The Unreliability of Network reliability measures

Journal of Spatial Science, 50(1), 91-4.

Inner constraints for 3-D survey networks Willie Tan Department of Building School of Design and Environment National University of Singapore Kent Ridge Crescent, Singapore 117566 e-mail: [email protected]

Abstract: This paper provides a simple and accessible derivation of the inner constraints for 3-D survey networks. There are three positional constraints, three rotational constraints, and one scale constraint. These constraints are often used in minimally constrained free net adjustments where error ellipses reflect the geometry of the network rather than the arbitrary imposition of constraints.

1 Introduction The purpose of this paper is to provide a simple and accessible derivation of the inner constraints for free net adjustment of 3-D survey networks. In the literature, these constraints are often merely stated rather than derived (Fraser, 1982; Papo and Perelmuter, 1982; Granshaw, 1980; Blaha, 1971). Even fairly advanced surveying textbooks such as Wolf and Ghilani (1997) and Wolf and Dewitt (2000) did not consider inner constraints. Kuang (1996) and Mikhail et al. (2001) only stated the constraints while Cooper (1987) derived inner constraints for a 2-D survey network. Dermanis (1994) provided an accessible but rather complex derivation of the photogrammetric inner constraints. The paper is organized as follows. Section 2 reviews the rationale for free net adjustment of survey networks. In Section 3, the positional, rotational and scale constraints for a 3-D photogrammetric or geodetic network are derived. The final section concludes the paper.

2 Free net adjustment Consider the linear model (1)

y = X + 

where y is an n x 1 vector of observations, X is the n x k non-stochastic design matrix (n > k),  is the k x 1 vector of parameters and  ~ N(0, 2I) is the n x 1 vector of error terms. In closerange photogrammetry, (1) may consist of the usual linearized collinearity equations or direct linear transformation (DLT) equations found in standard texts. Similarly, 2-D or 3-D geodetic

networks will consist of linearized distance, angular, astronomic, gravimetric, and other observation equations. For ease of exposition and without loss of generality, we shall restrict ourselves to the unweighted case where the errors are homoscedastic (i.e. i ~ N(0, 2)). Weighted least squares may be converted to the unweighted case by a simple transformation (see Tan, 1989). The normal equations are b = (XX)-1Xy

(2)

where b is the least squares estimator of  and the prime denotes the transpose of a matrix. In general, X is rank deficient with its image (i.e., column space) spanned by fewer than k linearly independent vectors. Thus, XX is singular and not invertible in the usual sense when there is a datum defect. In view of this rank deficiency, the least squares adjustment can proceed in several ways. A common technique is to impose the minimal q constraints just sufficient to define a datum, that is, we augment (1) with (3)

R = d

where R is a q x k matrix and d is a q x 1 vector of constants. In close-range photogrammetry, there are seven datum defects (three positional, three rotational, and one scale) so that q = 7. In a leveling loop network, only one height is required to define the datum so that q = 1. Combining (1) and (3), we have y   X  ε  d  R  β  u , i.e., y* = X* + *.      

(4)

Here u is a zero error vector, implying the constraints are fixed without error. In general, it is non-zero if one uses stochastic control (see Tan, 1989) and the estimator is biased if E[u]  0. The corresponding normal equations for the model in (4) become b* = (X*X*)-1X*y*.

(5)

Since rank(X*X*) = k, X*X* is now invertible and it is possible to solve for b*. Hence, (3) plays a key role in the solution. In conventional adjustments, the co-ordinates of selected stations are chosen as constraints. In a free net adjustment, the constraints are imposed on fictitious stations to reflect the geometry of the network (see Tan (2002) for a simple guide for readers who are unfamiliar with this logic). Typically, the centroid of the network is selected. As pointed out earlier, there are seven datum defects in a 3-D network and the constraints are: 

the centroid remains unchanged (three positional constraints);

 

the average bearing from the centroid to all points remain unchanged (three rotational constraints); and the average distance from the centroid to all points remain unchanged (one scale constraint).

These constraints are derived in the next section. Additional constraints may, of course, be imposed on the network. For example, if there are q minimal constraints and h additional constraints, then R is a (r + h) x k matrix and d is the corresponding (r + h) x 1 vector of constants. The network is then said to be over-constrained. Further, it is not necessary for R to consist of inner constraints; in fact, any q (or more) constraints will do as long as X*X* is invertible, that is, they must form an independent set. The main reason for preferring a minimal set of constraints is that error ellipses will then reflect network geometry rather than an arbitrary set of constraints. For example, in a leveling loop adjustment, any arbitrary point can be used as a fixed datum, and the standard errors will depend on which point is used. In particular, if a point A is used, it is held fixed and therefore assumed to have zero error. In contrast, in a free net adjustment, the mean reduced level (a fictitious station) of the entire network is held fixed.

3 Inner constraints for 3-D survey network

(a) Positional constraints Let (Xi, Yi, Zi) be the co-ordinates of an arbitrary point and there are m points (stations) in the network. We first derive the three positional constraints. The co-ordinates of the centroid G are (6)

XG =

(X

i

/ m) ;

YG =

i

 (Y / m) ; i

ZG =

 (Z

i

/ m) .

i

i

If we impose the condition that G does not move, dXG = dYG = dZG = 0, i.e., the three positional constraints are (7)

 dX i

i

= 0;

 dY = 0; i

i

 dZ

i

= 0.

i

(b) Rotational constraints The three rotational constraints may be derived by imposing the condition that the average bearing from G to all points remains unchanged.

Z P(Xi, Yi, Zi)

G r1



Y

X Figure 1. Bearing from centroid G to P.

Consider the bearing from G to an arbitrary point P. If P is projected onto the X-Y plane, by simple geometry, (8)

tan  =

Xi  XG = f(Xi, Yi, XG, YG). Yi  YG

Differentiating both sides and using Taylor series approximation, (9)

sec2 d =

f f f f dX i  dYi  dX G  dYG . X i Yi X G YG

The last two terms are zero since G must not move, i.e. dXG = dYG = 0. Hence, (10)

d = cos2 [ =

X  XG 1 dX i  i dYi ] Yi  YG (Yi  YG ) 2

1 [(Yi  YG )dX i  ( X i  X G )dYi ] r12

since cos = (Yi -YG)/r1. If the average bearing from G to all points remains unchanged, (11)

 d

i

 0.

i

That is, (12)

{(Y

i

i

 YG )dX i  ( X i  X G )dYi }  0

Substituting (7) into (12) gives the first of three rotational constraints, namely, (13)

 (Y dX i

 X i dYi )  0 .

i

i

The other two rotational constraints are derived in a similar manner by projecting the perpendicular from P onto the Y-Z plane and X-Z plane respectively. This gives (14)

 (Z dY  Y dZ )  0 ; and i

i

i

i

i

(15)

 (Z dX i

 X i dZ i )  0 .

i

i

(c) Scale constraint The distance from G to any point P is given by (16)

r2 = (Xi – XG)2 + (Yi – YG)2 + (Zi – ZG)2.

Differentiating both sides and using Taylor series approximation again, we have (17)

2rdr = 2(Xi – XG)dXi + 2(Yi – YG)dYi + 2(Zi – ZG)dZi.

Hence, (18)

dr = (1/r)[(Xi – XG)dXi + 2(Yi – YG)dYi + 2(Zi – ZG)dZi].

If the average distance from all points to G remains unchanged, (19)

 dr

 0.

i

i

This implies (20)

{( X

i

 X G )dX i  (Yi  YG )dYi  ( Z i  Z G )dZ i }  0 .

i

Substituting (7) into (20), we have (21)

 ( X dX i

i

 Yi dYi  Z i dZ i )  0 .

i

Putting (7), (13), (14), (15) and (21) together in the form R = d, we have the inner constraints for a 3-D survey network:

(22)

1 0  0  0  Z1   Y1 X  1

0 1

0 0

1 0

0 1

0 0

0 Z1 0  X1 Y1

1  Y1  X1 0 Z1

0 0 Z2 Y2 X2

0 Z2 0  X2 Y2

1  Y2  X2 0 Z2

 

1 0

 0  0  Zm  Ym  Xm

0 1 0 Zm 0  Xm Ym

 dX 1   dY  0  i   dZ  0   1  dX 2  1    dY2   Ym    0. dZ 2    Xm      0  dX   m Z m   dY   m  dZ m 

The first three rows are the positional constraints from (7). The next three rows are the rotational constraints from (13), (14) and (15). The scale constraint from (21) is found in the last row. The vector d in (22) consists of zeros. As noted in the previous section, additional constraints may be added to the rows of R in (22) but error ellipses will no longer reflect network geometry as long as there are more than the minimum q constraints.

4 Conclusion This paper provides a simple and accessible derivation of the inner constraints for a 3-D survey network. The derivation is not typically found in surveying textbooks, and most research papers tend to “derive” it in an incomplete or complicated way. This need not be the case.

References Blaha, G. (1971) Inner adjustment constraints with emphasis on range observations. Report No. 148, Department of Geodetic Science, Ohio State University. Cooper, M. (1987) Control surveys in civil engineering. London: Collin. Dermanis, A. (1994) The Photogrammetric inner constraints. ISPRS Journal of Photogrammetry and Remote Sensing 49(1), 25-39. Fraser, C. (1982) Optimization of precision in close-range photogrammetry. Photogrammetric Engineering and Remote Sensing 48(4), 561-70. Granshaw, S. (1980) Bundle adjustment methods in engineering photogrammetry. Photogrammetric Record 10, 181-207. Kuang, S. L. (1996) Geodetic network analysis and optimal design. Illinois: Sams Publishing. Mikhail, E., Bethel, J. and McGlone, C. (2001) Introduction to modern photogrammetry. New York: Wiley.

Papo, H. and Perelmuter, A. (1982) Free net analysis in close range photogrammetry. Photogrammetric Engineering and Remote Sensing 48(4), 571-76. Tan, W. (1989) Stochastic control. The Australian Surveyor 34(7), 728-36. Tan, W. (2002) In what sense a free net adjustment? Surveying and Land Information Science 64(4), 215-8. Wolf, P. and Dewitt, B. (2000) Elements of photogrammetry. New York: McGraw-Hill. Wolf, P. and Ghilani, C. (1997) Adjustment computations. New York: Wiley.