Digital Analysis of Geophysical Signals and Waves

4.1 Introduction to Filtering

Filtering is one of the most important steps in geophysical data recording and processing.  Often this step is explicitly called filtering; other times we easily recognize the data manipulation as a filtering operation.  Every form of Earth science data from space imaginary to airborne, surface, and borehole geophysical surveying undergoes filtering before an analysis and interpretation of the data are made.  In this sense, the very act of observing the data visually or graphically is filtering by our past experience and knowledge.  Behavior scientists would say that any living organism has a response to a certain stimulus.  This is filtering.  Engineers describe filtering as an input signal being modified or transformed into a new output signal. 

The engineering approach of treating filtering as a "black box" in which an input signal s(t) is modified to yield a new output y(t) is particularly revealing we think (Figure 4.1).  The only kind of filtering that we will consider is where the "black box" performs an operation called convolution.  In fact, we can consider filtering to be identical to convolution.

The convolution integral is fundamental to what is called linear filtering or linear systems analysis.  Fortunately, there is a very nice graphical way to visualize the convolution integral since the mathematical operation is not easy to recognize on first exposure.  So convolution with continuous or digital functions will not pose any problems even though the term convolution integral often conjures up something quite formidable.  In Section 4.6 we will also address a close relative of convolution namely correlation.

An added bonus in understanding filtering or the convolution integral is that a remarkable number of defining relations in geophysics can be recognized for what they really are, namely convolution operations.  We will reserve most of these revelations for Section 5 where we discuss actual geophysical results.

If you are new to the convolution concept, you will be pleasantly surprised when we take the Fourier transform of the convolution integral in time or space domain and discover that it becomes a simple multiplication in the frequency domain.  This fact was called "possibly the most important and powerful tool in modern scientific analysis" by Brigham (1974) shortly after the FFT revolutionized applied Fourier analysis.  So, obviously, this has immense consequences in digital geophysical analysis.