Seismic moment tensor
Moment tensors play an important role in mining seismicity. They aid in the visualization of seismic events, and provide quantitative data about the event. Although the moment tensor has some limitations, it is very useful and widely used.
A seismic moment tensor is a mathematical representation of the moments generated by a seismic event. A moment is an expression involving the product of a distance and a physical quantity.
The moment tensor is a square matrix composed of nine elements. An element is a place in a matrix, such as M12. The element M12 is the entry in the first row of a matrix, and in the second column. Each of the elements represent the moment generated by one of nine sets of two vectors. These pairs of vectors act on points in space and are called couples. The moment tensor depends on the seismic source characteristics and the fault orientation.
The moment tensor describes seismic activity according to a point source model. This means that shear displacements rotate around one point, and that P-wave displacements radiate outwards from the same point.
The moment tensor itself is a matrix of constants.
Symmetry of the Moment Tensor
The moment tensor is symmetric. This means that element Mkj is equal in magnitude to element Mjk. The kj subscript means “in the direction of k, and in the plane perpendicular to j”.
Element Mxx acts in the plane that is normal, or perpendicular, to x. This is the yz plane. The choice of element notation is very important, because it is related to fault orientation. The x y z indices are appropriate for use when the principal stresses σ_1, σ_2, and σ_3 are the x, y, and z axes. For faults of different orientations, it is more appropriate to use the generalized 1, 2, 3 indices to indicate directions.
The moment tensor is symmetric due to the conservation of angular momentum. This means that with no external force to act upon the system at a point in time, the original angular momentum of the system is conserved. This does not mean that the angular momentum of individual particles must be zero at any instant in time. If this was true, seismic waves would be unable to propagate.
Written in a different way, the total momentum of the system before the event must be equal to the total momentum of the system after the event for momentum to be conserved.
Newton’s third law of motion helps to explain why nine couples are required to describe the system, and why there are only six independent couples. Newton’s third law states that all forces between two objects exist in equal magnitudes and opposite directions. The diagonal terms in the moment tensor represent pairs of equal, opposite, collinear vectors. The vectors are collinear because they lie on the same straight line.
As such, the diagonal elements of the moment tensor represent both of the equal and opposite components of the system for the part of the motion that they represent.
Six more terms are required for the moment tensor, because they represent vectors that are offset from the centre point and act in a plane, rather than perpendicular to a plane. In order for a reaction to exist, another pair of vectors offset from the centre point must be defined. They must be in the same plane, equal in magnitude to the other pair of vectors, and in a direction that results in an opposite rotation in the plane.
These pairs of couples are arranged diagonally across from each other in the moment tensor, and are shown in the figure below.
The elements circled in grey correspond to motion in the zy plane. Together, these two pairs of points and vectors can describe the behaviour of the system according to Newton’s laws of motion.
Moment Tensor Visualization
Two ways of visualizing a moment tensor are through focal mechanisms and radiation patterns. Some focal mechanisms show only if the P wave amplitude is positive or negative at different points on the sphere. Others additionally have colour gradations to show the amplitudes of the waves at those points.
A more detailed way of visualizing the moment tensor is through radiation patterns. Radiation patterns exist for both P and S waves, and the P and S wave patterns can be plotted on the same graph so that they can be viewed simultaneously. Simultaneous viewing of P and S wave amplitudes can sometimes be beneficial in analysis. Radiation patterns are scalar functions of one or two variables, and thus respectively represent waves in two or three dimensions.
The figure below shows some focal mechanisms that are likely to be encountered in mining seismicity.
In the diagram, red represents expansion relative to the centre point, and blue represents compression relative to the centre point. The deviatoric example is also labelled as CLVD, which stands for compensated linear vector dipole.
It is important to note that there are really only three unique focal mechanisms in the diagram, and that each of the five mechanisms shown can be created by changing the orientation of one of the three unique mechanisms.
An explosion in an isotropic medium that exerts the same radial force everywhere will have no variation in the amplitude of the waves, and the first motion will be radially outwards. This creates an isotropic mechanism with a positive first P wave amplitude.
In figure 2, The mode 1 crack is a deviatoric mechanism tilted on its side and compressing in the opposite direction of the example shown.
The mode two and three shear cracks are double couples oriented ninety degrees apart.
More examples of focal mechanisms, also called beach balls, are provided in the figure below.
It is important to recognize that the square root of the sum of the squares of the elements of each of the moment tensors is equal to one. This attribute makes the moment tensors easier to work with mathematically, when combined into a larger equation.
Furthermore, notice that the bottom four focal mechanisms are different orientations of the deviatoric case, the six mechanisms above them are different orientations of the double couple case, and the top two mechanisms represent the isotropic cases of explosion and implosion.
Some earthquakes might involve a combination of dilatational and shear cracking. In this case, the perpendicular planes of the double couple mechanisms would become curved. Their curvature would depend on the ratio of dilation to shear.
A two dimensional radiation pattern is shown below. Although the moment tensor is three dimensional, sections of the moment tensor can be shown as two dimensional radiation patterns. The P wave radiation pattern shown below is for a double couple in the yz plane.
Note that in a real radiation pattern, the four regions are not actually ellipses. In a real radiation pattern, they are four lobes of a curve that intersects the origin. This diagram shows very well how the couples Mzy and Myz work together to generate the radiation pattern.
The top left and bottom right regions are compressing towards the centre point, so the vectors there would point inward. The top right and bottom left regions are expanding away from the centre point, so the vectors there would point outwards.
The vectors collinear with the P axis are the strongest compression vectors, and the vectors collinear with the T axis are the strongest tensile vectors.
The two half arrows show the total shear experienced by the system. The figure below shows the mathematically correct P and S wave radiation patterns for a two dimensional double couple. The true shape of the lobes can be observed. Note that the function intersects the origin.
The magnitudes of the vectors correspond to the radial distance from the origin to the radiation pattern curve, as shown. For the P waves, the vector direction is either towards or away from the centre. For the S waves, imagine a line and a circle, both centred at the origin. The vector direction where the line intersects the circle is tangential to the circle at the intersection point of the line and circle. The magnitude of the tangent vector at the intersection point is the distance along the line from the origin to the radiation pattern curve.
Since the moment tensor is three dimensional, it is important to be able to visualize three dimensional radiation patterns. The diagram below shows some three dimensional radiation patterns. They begin as double couples, and end as the deviatoric case.
The olive green vector shows how much of the radiation pattern’s shape is generated by a deviatoric moment tensor, and how much of the pattern’s shape is generated by a double couple moment tensor. The larger the green vector’s angle is with the horizontal, the more deviatoric the radiation pattern is. Note that when the green vector is perpendicular to the horizontal, the radiation pattern is completely deviatoric. More specifically, the radiation pattern is positive deviatoric with respect to the vertical components of the vectors in the pattern. This means that the vectors in the radiation pattern point inwards and upwards. Refer to figures 2 and 3 for clarification.
Significance of the Moment Tensor to Mining
In mining seismicity, the events are often relatively small compared to large earthquakes. If the event is not too close to the seismometers and the displacement isn’t too large, the point source approximation is reasonable enough to be used. If the location of a group of microseismic events is known, and the moment tensors can be determined, the whole group of focal mechanisms can be plotted in three dimensions and a pattern might be found.
If the focal mechanisms indicate motion in a similar direction, and if the mechanisms seem to exist on the same plane, it means that the seismic activity might be occurring on a fault.
If the microseismic activity is corresponding only to the adjustment of the stress field near a new excavation, the focal mechanisms might have a wider range or orientations and surround the new excavation.
How to Calculate the Moment Tensor: Moment Tensor Inversion
The process of calculating the moment tensor is called moment tensor inversion. Data from seismometers is used to calculate the elements of the moment tensor. Not all moment tensor inversion problems are able to be solved with the same degree of accuracy.
Inversion problems ask the question “Knowing some information about a system, how can I use this information to do some calculations that describe how the system created that information?”.
With regards to microseismic events, the information for the inversion comes from seismometers, and the goal of the inversion is to calculate the moment tensor that describes the event.
Limitations of the Moment Tensor
It is very important to understand the limitations of the moment tensor. Remember that the moment tensor is a point source approximation of a seismic event. Seismic events do not occur in one infinitesimally small point.
Rather, they are best described as occurring on surfaces, potentially with some volume change if there is tension.
Finite fault models use the physics of crack propagation to model seismic events, rather than a point source approximation.
The main difference between finite fault models and point source approximations is the concept of seismic wave directivity.
Seismic directivity describes how the seismic waves are strongest near the tips of the crack, and how the waves travel mostly in the direction of the crack propagation. Seismic directivity explains why most of the energy from a seismic event radiates in the direction of the crack propagation, and is not evenly distributed radially from a point source.