From: stead@skadi.CSS.GOV (Richard Stead)
Newsgroups: ca.earthquakes
Subject: Re: Richter scale question
Date: 9 Nov 92 15:18:45 GMT

In article <1992Nov6.180742.11228@ultra.com>, aviva@ultra.com (Aviva Garrett) writes:
> John McPhee makes the following statement (on page 90):
> 
>   Since I have no idea how the scale works, let me say only that
>   it is a mathematically derived combination of three scales
>   parallel to one another: a magnitude scale flanked by scales of
>   amplitude and distance. (Amplitude is the height of the mark
>   an earthquake produces on a seismogram.) Where a line drawn
>   between amplitude and distance crosses the central scale, it
>   registers magnitude. With each rising integer on the magnitude
>   scale, an earthquake's waves have ten times as much amplitude
>   and thirty times as much energy. Richter always insisted that
>   is was the Gutenberg-Richter scale.
> 
> (Earlier, John McPhee also says, "[Richter's] scale ... is
> understood by professors at Caltech and a percentage of the rest
> of the population too small to be expressible as a number" @:-)
> 
> This statement did not make sense to my husband, a physicist, so
> I am asking for some net.wisdom.
> 
> (1) Is the explanation of the three scales correct?

Naive, but correct.  McPhee has just described an old method of fast
computation - a nomograph.  It works a little like a slide rule.  Basically
any equation that has three variables can be set up as a nomograph.
The three variables are plotted on three lines.  (The lines are parallel,
but they do not have to be).  The values on each scale al not equally
spaced, but are designed so that geometrically, given two points on any two
of the scales, a straight line through the two known points intersects
the third scale at a value equivalent to a solution of the equation.

When calculators and computers were not common, nomographs were one of the
fastest ways of doing repetitive computations.

For "Richter" magnitude, the distance correction is an empirical curve,
so a nomograph is even more appropriate.  computer programs to compute
this employ a look-up table and interpolation to determine the distance
correction.

In general, magnitude is expressed as

     M = log(A/T) + q * log(D) + p

where M is magnitude, A is amplitude, T is period and D is distance.
q and p are factors determined empirically.  For example, the revised
MS scale of Vanek uses constants for p and q (q=1.66, p=3.3).
In general, however, p and q are not constants, and depend on distance
and period.  For the simplest form of the Richter magnitude, there is
no period dependence in p and q, so it can be expressed

     ML = log(A) + q(D)

And this can then be represented as a nomograph.

As a bit of trivia, the nomograph McPhee refers to was probably drawn by
John M Nordquist, who was responsible for network operations for a long time
at Caltech.  He produced several nomographs for routine computations
while he was there.  I don't know if any are published anywhere.
Richter did not use a nomograph when he computed magnitudes, as far as I
know.


> (2) Does the energy really increase by a factor of 30? My
>     husband though it would be 3.

Really 30.  Energy goes as amplitude squared.  Now that would imply for
each unit of magnitude, energy should go up by a factor of 100, since
amplitude goes up by a factor of ten.  However, this ignores the fact that
the spctral content of earthquake energy shifts as magnitude changes.
The long periods increase faster, while the short periods saturate, and
fail to increase.  Shorter periods are more energetic for the same
amplitudes.  Thus, energy does not increase as fast as amplitude, and
is only a factor of 30 instead of 100.

On the saturation of short periods - the original Richter magnitude saturates
and M=8.5.  This is because it was defined on a specific recording instrument -
the Wood-Anderson 0.8 second torsion.  This instrument is relatively
band-limited, and once an earthquake has grown to M8.5, all frequencies
recorded by this instrument have saturated.  There are several recorded
quakes with magnitudes larger than 8.5 that record a Richter magnitude
of 8.5.  Several other magnitude scales are not limited at 8.5, which
is how we know better magnitudes for these events.  One such scale is the
"open-ended Richter scale" which is often reported in news stories.  The
largest quake ever recorded is 1960 Chile - Mw=9.6 (an "energy" magnitude
scale measured at very long periods).



--
Richard Stead
Center for Seismic Studies
Arlington, VA
stead@seismo.css.gov
Newsgroups: ca.earthquakes