COMP ARISON OF DIFFERENT ALGORITHMS TO TRANSFORM GEOCENTRIC TO GEODETIC COORDINATES. Alireza Amiri Seemkooei

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1 Survey Review 36, 286 (October 2002) COMP ARISON OF DIFFERENT ALGORITHMS TO TRANSFORM GEOCENTRIC TO GEODETIC COORDINATES Alireza Amiri Seemkooei Dept. of Surveying Engineering The University of Isfahan, 81744, Isfahan, IRAN ABSTRACT The conversion from geocentr.ic to geodetic coordinates is among of the most important tasks in computational geodesy. This conversion is often complicated and time consuming. In order to deal with the large quantities of points, it is important to select the fastest possible algorithm without compromising the accuracy. From the numerical standpoint, the available conversion algorithms have to be solved either iteratively, through a linearization, or using an algebraic equation offorth degree. In this paper, five iterative algorithms and one closedform solutionfor transformationfrom geocentric to geodetic coordinates are compared for numerical efficiency. It is concluded that the simple iteration method implemented in Bowring's algorithm executes faster than the others. Only one iteration is sufficient to produce coordinates accurate to the comparable level of O.3. J0-16 rad, which exceeds the requirements of any practical application. INTRODUCTION By definition, a geocentric coordinate system is a system whose oflgin (0,0,0) coincides with the centre of mass, C, of the Earth, and whose axes are fixed by convention. The most common geocentric system used in geodesy is the Conventional Terrestrial (CT) system, which is oriented so that the z-axis points towards the Conventional International Origin(CIO), the x-axis lies in the Conventional Greenwich Meridian, and the y-axis makes, with the other two axes, a right-handed Cartesian triad [7]. Positions in the CT-system are sometimes given in Cartesian coordinates (x,y,z) and sometimes in curvilinear geodetic coordinates (<p,a,h), i.e., in geodetic latitude, longitude, and height. The use of curvilinear geodetic coordinates, however, requires the introduction of a geocentric reference ellipsoid with major semi-axis a, and minor semi-axis b. The geodetic height h, sometimes called the ellipsoidal height, is the height of a point above the surface of that reference ellipsoid. The conversion between these two coordinates is one of the most important tasks in computational geodesy. Curvilinear geodetic coordinates are easily transformed into geocentric coordinates by the following formulae [6]: X] [(N + h) coscpcosa] y = (N + h)coscp sina, [ z (Nb 2 / a 2 + h) sin cp where N, the prime vertical radius of curvature, is given by N= a (1 - e SIn cp 2 2 )112' where e 2 is the eccentricity of the ellipsoid and is given by 2 _ a 2 - b 2 e a (1) (2) (3) 627

2 CONVERTING GEOCENTRIC TO GEODETIC COORDINATES The inverse transformation is more complicated and it has to be solved either iteratively, through a linearization, or using an algebraic equation of forth degree. Several authors have suggested different iteration algorithms for doing the inverse conversion. When we are dealing with the large quantities of points, it is important to select the fastest possible algorithm without compromising the accuracy. In the following, we have evaluated different algorithms and compared their convergence speeds and accuracies. In all the algorithms, A is evaluated from the first two equations (1) as follows: There is another equation for longitude, equations (1) as well as [6] A = arctan I. (4) x which can be obtained from the first two A = 2 arctan y (5) X+~X2 + y2 The advantage of the above equation compared with the equation (4) is that it is not singular at x = O. TRANSFORMATIONS FROM GEOCENTRIC TO GEODETIC COORDINATES Author's Algorithm This approach employs the distance p from the minor axis, which equals or, from equations (1) p=~x2 +y2 p = (N + h)cosq>, (6) (7) z = (N(I- e 2 ) + h)sinq>, (8) Equation (8) can be rewritten as follows: z+ N e2sincp= (N + h) sin q>. (9) Equations (9) and (7) yield z + Ne 2 sinq> tan q> = -----, p (10) or (11) Rearranging the above equation gives or simply _ z + N e 2 sin 3 cp tanq>- 2 3 p- N e cos cp Therefore, q> is calculated using the following iterative procedure: (1) Compute an initial guess of <Po using the equations (7) and (8) (h=o) (12) (13) 628

3 A. A. SEEMKOOEI (2) Evaluate N using the formula z q> 0 = arctan[ 2 ]. p(l- e ) (14) (15) (3) Update the value of <p using _ z + N e 2 sin 3 q> (16) qj - arctan 2 3. p- N e cos q> The procedure iterates between steps 2 and 3 until <p converges. After obtaining <p, h can be calculated using equation (7), i.e., h=-p--n. (17) cos q> Bowring's Algorithm One example for the transformation of geocentric into geodetic coordinates without iteration but with an inherent approximation was proposed by Bowring [2]. In this algorithm the geodetic latitude and height can be derived by and where is an auxiliary quantity and z + be' 2 sin 3 e q> = arctan[ 2 3 ], p- ae cos e h = -p- - N cosq>, a z (J = arctan(-), bp (18) (19) (20) (21) is the second eccentricity. Bowring [2] shows that the algorithm needs no iteration within 400km of the earth surface. Borkowski's Algorithm In this algorithm, we compute the reduced latitude, ~, from the following equation [1]: where f(/3) = 2 sin(/3 - Q) - c sin 2/3= 0, (22) bz a 2 - b 2 Q= arctan(-), c= (23) a p ( )112. ap+bz Equation (22) can be solved for the reduced latitude, ~, using the famous Newton- Raphson method. After obtaining ~, the geodetic latitude and height can be computed using the following equations: 629

4 CONVERTING GEOCENTRIC TO GEODETIC COORDINATES a cp = arctan( - tan f3), b h = -p-- N. coscp (24) Lin and Wang's Algorithm In this algorithm the calculation of the latitude and height can be reduced to a geometry problem as follows. The earth can be modelled as an ellipsoid represented by [5] (25) where (Xe,ye,Ze)are geocentric coordinates on the surface of the ellipsoid, and (x,y,z) are the same as before. A straight line in space passing through the point (x,y,z) and normal to the surface of the ellipsoid can be described by the following parametric equations: x= x 1+ 2m/a' y= Y 1+ 2m/a' Z= z 1+ 2m/b' where m is a parameter and (X, Y,Z) are the coordinates of any point on the straight line. When m varies, (X, Y,Z) moves along the normal line to the ellipsoid. The objective now is to determine the parameter m such that (X,Y,Z) = (Xe,Ye,Ze). Replacing (X,Y,Z) by (Xe,Ye,Ze) and substituting for (Xe,Ye,Ze) in equation (25) from equations (26) yields 222 f(m ) = x + Y + z _ 1 = 0 (27) (a + 2m/a)2 (a + 2m/a)2 (b + 2m/b)2 ' or simply 2 2 f(m) = p + z - 1 = O. (a + 2m/a)2 (b + 2m/b)2 Equation (28) can be solved for the parameter m using the Newton-Raphson method: (1) Compute an initial guess of mo (26) (28) (29) (2) Update the value ofm using the formula where - f(mi- I) mi- mi-i- ---,1-,,..., ftmi-i) 2 2 ftm) = (-4)[ p 3+ Z ] a(a + 2m/a) b(b + 2m/b)3. (30) (31) Step 2 repeats until f(m) converges to zero. After obtaining m, the equation given by p, = 1(1 + im/ a2)1, (32) 630

5 A. A. SEEMKOOEI can be used to calculate (Xe,Ye,Ze), which is the point on the reference ellipsoid corresponding to (<p,a,o).therefore 2 q> = arctan(t)' a z (33 ) b P e and 2 ( 2 1/2 h = [(p - Pe) + Z - Z e)]. (34) Heiskanen and Moritz's Algorithm According to Heiskanen and Moritz [3], to compute the geodetic latitude, we can write the following equation, from equations (7) and (8), z e2n_} tanq>=-[i---]. (35) p N+ h This equation can be solved for the geodetic latitude, <p,using an iterative procedure: (1) Compute an initial guess of <pousing z q>o = arctan[ 2 ].. (36) P(1- e ) (2) Evaluate N using the formula a N= 1/2' (1- e 2 sin 2 q» (3) Compute the ellipsoidal height by h= -p-- N. cosq> (4) Compute an improved value for the latitude using (37) (38) Z 2 N -} q> = arctan-=-( 1- e --). P N +h The procedure iterates from step 2 until <pand h converge. Closed Form Solution The closed form solution uses equations (7) and (8) to obtain p tan q> - z = e 2 N sin q> (39) (40) In this equation, the only unknown is <p,n being a function of q>as well. Substitution of equation (2) in equation (40) yields _ ae2sinq> ptanq>- z- 1/2' (41) (1 - e 2 sin 2 q> ) Dividing the numerator and denominator of the right-hand side by cosq> and squaring the whole equation yields [6] (2 p tan qj - pz tan qj + P - a e) 2 pz z 0 Z + 2 tan qj tan qj =. (42) l-e l-e l-e This is a quartic equation in tanq> in which the values of all the coefficients are known. Standard procedures for solving quartic equations exist (see, e.g., [4]). Once a solution for tan<p is obtained, h is computed from equation (17). 631

6 CONVERTINGGEOCENTRICTO GEODETICCOORDINATES NUMERICALRESULTS In this section, we use the numerical examples to compare the different algorithms. The CPU time of each algorithm are measured until we get acceptable results. The performance of the six algorithms is tested on a set of points at different latitudes and heights. For each point, five heights (1m, 10m, 100m, 1000m, 10 OOOm)are used. Ninety test points in geodetic coordinates (<p,a.,h)are first chosen at 10 interval along the meridian, then geodetic coordinates are converted to the geocentric coordinates (x,y,z) using equations (1). Different algorithms are used to convert (x,y,z) to (<p,a.,h). The errors of the conversion are defined as the errors in computing cpo In order to compare the different algorithms, a program has been written in MATLAB language. Double precision arithmetic, allowing 16 significant digits, is used throughout. The source program is not listed. The CPU time (sec) of each algorithm has been measured until we get acceptable results. Neglecting the number of iterations, this CPU time is a measure for the speed of the algorithms. Tables 1 and 2 summarize the test results for those algorithms. As shown in Table 1 all the algorithms are converged after less than four iterations. The fractions of iterations in the tables are due to the averaging in different latitudes and/or heights. The results show that the number of iterations and processing time in Bowring's as well as Borkowski's algorithms are independent from the latitudes and heights. In the author's, Lin & Wang's, and Heiskanen & Moritz's algorithms, number of iterations and CPU time increase when increasing the heights. To compare the speed of the algorithms, it is clear that Bowring's algorithm is 35% faster than the author's, 50% faster than Borkowski's, 1200/0faster than Lin & Wang's, 800/0faster than Heiskanen & Moritz's, and 70% faster than the closed form algorithm. The last row of the Table 1 indicates that the computational errors in the closed form algorithm are much more than the iterative methods. It is concluded that the iterative algorithms when compared with the closed form are preferable because of rounding errors in that algorithm. CONCLUSIONS Tables 1 and 2 show that the number of iterations and processing time in Bowring's as well as Borkowski's algorithm are independent from the latitudes and heights; but in the other algorithms they are correlated with heights and latitudes except for Lin and Wang's algorithm which is independent from latitudes. As shown, the first five algorithms nearly have errors of the same magnitude. It is also concluded that the iterative algorithms when compared with closed form's are preferable because of rounding errors in that algorithm. Comparing the first two rows of the Table 1 with the other rows shows that Bowring's iterative algorithm is the most efficient and reliable technique. It is clear that this algorithm executes about 35% faster than the author's algorithm; and therefore, it should be the algorithm of choice in the class of other methods. It is concluded that the simple iteration method implemented in Bowring's algorithm executes faster than the others. Just one iteration is sufficient to produce coordinates accurate to the comparable level of rad., which exceeds the requirements of any practical application. 632

7 A. A. SEEMKOOEI TABLE 1. Comparison of computation times (sec), number of iterations, and average error for conversion of"geocentric to geodetic coordinates (latitude and height) in different heights using different algorithms. Algorithm Heights (metre) Average Error(* ) Ave. Bowring CPU time (sec) No. ofltt~ration CPU time (sec) Author No. of Iteration Borkowski CPU time (sec) No. of Iteration Lin & CPU time (sec) Wang No. of Iteration Heiskanen CPU time (sec) & Moritz No. of Iteration Closed CPU time (sec) Form No. of Iteration 250 TABLE 2. Number of iterations for conversion of geocentric to geodetic coordinates (latitude and height) in different latitudes using different algorithms Algorithm Latitudes (Degree) Ave. Bowring Nos. of It Author Nos. of It Borkowski Nos. of It Lin & Wang Nos. of It Heiskanen& Nos. of It Moritz Closed Form Nos. of It. ACKNOWLEDGMENTS Iwould like to thank the editor for his guidance during the reviewing and publication processes. I would also like to appreciate two anonymous reviewers for their meticulous reviewing and constructive comments. References 1. Borkowski, K.M., Accurate algorithms to transform geocentric to geodetic coordinates. Bulletin Geodesique, 63: Bowring, B., Transformation from spatial to geographical coordinates. Survey Review, XXIII, 181: Heiskanen, W.A., H. Moritz, Physical Geodesy. Freeman and Co., San Francisco. 4. Korn, G.A. and T.M. Korn, Mathematical Handbook for Scientists and Engineers. 2-nd edition, McGraw-Hill. 5. Lin, K.C., 1. Wang, Transformation from geocentric to geodetic coordinates using Newton's iteration. Bulletin Geodesique, 69: Vanieek, P., and E.1. Krakiwsky, Geodesy: the concepts. 2nd edition, North Holland, Amsterdam. 7. Vanieek,P., R.R. Steeves, Transformation of coordinates between two horizontal geodetic datums. Journal a/geodesy, 70: