Covariance and Uncertainty Realism in Space Surveillance and Tracking

Characterizing uncertainty in estimating the state of a resident space object is one of the fundamentals of many space surveillance tasks.

The characterization of uncertainty in the estimate of the state of a resident space object is fundamental to many space surveillance tasks including data association, uncorrelated track (UCT) resolution, catalog maintenance, sensor tasking and scheduling, as well as space situational awareness (SSA) missions such as conjunction assessments and maneuver detection. Generally, uncertainties are classified as either aleatoric, epistemic, or a mixture of both. Aleatoric uncertainty is the natural randomness or physical variability present in the system or its environment and is thus statistical in nature. In contrast, epistemic uncertainty is uncertainty that is due to limited data or knowledge.

With respect to some terms, covariance realism means that the uncertainty in the state of an object can be represented as a Gaussian random variable and that the estimated mean and covariance of said Gaussian are the true mean and true covariance, respectively. Since the underlying dynamical processes are not always linear nor Gaussian, one may generalize covariance realism to uncertainty realism described by a potentially non-Gaussian probability density function. Uncertainty realism requires that all cumulants (beyond a state and covariance) be properly characterized. The relationship between covariance realism and uncertainty realism is that the former is a necessary but not a sufficient condition for achieving the latter. The two definitions coincide if the process is Gaussian.

The achievement of covariance or uncertainty realism is a challenging problem due to the complex and numerous sources of uncertainty. To achieve a proper characterization of uncertainty, one must account for the uncertainty sources in the system and roll these up into the uncertainty in the estimate at each needed time. Generic sources of uncertainties for point objects include the following:

  1. Structural uncertainty or model bias in the model dynamics;

  2. Uncertain parameters found in the model dynamics (including space environment) and in the measurement equation relating the dynamics to the sensor measurements;

  3. Sensor level errors including measurement noise and sensor and navigation biases;

  4. Inverse uncertainty quantification including the statistical orbit determination and bias estimation uncertainty;

  5. Propagation of uncertainty;

  6. Algorithmic uncertainty or numerical uncertainty that comes from numerical errors and numerical approximations in a computer model;

  7. Cross-tag or misassociation uncertainty;

  8. Hardware and software faults/errors.

Additional sources of uncertainty occur for medium to large objects called extended body uncertainties. For example, an extended body covering several pixels may have an overly optimistic (too small) covariance if the uncertainty of the estimated state only covers the centroid of the body.

The goal of correctly characterizing or quantifying uncertainty is not unique to astrodynamics. Indeed, the currently active field of uncertainty quantification deals with the same problem in many other areas of engineering and science.

As stated above, the correct characterization of the uncertainty in the state of each object is fundamental to many space surveillance and space situational awareness missions. The following four examples demonstrate the importance of covariance and uncertainty realism:

  1. Computation of the probability of collision for conjunction assessment;

  2. Data or track association/correlation;

  3. Maneuver detection;

  4. Sensor tasking and scheduling.

This work was done by Aubrey B. Poore, Jeffrey M. Aristoff, and Joshua T. Horwood of Numerica Corporation for the Air Force Space Command. AFRL-0292

This Brief includes a Technical Support Package (TSP).
Document cover
Covariance and Uncertainty Realism in Space Surveillance and Tracking

(reference AFRL-0292) is currently available for download from the TSP library.

Don't have an account? Sign up here.