## PSN-L Email List Message

Subject: strain and tilt units
From: sean@...........
Date: Mon, 18 Sep 2000 14:15:24 -0500 (CDT)

```Mauro,

There is no such thing as a dumb question, and not responding could be
dangerous, as my parents often found out when I asked such obvious childhood
questions as how an alarm clock worked and they didn't respond, so I took
it apart to find out. (I really got into trouble with the tube radio).

I assume that you know that 10^-7 means 0.000 000 01, or one part in
1/10 th of a million.

Strain and tilt are generally considered dimensionless, although the
small angle approximation that sine(i) = i (in radians), means that
tilt is represented in microradians (a radian is 360 degrees/ (2* pi));

Strain is the change in length over a distance divided by that distance.
So a change of 1 micron over 1 meter is a strain of 10^-6, or 1 microstrain.
To measure it, some stable reference length, like quartz rods or laser
interferometers, are used to determine the distance, and some sensitive
transducer is used to compare the reference length with the actual length
of the piece of earth in question. One of my earliest projects involved
100 ft quartz tube strainmeters deep in a lead mine in Missouri. Some of
the best laser/optical strainmeters are run by UCSD at LaJolla, where
lasers in 750 meter vacuum tubes have resolutions of 10^-9 and annual
stability of 10^-6. The new LIGO (Laser Interferometer Gravitational
Wave Observatories) use multi-path laser interferometers 4 kilometers
per arm to try to detect gravitational waves from supernovas, colliding
black holes, etc., that will strain the earth at 10^-18 to 10^-21.
This megabucks NSF project has two LIGOs (Washington state and Louisiana),
to sort out non-deep-space noise.

Tilt has a similar definition of the change in elevation at one end
of a reference length divided by that length. So if I balance a 30 cm
ruler at the hole in the center, the length to the end is 150 mm. If I
can measure any change in height at the free end to within 0.1 mm, this
would be a tilt of 0.1/150 or 0.00067 radians, or 6.7 x 10^-4.
The usual units of tilt are in microradians, or 10^-6. The geodetic
tiltmeters that I had installed as part of the USGS "prediction" effort
in the '80s (in the Aleutians, at Parkfield, and almost at the Palmdale
bulge before Dave Jackson at UCLA found it to be a surveying artifact), have
resolutions of 10^-9, and annual stability if 10^-6, so the earth tides
at 10^-7 are a ready calibration signal for which we have accurate
computer models for comparison. The annual stability of 10^-6 was the
best we could get three co-sited instruments to agree, and was much
larger than what models of tectonic deformation indicated for pre-earthquake
deformation. The largest coherent signal was the annual rainfall hydrology.

I have submitted an abstract to the AGU meeting regarding the new dynamic
broadband tiltmeter that senses pure tilt but is not sensitive to horizontal
acceleration as a seismometer is, so it can be used to record and
remove tilt noise from broadband horizontal seismic data. The current
noises as acoustic gravity waves (from storms) at 50 nanoradians (10^-9)
amplitude. (I will post the abstract on m web site later in the week.)

Regards,
Sean-Thomas

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