Numerical accuracy of block adjustments

Photogrammetria, 32 (1976) 101--109 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

NUMERICAL

ACCURACY

OF BLOCK

ADJUSTMENTS*

H . EBNER and R. MAYER

Institute for Photogramrnetry, Stuttgart University, Stuttgart (F.R.G.) (Received September 10, 1976)

ABSTRACT Ebner, H. and Mayer, R., 1976. Numerical accuracy of block adjustments. Photogrammetria, 32: 101--109. A successful application of aerial triangulation presupposes adequate numerical accuracy of the adjusted data. The numerical accuracy is limited by rounding-off errors of the arithmetic operations and their propagation in the block adjustment algorithm. The present paper investigates the effect of block size, control distribution and overlap on the numerical accuracy of the adjusted terrain coordinates in a purely empirical way. For that, a large number of simulated block adjustments by independent models was performed at the CDC 6600 computer of Stuttgart University, using the computer program PAT-M43. The reduced normal equations are solved by a generalized Cholesky method. The simulations confirm, that a word length of 60 bit is sufficient to guarantee adequate numerical accuracy of the adjusted coordinates. Even with a free block adjustment of 1250 models 8 correct digits were obtained. Moreover, the results have shown that the numerical accuracy in x and y can be well approximated by a constant multiplied by n • i 3, with n denoting the block length and i the planimetric control spacing along the block perimeter, expressed in units of the base length. In first approximation the obtained results are also valid for bundle block adjustments with a Gaussian or a Cholesky solution.

INTRODUCTION A successful application of aerial triangulation presupposes adequate numerical accuracy of the adjusted data, particularly of the adjusted terrain coord i n a t e s . T h e n u m e r i c a l a c c u r a c y is l i m i t e d b y r o u n d i n g e r r o r s . U s i n g t h e n o t a t i o n o f B a r t e l m e a n d M e i s s l ( 1 9 7 5 ) a l o c a l r o u n d i n g e r r o r is o n e o c c u r r i n g during an individual arithmetic operation within the block adjustment algor i t h m a n d a g l o b a l r o u n d i n g e r r o r is t h e a c c u m u l a t e d e f f e c t o f t h e l o c a l r o u n d i n g e r r o r s o n t o t h e r e s u l t s o f t h e a d j u s t m e n t . T h e s e g l o b a l r o u n d i n g errors represent the numerical accuracy of the adjusted data.

* Presented Paper, Commission III,13th Congress of the International Society for Photogrammetry, Helsinki, 1976.

102

In geodesy the problem of rounding errors was already treated in some investigations. Whilst Breuer (1969) and Schmitt (1973) have used computer simulations to determine the numerical accuracy of the adjusted coordinates of networks, Bartelme and Meissl (1975) have analysed the rounding error propagation during the direct solution of geodetic normal equations of the leveling type theoretically. In photogrammetry Schenk (1972) has estimated the global rounding errors of adjusted block coordinates from the condition numbers of the respective normal equation matrices of bundle blocks. The extent of this investigation however, is too limited to allow for a prediction of the numerical accuracy of large practical blocks, which today can contain 1000 or more images. To close this gap a comprehensive study was initiated by the first author and performed within the scope of the diploma thesis of the second author (Mayer, 1975). This study investigates the effect of block size, control distribution and image overlap on the numerical accuracy of the adjusted terrain coordinates in a purely empirical way. For that purpose a large number of simulated block adjustments with up to 1250 independent models was performed at the CDC 6600 c o m p u t e r of Stuttgart University, using the computer program PAT-M43 (Ackermann et al., 1973). ASSUMPTIONS AND SCOPE OF THE INVESTIGATION

The study proceeds from schematized, square-shaped blocks. The terrain points within a block, which shall be determined by the adjustment, form a grid of n X n quadratic meshes, respectively. The length of one mesh is equal to the base length which was chosen to 10 km. The terrain was assumed as flat. Each model consists of two perspective centers and six regularly distributed model points, covering two of the just mentioned block meshes. The longitudinal overlap is 60% and the focal length is 153 mm (wide angle). The independent model coordinates were assumed as rigorously consistent with the schematized block coordinates. In that way the propagation of local rounding errors during the block adjustment can be investigated, independent of the influence of all other possible errors. The discrepancies between the adjusted block coordinates and the initial grid coordinates then directly represent the requested global rounding errors. The c o m p u t e r CDC 6600, used with the investigation has 60 bit words. The length of mantissa, being decisive for the numerical accuracy, amounts to 48 binary digits, which is equivalent to 14 complete decimal digits. The rounding procedure is a simple truncation. The used c o m p u t e r program PAT-M43 for independent model block adjustm e n t is based on planimetry height iteration and solves the reduced normal equations by a generalized Gauss--Cholesky method. To cover the full range, in which block triangulation is applied today, five

103 d i ff er en t block sizes were investigated: n = 10, 20, 30, 40, 50. Here n denotes the lengths o f the square-shaped blocks, expressed in units of the base length b -- 10 km. The largest bl ock then covers an area of 500 × 500 km 2. F u r t h e r o n , three di f f er e nt versions of image overlap were treated: 20% side lap, 60% side lap and 2 cross f or m a t i ons of 20% side lap each. In the following the th r ee overlap versions are characterized by q = 20%, q = 60% and q = 2 × 20%. With n denoting the block length the block contains n2/2 models in case o f q = 20%, n2+n models if q = 60% and n 2 models in case of q = 2 × 20%. The four investigated cont r ol versions are shown in Fig. 1. In pl ani m et ry the same co n tr o l configurations were used for all three overlap versions. In

Height q = 20 %

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Height q-60%, q=2x20%

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104

height, dense chains of control across strip direction were used in case of q = 20%, whilst quadratic grids of control points were applied with q = 60% and q = 2 )< 20%. Starting from the rather dense control version 1 the n u m b e r of control points is stepwise reduced until, with version 4, the case of a free block adjustment is reached. PRESENTATION AND DISCUSSION OF THE RESULTS

Results of pre-investigations Before the influence of block size, control distribution and overlap on the numerical accuracy of the adjusted block coordinates was studied in detail, several pre-investigations were performed, the results of which shall be presented briefly. (1) The global rounding errors of the adjusted block coordinates are fully reproducible. Repetition runs lead to exactly the same values. (2) The numerical accuracy changes only by a few percent, when the schematized models are slightly transformed by similarity transformations, before the block adjustment is executed. (3) The sequence, by which the individual models are treated within the adjustment has only a relatively small effect on the global rounding errors. At the m a x i m u m the errors changed by a factor 3. (4) A reduction of the weight of control point coordinates from 108 (practically error-free control) to 10-: (the control points are 10 times less accurate than the model points) increased the global rounding errors by less than a factor 4. (5) Additional iteration steps do n o t improve the numerical accuracy. This disagrees with the results of preceding investigations, where the so-called residuum iteration has reduced the global rounding errors significantly (Schmitt, 1973). The contradiction can be explained by the fact that the PAT-M 43 program does n o t compute corrections to the initial values of the terrain coordinates, but computes the u n k n o w n coordinates directly. (6) When the n u m b e r of models of the block and the control distribution is kept, but the base length is changed, the global rounding errors change proportionally to the base length. Numerical accuracy o f adjusted planimetric coordinates The obtained planimetric results are summarized in Table I. Here rx,y mean denotes the RMS value of all global rounding errors rx and ry of the adjusted x and y coordinates and rx,y max denotes the m a x i m u m value of all rx and ry. The poorest numerical accuracy appears with the very extreme case n = 50, i = 100, q = 20%. This block consists of 1250 models with 20% side lap, covers an area of 500 X 500 km 2 and contains only two planimetric control points, which represent the absolute minimum. Even here however, the global round-

105

TABLE I Global rounding errors of the adjusted block coordinates x , y in um in the terrain. Block length n

Control distance i

q--20% rx,y mean

q=60% rx,y max

rx,y mean

q=2X20% rx,y max

rx,y mean

rx,y max

10

2 5 10 20

0 0 0 0

0 0 0 1

0 0 0 1

0 0 1 3

0 0 0 2

0 0 0 3

20

2 10 20 40

0 1 6 49

0 2 14 109

0 0 3 16

0 1 5 35

0 0 3 25

0 1 6 55

30

2 15 30 60

0 5 62 546

1 13 138 1219

0 1 14 79

1 2 31 219

0 2 20 152

0 5 51 370

40

2 20 40 80

0 6 97 830

1 22 234 1883

50

2 25 50 100

1 22 380 3132

2 77 904 7100

i n g e r r o r s r e m a i n p l e a s i n g l y s m a l l : r x , y mean b e c o m e s 0 . 0 0 3 m a n d r x , y max a m o u n t s t o 0 . 0 0 7 m . T h i s m e a n s t h a t e v e n in t h i s e x t r e m e c a s e t h e f i r s t 8 decimal digits of the adjusted block coordinates are correct and only 6 decimal digits were lost due to rounding effects. I f w e c o m p a r e t h e n u m e r i c a l a c c u r a c y f i g u r e s r x , y mean a n d r x , y max, r e p r e s e n t e d in T a b l e I, w e o b t a i n t h e r a t h e r c o n s t a n t r a t i o : r x , y mean : rx,y max ~ 1 :

2.3

(1)

From (1) it follows that one of the two figures suffices to describe the numerical accuracy of the adjusted block coordinates. In the following we theref o r e o n l y u s e t h e R M S v a l u e rx, y m e a n o f t h e g l o b a l r o u n d i n g e r r o r s , w h i c h is b e t t e r d e t e r m i n e d t h a n t h e s i n g l e v a l u e r x , y max. A d e t a i l e d a n a l y s i s o f t h e o b t a i n e d r e s u l t s h a s s h o w n , t h a t r x z mean c a n b e well approximated by:

106

r x, y

mean

= a

•

(2)

n • i~

In eq. 2 a d e n o t e s a c o n s t a n t w h i c h d e p e n d s on the overlap version. If we express rx,y mean in ~m (in the terrain) we o b t a i n the f o l l o w i n g results f o r a and it's relative a c c u r a c y Oa/a: q = 20% q = 60% q = 2X20%

: a = 5 8 . 1 0 -6 pro, a a / a : a = 1 3 . 1 0 -6 pm, Oa/a i a = 2 3 . 1 0 -6 pm, e a / a

= 0.06 = 0.05 = 0.03

(3a) (3b) (3c)

Relations (3) s h o w t o which e x t e n t the n u m e r i c a l a c c u r a c y improves w h e n q = 20% is replaced by q = 60% or q = 2 X 20%. In first a p p r o x i m a t i o n the f o l l o w i n g ratios are valid: rx, y 20%

: rx, y 60% : rx, y 2x 20% ~ 1 : 0.2 : 0.4

(4)

The empirically o b t a i n e d m o d e l (2) indicates t h a t the n u m e r i c a l a c c u r a c y o f the adjusted p l a n i m e t r i c c o o r d i n a t e s is m u c h m o r e a f f e c t e d b y the c o n t r o l spacing i t h a n b y the b l o c k length n. This is in a g r e e m e n t with the findings o f S c h e n k (1972}. F u r t h e r o n it shall be m e n t i o n e d , t h a t the global r o u n d i n g errors o f adjoining b l o c k c o o r d i n a t e s are strongly (positive) correlated. T h e same result was o b t a i n e d b y B a r t e l m e and Meissl (1975). This characteristic p r o p e r t y is d e m o n strated b y Fig.2, which shows the global r o u n d i n g errors o f the adjusted x c o o r d i n a t e s f o r o n e o f the investigated blocks (x = flight direction). A c o m p a r i s o n o f the n u m e r i c a l a c c u r a c y o f a practical b l o c k (n ~ 20, c o n t r o l version 3 and 4, q = 20%) with the c o r r e s p o n d i n g results o f t h e present investigation s h o w e d a v e r y g o o d a g r e e m e n t . Even the figures rx,y max d i f f e r e d o n l y b y a f a c t o r 1.2 and 1.5. This a g r e e m e n t c o n f i r m s t h a t the pres e n t e d results are realistic estimates f o r the n u m e r i c a l a c c u r a c y o f practical blocks. Numerical

accuracy of adjusted height coordinates

T h e o b t a i n e d global r o u n d i n g e r r o r figures rz mean a n d rz max are considerably smaller t h a n the c o r r e s p o n d i n g p l a n i m e t r i c figures t r e a t e d above. T h e largest a m o u n t s again a p p e a r with the very e x t r e m e case n = 50 ( b l o c k size 500 X 500 km2), c o n t r o l version 4, q = 20% (see F i g . l ) . But rz mean only a m o u n t s t o 21 pm in the terrain and rz max o n l y t o 29 #m. Because o f the small size o f the o b t a i n e d global r o u n d i n g errors n o definite s t a t e m e n t s c o n c e r n i n g the d e p e n d e n c e o f the n u m e r i c a l a c c u r a c y o n b l o c k size and c o n t r o l spacing can be m a d e here. A c o m p a r i s o n o f the p r e s e n t h e i g h t results with t h e n u m e r i c a l h e i g h t a c c u r a c y o b t a i n e d with the a b o v e m e n t i o n e d practical b l o c k (n ~ 20, c o n t r o l version 3 and 4, q = 20%) s h o w e d large discrepancies. T h e practical figures

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block adjustment, have a direct effect on the terrain heights. In spite of this weak point of the present investigation, the statement, that the global rounding errors, resulting from an extreme control distribution in x, y and z (control version 3 or even 4) are larger in planimetry than in height, remains valid. With the mentioned practical block the ratio r x , y m a x : r z m a x is still in the order of 10 : 1. From there it follows that the numerical accuracy of spatial block adjustment by independent models usually will be limited by the global rounding errors of the planimetric coordinates and n o t of the block heights. CONCLUDING REMARKS

As stated in the 2nd section the present study is based on wide angle photography and 6 points per model. Other image angles and more points per model were n o t investigated here, because this was already done by Schenk {1972). Although Schenk's study is related to bundle blocks the corresponding results are transferable to our case, at least qualitatively. From an extension of the image angle a (slight) improvement of the numerical accuracy can be expected. The same is valid for an extension of the number of points per model. The results of the preceding section have shown, that a mantissa length of r = 48 binary digits {CDC 6600) guarantees adequate numerical accuracy. Even in the most extreme case the first 8 decimal digits of the adjusted terrain coordinates were correct. If a c o m p u t e r with another mantissa length r is used the results of the present study can be approximately converted to the new case, considering that the global rounding errors are proportional to 2 -r. This means, that a reduction of the mantissa length by k binary digits to r = 4 8 - k will increase the global rounding errors, obtained in the preceding section by the factor 2 k approximately. Whether the numerical accuracy then still suffices depends on the amounts of the block parameters n, i and q (see relations (2) and (3). Although the results of the present study were obtained with a certain computer program (PAT-M43), in first approximation they will remain valid, when other comparable programs with Gauss--Cholesky solution are applied. Moreover, for bundle block adjustments with the same solution type, similar numerical accuracies can be expected too. The model (2), describing the dependence of the numerical accuracy on the block length and on the control spacing, was derived in a purely empirical way. It would be desirable if future theoretical analyses would either confirm relation (2) or improve the model. The concept, used by Bartelme and Meissl {1975) seems to be most suitable for that.

109

REFERENCES Ackerraann, F., Ebner, H. and Klein, H., 1973. Block Triangulation with Independent Models. Photogramm. Eng., 1973: 967--981. Bartelme, N. and Meissl, P., 1975. Theoretical Analysis of Rounding Error Propagation During the Direct Solution of Geodetic Normal Equations of the Leveling Type. Paper presented at the General Assembly of IAG, Grenoble, 1975. Breuer, P., 1969. Rundungsfehler bei direkter AuflSsung geod~tischer Gleichungssysteme. DGK, Reihe C, Heft 135. Mayer, R., 1975. Empirische Untersuchung der Rechensch~fe bei r~iumlicher Blockausgleichung mit unabh~ingigen Modellen. Diplomarbeit am Institut fiir Photogrammetrie, Universit~t Stuttgart. Schenk, T., 1972. Untersuchungen zur Kondition yon Normalgleichungen der Blocktriangulation nach der Biindelmethode. Bildmessung Luftbildwesen, 1972: 276--282. Schmitt, G., 1973. Rounding-off Errors in Solving Linear Systems of High Order. Paper presented at the International Symposium on Computational Methods in Geometrical Geodesy, Oxford, 1973.