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 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?

Lexington West, KY, Quadrangle Toposheet: Latitude 38° 02’ 15’’ N 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. Longitude error between hand-calculation on the toposheet and GPS is 25 meters. GPS error estimated at 3 m. My hand-calculation error is based on my ability to measure in between graduation marks on a 0.1 cm graduated ruler. Maximum 25 m of error in latitude and 19 m of error in longitude. Latitude error between hand-calculation on the toposheet and GPS is 360 meters. This 300+ m difference comes from the difference between WGS84 and NAD27

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 (same as NAD27 – this is a datum for horizontal coordinates) National Geodetic Vertical Datum of 1929 (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) Prolate Spheroid Chocolate M&Ms 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 The geoid has a pear-shape. The Earth’s shape is not a perfect spheroid because there are lateral mass variations inside the Earth.

Actual geoid – reference spheroid = Geoid height anomaly 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 Reference Spheroid Seamount representing more mass compared to surrounding water (Basalt, density (rc) = 2.90 g/cc; Seawater, density (rw) = 1.03 g/cc) Sea bottom 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. Note for teachers: Nmax = maximum spherical harmonic degree used in modeling the gravity anomalies 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 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? 297m = 974.16 ft ~ 975 ft on the toposheet - fortunately in this location the vertical position of the geoid has not changed much from 1929 geoid reference and 1996 geoid reference. But in some other places, it may have changed. Why GPS units have errors is probably whole another lecture. And that is more conducive to higher level computer modeling assignment.