## 1.1 Introduction to Geophysical Analysis

Figure 1.2. Magnetic measurements recorded digitally using Geometrics 826 cesium vapor magnetometer.
Figure 1.1. Gravity measurements with "workhorse" LaCoste and Romberg G meter.

Geophysical field measurements begin a discovery process aimed at revealing hidden features in the Earth. Such measurements include gravity and magnetic observations along a survey line or over an area (Figures 1.1 and 1.2), and seismic and electromagnetic signals (Figures 1.3 and 1.4) produced by artificial and naturally occurring time-varying sources. Gravity and magnetic measurements are recorded in space; seismic and electromagnetic data are functions of both space and time. We refer to these measurements as being made in the space domain(x,y,z) or time domain (t). Figure 1.5 contains an example of space domain magnetic field data. Very few measurements are taken continuously in the space or time domains. Old-fashion pen chart recorders produced such records. Computers require discrete values for computation so geophysical data must be discretized (or digitized) for analysis. Gravity and magnetic measurements are inherently discrete because the instruments are moved from point to point where time is required to make a reading.

Seismic and electromagnetic sensors electronically measure time domain signals continuously; the digital output results from instrumental analog (continuous) to digital (A-to-D) conversion.

Figure 1.3. Students operate digital Sensor and Software Noggin 250 Plus ground penetrating radar (an electromagnetic system).
Figure 1.4. Digital shallow seismic recording with 48 channel Bison 9048 field seismograph.

Ideally, digitizing is performed at equal time or space intervals. This is easy to do with time domain signals but may not be so in the space domain where survey accessibility can be limited. Digital seismic data have been routinely recorded since the 1960’s but it has been within the last decade that everyday devices have "gone digital." Who isn’t constantly bombarded with advertisements touting digital sound, digital cell phones, and digital cameras?

Digitizing is not a trivial process since a host of potential pitfalls are lurking within this simple step. For example, questions such as how often should the digitizing interval be in space or time, for how long should the signal be recorded, and how fine should the signal amplitude be sampled, must be addressed properly.

Nothing is more important since improperly recorded digital data can be totally worthless or completely misleading. And, once the data have been improperly recorded digitally, the correct data can not be recovered! No one wants to return to the field to fill in a poorly sampled gravity or magnetic survey, or find out that they have missed the only opportunity to correctly record a rare seismic event. Section 2 of this web site explicitly addresses these considerations and relates them to the consequences inherent in transforming space and time domain data into the frequency domain, ordinary frequency, f (= 1/period, T) or angular frequency, ω (=2πf).

Figure 1.6. Visualizing spatial frequency components in space domain magnetic data.
Figure 1.5. Magnetic data in space domain.

...improperly recorded digital data can be totally worthless or completely misleading. Analysis of the frequency content of geophysical data is called spectral analysis which is addressed in Section 3. Everyone is familiar with temporal frequency, e.g., household electric current in most places has a frequency of 50 or 60 Hz (cycles/s). Geophysical data collected in the space domain have spatial frequencies which are expressed in cycles/distance, e.g., cycles/km or cycles/m.

One can usually visualize frequency components in geophysical data while still in the space or time domains. For example, we can easily identify three very different frequency components in the magnetic data shown in Figure 1.5. One is at approximately 1 cycle/4 m centered at about 5 m distance and at near 10 m distance there are spatial frequencies of 1 cycle/2 m and 1 cycle/1 m (Figure 1.6). There are many more frequency components in this magnetic signal but they are not as easy to identify with the eye. Fourier analysis (Section 3) describes how to obtain all of the spectral components. These are derived in Section 5.1 for the magnetic data in Figure 1.5. Fourier analysis works because geophysical signals (with very few exceptions) can be made up exclusively of trigonometric functions (sines and cosines). For some signals, the number of frequency components is only a few, for others it literally takes an infinite number of frequency contributions to construct the complex waveform. A complete description of the frequency domain components of a signal produces the spectrum of the continuous signal where the amplitude of the frequency components can be plotted versus frequency. There are many fine textbooks dealing with Fourier analysis. Some will be referenced when appropriate. Our intentions are to provide the basic fundamentals for applied geophysical analysis-strict mathematical rigor and proofs can be found in the references.

Figure 1.7. Magnetotelluric "sounding" in frequency domain containing high and low temporal frequency data (short and long period results, respectively).

Viewing geophysical data in the frequency domain can more clearly reveal the sought after features in the measurements. For example, gravity and magnetic data (Figures 1.5 and 1.6) with higher spatial frequency content are sensing smaller and/or shallower subsurface variations. Low temporal frequencies recorded by the magnetotelluric electromagnetic method (Figure 1.7) originate from deeper layers compared to high frequency data.

In Section 4 we describe how a remarkable amount of geophysical analysis amounts to filtering the data. Filtering can be accomplished in time, space, or frequency domains. So-called linear filtering or convolution is surely one of the most fundamental operations imbedded in the theory, processing, and interpretation of geophysical data. Once recognized and understood, this provides a powerful tool for a geophysicist.

The fifth and final section of our computer-aided learning program provides examples of digital recording, spectral analysis, and filtering of geophysical signals and waves. One example will be filtering of the ground magnetic data presented in Figures 1.5 and 1.6 in both space and frequency domains.

The need for this web based learning program was apparent from our experience with over 30 years of SAGE students, particularly, senior undergraduates. We found that there truly is a "digital divide." We hope that all SAGE (Summer of Applied Geophyisical Experience) students and students everywhere will find that our practical approach helps them appreciate the incomparable value of digital, spectral analysis and its beauty. A knowledge of undergraduate physics and mathematics through differential and integral calculus, including complex notation is assumed. No prior digital or spectral analysis background is needed. A course in geophysics is desirable but not required. The material is designed for senior level and graduate earth science students. Support for the development of interactive Digital Analysis of Geophysical Signals and Waves was received from the U.S. National Science Foundation and the Society of Exploration Geophysicists.