The omega-square hypothesis for the seismic source

📝 Abstract

The omega-square hypothesis assumes that the ground displacement u(t) in the far-field zone decays as the inverse square of frequency in the range ~1-30 Hz. This empirical fact remains theoretically unjustified. Our analysis of the problem is based on an integral representation of u(t) in terms of the source time function f and on the spectrum analysis of local features in f. The goal is to select the local features that will enable one to generate the omega-square behavior of u(t) on a large set of receivers. We found two appropriate fragments of f : first , f exhibits a local inverse-square-root behavior near the rupture front where the frontal surface is piecewise smooth (but not smooth and not rough ); and second, f is bounded near the rupture front where the frontal surface has a slight roughness (the Hurst parameter near 1). These facts can be useful for understanding the omega-square spectral behavior of the kinematic source models.

💡 Analysis

The omega-square hypothesis assumes that the ground displacement u(t) in the far-field zone decays as the inverse square of frequency in the range ~1-30 Hz. This empirical fact remains theoretically unjustified. Our analysis of the problem is based on an integral representation of u(t) in terms of the source time function f and on the spectrum analysis of local features in f. The goal is to select the local features that will enable one to generate the omega-square behavior of u(t) on a large set of receivers. We found two appropriate fragments of f : first , f exhibits a local inverse-square-root behavior near the rupture front where the frontal surface is piecewise smooth (but not smooth and not rough ); and second, f is bounded near the rupture front where the frontal surface has a slight roughness (the Hurst parameter near 1). These facts can be useful for understanding the omega-square spectral behavior of the kinematic source models.

📄 Content

1

                         The omega-square hypothesis  
                                                   G. Molchan 
                    Institute of Earthquake Prediction Theory and Mathematical Geophysics, 
                       84/32, Profsoyuznaya  Str., Moscow,117997, Russian Federation. 
                                                 E-mail: molchan@mitp.ru 

Summary. The omega-square hypothesis assumes that the ground displacement ) (t u in the far-field zone decays as the inverse square of frequency in the range ~1-30 Hz. This empirical fact remains theoretically unjustified. Our analysis of the problem is based on an integral representation of ) (t u in terms of the source time function f and on the spectrum analysis of local features in f . The goal is to select the local features that will enable one to generate the omega-square behavior of ) (t u on a large set of receivers. We found two appropriate fragments of f : first , f exhibits a local inverse-square-root behavior near the rupture front where the frontal surface is piecewise smooth (but not smooth and not rough ); and second, f is bounded near the rupture front where the frontal surface has a slight roughness (the Hurst parameter near 1). These facts can be useful for understanding the 2   spectral behavior of the kinematic source models.

.1 Introduction.
It is commonly thought that the ground displacement in the far-field zone decays as 2   in the range ~1-30Hz. This behavior is known as the 2   hypothesis and its justification is still a problem to be solved. The interest in a theoretical substantiation of this hypothesis peaked in the 1980s (Aki and Richards, 1980). However, the interest does not wane, because the problem is important for modeling strong motions for engineering purposes (Gusev 2013, 2014).

2 Our analysis of the 2  

• hypothesis is based on the following integral representation of the far-field displacements at a receiver site rec G (for simplicity, the signal is considered in the scalar form):

   d c G dist t f A t u rec ) /) , ( , ( ) ( g g        d c t t f A )) / , , ( 0 γ g g . (1) (see Aki and Richards, 1980). Here  is a broken area of the planar fault with coordinates
g  ) , ( 2 1 g g , f is a source time function, i.e., the local slip velocity ) , ( t u g   ,  γ is the orthogonal projection of the hypocenter-to-receiver direction γ onto the rupture plane,   , is the scalar product, с is wave velocity, ) , ( 0 rec G hypocenter dist c t  , 2 1dg dg d   is an element of and the A factor combines constant coefficients, geometric spreading, and the wave radiation pattern for unit force. In the framework of fracture mechanics, solutions ) (t u or ) , ( t f g are known in exceptional cases and, as a rule, for 2D rupture problem (i.e., the 1-D source model)(Kostrov,1964; Nielsen and Madariaga, 2003). Three-dimensional rupture problem require numerical computations that impede HF analysis (Madariaga, 1976; Madariaga et al., 1998). To justify the 2   hypothesis we follow the line of argument which goes back to Madariaga (1977) and Achenbach and Harris (1978). These authors assume that the slip velocity has a number of universal ‘topological features’ which are responsible for the high frequency generation. Proceeding on these lines, we need to choose the physically reasonable features that systematically create the 2   behavior with a large set of receivers. The last requirement is essential when we deal with the 3D rupture problem. Below we focus on a piecewise-smooth and fractal peculiarities of f . We find among them some acceptable solutions and show the difficulties in the way of justifying the 2   hypothesis.

2 Piecewise-smooth functions ) ,t f (g . The Fourier transform of (1) is

dt c t t f e d A u t i ) / ( ) (ˆ 0          γ g, g,  

ds s f e d Ae c s i t i ) ( / ( 0 g, Σ γ g,          ) , (ˆ 1 0        c f Ae t i γ (2)

3 where ) , (ˆ  k f is the Fourier transform of function ) ,t f (g that is extended as zero beyond its support.
In the spectral terms our problem looks as follows: describe analytical, but physically reasonable, features of piecewise smooth functions ) ,t f (g with a bounded support f  such that 1 ), (ˆ    p f has the omega-square asymptotics for a large set of vectors )1, / ( c    γ р . In applications, to simplify the inverse source problem or to generate the signal ) (t u with the appropriate properties, the models like the k -squared model of Herrero& Bernard (1994) are used. They are based on the postulate that
2 ) , (ˆ   k k const f  for large k . In this case the omega-square hypothesis is valid automatically a

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