Correction for Instrument Response

Abstract

The recorded signal from a seismic sensor will give a series of numbers which, in a given frequency range, will be proportional to velocity or acceleration. However the user wants to get the true ground motion in acceleration, velocity or displacement in the widest frequency band possible. This is also called correction for instrument response. For a given instrument, the amplitude frequency response function (gain of the instrument at different frequencies) for e.g. displacement can be determined such that for given harmonic ground displacement X(ω), the output Y(ω) can be calculated as Y ω ð Þ ¼ X ω ð Þ A ω ð Þ where ω is the frequency, Y(ω) is the recorded amplitude and A(ω) is the displacement amplitude response. In order to recover the displacement, X(ω) can simply be calculated as X ω ð Þ ¼ Y ω ð Þ=A ω ð Þ This response function can only be used for the amplitudes of a single sine wave at a given frequency. In order to make the complete instrument correction of the seismogram, the phase response must also be used. It turns out that, in general, the complete amplitude an phase response best can be described by a complex response function T(ω). In order then to calculate the corrected complex signal spectrum, X(ω), a complex Fourier transform is calculated of Y(ω) and the complex corrected spectrum is then X ω ð Þ ¼ Y ω ð Þ=T ω ð Þ of which the real part is the amplitude spectrum. In order to get the corrected complex signal in time domain, X(ω) is then converted back to time domain with an inverse Fourier transform and the corrected signal is then the real part of the converted signal. The response function T(ω) can be specified in different ways of which the most common are: discrete numbers of amplitude and phase, instrumental parameters like seismometer free period, damping, generator constant and digitizer gain or as a function described by poles and zeroes. The specification of anti-alias filters is also included in the response function.