The information content in threshold sensor data is significantly less than that in direct concentration measurements. Nevertheless, the sensor system can successfully reconstruct the source, at least in some circumstances. I demonstrate this with an example in which the concentration data have been converted to threshold data using a threshold level of 0.023, a sensor response time of 20 seconds, and without added error. The likelihood function was evaluated assuming a 5% error rate. I judge the sensor system performance by its ability to reduce the uncertainty of the probabilistic outcomes that describe the release location, mass, and duration parameters, and by the time required to do so.
Figure 4.3 depicts the time required to identify the release location. In this chapter, localization to either the intake of the return duct or to the room containing the return, Room 1.2a, is considered equally acceptable. At time zero, each zone is assumed to be equally likely as the release location. As sensor data arrive, the Bayes algorithm adjusts these probabilities, identifying the correct release location with greater than 90% probability within one minute. If rapid response hinges on locating a source quickly, this example suggests that threshold sensors under this network configuration and data quality might be acceptable for real-time monitoring.
Figure 4.3. Probability of source being in location indicated using threshold data.
Sensor characteristics: response time of 20 s, threshold level of 0.023, and without added error. The actual release location is Room 1.2a. Time is referenced to the instantaneous release event.
Figure 4.4 shows the time-resolved estimates of the release mass and duration parameters. These parameters are accurately estimated within tight uncertainty limits after ten minutes. Rather than emphasizing the accuracy at which these parameters are estimated, a more important observation is that threshold type data are capable of reducing the uncertainties that describe the release and thus, this type of sensor may contain information that is relevant to a system-level analysis and sensor-system objective.
Figure 4.4. Release mass and release duration estimated using threshold data.
Sensor characteristics: response time of 20 s, threshold level of 0.023, and without added error. The solid lines indicate 80 percent confidence intervals; dashed lines indicate medians. (Actual release mass was 20 g, released as an instantaneous puff.)
In addition to estimating key release parameters, it is useful to know the past and future uncertainty distibutions of concentrations in different zones. When the sensor signals are alarm type, rather than concentration signals, one may expect that the uncertainty descriptions of concentrations may not be significantly reduced. In contrast, if concentration data were available to the sensor system, it is more likely that the description of concentration distributions at current and future times will be represented by comparably more narrow confidence intervals. (Note that even in the case where sensors produce concentration signals, there is still uncertainty surrounding the true concentrations because of measurement error.)
Figures 4.5-4.7 show the uncertainty distributions for the concentrations in three zones based on three different conditions of data: no data (Figure 4.5), after analyzing 5 min of data (Figure 4.6), and after 10 min of data (Figure 4.7). One zone from each floor was selected: Rooms 1.2a and 2.2 and Stair 3. Figures 4.6-4.7 are both based on the use of threshold-type sensors, with one sensor per zone, and with sensor characteristics as described in the figure caption.
Figure 4.5. Prior confidence intervals of the concentration profiles for three zones with concentration data. Median (dashed line), 10% and 90% percentiles (solid lines), and concentration data from Experiment 1 (dotted line).
zero. The actual concentrations measured by in Experiment 1 are also shown in the figure. The measured concentrations lie well within the prior distributions for each zone. After 5 min of threshold data have been assimilated by the sensor system, an updated assessment of the uncertainty distributions of the concentrations in these zones is obtained (Figure 4.6). The original concentration data obtained from Experiment 1 are shown as well. Two types of information are obtained from this figure: the updated concentration distributions prior to the actual time (i.e., 5 minutes) and the updated concentration distributions for future time. The BMC algorithm substantially reduces the uncertainty with respect to both time periods, relative to the prior distributions. It is also worth noting that the original concentration data lie outside the 80% confidence intervals for the zones. This suggests that the confidence assigned to the measurements may not reflect the true model-to-measurement fit for all cases.
Figure 4.6. Confidence intervals of the concentration profiles for three zones after performing Bayesian updating using 5 min of alarm-type sensor data. Median (dashed line), 10th and 90th percentiles (solid lines), and concentration data (dotted line). Sensor signals were generated using a response time of 20 s and threshold level of 0.023.
Figure 4.7 shows the updated concentration distributions after assimilating 10 min of threshold data. The uncertainty regarding the concentrations is further reduced. The confidence intervals are now quite narrow. Two possible explanations for these narrow intervals exist. First, very high confidence is assigned to the model-to-measurement comparisons, resulting in very high posterior probabilties for a small number of realizations. In a real situation, these uncertainties are likely to be larger than the assumed 5% error in these simulations. Second, an improperly sampled library could result in negligible posterior probabilties for select realizations, leaving certain realizations, by default, with relatively high posterior probabilities. In this example, the first explanation is true. (Avoiding the second situation is the reason for a large library sampling size of 5000 simulations.) The purpose of these figures is to demonstrate the ability of the sensor system, using threshold signals, to reduce the uncertainty of the intpreted and predicted concentration distributions, and not to generate an optimal algorithm for this particular data set.
Figure 4.7. Confidence intervals of the concentration profiles for three zones after performing Bayesian updating using 10 min of data. Median, 10th and 90th percentiles of the estimated concentration profiles (solid line), which are coincident; concentration data (dotted line). Sensor signals were generated using a response time of 20 s and threshold level of 0.023.