## PSN-L Email List Message

Subject: seis dynamic range
From: S-T Morrissey sean@...........
Date: Sun, 9 Apr 2000 14:56:04 -0500 (CDT)

```Paul,

You correctly estimate of the wavelength of surface waves for a
20 second wave (0.05hz) with a velocity of 5 km/second as 100 km.
(using v = f * l; or wavelength = velocity * period)
Most people don't realize this. The longest waves seen are the
fundamental free oscillations of the earth at 51 minutes, where the
wavelength is the earths' circumference. Even near field P waves
of 15 hz at 7.5 km/sec have wavelenghts of half a kilometer.

Here is a repeat of a previous discussion on the subject. (August 99)

The range of ground motions expected from earthquakes as recorded
by a seismometer is hugh. Obviously if "you are there" at the fault
scarp, you will need an accelerometer with a range up to 2gs.

But most instrumental seismology follows general magnitude formulas
of the classic form:
M = log(A/T) + 1.66*log(distance) + constant.
The constant is a regional variable, usually about -0.18. The distance
is degrees (111 km per), A is the ground amplitude in nanometers,
(a nanometer = 10^-9 meter; in older data it was called a millimicron.)
and T is the period of the wave in seconds. For Ms calculations,
A is the sustained peak-peak surface wave amplitude.

You can plug in numbers for various scenarios. To answer your question
about what will happen in the seismometer, solve for A:
log(A/T) = Ms - 1.66*log(distance) + 0.18
A = T*log^-1(Ms -1.66*log(d) + 0.18.)

The first step though is to convert the velocity output of the seismometer
to displacement by dividing by omega, w = 2*pi/period.

For the Turkey quake: the record here (St. Louis) was 150 microns/second
at 24 seconds. 24 seconds is an w (omega) of 2*pi/24 = 0.262radian/sec.
The velocity is converted to displacement by dividing by w, so
150/0.262 = 573 microns or 5.73x10^5 nanometers. (0.573 millimeters)
(Every seismometer should have recorded this, and many clipped).

So Ms = log(5.73x10^5/24) + 1.66*log(81deg) -0.18 = 7.37

(solving for A for a maximum magnitude of 8.0 at that distance would
cause 2.46 millimeters of motion at St. Louis)

But what did I record for the Ms = 5.8 aftershock?

A = 24[log^-1(5.8 - 3.17 + 0.18) = 15 564 nanometers. (15.5 microns)

This is a velocity (at 24 seconds) of 4.07 microns/second.
With an output of 5.3mv/micron/sec, the record here was about 22
millivolts p-p, (or about 10 times the 6-second microseisms; at 0.5
microns p-p; at the time they were running more than 10 times that due to
the hurricane).

But what about a nearby quake? Say a Ms 4.0 in the next state? Putting
numbers in the formula with distance = 3 degrees (200 miles) and
T = 0.3 second, the seismometer will record 0.81 microns. For a
magnitude 3, it will only sense 0.081 microns or 81 nanometers.
(using the more accurate local Mblg(3hz) formula gives 43 nanometers)

If M = 4 and is 1000km away, the motion at 1 second will be 398 nanometers.
If M = 3 and is 1000km away, the motion at 1 second will be 40 nanometers.

For a threshold event, say a 2.5 at 110 km with T = 0.22 sec (4.5hz),
(this magnitude formula is revised for such local events, but we'll
use it anyhow) the amplitude is 105 nanometers. (using the local
Mblg formula gives 56 nanometers). To record these with useful
signal-to-noise ratio requires a resolution of 1 nanometer from
the seismometer.

SO.... we need a displacement dynamic range of 1 nanometer to 1
millimeter, or 10^6. This is one reason seismometers prefer a velocity
output. Looking at such numbers:.
At 10 hz, 1 nanometer is 63nm/sec. At 1 hz, it is 6.3nm/sec. At 20
seconds (a broadband instrument) 1 millimeter is 314 microns/second.
This is a velocity range of about 50 000 to 1, which is the same as
the voltage output range.

Until recently this range was difficult to handle with analog
electronics, so multi-level recording was used. A 16-bit digitizer
can realize this if noise is ignored. Providing for noise and
instrumental drift increases the required range by a factor of
100 to 1000. So modern broadband stations use 24-bit digitizers
with a resolution of 1 part on 16 777 216 with a least count value
of 1.2 microvolts. For a seismometer with a VBB velocity output of
2000 volts/meter/second, the LSB represents 1.67 nanometers/sec.
The maximum value is 10 millimeters/second  p-p.

Regards,
Sean-Thomas
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