Computing of robust and improved signal-in-space accuracy parameters in a regional or global navigation satellite system

1. A method for computing, via a processor, Signal-in-Space Accuracy (SISA) parameters in a regional or Global Navigation Satellite System (GNSS), the method comprising:

at least one of:

determining individual Signal-in-Space Error (SISE) vectors by mapping the individual SISE vectors to a service area;

accumulating sample sets;

processing density functions of estimations of accumulated sample sets;

individually overbounding density functions in one of an overbounding sense related to the regional or GNSS or in a paired overbounding with excess mass sense; and

selecting a worst case according to predefined requirements.

2. A method for computing, via a processor, Signal-in-Space Accuracy (SISA) parameters in a regional or Global Navigation Satellite System (GNSS), the method comprising:

at least one of:

determining individual Signal-in-Space Error (SISE) vectors by mapping the individual SISE vectors to a service area;

accumulating sample sets;

processing density functions of estimations of accumulated sample sets;

individually overbounding density functions in one of an overbounding sense related to the regional or GNSS or in a paired overbounding with excess mass sense; and

selecting a worst case according to predefined requirements,

wherein the mapping of the individual SISE vectors to the service area comprises:

projecting a four-dimensional individual SISE vector to a finite number of dedicated directions pointing into the service area such that the service area is covered sufficiently,

wherein the projecting utilizes a reference frame that is formed by one of:

a satellite fixed reference, so that the reference frame is as seen from the navigation satellite antenna and which is fixed to the satellite;

a first orbit fixed reference frame defined by a “satellite-Earth” vector, an orthogonal projection of an “in flight direction” vector of the satellite that is perpendicular to “satellite-Earth” vector, and a “cross flight direction” vector that is perpendicular to the “satellite-Earth” and “in flight direction” vectors; and

a second orbit fixed reference frame defined by the “in flight direction” vector, an orthogonal projection of the “satellite-Earth” vector that is perpendicular to the “in flight direction” vector, and the “cross flight direction” vector that is to the “satellite-Earth” and “in flight direction” vectors.

3. A method for computing, via a processor, Signal-in-Space Accuracy (SISA) parameters in a regional or Global Navigation Satellite System (GNSS), the method comprising:

at least one of:

determining individual Signal-in-Space Error (SISE) vectors by mapping the individual SISE vectors to a service area;

accumulating sample sets;

processing density functions of estimations of accumulated sample sets;

individually overbounding density functions in one of an overbounding sense related to the regional or GNSS or in a paired overbounding with excess mass sense; and

selecting a worst case according to predefined requirements,

wherein the accumulating of empirical sample sets comprises one of:

accumulating per direction, or

accumulating per dedicated conditions; and

wherein when more than one satellite is involved in the accumulating of empirical sample sets, accumulating per direction and per satellite.

4. The method of claim 3 , wherein the dedicated conditions comprise eclipse condition of satellite, clock type, signal type, or satellite (build) groupings.

5. The method of claim 3 , wherein the accumulating empirical sample sets further comprises, for paired overbounding with excess mass, widening of projected errors by:

estimating a mean for each set of accumulated samples;

subtracting a dedicated amount, reflecting a small deterministic error, from all samples of the set which are essentially smaller than the estimated mean; and

adding a same amount to all samples of the set which are essentially greater than the estimated mean.

6. The method of claim 3 , wherein the accumulating empirical sample sets further comprises, for overbounding related to the relative or GNSS, widening projected errors by:

estimating a mean for each set of accumulated samples;

calculating a SISA parameter for the samples of the set;

subtracting a dedicated amount, reflecting a small deterministic error, from all samples of the set, and then calculating the SISA parameter to identify a worst case calculated SISA parameter and the calculated SISA parameter preceding the identified worst case calculated SISA parameter; and

adding a same amount to all samples of an original set, and then calculating the SISA parameter to identify a worst case calculated SISA parameter and the calculated SISA parameter preceding the identified worst case calculated SISA parameter.

7. A method for computing, via a processor, Signal-in-Space Accuracy (SISA) parameters in a regional or Global Navigation Satellite System (GNSS), the method comprising:

at least one of:

determining individual Signal-in-Space Error (SISE) vectors by mapping the individual SISE vectors to a service area;

accumulating sample sets;

processing density functions of estimations of accumulated sample sets;

individually overbounding density functions in one of an overbounding sense related to the regional or GNSS or in a paired overbounding with excess mass sense; and

selecting a worst case according to predefined requirements,

wherein the processing density functions of estimations of accumulated sample sets comprises deriving an estimation of the probability density of a corresponding underlying random variable by applying one of a kernel density estimator or an adaptive kernel density estimator method.

8. The method of claim 7 , wherein a kernel k is selected from the group consisting of Epanechnikov, Biweight (Quartic), Triweight, Gaussian, and Cosine kernels.

9. The method of claim 7 , wherein a kernel k is selected from the group consisting of cardinal B-splines of any order and re-centered to zero.

10. The method of claim 9 , wherein the cardinal B-splines have an order of one of 3 or 4.

11. The method of claim 7 , wherein a bandwidth h is derived automatically via optimality criteria selected from the group consisting of:

a cross-validation based on minimizing the integrated squared error (ISE); a mean integrated squared error (MISE), or plug-in methods based on minimizing the asymptotic mean integrated squared errors (AMISE);

a least squares cross-validation that includes minimizing ISE;

a bandwidth factorized smoothed cross-validation that includes minimizing MISE; and

at least one of a Park & Marron Plug-In and the Sheather & Jones Plug-In, which are based on AMISE.

12. The method of claim 11 , wherein the individually overbounding in the overbounding sense related to the regional or GNSS comprises approximating a SISA parameter using the equation:

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wherein f is a kernel density estimator for the unknown probability density p, and an interval [0, L — 0] is discretized, where L — 0 is any number such that the integral ∫ −L L p(t)dt is numerically one, and wherein a corresponding maximal value of the quotient expressing SISA in equation (1) is sought.

13. A computer program stored on a non-transitory, tangible storage medium that stores a set of code that is executable by a processor to direct the processor to perform a method for computing Signal-in-Space Accuracy (SISA) parameters in a regional or Global Navigation Satellite System (GNSS), the computer program comprising:

code executable to determine individual Signal-in-Space Error (SISE) vectors;

code executable to map the individual SISE vectors to a service area;

code executable to accumulate empirical sample sets;

code executable to process density functions of estimations of the accumulated sample sets;

code executable to individually overbound density functions in one of an overbounding sense related to the regional or GNSS or in a paired overbounding with excess mass sense; and

code executable to select a worst case according to predefined requirements.

14. A non-transitory, tangible storage medium comprising the computer program of claim 13 .

15. An apparatus for computing Signal-in-Space Accuracy (SISA) parameters for a navigation system, comprising:

a navigation system comprising a navigation signal generator;

an individual Signal-in-Space Error (SISE) vector determiner associated with the navigation signal generator;

a service area mapper associated with the navigation signal generator configured to map the individual SISE vectors;

empirical sample sets accumulator for each clock/position signal generator in the navigation system;

a calculator structured to calculate density functions of estimations of the accumulated sample sets;

a device structured to individually overbound density functions in one of an overbounding sense related to the navigation system or in a paired overbounding with excess mass sense; and

a worst case selector structured to identify and select a worst case condition based upon predefined requirements.

16. The apparatus of claim 15 , wherein the navigation system comprises a regional or Global Navigation Satellite System (GNSS).

17. The apparatus of claim 15 , wherein the navigation system is not a satellite-based system.

18. The apparatus of claim 15 , wherein the navigation signals comprise at least one of position and clock data.

19. A method for computing, via a processor, Signal-in-Space Accuracy (SISA) parameters in a regional or Global Navigation Satellite System (GNSS) comprising:

determining individual Signal-in-Space Error (SISE) vectors;

mapping the individual SISE vectors to a service area;

accumulating empirical sample set according to a direction;

processing density functions of estimations of the accumulated empirical sample sets;

individually overbounding density functions in one of an overbounding sense related to the regional or GNSS or in a paired overbounding with excess mass sense; and

selecting a worst case direction.

20. The method of claim 19 , further comprising broadcasting the SISA parameter in the regional or GNSS.