Land Information Science What Does Height Really Mean? Part IV: GPS Heighting Thomas H. Meyer, Daniel R. Roman, and David B. Zilkoski ABSTRACT: This is the final paper in a four-part series examining the fundamental question, "What does the word height really mean?" The creation of this series was motivated by the National Geodetic Survey's (NGS) embarking on a height modernization program as a result of which NGS will publish measured ellipsoid heights and computed Helmert orthometric heights for vertical bench marks. Practicing sur- veyors will therefore encounter Helmert orthometric heights computed from Global Positioning System (GPS) ellipsoid heights and geoid heights determined from geoid models as their published vertical control coordinate, rather than adjusted orthometric heights determined by spirit leveling. It is our goal to explain the meanings of these terms in hopes of eliminating confusion and preventing mistakes that may arise over this change. The first paper in the series reviewed reference ellipsoids and mean sea level datums. The second paper reviewed the physics of heights culminating in a simple development of the geoid in order to explain why mean sea level stations are not all at the same orthometric height. The third paper introduced orthometric heights, geopotential numbers, dynamic heights, normal heights, and height systems. This fourth paper is composed of two sections. The first considers the stability of the geoid as a datum. The second is a review of current best practices for heights measured with the Global Positioning System (GPS), essentially taking the form of a commentary on NGS' guidelines for high-accuracy ellipsoid and orthometric height determination using GPS.
JULIET: And not impute this yielding to light love, Which the dark night hath so discovered. ROMEO: ... Lady, by yonder blessed moon I vow, That tips with silver all these fruit-tree tops-- JULIET: 0, swear not by the moon, th' inconstant moon, That monthly changes in her circle orb, Lest that thy love prove likewise variable. [William Shakespeare: Romeo and Juliet-The Balcony Scene (Act 2, Scene 2)]
Vertical Datum Stability Thomas H. Meyer, Department of Natural Resources Management and Engineering, University of Connecticut, Storrs, CT 06269-4087. Tel: (860) 486-2840; Fax: (860) 486- tability is a desirable quality for a datum, meaning that a datum ought not to change 5408. E-mail:
Surveying and Land Injbrmation Science, Vol. 66, No. 3, 2006, pp. 165-183 of the day is decreasing by about two milliseconds sphere and, occasionally, striking the Earth's per century and that there are seasonal varia- surface. Similarly, the Earth is constantly loosing tions (with periods on the order of a month) on mass as gaseous molecules too light to be bound the same order (Vanicek and Krakiwsky 1996, p. by gravity drift off into space (e.g., helium gas). 68). Consequently, the Earth's centrifugal force Neither the addition nor the removal of mass is likewise diminishing and variable. However, changes the Earth's gravity field enough to be of these variations are far too small (on the order concern in this paper. of 10•12 radians s-i) to change the Earth's cen- The Earth's mass is redistributed in various trifugal force at a discernable level in faster than ways including postglacial rebound, melting ice a geologic time frame. caps and glaciers, the Earth's fluid outer core, the The rotational axis of the Earth slowly traces oceans (Cazenave and Nerem 2002), and earth- a circle on the celestial sphere, the same motion quakes. For example, earthquakes can be caused that can be observed in a spinning top. This by the motion of tectonic plates along their mar- motion is called precession. The Earth's preces- gins, and this motion causes a change in the sion is caused by the equatorial bulges not align- Earth's shape. Earthquakes can cause a measur- ing in the plane of the ecliptic (the plane in which able change in the Earth's rotation velocity, and the Earth orbits the sun), thereby giving rise to thus its gravity, by changing one of its moments a torque from the gravitational attraction of the of inertia (Chao and Gross 1987; Smylie and sun (Vanicek and Krakiwsky 1996, p. 59). The Manshina 1971; Soldati and Spada 1999). The Earth's precession is slow, with its axis returning Sumatra, Indonesia, earthquake of December 26, to a previous orientation once in approximately 2004 was such an event. It decreased the length 25765 years, a period known as a Platonic year. of day by 2.68 microseconds, shifted the "mean Likewise, the equatorial bulges are not aligned North pole" about 2.5 cm in the direction of with the Moon's orbital plane, which is inclined 145 degrees East Longitude, and decreased the 5°11' to the ecliptic. The intersection of the Earth's flattening by about one part in 10 billion Moon's orbital plane with the ecliptic is known (Buis 2005). The uplift of plates due to tectonic as the nodal line, and the nodal line rotates once or postglacial activities affects ellipsoidal heights, in 18.6 years, the Metonic cycle. This constant as well as having a smaller gravity-based effect realignment of the Moon with the Earth also which changes the geoid. The National Geodetic affects the orientation of the Earth's rotational Survey is planning to engage in research which axis, causing a motion called nutation (Vanicek tracks the time-dependent changes of the geoid and Krakiwsky 1996; Volgyesi 2006, p. 61). due to these effects. There are additional, smaller perturbations as well. The motion of the Earth's rotational axis Tides in the celestial reference frame affects astro- nomic and satellite observations but not gravity People who have been at an ocean shore for because, although the direction of the centrifu- half a day or more have had the opportunity to gal force vector is changing, this change was watch the ocean advance inland and then retreat brought about by a motion of the Earth itself, back out to sea. This motion is caused primarily so the relative change is zero. However, actual by the gravitational attraction of the Moon and, movement of the rotational axis relative to the to a lesser degree, the Sun. Therefore, the defi- Earth's crust itself (known as "Polar Motion" or nition of tide found in NGS' Geodetic Glossary "Polar Wobble") does affect gravity, because the may be somewhat surprising. direction of the centrifugal force vector in this Tide (1) Periodic changes in the shape of case is changing relative to the Earth's crust. the Earth, other planets or their moons that These small changes are only on the order of a relate to the positions of the Sun, Moon, few nanoGals, well below the noise level of most and other members of the solar system. gravity measurements. Note that this definition is not about the oceans, per se. Instead, it speaks of, among other things, Changes in the Earth's Mass a change of the shape of the Earth itself, the Earth tide or body tide. It is commonplace knowledge The Earth's mass can increase or decrease, and it that the Moon moves the oceans; it deforms can be redistributed. Concerning the former, the them to set them in motion. But, what is prob- Earth does gain mass almost continuously due ably not so well known is that the Earth's core, to a stream of space debris entering the atmo- mantle, and crust have their shape deformed
and Land Information Science 166Surveying166 Su?-,eyi?zg and Land Inflorviation Science in a manner similar to the deformation of the so that the force is directed from the Earth's oceans. The NGS definition continues: center towards the moon's center); In particular, (2) those changes in the size V, m = the mass of the Earth and the Moon, and shape of a body that are caused by respectively; and movement through the gravitational field of FV, = the gravitational force vector produced by another body. The word is most frequently the Moon on the Earth. used to refer to changes in size and shape The gravitational force of the Earth exerted on of the Earth in response to the gravitational the Moon can be found simply by defining r to attractions of the other members of the have the opposite direction, so the magnitudes solar system, in particular, the Moon and of the two forces are equal. The gravitational sun. In such cases, three different tides attraction of the Earth on the Moon causes the are usually distinguished: the atmospheric Moon to orbit the Earth rather than to move off tide, which acts on the gaseous envelope into space. However, Equation (IV. 1) also means of the Earth; the earth tide, which acts on that the Earth is orbiting the Moon, but this motion the solid Earth; and the ocean tide (usually is much less obvious due to the difference in 2 7 simply called "the tide"), which acts on the masses of the two bodies. If we take 5.9742x10 hydrosphere. g to be the Earth's mass, 7.38x10 25 g to be the The effects of the tides are numerous and com- Moon's mass, and 3.84x108 m to be their mean plicated, so perhaps the first question to consider separation, then the barycenter of the Earth- is whether the tides cause enough of an effect to Moon system can be found to be at a point on a be of concern. Is the earth tide large enough to line connecting their two centers approximately affect the geoid in any practical way? It happens 4.69x 106 m from the Earth's center. This point is that there are two high and low earth tides each inside the Earth, being about 73.5 percent of the day, with the highest being on the order of a 50 length of the GRS 80 semimajor axis. cm displacement from its undeforined shape It is critical to understand the nature of the (Moritz 1980, p. 477)! So, the answer is "yes;" we motion of the Earth's orbiting the Moon. The must take tides into consideration. diurnal rotation of the Earth, the source of days Tides on the Earth arise due to the influences and nights, is a rotation around its axis, which is from all celestial bodies. The Sun and the Moon nominally the North Pole. Points on a rigid rotat- produce the largest effects by fat, but the other ing body that are on different radii move in dif- planets have a discernable affect, albeit too small ferent directions and at different instantaneous to impact GPS positioning (Wilhelm and Wenzel linear velocities (see Figure IVIa). However, a 1997, p. 11). All celestial bodies create tides in the rigid body can rotate around only one axis at same way, the only difference being the details any moment in time. Therefore, the Earth does of how these manifest themselves. Therefore, we not rotate about the Earth-Moon barycenter. To will consider the effects created by the Moon, understand this orbital motion, envision some- with the understanding that they apply to any one waxing a tabletop with a cloth by rubbing celestial body with the appropriate change of it in a circular motion, such that their fingers mass and distance variables. remain parallel to some wall in the room. If the circular motion of the cloth has a fairly small Tidal Gravitational Attraction and radius, then the point around which the cloth Potential is moving is always beneath the cloth, just as the motion of the Earth around the barycenter has According to Newton, force gives rise to motion its center at a point within the Earth. Now, it is by accelerating mass. The gravitational force apparent that every point on the cloth is actually of the Moon on the Earth itself is found using moving with the same velocity (same direction Equation (11.2): and speed). Similarly; the orbital motion of the Earth around the Moon gives rise to a constant GMmf 2 acceleration that is always directed opposite to FE G fmi (Iv.l) the line connecting the Earth's center to the Moon's. In particular, everywhere and everything on where: and in the Earth is accelerating away ftnom the Moon r = a vector from the Moon's center to the Earth's as if the Earth were moving in a straight line along the center (note the negative sign in Equation instantaneous axis between them; see Figure IV.lb. (IV.1) reversing the direction of the vector This acceleration gives rise to a component of
167 Vot.[lol. 66,66, No. 33 167 observable gravity that is at most 3.4 percent of the total acceleration (Vanicek and Krakiwsky 1996, p. 125). To the Moon The moon's gravita- tional attraction gives rise To the Moonl *b. to a force at any particular a. place on the Earth that is directed (approximately') Figure IV.1. Velocities of points moving on a rigid rotating body. Panel (a) presents along the line from the the instantaneous velocity vectors of four places on the Earth; the acceleration vec- point of interest to the tors (not shown) would be perpendicular to the velocity vectors directed radially Moon's center. In contrast, towards the rotation axis. The magnitude and direction of these velocities are func- the orbital acceleration tions of the distances and directions to the rotation axis, shown as a plus sign. Panel experienced at that place (b) presents the acceleration vectors of the same places at two different times of is always parallel to the the month, showing how the acceleration magnitude is constant and its direction is line connecting the Earth- always away from the Moon. Moon centers, so these forces are not generally Figure IV.3 shows the details of the vector addi- parallel to each other. Furthermore, places on tion of three points of interest from Figure IV.2. the side of the Earth opposi te the Moon expe- Orange vectors are the Moon's attraction; their rience a smaller attraction tban places on the non-parallelism with the orbital acceleration vec- same side as the Moon due toybeing closernto tors, shown in blue, is greatly exaggerated. The the Moon, giving rise to the asymmetry evident vector result of the addition of these two vectors in Figure IV.2. Each of the ve ctors in Figure 1V.2 is shown in black. Figure IV3a represents the indicates the force vector of the place located at situation at point a, which is located at the top of the tail of the vector resulting from the combina- the circle in Figure IV2. The Moon's attraction tion of the orbital acceleratioon and the Moon's is the most non-parallel with the orbital accel- attraction at that place. eration at this place and its antipodal counter- part. Given the roughly equal magnitude of the orbital acceleration and Moon attraction forces, their component in the direction of the Moon largely cancels ata, leaving a small result oriented Moon sharply towards the Earth's middle. Figure IV.3b represents the situation at b which is located at the point furthest from the Moon. The Moon's 0 attraction is parallel but opposite in direction with the orbital acceleration at this place. The orbital acceleration is moderately stronger than the Moon's attraction here, creating the force primarily responsible for the lower high tide of the day. Figure IV3c represents the situation at c Figure IV.2. Arrows indicate for ce vectors that are the which is located at the point closest to the Moon. combination of the moon's attractiion and the Earth's orbital The Moon's attraction is considerably stronger acceleration around the Earth-Mo on barycenter. This force here than the orbital acceleration, creating the is identically zero at the Earth ceenter of gravity. The two force that is primarily responsible for the higher forces generally act in opposite directions. Points closer tide of the day (see (Vanicek and Krakiwsky 1996, to the Moon experience more of the Moon's attraction p. 124; Bearman 1999, pp. 52-61). whereas points furthest from thie Moon primarily experi- The magnitude and direction of the Moon's ence less of the moon's attraction ;c.f. (Bearman 1999, pp. attraction is periodic due to the nature of its 54-56; Vanicek and Krakiwsky 19916, p. 124). orbit around the Earth. The situation is com-
1 The moon is too close to the Earth for this to be exact. The actual direction of the vector would be determined by triple integrating over the Moon's mass, and approximately end up pointing at the Moon's center of mass, approximate because the Moon is not a perfectly homogeneous sphere.
168 Surveying and Land Information Science Up to this point we have been concerned with gravity force. We a. 4 Result now consider how tides affect Moon's a ttraction gravity potential because, after all, the geoid (an equipotential ac:celeration surface) is a principle datum of b. Orbital interest, hence we must exam- ine how these temporal changes C._ come into play. The gravitational potential field created by the Moon at some point of interest Figure IV.3. Details of force combinations at three places of interest; c.f. can be expressed as an infinite Figure IV.2. series of which only the second term is important for tides. This plicated but made tractable by accounting for second termn W2 takes the form of the expression individual tidal constituents. It is possible to (Vanicek 1980, p. 5, Equation (12)): decompose the Moon's attraction into individual constituents, a constituent being a sinusoid with I V2sz Dlcos-oozScos 2wit+sin 20sm 2&cont +3(.mn20- 1/3)(.i.23 - 1/3j) a particular amplitude, frequency and phase that arises due to a particular phenomenon. As dis- (IV.2) cussed by Boon (2004), some of the prominent where: tidal constituents are caused by: D = Doodson's constant (Doodson 1922); "* The inclination of the Moon's orbital plane p= geocentric latitude; with respect to the ecliptic giving rise to the 6 = the declination of the Moon; and lunar declination (topic-equatorial) cycle; t = the Moon's hour angle (see any "*The Sun's attraction giving rise to the spring- standard work on celestial mechan- neap cycle; ics for exact definitions of 6 and t). "*The eccentricity of the Moon's orbit giving Doodson's constant is given by Vanicek (1980, rise to the perigean-apogean cycle; and p. 4, Equation (7)) as: "*The precession of the lunar nodes giving rise R to the metonic cycle. D = 3 Gm- 4 r.3 (IV 3)
Chesapeake Bay Bridge Tunnel Gladstone Dock, Liverpool where: S0.4 G = the universal gravitation con- S° ,I,A,' stant; 0.o 2 lllll I! '" " !lll llllll llll R = the mean (equivoluminous) radius of the Earth; and r = the mean distance to the Moon. Time (hour) Time (hour) D has a value of approximately 2.6277x10 7 cm mgal. Equation Figure IV.4. Two simulations of tide cycles illustrating the variety of possible (IV.2) consists of three terms within affects. the brackets. The first term con- tains sectorial constituents; the Simple ocean tide models include as few as six second term contains tesseral constituents, and constituents; complicated models can incorporate the third ten m contains zonal constituents. These more than 100 (Wilhelm and Wenzel 1997). These three compionents are shown in Figures IV.5-7 models produce tidal predictions such as those and their cc mbination in Figure IV.8. Sectorial shown in Figure IW4. The predictions in Figure constituents vary in longitude (time), much like fV4 use constituents from Boon (2004, pp. 97-102) the sectors o[fan orange, and give rise to the two and clearly show higher high water, lower high daily tides. 1-esseral constituents possess both lat- water, higher lower water, and lower low water, as itude and loirigitude components and give rise to well as many longer-period variations, patterns res embling the tessellation of a checker
Vol. 66. No. 3 169 Sectorial Potential 1. Zonal Potential 1- 11 J
Figure IV.5. The sectorial constituent of tidal potential, Figure IV.6. The zonal constituent of tidal potential. The The green line indicates the Equator. The red and blue red and blue lines are as in Figure IV.5; the equatorial lines indicate the Prime Meridian/International Date green line is entirely inside the potential surface. The line and the 90/270 degree meridians at some arbitrary zonal constituent to tidal potential gives rise to latitudinal moment in time. In particular, these circles give the tides because it is a function of latitude. viewer a sense of where the potential is outside or inside the geoid. The oceans will try to conform to the shape Therefore, no gravity observations are needed of this potential field and, thus, the sectorial constituent to determine the potential from the Moon; it all gives rise to the two high/low tides each day. falls out of the mathematics. The parameters that describe the response board. The zonal constituents do not vary in time of the Earth's shape and gravitational potential field to tidal forces and give rise to so-called "permanent" tides. are called Love and Shida numbers, which are empirically derived. They are used in equations similar to Equation (JV.2) Body Tides and sufficiently capture the deformation of the The first clear evidence of body tides came from Earth so that tidal affects may be removed from the measurement of ocean tides, which showed geoid models, gravity observations, GPS obser- that they were consistently about two-thirds as vations, and other geodetic quantities (Vanicek high as Newton's physics predicted. It was even- 1980). However, it should be noted that the per- tually shown that the missing one-third was due manent tides (those portions of the tidal equa- to deformation of the Earth itself, moving with tions which describe the non-time-varying, or the oceans (Melchior 1974). The tides of the "permanent" deformations) are not completely solid Earth behave in the same manner as the determinable empirically. There are two compo- ocean tides, but in a simpler manner because nents of this permanent tide: first, the perma- the Earth deforms like an elastic solid at the fre- nent deformation of Earth's geopotential field quencies of tides, rather than with all the free- due to the existence of the permanent (non-zero dom of a liquid, like the oceans. time-averaged) Sun and Moon and second, the It is remarkable that the effect of the Moon's permanent deformation of Earth's geopoten- potential field upon the Earth can be described tial field due to the existence of the permanent with such high accuracy by such a simple equa- deformation of Earth's crust (which, in turn, is tion as Equation (IV.2); compare this with the due to the existence of the permanent Sun and effort necessary to determine the geoid! The Moon). simplicity of Equation (IV2) is because (1) the The first part (called the "direct" component Moon is far enough away to be treated as a of the permanent Earth tide) is computable point mass, and (2) the motion of the Moon is empirically, as it deals solely with the Sun's and very accurately described by celestial mechanics. Moon's mass affecting the Earth's geopoteritial
170 Surveying and Land Information Science Tesseral Potential Total Potential
03 1 I : M
C_IIZI B=
Figure IV.7. The tesseral constituent of tidal potential. Figure IV.8. The total tidal potential is the combination of The green, red and blue lines are as in Figure IV.5. The the sectorial, zonal, and tesseral constituents. The green, tesseral constituent to tidal potential gives rise to both red and blue lines are as in Figure IV.5. The complicated longitudinal and latitudinal tides, producing a somewhat result provides some insight into why tides have such a distorted looking result, which is highly exaggerated in wide variety of behaviors. the figure for clarity. The tesseral constituent accounts for the Moon's orbital plane being inclined by about five liquid nature of the oceans allows dramatically degrees from the plane of the ecliptic. more complexity in their response to gravita- tional attraction and, consequently, its modeling field. The second part is not computable empiri- is likewise more complex. cally. This is because the permanent deformation of the Earth's crust can not be directly observed. The Earth's crust perpetually ("permanently") Global Navigation Satellite exhibits a deformation due to the permanent System (GNSS) Heighting existence of the Sun and Moon. Because we can not observe how the crust would react without a Global navigation satellite systems, such as the permanent Sun and Moon, we can not determine European Union's Galileo system, the Russian empirically how much permanent deformation Global'naya Navigatsionnaya Sputnikovaya Sistema actually exists (that is, we can not determine a (GLONASS), and the U.S. Global Positioning "zero degree Love number" for the Earth), and System (GPS) offer, in conjunction with a highly thus can not compute what the effect of this per- accurate model of the gravimetric geoid, the manent crustal deformation is on the Earth's potential of determining orthometric heights geopotential. with centimeter accuracy without conventional leveling. The prospect of establishing vertical control in remote locations without running Ocean Tides levels to established distant bench marks holds Ocean tides affect the geoid by redistributing great promises of time savings, and therefore, cost the mass of the oceans, which has the following savings. These savings are the reward for surveyors effects. First, the redistribution of the water in who practice GPS heighting and were a primary the oceans creates a discernible change in the motivation for this series. According to Zilkoski et geoid. Second, the weight of the water deforms al. (2000), "GPS-derived orthometric heights can the Earth below it, in addition to the tidal poten- now provide a viable alternative to classical geodetic tial also deforming the Earth (Vanicek 1980, pp. leveling techniques for many applications." 9-12). The deformation of the Earth due to tidal Deriving orthometric heights fi-om ellipsoid loading can also be modeled by certain Love heights is mathematically very simple. As explained numbers that parameterize Equation (1V.2). The in the previous papers, a geoid height is the geo-
11ol. 66. No. 3 171 metrical separation (distance) fi-om some reference tract the geoid height, which is provided by a ellipsoid to the geoid, an ellipsoid height is the geo- gravimetric geoid model, to obtain the approxi- metrical separation from some reference ellipsoid mate orthometric height. In practice, things to a point of interest, and an orthometric height are more complicated. This fourth paper now is the length of the plumb line from the geoid to presents a survey of GPS heighting error sources a point of interest. Were plumb lines straight lines and best practice guidelines put forth by NGS and if they were normal to the reference ellipsoid, and other authors in the peer-reviewed litera- these three definitions would immediately lead to ture. Although this paper depends in large part an exact relationship: on previous work by Zilkoski and others at the H=h-N J(V4) NGS (Zilkoski et al. 1997), it is not our inten- tion to restate that material verbatim (Zilkoski et where: al. 2000). Instead, this final paper will provide H = orthometric height; commentary on the guidelines and explanations N = geoid height; and why some of the recommendations were made. h = ellipsoid height. We will emphasize the key issues necessary for However, plumb lines are curved and not normal achieving the accuracies in those guidelines and to reference ellipsoids, in general. Therefore, we provide examples from the literature that illus- cannot be correct in using an equality relationship trate them, when possible. More detailed and and must instead write: comprehensive treatments include (Leick 1995; H-= h-N (IV.5) Hofmann-Wellenhof et al. 1997; Seeber 2003; Hofmann-Wellenhof and Moritz 2005). Although Equation (IV.5) is not exact, it is close enough for most practical purposes (Hein 1985; Henning et al. 1998; Vanicek et al. 1999; Error Sources Zilkoski 1990; Zilkoski and Hothem 1989). For Effective GPS heighting depends upon having example, an extreme case of a two-arc-minute an understanding of the measurement error deflection of the vertical would introduce less budget and acting in such a manner as to elimi- than two millimeters of error in the orthomet- nate or mitigate those errors. Error sources have ric height (Tenzer et al. 2005, p.89), based on been grouped in three main categories: satellite Equation (IV.5). position and clock errors, signal propagation Much of the information from this series is errors, and receiver errors (Seeber 2003). We contained within Equation (IV.5) (Hwang and will discuss these error sources and explain what, Hsiao 2003; Kao et al. 2000; Sun 2002). For if anything, can or should be done about them example, the choice of the reference ellipsoid is according to best practices reported in the cur- important. Local geodetic reference ellipsoids rent literature. Although it is beyond the scope are generally not geocentric, so their normal of this paper to review GNSS as a whole, the directions could differ significantly from those reader is referred to the large existing literature of ellipsoids that are geocentric insofar as was on the topic, such as-(Hofmann-Wellenhof et al. possible at the time of their creation. It is impor- 1997; Leick 1995; Seeber 2003; van Sickle 1996) tant not to mix heighting systems. The GPS sur- and collections of articles published by the U.S. veyor must therefore use a reference ellipsoid of Institute of Navigation (ION). However, before a datum that matches the reference ellipsoid of discussing these error sources, we present issues the gravimetric geoid model. In the U.S., NGS that arise due to the Earth itself. recommends using GEOID03 which is modeled relative to the NAD 83 datum (which uses the Geophysics GRS 80 ellipsoid). Therefore, for example, GPS There are several issues pertaining to the Earth heighting should not be done with GEOID03 and itself that factor into GNSS heighting. Most the WGS 84 datum. Also, because Equation (JV.5) is of these pertain to the dynamic shape of the an approximation rather than an equality (due to Earth but one arises simply because the Earth the non-parallelism of the equipotential surfaces), is opaque at the radio frequencies broadcast by dynamic/orthometric corrections will have to be GNSS satellites. applied to (the purely geometric) spirit leveling No Satellites Below measurements (Strang van Hees 1992). We begin by explaining why it is that GNSS posi- In theory, GPS heighting is simple: determine tioning cannot be expected to be as accurate an ellipsoid height with a GPS receiver and sub- for vertical coordinates as for horizontal ones.
172 Surveying and Iand Information Science Currently operational GNSS satellites, abbrevi- ence (Gabrysch and Coplin 1990), which com- ated as SV for "space vehicle," are stationed in plicates matters considerably. orbital planes inclined from the equator by 55 degrees (for GPS) or 64.8 degrees (for GLONASS). Satellite Position and Clock Errors Consequently, any place on Earth is always sur- rounded by SVs, above and below. However, the We now begin a discussion of the GNSS error Earth completely blocks signals from SVs below budget. Because GNSS positioning is accom- the horizon from reaching a receiver; the radio plished by a process similar to trilateration there signals cannot penetrate solid rock. Therefore, are two key pieces of information upon which receivers on the ground cannot detect signals GNSS positioning depends: signal propagation from SVs below the horizon. As a result, while time and the location of the SVs. Signal propa- it is possible to be surrounded on all azimuth gation time is used to infer the range from the points by SVs, one cannot be surrounded on SVs to a receiver antenna's phase center, and all zenith angles (essentially none greater than SV locations are used as the coordinates of 90 degrees). Consequently the local vertical is the known points in the trilateration scheme. not controlled as well as the local horizontal. As However, the signal propagation time is biased stated by Brunner and Walsh (1993), "We note due to an immeasurable time offset between that even without any tropospheric propagation GPS time and a receiver's internal clock; this errors, an inherent geometrical weakness exists results in a "pseudo-range" rather than the in the GPS baseline results that usually makes actual range. The implications of this will be the determination of height differences worse by discussed below. Any errors in locating a SV and a factor of about 3 compared with the horizon- any inconsistencies in the clocks on board the tal baseline components." Therefore, we cannot SVs that govern its operation result directly in expect the best GNSS heighting to be as accu- positioning errors. rate as the best GNSS horizontal positioning. Orbit Errors and Ephemerides Earth Tides, Ocean and Atmospheric Loading Knowing the position of the satellites at any GNSS post-processing software often includes given moment in time is a cornerstone of how tide corrections which remove these effects, cre- GNSS positioning works. The satellites them- ating a "tide-free" system. See the opening dis- selves should be perceived as being moving cussion for more elaboration. monuments because pseudo-range positioning (positioning using pseudo-ranges) is based on Crustal Motion trilateration: given three (or more) known loca- Plate tectonics constantly move the Earth's crust tions and a distance from those locations to the both horizontally and vertically. Horizontal point of interest, determine the coordinates of 2 motions can be accounted for by modeling the the point of interest. Therefore, since the satel- position and velocity of fiducial stations and lites are in motion, it does not suffice to pub- then interpolating to places of interest. The lish a single set of coordinates for them. Instead, NGS Horizontal Time-Dependent Positioning ephemerides are created for each SV so that the (HTDP) software (Snay 2003; Snay 1999) allows processing software can determine SV positions U.S. users to reconcile control coordinates pub- at the moment of transmission, which form the lished in the past with current position measure- basis for the trilateration. ments that have moved due to plate motion, In broad strokes, GNSS ephemerides come including earthquakes. Of particular note to in two types: broadcast and precise. Broadcast heighting, a vertical equivalent, VTDP, has been ephemerides, as the name implies, are broad- created for the lower Mississippi valley and the cast by the SVs and read by GNSS receivers as northern Gulf Coast (Shinkle and Dokka 2004). they operate. Broadcast ephemerides are essen- Vertical crustal motion includes both tectonic tially highly educated, physics-based guesses crustal motion and anthropogenic factors, such about the future locations of the SVs based on as liquid extraction resulting in ground subsid- their past locations and velocities. Precise eph-
In fact, three known locations and distances do not uniquely determine a three-dimensional position; the problem is reduced to a selection between two solutions. One of these solutions will either be deep inside the Earth or in outer space and can be discarded by inspection for terrestrial GNSS positioning. See Awange and Grafarend (2005) for novel solutions of this problem based on Groebner bases.
Vol. 66, No. 3 173 emerides are produced by observing the SVs and sphere and the troposphere. Both of these atmo- deducing their positions after-the-fact. Needless spheric layers delay the signals, thus introducing to say, broadcast ephemerides are not as accu- timing/ranging errors. rate as precise ephemerides. The accuracy of the broadcast ephemerides is currently around one Ionosphere Delay,s to three meters (Seeber 2003). The ionosphere is a high-altitude (roughly The International GNSS Service (IGS) pro- 50 km to 1000 km above the Earth's surface) vides three types of precise ephemerides, which part of the atmosphere that is composed of differ by how much time elapses before they are charged particles that have been ionized by available (IGS 2005). The most accurate are the solar radiation. The ionosphere refracts radio "final ephemerides" which are updated weekly signals in a manner similar to how water in a with a latency of about 13 days and have accu- glass refracts light, such that a pencil appears racy reported to be better than 5 cm (Seeber to have a sharp bend in it. It happens that the 2003). The "rapid ephemerides" are updated ionosphere refracts radio-frequency electromag- daily with a 17-hour latency and an accuracy netic waves of different frequencies differently. around 5 cm. Ultra-rapid ephemerides are Consequently, it delays the two GPS broadcast updated four times daily with a latency of either frequencies, Ll and L2, differently. This differ- 3 hours (observed halo or none (predicted half) ence can be detected by dual-frequency receiv- with an accuracy around 25 cm. It can be shown ers and subsequently virtually eliminated by that the error introduced into computed posi- post-processing. For more details consult, for tions varies by baseline length as a function of example, Brunner and Walsh (1993), Hofmann- ephemeris accuracy: the longer the baseline, the Wellenhofet al. (1997), Leick (1995), and Seeber more accurate the ephemeris needs to be (Eckl (2003). Single-frequency receivers cannot detect et al. 2002, Seeber 2003, p. 305). For high-accu- the ionosphere delays, but differencing process- racy GPS heighting, final precise ephemerides ing on short baselines can cancel out most of the are required by NGS guidelines (Zilkoski et errol, leaving errors on the order of 1 to 2 ppm al. 1997). of the interstation distance (Seeber 2003). The NGS guidelines require dual-frequency receiv- Satellite Clock Errors ers for baselines greater than 10 kin, and they Although GNSS satellites have onboard atomic are the preferred type of GPS receiver for all time standards that are highly accurate and observations (Zilkoski et al. 1997). precise, they are not perfect. Like all clocks, According to Jakowski et al. (2005, p. 3071), atomic clocks drift and experience unpredict- "The space weather is defined as the set of all able jumps, albeit very small ones (Diddams et conditions -on the Sun, and in the solar wind, al. 2004; Flowers 2004). GPS time is a weighted magnetosphere, ionosphere and the thermo- average of the clocks in the controlling station sphere-that can influence the performance and on Earth and the GPS satellite clocks. Each SV reliability of space-borne and ground-based tech- clock is monitored for its offset from GPS time, nological systems and can endanger human life." and this time bias estimate is included with the Space weather can significantly influence the ephemerides, both broadcast and precise, to be propagation of the SV transmissions through the accounted for in the positioning software. ionosphere, resulting in a degradation of posi- tioning quality (ibid). Dual-frequency receivers are not able to eliminate the problems caused by GPS Signal Propagation Delay Errors severe space weather; hence observations should GNSS ranges are inferred by measuring a not be performed during severe ionospheric (biased) elapsed time from the satellite to the storms. The National Oceanic and Atmospheric receiver; it is biased due to an immeasurable Administration (NOAA) includes space weather time offset between GPS time and a receiver's reporting from its Space Environment Center, internal clock. This elapsed time interval is which is part of the National Weather Service scaled to be a distance by multiplying by the (http://Nww.sec.noaa.gov/). speed of light. Although the speed of light is constant in a vacuum, electromagnetic waves Troposphere Delays propagating through media can be delayed and The troposphere is that part of the atmosphere refracted. GNSS signals propagate through the in which weather (in the ordinary sense) occurs. Earth's atmosphere and are affected by the iono- Atmospheric density gradients of the tropo-
174 Surveying and Land Information Science sphere, like the ionosphere, refract GNSS radio that following their guidelines should result in waves. However, the tropospheric delays do not 2 cm - 5 cm ellipsoid height accuracy. The dif- depend upon the frequency of the electromag- ference is the length of the baselines. Marshall's netic waves. No hardware exists today that can study had baselines not shorter than 60 kin, but directly measure the delay created by the tro- NGS requires lines no longer than 10 km. This posphere, so its affect must be accounted for by is an important difference because the unmod- modeling the troposphere or by treating it as an eled tropospheric delay error is spatially auto unknown nuisance variable determined using correlated, meaning that the closer two stations least squares techniques. are, the more likely they are to "see" the same The errors associated with the troposphere tropospheric delay. If the delays were exactly the are considered the most problematic member of same, they would be canceled by post-process- the GNSS heighting error budget. According to ing differencing. To what degree they are not the Seeber (2003), "[tropospheric delay]... is one of same, they do not cancel. the reasons why the height component is much According to the current literature, measuring worse than the horizontal components in precise weather parameters is not very helpful. Marshall GPS positioning." According to Brunner and et al. (2001) found that, "For session lengths Walsh (1993), "Tropospheric delay errors mainly greater than two hours, we conclude that suffi- affect the accuracy of height differences. Today ciently precise NAD [neutral atmospheric delay] this must be considered the main limitation of modeling for geodetic activities may be achieved the attainable accuracy using GPS, which seems by coupling nuisance parameter estimation with to be around 2.5 centimeters for height differ- the relatively crude seasonal model." This means ences of baselines longer than about 50 kilome- that weather measurements were not needed ters." to achieve sufficiently precise error models. Marshall et al. (2001) performed a detailed Brunner and Walsh (1993) note that: study of the affect of tropospheric modeling suc- "In general, the tropospheric delay models cessfulness at addressing the tropospheric delay using meteorological ground observations on baselines from 62 km to 304 km in length. have produced rather poor, and in most Based on their experiments conducted using the cases worse, results compared with the NGS Continuously Operating Reference Stations results from the default model values that (CORS), they show significant reductions in replaced the actual observed meteorological height standard deviations by increasing session values. We would like to comment on duration from one to four hours, and that the this surprising finding. Taking accurate choice of the tropospheric model has a strong meteorological observations is a somewhat influence on the precision and accuracy of the difficult task, and frequently large resulting heights. Some of these models depend observation errors can occur. In addition, upon measured tropospheric parameters such the closeness of the ground and very local as atmospheric pressure, atmospheric tempera- micrometeorological conditions severely ture, and relative humidity, i.e., the quantities affect meteorological observations." that determine the static density of the atmo- These comments appear to support the conclu- sphere and its density gradient. Others rely on sions found by Marshall et al. (2001). Recently, Ray standard models of the atmosphere and are et al. (2005) noted succinctly: "To the central ques- parameterized by latitude and day of the year. tion, whether measured surface met data can be Another approach is to treat the tropospheric used to improve geodetic performance, we find no delay as another unknown parameter and esti- such utility." Nevertheless, NGS guidelines require mate it using statistics from the GPS observables. meteorological data to be collected (Zilkoski et al. Marshall et al. (2001) concluded that, "Session 1997). It has been shown (Marshall et al. 2001) that lengths shorter than two hours contain insuf- "Weather fronts may cause the GPS signal delay to ficient GPS data to estimate both heights and vaiy by greater than 3 centimeters over a 1-hour nuisance parameters, and hence more accurate period, potentially leading to ellipsoidal height weather information is needed to obtain more errors exceeding 9 cm." Surface met data are not precise heights for these shorter sessions." The collected for modeling purposes. Rather, they are models showed a large amount of variability useful for a posteriori error detection as they help to among each other and all of them displayed sig- spot the passing of a weather front through the sur- nificant individual variability-more than 5 cm. veying network, something that could possibly This fact would appear to contradict NGS claims go unnoticed by the ground crews.
175 Vol. 66, No. 3 175 The affect of the troposphere increases with zenith angle. For this reason (amongst others) Phase Center Variation (mm) NGS recommends a 15 degree minimum eleva- 30 tion mask (Zilkoski et al. 1997).
Multipath 20 One of the two GNSS observables is carrier phase: "carrier" refers to the unmodulated radio 10 signal broadcast by the SVs and "phase" refers to the total number of cycles of the carrier waves from its transmission to its reception, includ- 0 ing a partial wavelength at the end. In relative positioning, baselines between phase centers -10 are deduced by differencing phase observations from multiple SVs; see Hofmann-Wellenhof et al. (1997) among many others for more details. -20 Multipath is the situation where GNSS radio sig- 0 20 40 60 80 100 nals arrive at the receiver via more than one path. Elevation (degrees) This happens by the signal reflecting from some Figure IV.9. This image depicts the location of a GPS surface such as a chain link fence, a building, a receiver's phase center as a function of the elevation car, or the ground. According to Seeber (2003), angle of the incoming GPS radio signal. "Multipath influences on carrier phase observa- tions produce a phase shift that introduces a sig- nificant periodic bias of several centimeters into interfere with the GNSS radio signals, electro- the range observation... Their propagation into magnetic noise requires some attention, too. height errors may reach +/- 15 cm (Georgiadou and Kleusberg 1988)". Multipath also affects Antenna Phase Center Variation pseudo-range derived positions, introducing The electrical phase center of a GNSS receiver errors potentially on the order of meters. antenna is a point in space where the antenna detects Multipath can be reduced by antenna design, the radio signal broadcast from the satellites; it is the principally choke rings and ground planes, and point whose coordinates are being deterimined. That by elevation masks. Multipath is more likely to is to say, unless the position is reduced to the antenna occur at low elevation angles so, again, NGS rec- reference point (ARP) or a surveying marker, the lati- ommends a 15 degree minimum elevation mask tude, longitude, and ellipsoid height reported by the (Zilkoski et al. 1997). Ground planes are known GNSS post-processing package are those of the phase to reduce multipath, especially spurious signals center. Interestingly, the phase center is not on the arriving at the receiver from below, perhaps physical surface of the antenna; indeed, it is not on being reflected off the ground. Likewise, choke or in the hardware at all. It is above the antenna ring antennas mitigate multipath by attenuat- and, furthermore, it is not a single locatio'n (see ing reflected signals. Therefore, NGS requires Figure fV9). Although most modern antennas ground planes for GPS antennas and recom- are azimuthally symmetric electrically, local mends choke rings (Zilkoski et al. 1997). There environmental conditions can produce depen- are also software techniques for multipath reduc- dences on azimuth. Therefore, phase centers can tion (e.g., Seeber et al. 1997) that are available change with the zenith and azimuth angle of the in some processing packages and, sometimes, in incoming signal. Additionally, the phase center the receiver itself (Townsend and Fenton 1994). for LI is typically different than that for L2 (Mader 1999). Because the phase center is the Receiver Errors and Interference position being determined by the receives, as the satellites move, the phase centers move, which No instrument is perfect, and GNSS receivers are is an effect called phase center variation (PCV). no exception. The receivers themselves cannot As a phase centers moves, its coordinates change. determine positions exactly, but we know the If left uncorrected, phase center variations can error sources associated with the receiver hard- introduce as much as a decimeter of error into ware. Also, since the presence of electromagnetic the vertical coordinate. The NGS antenna cali- noise in the environment has the potential to bration program has produced models of phase
Surveying aud Laud InJormation Sejeuce 176 Surveying an,d Lan.d In•jbrnialion Science center variation that are available for download- Also, GNSS observation files often allow for ing at http://www.ngs.noaa.gov/ANTCAIJ (NGS marker offsets. Some CORS base station RINEX 2005). These models can be entered into the observation files have offsets that reflect the phase post-processing software, which will adjust for center-ARP separation, typically a negative number the effect. a few centimeters in magnitude. Surveyors will National Geodetic Survey publishes several coor- need to zero these offsets if their processing soft- dinates for its CORS base stations. Coordinates are ware assumes the control coordinates refer to the currently given in the ITRFO0 (epoch 1997.0) and ARP and computes the offsets automatically via the NAD 83 (CORS96) datums for both the ARP and antenna geometry database. If they are not zeroed, software will account for the distance from the the LI phase center. Coordinates for the ARP and the phase center to the ARP twice, introducing a sev- the phase center are different by several centime- eral-centimeter blunder into the vertical control ters, typically. For example, the NAD 83 ellipsoid coordinate. Such a blunder can be extremely dif- height for the DE6429 NRME COOP CORS LI ficult to find if the processing package does not phase center is 163.027 m, whereas the ellipsoid give a complete account (report) of how the vertical height of the ARP is 162.951 m, a difference of coordinate was determined. The NGS processing 7.6 cm. Surveyors clearly need to be very careful in software, PAGES, does report all the offsets that go choosing their control coordinates and know what into determining the spatial location of the phase their post-processing software does with those coor- centel, so the surveyor knows whether all the con- dinates. trol coordinates and offsets are consistent. Some packages may assume that the vertical Additionally, as CORS stations are increasingly coordinate refers to some particular place, typi- being used in local surveys, it is likely that a mixture cally the ARP or the phase center; others allow of antenna types will occur in a single survey. Any azi- the user to specify to which place the control coor- muthal PCV inconsistencies among the antennas will dinates refer. Surveyors should take care to pick not cancel in the differencing processing unless the coordinates that match the expectations of their same inconsistency occurs for all antennas. Therefore, software or they will introduce systematic vertical it is important to orient all antennas in the survey to errors by accounting for the phase center-ARP the North so that any residual azimuthal effects are separation incorrectly. Furthermore, some pack- canceled. CORS antennas are already oriented to ages have antenna geometry databases to allow the North, which means that surveyors need only be concerned about their own antennas. the software to compute the distance from the ARP to the phase center. Surveyors should check Electromagnetic interference and signal attenuation the, values in such a database to verify they are The radio signals currently broadcast by the GPS correct by comparing with designs provided by satellites are relatively low power, around 50 watts. manufacturers or by information on the afore- Although GNSS signals occupy a protected fre- mentioned NGS antenna calibration website. quency band, nearby sources of broadband electro-
Error Remedy Use final precise ephemerides; Orbit errors and clock errors Double differencing of phase observations eliminates orbit and clock errors receivers; Ionospheric delay Use dual-frequency Can be reduced on short baselines by differencing phase observables Modeled or determined in post processing; Tropospheric delay Longer observation times yield better results; Can be reduced on short baselines by differencing phase observables Avoid multipath-prone locations; Multipath Use a ground plane or choke ring antenna Use antenna calibration models; Phase center variation Orient antennas to North;Check antenna offsets and antenna geometry databases to ensure consistency with control coordinates Electromagnetic noise L5 receivers; Electromagnetic__noise_Avoid problem sites if possible Table IV.1 Summary of error sources and recommended remedies.
Vol. 66, No. 3 177 magnetic noise can overwhelm them (Johannessen control point with respect to the geodetic 1997; Butsch 2002), thus causing decreased signal datum at the 95 percent confidence level. to noise ratios, increased difficulty or prevention of For NSRS network accuracy classification, GNSS signal acquisition, and loss of signal track- the datum is considered to be best ing (Seeber 2003, p. 320). Power transmission lines, supported by NGS. By this definition, the television and radio stations, and radar installations local and network accuracy values at CORS are possible examples of such noise sources. To help sites are considered to be infinitesimal, i.e., address this problem, the GPS modernization pro- to approach zero." (ibid) gram includes a third, higher-power frequency (L5) This section presents an overview of these which is expected to reduce this problem (Hatch guidelines and of currently available U.S. geoid et al. 2000). Unfortunately, new receivers will prob- models and how local geoid modeling is used in ably have to be purchased when enough satellites practice. have been placed in orbit to make using L5 prac- tical and to take advantage of its potential. In the Three Rules, Four Requirements, mean time, surveyors should occupy sites that are Five Procedures not directly below electromagnetic noise sources, if possible. Overhead vegetation that comes between The National Geodetic Survey created a series the receiver and the SVs can also attenuate or block of rules, requirements and procedures to derive the SV transmissions, causing the same problems as orthometric heights using GPS (Zilkoski et with decreased signal to noise ratios (Spilker 1996; al.199 7 ; Zilkoski et al.2000). We now review this Meyer et al. 2002). material.
Error Summary Three Rules Rule 1. Follow NGS' guidelines to establish GPS- Table IV 1 provides a summary of error sources derived ellipsoid heights (Zilkoski et al. 1997) and recommended remedies. when performing a GPS survey; Rule 2. Use NGS's latest National Geoid Model, i.e., GEOID03 (Roman et al. 2004), when com- NGS Guidelines for GPS puting GPS-derived orthometric heights; and Ellipsoid and Orthometric Rule 3. Use the latest National Vertical Datum, i.e., NAVD 88 (Zilkoski et al. 1992), height values Heighting to control the project's adjusted heights. NGS has guidelines and suggested practices that, We note that GEOID03 is a hybrid geoid if followed exactly, are intended to achieve ellip- model for the conterminous U.S. and, as such, soid / orthometric height network accuracies of has been custom-crafted to fit properly with the 5 cm (95 percent confidence level) and ellipsoid NAVD 88 level surface (Milbert 1991; Milbert / orthometric height local accuracies of 2 cm and and Smith 1996; Roman et al. 2004; Smith and 5 cm (95 percent) (Zilkoski et al.1997; Zilkoski Milbert 1999; Smith and Roman 2000; Smith et al.2000). The local accuracy of a control and Roman 2001; Smith 1998). Inferior results point is defined as: would likely result from using a geoid model that "... a value expressed in cm that represents had not been so fitted. There are many studies the uncertainty in the coordinates of the on how to apply local geoid models for survey- control point relative to the coordinates ing purposes; for example see Amod and Merry of the other directly connected, adjacent (2002), Corchete et al. (2005), Featherstone and control points at the 95 percent confidence Olliver (2001), Forsberg et al. (2002), Fotopoulos level. The reported local accuracy is an (2005), Luo et al. (2005), Pellinen (1962), Soycan approximate average of the individual and Soycan (2003), and Tranes et al. in press). local accuracy values between this control Some of these are studies were across very lim- point and other observed control points ited areas (Soycan and Soycan 2003; Tranes et al. used to establish the coordinates of the in press) in which the geoid could be adequately control point" (Zilkoski et al. 1997). modeled with simple polynomial models. The The network accuracy of a control point is others are local improvements over global defined as: models for regions as large as Iberia (Corchete "... a value expressed in cm that represents et al. 2005), Hong Kong (Luo and Chen 2002; the uncertainty in the coordinates of the Luo et al. 2005; Zhan-ji and Yong-qi 2001), the
178 Surveying and Land Information Science Caribbean Sea (Smith and Small 1999), Taiwan uplift, such as along the U.S. Gulf Coast or in (Hwang and Hsiao 2003), and the British Isles California, for example. (Featherstone and Olliver 2001; Forsberg et al. GNSS heighting can take advantage of four- 2002; Iliffe et al. 2000; Iliffe et al. 2003), where a dimensional markers, where they exist. The simple polynomial model will not suffice. These National Geodetic Survey has conducted "GPS- approaches depend upon absolute and relative on-bench-mark" field surveys as part of its height gravity measurements. modernization program, thereby establishing Although it can be shown that completely rig- many four-dimensional markers: geodetic lati- orous orthometric heighting also depends on tude, longitude, ellipsoid height, and Helmert such data (Tenzer et al. 2005), collecting them is orthometric height. For example, according impractical for most surveyors. Fortunately, U.S. to the data sheet for Y88 (PID LX3030) in surveyors need not resort to such efforts because Connecticut, Y88 is vertical First-Order, Class II; GEOID03 has been shown to be accurate at the Horizontal Order A and ellipsoid Fourth Order, 2 cm (95 percent confidence) level for the conti- Class I. Four-dimensional bench marks are very nental U.S (Roman et al. 2004). Although newer useful for GNSS adjustment software packages versions are planned to be released in the future, because they eliminate the need to estimate any GEOID03 is sufficient for GPS orthometric of the four coordinates (usually either ellipsoid heighting at the 2 cm and 5 cm accuracy levels or orthometric height) with a model. as put forth by NGS, thus eliminating the need Occupying bench marks at the bases and for U.S. surveyors to create their own gravimetric summits of mountains helps overcome error geoid models. sources in geoid models typically caused by a lack of gravity measurements at such places Four Requirements (Control) (Featherstone and Alexander 1996; Allister and Requirement 1 Featherstone 2001; Dennis and Featherstone 2002; Featherstone and Kirby 2000; Goos et al. GPS-occupy stations with valid NAVD 88 ortho- 2003; Kirby and Featherstone 2001; Zhang and metric heights; stations should be evenly distrib- Featherstone 2004). uted throughout (the) project. Requirement 2 Five Procedures For project areas less than 20 km on a side, sur- 1. Perform a 3-D minimum constraint least round project with valid NAVD 88 bench marks, squares adjustment of the GPS survey project, i.e., minimum number of stations is four; one in i.e., constrain one latitude, one longitude, each corner of the project. and one orthometric height value. Requirement 3 2. Using the results from the adjustment in procedure 1, detect and remove all data For project areas greater than 20 km on a side, outliers. keep distance between valid GPS-occupied Repeat procedures 1 and 2 until all outliers NAVD 88 bench marks to less than 20 km. have been removed. Requirement 4 3. Compute differences between the set of For projects located in mountainous regions, GPS-derived orthometric heights from the occupy valid bench marks at the base and minimum constraint adjustment (using the summit of mountains, even if distance is less latest national geoid model, i.e., GEOID03) than 20 kmi. from procedure 2 above and published NGS guidelines repeatedly stress the need to tie NAVD 88 bench marks. to valid NAVD 88 bench marks, although (unfor- 4. Using the result from procedure 3 above, tunately) the criteria for validity are not dis- determine which bench marks have valid cussed. Obviously, bench marks without NAVD NAVD 88 height values. This is the most 88 heights are not valid. This disqualifies NGVD important step in the process. Determining 29 heights or bench marks tied to tide gauges. A which bench marks have valid heights is valid bench mark is one that has been tied into critical to computing accurate GPS-derived NAVD 88 and has not been disturbed either by orthometric heights. natural and human forces in such a way as to 5. Using the results from procedure 4 above, render its published NAVD 88 height inconsis- perform a constrained adjustment fixing tent with the remainder of the network. Caution one latitude and one longitude value and all should be used in areas of ground subsidence or valid NAVD 88 height values.
179 Vol. 66, No. 3 179 Correctness is ascertained by repeatability in them, the geoid model must be referred to the GPS heighting. same-heighting system as the control. Currently, in the United States, GEOID03 is the correct model to use, although surveys over very small Discussion and Summary areas can also benefit from polynomial-based geoid models derived from GPS-on-bench-mark GNSS surveying is becoming more commonly observations. used for vertical control. GNSS heighting can The primary error factor is the difficulty to be attractive from a cost perspective because it measure and model wet zenith delay. In arid offers the possibility of reducing or eliminating regions the wet zenith delay is very small, and the need, for. leveling runs and trig-heighting, short occupations (even as short as 30 minutes) which are very costly. Although GNSS height- have been used successfully. In humid regions, ing is not a panacea,. the prospect of establish- this is seldom true. It has also been shown by ing high-quality vertical control in a remote site without running levels to distant bench marks is several investigators that collecting meteoro- logical measurements very attractive. for the purpose of tropo- Unfortunately, traditional training in level- spheric delay modeling is ineffectual. However, ing does not adequately prepare a surveyor these measurements should be collected for use to perform GNSS heighting because the two as evidence regarding which baselines need to techniques are nearly completely different. For be re-observed. It has been shown by Marshall example, different instruments are used for each et al. (2001) that "Weather fronts may cause technique; the concept of a leveling route does the GPS signal delay to vary by greater than 3 not exist in GNSS heighting; they have different centimeters over a 1-hour period, potentially error budgets; they reference different vertical leading to ellipsoidal height errors exceeding 9 datums; and they are even based on different cm." Therefore, weather observations are useful conceptualizations of height itself. not so much for tropospheric modeling as they This series presented concepts such as refer- are for detecting that a weather front may have ence ellipsoids, vertical datums, mean sea level, passed through unnoticed. The key for reducing level surfaces and the geoid, gravity and poten- tropospheric delay errors to acceptable levels tial, and orthometric vs. geometric vs. ellipsoid is to keep baselines very short, less than 10 km heights. From these concepts come applications in length. By doing so the delay at both ends is such as why some reference ellipsoids are suit- nearly the same, and it is subsequently removed able as vertical datums while others are not; what by post-processing differencing. The accuracy of is a GNSS receiver really doing when used for GNSS heighting on long baselines is currently heights and how to integrate its measurements limited by wet zenith delay errors. with those of a spirit level, and what is an ortho- The importance of antenna modeling cannot metric correction. Finally, this last paper pre- be overstated, as well. Ellipsoid height errors as sented practical aspects of GNSS heighting based much as 10 cm for certain antennas can be intro- on suggested practices given by the National duced simply by failing to include phase center Geodetic Survey in light of its height modern- variation correction models in the processing. ization program. This paper considered network It is critical to check the database of the post- design and control, observation strategies, the processing software to ensure that the antenna role and application of geoid models, and the geometry is entered correctly and that a PCV integration of leveled heights with GNSS-deter- model is used. Similarly, when using RINEX mined heights. observations, make sure that the offsets that may Although there are many issues affecting come with those data have correct signs for the GNSS-determined or thometric heights, we conventions of your software and -that they ulti- believe the key points are these. GNSS height- mately refer to your control coordinates, which ing depends on using consistent control, control can be either ARP or phase center. Any mistakes from a single, modern datum such as NAVD 88. here will introduce a several-centimeter bias in For example, mixing heights in NGVD 29 and all baselines with an endpoint at the receiver. those referenced to a mean sea level station with NAVD 88 heights would violate this rule. Since ACKNOWLEDGEMENTS orthometric heights are derived from ellipsoid The authors would like to acknowledge the care- heights by subtracting the geoid height from ful and constructive reviews of this series by Dr.
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