Colombo, C.

Di Lizia, P.

Losacco, M.

Romano, M.

Romano M.

Losacco M.

Colombo C.

Di Lizia P.

Archivio istituzionale della ricerca - Politecnico di Milano

Source: "http://www.redorbit.com/media/uploads/2013/01/shutterstock_105231248.jpg"Estimation of impact probability of asteroids and space debris through Monte Carlo Line Sampling and Subset SimulationKePASSA2017 – ESTEC/ESAMatteo Romano, Matteo Losacco, Camilla Colombo, Pierluigi Di Lizia▪ Problem of space debris and asteroids• Common approaches and alternative proposals▪ Advanced Monte Carlo techniques• Line Sampling• Subset Simulation▪ Application of the proposed methods• Definition of test cases• Results, comparisons and comments▪ Conclusions and future developments1 August 2017 KePASSA2017 – ESTEC/ESA 2Contents1 August 2017 KePASSA2017 – ESTEC/ESA 3INTRODUCTION1. Common approach to the problem of space impacts2. Proposed advanced methods▪ Impacts of asteroids and space debris with planets threaten the safety of human activity in space• Planetary protection sets special requirements to avoid the contamination of celestial bodies due to man-made debris in interplanetary missions• Time periods under study generally span up to 100 years▪ Estimating the impact probability for space objects with celestial bodies requires a large number of long-term orbital propagations with standard Monte Carlo Simulations, resulting in high computational cost• The amount of simulations required to estimate the probability within a given confidence level increases as the expected probability decreases1 August 2017 KePASSA2017 – ESTEC/ESA 4IntroductionCommon approach to the problem of space impactsMore efficient sampling methods may increase the precision of the probability estimate, or reduce the amount of propagations and the computational cost:▪ The Line Sampling method probes the impact region of the uncertainty domain by using lines instead of random points• The impact probability is estimated via analytical integration, resulting in a more accurate solution▪ The Subset Simulation method computes the impact probability as the product of larger conditional probabilities• The method progressively identifies intermediate conditional levels moving towards the impact event, reducing the overall number of samples1 August 2017 KePASSA2017 – ESTEC/ESA 5IntroductionProposed advanced methods1 August 2017 KePASSA2017 – ESTEC/ESA 6ADVANCED MONTE CARLO METHODS1. Line Sampling2. Subset simulationThe method follows 4 phases:1. Determination of the “reference direction”Starting from a reference solution, a Markov Chain is used to find a direction pointing toward the impact region of the domain2. Mapping onto the standard normal spaceEach sample is mapped from the physical coordinates to normalized ones, in order to associate a normal distribution to each of them3. Line SamplingFor each sample, a line following the reference direction is probed to identify the limits of the impact region4. Estimation of impact probabilityProbability is analytically estimated as the average of integrals of unit normal distribution obtained along each line1 August 2017 KePASSA2017 – ESTEC/ESA 7Advanced Monte Carlo methodsLine Sampling1 August 2017 KePASSA2017 – ESTEC/ESA 8Advanced Monte Carlo methodsLine Sampling𝑥1𝑥2Initial impact solution is foundx01 August 2017 KePASSA2017 – ESTEC/ESA 9Advanced Monte Carlo methodsLine Sampling𝑥1𝑥2Monte Carlo Markov Chain is used to cover the impact region1 August 2017 KePASSA2017 – ESTEC/ESA 10Advanced Monte Carlo methodsLine Sampling𝑥1𝑥2xG𝛼The approximate barycentre is taken as reference direction1 August 2017 KePASSA2017 – ESTEC/ESA 11Advanced Monte Carlo methodsLine Sampling𝜃1𝜃2𝛼Random samples are transformed from physical coordinate space into normalised standard spaceΦ θk = F xkxk → θk1 August 2017 KePASSA2017 – ESTEC/ESA 12Advanced Monte Carlo methodsLine Sampling𝜃1𝜃2𝛼𝜃k,⊥ckLines from each random sample are defined using parameter ck෨𝜃k = ckα + 𝜃k,⊥1 August 2017 KePASSA2017 – ESTEC/ESA 13Advanced Monte Carlo methodsLine Sampling𝜃1𝜃2𝛼𝜃k,⊥ckIntersections with impact region are found with limits values ҧ𝑐1k, ҧ𝑐2k where an objective function 𝑌 𝑐𝑘 = 01 August 2017 KePASSA2017 – ESTEC/ESA 14Advanced Monte Carlo methodsLine Sampling𝑥1𝑥2The boundary of the impact region is covered▪ The limit values are used to compute one-dimensional impact probability estimates ෡Pk(I) along each line as෡Pk I = Φ തc2k −Φ തc1kwhere Φ represents the CDF of the unit normal distribution▪ The total probability and the associated variance are estimated as෡P I =1NT෍k=1NT෡Pk(I)ෝσ2 ෡P I =1NT(NT − 1)෍k=1NT෡Pk I − ෡P I21 August 2017 KePASSA2017 – ESTEC/ESA 15Advanced Monte Carlo methodsLine SamplingThe method uses a series of MC Markov Chains in order to cover the initial uncertainty domain and move toward the impact region:1. Generate N samples at conditional level (CL) 0 by standard Monte Carlo2. Propagate each sample and identify its minimum distance from the Earth, and sort them according to their distance from Earth3. Identify samples belonging to the next conditional level• Consider fixed values of conditional probability and identify the threshold as the (1-𝑝0)Nth element of the list4. Generate (1- 𝑝0N) new samples belonging to that CL by means of Monte Carlo Markov Chain5. Go back to step 21 August 2017 KePASSA2017 – ESTEC/ESA 16Advanced Monte Carlo methodsSubset Simulation1 August 2017 KePASSA2017 – ESTEC/ESA 17Advanced Monte Carlo methodsSubset Simulation𝑥1𝑥2Standard MC (CL0)1 August 2017 KePASSA2017 – ESTEC/ESA 18Advanced Monte Carlo methodsSubset Simulation𝑃(𝐼1) = 𝑝0𝑥1Standard MC (CL0)𝑥21 August 2017 KePASSA2017 – ESTEC/ESA 19Advanced Monte Carlo methodsSubset Simulation𝑃(𝐼1) = 𝑝0𝑥1𝑥2Standard MC (CL0)𝐼1 (𝐶𝐿1)1 August 2017 KePASSA2017 – ESTEC/ESA 20Advanced Monte Carlo methodsSubset Simulation𝑃(𝐼2|𝐼1) = 𝑝0𝑥1𝑥2Standard MC (CL0)𝑃(𝐼1) = 𝑝0𝐼1 (𝐶𝐿1)1 August 2017 KePASSA2017 – ESTEC/ESA 21Advanced Monte Carlo methodsSubset Simulation𝐼2 (𝐶𝐿2)𝑥1𝑥2Standard MC (CL0)𝑃(𝐼2|𝐼1) = 𝑝0𝐼1 (𝐶𝐿1)𝑃(𝐼1) = 𝑝01 August 2017 KePASSA2017 – ESTEC/ESA 22Advanced Monte Carlo methodsSubset SimulationImpact region 𝐼𝑃(𝐼3|𝐼2) = Τ𝑁𝑛 𝑁𝑥1𝑥2Standard MC (CL0)𝐼2 (𝐶𝐿2)𝑃(𝐼2|𝐼1) = 𝑝0𝐼1 (𝐶𝐿1)𝑃(𝐼1) = 𝑝0▪ Procedure is stopped when the threshold for the current conditional level is lower than the minimum distance set for the impact condition▪ Given a sequence of intermediate impact regions 𝐼1 ⊃…⊃ 𝐼𝑛 = 𝐼, the impact probability is estimated as the product of larger conditional probabilities෠𝑃 𝐼 = ෠𝑃 𝐼1 ෑ𝑖=1𝑛−1൯෠𝑃(𝐼𝑖+1|𝐼𝑖 = 𝑝0𝑛−1 Τ𝑁𝑛 𝑁• ൯෠𝑃(𝐼𝑖+1|𝐼𝑖 − probability of 𝐼𝑖+1 conditional to 𝐼𝑖• 𝑝0 − predefined conditional probability (equal for every level)• 𝑛 − number of conditional levels• 𝑁 − number of samples per conditional level• 𝑁𝑛 − number of impacting samples at the last conditional level1 August, 2017 KePASSA2017 – ESTEC/ESA 23Advanced Monte Carlo methodsSubset Simulation1 August 2017 KePASSA2017 – ESTEC/ESA 24APPLICATION OF THE PROPOSED METHODS1. Case 1: definition, results and comparison2. Case 2: definition, results and comparisonThe comparison among the different techniques is performed by analysing the following parameters:▪ Number of initial random samples 𝑁𝑆▪ Total number of propagations 𝑁𝑃▪ Impact probability estimate ෠𝑃(𝐼)▪ Associated standard deviation ො𝜎 ෠𝑃 𝐼▪ Coefficient of variation 𝛿 = ො𝜎/ ෠𝑃(𝐼)▪ Figure of Merit 𝐹𝑂𝑀 = 1/( ො𝜎2 ∙ 𝑁𝑃)Notes:▪ The computational time was not considered due to the nature of the numerical integration and of the machine, and the number of simulations was preferred instead as reference parameter▪ The FOM parameter puts more emphasis on the accuracy of the results than on the computational burden [1]1 August 2017 KePASSA2017 – ESTEC/ESA 25Application of the proposed methodsPremise• 2010 RF12 is a small Near Earth Asteroid (NEA) with currently the highest probability of hitting the Earth (around 6%) • 2095 fly-by was chosen as a test case due to the high expected impact probability• LEO crossing (with LEO under 2000 km from Earth’s surface) was considered as main event for the application of the methods1 August 2017 KePASSA2017 – ESTEC/ESA 26Application of the proposed methodsCase 1: 2010RF12 – Definition2010 fly-by, initial data [4]2095 fly-by1 August, 2017 KePASSA2017 – ESTEC/ESA 27Application of the proposed methodsCase 1: 2010RF12 – Comparison of resultsVisualisation of initial uncertainty set and impact region in (𝛿𝑎, 𝛿𝑙) planeMCLSSS1 August 2017 KePASSA2017 – ESTEC/ESA 28Application of the proposed methodsCase 1: 2010RF12 – Comparison of results𝑵𝑺 𝑵𝐏 ෡𝑷(𝑰) ෝ𝝈 𝜹 𝑭𝑶𝑴 ෝ𝝈𝑴𝑪/ෝ𝝈𝑭𝑶𝑴/𝑭𝑶𝑴𝑴𝑪Std MCS 1e4 1e4 7.07e-2 2.56e-3 2.76e1 15.22 1.00e0 1.00e0LS 1e3 ~6e3 7.29e-2 1.13e-3 1.56e-2 122.17 2.26e0 8.03e0SS 1e3 1.9e3 8.01e-2 6.00e-3 7.50e-2 14.62 4.27e-1 9.61e-1▪ Both methods correctly identify the impact region▪ For high values of the expected probability, the two methods are comparable with the standard Monte Carlo in terms of accuracy1 August 2017 KePASSA2017 – ESTEC/ESA 29Application of the proposed methodsCase 1: 2010RF12 – Comparison of results𝑵𝑺 𝑵𝐏 ෡𝑷(𝑰) ෝ𝝈 𝜹 𝑭𝑶𝑴 ෝ𝝈𝑴𝑪/ෝ𝝈𝑭𝑶𝑴/𝑭𝑶𝑴𝑴𝑪Std MCS 1e4 1e4 7.07e-2 2.56e-3 2.76e1 15.22 1.00e0 1.00e0LS 1e3 ~6e3 7.29e-2 1.13e-3 1.56e-2 122.17 2.26e0 8.03e0SS 1e3 1.9e3 8.01e-2 6.00e-3 7.50e-2 14.62 4.27e-1 9.61e-1▪ Both methods correctly identify the impact region▪ For high values of the expected probability, the two methods are comparable with the standard Monte Carlo in terms of accuracy• 2036 close approach may present impacts with Earth depending on the uncertainty given by 2009 observations, and was chosen as a test case as more challenging for the application of the methods • GEO crossing was considered as main event for the application of the methods for computational reasons1 August 2017 KePASSA2017 – ESTEC/ESA 30Application of the proposed methodsCase 2: Apophis – Definition2036 resonant return2009 initial data [4]1 August, 2017 KePASSA2017 – ESTEC/ESA 31Application of the proposed methodsCase 2: Apophis – Comparison of resultsVisualisation of initial uncertainty set and impact region in (𝛿𝑎, 𝛿𝑙) planeMCLSSS1 August 2017 KePASSA2017 – ESTEC/ESA 32Application of the proposed methodsCase 2: Apophis – Comparison of results𝑵𝑺 𝑵𝐏 ෡𝑷(𝑰) ෝ𝝈 𝜹 𝑭𝑶𝑴 ෝ𝝈𝑴𝑪/ෝ𝝈𝑭𝑶𝑴/𝑭𝑶𝑴𝑴𝑪Std MCS 1e6 1e6 4.70e-5 6.86e-6 6.85e0 2.13e4 1.00e0 1.00e0LS 1e5 ~1e6 5.32e-5 3.45e-7 6.48e-3 8.46e6 1.99e1 3.97e2SS 1e3 4.6e3 6.10e-5 1.30e-5 2.10e-1 1.28e6 5.27e-1 6.05e1Both methods become more convenient as the expected probability decreases, outperforming standard MC either in computational time (SS)  or accuracy (LS)▪ SS provides an estimate of the probability using a much lower number of propagations, with significant reductions in the computational burden, but also a larger variance▪ LS is computationally heavier in its current implementation, but offers the highest accuracy, especially when the estimated probability is low1 August 2017 KePASSA2017 – ESTEC/ESA 33Application of the proposed methodsCase 2: Apophis – Comparison of resultsBoth methods become more convenient as the expected probability decreases, outperforming standard MC either in computational time (SS)  or accuracy (LS)▪ SS provides an estimate of the probability using a much lower number of propagations, with significant reductions in the computational burden, but also a larger variance▪ LS is computationally heavier in its current implementation, but offers the highest accuracy, especially when the estimated probability is low𝑵𝑺 𝑵𝐏 ෡𝑷(𝑰) ෝ𝝈 𝜹 𝑭𝑶𝑴 ෝ𝝈𝑴𝑪/ෝ𝝈𝑭𝑶𝑴/𝑭𝑶𝑴𝑴𝑪Std MCS 1e6 1e6 4.70e-5 6.86e-6 6.85e0 2.13e4 1.00e0 1.00e0LS 1e5 ~1e6 5.32e-5 3.45e-7 6.48e-3 8.46e6 1.99e1 3.97e2SS 1e3 4.6e3 6.10e-5 1.30e-5 2.10e-1 1.28e6 5.27e-1 6.05e11 August 2017 KePASSA2017 – ESTEC/ESA 34CONCLUSIONS1. Final considerations2. Future developments▪ Line Sampling• Advantages‒ Using the same number of random samples, LS can achieve a lower variance of the solution (higher accuracy) thanks to the use of exact integrals on each line‒ Larger efficiency with respect to standard techniques as the impact probability gets lower• Current limitations‒ Current implementation supposes a unique impact region with a regular shape‒ Current implementation uses extra evaluations to probe each line, thus decreasing the efficiency of LS1 August 2017 KePASSA2017 – ESTEC/ESA 35ConclusionsFinal considerations▪ Subset Simulation• Advantages‒ It drives the generation of samples towards the impact region of the initial uncertainty set‒ Larger efficiency with respect to standard techniques as the impact probability gets lower‒ Reduced computational burden with respect to standard MC and LS• Current limitations‒ Correlation among conditional samples: possible bias in the estimates‒ Larger variance with respect to LS method1 August 2017 KePASSA2017 – ESTEC/ESA 36ConclusionsFinal considerations▪ Line Sampling• Analysis of the influence of the main implementation degrees of freedom on the accuracy of the results• Obtain an analytical expression for the confidence interval in order to determine a minimum number of lines required for a given confidence level• Improvement of the zeros computation• Application to planetary protection studies as part of a Ph.D. research▪ Subset Simulation• Analysis of the influence of the main implementation degrees of freedom on the accuracy of the results• Optimization of the Monte Carlo Markov Chain for samples generation1 August 2017 KePASSA2017 – ESTEC/ESA 37ConclusionsFuture developments1. Enrico Zio, Nicola Pedroni, Subset Simulation and Line Sampling for Advanced Monte Carlo Reliability Analysis, Proceedings of the European Safety and RELiability (ESREL) 2009 Conference, 2009, pp.687-694. <hal-00721047>2. Murray Rosenblatt, Remarks on a Multivariate Transformation, The Annals of Mathematical Statistics, Vol. 23, No. 3 (Sep., 1952), pp. 470-4723. Zuev K. et al., Bayesian post-processor and other enhancements of Subset Simulation for estimating failure probabilities in high dimensions, Computers and Structures,92, 283-296, 20114. Initial conditions and uncertainty values taken from Near Earth Object Dynamic Site NEODyS (http://newton.dm.unipi.it/neodys)1 August, 2017 KePASSA2017 – ESTEC/ESA 38References1 August 2017 KePASSA2017 – ESTEC/ESA 39THANK YOU FOR YOUR ATTENTIONSource: "http://www.redorbit.com/media/uploads/2013/01/shutterstock_105231248.jpg"Estimation of impact probability of asteroids and space debris through Monte Carlo Line Sampling and Subset SimulationKePASSA2017 – ESTEC/ESAMatteo Romano, Matteo Losacco, Camilla Colombo, Pierluigi Di LiziaThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 679086 – COMPASS)

Estimation of impact probability of asteroids and space debris through Monte Carlo Line Sampling and Subset Simulation

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Estimation of impact probability of asteroids and space debris through Monte Carlo Line Sampling and Subset Simulation

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