CJwCH3J Professional Paper 7 Three-Dimensional Triangulation With Satellites HELLMUT H. SCHMID National Ocean Survey, Geodesy Di.vision ~ lAf U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary ~ NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION, Robert M. White, Administrator RockviUe, Md., October 1974 For sale by· the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402- Price $1.65 (paper cover) Stock Number 0321-00011 Contents Page Albstract ---------------------------------------------------- 1 1. Foundations of satellite geodesy and the creation of worldwide geodetic reference systems -------------------------------- 1 1.1 Geometric and geophysical aspects of satellite geodesy _____ _ 1 1.2 Development and organization of a geodetic satellite program for the creation of a worldwide geodetic reference system 3 1.3 References 5 2. Geometric solution of satellite geodesy 6 2.1 lptroductory considerations 6 2.2 Geometric foundations 7 2.3 Alstronomical reference system ---------------------------- 9 2.4 Meaning and measurement of time ------------------------ 16 2.5 Aldditional geometric and physical influences -------------- 19 2.6 Measuring procedures of the geometric method ------------ 24 2.6.1 Photogrammetric camera ---------------------------- 24 2.6.2 Camera shutters and their mechanical drives ---------- 25 2.6.3 Electronic control instrumentation -------------------- 27 2.6.4 Photogrammetric registration ------------------------ 28 2.6.5 Coordinate measurements on the comparator and their reduction ---------------------------------------- 29 2. 7 Numerical adjustment ------------------------------------ 32 2. 7.1 Introductory remarks ------------------------------ 32 2. 7.2 Mathematical aids ---------------------------------- 32 2. 7.3 Setting up the general photogrammetric observation equa- tions -------------------------------------------- 39 2. 7.4 Mathematical model for the photogrammetric camera __ 42 2. 7.5 Spatial triangulation ----------------------------"--- 53 2.8 References 60 3. Error theoretical considerations ------------------------------ 62 3.1 Error budget of geometric satellite triangulation ____________ 62 3.2 Alnalysis of the essential sources of error and the error propagation into the spatial triangulation --------------- 68 3.2.1 Alccuracy of the comparator measurements ------------ 70 3.2.2 Alccuracy of the reconstructions of the photo,raphic bun- dles and their orientations ------------------------ 71 iii 551-596 --------------------------------------------------------------- iv Contents Page 3.2.& Accuracy of the trace of the satellite orbit after the polynomial fit _ ----------------------------------- 73 3.2.4 ·Error propagation into the spatial triangulation -------- 75 3.~ Result of the worldwide ge.vmetric satellite triangulation ------ 78 3.4 Analysis of the triangulation adjustment ------.-------------- 81 3.5 References -------------------------------------------- 89 Acknowledgments---------------------------------------- '91 FIGURES Page Figure I.-Basic forces producing the satellite orbit -----------------­ 3 Figure 2.-Portion of the world net showing ties between four continents 6 Figure 3.-Geometry of satellite triangulation -----------------------­ 7 Figure 4.-Geometry of satellite triangulation ------------------------ 7 Figure 5.-Simultaneous observation of satellite from.three stations ___ _ 8 Figure 6.-Densification network in North Anierica 9 Figure 7.-Astronomic reference systems ---------------------------- 11 Figure 8.-Geocentric and topocentric coordinate systems 13 Figure 9.-0rientation of rotating astronomic system relative to assumed fixed system ---------------------------------------- 14 Figure 10.-Components of the aberration vector Ll 14 Figure 11.-Relations between the instantaneous pole and the Coven- tiona! International Origin ---------------------------- 18 Figure 12.-Schematic of star and satellite refraction 20 Figure 13.-The satellite phase correction 21 Figure 14.-Influence of diurnal aberration and Earth rotation when record- ing light flashes from satellite ------------------------ 22 Figure 15.-Influence of diurnal aberration and Earth rotation on light reflected from continuously illuminated satellite -------- 23 Figure 16.-Principle of time interpolation 24 Figure l7 .-Wild Bc-4 camera with external Henson shutter assembly __ 25 Figure 18.-0bservation site with camera shelter and recording equipment shelter ---------------------------------------------- 26 Figure 19.-Zeiss satellite tracking camera with recording unit -------- 26 Figure 20.-Henson capping shutter assembly (cover removed) -------- 27 Figure 21.-(a) BC-4 electronics control console, and (b) schematic 28 Figure 22.-Schema~ic of star and satellite images -------------------- 29 Figure 23.-Star plate with trails of two satellites 30 Figure 24.-Basic coordinate systems used in numerical evaluation 33 Figure 25.-Increment of a unit vector ------------------------------ 34 Figure 26.-Increment coplanar with a coordinate axis ---------------- 34 Figure 27.-The photogrammetric bundle vector p -------------------- 34 Contents v Page Figure 28.-Direction of exterior elements of orientation -------------- 34 Figure 29.-Diapositive as seen from center of projection -------------- 36 Figure 30.-Components of distortion ------------------------------ 36 Figure 31.-Earth ellipsoid parameters and coordinates 36 Figure 32.-Parameters for the transformation of ellipsoidal to Cartesian coordinates ------------------------------------------ 38 Figure 33.--Schematic of reduced normal equations system ------------ 48 Figure 34.-Basic normal equations system corresponding to observation equations Bv = il ------------------------------------ 64 Figure 35.-The normal equations system and additional constraints ____ 65 Figure 36.-Schematic of the portion of the satellite orbit observed and the typical location of observing stations ------------------ 67 Figure 37.-.Scalars connecting observing stations measured in the world net program ---------------------------------------- 69 Figure 38.-Comparison of measuring accuracies -------------------- 70 Figure 39.-Histogram of 1,291, 744 double measurement differences ____ 7l Figure 40.-500 photo.grams, _arranged according to increasing latitude of observmg station ------------------------------------ 72 Figure 41.-Histograms of x and y plate residuals for 25 typical single camera adjustments -------------------------------- 73 Figure 42.-Plot of the mean errors of individual polynomial fits to the satellite trail for each of the 500 plates of figure 40 ______ 74 Figure 43.-Average scintillation at observing stations ---------------- 75 Figure 44.-Section of error ellipsoid -------------------------------- 77 Figure 45.-Error propagation factors. for ellipsoid height, latitude, and longitude ------------------------------------------ 77 Figure 46.-Error propagation in geometric satellite triangulation 78 Figure 47.-Geographic distribution of stations and observations centered at 90°E -------------------------------------------- 78 Figure 48.-Geographic distribution of stations and observations centered at Greenwich meridian ------------------------------ 78 Figure 49.-Geographic distribution of stations and observations centered at 180° meridian ------------------------------------ 79 Figure 50.-Geographic distribution of stations and observations centered - at 90°W -------------------------------------------- 79 Figure 51.-Geographic distribution of stations and observations centered at North Pole ______ -------------------------------- 79 Figure 52.-Geographic distribution of stations and observations centered at South Pole -------------------------------------- 79 Figure 53.-Stations of the worldwide BC-4 photogrammetric satellite triangulation network -------------------------------- 80 Figure 54.-Plot of the increments in A cos E fcom pre- to post- calibration ----------------------------------------- _ 85 Figure 55. -Plot of the increments in elevation from pre- to post- calibration ------------------------------------------ 85 vi Contents Page Figure 56.-Differences in azimuth in seconds of arc between direc­ tions from single line adjustments and from the final adjustment ---------------------------------------- 86 Figur.e 57.-The elevation differences of the data in figure 56 ---------- 87 Figure 58.-Mean errors of unit weight u after single line adjustments __ 88 Figure 59.-Piate coordinate residuals for two-plate events ------------ Figure 60.-Piate coordinate residuals for three-plate events---------- 89 Figure 61.-R.M.S. of mean coordinate errors of adjusted station posi- tions ---------------------------------------------- 90 TABLES Table 1.-The world net stations 5 Table 2.-Most useful combinations of primary image sequence, expo- sure interval, and actual exposure sequence ------------ 26 Table 3.-Curve fit of 380 fictitious satellite images with various polynomials ---------------------------------------- 67 Table 4.-Identification of the 500-plate sample used in figure 40 ------ 72 Table 5.-Statistics for the 500-plate sample 76 Table 6.-Mean direction accuracies corresponding to image positions on a typical single plate ---------------------------------- 76 Table 7.-Adjusted Cartesian coordinates of world net stations ---~---- 81 Table B.-Comparison of measured baselines with the geometric triangu- lation result ---------------------------------------- 82 Table 9.-Corrections to baselines weighted 1:2,000,000