The Extended Source

The understanding of the earthquake source experienced a significant progress when it was recognized that their occurrence is tightly linked to tectonic activity (Süss, 1873, 1875). Indeed, the most prominent seismoactive zones are found along the midatlantic ridges, the mountain belts and along major fault systems in the continental plates, i. e., zones with intensive tectonic activity. Earthquakes are nowadays seen as a particular manifestation of geodynamical processes. In occasion of the 1906, San Francisco earthquake, Reid (1910) demonstrated that during earthquake process two crustal segments along the San-Andreas fault were dislocated with respect to each other. Following the hypothesis of Reid, which by now is generally accepted, earthquakes are caused by a shear fracture of crust material which had been deformed in the time before the occurrence of the earthquake until reaching a critical state.


The link with geotectonic movements, together with the physical evidence, that a finite (non-vanishing) amount of elastic energy cannot be concentrated in a point (Tsuboi, 1956) lead to the development of concepts describing the earthquake source as a volume with finite (i.e., non-zero) extensions.

The place where the earthquake dislocations occur is referred to as source plane or source area. Common models use source planes of rectangular or circular shape (see Fig. above). The characteristics of the seismic signal highly depend on form and extension of the source plane, together with the amount of dislocation occurring across the source plane. The extension of the source and the dislocation define the first important parameter used in SHAKYGROUND: the seismic moment M0.

The seismic moment is given by

1 formula The extended source

where lettera the exten..source is the shearing modulus [Pa], A the source area [m2], <do> the average dislocation [m]. The seismic moment can be determined from the seismic signal in the far-field using

2-Formula-Extended-Sou


where u(t) is the ground displacement to be integrated over the source duration T, t is the time variable, and C is a factor accounting for geometrical spreading and the radiation pattern R2-lettera-Extended-Source, i.e., for body waves with the propagation velocity c:

3-formula-Extende-Sourc

In simple words, the seismic moment is proportional to the area of the far-field source time function. Keeping the seismic moment fixed, the peak amplitude of the ground displacement is inversely proportional to the source duration T.

Most common source models claim an inverse relation between source dimensions (length, radius) and the duration of the source time function. With a given seismic moment a short source duration implies a small extension of the seismic source, and consequently, according to eq. (4.4), a high value for the average dislocation . Instead of directly relating to the source dimension, seismologists prefer to use the global (seismic) stress drop, which is proportional to the average dislocation. The global stress drop is the second parameter used in SHAKYGROUND to describe the source properties. We calculate the global stress drop with the formula

5 form Extended sou

In SHAKYGROUND t is given in bars according to common conventions in seismology. Note that 1 bar = 105 Pa = 105 N/m2. The next step is to understand how SHAKYGROUND uses the relations outlined here for the generation of synthetic accelerograms.

 

 

2 A more exact formula for the seismic moment, which takes into account the possible heterogeneities within the source volume is given by

4-Form.-Extended-sour

3 The formula 2.6 strictly holds for circular sources. In SHAKYGROUND eq. 2.6 is used to fix the lower frequency bound using a Brune (1970) source model. For non-circular source planes a good approximation can be achieved by taking the source area in m2 and calculating an equivalent source radius.