Computational Geology 27 Logarithmic Scales

Topics this issue-

Mathematics: geometric vs. arithmetic progressions; logarithms; antilogarithms; orders of magnitude; exponential function.

Geology: Richter magnitude; density; viscosity; Mohs hardness; Knoop hardness; solar system; geologic time scale.

INTRODUCTION

Asimov (1982, p. 15) tells of the first university. It was named The Academy for Academus, who once owned the grounds. The founder of the Academy was Plato (~427 B.C. to ~347 B.C.). Over the entrance, Plato inscribed the words, "Let no man ignorant of geometry enter here."

To Plato, mathematics (geometry) was a domain of idealized abstraction. Now, mathematics is studied by many in academia ("the academy") because of the insight it gives to the material world. To many, mathematics provides a way of conceptualizing the real world. This includes the real world of geologists.

The world of geologists involves big space and big time. Geologic variables vary over a huge range. This huge variation applies not only to the basic concepts of length and duration, but also to properties of geologic substances - properties such as density, viscosity and hardness. To conceptualize points along the axes or these variables, we often must take steps that are multiplicative rather than additive. We must expand our mind to think of geometric progressions, not just arithmetic ones.

The entrance to the geologic academy today could be marked with the sign "Let no person ignorant of geometric progressions enter here."

LOGARITHMIC SCALES

A logarithmic scale of a graph converts geometric progressions to arithmetic progressions. A geometric progression is a succession of numbers that increase (or decrease) by a common ratio; for example, 1, 2,4, 8, is an increasing geometric progression with a common ratio of 2. An arithmetic progression is a succession of numbers that increase (or decrease) by a common difference; for example 20, 18, 16, 14, is a decreasing arithmetic progression with a common difference of -2.

The spreadsheet of Figure 1 illustrates how a logarithmic scale converts geometric progressions to arithmetic progressions. Column C lists three geometric progressions: Progression 1 has a common ratio of 10; Progressions 2 and 3 have common ratios of 2 and 3, respectively.

The upper graph of Figure 1 has an arithmetic scale. Note how the numbers of each progression are spaced further apart as they increase. In Progression 3, for example, there is more space between the sixth and seventh numbers (32 and 64) than between the first and sixth (1 and 32).

The lower graph has a geometric scale. On this graph, the numbers in each progression are spaced equally. For each one of the geometric progressions, there is a common difference in the height or the numbers on the logarithmic chart. The heights of successive numbers form an arithmetic progression for each geometric progression.

For the first progression, with a common ratio of 10, the common difference on the log scale is the same as the spacing on the log axis. Call this distance 1 unit. Then the spacing between the numbers of Progression 2 on the logarithmic chart is 0.301 units, or log(2). Similarly, the spacing between the numbers of Progression 3 is 0.477 units, or log(3).

RICHTER MAGNITUDE

The Richter Magnitude, which quantifies the size of earthquakes, is probably the most well known logarithmic scale in geology.

Spacing along the log scale - The spreadsheet of Figure 2 lists Richter magnitudes (Column C) for six selected earthquakes (Col D). The magnitudes (Ms) range from 6.5 to 8.6. The lower graph of Figure 2 is a plot of these magnitudes. The upper graph is a plot of the antilogarithm of the magnitudes (Col F). Thus the ordinate (y-axis) of the upper graph is an arithmetic scale. The ordinate of the lower graph is a logarithmic scale.

In reading graphs with logarithmic scales such as the lower graph in Figure 2, one needs to remember that the numbers on the logarithmic scale represent steps along a feometric sequence. Thus the vertical differences etween 6, 7, 8 and 9 on this graph are not intended to portray a common difference; rather they portray a common ratio. One can anticipate that, although the difference in Richter Magnitude between the Anchorage and San Francisco earthquakes (the top two, with AM5 = 0.3) is smaller than the difference in Richter Magnitude between the Loma Prieta and San Fernando earthquakes (the bottom two, with AM5 = 0.6), the "sensed' (i.e., arithmetic) difference in size between the Anchorage and San Francisco earthquakes is much larger than that between the Loma Prieta and San Fernando earthquakes. The upper graph shows exactly that.

From Equation 3, the values in Column F of Figure 2 are larger than the amplitude of the surface wave in microns by a factor equal to 2000 DL66/T. For T = 20 sec, and D = 20° (2222 km), this factor is 14,445 microns. Thus the values in Column F and along the ordinate of the upper graph in Figure 2 are in units of 0,00007 microns. In other words, the common difference (1×10^sup 8^) of the arithmetic progression along the axis is equivalent to 7000 microns (or 7 mm) for the amplitude of a 20-sec surface wave 20° away from the epicenter.