# Understanding the frequency scale of a spectrogram

The graph below was derived from a raw seismogram recorded during an earthquake over a timespan from t=0 to t=1400 seconds (not shown on x-axis).

The original seismogram $s(t)$ is not shown, but it is known that the original signal is the convolution between the source wave $w(t)$, the effects of the medium $g(t)$, and the effects from instrument itself $i(t)$; that is,

$$s(t)=w(t)*g(t)*i(t)$$

The graph above is what remains after removing the instrument response as shown below:

I think this instrument response is telling me that the amplitudes of the original seismogram $s(t)$ at frequencies higher than $0.02Hz$ were amplified at a constant gain of approximately $1E9$, while amplitudes lower than $0.01Hz$ are effectively dampened (by gains significantly smaller than at those frequencies higher than $0.02Hz$).

But what exactly can I effectively say about the shape of the original seismogram from this information? If these frequencies are 'dampened' in the broadband representation above, what, if anything, can be said about the amplitudes and shapes observed in the original graph? Can I simply say that longer waves should be present in the original signal but not say anything about when they occur? Could I somehow convolve these graphs to generate the original signal (or at least closely approximate it)?

This is similar to the process of equalization used in communications systems. Since you have some knowledge of the system (in this case characterized by the impulse response $i(t)$ or the frequency response plot that you showed), you can attempt to undo the effects on the signal that are imposed by the measurement instrument. More generally, this type of operation is known as deconvolution: given an output of a convolution and optionally knowledge of one of the inputs to the convolution, one would like to calculate the other input. In this case, one of the convolution inputs is $i(t)$ and the output is the measurements reported by your instrument.
Edit: And, as to your questions about how you're interpreting the plot, your understanding seems correct. The system has a highpass characteristic with a flat passband starting at approximately $0.02\ \text{Hz}$, with the magnitude response decreasing by a factor of 10 per decade of frequency below $0.02\ \text{Hz}$. This could be modeled well as a single-pole highpass filter.
• They would have larger amplitudes relative to the high-frequency region. The plot you showed indicates that the instrument has a passband gain of $10^9$; even after the response has rolled off significantly, it still has a gain of $10^4$. So overall, the original signal will be much smaller in amplitude across the board. But, the high-frequency components of interest get amplified by the instrument much more than the lower frequencies. Oct 15, 2012 at 17:40