Microwave measurement of the probability of collision of low energy electrons in nitrogen technical report no. 17
Collision probability for momentum transfer of low energy electrons in nitrogen
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A Multistep Probability of Collision Computation Algorithm
The two-dimensional probability of collision calculation method is both widely used and computationally efficient. However, Monte Carlo simulations show that this method sometimes fails to provide sufficiently accurate estimates for Earthorbiting satellites. This study presents a multistep algorithm that calculates collision probabilities for all conjunctions, regardless if affected by curvilinear trajectory or time varying covariance dynamics. The algorithm sequentially applies increasingly accurate estimation methods, but only as required for efficiency. Evaluating usage violations for the two-dimensional probability of collision method represents one of the most important steps. Extensive testing demonstrates the efficiency and reliability of the multistep algorithm.
Doyle T. Hall↗
Development of a Nonlinear Probability of Collision Tool for the Earth Observing System
The Earth Observing System (EOS) spacecraft Terra, Aqua, and Aura fly in constellation with several other spacecraft in 705-kilometer mean altitude sun-synchronous orbits. All three spacecraft are operated by the Earth Science Mission Operations (ESMO) Project at Goddard Space Flight Center (GSFC). In 2004, the ESMO project began assessing the probability of collision of the EOS spacecraft with other space objects. In addition to conjunctions with high relative velocities, the collision assessment method for the EOS spacecraft must address conjunctions with low relative velocities during potential collisions between constellation members. Probability of Collision algorithms that are based on assumptions of high relative velocities and linear relative trajectories are not suitable for these situations; therefore an algorithm for handling the nonlinear relative trajectories was developed. This paper describes this algorithm and presents results from its validation for operational use. The probability of collision is typically calculated by integrating a Gaussian probability distribution over the volume swept out by a sphere representing the size of the space objects involved in the conjunction. This sphere is defined as the Hard Body Radius. With the assumption of linear relative trajectories, this volume is a cylinder, which translates into simple limits of integration for the probability calculation. For the case of nonlinear relative trajectories, the volume becomes a complex geometry. However, with an appropriate choice of coordinate systems, the new algorithm breaks down the complex geometry into a series of simple cylinders that have simple limits of integration. This nonlinear algorithm will be discussed in detail in the paper. The nonlinear Probability of Collision algorithm was first verified by showing that, when used in high relative velocity cases, it yields similar answers to existing high relative velocity linear relative trajectory algorithms. The comparison with the existing high velocity/linear theory will also be used to determine at what relative velocity the analysis should use the new nonlinear theory in place of the existing linear theory. The nonlinear algorithm was also compared to a known exact solution for the probability of collision between two objects when the relative motion is strictly circular and the error covariance is spherically symmetric. Figure I shows preliminary results from this comparison by plotting the probabilities calculated from the new algorithm and those from the exact solution versus the Hard Body Radius to Covariance ratio. These results show about 5% error when the Hard Body Radius is equal to one half the spherical covariance magnitude. The algorithm was then combined with a high fidelity orbit state and error covariance propagator into a useful tool for analyzing low relative velocity nonlinear relative trajectories. The high fidelity propagator is capable of using atmospheric drag, central body gravitational, solar radiation, and third body forces to provide accurate prediction of the relative trajectories and covariance evolution. The covariance propagator also includes a process noise model to ensure realistic evolutions of the error covariance. This paper will describe the integration of the nonlinear probability algorithm and the propagators into a useful collision assessment tool. Finally, a hypothetical case study involving a low relative velocity conjunction between members of the Earth Observation System constellation will be presented.
McKinley, David P.↗
Approaches to Evaluating Probability of Collision Uncertainty
While the two-dimensional probability of collision (Pc) calculation has served as the main input to conjunction analysis risk assessment for over a decade, it has done this mostly as a point estimate, with relatively little effort made to produce confidence intervals on the Pc value based on the uncertainties in the inputs. The present effort seeks to try to carry these uncertainties through the calculation in order to generate a probability density of Pc results rather than a single average value. Methods for assessing uncertainty in the primary and secondary objects' physical sizes and state estimate covariances, as well as a resampling approach to reveal the natural variability in the calculation, are presented; and an initial proposal for operationally-useful display and interpretation of these data for a particular conjunction is given.
PROBABILITY↗
Implementation Recommendations and Usage Boundaries for the Two-Dimensional Probability of Collision Calculation
The two-dimensional (2D) probability of collision (𝑃𝑐) estimation method relies on several assumptions that must be satisfied for accurate results. Monte Carlo analysis of ~44,000 conjunctions indicates that 2D-𝑃(sub 𝑐) pro-vides accurate estimates for most typical conjunctions, but occasionally underestimates 𝑃(sub 𝑐) significantly, indicating an assumption violation. A test to detect large-amplitude underestimation inaccuracies can be based on how much “offset-from-TCA” 2D-𝑃(sub 𝑐) values vary during a well-defined time interval bracketing closest approach. The test successfully detects all large-amplitude 2D-𝑃(sub 𝑐) underestimations found to date, but with a high false-alarm rate. The analysis also provides implementation recommendations and usage boundaries for the 2D-𝑃(sub 𝑐) method.
Hall, Doyle T.↗
Implementation Recommendations and Usage Boundaries for the Two-Dimensional Probability of Collision Calculation
The two-dimensional (2D) probability of collision (𝑃𝑐) estimation method relies on several assumptions that must be satisfied for accurate results. Monte Carlo analysis of ~44,000 conjunctions indicates that 2D-𝑃(sub 𝑐) pro-vides accurate estimates for most typical conjunctions, but occasionally underestimates 𝑃(sub 𝑐) significantly, indicating an assumption violation. A test to detect large-amplitude underestimation inaccuracies can be based on how much “offset-from-TCA” 2D-𝑃(sub 𝑐) values vary during a well-defined time interval bracketing closest approach. The test successfully detects all large-amplitude 2D-𝑃(sub 𝑐) underestimations found to date, but with a high false-alarm rate. The analysis also provides implementation recommendations and usage boundaries for the 2D-𝑃(sub 𝑐) method.
Hall, Doyle T.↗
Probability of satellite collision
A method is presented for computing the probability of a collision between a particular artificial earth satellite and any one of the total population of earth satellites. The collision hazard incurred by the proposed modular Space Station is assessed using the technique presented. The results of a parametric study to determine what type of satellite orbits produce the greatest contribution to the total collision probability are presented. Collision probability for the Space Station is given as a function of Space Station altitude and inclination. Collision probability was also parameterized over miss distance and mission duration.
Mccarter, J. W.↗
Metrics on Predictive Probability of Collision Assessments Reliant on Object State Updates
The probability of collision (Pc) metric is the dominant method of risk charac-terization for evaluating satellite close approaches, and has become of particular interest following major debris production incidents. However, confidence in the use of Pc for actionability has been questioned due in part due to its non-intuitive and often significant corrections following messaging updates. This paper assesses predictively computed probabilities and confidence bounds on Pc values corresponding to orbit determination updates by comparing these predic-tive values to observed operational values. This paper also presents use cases for this methodology in assessing event actionability.
conjunction assessment↗
Realistic Covariance Prediction For the Earth Science Constellations
Routine satellite operations for the Earth Science Constellations (ESC) include collision risk assessment between members of the constellations and other orbiting space objects. One component of the risk assessment process is computing the collision probability between two space objects. The collision probability is computed via Monte Carlo techniques as well as numerically integrating relative probability density functions. Each algorithm takes as inputs state vector and state vector uncertainty information for both objects. The state vector uncertainty information is expressed in terms of a covariance matrix. The collision probability computation is only as good as the inputs. Therefore, to obtain a collision calculation that is a useful decision-making metric, realistic covariance matrices must be used as inputs to the calculation. This paper describes the process used by NASA Goddard's Earth Science Mission Operations Project to generate realistic covariance predictions for three of the ESC satellites: Aqua, Aura, and Terra
Duncan, Matthew↗
Realistic Covariance Prediction for the Earth Science Constellation
Routine satellite operations for the Earth Science Constellation (ESC) include collision risk assessment between members of the constellation and other orbiting space objects. One component of the risk assessment process is computing the collision probability between two space objects. The collision probability is computed using Monte Carlo techniques as well as by numerically integrating relative state probability density functions. Each algorithm takes as inputs state vector and state vector uncertainty information for both objects. The state vector uncertainty information is expressed in terms of a covariance matrix. The collision probability computation is only as good as the inputs. Therefore, to obtain a collision calculation that is a useful decision-making metric, realistic covariance matrices must be used as inputs to the calculation. This paper describes the process used by the NASA/Goddard Space Flight Center's Earth Science Mission Operations Project to generate realistic covariance predictions for three of the Earth Science Constellation satellites: Aqua, Aura and Terra.
Duncan, Matthew↗
Effect of Cross-Correlation of Orbital Error on Probability of Collision Determination
This paper discusses the effect of global model error on probability of collision (Pc) determination. Modifications to the Pc formulation for cross-correlation of orbital error in prediction are developed and assessed for recent conjunctions. While specific geometries can be identified or constructed to produce significant change in Pc for the modified formulation, it is of operational interest to quantify the relative occurrence of such cases for satellite conjunction risk assessment. Such analysis is feasible per data collections in place over the past year.
Cross-correlation↗
Relative Velocity as a Metric for Probability of Collision Calculations
Collision risk assessment metrics, such as the probability of collision calculation, are based largely on assumptions about the interaction of two objects during their close approach. Specifically, the approach to probabilistic risk assessment can be performed more easily if the relative trajectories of the two close approach objects are assumed to be linear during the encounter. It is shown in this analysis that one factor in determining linearity is the relative velocity of the two encountering bodies, in that the assumption of linearity breaks down at low relative approach velocities. The first part of this analysis is the determination of the relative velocity threshold below which the assumption of linearity becomes invalid. The second part is a statistical study of conjunction interactions between representative asset spacecraft and the associated debris field environment to determine the likelihood of encountering a low relative velocity close approach. This analysis is performed for both the LEO and GEO orbit regimes. Both parts comment on the resulting effects to collision risk assessment operations.
Frigm, Ryan Clayton↗
Setting the Bar for the Replacement of the Probability of Collision Metric
To date, satellite conjunction assessment (CA) risk analysis has largely embraced the probability of collision (Pc) as the omnibus metric to evaluate collision likelihood, and its use in such assessments has mostly been straightforward: at the point at which a conjunction mitigation decision is required, the calculated Pc is compared to a threshold; and if the calculated Pc exceeds that threshold, then a mitigation action is warranted. With only minor variation, this approach is employed by major CA risk assessment centers (e.g., NASA, EUSST, CNES, JAXA) and is advanced as the preferred method in the published CA best practices handbooks. Despite this near unanimity of operational practice, there is a major strain of secondary literature critical of the Pc and willing to propose alternatives. Alfano (2005) pointed out the ability of the Pc to underrepresent the risk in certain situations and counselled a maximum Pc construct. Carpenter (2017, 2019) reiterated this criticism and proposed using instead a confidence interval on the miss distance. Balch et al. (2019) identified what they argued was a defect in the entire Bayesian Pc construct and believed that the use of a more conservative methodology based on covariance ellipsoid overlap was necessary. Delande (2022) introduced the framework of collision “plausibility” to the risk assessment process and sketched out how this might be used operationally. Elkantassi (2022) published a full development of the miss distance confidence interval approach and applied it to several worked examples. While these different approaches to collision risk assessment do differ in their details, they all converge on two central points: first, the Pc’s failure to give an adequate expression of the risk in dilution region situations is a fatal flaw; and second, a conjunction should be presumed risky and in need of mitigation until the evidence of the situation can establish otherwise. These criticisms, if correct, would counsel a number of modifications to current CA operational practice; as such, they force a re-examination of fundamental aspects of the CA problem, including the following: 1. Is the CA risk assessment a probability problem, a statistics problem, or something else? 2. If it is a statistics problem, does it lend itself naturally to a hypothesis test construction? 3. If it can be construed as a hypothesis test, what form should the null hypothesis take, to wit: what constraints exist on the choice of the null hypothesis, what selections are in best alignment with all of the attendant parameters of the problem, and what is implied philosophically by different choices? 4. What are the implications of using the different proposed risk assessment parameters for CA? This question should be answered both in determining how frequently the dilution region situation cited by the critics of the Pc actually appears in an operationally significant manner and the missed detection and false alarm rates of all of the proposed risk assessment metrics, compared both to the Pc and to each other. This paper explores and offers preliminary answers to the above questions, presenting a researched treatment of the philosophical nature of the CA problem and the null hypothesis choice that achieves the greatest consistency with all of the different aspects of operational CA conduct. It then profiles all of the different proposed risk assessment metrics enumerated in the earlier paragraph against an extremely large database of conjunction events at both the 550km and 700km altitudes. The combination of the philosophical exploration of the CA problem and the results of the profiling activity articulates what a risk assessment metric will need to demonstrate, in terms of both innate construction and performance, in order to be a true competitor to the Pc.
conjunction assessment↗
Trending in Probability of Collision Measurements via a Bayesian Zero-Inflated Beta Mixed Model
We investigate the performance of a generalized linear mixed model in predicting the Probabilities of Collision (Pc) for conjunction events. Specifically, we apply this model to the log(sub 10) transformation of these probabilities and argue that this transformation yields values that can be considered bounded in practice. Additionally, this bounded random variable, after scaling, is zero-inflated. Consequently, we model these values using the zero-inflated Beta distribution, and utilize the Bayesian paradigm and the mixed model framework to borrow information from past and current events. This provides a natural way to model the data and provides a basis for answering questions of interest, such as what is the likelihood of observing a probability of collision equal to the effective value of zero on a subsequent observation.
Conjuction Assessment↗
Evolution of the Debris Cloud Generated by the Fengyun-1C Fragmentation Event
The cloud of cataloged debris produced in low earth orbit by the fragmentation of the Fengyun-1C spacecraft was propagated for 15 years, taking into account all relevant perturbations. Unfortunately, the cloud resulted to be very stable, not suffering substantial debris decay during the time span considered. The only significant short term evolution was the differential spreading of the orbital planes of the fragments, leading to the formation of a debris shell around the earth approximately 7-8 months after the breakup, and the perigee precession of the elliptical orbits. Both effects will render the shell more "isotropic" in the coming years. The immediate consequence of the Chinese anti-satellite test, carried out in an orbital regime populated by many important operational satellites, was to increase significantly the probability of collision with man-made debris. For the two Italian spacecraft launched in the first half of 2007, the collision probability with cataloged objects increased by 12% for AGILE, in equatorial orbit, and by 38% for COSMO-SkyMed 1, in sun-synchronous orbit.
Pardini, Carmen↗
Earth Observing System Covariance Realism
The purpose of covariance realism is to properly size a primary object's covariance in order to add validity to the calculation of the probability of collision. The covariance realism technique in this paper consists of three parts: collection/calculation of definitive state estimates through orbit determination, calculation of covariance realism test statistics at each covariance propagation point, and proper assessment of those test statistics. An empirical cumulative distribution function (ECDF) Goodness-of-Fit (GOF) method is employed to determine if a covariance is properly sized by comparing the empirical distribution of Mahalanobis distance calculations to the hypothesized parent 3-DoF chi-squared distribution. To realistically size a covariance for collision probability calculations, this study uses a state noise compensation algorithm that adds process noise to the definitive epoch covariance to account for uncertainty in the force model. Process noise is added until the GOF tests pass a group significance level threshold. The results of this study indicate that when outliers attributed to persistently high or extreme levels of solar activity are removed, the aforementioned covariance realism compensation method produces a tuned covariance with up to 80 to 90% of the covariance propagation timespan passing (against a 60% minimum passing threshold) the GOF tests-a quite satisfactory and useful result.
Realism↗
Trending in Probability of Collision Measurements
A simple model is proposed to predict the behavior of Probabilities of Collision (P(sub c)) for conjunction events. The model attempts to predict the location and magnitude of the peak P(sub c) value for an event by assuming the progression of P(sub c) values can be modeled to first order by a downward-opening parabola. To incorporate prior information from a large database of past conjunctions, the Bayes paradigm is utilized; and the operating characteristics of the model are established through a large simulation study. Though the model is simple, it performs well in predicting the temporal location of the peak (P(sub c)) and thus shows promise as a decision aid in operational conjunction assessment risk analysis.
Conjunction↗
Effect of Cross-Correlation of Orbital Error on Probability of Collision Determination
This paper discusses the effect of global model error on probability of collision (Pc) determination. Modifications to the Pc formulation for cross-correlation of orbital error in prediction are developed and assessed for recent conjunctions. While specific geometries can be identified or constructed to produce significant change in Pc for the modified formulation, it is of operational interest to quanti- fy the relative occurrence of such cases for satellite conjunction risk assessment. Large-scale analysis is feasible per data collections in place over the past year.
Steve Casali↗