Statistical analysis

For the statistical analysis, we follow the modelling framework provided by the DBN model. Therein, unit time is the time for excavation of a segment with length . It is dependent on the construction method (i.e. on combination of ground class and geometry). For given construction method, the unit time is a stationary random process.

Following the DBN model framework, the unit time is furthermore dependent on the human factor. It is recalled that the human factor represents the deviation of the actual performance from the estimated performance. By definition, the human factor is constant throughout one tunnel. Based on the data alone, i.e. without knowledge of the original estimate, it is not possible to determine the human factor. Therefore, the dependence on human factor is not explicitly included in the data analysis.

In the analysis, a segment length was selected. Because the records were not made at the borders of the segments, the unit time observed in th segment of the tunnel, denoted as ̂, was calculated by linear interpolation of the observed data as illustrated in Figure 7.3.

Figure 7.3: Determining the observed unit times ̂ from the data with linear interpolation. The variability of the observed unit time ̂ per 5 m in different locations along the 1st _{tube of the}

tunnel TUN1, after excluding the extraordinary events, is depicted in Figure 7.4. The graphs of the observed unit time in other analysed tunnels are presented in Annex 6.

Figure 7.4: Observed unit time ̂per 5 m in different positions of the tube 1 of the tunnel TUN1 after excluding the extraordinary events.

The probabilistic model of the unit time must include both the normal performance (1) and small disturbances (2). Given normal performance, the unit time is described by a probability density function (PDF) . Given small disturbances it is described by a PDF . Furthermore, let be the probability of normal performance in a segment; consequently the probability of small disturbances is . Following the total probability theorem, the PDF of the unit time including both (1) and (2) is:

| |

(7.11)

From the data one cannot clearly distinguish between the normal performance and small disturbances. If this was possible, we could calculate the probability as the share of normal performance on the whole sample and fit common probabilistic models, and , to the classified data. To avoid this manual classification, we use a probabilistic approach and we fit directly a combined probabilistic model:

(7.12) where is here modelled as a lognormal PDF with parameters and as a beta PDF with parameters , bounded from 0 to 15 days. The parameters are estimated by means of the maximum likelihood method (see Section 6.1.2).

The left bounded lognormal distribution describes well the normal performance, which has mean close to zero, relatively small variance and is slightly skewed. The beta distribution is suitable for the small disturbances, which have much higher variance and following the definition in our model framework are bounded between 0 and 15 days. However, the model of Eq. (7.12) is also valid with other distribution types for and .

Example PFDs and CDFs for tunnel TUN1 and ground class 5 are shown in Figure 7.5. The distributions of unit time for other ground classes and analysed tunnels are presented in Annex 6.

Figure 7.5: Fitted PDFs and CDFs of unit time per 5 m for tunnel TUN1, ground class 5, for different excavation sequencing.

The means and standard deviations of the unit time for particular construction methods calculated directly from data are summarized in the first part of Table 7.3. The second and third part of Table 7.3 show the means and standard deviations of the two components of the unit time: the normal performance described by and small disturbances described by . These values are determined from the fitted distributions. The fourth part of the table shows the probability of normal performance and the last part of the table summarizes the number of tunnel segments with length , where the construction method was used (i.e. the sample size). Table 7.3: Statistical estimation of unit time per 5 m of the tunnel tube in [days] - summary of mean values and coefficients of variation (in brackets) for different ground classes, cross-section areas and excavation sequencing

Tunnel TUN1 TUN2 TUN3

Sequencing Horizontl Vertical Horizontl Vertical Full face Area Ground class ~60 m2 _{~30 m}2 _{~85 m}2 _{13 m}2 _{37 m}2 _{43 m}2 _{46 m}2 Both component s 3 1.4 (0.5) 1.9 (1.1) 1.9 (1.1) - 1.5 (0.4) 1.9 (0.5) 0.9 (0.1) 4 1.4 (0.5) 2.0 (1.2) 2.0 (0.2) 3.2 (0.6) 1.6 (0.1) 2.4 (1) 2.1 (0.7) 5 2.6 (0.5) 3.5 (0.5) - 3.7 (0.5) 2.1 (0.3) - 3.0 (1.0) Normal perform. 3 1.2 (0.2) 1.5 (0.2) 1.5 (0.2) - 1.4 (0.3) 1.4 (0.1) 1.0 (0.2) 4 1.3 (0.2) 1.5 (0.2) 2.0 (0.2) 2.9 (0.6) 1.6 (0.1) 1.6 (0.1) 1.5 (0.2) 5 2.3 (0.3) 3.3 (0.4) - 3.1 (0.3) 1.6 (0.0) - 1.2 (0.1) Small disturbanc es 3 3.2 (0.6) 11.3 (0.1) 12.3 (0.1) - 3.2 (0.3) 3.5 (0.3) 1.0 (0.6) 4 3.2 (0.5) 12.0 (0.4) 2.3 (0.4) 4.3 (0.25) 1.8 (0.1) 7.5 (0.5) 5.2 (0.2) 5 5.8 (0.3) 6.8 (0.4) - 8.1 (0.2) 2.7 (0.0) - 8.3 (0.2) Prob. of normal perf. ( ) 3 0.93 0.95 0.95 - 0.93 0.75 0.95 4 0.95 0.95 0.95 0.79 0.95 0.87 0.83 5 0.93 0.95 - 0.87 0.55 - 0.75 Sample size 3 ₄ 245 ₁₃₈ 28 ₄₅ 28 _{9 286} 0 19 _{3 47} 4 8 ₆ 5 88 95 0 41 4 0 4

The performance of the excavation in TUN1 and TUN3 is relatively similar. Even if the total cross- section area of the two tunnels is different, the leading tunnel heading allows utilization of high- performance machinery. A difference can be observed in the tunnel TUN2, where the excavation is significantly slower (i.e. the mean unit time is higher). TUN2 is excavated in very difficult geotechnical conditions requiring complicated excavation sequencing and support measures. The leading tunnel heading has only 13 m2, the utilized machinery is therefore not very efficient and the support measures are demanding.

The coefficient of variation (c.o.v) for the normal performance is in most cases in the range of 0.1 – 0.3. A higher c.o.v. can be observed in tunnel TUN2 and in case of vertical sequencing in

tunnel TUN1. This indicates that for demanding excavation technologies the variability of the performance is increased.

Excluding the cases for which the sample size is insufficient, we can conclude that the probability of normal performance, as expressed by the parameter , is in the order of 0.8 to 0.95. It is noted that the estimated values of correspond well to the observed frequencies of time between records shown in Figure 7.2.

For several construction methods, the data basis is not sufficient for reliably estimating the parameters of . Nevertheless, the analysis shows that the small disturbances can explain the difference between the c.o.v. of the observed unit time and the c.o.v. of the normal performance. The latter is the value that most experts would estimate.