1 Introduction

The transfer to displacement monitoring of the concept of Minimal Detectable Bias (MDB) introduced by Baarda (1968) was made by Pelzer (1972). The term network sensitivity with minimal detectable displacement (MDD) as its measure was introduced and later became an element of network analysis together with accuracy and reliability (Niemeier 1982; Niemeier et al. 1982). The concept of network sensitivity to displacements was elaborated in detail and extended onto deformation models by Heck (1986). In that paper, the MDD and a sensitivity ellipse for a single network point were first proposed. Some important findings concerning the theory of network sensitivity and datum invariance can be found in, e.g., Even-Tzur (2002, 2010), Xu (1995, 1997). The need for inserting the sensitivity criteria into the network optimization procedures is formulated in (Kuang 1991; Yazji 1997; Alizadeh-Khameneh et al. 2015).

The MDD concept is still commonly used in a priori analyses of monitoring network accuracy and sensitivity. The results of such joint analyses for horizontal networks are presented in the form of a confidence ellipse and a sensitivity ellipse for each network point. However, except for comparing the sizes, no other relationships between these ellipses are considered. Such a limited approach is most certainly due to the lack of a relevant theoretical basis. To our knowledge, in significance tests of computed displacements, the information contained in the sensitivity characteristics has not been taken into account as yet. Hence, the need to examine the possibility of using MDD as a source of supporting information both in a priori analyses of monitoring networks accuracy and in significance tests of displacements. If successful, that might extend the range of applications of MDD.

To respond to the above-mentioned need, the objectives of the present paper are the following:

  • to create a theoretical basis that would enable one to investigate the extent of MDD support in the analyses of monitoring network accuracy and also in significance tests of the computed displacements,

  • to examine the possibility of modifying the above-mentioned procedures so as to use the supporting information contained in the MDD.

It is necessary to emphasize that the paper concerns only the models with parameters being network points displacements. It does not deal with the models containing also strain parameters as used in network deformation analysis.

2 Brief review of basic concepts and notation

The review concentrates on minimum-constraint datum definition as commonly used in the monitoring of displacements and the measures of network sensitivity to displacements.

2.1 Minimum-constraint datum definitions

Let us consider a displacement monitoring model based on observation differences from two measurement campaigns

$$ {\mathbf{A}} \cdot {\mathbf{d}} = \Delta {\mathbf{l}} + {\mathbf{v}}_{{\Delta {\mathbf{l}}}} ,\quad {\mathbf{C}}_{{\Delta {\mathbf{l}}}} $$
(1a)
$$ {\mathbf{S}} \cdot {\mathbf{d}} = {\mathbf{0}} $$
(1b)

where \({\mathbf{A}}(n \times u)\)—design matrix, rank A = u − d (d—network defect), \({\mathbf{d}}(u \times {1})\)—vector of displacements, \(\Delta {\mathbf{l}}(n \times 1)\)—vector of observation differences, \({\mathbf{v}}_{{\Delta {\mathbf{l}}}} (n \times 1)\)—vector of random errors in \(\Delta {\mathbf{l}}\) (with opposite sign), \({\mathbf{C}}_{{\Delta {\mathbf{l}}}} (n \times n)\)—covariance matrix for \(\Delta {\mathbf{l}}\) (positive definite), \({\mathbf{S}}(d \times u)\)—coefficient matrix in datum constraints.

We shall denote by \({\mathbf{A}}_{{\text{s}}}\) and \((\Delta {\mathbf{l}})_{\rm s}\) the elements of the standardized model (1) corresponding to A and \(\Delta {\mathbf{l}}\), respectively.

The models as in (1) having point displacements as parameters are basically equivalent to those with point velocities.

The matrix S has the following properties

$$ {\text{rank}}\;\left[ {\begin{array}{*{20}c} {\mathbf{A}} \\ {\mathbf{S}} \\ \end{array} } \right] = u,\quad {\text{rank}}\;{\mathbf{S}} = d $$
(2)

The minimum-constraint datum definition eliminates the network defect without causing observation distortions. There are several types of such constraints, e.g., free net constraints.

Below, we present the least squares (LS) solution for the standardized model (1), distinguishing the following two ways in which the result of the global model test can be used in defining the covariance matrix for displacements

  1. (i)

    using the a priori variance factor \(\sigma_{\rm o}^{2}\) = 1, where \(\sigma_{\rm o}^{2}\) is a variance of the standardized observation difference \((\Delta \text{l})_{{\rm s},i}\) (i = 1,…,n);

    $$ {\mathbf{C}}_{{{\hat{\mathbf{d}}}}} = \sigma_{\rm o}^{2} {\mathbf{Q}}_{{{\hat{\mathbf{d}}}}} = {\mathbf{Q}}_{{{\hat{\mathbf{d}}}}} $$
    (3)
  2. (ii)

    using the estimator (unbiased) of \(\sigma_{\rm o}^{2}\), i.e., \(\hat{\sigma }_{\rm o}^{2}\), termed the a posteriori variance factor;

    $$ {\hat{\mathbf{C}}}_{{{\hat{\mathbf{d}}}}} = {\hat{\sigma }}_{{\text{o}}}^{{2}} {\mathbf{Q}}_{{{\hat{\mathbf{d}}}}} $$
    (4)
$$ {\text{Ad}}\;\left( {\text{i}} \right)\;\;{\hat{\mathbf{d}}} = \left( {{\mathbf{A}}_{\rm s}^{\text{T}} {\mathbf{A}}_{\rm s} } \right)_{{\mathbf{S}}}^{ - } {\mathbf{A}}_{\rm s}^{\text{T}} (\Delta \mathbf{l})_{\rm s} ;\;{\mathbf{C}}_{{{\hat{\mathbf{d}}}}} = \left( {{\mathbf{A}}_{\rm s}^{\text{T}} {\mathbf{A}}_{\rm s} } \right)_{{\mathbf{S}}}^{ - } \quad {\text{rank}}\;{\mathbf{C}}_{{{\hat{\mathbf{d}}}}} = u - d $$
(5)

for free net constraints \({\mathbf{C}}_{{{\hat{\mathbf{d}}}}} = \left( {{\mathbf{A}}_{\rm s}^{\text{T}} {\mathbf{A}}_{\rm s} } \right)_{{{\mathbf{S}}_{\rm o} }}^{ - } \equiv \left( {{\mathbf{A}}_{\rm s}^{\text{T}} {\mathbf{A}}_{\rm s} } \right)^{ + }\)

$$ {\text{Ad}}\;\left( {{\text{ii}}} \right)\;\;{\hat{\mathbf{d}}} = \left( {{\mathbf{A}}_{\rm s}^{\text{T}} {\mathbf{A}}_{\rm s} } \right)_{{\mathbf{S}}}^{ - } {\mathbf{A}}_{\rm s}^{\text{T}} (\Delta \mathbf{l})_{\rm s} ;\;{\hat{\mathbf{C}}}_{{{\hat{\mathbf{d}}}}} = \hat{\sigma }_{\rm o}^{2} \left( {{\mathbf{A}}_{\rm s}^{\text{T}} {\mathbf{A}}_{\rm s} } \right)_{{\mathbf{S}}}^{ - } ,\;{\text{rank}}\;{\hat{\mathbf{C}}}_{{{\hat{\mathbf{d}}}}} = {\text{rank}}\;{\mathbf{C}}_{{{\hat{\mathbf{d}}}}} $$
(6)

In the above formulas, \(\left( {{\mathbf{A}}_{\rm s}^{\text{T}} {\mathbf{A}}_{\rm s} } \right)_{{\mathbf{S}}}^{ - }\) is a reflexive g-inverse of \({\mathbf{A}}_{\rm s}^{\text{T}} {\mathbf{A}}_{\rm s}\), such that \({\mathbf{S}} \cdot \left( {{\mathbf{A}}_{\rm s}^{\text{T}} {\mathbf{A}}_{\rm s} } \right)_{{\mathbf{S}}}^{ - } = {\mathbf{0}}\), \(\left( {{\mathbf{A}}_{\rm s}^{\text{T}} {\mathbf{A}}_{\rm s} } \right)^{ + }\) is a pseudoinverse.

Also known is another datum definition, termed over-constraint datum (where \({\mathbf{S}}(w \times u)\), \({\text{rank}}\;{\mathbf{S}} = w\), \(w > d\)), fixing some network points. It eliminates the network defect but causes observation distortions. Its use is bounded by the restrictions on mutual displacements of the points being fixed and the types of measured quantities.

2.2 Network sensitivity to displacements

The measures of sensitivity to displacements are derived on the basis of hypotheses testing

$$ H_{\rm o} :\;{\rm E}\left( {{\hat{\mathbf{d}}}} \right) = {\mathbf{0}} $$
$$ H_{\rm a} :\;{\rm E}\left( {{\hat{\mathbf{d}}}} \right) \ne {\mathbf{0}} $$

where \({\hat{\mathbf{d}}}\) is the LS estimator of the vector of displacements as in (5) and (6).

If the covariance matrix \({\mathbf{C}}_{{{\hat{\mathbf{d}}}}}\) as in (3) is used, the test statistic is

$$ \Phi = {\hat{\mathbf{d}}}^{\rm T} {\mathbf{C}}_{{{\hat{\mathbf{d}}}}}^{ + } {\hat{\mathbf{d}}} $$
(7)

  • under \(H_{\rm o}\), \(\Phi\) has a central \( {\chi_{\rm h}^{2}}\) distribution, h is a rank of \({\mathbf{C}}_{{{\hat{\mathbf{d}}}}}\),

  • under \(H_{\rm a}\), \(\Phi\) has a non-central \( {\chi_{\rm h}^{2}}\) distribution, with a non-centrality parameter

    $$ \lambda = {\hat{\mathbf{d}}}^{\rm T} {\mathbf{C}}_{{{\hat{\mathbf{d}}}}}^{ + } {\hat{\mathbf{d}}} $$
    (8)

A vector \({\hat{\mathbf{d}}}\) is considered detectable if \(\lambda > \lambda_{{\rm h},\upalpha_{\rm o},\upbeta_{\rm o}}\), where \(\lambda_{{\rm h},\upalpha_{\rm o},\upbeta_{\rm o}}\) is a critical value of non-centrality parameter determined for the specified values of Type I and Type II error probabilities \(\alpha_{ o}\) and \(\beta_{ o}\).

If the covariance matrix \({\hat{\mathbf{C}}}_{{{\hat{\mathbf{d}}}}}\) as in (4) is used, the test statistic is

$$ \Phi_{\rm F} = \frac{{{\hat{\mathbf{d}}}^{\rm T} {\mathbf{Q}}_{{{\hat{\mathbf{d}}}}}^{ + } {\hat{\mathbf{d}}}}}{{h \cdot \hat{\sigma }_{\rm o}^{2} }} $$
(9)

where \({\mathbf{Q}}_{{{\hat{\mathbf{d}}}}}\) is a cofactor matrix, h is a rank of \({\mathbf{Q}}_{{{\hat{\mathbf{d}}}}}\), \(\hat{\sigma }_{\rm o}^{2}\) is the a posteriori variance factor

  • under \(H_{\rm o}\), \(\Phi_{\rm F}\) has a central \(F_{\rm h,r}\) distribution (r is a model redundancy; r = n − u + d),

  • under \(H_{\rm a}\), \(\Phi_{\rm F}\) has a non-central \(F_{\rm h,r}\) distribution, with a non-centrality parameter

    $$ \lambda_{\rm F} = \frac{{{\hat{\mathbf{d}}}^{\rm T} {\mathbf{Q}}_{{{\hat{\mathbf{d}}}}}^{ + } {\hat{\mathbf{d}}}}}{{\hat{\sigma }_{\rm o}^{2} }} $$
    (10)

A critical value of the non-centrality parameter is denoted here by \(\lambda_{ {\rm h,r}, \upalpha_{\rm o},\upbeta_{\rm o}}\). A vector \({\hat{\mathbf{d}}}\) is considered detectable if \(\lambda_{\rm F} > \lambda_{{\rm h,r},\upalpha_{\rm o},\upbeta_{\rm o}}\), where \(\alpha_{\rm o}\) and \(\beta_{\rm o}\) as above.

For a single point vector with a non-singular covariance matrix, h is a number of the vector components.

On the basis of the formulas (8) and (10), we can write the equations of h—dimensional sensitivity ellipsoids for the case of \({\mathbf{C}}_{{{\hat{\mathbf{d}}}}}\) and \({\hat{\mathbf{C}}}_{{{\hat{\mathbf{d}}}}}\), respectively, i.e.,

$$ {\mathbf{d}}^{\rm T} {\mathbf{C}}_{{{\hat{\mathbf{d}}}}}^{ + } {\mathbf{d}} = \lambda_{{\rm h},\upalpha_{\rm o},\upbeta_{\rm o}} $$
(11)
$$ \frac{{{\mathbf{d}}^{\rm T} {\mathbf{Q}}_{{{\hat{\mathbf{d}}}}}^{ + } {\mathbf{d}}}}{{\hat{\sigma }_{\rm o}^{2} }} = \lambda_{{\rm h,r}, \upalpha_{\rm o},\upbeta_{\rm o}} $$
(12)

Based on (11) and (12), we get the commonly used sensitivity measure as Minimal Detectable Displacement (MDD), correspondingly for the case of \({\mathbf{C}}_{{{\hat{\mathbf{d}}}}}\) and \({\hat{\mathbf{C}}}_{{{\hat{\mathbf{d}}}}}\) (Niemeier 1982, 1985)

$$ \left\| {\mathbf{d}} \right\|_{\min } = \sqrt {\frac{{\lambda_{{ {\rm h},\upalpha_{\rm o} ,\upbeta_{\rm o} }} }}{{\Lambda_{\max } }}} \quad \left\| {\mathbf{d}} \right\|_{\min } = \hat{\sigma }_{\rm o} \sqrt {\frac{{\lambda_{{{\rm h,r},\upalpha_{\rm o} ,\upbeta_{\rm o} }} }}{{\Lambda_{\max } }}} $$
(13)

where \(\Lambda_{\max }\) is a maximal eigenvalue of the matrix \({\mathbf{C}}_{{{\hat{\mathbf{d}}}}}^{ + }\) and \({\mathbf{Q}}_{{{\hat{\mathbf{d}}}}}^{ + }\), \(\hat{\sigma }_{\rm o}\) is a square root of the a posteriori variance factor.

According to (Hsu and Hsiao 2002), both the minimal and the maximal value of MDD should be taken into consideration. The maximal value is based on a minimal eigenvalue \(\Lambda_{\min }\).

Datum invariance of the sensitivity quadratic form as in (7) is an important issue in the subject of the paper since the MDD defined on its basis is applied in different types of datum definition. It is known that the form is invariant to the choice of the type of minimum-constraint datum definition. A concise proof, complementary to the existing proofs (e.g., Even-Tzur 2010), is presented in “Appendix A” section. It covers all possible types of datum definition.

A non-centrality parameter for a test statistic (10) in the case of the model (1) with weights of observation differences is considered in “Appendix B” section.

3 Investigating MDD support in accuracy analysis and in the testing of displacements

In investigations, we operate with h-dimensional displacement vectors (h \(\ge\) 1). For simplicity, the corresponding regions of confidence, significance and sensitivity, being line segments (h = 1), ellipses (h = 2), ellipsoids (h = 3) and hyper-ellipsoids (h > 3), will be termed for all h the ellipsoids, e.g., (Pope 2008).

The proposal covers the following two task cases:

  • Case I Confidence region determination supported by network sensitivity characteristics (variance factor option (i))

  • Case II Significance test of computed displacements supported by network sensitivity characteristics (variance factor option (i) and (ii)).

As a final result, the extent of the support is evaluated.

3.1 Creating of the theoretical basis

3.1.1 Variance factor option (i)

A property of the test statistic \(\Phi = {\hat{\mathbf{d}}}^{\rm T} {\mathbf{C}}_{{{\hat{\mathbf{d}}}}}^{ + } {\hat{\mathbf{d}}}\) under \(H_{\rm o}\) and \(H_{\rm a}\), i.e., \(\Phi\) = \(\lambda\) for h \(\ge\) 1 (see (7), (8)), as expressed in terms of the corresponding values of \(\Phi\) and \(\lambda\) (i.e., u and \(\lambda\), respectively), means the following equality

$$ u_{{\rm h},\upalpha } = \lambda_{{\rm h},\upalpha ,\upbeta } $$
(14)

For a given α, we get β from the above equality. We will say that such β is coordinated with a given α. Assuming an arbitrary value of \(\beta\) not coordinated with a given α and denoted by \(\tilde{\beta }\) will result in \(u_{{\rm h},\upalpha } \ne \lambda_{{{\rm h},\upalpha ,\tilde{\upbeta }}}\).

Let critical values of \(\Phi\) and \(\lambda\) be denoted by \(u_{\rm o}\) and \(\lambda_{\rm o}\), where \(u_{\rm o}\) = \(u_{{\rm h},\upalpha_{\rm o}}\), \(\lambda_{\rm o} = \lambda_{{\rm h},\upalpha_{\rm o},\upbeta_{\rm o}}\); \(\alpha_{\rm o}\) and \(\beta_{\rm o}\) are the values of α and β assumed in \(u_{\rm o}\) and \(\lambda_{\rm o}\), respectively. Putting, e.g., \(\alpha_{\rm o}\) = 0.05 and \(\beta_{\rm o}\) = 0.20, we get from (14) \(u_{\rm o}\) \(\ne \lambda_{\rm o}\), and hence, we can see that \(\beta_{\rm o}\) = 0.20 is not coordinated with \(\alpha_{\rm o}\) = 0.05. The value of \(\beta\) coordinated with \(\alpha_{\rm o}\) = 0.05 will be denoted by \(\overline{\beta }\).

The investigation of MDD support requires knowledge on the relationship between the sizes and probability features of the following three concentric and similar h-dimensonal ellipsoids, such as sensitivity ellipsoid \(\left( {E_{{{\text{sen}}}} } \right)\), confidence ellipsoid \(\left( {E_{{{\text{con}}}} } \right)\) (also termed as a limit-error ellipsoid) and significance ellipsoid \(\left( {E_{{{\text{sig}}}} } \right)\).

Let us consider equations of these ellipsoids, referring to the expression for a test statistic \(\Phi\) as in (7). The components of the vector d will be treated here as algebraic variables.

$$ E_{{{\text{sig}}}} \quad {\mathbf{d}}^{\rm T} {\mathbf{C}}_{{{\hat{\mathbf{d}}}}}^{ + } {\mathbf{d}} = u_{\rm o} ;\;u_{\rm o} = u_{{\rm h},\upalpha_{\rm o}} ;\;\alpha_{\rm o} {-\!\!-}{\text{significance}}\;{\text{level}} $$
(15)

\(u_{\rm o}\)—a critical value of the variable \(\Phi\) such that \(P\left\{ {\Phi > u_{{\rm h},\upalpha_{\rm o}} } \right\} = \alpha_{\rm o}\)

$$ E_{{{\text{con}}}} \;\;{\mathbf{d}}^{\rm T} {\mathbf{C}}_{{{\hat{\mathbf{d}}}}}^{ + } {\mathbf{d}} = u_{{\rm o}(\gamma )} ;\;u_{{\rm o}(\gamma )} = u_{{\rm h},\upgamma_{\rm o}} ;\;\gamma_{\rm o} {-\!\!-}{\text{confidence}}\;{\text{level}} $$
(16)

\(u_{{\rm o}(\upgamma )}\)—a critical value of the variable \(\Phi\) such that \(P\left\{ {\Phi < u_{{\rm h},\upgamma_{\rm o}} } \right\} = \gamma_{\rm o}\)

$$ E_{{{\text{sen}}}} \;\;{\mathbf{d}}^{\rm T} {\mathbf{C}}_{{{\hat{\mathbf{d}}}}}^{ + } {\mathbf{d}} = \lambda_{\rm o} ;\;\lambda_{\rm o} = \lambda_{{\rm h},\upalpha_{\rm o},\upbeta_{\rm o}} \;\left( {{\text{see}}\;\left( {{11}} \right)} \right) $$

\(\lambda_{\rm o}\)—a critical value of a non-centrality parameter of the variable \(\Phi\).

\(E_{{{\text{sen}}}}\) represents a detectability threshold for displacements. It is easy to prove that every vector that satisfies Eq. (11) is an MDD in this direction.

Putting \(\gamma_{\rm o} = 1 - \alpha_{\rm o}\) in (16), we get \(u_{{\rm h},\upgamma_{\rm o}}\) = \(u_{{\rm h},\upalpha_{\rm o}}\) which reduces the number of sizes to be analyzed to two, the corresponding parameters being \(u_{{\rm h},\upalpha_{\rm o}}\) and \(\lambda_{{\rm h},\upalpha_{\rm o},\upbeta_{\rm o}}\).

The values of \(u_{{\rm h},\upalpha_{\rm o}}\) were computed iteratively on the basis of the arguments of cumulative distribution function (CDF) for a \(\chi_{\rm h}^{2}\)—distributed random variable. For the computation of \(\lambda_{{\rm h},\upalpha_{\rm o},\upbeta_{\rm o}}\) use was made of the efficient and sufficiently accurate numerical program developed by Aydin and Demirel (2005) according to the algorithm and the associated theory given in (Gaida and Koch 1985). The results of the computation are presented in Table 1 and Fig. 1.

Table 1 Values of \(u_{\rm o}\) and \(\lambda_{\rm o}\) for different h (\(\alpha_{\rm o}\) = 0.05, \(\beta_{\rm o}\) = 0.20); \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\) = 7.3
Full size table
Fig. 1
figure 1

Parameters \(u_{\rm o}\) and \(\lambda_{\rm o}\) as functions of h; \(\alpha_{\rm o}\) = 0.05, \(\beta_{\rm o}\) = 0.20

Full size image

The symbol \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\) in Table 1 denotes a crossing point of the curves \(u_{\rm o} (h)\) and \(\lambda_{\rm o} (h)\) (see Fig. 1).

We can see in Fig. 1 that with the increase in h, both the curves increase monotonically, the former one at a higher rate.

At a certain value of h, denoted as \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\) (being equal to 7.3), these curves cross each other. For h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\), we have \(u_{\rm o}\) < \(\lambda_{\rm o}\), whereas for h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\) we have \(u_{\rm o}\) > \(\lambda_{\rm o}\). For a given h, the symbol \(\overline{\beta }\) denotes the value of β coordinated with \(\alpha_{\rm o}\) = 0.05. For \(u_{\rm o}\) < \(\lambda_{\rm o}\), we have \(\overline{\beta }\) > \(\beta_{\rm o}\) and for \(u_{\rm o}\) > \(\lambda_{\rm o}\), \(\overline{\beta }\) < \(\beta_{\rm o}\). We can also notice that the values of \(\overline{\beta }\) decrease with the increase in h, which is consistent with the findings in (Aydin 2011) focused on the power of the test.

The properties as presented above for \(\alpha_{\rm o}\) = 0.05, \(\beta_{\rm o}\) = 0.20 can be observed for other values of \(\alpha_{\rm o}\) and \(\beta_{\rm o}\).

For single network points, we have h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\), so the size of \(E_{{{\text{sen}}}}\) is greater than that of \(E_{{{\text{sig}}}}\) and \({\text{E}}_{{{\text{con}}}}\) (with \(\gamma_{\rm o} = 1 - \alpha_{\rm o}\)). For networks taken as a whole or for the subsets of their points, such that h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\), we have the opposite situation.

3.1.2 Variance factor option (ii)

From the formulas (9) and (10), it follows that the property analogous to that in the variance factor option (i), i.e., \(\Phi\) = \(\lambda\), does not hold directly for \(\Phi_{\rm F}\). We can get this effect by introducing a rescaled variable \(\Phi_{\rm F}\), defined by

$$ \Psi = h \cdot \Phi_{\rm F} $$
(17)

Hence,

$$ \Psi = \frac{{{\hat{\mathbf{d}}}^{\rm T} {\mathbf{Q}}_{{{\hat{\mathbf{d}}}}}^{ + } {\hat{\mathbf{d}}}}}{{\hat{\sigma }_{\rm o}^{2} }} $$
(18)

The values of \(\Psi\) will be denoted by “w”.

The variable does not have an identifiable probability distribution but has a property useful for the present derivations, i.e.,

Property

If r \(\to \infty\), the probability distribution of the variable \(\Psi \to \chi_{\rm h}^{2}\) (the property is well known in mathematical statistics).

So, since with r\(\to \infty\) \(\hat{\sigma }_{\rm o}^{2} \to \sigma_{\rm o}^{2}\), the variable \(\Psi \to \frac{{{\hat{\mathbf{d}}}^{\rm T} {\mathbf{Q}}_{{{\hat{\mathbf{d}}}}}^{ + } {\hat{\mathbf{d}}}}}{{\sigma_{\rm o}^{2} }} = {\hat{\mathbf{d}}}^{\rm T} {\mathbf{C}}_{{{\hat{\mathbf{d}}}}}^{ + } {\hat{\mathbf{d}}} \equiv \Phi\).

As was assumed in Sect. 2.1, \(\sigma_{\rm o}^{2}\) = 1.

Now, we find the critical value of \(\Phi\) for a given significance level α. Denoting the values of \(\Phi_{\rm F}\) by “z”, we shall write

$$ P\left\{ {\Phi_{\rm F} > z_{{\rm h,r},\upalpha } } \right\} = \alpha $$

Substituting (17) and duly modifying the internal inequality, we obtain finally

$$ P\left\{ {\Psi > w_{{\rm h,r},\upalpha } } \right\} = \alpha $$
(19)

where \(w_{{\rm h,r},\upalpha } = h \cdot z_{{\rm h,r},\upalpha }\).

The variable \(\Psi\), being compatible with \(\Phi\) as regards properties and scale, satisfies the equality \(\Psi\) = \(\lambda_{\rm F}\) for h \(\ge\) 1 and r \(\ge\) 1. So, the values of \(\Psi\) and \(\lambda_{\rm F}\), i.e., w and \(\lambda_{\rm F}\) are bound by the equality

$$ w_{{\rm h,r},\upalpha } = \lambda_{{\rm h,r},\upalpha ,\upbeta } $$
(20)

In analogy to (14), \(\beta\) is coordinated with a given value of \(\alpha\).

Let critical values of \(\Psi\) and \(\lambda_{\rm F}\) be denoted by \(w_{\rm o}\) and \(\lambda_{{\rm F,o}}\), where \(w_{\rm o} = w_{{\rm h,r},\upalpha_{\rm o}}\), \(\lambda_{{\rm F,o}} = \lambda_{{\rm h,r},\upalpha_{\rm o},\upbeta_{\rm o}}\) (\(\alpha_{\rm o}\) = 0.05, \(\beta_{\rm o}\) = 0.20). We may check that \(w_{\rm o}\) \(\ne \lambda_{{\rm F,o}}\), and hence, \(\beta_{\rm o}\) = 0.20 is not coordinated with \(\alpha_{\rm o}\) = 0.05. The value of \(\beta\) coordinated with \(\alpha_{\rm o}\) = 0.05 will be denoted by \(\overline{\beta }\) (for simplicity of notation, the same symbol as for variance factor option (i) is used).

For Case II (variance factor option (ii)), we need to acquire knowledge on the relationship between the sizes of the corresponding significance ellipsoid and the sensitivity ellipsoid. We shall denote these ellipsoids by \(\hat{E}_{{{\text{sig}}}}\) and \(\hat{E}_{{{\text{sen}}}}\), respectively. We write their equations without additional explanations

$$ \hat{E}_{{{\text{sig}}}} \;\frac{{{\mathbf{d}}^{\rm T} {\mathbf{Q}}_{{{\hat{\mathbf{d}}}}}^{ + } {\mathbf{d}}}}{{\hat{\sigma }_{\rm o}^{2} }} = w_{\rm o} ;\quad w_{\rm o} = w_{{\rm h,r},\upalpha_{\rm o}} $$
(21)
$$ \hat{E}_{{{\text{sen}}}} \;\frac{{{\mathbf{d}}^{\rm T} {\mathbf{Q}}_{{{\hat{\mathbf{d}}}}}^{ + } {\mathbf{d}}}}{{\hat{\sigma }_{\rm o}^{2} }} = \lambda_{{\rm F,o}} ;\;\;\lambda_{{\rm F,o}} = \lambda_{{\rm h,r},\upalpha_{\rm o},\upbeta_{\rm o}} $$
(22)

To the above-mentioned property of \(\Psi\) concerned with r increasing to infinity, we may add the following

  • since with r \(\to \infty\) \(w_{{\rm h,r},\upalpha } \to u_{{\rm h},\upalpha }\) and \(\lambda_{{\rm h,r},\upalpha ,\upbeta } \to \lambda_{{\rm h},\upalpha ,\upbeta }\), so \(\hat{E}_{{{\text{sig}}}} \to E_{{{\text{sig}}}}\) and \(\hat{E}_{{{\text{sen}}}} \to E_{{{\text{sen}}}}\).

The values \(w_{\rm o}\) were obtained by rescaling the values \(z_{{\rm h,r},\upalpha_{\rm o}}\), computed iteratively based on the arguments of CDF for \(F_{{\rm h,r}}\)—distributed variable. Like the values of \(\lambda_{\rm o}\) in option (i), the values of \(\lambda_{{\rm F,o}}\) were computed with the use of a numerical program developed by Aydin (2011). The results of the computation are presented in Table 2 and Fig. 2.

Table 2 Values of \(w_{\rm o}\) and \(\lambda_{{\rm F,o}}\) for different h and r (\(\alpha_{\rm o}\) = 0.05, \(\beta_{\rm o}\) = 0.20); \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\)
Full size table
Fig. 2
figure 2

Parameters \(w_{\rm o}\) and \(\lambda_{{\rm F,o}}\) as functions of h, for r = 10 and r = 50 (\(\alpha_{\rm o}\) = 0.05, \(\beta_{\rm o}\) = 0.20)

Full size image

The symbol \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\) in Table 2 denotes a crossing point of the curves \(w_{\rm o} (h)\) and \(\lambda_{{\rm F,o}} (h)\) for some r (see Fig. 2).

Figure 2 presents the curves \(w_{\rm o} (h)\) and \(\lambda_{{\rm F,o}} (h)\) for r = 10 and r = 50. The curves are generally similar to their equivalents in Fig. 1. We can see in Table 2 and Fig. 2 that the values of h where these curves intersect (denoted here as \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\)) increase with the decrease in r. For r = 10, the intersection of the curves falls outside the diagram.

Interpretation of the cases h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\) and h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\) is like that given for option (i). The values of \(\overline{\beta }\) decrease with the increase in h.

3.2 Analysis for Case I

3.2.1 Variance factor option (i)

Usually, both the confidence ellipse and the sensitivity ellipse are presented together as a result of network quality analysis. We provide the description of characteristic displacement vectors and some probabilistic features of each of the ellipses (here—ellipsoids).

The values of \(\overline{\beta }\) in Fig. 3 are computed for h = 2 (h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\)), h = 15 (h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\)).

Fig. 3
figure 3

Presentation of ellipsoids for analysis in Case I (1, 2, 3—vectors of characteristic location, \(\gamma_{\rm o} = 1 - \alpha_{\rm o}\))

Full size image

Since we consider the a priori accuracy analysis, the vectors 1, 2, 3 represent potential (not determined) displacement vectors.

Deterministic description of the characteristic vectors:

for h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\)

for h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\)

1. error-affected displacements displ. under detectability threshold

1. displ. under detectability threshold error-affected displacements

2. error-free displacements displ. under detectability threshold

2. detectable displacements error-affected displacements

3. error-free displacements detectable displacements

3. error-free displacements detectable displacements

For a zone between the ellipsoids (see vector 2), we observe some inconsistency in the above description. It is due to the fact that the confidence related and the sensitivity related qualifications concern different assumptions (i.e., hypothesis \(H_{\rm o}\) and hypothesis \(H_{\rm a}\)), with Type I and Type II error probabilities not mutually coordinated. The detectability threshold \(\lambda_{\rm o}\) is different from the error-limit \(u_{{\rm o}(\gamma )}\).

3.3 Analysis for Case II

3.3.1 Variance factor option (i)

For the analysis of significance combined with sensitivity, we present the significance ellipsoid and the sensitivity ellipsoid together with the description of characteristic displacement vectors and some probabilistic features of each of the ellipsoids.

The values of \(\overline{\beta }\) in Fig. 4 are computed for h = 2 (h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\)), h = 15 (h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\)).

Fig. 4
figure 4

Presentation of ellipsoids for analysis in Case II, option (i) (1, 2, 3—vectors of characteristic location)

Full size image

The deterministic description is as follows

for h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\)

for h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\)

1. insignificant displacement

1. displ. under detectability threshold

2. significant displacement displ. under detectability threshold

2. detectable displacement insignificant displacement

3. significant displacement detectable displacement

3. detectable displacement significant displacement

Like in Case I, for a zone between the ellipsoids (see vector 2), we observe some inconsistency in the above description. The cause of inconsistency is as that explained in Case I. Here, the detectability threshold \(\lambda_{\rm o}\) is different from the significance threshold \(u_{\rm o}\).

We may readily classify the computed h-dimensional displacement vector \(\hat{\bf{d}}{\kern 1pt}\) either as 1, 2 or 3, respectively, by means of the following checks based on the values u of the test statistic \(\Phi\), i.e.,

$$ {\text{for}}\;h < h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*} :u \le u_{\rm o} ,\;u_{\rm o} < u \le \lambda_{\rm o} ,\;u > \lambda_{\rm o} $$
(23)
$$ {\text{for}}\;h > h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*} :\;u \le \lambda_{\rm o} ,\;\lambda_{\rm o} < u \le u_{\rm o} ,\;u > u_{\rm o} $$
(24)

3.3.2 Variance factor option (ii)

Since the deterministic description of the displacement vectors is analogous to that in option (i), we only present both the ellipsoids together with their probabilistic features (Fig. 5).

Fig. 5
figure 5

Presentation of ellipsoids for analysis in Case II, option (ii) (1, 2, 3—vectors of characteristic location)

Full size image

To classify the computed h-dimensional displacement vector \(\hat{\bf{d}}{\kern 1pt}\) either as 1, 2 or 3, respectively, we use here the following checks based on the values w of the variable \(\Psi\) and \({\uplambda }_{{\text{F}}}\) as in (21) and (22)

$$ {\text{for}}\;h < h_{\upalpha_{\rm o},\upbeta_{\rm o},r}^{*} :\;w \le w_{\rm o} ,\;w_{\rm o} < w \le \lambda_{{\rm F,o}} ,\;w > \lambda_{{\rm F,o}} $$
(25)
$$ {\text{for}}\;h > h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*} :\;w \le \lambda_{{\rm F,o}} ,\;w_{{\rm f,r},\uplambda} < w < w_{{\rm f,r},\upalpha} ,\;w > w_{\rm o} $$
(26)

For this variance factor option, the greater the system redundancy, the lower is the detectability threshold, and the smaller are the displacement vectors that can be considered as insignificant. It is therefore recommended to always ensure an appropriately high redundancy level.

The values of \(\overline{\beta }\) in Fig. 5 are computed for r = 20 and h = 2 (h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\)), h = 20 (h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\)).

3.4 Evaluating the MDD support in task Cases I and II

The MDD support is understood here as providing grounds for verifying the correctness of the significance (or confidence) threshold. The support is the greater the closer is the sensitivity ellipsoid to significance (or confidence) ellipsoid.

The value of \(\overline{\beta }\) obtained from the equality (14) with \(\alpha_{\rm o}\) = 0.05 (or \(\gamma_{\rm o} = 1 - \alpha_{\rm o}\)) depends except h only on the value of \(\alpha_{\rm o}\). Hence, \(\overline{\beta }\) is only partially connected with the detectability threshold \(\lambda_{\rm o}\), which is defined on both \(\alpha_{\rm o}\) and \(\beta_{\rm o}\). The difference \(\left| {\lambda_{\rm o} - u_{\rm o} } \right| \ne 0\) is a main cause of inconsistency in the description of vector qualification discussed in Sect. 3.2 (Fig. 3) and Sect. 3.3 (Figs. 4 and 5). The smaller is \(\overline{\beta }\), and the smaller is \(\left| {\lambda_{\rm o} - u_{\rm o} } \right|\), the greater is MDD support. The analysis of the above-mentioned relationships led to introducing of the following directly-proportional measure of MDD support, defined by

$$ {\text{for}}\;h < h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*} \quad Q\left( {\underline{\beta } ,\frac{{u_{\rm o} }}{{\lambda_{\rm o} }}} \right) $$
(27)
$$ {\text{for}}\;h > h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*} \quad Q\left( {\underline{\beta } ,\frac{{\lambda_{\rm o} }}{{u_{\rm o} }}} \right) $$
(28)

where \(\underline{\beta }\) = 1 − \(\overline{\beta }\) is a power of the test for \(u_{\rm o} = \lambda_{\rm o}\), the second components in Q are equal to 1 for h = \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\).

Since both the parameters in Q are mutually incomparable, in the case of opposite trends in their functions of h, ranking of Q will not be possible. Therefore, the ranking of the MDD support will then have to be carried out with respect to each of the parameters separately.

The measure of MDD support as in (27) and (28) is constructed on the basis of parameters used in Table 1 and diagrams in Fig. 1. Hence, variability of Q values is some other representation of the above-mentioned diagrams.

The values of Q for some h are presented in Table 3 and Fig. 6.

Table 3 Values of Q for some h (\(\alpha_{\rm o}\) = 0.05, \(\beta_{\rm o}\) = 0.20); italics concerns h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\)
Full size table
Fig. 6
figure 6

Plot of Q values for some h (\(\alpha_{\rm o}\) = 0.05, \(\beta_{\rm o}\) = 0.20); Q points are labeled with the values of h; h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\) in blue, h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\) in red

Full size image

The results in Table 3 and Fig. 6 apply to Task Case I and Task Case II (variance factor option (i)). We may conclude that the MDD support is the smallest for single point displacements in 1D networks (\(\underline{\beta }\) = 0.5, \(\frac{{u_{\rm o} }}{{\lambda_{\rm o} }}\) = 0.49) and becomes successively greater for 2D (\(\underline{\beta }\) = 0.58, \(\frac{{u_{\rm o} }}{{\lambda_{\rm o} }}\) = 0.62) and 3D networks (\(\underline{\beta }\) = 0.64, \(\frac{{u_{\rm o} }}{{\lambda_{\rm o} }}\) = 0.72). For h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\) (i.e., for global network displacements), in terms of \(\underline{\beta }\) the support is increasing, whereas in terms of \(\frac{{\lambda_{\rm o} }}{{u_{\rm o} }}\) it is decreasing. For h = 50, \(\underline{\beta }\) is approaching 1 while \(\frac{{u_{\rm o} }}{{\lambda_{\rm o} }}\) falls down to 0.45.

In analogy to the definition in (27) and (28), the measure of MDD support for Task Case II (variance factor option (ii)) is defined as follows

$$ {\text{for}}\;h < \;h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*} \quad Q\left( {\underline{\beta } ,\frac{{w_{\rm o} }}{{\lambda_{{\rm F,o}} }}} \right) $$
(29)
$$ {\text{for}}\;h > \;h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*} \quad Q \, \left( {\underline{\beta } ,\frac{{\lambda_{{\rm F,o}} }}{{w_{\rm o} }}} \right) $$
(30)

where \(\underline{\beta }\) = 1 − \(\overline{\beta }\) is a power of the test for \(\lambda_{\rm F} = w_{\rm o}\), \(\overline{\beta }\) is obtained from the equality (20), i.e., \(w_{{\rm h,r},\upalpha_{\rm o}}\) = \(\lambda_{{{\rm h,r},\upalpha_{\rm o},\overline{\upbeta }}}\), the second components in Q are equal to 1 for h = \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\). The MDD support is presented in Table 4 and Fig. 7.

Table 4 Values of Q for some h and r = 20 (\(\alpha_{\rm o}\) = 0.05, \(\beta_{\rm o}\) = 0.20); italics concerns h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\)
Full size table
Fig. 7
figure 7

Plot of Q values for some h and r = 20 (\(\alpha_{\rm o}\) = 0.05, \(\beta_{\rm o}\) = 0.20); Q points are labeled with the values of h; h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\) in blue, h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\) in red

Full size image

Like in Fig. 6, the MDD support is the smallest for single point displacements in 1D networks and becomes successively greater for 2D and 3D networks. And also, for h > \(h_{\upalpha ,\upbeta ,{\rm r}}^{*}\) (i.e., for global network displacements), in terms of \(\underline{\beta }\) the support is increasing, whereas in terms of \(\frac{{\lambda_{\rm o} }}{{u_{\rm o} }}\) it is decreasing.

According to the property presented in Sect. 3.1 (variance factor option (ii)), with the increase in r, the MDD support becomes closer and closer to that in Fig. 6 for Task Case II (variance factor option (i)).

The diagrams in Figs. 6 and 7 confirm the theory in Sect. 3.1 in that the MDD support for both the variance factor options increases up to the intersection point (\(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\) or \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\)), and behind this point it increases in terms of the first parameter but decreases in terms of the second parameter. Since the smaller is r (variance factor option (ii)), the intersection point falls for greater h, a zone of the increasing MDD support becomes longer. This effect was observed on a diagram for r = 10 where \(h_{\upalpha_{\rm o},\upbeta_{\rm o},{\rm r}}^{*}\) = 49.9, not shown in the present paper.

3.5 Investigating the possibility of thresholds modification

The question now arises whether the MDD support gives grounds for advantageous modification of the thresholds in the significance test of displacements (or confidence region determination) and detectability for h = 1, 2, 3. The following two possibilities can be taken into account:

  • lowering the detectability threshold while keeping the sensitivity (or confidence) threshold fixed, i.e., \(E_{{{\text{sen}}}}\) is brought toward \(E_{{{\text{sig}}}}\) in terms of size,

  • raising the sensitivity (or confidence) threshold while keeping the detectability threshold fixed, i.e., \(E_{{{\text{sig}}}}\) is brought toward \(E_{{{\text{sen}}}}\) in terms of size.

From the analyses carried out for the first possibility, it follows that such a modification would result in an unacceptable increase in the probability of the Type II error. Maximum values of this probability, expressed in terms of the power of the test \(\underline{\beta }\) are presented in Table 3 and Fig. 4. So, we pass to the second possibility.

Below, we present the analysis for two reasonable options of modifying the significance threshold for both the variance options, i.e., (i) and (ii). In Option 1, we assume \(u_{{\rm o,m}}\) = \(0.5(u_{\rm o} + \lambda_{\rm o} )\), while in Option 2 \(u_{{\rm o,m}}\) = \(\lambda_{\rm o}\), where the subscript “m” denotes the modified quantity. In Option 1, \(E_{{{\text{sig,}}{\rm m}}}\) lies in between \(E_{{{\text{sig}}}}\) and \(E_{{{\text{sen}}}}\), while in Option 2 \(E_{{{\text{sig}},{\rm m}}} \equiv E_{{{\text{sen}}}}\).

The data for the analysis are presented in Tables 5 and 6. The symbol \(\overline{\beta }_{m}\) denotes \(\beta\) coordinated with \(\alpha_{{\rm o,m}}\), and as in Sect. 3.1 (variance factor option (i)), the symbol \(\overline{\beta }\) denotes \(\beta\) coordinated with \(\alpha_{\rm o}\) (here \(\alpha_{\rm o}\) = 0.05).

Table 5 Data for Option 1; variance factor option (i)
Full size table
Table 6 Data for Option 2; variance factor option (i)
Full size table

Ad option 1 (variance factor option (i))

For h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\), the Type I error probability \(\alpha\) is decreased from the level of 0.05 to an average level of 0.02 (for h = 1, 2, 3), whereas the coordinated Type II error probability \(\beta\) displays a slight increase of about 0.01 from the average level of about 0.43. For h > \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\), the Type I error probability \(\alpha\) is increased over the level of 0.05, whereas the coordinated Type II error probability \(\beta\) is decreased below 0.20.

Ad option 2 (variance factor option (i))

The changes, being analogous to those in Option 1, are greater. For h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\), the probability \(\alpha\) is decreased to an average level of 0.01, while the coordinated Type II error probability \(\beta\) displays a slight increase of about 0.02.

Based on the above results, it seems reasonable to apply modification only for h < \(h_{\upalpha_{\rm o},\upbeta_{\rm o}}^{*}\), and thus for single point displacements. Either of the Options, as advantageous in terms of testing errors probabilities, is acceptable. Instead of the diversified level of \(\alpha_{{\rm o,m}}\) for different h, the averaged common values 0.02 and 0.01 can be recommended for Option 1 and Option 2, respectively. However, since it was found in computations that for h = 3 in Option 2 (with \(\alpha_{{\rm o,m}}\) = 0.01) \(E_{{{\text{sig,}}{\rm m}}}\) lies slightly outside \(E_{{{\text{sen}}}}\), the significance level \(\alpha_{{\rm o,m}}\) = 0.015 is proposed for Option 2. With this level \(E_{{{\text{sig,}}{\rm m}}}\) lies inside \(E_{{{\text{sen}}}}\), i.e., \(u_{{\rm o,m}}\) < \(\lambda_{\rm o}\).

It was verified that the above conclusions also apply to the variance factor option (ii) the more, the greater the r. The computations showed that for h = 3 in Option 2 with \(\alpha_{{\rm o,m}}\) = 0.015 \(\hat{E}_{{{\text{sig}},{\rm m}}}\) lies inside \(\hat{E}_{{{\text{sen}}}}\) (i.e., \(w_{{\rm o,m}}\) < \(\lambda_{{\rm F,o}}\)) for r \(\ge\) 23. For h = 2 and h = 1, we have r \(\ge\) 10 and r \(\ge\) 7, respectively.

If the above-mentioned restrictions on the value of r are acceptable, the significance level \(\alpha_{{\rm o,m}}\) = 0.015 can be proposed for both options (i) and (ii) in Option 2. Otherwise, the value of \(\alpha_{{\rm o,m}}\) for option (ii) should be increased accordingly. The data for variance factor option (i) are presented in Table 7.

Table 7 Data for Option 2, variance factor option (i), \(\alpha_{{\rm o,m}}\) = 0.015
Full size table

A similar modification can be carried out for confidence area determination (variance factor option (i)). The modified confidence level \(\gamma_{{\rm o,m}}\) would be 0.98 for Option 1 and 0.985 for Option 2.

The choice of Options for both Task Cases I and II can be made depending on the type of practical applications.

The determined measures of MDD support and the proposed threshold modifications have a general character and are valid for any realization of Task Case I or Task Case II. Therefore, enclosing a numerical example did not seem to be necessary.

4 Concluding remarks

Theoretical findings of the research can be summarized as follows:

  • determining Type II error probability \(\overline{\beta }\) corresponding to Type I error probability \(\alpha_{\rm o}\),

  • introducing a rescaled Fisher-distributed test statistic which made it possible to construct a significance ellipsoid comparable with the sensitivity ellipsoid for the case when the a posteriori variance factor is used,

  • proposing a 2-parameter measure Q of MDD support in the accuracy analysis and significance test of displacements.

The computations show that the MDD support both in confidence region determination and significance test of single point displacements is the smallest for 1D networks. It increases successively for 2D and 3D networks. For global network displacements, with the increase in h, the MDD support increases with regard to \(\underline{\beta }\) much more than for single point displacements but decreases with regard to the second parameter in Q. Based on the analysis of MDD support, two options of modifying the significance and confidence thresholds for single point displacements are proposed. The modified thresholds being closer to the detectability threshold, correspond to significance level \(\alpha\) = 0.02 or \(\alpha\) = 0.015 and the confidence level γ = 0.98 or γ = 0.985. The advantage of these modifications is that the proposed decrease in the significance level \(\alpha\) (or increase in confidence level γ) results in a negligibly small increase in the Type II error probability β.

In future research, it is planned to consider other types of displacement monitoring models and also, based on publications such as, e.g., (Xu et al. 2000) to examine the possibility of extending the findings of the present paper onto geometrical deformation approach.