Alejandro Cano Sanchez, Universidad Carlos III de Madrid; Manuel Sanjurjo-Rivo, Universidad Carlos III de Madrid; Joaquin Míguez, Universidad Carlos III de Madrid; Alejandro Pastor, GMV; Diego Escobar, GMV
Keywords: uncertainty quantification, covariance realism, consider analysis, covariance determination, LEO, Mahalanobis distance, chi-square distribution, catalogue, Cramer-von-Mises
Abstract:
In the increasingly over-populated space environment, the provision of Space Surveillance and Tracking (SST) and Space Traffic Management (STM) services is becoming the cornerstone for safe and efficient spacecraft operations. Many of these services, such as conjunction risk assessment, manoeuver detection, fragmentation analysis or catalogue build-up and maintenance require reliable estimations of the state of the Resident Space Objects (RSOs) and their associated uncertainty. The quality of this uncertainty is generally measured by how closely it represents reality, also known as uncertainty realism.
Under the assumption that the state of an orbiting object can be represented by Gaussian random variables, the uncertainty realism problem is reduced to covariance realism, representing the state of the RSO with its two first statistical moments (i.e. the mean and variance). This is common in most operational scenarios, where the uncertainty of the catalogued RSOs is represented by a covariance matrix, generally obtained as part of an Orbit Determination (OD) process. However, typical OD processes, based on batch least-squares algorithms, normally consider the measurements noise as the only source of uncertainty, being called the noise-only covariance. This results in the estimated covariance matrices lacking realism, being overly optimistic and jeopardizing enormously products such as collision risk assessment or catalogue maintenance. This loss of covariance realism is caused by an improper characterization of some of the uncertainty sources present in the space environment. Among them, the lack of uncertainty measures of the underlying dynamical models used to represent the atmosphere outstands as one of the main sources.
Simple but effective methods to enhance covariance realism are required in operational environments. Many operators resort to artificially scaling the covariance as a safe-margin approach. However, this hinders the physical interpretation of the uncertainty and may lead to covariance over-sizing and its consequent impact on SSA products such as an increased rate of false collision alarms. A classical approach to include the uncertainty of the dynamical models is the theory of consider parameters. In this approach, a set of additional (random, zero-mean) parameters are included in the underlying dynamical model, in such a way that the model uncertainty is represented by the variances of these parameters. However, the main drawback of this theory is that realistic variances of these consider parameters are not known a-priori.
This work applies our recent methodology to infer the variance of consider parameters based on the observed distribution of the Mahalanobis distance of the orbital differences between predicted and estimated orbits, which theoretically should follow a chi-square distribution under Gaussian assumptions. Focusing on Low Earth Orbits (LEO), the main source of uncertainty arises from the interaction of the atmosphere with the satellite motion, the atmospheric drag. This force is not only influenced by satellite dependent factors such as the cross-sectional area or the mass, but also by external factors such as the solar or geomagnetic activity that govern the atmospheric density evolution, which to this day remain difficult to model. This work proposes a consider parameter model for the atmospheric drag uncertainty, applying Empirical Distribution Function statistics such as the Cramer-von-Mises and the Kolmogorov-Smirnov distances to determine optimum variances of such parameters for covariance realism.
Following the operational goals of this work, the analysis is conducted in a catalogue scenario using simulated data of LEO satellites. The applied methodology relies on a set of orbital data to infer the consider parameter variances. However, complete and extended datasets for single objects are not generally available in most SST operational environments. For this reason, this work analyses the covariance determination methodology in a catalogue scenario, in which orbital datasets from multiple objects are accessible routinely. Thus, instead of estimating consider parameter variances tailored to a single object during a long period, the objective of this work is to determine variances that allow improving the covariance realism for different clusters of catalogued objects (i.e. different altitudes or ballistic coefficient). Altitude-dependent models of the atmospheric density uncertainty, based on statistical analysis of historic space weather data, are applied for realistic simulations together with space-correlated density perturbations. In this paper, we review the recent covariance determination methodology and develop its application to a catalogue scenario, detailing the processing chain and clustering process. Results are presented focusing on the physical interpretation of the determined consider parameter model variances and their effectiveness for improving the covariance realism in the different objects clusters of the catalogue.
Date of Conference: September 27-20, 2022
Track: Astrodynamics