__mean sea level, gps, and the geoid by witold fraczek

8/12/2019 __Mean Sea Level, GPS, And the Geoid by Witold Fraczek

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Mean Sea LevelGPS and the Geoid

By Witold Fraczek, ESRI Applications Prototype LahFrequently research and technology endeavorshave unforeseen but positive outcomes. WhenEuropean explorers set out to find a shortcut toIndia, they discovered the New World. When astaphylococci bacteria culture was mistakenlycontaminated with a common mold, the cleararea between the mold and the bacterial colonyled to the conclusion that the mold, Penicillinnotatum, produced a compound that inhibitedthe growth of bacteria.1hls chance discovery ledto the development of the antibiotic penicillin.

That the earth does not have a geometricallyperfect shape is well established, and the geoid isused to describe the unique and irregular shapeof the earth. However, only recently have themore substantial irregularities in the surfacecreated by the global mean sea level (MSL)been observed. These irregularities are an orderof magnitude greater than experts had predicted.Controlled by the gravitational potential of theearth, these irregularities form very gentle butmassive hills and valleys. 1hls astonishingfinding was made possible through the useof GPS, a technology designed by the United

h=H N

States Department of Defense to revolutionizenavigation for the U.S. Navy and Air Force.GPS has done that and a lot more.

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Model of the arth

~ -- '/,1 --Ellipsoid

The geo id approximates. mean sea level.The shape of he ellipsoid was calculatedbased on the hypothetical equipotentialgravitational suiface. A significant differenceexists between this mathematical modeland the real object. However, even the mostmathematically sophisticated geoid can onlyapproximate the real shape of he earth.

Topo surface (earth surface or GPS antenna)

Geoid (NSL) Ellipsoid

h=ellipsoid heightH=orthometric heightN=geoid heightThe accuracy of GPS height measurements depends on several factors but the most crucial oneis the impeifect ion of he earth's shape. Height can be measured in two ways. The GPS usesheight (h) above the reference ellipsoid that approximates the earth's suiface. The traditional,orthometric height (H) is the height above an imaginary suiface called the geoid, which isdetermined by the earth's gravity and approximated by MSL. The signed difference between thetwo heights the difference between the ellipsoid and geoid is the geoid height (N). The figureabove shows the relationships between the different models and explains the reasons why thetwo hardly ever match spatially.

36 rcUser July-September 2003

El graviLanomalValueI igh: 87Low: 105

El hs_anomaliesValueI igh: 254Low: 0

El ellipsoid_demValueI igh: 63KLow: 6352El hs_radiussValueI igh: 254Low: 0

[) 0 world_1 k

What Is ean Sea Level?For generations, the only way to expresstopographic or bathymetric elevation was torelate it to sea level. Geodesists once believedthat the sea was in balance with the earth'sgravity and formed a perfectly regular figure.MSL is usually described as a tidal datumthat is the arithmetic mean of hourly waterelevations observed over a specific 19-yearcycle. 1hls definition averages out tidal highsand lows caused by the changing effects of thegravitational forces from the moon and sun.

MSL is defined as the zero elevation fora local area. The zero surface referencedby elevation is called a vertical datum.Unfortunately for mapmakers, sea level is not asimple surface. Since the sea surface conformsto the earth's gravitational field, MSL also hasslight hills and valleys that are similar to theland surface but much smoother. However, zeroelevation as defined by Spain is not the samezero elevation defined by Canada, which is whylocally defined vertical datums differ from eachother.

The MSL surface is in a state of gravitationalequilibrium. It can be regarded as extendingunder the continents and is a close approximation

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http://www.wou.edu/las/physci/taylor/g492/geoid.pdf


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Special Section

The surface ofglobal undulations was calculated bafed on altimetric observationsand very precise (up to two centimeters) measurementstaken from the TOPEX/POSEIDON satellite. This dara was represented in the Earth Geodet ic Model (EGM96), which is also referred to as thespherical harmonic model of he earth s gravitationdl potential.of the geoid. By definition, the geoid describes r e v ~ l u t i o n was intended to replicate the MSLthe irregular shape of the earth and is the true as the main geodetic reference or vertical datum.zero surface for measuring elevations. Because f is ellipsoid vertical datum is used, heightthe geoid surface cannot be directly observed, abdve the ellipsoid will not be the same as MSLheights above or below the geoid surface can't andl direct elevation readings for most locationsbe directly measured and are inferred by making will be embarrassingly off. This is caused; ingravity measurements and modeling the surface patj, because the OPS definition of altitude doesmathematically. Previously, there was no way to no refer to MSL, but rather to a gravitationalaccurately measure the geoid so it was roughly surface called the reference ellipsoid. Becauseapproximated by MSL. Although for practical the reference ellipsoid was intended to closelypurposes, at the coastline the geoid and MSL approximate the MSL, it was surprising whensurfaces are assumed to be essentially the same, the two figures differed greatly.at some spots the geoid can actually differ from The TOPEX/POSEIDON satellite, launchedMSL by several meters. in 1992, was specifically designed to performiffering easurements

GPS has transformed how altitude at any spotis measured. OPS uses an ellipsoid coordinatesystem for both its horizontal and verticaldatums. An ellipsoid-or flattened sphere-isused to represent the geometric model of theearth.

vet)' precise altimetric observations. Thesem ~ u r m n t s have demonstrated that neitherhUJJ;J an error norOPS inaccuracies are responsiblefor the sometimes substantial discrepanciesbet 'een ellipsoid and MSL measurements.In fact, the three-dimensional surface createdby lthe earth's sea level is not geometricallycoqect, and its significant irregularities could

Conceptually,ellipsoid, called

this precisely calculated not:,be mathematically calculated; this explainsan oblate ellipsoid of the idifference between the ellipsoid-based GPS

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elevation readings and elevations shown onaccurate topographic maps.

A brief examination of elevation readingsfor ESRI headquarters in Redlands, California,demonstrates these differences. The campuselevation is shown on topographic quadranglemaps and high-resolution digital elevationmodels (DEMs) for the area as approximately400 meters above MSL. However, a precise,nonadjusted OPS reading for the same locationtypically shows the elevation as 368 meters.

Why is there a 32-meter difference? The OPSreceiver uses a theoretical sea level estimated bya World Geodetic System (WGS84) ellipsoid,which does not perfectly follow the theoreticalMSL. The MSL, approximated by an ellipsoid,is related to gravity or the center of mass ofthe earth. Discrepancies between a WGS84ellipsoid, and the geoid vary with location. Tocontinue with this example, elevation readingsfor Yucaipa, a city located less than 10 mileseast of Redlands, differ by 31.5 meters.

The geoid and ellipsoid intersect at the geoidContinued on page 8

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Mean Sea Level GPS and the GeoidContinued from page 7

LayersEl Ii counby

l Ii Below opec iejValue II High:OLow:105El Ii graYiLanomalValueI igh: 87Low: 105El Ii hs_ anomaliesValueI igh: 254Low:OEl Ii ellipsoid_demValue

The map shows the areas of he globe that would have a sea level below the theoretical surface of he WGS84 ellipsoid, or the theoretical andgeometrically correct sea level (shown in blue). The sharp contrast between the blue and green indicates where the ellipsoid and geoid intersect.With the continents displayed as opaque, the remaining area covered by water reveals where sea level is actually at zero elevation relative to theWGS84 ellipsoid.undulations. Undulations result from severalphenomena, the most significant o whichis the existence o gravitational anomaliescaused by the nonhomogeneous nature o theearth. The density o magma in the earth'scrust is unevenly distributed. In areas ogreater density, it can be significantly higherand, consequently, cooler. Less dense areasare correspondingly lower and hotter. Densemagma exerts a stronger pull, which causesthe accumulation o water masses. Not much isknown about whether these volumes relocate orhow fast they move. f hese locations do move,their movement would be at the same pace asother geologic events (i.e., extremely slow).

Precise measurements taken from space areapplied to PS readings. These measurementsare based on an ellipsoid surface that is amathematically generated model o the earthobtained from a three-dimensional Cartesiancoordinate system. The PS receivers canprovide only the ellipsoid (geometric) height.

However, most users expect accurateelevation readings that are related to MSL.Consequently, newer PS devices outputorthometric (geoid) height measurements asa product o behind the scenes calculationsbased on a combination o formulas, tables,and matrices that use geographic coordinatesas inputs. The appropriate height for thegeographic location taken from a coarse orfine DEM matrix is provided instead o a realmeasurement o the z value (or height). Some

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Hlgh:73 2I ow : ~ 1 7 3 5J hs dem

This map shows a world with no gravitational anomalies. A hypothetically homogeneousd i s t r i ~ u t i o n ofmass within the earth would change the delineation ofoceans and landmasses.This i iew ofSoutheast Asia depicts such a s p e c u l t i ~ e shoreline. Note the differences betweenthe current and the theoretical outlines of he wor ld ocean.recei\trs use approximations o the geoidheight to estimate the orthometric height fromthe ellipsoid height. Still other units, based onan ol4er technology, provide direct readings othe z talue based on the ellipsoid.A Glqbally Defined Geoid Geoid99A g l o ~ l l y defined geoid was required so that

PS 1receivers could calculate the correct

z value needed as a reference surface for aglobal vertical datum. Geoid99, a model witha submeter level o accuracy, was developedby the National Geodetic Survey. It is used as azero surface to establish consistent and accurateelevations worldwide. However, despite thisimpressive level o accuracy, portions o theGeoid99 deviate from MSL due to gravityeffects.

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Layers13 Below specified EValueI High:Olow: 2058013 c o u n ~ yD13 spheroid_topo

ValueI igh : 14564Low: 2058013 hs_demValueI igh: 254Low: 013 spheroid_demValueI igh: 6374816'Low: 6356n413 D spheroid

VALUE

Special Section

This map shows the hypothetical approximate delineation of landmass and ocean based on a spherically shaped earth, which assumes that thecurrent relie f would be preserved.

The Earth Geodetic Model (EGM96),developed in a collaborative effort by NASAGoddard Space Flight Center, the NationalImagery and Mapping Agency (NIMA),and Ohio State University, has been used tocompute geoid undulations accurate to betterthan one meter (ex..:ept in areas lacking accuratesurface gravity data). The values of this surfaceshow at every location how much MSL variesfrom the ellipsoid used as the reference for GPSelevation readings. In other words, EGM96shows how uneven the gravitational potentialof the ocean's surface really is; The maximumrange of the Geoid99 undulations with respectto the WGS84 ellipsoid is 192 meters. Thebiggest anomaly was discovered southeast ofIndia where the geoid is 105 meters below theellipsoid, and its highest swelling was observedin eastern Indonesia.''What If SimulationsHow would continental coastlines change ifthere were no gravitational anomalies? Theshort answer would be that they would appearthe same as if the earth were an ellipsoid.The biggest undulations are concentrated inthe northern Indian Ocean and within theIndonesian Archipelago. Consequently, areascurrently covered by the Arafura Sea andCarpenteria Bay, both north ofAustralia, wouldemerge. Conversely other areas, such as thelowlands formed by the deltas of some of themain South Asian rivers such as the Gangesand Brahmaputra as well as those of Indus,Irrawaddy, and Mekong, would sink.

Voluo~ : 6 3 7 6low :635225

621COIJl ltry ;stherfromEarthceoterl. :

YalueI ''' 'low:6371146hs j dem53.750000

Low :0.000000D o1j' -'V ' """''""'"low 6352258'9 '-' 'IWhqt i he earth actually stood still? The data approximates a theoretical elevation model

withlthe elevations relative to the distance from the ear th s center. The GPS refers its directelevbtion readings to this shape. The cell values of he above ellipsoid_dem grid are the directreadings ofGPS. The displayed GPS readings are based on the added gravitational correctionas the shape of he geoid is based on gravitational potential (both positive and negative).

As an intellectual exercise, imagine for amoment that the earth has the geometricallyp e r f ~ c t shape of a spheroid and then anellipsoid. To visualize these what if scenarios,a cqmputer simulation was generated usingrasters that represented distances to the earth'sc e n t ~ r and were generated with the RasterCalculator in the ArcGIS Spatial Analyst

e x t e ~ s i o n A standard digital elevation modelfor the globe was modified by removing the

gravitational anomalies identified by EGM96.These geometrically correct representations ofa spheroid and ellipsoid were used to produceGIS simulations called the SPHEROID DEMand ELLIPSOID DEM. The accompanyingillustrations show the effects these shapes, aswell as other scenarios, have on the distributionof landmass and ocean.

Continued on page 4rcUser July September 2003 39


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Mean Sea Level GPS and the GeoidContinued from page 9

t J ayersB countryDB Farther from Earth ce

ValueI igh : 6383743Low : 6380500B geoid_demValueI igh : 6383743Low : 6352243

B hs_demValueI igh: 254Low: OB D geoid_topoValueI igh: 7343Low : -1075 9

The map shows distances to the earth s center ofmass, The areas highlighted in red represent mountains that are the farthest from the center ofmass on the earth and are the areas of he lowest grav(tational pull on the earth s surface.SPHEROID DEMThe SPHEROID DEM was created based onthe assumption that the earth s shape, or moreprecisely, the shape of the earth s surface asrepresented by MSL, is spherlcal. In otherwords, the globe s radius connecting the centerof mass with the hypothetical equipotentialgravitational surface is identical everywhereon the planet. The radius was set to 6,367,473meters, which is the distance to the earth scenter from the ellipsoidal reference surface at45 degrees latitude.

This change in geometry to a spheroidgeometry with its altered gravity would causethe global ocean to change. The polar zoneswould be relatively farther from the center of theearth, and these new higher altitudes would forceseawater toward the equator. By the same token,the equatorial region would be relatively closerto the earth s center and would be more stronglyaffected by gravity. Increased gravitational forcein the equatorial zone would pull oceanic watertoward the equator and form a global equatorialocean.

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This map illustrates what would happen i all the glaciers melted. he cell values of thisorthornetric surface grid are related to MSL

E L L l ~ S O I DEMThe JELLIPSOID DEM depicts the earth as ane l l i p s ~ i d Although distances from the surface ofan elliiE soid to the earth s center along each latitudeare idntical, each latitude has its own unique value

increasing from each pole toward the equator.To look at this ellipsoid earth from yet anotherperspective-its physical shape characterized bythe distance of any point on the surface to thecenter of the earth s mass--a grid was generated

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using the WGS84 datum definition. Each grid cellvalue represents the distance in meters from thesurface to the earth's center of mass. A complexcombination of trigonometric functions wasapplied to create the representation of he ellipsoid.Then the DEM representing the earth's currentreliefwas added to the ellipsoid raster.The difference between the major and minoraxes of the WGS84 ellipsoid is 42,770 meters. Thedifference between the length of the radius at theequator and one of the poles is 21,385 meters-only 0.33 percent of the radius. The ''flatteningof the earth is, geometrically speaking, relatively

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insignificant but in tenns of geography, it has atremendous impact.If the Earth Stood StillWhat would happen if he earth stopped spinningand the centrifugal effect ceased to force oceans toaccumulate around the equator? It appears that theworld's ocean would split into two polar oceansand leave the equatorial area totally dry. To modelthis hypothesis, a value of 6,371,146 m t ~distance from the earth's center indicates theapproximate elevation of the sea level on thereference ellipsoid-was specified to separate

water from land. For this what i f simulation,the elevation of the sea level was based on theassumption that the volume of ocean water wouldbe about the same as it is today.Melting GlaciersReturning to a geoid representation of the earth,one more simulation models an earth where allthe glaciers have melted. This simulation mightpredict a future that is only a few hundred yearsdistant if the large glaciers of Antarctica andGreenland, which currently cover approximately10 percent of all land, melted as a result of globalwarming. I f all the water in these glaciers werereleased, the MSL would rise about 80 metersabove its current level.onclusion

GIS makes it possible to explore the effects ofdifferent conceptualizations of the earth's shapeand model a variety ofglobal conditions.

cknowledgmentsThe author would like to thank his colleagues-Melita Kennedy, Lenny Kneller, Corey LaMar,and Marcelo Villacres--for their significantcontributions to this article.

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__mean sea level, gps, and the geoid by witold fraczek

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