SIGNED ROOT COMPRESSION

Seismograms which have "large" surface waves may obscure smaller earthquakes
for one of the following reasons:

 1. To have the surface waves entirely within the seismogram frame, the
    magnification must be lowered to the point that most of the seismogram
    (including "smaller" earthquakes) are straight lines. 

 2. The large surface waves mask other surface waves.

Signed logarithm (arc sinh) compression can be used, but the compression is
so extreme that all waves are small square waves. Reducing each point to it's
square root seems an obvious compression, but all numbers below the 
centerline of each trace are negative numbers, and negative numbers do not 
have a real square root. (Negative numbers have an "imaginary" square root
but this is used only in complex number analysis and can not be plotted on
these seismograms with only real axes).

This problem was solved by calculating a file of signs (positive or negative)
of each number of the seismogram. Next, all numbers are changed to their
absolute values (made positive), after which all numbers are converted to 
their square roots. Finally, the file of signs is used to restore negative
signs to all square roots of numbers which originally negative.

In practice signed square root compression is usually too extreme. The
signed root process was generalized to allow fractional roots. Roots generally
run from the 1.1th root to the 2.0th root (square root) and rarely from the
2.1th root to the 3.0th root (cube root). The 1.1th root slightly reduces
higher amplitudes, and can not be spotted except for the label on the
seismogram. Higher roots compress more and more. by the In the 1.5 to 2.0
root the seismogram has obvious oval or square peaked waves.

The magnification looses its meaning. The multiplier times 10000 gives
approximate magnification for very small motion, but magnification falls
off as the motion becomes progressively larger.

In the Lawrence Livermore National Labs Seismic Analysis Code, the signed
root compression is produced as follows:

*=comments

* X is the series of numbers making up the seismogram
write X
* take absolute values of X (every number is now positive)
abs X
* divide the absolute values of X by the original values. This gives
* the signs (all +1 or -1) of X in file signX
divf X
write signX
*recover the original numbers
read X
* make all the original numbers positive (their absolute value)
abs X
* Take root "%bbroot" of each number. In this case the log is the natural log
log
div %bbroot
exp
* multiply all the roots by their original signs
mulf signX


SIGNED POWER EXPANSION

If all the earthquakes on a 24 hour seismogram are very small,
it is useful to expand the earthquake waves without expanding
the noise background. This is done with signed power expansion
in which the deflection of the trace is raised to a power (for
example the power 2.0 is the same as squaring the trace
deflection and then restoring the positive or negative sign).
In general powers of 1.2 to 1.8 are most useful. At powers of     
2.0 or greater, the wave peaks are spike like. 

The sac algorithm is the same as for signed root compression,
except that "div %bbroot" is replaced by "mul %bbpower".

The ordinary or linear seismogram can be seen as a special 
case of signed root compression with root 1.0, or as a 
special case of signed power expansion with power 1.0.

 

  SIGNED ROOT COMPRESSION  peaks reduced more and more as they
                           become larger.
                           at high compression peaks are noticably
                           flattened

  LINEAR                   peaks are the same as the signal from
                           the seismometer, which is similar to
                           earth velocity. This is what all or
                           almost all paper recorded seismograms
                           record. Peaks are usually rounded and
                           frequently off the display.

  SIGNED POWER EXPANSION   peaks expanded more and more as they
                           become larger.
                           at high expansion peaks are noticably
                           spiked.