Firstly, the reference value a_g is controlled by the seismicity of the zone. Its choice reflects the strength of the design earthquake, whose parameters may vary with respect to the importance assigned the structure. In the EC 8 a_g is additionally modified by a soil parameter. Both the EC 8 and the US building code prescribe that the flat part of the spectrum is 2.5 times the value of a_g.

The form of the spectrum, i. e., the position of the upper and lower boundary frequencies of the flat part of the spectrum, depend on the soil conditions. The 5% elastic pseudoacceleration response spectrum of the EC 8 (preliminary version [5])is constructed with the relations:

And is the amplification of the response spectrum with respect to peak acceleration with a proposed value of 2.5. depends on the damping after

and is 1 for = 5%. The soil classes A, B, C in the EC 8 (see Tab. 4.1) are defined with respect to their S-wave velocity and their lithological description. Roughly speaking class A represents material of relatively high velocities (800 m/s and higher) and a favorable lithology in the sense of stiffness, class B corresponds to an intermediate situation, class C stands for weak material and an S-wave velocity of 200 m/s or less in the uppermost 20 m. For more details with respect to the definition of soil classes see the original text of the EC 8.

In the US building code only formulae (4.18a) to (4.18c) are used. Its main difference resides in the parameter definition in the table. Similarly to the EC 8 there are 3 soil classes, defined with respect to the S-wave velocities and stiffness. Type 1 (see Tab. 4.2) corresponds to “Rock and stiff soils”, Type 2 to “deep cohesionless or stiff clay soils” and Type 3 represent “soft soil to medium clays and sands”. At a fist glance the US building code looks more conservative than the EC 8, particularly with respect to the treatment of unfavorable soil conditions. On the other hand in the EC 8 the definition of a_g is left to the analyst whereas the US building code gives strict rules how to obtain a_g. The values indicated in table for the EC 8, moreover, represent proposals which are subject to possible modifications by national authorities. The EC 8, reflecting the differing necessities of the EU members, may be understood in some way as a strategy of how to construct a norm spectrum rather than a rigid regulation as it is the US building code.

A more sophisticated rule is given by the seismic code of Japan. First of all it should be noted that the Japanese regulation is based on two families of four response spectra related to the soil categories “A” (“tertiary or older or rock”), “B” (“Dilluvium”), “C” ( . Contrary to the US building code the shape of the response spectra depends not only on the soil class but has to be adjusted also with respect to the seismic zone where the site belongs to (see Tab. 4.3 and 4.4). This renders it rather difficult to apply the Japanese regulation outside Japan, because one had to make sure that both soil conditions and seismotectonic setting are in agreement with the corresponding situation in Japan.

From a geophysical point of view, however, it seems to be the best founded one out of the three presented here. Note that in the Japanese code the amplification of the response with respect to peak ground acceleration is assumed to depend on the soil conditions. This is certainly realistic and can be easily checked carrying out simulations with SHAKYGROUND. It is also easy to understand that the shape of the response spectra should depend on the seismotectonic setting, in other words the type earthquakes occurring close to the site. The definition of the shape of the response spectra in the Japanese regulation differs particularly in the ascending part. First of all, the spectra are flat between T_n = 0 and 0.05 and has a value corresponding to a_g * S. Beyond the period of 0.05 s the spectra can be calculated after

[5] The actual version of the EC 8 has been accepted by the responsible institutions of the EU, but is still subject to appeal.

[6] Besides the parameter S all values in Tab. 4.3 and 4.4 have been picked from the graphical representation of the code which may introduce a slight degree of inexactness during the calculation of the response spectra with the formulae 4.19

Here C_t is a factor between 0.05 and 0.085 depending on the characteristics of the structure, and H being its height in m. After this formula the 1^st eigenfrequency of a three story building with a height of ca. 10 m and brick walls (i. e., C_t = 0.05) should be expected at around 3.5 Hz.

For the calculation of response spectra for a given ground motion first eigenfrequency and damping of the oscillator have to be selected. The next step consists in the compuation the oscillators displacement with respect to ground. The displacement response u(t) to a ground acceleration a(t) is given, for small damping values by the Duhamel integral

Instead o the Duhamel integral (2.17) which is the exact mathematical formulation but slows down considerably the processing, SHAKYGROUND uses a fast recursive filter algorithm for the calculation of the displacement response. Note that only the maximum value of the oscillators response is kept. The procedure is repeated for a set of eigenfrequencies in order to obtain a “spectrum of maximum responses”. Standard seismic regulations usually require that response spectra are given in terms of acceleration. The correct way to obtain acceleration response spectra would be to differentiate twice the displacement response of the oscillator. For historical reasons, however, the velocity and acceleration response spectra are obtained by simply multiplying the maximum displacement by and The resulting spectra are called pseudo response spectra of velocity or acceleration. In fact, many seismic regulation codes were originally designed in times when computing capacities were low and response spectra were partly obtained from Fourier spectra of acceleration time series. In order to maintain compatibility with the older seismic codes the convention prescribes the use of pseudo response spectra instead of “true” velocity or acceleration response spectra.

The response spectra, though representing a highly simplified description of structure response, have proven to be an important tool for the understanding of the damages caused by earthquakes. First of all, they are at the base of standard calculation of the structure resistance with respect to dynamic loading. In particular, the structure response of towers or similar high buildings can by simulated in a straight forward way using acceleration response spectra. From the seismological point of view response spectra, contrary to mere peak ground acceleration, provide useful information about the frequency intervals where maximum hazard has to be expected. Particularly dangerous situations arise when the eigenfrequencies of the structures happen to be in the frequency band of major seismic radiation. In microzonation studies land use planners may identify sites where specific facilities would be at particular risk, emergency planners may recognize damage prone areas and develop their scenarios with respect to these information.

where A(f) is the (complex) spectrum of the (synthetic) accelerogram, S(f) is the source spectrum (in terms of acceleration density) and T(f) represents the transferfunction of the propagation medium, or, in other words, the spectrum of the Green’s function.

Both the whole path effects as well as the site effects are crucial for the estimation of possible seismic loading. Besides the geometrical spreading, already mentioned in chapter 4.1, the waves undergo attenuation due to absorption. The absorption properties of a medium are described with a factor Q which is defined as energy decay per wave length, i. e.

The bulk attenuation effects of the medium are expressed by a factor k, given by

If we simply apply the bulk absorption, each frequency value in S(f) would be diminuished by a term e . In SHAKYGROUND the absorption is represented using complex wave number approaches. Besides an acausal approach corresponding to a viscoelastic behavior of the material, SHAKYGROUND offers a causal representation with velocity dispersion after Futterman (1962). Whatever of the two options will be the chosen by the user, the effect on the final result for most common situations will be of minor importance for engineering seismology purposes. The most important feature of the complex wavenumber approach is that multiple reflections are damped according to their way within the layer stack. This is certainly more realistic than simply applying the bulk absorption term to the spectrum S(f). Nevertheless the overall spectral shape is in agreement with the k as calculated for the geological model.

It is now commonly accepted that the geological site conditions, or, in other words, the geotechnical parameters of the uppermost several hundreds of meters underneath the receiver, have a strong impact on possible damage caused by earthquakes. There is a long list of case studies which prove this fact. We do not repeat the results of these studies here, the interested reader may find them in textbooks like Reiter (1991). The most obvious and important site effects are caused by impedance contrasts which are most pronounced close to the earth’s surface, particularly when a layer with low impedance (i. e., low seismic wave velocity and/or low density) overlies a high impedance layer.

From the principle of energy conservation it becomes immediately clear, that the amplitudes of an upward travelling wave are amplified corresponding to the impedance contrast between the two layers. Assuming a vertically incident wave (see Fig. 4) and Q the transferfunction of a medium with one layer over a half space in the frequency domain is given by

with

Extremals of amplitude amplification occur for

which become maxima if < and minima if >. Impedance contrast of up to 5 or even more are quite frequently found in layers close to the surface. The impact of those situations on damaging effects during earthquakes obvious. On the other hand one easily learns from eq. (4.14) that high impedance layers overlying structures with low impedance act like a “shield”, protecting buildings and other facilities from seismic waves.

In reality the geological models are more complicated than outlined here. The model considered in SHAKYGROUND consists of a stack of plane layers with horizontal interfaces. These layers overly a half space where the source is located. The layer stack may actually contain up to 100 layers. Each layer is characterized by its geotechnical parameters, i. e., the thickness, the S-wave velocity, the density and the factor Q for S-waves. SHAKYGROUND accounts for SH-waves (horizontally polarized shear waves). These are the most important ones in engineering seismology since they yield the dominant contribution on strong motion records. Furthermore, most buildings are more sensitive to horizontal than to vertical loading, because they are, by standard, constructed to sustain at least the gravity forces. Reflection and refraction of the waves are treated with Thomson-Haskell matrices (see Haskell, 1953, 1960) assuming plane wavefronts. The effect of reflection and refraction of the layer stack multiplying a series of “layer matrices”, i. e.

with k being the wave number, i the imaginary unit, a_n the incidence angle of the ray at the n-th layer interface, h_n the thickness and G_n the shearing modulus of the n-th layer. SHAKYGROUND carries out a raytracing for the direct wave travelling from the source to the receiver at the surface and calculates the incidence angles at each layer interface. A specific feature of SHAKYGROUND is the possibility to calculate the wavefield for a receiver position at some position within the layer stack. This option can be useful if the user wants to perform his simulations at some depth, e. g., at the base of the foundations of his construction.

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.

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.

Global source parameters are accounted for by applying a band-pass filter to the Gaussian white noise, i.e. C M₀ S(f,f₀) P(f, fmax) where C is a constant for geometrical spreading and radiation pattern, M₀ the seismic moment of the event, f₀ the corner frequency, S(f,f₀)=f²/(1+(f/f₀)²), P(f,fmax)=(1+(f/fmax)^2q)^-1/2, q the parameter of the steepness of the high frequency decay (here q=4). The corner frequency f₀ can be related to the size of the source (its radius r₀) after Brune (1970) by: f₀=0.372 c/r₀, where c is the shear-wave velocity. Finally the seismic stress drop is computed as 7M₀/(16r₀³).

The strength of the synthetic approach resides in the possibility to account for the specific geological site conditions, the effects of wave propagation and the characteristics of the seismotectonic zone. According to the strategy chosen in SHAKYGROUND the user’s experience is exploited for establishing the model parameters rather than for searching of a suitable, experimental example seismogram. Given the stochastic nature of the source model, The program performs a number of simulations and calculates average, standard deviations and peak hold values of engineering seismological parameters.
Furthermore, it allows to randomly change the geotechnical and source parameters, thus enabling the statistical evaluation of the stability of the results. Finally SHAKYGROUND produces a report of relevant simulated engineering parameters of seismological signal together with response spectra which can be directly compared to standard seismic codes.

]]> http://www.shakyground.biz/estimation-of-strong-ground-motion/feed/ 0 http://www.shakyground.biz/the-extended-source-2/ http://www.shakyground.biz/the-extended-source-2/#comments Sun, 10 Sep 2006 14:48:29 +0000 info http://www.shakyground.biz/wordpress/?p=15 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

where 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

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:

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.

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_max is 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 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.