Journal of African Earth Sciences 60 (2011) 213–221

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Journal of African Earth Sciences

journal homepage: www.elsevier .com/locate / ja f rearsc i

A new gravimetric geoid model for Sudan using the KTH method

Ahmed Abdalla a,⇑, Derek Fairhead b

a National School of Surveying, Faculty of Sciences, University of Otago, 310 Castle Street, Dunedin, New Zealandb GETECH Group plc., Elmete Hall, Elmete Lane, Roundhay, Leeds LS8 2LJ, UK

a r t i c l e i n f o

Article history:Received 16 March 2010Received in revised form 10 November 2010Accepted 28 February 2011Available online 12 March 2011

Keywords:Sudanese geoidAdditive correctionTerrestrial gravity dataGeopotential modelLeast-squares modificationStokes formulaKTH methodKTH-SDG08

1464-343X/$ - see front matter � 2011 Elsevier Ltd.doi:10.1016/j.jafrearsci.2011.02.012

⇑ Corresponding author. Tel.: +64 226003753; fax:E-mail addresses: abdah305@student.otago.ac.nz, a

la), jdf@getech.com (D. Fairhead).

a b s t r a c t

Sudan is characterized by vast area with flat terrains in most of the country’s regions and therefore theexistence of a high-resolution geoid model is considered very important especially with the widespreadof the GPS technology in the country. A few studies were previously conducted to compute a geoidmodel for Sudan, the current study is the second of its kind to compute the Sudanese geoid model bymeans of gravimetric method. In this study, a new gravimetric geoid model (KTH-SDG08) is computedfor Sudan, we apply the method developed at the Royal Institute of Technology (KTH) Stockholm-Sweden.The method utilizes the least-squares modification (LSM) of Stokes formula providing three stochasticsolutions (biased, unbiased and optimum). The three solutions are quite similar however in this studywe select the optimum solution which provides the best agreement with GPS-levelling data. The modi-fied Stokes formula combines the regional terrestrial gravity data together with the long-wavelengthgravity information from a global gravitational model (GGM). We use two sets of regional gravity data:the first set is provided by the GETECH, the points of this set cover discretely some parts of Sudan whilethe rest of the country’s area are still not covered, the second set was provided by BGI contains a fewpoints scattered over the neighbouring countries. Both datasets (GETECH and BGI) are unified togetherinto one dataset and evaluated by Cross validation technique. Due to the lack of gravity observationswe construct our final grid for geoid computation with spatial resolution of 5 � 5 arc-min. The long-wavelength contribution is donated by two geopotential models, EIGEN-GRACE02S satellite-only modelis employed in the modified Stokes formula whereas the EIGEN-GL04C combined model is used to enrichthe local gravity coverage over the areas with missing data. The Digital Elevation Model (DEM), SRTMgenerated by NASA and the National Imagery and Mapping Agency (NIMA) is used to compute topogra-phy effects on the geoid. Four additive corrections are computed over the entire target area and applied tothe approximate geoid heights obtaining the final geoid solution. The new Sudanese gravimetric geoidmodel KTH-SDG08 is computed on a 5 � 5 arc-min geographical grid over the computation area boundedby the parallels of 4 and 23 arc-deg northern latitude, and the meridians of 22 and 38 arc-deg of easternlongitude. The gravimetric geoid is validated using GPS-levelling information at 19 points distributedover the whole country. The results show that the standard deviation (STD) of differences between thegravimetric and geometric geoid heights at 19 GPS-levelling points is about 0.3 m.

� 2011 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2142. Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

2.1. Terrestrial gravity data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2152.2. Digital Elevation Model (DEM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2162.3. GPS-levelling data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2162.4. Global gravitational model (GGM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

2.4.1. EIGEN-GRACE02S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2162.4.2. EIGEN-GL04C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

3. KTH method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

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+64 3 4797586.hmedmaa@kth.se (A. Abdal-

214 A. Abdalla, D. Fairhead / Journal of African Earth Sciences 60 (2011) 213–221

3.1. The additive corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

4. The new gravimetric geoid model (KTH-SDG08) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2185. The geoid validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

5.1. Verification of geoid in absolute sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2205.2. Verification of geoid in relative sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

6. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

1. Introduction

The geoid is one of the most fundamental concepts in geodesy,it is defined as an equipotential surface that closely coincides withthe mean sea level (MSL) and extends below the continents. Thegeoid surface is much smoother than the natural Earth surface de-spite of its global undulations (changes). In Contrast, it is very closeto an ellipsoid of revolution, but more irregular, therefore it isapproximated by the ellipsoid. The geoid height or the geoidalundulation N is described by the separation of the geoid from theellipsoid of revolution. However due to the spatial irregularitiesof the geoid surface, it cannot be described by a simple mathemat-ical function. High-resolution geoid model is valuable to geodesy,surveying, geophysics and several geosciences, because it repre-sents datum to height differences and gravity potential field. More-over, the geoid is very important for connection between localdatums and the global datum, for purposes of positioning, level-ling, inertial navigation system and geodynamics.

On the other hand, the impact of wide and rapid use of the Glo-bal Positioning System (GPS) has revolutionized the fields of sur-veying, mapping, navigation, and Geographic InformationSystems (GIS) and replaced the time-consuming traditional ap-proaches. In particular, GPS offers a capability of performing geo-detic measurements with efficient accuracy which formerlyrequires ideal circumstances and inevitable conditions. The GPSis a three-dimensional System and this implies that it suppliesheights as well as horizontal positions. The supplied height in thissystem is computed relative to the ellipsoid hence it is called theellipsoidal height. While the height derived from spirit levellingis related to the gravity field of the Earth, it is called the orthomet-ric height. The geoid height is the difference between the ellipsoi-dal and the orthometric height. It is well known that theorthometric height can be obtained without levelling by usingthe ellipsoidal and geoidal height. In this case the obtainable ortho-metric height is supposed to be accurate. Therefore, the determina-tion of a high-resolution geoid has become a matter of greatimportance to cope possibly with accuracy level of height fromGPS. Hence, it is possible to say that gravimetric geoid models offerthe third dimension to the GPS system by means of orthometricheight.

The first attempt to compute the geoid model in Sudan wasdone by Adam (1967), Cornell University in USA, who used 46 As-tro-geodetic stations to compute the deflections of the vertical andthe separation between the geoid and the reference ellipsoidClarke 1880. Due to the lack of data from neighbouring countriesand large un-surveyed areas in the Northwest and Southwest partsof Sudan, he concluded that the information was insufficient todetermine accurate geoid unless more measurements are to be car-ried out over the un-surveyed areas.

Salih (1983, 1985) computed the geoid shape in Sudan by theusing astrogeodetic and Satellite methods. The dataset for astro-geodetic method was collected in 1937 while the satellite methodsemployed dataset collected from observations conducted in 1975,1978 and 1981 using a Doppler receiver. Results by the two meth-ods were validated versus each other and the comparison shows

significant discrepancies from east to west while a better fit isshown from north to south.

Salih et al. (1990) determined the shape of the geoid in Sudanusing consecutive Doppler observations collected over a decade(1975 to 1986) from Transit Satellite. The use of Doppler observa-tions was initially suggested to augment the existing horizontalnetwork. However, the augmentation became limited to the scaleand orientation, for more details we refer readers to Fashir andAbdalla (1991). Salih concluded that the geoid in Sudan rises fromwest to east and from north to south with respect to the local(Adindan) datum based on the Clarke 1880 Spheroid. In both direc-tions the surface of the geoid follows the overall surface of theearth.

Another study was conducted by Fashir (1991), who computeda gravimetric geoid model for Sudan using heterogeneous data.This model was computed over the area bounded by the parallelsof 5 and 22 arc-deg northern latitude, and the meridians of 22and 38 arc-deg of eastern longitude, using the modified Stokeskernel and GEM-T1 global geopotential model complete to degreeand order 36. The direct and indirect atmospheric and topographiceffects were examined and the ellipsoidal correction to the geoidundulation was taken into account. The choices for the maximumdegree of GEM-T1 model and the spherical cap radius were consid-ered as 7 and 36 arc-deg, respectively. The model was evaluated by89 stations referred to different Doppler datums in Sudan. Further-more, the Doppler geoid heights were derived on the 89 stationsrelying on the co-located corresponding orthometric heights andthen transformed to GRS80. The gravimetric geoid heights werealso computed on the same stations for which the Doppler geoidheights were derived. A comparison between Doppler geoidheights and gravimetric geoid heights was performed to showthe accuracy of the geoid model, the result showed the followingstatistics (mean = 0.43 m, standard deviation = 1.88 m). It isconcluded by Fashir (1991) that for more accurate geoid modelin Sudan, densification of regional gravity data should be raisedat least up to 144 point values within area of 1 � 1 arc-deg, in otherwords a density of average spacing of 5 arc-minute (in the finalcomputation grid) will definitely help improving accuracy ofgravimetric geoid model in Sudan.

In this study, a new geoid model for Sudan is computed usingthe method of the Royal Institute of Technology (KTH), whichwas developed by Sjöberg (1984, 1991, 2003a,b). This method isbased on the least-squares modification of Stokes formula (LSMS).The principle of modifying the Stokes Formula was introduced byMolodensky et al. (1960) to minimize the truncation errors. Theoriginal Stokes formula is supposed to be used over the wholeearth surface assuming that the earth surface is well covered by lo-cal gravity. In practice, this assumption is very difficult due to thepoor coverage of the local terrestrial gravity over the earth surface.Therefore the integration cap around the computation point is usu-ally shortened to a few hundreds of kilometres neglecting gravityinformation from far zones. Consequently, the omission of the farzone contribution causes the truncation errors. The most impor-tant step in the KTH method is the determination of the least-square parameters which minimize the global square error and

A. Abdalla, D. Fairhead / Journal of African Earth Sciences 60 (2011) 213–221 215

reduce the influence of any source of errors in geoid modelling,Ellmann (2005a). In contrast to the other methods of the gravimetricgeoid solution based on gravity reduction, the KTH methods usesthe gravity anomalies directly without reduction and instead ap-plies some corrections to the approximate geoid solution Eq. (4).There are four additive corrections usually applied to the approxi-mate geoid heights derived by the KTH method (see Eqs. (4) and(5)) in order to account for the effect of topography, atmosphere,downward continuation, and the ellipsoidal approximation of theEarth’s shape. The KTH method was successfully used for a deter-mination of several regional geoid models in Sweden, the Balticcountries, Iran, Zambia, Ethiopia, Greece, and Tanzania; for moredetails we refer readers to Nahavandchi (1998), Ågren (2004),Ellman (2001, 2004), Nsombo (1996), Hunegnaw (2001), Kiamehr(2006), Daras (2008), Abdalla (2009), and Ulotu (2009).

The input data are described in Section 2. The mathematical as-pects of the KTH method and the additive corrections are recapit-ulated in Section 3. The results and the analysis of accuracy aregiven in Sections 4 and 5. The conclusions are given in Section 6.

Fig. 2. Histogram of the absolute values of differences between predicted andobserved Bouguer gravity anomalies.

2. Input data

2.1. Terrestrial gravity data

The terrestrial gravity data used for modelling the gravimetricgeoid for Sudan are provided by the Geophysical Exploration Tech-nology (GETECH) group from the University of Leeds, UK. The grav-ity database comprises 23,509 gravity points over the area ofstudy. The additional data set comprises 2645 gravity observationscovering some parts of neighbouring countries. The total numberof gravity data is 26,154, and they cover about 33% of the compu-tation area. This makes abundantly clear that there is data shortage(see Fig. 1) in terrestrial observations. Moreover, the data has beencollected by different organizations (Ibrahim, 1993) with differentintentions. Therefore, gravity data need to be validated carefullybefore they are used in the geoid computation. Both GETECH andthe additional data are used as one dataset for cleaning of the grav-ity anomaly to avoid any data corruption in geoid result.

Fig. 1. Distribution of the gravity data provided by GETECH (red) and the additionaldata (blue). (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

Two cross validation tests are implemented to detect and elim-inate gross errors from our gravity data. The cross-validation meth-od is introduced by Geisser and Eddy (1979). It is an establishedtechnique for assessing the data accuracy. The Leave-one-out crossvalidation (LOOCV) is used in this paper to predict new gravity val-ues. In this technique we omit a single observation from the origi-nal dataset and call it the test data while we call the rest of datasetas the training data. The training data are used to interpolate thevalue of the test data giving the predicted value Dgpre, thereafterwe put the tested observation back to the training data and weomit a new one instead. This process is rotationally repeated untilall dataset are one by one tested. The original and the predictedvalues of the same point are compared for all points and thenthe residual dDg is derived as the interpolation error as:

dDg ¼ Dgpre � Dgobs ð1Þ

In the first test we consider the Bouguer gravity anomalies as theoriginal dataset and then we apply LOOCV comparing the predictedvalues of the Bouguer anomalies against the original values yieldingthe residuals. The residuals absolute values are clearly representedon the histogram in Fig. 2. As result, 213 points were detected asoutliers and removed from the dataset according to the tolerancevalue of suspicious gross errors which was set to reject all error val-ues larger than 60 mGal.

Fig. 3. Histogram of the absolute values of differences between Molodensky’s andEIGEN-GL04C free-air gravity anomalies.

216 A. Abdalla, D. Fairhead / Journal of African Earth Sciences 60 (2011) 213–221

In addition, 20 points have been deleted in the second test thatwas conducted in order to compare the Molodensky’s gravityanomalies with the gravity anomalies obtained from the EIGEN-GL04C gravitational model (see Fig. 3). Gravity anomaly of Molo-densky can be defined as the difference between the actual gravityon the Earth surface and the normal gravity on the telluroid.

DgA ¼ gA � cB ð2Þ

where point A lies on the surface of the Earth and point B along theellipsoidal normal, at the telluroid. Numerically DgA is close to thefree-air gravity anomaly on the geoid. If point A has normal heightHA, then the normal gravity on the telluroid cB can be calculatedfrom the normal gravity cQ on the ellipsoid:

cB ¼ cQ � 2ceHA

a

� �1þ f þmþ �3f þ 5

2m

� �sin2 /

� �þ 3ce

HA

a

� �2

; ð3Þ

where a is the equatorial radius, ce is the normal gravity at theequator, cQ is computed by Somigliana’s formula as follows

cQ ¼ ce1þ k sin2 /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e2 sin2 /

q where;k ¼bcp

ace� 1 and e ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 � b2

a2

s;

m = 0.003449786000308 (constant for GRS80) and / the geodeticlatitude of the point, f the geometric flattening of the referenceellipsoid.

For the sake of constructing our grid, interpolation is applied toput data in regular grid. Kriging algorithm is used to construct thefinal 5 � 5 arc-min gravity anomaly grid with 3 arc-deg offset out-side the computation area (outer zone). The gaps in grid due todata shortage were filled by the free-air gravity anomalies fromthe EIGEN-GL04C combined gravitational model. It should be men-tioned that three interpolation algorithms (Kriging, Inverse Dis-tance Weighting, and Nearest Neighbour) have been tested.Among the three methods, Kriging has been selected by Abdalla(2009) as the method for the final gridding.

2.2. Digital Elevation Model (DEM)

The 30 � 30 arc-sec digital elevation data used in this study areprovided by Shuttle Radar Topography Mission (SRTM) released bythe National Aeronautics and Space Administration (NASA) in2003. The DEM extends to cover the target area with the proposed3 arc-deg adapted offset as re-sampled to 5 � 5 arc-min to meetagreement in resolution with the point within area of computationand the gravity anomaly grid. Since that the data coverage is in aglobal sense, missing data appears in some regions due to the lackof contrast in the radar image, presence of water, or excessiveatmospheric interference. Many global topography datasets havebeen produced after the appearance of the satellite imagery, thisprovides better resolution, from 10 arc-min (approximately18 km at the equator) to 30 arc-seconds (approximately 1 km atthe equator). The DEM data are used in KTH method in computa-tion of the topographic and downward continuation corrections(Kiamehr and Sjöberg, 2005).

2.3. GPS-levelling data

The first geodetic work in Sudan was established in 1903according to the recommendation of the International GeodeticAssociation (IAG), by continuation of the arc of the 30th meridianfrom Greece across the African continent starting in Egypt. Theproject was postponed due to various reasons and finally startedin 1935. The Egyptian network was extended up to Adindan station

at the northern Sudan. Thereafter, the 30th meridian became thefoundation for the geodetic work in Sudan, and the work continuedfurther to the south to the border with Uganda with a number offirst and second order networks. Between 1961 and 1966, the cen-tral and north-east parts of Sudan were covered with measure-ments. The national reference system in Sudan is realized by theAdindan planimetric datum. The fundamental point (tide-gaugestation) for a realization of the height system in Sudan is Alexan-dria in Egypt (cf. Adam, 1967).

In this study, we used GPS-levelling information at 19 points. Thelevelling data are extracted from the old geodetic networks of 1st to2nd or 3rd order. The GPS data were observed during different indi-vidual projects between 2005 and 2008. These GPS-levelling dataare used to estimate the accuracy of the gravimetric geoid.

2.4. Global gravitational model (GGM)

2.4.1. EIGEN-GRACE02SEIGEN-GRACE02S is a medium-wavelength gravity field model

which is calculated from 110 days of GRACE tracking data andwas released in February 13, 2004. The EIGEN-GRACE02S solution,resulting from the least-squares adjustment, was derived onlyfrom GRACE inter-satellite observations and it is thus independentfrom terrestrial gravity data. This model that resolves the geoidwith the accuracy better than 1 mm at a half-wavelength resolu-tion of about 1000 km is about one order of magnitude more accu-rate than the latest CHAMP derived global gravity models and itprovides the coefficients complete to degree and order 120 (cf.Reigber et al., 2004). In practice, the GGM and terrestrial gravitydata are often correlated especially when computing the combinedGGMs from satellite, terrestrial and altimetry data. This correlationcan be avoided when using satellite-only harmonic coefficients inlow degrees of the geopotential model, cf. (Sjöberg and Hunegnaw,2000) and (Hunegnaw, 2001). Abdalla (2009, Table 5.1, p. 39)showed that results computed using the EIGEN-GRACE02S satel-lite-only model are more accurate than the results obtained usingthe EGM96, EIGEN-GL04C and GGM02S combined models.

2.4.2. EIGEN-GL04CThe combined gravitational model EIGEN-GL04C was released on

March 31, 2006; it is an upgrade of EIGEN-CG03C. It is a combinationof GRACE and LAGEOS mission with high resolution 0.5 � 0.5� gra-vimetry and altimetry surface data. The satellite data have been ana-lyzed by GFZ Potsdam and GRGS Toulouse. All surface gravity dataare alike those of EIGEN-CG03C excluding the geoid undulationsover the oceans derived from a new GFZ mean sea surface height(MSSH) model minus the ECCO sea surface topography (EIGEN-CG03C: CLS01 MSSH minus ECCO). EIGEN-GL04C is complete to de-gree and order 360 in terms of spherical harmonic coefficients andthus resolves geoid and gravity anomaly wavelengths of 110 km.High-resolution combination gravity models are important for allapplications that require precise knowledge of the static gravity po-tential and its gradients is needed in the medium and short wave-length spectrum. EIGEN-GL04S1 represents the satellite-only partof EIGEN-GL04C; this part can be derived by reduction of the terres-trial normal equation system and is complete up to degree and order150. The EIGEN-GL04C model complete to degree and order 360 isutilised in our study to fill-in the gabs in the gravity anomaly grid.

3. KTH method

The Stokes formula presupposes that the disturbing potential isharmonic outside the geoid. This simply implies that there are nomasses outside the geoid surface, and that must be moved insidethe geoid or completely removed in order to apply Stokes formula.

Fig. 4. The Digital Elevation Model (DEM) and the GPS-levelling points over Sudan.

A. Abdalla, D. Fairhead / Journal of African Earth Sciences 60 (2011) 213–221 217

This assumption of the forbidden masses outside the geoid (bound-ing surface) is necessary when treating any problem of physicalgeodesy as a boundary-value problem in potential theory (seee.g., Heiskanen and Moritz, 1967).

In the KTH computational scheme (Sjöberg, 2003a), the gravityanomalies and GGM are used to determine the approximate geoidheights eN , and then all corrections are added to eN separately asshown in Eq. (1). In contrast to conventional methods by meansof gravity reductions, the forbidden masses are treated beforeusing Stokes formula which is the purpose of various gravityreductions.

The computational procedure of the KTH scheme for a determi-nation of the gravimetric geoid height is given by the followingexpression

N ¼ eNj þ dNTopocomb þ dNDWC þ dNa

comb þ dNe ð4Þ

where dNTopocomb is the combined topographic correction, which in-

cludes the sum of the direct and indirect topographical effects onthe geoid, dNDWC is the downward continuation correction, dNa

comb

is the combined atmospheric correction, which includes the sumof the direct and indirect atmospheric effects, and dNe is the ellipsoi-dal correction for the spherical approximation of the geoid in Stokesformula.

The approximate geoid heights are computed from (Sjöberg,2003b)

eN ¼ c2p

ZZro

SLðwÞDgdrþ cXL

n¼2

bnDgEGMn ; ð5Þ

where c ¼ R2c, R is the mean earth radius, c is the mean normal grav-

ity on the reference ellipsoid, Dg is the terrestrial gravity anomaly,bn is the vector of the least-squares parameters for the optimumsolution, bn ¼ ðQL

n þ s�nÞ cncnþdcn

for 2 6 n 6 M, DgEGMn is the Laplace

harmonics of degree n and it is calculated from the GGM (cf. Heiska-nen and Moritz, 1967, p. 89).

SLðwÞ ¼ SðwÞ �XL

n¼2

2nþ 12

snPnðcos wÞ ð6Þ

where SL(w) is the modified Stokes function to the modification lim-it L, S(w) is the Stokes original function, w is the spherical distancefrom the computation point ð/; kÞ to the block centre of r0 (the sur-face element). The spherical distance w can be computed over thecomputation grid by the following formula:

cos wij ¼ sin / sin /i þ cos / cos /i cosðk� kjÞ; ð7Þ

DgEGMn ¼ GM

a2

ar

� �nþ2ðn� 1Þ

Xn

m¼�n

CnmYnm; ð8Þ

where GM is the geocentric gravitational constant, a is the semima-jor axis of the reference ellipsoid, r is the geocentric radius, Cnm arethe fully normalised spherical harmonics coefficients of disturbingpotential from the normal gravity field referring to the sphere of ra-dius a and Ynm are the fully normalised surface spherical harmonicsdefined by (Heiskanen and Moritz, 1967, p. 31)

The truncation coefficients Q Ln are computed by

Q Ln ¼ Q n �

XL

k¼2

2kþ 12

skenk; ð9Þ

where Qn denote Molodensky’s truncation coefficients:

Q n ¼Z p

w0

SðwÞPnðcos wÞ sinðwÞdw; ð10Þ

and enk are functions of the spherical distance wo of the lower inte-gration bound:

enk ¼Z p

w0

Pnðcos wÞPkðcos wÞ sin wdw: ð11Þ

The least-squares modification (LSM) parameters are computedsolving the following system of linear equations (cf. Sjöberg, 2003b)

XL

r¼2

akrsr ¼ hk; k ¼ 2;3; . . . ; L; ð12Þ

where

akr ¼X1n¼2

EnkEnrCn þ dkrCr � EkrCk � EkrCr ; ð13Þ

and

hk ¼ Xk � Q kCk þX1n¼2

ðQ nCn �XkÞEnk; ð14Þ

where

Xk ¼2r2

k

k� 1; ð15Þ

dkr ¼1 if k ¼ r

0 otherwise

ð16Þ

Ck ¼ r2k þ

ckdck=ðck þ dckÞ if 2 6 k 6 M

ck if k > M

; ð17Þ

Enk ¼2kþ 1

2enkðw�Þ; ð18Þ

The system of equations in Eq (12) in the biased solution iswell-conditioned (cf. Sjöberg, 1984) it can be obtained by normalmatrix inversion. In contrast, the system in Eq (12) is ill-conditionedfor the optimum and unbiased (LSM) solutions (cf. Sjöberg, 1991,

Fig. 5. The new gravimetric geoid model KTH-SDG08 for Sudan. The geoid heightsare with respect to the GRS80 reference ellipsoid. The units are in metres (contourinterval is 1 m).

218 A. Abdalla, D. Fairhead / Journal of African Earth Sciences 60 (2011) 213–221

2003b), this causes least-squares parameters (bn and sn) to oscillatewithin the range of ±105 which makes the parameters aimless.Due to the fact that the good solution is not expected as well asit is not possible to be obtained from extremely ill-conditionedsystems, Ågren (2004) and Ellmann (2005b) used the singularvalue decomposition (SVD) procedure provided by Press et al.(1992) to avoid limitation and drawback of ill-conditioned cases.A software programme by Ellmann (2005b) has been used to de-rive the least-squares parameters by regularizing the problematicsystem in Eq. (12) by applying truncated singular value decompo-sition (T-SVD) and truncated total least-squares (T-TLS) methods,for more information about regularization methods readers are re-ferred to Hansen (1998, 2007).

3.1. The additive corrections

The terms dNTopocomb, dNdwc, dNa

comb and dNe on the right-hand side ofEq (4) are called the additive corrections. The additive correctionsare computed one by one before they are added to the approximategeoid eN as the followings:

The combined topographic correction is computed as follows(Sjöberg, 1997, 2000):

dNTopocomb ¼ dNdir þ dNindir � �

2pGqc

H2; ð19Þ

where q ¼ 2:67g=cm3 is the mean topographic density, and H is theorthometric height.

The downward continuation correction (dwc) is computed usingthe following expression (cf. Ågren, 2004)

dNdwcðPÞ ¼ dNð1ÞdwcðPÞ þ dNL1;Fardwc ðPÞ þ dNL2

dwcðPÞ; ð20Þ

where

dNð1ÞdwcðPÞ ¼DgðPÞ

cHP þ 3

f0P

rPHP �

12c@Dg@r

P

H2P ; ð21Þ

and f0P denotes the approximate value of the height anomaly. Due to

diminutive value of dNdwc(P) = 1 mm that corresponds to an error ofabout 1 m for the height of computation point of HP = 2 km andrP = 6375 km, it is convenient to adopt

f0P �

c2p

ZZro

~SðwÞDgdrþ cXM

n¼2

ðsn þ QLnÞDgEGM

n ; ð22Þ

dNLð1Þ;Fardwc ðPÞ ¼ c

XM

n¼2

ðsn þ Q LnÞ

RrP

� �nþ2

� 1

" #DgnðPÞ; ð23Þ

and

dNLð2ÞdwcðPÞ ¼

c2p

ZZr0

~SðwÞ @Dg@r

P

ðHP � HQ Þ� �

drQ ; ð24Þ

where rP = R + HP, r0 is a spherical cap with radius w� centredaround the computation point P and it should be the same as usedin modified Stokes formula, HP is the orthometric height of point P.The linear gravity gradient @Dg

@r

P at point P is computed based on

Heiskanen and Moritz (1967, p. 115):

@Dg@r

P

¼ R2

2p

ZZr0

DgQ � DgP

l30drQ �

2R

DgðPÞ; ð25Þ

where lo ¼ 2R sin wPQ2 .

The approximate ellipsoidal correction is computed approxi-mately using the following expression (cf. Sjöberg, 2004):

dNe � wo½ð0:12� 0:38 cos2 hÞDg þ 0:17eN sin2 h� ð26Þ

where wo is the cap size (in units of degree of arc), h is the geocen-tric co-latitude, Dg is given in mGal and eN is in metre.

The combined atmospheric effect dNacomb is computed from

(Sjöberg, 1997, 2000):

dNacombðPÞ ¼ �

2pRq�c

PMn¼2

2n�1� sn � QL

n

� �HnðPÞ

� 2pRq�c

P1n¼Mþ1

2n�1� nþ2

2nþ1 QLn

� �HnðPÞ;

ð27Þ

where q� = q� � G, q� is the atmospheric density at the sea levelðq� ¼ 1:23� 10�3g=cm3Þ; G is the gravitational constantðG ¼ 6:673� 10�11m3=kg1s2Þ; and Hn is the Laplace harmonic of de-gree n for the topographic height.

4. The new gravimetric geoid model (KTH-SDG08)

The new geoid model for Sudan KTH-SDG08 is shown in Fig. 5,the new model is depreciating from the north-west to south-eastas well as changing its sign from positive to negative in the samedirection. KTH-SDG08 reaches large values 17.2 m and �16.8 min the north-west and south-east corners, respectively. Abdalla(2009) tested the biased (Sjöberg, 1984), unbiased (Sjöberg,1991) and optimum (Sjöberg, 2003b) LMS stochastic methods in fi-nal geoid compilation. The optimum LSM method provides the bestfitting KTH-SDG08 gravimetric geoid model with GPS-levellingdata.

The additive corrections for (topography, downward continua-tion, ellipsoid approximation and atmosphere effects) are com-puted and included into the final geoid heights (Fig. 5) are shownin Figs. 6–9.

Fig. 8. The ellipsoidal correction for the new KTH-SDG08 geoid model in [mm].

Fig. 9. The combined atmospheric correction for the new KTH-SDG08 geoid modelin [mm].

Fig. 7. The downward continuation effect on the new KTH-SDG08 geoid model in[m].

Fig. 6. The combined topographic correction on the new KTH-SDG08 geoid modelin [m].

A. Abdalla, D. Fairhead / Journal of African Earth Sciences 60 (2011) 213–221 219

220 A. Abdalla, D. Fairhead / Journal of African Earth Sciences 60 (2011) 213–221

5. The geoid validation

The GPS-levelling data are commonly used to validate the gravi-metric geoid. The ellipsoidal heights h above the WGS84 referenceellipsoid are provided from GPS measurements. The orthometricheights H are the product of spirit levelling and gravity measure-ments along the levelling lines. The orthometric heights subtractedfrom the ellipsoidal heights give the geometric geoid heights N asshown in Eq. (28). The geometric geoid heights are used tradition-ally to evaluate the gravimetric geoid models. The relationship be-tween N, h, and H reads:

N � h� H ð28Þ

5.1. Verification of geoid in absolute sense

Systematic errors, distortions, and datum inconsistencies be-tween orthometric, ellipsoidal and geoid heights can be to someextend modelled by fitting the GPS-levelling derived geoid heightsto the gravimetric geoid heights using the least-squares adjust-ment and using several models, four, five and seven parametermodel.

DNi ¼ NGPSi � Ni � hi � Hi � Ni ¼ aT

i xþ ei ð29Þ

where Ni is the interpolated geoidal height value for ith GPS point,considering points from the geoid model that exist in the neigh-bourhood, x is a n � 1 vector of unknown parameters (where n isthe number of the GPS-levelling points), ai is a n � 1 vector ofknown coefficients, and ei denotes a residual random noise term.The parametric model aT

i x is supposed to describe the systematic er-rors and inconsistencies inherent in the different height data sets.7-parameter model reads

ai ¼

cos /i cos ki

cos /i sin ki

sin /i

cos /i sin /i cos ki=Wi

cos /i sin /i sin ki=Wi

sin2 /i=Wi

1

0BBBBBBBBBBB@

1CCCCCCCCCCCAand x ¼

x1

x2

x3

x4

x5

x6

x7

0BBBBBBBBBBB@

1CCCCCCCCCCCA; ð30Þ

where ð/; kÞ are the horizontal geodetic coordinates of the GPS-lev-elling points, and

Wi ¼ ð1� e2 sin2 /iÞ1=2; ð31Þ

where e is the first eccentricity of the reference ellipsoid. We thenobtain the following system of observation equations:

Ax ¼ DN� e; ð32Þ

where A is the design matrix composed of one row aTi for each

observation DNi. The least-squares adjustment to this equation,which utilize the mean squares of the residuals ei, becomes

x ¼ ðATAÞ�1ATDN; ð33Þ

The least-squares residuals are computed from

e ¼ DN � Ax ¼ I� ðATAÞ�1ATh i

DN; ð34Þ

The standard deviation r can be estimated from the residuals eusing the following equation

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffieT e

n�m

r; ð35Þ

where n is the number of GPS-levelling points, and m is the numberof estimated parameters.

5.2. Verification of geoid in relative sense

In case of the geoid evaluation in relative sense, the orthometricand ellipsoidal height differences must be known. The main advan-tage of differencing is that any errors related to the baseline arecancelled, e.g. errors of vertical datum in long baselines; this couldhappen mainly over short distances of recent vertical datums (seeFeatherstone, 2001). The relative verification is basically used toassess the precision of the geoid gradient, which is given by sub-tracting the difference in orthometric height from the differencein ellipsoidal height. For this purpose, Eq (28) is rearranged intothe following form

DHGPS-Geoid � DhGPS � DNGeoid ð36Þ

where in DNGeoid the gravimetric geoid values after the 7-parameterfitting are used. Hence

DNGeoid � ðN2 � N1Þ þ ðe2 � e1Þ ð37Þ

The differences between two different orthometric height dif-ferences, i.e. from levelling (DHOrtho) and from GPS minus geoid(DHGPS-Geoid), are then derived to be

dDHGeoid-Level � DHGPS-Geoid � DHLevel ð38Þ

The relative differences between the GPS-geoid heights and thelevelling heights becomes in part per million (ppm):

ppm ¼ meanðdDHGeo-LevelÞmm

DijðkmÞ

; ð39Þ

where Dij is the length of the baseline. The average distance betweenthe 19 GPS/levelling points is about 1120 km, the relative accuracybased on DHGPS-Geo against the levelling data shows that the fittingbetween the new geoid and GPS-levelling is about 0.5 ppm.

6. Conclusions

The KTH method has been used to compute a new geoid modelfor Sudan. The KTH-SDG08 geoid model is available on a regular5 � 5 arc-min geographical grid over the area bounded by the par-allels of 4 and 23 arc-deg northern latitude, and the meridians of22 and 38 arc-deg eastern longitude. The cross validation tech-nique is utilized to detect and reduce the gross errors in the gravitydata, and therefore a number of 233 observations is excluded fromthe gravity dataset before the construction of the final grid. Inter-polation has been used to build up the gravity anomaly grid usingKriging method. Due to the lack of the gravity measurements overthe country, areas with no data have been filled by the gravityanomalies produced by EIGEN-GL04C (combined geopotentialmodel) complete to degree and order 360.

The modified Stokes formula is applied in the KTH method bymeans of least-squares, integrating the constructed gravity anomalygrid and the long-wavelength contribution from EIGEN-GRACE2S(satellite-only model) complete to degree and order 120 yieldingthe approximate geoid height. The additive corrections for thetopography, downward continuation, atmosphere, and ellipsoidapproximation are respectively added to the approximate geoidsolution. The additive corrections (Sjöberg, 2003a) have improvedthe absolute accuracy of the fitting between the geometric and gravi-metric geoid heights at 19 GPS-levelling points to become 0.30 m in-stead of 0.42 m in the approximate geoid heights. On the other hand,the relative accuracy based on the differencing between the ortho-metric heights derived from the GPS-geoid heights and the observedorthometric heights from the levelling points is found to be 0.5 ppm.Therefore KTH-SDG08 can provide results with a good quality overlarge areas of the country due to the fact that the topography inSudan is mostly flat as seen in Fig. 4.

Fig. 10. Gravimetric geoid heights with the contribution of EIGEN-GRACE02S andthe derived geoid heights by 19 GPS-levelling points.

Table 1Statistics of differences between the approximate gravimetric geoid heights and thegeometric geoid heights at 19 GPS-levelling points.

Before fitting After fittingDN (m) e (m)

Min �0.61 �0.88Max 1.40 0.69Mean 0.64 0.00r 0.59 0.42

Table 2Statistics of differences (in cm) between the gravimetric and the geometric geoidheights at 19 GPS-levelling points.

Before fitting After fittingDN (m) e (m)

Min 0.37 �0.52Max 2.45 0.64Mean 1.62 0.00r 0.58 0.30

A. Abdalla, D. Fairhead / Journal of African Earth Sciences 60 (2011) 213–221 221

This study represents a preliminary result in geoid modelling to-wards obtaining a high precession geoid model for Sudan. With re-gard to achieving better results in the future, the terrestrial gravitydata should be enhanced and intensified by filling the areas withno data (Fig. 1). More GPS measurements are needed to be co-locatedwith the existing levelling networks Fig. 10, Tables 1 and 2.

Acknowledgements

The principal author wishes to thank the following people fortheir support which is greatly appreciated Dr. Huaan Fan for pro-viding the additional gravity data around Sudan, Mr. MubarakElmotasim and Mr. Suliman Khalifa for providing the GPS-levellingdata. Dr. Robert Tenzer is gratefully acknowledged for his construc-tive comments and scientific review of this paper. Professor J.LVingeresse and anonymous reviewer are greatly acknowledgedfor their critical review and valuable comments on the manuscript.

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