Probability of collision topology

Systems, methods, devices, and non-transitory media of the various embodiments provide for a three dimensional tool for depicting the variability of probability of collision (Pc) inputs to Pc estimates. The depictions generated by the various embodiments may be used to quantify the quality of Pc input data required to yield actionable Pc estimates. Various embodiments may provide a graphical user interface (GUI) for a computing device that may display a three dimensional depiction of the variability of Pc inputs to Pc estimates.

BACKGROUND

Satellite and space vehicle operators continually face the risk of conjunctions (e.g., collisions) between objects in space and their satellites and space vehicles. The potential for damage to satellites and space vehicles drives the need to identify conjunction threats and maneuver the satellites and space vehicles accordingly to mitigate the risk of conjunctions.

Even when a risk of conjunction is identified, many satellite and space vehicle operators are uncertain of the level of risk posed by the conjunction and whether the risk is such that an at-risk satellite and/or space vehicle should be maneuvered. Probability of collision (Pc) is a tool that operators could use to assess conjunction threats, but many satellite and space vehicle operators don't fully understand Pcincluding not understanding how Pcis estimated, what variables can be used to estimate Pc, how Pcis sensitive to changes in those variables, and how trends in Pccan be used to make maneuver selections. As such, tools for depicting Pcare needed to reduce the risk of conjunctions for satellites and space vehicles and to improve the maneuver planning and execution for satellites and space vehicles.

SUMMARY

The systems, methods, devices, and non-transitory media of the various embodiments provide for a three dimensional tool for depicting the variability of probability of collision (Pc) inputs to Pcestimates. The depictions generated by the various embodiments may be used to quantify the quality of Pcinput data required to yield actionable Pcestimates. Various embodiments may provide a graphical user interface (GUI) for a computing device that may display a three dimensional depiction of the variability of Pcinputs to Pcestimates.

DETAILED DESCRIPTION

The term “computing device” as used herein refers to any one or all of cellular telephones, smartphones, personal or mobile multi-media players, personal data assistants (PDA's), laptop computers, personal computers, servers, tablet computers, smartbooks, ultrabooks, palm-top computers, multimedia Internet enabled cellular telephones, and similar electronic devices that include a memory and a programmable processor. While specific examples are listed above, the various embodiments are generally useful in any electronic device that includes a processor and executes application programs.

Probability of collision (Pc) may be used as a tool to assess conjunction threats. Unlike other singular methods for assessing conjunction threats, such as Cartesian distance, Mahalanobis distance, maximum probability, or ellipsoids-touching tests, Pc-based methods and action thresholds may be preferable because Pcincorporates miss distance, covariance size and orientation and the sizes of the conjuncting objects in a mathematically rigorous fashion.

Various embodiments provide Pcestimation techniques and visualization user interfaces. Various embodiments provide three-dimensional, interactive tools for depicting the variability of the Pcinputs (miss distance, covariance size, and object size) to Pcestimates. Various embodiments may enable the integration of Mahalanobis space (i.e. Sigma space) and Pcto be presented to a user and used to derive bounding values for Type I (false alarm) and Type II (missed alarm) errors for prescribed screening thresholds.

Various embodiments provide three-dimensional, interactive visualizations and determined bounds of false and missed alarms for prescribed screening metrics. Various embodiments enable the depiction of inter-variability (such as smaller object size coupled with larger covariance). Various embodiments may provide a graphical user interface (GUI) for a computing device that may display a three dimensional depiction of the variability of Pcinputs to Pcestimates. The various embodiment GUI displays may provide useful information regarding miss distance, covariance size, and object size in to a single visualization. In various embodiments, the user may be enabled to zoom in and out, reorient, change the Pcthreshold plane and limits of variability, as well as background color. Various embodiments may provide GUIs for satellite tracking systems to display a three dimensional depiction of the variability of Pcinputs to Pcestimates to satellite and space vehicle operators.

The various embodiments provide a tool that may rapidly and easily show the variability of the inputs to computing collision probability. The various embodiments may show the evolution of a specific conjuncting pair over time. The various embodiments may also show historical trends for an orbit regime, a specific satellite, or a specific sensor used for the orbital computation. The various embodiments may show the ‘goodness’ of the probability calculation by relating it visually to the maximum probability. The various embodiments may be extremely useful in assessing a conjunction. The visual nature of the various embodiments, may enable rapid assessment.

The various embodiments may relate false and missed alarms threshold to their respective error bounds. This may provide for setting/analyzing screening thresholds and may enable an answer to the question “How far out should I start looking/worrying?”

The various embodiments may explore Pcsensitivities to input errors and derive required input data qualities necessary to yield actionable, decision-quality Pcmetrics in the collision avoidance maneuver process.

In various embodiments, collision probability metrics may be compared on an equal footing with other failure scenario probabilities, such as the probability that a thruster would “stick open.”

FIG. 1illustrates an embodiment satellite tracking system100including a first satellite101and a second satellite105, in orbit around the Earth150. While illustrated with only two satellites105and106, the system100may track more than two satellites or other space based objects. The system may also include a communication network170, such as the Internet, allowing a computing device180to communicate with a server190and tracking station160. The computing device180may receive inputs from the server190and/or tracking station160to determine information associated with the orbits of the satellites105and106. The computing device180may perform operations of various embodiments to output and display a three dimensional tool (e.g., a graphical user interface (GUI)) for depicting the variability of Pcinputs to Pcestimates. The depictions generated by the various embodiments may be output in various three dimensional tools (e.g., a graphical user interface) to quantify the quality of Pcinput data required to yield actionable Pcestimates.

En masse adoption of Pcby operators is ill-advised without first having a firm grasp of the input data accuracies required to meet desired Pcaccuracies. Employing Pcas one's collision avoidance maneuver decision criteria would not make sense if the inputs were so erroneous that the resulting Pcestimates are not of decision quality. Various embodiments leverage Pcsensitivities to input errors and then derive required input data qualities necessary to yield actionable, decision-quality Pcmetrics in the collision avoidance maneuver process. Various types of Pcestimation techniques are provided, such as (1) short (“linear”) contrasted with low-velocity “non-linear” encounters, (2) non-spherical object collision probability and (3) maximum probability. The inputs to these techniques that may be required include: (1) object size/shape/orientation; (2) uncertainty size/shape/orientation; and (3) nominal miss distance geometry and magnitude. In the context of the Pcestimation techniques and their resulting topographies, the sensitivity of Pcestimates to errors in each of the input types may be examined. To better understand how these parameters affect the calculation, three-dimensional tools that depict the variability of Pcestimates to variability in Pcinput parameters may be provided by the various embodiments. These depictions are then used to quantify Pcinput data quality required to yield actionable Pcestimates. This promotes informed decisions by displaying all possibilities for a given collision scenario, leaving the analyst to determine the bounds of reasonableness to apply to the resulting Pctopography.

Variations in any of the input parameters affect the Pcestimate. The accuracy of positional covariance resulting from orbit determination and subsequent propagation is often questionable. Sizes of objects is typically handled by simplistically modeling the objects as spheres by circumscribing their largest dimension, resulting in an overestimation of probability. Miss distance itself can vary dramatically due to force mismodeling, inclusion of additional observations and the presence of unknown maneuvers. Next, Pcsensitivities are “inverted” to determine the accuracies required of each of the Pcinputs in order to produce resulting Pcestimates that are accurate to within a specified percentage of the median value. This inversion yields a multi-dimensional boundary (sheet) governing input accuracy. As a worst-case allowable error, one can simply set all other error components to zero for a conjunction of interest in order to determine what the worst-case error of any one component is allowed to be. The resulting worst case errors thus obtained may then be compared to estimated errors in each input component, allowing conclusions to be made regarding: (a) how accurate the orbit solution nominal trajectory and position along that trajectory needs to be as a function of time; (b) how accurate the nominal trajectory is at OD epoch and how that accuracy degrades with time (covariance and its propagation and scaling); (c) how accurately the object size, shape and orientation must be known.

The worth of a Pcmeasurement is only as good as the inputs and assumptions that produce it. For short-term encounters the simplifying assumptions are well defined and summarized as follows: 1) The relative motion of the conjuncting objects is fast enough to be considered linear; 2) the positional errors are zero-mean, Gaussian, and uncorrelated; 3) the covariance (and its derived size, shape and orientation) is assumed constant due to the relatively short encounter period; and 4) the objects themselves are modeled as spheres.

There are a wide variety of Pcassessment methods, and in the following discussion of the various embodiments, spherical conjuncting objects and high-velocity encounters are assumed as examples. For more challenging conditions, more detailed Pcassessment methods may be substituted for the examples discussed herein, such as non-linear conjunctions and non-spherical hardbody shapes.

Satellite positional errors are shown as error ellipsoids inFIG. 2. Because the covariances which describe these error ellipsoids are expected to be uncorrelated, they are simply summed to form a single combined covariance ellipsoid centered on the primary object's location. The secondary object passes quickly through this space creating a cylindrical path that is commonly called a collision tube. A physical overlap occurs if the secondary sphere comes within a distance equal to the sum of the two radii. Thus, a condition exists for collision. As shown inFIG. 3, the volume swept out by the combined spherical object is a cylinder passing through the combined covariance (represented pictorially as an ellipsoid shell) whose axis is aligned along the relative velocity vector.

Pcis obtained by evaluating the integral of the three-dimensional probability density function (PDF) within the collision tube. Orbital bending effects reflected in the relative motion are typically negligible for high relative velocity conjunctions. This means the tube is straight, allowing the decoupling of the dimension associated with its path. Projecting the tube onto the encounter plane perpendicular to relative velocity produces a circle whose radius is the sum of the radii of the two spherical objects. The projected covariance ellipsoid becomes an ellipse as shown inFIG. 4. This representation allows us to reduce the computational dimensionality from three to two. Pcis found by evaluating the integral of the resulting two-dimensional PDF within a circle on the encounter plane perpendicular to the relative velocity at the time of closest approach.

The resulting encounter plane's two-dimensional probability equation is given in Cartesian space as in Equation (1):

where r is the combined object radius, x lies along the covariance ellipse minor axis, y lies along the major axis, xm and ym are the respective components of the projected miss distance, and σx and σy are the corresponding standard deviations. There are numerous methods to evaluate this integral. The methodology that follows is not limited to any particular computational method.

While there are many reasons to use collision probability as a collision avoidance decision metric, it must be recognized that errors in the independent variables which Pcdepends on can degrade or invalidate the estimated Pcresult. Sensitivity of estimated actual collision probability to variations in such input data (object physical size, covariance size and covariance shape) are now explored.

Covariances generated from Orbit Determination (OD) processes are often unrealistic, requiring “scale factors” to properly reflect reality. This is especially true for OD systems that are either batch least squares-based or which do not properly account for error sources via Physically Connected Process Noise (PCPN) techniques. Flight experience, comparisons with positionally well-known reference orbits, and overlap (consistency) checks often indicate that the error covariances need to be “scaled” (occasionally smaller, but typically larger) to represent the actual errors in the state estimation. Such scaling is applied to the one-sigma eigenvectors corresponding to the inner ellipsoid503inFIG. 5, yielding the scaled ellipsoid502. Typically, a single scaling factor is consistently applied to all three eigenvector directions, but in principle each eigenvector could have its own scaling value. Such scale factors can help match the OD covariance with reality, and yet: 1) is scale factor applied to covariance at OD epoch and then propagated; 2) or is scale factor applied to covariance after propagation; 3) is it rigorous to scale covariance equally in all directions; 4) how does one know what scale factor to use for each Resident Space Object (RSO); and 5) is it rigorous to assume that the alignment (eigenvectors) of the scaled covariance are the same as the non-scaled one?

An alternative, and perhaps slightly more proper, approach is to insert “consider parameters” for unmodeled, or poorly modeled, forces such as drag, Solar Radiation Pressure (SRP), etc. While such consider parameters give the orbit analyst an additional degree of freedom to try to align the covariance with observed overlap, reference orbit or independent sensor metrics, consider parameters represent an assumed forcing function which may, or may not, be present. As such, they are only effective in the orbit regimes where such forces are present (i.e. drag consider parameters are probably not overly effective above 800 km or so, and SRP consider parameters are probably not meaningful in LEO). These reasons may explain the relatively large scale factors still required in GEO to bring Batch Least Squares covariances in line with independent “truth” measurements.

As an example of covariance scaling using the fictitious conjunction case illustrated inFIG. 2, the ellipsoid corresponding to the combined error covariances for the two spacecraft is depicted as ellipsoid503inFIG. 5. A combination of reference orbit comparisons, statistical overlap tests and flight experience can be used to infer that the combined error ellipsoid must be scaled (occasionally smaller but typically larger) as depicted by the wireframe ellipsoid502shown inFIG. 5.

Variations in position covariance size can have a considerable effect on the probability calculation. The results of these variations are illustrated inFIG. 6, showing a nominal relationship for a sample case between Pcand covariance size for a fixed Cartesian miss distance and fixed combined object radius. The scaled, combined covariance that produces the maximum probability (Pcmax) defines the dilution region boundary. To the left of this boundary (vertical line inFIG. 6), lesser positional uncertainty (smaller σ) decreases collision probability. To the right of the boundary, larger positional uncertainty (greater σ) also decreases collision probability. As shown, both small and large uncertainties can produce the same mathematically correct probability (1.e-6 is given as an example here). A very large uncertainty indicates little confidence in the predicted miss distance and will result in a low probability, deceptively implying that satellite safety increases as data quality decreases.

Such a characterization of the Pcdependency on combined error covariance can be used to explore the Pcsensitivity to the aggregate orbit solution quality at the Time of Closest Approach (TCA). A full understanding of this functional dependency is crucial to understanding, again for a fixed miss distance and combined object radius, how “good” (based on the combined covariance) the orbit solutions need to be to provide an actionable Pcresult. WhileFIG. 6provides a 2-D understanding,FIG. 11discussed below illustrates a 3-D topographical representation of probability contours that may provide such an understanding of this functional dependency.

For example, in the probability dilution region (right-hand side) in this sample case, note that an increase in a covariance's corresponding eigenvalues by one Order Of Magnitude (OOM) asymptotically reduces Pcby 2 OOMs, and a factor of five eigenvalue increase maps to a Pcdecrease by a factor of fifty.

Outside the dilution region, increasing covariance by one OOM asymptotically increases Pcby about four OOMs, and a factor of five increase yields an increase in Pcby five thousand.

If the estimated actual probability was below an action threshold, the user could be led to conclude the encounter does not pose a worrisome threat. With great surety one can infer this when outside the dilution region provided, of course, the positional uncertainty is properly represented. This is because the low probability is coupled with great confidence in the predicted miss distance. Such would not be the case in the dilution region where there is little confidence in the predicted miss distance, yielding an unactionable result.

Faced with a low probability below an actionable threshold, one could use the dilution region boundary as a discriminator. If outside the dilution region then no remedial action is warranted. If inside the dilution region, rather than dismiss the conjunction as non-threatening, one should consider getting better (more current) data and re-evaluating Pc. This will help to ensure that a decision maker is not lulled into a false sense of security by a low probability calculation that may be specious.

A space object's physical size also plays an important role in the estimation of actual Pc. In a perfect world, such information would be supplied by satellite operators as a function of vehicle attitude and the vehicle's corresponding attitude flight rules. Typically such object size information is unavailable for operating spacecraft, and for debris it is almost always unavailable. Instead, object size is estimated from Radar Cross Section (RCS) measurements using a matched multi-modal/multi-regime radar-scattering cross-section-to-size model.

One such model is the Stanford Research Institute (SRI) RCS model, which assumes metallic conducting spheres with diameter d=2r to relate object detectability to object size. This simplified radar cross-section model models three regimes—the optical, Rayleigh, and Mie scattering regions—where the cross-sections are given by Equations (2), (3), and (4):

Note that there is no attempt to model the detailed interference pattern in the Mie region; rather, the Mie cross-section defines the expected maximum and minimum cross section, and the region between the end of the Rayleigh regime and the start of the Mie region is obtained via interpolation.

To examine the accuracy of such an RCS-to-size estimation, four reference spheres of known size are selected, the POPACS, LARES, LARETS, and Stella orbiting reference spheres. Each of these reference spheres is of known size, with POPACS=10 cm, LARES=37.6 cm, LARETS=21 cm and Stella=24 cm radii, respectively.

The RCS model's cross-section normalized to the optical cross-section as a function of normalized radius (κr) is shown inFIG. 7. The RCS of each sphere was then measured by Poker Flat Incoherent Scatter Radar (PFISR). The vertical lines connecting the ‘x’ symbols show the range of measured RCS values obtained from PFISR, and the filled in circle symbols show the corresponding minimum and maximum predicted values using the SRI RCS model.

Next, the PFISR RCS measurements and the equivalent time history of Joint Space Operations Center (JSpOC)-calibrated RCS values were mapped to estimated physical size using the SRI RCS model and the NASA Size Estimation Model, respectively. These size estimates could then be compared to the known actual size of each reference sphere as shown inFIGS. 8A, 8B, 8C, and 8D.FIG. 8Ais a comparison of STELLA size estimates derived from Commercial Space Operations Center (ComSpOC) and JSpOC RCS measurements.FIG. 8Bis a comparison of LARETS size estimates derived from ComSpOC and JSpOC RCS measurements.

FIG. 8Cis a comparison of LARES size estimates derived from ComSpOC and JSpOC RCS measurements.FIG. 8Dis a comparison of POPACS size estimates derived from ComSpOC and JSpOC RCS measurements. As the figures show, PFISR and JSpOC size estimates are of roughly comparable accuracy but both estimates (depending upon the radar regime assumed and viewing geometry or object asymmetry-based RCS variations) can easily be inaccurate by a factor of between 0.25 and 4.0

How do such uncertainties in object size affect Pc estimates? The parametric evaluation of Equation 1 as shown inFIG. 9indicates that increasing or decreasing the combined object radius by a factor of four increases or decreases the resulting Pc by a factor of sixteen. This “squaring” relationship is due to the fact that the combined object area in the encounter plane is πr2. Conclusion: The use of accurate object size information is critical to the accurate estimation of collision probability.

A work-around to such inaccuracies in estimated object size is to simply assume a “nominal” value and then notify decision makers of that assumption. An example of this is the20massumption used in some current Conjunction Data Message (CDM) screening processes. For the spheres analyzed here, such an assumption could result overestimation of combined object size by a factor of twenty.FIG. 9indicates that such an object size error yields Pc values that can be several orders-of-magnitude off.

The individual functional relationships in the above covariance realism and hardbody radius sections indicate a joint functional association between combined object size, combined positional error and maximum probability, as characterized inFIG. 10. This sample association assumes a combined covariance ellipsoid aspect ratio of 2.0 when projected on to the encounter plane; the association would be somewhat different for other aspect ratio values.

To use this figure, the user selects the combined hardbody radius of interest and the operator's desired Pc threshold. The intersection of the vertical line1002(hardbody radius) with the desired Pc threshold (e.g. a Pc of one in 10,000) yields line1003a maximum allowable combined covariance major one-sigma eigenvalue of approximately 600 meters, or a three-sigma equivalent of 1.8 km. If one were to assume that both objects had equivalent contributing error covariances, then this indicates that each Resident Space Object (RSO) would need to have 1.8 km combined, /√2=1.273 km maximum three-sigma eigenvalue to EVER yield an estimated collision probability of the desired Pc threshold of one in 10,000.

Probability Contour Visualization

The approach that producedFIG. 6may be extended to show a nominal relationship between Pcand covariance size for a range of miss distances while holding the covariance shape, orientation and combined hardbody object size fixed.FIGS. 11-15illustrate aspects of embodiment GUIs that may display a three dimensional depiction of the variability of Pcinputs to Pcestimates to a user of a computing device.FIG. 11shows an embodiment topographical representation of probability contours for fixed hardbody radius r in the true space. A grid is created by scaling the distance components by siwhile also scaling the standard deviations by Cj. The results are then plotted to show how probability varies with miss distances and covariance size through the following relationships in Equations (5) and (6):
Pci,j=Pc(r,sixm,siym,Cjσx,Cjσy)  (5)
di=si√{square root over (xm2+ym2)}  (6)

For a given diit is also possible to capture and store the corresponding C1that yields the maximum probability Pcmaxi. Specifically, the display output as shown inFIG. 11indicates where data of insufficient quality is present1103, the operator's Pcthreshold1102, where data of sufficient quality is present1104, and a maximum probability “ridge” line1101.

FIG. 12was formed by an HTML script that invokes a three-dimensional surface plot1200. Once created, the viewer is enabled to interactively reorient the three-dimensional plot and/or zoom in/out using any browser. To generateFIG. 12a single, random conjunction in Systems Tool Kit (STK) is simulated. The next step was to invoke STK's Advanced Conjunction Analysis Tools (AdvCAT) and then have a MATLAB program read in the data from its provider. This MATLAB program then varied s and C to create grid points through Equations 5 and 6 to generate the HTML file. Once created, STK and MATLAB were no longer needed. While STK and MATLAB are discussed in relation toFIG. 12, any conjunction analysis tool that can provide relative position and velocity vectors, associated uncertainty, and hard-body radius may be substituted for STK and any tool that can perform matrix algebra may be substituted for MATLAB and used to process the data and create the HTML file.

To visually assist the reader, a maximum probability ridge line (Pcmaxi)1201is superimposed on this surface. Also included is the original, unscaled Pcsingle-point value1205of 0.0002 at 1280 meters generated by AdvCAT (seen as a dot to the right of the ridge line) and a translucent horizontal plane1202at P=10−4. For illustration purposes the plane1202represents an acceptable probability action threshold. If below this plane1202and outside the dilution region (indicating that data is of sufficient quality) then no remedial action is suggested. It must be emphasized that a threshold of 10−4is not a recommendation or endorsement; it is the responsibility of each satellite owner/operator to decide what their acceptable probability threshold should be.

The utility of such a visualization as shown inFIG. 12becomes immediately apparent. This example shows the probability computation1205is inside the dilution region (right of maximum probability ridge line1201) and above the threshold plane1202, implying the need for further attention and better quality input data. Also, there is a specific distance where the maximum probability ridge line1201crosses the translucent horizontal plane1202. Holding all other variables equal, this distance can be used to represent an alternative screening threshold. If the distance is exceeded then the conjunction will always be below the probability action threshold; this depiction gives an indication of how far the satellite might need to maneuver.

The topology shown inFIG. 12is unique to all the inputs affecting the probability calculation: combined object size, miss distance components, and covariance size, shape, and orientation. It is perhaps more useful to construct a common, “normalized” version of this surface projection by taking the following steps. First, all conjunction probability computations are referenced to the maximum probability of each. A reference contour of the user's choice is created as just described forFIG. 12. The adjusted probability Pc′ is then mapped on to this surface by the components of its true miss distance and a transformed covariance scale factor C′. For the kth conjunction, Pck′ can be computed by scaling the estimated actual value Pckto the ratio of the contour's reference maximum Pmaxrefas follows in Equations (7), (8), and (9):

Pcmaxkcan be found by evaluating Equation 7 for different scalings of the σ values. Undoubtedly, dk′ will not match one of the referenced divalues precisely. Pcmaxrefin Equation 8 can be found for dk′ through interpolation of the Pcmaxipoints associated with di. Knowing Pk′ and dk′, the remaining step is to determine the transformed scale Ck′ that corresponds to the reference contour; this is done through bilinear interpolation of the Pci,jpoints to match Pck′ at dk′. By using this approach,FIG. 13shows that the interpolation forces all Pck′ to lie on the reference surface while preserving the ratio of Pckto its maximum for all variations of its inputs.

Variations in covariance shape and orientation will affect the calculation of Pckas well. Although the contour was created by only varying covariance size, the computation of adjusted probability Pck′ accounts for size, shape, and orientation. Its mapping on to the prime-space contour preserves the ratio of Pckto Pcmaxkwhere changes in any or all of the three variations will change the position of Pck′ on the surface. The depiction is therefore suitable for assessing probability dilution.

Regrettably, the effects of combined object radius r variations are insignificant in the prime space. The utility of examining those variations is marginal because Pck′ will remain relatively unchanged in this space. Increasing r by scale factor α causes Pckto approximately increase by a factor of α2with a nearly identical α2increase occurring for Pcmaxk. By definition, Pck′ is determined from the ratio of Pckto Pcmaxk(Equation 8) meaning the change on this surface will be negligible (imperceptible) for most cases. Also, because of this ratio, presenting a probability threshold plane in this space is not beneficial because each and every Pck′ will have its own transformed plane associated with it.

Because neither the true space nor the prime space gives a complete portrayal, a hybrid approach is taken to simultaneously display the estimated actual probability and its representative projection on a reference contour. In this space both Pckand Pck′ are displayed as functions of dk′ and Ck′. Since differing conjunctions will produce different contours, it is not constructive to present those contours in the hybrid space. Instead, a single representative contour is chosen and displayed in this space, and all Pck′ probabilities mapped to it. The ratio of Pckto Pcmaxkfor all k is preserved because the different contours are morphed into a single surface in this space. As seen inFIG. 14, each probability instance Pck1402is connected by a line to its corresponding Pck′ on the surface associated with dk′ and Ck′. In this manner a probability threshold plane can be properly shown along with estimated actual probability for all variations of its inputs while also depicting the ratio of Pckto its maximum for assessing probability dilution.

In this hybrid space the estimated actual probabilities are displayed as dots1402above the contour; for enhanced visualization the dots are representatively sized and color-coded (shown in grey scale inFIG. 14) according to the figure's Pclegend on the right. The Pckvalues are accurately depicted but the contour is not. The surface is merely meant to show where those probabilities rest relative to the maximum probability “ridge”1403. This also means that it is possible, for a given miss distance, to have the estimated actual probability appear higher than the maximum ridge line1403depending on the chosen reference contour. If such a depiction is of concern, one should display the highest contour produced by all the probability points considered. For viewing purposes it is advisable to choose the lowest contour so that all those points are above the surface and projected down.FIG. 14also has two probability points1402encircled with circle1406that reference the same spot on the contour's surface, thus demonstrating the insignificance of singular variations of combined object radius r when projected on to the contour.

Any and all factors that go in to the probability calculation can now be varied and plotted in the true, prime, or hybrid spaces. The hybrid depiction is valid even with data points having different covariance aspect ratios, due to the scaling of actual value Pckto the ratio of the contour's reference maximum Pmaxref. Mapping conjunctions into these spaces makes these visualization tools suitable for examining variations in combined object size, miss distance components, and covariance size, shape, and orientation. As previously illustrated, probability and miss distance threshold planes can also be included.

The intuitive, three-dimensional visualization tools introduced above allow the analyst to project and examine either a time sequence, or a filtered set of samples (for example, all LEO conjunctions over the past year) of conjunction probabilities on to a common surface. These depictions indicate the usability (soundness) of data feeding a conjunction screening process. Although Gaussian distributions were used for the topological depictions, any distribution can be used. Comparing the probability predictions from different sources and epochs can be easily characterized. One can discover how deep into the dilution region the conjunctions are and/or examine the progression of updates relative to the maximum probability ridge line.

FIG. 15is an example of what one might expect when looking at the time progression of a conjuncting satellite pair. The line “Y” connecting the true point to the prime plane (far right, #1) depicts the first conjunction prediction 4.5 days before TCA. The progression of the horizontal lines B shows the evolution of orbital conjunction updates; as one might expect, the predicted miss distance is varying with each update while the covariance at TCA simultaneously (and typically) improves. Depicted to the far left (#7) is the final update 8 hours prior to TCA showing a low probability well outside the dilution region, indicating that the collision risk is low and the inputs feeding the Pcanalysis yield an actionable Pcmetric.

Although Gaussian distributions were used for the topological depictions, any distribution can be used.

The various embodiment methods may also be performed partially or completely on a server. Such embodiments may be implemented on any of a variety of commercially available server devices, such as the server1600illustrated inFIG. 16. Such a server1600typically includes a processor1601coupled to volatile memory1602and a large capacity nonvolatile memory, such as a disk drive1603. The server1600may also include a floppy disc drive, compact disc (CD) or DVD disc drive1604coupled to the processor1601. The server1600may also include network access ports1605coupled to the processor1601for establishing data connections with a network1606, such as a local area network coupled to other broadcast system computers and servers. The processor1601may be any programmable microprocessor, microcomputer or multiple processor chip or chips that may be configured by software instructions (applications) to perform a variety of functions, including the functions of the various embodiments described above. Typically, software applications may be stored in the internal memory1602,1603before they are accessed and loaded into the processor1601. The processor1601may include internal memory sufficient to store the application software instructions.

The various embodiments described above may also be implemented within a variety of computing devices, such as a laptop computer1700illustrated inFIG. 17. Many laptop computers include a touchpad touch surface1717that serves as the computer's pointing device, and thus may receive drag, scroll, and flick gestures similar to those implemented on mobile computing devices equipped with a touch screen display and described above. A laptop computer1700will typically include a processor1711coupled to volatile memory1712and a large capacity nonvolatile memory, such as a disk drive1713of Flash memory. Additionally, the computer1700may have one or more antennas1708for sending and receiving electromagnetic radiation that may be connected to a wireless data link and/or cellular telephone transceiver1716coupled to the processor1711. The computer1700may also include a floppy disc drive1714and a compact disc (CD) drive1715coupled to the processor1711. In a notebook configuration, the computer housing includes the touchpad1717, the keyboard1718, and the display1719all coupled to the processor1711. Other configurations of the mobile computing device may include a computer mouse or trackball coupled to the processor (e.g., via a USB input) as are well known, which may also be used in conjunction with the various embodiments.