The procedure presented in the previous Section is stated in a general form, and applies to any type of monitoring problem involving structural and environmental measurements. Below, we explain in more detail how this procedure applies to the Portogruaro Civic Tower. In this case, the problem is to understand as soon as possible if the Tower is still tilting. Using the formal approach introduced in the previous Section, two scenarios are possible: according to the first scenario, S1, the Tower inclination basically does not change with time, any shift from the mean position being due to daily and annual temperature changes; according to the second, S2, the Tower tilt is increasing, with a trend we can assume linear.
In scenario S1 the compensated inclination px(t) of the tower in xdirection (i.e. east-west) is constant and equal to Vo,x. Conversely, in scenario S2 the compensated inclination is a linear function trend Wj and an offset Vo,x. In equations, we can write for direction x (i.e. West-East)
and similarly for direction y (i.e. North-South). Response measurements are in this case the out of line components, x and y, measured at the base of the pendulum while vector 0 collects the 4 thermocouple measurements.
According to Eq.(1), we can see the out of line measured by the pendulum as the sum of a term depending on tower inclination, a term linearly correlated with the temperatures recorded and a noise:
where H = 29.90 m is the reference level for the out of line measurement, ax is the linear transformation that correlates the temperatures to the out of plumb, while indices 1 or 2 indicate that this vector generally assumes different values in different scenarios. As stated in the equation, the linear relation between temperature and wire measurements is independent of the scenario, and vectors aj, are considered as nuisance parameters, that is, to be marginalized to gain information on the relevant parameters Wx and Wy. Random noise e is assumed to be Gaussian with zero mean and standard deviation n(ffe)j equal to 10 mm for every sensor and scenario: this value, much larger than that strictly related to the instrumental noise, also takes into account the uncertainties of the correlation model.
It is worth noting that the two scenarios have differing degrees of complexity: scenario S2 involves parameters Wx and Wy, so it can follow the measurements more closely. Furthermore, scenario S1 can be regarded as a special sub-case of scenario S2, when wx and Wy are null. The reader might argue that, because of this, the probability of scenario S2 will always be greater than that of S1. Actually, this is not necessarily the case. Indeed, by tuning the parameters of scenarioS2, we can always obtain a better fit than that related to scenario Sj. However, according to Bayesian logic, the ratio between the best and the average fit (the so-called Ockham factor) plays a fundamental role too (Gregory 2005).
To implement the Bayesian updating procedure, we have to define quantitatively the prior knowledge. As mentioned, in October 2003, the probability of the Tower tilting further was reputed very low: we can formalize this initial perception assuming for the tilting scenario a prior probability equal to prob(S2) = 1/1000, and therefore prob(Si) = 99.9% for the no-trend scenario. We must also define the prior estimate of distribution of trends wx and wy. Based on the limited documentation available in October 2003, the only hint is that at the time of construction, say XIV century, the body Tower was reasonably straight. Compared to Busetto's measurements, carried out in 2001, and assuming a linear trend, we can roughly estimate average shifts of -1.5 mm-year-1 in both directions, corresponding to inclination trends of -40.8 x 10-6rd-year-1. In summary, given the scarce information available at that time, this was assumed as the most likely value of current inclination trend, if any. Of course, this information is very imprecise: to account for this uncertainty, we can assume a prior scatter of, say, 4 mm-year-1, corresponding to an angular trend of 108.9 x 10-6rd.year-1. Note that this large scatter simply reflects our almost total prior ignorance of the trends: the specific value selected is not critical, in view of the final result of the updating process.
Based on this prior information (i.e., without considering the historical information discovered in 2005) we can apply Eq. (7) to update our knowledge using the data acquired real-time by the monitoring system. The light plot on Figure 4e shows how the monitoring data modifies the system perception of having a trend. We can see that during the first two years of monitoring the probability of the trend scenario is always close to zero. Only during the third year the monitoring information starts overturning the initial perception, to the point that in April 2006 the data is sufficient to convince the system that the Tower is tilting.
Similarly, the light plots of Figures 4c and 4d show the evolution of the distributions of trends wx and wy: we see that the trend estimates, which are very uncertain during the first two years, rapidly converge to more reliable values.
The documentation acquired in 2005 radically changed the initial judgment on the stability of the Tower. Anyone with common sense can understand that this new information supports the idea that the Tower is tilting. However, the problem is how to quantify the impact of this information on the probability of there being further tilt.
The approach we followed is to cluster the historical information in a number of separate datasets, each associated with a specific distribution, and use recursively the same Eq. (7) to update the likelihood of Scenario 2. The first dataset identified (labelled A) consists in the three out of plumb measurements taken on years 1962, 1997 and 2001. In these cases the measurement techniques adopted suggest we consider a noise of 20 mm. A second set (B) includes the comparison of inclination measurements of the spire taken in 1962 and 2001; in this case a scatter of 50 mm is assumed. The third data set (C) is basically the
notion that the spire was rebuilt vertical in 1879; this conjecture is assumed true with a likelihood of 50%; if true, the standard deviation of the original inclination is assumed equal to 8 x 10-3rd.
Table 2 summarizes the outcomes of the prior information update using recursively the three historical data sets. With respect to the initial judgment, prob(52) increases up to 3.7% when all the historical datasets are available, while the standard deviations of the angular trend decrease to 17.3 x 10-6rd year-1.
Mixing this new prior knowledge with the monitoring data, we obtain, day by day, the posterior trend distributions and the probability of scenario S2, plotted in bold in Figures 4c, 4d and 4e.
In detail, Figure 4d illustrates that, considering the historical data, the system would theoretically become
wx [rd year 1 ]
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