Abstract This paper aims to provide a precise transfor-mation model for the geoid heights between earth geopo-tential models and regional vertical datum in geodeticapplications. With this purpose, Earth Gravitational Model2008 (EGM08), which is ultra-high-resolution geopotentialmodel, that is complete spherical harmonic degree and order2190 is considered and fitted to the local vertical datum inthree local test areas in west and south of Turkey. The obser-vations at GPS/leveling benchmarks were used for fittingthe EGM08 to the local height reference surface and a cou-ple of mathematical methods were evaluated, namely sim-ilarity transformation model (STM), polynomial regressionmodel (PRM), least squares collocation (LSC), spline inter-polation model (SIM). They are commonly used for correc-tor surface fitting approaches in height datum studies. Theirperformances were assessed depending on the statistics attest benchmarks. The results show that fitting of the dif-ferences between the GPS/leveling, EGM08-derived geoidheights with LSC, and SIM supplies more accurate solutionsthan STM and PRM. The accuracies of the achieved models(improved from EGM08 global model) are tested in the sev-eral GPS/leveling benchmarks throughout Turkey; the exter-nal model accuracy has been estimated as ±4–5 cm. In thisaspect, this study will provide benefits as both representingthe performance of EGM08 model in given local areas andclarifying the modeling performances of the applied trans-formation approaches for fitting of EGM08 geoid models toGPS/leveling defined vertical datum.
M. Soycan (B) · A. SoycanGeomatics Engineering Division, Yıldız Technical University CivilEngineering Faculty, Istanbul, Turkeye-mail: [email protected]
The improvement in GPS technology makes it essential todetermine precise geoid or similar surface model referring toa global geocentric datum. Geoid is defined as the equipo-tential surface of the Earth’s gravity field, which best fits, ina least-squares sense, global mean sea level. It is regarded
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as the reference surface for the vertical control surveys ingeodesy [1,2]. Recently, the new gravity satellite missionsprovide new global solutions that allow modeling the longand medium wavelengths of the Earth’s gravity field. Thedevelopment of the EGM08 is a significant contributionfor modeling the Earth’s gravity and geoid. The NationalGeospatial-Intelligence Agency (NGA) EGM DevelopmentTeam has publicly released the official Earth GravitationalModel EGM08. This gravitational model is complete tospherical harmonic degree and order 2159, and contains addi-tional coefficients extending to degree 2190 and order 2159.It consists of a total of ∼4.8 million spherical harmoniccoefficients. The model was computed from a global 5 arc-minute grid of gravity anomalies from land and satellite-based sources. The model is provided complete to sphericalharmonic degree and order 2159, which equates to a grid sizeof approximately 6.5 km. The EGM08 geopotential model isavailable free-of-charge together with several products suchas a spherical harmonic model of Earth’s topography, grids ofcommission error estimates for different gravity field func-tionals and a high-degree synthesis software. EGM08-basedfunctionals of the gravity field are obtained through harmonicsynthesis of the model coefficients for practical applications[3–6]. Computations using harmonic synthesis are mainlyaffected by:
(a) Commission (or propagated) error, which arises becausethe model coefficients can never be perfect. This errordepends primarily on the accuracy of the data used todevelop the model.
(b) Omission (or truncation) error, which arises because themodel coefficients can never extend to infinity. This errordepends primarily on the resolution of the data used todevelop the model and on the maximum degree used inthe computations.
Because of the inhomogeneous and incomplete global cov-erage by surface gravity observations, the accuracy of theEGM08-derived geoid heights vary over different parts ofEarth. The most accurate regions have high-quality terres-trial gravity data sets that are available for its construction.
For the areas with rather scarce surface gravity coverage(for instance, parts of Africa, South America and Asia) thecommission errors for EGM08 geoid heights are estimatedat the level of ∼15 cm with maximum uncertainties encoun-tered in the mountainous parts of Asia and South America(around ∼30–40 cm) and Antarctica (∼100 cm). In contrastto this, the lowest commission errors are found over mostparts of Europe, Oceania, North America and the oceans(see Fig. 1). For the regions which have high-quality surfaceand altimetry-derived gravity the EGM08 geoid commissionerrors are mostly at the level of ∼5 cm [7–9].
The evaluation and quality assessment of the EGM08 areimportant for use in various geodetic and other scientificapplications at global and regional scales. Many studies havebeen carried out by researchers for the refinement of EGM08and other global geoids that use gravimetry, GPS and levelingdata [10–16]. Most of these studies use local terrestrial grav-ity data, which assume including all frequencies of local grav-ity field, for correction to the local reference surface heights.On the other hand, the fitting of EGM08 to GPS/levelingdatum with the observations at GPS/leveling benchmarksby modeling the geoid differences (�N ) between EGM08-derived geoid height (N EGM08) from a geopotential modeland GPS/leveling (N ∼= h − H ) data by using several fittingmethods can be considered as an alternative solutions forpractical applications. This procedure is described as eithera correction or corrector surface or a height transformation.Although it has a practical usage, it does not improve thegeoid model but it minimizes the discrepancies by modifica-tion of the model to fit a GPS/leveling defined vertical datum[17,18]. Moreover, it may distort the EGM08 and also intro-duce some errors both in the global model and in the GPS andleveling measurements. Gross, random, and systematic errorsin the three parameters (N EGM08, h, H ) affect the differencein the geoid heights (�N ) from the two independent sources.The systematic datum differences between the EGM08 andthe GPS/leveling data, and possible long-wavelength errorsof the geoid, are removed by applying a simple correctionmodel by this procedure [10–16]. This helps to make theEGM08 fit better the GPS/leveling data. The successful useof this kind of improved global geoid models versus geo-metric or gravimetric model for local areas depends on theminimization random and systematic errors, biases and dis-tortion effects.
In fact, it is not a common way of correcting a global modelin order to derive a height transformation model. Since theglobal spherical harmonic models do not include all the fre-quencies of local gravity field, they give the low-frequencycontent; so global models contain only long wavelength prop-erties of the gravity field depending on its expansion degreeand order [20–22]. Therefore, the spherical harmonic modelswith an optimal degree of expansion are combined with ter-restrial gravity and terrain data to express the gravity field inevery frequency in the spectrum. In this manner one can claimthat EGM08 model is the exception because it is ultra-highresolution model and with its 2190 degree/order of expan-sion and that is why it should be different from other earthgeopotential models. Hence we must focus on the ultra-high-frequency band of EGM08 (360 < n < 2190) in orderto claim that the EGM08 model is good in representinglocal gravity field in a region. As the results of the inves-tigation of spectral performance of EGM08 in the high andultra-high degrees of expansion, it is easily recognized thatthe major contribution comes from the ultra-high-frequency
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Number of Values 9331200Percentage of Area 100
Minimum Value 3.045Latitude of Minimum 54.542
Longitude of Minimum 0.125Maximum Value 102.194
Latitude of Maximum -84.042Longitude of Maximum 167.042
band of EGM08 (360 < n < 2190) which enhances thedegree variances in terms of geoid height and spatial reso-lution (Fig. 2). They are very important indicator togetherwith propagated errors with a view to defend these claimsand proposed approach. However, these indicators refer tothe internal accuracy of the EGM08 and they may not repre-sent the characteristics of the local study areas investigatedin this study.
For this purpose, we need independent external datasets;such as, GPS/leveling observations, airborne and surfacegravity data, and sea surface topography, sea surface heightsfrom altimetry, tide gauge data, and deflections of the verti-cal. These can be used for external evaluation procedures.
2 Description of Test Areas and Methodology
We mainly used GPS/leveling-derived geoid heights in localtest areas to perform comparisons of accuracy of the EGM08.Hundreds of benchmarks complied with their precise ellip-soidal geographic coordinates (ϕ, λ, h) and the orthometricheights (H) for three different test areas in Istanbul, Izmir andAntalya cities of Turkey (Fig. 3). Some benchmarks fromthese data sets were identified as reference points. Thesepoints were used to construct corrector surface for correct-ing of the systematic effects, the medium and the long wave-length errors and local distortions by several models. Theremaining benchmarks were considered as checkpoints to
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Fig. 3 Test areas and data used in numerical study
evaluate accuracy of the achieved corrected and improvedmodels (Fig. 3). In the Fig. 3, the background reflects thetopography, the contours represent the variation of EGM08geoid heights and the points indicate the reference and thecheck benchmarks, respectively. In principle, the test areaswere selected from different parts of the Turkey. The firstdata set was chosen in the West part of Turkey (near theAegean Sea) between the 38◦.00–38◦.40 latitudes and the27◦.10–27◦.50 longitudes in the Izmir city with an area of1,537 km2. The examinations show that geoid height variesapproximately 1.315 ± 0.325 m in test area (approximately115 km × 112 km). This variation is about a 2 cm/km throughthe northwest direction. The first test area has rather uneventopography and significant differences even in the local scale.The topography involves wavy, level and highland fragmentswith elevation spanning from 5 to 1,391 m with 200 m stan-dard deviation (See Fig. 3). 247 benchmarks were compiledfrom the common points of GPS and leveling network of theIzmir Municipality. 16 benchmarks were selected as refer-ence for computation of corrector surface and the other oneswere selected as checkpoints.
The second data set with 16 reference and 206 check-points and with area of 833 km2 (approximately 115 km ×112 km) was chosen on the European side of Istanbul cityalong the Black Sea to the Marmara Sea. This area is locatedbetween the 41◦.04–41◦.34 latitudes and the 28◦.42–28◦.72longitudes. Geoid heights vary between 36.778 and 37.642 mwith 0.155m standard deviation. Geoid surface inclines in theeast–west direction about 1 cm/km. Although the topographyis rather smooth as the first frame, it has local fluctuationsin some regions. The elevations span 7–284 m with 89 mstandard deviation.
The third data set, was chosen in the south part of Turkey(near the Mediterranean Sea), which is more stable and flat,and compared to the previous test areas. 9 benchmarks wereselected as reference and 177 benchmarks were selected ascheckpoints. The elevation values span 28–303 m with 101 mstandard deviation. This area involves an area of approxi-mately 115 km × 112 km between the 36◦.90–37◦.10 lat-itudes and the 30◦.60–30◦.80 longitudes (the area of thisframe is about 391 km2). The geoid heights vary between26.792 and 29.597 m with 0.751 m standard deviation. The
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Fig. 4 Geoid height differences between EGM08 and GPS/leveling in checkpoints
geoid is slope in this area (10 cm in 1 km along the northwestdirection).
The ellipsoidal geographic coordinates were derived fromthe GPS networks, which were calculated via hierarchicalnetwork densification in C1 and C2 order based on Turkey’sB order National Fundamental GPS Network (TUTGA). Theachieved ellipsoidal heights were referred to ITRF96 datum,2005 epoch, and their standard deviations varied from 0.2to 7.6 cm with an average of 1.3 cm. The standard devia-tions of latitude and longitude components were about 1–2 cm. On the other hand, the co-located orthometric heightsof benchmarks were estimated from leveling measurementsfrom the second order leveling networks which was createdbased on Turkey’s first order National Vertical Control Net-work (TUDKA). Their standard deviations varied 3 mm to7 cm with a mean standard deviation of 4 cm with increas-ing distance from the fixed benchmark in Antalya tide gaugestation [23]. EGM08 Tide Free Model with zero-degree termcorrection were used to achieve EGM08-derived quantitiesby using a calculation service that was provided by Inter-national Centre of Global Earth Models (ICGEM) at GFZ,which is a component of IGFS) [24]. Thus, the final datasets for evaluation were obtained as latitude, longitude, geoidheight from GPS/leveling data and EGM08 model for threetest areas.
3 Pre-Analysis of EGM08 in Local Test Areas
As mentioned in the introduction section a few gravity datafrom Bureau Gravime’trique International (BGI) have beenincluded in the development of EGM08 in some countries.In those regions, direct use of EGM08 (DUEGM08) maynot guarantee an accurate transformation of the ellipsoidal
heights to the orthometric heights. Our pre-analysis resultsin test areas show that, EGM08 generally contains errors ofdm-level on the long wavelengths due to the lack of Turk-ish proprietary gravity and GPS/leveling data in the EGM08computations [25–27].
A detailed contour map of the geoid height differencesbetween EGM08 and GPS/leveling over three test areas isshown in Fig. 4, where the differences range from −8.1to −76.7 cm, with mean values of −21.4 to −56.3 cm.Although the standard deviations are about 6–7 cm, theRMS errors are reached 22.6, 42.2 and 56.8 cm for Izmir,Istanbul and Antalya areas, respectively, due to the biases.
The differences in the estimated bias obtained from eachtest area also indicate that the existence of systematic regionaloffsets among the EGM08 geoids that are mainly caused bylong/medium-wavelength commission and additional omis-sion errors due to their limited spectral resolution. Somestrong localized tilts and systematic oscillations can be recog-nized in the �N differences, mainly due to larger com-mission errors and significant omission errors. Moreover,some important factors also cause both systematic biases andunsystematic deviations in the �N differences. This can bebriefly listed as follows:
• The datum inconsistencies among the geoid heights, eachof them refers to a different reference surface and longwavelength systematic errors in N.
• Distortions in the orthometric height datum due to an over-constrained adjustment of the leveling network and thephysical effects on the networks by time.
• Improper or non-existent terrain/density modeling in thegeoid modeling and orthometric heights and negligence ofthe sea surface topography at the tide gauges.
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Table 1 Statistics for �N differences between EGM08 and GPS/leveling in checkpoints
Number of values Minimum Maximum Mean Median Standard deviation Root mean square error
Izmir 231 −76.7 −37.4 −56.3 −57.2 7.1 56.8
Istanbul 206 −56.9 −26.6 −41.7 −41.5 6.1 42.2
Antalya 177 −38.5 −8.1 −21.4 −19.3 4.5 22.6
• Poorly modeled GPS errors (i.e., tropospheric effects);also assumptions and theoretical approximations in theprocess of the observed data, etc.
These inconsistencies must be corrected for the use ofEGM08 in practice. Otherwise, the direct use of EGM08does not guarantee an accurate transformation of the ellip-soidal heights to the orthometric heights.
4 Fitting EGM08 Geoid to the Local Vertical Datum
The differences (�N ) between EGM08-derived geoid height(N EGM08) from a geopotential model and GPS/leveling (N ∼=h − H ) data generally contain uniform, low-frequency andsmoothly varying distortions which make them easier to bemodeled by using simple mathematical equations. It can beconcluded from technical literature that successful resultscan be obtained by using the methods, such as least squaresadjustment with a four, five or seven-parameter STM, leastsquares or robust estimations with PRM, LSC, finite ele-ment method (FEM), the Fourier series and similar fittingmethods [10–14,28–30]. This study alternatively focuses onelastic corrector surface fitting by SIM and probes into leastsquares collocation (LSC) methods versus STM and PRM.All of the methods were compared with the solution of the fit-ting problem between the EGM08 and GPS/leveling datum.In the following section, their mathematical models werethoroughly explained.
4.1 Similarity Transformation Models (STM)
Several similarity equations including one to seven parame-ters can be used for complicated surface correction techniquein the combination of GPS/leveling data and geoid modelsby this procedure. The STM is traditionally based on thefollowing equation [10,14,31–34]:
f (ϕ, λ) = �Ni = hi − Hi − N EGM08i
= N GPS_LEV.i − N EGM08
i = aTi .x + vi ,
(1)
where
�M : Geoid height difference betweenGPS/leveling and EGM08 model
hi : GPS-derived ellipsoidal heightHi : Leveling-derived orthometric heightN EGM08
i : EGM08-derived geoid heightN GPS_LEV
i : Geoid height from GPS/levelingai : Vector of known coefficientsx: Vector of unknown parameters
V i : Residuals random noise term,
where aTi ·x is supposed to describe the systematic errors and
datum inconsistencies inherent in the different height datasets, ϕi and λi are ellipsoidal geographical coordinates (lati-tude and longitude), e is the first eccentricity of the referenceellipsoid. One-parameter model (OPM) comprises a singlebias parameter (a0) and it can be easily adopted by a constantoffset. On the other hand, the three-parameter model (TPM)involves two additional terms (a1 and a2) for the north–southand east–west components of an average spatial tilt betweenthe EGM08 and GPS/leveling datum. It is clear that the dif-ference between the OPM and TPM lies in the assumptionof tilt factor. In the OPM, the surface correction is a sin-gle unique value across all reference points. However, TPMallows difference surface correction by a tilt factor in the twoaxis directions. As for the four-parameter (a0, a1, a2 and a3)
model (FPM), it corresponds to 3D spatial shift parametersand an approximate uniform scale change of the EGM08’sreference frame with respect to the reference frame of theGPS/leveling-derived geoid heights. Finally, a4, a5 and a6
are the most complicated coefficients for seven-parametermodel (SPM) (Eq. 2).
�Ni = aTi · x + vi = a0 + a1 · cos ϕi · cos λi
+ a2 · cos ϕi · sin λi
+ a3 · sin ϕi + a4 ·(
cos ϕi · sin ϕi cos λi
w i
)
+ a5 · ((cos ϕi . sin ϕi sin λi )/wi )
+ a6 · ((sin2 ϕi ·)/wi ) (2)
wi =√
(1 − e2 sin2 ϕi ) (3)
The STM that includes one, three, four or five parameterscan be constituted from shortened version of equation 2.The unknown coefficients of the models with their covari-ance information are simply determined according to leastsquares estimation (LSE) principles. The matrix system of
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the observation equations can be defined as:
v = A · x − �N , (4)
where A is called as design matrix composed of one rowaT
i for each observation �Ni . The unknown parameters areestimated by utilizing the sum of squares of the residuals vi
as follows:
x = (AT A)−1 AT �N (5)
yielding the residuals
v = �N − Ax =[
I − A(AT A)−1 AT]�N . (6)
4.2 Polynomial Regression Model (PRM)
This approach is based on the approximate function, whichattempts to model the varying differences between GPS/leveling data and EGM08 model as a surface described by apolynomial function. It is also known as polynomial or mul-tiple regression in literature [29,30,35–37]. The systematicpart of distortions can be easily modeled by an STM or loworder PRM with two or three orders. The general equationof PRM to estimate unknown coefficients can be given byEq. 7. The relationship between observation (�Ni ) and itsresiduals (vi ) are described with the function of unknownsas follows:
�N =aTi · x + vi =
k∑i=0
i∑j=0
ai j (ϕ − ϕ0)j (λ − λ0)
i− j , (7)
where ϕ0 and λ0 represent the position coordinates as themean values of the latitudes and longitudes of the data set, ai j
symbolize the polynomial coefficients, and k is the degree ofthe polynomial. The polynomial coefficients are determinedalso according to the LSE method by Eqs. 4, 5, 6, respectively.The above model with the one, four, six and ten-parametersolutions represent average plane, bilinear surface, quadraticand cubic polynomial surface, respectively. A first order poly-nomial (affine) fits a flat plane to the input points. Dependingon the degree of variability in the distortion, second or thirddegree polynomial functions should be used in practice. Asecond order PRM fits a somewhat more complicated sur-face to the input reference points. The third order PRM maybe adequate to absorb non-periodic large-scale systematicerrors, such as systematic leveling errors. It fits a more com-plicated surface to the input reference points. The higherdegree polynomial may result in undesirable stretching orsqueezing and it may perform badly at extrapolation.
4.3 Least Squares Collocation (LSC)
STM and low order PRM are useful for creating smooth cor-rection surfaces and identifying long-range trends in the data.
However, here, the variable of interest usually has short-range variation in addition to long-range trend. When thedata exhibits short-range variation, LSC can be consideredamong the several possible methods to identify the short-range variation. The LSC theory includes two kinds of ran-dom quantities as noise (residuals in classic least squares)and signal (related by a known covariance function). Thesystematic biases may be treated as signals to be estimatedin LSC approach. The LSC model is given for fitting of thedifferences between the EGM08 and GPS/leveling data bythe following form [10–12,16,38],
�N = aTi · x + vi = a0 + a1 · ϕi + a2 · λi + si + ni (8)
The collocation method combines adjustment of model para-meters in the conventional least squares sense with estimatedof the signals. It depends on assumption of the a priori covari-ance function between signals and observations. All opera-tions are carried out in three stages:
• Determination of trend (a0, a1, a2, . . . parameters),• Experimental calculation of covariance,• Estimation of signals.
First, the residuals of observations should be separated in twocomponents as the signal and the noise:
(s + n) = A · x − �N , (9)
where s is the signal vector, n is the noise vector, A is thecoefficient matrix and x is the vector of unknowns. The cor-rector surface comprises the trend part Ax and the signal parts. Then, the functional models of unknown parameters aredetermined and unknown parameters and mean errors arecalculated for observations. The following unbiased estima-tors can be used for the solutions:
x = (AT (Css + Cnn)−1 A)−1 AT (Css + Cnn)
−1�N (10)
s = Css(Css + Cnn)−1(Ax − �N ) (11)
n = Cnn(Css + Cnn)−1(Ax − �N ). (12)
The signals at any reference points can be estimated fromall the difference values through a covariance function. Thesignal component sP at a point p can be obtained by thefollowing equation,
sP = cTP (Css + Cnn)
−1(Ax − �N ). (13)
The final estimation is performed for a new point by the leastsquares principle. The matrix Css denotes the covariancematrix of signals among the reference points, and the vectorcp contains the signal covariances between the estimationpoint and the reference points. Both of them are determinedby an empirical covariance function. The difficulty of theLSC lies in the selection of an appropriate covariance func-tion. The empirical method can be selected, but needs large
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amount of data to get a meaningful and reliable covariancefunction. Alternatively, one may arbitrarily select a function.However, the results will vary with this selection. We selectedGauss function as covariance function in this study.
4.4 Elastic Surface Fitting by Spline Interpolation Model(SIM)
We can create a continuous corrector surface from the �Ndifferences using some interpolation techniques to identifythe short-range variation and the distortions of the model.Scientific literature, application software and research sup-ply that various interpolation methods are available for inter-polation [39–44]. The diversities of interpolation methodslead to the conclusion that no method is the best or theworst when the application area, surface features, data types,accuracy and ease of calculation are taken into considera-tion. A method that offers ideal results for one applicationmay not show the same performance in other applications orwith different data types. Different interpolation techniqueswere considered for the creation of the corrective surface inthis research. According to our previous experiences, pre-analysis results show that some interpolation methods suchas Kriging, multiquadric interpolation function, Triangula-tion with linear interpolation or SIM have come to the forein all cases based on the accuracy criteria and the ability torepresent the local variations.
The SIM fits a flexible surface as if it were stretching arubber sheet across all the reference point values. It has beenused to interpolate irregularly spaced data using a mathemat-ical function that minimizes overall surface curvature whichinitially computes the signed-distance function to generatethe spline interpolant [39–41,45–47]. The basic form of thespline interpolation imposes the following two conditions onthe interpolant:
• The surface must pass exactly through the reference datapoints.
• The surface must have minimum curvature.
It means that the SIM is based on the assumption that theinterpolation function should pass through (or close to) thedata points and, at the same time, should be as smooth aspossible. The estimated surface generated by spline is anal-ogous to a thin and linearly elastic plate passing througheach of the data values with a minimum amount of bend-ing. These requirements are combined into a single conditionof minimizing the sum of the deviations from the referencepoints and the smoothness seminorm of the spline function[40,43,44,49,50].
n∑i=1
∣∣�Ni − F(ri)∣∣2
wi + w0 I (F) = minimum, (14)
where �Ni is the value measured at reference points;wi , w0 are positive weights and I(F) denotes the smoothnessseminorm; their corresponding smoothness seminorms andEuler–Lagrange equations were examined in [48] in detail,e.g., bivariate spline functions. The solution of Eq. (14) canbe expressed as a sum of two components. Thus, the fol-lowing general equations can be summarized for the splinealgorithm in surface interpolation.
F(r) = T (r) +n∑
i=1
ci · R(r, ri ), (15)
where
ci : The coefficients that are estimated from thesolution ofa system of linear equations
ri : The distance from the point to the i. pointT(r): Trend functionR(r, ri ): Spline function which has a form dependent
on the choice of I(F).
In this research, two fundamental SIMs considered as reg-ularized and tension [44,45,47]. The regularized method cre-ates a smooth and gradually changing surface with values thatmay lie outside the data range of reference points. The ten-sion method controls the stiffness of the surface according tothe character of the modeled phenomenon. It creates a lesssmooth surface with values more closely constrained by thedata range. However, different types of spline functions suchas thin plate splines, polyharmonic splines, and naturel cubicsplines can be also used depending on the surface and datacharacteristic. Each spline function has a different shape andRESULTS in a different interpolation surface. In Eq. (15),T(x, y) and R(r) are defined differently depending on thespline function. For the regularized SIM,
T (x, y) = a0 + a1 · x + a2 · y (16)
R(r) = 1
2π
(r2
4
(ln
( r
2τ
)+ c − 1
)+ τ 2
(K0
( r
τ
)
+ c + ln( r
2π
)))(17)
For the SIM with tension,
T (x, y) = a0 (18)
R(r) = − 1
2πϕ2
(ln
(rϕ
2
)+ c + K0(rϕ)
)(19)
where τ 2 and ϕ2 are the weight parameters, Ko is the modi-fied Bessel function and c = 0.577215 is the Euler constant.The weight parameter (τ 2) for the regularized SIM definesthe weight of the third derivatives of the surface in the cur-vature minimization expression. The typical values that maybe used are 0, 0.001, 0.01, 0.1, and 0.5. Higher values of thisterm lead to smoother surfaces. On the other hand, the weightparameter (ϕ2) defines the weight of tension for the tension
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SIM. The typical values are 0, 1, 5, and 10. Using a largervalue of weight reduces the stiffness of the plate. Therefore,we can say that the tension SIM produces a surface morerigid in character, while the regularized SIM creates one thatis more elastic. We focused on elastic surface fitting by reg-ularized SIM as flexible interpolation methods that is widelyused in applications [51]. The interpolation function given byEq. (15) requires solving a system of n linear equations. Themodel for fitting of the differences between the EGM08 andGPS/leveling data can be defined as below with the followingequilibrium constraints as [32,40–42];
aTi · x + vi = T (ϕ, λ) +
n∑i=1
ci · R(r j ) (20)
Although we have only n constraints, the Eq. (20) has n + 1(c1, c2, . . . , cn and a0) or n + 3 (c1, c2, . . . , cn and a0, a1, a2)
unknown values which come from an orthogonality condi-tion, respectively, for tension and regularized spline types.The coefficients in the approximation are determined byrequiring that the function interpolate and the certain con-ditions on the coefficients are satisfied. In addition, the con-straint equations may be considered as “equilibrium” condi-tions on the plate; the sums of the forces and moments mustbe zero. The additional conditions are related to the polyno-mial terms for trend function.
n∑i=1
ci = 0 orn∑
i=1
ci =n∑
i=1
ci · ϕi =n∑
i=1
ci · λi = 0 (21)
A surface created with SIM passes through each referencepoint and may exceed the value range of the reference pointset. This is the most important advantage of the method andthis makes the SIM good for estimating lows and highs wherethey are not included in the data. It is useful if we want esti-mated values that are below the minimum or above the max-imum values found in the data. On the other hand, SIM doesnot work well with reference points that are close togetherand have extreme differences in value. It is because of thiscondition that the spline uses slope calculations (change overdistance) to figure out the shape of the flexible rubber sheet.It is also possible to provide a smoothing parameter for mod-eling noisy data. The surface does not pass through the refer-ence points in case of using the smoothing parameter so theinterpolation will not be exact.
5 Analysis of Methods and Validation of AchievedGeoid Models
The EGM08 model for the three local test areas were fittedinto the GPS/leveling datum by using six different modelsin this research. The fitting were performed by using sin-gle bias parameter with a constant offset (OPM), STM with
4 parameter (FPM) and 7 parameter (SPM), second orderPRM (SOPRM) LSC and SIM transformation, respectively.The achieved transformation parameters were used for fittingof EGM08 to the GPS/leveling datum. If the transforma-tion parameters are used to transform of the actual referencepoints, the transformed values will not match with their truevalues. A difference that is called as residual error occurred inthe original input data values. The standard deviation of theadjusted values for the residuals is traditionally considered asthe internal absolute accuracy of the model. By applying thefitting approach with STM, PRM and LSC (Eqs. 5,7 and 10),the adjusted values for the residuals (vi ) are easily obtainedbased on the least-square model (Eqs. 6, 9). Although it is agood assessment of the transformations accuracy, low resid-uals does not mean that the EGM08 will be perfectly trans-formed into GPS/leveling datum with an expected accuracy.The transformation may still contain significant errors due toan incorrect reference point. Besides, some methods such asSIM give residuals of nearly zero or zero. Since the correc-tive surface passes through the given reference points, thesepoints have no residuals. Generally, cross-validation statis-tic can be considered as a measure of the surface error forassessing the quality of the models. These are very impor-tant indicators to evaluate the appropriateness of the trans-formation models, but they reveal only the inner accuracyrelating to the model and do not provide any informationabout the external accuracy for the achieved model. In orderto determine the quality of the applied models, a more com-prehensive statistical analysis must be performed with vali-dation data. The validation procedure allows evaluating thepredictions using a dataset that is not involved in creatingthe prediction model. For a better measurement of the accu-racy of the model, the discrepancies between fitted EGM08and GPS/leveling-derived geoid heights were calculated andevaluated for each fitting model. Then, some analyses wereperformed in our local test areas. The achieved results aregiven and discussed in detail in the next section of the paper.
The accuracies and compatibilities of the models areassessed using serial statistical tests. Statistical informa-tion, such as, mean error (ME), root mean square error(RMSE), standard deviation (STD), mean absolute error(MAE), etc., can be used for understanding of goodness ofthe model. Results are generally summarized as ME, MAEand RMSE. ME can be described as the average differ-ence between the measured and estimated values. RMSEindicates how closely model predicts the measured values.The smaller this error, the better predictions could be per-formed [44,46,47,50]. Our evaluation results are summa-rized in Table 2. The performances of models were mea-sured by considering RMSE of discrepancies and residualsindicate how good the derived transformation is. Among theevaluated transformation model, the LSC and SIM methodsgive the minimum RMSE and minimum spread of residuals
Fig. 5 Geoid height discrepancies in checkpoints for FPM
versus the others. The detailed improvement process can bealso seen in the contour plots of the discrepancies (Figs. 5,6, 7, 8). Although the major part of inconsistencies can beminimized by STM and PRM sometimes, the high-frequencydistortions are often too complicated.
STM and PRM for Izmir area give us similar accuracywith LSC and SIM unlike other areas. Nevertheless, these
approaches may not always produce the expected results forall cases. The magnitude of discrepancies and the high RMSEvalues show that LSC and SIM works well for Istanbul andAntalya areas versus STM and PRM. According to the resultsillustrated in Fig. 9, there is a significant improvement forelimination of distortion effects with LSC and SIM for Istan-bul and Antalya areas.
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Fig. 6 Geoid height discrepancies in checkpoints for second order PRM
Fig. 7 Geoid height discrepancies in checkpoints for LSC model
LSC and SIM approaches are slightly precise than theSTM and PRM for all test areas according to standard devi-ation (about 4–5 cm). But, there is no significant differencebetween LSC or MSC and the STM or PRM in terms of stan-dard deviations. Standard deviations of the discrepancies var-ied between 6 and 7 cm, respectively, for DUEGM08, OPM,FPM, SPM and SOPRM models. This is also clearly seenin Table 2. Besides, the mean values of discrepancies areremarkable. The mean value of discrepancies ranged from0.3 to 10 cm. This value can be considered as bias. Basedon the RMSE values, it may be seen that the LSC and SIMmodels give an accuracy of about 4–5 cm, SPM gives anaccuracy of about 6 cm, SOPRM and FPM give an accuracyof about 7 cm, and finally OPM gives an accuracy of about8–10 cm.
The discrepancies shown in Fig. 9 are compared betweendirect use of EGM08 and the SIM model. In a sense, the bestand the worst results are compared here. This comparisonis extremely important to emphasize of the success of theSIM model in terms of distortion modeling. As a result, thediscrepancies varied between −0.15 and 0.08 m, the averagewas −1.4 cm, the standard deviation was 0.039 m and theRMSE was 0.041 m for Antalya. For the Istanbul test areas,the RMSE was obtained as 0.049 in the range of −0.139 to0.112 maximum and minimum values with 4.9 cm standarddeviation and 0.3 cm mean. In Izmir area, the discrepanciesvaried between−0.15 and 0.18 m, the average was 0.6 cm, thestandard deviation was 0.052 m and the RMSE was 0.052 m.
According to the above results, it can be concluded that thecommonly used low-order models (OPM, FPM and SOPRM)
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Fig. 8 Geoid height discrepancies in checkpoints for MSC model
Fig. 9 Distributions of geoid height discrepancies for checkpoints in 3 test areas
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0.0
30.0
60.0
4.1 4.9 5.2 4.7 4.8 5.8 5.5 5.4
5.6 7.2 6.1 6.3 6.5 7.2 6.1 6.6
7.4 7.5 6.2 7.0 8.4 11.3 7.7 9.1
22.6
42.2
56.8
40.5
SIMLSCSPMSOPRMFPMOPMDUEGM08
Fig. 10 RMSE for all methods examined in test study
do not offer any significant improvement for the fittingbetween the EGM08 and the GPS/leveling data over the localtest areas. On the other hand, SIM and LSC methods enhancethe fit between the EGM08 and GPS/leveling heights by 4.7and 5.4 cm, respectively (i.e., compared to the performanceof the DUEGM08) (Fig. 10).
After datum-related biases and distortion effects are cor-rected, the standard deviations of the discrepancies will bemore informative and the model accuracy in terms of RMSEwill be increased at the level of 4–5 cm for EGM08.
6 Conclusion
Despite the fact that, gravimetric or geometric geoid model-ing with GPS and leveling data have great significance fortransformation of GPS-derived ellipsoidal heights [52–55],it is difficult to determine the orthometric heights of the ref-erence points by using precise geometric or precise trigono-metric leveling in challenging topography such as Turkeyand similar regions.
When an adequate number of properly spread out refer-ence points are used for surface modeling, accuracies of thelocal models used in Turkey about ±4–5 cm. The actual accu-racy, of course, depends on the available data, their accuracy,and their spatial distribution. In addition, the ellipsoidal andorthometric height accuracies and interpolation method areother significant factors for the accuracy of modeling. Thepossible vertical deformation in the leveling network andthe vertical datum also play an important role on the accu-racy. On the other hand, Regional height datum of Turkeyis created based on the Antalya tide gauge station. Helmertorthometric heights are often used in practical applications.The accuracy of GPS-derived ellipsoidal heights of referencepoints in the network is 3 cm approximately.Also, the accu-racy of orthometric heights depends on the distance of theregion to Antalya tide gauge station, it varies between 0.3
and 9.0 cm throughout Turkey. When considering distancefrom station of the project area, orthometric height accu-racy of the GPS/leveling reference points can be regardedas 2 cm. So, according to the rule of error propagation, theGPS/leveling data provides about 3–4 cm accuracy for perbenchmarks. Consequently, the accuracy provided by localgeometric models in the test areas of Istanbul, Antalya andIzmir is about ±4–5 cm. The other option is use of nationalgeoid model called TG03 with an accuracy about ±10 cmfor the whole of Turkey.
Moreover, the absolute and relative accuracy of theimproved EGM08 model considered as 5 cm and 1.8 ppm,respectively, by the approach that is proposed in this study.The tolerance value for leveling is 12–20 km2 for the thirdorder leveling application in Turkey. It can be convertedinto relative accuracy as 3.8–1.2 ppm for distances between10 and 100 km. So, the resulting geoid models will meetthis requirement for GPS leveling application in test areas.This kind of models for these regions can be replaced ver-sus the geometric geoids for intermediate accuracy level(about 5 cm). This is a very good performance for improvedEGM08. It is also equivalent to the accuracies of regionaland local quasi-geoid models used in test areas. Of course,the final accuracy of the transformation of any geoid mod-els to GPS/leveling datum based on corrector surface fittingtechniques depends on the distortion, the size of the area,the height variations, distribution of reference points andmodel. When statistical evaluation of discrepancies for alltransformation techniques is taken into consideration, it canbe seen that SOPRM, SPM, LSC and SIM have very reason-able and accurate solutions. Especially, very smooth correc-tor surfaces can be fitted and the distortions on the referencepoints can be highly eliminated by LSC and SIM. However,the mathematical models are complex and their computationis difficult over other methods.
In this respect, the proposed approach in this research willprovide positive contributions for application of GPS level-ing in terms of speed, time, cost and economy according togeometric or gravimetric methods.
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