# [time-nuts] Averaging Location for Position Hold

J. Forster jfor at quikus.com
Wed Sep 14 16:37:40 UTC 2011

```>From a friend:

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> Any guess as to how accurate these are?

1. They vary.  You have to look up the info on the particular benchmark.
Some benchmarks and geodetic monuments (see below) receive a lot of
attention and are known as "First Order" points.  Many more receive less
attention and are known as "Second Order" points.  _Most_ receive _little_
attention and are known as "Third Order" points.  Basically, the
positions/heights of third order points are determined with respect to
nearby second order points; and second order points are determined with
respect to the nearest first order points.  First order points are few and
far between, but great care is devoted to determining their relative
positions/heights.

2. Traditionally, only the _height_ of a "benchmark" with respect to a
"vertical datum" (approximating mean sea level) has been determined
accurately.  Within the USA, the US Geological Survey (USGS) is legally
responsible for determining heights of benchmarks.  Height differences are
determined by means of "leveling."  Level surfaces are determined by
liquids in stationary tubes, and by tide gauges.  A tide gauge senses the
constantly varying height of open water.  Of course much time-averaging
must be done with tide-gauge data.  My point is that the reference surface
is an equipotential in the combined gravitational and centrifugal (due to
the rotation of the Earth with respect to an inertial frame) field of the
Earth.  The reference surface has lumps and bumps because the Earth's mass
distribution is not uniform or symmetrical.  The reference surface is
_not_ a regular mathematical-geometric figure such as an ellipsoid.

Whereas, only "horizontal" position coordinates (basically, latitude &
longitude) have been determined accurately for "geodetic" monuments.
Within the USA, the National Geodetic Survey (NGS) is legally
responsible for determining horizontal positions.  Lat. & long. are
determined with respect to a "geodetic datum," by means of geometric
measurements such as distances and "horizontal" angles.  The datum, or
reference coordinate system, is defined geometrically, by means of
monuments planted in hopefully stable ground, and mathematical formulas
that attempt to account for known systematic motions such as
continental drift and solid-earth tides.

Benchmarks are much more abundant than geodetic monuments, basically
because height differences determine drainage, and drainage problems
are more common than horizontal-position problems.

It is significant that the USGS is part of the US Dept. of the
Interior, whereas the NGS is part of the US Dept. of Commerce.  To
determine how this plays out in practice is left as an exercise for the
student. :-)

3. The advent of GPS has changed the game greatly in practice, although
not so much in theory.  GPS-based geodetic measurements are
three-dimensional, and they are geometric; that is, they do not determine
position with respect to an equipotential surface.  The fundamental
measurements are of double-differences between the times of arrival of
different points on Earth.  The origin of the fundamental reference frame
is the center of mass of the Earth, about which the satellites orbit; and
the directions of the coordinate axes are determined by the average spin
angular velocity vector of the Earth, and an artifact that marks the
"prime meridian" or origin of longitude.  This artifact is not a single
marker but an internationally agreed set of markers distributed around the
world.  These markers are rocks and/or concrete set into hopefully stable
ground.  Their relative positions have been well-established by great
numbers of observations by many people and organizations.  One might call
these markers "Zero Order" points, although I don't believe that this
terminology is official.

GPS does not determine height with respect to an equipotential surface
or "vertical datum."  However, its 3-D geometrical position
determinations include determinations of "height" with respect to a
mathematically-geometrically defined ellipsoid whose center is at the
origin of the GPS coordinate system.  "Ellipsoidal height"
determinations by GPS are good enough for most practical purposes.

A good GPS geodetic surveyor utilizing good equipment and techniques
can determine the ellipsoidal-height differences between points a few
kilometers apart within a few millimeters.  A _very_ good GPS geodetic
surveyor utilizing very good equipment and techniques can determine the
ellipsoidal-height differences between points 100 kilometers apart
within a few millimeters.  An excellent GPS geodetic surveyor utilizing
excellent equipment and techniques can determine the ellipsoidal-height
differences between points 1000 kilometers apart within a few
millimeters.

The uncertainties of GPS determinations of ellipsoidal-height
differences are typically three times greater than the uncertainties of
GPS determinations of horizontal-coordinate differences, for two main
reasons:  (1) A GPS receiver cannot observe satellites below the
horizon, although it can observe satellites in both the northern and
southern sky, and it can observe satellites in both the eastern and
western sky.  (2) Uncertainty in the radio-signal propagation
time-delay added by the troposphere, especially by the water vapor in
the troposphere, affects mainly the height determination, because the
troposphere is entirely above a ground-based receiver and, to first
order, the troposphere is horizontally stratified.

4. You can probably find your local benchmark in a USGS database on the
Web.  (Your state or county may also list your benchmark in a database.)
See how your benchmark is classified, as first-, second-, or third-order;
and look at the accuracy standards in the definitions of these orders.

If you're lucky, your benchmark has been visited by GPS receivers and
has been incorporated into the NGS database of geodetic monuments.  So
look at the NGS' website(s), too.  (Your state or county may have made
GPS observations, and these may or may not have made it into a national
database.)

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Best,

-John

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>> I hear that there is a hobby where people photograph these and collect
>> the photos.  Seems pointless in the city but many are on mountain
>> peaks and other places with good views.  The hobby is more reasonable
>> if you find them without using GPS.  The USGS maps are good enough o
>> put you within 10 to 20 feet of the BM without using a GPS.
>>
>> Chris Albertson
>> Redondo Beach, California

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