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Relative accuracy and independent geodetic control in strip triangulation
Photogrammetria - Elsevier Publishing Company, Amsterdam - Printed in The Netherlands
R E L A T I V E A C C U R A C Y AND I N D E P E N D E N T G E O D E T I C C O N T R O L IN STRIP T R I A N G U L A T I O N l A. J. VAN DER WEELE
International
Training Centre for Aerial Survey, Delft (The Netherlands)
(Received February 15, 1966)
SUMMARY
In this paper the role of photogrammetry in surveying is described as the technique to avoid fieldwork. This is in particular important as far as the application of aerial triangulation is concerned. Progress in this field has been made in several directions but, apart from using airborne camera orientation and positioning equipment a limit seems to present itself in our present knowledge of the properties of aerial photography. Two ideas are taken up here which might contribute to a further saving on fieldwork. They are not new but the author feels that they have not got sufficient attention up until now. The first deals with the definition of tolerances. For the majority of applications of photogrammetry it is sufficient to require a good relative accuracy of points. It is shown that the error in relative position of points as a function of the strip length accumulates much more favourably than the absolute accuracy. This indicates that greater distances can be bridged so that a saving of fieldwork can be obtained. The results shown here are not yet complete und further research in this field will be required to give a sound basis for planning. The second point deals with the use of independent ground control. Here again the provisional results obtained so far indicate that in comparison to cantilever strips a gain in accuracy can be obtained, which is growing with an increasing density of these baselines. The analogy with the use of camera orientation equipment is mentioned as a limit where each model is controlled by an independent additional observation. INTRODUCTION
Photogrammetry has once been described as the technique to avoid computations. 1 Paper presented at the International Symposium on Spatial Aerotriangulatiom February 28_ March 4, 1966, University of Illinois, Urbana, Ill. (U.S.A.).
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A. J. VAN DER W E E L E
From the point of view of the man who had to restitute aerial photographs and to produce maps, this characterisation may have been correct; however, it certainly does not reflect the reason why photogrammetry has penetrated in so many fields of the surveying profession and why it is still making progress to further applications. Not only have the modern developments made the description obsolete in view of the fact that the recent progress of photogrammetry is based on the wide application of modern computational means. Much more important has been the fact that photogrammetry has proven to be a more economical procedure for producing maps and, in general, for providing measuring data. This economical value is, to a great extent, based on the property that the application of aerial photographs allows a considerable saving on the extensive fieldwork that formerly was an integral part of every surveying job. On this basis it will be more appropriate to describe photogrammetry as the technique to avoid fieldwork. The saving in fieldwork has been achieved along two lines. First of all the use of pairs of photographs for the plotting of topographical details (planimetry and heights) has replaced a great part of the cumbersome detail surveying. Secondly the application of aerial triangulation has given a means to replace lower-order triangulation and traversing as well as levelling to an extent that has proven to be of considerable value. The principles of both fields of application have been known already for about half a century and the improvements made during that time are mainly limited to three aspects, viz.: (l) Improvements in the quality of photographs and instruments, leading to more complete and more accurate restitutions. (2) Improvements in methods of adjustment, in particular by the development of block-adjustment procedures (analogue as well as digital). (3) A third development comprises a number of airborne camera-positioning instruments such as statoscopes, horizon cameras, gyroscopes, and different types of radar positioning. They have reached a stage of operational possibility but their economical advantages are so far, only proven under special circumstances. In this paper the use of aerial photographs for plotting will not be discussed. The present developments, aiming at the extraction of more and better information from the photographs may improve the economy of photogrammetry, but will add nothing to the principles on which the restitution is based. In the field of aerial triangulation a similar influence can be observed. Better accuracies than where formerly possible are being obtained nowadays by using better photography, better instruments and better adjustment procedures but apart from their quality there is not much really new in these aspects. The camera-positioning instruments have the common property that they are still less accurate than the usual procedures of relative orientation so that they only contribute to control the accumulation of errors in an aerial triangulation. Roughly speaking, we can state that we have reached a phase where the limits of accuracy obtained by using the best-known instruments and procedures Photogrammetria, 21 11966) 43 55
ACCURACY AND GEODETIC CONTROL IN STRIP TRIANGUI,ATION
45
is lying in the photography itself and that further progress depends on our ability to improve our knowledge about the image deformations in the widest sense, including the properties of the atmosphere, the lens, the camera, the emulsion and its support, the photographic process and finally the process of observation and identification. In the context of the reasoning presented here it will be clear that the importance of the improvements is that they will allow a further decrease in fieldwork or, in other words, that bridging distances between ground control can be increased. This possibility will yield an additional saving and a strengthening of the economical justification of photogrammetry. Apart from the technical improvements, mentioned above, there are two other concepts which may contribute in the same sense but which, so far, have not received sufficient attention. They are certainly not new, but I am convinced that they deserve a closer examination since their influence on the saving on fieldwork may well be of the same order of magnitude as we can expect in the near future from our improved knowledge of the properties of our photographs. The first concept is that for practical purposes tolerances can be expressed in terms of relative accuracy instead of absolute accuracy. The second concept is that relative positions of ground control points (lengths and azimuths) can be determined easier than their absolute coordinates. Both these subjects will now be examined in more detail.
RELATIVE ACCURACY
Definition of tolerances The introduction of relative accuracy as a criterion for the quality of a survey is based on the fact that the majority of human activities, where measurements are carried out, are limited to restricted areas only. A simple example may be quoted to illustrate the idea. If a road has to be designed between two points over a distance of say 50 km for which maps on the scale 1 : 2,000 are required, one will, in the first instance, be inclined to fix the tolerances for the survey in relation to the map scale. For planimetry this would be about 40 cm in this case, assuming that 0,2 m m or 0,01 inch is the accuracy required on map scale. To determine a point in the middle between the end points with a tolerance of 0,4 m means in terms of a ratio, a tolerance of 1 : 6 0 , 0 0 0 or a standard error of the order of 1 : 1 2 0 , 0 0 0 . To achieve this will require quite some effort. If a traverse would be measured with reasonably good equipment so that the standard errors of distances are about 1 : 10,000 and of the angles about 0,5 centigrades, then it would take a traverse length of about 2 km before the tolerance of 40 cm is being trespassed. Extending the traverse to 50 km will not change essentially the internal or relative accuracy within each section of 1 or 2 kin. Photogrammetria.
21 (1966) 43 55
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A . J . VAN DER WEELE
For all practical purposes this will be quite sufficient provided that care is taken that the staking-out of the design is done with reference to the same points, that have been used to construct the map. However, the error in the middle of the traverse between the two end points may easily be several meters if compared with the result of primary or secondary triangulation points. What might happen is, for example, that a part of the road which has been designed as a straight line will be slightly curved in reality. It would require someone with a good theodolite to show this deviation and none of the road users will ever worry about it. The most important feature is that the road is constructed exactly where it has been planned "'with respect to the terrain details". Although it has been sufficiently demonstrated with this example that a requirement for relative precision is quite sufficient in this case, and although it is hoped that, in addition, it will be accepted that this same concept can be applied in a lot of other cases, one remark should be added to anticipate questions that might arise. To replace the traverse mentioned in the example by a photogrammetric triangulation of a strip of photographs is a very obvious step. Everything which is said about the traverse applies also to this strip, although there may be a difference in the precision of the basic elements. To get an idea about the length of a strip that can be accepted if the requirements for accuracy are put in terms of relative accuracy, M. B. Osman from Egypt has carried out a number of computations in the frame of his work for a M. Sc. degree at the I.T.C. in Delft. In the next section a few of his results will be quoted.
Derivation of formulae The following assumptions have been made: (1) The first model of the strip (Fig.l, 0-1) has a perfect absolute orientation. (2) The triangulation of the strip is carried out in an analogue instrument using the numerical relative orientation method (elements of new projector). (3) Scale transfer is done by equating heights of the common nadir point of consecutive models.
Y x !
2
3
Fig. 1. Definition of coordinate axes and m o d e l n u m b e r s in a strip.
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ACCURACY AND GEODETIC CONTROL IN STRIP TRIANGULATION
47
The coordinate errors of an arbitrary point in the strip can now be expressed as a function of the observations (consisting of Y-parallaxes, and X-, Yand Z-coordinate settings) and the law of propagation of errors can be applied to compute the cofactors of these coordinates. Taking into account that the coordinates of neighbouring points in a strip are highly correlated, it is also possible to compute the cofactors of the distance between two of those neighbouring points. On this basis a few examples have been computed. In these computations the standard error of eliminating a Y-parallax has been chosen as the standard error of unit weight so that Q~'~'~ = 1. Further assumptions have been: Q~'"p 1, Q ..... 1, Q."." = l, where Q ...... and Q~" are the cofactors of the setting errors in x and y in the photogrammetric model. The complete derivations will not be given here. They would cover many pages but since they are based on well-known principles which have been published elsewhere they will be limited here to show the type of formulae that is found and in the next section some of the results will be shown. In Fig.2 the distances that have been chosen are indicated by numbers 1-18. For the distance between the points 3 and 7 which is a distance in X-direction, we have in the case of the end-free strip: Q,hJ, - Qx:,x:~ + Qx~x~__ 2Qx:~x~. From the derivation we have found:
QX3X3
Qp_,:px+ (i ~ QxTxr
- 1 " - - 1) (4i ~ ÷ 4i + 1
Qx:,x; = E 2- 4 1 ( '.-
QPYPY+
1) (4i'-' -- i + 16 )~ QPYPY+
1)Qxx
6 6 [li(i--1)(4i+
~)~Qpxpx~ Q''
where i is the number of the model in which the distance has been chosen. In the case that it is assumed that the strip is connected to two points, one at the beginning and one at the end of the strip, the computation of Q,l,,t, would require the knowledge of the full cofactor-matrix as follows: m
Q.~,~,~
Q.,',,u,,
Q~,,.,,,,
Q J>,".,
Q,,.,,.,'7
Q,,,,,,,,.
Q,,,,.,:;
Q,,,,,;
Q,,':m~
Q~:<7
Q ~'7a:7
PhotograntmetHa, 21 (1966) 43 55
48 "3
8
A. J. VAN I ) E R W E E L E
~
,o.
i
i.
'I '.5
r,
D/s~ence l b e / ' w o e n po/n,~s
2 ¢T 4 5 6
'"
i 1
~9
L,
12
2~ 16_.
20
6
2,
7 8 9 1£7 II IZ 15 14 15 16' 17 18
I'1 •
,,
3-7
J-4 1-12 I-2 5-8 5-1 3-9 5-5 I0-II 8-9 4 -14 15-162-18 4 -17 I0 -16 4 -Ig 4 -20 4 -21
Fig.2. Definition of distances for which cofactors are computed. in which b = point number at beginning of the strip, and n -- point number at end of the strip. When the adjustment of the strip is to be based on more points the required cofactor matrix will grow accordingly and this will be for the present moment the excuse for not giving results here for these cases.
Results For the eighteen distances, indicated in Fig.2, the cofactors have been computed for different lengths of strips, and for the model or models at the end of the strip. The ratio between the standard error of the distance and the standard error of unit weight has been used as ordinate in Fig.3 to show the relation with the strip length ( ~ / ~ o - VQ'~"). Where the same line in the graph is indicated as belonging to a combination of distances, it is not so that the computed values are exactly the same, but the differences are so small that they could hardly be shown on the chosen scale. It looks as if there is quite a range between the best and the worst distance indicated. The graphs come much closer together when the ratio 1/Q't"/d is considered. Much more important is, however, the comparison between distances and coordinates. In Fig.4 a presentation is given of the standard error at the end of a cantilever strip in terms of VQ'"+ Q"" in comparison with the data of Fig.3 which are here concentrated in a band. It is obvious that the difference is considerable and that much longer strips can be accepted to fulfil requirements for relative accuracy than for absolute accuracy. This presentation is clearly inadequate as a basis for planning. The mathematical as well as stochastical model which form the basis of the present derivations still lack sufficient confirmation from practical tests. As far as the mathematical model is concerned a connection of the strip to the two ends will be more representative for the usual practical arrangements. P h o t o ~ r a m m e t r i a . 21 ( 1 9 6 6 ) 4 3 - - 5 5
ACCURACY AND GEODETIC CONTROL IN STRIP TRIANGULATION
49
13-" 1
~o
~14
9 8 7
6 5
/ /
~16
/
~15t17 ~2,5,6,9,10
4
/ /
3
/
/
/
J
~11,12,13
I
/
2
I
I
I
4
8
10
12
L---I~
14
i
Fig.3. S t a n d a r d errors of distances as a f u n c t i o n of strip l e n g t h (i).
Indications from a few computations carried out so far lead to the impression that the standard error in the distances will not improve essentially from this connection. The maximum standard error of the strip coordinates (in the middle of the strip) will be about four times better than that at the end of a cantilever strip. If this impression is confirmed by further research the difference between absolute and relative accuracy in the strip will be much less than Fig.4 demonstrates now. As far as the present estimations go, there remains a difference in favour of the relative accuracy and consequently the expression of specifications in terms of relative accuracy will anyhow be a contribution to a saving in fieldwork and to a greater economy of the application of photogrammetry.
~r" % 6O
/
50 40
/
30 20 10 0
2
4
6
8
I0
12
14
Fig.4. C o m p a r i s o n between absolute Ca) a n d relative a c c u r a c y (b) as a f u n c t i o n of the l e n g t h of a strip. C u r v e a =
1 / Q ~'x +- QSJ~,; area b ~
V Q 'ta. Photogrammetria,
21 (1966) 4 3 - 5 5
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A . J . VAN DER WEELE
INDEPENDENT GROUND CONTROl,
The concept to use relative positions of ground control points to strengthen the adjustment of strips (or blocks) has been treated in literature under the headings: independent bases, local control, independent ground control, crossbases, etc. These terms have in common the fact that they are meant to express that a ground-control figure (which may consist of two or more points) is used, in which the coordinates of the points are known in a local system only. In its simplest form this may be the length, azimuth and inclination of a line connecting two points only, but the figure can also exist of a group of points distributed systematically or arbitrarily over one or more models of a strip (or block). The improvements that can be obtained by including these groups of points in the adjustment of a strip depend on several circumstances of which the most important are:
(1) The distances between points. (2) The distribution of the bases over the strip. (3) The number of additional points of which absolute coordinates are known. (4) The identification errors of the points. (5) The accuracy of the terrestrial measurements. (6) The adjustment procedure used for determining final coordinates. It will be clear that a treatment of all those parameters and their influence on the accuracy of strip coordinates separately or in different combinations requires the execution of a vast program of computations. In the present paper no attempt will be made to cover the complete gamma of possibilities, but the treatment will, for the present time, be limited to a few points only. The computations which have been carried out so far, partly by Ackermann and partly by Gordon in connection with his study at the I.T.C. have already made sufficient progress to demonstrate a few important properties of the concept. Contrary to the developments on which the derivations, used in the section "Relative accuracy", were based and which start from the parallax and coordinateobservations themselves, here a simplification has been accepted, on the same lines as explained by ACKERMANN (1962). The implifications include the assumption that all coordinate errors in a strip are caused by errors in the connection of subsequent models and that the models themselves are perfect. This assumption means that in each connection of one model to the foregoing one seven elements are required viz., three shifts, three rotations and a scale factor. The error accumulation in the strip leads to a separation of these seven elements in two groups in relation to the fact of whether their influence on the coordinates can be computed as a single or as a double summation. When long strips are considered (e.g., with more than six or ten models) Photogrammetria, 21 (1966) 43 55
ACCURACY AND GEODETIC CONTROL IN STRIP TRIANGULATION
51
the errors giving rise to a double summation are predominant and the other can practically be neglected. Based on this reasoning the general properties of the error propagation in strips can be studied on the basis of a few error sources only viz., the scale transfer error with respect to the X-coordinates, the azimuth transfer error with respect to )'-coordinates and the longitudinal tilt with respect to Z-errors on the strip axis. From this reasoning it follows that there is no difference in general attitude between X-, Y- and Z-coordinates with the exception that the standard error in the transfer element, which will be chosen as the standard error of unit weight, may have different values in the three cases. It can be shown, at least for the usual method of aerial triangulation in an analogue instrument, that the subsequent transfer errors in a strip can be considered to be uncorrelated which simplifies their use conveniently (see e.g., VERMEIR, 1954). To be able to study the influence of errors in the given control points the weight matrix will be supplemented with a unit matrix 7 [1] where for 7 values of 0, 1, 4 and 9 respectively will be used. The use of this unit matrix means that we consider the coordinates of the control points to be uncorrelated and equally accurate. This choise may be based in the first place on the absence of information to the contrary, but it is supported by the fact that it enables us to assume that setting errors or identification errors may be considered to be included. In the case of treating Z-coordinates on the strip axis it means that the standard error of unit weight o,, corresponds approximately with oz so that 7 - 1 means that just the setting error is taken into account. In the case of X- and Y-coordinates the relations will be slightly different, but this is of no importance here since the aim of this paper is only to show some general properties of the problem. The choice of numerical values for the standard error of the observations has anyhow to be decided on the bases of the material available in each particular case. The adjustment of the strips is carried out according to a rigorous least squares method (in fact according to standard problem IV of TIENSTRA (1956) which allows the combined determination of the transformation parameters of the strip coordinates (shift and rotation) and the corrections to the observations according to the least square principle. This adjustment is defined as rigorous since the mathematical model is exactly defined and the corresponding weight matrix is being used according to the formula (for Z-coordinates and points on strip axis): i
G~' -= g " " X
1
( x ~ " - - x , ) ( x ~ - - x , , ) + )'~-~
where # o - - 1 . . . i ; ~ ? < o ; 6 e° = 0 f o r ~27d°;6'-''' = l for~2-- doandg'~'J -- I. The arrangements of control points have been chosen as illustrated in Fig.5. Control points are assumed to lie on the axis of the strip. At the beginning two points
Photogrammetria, 21 (1966) 4 3 - 5 5
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A . J . VAN DER W E E L E
o
B
Fig.5. A r r a n g e m e n t of i n d e p e n d e n t c o n t r o l a b s o l u t e c o o r d i n a t e s are k n o w n .
in a strip. T r i a n g l e s i ndi c a t e points of w hi c h
are assumed to be known in absolute coordinates to fix the axis system in which the computations are done. The length of each basis is a whole number of base lengths; so B = kb, where k = 1, 2, 3, etc. All bases considered in one computation have equal length. The results will be expressed in terms of ,~, for which in itself no numerical value will be fixed.
Results Influence of one base at the end of the strip In Fig.6 the maximum standard error of a point of the strip (which is at the end point in this case) is shown as a function of the number of models for several arrangements of bases. From this graph the following conclusions can be drawn: (1) The influence of a base at the end gives an improvement of the order of 30% in the standard error at the end. (2) The use of longer base-lengths does not give an additional accuracy. That curve c is sligthly lower than curve b is due to a shift to the right over three
L O-ma x CF'o
100
. . . .
.,,e,._~ a w
80
/
/
60
'
///
×
40
-~--- b 4~----
C
~--d
20
0
10
20
30
40
Fig.6. T h e m a x i m u m s t a n d a r d e r r o r of a m o d e l s (n). a = c a n t i l e v e r strip w i t h 7 -f or w h i c h 7 -- 0, k -- 1; c -- strip w i t h k 4; d -- strip with one base at the end
p o i n t of the strip as a f u n c t i o n of the n u m b e r of 0; b = strip w i t h in a d d i t i o n one base at the e nd in a d d i t i o n one base at t he e nd for w h i c h 7 := 0, a n d a d d i t i o n a l base in t he m i d d l e of the strip.
Photogrammetria, 21 (1966) 4 3 - 5 5
ACCURACY AND GEODETIC CONTROL IN STRIP TRIANGULATION
53
models since the control point with o = 0 at the beginning is formed at n = 4 instead of at n = 1 for curve b. Influence of errors in ground control For an end-free strip the errors in ground control have, in addition to the error propagation in the strip, an influence on o....... which grows proportional to the strip length. Fig.7 demonstrates this influence for a particular example where n = 15 and one base at the end of the strip is available. The standard error at the end is shown as a function of 7 and of k. According to the second conclusion drawn from Fig.6 the line for ), = 0 should be horizontal. The slight decrease of ~..... with growing values of k reflects the fact in reality the effective length of the strip is shorter. O-ma x o--0
n=15
SO
40
30
20
10
b-k Fig.7. Influence o f errors in control points (7) and of varying base length on ~m.x for a strip of fifteen models.
The conclusion from this graph is that the influence of ), is considerable and that the accuracy of the result can be improved by using the longest feasable base length as well as by aiming at the lowest possible value of 7. Since ~, is supposed to include terrestrial as well as photogrammetrical errors the photogrammetrist can contribute to an effective lower ;, by using multiple points of the end of each base and choosing them carefully with respect to identification errors. A group of signalised points seems to be the best solution. Strips with one base in the centre The computation of o, .... for strips having one base in the centre and two single points at the end which are given in absolute coordinates (7 = 0), shows that this base gives no contribution to the improvement of the accuracy. The length of the base line has neither any influence and it can be stated that the case of a base line with a length equal to the strip length is just the limiting situation where the effect is the same as if no base existed at all.
Photogrammetria, 21 (1966) 4 3 - 5 5
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A. J. VAN DER W E E I , E
Strips with bases on equal distances
Let us assume that the strip consists of l sections, each with m models between consecutive bases. The computations have shown that if o....... for a strip of i models is equal to S (given in curve b of Fig.6) that ~...... at the end of t sections is equal to S 1/ I. Curve b (Fig.6) can (for larger numbers of n) be described with the formula o2,..x = a~n3oo2 of i~lllltX a n V n oo. Hence for one section of m models S == a m V m oo, and for a strip of l sections of m models each S ...... = S 1/ l = a r n 1/m Ioo. If the total number of models m l is again equal to n we have: Sm~x = a m l / n o o Compared with a strip with one base at the end it is found: =
Omax S,,,,,x
a n V n o0 - - a m 1 / n o~,
n m
The accuracy of the strip can obviously be increased by making m smaller. The extreme is that we have a ground base in each model, which would obviously be very close to a complete terrestrial triangulation of the strip. However, in the context of this paper we can say that for the X-coordinates in the strip the same effect is obtained by a scale check in every model. This can easily be provided by an A.P.R. measuring ground clearances. For the Z-coordinates the same effect is obtained by measuring .; e.g. with a horizon camera, a gyroscope or a solar periscope. For the Y-coordinates a similar obvious means of continuous control is not readily available although the forward looking oblique camera as proposed by BLACnUT (1957/1958, p.38), comes very close. The actual effect of this control in each model is one of the subjects treated by Dr. JERIE (1966) on this symposium. It should be added that provisional computations show that the properties mentioned in this section do not hold completely when ~, ~ 0. The tendency however, is still the same.
CONC LUSION
The general conclusion from the foregoing considerations should be, that in a complete set of tools which is used as a base for planning an aerial survey, the properties of both ideas should be included. Whether their application is justified depends on the specifications and circumstances of each particular case. It is hoped that the indications in this paper will be sufficient to prove that both ideals have potential advantages and that it is worthwhile to continue research in this direction, both in theory and in practice.
Photogrammetria,
21 (1966) 4 3 - 5 5
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REFERENCES
ACKERMANN, F. E., 1962. Analytical Strip Adjustment. I.T.C. Publ. A.17, 24 pp. BLACHUT, T. J., 1957/1958. Use of auxiliary data in aerial triangulation over long distances. Photograrnmetria, 14(I) : 38-45. .IERIE, H. G., 1966. Theoretical height accuracy of strip and block triangulation with and without use of auxiliary data. Photogramrnetria, in preparation. TIENSTRA, J. M., 1956. Theory o/ the Adjustment o/ Normally Distrihuted Observations. Argus, Amsterdam, 232 pp. VV~RMEIR, P. A., 1954. k a triangulation agrienne. Bull. Beige Photograrnmetrie, 1953(35): 17-57.
Photograrnmetria, 21 (1966) 4 3 - 5 5
R E L A T I V E A C C U R A C Y AND I N D E P E N D E N T G E O D E T I C C O N T R O L IN STRIP T R I A N G U L A T I O N l A. J. VAN DER WEELE
International
Training Centre for Aerial Survey, Delft (The Netherlands)
(Received February 15, 1966)
SUMMARY
In this paper the role of photogrammetry in surveying is described as the technique to avoid fieldwork. This is in particular important as far as the application of aerial triangulation is concerned. Progress in this field has been made in several directions but, apart from using airborne camera orientation and positioning equipment a limit seems to present itself in our present knowledge of the properties of aerial photography. Two ideas are taken up here which might contribute to a further saving on fieldwork. They are not new but the author feels that they have not got sufficient attention up until now. The first deals with the definition of tolerances. For the majority of applications of photogrammetry it is sufficient to require a good relative accuracy of points. It is shown that the error in relative position of points as a function of the strip length accumulates much more favourably than the absolute accuracy. This indicates that greater distances can be bridged so that a saving of fieldwork can be obtained. The results shown here are not yet complete und further research in this field will be required to give a sound basis for planning. The second point deals with the use of independent ground control. Here again the provisional results obtained so far indicate that in comparison to cantilever strips a gain in accuracy can be obtained, which is growing with an increasing density of these baselines. The analogy with the use of camera orientation equipment is mentioned as a limit where each model is controlled by an independent additional observation. INTRODUCTION
Photogrammetry has once been described as the technique to avoid computations. 1 Paper presented at the International Symposium on Spatial Aerotriangulatiom February 28_ March 4, 1966, University of Illinois, Urbana, Ill. (U.S.A.).
Photogrammetria, 21 (1966) 43 55
44
A. J. VAN DER W E E L E
From the point of view of the man who had to restitute aerial photographs and to produce maps, this characterisation may have been correct; however, it certainly does not reflect the reason why photogrammetry has penetrated in so many fields of the surveying profession and why it is still making progress to further applications. Not only have the modern developments made the description obsolete in view of the fact that the recent progress of photogrammetry is based on the wide application of modern computational means. Much more important has been the fact that photogrammetry has proven to be a more economical procedure for producing maps and, in general, for providing measuring data. This economical value is, to a great extent, based on the property that the application of aerial photographs allows a considerable saving on the extensive fieldwork that formerly was an integral part of every surveying job. On this basis it will be more appropriate to describe photogrammetry as the technique to avoid fieldwork. The saving in fieldwork has been achieved along two lines. First of all the use of pairs of photographs for the plotting of topographical details (planimetry and heights) has replaced a great part of the cumbersome detail surveying. Secondly the application of aerial triangulation has given a means to replace lower-order triangulation and traversing as well as levelling to an extent that has proven to be of considerable value. The principles of both fields of application have been known already for about half a century and the improvements made during that time are mainly limited to three aspects, viz.: (l) Improvements in the quality of photographs and instruments, leading to more complete and more accurate restitutions. (2) Improvements in methods of adjustment, in particular by the development of block-adjustment procedures (analogue as well as digital). (3) A third development comprises a number of airborne camera-positioning instruments such as statoscopes, horizon cameras, gyroscopes, and different types of radar positioning. They have reached a stage of operational possibility but their economical advantages are so far, only proven under special circumstances. In this paper the use of aerial photographs for plotting will not be discussed. The present developments, aiming at the extraction of more and better information from the photographs may improve the economy of photogrammetry, but will add nothing to the principles on which the restitution is based. In the field of aerial triangulation a similar influence can be observed. Better accuracies than where formerly possible are being obtained nowadays by using better photography, better instruments and better adjustment procedures but apart from their quality there is not much really new in these aspects. The camera-positioning instruments have the common property that they are still less accurate than the usual procedures of relative orientation so that they only contribute to control the accumulation of errors in an aerial triangulation. Roughly speaking, we can state that we have reached a phase where the limits of accuracy obtained by using the best-known instruments and procedures Photogrammetria, 21 11966) 43 55
ACCURACY AND GEODETIC CONTROL IN STRIP TRIANGUI,ATION
45
is lying in the photography itself and that further progress depends on our ability to improve our knowledge about the image deformations in the widest sense, including the properties of the atmosphere, the lens, the camera, the emulsion and its support, the photographic process and finally the process of observation and identification. In the context of the reasoning presented here it will be clear that the importance of the improvements is that they will allow a further decrease in fieldwork or, in other words, that bridging distances between ground control can be increased. This possibility will yield an additional saving and a strengthening of the economical justification of photogrammetry. Apart from the technical improvements, mentioned above, there are two other concepts which may contribute in the same sense but which, so far, have not received sufficient attention. They are certainly not new, but I am convinced that they deserve a closer examination since their influence on the saving on fieldwork may well be of the same order of magnitude as we can expect in the near future from our improved knowledge of the properties of our photographs. The first concept is that for practical purposes tolerances can be expressed in terms of relative accuracy instead of absolute accuracy. The second concept is that relative positions of ground control points (lengths and azimuths) can be determined easier than their absolute coordinates. Both these subjects will now be examined in more detail.
RELATIVE ACCURACY
Definition of tolerances The introduction of relative accuracy as a criterion for the quality of a survey is based on the fact that the majority of human activities, where measurements are carried out, are limited to restricted areas only. A simple example may be quoted to illustrate the idea. If a road has to be designed between two points over a distance of say 50 km for which maps on the scale 1 : 2,000 are required, one will, in the first instance, be inclined to fix the tolerances for the survey in relation to the map scale. For planimetry this would be about 40 cm in this case, assuming that 0,2 m m or 0,01 inch is the accuracy required on map scale. To determine a point in the middle between the end points with a tolerance of 0,4 m means in terms of a ratio, a tolerance of 1 : 6 0 , 0 0 0 or a standard error of the order of 1 : 1 2 0 , 0 0 0 . To achieve this will require quite some effort. If a traverse would be measured with reasonably good equipment so that the standard errors of distances are about 1 : 10,000 and of the angles about 0,5 centigrades, then it would take a traverse length of about 2 km before the tolerance of 40 cm is being trespassed. Extending the traverse to 50 km will not change essentially the internal or relative accuracy within each section of 1 or 2 kin. Photogrammetria.
21 (1966) 43 55
46
A . J . VAN DER WEELE
For all practical purposes this will be quite sufficient provided that care is taken that the staking-out of the design is done with reference to the same points, that have been used to construct the map. However, the error in the middle of the traverse between the two end points may easily be several meters if compared with the result of primary or secondary triangulation points. What might happen is, for example, that a part of the road which has been designed as a straight line will be slightly curved in reality. It would require someone with a good theodolite to show this deviation and none of the road users will ever worry about it. The most important feature is that the road is constructed exactly where it has been planned "'with respect to the terrain details". Although it has been sufficiently demonstrated with this example that a requirement for relative precision is quite sufficient in this case, and although it is hoped that, in addition, it will be accepted that this same concept can be applied in a lot of other cases, one remark should be added to anticipate questions that might arise. To replace the traverse mentioned in the example by a photogrammetric triangulation of a strip of photographs is a very obvious step. Everything which is said about the traverse applies also to this strip, although there may be a difference in the precision of the basic elements. To get an idea about the length of a strip that can be accepted if the requirements for accuracy are put in terms of relative accuracy, M. B. Osman from Egypt has carried out a number of computations in the frame of his work for a M. Sc. degree at the I.T.C. in Delft. In the next section a few of his results will be quoted.
Derivation of formulae The following assumptions have been made: (1) The first model of the strip (Fig.l, 0-1) has a perfect absolute orientation. (2) The triangulation of the strip is carried out in an analogue instrument using the numerical relative orientation method (elements of new projector). (3) Scale transfer is done by equating heights of the common nadir point of consecutive models.
Y x !
2
3
Fig. 1. Definition of coordinate axes and m o d e l n u m b e r s in a strip.
Photogrammetria,
21 (1966) 4 3 - 5 5
ACCURACY AND GEODETIC CONTROL IN STRIP TRIANGULATION
47
The coordinate errors of an arbitrary point in the strip can now be expressed as a function of the observations (consisting of Y-parallaxes, and X-, Yand Z-coordinate settings) and the law of propagation of errors can be applied to compute the cofactors of these coordinates. Taking into account that the coordinates of neighbouring points in a strip are highly correlated, it is also possible to compute the cofactors of the distance between two of those neighbouring points. On this basis a few examples have been computed. In these computations the standard error of eliminating a Y-parallax has been chosen as the standard error of unit weight so that Q~'~'~ = 1. Further assumptions have been: Q~'"p 1, Q ..... 1, Q."." = l, where Q ...... and Q~" are the cofactors of the setting errors in x and y in the photogrammetric model. The complete derivations will not be given here. They would cover many pages but since they are based on well-known principles which have been published elsewhere they will be limited here to show the type of formulae that is found and in the next section some of the results will be shown. In Fig.2 the distances that have been chosen are indicated by numbers 1-18. For the distance between the points 3 and 7 which is a distance in X-direction, we have in the case of the end-free strip: Q,hJ, - Qx:,x:~ + Qx~x~__ 2Qx:~x~. From the derivation we have found:
QX3X3
Qp_,:px+ (i ~ QxTxr
- 1 " - - 1) (4i ~ ÷ 4i + 1
Qx:,x; = E 2- 4 1 ( '.-
QPYPY+
1) (4i'-' -- i + 16 )~ QPYPY+
1)Qxx
6 6 [li(i--1)(4i+
~)~Qpxpx~ Q''
where i is the number of the model in which the distance has been chosen. In the case that it is assumed that the strip is connected to two points, one at the beginning and one at the end of the strip, the computation of Q,l,,t, would require the knowledge of the full cofactor-matrix as follows: m
Q.~,~,~
Q.,',,u,,
Q~,,.,,,,
Q J>,".,
Q,,.,,.,'7
Q,,,,,,,,.
Q,,,,.,:;
Q,,,,,;
Q,,':m~
Q~:<7
Q ~'7a:7
PhotograntmetHa, 21 (1966) 43 55
48 "3
8
A. J. VAN I ) E R W E E L E
~
,o.
i
i.
'I '.5
r,
D/s~ence l b e / ' w o e n po/n,~s
2 ¢T 4 5 6
'"
i 1
~9
L,
12
2~ 16_.
20
6
2,
7 8 9 1£7 II IZ 15 14 15 16' 17 18
I'1 •
,,
3-7
J-4 1-12 I-2 5-8 5-1 3-9 5-5 I0-II 8-9 4 -14 15-162-18 4 -17 I0 -16 4 -Ig 4 -20 4 -21
Fig.2. Definition of distances for which cofactors are computed. in which b = point number at beginning of the strip, and n -- point number at end of the strip. When the adjustment of the strip is to be based on more points the required cofactor matrix will grow accordingly and this will be for the present moment the excuse for not giving results here for these cases.
Results For the eighteen distances, indicated in Fig.2, the cofactors have been computed for different lengths of strips, and for the model or models at the end of the strip. The ratio between the standard error of the distance and the standard error of unit weight has been used as ordinate in Fig.3 to show the relation with the strip length ( ~ / ~ o - VQ'~"). Where the same line in the graph is indicated as belonging to a combination of distances, it is not so that the computed values are exactly the same, but the differences are so small that they could hardly be shown on the chosen scale. It looks as if there is quite a range between the best and the worst distance indicated. The graphs come much closer together when the ratio 1/Q't"/d is considered. Much more important is, however, the comparison between distances and coordinates. In Fig.4 a presentation is given of the standard error at the end of a cantilever strip in terms of VQ'"+ Q"" in comparison with the data of Fig.3 which are here concentrated in a band. It is obvious that the difference is considerable and that much longer strips can be accepted to fulfil requirements for relative accuracy than for absolute accuracy. This presentation is clearly inadequate as a basis for planning. The mathematical as well as stochastical model which form the basis of the present derivations still lack sufficient confirmation from practical tests. As far as the mathematical model is concerned a connection of the strip to the two ends will be more representative for the usual practical arrangements. P h o t o ~ r a m m e t r i a . 21 ( 1 9 6 6 ) 4 3 - - 5 5
ACCURACY AND GEODETIC CONTROL IN STRIP TRIANGULATION
49
13-" 1
~o
~14
9 8 7
6 5
/ /
~16
/
~15t17 ~2,5,6,9,10
4
/ /
3
/
/
/
J
~11,12,13
I
/
2
I
I
I
4
8
10
12
L---I~
14
i
Fig.3. S t a n d a r d errors of distances as a f u n c t i o n of strip l e n g t h (i).
Indications from a few computations carried out so far lead to the impression that the standard error in the distances will not improve essentially from this connection. The maximum standard error of the strip coordinates (in the middle of the strip) will be about four times better than that at the end of a cantilever strip. If this impression is confirmed by further research the difference between absolute and relative accuracy in the strip will be much less than Fig.4 demonstrates now. As far as the present estimations go, there remains a difference in favour of the relative accuracy and consequently the expression of specifications in terms of relative accuracy will anyhow be a contribution to a saving in fieldwork and to a greater economy of the application of photogrammetry.
~r" % 6O
/
50 40
/
30 20 10 0
2
4
6
8
I0
12
14
Fig.4. C o m p a r i s o n between absolute Ca) a n d relative a c c u r a c y (b) as a f u n c t i o n of the l e n g t h of a strip. C u r v e a =
1 / Q ~'x +- QSJ~,; area b ~
V Q 'ta. Photogrammetria,
21 (1966) 4 3 - 5 5
50
A . J . VAN DER WEELE
INDEPENDENT GROUND CONTROl,
The concept to use relative positions of ground control points to strengthen the adjustment of strips (or blocks) has been treated in literature under the headings: independent bases, local control, independent ground control, crossbases, etc. These terms have in common the fact that they are meant to express that a ground-control figure (which may consist of two or more points) is used, in which the coordinates of the points are known in a local system only. In its simplest form this may be the length, azimuth and inclination of a line connecting two points only, but the figure can also exist of a group of points distributed systematically or arbitrarily over one or more models of a strip (or block). The improvements that can be obtained by including these groups of points in the adjustment of a strip depend on several circumstances of which the most important are:
(1) The distances between points. (2) The distribution of the bases over the strip. (3) The number of additional points of which absolute coordinates are known. (4) The identification errors of the points. (5) The accuracy of the terrestrial measurements. (6) The adjustment procedure used for determining final coordinates. It will be clear that a treatment of all those parameters and their influence on the accuracy of strip coordinates separately or in different combinations requires the execution of a vast program of computations. In the present paper no attempt will be made to cover the complete gamma of possibilities, but the treatment will, for the present time, be limited to a few points only. The computations which have been carried out so far, partly by Ackermann and partly by Gordon in connection with his study at the I.T.C. have already made sufficient progress to demonstrate a few important properties of the concept. Contrary to the developments on which the derivations, used in the section "Relative accuracy", were based and which start from the parallax and coordinateobservations themselves, here a simplification has been accepted, on the same lines as explained by ACKERMANN (1962). The implifications include the assumption that all coordinate errors in a strip are caused by errors in the connection of subsequent models and that the models themselves are perfect. This assumption means that in each connection of one model to the foregoing one seven elements are required viz., three shifts, three rotations and a scale factor. The error accumulation in the strip leads to a separation of these seven elements in two groups in relation to the fact of whether their influence on the coordinates can be computed as a single or as a double summation. When long strips are considered (e.g., with more than six or ten models) Photogrammetria, 21 (1966) 43 55
ACCURACY AND GEODETIC CONTROL IN STRIP TRIANGULATION
51
the errors giving rise to a double summation are predominant and the other can practically be neglected. Based on this reasoning the general properties of the error propagation in strips can be studied on the basis of a few error sources only viz., the scale transfer error with respect to the X-coordinates, the azimuth transfer error with respect to )'-coordinates and the longitudinal tilt with respect to Z-errors on the strip axis. From this reasoning it follows that there is no difference in general attitude between X-, Y- and Z-coordinates with the exception that the standard error in the transfer element, which will be chosen as the standard error of unit weight, may have different values in the three cases. It can be shown, at least for the usual method of aerial triangulation in an analogue instrument, that the subsequent transfer errors in a strip can be considered to be uncorrelated which simplifies their use conveniently (see e.g., VERMEIR, 1954). To be able to study the influence of errors in the given control points the weight matrix will be supplemented with a unit matrix 7 [1] where for 7 values of 0, 1, 4 and 9 respectively will be used. The use of this unit matrix means that we consider the coordinates of the control points to be uncorrelated and equally accurate. This choise may be based in the first place on the absence of information to the contrary, but it is supported by the fact that it enables us to assume that setting errors or identification errors may be considered to be included. In the case of treating Z-coordinates on the strip axis it means that the standard error of unit weight o,, corresponds approximately with oz so that 7 - 1 means that just the setting error is taken into account. In the case of X- and Y-coordinates the relations will be slightly different, but this is of no importance here since the aim of this paper is only to show some general properties of the problem. The choice of numerical values for the standard error of the observations has anyhow to be decided on the bases of the material available in each particular case. The adjustment of the strips is carried out according to a rigorous least squares method (in fact according to standard problem IV of TIENSTRA (1956) which allows the combined determination of the transformation parameters of the strip coordinates (shift and rotation) and the corrections to the observations according to the least square principle. This adjustment is defined as rigorous since the mathematical model is exactly defined and the corresponding weight matrix is being used according to the formula (for Z-coordinates and points on strip axis): i
G~' -= g " " X
1
( x ~ " - - x , ) ( x ~ - - x , , ) + )'~-~
where # o - - 1 . . . i ; ~ ? < o ; 6 e° = 0 f o r ~27d°;6'-''' = l for~2-- doandg'~'J -- I. The arrangements of control points have been chosen as illustrated in Fig.5. Control points are assumed to lie on the axis of the strip. At the beginning two points
Photogrammetria, 21 (1966) 4 3 - 5 5
52
A . J . VAN DER W E E L E
o
B
Fig.5. A r r a n g e m e n t of i n d e p e n d e n t c o n t r o l a b s o l u t e c o o r d i n a t e s are k n o w n .
in a strip. T r i a n g l e s i ndi c a t e points of w hi c h
are assumed to be known in absolute coordinates to fix the axis system in which the computations are done. The length of each basis is a whole number of base lengths; so B = kb, where k = 1, 2, 3, etc. All bases considered in one computation have equal length. The results will be expressed in terms of ,~, for which in itself no numerical value will be fixed.
Results Influence of one base at the end of the strip In Fig.6 the maximum standard error of a point of the strip (which is at the end point in this case) is shown as a function of the number of models for several arrangements of bases. From this graph the following conclusions can be drawn: (1) The influence of a base at the end gives an improvement of the order of 30% in the standard error at the end. (2) The use of longer base-lengths does not give an additional accuracy. That curve c is sligthly lower than curve b is due to a shift to the right over three
L O-ma x CF'o
100
. . . .
.,,e,._~ a w
80
/
/
60
'
///
×
40
-~--- b 4~----
C
~--d
20
0
10
20
30
40
Fig.6. T h e m a x i m u m s t a n d a r d e r r o r of a m o d e l s (n). a = c a n t i l e v e r strip w i t h 7 -f or w h i c h 7 -- 0, k -- 1; c -- strip w i t h k 4; d -- strip with one base at the end
p o i n t of the strip as a f u n c t i o n of the n u m b e r of 0; b = strip w i t h in a d d i t i o n one base at the e nd in a d d i t i o n one base at t he e nd for w h i c h 7 := 0, a n d a d d i t i o n a l base in t he m i d d l e of the strip.
Photogrammetria, 21 (1966) 4 3 - 5 5
ACCURACY AND GEODETIC CONTROL IN STRIP TRIANGULATION
53
models since the control point with o = 0 at the beginning is formed at n = 4 instead of at n = 1 for curve b. Influence of errors in ground control For an end-free strip the errors in ground control have, in addition to the error propagation in the strip, an influence on o....... which grows proportional to the strip length. Fig.7 demonstrates this influence for a particular example where n = 15 and one base at the end of the strip is available. The standard error at the end is shown as a function of 7 and of k. According to the second conclusion drawn from Fig.6 the line for ), = 0 should be horizontal. The slight decrease of ~..... with growing values of k reflects the fact in reality the effective length of the strip is shorter. O-ma x o--0
n=15
SO
40
30
20
10
b-k Fig.7. Influence o f errors in control points (7) and of varying base length on ~m.x for a strip of fifteen models.
The conclusion from this graph is that the influence of ), is considerable and that the accuracy of the result can be improved by using the longest feasable base length as well as by aiming at the lowest possible value of 7. Since ~, is supposed to include terrestrial as well as photogrammetrical errors the photogrammetrist can contribute to an effective lower ;, by using multiple points of the end of each base and choosing them carefully with respect to identification errors. A group of signalised points seems to be the best solution. Strips with one base in the centre The computation of o, .... for strips having one base in the centre and two single points at the end which are given in absolute coordinates (7 = 0), shows that this base gives no contribution to the improvement of the accuracy. The length of the base line has neither any influence and it can be stated that the case of a base line with a length equal to the strip length is just the limiting situation where the effect is the same as if no base existed at all.
Photogrammetria, 21 (1966) 4 3 - 5 5
54
A. J. VAN DER W E E I , E
Strips with bases on equal distances
Let us assume that the strip consists of l sections, each with m models between consecutive bases. The computations have shown that if o....... for a strip of i models is equal to S (given in curve b of Fig.6) that ~...... at the end of t sections is equal to S 1/ I. Curve b (Fig.6) can (for larger numbers of n) be described with the formula o2,..x = a~n3oo2 of i~lllltX a n V n oo. Hence for one section of m models S == a m V m oo, and for a strip of l sections of m models each S ...... = S 1/ l = a r n 1/m Ioo. If the total number of models m l is again equal to n we have: Sm~x = a m l / n o o Compared with a strip with one base at the end it is found: =
Omax S,,,,,x
a n V n o0 - - a m 1 / n o~,
n m
The accuracy of the strip can obviously be increased by making m smaller. The extreme is that we have a ground base in each model, which would obviously be very close to a complete terrestrial triangulation of the strip. However, in the context of this paper we can say that for the X-coordinates in the strip the same effect is obtained by a scale check in every model. This can easily be provided by an A.P.R. measuring ground clearances. For the Z-coordinates the same effect is obtained by measuring .; e.g. with a horizon camera, a gyroscope or a solar periscope. For the Y-coordinates a similar obvious means of continuous control is not readily available although the forward looking oblique camera as proposed by BLACnUT (1957/1958, p.38), comes very close. The actual effect of this control in each model is one of the subjects treated by Dr. JERIE (1966) on this symposium. It should be added that provisional computations show that the properties mentioned in this section do not hold completely when ~, ~ 0. The tendency however, is still the same.
CONC LUSION
The general conclusion from the foregoing considerations should be, that in a complete set of tools which is used as a base for planning an aerial survey, the properties of both ideas should be included. Whether their application is justified depends on the specifications and circumstances of each particular case. It is hoped that the indications in this paper will be sufficient to prove that both ideals have potential advantages and that it is worthwhile to continue research in this direction, both in theory and in practice.
Photogrammetria,
21 (1966) 4 3 - 5 5
~CCURACY AND GEODETIC CONTROL IN STRIP TRIANGULATION
55
REFERENCES
ACKERMANN, F. E., 1962. Analytical Strip Adjustment. I.T.C. Publ. A.17, 24 pp. BLACHUT, T. J., 1957/1958. Use of auxiliary data in aerial triangulation over long distances. Photograrnmetria, 14(I) : 38-45. .IERIE, H. G., 1966. Theoretical height accuracy of strip and block triangulation with and without use of auxiliary data. Photogramrnetria, in preparation. TIENSTRA, J. M., 1956. Theory o/ the Adjustment o/ Normally Distrihuted Observations. Argus, Amsterdam, 232 pp. VV~RMEIR, P. A., 1954. k a triangulation agrienne. Bull. Beige Photograrnmetrie, 1953(35): 17-57.
Photograrnmetria, 21 (1966) 4 3 - 5 5