## George R. Jiracek - San Diego State University

## John F. Ferguson - University of Texas (Dallas)

## Lawrence W. Braile - Purdue University

## Bernard Gilpin - Golden West College

## 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.