Amplitude pitfall

In geophysics we often discuss signal amplitude versus offset distance. It sounds easy, but there are some serious pitfalls. Such pitfalls are one reason why mathematicians often use nonintuitive weasel words. The best way for you to appreciate the pitfall is for me to push you into the pit.

Suppose we are writing a seismogram modeling program and we wish to model an impulsive plane wave of unit amplitude. Say the signal seen at x is $(\cdots,0, 0,1,0,0, \cdots )$.At $x+\Delta x$ the plane wave is shifted in time so that the impulse lies half way between two points, say it is $(\cdots,0, 0,a,a,0,0, \cdots )$.The question is, ``what should be the value of a?'' There are three contradictory points of view:

1.

The amplitude a should be 1 so that the peak amplitude is constant with x.

2.

The amplitude a should be $1/\sqrt{2}$ so that both seismic signals have the same energy.

3.

The amplitude a should be 1/2 so that both seismic signals have the same area.

Make your choice before reading further.

What is important in the signal is not the high frequencies especially those near the Nyquist. We cannot model the continuous universe with sampled data at frequencies above the Nyquist frequency nor can we do it well or easily at frequencies approaching the Nyquist. For example, at half the Nyquist frequency, a derivative is quite different from a finite difference. What we must try to handle correctly is the low frequencies (the adequately sampled signals). The above three points of view are contradictory at low frequencies. Examine only the zero frequency of each. Sum over time. Only by choosing equal areas a=1/2 do the two signals have equal strength. The appropriate definition of amplitude on a sampled representation of the continuum is the area per unit time. Think of each signal value as representing the integral of the continuous amplitude from $t-\Delta t/2$ to $t+\Delta t/2$.Amplitude defined in this way cannot be confounded by functions oscillating between the sampled values.

Consider the task of abandoning data: We must reduce data sampled at a two millisecond rate to data sampled at a four millisecond rate. A method with aliasing is to abandon alternate points. A method with reasonably effective antialiasing is to convolve with the rectangle (1,1) (add two neighboring values) and then abandon alternate values. Without the antialiasing, you could lose the impulse on the $(\cdots,0, 0,1,0,0, \cdots )$ signal. A method with no aliasing is to multiply in the frequency domain by a rectangle function between $\pm$ Nyquist/2 (equivalent to convolving with a sinc function) and then abandoning alternate data points. This method perfectly preserves all frequencies up to the new Nyquist frequency (which is half the original).