The seismic moment tensor

Since the work of Reid [1910] on the San Andreas fault before and after the 1906 San Francisco earthquake, it has been generally recognized that most earthquakes are caused by slippage on active geological faults. This process can be mathematically described either as a displacement discontinuity in the medium or in terms of body forces in an intact medium, which are applied to certain elements in the source region. The first approach can, however, be incorporated into the second by the use of body-force equivalents, i.e. outside the source region the seismic waves set up by slip on a fault are identical to the waves generated by a distribution of certain forces with canceling moments on the same fault [Aki and Richards, 1980].

Figure: Distribution of body-forces equivalent to fault slip. The fault plane is the $x_3$ = 0 plane. A) Two single-couples (a double-couple). B) One single-couple and a system of single-forces. Modified after Aki and Richards [1980].

Close to the source the force equivalents are non-unique, fault slip is equivalent either to two single-couples (a double-couple), Figure 8A, a single-couple plus a single-force system, Figure 8B, or a single-couple with any combination of single-couple to single-force system, all of which give no net force or net couple outside the fault region [Aki and Richards, 1980]. Far away from the source, where we usually only observe waves with wavelengths greater than the fault's linear dimension, the fault acts as a point source and the body-force equivalent to shear faulting is a double-couple. The strength of the couple, the seismic moment $M_0$, is defined as $M_0 = \mu\bar{u}A$ where $\mu$ is the shear modulus and $\bar{u}$ the average displacement over the active fault area $A$. In Figure 8A our hypothetical fault lies in the $x_3$ = 0 plane and slip occurs in the $x_1$ direction. In the point source approximation, where we only have one double-couple, the equivalent body-force description for a fault in the $x_1$ = 0 plane, with slip in the $x_3$ direction, is identical to the one above and we have the well known ambiguity between fault plane and auxiliary plane. There is no information in the radiation from an effective point source of slip that enables one to distinguish the fault plane from the auxiliary plane.

Figure: The nine generalized couples of the seismic moment tensor. Modified after Aki and Richards [1980].

If the double-couple description is generalized to arbitrary orientations in space and if we allow for discontinuities in displacements normal to the fault (this incorporates apparent expansions or compressions), there will be nine generalized couples forming the seismic moment tensor $\underline{M}$, as shown in Figure 9. The tensor depends on the source strength and fault orientation and it characterizes all information about the source that can be obtained from observing waves with longer wavelength than the source dimensions [Aki and Richards, 1980]. The moment tensor is symmetric due to conservation of angular momentum, it is time dependent (following the temporal development of slip, see e.g. Aki and Richards [1980]) and can be written

$$M_{pq} = \mu A(\bar{u}_p\nu_q + \bar{u}_q\nu_p)$$ (49)

where $\bar{u}_p$ is the average slip in the $x_p$ direction on a plane with normal $\nu_q$ in the $x_q$ direction and vice versa. The components of $M_{pq}$ are, hence, the representations of forces in the $x_p$ directions with moment arms in the $x_q$ directions. In terms of the seismic moment and with the choice of coordinate axis made in Figure 8, the moment tensor for the double-couple is

$$\begin{pmatrix}0 & 0 & M_0 \\ 0 & 0 & 0 \\ M_0 & 0 & 0 \end{pmatrix}$$ (50)

As the moment tensor is real and symmetric, it can be rotated into a principal system where all off-diagonal elements are zero

$$\begin{pmatrix}0 & 0 & 0 \\ 0 & -M_0 & 0 \\ 0 & 0 & M_0 \end{pmatrix}$$ (51)

This tensor shows the characteristics of a double-couple moment tensor; one eigenvalue is zero and the trace of the tensor is zero. We see that the double-couple is now represented by two force dipoles without shear, c.f. Figure 9, one compressive (the negative eigenvalue) and one tensile (the positive eigenvalue). The corresponding eigenvectors are $(0,1,0)$ for the null eigenvalue, also called the B axis, $1/\sqrt{2}(1,0,-1)$ for the $-M_0$ eigenvalue, or the P (pressure) axis, and $1/\sqrt{2}(1,0,1)$ for the $M_0$ eigenvalue, or the T (tension) axis, see Figure 10.

Figure: A) The double-couple force system and the equivalent force dipoles, the P and T axes. B) The ``beach ball'' representation of the double-couple mechanism in A. Quadrants where the first motion of the P-wave is compressional are black, the P and T axis directions are indicated with small squares.

The P and T axes are always at ${45}^{\circ}$ to the fault plane/auxiliary planes and always in the plane of the nodal plane normals, thus, in the plane of the double-couple force system.

Figure: Definition of the angles used to describe a fault plane solution. $\phi _S$ is the azimuth of the fault strike with respect to North, $0\leq \phi _S \leq 360$, $\delta$ is the dip of the fault plane from horizontal, $0\leq \delta \leq 90$ and $\lambda$ is the rake, or direction of slip, in the fault plane with respect to the horizontal, $-180\leq \lambda \leq 180$. The rake is the direction of motion of the hanging wall with respect to the foot wall.

The slipping fault is specified by three angles, see Figure 11. I use the convention established by Aki and Richards [1980], p. 106, for the angles of the strike, dip and rake of the fault plane solution. These angles can be used to parameterize the fault normal, ${n}$, and slip, ${\bar{u}}$, vectors, the moment tensor components $M_{pq}$ and the P and T axes, i.e. the geometrical properties of a double-couple source is adequately described by the strike, dip and rake.

The seismic moment tensor is a convenient tool for calculating the displacement field from a seismic source, since once the Earth responses (the Green's functions) to the different moment tensor components have been computed, actual ground motion is just a linear combination of the responses weighted by the moment tensor components [Aki and Richards, 1980]. The moment tensor can accommodate other types of sources such as tensile faults, volume sources such as explosions or rapid phase transitions and also non-double-couple mechanisms, where there might be a net force, such as in landslides, volcanic eruptions or unsteady-fluid flow [Julian et al., 1998]