1 Introduction
Space Situational Awareness (SSA) concerns itself with having the actionable knowledge to predict, deter, avoid, operate through, recover from, or attribute cause to the loss, disruption, degradation, or denial of space services, capabilities, or activities. Maintaining a catalog of socalled RSOs can be cast as a typical multiobject detection and tracking problem. The individual state of each object consists of kinematic and relevant model parameters to the construction of the catalog and/or its exploitation for various activities related to space, such as mission planning, conjunction analysis to prevent collisions, etc. Typically, the individual state of an orbiting object describes a kinematic relationship (Kepler elements, or position/velocity coordinates), but may include additional characteristics such as its attitude, its ballistic coefficient, etc.
The main challenge to the resolution of most detection and tracking problems is to appropriately address the various sources of uncertainty involved in its different components, due to the limited knowledge in the signal processing chain affecting the acquisition of information on the population of objects. The SSA problem is no different, and the relevant sources of uncertainty can be sorted in two broad categories:

To first order, the perturbing accelerations on the RSO population are known, gravity being the dominant perturbing source. However, no orbital propagator is able to produce the exact dynamical state of an RSO, since the various physical perturbations affecting orbiting trajectories are either approximated, discarded on purpose for the sake of algorithmic efficiency, or simply unknown to the space analyst. In addition, satellites owned by others may transition to new orbital paths unknown to the analyst. New RSOs entering some specific orbital region, either through a collision or a new launching event, are not all accounted for and thus the number of RSOs remain uncertain.

The mechanisms through which observations (or opinions) are produced by the data sources are partially known, at best. Traditional, physicsbased sensors like radars, telescopes, or cameras are subject to missed detections, false positives, observation noise, from which some statistical description is usually available. On the other hand, the TLEs produced by the U.S. Strategic Command (USStratCom) are publicly available, but no information regarding the accuracy of the orbital elements or the confidence in the object labeling are available.
The Bayesian estimation framework is a very popular approach to solve multiobject detection and tracking problems, and the origin of the overwhelming majority of modern tracking algorithms. Its key features are that

the uncertainty on each component of the system is modeled with a random variable and characterized with a probability distribution, and

a probability distribution can be updated sequentially, or filtered, with the availability of new information regarding the corresponding random variable.
Bayesian filters then maintain a probabilistic description of the population of objects and propagate it through time, updating it whenever new observations are collected from a data source, also described through a probabilistic model. The characteristics of a specific Bayesian filter primarily depends on the nature of the probabilistic description representing the population of objects; most notably, through individual tracks (Blackman, 2004), labeled random finite sets (Vo and Vo, 2013), or stochastic populations (Houssineau and Clark, 2016; Delande et al., 2016). Applications of these various multiobject filtering techniques to the context of SSA can be found in DeMars et al. (2012), in Jones and Vo (2015); Jones et al. (2016), and in Delande et al. (2018, 2017), respectively.
One of the key challenges of Bayesian estimation is to maintain an appropriate representation of the uncertainty on the components of the system, especially when information about their state is scarce. A typical example in the context of SSA is the initial orbit determination procedure through which the possible values for the dynamical state (position, velocity coordinates) of a newlydetected RSO are restricted to a specific subspace known as the admissible region in DeMars and Jah (2013). The physical considerations behind the initial orbit determination procedure do not provide information on whether any dynamical state within the admissible region is likelier than the other, but it does not indicate that all states within the admissible region are equally likely either^{1}^{1}1It turns out that they are not, since a more restrictive initial orbit determination procedure can lead to a constrained admissible region (DeMars et al., 2012).. However, a probabilistic interpretation of the RSO’s initial state can lead to a uniform probability distribution on the admissible region, thus producing a description of the RSO’s state that is not inferred from the information available to the analyst. It thus appears that probability density functions are inappropriate tools to describe admissible regions, as argued in a recent study (Worthy III and Holzinger, 2017b). Another example is the exploitation of humanbased or semantic data sources such as TLEs or natural languages statements: they could provide a wealth of information regarding RSOs, but the lack of statistical information on their accuracy/truthfulness makes their probabilistic representation, and thus their integration to a Bayesian tracking filter, difficult and largely unexplored to this day.
Alternatives to the standard probabilistic representation of uncertainty exist, such as fuzzy logic, imprecise probabilities, possibility theory, fuzzy random sets, and DempsterShafer theory (Zadeh, 1965; Walley, 1991; Dempster, 1967; Shafer, 1976; Dubois and Prade, 1983; Yen, 1990; Friedman and Halpern, 2001); recently, the DempsterShafer theory has been exploited to approach the initial orbit determination problem with admissible regions in Worthy III and Holzinger (2017a). While these methods have been widely used to describe the uncertainty referring to a fixed unknown state, their exploitation to the estimation of dynamical systems is less straightforward, limiting their applicability to sequential estimation problems – that is, to the design of detection and tracking filters.
A recent alternative is given by the outer probability measures as introduced in Houssineau and Bishop (2017). They aim at proposing a “prejudicefree” representation of the uncertainty on a system through a natural construction that is derived from the available information, and nothing more. Built from fundamental tools of measure theory, o.p.m.s are compatible with the Bayesian filtering framework and can be integrated in a complex system where some uncertain components are described with pdfs, while others are described with o.p.m.s.
In this paper, we will focus on the representation of uncertainty for SSA data sources supporting a RSO tracking algorithm. More specifically, we will show that radar observations (physicsbased information) and TLEs (humanbased information) can both be represented by o.p.m.s and integrated into a Bayesian filtering algorithm, where the information on the RSO’s state is maintained with a probability distribution. Sec. 2 presents the concepts of o.p.m. and possibility function, and Sec. 3 describes the filtering equations of the singleobject Bayesian tracking algorithm exploiting possibility functions. Sec. 4 focuses on the modeling of the data sources relevant to this paper, i.e., a radar with Doppler effect and a “TLEgenerator”. Then, Sec. 5 describes the construction of a target tracking scenario with realisticallysimulated radar observations and real TLE data provided by the USStratCom, and gives a detailed implementation of the singletarget tracking algorithm. Sec. 6 presents the filtering results. A discussion on future works follows in Sec. 7, and Sec. 8 concludes.
2 Uncertain variables and o.p.m.s
2.1 Random and nonrandom uncertainty
Assume some system, whose state is described by some state space , on which some operator – say, a space analyst – possesses some knowledge. We shall decompose the uncertainty on the system’s state in two different parts:

The random uncertainty is the inherent component, that exists as a natural disposition of the system and independently of the analyst, sometimes known as the aleatoric uncertainty,

The nonrandom uncertainty is the extraneous component, that exists only with respect to (w.r.t.) the information possessed by the analyst, sometimes known as the epistemic uncertainty.
The random component can be seen as the remaining uncertainty on a system’s state, once the analyst possesses all the information on the system that could be acquired. It is unclear if physical systems described on the macroscopic level, such as an RSO’s dynamical state, can be anything but deterministic, or if there are random effects that cannot be explained assuming perfect knowledge of the physical laws governing the orbital motion model. It is, in the end, irrelevant to the estimation problem that focuses on reducing the nonrandom – or epistemic – uncertainty.
2.2 Possibility functions
The random uncertainty, inherent to the system, shall be characterized by a probability distribution on the state space .^{2}^{2}2For the sake of simplicity, we shall assume that admits a density w.r.t. some reference measure on , also called , and we shall handle pdfs rather than probability distributions, whenever applicable, in the rest of the paper. The nonrandom uncertainty reflects the information on the system’s state possessed by the analyst. In the simplest case, it is maintained by a single function satisfying
(1) 
for any appropriate^{3}^{3}3The subset must be measurable, for the integral to be welldefined. For the sake of simplicity this requirement on subsets, whenever applicable, will be omitted in the rest of the paper. subset of . The function is assumed to have maximum for the sake of convenience. In other words, while the pdf describing the system’s state is inaccessible to the analyst, they maintain an upper bound reflecting the information they possess about the system. A simple example of upper bound is given in Fig. 1.
The nonnegative, measurable functions on with maximum are also called possibility functions (Houssineau, 2015), or possibility distribution as in Dubois and Prade (1983). Boxshaped possibility functions such as illustrated in Fig. 1 are convenient to represent negative evidence of the system’s state, i.e., to exclude regions of the state space as possible values without inferring anything else – most notably, without assuming that all the remaining values are equally probable.
In many situations, such as the modeling of a physicsbased sensor, more refined possibility functions are necessary. Similarly as for probability distributions, we define Gaussian possibility functions as functions on , , of the form
for some and some positivedefinite matrix , where denotes the matrix transposition. For the sake of convenience, we will refer to and
as the mean and variance of
, although these concepts do not bear the statistical meaning of their namesakes for probability distributions. A family of Gaussian possibility functions is illustrated in Fig. 2.Fig. 2 also illustrates two particular possibility functions. The indicator function denotes full knowledge about the system, known to have state almost surely, since the pdf with mass concentrated in is the only pdf on satisfying (1) when . Conversely, the indicator denotes the total absence of information about the system, since every pdf on satisfy (1) when . The latter possibility represents a truly uninformative uncertainty, for one does not infer anything regarding the system’s state, and in particular does not infer that all the states in are equally probable: indeed, the uniform probability distribution on does satisfy (1) when , but so do any other probability distribution on .
2.3 Uncertain variables and o.p.m.s
In the general case, the information on some system’s state maintained by an analyst can assume a complex form built from a combination of possibility functions. More formally, it can be represented through a probability distribution on the space of possibility functions such that
(2) 
for any subset of . In practical problems, a finite combination of possibility functions is often sufficient to describe the uncertainty regarding many systems; in the scope of this paper, we will see in Sec. 4 that a single possibility function is enough to describe various data sources relevant to the context of SSA.
The information given by induces an outer probability measure on given by
(3) 
for any subset of . The scalar reads as the possibility that the system’s state is in . This is to be contrasted with the probability measure characterizing the inherent randomness of the object’s state, independently of the information possessed by the analyst, and given by
(4) 
From (2) it follows that
(5) 
for any subset of , where denotes the set difference. In other words, an analyst with information represented by cannot (in the general case) evaluate the probability for the object to lie within , but they may bound its value.
Note that an o.p.m. represents the uncertainty of an analyst in the state of some system that might not be random. In that case, the system has a fully deterministic behavior and the mass of the pdf in (4) is concentrated to some state – unknown to the analyst, as they are unaware of the nature of . For this reason, the notion of uncertain variables is introduced as a replacement for random variables to describe uncertain systems (Houssineau and Bishop, 2017).
2.4 A simple example of o.p.m.s
We shall describe two distinct levels of information, supported by the same subsets , of the state space illustrated in Fig. 3.
Suppose first that the information maintained by the analyst is based solely on the possibility function – that is, the probability measure in (3) gives weight to the singleton . Following (3), the information is represented by the o.p.m.
(6) 
where is the Iverson bracket^{4}^{4}4 is the event is true, and zero otherwise.. The analyst wishes to gauge the probability that the object’s state lies within : from Eq. (6) we have
and from Eq. (5) the analyst concludes that
that is, the object lies in almost surely. The analyst then wishes to inquire about the presence of the object in some smaller region : from Eq. (6) we have
and from Eq. (5) the analyst concludes that
i.e., the analyst is clueless about the distribution of the object within . In conclusion, the analyst infers that the object is in with certainty, but nothing more.
Suppose now that the information maintained by the analyst is based on the two possibility functions , , with respective weights and , where . Following (3), the information is now represented by the o.p.m.
(7) 
The analyst wishes to gauge the probability that the object’s state lies within : from Eq. (7) we have
and from Eq. (5) the analyst concludes that
that is, the object lies in with probability . The analyst then wishes to inquire about the presence of the object in some smaller region : from Eq. (7) we have
and from Eq. (5) the analyst concludes that
i.e., the analyst is clueless about the distribution of the object within . A similar reasoning applies in region ; in conclusion, the analyst infers that the object is in with probability , in with probability , but nothing more.
2.5 From o.p.m.s to pdfs
We see in the example illustrated in the previous section that provides a more refined level of information on the system’s state than , because it imposes tighter constraints on the system’s underlying pdf . In the extreme case where an o.p.m. imposes constraints on all the singletons of the state space, i.e., when the associated probability measure in (3) has support in the possibility functions of the form , where , then it can be shown that induces a pdf on such that
(8) 
for any subset of . That is, the o.p.m. is equivalent to the probability measure in (4), and thus the uncertainty of the analyst in the system’s state reduces to the system’s inherent random component characterized by . In other words, the analyst has nothing more to learn about the system’s state. In particular, if has its mass concentrated in some point , the analyst concludes that the system is deterministic and has state almost surely.
This connection between o.p.m.s and pdfs is of practical importance. It highlights the fact that the distinction between random and nonrandom uncertainty in some component of a complex system can be purposely ignored for the sake of simplicity; this component is then represented with a usual random variable and described with a level of information that amounts to a probability distribution. As we shall see in Sec. 3, it allows for the representation of each uncertain component of a detection and tracking problem – object’s state, prediction model, observation model specific to each sensor – with an uncertain variable, that may or may not reduce to a random variable, in a single coherent Bayesian filtering framework.
3 Bayesian filtering with uncertain variables
3.1 Conditional possibility functions and o.p.m.s
The formulation of Bayesian inference algorithms rely on the definition of appropriate conditional
pdfs describing the motion and the observation of the object of interest. It is then natural to introduce conditional possibility functions of the form describing an uncertain variable given the realization in a state space of another uncertain variable and verifyingfor any . Conditional possibility functions verify the same sort of property as conditional pdfs: letting be a possibility function describing , it holds that
(9) 
is a possibility function describing the state of in , and that
(10) 
is a possibility function describing the state of given a realization of . Note that Eq. (9) is the analogue of the prediction formula, Eq. (10
) is the analogue of Bayes’ theorem, for possibility functions. Following the approach considered in
Houssineau (2018), conditional o.p.m.s can also be defined aswhere is a probability measure on the space of conditional possibility functions^{5}^{5}5The conditioning on indicates the nature of the possibility functions in the support of but does not make it a function of ..
3.2 Bayesian filtering equations for o.p.m.s
Suppose that an analyst studies some object with state in some space , described at time ( respectively (resp.) ) by the uncertain variable (resp. ). Besides, some sensor with observations^{6}^{6}6Since TLEs are not generated by a physicsbased sensor, a more accurate term to describe one TLE point would perhaps be “opinion”. For the sake of simplicity, the terms “sensor” and “observation” are usually employed throughout this paper regardless of the nature of the data source. in some space observes the scene, and is described by the uncertain variable .
Suppose that the information that the analyst possesses on the object’s state at time is represented by an o.p.m. , and the information they possesses on the object’s evolution between time and is represented by a conditional o.p.m. , depending on a realization of . The predicted information the analyst possesses on the object’s state at time is then described by the o.p.m.
(11) 
Suppose that the information that the analyst possesses on the sensor’s observation process is represented by a conditional o.p.m. , depending on a realization of , and that an observation is collected from the sensor. The updated information the analyst possesses on the object’s state at time is then described by the o.p.m.
(12) 
The equations (11), (12) are the analogues of the time prediction and data update equations for pdfs, but for o.p.m.s; that is, they are the Bayesian filtering equations for an estimation problem in which the three sources of uncertainty – the object’s state, the object’s evolution, the sensor’s observation process – are described with arbitrary uncertain variables, that may or may not reduce to random variables.
Detection and tracking problems in the context of SSA present many challenges, whose resolution with Bayesian filters may benefit from approaching these three types of uncertainty with uncertain variables (this is briefly discussed in Sec. 7). In the context of this paper, however, we shall simply illustrate the benefit from modeling the sensors’ observation process with a single possibility function; in particular, both the information on the object’s state and the object’s evolution will be represented with usual pdfs.
3.3 A simpler case of Bayesian filtering equations
We shall suppose, then, that the o.p.m. on the object’s state at time reduces to a pdf on , and the o.p.m. on the object’s evolution reduces to a transition kernel on , depending on a realization of . The prediction formula (11) then reduces to
(13) 
where is the pdf describing the predicted information on the object’s state. As expected, we obtain the usual prediction formula for random variables.
We shall also suppose that the o.p.m. , representing the information the analyst possesses on the sensor’s observation process, is based on a single possibility function on . The update formula (12) then reduces to
(14) 
where is the pdf describing the updated information on the object’s state^{7}^{7}7The explicitly dependency of over past observations is omitted here for the sake of simplicity. As expected, we obtain an update equation with a very similar structure than the usual Bayes’ update rule, except that the observation process is described by a possibility function on rather than a likelihood function , i.e., a pdf on .
4 Modeling of Ssa data sources with o.p.m.s
In this section, we shall focus on the modeling of two data sources relevant to the context of SSA, namely, a radar ground station and USStratCom’s catalog. Both data sources provide pointlike observations in a welldefined observation space. They differ, however, on the available information about their observation process: a statistical description of the accuracy of a radarlike sensor is usually available, while no measure of uncertainty is provided (at least publicly) alongside TLEs. We will see in that section that both data sources can be represented in a natural way with a single possibility function.
4.1 Radar observations
We shall simulate, in the scope of this paper, a radar ground station with Doppler effect. Suppose that a radar is exploited at some time relevant to the scenario. The observation space describes a radarlike return , providing range, azimuth, elevation, and range rate coordinates in the topocentric frame centered on the radar ground station on the surface of Earth.
As usual for radar or telescopelike sensors, the analyst is assumed to have access to some statistical description of the accuracy of the observation process, given by some positivedefinite covariance matrix . The possibility function on in Eq. (14) is simply given by the Gaussian possibility
(15a)  
(15b) 
where is the mapping transforming coordinates from the target state space to the observation space . Note that this possibility function does not characterize the residual error of the radar observation process, a complex phenomenon that involves physical properties of the radar that may be known from the manufacturer, but also unknown perturbations due to the atmosphere, the object’s shape, etc. Perhaps the radar observation is fully deterministic, or perhaps it possesses some aleatoric uncertainty that even an observer with perfect knowledge of the radar, the object, and the middle between them would still be unable to eliminate. In any case, a characterization of the radar’s noise profile is unavailable to the analyst, who is only able to bound the error in the observation process with (15).
Note that the possibility (15) is proportional to the likelihood
(16) 
i.e., the probabilistic description usually adopted for a radar described by , characterizing the radar’s noise profile. The difference is important: while the possibility is a dimensionless function on , the likelihood is a pdf on and has dimension as the inverse of the unit volume of the observation space . The value of the likelihood in Eq. (16) thus scales with the changes on the reference measure on – say, if the azimuth and elevation angles are measured in arcseconds rather than in degrees – while the possibility in Eq. (15) does not (Delande et al., 2017). This has important consequences in the context of multitarget detection and tracking problems, as briefly discussed in Sec. 7.
4.2 TLEs
We shall query, in the scope of this paper, real TLE data produced by the USStratCom^{8}^{8}8Available at https://www.spacetrack.org.. For the sake of simplicity, we shall convert the original orbital elements of the TLEs to Cartesian coordinates in the reference EarthCentered Inertial (ECI) frame, so that a TLE is considered as a pointlike observation in the observation space . Unlike radarlike observations, little is known about the generation process behind the TLEs, let alone a statistical description of their residual error, i.e., the discrepancy between a RSO’s dynamical state and the generated TLE .
In order to produce some quantitative description of the TLEs’s accuracy, we performed a brief analysis on the satellite 0E0E operated by Planet Labs^{9}^{9}9https://www.planet.com/., on the daylong period starting from Sep 2, 2017 at midnight. Using a batch leastsquare method^{10}^{10}10Details on the orbital propagator employed are given in the description of the tracking algorithm in Sec. 5., an orbital trajectory for that satellite was produced from Global Positioning System (GPS) points provided by Planet Labs, considered as ground truth for the analysis to come. We then queried the USStratCom for the TLE points available relating to that satellite over that period, and studied the discrepancies of the TLEs as explained in the following paragraphs.
The natural observation space of a TLE is sixdimensional, since it provides information on the full dynamical state of the observed RSO. In this paper, we shall propose a simple possibility function of the observation process that aggregates the information extracted from a TLE into two meaningful dimensions, capturing salient features of the RSO’s orbital regime. Both the specific angular momentum and the specific orbital energy are conservative quantities of an orbital trajectory, in the twobody problem and assuming Keplerian motion. Denoting by (resp.
) the vector of position (
resp. velocity) coordinates of the RSO with state , and are given by(17a)  
(17b) 
where denotes Earth’s gravitational constant.
Because TLEs are expected to provide accurate information on the RSO’s orbital plane, the first component on which we shall build the possibility function is the normalized dot product
(18) 
quantifying the angle offset between the orbital planes described by some TLE and some RSO’s state . The study of the angle offset (18) on the training data is depicted in Fig. 4.
The analysis of the angle offset (18) on the training period (Fig. 4) suggests that the TLEs do indeed provide accurate information on the RSO’s orbital plane, with an extremely small deviation of magnitude below the nominal (and maximal) value of . We can then represent this information with a possibility function on the interval as shown in Fig. 5.
The possibility function in Fig. 5 reflects our acquired information on the observation process; since it is limited in scope, we follow a cautious approach and model a possibility that has the maximum value of around the values consistent with the training data, but decreases linearly towards zero over a larger region that covers five times the deviation observed on the training data.
Similarly to the radar model, this possibility function does not characterize the residual error in the RSO’s orbital plane as given by a TLE point, like a typical likelihood function would. Such a descriptive model remains largely inaccessible through the modest analysis in Fig. 4, limited to a small period of time only and to a single satellite only. Whether a statistical profile of a TLE’s residual error could be inferred from a large scale analysis is an interesting question beyond the scope of this paper.
A similar analysis can be performed on the specific orbital energy in Eq. (17b). The second component on which we shall build the possibility function is the energy offset
(19) 
which represents the difference in specific orbital energies between the orbits described by some TLE and some RSO’s state . The study of the energy offset (19) on the training data yields the results in Fig. 6.
The energy offset on the training period suggests an underestimation from the collected TLEs w.r.t. to the (supposed) ground truth, that amounts to some , with a deviation of about . Similarly to the study of the angle offset, we can represent this information with a possibility function on as shown in Fig. 7.
Interpreting the information produced by a TLE solely through the orbital plane and the specific orbital energy of the observed RSO, we can then design a possibility function for a “TLEgenerator” as
(20) 
It is easy to see that in Eq. (20) is indeed a possibility function; in particular, it reaches the maximum value of on the TLE space .
5 Scenario implementation
This section details the implementation of the Bayesian estimation scenario on which the fusion of TLEs and radarlike observations will be exploited for the detection and tracking of a single orbiting RSO.
5.1 Ground truth and observations
As for the training data on which the analysis of the TLEs was performed in Sec. 4, the reference RSO is Planet Labs’ 0E0E satellite (the corresponding CAT ID assigned by the USStratCom is 41609). The filtering period covers approximately one day, from Sep 6, 2017 00:11:47 (UTC) to Sep 7, 2017 00:00:00 (UTC). A ground truth trajectory was produced for that period from a batch leastsquare method, exploiting the GPS points provided by Planet Labs.
The TLE points relevant to the filtering period were queried from the USStratCom; their state are given in Tab. 1. A radar with Doppler effect is simulated in order to generate radar observations, following the characteristics of LeoLabs’ PFISR given in Nicolls et al. (2017). Its Field of View (FoV) is set as the ball centered on Fairbanks (Alaska, U.S.A.) with a radius of , and the RSO is observed during eight brief windows over the filtering period (see Fig. 8). The noise covariance matrix of the radar is set as the diagonal matrix
, where the standard deviations are set to the values
, , , .1st TLE (20170906 03:58:22)  
311.18  97.45  144.12  11.07e4  11.95e4  216.09 
2nd TLE (20170906 11:51:33)  
311.51  97.45  143.20  11.07e4  11.96e4  217.00 
3rd TLE (20170906 13:26:11)  
311.57  97.45  143.01  11.07e4  11.97e4  217.20 
4th TLE (20170906 18:10:06)  
311.77  97.45  142.49  11.07e4  11.99e4  217.72 
5th TLE (20170906 21:19:22)  
311.90  97.45  141.88  11.07e4  12.01e4  218.33 
5.2 Filter implementation
The object state space describes the RSO’s position and velocity coordinates in the reference ECI frame. The time flow of the filtering scenario is built as follows: the filtering period is split in even time lapses of , to which the five collection dates of TLEs are added. The resulting time flow is indexed with with
corresponding to the scenario’s initial date. In addition, the duration since epoch J2000 (in
) is given at time by .5.2.1 Time prediction step
Since the law on describing the RSO’s state is not parameterizable in a straightforward manner, the practical implementation of the Bayesian filtering equations follows a Sequential Monte Carlo (SMC) approach. At time , the posterior pdf is approximated by a set of weighted particles such that
where is the Dirac function at , and with .
The prediction kernel in Eq. (13) aims at describing a LowEarth Orbit (LEO) trajectory between epochs and , and is constructed as follows. Assuming the object has state at some epoch , the acceleration vector is given by
where the mapping computes the orbital acceleration terms modeled in the scope of this paper. In addition to the central term of Earth’s gravitational pull, it includes the following perturbations: the zonal/tesseral effects up to order and degree , the gravitational pull of the Sun and the Moon, the solar radiation pressure (assuming an AreatoMass Ratio (AMR) of , and a spherical shape with a radiation pressure coefficient of ), and a drag term based on the MSIS86 atmospheric model (assuming a ballistic coefficient of ). An additional acceleration term, accounting for the unmodeled perturbations, is built as
where is a zeromean Gaussian noise with standard deviation on each component in the object’s RadialIntrackCrosstrack (RIC) frame, and is the mapping that transforms a vector in the object’s RIC frame to the reference ECI frame. Denoting by
the total acceleration term describing the orbital dynamics, the timederivative of the target state is then given by
Given a particle , the predicted particle is then computed through
(21) 
Since the stochastic component is constant throughout the time interval , Eq. (21) can be solved with a usual numerical integrator (we used Matlab^{®}’s ode45).
5.2.2 Data update step
At time , the predicted pdf is thus approximated by the set . Three cases are now to be considered, depending on the availability of corrective data:
Case 1. No observation is available
This is by far the most frequent case, and also the most straightforward to process. Since no additional information is available on the RSO the prior is set as the posterior , i.e.
for any .
Case 2. A TLE is available
This is also straightforward to process. The update equation (14) becomes
for any .
Case 3. A radar observation is available
The data update equation in this case is similar to the TLE’s above, except that the radar possibility is substituted to the TLE possibility . However, because of the high accuracy of radar observations, implementing the data update mechanism leads to quick degeneracy among the particles, especially when the particle cloud has spread significantly after a long period without observations (Delande et al., 2017). Our approach is to find a parametrization of the predicted pdf leading to a more robust data update mechanism unaffected by the scarcity of particles.
Since pdfs representing orbital states can hardly be parametrized in a simple manner in the Cartesian coordinates , we exploit an alternative space – namely, the spherical coordinates in the sensor’s local frame – in which a Gaussian approximation is more valid. The data update procedure can be summarized as follows (Delande et al., 2017):

Transform in the spherical frame : ,

Approximate as a Gaussian distribution:
, 
Update Gaussian distribution, using observation : ,

Sample resulting distribution: ,

Transform back in the object space : .
Note that the data update step in the spherical frame is a simple linear Kalman update, since the radar is modeled with a Gaussian possibility in Eq. (15).
In any case, the particles are resampled, following the data correction step, if the efficiency ratio falls below a threshold set at .
5.2.3 Initial orbit determination
No prior information is assumed on the RSO’s state prior to its first observation, and an initial orbit determination procedure is implemented in order to determine the posterior distribution following the first radar observation. We followed the admissible region approach developed in DeMars and Jah (2013), where admissible values for the unobserved angular rates are established from the initial radar observation , exploiting internal energy constraints (Tommei et al., 2007; Farnocchia et al., 2010). We employed a similar SMC implementation of the admissible region approach as in our previous works; more details can be found in Delande et al. (2017).
6 Filtering results
In order to illustrate the exploitation of TLEs as a complementary data source, we ran the Bayesian tracking filter described in Sec. 5 on two parallel scenarios based on the same ground truth trajectory: one exploits the simulated radar observations only, the other exploits the radar observations and the TLE points collected from the USStratCom’s catalog. The filters’ outputs are averaged over 20 runs, where the observations generated for each run follow the radar model presented in Sec. 5.
We first compare the output of the filter through a Maximum A Posteriori (MAP) estimate of the RSO
’s state. Because of the frequent resampling and the scarcity of data corrective steps, the posterior particle weights are often uniformly distributed; added to the fact that the statistical moments of a
pdf on have little sense in the context of SSA^{11}^{11}11For example, even if the particle cloud spreads alongside a ray of close orbital trajectories, the weighted mean of the particle distribution may belong to an entirely different orbit. it is not straightforward to derive a MAP estimate directly from the posterior pdf . Similarly to the processing of radar observations presented in Sec. 5, we use the expression of the posterior pdf in spherical coordinates (still in the reference ECI frame), assumed Gaussian distributed, to derive a MAP estimate of the RSO’s state. The procedure can be summarized as follows:
Transform to spherical coordinates: ,

Compute the MAP in spherical coordinates as the weighted mean of the resulting distribution: ,

Transform back the MAP estimate to Cartesian coordinates: .
Note that the resulting point is not, strictly speaking, the MAP estimate of the RSO’s state, but will be considered as such for the purpose as this analysis. The MAP estimate of the filter’s output is depicted in Fig. 8.
We also compare the output of the filter in the RIC frame of the ground truth trajectory; the result is depicted in Fig. 9 for position coordinates, and in Fig. 10 for velocity coordinates.
By construction, the output of the two filters are nearly identical, from the first pass of the RSO in the radar FoV to the first data correction with a TLE point. Since the radar does not provide information on the RSO’s angular rate, the initial estimation on the RSO’s state is poor and the associated uncertainty grows significantly as soon as the RSO leaves the radar FoV and observations are unavailable. As expected, the following data correction steps with a TLE improve the quality of the MAP estimate (see Fig. 8), and sharpen the distribution significantly (see Figs 9, 10).
Due to the cautious approach in the modeling of the TLEs (see Sec. 4), the information gain of the following TLE points is much less significant, though not inexistent, once the RSO has entered the radar FoV for the third time (from around time step , onwards). This suggests that a more informative TLE model than presented in Sec. 4, built on a larger set of training data and/or on different physical quantities, may benefit the estimation to a larger extent.
7 Future works
The introduction of o.p.m.s to the context of SSA opens a wealth of possibilities for the development of Bayesian estimation algorithms, as the modeling of uncertain components in a typical SSA target tracking scenario can be revisited through o.p.m.s (and the associated possibility functions) rather than pdfs.
Many differences between the two approaches are apparent in multitarget detection and tracking scenarios, where marginalized quantities play an important role in the computation of the relative weight (i.e., probability of existence) of individual tracks. For example, because possibility functions are dimensionless, a marginalization term such as the denominator in Eq. (14), assessing the matching between the estimated object and the observation , does not scale whether the distances in the radar observation space are measured in meters or kilometers, the angles in degrees or radians, etc. The usual denominator does, however, since the pdf in Eq. (16) has units and thus scales with the change of reference measure on the observation space. This feature is often overlooked, but raises issues when such a data association event must be compared to another uncertain event that either scales differently (e.g., the association of the same estimated object to a telescopelike observation ), or does not scale at all (e.g., a missed detection event) with arbitrary changes in physical units (Delande et al., 2017).
A natural followup on our current work would be to refine the modeling of the TLEs as an uncertain data source. The analysis presented in this paper could be extended to larger period of time and to a wider range of RSOs covering different orbital regimes; studies on the accuracy of TLEs such as in Kelso (2007); Frueh and Schildknecht (2012) would also provide ground for a further refinement of the TLE model. Another natural followup would be to propose an o.p.m.based representation of other data sources already exploited in the context of SSA, such as telescopes, cameras, etc. Less conventional data sources can be found in the numerous natural language statements related to specific launch events, the illformatted data collected from unknown/untrusted amateur telescopes, etc. Natural language statements have already been approached with o.p.m.s in the design of target tracking algorithms in Bishop et al. (2018+), and this study could spearhead the integration of a range of data sources relevant to the context of SSA.
Another promising lead to follow is to explore an o.p.m.based representation of the propagated information, currently represented with a usual pdf in Eqs (13), (14). The initial orbit determination procedure (DeMars and Jah, 2013) leaves no real knowledge of the distribution of the RSO’s state within the admissible region: this is a straightforward example where the information possessed by the analyst is represented in a more natural and less prejudiced manner with a possibility function equal to one within the admissible region and zero outside, rather than a uniform pdf supported by the admissible region.
Orbital propagators designed for practical SSA tracking algorithms always operate on limited information. They approximate physical effects affecting the orbital trajectory (e.g. limited order in the zonal/tesseral components of the Earth’s gravitational pull), discard them on purpose (e.g. the Earth’s shadow is ignored) or, less conspicuously, ignore perturbation effects whose existence is simply unknown to the space analyst. Given the complexity of the orbital dynamics and the many sources of uncertainty shaping the design of an orbital propagator, a probabilistic representation of the prediction model appears overdescriptive and could benefit from an o.p.m.representation as well.
8 Conclusion
In this paper, we introduce a new representation of uncertainty for Bayesian detection and tracking algorithms in the context of SSA, based on outer probability measures rather than probability density functions. Less descriptive than pdfs, o.p.m.s do not characterize the distribution of an uncertain system’s state; rather, they aim at proposing a prejudicefree representation of the system’s state, matching the information possessed by some operator – say, a space analyst – about that system and nothing more. We show that each uncertain component of a single object Bayesian estimation problem – the estimated state, the prediction model, the data source(s) generating observations/opinions – can be represented either with an o.p.m. or a pdf in a single coherent Bayesian estimation framework.
More versatile than pdfs, o.p.m.s allow for the modeling of uncertain components for which statistical information is scarce or inexistent. In particular, we show that TLEs can be treated as a data source in a Bayesian estimation problem through a simple o.p.m.based model, and fused with radar observations in a RSO tracking algorithm. This concept is then illustrated on a scenario where TLE points queried from the USStratCom catalog and simulated radar observations are exploited in order to track a LEO satellite.
Acknowledgements
The authors wish to thank Vivek Vittaldev (Planet Labs) for his help in accessing and intepreting GPS data collected by Planet Labs, and Marcus Bever (University of Texas at Austin) for the extraction of USStratCom’s TLEs and Planet Labs’ GPS points, and their formatting into a common reference frame.
References
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