Absolute Heights and the Elusive 1 cm Geoid Dr. Dru Smith Chief Geodesist, NOAA/NGS

1. Absolute Heights and the Elusive 1 cm GeoidDr. Dru Smith Chief Geodesist, NOAA/NGS NRC - National Academies’ Mapping Science Committee Meeting Washington, D.C. September 11, 2007

2. Defining “Height” • Isn’t it intuitive? Don’t we already “know” what it means? • Generally…yes • Specifically…no (and it’s important!) • These statements keep geodesists awake at night: • What is the height of __________? • How accurately can we know a height? • Where will water flow if this region is flooded? • How fast are heights changing?

3. Defining “Height” • Height is… • Some length • (usually)* • along some path • between two points • in some specified “up” direction. ? B A * = More on this later

4. Dominant Height Systems in use in the USA • Orthometric • Colloquially, but incorrectly, called “height above mean sea level” • On most topographic maps • Is a >99% successful method to tell which way water will flow • Ellipsoid • Almost exclusively from GPS • Poor at determining water flow anywhere “non mountainous” • Dynamic • Directly proportional to potential energy : always tells which way water will flow • Dynamic heights are not lengths! • More on this later…

5. Orthometric Height (H) • The distance along the plumb line from the geoid up to the point of interest H

6. Ellipsoid Height (h) • The distance along the ellipsoidal normal from some ellipsoid up to the point of interest h h h

7. Some definitions are required… • “the geoid” • is the one equipotential surface surrounding the Earth which best fits to global mean sea level in a least squares sense.

8. Orthometric Height (H) • The distance along the plumb line from the geoid up to the point of interest W=W3=Constant H W=W2=Constant W=W1=Constant W=W4=Constant The geoid. Its gravity potential energy (W) is constant at all points on itself. That is W = W0 = Constant. There are an infinitude of such surfaces where W=Constant…

9. So…which one is the geoid? Earth’s Surface C…correct! Why? W=WA W=WB Mean Sea Level W=WC W=WD W=WE W=WF

10. Earth’s Surface Let’s take a closer look at what happens right at the coastline… Mean Sea Level W=WC

11. Q = Reference point for a tide gage Earth’s Surface hQ= Distance above Local Mean Sea Level (LMSL) HQ= Orthometric Height Q HQ hQ The Geoid eQ Mean Sea Level eQ= Error in assuming MSL = geoid at this tide gage

12. Absolute vs. Relative Heights • Determining heights at the highest accuracy is mostly relative • Assume some known absolute (=true) height at point A (HA) • Determine height differences between A and B (DHAB = HB-HA) • Compute height at B: • HB = HA + DHAB • Generally true for accurate Orthometric, Ellipsoidal and Dynamic heights

13. Examples of relative heights • Leveling • Measure geometric changes point to point • Correct for multiple physical effects • Attempts to yield differential geopotential (energy) levels • Convert from geopotential to dynamic height or orthometric height • Very time consuming and tedious

14. Examples of relative heights • DGPS • Begin with a known (often permanent) GPS station (pt A) • (Even this is “known” from a global relative adjustment of stations and orbits) • NGS manages a network of such stations: CORS • Set up a temporary GPS receiver over point “B” • Take enough measurements (15+ minutes) to drive GPS inaccuracies out of the equation • Voila! DhAB without any line of sight between A & B (Dlatitude, Dlongitude, Dh)

15. What does this mean so far? • Orthometric heights are the most used / most needed for mapping applications • Determining orthometric heights from leveling is time consuming! • Determining ellipsoid heights is fast and easy, but they aren’t as useful • If only there was some way to get “accurate” and “fast and easy”…

16. Geoid Undulation (N) • The distance along the ellipsoidal normal from some ellipsoid up to the geoid h H ≈ h-N H The Geoid N A chosen Ellipsoid

17. H ≈ h-N • Good to sub-mm over most of the world • Good to < 1 cm anywhere in the USA • If determining N were fast (it is) and accurate (well…) then H can be determined from GPS! • That brings us to…

18. The elusive 1 cm geoid • Can we know the geoid to 1 cm absolutely? • Probably • Basics go back to 1888 • With global surface gravity measurements, the equations exist to approximate the geoid’s location • Refinements over decades • “GPS-for-H” drove this from an academic question to a practical one in the last 20 years • Without consideration of “1 cm” just yet, NGS embarked upon “geoid modeling” in 1990 as a service to the people of the USA

19. NGS and the geoid • 1990 / 1993 • First attempts to get N • Geocentric ellipsoid (shape was “GRS-80”) • Best global MSL fit for geoid • Problem: • h in USA is h(NAD 83) which is non-geocentric • H in USA is H(NAVD 88) which isn’t fit to MSL

20. NGS and the geoid H (NAVD 88) h (NAD 83) H NAVD 88 reference level (W = constant????) h The NAD 83 ellipsoid The Geoid N (GEOID96) N (GEOID93) A chosen Ellipsoid (Geocentric, GRS-80) H (NAVD 88) ≈ h (NAD 83) – N (GEOID96)

21. NGS and the geoid • From 1996 on, NGS created “hybrid geoids” to convert from h (NAD 83) to H (NAVD 88) • For 10 year has done its job well: • “To convert one erroneous datum into another erroneous datum” • Has never given people “absolute orthometric heights” • Problems: • NAD 83 is non-geocentric • NAVD 88 has systematic errors (especially in mountains) • Relies on GPS surveys on passive NAVD 88 monuments • Vulnerable, sparse and moving in time • Requires re-leveling to get updated NAVD 88 heights • Requires re-DGPS to get updated NAD 83 heights

22. NGS and the geoid • The NGS 10 year plan (2007-2017) • Recognizes a better way of doing business • Remove the non-geocentricity of the ellipsoidal datum • Define the vertical datum reference surface as being the geoid • Compute the geoid accurately, and track its changes in time using sparse gravity resurveys • No re-leveling, no re-DGPS • If we know changes to “g” we know changes to “N”

23. Can we know the geoid to 1 cm? • Again, “probably” • What stands in the way? • Aged and aging gravity data • 1000’s of surveys, dozens of years • No existing model for gravity change • Existing theory has “a few cm” of approximations still built in

24. How will NGS achieve a 1 cm geoid? • Snapshot of gravity • A country-wide airborne survey spanning a few years and focusing on accuracy and self-consistency • Temporal gravity tracking • Using both GRACE and episodic absolute gravity surveys, model g(latitude, longitude, time) • Improve theory • Chairing an international collaboration of theorists to drive the last few cm of approximations out of existing computational methods

25. Gravity Survey Plan • Airborne • 10 km spacing over the USA and territories • One time survey • Estimated cost: 5-8 years and $30-50M • Absolute • Cyclical for episodic checks in fixed locales • Two field meters plus one fixed Superconducting Gravimeter • Relative • More frequently attached to “Height Mod” surveys • For field checking aged data against new surveys

26. Theoretical Improvements • New International Association of Geodesy study group devoted to finding this: • Mathematical equations which, if perfect data were applied, would yield the location of the geoid to sub-cm accuracy • Estimated time frame: 5-7 years

27. Summary • The use of passive monuments as the method to define and provide access to absolute heights in a dynamic world with access to GPS is no longer appropriate • A better way, involving a one-time airborne survey followed by low-cost gravity tracking and low-cost GPS-CORS is the best method for delivering accurate absolute orthometric heights quickly • By 2017, NGS expects to implement these full changes and deliver a new ellipsoidal (“horizontal”) and geopotential (“vertical”) datum • And be able to sustain their absolute accuracy long into the future

28. Questions/Comments? • Dr. Dru Smith • Chief Geodesist, National Geodetic Survey • Dru.Smith@noaa.gov • 301-713-3222 x 144