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Geoid Surfaces and Theory

Session B of Datums, Heights and Geodesy

Presented by Daniel R. Roman, Ph.D. 

Of the National Geodetic Survey

-ellipsoid only loosely tied to the Earth -(4 parameters: a, f, GM, and ω)

-great for tracking satellites - but not flood plains

-mass variations cause gravity and geoid variations

-explain about geoid undulations/heights (+/- 100 m)

-cover reference feidl models, such as EGM96

Emphasize that geoid height models are transformation models between ellipsoids and vertical 

datums (geoids).



Definitions: GEOIDS versus GEOID HEIGHTS

• “The equipotential surface of the Earth’s gravity field which 

best fits, in the least squares sense, (global) mean sea 

level.”*


• Can’t see the surface or measure it directly.

• Can be modeled from gravity data as they are mathematically 

related.

• Note that the geoid is a vertical datum surface.

• A geoid height is the ellipsoidal height from an ellipsoidal 

datum to a geoid.

• Hence, geoid height models are directly tied to the geoid and 

ellipsoid that define them (i.e., geoid height models are not

interchangeable).

*Definition from the Geodetic Glossary, September 1986

-If you recall, this was shown previously to emphasize the nature of the geoid 

as a vertical datum

--now we’ll focus on the geoid height model as a transformation between an 

ellipsoidal datum and a vertical datum




In Search of the Geoid…

Courtesy of Natural Resources Canada  www.geod.nrcan.gc.ca/index_e/geodesy_e/geoid03_e.html

-The geoid extends everywhere and provides a common datum for all.

… but where is it and how do you find it …




http://www.csr.utexas.edu/grace/gravity/

-The GRACE satellite mission determined a global gravity field model by 

circling the Earth and tracking the orbital perturbations of the satellite pair.

-This model is incomplete because significant short wavelength signal was 

omitted. Recall that the father above the Earth you are, the more the gravity 

field resembles an ellipsoid because all these humps and bumps attenuate.

-Hence, the GRACE model by itself can provide accurate estimates for the 

geoid and gravity field changes – but only sufficiently large regions – such as 

the entire Colorado Plateau.

-NGS models add back the missing shorter wavelengths to give a more 

complete model for surveying, engineering and scientific applications.

-This also brings up another point: that the variations in the mass of the 

Earth cause these features. This point will become clearer shortly.



geoid

C

0



C

1

C



2

C

3



C

4

C



5

g

1



g

2

H



1

H

2



g*

1

g*



2

g

1



= gravity on geoid at station 1     g

1

= average gravity from g



1

to g*


1

g*

1



= surface gravity at station 1      g

2

= average gravity from g



2

to g*


2

g

2



= gravity on geoid at station 2  

g*

2



= surface gravity at station 2

H

1



= orthometric height to station 1

H

2



= orthometric height to station 2

g

1



> g

2

g*



1

> g*


2

g

1



> g

2

H1 = C5/g



1

H2 = C5/g

2

H

1



< H

2

Note that surface location of station 1 is closer to the geoid than station 2.



A steep gradient of geops indicates higher gravity – less steep indicates lower gravity.

The geops being farther apart beneath station 2 to reflect lower local mass and gravity.

Hence, H1 should be less than H2 – even though both have the same geopotential.

Station 1

Station 2

-Heights, gravity and the geoid are all related.

-As see here, both stations have different heights but the same geopotential. Since there is not 

geopotential difference, water will not flow even though one point is “higher” than the other. This 

is why dynamic heights are favored over orthometric heights in local areas dealing with 

hydrographic issues.

-However, you can’t see the geoid. The ocean surfaces are close, but ocean currents and weather 

disturb the ocean surface from what it would be … and you definitely can’t “see” it under the 

land.

-Hence, another means must be found to define this surface – possibly in relation to one already 



established.


From Figure 2-12, p.83 of Heiskanen and Moritz, 1967, Physical Geodesy

n’

n



Reference Ellipsoid

U = W


Q

Geoid


W = W

0

N



Q

P

P



g

Q

γ

-As discussed earlier, the international ellipsoid (GRS-80) can be used to define most of the 



Earth’s gravity field and geospatial information. The normal gravity (γ) from the GRS-80 model is 

even used to define some heights.

-The surface of the ellipsoid is not only a datum from which to measure heights – it also is a 

sphero-potential surface. The potential of normal gravity varies smoothly towards the poles but is 

constant along that datum surface (U). The geoid is also a constant potential of the real gravity 

field (W) and undulates about the ellipsoid. The separation can be +/- 100meters – and these 

differences are geoid heights that are a function of the difference in the potentials (T = W – U).

-Hence, the ellipsoid can be used to define most of the Earth’s gravity signal as well as a 

coordinate system. Then geoid heights can be used to define the geoid surface away from the 

ellipsoid surface.




Well, OK, maybe a few formulas …

( )


σ

ψ

π



σ

d

gS

G

R

N

∫∫



=

4

( )







+



+









=



2

sin


2

sin


ln

cos


3

cos


5

1

2



sin

6

2



sin

1

2



ψ

ψ

ψ



ψ

ψ

ψ



ψ

S

P

P

U

W

T

=



Stokes’ Formula:

where:


Disturbing Potential:

γ

T



N

=

Bruns Formula:



Height Relationships:

N

H

h

+

=



R

T

r

T

g

g

Q

P

2





=

=



γ

Gravity Anomalies:



-The undulations of the geoid surface reflect lateral and vertical variations in the mass of the 

Earth. These mass variations cause a disturbing potential (T). The actual geopotential (W) differs 

from that predicted by the ellipsoid model (U) by the disturbing potential (T).

-This disturbing potential can then be related to both geoid undulations and gravity anomalies, 

both of which represent deviations away from some standard. Hence all three are and can be 

mathematically related.

-While the geoid surface and the disturbing potential are very difficult to observe, gravity 

observations are fairly easy. Removing the normal gravity yields gravity anomalies that can be 

used to determine the geoid surface.

-That’s the big takeaway from this slide. The reason we are trying to invigorate a gravity 

observation program is because the quality of the gravity data will directly relate to the quality of 

the geoid height models produced.

-Also note that the geoid height (N) is scaled by the normal gravity. This is appropriate because a 

geoid height is measured along the ellipsoidal normal (i.e., perpendicular to the ellipsoid – not 

the geoid). It measures the distance from the ellipsoid to the geoid. The “ h = H + N” formula is 

true only in so far that the DoV can be neglected. This has sub-cm impact for the most part.

-This all seems straight forward except that these formulas shown here are all 

spherical

approximations. Ellipsoidal formulations are similar but more complicated to implement. These 

approximations represent one area where theoretical improvements should directly contribute to 

improved geoid height models.



• Earth Gravity Model of 1996 (EGM96)

• 2.6 million terrestrial, ship-borne, and 



altimetric gravity measurements

• 30 arc second Digital Elevation Data

• 3 arc second DEM for the Northwest USA

– Decimated from 1 arc second NGSDEM99

• Computed on 1 x 1 arc minute grid spacing

• GRS-80 ellipsoid centered at ITRF97 origin

Long Wavelength     

- global


High Resolution Geoid Models

G99SSS (Scientific Model)

Medium Wavelength 

- regional

Short Wavelength    

- local

- G99SSS (scientific model) - determined from 2.6 million gravity measurements taken over the



last century and stored in NGS database.  Major improvements over previous models; more data.

- 30 arc second, 1 km x 1 km, grid of digital elevation data; with updated Canadian Rockies data 

from G96SSS.

- 3 arc second DEM - northwestern portion of U.S. which has history of poor geoid interpretation.

- EGM96 (Earth Gravity Model of 1996) developed by National Imaging and Mapping Agency 

(NIMA) - global model of gravity and geoid undulations; good in large areas but not as good for 

smaller areas.

- 1-D spherical Stokes’ FFT (Fast Fourier Transformation) - “remove-compute-restore” (fix and put 

back into model - iterative process).

Final outcome - 1 x 1 arc minute spacing interpolation of the geoid model; 2' x 2' in Alaska.  1 

arc minute equals about 1.8 km on ground.

ITRF97 (1997.0) COM (Center Of Mass) based origin for model.




High Resolution Geoid Models

USGG2003 (Scientific Model)

• 2.6 million terrestrial, ship, and altimetric gravity 



measurements

– offshore altimetry from GSFC.001 instead of KMS98

• 30 arc second Digital Elevation Data

• 3 arc second DEM for the Northwest USA

– Decimated from 1 arc second NGSDEM99

• Earth Gravity Model of 1996 (EGM96)

• Computed on 1 x 1 arc minute grid spacing

• GRS-80 ellipsoid centered at 



ITRF00

origin

Offshore altimetry from a different source => better defines ocean topography.

ITRF00 used instead of ITRF97.



Gravity Coverage for GEOID03

This shows the extent of the gravity data that went info the USGG2003 geoid 

model.

Terrestrial data onshore; shipborne and altimetric data offshore.




-although GSFC00.1 created significant changes (50 cm or more in some regions 

when looking at the difference between G99SSS and USGG2003, the magnitude of 

those differences is dwarfed by the total signal

This model is tied to a GRS-80 ellipsoid centered at ITRF00 and the Wo value of 

EGM96 … but that isn’t what we need to use it with level data.



• We must have a consistent and seamless gravity field at least 

along the shorelines if not across all the U.S.

– Use GRACE data to test long wavelength accuracy.

– Use aerogravity to locate and possibly clean systematic 

problems in terrestrial or shipborne surveys (biases, etc.).

– Determine and remove any detected temporal trends in 

the nearly 60 years of gravity data held by NGS. Ensure 

consistency of datums, corrections and tide systems.

– This solves problems of current remove-compute-restore 

approach, which honors terrestrial data over EGM’s.

• Exploration of utility of coastal/littoral aerogravity

– Need a consistent gravity field from onshore to offshore.

– Aids in database cleansing; also fills in coastal gaps.

– Ties to altimetric anomalies in deeper water.

– In conjunction with tide gauges & dynamic ocean 

topography models, this will aid in determining the 

optimal geopotential surface for the U.S. (Wo).

Ongoing research areas

-Need seamless gravity data to reduce errors in gravity to geoid modeling

-Need additional gravity outside of U.S. areas – altimetric, neighboring countries

-Also need other data such as density anomalies and terrain data

-Current approach uses many simplifications – a more rigorous approach will reduce errors

-aerogravity fills in gaps and identifies systematic problems in gravity data (shipborne and 

terrestrial)



• Must acquire data and models for outlying regions.

– Definitely need surface gravity (terrestrial and shipborne) 

and terrain models for Guam, CNMI, American Somoa.

– Desire to get such for nearest neighbors including Mexico, 

Caribbean nations, Central American nations, etc.

– Also need to get any available forward geophysical models 

for all regions (such as ICE-5G for modeling the Glacial 

Isostatic Adjustment).

• GPS/INS evaluation of the gravity field.

– GPS & IMU information were also collected on flights.

– This data can be used to derive gravity disturbances and 

to estimate gravity anomalies.

– It may be useful in benign areas for determining the 

gravity field. Possibly cheaper and more cost-effective 

than aerogravity (run with other missions?).

Ongoing research areas (cont.)

-Need seamless gravity data to reduce errors in gravity to geoid modeling

-Need additional gravity outside of U.S. areas – altimetric, neighboring countries

-Also need other data such as density anomalies and terrain data

-Current approach uses many simplifications – a more rigorous approach will reduce errors

-aerogravity fills in gaps and identifies systematic problems in gravity data (shipborne and 

terrestrial)



• Geodetic theory improvements.

– Downward continuation of high altitude gravity 

observations.

– Merging of gravity field components.

• Current approach is remove-compute-restore.

• Spectral merging of EGM, gravity and terrain data.

• Would honor long wavelength (GRACE).

• Retain character of the terrain and observed data.

– Determination of geoid height using ellipsoidal coordinates 

instead of the spherical approximation.

– Resolution of inner and outer zone effects from terrain on 

gravity observations.



Ongoing research areas (cont.)

-Need seamless gravity data to reduce errors in gravity to geoid modeling

-Need additional gravity outside of U.S. areas – altimetric, neighboring countries

-Also need other data such as density anomalies and terrain data

-Current approach uses many simplifications – a more rigorous approach will reduce errors

-aerogravity fills in gaps and identifies systematic problems in gravity data (shipborne and 

terrestrial)



Geoid

Ellipsoid

Earth’s

Surface


Coast

Ellipsoid Ht

From GPS

How “high above sea

level” am I? (FEMA, 

USACE, Surveying 

and Mapping)

Ocean


Surface

From 


Satellite Altimetry

How large are near-shore

hydrodynamic processes?

(Coast Survey, CSC,

CZM)

Gravity measurements help answer two big questions…

Geoid Height

From Gravity

Orthometric Ht

From Leveling

This slide is an idealized situation demonstrating the usefulness of near-shore gravity, specifically the use of such gravity to 

accurately compute a gravimetric geoid model.  (The geoid is the one unique equipotential surface of the Earth’s gravity field 

which best fits, in a least-squares sense, global mean sea level).  

On land, the use of the Global Positioning System (GPS) for fast, accurate surveying is well documented.  However the height 

determined by GPS refers to a highly idealized, mathematically simple surface known as the ellipsoid.  Thus GPS heights are 

“ellipsoid heights”.  Unfortunately, ellipsoid heights are non-intuitive in many ways.  First, they can be dozens of meters 

different from heights above the geoid (known as orthometric heights, or more colloquially, but less correctly as “heights above 

sea level”).  Secondly, ellipsoid heights do not give an accurate portrayal of how water will flow, due to their complete 

disconnect from the gravity field of the Earth.  If one has an accurate model of geoid undulations (the separation between the 

geoid and ellipsoid), then one can take the GPS-derived ellipsoid heights, remove the geoid undulation, and arrive at your 

orthometric height (a.k.a. “height above sea level”).  This is a significantly faster method of determining orthometric heights 

compared to traditional surveying methods of spirit leveling.  

On the ocean, satellite altimetry is used to monitor sea level.  However, because the satellite orbits are known relative to the

ellipsoid, and the altimetric measurement is from satellite to sea surface, the resulting information is the height of the sea 

surface above the ellipsoid, often called the “sea surface height” or SSH.  If one is interested, however, in the impact of 

winds, heating, currents and other phenomena on the ocean surface, then one must remove the gravitational impact on the 

ocean surface.  That is, one must know the separation between the ellipsoid and the geoid over the ocean in order to remove 

this signal, and arrive at the residual value.  This residual, the distance from the geoid up to the sea surface, is known by 

many names, the most common being “sea surface topography” (SST) or “dynamic ocean topography”.  

Coastal areas prone to flooding would benefit both by knowing accurate land elevations (to aid in construction planning, 

floodplain mapping, etc) as well as knowing accurately how the near shore ocean processes work (to better model responses to 

hurricanes and other dynamic ocean phenomena).  It is an unfortunate fact that the largest areas missing accurate gravity 

measurements are those areas immediately offshore.  This is because ships with gravimeters can not get into these shallows, 

altimetrically derived gravity models are unreliable due to the inability to accurately compute near-shore tides, and terrestrial 

gravimeters can not measure in the water.  The only practical option to collecting the near-shore ocean gravity data is through 

airborne based technologies.

Dr. Dru A. Smith

Chief Geodesist

NOAA, National Geodetic Survey

Dru.Smith@noaa.gov

1/11/2006




Relationships

• Geoid = global MSL

– Average height of ocean globally 

– Where it would be without any disturbing forces (wind, 

currents, etc.).

• Local MSL is where the average ocean surface is with the 

all the disturbing forces (i.e., what is seen at tide gauges).

• Dynamic ocean topography (DOT) is the difference 

between MSL and LMSL: 

LMSL = MSL + DOT

• Hence: 

error = TG – DOT - N

ellipsoid

LMSL


geoid

N

Tide gauge height



DOT

NAVD 88


-Additionally, lidar observations on the open ocean can be reduced as well in VDatum areas 

where the tides are known. 

-Only temporal effects due transient weather will remain unmodeled – and generally you don’t fly 

in adverse weather.




M1

M2

M3



M4

M5

M6



M7

M8

M1



0

M9

M1



1

M1

2



M1

3

M1



4

M1

5



M1

6

M1



7

M1

8



M2

0

M1



9

M2

1



M2

2

M2



3

M2

4



M2

5

M2



6

M2

7



M2

8

M3



0

M2

9



J1

J2

J3



J4

J5

J6



J7

J8

J1



0

J9

J1



1

J1

2



J1

3

J1



4

J1

5



J1

6

J1



7

J1

8



J2

0

J1



9

J2

1



J2

2

J2



3

J2

4



J2

5

J2



6

T1

T2



T3

T4

T5



T6

T7

T8




Extent of Gravity and Data Collection Flights

-Surface gravity data are from NGS database: terrestrial, shiptrack and altimetry 

farther offshore (500 m depths)

-Flights extend from 50-100 km onshore (to stable regions) to well offshore 

(trustworthy ocean models & altimetry)

-Note lack of crossovers and actual data in Northeast corner => this will affect 

results later on




-Unresolved gravity signal differences between airborne and terrestrial data at elevation (10 km)

-Note the stripes – residual biases in aircraft lines

-Also note the features that have significant lateral extents between the lines – particularly in the 

NW corner.

-This implies a significant problem in our terrestrial data. 

-Additional collections further North and West will be needed to verify and analyze this.

-These differences must be accounted for because they imply significant errors in a geoid model 




Implied Geoid Changes

-Unresolved gravity signal differences between airborne and terrestrial data at elevation (10 km)

-Note the stripes – residual biases in aircraft lines

-Also note the features that have significant lateral extents between the lines – particularly in the 

NW corner.

-This implies a significant problem in our terrestrial data. 

-Additional collections further North and West will be needed to verify and analyze this.

-These differences must be accounted for because they imply significant errors in a geoid model 

as can be seen here. 

-Note the decimeter level feature in the NW corner 9in the eastern part of the Louisiana 

subsidence area.

-Note the range of values from +18 cm to -2 cm along the shoreline.

-The aerogravity are consistent (agreement was at the +/- 2 mgal level with no systematic effects 

accept biases).

-Hence, the differences seem valid and imply a significant error in the terrestrial data.

-These must be resolved before proceeding with geoid generation.




Once a satisfactory geoid has been made, it can be added to the DOT to simulate SSH’s, which 

can be compared to TBM’s.




tidal benchmarks with a NAVD88 tie

tidal benchmarks without a NAVD88 tie

Numerous TBM’s exist where the NAVD 88 value and ocean surface (MDT+geoid) 

are both known. These sites could suffice, providing that suitable gravimetric geoid 

and MDT models are generated.




Geoid Heights Compared 

with Tide Gauges

• Top image shows 

USGG2003 comparison

• Triangles note the 

locations of Tide Gauges 

within the study area

• Note that the dm-level 

trend along shore

• Bottom image shows 

aerogravity enhanced 

model comparison

• Note there is very little 

along shore trend

• Inclusion of aerogravity 

seems to have improved 

the geoid height model 

with respect to MSL

The bias shown here is around -1.00 meters. Subsequently, a datum bias was determined that 

moved theoffset to +0.35 m. This remaining bias is though to relate to possible errors in the 

generation of the reference gravity & geoid fields. 

Which proves exactly why we are performing these studies – to resolve such differences and 

better estimate the accuracy of geoid height models.




GPS/INS-Derived Aerogravity vs. Surface Point Gravity in CA

-Note the agreement  between tracks at crossovers (internally consistency)

-Note the larger number of surface points in the flat plains and lack of points in mountains

-Note the increased disagreement of surface points in mountains WRT the aerogravity

-Given the reduced amount of data in the mountains and greater likelihood of error in it, the 

result is that geoid errors will be greater in and near the edges of the mountains – where flooding 

can more readily occur from runoff.



• A Consistent vertical datum between all U.S. states and 

territories  as well as our neighbors in the region. 

– Reduce confusion between neighboring jurisdictions.

– Local accuracy but national consistency.

• This provides a consistent datum for disaster management.

– Storm surge, tsunamis, & coastal storms.

– Disasters aren’t bound by political borders.

• Heights that can be directly related to oceanic and hydrologic 

models (coastal and inland flooding problems). 

• The resulting improvements to flood maps will better enable 

decision making for who does & doesn’t need flood insurance.

• Updates to the model can be made more easily, if needed, to 

reflect any temporal changes in the geoid/gravity.

• Finally, offshore models of ocean topography will be improved 

and validated. These models will provide better determination 

of offshore water flow (useful for evaluating the movement of 

an oil slick).

Expected Results



QUESTIONS

?

Geoid Research Team:

Dr. Daniel R. Roman, research geodesist 



dan.roman@noaa.gov

Dr. Yan Ming Wang, research geodesist



yan.wang@noaa.gov

Jarir Saleh, ERT contractor, gravity database analysis

William Waickman, programming & database access

Ajit Sing, Datum Transformation Expert

Website: http://www.ngs.noaa.gov/GEOID/

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