Detection and Estimation Final Project Report: Modeling and the K-Sigma Algorithm for Radiation Detection

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1 Detection and Estimation Final Project Report: Modeling and the K-Sigma Algorithm for Radiation Detection Sijie Xiong, Department of Electrical and Computer Engineering Rutgers, the State University of New Jersey Abstract In this paper, we examine simple yet insightful models and the K-Sigma algorithm which characterize the problem of radiation detection in sensor networks. Our goal is to detect dangerous radiation sources as soon as possible with a relatively low probability of false alarm and miss. Simulation results presented the tradeoff relationship between discrimination threshold, time to detect, probability of false alarm and sensor deployment strategies. I. INTRODUCTION Detecting dangerous radiation sources as well as many other chemical and biological threats is of primary concern to fight against terrorism and violence around the world. We aim to minimize the potential cost by detecting, localizing and neutralizing threats correctly and rapidly enough. A large body of detection algorithms and sensor deployment strategies has been developed to accomplish the goal in many detection scenarios. For example, at the entries and exits of train stations and airports, large portal styled sensors are placed to detect whether people are carrying dangerous materials. However, they are not practical to utilize in more general urban settings such as an open park, a trash can, a river and so on. In these cases, distributed sensor network (DSN) is preferred because of its flexibility in deployment. In this paper, we only consider a point radiation threat. In Section II, we examine the radiation detection problem in two simple models and analyze the tradeoff relationship between detectors performance and detection time. The K-Sigma algorithm will be presented in Section III. Finally, we make a brief summary and some prospective ideas. II. MODELS The detecting sensors record the number of photons striking the sensors. The source

2 radiating and the sensor recording process can both be modeled under poisson distribution. Poisson Distribution Let poisson(r,t,n) be the probability of n occurances in time interval t when the events are generated under a Poisson distribution with parameter r: A. A Single Sensor poisson(r, t, n) = (rt)n e rt n! We first analyze the case when there is only a point radiation source present and a sensor r meters away from the source. The radiation source generates photons at a rate of µ photons per unit time. The sensor detects photons at a rate decreasing as 1/ r 2 if there is no absorption in the air, and a multiplicative term e αr (α is the absorption rate per unit distance) should be added if absorption is present. Then the expected rate of photons detected at the sensor denoted by Λ is: Λ = Aµe αr r 2 where A is a constant depending exclusively on the type of detector. Given a time period T, the sensor records the number of photons striking the sensor as n. If n exceeds a predefined threshold B, then the system issues an alarm that a threat is detected, which is called a true positive. However, the background noises also produce photons at the sensor, let the rate be denoted as Γ, then if the number of photons detected from the background is no less than B, the system will also issue an alert, which is called a false positive. From the poisson distribution, we can compare two estimates of probabilities in the above scenario. Let pfp be the probability of a false positive when there is actually no source present, and ptp be the probability of a true positive when the source is present. We can write the following equations, pfp = ptp = n B n B poisson(γ, T,n) poisson(γ + Λ, T,n) If we fix Γ, Λ and T, then by varying the threshold B, we can plot ptp as a function of pfp, which is called the ROC (Receiver Operating Characteristic) curve. Figure 1 shows different ROC curves under different time intervals with two pair of Γ and Λ settings. Γ =8, Λ =2 for the left figure and Γ =8, Λ =32 for the right figure. It is easy to conclude that with increasing observing time T, the ROC curves converge to the left-top corner (lower probability of false alarm and higher probability of detection)

3 Fig. 1. ROC curves of the probability of detection versus false alarm. Fig. 2. ROC curves when background intensity is increased. the Area under ROC Curves (AUC) is increasing which indicates that the overall performance of the detection system is improved. In Figure 2, we set Γ =8, Λ =4 and Γ =16, Λ =4 for the two figures respectively. The result shows that increasing the background intensity Γ reduces the overall performance of the detection system since the ROC curves move toward the right-bottom corner. Tradeoff Relationship between Threshold B, True Positives, False Positives, Time and Distance Firstly, along a ROC curve, as B varies, the probabilities of false alarm and detection also change. We can examine the relationship between them by looking at the probability distributions of detection and false alarm. If the two distributions don t overlap, the presence of radiation source can be easily inferred without error. However, if they overlap, intuitively, the probabilities of detection and false alarm will both be 1 when B = 0 and both be 0 when B =. And if B falls in between, then ptp and pfp are the areas (Cumulative Distribution Function) under the detection and false alarm probability distribution exceeding B, respectively. Therefore, increasing B will result in

4 lower probability of false alarm as well as lower probability of detection. Secondly, we know that a Poisson distribution can be approximated with small probability of error by a normal distribution when the expected occurrences of events is 10 or more. Let σ source and σ null denote the standard deviations of the detected number of photons at a sensor when the radiation source is present and not present respectively. Then, σ source = (Γ + Λ)T σ null = ΓT Let bfp and bfn be the upper bounds on the probabilities of false positives and false negatives (no alarm when source is present), z FP and - z FN be the corresponding value in a standard normal distribution resulting in bfp and bfn. For instance, when bfp = bfn = 0.01, then z FP = 2.33 and - z FN = We can get, B ΓT z FP σ null (Γ + Λ)T B z FN σ source Our goal is to get lower probability of false alarm and miss. Let z = z FP = z FN, then, B ΓT = z σ null (Γ + Λ)T B = z σ source (Γ + Λ)T ΓT = z (σ null +σ source ) z = = ΛT σ null +σ source Λ T Γ + (Γ + Λ) Therefore, the higher z is the better the quality of ROC curves and overall performance of detection system. Now we ignore the absorption of photons in the air and substitute the equation Λ = Aµ / r 2 in the above result to get, when Λ Γ : z Λ T / 2 Γ when Aµ / Γ r 2 : z Aµ T / 2r 2 Γ when Λ Γ : z ΛT when Aµ / Γ r 2 : z AµT / r Consider the first case, if we fix the bounds on the probabilities of detection and false alarm, which is often determined by design requirements, then the value of z and the ROC curves are fixed. Then, A 2 µ 2 T = 4 z 2 r 4 Γ is the key tradeoff relationship between different design parameters. We can see that if the distance between source and sensor doubles, the time to detect increases by a factor of

5 Fig. 3. Tradeoff relationship between detection time (T) and distance (r). 16 or A must increase by a factor of 4 with other parameters unchanged. Figure 3 shows the tradeoff relationship between distance and detection time under different settings of probabilities of false positives. B. Multiple Sensors Now consider a simple case of 4 distributed sensors located at [0, 0], [0, 30], [30, 0], [30, 30]. The detecting processes at each sensor are independent Poisson process, thus the aggregate count of photons by 4 sensors also form a Poisson process, the system issue an alarm when the total count exceeds B. Figure 4 shows the time T to detect as a function of radiation source location [x, y]. We can see that when source is further away from sensors, the time to detect increases. And when the source is at the center [15, 15], the elapsed time for a true positive is the longest (i.e., the worst performance). So we can actually utilize mobile sensors and dynamic redeployment strategy to improve the performance. Fig. 4. Contour plot of detection time T on the 2-D plane.

6 III. THE K-SIGMA ALGORITHM The K-Sigma algorithm is quite similar to the previous method, with an extension into a more general distributed sensor network and another form of the threshold. We first group the distributed sensor network in detector groups, and each consists of 1 (singleton), 2 (edge), 3 (triangle), or 4 (quad) sensors. The aggregate radiation count rate in each group is N, the expected aggregate count rate from only the background is Γ. Then, ksigma = N at Γ a T Γ a T denotes the number of standard deviations the aggregate group count has from the total expected count only from the background in time T. Detection alert is issued if ksigma exceeds a specific threshold K. By varying K, with µ and Γ a fixed, we can also get ROC curves at different values of detection time T. Figure 5 shows a simple example of the ROC curves applying the K-Sigma algorithm. Here, we use the same sensor deployment as in the multiple sensors case. The detection group consists all 4 sensors. The radiation source is randomly located in the field. Γ a = 4 8 counts per second. The ROC curves are averaged over 1,000 random initializations. From Figure 5, we can also see that with increasing time T to detect, the overall performance of the detection system, also the quality of ROC cures improves. Note that the quality of the ROC curve when the source is located at the center is relatively lower, compared to the curve under the same elapsed time (10s), which is also stated at the end of Section II B. IV. CONCLUSION We have described the models and the K-Sigma algorithm to detect and localize dangerous radiation sources in the presence of background noises. The ROC curves showed that given sufficient time, we can detect the targets with infinite accuracy. In practice, however, it is very important to protect the public from being hurt in a very short time, which requires us to find a balance among different settings so that the detection system can be better designed. In this paper, we only considered the case that one single point stationary radiation threat is present, and the sensor network consists of only four static detectors located at the vertices of a square plane. We also assumed no absorption of photons in the background. In fact, the radiation source may be moving or shielded and there may be

7 Fig. 5. ROC curves for the K-Sigma algorithm, at different elapsed times. multiple sources. The background may be heterogeneous and absorb photons. These can result in more complicated modeling strategies. Also, we can use more sensors or sensors with limited or free mobility cooperatively to detect the radiation threats. What s more, we can combine the K-Sigma algorithm with other probabilistic detection algorithms such as Bayesian algorithm [2] to get a much better performance. REFERENCES [1] Chandy, K. Mani, Julian Bunn, and A. Liu. "Models and algorithms for radiation detection." Modeling and Simulation Workshop for Homeland Security [2] Liu, Annie H., Julian J. Bunn, and K. Mani Chandy. "Sensor networks for the detection and tracking of radiation and other threats in cities." IPSN [3] Srinath, Mandyam Dhati, P. K. Rajasekaran, and Ramanarayanan Viswanathan. Introduction to statistical signal processing with applications. Prentice-Hall, Inc., [4] Brennan, Sean M., et al. "Radiation detection with distributed sensor networks." Computer 37.8 (2004):