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I'm doing a project in which I'm analyzing earthquake seismogram waves. I used a program to graph the exact amplitudes and how they changed over the course of a single earthquake. For the project I need to incorporate calculus, so I was wondering, what does finding the derivative of amplitude do? And what does finding the integral get you?

The integral is the area under the entire wavelength, but I was wondering if it was equivalent to energy released, or some sort of geology/physics thing I didn't understand. Thank you.

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This is a very simple answer, and it depends on the what the seismometer is measuring. I'm assuming you mean the time derviative/intdegral $\frac\partial{\partial{}t}$ or $\int{dt}$

Most seismometers measure displacement over time or velocity over time (series). So taking the time derivative will give you the spectral velocity or the spectral acceleration respectively when you take their derivatives.

When you take the integral of displacement, you calculate the total area displaced over time, and when you take the integral of velocity, you calculated the displacement.

It's all physics 101 really!

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It really depends on what type of seismometer you are using. Seismometers are mechanical/electronic instruments intended to sense ground motion, but the trace recorded on the seismogram depends on the design of the seismometer. Classical modern seismometers are actually a driven damped oscillator. You need to know the response of the seismometer to reduce it to ground displacement, velocity, or acceleration. Seismometers are orientation-dependent, and typically three seismometers are required to measure all three components of ground motion. Seismometer response is frequency dependent.

Like the development of microscopes, and telescopes, the development of seismic monitoring instruments has a long history and improvements in design have led to advances in the science. For more information about the development of seismometers through history, see this article about Earthquake Monitoring.

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The notion that an integral is the area under a curve, and that differentiation is the slope doesn't really provide a lot of physical insight for seismograms. Such signals are oscillatory, meaning that the curve moves above and below (the peaks and troughs) a mean value of zero; summing up these areas (integration) will give you zero.

However, what we see on a sample by sample basis is that summation (integration) and differencing (differentiation) both look like 90 degree phase rotations of the original signal; but in opposite directions. Just as the derivative of sine is cosine, and a sine-wave shifted by 90 degrees is also a cosine; calculus on periodic signals changes the phase of the signal.

Furthermore, the signal envelope, which is a positively valued function, might be a better description of the amplitude intensity versus time. Here is an illustration of signal envelopes on seismic traces.

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Most seismometers measure velocity of ground motion.

If you integrate velocity you get displacement. So, integrating tells you the displacement of the ground relative to its original position.

If you differentiate velocity you get acceleration. So, differentiating tells you how much the ground is accelerating at any given moment.

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