The concept of moment tensor provides a complete description of equivalent forces of a general seismic point source. A source can be considered a point source if both the distance D of the observer from the source and the wavelength l are much greater than the linear source dimension. A special case is the shear dislocation along a planar fault which is described by the double couple source model. This model has proven to be able to explain well the energy and polarity pattern of seismic P -, S- and surface waves radiated by tectonic earthquakes. In the following, we briefly outline the relevant relations (in a first order approximation) between moment tensor of a seismic source and the observed seismogram. The latter may be either the complete seismogram, one of its main groups (P-or S-waves, surface waves), or specific features of seismograms such as peak-to-peak amplitudes of body waves, amplitude ratios or spectral amplitudes. Then we sketch a linear inversion scheme for the moment tensor using waveform data in the time domain. Finally, we will give an overview of some useful programs for moment tensor analysis. Applications of moment tensor inversions to the rapid (i.e., generally within 24 hours after the event) determination of source parameters after significant earthquakes will also be described.
Following Jost and Herrmann (1989), the displacement d on the Earth surface at station s can be expressed as a linear combination of time-dependent moment tensor elements Mkj(x,t) that are assumed to have the same time dependence convolved (indicated by the star symbol) with the corresponding Green’s functions Gsk,j(x,x,t):
In this expression, the higher order terms of the Taylor expansion around the source point of the Green's functions Gsk,j(x,x,t) have been neglected. Note that the source-time history s(t) (see Manual section Introduction to seismic sources and source parameters, Figs 4 and 7), which describes the time dependence of moment released at the source, is contained in Mkj(x,t). If we assume that all the components of Mkj(x,t) have the same time dependence s(t) the equation can be written as:
When determining Mkj(x , t) from seismic records, ds(x, t) is calculated by convolution of the observed seismogram y(x, t) with the inverse of the seismograph's displacement response function i(t):
ds(x, t) = y(x, t) * i(t)-1
In the frequency domain convolution is replaced by multiplication:
Ds(x, w ) = Y (x, w )I(w )-1
where w is circular frequency. The Ds(x, w ), Y (x, w ), and I(w )-1 are the respective Fourier transforms of the time series ds(x, t), y(x, t), and i(t)-1 (see section theory in the chapter "Seismic sensors and their calibration" where I(w )-1 is denoted as Td(w )-1). The Einstein summation notation is applied in equation (5), i.e., the repeated indices k and j = 1; 2; 3 imply summation over x1, x2 and x3. The Green’s function represents the impulse response of the medium between source and receiver and thus, it contains the various wave propagation effects through the medium from source to receiver. These include energy losses through reflection and transmission at seismic discontinuities, anelastic absorption and geometrical spreading. The Mkj(x , t) from eq.(5) completely describes the forces acting in the source and their time dependence.
In the following we assume that the source-time function s(t) is a delta function (i.e. a "needle" impulse). Then, Mkj(x , t) = Mkj(x )× d (t), and the right side of eq.(6) simplifies to Mkj(x )Gsk,j(t). The derivative of the Green's function Gsk,j(t) can be regarded as a seismogram recorded at x which is caused by a force couple at the source with strength Mkj(x ) (e.g. Aki and Richards, 1980, Lay and Wallace, 1995). Thus, the derivative of Gsk with regard to the source coordinate x j is equivalent to a single couple with its lever arm pointing in the x j direction (Fig. 20). For k = j we obtain a vector dipole; these are the couples (x,x), (y,y), and (z,z) in Fig. 20. A double couple source is characterized by a moment tensor where one eigenvalue of the moment tensor vanishes (equivalent to the Null or B axis), and the sum of eigenvalues vanishes, i.e. the trace of the moment tensor is zero. Physically, this is a representation of a shear dislocation source without any volume changes. Using the notation of Fig. 20, double couple displacement fields are represented by the sum of two couples such as (x,y)+(y,x), (x,x)+(y,y), (y,y)+(z,z), (y,z)+(z,y) etc. An explosion source (corresponding to M6 in eq. (7) and Fig. 21) can be modelled by the sum of three vector dipoles (x,x) + (y,y) + (z,z). A compensated linear vector dipole (CLVD, see section Decomposition of the moment tensor below) can be represented by a vector dipole of strength 2 and two vector dipoles of unit strength but opposite sign in the two orthogonal directions.
Figure 20: The nine generalized couples representing Gsk,j(x, x , t) in eq. (1) after Jost and Hermann (1989). Note that force couples acting on the y axis in x direction or vice versa (i.e. (x,y) or (y,x)) will cause shear faulting in the x and y direction, respectively. Superimposition of vector dipoles in x and y direction with opposite sign, e.g. (x,x) + (-y,-y) will also cause shear faulting but 45° off the x and y direction, respectively. Both representations are equivalent.
The seismic moment tensor M has, in general, 6 independent components which follows from the condition that the angular momentum for the equivalent forces in the source must vanish. For vanishing trace, we have 5 independent components that describe the deviatoric moment tensor. The double couple source is a special case of the deviatoric moment tensor with the constraint that the determinant of M is zero.
For a double couple source, the cartesian components of the moment tensor can be expressed in terms of strike f , dip d and rake l of the shear dislocation source (fault plane), and the scalar seismic moment Mo (Aki and Richards, 1980):
(7) Mxx = - Mo(sind cosl sin2f + sin2d sinl sin2f )
Mxy = Mo(sind cosl cos2f + 0.5 sin2d sinl sin2f )
Mxz = - Mo(cosd cosl cosf + cos2d sinl sinf ) (7)
Myy = Mo(sind cosl sin2f - sin2d sinl cos2f )
Myz = - Mo(cosd cosl sinf - cos2d sinl cosf )
Mzz = Mo sin2d sinl
As the tensor is always symmetric it can be rotated into a principal axis system such that all non-diagonal elements vanish and only the diagonal elements are non-zero. The diagonal elements are the eigenvalues of M, the associated directions are the eigenvectors. Linear combination of the principal moment tensor elements completely describes the radiation from a seismic source. In the case of a double couple source, for example, the diagonal elements of M in the principal axis system have two non-zero eigenvalues M0 and -M0 (with M0 the scalar seismic moment) whose eigenvectors give the direction at the source of the tensional (positive) T axis and compressional (negative) P axis, respectively, while the zero eigenvalue is in the direction of the B (or null) axis of the double couple.
Equation (6) describes the relation between seismic displacement and moment tensor in thetime domain. If the source-time function is not known or the assumption of time-independent moment tensor elements is dropped, e. g. for reasons of source complexity, the frequency-domain approach is chosen:
where f denotes frequency. Procedures for the linear moment tensor inversion can be designed in both the time and frequency domain using (6) or (8). We can write (6) or (8) in matrix form:
In the time domain, the d is a vector containing n sampled values of observed ground displacement at various times, stations and sensor components, while G is a 6 ¥ n matrix and the vector `m contains the 6 independent moment tensor elements to be determined. In the frequency domain, d contains m complex values of the displacement spectra determined for a given frequency f at various stations and sensor components. G is a 6 ¥ m matrix and is generally complex like `m. For more details on the inversion problem (9) the reader is referred to chapter 6 in Lay and Wallace (1995) or Aki and Richards (1980), chapter 12.
To invert (9) for the unknown `m, one has to calculate the Green's functions for the corresponding moment tensor components. The calculation of the Green's functions constitutes the most important part of any moment tensor inversion scheme. A variety of methods exists to calculate synthetic seismograms (e.g., Dornboos, 1988; Kennett, 1988; Müller, 1975). Some of the synthetic seismogram codes allow calculations for the moment tensor elements as input source while others allow input for double couple and explosive point sources. The general moment tensor can be decomposed in various ways using moment tensor elements of double couple and explosive sources so that synthetic seismogram codes employing these source parametrizations can also be used in the inversion of (9).
In this section, we will describe at some detail the moment tensor inversion algorithm of Kikuchi and Kanamori (1991) where the moment tensor is decomposed into elementary double couple sources and an explosive source. Adopting the notation used by Kikuchi and Kanamori (1991), the moment tensor Mkj is represented by a linear combination of Ne = 6 elementary moment tensors Mn (Fig. 21):
with M1 :
; M2 : 
; M3 : 
M4 : 
; M5 : 
; M6 : 
The M1 and M2 represent pure strike slip faults; M3 and M4 represent dip slip faults on vertical planes striking N-S and E-W, respectively, and M5 represents a 45° dip slip fault. The M6 represents an isotropic source radiating energy equally into all directions (i.e., an explosion).
Figure 21: Elementary moment tensors used in the inversion of the full moment tensor (after Kikuchi and Kanamori, 1991)
A pure deviatoric moment tensor (trace(Mkj) = 0) is entirely represented by the five elementary moment tensors M1 to M5. The following brief description of the linear inversion for the moment tensor (Kikuchi and Kanamori, 1991) is an example of an inversion performed in the time domain. It can be easily adopted also for an inversion in the frequency domain by replacing the time series by their spectra. Let wsn(t) denote the Green's function at station s in response to the elementary moment tensor Mn, and let xs(t) be the observed ground displacement as function of time at station s. The best estimate for the coefficients an in eq. (10) can be obtained from the condition that the difference between observed and synthetic displacement functions be zero:
The Ne is the number of elementary moment tensors, and Ns is the number of displacement records used. The other terms in (11) are given by:
Integration is carried out over selected portions of the waveforms.
Evaluating ¶ D / ¶ 
= 0 for n = 1,..., Ne yields the normal equations
with n ranging from 1 to Ne. The solution for an is given by:
The inverse 
of matrix Rnm can be obtained by the method of
generalized least squares inversion (e.g. Pavlis, 1988).
The resultant moment tensor is then given by
The variance of the elements an can be calculated under the assumption that the data are statistically independent:
where
is the variance of the data Gn. In the case where the variance
of the data is not known,
can be used as relative measure for the uncertainty.
Except for the volumetric and deviatoric component, the decomposition of the moment tensor is not unique. Useful computer programs for this purpose were written by Jost and distributed in Volume VIII of the Computer Programs in Seismology by Herrmann of Saint Louis University (http://www.eas.slu.edu/People/RBHerrmann/ComputerPrograms.html or e-mail to R. W. Herrmann: rbh@slueas.slu.edu). The first step in the decomposition is the calculation of eigenvalues and eigenvectors of the seismic moment tensor with program mteig. This performs rotation of the moment tensor M into the principal axis system. The eigenvector of the largest eigenvalue gives the T (or tensional) axis; the eigenvector of the smallest eigenvalue gives the direction of the P (or compressional) axis, while the eigenvector associated with the intermediate eigenvalue gives the direction of the Null axis. The output of mteig is the diagonalized moment tensor
(15) 
whose elements are input to program mtdec which performs a moment tensor decomposition. First, the moment tensor is decomposed into an isotropic and a deviatoric part:
(16) 
with tr(M) = m1 + m2 + m3 being the trace of M. The isotropic part of M is important in quantifying volume changes of the source, but it is usually difficult to resolve so that isotropic parts of less than 10% are often not considered to be significant. The deviatoric part of the moment tensor can be further decomposed. Options include decompositions into three vector dipoles, into three double couples, into 3 CLVD sources, into a major and minor double couple, and into a best double couple and a CLVD having the same principal axis system. The source mechanisms reported by Harvard and USGS are based on the decomposition of the moment tensor into a best double couple and a CLVD. In addition to the best double couple they also provide the moment tensor elements. To estimate the double couple contribution to the deviatoric moment tensor, the parameter
is used (Dziewonski et al., 1981) where mmin and mmax are the smallest and largest eigenvalues of the deviatoric part of M, respectively, both in absolute terms. For a pure double couple source, e = 0 because mmin = 0; for a pure CLVD, e = 0.5. The percentage double couple contribution can be expressed as (1-2e )¥100. Significant CLVD components are often reported for large intermediate-depth and very deep earthquakes. In many cases, however, it can be shown that these are caused by superposition of several rupture events that have different double-couple mechanisms (Kuge and Kawakatsu, 1990; Frohlich, 1995; Tibi et al., 1999).
Harvard and USGS publish the moment tensors using the notation of normal mode theory. It is based on coordinates (r;Q ;F ) where r is the radial distance of the source from the Earth's centre, Q is co-latitude, and F is longitude of the point source. The 6 independent moment tensor elements in the (x, y, z) = (north, east, down) coordinate system are related to the components in (r;Q ;F ) by
Mrr = Mzz
MQ Q = Mxx
MF F = Myy
MrQ = Mzx
MrF = -Mzy
MQ F = -Mxy
Generally, the quality of moment tensor inversion depends to a large extent on the number of data available and the azimuthal distribution of stations about the source. Dufumier (1996) gives a systematic overview for the effects caused by differences in the azimuthal coverage and the effects caused due to the use of only P-waves, P- plus SH-waves or P- and SH- and SV-waves for the inversion with body waves.
A systematic overview with respect to the effects caused by an erroneous velocity model for the Green function calculation and the effects caused due to wrong hypocenter coordinates can be found in Sileny et al. (1992), Sileny and Psencik (1995), Sileny et al. (1996) and Kravanja et al. (1999).
The following is a general outline of the various steps to be taken in a moment tensor inversion using waveform data:
The inversion may be done in the time domain or frequency domain. Care must be taken to match the synthetic and observed seismograms. Alignment of observed and synthetic waveforms is facilitated by cross-correlation techniques. In most moment tensor inversion schemes, focal depth is assumed to be constant. The inversion is carried out for a range of focal depths, and the best solution is taken as the one where variance of the estimate reaches a minimum.
This is an effort by the US National Earthquake Information Center (NEIC) in co-operation with the IRIS Data Managment Center to produce rapid estimates of the seismic moment tensor for earthquakes with body wave magnitudes > 5.7. Broadband waveform data are quickly retrieved from open" IRIS stations and transmitted to NEIC by Internet. These data at teleseismic distances contain the P waveforms that are used to improve the earthquake locations over what was reported initially and to compute a seismic moment tensor using a technique based on optimal filter design (Sipkin, 1982). The solution is then disseminated by e-mail to a list of subscribers. To register send a request by e-mail to sipkin@usgs.gov. More information under http://gldss7.cr.usgs.gov/ neis/FM/fast_moment.html.
The Harvard group maintains the most extensive catalogue of centroid moment tensor (CMT) solutions for strong (mainly M > 5.5) earthquakes over the period from 1976 till present. This and quick CMT solutions of recent events can be viewed at http://www.seismology.harvard.edu/projects/CMT/. The Harvard CMT method makes use of both very long-period (T > 40 s) body waves from the P wave onset until the onset of the fundamental modes and so-called mantle waves at T > 135 s that comprise the complete surface wave trains. Besides the moment tensor the iterative inversion procedure seeks also a solution for the best point source location of the earthquake. This is the point where the system of couples is located in the source model described by the moment tensor. It represents the integral of the moment density over the extended rupture area. This centroid location may, for very large earthquakes, significantly differ from the hypocentre location based on arrival times of the first P wave onsets. The hypocentre location corresponds to the place where rupture started. Therefore, the offset of the centroid location relative to the hypocentral location gives a first indication on rupture directivity. In case of the August 17, 1999 Izmit (Turkey) earthquake the centroid was located about 50 km east of the "P-wave" hypocentre. It coincided with the area where the maximum surface ruptures were observed.
This method uses a grid search algorithm to derive within 24 hours after the event fault plane solutions and seismic moment of earthquakes (M > 5.5) in the European- Mediterranean area. This is an initiative of the European-Mediterranean Seismological Centre (Bruyeres-le-Chatel, France, http://www.emsc-csem.org/) and the GEOFON Programme at the GeoForschungs- Zentrum Potsdam (http://www.gfz-potsdam.de/geofon/). The data used are P- and S-wave amplitudes and polarities. Figure 22 shows and example of the kind of output data produced. More information through http://www.gfz-potsdam.de/pb2/pb24/emsc/emsc.html.
| European-Mediterranean Seismological
Centre Centre Sismologique Euro-Mediterraneen Double couple solution provided by GFZ Potsdam EMSC event parameters: 21-JUN-2000_00:51:46.6
63.88 N 20.69 W (Iceland)
Depth = 10 km (adopted in inversion)
Depth = 9 km (based on 32 depth phases)
32 stations used in inversion: Station Delta Azimuth Takeoff Polarity -------------------------------------- adk 63.06 343.55 20.3 C aqu 29.12 121.60 27.6 C biny 38.06 262.06 26.1 x brg 22.41 109.49 33.8 C cart 28.87 146.49 27.7 C cmb 60.57 296.59 20.9 C cmla 26.31 188.62 28.2 D cor 56.07 302.72 22.0 C css 43.52 105.32 24.9 C dug 55.68 291.95 22.1 C eil 48.77 107.33 23.7 C ffc 39.71 296.00 25.8 C furi 68.86 114.32 19.0 C hgn 19.27 120.81 34.9 C incn 75.70 26.33 17.4 D kev 19.21 51.83 34.9 D kmbo 77.57 119.84 17.0 C kbs 17.85 20.07 40.1 D kwp 27.08 101.36 28.0 C morc 24.75 106.92 28.4 C mrni 46.08 104.64 24.3 C mte 24.76 155.63 28.4 C pas 63.05 292.64 20.3 C pet 58.94 331.77 21.3 C rgn 19.51 103.10 34.8 C selv 28.58 151.01 27.7 C sfuc 28.62 155.24 27.7 C sjg 55.09 235.67 22.2 D sspa 40.16 262.47 25.7 D suw 24.19 93.73 28.5 C tns 20.68 118.00 34.5 C tuc 61.55 285.54 20.7 C Data provided by: IRIS/USGS, MedNet, USNSN, GRSN, UCM/SFO/GEOFON, IRIS/IDA, GEOFON, GII/GEOFON, KNMI, IRIS/GEOFON, IRIS/AWI/GEOFON, TERASCOPE, GRSN/GEOFON, IAG, GTSN, U. Arizona |
Corner frequencies of bandpass filter: 0.020 and 0.100
Hz First fault plane: Strike = 358 degrees
Rake = 185 degrees
Dip = 85 degrees
Second fault plane: Strike = 268 degrees
Rake = -5 degrees
Dip = 85 degrees
Mo = ( 4.3 +/- 2.1)*10**18 N*m
Mw = 6.4
Source duration = 4 s (from BB displacement seismograms)
Principal axes Trend Plunge
-----------------------------------
P 223 7
N 43 83
T 313 0
N
|
########---------
##########----------
T ##########------------
# ##########-------------
###############--------------
################---------------
################---------------
#################----------------
#############----###############-
------------------###############
------------------###############
-----------------##############
-----------------##############
--- ----------#############
-- P ----------############
- ----------###########
-----------##########
---------########
|
S
Done by G. Bock (bock@gfz-potsdam.de) |
Figure 22: Example of output data produced by the routine procedure for rapid EMSC source parameter determinations by the GEOFON group at the GFZPotsdam.
Especially for the inversion of local events so called relative moment tensor inversion schemes have been developed (Oncescu, 1986; Dahm 1993, 1996). If the sources are separated by not more than a wavelength, the Green´s functions can be assumed to be equal with negligible error. In this case it is easy to construct a linear equation system, which relates the moment tensor components of a reference event to those of another nearby event. This avoids the calculation of high-frequency Green´s functions necessary for small local events and all problems connected with that (especially the necessity of modelling site transfer functions in detail).
This is a very useful scheme for the analysis of aftershocks if a well determined moment tensor of the main shock is known. Moreover, if enough events with at least slightly different mechanisms and enough recordings are available, it is also possible to eliminate the reference mechanism from the equations (Dahm 1993, 1996). This is interesting for volcanic areas, where events are swarm-like and of similar magnitude, and where a reference moment tensor cannot be provided (Dahm 1997).
Date created: September 5, 2000
Last modified: September 14,
2000
Copyright © 2000, Global Seismological
Services
Maintained by: Eric Bergman bergman@seismo.com
