Assessment of inner reliability in the Gauss-Helmert model

Andreas Ettlinger; Hans Neuner

doi:10.1515/jag-2019-0013

Published by De Gruyter October 19, 2019

Assessment of inner reliability in the Gauss-Helmert model

Andreas Ettlinger and Hans Neuner

From the journal Journal of Applied Geodesy

https://doi.org/10.1515/jag-2019-0013

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Abstract

In this contribution, the minimum detectable bias (MDB) as well as the statistical tests to identify disturbed observations are introduced for the Gauss-Helmert model. Especially, if the observations are uncorrelated, these quantities will have the same structure as in the Gauss-Markov model, where the redundancy numbers play a key role. All the derivations are based on one-dimensional and additive observation errors respectively offsets which are modeled as additional parameters to be estimated. The formulas to compute these additional parameters with the corresponding variances are also derived in this contribution. The numerical examples of plane fitting and yaw computation show, that the MDB is also in the GHM an appropriate measure to analyze the ability of an implemented least-squares algorithm to detect if outliers are present. Two sources negatively influencing detectability are identified: columns close to the zero vector in the observation matrix B and sub-optimal configuration in the design matrix A. Even if these issues can be excluded, it can be difficult to identify the correct observation as being erroneous. Therefore, the correlation coefficients between two test values are derived and analyzed. Together with the MDB these correlation coefficients are an useful tool to assess the inner reliability – and therefore the detection and identification of outliers – in the Gauss-Helmert model.

Keywords: Least-Squares Adjustment; Gauss-Helmert model; Inner Reliability

Appendix A General formulation of systematic observation errors

The observation model with non-linear systematic observation errors can be formulated as

(43)gl(l,∇)=E{l}−v.

gl() is a function which gives calibrated observations depending on the – in general – non-linearly included systematic observation errors ∇∈Rm×1, treated as deterministic additional parameters to be estimated. By setting ∇=∇∗ such that gl(l,∇∗)=l, one arrives at the linearized functional model of the GHM (6). Hence, a GHM adjustment based on l can be solved with the equations (8)–(13) if the null hypothesis

(44)H0:∇−∇∗=0

is correct. To derive appropriate test values for checking the validity of H0, the estimated additional parameters ∇ˆ have to be computed. This is done with Lagrangian optimization (as also done in section 2), which corresponds to the procedure of deriving ∇ˆ in the GMM (as shown in [16] p. 184 ff.). The linearization of the functional model with additional parameters is done with

(45)0=h(xˆ′,lˆ′)=h(xˆ′,gl(l,∇ˆ)+vˆ′)≈h(x(0),gl(l,∇(0))+v(0))+∂h(xˆ′,lˆ′)∂xˆ′|x(0),gl(l,∇(0))+v(0)·dxˆ′+∂h(xˆ′,lˆ′)∂vˆ′|x(0),gl(l,∇(0))+v(0)·(vˆ′−v(0))+∂h(xˆ′,lˆ′)∂∇ˆ|x(0),gl(l,∇(0))+v(0)·(∇ˆ−∇(0))≈h(x(0),gl(l,∇(0))+v(0))+∂h(xˆ′,lˆ′)∂xˆ′|x(0),gl(l,∇(0))+v(0)·dxˆ′+∂h(xˆ′,lˆ′)∂lˆ′·∂lˆ′∂vˆ′|x(0),gl(l,∇(0))+v(0)·(vˆ′−v(0))+∂h(xˆ′,lˆ′)∂lˆ′·∂lˆ′∂gl(l,∇ˆ)·∂gl(l,∇ˆ)∂∇ˆ|x(0),gl(l,∇(0))+v(0)·(∇ˆ−∇(0))≈h(x(0),gl(l,∇(0))+v(0))+A′dxˆ′+B′I·(vˆ′−v(0))+B′IA∇d∇ˆ≈w′+A′dxˆ′+B′vˆ′+B′A∇d∇ˆwithw′=h(x(0),gl(l,∇(0))+v(0))−B′v(0).

In (45) A∇∈Rb×m is the Jacobi matrix, resulting from the partial derivatives of gl(l,∇ˆ) with respect to ∇ˆ and having full column rank. By using quotation marks as superscripts, it should be clarified, that w′, A′ and B′ are different to the corresponding quantities in section 2, as the partial derivatives are evaluated now additionally at the approximate, additional parameters ∇(0). By setting ∇(0)=∇∗ (what will be assumed from here on), one can again set w′=w, A′=A and B′=B. xˆ′ and vˆ′ (and therefore dxˆ′ and lˆ′) as well as kˆ′ will differ from its corresponding quantities in section 2, due to the incorporation of ∇ˆ.

The extended normal equation system becomes:

(46)NABA∇AT00A∇TBT00kˆ′dxˆ′d∇ˆ=−w00

As stated e. g. in [16] p. 185, d∇ˆ is not unambiguously estimable if m exceeds the redundancy r=b−u, i. e. overall more parameters (m+u) have to be estimated than conditions are given and the LS-problem gets under-determined. The Lagrangian multipliers kˆ′ are determined from the first row of (46)

(47)kˆ′=−N−1(w+Adxˆ′+BA∇d∇ˆ).

Inserting (47) into the second row of (46) and rearranging this expression gives

(48)dxˆ′=−(ATN−1A)−1ATN−1(w+BA∇d∇ˆ)=dxˆ−(ATN−1A)−1ATN−1BA∇d∇ˆ.

Now one can insert this into (47), which equals

(49)kˆ′=−N−1(w+Adxˆ+(I−A(ATN−1A)−1ATN−1)BA∇d∇ˆ)=kˆ−N−1(I−A(ATN−1A)−1ATN−1)BA∇d∇ˆ

After setting I−A(ATN−1A)−1ATN−1=I−AA+=RW for convenience, the residuals are calculated with

(50)vˆ′=QllBTkˆ′=QllBT(kˆ−N−1RWBA∇d∇ˆ)=vˆ−QllBTN−1RWBA∇d∇ˆ.

The result for d∇ˆ is obtained by inserting (49) into the third row of (46) and can be expressed in terms of w, kˆ or vˆ

(51)d∇ˆ=−(A∇TBTN−1RWBA∇)−1A∇TBTN−1RWw=(A∇TBTN−1RWBA∇)−1A∇TBTkˆ=(A∇TBTN−1RWBA∇)−1A∇TBTN−1Bvˆ.

As N corresponds to the cofactor matrix of w, Q∇ˆ∇ˆ is derived by using error propagation, the fact that RW is idempotent and the identity NRWTN−1=RW. It can also be expressed in terms of Qkˆkˆ or Qvˆvˆ

(52)Q∇ˆ∇ˆ=(A∇TBTN−1RWBA∇)−1A∇TBTN−1RWNRW′TN−1BA∇·(A∇TBTN−1RWBA∇)−1=(A∇TBTN−1RWBA∇)−1=(A∇TBTQkˆkˆBA∇)−1=(A∇TBTN−1BQvˆvˆBTN−1BA∇)−1.

$Figure 13 MDBs of point coordinates when applying translation of 200m200\hspace{0.19em}\text{m} in x- and y-direction.$

Figure 13

MDBs of point coordinates when applying translation of 200m in x- and y-direction.

Appendix B Numerical example: 2D similarity transformation

In the numerical example of 2D similarity transformation, nine points ps,j=ys,jxs,jT are considered which are arranged in the same way as in the plane fitting example. s=1,2 is the index for the coordinate system and j=1,…,9 specifies the point-ID. In 2D similarity transformation, the transformation parameters are the translation vector components ty and tx, the rotation ψ and the scale m. For each point, one can set up a pair of condition equations

(53)0=y2,j−ty−(1+m)(y1,jcosψ+x1,jsinψ)0=x2,j−tx+(1+m)(y1,jsinψ−x1,jcosψ)

to estimate the transformation parameters in a GHM adjustment. For the numerical evaluations, the adjustment procedure is repeated again 1000 times, where in each repetition white Gaussian noise with zero mean and standard deviation of σp=0.25m is added to the point coordinates (i. e. the observations) in both systems.

Figure 14

Absolute values of correlation coefficients.

In the first evaluation, only a translation of 200m in both coordinate directions is applied. The resulting MDBs ∇ˆ0,i are shown in figure 13. The structure is the same in both coordinate systems as well as in the plane fitting example. The point in the middle is the best controlled one, whereas the MDBs of the corner points are considerably higher. The correlation coefficients ρij of the test statistics TN(∇i) in figure 14a show, that the corresponding coordinates in both coordinate systems are highly correlated. Hence, by only using the condition equations (53), one can’t distinguish if the outlier influences the coordinate in system 1or2.

In another evaluation only a rotation of 45∘ is applied. The MDBs show the same structure as in the first evaluation (figure 13) but the behavior of the ρij is slightly different (figure 14b). Because of the rotation of 45∘, an outlier in one coordinate in system 1 appears to be an outlier with equal extent in both coordinate directions in system 2. Varying the scale parameter m from 1ppm to1000ppm, shows no influence on the MDBS and the correlation coefficients.

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Received: 2019-03-28

Accepted: 2019-10-09

Published Online: 2019-10-19

Published in Print: 2020-01-28

Assessment of inner reliability in the Gauss-Helmert model

Abstract

Appendix A General formulation of systematic observation errors

Appendix B Numerical example: 2D similarity transformation

References

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