Absolute Heights Absolute Heights and the Elusive and the Elusive

1 cm Geoid1 cm Geoid

Dr. Dru Smith Dr. Dru Smith Chief Geodesist, NOAA/NGSChief Geodesist, NOAA/NGS

NRC - National Academies’ NRC - National Academies’ Mapping Science Committee Mapping Science Committee

MeetingMeeting

Washington, D.C.September 11, 2007

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?

Defining “Height”

• Height is…

• Some length• (usually)* • along some path • between two points • in some specified “up”

direction.

?

* = More on this later

A

B

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…

Orthometric Height (H)• The distance along the plumb line from the geoid

up to the point of interest

H

Ellipsoid Height (h)• The distance along the ellipsoidal normal from

some ellipsoid up to the point of interest

h

hh

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.

Orthometric Height (H)• The distance along the plumb line from the geoid

up to the point of interest

H

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…

W=W1=Constant

W=W2=Constant

W=W3=Constant

W=W4=Constant

Earth’s Surface

Mean Sea Level

W=WA

W=WE

W=WD

W=WC

W=WB

W=WF

So…which one is the geoid?

C…correct! Why?

Earth’s Surface

Mean Sea LevelW=WC

Let’s take a closer look at what happens rightat the coastline…

Earth’s Surface

Q

Q= Distance above Local Mean Sea Level (LMSL)

Q

Q = Reference point for a tide gage

HQ= Orthometric Height

HQ

Mean Sea Level

The Geoid

eQ

eQ= Error in assuming MSL = geoid at this tide gage

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 (HAB = HB-HA)

• Compute height at B:– HB = HA + HAB

• Generally true for accurate Orthometric, Ellipsoidal and Dynamic heights

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

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! hAB without any line of sight between A & B

(latitude,longitude,h)

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”…

Geoid Undulation (N)• The distance along the ellipsoidal normal from

some ellipsoid up to the geoid

h H

N

The Geoid

A chosen Ellipsoid

H ≈ h-N

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…

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

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

NGS and the geoid

h

H

N (GEOID93)

The Geoid

A chosen Ellipsoid(Geocentric, GRS-80)

h (NAD 83)

NAVD 88 reference level

(W = constant????)

H (NAVD 88)

The NAD 83 ellipsoidN (GEOID96)

H (NAVD 88) ≈ h (NAD 83) – N (GEOID96)

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

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”

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

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

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

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

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

Questions/Comments?

• Dr. Dru Smith• Chief Geodesist, National Geodetic Survey

• Dru.Smith@noaa.gov• 301-713-3222 x 144