Advances in Moment Tensor Inversion for Civil Engineering
Bundesanstalt für Materialforschung und -prüfung

International Symposium (NDT-CE 2003)

Non-Destructive Testing in Civil Engineering 2003
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Advances in Moment Tensor Inversion for Civil Engineering

Florian Finck, Jochen H. Kurz, Christian U. Grosse, Hans-Wolf Reinhardt


In the field of investigating fracture mechanisms, an intensive co-operation between earth seismologists and civil engineers has been existing for more than ten years. Numerous procedures of earthquake analysis and other seismological phenomena were transferred to material sciences, since the parallels to acoustic emission analysis are evident. Besides an accurate localization of fracturing an inversion on the moment tensor is performed. The moment tensor contains all information about the stress release during a rupture process, like the total energy and the orientation of rupture planes. Several methods for an inversion on this tensor are in use by seismologists today. After a suitable decomposition of the moment tensor, the type of the rupture process can be quantified and visualized using ostensive crack models, like those for shear and explosion. However, common methods of moment tensor inversion from acoustic emission data face various limitations, though relative methods were successfully applied before. In this article some theoretical principles on the decomposition of the moment tensor are reviewed. An adaptation of the physical theory to common crack models in material sciences, is discussed. Furthermore, a major influence of size effects caused by the dimensions of the specimen, the relative distance between source and receiver, or the size of the crack surface itself are evaluated in the presentation.

Keywords: moment tensor, decomposition of MT, near-field


Before the concept of the moment tensor as a physical representation of an earthquake was invented in the early 1970’s [e. g. GILBERT, 1970], focal solutions, based on the double couple model, were estimated by polarities of first p-wave (i. e. the primary longitudinal wave, arriving at first at a receiver) arrivals. This simple model was sufficient to describe the mechanics of tectonic processes within the earth’s crust. However, for a more detailed analysis of internal geological and mineralogical processes as well as the supervision of a nuclear test ban treaty, additional information must be derived from the seismograms [e. g. DAHLEN & TROMP, 1998]. The moment tensor as a mathematical description of equivalent forces and moments in a point source, is very well suited to investigate source processes with a better resolution. Using an eigenvalue analysis, the source mechanism is decomposed into basic models of fracture mechanics [JOST & HERRMANN, 1989]. A complete inversion on the seismic moment tensor implies a broad knowledge of the source time function, the size of the source, the Green’s functions of the medium and the displacement at the point of observation [AKI & RICHARDS, 1980; BEN-MENAHEM & SINGH, 1981]. Since the mathematical description of this inverse problem is extremely complex and the transfer functions of various components within the travel path of seismic energy is underdetermined, simplifications are performed to calculate moment tensors of seismic sources. The acoustic waves are assumed to be observed in the far-field of a time invariant point source. The onset of the analysed phases must be well determined with a reasonable signal to noise ratio. Depending on the inversion method, additionally the transfer functions of the sensors and effects of inhomogeneous or anisotropic media must be considered.

Theoretical Principles

A general elastodynamic source within a volume V can be represented by a sum of single forces fk dependent on location r and time t . The observed displacement dn at position x and time t is then calculated from the integral of the source time function and the Greens’s functions Gnk [AKI & RICHARDS, 1980]:


The Green’s functions contain all propagation effects of the medium on the elastic waves. The spatial integration in (1) is simplified on the assumption of only smooth variations of the Green’s functions within the source region. Then, they can be expanded into a Taylor series around a reference point, the so called centroid r = ξ. The physical source region is characterized by a set of equivalent forces. These forces arise due to differences between the model stress and the actual physical stress (stress glut). Using a temporal convolution of the Taylor expansion of the Green’s functions and the introduction of the time dependent moment tensor, (1) can be expressed in the form [JOST & HERRMANN, 1989]:


Further assumptions lead to the common form of the forward problem. The consideration of a point source, the neglection of all external forces and a time invariant moment tensor yields


The constants Mkj represent the components of the moment tensor M, s( t ) the source time function. Mkj with the unit of a moment of force is a symmetric 3x3-tensor containing six independent elements:


Various absolute and relative methods of an inversion on the moment tensor are in use. Detailed descriptions can be found in DAHM [1993], ANDERSON [2002]. The physical meaning of the moment tensor elements is illustrated in figure 1. The components of the moment tensor can be thought of dipoles (see the joints) oriented in the three spatial dimensions (columns of Mkj) on which ends forces (see the arrows) act in the three spatial dimensions (rows of Mkj). On the diagonal, the axes of the joints are parallel to the forces. For the other elements, the forces lead to a torque around the axis perpendicular to the plane containing the force and the dipole [AKI & RICHARDS, 1980].

Figure 1: The elements of the moment tensor as dipoles in the spatial dimensions and the equivalent forces, which lead to moments of force.

The symmetry of the tensor yields a total zero moment of forces. This leads to the model of the double couple. In figure 2 the double couple concept is illustrated for the two highlighted elements Myz and Mzy.

Figure 2: The p-wave radiation pattern of a pure shear event with the corresponding pair of a double couple. The symmetry of the tensor yields a zero total torsion.

The two elements Myz and Mzy build a double couple, which leads to a pure shear failure when the stress in the material exceeds a critical mechanic threshold. Dependent on the global stress regime or the constitution of the material, a crack will occur in the xy-plane or the xz-plane. These two planes are called the nodal planes. From the radiation pattern alone, it can not be resolved if the stress was released by a relative slip of the upper part to the right or a relative slip of the right part upwards. To solve this ambiguity, more information about the stress regime or an analysis of other acoustic events is necessary. Along the nodal planes the amplitudes are zero, parallel to the T-axis (Tension) the amplitudes are positive (compressional) and along the P-axis (Pression) the amplitudes are negative (dilatational). This nomenclature appears confusing at a first glance. It originates from different points of view, the source with its particle motion following the stress regime (P- and T-axis and vectors in figure 2), and the inverse corresponding amplitudes at a receiver.

Interpretation of the moment tensor

With the introduction of the double couple we already began a mechanical interpretation of the moment tensor. In figure 3 some basic source models are introduced and correlated with the estimated radiation pattern of seismic p-waves. The upper part contains a collection of seismic sources and failure mechanisms which could play a major role in material sciences. The coordinate systems on the left help to connect these models with the radiation pattern images in the lower part of figure 3, where the respective columns belong to the same models above. The illustration of the radiation pattern as a projection into the Schmidt net (FOWLER, 1990) gives all important information of the 3d-spatial distribution of seismic energy on a fictive focal sphere around the source point in a view from above (indicated by a circle). In the lower row only the polarities of the amplitudes are plotted in black and white, above that, the normalized amplitudes.

The simplest model is a pure isotropic source with constant energy in all directions, comparable to an ideal explosion (positive amplitudes) or implosion (negative amplitudes). Uni-axial tension (mode 1), displayed in the second model from left leads to an opening crack. Here, the mean particle motion is parallel to the tension axis. In the plane perpendicular to this axis some small energy might be radiated with positive or negative amplitudes, depending on the elastic properties of the medium. A change of volume will be remaining. A similar mechanism is the so called compensated linear vector dipole (CLVD), shown in the middle. Here, the change of volume is compensated by the particle motion in the plane parallel to the largest stress. The CLVD was suggested as a possible deep earthquake mechanism due to mineralogical processes [KNOPOFF & RANDALL, 1970]. A second pure deviatoric mechanism can be formulated by a double couple. Shear is the corresponding failure mechanism. Mode 2 and mode 3 can de distinguished considering the direction of the crack growth relative to the particle motion. For the first, the directions are parallel, for the latter, perpendicular to each other. In reality, the stress field and homogeneities in the material lead to a superposition of these basic mechanisms. The basis of a physical interpretation of the real and symmetric moment tensor is a transformation into the system of its principal axes by an eigenvalue analysis [JOST & HERRMANN, 1989]. From the eigenvalue decomposition of the moment tensor


we get the diagonal matrix mcontaining the eigenvalues mi, and the Matrix A, containing the corresponding eigenvectors ai. The eigenvectors are parallel to the principal stress axes, and the eigenvalues quantify the stresses. As a measure of the total seismic energy released, the scalar seismic moment can be calculated from the eigenvalue


In a first step, the isotropic component mISO is extracted from the diagonalized moment tensor m calculating the trace of M. This reveals the change in volume due to an explosion or implosion. The last term on the right defines the pure deviatoric diagonalized tensor:


The decomposition of this deviatoric part of the moment tensor with the deviatoric eigenvalues mi* can be performed into various combinations of the source models shown in figure 3. The most popular decomposition of the remaining deviatoric part, which is also applied on acoustic emission data by several authors [e. g. OHTSU et al., 2002, GROSSE, 1999]CLVD:is yielding a mean double couple mDC and the compensated linear vector dipole mCLVD:


Assuming that the same principal stresses produce the radiation of the double couple and the CLVD, with


and F = -m1* / m3, mDeviatoric is decomposed:


The deviation of the seismic source from the model of a pure double couple can be estimated by the parameter


[DZIEWONSKI et al., 1981]. For a pure double couple source, ε vanishes, for a pure CLVD, ε= 0,5. To measure the magnitudes of the mechanisms involved, percentage decompositions can be calculated [ANDERSEN, 2002]:


The model of the CLVD is not undisputable, since a Poisson’s ratio of 0,5 would be required for its realization, even under an idealized stress regime. In construction material sciences, the CLVD alone has no great significance as a mechanism of failure. In combination with a mayor positive isotropic component it can be seen as an opening crack (mode 1). One possibility to distinguish between various different focal mechanisms is simply to plot the isotropic component over the double couple component. As a very ostensive way HUDSON et al. [1989] introduced their “source type plot” on the basis of the two parameters T and k. Parameter k is a relative measure of the dilatational component (k = 1 for an explosion, k = -1 for an implosion and zero for no volume change). The deviatoric component is given by (1-|k|) and Parameter T defines its geometry. In the middle of the plot, with T,k = 0, the double couple is situated. With the extreme values for T, CLVD’s are realized.

Figure 4: Equal-area source type plot, showing key points and the positions of key source types in terms of (T,k) coordinates. The dotted lines denote the zone within which nodal surfaces exist in the p-wave radiation pattern. Outside this zone, P radiation is all of the same polarity [after HUDSON et al., 1989].


In previous investigations [e. g. FINCK et al., 2002], moment tensors were inverted from acoustic emission data, revealing very good results. The orientation of the radiation pattern, as well as the failure mechanism could be correlated with the results from FE-simulations and macroscopic observations. In other experiments the stability of the moment tensor solutions was poor. Possible explanations and sources of errors are expanded crack surfaces and influences of the near-field, but also technical aspects. An important role in the measuring chain have the ultra sonic transducers. Their transfer functions have to be examined and taken into account more precisely. These points are part of our investigations.

Currently, a series of compression tests is being performed at our institute following the KIIctests
suggested by REINHARDT et al. [1997]. The setup of the test and the stresses acting within the specimen produce four well determined clusters of acoustic emissions. This was evaluated from first localization results (see figure 5). Moment tensors of these data will be calculated soon and the source types of these events are expected to vary significantly, corresponding to shear, tensile and compressive stress. This test yields a good possibility to verify the inversion results. On the basis of the theoretical concepts reviewed in the previous sections, results from the moment tensor inversion can be presented using ostensive visualization techniques. Hereby, the T-k-Plot is a good completion to the radiation patterns illustrated in figure 3.


These studies were performed within the collaborative research centre SFB 381, supported by the Deutsche Forschungsgemeinschaft (DFG).
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