Testing deformation hypotheses by constraints on a time series of geodetic observations

Hiddo Velsink

doi:10.1515/jag-2017-0028

Published by De Gruyter December 8, 2017

Testing deformation hypotheses by constraints on a time series of geodetic observations

Hiddo Velsink

From the journal Journal of Applied Geodesy

https://doi.org/10.1515/jag-2017-0028

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Abstract

In geodetic deformation analysis observations are used to identify form and size changes of a geodetic network, representing objects on the earth’s surface. The network points are monitored, often continuously, because of suspected deformations. A deformation may affect many points during many epochs. The problem is that the best description of the deformation is, in general, unknown. To find it, different hypothesised deformation models have to be tested systematically for agreement with the observations. The tests have to be capable of stating with a certain probability the size of detectable deformations, and to be datum invariant. A statistical criterion is needed to find the best deformation model. Existing methods do not fulfil these requirements. Here we propose a method that formulates the different hypotheses as sets of constraints on the parameters of a least-squares adjustment model. The constraints can relate to subsets of epochs and to subsets of points, thus combining time series analysis and congruence model analysis. The constraints are formulated as nonstochastic observations in an adjustment model of observation equations. This gives an easy way to test the constraints and to get a quality description. The proposed method aims at providing a good discriminating method to find the best description of a deformation. The method is expected to improve the quality of geodetic deformation analysis. We demonstrate the method with an elaborate example.

Keywords: Geodetic deformation analysis; Time series analysis; Best deformation description; Testing constraints; Nonstochastic observations; Least-squares adjustment

Acknowledgment

The support of research programme Maps4Society is gratefully acknowledged.

Appendix A Overview: adjustment

Model (1) has a rank deficient (singular) covariance matrix. It may have a rank deficient coefficient matrix, and therefore, elements in the parameter vector x may not be estimable. Let pTx be any unbiasedly estimable (u.e.) linear function of x under model (1), cf. [30, p. 137]. We mention below three methods to compute a best linear unbiased estimator (BLU-estimator) of pTx. This estimator is also the least-squares estimator. It yields BLU-estimators for the adjusted observations (which are u.e. functions of x) and also for those elements of x that are u.e. themselves.

The first method transforms model (1) into a model with full rank and a regular covariance matrix [30, pp. 149, 144]. In a first step the observations are orthogonalised relative to the covariance matrix. The result is a subvector of observations with a scaled unit matrix as covariance matrix, and a subvector of nonstochastic observations. The latter are considered constraints on the parameters, which implies that a model with less parameters is possible. In a second step such a model is derived. The result is a model with full rank and a regular covariance matrix, and no constraints.

The second method uses the fact that the BLU-estimator in model (1) for pTx is given by a minimum Qy-norm solution of ATf=p as [30, equation (7.4.2)]:

(A.24)fTy_=pT[(AT)n(Qy)−]Ty_.

where (AT)n(Qy)− is a minimum Qy-norm generalised inverse (g-inverse) of AT. Expressions to compute this g-inverse are given in [30, p. 148] and [29].

The third method is closely related to the use of a minimum Qy-norm g-inverse. This g-inverse is, in fact, a solution of an extended system of normal equations, in which Lagrange multipliers are used [30, eq. 7.4.10^[1]]. So formulating this system and solving it numerically, provides the desired solution of x.

Model (1) is often formulated after linearisation. In that case iteration is needed to arrive at its solution.

Appendix B Overview: testing

To test a model of observation equations like model (1), a null hypothesis H0 and an alternative model Ha are formulated:

(B.25)H0:y_=Ax+e_,

(B.26)Ha:y_=Ax+C∇+e_,

with C a known (m×q)-coefficient matrix, and an unknown q-vector ∇. The product C∇ describes the bias in the functional model. Both hypotheses have the same stochastic model.

The null hypothesis is tested against the alternative hypothesis with test statistic T_q. If Qy is an invertible matrix, T_q is [33, p. 78]:

(B.27)T_q=1σ2e_ˆTQy−1C(CTQy−1QeˆQy−1C)−1CTQy−1e_ˆ.

T_q is the square of the norm of the difference between the vector of adjusted observations under H0 and the same vector under Ha. The norm is determined by the metric of the vector space, which is defined by Qy−1 [33, p. 85].

Let the reciprocal residualsr_ˆ be defined by r_ˆ=Qy−1e_ˆ, if Qy is not singular. Let their cofactor matrix be called Qrˆ. Equation (B.27) becomes:

(B.28)T_q=1σ2r_ˆTC(CTQrˆC)−1CTr_ˆ.

Deformation analysis model (1) has a singular cofactor matrix Qy. Therefore, we need another, more general, definition of the reciprocal residuals. For this, we switch to the model of condition equations. It is defined as:

(B.29)BTy_=t_;E{t_}=0,withBTA=0.

Matrix B is the (m×b)-coefficient matrix. t_ is the b-vector of misclosures. b=m−n is the number of conditions. The model gives the same least-squares solution as model (B.25). Its solution, using a positive semidefinite Qy, is given by solving the normal equations. They have the same form as when using a positive definite (regular) Qy. The parameters to be solved are the correlatesk. They are used to define more generally the reciprocal residuals r_ˆ.

(B.30)BTQyBk_ˆ=t_normal equations,

(B.31)r_ˆ=Bk_ˆreciprocal residuals,

(B.32)e_ˆ=Qyr_ˆleast-squares residuals,

(B.33)Qeˆ=QyQrˆQycofactor matrix ofe_ˆ.

It is assumed that (BTQyB) is invertible (which implies that the conditions are linearly independent), but Qy can be singular. The reciprocal residuals and their cofactor matrix are:

(B.34)r_ˆ=B(BTQyB)−1t_;Qrˆ=B(BTQyB)−1BT.

Test statistic T_q is computed with equation (B.28).

To switch from model (B.25) to (B.29) B is computed as a base matrix of the nullspace of AT. It means solving the equation ATB=0 for B, e.g. by singular value decomposition.

The probability density function of T_q is a χ2-distribution, with E{T_q}=q. After choosing a significance level α, the critical value is computed, and it is determined, whether the critical value is exceeded by the computed value of T_q. In that case the null hypothesis is rejected.

B.1 Overall model test

Testing the adjustment results begins with the overall model test [36]. It uses the reciprocal residuals r_ˆ of equation (B.34) and the test statistic of equation (B.28) with q=m−n (m is the number of observation, n the number of parameters):

(B.35)T_m−n=1σ2r_ˆTQyr_ˆ.

The test statistic is χ2-distributed and its critical value is computed with the B-method of testing, after choosing the one-dimensional test significance level (often 0.1%) and the power (often 80%). The significance level of the (m−n)-dimensional test is then derived.

B.2 w-tests

If the overall model test doesn’t lead to rejection of the adjustment model (the null hypothesis), more specific tests are not needed, if the B-method of testing is used. The B-method of testing uses the principle that if a hypothetical reference bias is present, the overall model will find it with the same statistical power as more specific tests, that have a smaller degree of freedom q [2, p. 33]. A more specific test can be the one-dimensional test that the alternative hypothesis has only one additional parameter ∇ that affects only one observation (conventional w-test [2, p. 15]). As reference bias in general the minimal detectable bias of the conventional w-test is taken.

If the overall model test leads to rejection of the null hypothesis, more specific alternative hypotheses are formulated by specifying matrix C in equation (B.26). For the conventional w-tests we have C=0,…,0,1,0,…,0T.

B.3 Tests of specific deformation hypotheses

In deformation analysis one may expect certain deformations of subsets of points, like the gradual subsidence of a few points in the course of several epochs, or the temperature induced fluctuation of other points. Such expectations may be tested by formulating alternative hypotheses, using appropriate matrices C. Examples are given in Section 4.

Appendix C S-transformation invariance

In this section it is shown that test statistic T_q and the m.d.b.’s are S-transformation invariant. Use the following definitions:

(C.36)y_a=y_szd;Aa=As0ZdZ∇;e_a=e_s0;

(C.37)G=Zg0,

to write model (1) as:

(C.38)y_azg=AaGx+e_a0,

The S-system is defined by the second row: zg=Gx. It solves the rank deficiency of Aa. If N is a (n×(n−nG)) base matrix of the null space of G, we can write [30, p. 24]:

(C.39)x=G−zg+Nλ.

λ is a vector of (n−nG) parameters and G− any generalised inverse of G (i.e. it is defined by: GG−G=G). We can insert equation (C.39) into model (C.38). For zg we get

(C.40)zg=GG−zg+GNλ=zg+0λ,

which is valid for any λ. It can be left out of the model. Model (C.38) becomes therefore:

(C.41)y_a−AaG−zg=AaNλ+e_a,

Because G solves the rank deficiency of Aa, the product AaN has full rank. The parameters λ are solved from equation (C.41) by least squares. Then they are inserted into (C.39) to get the estimated parameters x_ˆ, relative to the S-basis, defined by the equation zg=Gx.

Let the range space of any matrix U be written as R(U). Then R(AaN)=R(Aa), because rank(AaN)=rank(Aa).

If we switch to another S-basis, we have another z‾g, G‾ and N‾. We have, however:

(C.42)R(AaN‾)=R(AaN)=R(Aa).

Let us switch to model (B.29), the model of condition equations. We have, with ⊕ indicating the direct sum of two vector spaces, and Rma the ma-dimensional Euclidian space, where ma is the number of elements in y_a:

(C.43)R(B)⊕R(Aa)=Rma.

Because of (C.42) we have

(C.44)BTAaN=BTAaN‾=0;

(C.45)R(B)⊕R(AaN)=R(B)⊕R(AaN‾)=Rma.

So, if we use equation (C.41) using N, and premultiply it with BT; or we use the same equation (C.41), but with another N‾, and premultiply it again with BT, we get the same model:

(C.46)BTy_a=BTe_a,E{BTe_a}=0.

This is a complete model to use for adjustment, irrespective of the choise of N, i.e. irrespective of the S-system. Therefore r_ˆa and Qrˆa are S-transformation invariant. Then also test statistic T_q and the m.d.b.’s are S-transformation invariant, as long as matrix C contains zeros for the rows that pertain to S-basis elements (i.e. as long as the S-basis elements are not tested, which would be, by the way, meaningless). □

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Received: 2017-7-27

Accepted: 2017-10-10

Published Online: 2017-12-8

Published in Print: 2018-1-26

Testing deformation hypotheses by constraints on a time series of geodetic observations

Abstract

Acknowledgment

Appendix A Overview: adjustment

Appendix B Overview: testing

B.1 Overall model test

B.2 w-tests

B.3 Tests of specific deformation hypotheses

Appendix C S-transformation invariance

References

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