Shape of the Earth, Geoid, Global
Positioning System, Map Coordinate
Systems, and Datums
Or how you can impress your friend
on a hike
D. Ravat
University of Kentucky
May 2012
Motivation for the Exercise
• Most students and instructors are unfamiliar
about the reasons for the discrepancies in
latitudes and elevations from measurements
made by a handheld GPS unit, a smartphone,
Google Earth, and a USGS toposheet.
• Many correctable errors are being made by
field scientists.
• There are two basic reasons for these
discrepancies: 1) different coordinate
reference frames; and 2) observational errors.
Background Preparation for Students
• Understanding of latitude and longitude
space and maps are two dimensional
projections of a three dimensional world
• Ability to convert, using a handheld
calculator, the latitude and longitude in
degree minutes seconds format to the
decimal degrees and vice versa.
• Ability to convert between meters and
feet
My GPS has gone bad; it is not
getting heights right ! (or worse, My
toposheet is wrong!)
• You have a USGS toposheet, a GPS unit,
and a smartphone with GPS with you when
you are on a hike and find that the elevation
on the toposheet is not the same as the one
you get from the GPS unit or the smartphone.
• What are the factors leading to these
discrepancies?
Consider these observations at the Bench Mark at the Univ of
Kentucky Main Building with my brand new GPS unit, a
smartphone, and an old faithful U.S. Geological Survey toposheet:
Lexington West, KY, Quadrangle
Toposheet:
Latitude
38° 02’ 15’’
N
38° 02.252’
N
38.037532637° N
Longitude
84° 30.330
W
84.505500°
W
Elevation
975 ft.
GPS Unit (stated accuracy 10 ft):
Latitude
38° 02.446’
N
38.04076667° N
Longitude
84° 30.308’
W
Elevation
1006 ft.
Comparison of a GPS unit, an iphone, and Google Earth
for the same benchmark:
iphone (horizontal accuracy 5 m, vertical accuracy 4 m):
Latitude
38.038994° N
Longitude
84.504924° W
Elevation
303.0 m => 994±13 ft.
GPS Unit (stated accuracy 10 ft):
Latitude
38.04076667° N
Longitude
84.505133’
W
Elevation
1006 ft. (within the accuracy of the iphone specs.)
Google Earth (accuracy based on the pointer location):
Latitude
38.039149° N
Longitude
84.505138° W
Elevation
297 m => 974 ft.
Questions to students
• How do you determine error bars in
measurements? How do you use them to
determine if two observations are similar or
different?
• Are latitudes similar to one another for all four
methods of measurements? Which ones are?
How few groups could one make? Is one group
south or north compared to the others? By how
many meters?
• Are longitudes similar to one another for all four
methods of measurements?
• Are elevations similar to one another? How many
distinct groups exist? Which one is the lowest in
this case?
Observations students should make:
• Each kind of measurement has its intrinsic
measurement error
• All longitudes are similar to one another
• GPS latitude, iphone latitude, Google Earth
latitudes are similar within their error bars
• Toposheet latitude is southward by 300+ m in
comparison to the other methods
• Toposheet elevation is similar to the Google
Earth Elevation but different than the GPS
unit or the iphone
• Elevation from the GPS and iphone are similar
within their error bars
What does an old toposheet say?
Polyconic Projection
1927 North American Datum
National Geodetic
Vertical Datum of 1929
(same as NAD27 – this is a
datum for horizontal coordinates)
(same as NGVD 29 – a
datum w.r.t. sea level)
How did people know where
they were before GPS?
• Stars were used to
calculate latitude and
longitude is based on
time differences with
respect to an arbitrary
reference
10
The rest is geometry!
Mathematical
Shape of the Earth
Spheroid (Ellipsoid of
Revolution)
Oblate Spheroid
(Earth’s Ellipsoid of
Revolution)
Chocolate M&Ms
Prolate Spheroid
Peanut M&Ms
Many different ellipsoids are used - they work better for their own regions
http://en.wikipedia.org/wiki/Figure_of_the_Earth
Geoid – an equipotential surface that best represents the Earth
(represented by the mean sea level - in a point by point sense on the oceans, and on the land from the observed gravity field)
These heights are
with respect to a
reference spheroid,
e.g., WGS84, that
best fits the Earth
Geoid vs. Reference Spheroid/Ellipsoid for the
Earth in a cross-section
Ellipsoid
Geoid
The geoid has a pear-shape. The Earth’s shape is not a perfect spheroid
because there are lateral mass variations inside the Earth.
Earth’s Gravity and Shape are related:
Actual geoid – reference spheroid = Geoid height anomaly
Example: Water collects over a region of excess mass. A positive
geoid height anomaly indicates mass excess in the subsurface (and
vice versa)
Sea surface shape
Sea bottom
Reference Spheroid
Seamount representing more mass
compared to surrounding water
(Basalt, density (rc) = 2.90 g/cc;
Seawater, density (rw) = 1.03 g/cc)
Geoid height is measured in meters and a common unit of
gravitational acceleration is milliGal. 1mGal = 1x10-5 m/s2
Can calculate gravity anomalies from geoid anomalies & vice versa, on
the interconnected water bodies: Satellite radar altimetry of the sea water
gives us a high resolution data of sea surface topography. This is how
intermediate resolution gravity anomalies on the ocean are calculated.
mGal
Nothing about Earth’s coordinate systems is simple!
But we still have to use them, and so we must understand at least the
simplest basics. Because the shape of the Earth is determined from
mass variations inside the Earth, it makes sense to choose the
Earth’s Center of Mass as the coordinate system’s center. But a
giant earthquake, mantle convection, plate tectonics can move things
around in the Earth so even the Center of Mass changes over time.
Also, because each reference
system was designed to fit
different large or small region of
the world and they all can have
different origins, they fit that
particular region the best, and
can poorly fit the rest of the
Earth’s surface. WGS84 is the
overall best fitting spheroid
agreed upon by geodesists in
1980s and revised in 2004.
Elevations on a toposheet
Toposheet
Elevations on a toposheet are generally determined from leveling surveys. A
properly leveled instrument sight is parallel to the equi-potential surface.
And so the resulting elevation is referenced to (or is with respect to) the local
geoid. (These elevations are called Orthometric Heights if it ever comes up in
specifications of instruments/Google Earth, etc.)
Ellipsoid
Ellipsoid height (A reference used commonly for GPS, like WGS84)
(or also called geoid height anomaly)
Google Earth’s Coordinates
• Its horizontal coordinate system is
WGS84.
• Its vertical datum (as of 2012) is EGM96
geoid (i.e., height above mean sea
level).
• KML standard refers to this as “WGS84
EGM96” coordinate system.
Handheld GPS units
• Have a choice of selecting horizontal
and vertical datums – a common default
is WGS84 for both
• Geoid is not a commonly available
datum in handheld GPS units
iphone (OS5.1)
• Horizontal datum: WGS84
• States vertical datum to be sea level;
however, it is not geoid, it’s closer to
WGS84 (within observational error and
not close to the height w.r.t. sea level)
Can you now explain why elevations on my toposheet and
Google Earth match, but they don’t match with the iphone or
a GPS unit? Also the latitudes in the US should be roughly
200 m southward with the horizontal coordinate system of
the toposheet?