Estimating the Influence of Point Heights in Computing Datum Transformation Parameters Dejan VASI Ć1* 1Republic Geodetic Authority, Belgrade, Serbia
Geonauka Vol. 2, No. 3 (2014)
UDC: 528.21/528.232 DOI: 10.14438/gn.2014.18 Typology: 1.01 Original Scientific Article
Estimating the influence of point heights in computing datum transformation parameters Dejan VASI Ć1* 1Republic Geodetic Authority, Belgrade, Serbia
Abstract . Physical heights in geodesy are expressed through the theory of potential, and their definition is based on the concept of geoid and equipotential surfaces. When defining the conceptually simpler ellipsoidal heights, only the geometric aspects of the Earth’s body are considered. In recent years it becomes relatively easy to determine the ellipsoidal heights by using modern technologies of global positioning. However, these heights are measured from the geocentric ellipsoid, while, for the determination of the ellipsoidal height related to the reference ellipsoid, it is necessary to know the geoid undulation relative to the considered ellipsoid. The results of the global navigation satellite system measurements must be transformed into the state coordinate system. This procedure is called the datum transformation. To carry out the transformation it is necessary to know the ellipsoidal heights in both systems. As these heights are often not known in the state coordinate system, they are replaced by known values of the physical heights. The datum transformation based on the indicated approximation certainly changes the values of the determined transformation parameters, as well as the values of the transformed coordinates. The intensity of these changes depends on several factors, but the change of the transformation results itself affects geodetic computations in different ways. Keywords : ellipsoid, undulation, physical heights, ellipsoidal heights, coordinates, geodetic datum
* Dejan Vasi ć> [email protected] 17
Geonauka Vol. 2, No. 3 (2014) or so-called the reference ellipsoid. The reference 1 Introduction ellipsoid approximates the Earth best in a limited area. In this paper the usage of the physical heights in The ellipsoids of this type have been used in the past the datum transformation is discussed, in terms of its as the datum of the horizontal networks, and often influence on the calculated transformation parameters different ellipsoids were used by the individual states and the transformed coordinates. These heights are when they were developing their own networks. In often used as a substitute for the unknown ellipsoidal that case, the terrestrial methods were used for heights in many surveying tasks. The same is in the topocentrical ellipsoid positioning. The Earth’s global case of the datum transformation, where the ellipsoid should approximate the entire Earth in terms ellipsoidal heights should be originally used, but often of the method of the least squares, i.e. the volume of are approximated with different types of the available the ellipsoid should be equal to the volume of the physical heights. Earth. Such ellipsoid is geocentric, and its positioning The height of a point on the physical surface of is made by the extraterrestrial methods. the Earth, in terms of geodesy, means the distance between the equipotential surface of the point and the 2 Height systems in geodesy equipotential surface that is accepted as a vertical The basic classification of the heights in geodesy geodetic datum [9]. The height difference between considers two categories: the ellipsoidal (geometrical) two points is the distance between their equipotential and the physical heights [8]. A conceptual difference surfaces. The geodetic datum is the surface on which between these heights is manifested in the fact that, in the coordinates are calculated, namely, it is a set of the defining the physical height systems, the physical parameters and the constants presenting a aspects of the Earth’s body are taken into account, mathematical definition of the surface on which the while the determination of the ellipsoidal heights does point coordinates are calculated. In the altitudinal not follow the natural, but only the geometrical geodetic networks the datum is, in fact, a reference characteristics of the Earth’s body. Modern surface from which the heights are calculated. Only technologies used in geodesy provide the opportunity the ellipsoid can be used as the datum for the of a relatively simple determination of the ellipsoidal horizontal networks, while the altitudinal networks point heights. However, considering that all geodetic can use a geoid, a quasigeoid and a zero equipotential works are, on one way or another, related to the surface as the geodetic datum [2]. physical body of the Earth, for practical purposes it is The geoid is a gravity equipotential surface, necessary to define the physical height systems that which at the sea coincides with a perfect calm level of would fit the heights into the real world. There is no the sea and on the mainland goes underneath the universal system of the heights that would meet all the continents. It is the best approximation of the Earth in criteria of the theoretically ideal height system. the terms of a size and a shape. Because the Therefore, there are more physical systems in use: equipotential surfaces are virtually everywhere geopotential elevations, dynamic, orthometric, normal horizontal, they have a very strong physical meaning and spheroidal heights [2]. These systems differ of horizontality and therefore the meaning of according to how the gravity acceleration is treated verticality that is perpendicular to the equipotential [5]. Which one of the systems will be used, depends surface. Therefore, the study of the equipotential primarily on the amount of the available data and the surfaces has received much attention, because the type of a task. concepts of vertical and horizontal can be defined in Let the ellipsoidal normal through a given point P physical terms only through the theory of the potential on the physical surface of the Earth, to penetrate and the equipotential surfaces. Although regarded as a surface of the ellipsoid at point Q (Fig. 1). A segment mathematical representation of the Earth, there is no of the ellipsoidal plumb-line from the point P to the geoid equation with closed – form solution, because point Q is called the ellipsoidal (geodetic) height of of its irregular shape. The geoid can only be the point P and is denoted by h. The distance between mathematically expressed through functional series the geoid and ellipsoid, calculated over the ellipsoidal with the infinite number of coefficients [3]. plumb-line is called the geoid undulation or the The simplest mathematical regular body that geoidal height and is denoted by N [8]. The represents a fairly good geometric approximation of undulation can be positive (the geoid above the the general shape of the Earth is an oblate ellipsoid of ellipsoid) and negative (the geoid below the ellipsoid). revolution, which can be the Earth’s global ellipsoid The departures of the geoid from the best fitting 18
Geonauka Vol. 2, No. 3 (2014) geocentric ellipsoid range approximately between the Earth’s global ellipsoid, not the reference −100 m and +100 m globally [10]. The ellipsoidal ellipsoid. The i ndirect methods include determination plumb-line is perpendicular to the ellipsoid , and in the of the ellipsoidal heights indirectly through the normal gravity field represents the equivalent to the physical heights. It is b ased on the division of the true plumb-line, which is perpendi cular to the geoid. ellipsoidal heights into two mutually independent components: the phy sical height and the distance between the ellipsoid and the reference surface of the physical heights. In this regard, the orthometric and the normal heights are used. Although the physical and the ellipsoidal heights refer to the different spatial curves, because the true and the ellipsoidal plumb-line do not match completely (Fig. 1), due to the small angular values of the vertical deflections, the equations (1) are valid with the sufficient accuracy. (1) Figure 1. Relation between geoid and ellipsoid In equations (1), HO represents the orthometric N The angle ( difference in direction s) between the height, H is the normal height, and denotes the true and the ellipsoidal plumb-line is called the quasigeoidal height (a.k.a. a height anomaly) which is vertical deflection, and it is denoted by θ . The the vertical distance between the quasigeoid and the vertical de flections are very small angles that can ellipsoid. The q uasigeoid is the reference surface for reach the order of magnitude of several tens of the normal hei ghts, but it is not the physically arcseconds, which means that the true and the meaningful equipotential surface [3]. The orthometric ellipsoidal plumb-line almost coincide . This and the normal heights are obtained by leveling assumption about the coinciding of these two plumb- (measurements), while the geoidal and the lines comes to the fore d uring the transform ation of quasigeoidal height s can be determined only by the heights from the physical to the ellipsoidal system computational techniques based on the data of the and vice versa. The vertical deflections , although gravimetric measurements . In recent years, there are small in numerical values, have a great conceptual global geopotential model s of the Earth in use, which significance in geosciences and astronomy because enable easy calculation of the geoidal height in the they virtually define the slope of the geoid relative to relation to the global ellipsoid. These models are the ellipsoid. Therefore, the vertical deflections are obtained by solving the b oundary-value problems of the basic components of the geoid and together with the spheres gravi tational potential , and consist of the the undulations define its position relative to the coefficients related t o a progression of the Earth’s ellipsoid. gravi ty acceleration potential into spherical harmonic The ellipsoidal heights can be determined b y the series. The p hysical height s are determined with the direct and the indirect methods [2]. The direct method accuracy of 1 - 2 cm, while the geoidal and the means measuring of the global navigation satellite quasigeoidal height s can be determined with the system (GNSS). Based on the GNSS measurements, accuracy of 1 - 2 m, because of the hypotheses about the rectangular coordinates in the three -dimensional the Earth’s internal structure. Cartesian coordinate system related to the global ellipsoid are obtained. These rectangular coordinates 3 Datum transformations are then transformed into the curvilinear geodetic When the coordinates of the geodetic points are coordinates: the geodetic latitude, the geodetic computed in the different coordinate systems, there is longitude and the ellipsoid height, denoted with, B, L often a need to transform them from one coordinate and h, respectively. In this way, the ell ipsoidal heights system to another. This implies the determination of are determined with an accuracy of 100 m using the appropriate coordinate values in another absolute positioning, or 1 cm using relative coordinate system for a group of the points whose positioning. The GPS technology enables the coordinates are known i n the original coordinate determination of the ellipsoidal heights only related to system. Such a transition from one set of the
19
Geonauka Vol. 2, No. 3 (2014) coordinates to another, apropos recalculation addition to the different position, the ellipsoids differ (conversion) of the coordinates is called a coordinates in the shape and the size. The main parameters that transformation. The connection between the two define the shape and the size of the ellipsoid as a coordinate systems in order to transform the geometric body, and which are included in the coordinates can be achieved through transformation definition of the geodetic datum, are a semi–major parameters, and it is possible if there is the necessary and a semi–minor axis, a and b, respectively. On the number of common (identical) points whose basis of these, other important geometric parameters coordinates are known in both systems [6]. It is of the ellipsoid can be derived, like in (2). The preferred that there are more identical points in order ellipsoid is often given with its semi–major axis, and to be able to determine the transformation parameters one of the derived parameters instead of the semi– with greater accuracy and reliability. In effect, these minor axis. Commonly used derived parameters are parameters are the differences between the coordinate the first ellipsoid flattening f, and the first numerical systems. Two coordinate systems may differ in the eccentricity e: translation (location), the orientation, the scale, and , (2) theoretically in the polarity. The datum transformation is the type of the coordinate transformation that involves the transformation between two geodetic Table 1 shows the definition parameters of some datums. Often the coordinates referring to one datum significant ellipsoids in use. must be converted into the coordinates related to the For the complete datum transformation (three- other datum, because, through the past, various dimensional) it is necessary to determine seven datum datums were developed and used, many of which are transformation parameters: three translations over the still in use today. In addition, the technology of the coordinate axes, three angles of rotation around the satellite positioning is related to the geocentric axes and one scale parameter. The coefficients of (absolute) datum, while the majority of the datums progression of the Earth's gravity field potential and used for the national horizontal networks are the fundamental constants such as the speed of light in topocentric. The modern technology allows the vacuum, the Earth’s rotation speed or the geocentric geocentric positioning of the Earth's global ellipsoid, gravitational constant, can also be parts of the datum which is the absolute datum. The reference ellipsoids transformation, especially if the information about the should also be geocentric, but in praxis they are not. datum are received through the analysis of the satellite Most of the countries established their national orbits. Such a datum is, for example, World Geodetic geodetic datum at a time when geocentrical System WGS84. The origin of this system is located, positioning of the oblate ellipsoid of revolution and its by definition, in the Earth’s centre of mass, the z axis preferred orientation could have been provided only passes through the Conventional Terrestrial Pole approximately. Therefore, the classical definition of (CTP), the x axis passes through the Greenwich datum mainly consisted in requiring that the chosen meridian and the y axis complements the three- ellipsoid, with its location, shape and size represents dimensional Cartesian coordinate system of the right- the best approximation of Earth on a limited national handed rule orientation. territory. The result of this approach is the fact that the Processing the results of the GPS measurements, local geodetic datums, which are still in use, do not the rectangular coordinates of the points are obtained, match the position, orientation and scale of either the referring to the very system WGS84. In order for absolute datum, or mutually. The connection between these results to be suitable for use in some geodetic these datums is carried out through the datum tasks, it is necessary to transform the rectangular transformation. In praxis, the track of the geodetic coordinates X, Y and Z into the geodetic coordinates: networks continuity is kept. That means as more as the geodetic latitude B, the geodetic longitude L and possible the old network points should be used during the ellipsoidal height h. the developing process of the new networks. In this way the datum transformation between the networks becomes possible, which is conditioned by the existence of the identical (common) points.
The datum of the horizontal geodetic network is the bi–axial (oblate) ellipsoid of revolution, generated by rotating an ellipse around its minor axis. In 20
Geonauka Vol. 2, No. 3 (2014) Table 1. Definition parameters of some commonly used ellipsoids Ellipsoid Sem i–major axis a [m] Reciprocal flattening 1/ f Bessel 1841 6 377 397, 155 299, 1528128 Clarke 1880 6 378 249, 145 293, 465 Krassovsky 1940 6 378 245, 000 298, 3 GRS 1980 6 378 137, 000 298, 3 WGS72 6 378 135, 000 298, 26 WGS84 6 378 137, 000 298, 257223563
Dimension N is called the radius of the curvature in the prime vertical, and it can be calculated as: These geodetic coordinates relate to the oblate (4) ellipsoid (datum) of revolution, associated with the . system WGS84, with the parameters shown in Table 1. Coordinates B, L and h are better suited for the The seven-parameter transformation is usually in perception of the points posit ion than the rectangular use, named Helmert by the German surveyor coordinates. Namely , unlike the rectangular, geodetic (Friedrich Robert Helmert), who affirmed it. It is coordinates indicate intuitively correctly the widely used in surveying to determine the approximate position of the point on the Earth. But transformation parameters in the three -dimensional their lack is reflected in the fact that due to their polar datum transformation. In t hat transformation three singularities do not cover the entire Earth's surface. parameters of translation, three parameters of rotation The poles are two singular points for which the around the axes and one scale parameter are geodetic longitude is not defined. Therefore, the determined (Fig. 3). Only one scale parameter means datum transformation is operatio nally performed that after executed transformation the size of the using the corresponding rectangular coordinates. coordinate system changes , while the shape remains There is a mathematical relationship between the the same [6]. The specificity of the datum rectangular and the geodetic coordinates (Fig. 2). transformation is related to also very small rotation angles, as a mitigating factor during the calculation process. The reference ellipsoids were positioned to approximate the best Earth in some territory, but despite the limited technology they were not positioned quite arbitrarily, but certain principles were respected. Care has been taken to create appr oximately geocentric ellipsoid such that the minor axis coincides with the Earth ’s axis of rotation, etc.
Figure 2. Rectangular and geodetic coordinates The rectangular coordinates calculation is based on the equations:
cos cos , cos sin , 3 ∙ 1 sin . Figure 3. Helmert three-dimensional transformation Theoretically, to determine seven transformation parameters, it is necessary to create the system of the 21
Geonauka Vol. 2, No. 3 (2014) seven equations. For that purpose, two identical points (10) and another coordinate of the third identical point = , should be known. This means that at least three points where the number of degrees of freedom in (10) is whose coordinates are known in both systems should = − be known. The transformation parameters are then r3 n 7 , and n is the number of the identical points. evaluated through the adjustment by the method of the The experimental standard deviations of the unknown least squares. This method is based on the principle parameters are calculated as: that the sum of the squares of the measurement result (11) corrections (residuals) is minimal, i.e. = ∙ n In (11) are diagonal members (on the main T= 2 = v v ∑vi min . In this case, the differences diagonal) of the cofactor matrix . = i 1 The described general transformation model is in between the given coordinates of identical points and the literature often referred to the model Bursa – Wolf. their coordinates obtained by transformation are If this model is applied to the geodetic networks of a minimized. In this way the optimal values of datum smaller scope, the translation and the rotation transformation parameters are obtained. parameters are highly correlated. An alternative model For each identical point the equations of is Molodensky – Badekas model that uses a set of corrections can be written: identical points centroid coordinates in computation, − ∙ + ∙ + ∙ + − given by: = − ∙ + ∙ + ∙ + , .(12) = ∑ , = ∑ , = ∑ = + ∙ − ∙ + ∙ + − = When it comes to the final results, the two = + ∙ − ∙ + ∙ + (5) models give different parameters of translation, but = − ∙ + ∙ + ∙ + − = the rotation parameters and the scale factor remain the = − ∙ + ∙ + ∙ + . same [1]. where the coordinates x' , y' and z' are formally considered as measurements. Equations (5) in matrix 4. Processing the test example form are as follows: A practical research was conducted in the (6) = + network of Belgrade city. The points in this network In (6) v is a vector of corrections, A is the design have the known coordinates in the state coordinate matrix, x is the vector of the unknown transformation system. The horizontal coordinates in the state parameters, and f is the vector of the equations free coordinate system of Serbia are reduced rectangular members. If the weight matrix P meets the coordinates x and y in the Gauss – Krüger requirement of P=E (the same weights for all transverse cylindrical conformal projection of three- measurements), the estimated values of the unknown degrees meridian zones [4]. The datum of this transformation parameters can be obtained by solving network is the Bessel ellipsoid, with the definition the system of the normal equations: parameters shown in Table 1. The height reference
(7) system is the physical system, similar to the = −( ) = − = − . spheroidal heights. For this research, it can be When these estimated values of the unknown assumed with the sufficient accuracy that the geoid is parameters are included in (6), the corrections of the the reference surface for those heights. At the points measurements or, in this case, deviations on the of the mentioned network, the GPS measurements identical points can be obtained: were made, and the rectangular geocentric coordinates (8) in WGS84 system were obtained. The definition = + . parameters of the oblate ellipsoid related to this The control of the calculated unknown system are also given in Table 1. Between these two parameters can be done by testing the equality: datums Helmert transformation (model Molodensky – (9) Badekas) is carried out, i.e. seven transformation = + . parameters are calculated. The experimental reference standard deviation S0 is determined by the equation: To perform the transformation in a three- dimensional Cartesian coordinates, the positions of the points are expressed in the appropriate 22
Geonauka Vol. 2, No. 3 (2014) coordinates, XB, YB and ZB in the spatial Cartesian indicators of accuracy. The angles of the rotation ε ε ε coordinate system refered to the Bessel ellipsoid. For around the coordinate axes x , y , and z in radians that purpose, the rectangular horizontal coordinates are converted into the rotation angles α , β and γ , from the state coordinate system are first transformed respectively, expressed in arc seconds (sexagesimal into the geodetic coordinates on the Bessel ellipsoid units of measure). [1]. Subsequently, their transformation into the spatial It is indicative that all transformation parameters coordinates in the Cartesian coordinate system refered obtained by using the ellipsoidal heights have lower to the Bessel ellipsoid is following, according to the accuracy compared to the corresponding parameters equations (3). This step is crucial in the entire obtained on the basis of only the physical heights. The research. In the equations (3) figures the ellipsoidal reason is probably the accuracy of the input values, height h, which in this particular case relates to the specifically the heights, in the adjustment process. reference ellipsoid. This height should be obtained The ellipsoidal heights are obtained by summing the from (1), i.e. by summing the physical height of the physical heights and the geoid undulations, so point and the height of the reference surface of the according to the law of error distribution, the physical height system over the ellipsoid. Therefore undulation determination error is also added to the the ellipsoidal height is often not known because of error of the physical height determination. In other the lack of the proper geoid undulation values or the words, the approximate ellipsoidal height is more height anomalies. Hence in practice, for the purpose accurate than the right one obtained by including the of the coordinates transformation and other undulations in the calculation, regardless of their calculations, the approximation that replaces absolute values. Testing the significance of the ellipsoidal heights with known physical heights is differences of the transformation parameters is used often. The introduced assumption is that H = h , performed by comparing the differences of the or N = 0. This approach in the datum transformation parameters obtained in two variants of the inevitably entails a change in the estimated values of ∆ =H − h the transformation parameters, and thus indirectly transformation x x x and double accuracy of reflects on the values of the transformed coordinates the differences, obtained by applying the law of error of the points. The practical part of the research is done distribution: precisely in order to determine the intensity of the (13) mentioned impact, and whether the change in the ∆ + values of the transformed coordinates of the points In order to consider that the difference between associated with the use of physical instead of two transformation parameters is statistically ellipsoidal heights in datum transformation is insignificant, a condition must be fulfilled: < significant for the works in geodesy. First the . The testing results are shown in Table 3. 2 Δ transformation parameters were determined with the The testing results show that the transformation introduction of the aforementioned approximation, by parameter of the scale q does not differ significantly using the physical instead of the ellipsoidal heights in from its respective equivalent, nor the rotation angle (3). After that, the adjustment was made and the around the x axis. For other parameters significant transformation parameters calculated with the differences are ascertained. However, these ellipsoidal heights used in (3). Necessary ellipsoidal differences between the individual transformation heights are determined on the basis of the known parameters could be a sort of an illusion, because the physical heights and the map of the geoid undulations differences between the coordinates in the state related to the Bessel ellipsoid [7]. coordinate system (SCS) obtained through the A part of Belgrade network used for this research transformation are more important. These coordinates covers an area of about 453 km 2 and includes 19 essentially represent the final product of the datum points whose coordinates are known in the state transformation procedure. Their analysis provides an coordinate system and in which the GPS insight into what are the average and the maximum measurements are made. The average mutual distance differences of the transformed coordinates and the between the points is 5.6 km. The average physical heights obtained by using different transformation height of the network points is 131.3 m, and the parameters on the identical points. The results are average geoid undulation at identical points is -0.67 given in Table 4. m. The estimated transformation parameters are given in Table 2, with their mutual differences and 23
Geonauka Vol. 2, No. 3 (2014) Table 2. Transformation parameters for Belgrade network Ellipsoidal heights h Physical heights H used used H h H h S H S h − Parameter x x x x x x t x -690.167 m 1.03 cm -690.613 m 1.48 cm 0.45 m t y 197.589 m 1.03 cm 197.423 m 1.48 cm 0.17 m t z -483.309 m 1.03 cm -483.782 m 1.48 cm 0.47 m α 5.31" 0.45" 6.46" 0.64" -1.15" β 1.48" 0.29" 3.60" 0.42" -2.13" γ -14.65" 0.40" -16.46" 0.58" 1.81" 1.73 0.090 q 0.99999391 1.21 mm/km 0.99999382 mm/km mm/km Table 3. Results of statistical test for Belgrade network
Parameter ∆x Relation 2m α m α t x 44.60 cm > 3.60 cm 1.80 cm t y 16.65 cm > 3.60 cm 1.80 cm t z 47.29 cm > 3.60 cm 1.80 cm α 1.15" < 1.57" 0.79" β 2.13" > 1.02" 0.51" γ 1.81" > 1.41" 0.71" q 0.09 mm/km < 4.23 mm/km 2.11 mm/km Table 4. Characteristical values of transformed coordinates differences in Belgrade network Difference Average Maximum SCS SCS y()()− y h H 0.61 mm 2.16 mm SCS SCS x()()− x h H 0.40 mm 1.50 mm H()()− H h H 5.36 cm 10.72 cm
It is evident that the largest deviations are in the recalculating from one coordinate system to another is heights, and the same amount to several centimeters, named the coordinate transformation. The special or in extreme cases reach an order of a magnitude of form of this transformation is the datum decimeter. This epilogue of calculation and the results transformation which implies that the coordinate analysis are somewhat expected, since the systems are based on different geodetic datums, as in transformation parameters change because of the the case of the coordinate transformation from the previous changes just in heights, that are used for the geocentric to the local geodetic datum. The three- auxiliary coordinate transformations. On the other dimensional transformation is the most frequently hand, the average change of the horizontal coordinates used, specifically the Helmert transformation. The is on the sub-millimeter level, or in the extreme cases ellipsoidal heights of the points figurate in the several millimeters. calculating process. As they are not well known, they are often replaced with available physical heights. The 5. Conclusion difference in these heights inevitably leads to the The method of determining the transformation changes in the final results. Talking about the datum parameters for the purpose of the points coordinate transformation, the transformation parameters and the
24
Geonauka Vol. 2, No. 3 (2014) ransformed coordinates are those that are changed. Estimating how the different heights influence the datum transformation is presented in the test-field of the Belgrade city network. The points which positions were determined using the GPS techonology have been transformed into the state coordinate system. After the changes of heights in both networks, the different digits has appeared. The biggest deviations appeared right in heights. There is no importance in the result differences for a geodetic survey, but for developing the geodetic networks. The differences in the heights are relatively big - ten centimeters, but only if the heights in the networks are determined through GPS measuring and transformation. The mentioned differences have no importance for the reduction of the measuring results to the computing surface.
References [1] Aleksi ć, I., Gu čevi ć, J., Popovi ć, Ј.: „Geodetski premer – zbirka rešenih zadataka“, Gra đevinski fakultet Univerziteta u Beogradu, 2009. [2] Del čev, S.: „Geodetske referentne mreže – zbirka rešenih zadataka“, Gra đevinski fakultet Univerziteta u Beogradu, 2009. [3] Heiskanen, W., Moritz, H.: „Fizi čka geodezija“, Gra đevinski fakultet Univerziteta u Beogradu, Institut za geodeziju, 2000. [4] Jovanovi ć, V.: „Matemati čka kartografija“, Vojnogeografski institut, Beograd, 1983. [5] Markovinovi ć, D., Špodnjak, T., Bjelotomi ć, O.: „Utjecaj sile teže u geometrijskom nivelmanu“, Ekscentar, br. 14, pp. 61–65, 2011. [6] Mihailovi ć, K., Aleksi ć, I.: „Koncepti mreža u geodetskom premeru“, monografija, Privredno društvo za kartografiju Geokarta, Beograd, 2008. [7] Muminagi ć, A.: „Preliminary investigation in establishment of the Yugoslav first order triangulation network: orientation, isostatic corrections, astronomic leveling and the figure of the geoid“, Beograd, 1967. [8] Odalovi ć, O.: „Fizi čka geodezija“, Gra đevinski fakultet Univerziteta u Beogradu, 2010. [9] Vani ček, P., Krakiwsky, E.: „Geodezija, koncepti“, Savez geodeta Srbije – Geodetski žurnal, Beograd, 2005. [10] Vani ček, P., Kingdon, R., Santos, M.: „Geoid versus quasigeoid: a case of physics versus geometry“, Contributions to Geophysics and Geodesy, Vol. 42/1, pp. 101–117, 2012.
25