This paper presents a method for incorporating information from waterborne debris that has been r... more This paper presents a method for incorporating information from waterborne debris that has been recovered at a known time and place into the object location distribution. The method involves calculating a likelihood function which can be combined with the prior distribution on the location of a lost object, such as a ship or aircraft lost at sea, to produce, in a Bayesian fashion, a posterior distribution that reduces uncertainty in the location of the loss. When the search operation involves a person or vessel adrift at sea, incorporating debris information in this fashion will allow the search planner to produce better predictions for the location of the search object at times of future searches and enable more effective deployment of search assets. Use of this method will save time, money, and lives.
The long-term goal is to develop methods to characterize and display the uncertainty in target st... more The long-term goal is to develop methods to characterize and display the uncertainty in target state estimates that result from uncertainty in environmental estimates.
Proceedings of the 16th International Conference on Information Fusion, 2013
In this paper we show that the standard definition of likelihood function used in Bayesian infere... more In this paper we show that the standard definition of likelihood function used in Bayesian inference simply and correctly handles imprecise likelihood functions and quantized measurements. Some recent papers have stated or implied that methods involving random sets, fuzzy membership functions, generalized likelihood functions, or Dempster-Shafer concepts are required to handle imprecise likelihood functions and quantized measurements. While it is true that one can use these methods, employing them adds unnecessary complication and possibly confusion to the solution of a simple problem. In the spirit of Occam's razor, we feel the simplest correct solution is the best.
2010 13th International Conference on Information Fusion, 2010
In 1857, while carrying passengers and gold from California to New York, the SS Central America s... more In 1857, while carrying passengers and gold from California to New York, the SS Central America sank in a hurricane, taking some three tons of gold bars and coins to the ocean bottom almost 8,000 ft below. Some 425 people, including the captain of the ship, lost their lives. In 1989, after three summers of effort at sea, the Columbus America Discovery Group recovered one ton of gold bars and coins from the wreck. This paper reviews the analysis that was performed to produce the probability distribution used to plan the successful search for the wreck and critiques that analysis based on information obtained in the years since the original 1992 article was written on this topic. This paper also provides an update on the legal battles and the disposition of the recovered gold.
I thank Professor Welsh for his very kind comments about the AF 447 paper. He makes a number of e... more I thank Professor Welsh for his very kind comments about the AF 447 paper. He makes a number of excellent points. One is that Bayesian analysis is a tool and that must be used carefully and thoughtfully in order to obtain good results in a complicated problem such as the search for AF 447. While this is true, the use of Bayesian analysis is required to incorporate the necessary subjective judgments into the analysis of the AF 447 search. As Welsh notes, Bayesian analysis allowed us to propagate these judgments and uncertainty distributions into the probability distribution on the location of the wreck in a logical and correct fashion. Classical statistics does not provide a framework for doing this. Bayesians should celebrate this advantage. The power of Bayesian analysis as a tool is further illustrated by the U.S. Coast Guard's Search and Rescue Optimal Planning System (SAROPS). SAROPS is a Bayesian search planning program used by the Coast Guard every day for planning searches for people and boats lost at sea. It is run by Coast Guard officers who are trained to use the program but are by no means experts in Bayesian analysis. The Coast Guard considers it one of their best operational computer programs. Welsh suggests that the use of data from nine somewhat similar situations casts doubt on the claim that the use of subjective probabilities is required for the AF 447 analysis. However, the availability of this data does not mean we could reasonably have produced the AF 447 distribution without the use of subjective probabilities. The use of subjective probabilities is one characteristic that distinguishes Bayesian statistics from classical statistics where decisions are supposed to be made solely
Signal and Data Processing of Small Targets 2003, 2003
This paper provides a brief history of some operational particle filters that were used by the U.... more This paper provides a brief history of some operational particle filters that were used by the U. S. Coast Guard and U. S. Navy. Starting in 1974 the Coast Guard system provided Search and Rescue Planning advice for objects lost at sea. The Navy systems were used to plan searches for Soviet submarines in the Atlantic, Pacific, and Mediterranean starting in 1972. The systems operated in a sequential, Bayesian manner. A prior distribution for the target's location and movement was produced using both objective and subjective information. Based on this distribution, the search assets available, and their detection characteristics, a near-optimal search was planned. Typically, this involved visual searches by Coast Guard aircraft and sonobuoy searches by Navy antisubmarine warfare patrol aircraft. The searches were executed, and the feedback, both detections and lack of detections, was fed into a particle filter to produce the posterior distribution of the target's location. This distribution was used as the prior for the next iteration of planning and search.
This paper discusses the operations analysis in the underwater search for the remains of the subm... more This paper discusses the operations analysis in the underwater search for the remains of the submarine Scorpion. The a priori target location probability distribution for the search was obtained by monte-carlo procedures based upon nine different scenarios concerning the Scorpion loss and associated credibility weights. These scenarios and weights were postulated by others. Scorpion was found within 260 yards of the search grid cell having the largest a priori probability. Frequent computations of local effectiveness probabilities (LEPs) were carried out on scene during the search and were used to determine an updated (a posteriori) target location distribution. This distribution formed the basis for recommendation of the current high probability areas for search. tions Research 5,613-626 (1957). publication.
The optimal survivor search problem is to maximize the probability of finding a moving target ali... more The optimal survivor search problem is to maximize the probability of finding a moving target alive. The target may be a person missing at sea or lost in a wilderness area. In a multistate target problem, the target is moving and changing states stochastically. Each state may imply different motion and detectability assumptions. We find necessary and sufficient conditions for optimal detection plans for multistate targets. Using these conditions, we extend the algoritnms of Brown and Stone for finding optimal detection search plans for moving targets to multistate target problems, and show that the optimal survivor search problem may be solved as a special case of the multistate target problem.
This paper shows that solving the optimal whereabouts search problem for a moving target is equiv... more This paper shows that solving the optimal whereabouts search problem for a moving target is equivalent to solving a finite number of optimal detection problems for moving targets. This generalizes the result of Kadane (Kadane, J. B. 1971. Optimal whereabouts search. Opns. Res. 19 894–904.) for stationary targets.
2010 13th International Conference on Information Fusion, 2010
In 1974 the U.S. Coast Guard put into operation its first computerized search and rescue planning... more In 1974 the U.S. Coast Guard put into operation its first computerized search and rescue planning system CASP (Computer-Assisted Search Planning) which used a Bayesian approach implemented by a particle filter to produce probability distributions for the location of the search object. These distributions were used for planning search effort. In 2003, the Coast Guard started development of a new decision support system for managing search efforts called Search and Rescue Optimal Planning System (SAROPS). SAROPS has been operational since January, 2007 and is currently the only search planning tool that the Coast Guard uses for maritime searches. SAROPS represents a major advance in search planning technology. This paper reviews the technology behind the tool.
On 1 June 2009 Air France Flight 447, with 228 passengers and crew aboard, disappeared over the S... more On 1 June 2009 Air France Flight 447, with 228 passengers and crew aboard, disappeared over the South Atlantic during a night flight from Rio de Janeiro to Paris. An international air and surface search effort located the first floating debris during the sixth day of search. Three phases of unsuccessful search for the underwater wreckage ensued. Phase I was a passive acoustic search for the aircraft's underwater locator beacons. Phases II and III were side-looking sonar searches scanning the ocean bottom for the wreckage field. In July of 2010 the French Bureau d'Enquêtes et d'Analyses tasked Metron to review the searches and produce posterior probability maps for the location of the wreckage. These maps were used to plan the next phase of search beginning in March 2011. On April 3, after one week of search, the wreckage was located in a high probability area of the map.
In the early morning hours of June 1, 2009, during a flight from Rio de Janeiro to Paris, Air Fra... more In the early morning hours of June 1, 2009, during a flight from Rio de Janeiro to Paris, Air France Flight AF 447 disappeared during stormy weather over a remote part of the Atlantic carrying 228 passengers and crew to their deaths. After two years of unsuccessful search, the authors were asked by the French Bureau d'Enquêtes et d'Analyses pour la sécurité de l'aviation to develop a probability distribution for the location of the wreckage that accounted for all information about the crash location as well as for previous search efforts. We used a Bayesian procedure developed for search planning to produce the posterior target location distribution. This distribution was used to guide the search in the third year, and the wreckage was found with one week of undersea search. In this paper we discuss why Bayesian analysis is ideally suited to solving this problem, review previous non-Bayesian efforts, and describe the methodology used to produce the posterior probability distribution for the location of the wreck.
We consider the problem of finding a plan that maximizes the expected discounted return when extr... more We consider the problem of finding a plan that maximizes the expected discounted return when extracting a nonrenewable resource having uncertain reserves. An extraction plan specifies the rate at which the resource is extracted as a function of time until the resource is exhausted or the time horizon is reached. The return per unit of resource extracted may depend on the rate of extraction, time, and the amount of resource previously extracted. We apply a new method called the generalized search optimization technique to find qualitative features of optimal plans and to devise algorithms for the numerical calculation of optimal plans. 1.1. Description of Results The optimal resource extraction problem considered in this article is similar to the one considered by Loury [8] and Liu [7]. One important difference is that our results apply to utility or return functions that have finite derivative at 0, whereas Loury and Liu require an infinite derivative at 0. Consequently, our results apply to problems in which utility is measured in terms of money, where marginal rates of return are finite. In contrast, the results of Loury and Liu apply to problems involving the social utility of a critical resource. In addition,
This paper presents a method for incorporating information from waterborne debris that has been r... more This paper presents a method for incorporating information from waterborne debris that has been recovered at a known time and place into the object location distribution. The method involves calculating a likelihood function which can be combined with the prior distribution on the location of a lost object, such as a ship or aircraft lost at sea, to produce, in a Bayesian fashion, a posterior distribution that reduces uncertainty in the location of the loss. When the search operation involves a person or vessel adrift at sea, incorporating debris information in this fashion will allow the search planner to produce better predictions for the location of the search object at times of future searches and enable more effective deployment of search assets. Use of this method will save time, money, and lives.
The long-term goal is to develop methods to characterize and display the uncertainty in target st... more The long-term goal is to develop methods to characterize and display the uncertainty in target state estimates that result from uncertainty in environmental estimates.
Proceedings of the 16th International Conference on Information Fusion, 2013
In this paper we show that the standard definition of likelihood function used in Bayesian infere... more In this paper we show that the standard definition of likelihood function used in Bayesian inference simply and correctly handles imprecise likelihood functions and quantized measurements. Some recent papers have stated or implied that methods involving random sets, fuzzy membership functions, generalized likelihood functions, or Dempster-Shafer concepts are required to handle imprecise likelihood functions and quantized measurements. While it is true that one can use these methods, employing them adds unnecessary complication and possibly confusion to the solution of a simple problem. In the spirit of Occam's razor, we feel the simplest correct solution is the best.
2010 13th International Conference on Information Fusion, 2010
In 1857, while carrying passengers and gold from California to New York, the SS Central America s... more In 1857, while carrying passengers and gold from California to New York, the SS Central America sank in a hurricane, taking some three tons of gold bars and coins to the ocean bottom almost 8,000 ft below. Some 425 people, including the captain of the ship, lost their lives. In 1989, after three summers of effort at sea, the Columbus America Discovery Group recovered one ton of gold bars and coins from the wreck. This paper reviews the analysis that was performed to produce the probability distribution used to plan the successful search for the wreck and critiques that analysis based on information obtained in the years since the original 1992 article was written on this topic. This paper also provides an update on the legal battles and the disposition of the recovered gold.
I thank Professor Welsh for his very kind comments about the AF 447 paper. He makes a number of e... more I thank Professor Welsh for his very kind comments about the AF 447 paper. He makes a number of excellent points. One is that Bayesian analysis is a tool and that must be used carefully and thoughtfully in order to obtain good results in a complicated problem such as the search for AF 447. While this is true, the use of Bayesian analysis is required to incorporate the necessary subjective judgments into the analysis of the AF 447 search. As Welsh notes, Bayesian analysis allowed us to propagate these judgments and uncertainty distributions into the probability distribution on the location of the wreck in a logical and correct fashion. Classical statistics does not provide a framework for doing this. Bayesians should celebrate this advantage. The power of Bayesian analysis as a tool is further illustrated by the U.S. Coast Guard's Search and Rescue Optimal Planning System (SAROPS). SAROPS is a Bayesian search planning program used by the Coast Guard every day for planning searches for people and boats lost at sea. It is run by Coast Guard officers who are trained to use the program but are by no means experts in Bayesian analysis. The Coast Guard considers it one of their best operational computer programs. Welsh suggests that the use of data from nine somewhat similar situations casts doubt on the claim that the use of subjective probabilities is required for the AF 447 analysis. However, the availability of this data does not mean we could reasonably have produced the AF 447 distribution without the use of subjective probabilities. The use of subjective probabilities is one characteristic that distinguishes Bayesian statistics from classical statistics where decisions are supposed to be made solely
Signal and Data Processing of Small Targets 2003, 2003
This paper provides a brief history of some operational particle filters that were used by the U.... more This paper provides a brief history of some operational particle filters that were used by the U. S. Coast Guard and U. S. Navy. Starting in 1974 the Coast Guard system provided Search and Rescue Planning advice for objects lost at sea. The Navy systems were used to plan searches for Soviet submarines in the Atlantic, Pacific, and Mediterranean starting in 1972. The systems operated in a sequential, Bayesian manner. A prior distribution for the target's location and movement was produced using both objective and subjective information. Based on this distribution, the search assets available, and their detection characteristics, a near-optimal search was planned. Typically, this involved visual searches by Coast Guard aircraft and sonobuoy searches by Navy antisubmarine warfare patrol aircraft. The searches were executed, and the feedback, both detections and lack of detections, was fed into a particle filter to produce the posterior distribution of the target's location. This distribution was used as the prior for the next iteration of planning and search.
This paper discusses the operations analysis in the underwater search for the remains of the subm... more This paper discusses the operations analysis in the underwater search for the remains of the submarine Scorpion. The a priori target location probability distribution for the search was obtained by monte-carlo procedures based upon nine different scenarios concerning the Scorpion loss and associated credibility weights. These scenarios and weights were postulated by others. Scorpion was found within 260 yards of the search grid cell having the largest a priori probability. Frequent computations of local effectiveness probabilities (LEPs) were carried out on scene during the search and were used to determine an updated (a posteriori) target location distribution. This distribution formed the basis for recommendation of the current high probability areas for search. tions Research 5,613-626 (1957). publication.
The optimal survivor search problem is to maximize the probability of finding a moving target ali... more The optimal survivor search problem is to maximize the probability of finding a moving target alive. The target may be a person missing at sea or lost in a wilderness area. In a multistate target problem, the target is moving and changing states stochastically. Each state may imply different motion and detectability assumptions. We find necessary and sufficient conditions for optimal detection plans for multistate targets. Using these conditions, we extend the algoritnms of Brown and Stone for finding optimal detection search plans for moving targets to multistate target problems, and show that the optimal survivor search problem may be solved as a special case of the multistate target problem.
This paper shows that solving the optimal whereabouts search problem for a moving target is equiv... more This paper shows that solving the optimal whereabouts search problem for a moving target is equivalent to solving a finite number of optimal detection problems for moving targets. This generalizes the result of Kadane (Kadane, J. B. 1971. Optimal whereabouts search. Opns. Res. 19 894–904.) for stationary targets.
2010 13th International Conference on Information Fusion, 2010
In 1974 the U.S. Coast Guard put into operation its first computerized search and rescue planning... more In 1974 the U.S. Coast Guard put into operation its first computerized search and rescue planning system CASP (Computer-Assisted Search Planning) which used a Bayesian approach implemented by a particle filter to produce probability distributions for the location of the search object. These distributions were used for planning search effort. In 2003, the Coast Guard started development of a new decision support system for managing search efforts called Search and Rescue Optimal Planning System (SAROPS). SAROPS has been operational since January, 2007 and is currently the only search planning tool that the Coast Guard uses for maritime searches. SAROPS represents a major advance in search planning technology. This paper reviews the technology behind the tool.
On 1 June 2009 Air France Flight 447, with 228 passengers and crew aboard, disappeared over the S... more On 1 June 2009 Air France Flight 447, with 228 passengers and crew aboard, disappeared over the South Atlantic during a night flight from Rio de Janeiro to Paris. An international air and surface search effort located the first floating debris during the sixth day of search. Three phases of unsuccessful search for the underwater wreckage ensued. Phase I was a passive acoustic search for the aircraft's underwater locator beacons. Phases II and III were side-looking sonar searches scanning the ocean bottom for the wreckage field. In July of 2010 the French Bureau d'Enquêtes et d'Analyses tasked Metron to review the searches and produce posterior probability maps for the location of the wreckage. These maps were used to plan the next phase of search beginning in March 2011. On April 3, after one week of search, the wreckage was located in a high probability area of the map.
In the early morning hours of June 1, 2009, during a flight from Rio de Janeiro to Paris, Air Fra... more In the early morning hours of June 1, 2009, during a flight from Rio de Janeiro to Paris, Air France Flight AF 447 disappeared during stormy weather over a remote part of the Atlantic carrying 228 passengers and crew to their deaths. After two years of unsuccessful search, the authors were asked by the French Bureau d'Enquêtes et d'Analyses pour la sécurité de l'aviation to develop a probability distribution for the location of the wreckage that accounted for all information about the crash location as well as for previous search efforts. We used a Bayesian procedure developed for search planning to produce the posterior target location distribution. This distribution was used to guide the search in the third year, and the wreckage was found with one week of undersea search. In this paper we discuss why Bayesian analysis is ideally suited to solving this problem, review previous non-Bayesian efforts, and describe the methodology used to produce the posterior probability distribution for the location of the wreck.
We consider the problem of finding a plan that maximizes the expected discounted return when extr... more We consider the problem of finding a plan that maximizes the expected discounted return when extracting a nonrenewable resource having uncertain reserves. An extraction plan specifies the rate at which the resource is extracted as a function of time until the resource is exhausted or the time horizon is reached. The return per unit of resource extracted may depend on the rate of extraction, time, and the amount of resource previously extracted. We apply a new method called the generalized search optimization technique to find qualitative features of optimal plans and to devise algorithms for the numerical calculation of optimal plans. 1.1. Description of Results The optimal resource extraction problem considered in this article is similar to the one considered by Loury [8] and Liu [7]. One important difference is that our results apply to utility or return functions that have finite derivative at 0, whereas Loury and Liu require an infinite derivative at 0. Consequently, our results apply to problems in which utility is measured in terms of money, where marginal rates of return are finite. In contrast, the results of Loury and Liu apply to problems involving the social utility of a critical resource. In addition,
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Papers by Lawrence Stone