ShakyGround » Seismic Source

Stochastic description of the seismic source

info — Tue, 09 Jan 2007 16:24:19 +0000

The description of the seismic source follows an approach proposed by Boore (1983). Boore’s method starts with the generation of a gaussian white noise. A window is applied to the noise which has the form

a is set to 1 since the scaling is done in the frequency domain, H(t) is the Heaviside function, and b is given by the relation

and

where

It reaches its maximum at the fraction e of the duration T_d. A commonly used value for is e = 0.2. At the time t=T_d the window amplitude has decreased to the threshold (for example h= 0.05). The windowed noise is filtered in the frequency domain with a bandpass filter given by

where

C* constant for geometrical spreading and radiation pattern

M₀ seismic moment

S(f_, f₀) f²/(1+(f²/f₀)²)
P(f, f_max) (1+(f/f_max)^2q)^-1/2

q parameter of the steepness of the high frequency decay (here q=4).

The filter is zerophase and therefore has an acausal response. For engineering seismology the acausal part of the signal is negligible. The spectra of the windowed random sequences are scaled with respect to their flat part between f₀ and f_max, to a value of M₀4p²f₀²/C.

The corner frequency f₀ is obtained from the source radius r₀ in terms of Brune’s (1970, 1971) model, i e.,

whereas f_max has to be specified according the ideas of the user. Often a value f_max = 20 Hz is recommended.

Spectral representation of the seismic source

info — Mon, 08 Jan 2007 16:48:46 +0000

Seismic source spectra in terms of displacement density[4] can be described in terms of a low pass filter, i. e., with a flat part at low frequencies, an intermediate part where the spectrum starts to decrease, and the high frequency part where we note a steep decay, often proportional f^-^g, where the spectral roll off g is a real number (Fig. 3a). The low frequency part can be directly related to the seismic moment using eq. 2.6. The intermediate part is characterized by the so-called corner frequency f₀. It is determined by the intersections of the tangents applied to the flat part and the high frequency part of the spectrum. The spectral roll off g in the high frequency part of the source spectrum typically has a value of 2 in a wide range. The shape of the source spectrum has induced the development of the stochastic source model which forms the base of SHAKYGROUND. Redesigning the spectrum sketched in Fig. 3a in terms of acceleration density (Fig. 3b), one notes a band-limited white spectrum with its lower bound formed by the corner frequency f₀. The upper limiting frequency, f_max, is necessary in order to obey the law of energy conservation. For this reason the radiation of seismic waves by the source must have an upper frequency limitation. Furthermore, the attenuation of the waves on their path from the source to the receiver causes additional band limiting effects.

The acceleration density spectrum shown in Fig. 3b can be interpreted as the spectrum of a band-limited white noise. In fact, this description fits well to the highly irregular and complicated waveforms of observed acceleration seismograms which are considered to be a consequence of heterogeneities (“Barriers” or “Asperities”) within the seismic source. Sometimes the heterogeneities are large enough to form “sub-events”, which can be localized separately. In most cases, however, the heterogeneities cannot be identified explicitly, but contribute anyway to the high frequency radiation. The enriched high frequency radiation is particularly visible in acceleration seismograms (see, e. g., Papageorgiu and Aki, 1983, Brüstle ,1985). The stochastic model, which outline below, accounts for the irregular and complicated waveforms of acceleration seismograms in a straightforward way.

[4] We are considering the far field contribution of seismic radiation. This is justified since, for the frequency range of technical interest, the near field terms can be neglected.

The Extended Source

info — Sun, 10 Sep 2006 14:48:29 +0000

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 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

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

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 R, i.e., for body waves with the propagation velocity c:

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

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

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.

The Point Source parameters

info — Sat, 09 Sep 2006 13:33:24 +0000

The simplest description of the seismic source is the concept of a point source (see Fig. 1). Even though in contrast with common physical and geological concepts the point source is still used for certain purposes, such as earthquake geographics or earthquake statistics. The parameters for the point source essentially are:

Further parameters concern the description of the energy released from the seismic source. A common measure is the magnitude which is obtained from the seismogram applying suitable corrections for the effects of attenuation due to geometrical spreading and absorption. The local magnitude Ml or MWA is obtained from a record on a WOOD ANDERSON seismometer with an eigenperiod of 0.8 s and a damping coefficient of 65% of critical. Ground motion and MWA are related to each other after:

where U_maxis the maximum horizontal amplitude of ground motion, V_max = 2800 is maximum amplification of the WOOD ANDERSON seismometer, s the hypocentral distance expressed in km and k(s) a factor which increases with distance. The values of k(s) or the product k(s) log(s) are given in most seismological textbooks (e. g, Richter, 1958). For small distances (0 < s < 30 km) relation (4.1) can be approximated with

MWA = log (U_max [mm] * V_max) + 1.4 log₁₀ s[km] + 0.1 (4.2)

The original definition of MWA by Richter was carried out using earthquakes from California (Richter, 1935, 1958). Even though the relations for calculating the local magnitude are applied world wide, one should be aware that they reflect in principle the characteristics of the California earthquake zones[1]. In general the application of the magnitude MWA is limited to hypocentral distances less than 1000 km and focal depths should not exceed values of ca. 20 km. Note that MWA tends to saturate for large earthquakes at a value of approx. MWA = 7.
In the 1940s Gutenberg and Richter extended the local magnitude scale to include more distant and larger earthquakes. They defined the surface wave magnitude M_S as

M_S = log A + k(s) log (s) + const. (4.3),

where A is the maximum combined horizontal ground motion amplitude for surface waves with a period of 20 sec. Tables with the values of the product k(s) log(s) can again be found, for example, in Richter (1958, pp. 345-347). Depending on the properties of earthquake scaling laws, the surface magnitude saturates at values of approx. M_S = 8.2.

For the sake of completeness we mention also the macroseismic scales or intensities as a measure for the quantification of earthquake radiation. The most common macroseimsic scales (Mercalli-Cancani-Sieberg MCS or Medvedev-Karnik-Sponheuer MKS) consist of 12 degrees corresponding to the effects caused by the earthquake at the surface. It is clear that intensities cannot be used directly for the quantification of earthquake energy since a weak event close to the surface may have the same effect as a strong one with the source situated at greater depth.

[1] For instance, the amplitude decay laws used in Richter’s formulae, in some zones produced distance-dependent local magnitudes, which is certainly undesired.

The extended source

info — Sat, 26 Aug 2006 15:30:37 +0000

The understanding of the earthquake source experienced a significant progress when it was recognized that their occurrence is tightly linked to tectonic activity (SÃƒÂ¼ss, 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) 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

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

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 R(q,f<), i.e., for body waves with the propagation velocity c:

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

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

[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.