1. LECTURE 6: SEISMIC MOMENT TENSORS
Represent other types of seismic sources as well as slip on a fault
Give additional insight into the rupture process
Simplify inverting (rather than forward modeling ) seismograms to estimate source parameters
Used to produce global data set of great value for tectonics

2. EQUIVALENT BODY FORCES REPRESENTING SEISMIC SOURCES
SINGLE FORCE - Landslide (Grand Banks slump) or Explosion (Mt. St. Helens)
SINGLE COUPLE - add 3 for isotropic explosion
DOUBLE COUPLE - slip on fault
Stein & Wysession, 2003

4. MT SAINT HELENS 1980 ERUPTION
Seismic source modeled by southward-pointing single force, opposite direction of the north-directed explosion and northward-flowing landslide. Modeling gives estimates of the force involved in the landslide and explosion, which devastated more than 250 square miles (640 km 2) on the north side of the mountain.
This explosion is equivalent to an Ms 5.2 earthquake, significantly bigger than the smaller earthquakes often associated with magma movements within volcanoes.
USGS

5. Ms 7.2 earthquake, one of the largest in a minor zone of
seismicity along the Atlantic continental margin of Canada
Large underwater slump (landslide in which mass of rock moves as a coherent body)
generated powerful sediment flows, known as turbidity currents, which ruptured telephone cables and hence provided important evidence on the speed and force of such currents. S waves are reasonably well modeled by synthetic seismograms for a horizontally-oriented single force, implying that the slump itself was the seismic source. However, another study found that the seismograms were well modeled by a double-couple earthquake at about 20 km depth, which triggered the slump.
Whether earthquakes are needed to generate such slumps is interesting because such mass movements, which might occur on many heavily sedimented continental margins, can also generate significant tsunamis. The tsunami for this earthquake caused 27 deaths along the Canadian coast, and a slump following an Ms 7 earthquake is thought to have caused the devastating 1998 New Guinea tsunami which caused over 2,000 deaths.

6. General representation of seismic source using 9 force couples
SEISMIC MOMENT TENSOR
General representation of seismic source using 9 force couples
Stein & Wysession, 2003

7. REPRESENTING EARTHQUAKE WITH MOMENT TENSOR
Simple representation yields seismic waves produced by a complex rupture involving displacements varying in space and time on irregular fault
First, approximate rupture with a constant average displacement D over a rectangular fault
Approximate further as a set of force couples.
Approximations are surprisingly successful at matching observed seismograms.
Stein & Wysession, 2003

8. FOR FAULT ORIENTED NORMAL TO COORDINATE AXIS, MOMENT TENSOR IS
Mo is scalar moment

9. EARTHQUAKE (DOUBLE COUPLE) MOMENT TENSOR
Interchangeability of n and d makes the tensor symmetric (Mij = Mji). Physically, this shows that slip on either the fault plane or the auxiliary plane yield the same seismic radiation patterns.

10. TENSOR PROPERTIES OF Mij
COMPARE TO STRESS TENSOR
The moment tensor for a double couple transforms this way, because it is a physical entity relating the normal and slip vectors in a way similar to the way the stress tensor relates the normal and traction vectors.

12. FIND FAULT GEOMETRY FROM MOMENT TENSOR

16. EXPLOSION IMPLOSION EARTHQUAKES (DOUBLE COUPLE) OTHER SOURCES (CLVD)
Dahlen and Tromp, 1998

19. EXPLOSION IMPLOSION EARTHQUAKES (DOUBLE COUPLE) OTHER SOURCES (CLVD)
Dahlen and Tromp, 1998

20. Nettles and Ekstrom, 1998

23. MOMENT TENSOR ADVANTAGES
FOR SOURCE STUDIES:
Analyze seismograms without assuming that they result from slip on a fault. In some applications, such as deep earthquakes or volcanic earthquakes, we would like to identify possible isotropic or CLVD components.
Makes it easier to invert seismograms to find source parameters, because seismograms are linear functions of components of the moment tensor, but are complicated products of trigonometric functions of the fault strike, dip, and slip angles. This is not a problem in forward modeling, but makes it hard to invert the seismograms to find the fault angles.

24. MOMENT TENSOR INVERSION

38. MOMENT TENSOR DATA FOR TECTONIC STUDIES
Globally-distributed broadband digital seismometers permit reliable focal mechanisms to be generated within minutes after most earthquakes with Ms > 5.5 and made available through the Internet.
Several organizations carry out this service, including the Harvard CMT (centroid moment tensor) project.
CMT inversion yields both a moment tensor and a centroid time and location. This location often differs from that in earthquake bulletins, such as that of the International Seismological Centre (ISC), because the two locations tell different things. Bulletins based upon arrival times of body wave phases like P and S give the hypocenter: the point in space and time where rupture began. CMT solutions, using full waveforms, give the centroid or average location in space and time of the seismic energy release.
The availability of large numbers of high-quality mechanisms (Harvard project has produced over 17,000 solutions since 1976) is of great value in many applications, especially tectonic studies.