Digital Analysis of Geophysical Signals and Waves

2.4 Finite Length Data

Before we leave this section on Digital Recording let us consider a point made above when we first represented digitizing as a sequence of numbers, s(nDt) where n went from - infinity to + infinity (equation 2.3.2). We correctly stated that this is an impossible task. When we actually record geophysical data we measure over a finite length of time or space. So, we don't know what happened before or after the time we operated our equipment (say for seismic recording) or what readings would be present at locations adjacent to where we started or ended a gravity survey. Experimental data must be truncated to yield a limited, finite-length record which is a portion of a much longer (possibly infinite) signal.

A very clever way to view this truncation is to think of the process as the viewing of the unlimited signal through a window that blocks the data where we don’t record it. The simplest window is one that passes everything unchanged within the view of the window (a multiplication by 1 everywhere we measure). And, a multiplication by zero where the signal is not observed.

Let us say that in the time domain the signal is observed over a time interval T and in the space domain, profile data are measured over the x space interval L_x. The function (window) that performs the multiplication by 1 over T or L_x (and multiplies by 0 outside of these intervals) is a rectangle function. Assuming that the interval we record goes from -T/2 to +T/2 (or -L_x/2 to +L_x/2) we define the rectangular function using the symbol II in the time domain as

or in the space domain by exchanging x for t and L_x for T. The long boxy appearance of the rectangle function (Figure 2.6) leads to the alternative, more colorful name of boxcar function. We do not believe that this name originated from a hobo studying digital analysis while riding a boxcar.

Figure 2.7 illustrates the windowing of the LANL magnetic signal using a boxcar function window. Sometimes the boxcar window is referred to as "no window" or as a "do-nothing" window because that’s just what is does to recorded data. This hints at the existence of windows that actually do something; there are many and we will discuss some of them in Section 4 on Filtering.

In the following section on Spectral Analysis we will discover why the length of the data recording (T or L_x) determines the discrete values of frequency that will be present in digital spectral analysis. These frequency values are integer multiples of 1/T (or 1/L_x) up to the Nyquist frequency, 1/2Dt (or 1/2Dx). So, the total interval over which we record our discrete data dictates the spectral frequency resolution, 1/T or 1/L_x; i.e., the spectral frequencies will be at discrete points with Df = 1/T or Df_x=1/L_x. There's more to this story which we will tell you when we revisit windows in Section 4.