DERIVATION OF ORTHOMETRIC HEIGHTS FROM GPS MEASURED HEIGHTS USING GEOMETRIC TECHNIQUE AND EGM 96 MODEL

Opaluwa Y. D. and Adejare Q. A. Department of Surveying & Geoinformatics, School of Environmental Technology, Federal University of Technology, Minna, Nigeria. E-Mail: geopaldy_xy@yahoo.com, Phone: +2348052292076 E-Mail: quadriadejare@yahoo.com, Phone: +2348032156620 Abstract As a result of wide spread use of satellite based positioning techniques, especially Global Positioning System (GPS), a greater attention has been focused on precise determination of geoid models with an aim to replace the classical leveling with Global Navigation Satellite System (GNSS) measurements. In this study, geometric technique of deriving orthometric height from GPS survey along a profile and the use of EGM 96 geoid model for deriving orthometric height from GPS data (using GNSS solution software) are discussed. The main focus of the research is to critically examine the potentials of these methods with a view to establishing the optimum technique as an alternative to classical differential levelling. From the results obtained, the standard errors are 1.453m and 1.450m for EGM 96 model and the geometrical approach respectively. From the graphical representation of the residuals from the two methods, it was observed that the two curves suddenly became sinusoidal from station 9 (corresponding to SB08 in the tables). This similarity pattern of the residuals makes it difficult to draw a conclusive judgment between the two methods examined; however, from the standard errors, it could be inferred that the geometrical technique gave a better result over EGM 96 model.


Introduction
The classical Vertical control is composed of several hierarchical networks which follow the principle of "working from whole to parts".The primary Vertical Control Network contains loops of first order precise leveling of some hundreds of kilometers in length.The accuracy of precise leveling (high precision leveling) should be at the millimeter-level per kilometer (Bomford, 1980).The other subnets in the network are densifications of the primary one, according to the needs.It is worthy to note that it is reasonable to establish a third-order network in the densely populated area.However, the difficulties involved in precise leveling are well-known; for example, Eriksson et al, (2002) observed that even with the most advanced technology of motorized leveling, it took some 25 years to accomplish the first-order network in Sweden.Due to such difficulties, it is actually impossible to get heights for lower-order networks with absolute accuracy (relative to the higher-order) better than 5-10 cm (Steinberg and Even-Tzur, 2005).
Because there was no alternative to precise leveling, as a tool, to achieve the objectives of primary leveling networks prior to the GPS era, it was naturally referred to as vertical control.However, current researches have shown that GNSS measurements are quite more effective for monitoring vertical tectonic changes over a wide area.Consequently, the ideas of Vertical Ellipsoidal Control, or 3-D Geodetic Control, based PDF Creator -PDF4Free v3.0 http://www.pdf4free.comon Permanent GNSS Networks (called Continuous Operating Reference Stations -CORS) are becoming popular (Steinberg and Papo 1996, 1998, 1999, Meyer et al 2004, Wonnacott 2005, etc).The advantage of GNSS networks over precise leveling is quite obvious.The major question, therefore, is whether Ellipsoidal Height Control Networks can replace the Orthometric ones.However, Steinberg and Even-Tzur (2005) observed that ellipsoidal Control is the imminent replacement for the orthometric control.For instance the Survey of Israel is already moving towards 3D Geodetic Control, based on the Israeli Permanent GPS Network.Generally speaking, Steinberg and Even-Tzur (2005) further observed that Vertical Ellipsoidal (Geometric) Control should be based on Permanent GNSS Network as one part of the 3-D Geodetic Control.The Permanent GNSS Network is the first order of the 3-D Control; by its nature (operating cost), Permanent Stations are quite far from each other.Due to the dependency of GPS accuracy on the length of the baselines (which can be compensated by longer measuring times), especially in the vertical direction, it is recommended to densify the first-order control by more orders, according to actual needs.This densification should be accomplished, of course, by GPS measurements.In Israel, it was decided that the accuracy of the Second-Order network will be 1cm (2ó), and that of the third-order, 2cm, relative to the nominal heights of the Permanent GPS stations.
In an effort towards realizing this goal, Steinberg and Even-Tzur, (2005) carried out GPS survey over eight points in Israel and suggested that any available geoid model should be accepted as the official geoid model no matter the accuracy.The results of the research indicated that accuracy of 25ppm was realized for benchmarks of 1km apart in the fourth order leveling network which is the same as the accuracy of the existing height difference of the benchmarks.
The biggest constraint in using this alternative in Nigeria is the lack of an existing official geoid model as well as primary GPS (CORS) stations.Therefore, representing geoid heights as mathematically formulated surface and calculating the geoid heights in new measured points according to GPS technique constitutes the idea in this study.

Height Relationship
The procedure of geodetic leveling provides a height that is commonly known as a height above Mean Sea Level.The process gives level difference between two consecutive benchmarks.The orthometric heights so derived reflect local variation in gravity as well as topographic gradients.The reference datum for orthometric heights, ellipsoid and the geoid, is approximated by Means Sea Level (MSL).Basically one has to establish a relationship between the Orthometric height obtained from geodetic leveling and GPS derived ellipsoidal heights using a common reference datum.The technique is often called geometrical approach for "height basis" estimation (Seker and Yildirim, 2002).The basic equation which relates the orthometric and ellipsoidal height is: Where H = Orthometric height, measured along curved plumbline.h = Ellipsoidal height measured along the ellipsoidal normal N = Geoid height, the separation between geoid and ellipsoid.Theoretically, since the ellipsoidal height and orthometric height are measured along the normal to ellipsoid and along the direction of the plumbline respectively, the PDF Creator -PDF4Free v3.0 http://www.pdf4free.comrelationship defined in equation (1.0) is only an approximation but serve the purpose for most of the engineering application.

Statement of Problem
The determination of orthometric heights from spirit levelling is known to be time consuming and cumbersome, especially in a large and very rough terrain.In fact, apart from the complexity in its field measurement, a lot of time and energy is spent in the stage of data reduction and adjustment thereby making it highly capital intensive to establish a country-wide high-resolution levelling network.Consequently, the availability of this data in most developing countries and particularly in Nigeria is inadequate.Also, there is lack of gravity data required to properly adjust the observed heights to yield the orthometric heights.However, advances in space technology has enhanced simultaneous determination of 3-D positioning referenced to the global geocentric ellipsoid (WGS 84 ellipsoid), Unfortunately the geoid for Nigeria has not been accurately determined and the geopotential model geoid for Nigeria determined in 2006 to 1m accuracy is yet to officially receive wide acceptability for application.Therefore, there is a serious problem of data transformation/conversion between the local and global reference datums, hence, a serious limitation to GPS usage in Nigeria.
In order to proffer solution to these problems, this research critically examined the potentials of geometric technique and EGM 96 geoid model for deriving orthometric heights from GPS, with a view to establishing the optimum technique to complement classical differential levelling.

Aim and Objectives of the Research
The aim of this research is to compare geometrical technique of deriving orthometric height from GNSS survey and the use of EGM96 model for processing orthometric height from GNSS survey.
The objectives of this research therefore are: (i).To acquire 3-D positional data of some benchmarks with the GPS along a level profile (ii).To compute the relative geometric geoid along the level profile by combining the ellipsoidal heights from GPS with the existing orthometric heights of observed benchmarks.(iii) To compute orthometric height along the profile using linear interpolation technique and EGM 96 geoid model respectively.
(iv) To compare the computed orthometric height in each case with the provisional orthometric height from differential levelling of the observed points.

Study Area
The study area lies between the Standard Bench Mark (SBM) at Limawa primary school in Kpakungu and the Standard Bench Mark (SBM) at Garatu Primary School in Bosso Local Government Area of Niger State.It covers a total distance of 18km and lies within latitude (ϕ ) 9 0 29' 20'' and 09 0 36' 14.14'' North of the equator and longitude(ë)  6 0 26' 24.82'' and 6 0 31' 46.72''East of the meridian.

GPS dericed orthometric heoght
One of the primary applications of a gravimetric geoid model is for converting GPSderived ellipsoid heights to orthometric height on the local vertical datum (Opaluwa, 2008).
The process of deriving elevations on a local height datum from GPS measurements has been well documented in Gilliland (1986); Kearsley (1988); Collier & Croft (1997); Featherstone et al. (1998).Since the main applications of a geoid model is to convert GPS-derived ellipsoid heights to gravity related elevations above a local height datum, central to this problem is the knowledge of geoid-ellipsoid separation relative to the GPS reference ellipsoid (Featherstone 1998, p.274).GPS-derived ellipsoidal heights can be converted to approximate height datum elevations in either an absolute or relative sense, depending on observation technique.The absolute case is a situation where an ellipsoidal height can be converted to an approximate orthometric height by algebraically subtracting the geoid-ellipsoid separation at a discrete point using equation (2.0).H = h -N (2.0 N is the geoid-ellipsoid separation (also known as geoid height) measured along the ellipsoid normal to the geoid.If the geoid is above the ellipsoid, N is positive.If the geoid is below the ellipsoid, N is negative.It is important to note that the ellipsoid height (h) and the geoid height (N) must refer to the same reference ellipsoid for the relationship to hold.Featherstone et al (1998, p.279) suggested that as the most accurate GPS applications are performed in the relative mode, equation (2.0) is not very practical for GPS height conversion.Rather, for the majority of surveying applications equation (2.0) can be rearranged to accommodate the relative situation, where an elevation is transferred from a known point, A, to an unknown point, B, via the following relationship Featherstone et al (1998): HB = HA + (hB -hA) -(NB -NA) (3.0)According to Kearsley (1988) and Featherstone et al. (1998), equation 3.0 can be reduced to: ÄHAB = ÄhAB -ÄNAB (4.0)where Ä denotes 'change in'.
Nevertheless, Sideris et al (1992) asserted that the determination of orthometric heights by traditional techniques, such as spirit levelling, is a difficult task, moreover levelling over areas with rough terrain is very strenuous and time consuming.On the other hand the combined use of GPS and geoid heights presents an alternative potential to the classic geometric levelling.Detail research on Geoid and GPS/Levelling differences can be found in Forsberg and Madsen (1990), Mainville et al. (1992), Kearsley et al. (1993), Featherstone and Kirby (1998), Erol and Celik (2004) and Fotopoulos et al. (1999b) e.t.c.

Strategy for GPS height survey
The accuracy specifications for GPS survey indicate in part that the GPS survey techniques and observables that must be used (Featherstone et al, 1998).They further affirmed that the accuracy of GPS techniques is near-proportional to the cost of GPS equipment and survey logistics involved.Therefore, the three main classes of GPS survey techniques and approaches to be used to model the geoid are as shown in table 1.0; while table 1.1 shows the list of the approximate accuracy of GPS survey mode.

Methodology Geometrical Technique
Featherstone ( 2004) noted that the standard approach for gravimetric geoid model validation is by comparison with GPS and levelling data observed at co-located points.Therefore, from the fundamental relationship between geoidal height, ellipsoidal height and orthometric height as shown in figure 1.0, discrete empirical geoid heights can be computed at each co-located point by re-arranging equation (1.0) to form: N= h -H (5.0) while relative empirical geoid height differences can be computed by re-arranging equations 1.0 and 3.0 to give: NB = NA + (hB -hA) -(HB -HA) (6.0)This can however, be reduced to: ÄNAB = ÄhAB -ÄHAB (7.0)As this is the most accurate GPS survey method, the most accurate geoid modeling method should be used.The options include a gravimetric geoid alone, geometrical interpolation alone or combined method.It is strongly recommended that each is tested to determine optimal approach in each particular survey area.The preceding calculations result is an empirical geoid model that can be used in comparisons with geoid heights interpolated from gravimetrically computed geoid models, subject to the errors in the GPS and levelling data (McDonald, 2004).If the geoid is assumed to be approximated by a flat surface, which is usually sufficient over small areas, linear interpolation can be used to estimate the geoid-ellipsoid separation (Featherston et al, 1998).Using two benchmarks which have both been occupied with GPS, the ellipsoidal height at an intermediate station X can be transformed to an orthometric height using (Featherstone et al, 1998).Hi = HA + ÄhAi -(lAi / SAB) * ÄNAB (8.0) Where; SAB = separation (length) of benchmarks A and B lAi = distance of the desired point X from point A.

Geopotential Geoid Model
PDF Creator -PDF4Free v3.0 http://www.pdf4free.com The geopotential geoid model as one of the global geoid models, represents the long wavelength part of the gravity field and is obtained from global geopotential solutions which are given as a set of spherical harmonic coefficients (Opaluwa, 2008).Different datasets are used to determine these coefficients, ranging from satellite observations (which give the so-called satellite only solutions) to models which incorporate satellite altimetry and surface gravity data, thus usually containing more coefficients (Sideris et al, 1992).The expression for computing geoid undulation (N) from such set of spherical harmonic coefficients is given by Heiskanen and Moritz (1967) as; = fully normalized associated Legendre functions, R = mean radius of the earth λ φ, = geodetic latitude and longitude.
Therefore, using the EGM96 geopotential coefficients, the geopotential model geoid undulation (N GM ) at any point on the earth's surface can be computed using equation (9.0).This concept of geopontential model geoid has been programmed into GNSS solution software as a default vertical datum therefore, given the 3-D coordinate of any point using DGPS receivers.It automatically transforms the ellipsoidal height to approximate orthometric height during post processing.

Procedure
For the purpose of this study, a level route bounded by two standard benchmarks (SBM Kpakungu and SBM Garatu) was identified along Minna-Bida road in Niger State.The route span for a distance of 18km with sixteen intermediate benchmarks and a primary cadastral points which were observed with a single frequency differential GPS using the cadastral point (CSN 128P) as the base for the observation.The stability of CSN 128P was first confirmed by observing on it using L40 as the base station.Each of the point including the SBMs was occupied for a minimum of 20 minutes in static survey mode.
As a precaution, the value for the epoch rate in a static survey must be the same for all receivers during the survey; this rate was set to be 5sec for this project, this was done to minimize the number of observations and thus the data storage requirements.All the receivers were connected to controllers that have internal memories as well as memory card (external device) for storing the observed data.However, the orthometric heights of the two Standard Bench Marks (SBMs) were obtained from the Office of the Surveyor General of the Federation, Minna Area office, the orthometric heights of the intermediate points were obtained from precise leveling exercise conducted by the Department of Surveying and Geoinformatics, FUT Minna in 2008; while the 3-D coordinate of the base (CSN 128P) was sourced from Niger State Surveyor General's Office.

Data Processing
After GPS observation, the observed data were post processed using GNSS solution software in two phases.The first phase involved the post processing of acquired data to the reference ellipsoid without using any vertical datum.However, the second phase of data processing involved the selection of the default vertical datum (EGM96 geoid model).
The GNSS solution computes baseline vectors as changes in X, Y, Z between the base station and the rover stations.If base station (A) has known coordinates, then the coordinates of rover stations (B) can be computed according to Wolf and Ghilani (2006) as: Where (XA,YA,ZA) are the geocentric coordinates at the base station A, (XB,YB,ZB) are the unknown station B, while X, Y, Z are the computed baseline vector components.These yields the ellipsoidal 3-D coordinate of all the occupied stations.The reduced level of the sixteen intermediate benchmarks as well as the primary traverse point (CSN128P) were adjusted using least square technique.
The processed ellipsoidal height from above was combined with the orthometric heights of the two SBMs (obtained from the Office of the Surveyor General of the Federation) to derive the relative geoid undulation along the profile using equation (7.0).Then, equation (8.0) was used to derive the approximate orthometric height of all the intermediate points by the combination of the derived relative geoid undulation with their respective ellipsoidal height.Since observation commenced from SBM Kpakungu, the computation originated from there.In order to derive the height of the observed points directly from the GPS data, the observed vectors were re-processed by choosing EGM 96 Geoid model as the height datum.

Numerical Evaluations
In the numerical investigations, we closely examined the variations in the two sets of approximate orthometric heights from GPS by comparing each with the height of the respective points from classical (differential) leveling.The heights as obtained from the various approaches discussed in this research are as shown in table 1.2 to 1.6, while table 1.7 shows their differences from classical leveling.The standard error of residuals was computed using SPSS 15.0 for windows; this is as shown in table 1.8 (the residual statistics).

Analysis of Results
From the numerical evaluations and the tables of results, it could be inferred from the residuals in table 1.7 and the residual statistics in table 1.8 that the maximum and minimum residuals from EGM 96 model are 12.575m and -12.02m respectively, while the geometrical interpolation method gave maximum and minimum residual values of 14.720m and -8.207m respectively.However, the standard errors of residuals for the two cases are 1.453m and 1.450m respectively for EGM 96 model and the geometrical approach.Also, the pearson correlation statistics gave a value of 95, which is an indication that the two sets of orthometric height derived are significantly correlated at 0.05 significant level.The graphical representation of the two residuals is as shown in figure 3.0.Furthermore, a close look at table 1.7 indicates that the geometric technique gave a zero residual for SBM GA which is one of the higher order height controls, while very large residuals values were noticed at points SB11, SB12, SB14 and CSN128P with fairly large ones at SB08, SB13 and SB15.Similarly for EGM96 model, very large residuals were observed at SB11, SB13, SB14 and CSN128P with fairly large ones at SB09, SB10, SB 12, SB16 and SBM GA.These can also be clearly discerned from the graph in figure 3.0 as the two curves suddenly became sinusoidal from station 9 (corresponding to SB08 in the tables).This similarity pattern of the residuals further confirms the results of the correlation statics above and this makes it difficult to draw a conclusive judgment between the two methods.However, from the standard error of residuals (table 1.8), it could be inferred that the geometrical method gave a better result over EGM 96 model.

Conclusion
Derivation of orthometric height from GPS Survey along a level profile has been discussed.The possibility of adopting geometrical interpolation procedure for profile GPS survey in relative static mode and the use of EGM 96 model as a reference datum for height in GPS data post processing was also examined.The two techniques were evaluated with a view to identifying the preferred approach for GPS leveling in the study area.From the performance evaluation of the two methods, arriving at a conclusive judgment was difficult because they are significantly correlated having almost equal standard errors of residuals (about 1.5m).However, the technique whose standard error was slightly less (1.450m) was seen to have an edge over EGM96 model with standard error of (1.453m).Therefore, in view of our observation of the results obtained, it will be worthwhile to further examine the integrity of the EGM96 model as height datum for converting ellipsoidal height from GPS to height above the geoid and the use of geometrical technique, considering a network situation rather than profile.This is expected to yield a better result that will lead to a conclusive remark.

Fig. 2
Fig. 2 Map of Niger State Showing the Study area.