often an assumption for earthquakes.

Anyway, hopefully that clears up at least some of the mystery around moment tensors. Feel free to contact support with any questions. For those looking to read up further I recommend this manual by Dahm and Krüger (2014) and the references therein. They go into much more detail on alternate decompositions and the moment tensor inversion process.

 

Moment Tensors in General Analysis

Moment tensors have been added to the General Analysis application in the recent update. Beach balls and principal axes can be viewed in the General Analysis 3D view. There is also a separate Moment Tensor window with a number of stereonets and mechanism charts. Two new training videos have been uploaded to the General Analysis page that walkthrough the new tools.

IMS sites should have moment tensors loaded in with the events table automatically. ESG sites can add moment tensors from CSV files in the Events Import app.

 

 

 

To a/b, or not to a/b

The a/b value is sometimes used as a measure of seismic hazard but there are some common mistakes made with this analysis and interpretation.

What is a/b?

The Gutenberg-Richter distribution is a statistical model that describes a log-linear relationship between the number of events, N, exceeding magnitude, M.

 log10 N = a – bM

At N = 1, M = a/b. The figure below shows an example of a frequency-magnitude chart with the a/b value highlighted.

Does a/b mean anything?

It is important to distinguish between properties of the dataset and properties of the statistical model. The a/b value is a property of the Gutenberg-Richter statistical model but it is defined at a particular data point (N = 1). The a/b value does have some meaning, but that’s really only because the a and b value both mean something (although I’ll come back to the a-value later). In terms of seismic hazard, the activity rate and b-value are the two primary inputs required.

The focus on the magnitude where N = 1 is somewhat arbitrary. The statistical model describes the relative frequency for all magnitudes. It is just as valid to normalise the frequency axis to a percentage i.e. express N as a percentage of the number of events at M = Mmin. So in the figure below, at Mmin, the frequency is 100% and events over M = 1 represent 0.1% of all events over Mmin. Note the a/b magnitude represents approx 0.006% of events. So the magnitude at N = 1 loses its significance. Asking what is the significance of a/b is like asking the significance of the magnitude of the top 0.1% of events? Why not the top 0.01% or 0.001%?

The normalisation trap (or the non-normalisation trap)

The reason the a/b value doesn’t mean much for seismic hazard is because the a-value by itself is meaningless. The number of events, by itself, doesn’t tell you anything about hazard because it has no associated time and space units. It should be pretty easy to understand the importance of normalisation to regular time and space units. If I tell you there has been 100 events, you don’t know anything about what seismic hazard that represents. It could be 100 events in a very small volume, in a very small time period; this would be a high hazard. It could be 100 events in a very large volume over a very long time period; this would be a low hazard. So the important thing for seismic hazard estimates is the event rate density, i.e. the number of events, per unit time, per unit volume. Only then can you compare apples with apples.

One final point. A constant event rate density, and a constant b-value over time represents a constant hazard state. The problem is that the a/b value without normalisation is entirely dependent on how long you have recorded this constant hazard state. The total number of events (i.e. the a-value) continuously grows and so does the a/b value, even though the hazard state is not changing. This is why without normalisation, the a/b is not a measure of hazard.

If you normalise the event count based on the event rate density and a standard time and volume, the a/b value can be a measure of hazard. However, in terms of probabilistic seismic hazard, the probability that the largest event in the database will exceed the a/b value is ≈ 63%, assuming an open-ended Gutenberg-Richter distribution or a very high MUL (MUL >> a/b).

 

Conclusions

  • The a/b value is a property of the Gutenberg-Richter model, not of the dataset
  • There is no special significance to the magnitude where the Gutenberg-Richter model crosses N = 1
  • The a/b value is a function of the number of events
  • Without space and time information, the a/b value (and the a-value) are not indicative of hazard
  • When comparing different times and zones using a/b, you must normalise using the event rate density and a standard time and volume
  • The probability of the largest event exceeding a/b is ≈ 63%

Frequency-Magnitude Chart Anatomy

When you are using the Frequency-Magnitude chart, it can be easy to forget it is log scale and this can distort a few things. Consider the chart below, have you ever thought the Gutenberg-Richter distribution doesn’t look right? Think it isn’t matching the large events very well?

The Gutenberg-Richter distribution is a statistical model of the data. Consider what the chart looks like in linear scale rather than log scale. The difference at the tail of the distribution (largest events) seems much less significant right? The other interesting point is the relative proportion of events above and below the Mmin. There is roughly only 20% of events in you database that are above the magnitude of completeness.

Obviously in linear scale, you can’t see what’s happening at the tail very well, that’s why we use the log scale in the first place :)

Event Tags and Comments

There are many reasons you might want to store a short snippet of text associated with an event. There are two ways to do this in mXrap; event tags and event comments.

Event tags can be used to group events into categories. Example tags might be “suspected blast”, “damage occurred”, “suspect location”, “outlier” or “likely crusher noise”. These tags can be used in event filters to quickly show or hide particular categories.

Event comments are a second option to assign user text to events. Each event comment can be unique and about anything. They have no effect on event filters.

You can find videos on “Event tags” and “Event comments” at the training video page below. Both event tags and comments are shown in the main events table in General Analysis.

https://mxrap.com/training-videos/ga-3d/

The event tags system has been modified recently. If your mXrap looks different to the video, you might need a root update. This process is now quick and easy with mXsync. We just need 5-10 minutes to connect via teamviewer / webex / gotomeeting.

Contact support@mxrap.com for assistance.

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