extendable linearised adjustment model for deformation

Extendable linearised adjustment model fordeformation analysis

H. Velsink*1,2

This paper gives a linearised adjustment model for the affine, similarity and congruence

transformations in 3D that is easily extendable with other parameters to describe deformations.

The model considers all coordinates stochastic. Full positive semi-definite covariance matrices

and correlation between epochs can be handled. The determination of transformation

parameters between two or more coordinate sets, determined by geodetic monitoring

measurements, can be handled as a least squares adjustment problem. It can be solved

without linearisation of the functional model, if it concerns an affine, similarity or congruence

transformation in one-, two- or three-dimensional space. If the functional model describes more

than such a transformation, it is hardly ever possible to find a direct solution for the

transformation parameters. Linearisation of the functional model and applying least squares

formulas is then an appropriate mode of working. The adjustment model is given as a model of

observation equations with constraints on the parameters. The starting point is the affine

transformation, whose parameters are constrained to get the parameters of the similarity or

congruence transformation. In this way the use of Euler angles is avoided. Because the model

is linearised, iteration is necessary to get the final solution. In each iteration step approximate

coordinates are necessary that fulfil the constraints. For the affine transformation it is easy to get

approximate coordinates. For the similarity and congruence transformation the approximate

coordinates have to comply to constraints. To achieve this, use is made of the singular value

decomposition of the rotation matrix. To show the effectiveness of the proposed adjustment

model total station measurements in two epochs of monitored buildings are analysed.

Coordinate sets with full, rank deficient covariance matrices are determined from the

measurements and adjusted with the proposed model. Testing the adjustment for deformations

results in detection of the simulated deformations.

Keywords: Deformation analysis, Euler angles, Observation equations with constraints, Rank deficient covariance matrix, Affine, Similarity and congruencetransformation

IntroductionGeodetic deformation analysis is about analysing thegeometric changes of objects on, above or under the earth’ssurface or changes of this surface itself. The objects aregenerally discretised by points, whose coordinates areregistered at two or more epochs.

Affine, similarity and congruence transformations play animportant role in geodetic deformation analyses, mainly fortwo reasons. First objects may undergo deformations thatare well described by such transformations. This is the case, ifthe deformations comprise translations, rotations, shears andchanges of size. The second reason is that objects are often

described with geodetically determined coordinates that aredefined by geodetic datums. To transform coordinates to acommon datum, transformations are necessary. The threementioned transformation types are often adequate for thispurpose.

Deformations of objects may comprise much morecomplicated patterns than can be described bycongruence, similarity or affine transformations.Therefore extended functional models have to bebuilt to describe the relations between coordinate setsof two or more epochs of geodetic measurements.These extended models can often be described asextensions to the congruence, similarity or affinetransformation or to combinations thereof. If neces-sary the models are complemented by covariancefunctions (collocation) or variograms (kriging) tocapture the systematic effects that are not describedby the functional model.

1Hogeschool Utrecht (University of Applied Sciences), Utrecht, TheNetherlands2Delft University of Technology, Delft, The Netherlands

*Corresponding author, email [email protected]

� 2015 Survey Review Ltd

Received 15 March 2014; accepted 19 September 2014DOI 10.1179/1752270614Y.0000000140 Survey Review 2015 VOL 47 NO 345 397

MORE OpenChoice articles are open access and distributed under the terms ofthe Creative Commons Attribution -Commercial License .0Non- 3

Problem definitionThe problem addressed in this paper is the problem offinding the relation between two coordinate sets in 3D,describing each object by means of the same points butat different moments in time. This relation describes thedeformation that the object is subject to and a possibledifference in geodetic datum. The coordinate sets aresupposed to be Cartesian with full covariance matrices,reflecting the precision by which the coordinates havebeen acquired. Testing the correctness of the deforma-tion model should be possible. If it is not correct, itshould be possible to find least squares estimates of thedeformations. These estimates can be used to extend theadjustment model.

Approach to solutionThe problem is addressed by first finding the optimalaffine transformation, or its special cases: the similarityand congruence transformation. Subsequently the trans-formation model is extended by additional parametersto find the best fitting solution.

The affine transformation in 3D involves a rotation.This rotation is often described by Euler angles. Theyhave, however, the disadvantage that gimbal lock mayoccur, i.e. the impossibility to determine the three Eulerangles, if a specific one of them is 0 or 90u. Which anglegives this problem and at what value, depends on thesequence in which Euler angles are applied to perform arotation. If the problematic angle is close to thedangerous value, this may still cause problems todetermine the other angles with sufficient precision.Therefore finding the deformation should avoid the useof Euler angles, because angles of 0 and 90u can occur indeformation problems.

The approach to solve the posed problem is setting upan adjustment model and solving the parameters of thismodel by means of the method of least squares. It is anadjustment model, because there are in general morecoordinate differences than transformation parametersto be estimated.

The adjustment model describes a null hypothesis, inwhich it is supposed that no deformation occurred. Itshould be possible to extend the model with extraparameters to formulate alternative hypotheses, whichdescribe possible deformations, and to test them againstthe null hypothesis. A linearised adjustment modelmakes it relatively easy to formulate many types ofhypotheses. Linearisation of the model makes it easy aswell to take account of full covariance matrices ofcoordinate sets. Such full covariance matrices are to beexpected when the coordinate sets are the result of theadjustment of geodetic measurements (terrestrial, air-borne or satellite).

OverviewThe setup of this paper is as follows.

First the general adjustment model for a transforma-tion is given. Determining the transformation para-meters between two or more coordinate sets is a problemthat can be solved without linearisation of the functionalmodel, if the transformation is one-, two- or three-dimensional and an affine, similarity or congruencetransformation. Direct solutions to the general adjust-ment model are referenced.

The paper continues with the linearised adjustmentmodel. If the deformations are more than just an affine,similarity or congruence transformation, it is hardlyever possible to find a direct solution for thetransformation parameters. Linearisation of the func-tional model and solving the linearised model isnecessary in that case. The linearised adjustment modelis given as a model of observation equations withconstraints on the parameters. The starting point is theaffine transformation, whose parameters are con-strained to get the parameters of the similarity orcongruence transformation. In this way the use of Eulerangles is avoided.

The least squares solution of a linearised adjustmentmodel is reviewed. It is shown how to handle a singularcofactor matrix. Methods to solve an adjustmentproblem with constraints are given.

Then the model for the affine transformation iselaborated upon, followed by the congruence transfor-mation. A linearised adjustment model needs approx-imate values for the unknown parameters. Theirdetermination and the iteration to arrive at the finalsolution are treated. In each iteration step approximateparameters are necessary that fulfil the constraints. Forthe affine transformation it is easy to get approximatecoordinates. To make subsequently the approximatecoordinates comply to the constraints, use is made ofthe singular value decomposition of the rotationmatrix.

Finally the similarity transformation is treated and anexperimental validation is given.

Recent literature mentions total least squares as amethod to find a least squares solution for transforma-tion problems, where all coordinates are consideredstochastic (Fang, 2011; Snow, 2012). In this paper it isshown that it is very well possible to construct anadjustment model, where all coordinates are consideredstochastic and the coefficient matrix does not containstochastic elements. This makes application of the totalleast squares method unnecessary. Since the standardmethod of least squares is used, its extensive body ofknowledge can be used.

General adjustment model fortransformationA general adjustment model for solving the transforma-tion parameters between two sets of coordinates istreated in this section.

Let a point field be described by a set of Cartesiancoordinates in reference system ra, taken together in avector a, and by another set of Cartesian coordinates inreference system rb, taken together in a vector b. Thevector names are underlined to indicate that they arerandom variables. It is supposed that vector a differsfrom vector b because of a deformation, a difference ingeodetic datum and of stochastic noise in both vectors.Let the vector c contain the mathematical expectationsof a, i.e.

c~Efag (1)

where E{ } indicates the mathematical expectation.Let a transformation, represented by the vector

function t, transform the vector b into the vector a bymeans of transformation parameters, taken together invector f. The transformation is supposed to describe the

Velsink Extendable linearised adjustment model for deformation analysis

Survey Review 2015 VOL 47 NO 345398

deformation and the difference in geodetic datum. Themathematical expectation of a and of the transformedvector b should then be equal

Efag~Eft(b,f)g (2)

The transformation t is non-linear in the elements ofvector b and of vector f for the affine, congruence andsimilarity transformation. If, however, vector b isconsidered non-random, i.e. a constant vector, transfor-mation t is linear for the affine transformation. Its directleast squares solution is given in section ‘Step 1: affinetransformation done simply’.

For the general case transformation t is non-linear,which means that a solution can be found by a directsolution, treated in the next section, or by solvingiteratively a linearised model, treated subsequently. Thispaper proposes a solution with a linearised model, whichis elaborated upon in the section ‘Linearised adjustmentmodel’.

Direct solutionsA direct solution of the transformation parameters in (2)is, as mentioned above, straightforward in the case of anaffine transformation if b is considered a constantvector. The stochastic behaviour of b cannot be takenaccount of, unless it is possible to have it included in thestochastic behaviour of vector a.

In the case of a congruence transformation thetransformation consists of a rotation around a certainaxis and a translation in a certain direction. It can beconsidered a special case of the similarity transforma-tion, treated subsequently.

The similarity transformation in 3D consists of arotation around a certain axis, a translation in a certaindirection and a change of scale. It is also called a seven-parameter transformation or a Helmert transformation(Awange et al., 2004; Krarup, 2006).

Much literature is devoted to finding direct solutionsfor the parameters of these transformations, wherefinding the three rotation parameters is most difficult.

Menno Tienstra writes (Tienstra, 1969) that the firstmethod published was by Thompson (1959) and thatSchut (1960/61) gave a more elegant derivation of thesame method. Tienstra himself gives a different method(Tienstra, 1969). At least eight solutions of more recentdate (from 1981 up to 2006) can be found in literature.They are the solutions of Hanson and Norris (1981),Bakker et al. (1989, p.55), Awange et al. (2004),Teunissen (1985, p.148), Arun et al. (1987), Hinsken(1987), Horn et al. (1988), and Krarup (2006).

Of the first three solutions a short description is givento illustrate the different approaches that are possible.

Hanson and Norris consider in their paper (Hanson andNorris, 1981) the estimation of the transformationparameters of the congruence and similarity transforma-tions. Their application is quality control of manufacturedparts, where points on parts are matched with points on adrawing. They prove that the least squares estimation ofthe rotation can be performed independently from thetranslation, if the elements of a and b are mutuallystochastically independent and the x, y and z-coordinate ofeach point i have the same weight ci. A procedure is givento compute the direct least squares estimates of therotation parameters by means of singular value decom-

position of a matrix that is computed from the elements ofa and b. Also a procedure is given to compute the directleast squares estimate of the change of scale of thesimilarity transformation. The direct least squares estimateof the translation is the difference between the centres ofgravity of a and the rotated and scaled b.

Bakker et al. (1989) consider the points of a and b as adistribution of mass points with unit mass. Theycompute for both a and b the centre of mass, the setof body-axes, the mean radius of gyration and the inertiatensor. After translating to the centre of mass, rotatingwith the inertia tensor and scaling both a and b, thetransformed vectors are equated and a formula for thetransformation parameters derived. The stochasticcharacteristics of a and b are not taken into account,although it is mentioned by Bakker et al. (1989) that anadjustment is easy, especially if the coordinates haverotationally symmetric variances.

Awange et al. (2004) give a procedure to compute theleast squares estimates of the parameters of thesimilarity transformation (seven-parameter transforma-tion), based on finding the roots of univariate poly-nomials using a Groebner basis.

Solutions by linearisationThe transformation parameters in (2) can be solved bylinearising the equation and using the standard leastsquares algorithm. The advantage of this approach is thepossibility of solving adjustment models, for whichdirect solutions are not known. The disadvantages arethe need to have approximate values for all unknownparameters and the need to iterate the computation withthe risk of divergence, if the approximate values are notchosen well.

The linearisation of the affine transformation issimplest and is given first. For the linearisation of thecongruence and similarity transformation, the rotationhas to be parameterised. This is generally done, see e.g.Hofmann-Wellenhof et al. (2001, p. 294), by usingthree angles that describe the rotation around threecoordinate axes, the so called Euler angles. Asdescribed in the introduction, the use of Euler anglesmay result in gimbal lock. This problem occurs forexample for the rotation parameterisation of Hofmann-Wellenhof et al. (2001, p. 294), if the second rotationangle a2 is a right angle (a25100 gon), which yields therotation matrix R

R~

0 sin(a1za3) {cos(a1za3)

0 cos(a1za3) sin(a1za3)

1 0 0

0B@

1CA (3)

Only the sum (a1za3) appears in the matrix, andangle a1 nor angle a3 can be determined separately,although the rotation matrix itself is well defined. This iscalled ‘gimbal lock’ after the equivalent problem inmechanical engineering, when two of three gimbalsbecome parallel to each other and one rotationpossibility is lost. An additional disadvantage of Eulerangles is that approximate values for them have to bedetermined in some way. In this paper a differentapproach is chosen, where the rotations are notparameterised by three angles, but by the nine elementsof the rotation matrix. By imposing six constraints on


Survey Review 2015 VOL 47 NO 345 399

the nine elements, the matrix is forced to describe onlyrotations.

The use of the direct solutions of the previous sectionhas the advantage that no approximate values and noiteration are needed. The advantage of a linearisedadjustment model, however, is that extending the modelwith other parameters is easy. There are two reasonswhy the model should be extended:

(i) errors may be present in the coordinates, causedby errors in the measurements that were used tocompute the coordinates, or caused by e.g.identification errors. To test for these errors themodel is extended by parameters, describing theseerrors. The extended model is tested as analternative hypothesis against the null hypothesisof no errors (Teunissen, 2006, p.78).

(ii) the description by means of 6, 7 or 12 parametersof one congruence, similarity or affine transfor-mation may be not adequate. Maybe not one butmore such transformations are needed, forexample each describing a subset of points. Ormore complex transformations with more orother parameters are needed.

Extending or changing the model is easier for alinearised model than for a non-linear one.

Linearised adjustment modelLinearisation of general modelTo solve the problem of finding the transformationparameters, a system of observation equations is set up.Constraints are added to this system. This constrainedsystem is solved by means of the method of leastsquares. As the equations of this system are not linear,they are linearised.

First the system is given for an affine transformation.The affine parameters can be obtained without theaddition of constraints. Then the system is augmentedby constraints that force the parameters of the affinetransformation to become the parameters of a con-gruence transformation. Finally it is shown how theparameters of a similarity transformation are obtained.

A linearised adjustment model is constructed bydifferentiating the function t of (2) relative to theelements of the vectors b and f (Teunissen, 1999, p.142).The resulting equations stay simpler, if vector b is firsttransformed to coincide approximately with vector a.For this transformation approximate transformationparameters are needed. Their computation is treatedlater on.

The approximately transformed vector is

b’~t’(b,f’) (4)

The cofactor matrix of b9 is determined by applyingthe law of propagation of cofactors. To do this,equation (4) is linearised by a Taylor expansion,neglecting second and higher order terms. The expan-sion is relative to the elements of vector b.

Equation (2) now becomes

Efag~Eft(b’,f)g (5)

Define two matrices B and F of partial derivatives asfollows

B~Lt

Lb’

� �0

; F~Lt

Lf

� �0

(6)

The sub index 0 indicates that approximate values ofresp. b9 and f have to be entered into the matrices.

Assuming that B is square and invertible, which is thecase in the situations treated in this paper, theadjustment model can be constructed as

EDa

Db0

� �� ~

I 0

B{1 {B{1F

� �Dc

Df

� �(7)

where Da~a{a0; Db0~b0{b00; Dc~c{c0; Df~f{f0;

a0,b00,c0,f0 are approximate values; I is the unit matrix; 0

is the zero matrix; c05a0 and thus E Daf g~DcOn the left hand side of (7) the vector of observations

can be found, which consists of the two vectorscontaining the coordinates. The right hand side containsthe matrix of coefficients and the vector of unknownparameters. The parameters are the mathematicalexpectations of the coordinates in reference system ra

and the transformation parameters.The vector of observations is assumed to have a

normal distribution, described by a covariance matrixthat is the product of a scalar variance factor and thecofactor matrix. The cofactor matrix is

Qfa

b0

� �g~

Qa Qab0

Qb0a Qb0

� �(8)

where Qaand Qb0 are the cofactor matrices of a and b9.

Qab0~(Qb0a)T gives the cofactors that describe thecorrelation between a and b9. It equals the zero-matrixif a and b9 are supposed to be not correlated mutually. Itis, however, possible to use non-zero matrices, forexample to describe temporal correlation using acovariance function. The variance factor is chosen insuch a way that the elements of the cofactor matrix havecomputationally convenient values. The situation thatQa or Qb0 or both are not regular, but positive-semidefinite matrices, is treated later on.

Reduced general modelThe system can be reduced to minimise the amount ofunknown parameters, which is advantageous in casethere is a large number of coordinates. The solution ofthe system of normal equations can be a computationalburden with many coordinates, especially if full covar-iance matrices are involved. The reduced system isacquired by premultiplying the second row of (7) with Band subtracting it from the first row

E Da{BDb0f g~FDf (9)

On the left hand side appears the vector of remainingcoordinate differences (Da{BDb0) as vector of observa-tions. On the right hand side the only unknownparameters are the transformation parameters. The leastsquares residuals of the coordinates can be computed bymeans of the stochastic correlation of the adjustedcoordinates with the estimated transformation para-meters, because the adjusted coordinates are so calledfree variates (Teunissen, 1999, p. 75).

In this paper use is made of the model as defined by(7), because it shows more clearly the structure, because



constraints on the coordinates can be added, e.g. todescribe more complex deformation behaviour, andbecause it can readily be extended to more than twoepochs.

Least squares solutionThe system of observation equations (7) is overdeter-mined if the number of parameters is less than thenumber of coordinates and can be solved by the methodof least squares. A weighted least squares solution isobtained, if account is taken of cofactor matrix (8).

A system of linearised observation equations, like (7),has the following general structure

E ‘f g~Ap; Df‘g~s2Q‘ (10)

where ‘ is the m-vector of observations, A is the (m6n)-matrix of coefficients and p is the n-vector of unknownparameters. The equation behind the semicolondescribes the stochastic model by giving the covariancematrix Df‘g, the variance factor s2 and the cofactormatrix Q‘. The least squares solution of system (10) isgiven in the appendix. There the equations for testing ofthe results are given as well.

For each system of linear observation equations anequivalent system of linear condition equations exists

KT Ef‘g~0; Df‘g~s2Q‘ (11)

where KT is the [(m-n)6m]-matrix of conditions, forwhich holds

KTA~0 (12)

The least squares solution of system (11) is given inthe appendix. The equations for testing are the same asthose given for the model of observation equations.

Positive semidefinite cofactor matrixThe vector ‘ of observations of system (7) containscoordinates, which may have resulted from geodeticmeasurements. In that case it can easily happen that thecofactor matrix Q, is positive semidefinite, because thecoordinates are defined relative to a geodetic datum,defined by some of the points.

A solution for handling a positive semidefinitecofactor matrix of the observations is given for thereduced general model by Teunissen et al. (1987) for thesimilarity transformation. Matrix Q, is regularised byadding a term to it, which does not change the leastsquares solution of the parameters p. It is possible togeneralise regularisation to any model of observationequations, if the coefficient matrix A is of full rank.

Matrix Q, of (10) can be regularised by adding to it amatrix l AAT, with A from (10) and l any real scalarwith l.0

Q’‘~Q‘zlAAT (13)

Matrix A is a real matrix of full rank and thereforeAAT is positive definite. Multiplying it with a positivefactor l does not change its positive definiteness. Q’‘ isthe sum of a positive semidefinite matrix and a positivedefinite matrix and is therefore positive definite. Thismeans that its inverse (Q’‘)

{1 exists.

Using Q’‘ instead of Q, for computing the leastsquares solution yields the same adjustment and testingresults. This can be seen by switching to the model ofcondition equations (11). The equations in the appendixshow that vector r and its cofactor matrix Q r are used toget the adjustment and testing results. In the equationsQ’‘ appears only in the product Q’‘K (or its transpose),for which we have

Q’‘K~(Q‘zlAAT)K~Q‘KzlAATK (14)

The second term is zero because of (12). Therefore

Q’‘K~Q‘K (15)

The conclusion is that the adjustment and testingresults do not change by using Q’‘ instead of Q,. Owingto the equivalence of using the model of observationequations and the model of condition equations, Q’‘ canbe used as well to get the least squares solution for themodel of observation equations.

Constraining parametersThe parameters of an adjustment model can beconstrained to satisfy certain linear or linearisedrelations

Cp~0 (16)

where matrix C is a matrix of coefficients and 0 a zerovector. In the following sections constraints are used toforce the affine transformation parameters to changeinto the parameters of a congruence or similaritytransformation.

The least squares solution for the parameters p from(10) has to be found under the condition that they fulfilthe constraints (16). A method is the extension of thesystem of normal equations with the constraints asfollows (Tienstra, 1956, section 7.3)

ATQ{1‘ A CT

C 0

!p

k

� �~

ATQ{1‘ ‘

c

!(17)

where k contains the Lagrange multipliers and 0 is a zeromatrix. Solving this system of normal equations deliversa solution for p and k. Vector k plays no further role inthe considerations of this paper.

If Q, is positive semidefinite, it has to be regularised.To see how this can be done, another method to solve(10), taking account of the constraints (16), is given.Take the null space of the space R(C), spanned by thecolumns of matrix C of (16). If NC is a base matrix ofthis null space, we can write

p~NCl (18)

with l a vector of (n–nc) parameters (nc is the amount ofconstraints). We can now write (10) as

Ef‘g~Arl; Df‘g~s2Q‘ (19)

with Ar5ANC. These are observation equations, forwhich a least squares solution is found in the normalway. It is the same solution as the one that follows from(17). The determination of Ar can be seen as theelimination of nc parameters from (10), which can bedone with the Gaussian algorithm or the Choleskymethod (Wolf, 1982). If Ar is determined as Ar5ANC,



the base matrix NC can be determined e.g. by theMatlab-command ‘null(C)’.

For the regularisation of Q, we now use

Q’‘~Q‘zlArATr (20)

Model for affine transformationIf an object is subject to a force, the material and the

force may be such that applying and releasing the forcecauses respectively an elastic deformation and thedisappearance of the deformation. Such a deformationis often linear and can be described by an affinetransformation. In this section the adjustment modelfor the affine transformation is given by defining thecontent of the matrices B and F of (6).

Let x, y, z be the vectors with the x-, y- and z-coordinates of vector a, as described in the appendix.Let u, v, w be the vectors with the x-, y- and z-coordinates of the same points in vector b9.

Let the vectors a1, a2, a3 and the vector t be defined as

a1~

a11

a12

a13

0B@

1CA, a2~

a21

a22

a23

0B@

1CA, a3~

a31

a32

a33

0B@

1CA, t~

tx

ty

tz

0B@

1CA

Let the matrix R be defined as

R~

a11 a12 a13

a21 a22 a23

a31 a31 a33

0B@

1CA~

aT1

aT2

aT3

0B@

1CA

The equation for the affine transformation reads then

xT

yT

zT

0B@

1CA~R

uT

vT

wT

0B@

1CAzteT

where e~ 1,1,:::,1ð ÞTThe parameters tx, ty, tz describe a translation and theparameters a11,..,a33 describe the rotation, scaling andshearing of the affine transformation.

To get the matrices B and F of (6) the coordinates x, y, zhave to be differentiated relative to the coordinates u, v, wand to the parameters tx, ty, tz and a11,..,a33.

Matrix B has the following structure in the case of anaffine transformation

B~

a011I a0

12I a013I

a021I a0

22I a023I

a031I a0

32I a033I

0B@

1CA

where a0ij, with i,j51,2,3, are the approximate values of

the parameters of (21) and I is the (n6n) unit matrix.Since with (4) vector b9 is already approximately equal to

a, the approximate values of (24) can be chosen as follows

a0ij~dij

with i,j51,2,3 and dij the Kronecker delta. Thenmatrix B becomes a (3n63n) unit matrix.

During iteration of the least squares adjustment, theapproximate values have to be adjusted. If this is doneby adjusting transformation (4), matrix B does not needto be adjusted and can stay a unit matrix.

To give a simple structure for matrix F the vector oftransformation parameters Df is divided as follows

Df~

Da1

Da2

Da3

Dt

0BBB@

1CCCA

with the D-quantities defined as described in theappendix (Conventions).

Matrix F has now the following structure

F~

b 0 0 e1

0 b 0 e2

0 0 b e3

0B@

1CA

where b, e1, e2, e3 and 0 are all (n63) matrices, asfollows

b5(u0, v0, w0); u0,v0,w0: approximate values of u, v, w

e1~

1 0 0

1 0 0

..

. ... ..

.

1 0 0

0BBBB@

1CCCCA, e2~

0 1 0

0 1 0

..

. ... ..

.

0 1 0

0BBBB@

1CCCCA, e3~

0 0 1

0 0 1

..

. ... ..

.

0 0 1

0BBBB@

1CCCCA

0 is the (n63) zero matrix.The adjustment model for the affine transformation

can be constructed with (7), with B and F according to(24) and (26). Solution of this model by the method ofleast squares follows by generating the normal equationsand solving them.

Model for congruence transformationOne of the simplest deformations an object canundergo on, under or above the earth’s surface is amovement that consists of a shift and a rotation, i.e. acongruence transformation (also called a rigid bodytransformation).

The congruence transformation is a special form of anaffine transformation, in which no central dilatations(changes of scale) and no shears occur, only translationsand rotations. It has less parameters than the affinetransformation. It involves a translation, described bythree parameters tx, ty, tz, and a rotation, described bythree Euler angles, or by one angle and a unit vector,around which the rotation occurs.

We can write down matrix R in (22) with only threeparameters: the three Euler angles. Differentiating sucha system results in matrices B and F that can be used toconstruct adjustment model (7). There are two dis-advantages to such an approach:

(i) determining the parameters to perform theapproximate transformation (4) is no easy task(but it is easy for an affine transformation, as willbe shown later on)

(ii) the Euler angles may cause problems, as it wasdiscussed before.

The approach chosen in this paper, is to use the model asit was derived for the affine transformation in theprevious section, and constrain the parameters in such a

(22)

(21)

(23)

(24)

(25)

(26)



way that they become the parameters of a congruencetransformation.

Applying constraints to affine transformationFor a congruence transformation matrix R of (22) has tobe an orthogonal matrix. This means that the rows of Rare orthogonal and each row has length 1 (Strang, 1988,p. 166). So six conditions have to be satisfied, i.e. threeorthogonality conditions (e.g.: row 1 is orthogonal torow 2 and to row 3, and row 2 is orthogonal to row 3)and three length conditions (three rows have length 1).As an equation

aTi aj~dij (27)

with i51,2,3; j51,2,3; j§i; ai defined as in (21) and dij

the Kronecker delta.The six conditions can be linearised and added as

constraints on the parameters to system (7). The sixlinearised constraints are

a0T

2 a0T

1 0

a0T

3 0 a0T

1

0 a0T

3 a0T

2

a0T

1 0 0

0 a0T

2 0

0 0 a0T

3

0BBBBBBBBBB@

1CCCCCCCCCCA

Da1

Da2

Da3

0B@

1CA~

0

0

0

0

0

0

0BBBBBBBB@

1CCCCCCCCA

(28)

where 0 is the (163) zero vector and a0T

i with i51,2,3 is

the transposed vector of approximate values of theparameters as defined in (21). As mentioned before, as

approximate values can be chosen a0ij~dij, with

i,j51,2,3, from which follows

a01,a0

2,a03

� �~I3 (29)

with I3 the (363)-unit matrix.

Determining approximate valuesTo determine the transformation parameters of thecongruence transformation, adjustment model (7) isconstructed, the matrices B and F are determined with(24) and (26), and the constraints are added with (28).Two methods to determine a solution of the adjustmentmodel with constraints have been described before.

In the matrices B and F, in the constraints and in theD-quantities, however, approximate values have to beentered. The case of matrix B has been treated alreadywhen the adjustment model for the affine transforma-tion was constructed: a unit matrix can be used. Inmatrix F the approximate values u0, v0, w0 are needed.Here the observed values of u, v, w must be transformedwith (4) and can then be used as u0, v0, w0.

For the transformation parameters of (4) approximatevalues are needed for all nine elements of matrix R andfor the three translation parameters tx, ty, tz.

The methods to get direct solutions of the transfor-mation parameters in (2) have been treated before andcan be used to get approximate values for the nineelements of matrix R and the three translation para-meters.

Instead of using one of the treated direct methods atwo-step procedure is described here to arrive atapproximate transformation parameters. It is given toshow its usability. The choice to use this method or one

of the direct solutions depends on considerations likecomputational suitability.

In the two-step procedure approximate values for thetranslation parameters and the elements of matrix R aredetermined by first using a simplified version of theadjustment model of the affine transformation andsecondly using the singular value decomposition ofmatrix R.

Step 1: affine transformation done simplyIf in (2) vector b9 is considered a non-random vector (i.e.b9 without an underscore), and the equations of an affinetransformation are used, the elements of vector E{a} area linear function of the transformation parameters

Efag~Ff (30)

where F is the matrix from (26) and f the vector ofparameters

f~

a1

a2

a3

t

0BBB@

1CCCA (31)

The least squares estimator of f, indicated as f, withall observations considered as having the same varianceand not being correlated, is

ff~(FTF){1FTa (32)

With (32) approximate values for all transformationparameters can be acquired.

Step 2: singular value decomposition of RThe approximate values, acquired with (32), are,however, not usable in transformation (4) and adjust-ment model (7), using the matrices B and F from (24)and (26), and the constraints from (28). The reason isthat the approximate values must fulfil the constraints(27), which is a consequence of the linearisation processof the constraints by means of a first order Taylorexpansion.

If the approximate values, computed with (32) areentered in matrix R from (22) the result should be anorthogonal matrix. In general this will not be the case.Changing R into an orthogonal matrix can be accom-plished by performing a singular value decomposition ofR. The result is three matrices, for which holds (Strang,1988, p. 443)

R~Q1SQ2T (33)

where Q1 and Q2 are (363) orthogonal matrices and Sis a (363) diagonal matrix that contains the threesingular values on the main diagonal. How thesematrices are computed can be found in textbooks onlinear algebra, e.g. Strang (1988). Computation routinesare available in mathematical software like Matlab.

Changing R into an orthogonal matrix is done bychanging all singular values to a value of 1, i.e. byremoving matrix S and computing the changed matrixR9 as follows

R’~Q1Q2T (34)



Since Q1 and the transpose of Q2 are orthogonalmatrices also their product R9 is an orthogonal matrix.A proof that the elements of R9 are as close as possibleto the analogous elements of R is given by (Higham,1989). Closeness is defined with the Frobenius norm.

By using the results of (32) to construct matrix R andusing singular value decomposition to arrive at theorthogonal rotation matrix of (34), approximate valuesfor all transformation parameters of the congruencetransformation can be computed.

IterationSince both adjustment model (7) and the addedconstraints (28) are linearised, solving the model bythe method of least squares needs iteration. In eachiteration step the estimates of the coordinate correctionsDc and the corrections to the transformation parametersDf are used to compute new approximate values. Theadjustment model with its added constraints is thensolved again. The iteration continues until the differencebetween the newly computed approximate values andthose from the previous step is less than a preset limit.

In each iteration step the new approximate transfor-mation parameters should again fulfil the constraints(27). That means that in each iteration step theadaptation of matrix R by means of a singular valuedecomposition, as described in the previous section, hasto be repeated.

As mentioned before, (4) is used in each iterationstep to compute new coordinates b9. Their cofactormatrix is determined by applying the law of propaga-tion of cofactors. This guarantees that matrices B andF and the linearised conditions (28) keep their simplestructure.

Model for similarity transformationA similarity transformation is a congruence transforma-tion with an additional parameter to account for achange of scale (change in the unit of length). Thisadditional scale parameter may be necessary for exampleif the two coordinate sets have been determined bymeasuring techniques that cannot be guaranteed to useexactly the same unit of length.

Because of the additional transformation parameter,equation (23) receives an additional parameter l as follows

xT

yT

zT

0B@

1CA~lR

uT

vT

wT

0B@

1CAzteT (35)

Again matrix R has to be orthogonal.Linearisation is done relative to the coordinates of b9

and to thirteen transformation parameters f (of whichseven remain as independent parameters, because thereare six constraints)

f~

a1

a2

a3

t

l

0BBBBBB@

1CCCCCCA

(36)

Linearisation gives matrices B and F that resemblevery much the ones of the congruence transformation.Matrix B is the same as in (24), matrix F has one columnmore than matrix F in (26)

F~

b 0 0 e1 ba1

0 b 0 e2 ba2

0 0 b e3 ba3

0B@

1CA (37)

A slight disadvantage of this matrix F is that it makesthe adjustment model singular without the constraints.Another approach is possible that does not have thisdisadvantage. To arrive at rotation matrix R sixconstraints are put on matrix R of (22). These sixconstraints were relaxed by allowing an extra parameterl. It is also possible to put only five constraints onmatrix R. The three constraints that the lengths of thethree rows are equal to 1, are replaced by twoconstraints: the length of the first row equals the lengthof the second, and it equals the length of the third row.Matrix F now stays the one of (26). The linearisedconstraints are

a0T2 a0T

1 0

a0T3 0 a0T

1

0 a0T3 a0T

2

a0T1 {a0T

2 0

a0T1 0 {a0T

3

0BBBBBB@

1CCCCCCA

Da1

Da2

Da3

0B@

1CA~

0

0

0

0

0

0BBBBBB@

1CCCCCCA

(38)

Approximate values are computed in the same way asfor the congruence transformation. An approximatevalue for l can be determined from matrix S of thesingular value decomposition by taking the mean of thesingular values, i.e. the mean of the three values onthe main diagonal of S.

Barycentric coordinatesThe numeric values of the coordinates can be very large,for example if coordinates in a national grid are used. Itis advisable to switch in such a case to barycentriccoordinates xb, yb, zb

xTb

yTb

zTb

0B@

1CA~

xT

yT

zT

0B@

1CA{

1

n

xTe

yTe

zTe

0B@

1CA eT (39)

where e~(1,1, . . . ,1)T. Likewise barycentric coordinatesub, vb, wb are defined. The cofactor matrix should beadapted accordingly, but it is possible to consider thesecond term on the right hand side of (39) as a constantterm (a non-stochastic shift). The cofactor matrix thenremains unchanged. If barycentric coordinates are used,the last column of (26) or the last-but-one column of(37) can be left out, because the pertinent parameters arezero, also after adjustment, as long as a and b are notcorrelated mutually.

Use of adjustment modelGiven two vectors of coordinates a and b, whichdescribe the same points in 3D, the relation betweenboth is searched by estimating the parameters of atransformation between them. To solve this problem in



case of a congruence or similarity transformation, theadjustment model (7) is set up and constrained by (28) or(38). The matrices B and F in this model have beendefined for the affine, congruence and similaritytransformation (resp. (24), (26) and (37)). The adjust-ment model can be used in the following ways:

(i) suppose that the two vectors of coordinates aand b are defined in different geodetic datums,i.e. reference system ra is different from systemrb. Adjustment model (7) and the constraints (28)or (38) can be used to transform vector b intosystem ra. Coordinates in vector b that have noanalogous coordinates in vector a can betransformed to reference system ra as freevariates (Teunissen, 1999, p. 75). Coordinatesin vector a that have no analogous coordinates invector b can be adjusted as free variates in thesame way

(ii) if the two vectors of coordinates a and b differfrom each other because of a deformation thatcan be described by an affine, congruency orsimilarity transformation, adjustment model (7)and the constraints (28) or (38) can be used toestimate the deformation

(iii) if a combination of both datum differences and adeformation describes the relation between a andb, adjustment model (7) and the constraints (28)or (38) can be used to estimate both datumdifferences and the deformation.

Using adjustment model (7) in combination with theconstraints (28) or (38) has the following characteristics:

(i) positive-semidefinite, full covariance matrices ofa and b, and the correlation between both, canbe taken account of, which is especially useful ifa and b stem from geodetic measurements, inwhich case such covariance matrices are to beexpected

(ii) testing for biases in the coordinates can be easilydone by adding parameters to the model thatdescribe the biases (Teunissen, 2006, p.71)

(iii) testing for several simultaneous deformationscan be easily done by extending the model withextra parameters and constraints that describethose deformations

(iv) suppose that a test shows that added parametersare significant. These parameters can easily beadded to the adjustment model

(v) extending the model to more than two epochscan be readily accomplished by adding thecoordinates of a new epoch to the vector ofobservations and adding additional parametersto describe the datum of the new epoch and thedeformation of this new epoch relative to theother epochs.

Model (7) contains a transformation. The model could beconstructed without such a transformation, but thetransformation fulfils a fundamental function: it takescare that the vectors a and b, including their cofactormatrices, are compared with each other only by means oftheir intrinsic geometric information [the information thatcan be extracted from the underlying measurements withenough ‘sharpness’, a word used by Baarda (1995, p.1)].Information about the geodetic datum and the precisionof its definition are eliminated from the deformationanalysis. Tests of the deformation measurements are

therefore more accurate. Also the description of theresulting precision and reliability is better. The necessity toeliminate the influence of the geodetic datum is closelyrelated to the search of Baarda (1995, p.1) for dimension-less quantities and the call of Xu et al. (2000) for invariantquantities.

Experimental validationTo show the effectiveness of the proposed adjustmentmodel for deformation analysis, a deformation analysistask, as it is encountered in professional practice in theNetherlands, is used. For this task fictitious observationswere generated (assuming a normal distribution) andtwo different deformations simulated.

Three monumental buildings are monitored by totalstation measurements. Because of underground works,movements of the buildings might occur. Fifteen pointsare monitored (Fig. 1) from an instrument point that isnot monumented and varies from epoch to epoch. Twoepochs are considered (99 and 100 are the instrumentpoints). A state-of-the-art high precision total station issupposed to have been used. The standard deviation ofhorizontal and vertical direction measurements is 0.3mgon, of distance measurements 1 mm. The precisionwith which a point is defined (idealisation precision) issupposed to be 0.5 mm. The generated observations arelisted in the appendix. The second epoch has twodifferent lists of observations. In the first one (calledcase 1 hereafter) a deformation of point 1 is intention-ally introduced. In the second one (case 2) the first fivepoints, belonging to one building, are deformed.

The generated observations were processed by thecommercial software package MOVE3 (2014), version4.2.1(x64). 3D coordinates and their full covariancematrix were computed. The covariance matrices of bothepochs were defined relative to the approximate x-, y-, z-coordinates of point 1 and 14 and z-coordinate of point26. Note, however, that the testing results of theadjustment model of this paper are invariant to achange of base points. This was confirmed by computa-tions with other base points. The approximate coordi-nates of the first epoch were in the national grid, those ofthe second epoch in a local system that was rotated onpurpose over 100 gon relative to the national grid.

The adjusted coordinates and their covariancematrix, of both epochs, were transformed to the systemof the first epoch and adjusted and tested with the

1 Location of points



http://www.maneyonline.com/action/showImage?doi=10.1179/1752270614Y.0000000140&iName=master.img-000.jpg&w=226&h=157

adjustment model of this paper. A similarity transfor-mation was used, in accordance with the degrees offreedom of the adjustments of each epoch in MOVE3.To do the computations a specifically designed Matlabprogramme was used. The results are shown in theappendix.

The covariance matrices of both epochs are rankdeficient, because they are defined relative to a subset ofthe points. This results in a rank deficiency of 14 for thecofactor matrix of (8). Regularisation of the cofactormatrix has been used to handle the rank deficiency.

To estimate the parameters of the 3D similaritytransformation and to adjust the coordinates, threecomputations (i.e. two iterations) were needed. In thelast iteration the absolute value of the largest correctionto the estimated parameters was less than 10212.

Testing of case 1In case 1 a deformation of point 1 of 5.2 mm wasinduced by giving the x-, y- and z-coordinate each a biasof 3 mm.

The overall model test (F-test) of the adjustmentyielded an F-value of 1.41, which was more than thecritical value of 1.18 (computed with a one-dimensionalsignificance level of 0.1%, a power of 80% and using theB-method of testing). Conventional w-tests were per-formed, using (60) and (61). None led to any rejection.Also point tests were performed, using (62). A point testis a three-dimensional test, where the alternativehypothesis is, that three independent biases are presentfor respectively the x-, y- and z-coordinate of a point.Point 1 was rejected with estimated errors (computedwith (58)) of 4, 2 and 3 mm in resp. the x-, y- and z-direction of system ra. This shows that using weightedleast squares with full covariance matrices and applyinga three-dimensional point test is capable of detectingdeformed single points.

Point 1 is one of the base points that were held fixedon their approximate values in the epoch adjustments.Testing of this point, however, can be done with themodel of this paper like the testing of the non-basepoints, without any additional action. This is possible,because the model includes a transformation. Theestimated least squares residuals from e of (46) are zerofor the base points, but the reciprocal least squaresresiduals from r of (47) are not. The reciprocal residualsare used for testing ((56), (59) and (60)).

Testing of case 2In case 2 the first five points, belonging to one building,are deformed, all with the same deformation: 3 mmalong the x- and y-axis, 22 mm along the z-axis. The x-,y- and z-axis are those of the local system of the secondepoch.

The overall model test (F-test) of the adjustmentyielded an F-value of 2.20, which was more than thecritical value of 1.18. Both conventional w-tests andpoint tests were performed. Only the w-tests of the x-coordinate of point 5 and the y-coordinate of point 2 ledto rejection. Points 2 and 5 were rejected by the pointtests. If, however, the deformation of this one buildingwas tested by formulating an alternative hypothesis thatthe five points of this building had undergone the samedeformation, the pertinent test led to rejection with atest statistic that was 2.01 times larger than the criticalvalue. This was larger than the same ratio for any other

alternative hypothesis that was formulated. The esti-mated deformation was 3, 3 and 24 mm in the directionof respectively the x-, y- and z-axis in system rb. Thisshows that using weighted least squares with fullcovariance matrices and applying more-dimensionaltests gives the possibility to detect deformations thatare below the noise level of individual points.

ConclusionsThe problem of finding the relation between two Cartesiancoordinate vectors that pertain to the same points of anobject under deformation, is addressed. If the two vectorsrefer to two different epochs, the relation between them isdetermined by a possible difference in geodetic datum, by apossible deformation, which may have occurred betweenboth epochs, and by measurement noise. In this paper therelation is considered as describable by in principle one ormore affine, congruence or similarity transformations, tobe extended by other parameters.

An adjustment model is given to estimate theparameters of an affine, a congruence or a similaritytransformation. The congruence and similarity transfor-mation are formulated as an affine transformation withconstraints. That makes it possible to avoid the use ofEuler angles.

To compute approximate values for the parameters ofa congruence and similarity transformation it is possibleto compute first approximate values for an affinetransformation and subsequently to change these valuesto those of a congruence or similarity transformation byapplying a singular value decomposition of the rotationmatrix. This computation of approximate values has tobe repeated in each iteration step of solving thelinearised adjustment model.

Applying the proposed linearised adjustment modelwith constraints makes it easy to extend the model asfollows1. Several transformations can be combined.2. Different geodetic datums are taken account of.3. Testing for biases in the coordinate vectors is made

possible.4. Testing alternative deformation models is possible.The proposed linearised adjustment model can be solvedto get a weighted least squares solution, where fullcovariance matrices of the coordinates of both epochsare taken account of. The covariance matrices may besingular positive semidefinite matrices.

The effectiveness of the proposed adjustment model isdemonstrated in an experiment, where artificially addeddeformations are successfully detected.

Extending the proposed linearised adjustment modelto more than two epochs and to more complexdeformation models is straightforward.

Appendix

ConventionsIn this paper use is made of the following conventions:

N T indicates the transpose of a matrix

N approximate values of a scalar or vector are indicatedby a sub or super script 0, e.g. a scalar s0 or a vector v0

N if v is a certain vector, then Dv is the vector ofdifferences of v with its vector v0 of approximatevalues



Dv~v{v0

For a scalar an analogous equation holds.

N a vector that contains the coordinates of n pointscontains first all x-coordinates from point 1 up topoint n, then all y-coordinates and finally all z-coordinates

N the vector a consists of three sub vectors x, y and z,and the vector b9 of three sub vectors u, v and w, suchthat

a~

x

y

z

0B@

1CA and b’~

u

v

w

0B@

1CA (40)

N vector x contains for each point the x-coordinates. Ifthere are n points, then

x~

x1

x2

..

.

xn

0BBBB@

1CCCCA (41)

In the same way y, z and u, v, w are defined.

Adjustment equationsMost of the following equations can be found in(Teunissen, 1999) and (Teunissen, 2006). To write theequations more concisely, and because of their centralrole in testing, the reciprocal least squares residuals r areintroduced.

A system of linear or linearised observation equationshas the following general structure

E ‘f g~Ap ; Df‘g~s2Q‘ (42)

where ‘‘ is the m-vector of observations, A is the (m6n)-matrix of coefficients and p is the n-vector of unknownparameters. The equation behind the semicolondescribes the stochastic model by giving the covariancematrix Df‘‘‘g, the variance factor s2 and the cofactormatrix Q‘. Both A and Q, are considered in the

following equations to be regular matrices. The situationthat Q‘ is singular is treated in the paper.

The least squares solution is given by

p~(ATQ{1‘ A){1ATQ{1

‘ ‘‘ (43)

Qp~(ATQ{1

‘ A){1 (44)

with p the vector of estimated parameters and Qp

its

cofactor matrix. We have also

‘~Ax; Q‘~AQ

xAT (45)

e~‘{‘; Qe~Q

‘{Q

‘(46)

r~Q{1‘ e; Q

r~Q{1

‘ QeQ{1‘ (47)

with ‘ the adjusted observations, e the least squares

residuals and r the reciprocal least squares residuals.The Q-matrices are their cofactor matrices. Thereciprocal least squares residuals are used in the testingequations.

For each system of linear observation equations anequivalent system of linear condition equations exists

KT Ef‘g~0; Df‘g~s2Q‘ (48)

where KT is the ((m-n)6m)-matrix of conditions, forwhich holds (Teunissen, 1999, p. 63)

KTA~0 (49)

Matrix K is considered a regular matrix. Q‘‘ isconsidered a positive semidefinite matrix, so it may besingular.

Define the vector of misclosures t and its cofactormatrix Qt as

t~KT ‘; Qt~KTQ‘K (50)

The least squares solution is

r~KQ{1t t; Q

r~KQ{1

t KT (51)

e~Q‘r; Qe~Q‘Qr

Q‘ (52)

‘~‘{e; Q‘~Q‘{Q

e(53)

Testing equationsConsider models (42) or (48) as the null hypothesis. Letan alternative hypothesis be defined as

Ef‘g~ApzG=; Df‘g~s2Q‘ (54)

with G a (m6q)-matrix of known coefficients and = a q-vector of unknown bias parameters, with m the amountof observations and q the amount of bias parameters.The formulation for the model of condition equations is

KTEf‘g~KTG=; Df‘g~s2Q‘ (55)

To test this alternative hypothesis against the nullhypothesis (model (42) or (48)) use is made of thefollowing test statistic (Teunissen, 2006, p.78)

Vq~rT

G(GTQrG){1GTr (56)

If Fq,?~Vq

qs2wFcrit (57)

with Fcrit the critical value, the null hypothesis is rejectedin favour of the alternative hypothesis.

If the null hypothesis is rejected, an estimate of thebiases and its cofactor matrix can be determined as(Teunissen, 2006, p. 76)

=~(GTQrG){1GTr; Q

=~(GTQ

rG){1 (58)

For q we have 1#q#m – n. If q.m – n the adjustmentmodels (54) and (55) are underdetermined and cannotbe solved. For the limiting case q5m–n the test is equalto the overall model test and we have

Vm{n~eT

Q{1‘ e~r

TQ‘r (59)

The rightmost expression can be used if the system ofcondition equations is used and Q, is singular.



The limiting case q51 is the test of w-quantities(Baarda, 1968, p.13). It is called a w-test

If w~gTr

s gTQrg

� �1=2wwcrit (60)

reject the null hypothesis. Here wcrit is the critical valueand g is the matrix G, but written as a lower case letter,because it has only one column: it is a vector. The w-quantity has a normal distribution with an expectationof 0 and a standard deviation of 1. A conventional w-test (Baarda, 1968, p.15) is the test of one observationbeing biased and all other observations being withoutbias. The vector g is

g~(0 � � � 010 � � � 0)T (61)

with the 1 corresponding to the biased observation.

A point test is a three-dimensional test (q53) of the x-,y- and z-coordinate of one point (used as observations).The matrix G is

G~

0 � � � 0 1 0 0 0 � � � 0

0 � � � 0 0 1 0 0 � � � 0

0 � � � 0 0 0 1 0 � � � 0

0B@

1CA

T

(62)

with the ones corresponding to the coordinates that aresupposed to be biased.

Results case 1Point 1 is biaseddx, dy, dz 3 mm, -3 mm, -3 mmin x-system

Number_of_computations 3Max_criterion 5 5.0e-013Scale_factor 5 1Translation_x_y_z_in_m 5

81475.939, 455202.030, 2.000Alpha_Beta_Gamma_in_gon 5

0, 0, -100Overall model test:

Number_of_conditions 5 38Ftest 5 1.41Critical_value 5 1.18

w-test: Critical value is 3.29No rejections

Point test: Critical value 5 4.21Point 001 6.2758Est.def. x Est.def. y Est.def. z3.6 mm -2.4 mm -2.5 mm

Data of experimental validation

Observations of the first epoch

pnt pnt Direction Distance Zenith angle

99 1 307.7820765 24.5060074 99.805545599 2 319.7344105 20.0876102 98.707071799 3 331.5296877 17.0625166 105.432748199 4 352.4563301 14.7326649 106.634975699 5 388.3056469 15.4070803 101.338012299 14 29.0918416 17.8292985 99.787155799 15 46.7749502 15.3559925 96.612983899 16 59.3812951 14.5848606 99.600197599 17 79.1592740 14.9923557 105.71582999 18 89.8716178 17.9265803 106.559807399 19 93.2971810 21.2104291 98.482051599 23 186.8788155 19.3414323 103.940822399 24 200.1056248 20.9264019 105.369281699 25 212.6876307 23.5421679 99.055320899 26 224.8455986 28.3270008 98.4196188

Observations of the second epoch, point 1 is biased


100 1 13.7372731 23.0672793 99.7859602100 2 27.7594635 19.1262636 98.6398106100 3 41.4401419 16.6097400 105,582282100 4 63.6494039 15.1749445 106.4430463100 5 96.2060486 17.1252428 101.2017896100 14 131.4346394 20.3808279 99.8127807100 15 146.8624786 18.0020637 97.1040642100 16 157.5422153 17.1916389 99.6580831100 17 174.4918375 17.3543574 104.9366243100 18 184.6390687 20.0631725 105.8611633100 19 188.5080762 23.2655969 98.6169971100 23 279.1965362 17.9334770 104.2543189100 24 294.0907282 19.0647202 105.8937977100 25 308.5860882 21.3237391 98.9581652100 26 322.5863556 25.8583270 98.2684568



5

5

F_q

Differences between x and adj. coord. in mm

1 0 0 0

2 -1.1 0.1 0.9

3 -1.6 1.5 1.2

4 -1.1 -0.2 1.1

5 -0.7 0.2 0.4

14 0 0 0

15 -1.1 -1.5 -1.1

16 -0.3 -0.6 -0.5

17 -0.9 -0.8 -0.4

18 -1.2 -1.7 -0.4

19 -2.1 -1.4 -0.6

23 -2.5 -1.1 0.0

24 -3.2 -1.5 -0.6

25 -2.9 0.9 0.0

26 -3.2 -0.5 0

Results case 2Points 1, 2, 3, 4, 5 are biased

dx, dy, dz 5 3 mm, -3 mm, 2 mm

Observations of the second epoch, points 1 up to 5 are biased


100 1 13.7372731 23.0672631 99.7997607100 2 27.7642973 19.1302064 98.6467483100 3 41.4423295 16.6141034 105.5885051100 4 63.6455986 15.1792709 106.4496345100 5 96.1955807 17.1284537 101.2089988100 14 131.4346394 20.3808279 99.8127807100 15 146.8624786 18.0020637 97.1040642100 16 157.5422153 17.1916389 99.6580831100 17 174.4918375 17.3543574 104.9366243100 18 184.6390687 20.0631725 105.8611633100 19 188.5080762 23.2655969 98.6169971100 23 279.1965362 17.9334770 104.2543189100 24 294.0907282 19.0647202 105.8937977100 25 308.5860882 21.3237391 98.9581652100 26 322.5863556 25.8583270 98.2684568

Approximate coordinates of the first epoch

pnt x y z

099 83 277.7360 457 303.9790 2.0000001 83 253.4140 457 306.9670 2.0740002 83 258.6090 457 310.1060 2.4080003 83 262.7790 457 312.0580 0.5460004 83 267.7820 457 314.7330 0.4670005 83 274.9220 457 319.1230 1.6760014 83 285.6020 457 319.9780 2.0600015 83 288.0160 457 315.3580 2.8170016 83 289.4500 457 312.6660 2.0920017 83 291.8750 457 308.7800 0.6560018 83 295.3420 457 306.8040 0.1560019 83 298.8230 457 306.2070 2.5050023 83 281.6860 457 285.0840 0.8030024 83 277.7020 457 283.1290 0.2370025 83 273.0750 457 280.9050 2.3490026 83 266.9620 457 277.7910 2.7030

Approximate coordinates of the second epoch

pnt x y z

100 0.0000 0.0000 0.00001 4.9360 22.5280 0.07402 80.750 173.330 0.40803 100.270 131.630 214.5404 127.020 81.600 215.3305 170.920 10.200 20.324014 17.9470 29.6600 0.060015 133.270 2120.740 0.817016 106.350 2135.080 0.092017 67.490 2159.330 213.44018 47.730 2194.000 218.44019 41.760 2228.810 0.505023 2169.470 257.440 211.97024 2189.020 217.600 217.63025 2211.260 28.670 0.349026 2242.400 89.800 0.7030



in x-system

Number_of_computations 5 3Max_criterion 5 5.2e-012Scale_factor 5 1Translation_x_y_z_in_m 5

81475.940, 455202.029, 2.001Alpha_Beta_Gamma_in_gon 5

0, 0, -100Overall model test

Number_of_conditions 5 38Ftest 5 2.20Critical_value 5 1.18

w-test: Critical value is 3.29x-coordinate

Point Ratio w Est.error005 1.1582 3.8106 3.0011

y-coordinatePoint Ratio w Est.error002 1.1816 3.8875 -3.2984

z-coordinateno rejections

Point test: Critical value 5 4.21Point F_q002 5.8141Est.def. x Est.def. y Est.def. z2.3 mm -3.9 mm -0.4 mm

More-dimensional testq53, points 001, 002, 003, 004, 005F_q5.8141Est.def. x Est.def. y Est.def. z2.8 mm -3.1 mm 3.7 mm

Differences between x and adj. coord. in mm1 0 0 2 0.2 -1.2 -0.23 0.0 0.1 0.34 0.7 -1.7 0.65 1.0 -1.6 0.4

14 0 0 015 -1.2 -1.5 -0.616 -0.3 -0.7 0.217 -0.8 -0.9 0.718 -1.0 -1.9 1.119 -2.3 -1.4 1.223 -2.4 -1.2 1.124 -3.1 -1.7 0.225 -3.0 0.9 0.426 -3.4 -0.5 0

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extendable linearised adjustment model for deformation

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