Wave propagation and site effects

On their way to the surface seismic waves undergo modifications through effects of geometric and inelastic attenuation as well as through reflection and refraction. In seismology these phenomena are treated mathematically with the so-called Green’s function. In the context of synthetic seismogram generation the simulated seismogram is obtained as the convolution of the source time function (in our case the bandpass filtered and windowed gaussian noise) and the Green’s function of the propagation medium. In the frequency domain this reads as

1 wave propagation

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 p greco K. 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 qfreccia 8 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 p1c1 < p2c2 and minima if p1c1 >p2c2. 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, an the incidence angle of the ray at the n-th layer interface, hn the thickness and Gn 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.