Quality Assessment of Adjustment Results

Σ_vv =Q_LLB^TM⁻¹ AN⁻¹A^TM⁻¹+I B Σ_LL

| {z }

=σ²₀Q_LL

B^T

M⁻¹ AN⁻¹A^TM⁻¹+IT

BQ_LL

=Q_LLB^TM⁻¹ AN⁻¹A^TM⁻¹+I

Bσ₀²Q_LLB^T

M⁻¹ AN⁻¹A^TM⁻¹+IT

BQ_LL

=σ₀²Q_LLB^TM⁻¹ AN⁻¹A^TM⁻¹+I

BQ_LLB^T

| {z }

=M

M⁻¹ AN⁻¹A^TM⁻¹+IT

BQ_LL

=σ₀²Q_LLB^TM⁻¹ AN⁻¹A^TM⁻¹+I

M AN⁻¹A^TM⁻¹+IT

M⁻¹T

BQ_LL

=σ₀²Q_LLB^T M⁻¹AN⁻¹A^TM⁻¹M

| {z }

=I

+M⁻¹M

| {z }

=I

M⁻¹AN⁻¹A^T+I

M⁻¹BQ_LL

=σ²₀Q_LLB^T M⁻¹AN⁻¹A^T+I

M⁻¹AN⁻¹A^TM⁻¹+M⁻¹ BQ_LL

=σ²₀Q_LLB^T M⁻¹AN⁻¹A^TM⁻¹A

| {z }

=−N

N⁻¹A^TM⁻¹+M⁻¹AN⁻¹A^TM⁻¹+ +M⁻¹AN⁻¹A^TM⁻¹+M⁻¹

BQ_LL

=σ²₀Q_LLB^T

−M⁻¹A N⁻¹N

| {z }

=I

N⁻¹A^TM⁻¹+M⁻¹AN⁻¹A^TM⁻¹+ +M⁻¹ AN⁻¹A^TM⁻¹+I

BQ_LL

=σ²₀Q_LLB^T

−M⁻¹AN⁻¹A^TM⁻¹+M⁻¹AN⁻¹A^TM⁻¹

| {z }

=0

+ +M⁻¹ AN⁻¹A^TM⁻¹+I

BQ_LL

=σ²₀Q_LLB^TM⁻¹ AN⁻¹A^TM⁻¹+I

BQ_LL. (3.61)

The variance-covariance matrix of the unknowns and of the residuals,Σ_XˆXˆ andΣ_vv, are utilized as measures for the quality assessment of the adjusted results.

where the transfer matrixR=Q_vvPis introduced. Observe if any one componentdL_iindLcontains an error, and, in that case how this error will influence all residuals bydvdue to the transfer matrixR. The diagonal element R_iiinRis a transfer factor that shows the (partly) impact of the erroneous componentdL_ion the corresponding residualv_iinv. This transfer factor is knowns asredundancy numberof the observationL_i

r_i =R_ii. (3.64)

Each redundancy numberr_ihas a value between0and1. In the extreme case ofr_i = 1, an error in the observation l_ican be completely detected in the residualsv_i, while in the other extreme case ofr_i = 0, an error in the observa-tionl_iis undetectable. Therefore, it is required that the sum of all redundancy numbers, the total redundancyr, is large enough to detect blunders in the observations. The trace of the matrixRleads to the total redundancyras follows

trace R

=

N

X

i=1

r_i =r . (3.65)

The redundancy number can be represented as a percentageEV that is known asinfluence on the residuals(German:

Einfluss auf die Verbesserung)

EV =r_i100 %. (3.66)

The following rating scale to evaluate the redundancy numbers has gained ground in the practise:

0 % ≤ EV < 1 % observation is not controlled, 1 % ≤ EV < 10 % observation is poorly controlled, 10 % ≤ EV < 30 % observation is sufficiently controlled, 30 % ≤ EV < 70 % observation well controlled,

70 % ≤ EV < 100 % observation can be removed without loss of reliability.

Blunders Detection and Localisation

In order to prevent incorrect adjustment results due to outliers, blunders have to be detected and removed from the observations. Aglobal testis performed as follows to find out if the observations contains blunders. According to Nuzzo (2014), the original purpose of ahypothesis testby Fischer is to study if the adjusted results are predomi-nately occur due to randomness of the observations (null hypothesis). If it is not the case, the mathematical model might be incorrect or the observations might contain blunders (alternative hypothesis). Before we question the mathematical model, we have to be sure that the observations contain no blunders by examining if theempirical reference standard deviationafter the adjustment

s₀=  

v^TPv

r . (3.67)

coincides with the theoretical reference standard deviationσ₀before the adjustment. We can state the null hypoth-esisH₀ as

H₀ : E s²₀

=σ₀². (3.68)

The alternative hypothesisH_Acan take one of the following forms H_A : E s²₀

6=σ₀², (3.69)

H_A : E s²₀

> σ₀², (3.70)

H_A : E s²₀

< σ₀². (3.71)

Now, we have to consider the statistical distribution ofs²₀. We rearrange Eq. (3.67) as follows s²₀ = v^TPv

r = σ²₀ σ²₀

v^TQ⁻¹_LLv

r =σ₀²v^TΣ⁻¹_LLv

r ⇒ rs²₀

σ₀² =v^TΣ⁻¹_LLv. (3.72)

The residualsvare assumed to be normal distributed random variables. Then, the squared residuals are weighted by the inverse of the variance-covariance matrixΣ⁻¹_LL. This in turn yields residuals that are divided by their cor-responding variances. Consequently, they are variables that follow a standard normal distribution. According to Pearson, the sum of their squares is conforming toχ²-distribution withrdegrees of freedom. In other words, the variable

χ²_r =rs²₀

σ₀² (3.73)

isχ²-distributed. Therefore, we choose the above variable astest statistic. To determine thethreshold valueχ²_r,α, an arbitrary chosen error probabilityαrespectively confidence levelS = 1−αhas to be considered. Then, we compare the test statistic with the threshold value for the following four statements

χ²_r < χ²_r,1−^α

2 andχ²_r> χ²_r,^α

2

⇒ RejectH₀in favor ofH_A : E s²₀

6=σ₀², (3.74) χ²_r > χ²_r,1−α ⇒ RejectH₀in favor ofH_A : E s²₀

> σ₀², (3.75) χ²_r < χ²_r,α ⇒ RejectH₀in favor ofH_A : E s²₀

< σ₀², (3.76) otherwise we fail to reject the null hypothesisH₀. But, in case we reject the null hypothesis, we have to look for individual blunders in the observations. To introduce a measure for removing a single observation that likely con-tains gross error, we assume that the error in an observation mainly affects its corresponding residual. The measure standardised residual(German:Normierte Verbesserung) is defined as

NV_i = v_i

σ_v

i

. (3.77)

The computational purpose, we can find an alternative way for computingNV_i. We rewrite Eq. (3.61) by means of Eq. (3.62) and the transfer matrix as

Σ_vv =σ₀²Q_vvPQ_LL=RΣ_LL. (3.78)

It is often the case that the variance-covariance matrixΣ_LLis a diagonal matrix. In this case, we can determine the standard deviation of the residualv_ias follows

σ_v

i =√ r_iσ_l

i. (3.79)

Inserting the above into Eq. (3.77) yields

NV_i = v_i

√ r_iσ_l

i

. (3.80)

The standardised residualNV_i follows a standard normal distribution due to the division of the normal distributed residualv_i by its corresponding standard deviationσ_v

i. A standardised normal test or alocal test has to be per-formed to evaluate if an observation contains gross error. But in practise, we can use the following rating scale to assess the standardised residual:

2.5 < NV ≤ 4 gross error possible, NV > 4 gross error most likely.

After we identified an observation as blunder, we can introduce a measure to quantify the magnitude of the gross errorε_i,potential blunder(German:Grober Fehler)

GF_i = ε_i

r_i = −v_i

r_i . (3.81)

This ratio describes how the diagonal componentr_iof the transfer matrixRaffects its corresponding gross error ε_irespectively−v_i. Another question that arises is how large a blunderGF_i must be that we are able to detect it.

A measure of detectability of blunder, theboundary value(German:Grenzwert) is introduced as follows GRZW_i = σ_l

iδ₀

√r_i , (3.82)

64 SHI TESTING|ADJUSTMENT CALCULATION

whereδ₀ = 4.13is the non-centrality parameter; for the derivation see Baarda (1968). Consequently, errors that are smaller than theGRZWhas to be regarded as random and therefore they are undetectable.

The workflow of the blunders detection by means of a global test as well as the localisation and removal of blunders as known asdata snooping can summarise as follows. First, after the adjustment, a global test respectively aχ² -test is performed. If we fail to reject the null hypothesisH₀ in favour of the alternative hypothesisH_A, it means that the final result is obtained. Otherwise, we have to pinpoint the blunders in the observations by means of the standardised residuals. Second, the observationl_iwith largest standardised residualNV_i has to be removed from the observation vectorL. Third, an adjustment calculation has to be performed and start this workflow from the beginning until we fail to reject the null hypothesis.

The influence of observation error on the parameters

Detectable and removable blunders and their influences are examined in theinternal reliability. The affection of non-detectable blunders on the unknowns analysed in theexternal reliability. A measure that describes the impact of the boundary value on the coordinates of the corresponding point (German:Einfluss des Grenzwertes auf die Koordinatender berührenden Punkte) is introduced as

EGK_i = 1−r_i

GRZW_i . (3.83)

The other measure that describes the impact of a potential blunder on a point corresponding to the measurement (German:Einfluss eines eventuellen groben Fehlers auf den die Messung berührenden Punkt) is introduced as follows

EP_i = 1−r_i

GF_i . (3.84)

For the derivation refer to Baarda (1968).

Im Dokument The measurement- and model-based structural analysis for damage detection (Seite 72-75)