1.1 Introduction
Seismologists now know that one of the important parameters controlling earthquake damage is the fault rupture speed, and changes in this rupture speed (Madariaga ) by studying directivity effects and/or spectra of very long wave length surface waves.
In the early 1970s, Wu et al. (). Once the theoretical result was established, scientists interpreting observations became more inclined to believe results showing supershear fault rupture speeds, and at the same time the data quality and the increase in the number of broadband seismometers worldwide, required to obtain detailed information on fault rupture started becoming available. Thus, the theory spurred the search for supershear earthquake ruptures.
The first earthquake for which supershear wave rupture speed was inferred was the 1979 Imperial Valley, California earthquake which had a moment-magnitude (Mw) of 6.5, studied by Archuleta () using strong motion accelerograms. But since the distance for which the earthquake propagated at the high speed was not long, the idea was still not accepted universally. And then for nearly 25 years there were no further developments, perhaps because earthquakes which attain supershear speeds are rare, and none are known to have occurred. This provided ammunition to those who resisted the idea of supersonic earthquake rupture speeds being possible.
Then, in the late 1990 to early 2000s, there were two major developments. Firstly, a group at Caltech, led by Rosakis, measured earthquake speeds in the laboratory, not only exceeding the shear wave speed (Rosakis et al. ) showed that not only did the rupture speed exceed the shear wave speed of the medium; it reached the compressional wave speed, which is about 70 % higher than the shear wave speed in crustal rocks.
Once convincing examples of supershear rupture speeds started to be found, theoretical calculations were carried out (Bernard and Baumont shows a schematic illustrating that formulae from acoustics cannot be directly transferred to seismology. The reason is that many regions of the fault area are simultaneously moving at these high speeds, each point generating a Mach cone, and resulting in a the Mach surface. Moreover, different parts of the fault could move at different supershear speeds, again introducing complexity into the shape and amplitudes of the Mach surface. Finally, accounting for the heterogeneity of the medium surrounding the fault through which these Mach fronts propagate would further modify the Mach surface. There could be special situations where the individual Mach fronts comprising the Mach surface could interfere to even lower, rather than raise, the resulting ground shaking. Such studies would be of great interest to the earthquake engineering community.
Fig. 1.1
Schematic representation of the leading edges of the multiple S-wave Mach cones generated by a planar fault spreading out in many directions, along the black arrows , from the hypocenter ( star ). The pink shaded region is the region of supershear rupture. The thick black arrows show the direction of the applied tectonic stress across the xy plane. Supershear speeds cannot be reached in the y - direction (that is, by the Mode III or the anti-plane shear mode). The higher the rupture speed, the narrower each cone would be. Dunham and Bhat () showed that additional Rayleigh wave Mach fronts would be generated along the Earths surface during supershear earthquake ruptures
1.2 Theory
Since damaging high-frequency waves are generated when faults change speed (Madariaga ) suggested that even for such 2-D in-plane faults which start from rest and accelerate to some terminal velocity, such a forbidden zone does exist.
Fig. 1.2
The linear slip-weakening model, relating the fault slip to the stress at the edge of the fault. The region between 0 to do is called the break-down zone, where the earthquake stress release occurs. Cruz-Atienza and Olsen () estimated do to be ~2 m for the 1999 Izmit, Turkey and 2002 Denali, Alaska earthquakes
Recent work of Bizzari and Das (). To reiterate, this is important as this smooth transition from sub- to super- shear wave speeds would reduce damage.
1.3 Seismic Data Analysis
The inverse problem of earthquake source mechanics consists of analysing seismograms to obtain the details of the earthquake rupture process. This problem is known to be unstable (Kostrov ) here.
By modifying the representation theorem (e.g., equation (3.2) of Aki and Richards () for the 2001 Mw 8.4 Peru earthquake.
Thus, the inverse problem is the solution of the linear system of equations under one or more constraints, in which the number of equations m is equal to the total number of samples taken from all the records involved and the number of unknowns n is equal to the number of spatial cells times on the fault times the number of time steps at the source. Taking m > n , the linear system is over determined and a solution x which provides a best fit to the observations is obtained. It is well known that the matrix A is often ill-conditioned which implies that the linear system admits more than one solution, equally well fitting the observations. The introduction of the constraints reduces the set of permissible (feasible) solutions. Even when an unique solution does exist, there may be many other solutions that almost satisfy the equations. Since the data used in geophysical applications often contain experimental noise and the models used are themselves approximations to reality, solutions almost satisfying the data are also of great interest.
Finally, for the system of equations together with the constraints to comprise a complete mathematical problem, the exact form of what the best fit to observations means has to be stated. For this problem, we have to minimize the vector of residuals, r = b A x , and some norm of the vector r must be adopted. One may choose to minimize minimize the 1, the 2 or the norm (see Tarantola ) has discussed the positivity constraint in detail.
