Some technical issues regarding wave amplitudes and digital seismographs
Amplitudes in frequency domain
The waveform seen in a seismograph for a particular phase , S(t) , can be theoretically synthesized as the convolution of several functions:
S(t) = STF(t) * G(t) * Q(t) * R(t)
where "*" denotes convolution, STF is the earthquake source time function, G is the Green's function which includes all effects of elastic wave propagation and focal mechanism, Q is the attenuation filter, and R is the instrument response which here includes both the analog and digital aspects.
To understand certain aspects of seismograms, it may be useful to transfrom the above time-domain expression into the frequency-domain. One of the well-known properties of the Fourier transform is that it converts a chain of convolutions into a chain of simple multiplications:
After Fourier transform -> s(f) = stf(f) g(f) q(f) r(f)
where f is frequency and lower case functions indicate the Fourier transforms of the above time domain functions. We then commonly "throw out" the phase information and just look at spectral amplitudes. When we are concerned about the high-freq roll-off in our seismograms, this frequency-domain spectral view is just what we need.
stf(f) contains the earthquake information we seek. The seismic moment determines the basic small-freq level of stf and earthquake duration determines the "corner frequency" where the high-freq decay begins. We should try to measure spectral amplitude at the smallest frequency so that our magnitude is the best measure of earthquake size rather than some combination of size and duration.
g(f) is highly variable as it depends on the phase and distance. The one sketched here is a simple one (as for a body wave phase) that has a couple of spectral peaks & zeroes that are caused by reverberations and earthquakes depth.
Below, we will consider the attenuation and instrument response functions.
Wave attentuation at higher frequencies (f>1 Hz) can be highly variable, yet control the largest individual peak amplitude in a seismogram. If we use a magnitude prescription that uses a 10 Hz peak in one seismogram (i.e. period of 0.10 sec) and a 1 Hz peak in another seismogram, then we must have a detailed specific attentuation correction in our magnitude formula. This is the approach in the Hermann & Kijko formulation. Alternatively, we can use spectral amplitude at a smaller frequency that is not so senstive to attenuation variations.
The response of modern seismometers like the S102 is "flat" to ground velocity out to a corner frequency, and then follows a (f^-2) high-freq decay. One of the key determinants in the appearance of seismograms is this instrument frequency corner. For the S102, the instrument reponse is down to:
- a factor of 0.77 at a freq of o .67 Hz (LLg period of 1.5 sec),
- a factor of 0. 45 at a freq of 1.0 Hz,
- a factor of 0.12 at a period of 2.0 Hz,
- a factor of 0.005 at a period of 10.0 Hz,
Thus, the S102 strongly filters the highest frequencies that can be seen in some USNSN seismograms. However, it turns out that these high frequencies must be filtered even more before we can digitize the signal -> see next section!
Aliasing & Digital Sampling
Aliasing occcurs if an anlog signal contains significant spectral amplitude at frequencies greater than the Nyquist frequency of digital sampling. The basic effect is illustrated below:
There are two ways to avoid aliasing:
-> analog filters before digitizing
-> digital over-sampling, then digital filter before final sampling.
The second way is best, and that method is used by the SeismoGraf software for the OhioSeis stations. In detail, there are several digital filters that can be used. Part of the consideration is the dynamic range of the A-D board, the other part is the nature of the seismological signal. The SeismoGraf filter is somewhat conservative, there is very little amplitude in the seismograms at frequencies higher than 2.0 Hz.
If we adopt the magnitude procedure to measure the largest isolated peak amplitude, even if it's frequency is close to the Nyquist frequency, then our magnitude determinations are not only sensitive to the seismological influences on the high-freq cut-offs, but our magnitude estimates might also change as we change sampling rates or small details of the anti-aliasing filter.
We should try to avoid amplitude measurements on peaks with pulse widths close to sampling interval!