Three-Dimensional Geodesy for Terrestrial Network Adjustment

Transcription

1 VOL. 77, NO. $ 3OURNAI, OF GEOPHYSICAL RESEARCH FEBRUARY 10, 1972 Three-Dimensional Geodesy for Terrestrial Network Adjustment D. M. J. Battelle Columbus Laboratories 505 King Avenue, Columbus, Ohio A rigorous practical application of three-dimensional geodesy to the djustment of a terrestrial network was investigated to determine (1) the pre(.ision obtainable from a given set of measurements and data oua ity: (2) the roles. the types. and the number of geodetic observations required; and (3) the implications of the type of geodetic coordinate system involved and stability problems in the matrix s.vstems for the weighted least squaresolutions. Since the 1962 International Symposium on Three-Dimensional Geodesy, the practical application of three-dimensional geodesy to the adjustment of terrestrial networks has been investigated by authors such as Bacon [1966], Henderson [1968], and Hradilek [1968_]. None of them utilized completely rigorously Hotine's [1959, 1969] formulations of three-dimensional geodesy. They either imposed inadmissible constraints or introduced undesirable classical concepts to achieve a solution. This is a summary of methods and.results of the investigation into a rigorous practical application of three-dimensional geodesy to the adjustment of a terrestrial network, completed in August By simulated observations. the roles and the necessary number of measured distances, horizontal directions, vertical argles, astronomical azimuths, latitudes, and longitudes in this type of adjustment were investigated. A test net with real observations was also adjusted, using a geometric and statistical model as in the simulated net to compare the conclusions based on a purely theoretical analysis to what can be obtained from a network with real observations. The conclusions, which heavily relied on extensive use of statistical tests, are of course dependent on the network configuration, the azimuth of its expanse, and the observations and precision estimates assigned to them, among other factors. For obvious reasons, more than 90% of the tables and graphs of results on which the conclusions are based are not given here, but are given by Fubara [1969]. THE SIMULATED NETWORK The normal meas, rements involved in terrestrial networks are astronomic azimuths, latitudes and longitudes, horizontal angles or directions, lengths, and zenith distances of the lines. In the absence of a network with all these data actually measured, it was necessary to simulate one for this investigation. The simulated data of the network totaled 195 values and consisted of 15 each astronomic latitudes, longitudes, and azimuths (one per station), 66 horizontal directions, 66 vertical angles, and 33 lengths (Fig, re 1). At each station, 7 'param- eters,' 3 geodetic coordinates, 2 astronomic coordinates, 1 station orientation for the horizontal direction, and 1 coefficient of refraction for all lines emanating from the station were solved for. At each point with an 'observed' astronomic azimuth, the station orientation was made zero. All the simulated adjustments were carried out in curvilinear coordinates' geodetic latitude b, longitude,, and ellipsoidal height h, but the real test net was adjusted in both curvilinear and Cartesian coordinates. The relevant equations for the 'observed' data, given by Fubara [1969], are slightly modified rederivations of formulations given by Hotthe [1959, 1969] and Wolf [1963a]. The coefficient of refraction was made a function of the station for reasons developed by Fubara [1969] and based on the fact that the curvature of the line of sight [Bornford, 1962; Hotthe, 1969] is given by Copyright 1972 by the American Geophysical Union. 796

2 T/:IREE-DIMENSIONAL GEODESY N 797 where $ and V are the meridian and prime vertical components of the deflection of the vertical. Equation 3 is the classical Laplace's equation for explicitly correcting astronomic azimuths to geodetic. This explicit correction was therefore not required in this investigation. THE STATISTICAL MODEL The variances (and covariances if any) of the observations constitute that component of the mathematical model termed the statistical model. Each observation was weighted inversely proportional to its assigned variance through the use of a unitless variance factor. In the method of intrinsic parameters [Fubara, 1969], where the functional parameters can also be weighted, the variances and covariances of such parameters are included in the statistical model if they are known. The following standard deviations, which are about the average values obtainable from firstorder triangulation measurements, were used: 2 I Fig. 1. Planimetric sections of concentration ellipsoid: scale, 1 cm cm; simulated net scale, 1 cm _-- 9 kin. where K is a constant varying with humidity and dependent on the wavelength of light and the gas of the medium. /? is altitude of line of sight. z is the refractive index of the gas of the medium. c is a constant dependent on and inversely proportional to the molecular weight of the gas. g is acceleration due to gravity. dt/dh is the temperature gradient. The following equations are intrinsically inherent in the formulated equations for applying three-dimensional geodesy to terrestrial networks, as has been shown in different ways by Wolf [1963b], Heiskanen and Moritz [1967], and Fubara [1969]. a' = a-[- vtan ' = / - cos c - V sin c (2) -[- ( sin a -- V cos a) tan/ (3) :t:0.75" for an astronomical azimuth :ko.2" for the astronomicalatitude and longitude (0.035m -]- 3. s X 10-6) for a length :ko.7" for a horizontal direction +[2.0 -]- (D -- 10)/10]" for a vertical angle where s in meters and D in kilometers are the lengths of the corresponding lines, and 10 km < D < 50km. Weighting the vertical angles in this manner accommodates the fact that, beyond 10 km, the precision of observed vertical angles decreases with increase in length. Since the vertical angles are thus the weakest link in the data, the influence of varying their precision from about --+4" to -+1" was investigated. Besides, the vertical angle implied in the adjustment is the observed value corrected for refraction. Its precision estimate must therefore reflect the inaccuracy in refraction correction. ANALYSES OF RESULTS Table 1 gives the standard errors of the adjusted functional parameters when all the 195 'observations' were simultaneously adjusted and represents the control of the experiment. For the geodetic coordinates of each point, the corresponding 3 X 3 submatrix of the weight eoefl%ient matrix was used to compute the

3 798 D. M. J. FUBARA TABLE 1. Standard Errors of Functional Parameters (x 10-4) ß One Point of the Net Held Fixed,, h,, A, Refraction Station arc sec arc sec meter arc sec arc sec Coefficient ! parameters and orientation of the confidence range between -+-2" and +4". As is shown in ellipsoid or ellipsoid of concentration [Cramer, Table 2, by assigning a variance of 1 sec to 1946; Linnik, 1961], schematically shown in the vertical angles, the standard errors of the Figure 1. As developed by Fubara [1969], the derived geodetic heights fell to between cm probability that the point to be determined and 14 cm. For the derived coefficient of lies within the ellipsoid of concentration is refraction, the standard errors varied from 99.95%. This high probability was chosen to to when the precision of the indicate the worst inaccuracy to be expected of vertical angles ranged between and the adjusted geodeti coordinates. The stand- For a vertical angle precision of ---+1, the ard errors of the determined geodetic:latitudes largest standard error was and longitudes ranged from to cm, the The simulated network involved 195 simurange for the height is _10 cm to cm lated observations. All the possible permutawhen a point at one end of the net is held tions and combinations of the 195 were not infixed. In these estimates, the values of the stand- vestigated. The approach was to analyze the ard deviation assigned to the vertical angles influence of varying or even completely elimi- TABLE 2. Standard Errors of Functional Parameters (x 10-4) ß Variance of 1 sec 2 Assumed for Each Vertical Angle d), ;, h,, ^,. Refraction Station arc sec arc sec meter arc sec arc sec Coeff

4 Fig. 2. (2,3,5) Station Numbers Effect of measured lengths on geodetic' latitudes. nating a particular type or groups of observations. From such analyses, a conclusion is made on the possible trends in observed data requirements in three-dimensional adiustment of terrestrial networks of the type, size, and fielddata precision used. Theoretically, one measured length should be suffcient to furnish scale control, provided that enough horizontal angles or directions are observed in the network to enable derived lengths to be computed at any part of the network. The investigations indicated that the only parameters significantly affected by variations in the number of measured side lengths were the geodetic latitudes and longitudes. The influence on the geodetic latitudes and longitudes is graphically shown in Figures 2 and 3. The numbers on the graphs show how many lengths were used in each cycle of the adjustment. Even though the measured lengths are a function of the geodetic coordinates, it is remarkable to note that the use of more than one length had no visible influence on the standard errors of THREE-DIMENSIONAL GEODESY 799 the geodetic heights. This and a later investigation on the effects of observed vertical angles indicate that, without precise vertical angles, measured lengths alone should not be expected to give reliable geodetic heights. The steepest slope in this work was about 4 ø, with side lengths between 10 km and 40 km. Hradilek [1968] does, however, suggest that, for lines inclined at more than 16 ø, reliable geodetic heights can be computed from measured lengths. The approach and assumptions in that investigation were quite different, and the network involved limited the side lengths to between 5 and 15 km. The effects of measured or absence of measured astronomic latitude and longitude at each station on the derived geodetic heights, the astronomic latitudes, and longitudes are shown in Figures 4, 5, and 6. On examining all the results of simulation investigations, it was obvious that all the 195 simulated observations were not required to uniquely define the network. The next logical step was to estimate how many and which types of observations are sufficient to give the network a unique definition. The two guide fines given below are based on the results of the various adjustments. These empirical recommendations are not necessarily the absolute minimum requirements. The results of the investigation suggesthat (1) all vertical angles need to be measured o.oz = o o.ol Fig Station Numbers Effect of measured lengths on geodetic longitudes. 1

5 800 I). M. J. FUBARA vertical angles, 100%; astronomic latitudes, 100%; horizontal directions, 100%; one astronomic azimuth per station, and one astronomic longitude for the entire net. The number of functional parameters remained the same as in case 1, but the number of redundancies for case 2 was 92. In spite of the greater redundancy in case 2, its results (Table 4) were in fact no different from those of case 1 (Table 3) and were similar to the control (Table 1). Case 1 is a very attractive proposal, depending of course on which set is easier to measure: 50% of the lengths or 100% of the horizontal directions. However, the danger in using it is that its efficiency depends on flawless astronomic observations especially void of systematic errors. A TERRESTRIAL NETWORK of R,A;, I)ATA Station Numbers The network (Figure 7) consisted of 6 stations, 11 distances, 22 directions, 22 vertical angles, 4 astronomicalatitudes, and 4 astronomical longitudes. The astronomical values should refer to the actual terrain points of ob- Fig. 4. E e½ of number oœ measured s ronomic latitudes and longitudes on derived geodetic heights. lo from both ends, not necessarily simultaneously, but the measurements at each station should be made together in as rapid a succession as possible; (2) when all lengths are measured and at least one astronomical azimuth is observed per station such that each station is connected to at least one other station by an astronomical azimuth, then horizontal directions need not be measured; (3) if at least one astronomical azimuth is observed per station, it is not necessary to measure more than one astronomic longitude; (4) the astronomic latitude of each station must be measured. On the basis of these general trends, an adjustment for each of the following cases was made for the simulated network. Case 1 consisted of lengths, 100% of the total observable data; vertical angles, 100%; astronomic latitudes, 100%; one astronomic azimuth per station, and one astronomic longitude for the entire net. Thus, it had 120 observations, 87 functional parameters, and hence 33 redundancies. For case 2, the composition was lengths, 50%; 7 o lo Station Numbers Fig. 5. Effect of number of measured astronomic latitudes and longitudes on derived tronomie latitudes.

6 0.8] o TI-IREE-DllVlEN SlONAL GEODES¾ network appear to have complied with these requirements except for polar motion and the time system. The people who furnished the data were not sure whether they used UT0, UT1, UT2, or UTC. A few values, including the only astronomical azimuth in the net, were corrected for polar motion. A computer program written in Fortran 4G was designed to carry through all the necessary computations to furnish the elements of A, B, C, L, V, and W of the linearized correction equations of type AVad-BVbd- CVcd- W= 0 (4) used by Fubara [1969] and based on the work of $chmid and $chmid [1964] for the adjustments in both curvilinear and Cartesian co , ordinates. The rest of the adjustment and the S... bers computing of eigenvalues and eigenvectors were Fig. 6. Effect of number of measured astro- done in IBM Matlan language. nomic latitudes and longitudes on derived astronomic longitudes. ANALYSIS Or RESULTS The variance estimate s; with dx degrees of servation without being reduced to the geoid. freedom and another independent estimate s The right ascension and declination of each with d degrees of freedom of the same varistar should be the apparent values for the ance, are each distributed as X square. The instant of observation corrected also for diurnal aberration. All astronomical values require cor- rection for polar motion. The time signal used nondimensional ratio (812/dl) = Fa (5) (s /d ) for computing the longitude should refer to UT1. Stars used for longitude determination is tabulated in the Fisher's distribution table as should have their coordinates on the FK-4 a function of d, d, and (1 -- a), the confisystem. The astronomical data used in this test dence coefficient. The applications of the vari- TABLE 3. Standard Errors of Functional Parameters for Case 1 Station %, t, h,, A, Refraction x 10 - x 10 - x 10-3 x 10-3 x 10-3 Coeff., arc sec arc sec meter arc sec arc sec X 10 -l

7 802 D. M. J. FUBARA TABLE 4. Standard Errors of Functional Parameters for Case 2 Stati yn x 10 - arc sec X, h,, A, Refraction x 10-4 x 10-3 x 10-3 x 10-3 Coeff., arc sec meter arc sec arc sec x $ ance factor ratio and x-square tests in leastsquares adjustment, as discussed by Hamilton [1964] and Richardus [1966], were used. The chosen confidence coefficient for all statistical tests except for the confidence ellipsoid is At this level, the tests showed that both the observations and the mathematical models were satisfactory in the curvilinear as well as Cartesian coordinate adjustment. The assessment of the parameters (Tables 5, 6, and 7) derived from adjustments in both coordinate systems was based on the comparison of standard errors m, mx, m, and mx, m, m of the adjusted curvilinear and Cartesian co- Fig. 7. PlaMmetric secti.ons of concentration ellipsoid: scale, I cm km, test network; Scale of ellipses, I cm cm. A triangle marks the station held fixed, a d circles represent astronomicm coordinates observed. ordinates, respectively, through the use of equations 6, 7, and 8 [Fubara, 1969]. m - +[(m 2 cos 2 k - m 2 sin 2 X) sin 2 b + m 2 cos 2 b + mx sin 2X sin 2 b -- (mx cos X q- m sin X) sin 2qb] / Via (6) mx= q-[m 2sin 2xq- m 2 cos " m where -- m sin 2X] /2(W see )/a (7) q-[m 2 cos X + m sin 2x) cos qb 2 2 q- m, 2 sin 2 qb q- m sin 2X cos qb q- (m, cos X q- m, sin X) sin 2rk] /2 (8) V -- (1 + e '2 cos 2 qb)l/2 W = (1 -- e sin 2qb) 1/2 e and e' are first and second eccentricities of the reference ellipsoid, and a is the semimajor axis of reference ellipsoid. Tables 5 and 7 show the adjusted functional parameters and the standard errors as derived from the adjustment in curvilinear and Cartesian coordinates, respectively. The corresponding values are of course not exactly the same. However, the question is, are they significantly different? For instance, a standard error from one coordinate system falls well within a 2a confidence interval [Linnik, 1961] of the corresponding standard error from the other system.

8 THREE-DIMENSIONAL GEODESY 803 TABLE 5. Adjusted Values of Functional Parameters Test ß Net Adjustment in Curvilinear Coordinates Station Latitude N, Std. Error G eode t i c As t ronomi c Refraction Height, Coeff. Longitude W, meters Latitude N, Longitude W, Std. Error Std. Error Std. Error Std. Error Std. Error x *7 48o " o o o o " 119o " " 119o " " 119o " " 119o " " 119o " " ø57'23.864" " " ø03'33.134".0040" " " o02'56.806".0028" " " ø02'50.310".0029" " " ø08'10.600" ß 0037" " " o13'25.474".0052" " 119ø46'38.132" " o24'09.502" " o34'05.448" " d51'10.416" " o40'02.677" " ø30'59.186" " 83 *Station held fixed. Station without observed astronomic data. On this premise, the standard errors from our adjustment in either of the two coordinate systems are not significantly different from the other. CHOICE OF COORDINATE SYSTEM BY R-FACTOR RATIO TEST Cartesian and curvilinear coordinates have exact mathematical relations to transform one into the other. The question of the choice of one of them in preference to the other arises only because the observations / involved are not linear functions of the functional param- eters in either case. In such a case, a Taylor series expansion is used, leading to the familiar linear form + = A is the matrix of the first partial derivatives of the observations with respect to the parameters. All other terms from the second to higher orders are dropped off as being insignificant. If the omitted partial derivatives are significant, the nonlinear relationship between the observations and their functional parameters has not been successfully linearized as given by equa- TABLE 6. Adjusted Values of Functional Parameters Test: Net Adjustment in Cartesian Coordinates X Station Std. Error Geodetic, meters Astronomic [Y, Std. Error Std. Error Latitude N Longitude W Refraction Coeff. x * t o o o o o o " 119o " " 119o24.284" " 119o34.309" " 119o " " 119o40.311" " 119o30.146" " 9.44" " ß 183" ".131" " ".130" ".252" *Station held fixed. tstation without observed astronomic data.

9 8O4 D. M. J. FvrAaA TABLE 7. Curvilinear Coordinates Derived from the Adjustment in Cartesian Coordinate Test Net G eodet i c Astr onomi c Refraction Height, Coeff. Latitude N, Station Std. Error Longitude W, meters, Latitude N, Std. Error. Std. Error Std. Error Longitude W, Std. Error Std. Error x *ñ 48ø57' " 119ø47' " '23.032" 119ø46'40.146" " 9.440" ø03, ,, 119o23' " ø03'33.160" '09.460" ".0032" " 0.183" ø02, ,, 119ø34' " ø02'56.790" 119o34'05.540" ".0031" " 0.131" ñ 49ø02, ,, 119ø52' " o02'49.068" 119ø51'11.765" 358 ß 0010".0009" " " ø08, ,, 119ø40' " ø08'10.620" 119o40'02.670" ".0032" " 0.130" ø13, ,, 119ø30' " o13'25.470" 119 ø30'59.180" 't.0033" " 0.252" 47 *Station held fixed. ñstation without observed astronomic data. tion 9. The choice between the Cartesian and the curvilinear coordinate systems is a question of in which system are the omitted partial derivatives more significant and how significant is the significanceß This is necessarily so, because the same geometric and statistical models were used in each mathematical modelß The computational accuracy in each case was the same. The fundamental basis is that, in one coordinate system, to be called X, the adjusted coordinates of two points are X{, Y{, Z,, and Xj, Yj, and Zj. In another system, Xs, using exactly the same observations, the adjusted coordinates of the same two points are.qb, X, hx, and s, is, h2. The number of degrees of freedom is the same. Let f jo be one of the observations involved in defining the two points. From the adjusted X, Y, Z coordinates, which should be unbiased estimates, an f j is computed. Similarly, from the, X, and h, f? is computed. f and f / are independent estimates of f{ o ],i '= g (X ) (10) ],, = ga(x ) (11) = - V = i, o_ i, (13) R is a function of ¾ P¾ and r degrees of freedom, and R is a function of V/'PV._ and r degrees of freedom. It is obvious, therefore, that, under all these identical conditions, the departure of R from Rs is a measure of the departure from acceptable lineartry of one system from the other. From the R-factor ratio test [Hamilton, 1964], if R and R are identical, then both systems are either equally badly linearized or equally well linearized. If statistically they differ, one of them is better linearized and should be chosen in preference to the other. Computing the magnitude of the sum of the omitted second partial derivatives is not the only way to test the degree of linearity. In fact, one disadvantage of the use of the second partial derivatives is that one must also define, in addition, the range of initial values over which the second derivatives are significant or not significant when omitted. In the absence of this defined range of sensitivity, the second derivative test is incomplete. Testing with the R- factor ratio does not have the above limitation. Let the estimates from X and X._ as defined above be independent estimates of X from another system. R = (f,,- rp([,,,)' (14) = [_(f,, ) rp(,, )..[ (15) As in Hamilton [1964], R... is obtained from the R-factor tables and compared to R

10 from R = R,.IR (16) If X2 is a better estimator of X than X, then THREE-DIMENSIONAL GEODESY $05 > (x7) where a is the confidence level. The test indicates that the hypothesis that X and X2 are unbiased estimators of the same parameters can be accepted at the 5% significance level. Therefore, it does not matter, in terms of the adjusted functional parameters, whether the adjustment is done in Cartesian or curvilinear coordinates. DERIVED ASTRONOMIC QUANTITIES AND COEFFICIENT OF REFRACTION hence the astronomic coordinates too. For the low vertical angles, less than 4 ø, the precision estimates and network setup used herein, the results suggesthat the determination of reliable astronomic coordinates must still depend on reliable direct measurements that themselves are never made to stars whose altitudes are less than 15 ø. Fortunately, these unacceptable derived astronomic values did not render the geodetic coordinates of the corresponding st.ations less precise than at other stations. This is very important. The reason is of course the fact that the lines from the other stations to these two stations were oriented accurately, and their vertical angles were well corrected at those ends where astronomic coordinates were observed. This is in perfect agreement with the conclu- The simulated adjustment showed that, at any station with no observed astronomicosion from the simulated net. At the stations where the directions of the ordinates, the derived astronomic latitude is gravity vector were defined by observed astrohighly unreliable, the astronomic longitude could nomical coordinates or azimuth, or both, the standard errors of the derived coefficient of be acceptable if an astronomic azimuth was observed at the station, and vice versa. In the refraction ranged from ----_0.004 to ----_0.008, the test net, there was only one astronomic azimuth. largest magnitude occurring at the station with The astronomic coordinates at two of the sta- the least number of observations. At the stations were not observed, and no azimuth obsertions without observed astronomical data and vation was made at those stations too. The standard errors of the derived astronomic coat which the gravity vector direction was not stipulated by any other type of observations, the ordinates at those two stations range from 4" standard errors were ñ0.016 and The to --+'_20". But of great significance is the fact magnitude of the correlation coefi%ient for the that, although the corresponding standard errors coefi%ient of refraction and any of the astroof the derived astronomic coordinates from the nomical data at the same station is approxicurvilinear and Cartesian coordinate adjustments differed by between 2" and 7", the coordinates actually derived differed only by bemately unity; for the ellipsoidal height it was always less than 0.3, whereas for the remaining parameters it was about At the two statween 0.3" and 2". This was not surprising, tions where the standard errors of the derived because in a correct least squares solution the coefi%ient of refraction were largest, it was derived parameters and their variances are not statistically found through the computation and only unbiased, but also are independent of each comparison of confidence intervals that the other [Hamilton, 1964]. At the stations with suspected parameters could not be rejected. observed astronomic longitudes, the standard error of a derived astronomic azimuth ranged from --+'--0.6" to --+'_1.4". This is about the type CONCLUSIONS The conclusions have qualitative and quanof precision obtainable from actually observed titative characters. Naturally, the latter more astronomic azimuths. The mathematical development of threedimensional geodesy suggests that from the geohighly depends on the network configuration and assigned data quality. The generality of the following conclusions is therefore subject to metric quantities (horizontal directions and reservation. A terrestrial network can be and vertical angles measured by theodolites, which are therefore influenced by the gravity field) it is possible to determine the direction of the gravity vector at the observation point, and was rigorously adjusted three-dimensionally with proper network orientation correction. Restriction on side lengths is not necessary, and the components of the deflection of the vertical

11 806 D.M. J, FUBARA and vertical angle refraction correction can be derived simultaneously with the three geodetic coordinates at each station subject to the following conditions' (1) the accurate determination of ellipsoidal heights requires the observation of precise astronomic latitude at each network station; (2) for a network of the type in Figure 1, it is sufiqciento observe at each It was also found that there is no significant difference between the adjustment in curvilinear and Cartesian coordinates. Scaling, or equilibrating [Fox, 1965], was applied to the design matrix and by comparison of adjusted parameters from the scaled and unscaled systems, no instability in the solution was detected. Because of the low precision of the vertical angles station where horizontal directions have been used, the geodetic heights were the most weakly measured one precise astronomic azimuth or longitude, but at least at one point of the geodetic system both the astronomic azimuth and longitude should be observed; (3) if all horizontal directions in the network are observed, between a quarter and a third only of all the side lengths need be measured in a manner well distributed throughout the network; (4) if all determined. The remedial efiqciency of including spirit leveled heights and gravity observation was recognized and is being investigated. Two of the most frequent deficiencies in the data of most terrestrial 'triangulation' nets are reliable observed vertical angles and astronomic latitudes. These two seem indispensable for the type of adjustment investigated here. By the the side lengths in the network are measured, use of such instruments as the Zeiss Ni 2 it is sufiqcient to observe horizontal directions Astrolabe [Fubara, 1967], obtaining economiat only about half the stations of the network such that each line has at the least one azimuth equation in the adjustment; (5) the observation of horizontal directions can be totally eliminated if all the lengths, the vertical angles, and astronomic latitudes are measured together with an astronomic azimuth at each station such that cally and speedily reliable astronomic latitude and longitude at every triangulation point is no longer an arduous task. The question may arise as to why one should not resort to classical methods if at every point either the astronomic latitude and longitude or the astronomic latitude and azimuth of one line every station is connected to one other station are available. When lines of 10 to 40 km or by an observed azimuth; (6) when all lengths are measured, it is necessary to observe all vertical angles at least in one direction. When this minimum is used and if the coefiqcient of refraction is treated as a function of the station, the vertical angle measurementshould be so organmore are involved in the classical method, then lengths must be properly reduced to the computing surface and not the geoid. Assuming that orthometric heights are known, the method also requires the geoidal undulations. Before the and 7, the meridian and prime vertical comized that the minimum number of stations is ponents of the deflection of the vertical, reoccupied and also that the number of observations at the station minus the number of unknown functional parameters is 6 or more. For practical expediency, vertical angle of each line should be observed from both ends, not necessarily simultaneously. But in the case, as applied here, in which the refraction correction is made a function of the station, the vertical angles of all lines at each station should be observed 4 to 8 times (each a mean of face left and right) in rapid succession at a time when vertical refraction is a minimum. The observations should be repeated about 2 hours later or on different nights; (7) for long lines with small slopes, all vertical angles must be precisely measured (standard deviation of arc sec or better), because measured side lengths alone cannot furnish reliable geodetic heights. quired to compute the geoidal undulation, can be obtained, and, if the necessary astronomic quantities are available, some reliable geodetic coordinates must be computed for the new points in the network. It must be established how precise these 'approximate' values of the geodetic coordinates must be so that systematic errors are not introduced. The separation of horizontal and vertical adjustments, as in the classical method, invariably reduces the number of degrees of freedom in each adjustment. Skew normal corrections should be applied, when necessary, to observed azimuths and horizontal directions [Bom[ord, 1962; Zakatov, 1962]. In view of all these ramifications, iterative procedures, the discussion by Levallois [1962], and the advantages of a three-dimensional adjustment outlined by Hotine [1959], the choice

12 of classical or nonclassical methods becomes a matter of the taste of individual geodesists. The role and required accuracy of observed astronomical coordinates is defined by equations 2 and 3, which are intrinsic in terrestrial threedimensional geodesy formulations. They enter as direction parameters correcting theodolite-observed vertical angles and horizontal directions to their geodetic equivalents. Since they are 'correction' parameters, their precision should be such that M > MA tan M > (M cos a q- MA sin s a)'/3 where M,, Mp, M, and M are the standard deviations of the observed horizontal directions, vertical angles, astronomic latitudes, and longitudes, respectively. These two inequahtieset the desirable limits for the accuracy of observed astronomical coordinates for any set accuracy limits of the azimuth and vertical angles of terrestrial network sides when theodolite measurements are employed. Acknowledgment. This work, completed in Augus 1 9, was supported in parc by the University of ew Brunswick, Shell BP of igeria, and he Geode ic Survey of Canada, who furnished M1 the tes da a. The paper is a summary of the author's Ph.D. di er afon published by the University of New Brunswick, Fredericton, N., Canada. The author is indebted to Dr. R. H. Rapp, who helped to initiate this research, and members of he dissertation reading committee, Drs. J. S. Allman, L. Hradilek, R. Kaiser, W. Knight, G. Konecny, E. J. Krakiwsky, Professor R.. Sanchez, and Mr. L. A. Gale for useful criticisms. ) EFEREN CES Bacon, C. J., Deflections of the vertical from mountain net adjustment, M.Sc. thesis, Univ. of New Brunswick, Fredericton, N.B., Bornford, G., Geodesy, Oxford University Press, New York, THREE-DIMENSIONAL GEODESY /{07 Cramer, H., Mathematical Methods of Statistics, Princeton Unœversity Press, Princeton, N.J., Fox, L., Introduction to Numerical Linear Algebra, Clarendon Press, Oxford, Fubara, D. M. J., The Use of Zeiss Ni2 Astrolabe for Geodetic Astrofix, M.S. thesis, Ohio State Univ., Columbus, Ohio, Fubara, D. M. J., Three-dimensional geodesy applied to terrestrial networks, Ph.D. dissertation, Univ. of New Brunswick, Fredericton, N.B., Hamilton, W. C., Statistics in Physical Science, Ronald Press, New York, Heiskanen, W. A., and H. Moritz, Physical Geodesy, W. H. Freeman, San Francisco, Henderson, J.P., Three-dimensional adjustment of geodetic triangulation, M.Sc. thesis, Univ. of New Brunswick, Fredericton, N.B., Hotinc, M., A primer of non-classical geodesy, paper presented at the First Symposium on Three-Dimensional Geodesy, IAG, Venice, Hotinc, M., Mathematical Geodesy, U.S. Dep. of Commer., Washington, D.C., Hradilek. L., Trigonometric levelling and spatial triangulation in mountain regions, Bull. Geod., 87, 33, Levallois, J. J., La rehabilitation de la geodesique classique et la geodesic tridimensionelle, Bull. Geod., 68, 193, Linnik, Y. V., Method o Least Squares and Prin- ciples o the Theory ol Observations, Pergamon, Macmillan, New York, Richardus, P., Project Surveying, North-Holland, Amsterdam, Schmid, H. H., and Erwin Schmid, A generalized least squares solution for hybrœd measuring systems, paper presented at Second Geodesy Symposium, Univ. of New Brunswick, Fredericton, N.B., Wolf, H., Die Grundgleichungen der Dreidimensionalert Geod isie in elementarer Derste Llung, Ze. Vetmess., 88, 225, 1963a. Wolf, H., Die Laplacesche Azimut-Gleichung in der Drei. dimensionalen Geodasie, Ze. Vetmess., 88, 246, 1963b. Zakatov, P.S., A Course in Higher Geodesy, 390 pp., U.S. Dep. of Commer., Washington, D.C., 1962 (translated from Russian by Israel Program for Scientific Translations, Jerusalem, 1962). (Received February 17, 1971; revised September 16, 1971.)

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