Abstract:
A method for determining overbounds comprises the steps of determining conservative overbounds (q i ) of at least one error (ε i ) in a first phase space, multiplying the conservative overbounds (q i ) of errors (ε i ) in the first phase space by a first parameter (θ(−x)·2) and a second parameter (θ(x)·2), and determining an upper bound for the integrity risk at the alert limit (p w,int (AL)) in a second phase space using overbounds (q i ) of errors (ε i ) in the first phase space by the first parameter (θ(−x)·2) and the second parameter (θ(x)·2).

Description:
BACKGROUND OF THE INVENTION 
     This application claims the priority of European patent document 06 004 754.5, filed Mar. 8, 2006, the disclosure of which is expressly incorporated by reference herein. 
     The invention relates to a method and apparatus for transferring a Galileo overbound or an ICAO overbound to a pairwise overbound with excess mass (POEM). 
     For Global Navigation Satellite Systems (GNSS) based navigation systems for aviation, it must be assured that the position the system provides has sufficient integrity. This means that the probability that the navigation system supplies hazardously misleading information (HMI) should be proven to remain extremely small under all circumstances. The problem of trying to guarantee that such a system offers sufficient integrity is known as the overbounding problem, because practical solutions are necessarily conservative (bounding) with respect to the performance that is actually obtained. Further, Safety-of-life (SoL) GNSS augmentation systems must provide bounds on the probability that hazardous navigation errors may occur. 
     The integrity information sent to the user contains no explicit provisions for protecting against biases. Instead users are sent protection factors that correspond to zero-mean error distributions. The users combine the received protection factors using their own local knowledge to calculate protection levels that correspond to their position estimate. The broadcast protection factors must be sufficient such that any individual user has only a small probability (e.g. less than a one in ten million), for each approach, that their true position error exceeds the calculated protection level. The ground system for instance must guarantee these protection factors without knowing precisely where the users are, or which satellite they observe. 
     For determining the system&#39;s integrity, errors in the range domain are transformed into errors in the position domain. During the transformation of the errors in the range domain into the errors in the position domain, the corresponding error statistics (probability distribution functions of the errors) are transformed by the convolution which is necessary for such transformation. 
     In the literature, several different overbounding concepts are known. One of these concepts is the Galileo overbounding as defined by the Galileo requirements. Another concept is the paired overbounding with excess mass (POEM). The Galileo overbounding has the disadvantage that it is not preserved during convolution, whereas the POEM is preserved during convolution. That means that the convolution of two excess mass overbounding distributions will overbound the convolution of the two original convolutions. Thus, the paired overbounding concept effectively relates range domain and position domain overbounding. 
     SUMMARY OF THE INVENTION 
     One object of the present invention is to define a process and an apparatus that transforms an overbound that is not preserved during convolution into an overbound that is preserved during convolution. 
     This and other objects and advantages are achieved by the method and apparatus according to the invention, in which the Galileo Overbounding definition of a distribution is used to define parameters for a paired overbounding with excess mass (POEM) of the same distribution. This is necessary as the properties of the Galileo overbounding definition are not preserved during convolutions of distributions, whereas the paired overbounding with excess mass properties are preserved during convolutions of distributions. The convolution is necessary for the transformation from the range domain to the position domain. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING FIGURES 
         FIG. 1  illustrates a method in accordance with the present invention. 
         FIG. 2  are graphs illustrating two functions. 
         FIG. 3  is a graph illustrating the overbounding condition. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Before describing several embodiments of the invention several overbounding definitions are stated. 
     Galileo Overbounding Definition 
     For Galileo the probability density p is overbounded by a function q if the equation 
                           ∫     -   ∞       -   y       ⁢       p   ⁡     (   x   )       ⁢     ⅆ   x         +       ∫   y   ∞     ⁢       p   ⁡     (   x   )       ⁢     ⅆ   x           ≤         ∫     -   ∞       -   y       ⁢       q   ⁡     (   x   )       ⁢     ⅆ   y         +       ∫   y   ∞     ⁢       q   ⁡     (   x   )       ⁢     ⅆ   x             ⁢           ⁢     
     ⁢       for   ⁢           ⁢   all   ⁢           ⁢   y     ≥   0             (   0.1   )               
holds true.
 
     It has to be noted further, that for Galileo it is foreseen to use as the overbounding distributions only Gaussian distributions of the form 
     
       
         
           
             
               
                 
                   
                     q 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ≡ 
                   
                     
                       1 
                       
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         σ 
                       
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           - 
                           
                             
                               t 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                 σ 
                                 2 
                               
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   0.2 
                   ) 
                 
               
             
           
         
       
     
     It is worthwhile to note, that q is symmetric zero mean
 
 q ( t )= q (− t ),  (0.3)
 
zero mean
 
                         ∫     -   ∞     ∞     ⁢       t   ·     q   ⁡     (   t   )         ⁢     ⅆ   t         =   0     ,           (   0.4   )                     ∫     -   ∞     0     ⁢       q   ⁡     (   t   )       ⁢     ⅆ   t         =         ∫   0   ∞     ⁢       q   ⁡     (   t   )       ⁢     ⅆ   t         =   0.5       ,           (   0.5   )               
and that
 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       a 
                       b 
                     
                     ⁢ 
                     
                       q 
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                   
                   ≥ 
                   
                     0 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     for 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     any 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     b 
                   
                   ≥ 
                   
                     a 
                     . 
                   
                 
               
               
                 
                   ( 
                   0.6 
                   ) 
                 
               
             
           
         
       
     
     It is worthwhile to note that 
                       ∫   a   b     ⁢     p   ⁡     (   x   )         ≥     0   ⁢           ⁢   for   ⁢           ⁢   any   ⁢           ⁢   b     ≥   a           (   0.7   )               
and that
 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∫ 
                         a 
                         b 
                       
                       ⁢ 
                       
                         p 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                     
                     ≤ 
                     
                       1 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       any 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       b 
                     
                   
                   , 
                   a 
                 
               
               
                 
                   ( 
                   0.8 
                   ) 
                 
               
             
           
         
       
     
     It is known from literature, that the property defined by equation (0.1) is not preserved during convolution of distributions. 
     Paired Overbounding with Excess Mass Definition 
     A probability density p is paired overbounded by the functions q L  and q R , if the equation 
                       ∫     -   ∞     y     ⁢         q   L     ⁡     (   x   )       ⁢     ⅆ   x         ≥       ∫     -   ∞     y     ⁢       p   ⁡     (   x   )       ⁢     ⅆ   x         ≥     1   -       ∫   y   ∞     ⁢         q   R     ⁡     (   x   )       ⁢     ⅆ   x     ⁢           ⁢   for   ⁢           ⁢   all   ⁢           ⁢   y                 (   0.9   )               
holds true. The functions q L/R  have to fulfil the following requirements.
 
     
       
         
           
             
               
                 
                   
                     
                       
                         q 
                         
                           L 
                           / 
                           R 
                         
                       
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     ≥ 
                     
                       0 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       for 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       all 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       x 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   0.10 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       ∞ 
                     
                     ⁢ 
                     
                       
                         
                           q 
                           
                             L 
                             / 
                             R 
                           
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                   = 
                   
                     
                       K 
                       
                         L 
                         / 
                         R 
                       
                     
                     ≥ 
                     1 
                   
                 
               
               
                 
                   ( 
                   0.11 
                   ) 
                 
               
             
           
         
       
     
     It is known that the property defined by equation (0.9) is preserved during convolution and scaling. To ensure that the convolutions can be performed analytically it is convenient to define q L  and q R  as follows: 
     
       
         
           
             
               
                 
                   
                     
                       q 
                       L 
                     
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         K 
                         L 
                       
                       
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           σ 
                           L 
                         
                       
                     
                     ⁢ 
                     
                       ⅇ 
                       
                         - 
                         
                           
                             
                               ( 
                               
                                 x 
                                 - 
                                 
                                   b 
                                   L 
                                 
                               
                               ) 
                             
                             2 
                           
                           
                             2 
                             ⁢ 
                             
                               σ 
                               L 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.12 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       q 
                       R 
                     
                     ⁡ 
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         K 
                         R 
                       
                       
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           σ 
                           R 
                         
                       
                     
                     ⁢ 
                     
                       ⅇ 
                       
                         - 
                         
                           
                             
                               ( 
                               
                                 x 
                                 - 
                                 
                                   b 
                                   R 
                                 
                               
                               ) 
                             
                             2 
                           
                           
                             2 
                             ⁢ 
                             
                               σ 
                               R 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.13 
                   ) 
                 
               
             
           
         
       
     
     If the individual contributions of the range errors ε i  are paired overbounded with excess mass by the functions 
                         q     L   ,   i       ⁡     (   x   )       =         K     L   ,   i             2   ⁢           ⁢   π       ⁢     σ     L   ,   i           ⁢     ⅇ     -         (     x   -     b     L   ,   i         )     2       2   ⁢     σ     L   ,   i     2                 ⁢     
     ⁢   and           (   0.14   )                   q     R   ,   i       ⁡     (   x   )       =         K     R   ,   i             2   ⁢   π       ⁢     σ     R   ,   i           ⁢     ⅇ     -         (     x   -     b     R   ,   i         )     2       2   ⁢     σ     R   ,   i     2                       (   0.15   )               
and if the errors in the range domain ε i  are mapped onto the error in the position domain ε pos  by
 
                     ɛ   pos     =       ∑     i   =   1     n     ⁢       M     w   ,   i       ·     ɛ   i                 (   0.16   )               
an upper bound for the integrity risk at the alert limit p w,int  (AL) in the direction w is given by
 
                         p     w   ,   int       ⁡     (   AL   )       ≤           K     L   ,     M   w         +     K     R   ,     M   w           2     -         K     R   ,     M   w         2     ⁢           ⁢   erf   ⁢           ⁢     (       AL   -     b     R   ,     M   w               2     ⁢     σ     R   ,     M   w             )       +         K     L   ,     M             ⁢   w           2     ⁢           ⁢   erf   ⁢           ⁢     (         -   AL     -     b     L   ,     M   w               2     ⁢     σ     L   ,     M   w             )           ⁢     
     ⁢   with           (   0.17   )                 g   ⁡     (   α   )       =     {           R   ,             if   ⁢           ⁢   α     &gt;   0               L   ,             if   ⁢           ⁢   α     &lt;   0                     (   0.18   )                 k   ⁡     (   α   )       =     {           L   ,             if   ⁢           ⁢   α     &gt;   0               R   ,             if   ⁢           ⁢   α     &lt;   0                     (   0.19   )                 K     R   ,     M   w         =       ∏     i   =   1     n     ⁢     K       g   ⁡     (     M     w   ,   i       )       ,   i                 (   0.20   )                 K     L   ,     M   w         =       ∏     i   =   1     n     ⁢     K       k   ⁡     (     M     w   ,   i       )       ,   i                 (   0.21   )                 b     R   ,     M   w         =       ∑     i   =   1     n     ⁢       M     w   ,   i       ⁢     b       g   ⁡     (     M     w   ,   i       )       ,   i                   (   0.22   )                 b     L   ,     M   w         =       ∑     i   =   1     n     ⁢       M     w   ,   i       ⁢     b       k   ⁡     (     M     w   ,   i       )       ,   i                   (   0.23   )                 σ     R   ,     M   w         =         ∑     i   =   1     n     ⁢       (       M     w   ,   i       ⁢     σ       g   ⁡     (     M     w   ,   i       )       ,   i         )     2                 (   0.24   )                 σ     L   ,     M   w         =         ∑     i   =   1     n     ⁢       (       M     w   ,   i       ⁢     σ       k   ⁡     (     M     w   ,   i       )       ,   i         )     2                 (   0.25   )               
ICAO Overbounding
 
     For ICAO the probability density p is overbounded by a function q if the equations 
                       ∫     -   ∞       -   y       ⁢       p   ⁡     (   x   )       ⁢     ⅆ   x         ≤       ∫     -   ∞       -   y       ⁢       q   ⁡     (   x   )       ⁢     ⅆ   y     ⁢           ⁢   for   ⁢           ⁢   all   ⁢           ⁢   y       ≥   0           (   0.26   )                   ∫   y   ∞     ⁢       p   ⁡     (   x   )       ⁢     ⅆ   x         ≤       ∫   y   ∞     ⁢       q   ⁡     (   x   )       ⁢     ⅆ   y     ⁢           ⁢   for   ⁢           ⁢   all   ⁢           ⁢   y       ≥   0           (   0.27   )               
hold true.
 
     The ICAO overbounding definition implies directly the Galileo overbounding definition. This can be seen by a simple addition of the defining inequalities. The opposite is not valid in general. 
     It has to be noted further, that for ICAO it is foreseen to use as the overbounding distributions only Gaussian distributions of the form 
     
       
         
           
             
               
                 
                   
                     q 
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   ≡ 
                   
                     
                       1 
                       
                         
                           
                             2 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         σ 
                       
                     
                     ⁢ 
                     
                       
                         ⅇ 
                         
                           - 
                           
                             
                               t 
                               2 
                             
                             
                               2 
                               ⁢ 
                               
                                 σ 
                                 2 
                               
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   0.28 
                   ) 
                 
               
             
           
         
       
     
     It has to be further noted that ICAO states that for the receiver contribution to the error it can be assumed that
 
 p ( x )= p (− x ),  (0.29)
 
 p (sign( x )(| x |+ε))≦ p ( x ) for all  x  and for all ε≧0  (0.30)
 
that is probability density p is symmetric and monotonically increasing up to a single mode in x=0 and then monotonically decreasing (p is also called unimodal).
 
Process for Mapping of Galileo Overbounding to Poem
 
     Mapping without bias. As a first embodiment the process for mapping of Galileo overbounding to POEM for the case without bias is described. 
     For a mapping of the Galileo overbounding to POEM without bias we define
 
 q   L/R ≡2 q   (0.31)
 
     We then compute, observing that q is symmetric, 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         
                           q 
                           L 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         y 
                       
                       ⁢ 
                       
                         2 
                         ⁢ 
                         
                           q 
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           x 
                         
                       
                     
                     = 
                     
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           y 
                         
                         ⁢ 
                         
                           
                             q 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             - 
                             y 
                           
                           ∞ 
                         
                         ⁢ 
                         
                           
                             q 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ⅆ 
                               x 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.32 
                   ) 
                 
               
             
           
         
       
     
     For y≦0 it follows now from (0.32), (0.1), and (0.7) that 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         
                           q 
                           L 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                   ≥ 
                   
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         y 
                       
                       ⁢ 
                       
                         
                           p 
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           y 
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           - 
                           y 
                         
                         ∞ 
                       
                       ⁢ 
                       
                         
                           p 
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           x 
                         
                       
                     
                   
                   ≥ 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         p 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ⅆ 
                           y 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.33 
                   ) 
                 
               
             
           
         
       
     
     For y≧0 it follows now from (0.32), the first part of (0.5), and (0.8) that 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             y 
                           
                           ⁢ 
                           
                             
                               
                                 q 
                                 L 
                               
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               x 
                             
                           
                         
                         = 
                         
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               0 
                             
                             ⁢ 
                             
                               
                                 
                                   q 
                                   L 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                           + 
                           
                             
                               ∫ 
                               0 
                               y 
                             
                             ⁢ 
                             
                               
                                 
                                   q 
                                   L 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             1 
                             + 
                             
                               
                                 ∫ 
                                 0 
                                 y 
                               
                               ⁢ 
                               
                                 
                                   
                                     q 
                                     L 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                           
                           ≥ 
                           1 
                           ≥ 
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               y 
                             
                             ⁢ 
                             
                               
                                 p 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 
                                   ⅆ 
                                   y 
                                 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.34 
                   ) 
                 
               
             
           
         
       
     
     So it has been shown that with the definition given in (0.31) and provided that (0.1) hold true the following condition holds true for all y 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         
                           q 
                           L 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                   ≥ 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         p 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         y 
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.35 
                   ) 
                 
               
             
           
         
       
     
     We now compute, observing that q is symmetric, 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               y 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 
                                   q 
                                   R 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                         = 
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               y 
                               ∞ 
                             
                             ⁢ 
                             
                               2 
                               ⁢ 
                               
                                 q 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               
                                 - 
                                 y 
                               
                             
                             ⁢ 
                             
                               
                                 q 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                           - 
                           
                             
                               ∫ 
                               y 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 q 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.36 
                   ) 
                 
               
             
           
         
       
     
     For y≧0 it follows from (0.36), (0.1), and (0.7) 
     
       
         
           
             
               
                 
                   
                     1 
                     - 
                     
                       
                         ∫ 
                         y 
                         ∞ 
                       
                       ⁢ 
                       
                         
                           
                             q 
                             R 
                           
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           x 
                         
                       
                     
                   
                   = 
                   
                     
                       1 
                       - 
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           
                             - 
                             y 
                           
                         
                         ⁢ 
                         
                           
                             q 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                       - 
                       
                         
                           ∫ 
                           y 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             q 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                     
                     ≤ 
                     
                       1 
                       - 
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           
                             - 
                             y 
                           
                         
                         ⁢ 
                         
                           
                             p 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                       - 
                       
                         
                           ∫ 
                           y 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             p 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                     
                     ≤ 
                     
                       1 
                       - 
                       
                         
                           ∫ 
                           y 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             p 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.37 
                   ) 
                 
               
             
           
         
       
     
     For y≦0 if follows from the first part of (0.36), (0.5), (0.6), and (0.8) 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               y 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 
                                   q 
                                   R 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                         = 
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                             ⁢ 
                             
                               2 
                               ⁢ 
                               
                                 q 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                           - 
                           
                             
                               ∫ 
                               y 
                               0 
                             
                             ⁢ 
                             
                               2 
                               ⁢ 
                               
                                 q 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                               
                                 ∫ 
                                 y 
                                 0 
                               
                               ⁢ 
                               
                                 2 
                                 ⁢ 
                                 
                                   q 
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                           
                           ≤ 
                           0 
                           ≤ 
                           
                             1 
                             - 
                             
                               
                                 ∫ 
                                 y 
                                 ∞ 
                               
                               ⁢ 
                               
                                 
                                   p 
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.38 
                   ) 
                 
               
             
           
         
       
     
     So it has been shown with (0.37) and (0.38) that for any y the following holds true: 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       - 
                       
                         
                           ∫ 
                           y 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             
                               q 
                               R 
                             
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                     
                     ≤ 
                     
                       1 
                       - 
                       
                         
                           ∫ 
                           y 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             p 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         p 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.39 
                   ) 
                 
               
             
           
         
       
     
     Combing (0.35) and (0.39) we finally get 
                       ∫     -   ∞     y     ⁢         q   L     ⁡     (   x   )       ⁢     ⅆ   x         ≥       ∫     -   ∞     y     ⁢       p   ⁡     (   x   )       ⁢     ⅆ   y         ≥     1   -       ∫   y   ∞     ⁢         q   R     ⁡     (   x   )       ⁢     ⅆ   x                   (   0.40   )               
for any y.
 
     Assuming now m satellites and on each range two different types of contributions of range errors, says ε i  and ε m+i  on range iε{1, . . . , m}, e.g. one type due to the system in space (errors from orbit, satellite, clock, etc.) and one due to the local effects at the receiver location (errors from atmosphere, receiver noise, multipath, etc.). Let n=2·m. Then using definition (0.31) results in
 
 K   L,i   =K   R,i =2 for all  i =1 , . . . , n   (0.41)
 
 b   L,i   =b   R,i =0 for all  i =1 , . . . , n   (0.42)
 
σ L,i =σ R,i =σ i   =SISA   i  for all  i =1 , . . . , m  and σ L,i =σ R,i =σ i =σ u,i−m  for all  i=m +1 , . . . , n   (0.43)
 
for the fault free case.
 
     Taking (0.41), (0.42), (0.43) and the equality −erƒ(x)=erƒ(−x) into account, (0.17) simplifies to 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       
                         w 
                         , 
                         int 
                       
                     
                     ⁡ 
                     
                       ( 
                       AL 
                       ) 
                     
                   
                   ≤ 
                   
                     
                       2 
                       n 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           erf 
                           ( 
                           
                             AL 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     n 
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           M 
                                           
                                             w 
                                             , 
                                             i 
                                           
                                         
                                         ⁢ 
                                         
                                           σ 
                                           i 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   0.44 
                   ) 
                 
               
             
           
         
       
     
     Observing M i =M m+i  for all i=1, . . . , m because these factors depend only on the geometry given by the m satellites and the receiver location but not on the special type of error contribution, this formular can be written as 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       
                         w 
                         , 
                         int 
                       
                     
                     ⁡ 
                     
                       ( 
                       AL 
                       ) 
                     
                   
                   ≤ 
                   
                     
                       4 
                       m 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           erf 
                           ( 
                           
                             AL 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     m 
                                   
                                   ⁢ 
                                   
                                     
                                       M 
                                       
                                         w 
                                         , 
                                         i 
                                       
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           SISA 
                                           i 
                                           2 
                                         
                                         + 
                                         
                                           σ 
                                           
                                             u 
                                             , 
                                             i 
                                           
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   0.45 
                   ) 
                 
               
             
           
         
       
     
     For the faulty case, satellite jε{1, . . . , m} is faulty, we get 
     
       
         
           
             
               
                 
                   
                     
                       K 
                       
                         L 
                         , 
                         i 
                       
                     
                     = 
                     
                       
                         K 
                         
                           R 
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           for 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           all 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           i 
                         
                         = 
                         1 
                       
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   n 
                 
               
               
                 
                   ( 
                   0.46 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         b 
                         
                           L 
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           b 
                           
                             R 
                             , 
                             i 
                           
                         
                         = 
                         
                           
                             0 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             for 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             all 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             i 
                           
                           = 
                           1 
                         
                       
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       
                         n 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         with 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         i 
                       
                       ≠ 
                       j 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       and 
                       ⁢ 
                       
                           
                       
                       - 
                       
                         b 
                         
                           L 
                           , 
                           j 
                         
                       
                     
                     = 
                     
                       
                         b 
                         
                           R 
                           , 
                           j 
                         
                       
                       = 
                       
                         TH 
                         j 
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.47 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         σ 
                         
                           L 
                           , 
                           i 
                         
                       
                       = 
                       
                         
                           σ 
                           
                             R 
                             , 
                             i 
                           
                         
                         = 
                         
                           
                             σ 
                             i 
                           
                           = 
                           
                             
                               
                                 SISA 
                                 i 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               for 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               all 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               i 
                             
                             = 
                             1 
                           
                         
                       
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       
                         m 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         with 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         i 
                       
                       ≠ 
                       j 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         and 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           σ 
                           
                             L 
                             , 
                             i 
                           
                         
                       
                       = 
                       
                         
                           σ 
                           
                             R 
                             , 
                             i 
                           
                         
                         = 
                         
                           
                             σ 
                             i 
                           
                           = 
                           
                             
                               
                                 σ 
                                 
                                   u 
                                   , 
                                   
                                     i 
                                     - 
                                     m 
                                   
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               for 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               all 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               l 
                             
                             = 
                             
                               m 
                               + 
                               1 
                             
                           
                         
                       
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     n 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         σ 
                         
                           L 
                           , 
                           j 
                         
                       
                     
                     = 
                     
                       
                         σ 
                         
                           R 
                           , 
                           j 
                         
                       
                       = 
                       
                         
                           σ 
                           j 
                         
                         = 
                         
                           SISMA 
                           j 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     0.47 
                     ′ 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       p 
                       
                         w 
                         , 
                         int 
                       
                     
                     ⁡ 
                     
                       ( 
                       AL 
                       ) 
                     
                   
                   ≤ 
                   
                     
                       2 
                       n 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           erf 
                           ( 
                           
                             
                               AL 
                               - 
                               
                                 
                                    
                                   
                                     M 
                                     
                                       w 
                                       , 
                                       j 
                                     
                                   
                                    
                                 
                                 ⁢ 
                                 
                                   TH 
                                   j 
                                 
                               
                             
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     n 
                                   
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           M 
                                           
                                             w 
                                             , 
                                             i 
                                           
                                         
                                         ⁢ 
                                         
                                           σ 
                                           i 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   0.48 
                   ) 
                 
               
             
           
         
       
     
     Again observing M i =M m+i  for all i=1, . . . , m this formular can be written as 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       
                         w 
                         , 
                         int 
                       
                     
                     ⁡ 
                     
                       ( 
                       AL 
                       ) 
                     
                   
                   ≤ 
                   
                     
                       4 
                       m 
                     
                     ⁢ 
                     
                       ( 
                       
                         1 
                         - 
                         
                           erf 
                           ( 
                           
                             
                               AL 
                               - 
                               
                                 
                                    
                                   
                                     M 
                                     
                                       w 
                                       , 
                                       j 
                                     
                                   
                                    
                                 
                                 ⁢ 
                                 
                                   TH 
                                   j 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       
                                         2 
                                         ⁢ 
                                         
                                           
                                             ∑ 
                                             
                                               
                                                 i 
                                                 = 
                                                 1 
                                               
                                               , 
                                               
                                                 i 
                                                 ≠ 
                                                 j 
                                               
                                             
                                             m 
                                           
                                           ⁢ 
                                           
                                             
                                               M 
                                               
                                                 w 
                                                 , 
                                                 i 
                                               
                                               2 
                                             
                                             ⁢ 
                                             
                                               ( 
                                               
                                                 
                                                   SISA 
                                                   i 
                                                   2 
                                                 
                                                 + 
                                                 
                                                   σ 
                                                   
                                                     u 
                                                     , 
                                                     i 
                                                   
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                         
                                           M 
                                           
                                             w 
                                             , 
                                             j 
                                           
                                           2 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               SISMA 
                                               j 
                                               2 
                                             
                                             + 
                                             
                                               σ 
                                               
                                                 u 
                                                 , 
                                                 j 
                                               
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   0.49 
                   ) 
                 
               
             
           
         
       
     
     Mapping with Bias. 
     As a second embodiment the process for mapping of Galileo overbounding to POEM for the case with bias is described. 
     For a mapping of the Galileo overbounding to POEM with bias we define
 
 q′   L ( x )≡θ(− x )2 q ( x )  (0.50)
 
 q′   R ( x )≡θ( x )2 q ( x )  (0.51)
 
where θ is a function, defined for real values x by
 
     
       
         
           
             
               θ 
               ⁡ 
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     
                       1 
                       , 
                     
                   
                   
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         x 
                       
                       ≥ 
                       0 
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       otherwise 
                       . 
                     
                   
                 
               
             
           
         
       
     
     For y≦0 we then compute, observing that q is symmetric, 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         
                           q 
                           L 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         y 
                       
                       ⁢ 
                       
                         2 
                         ⁢ 
                         
                           q 
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           x 
                         
                       
                     
                     = 
                     
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           y 
                         
                         ⁢ 
                         
                           
                             q 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                       + 
                       
                         
                           ∫ 
                           
                             - 
                             y 
                           
                           ∞ 
                         
                         ⁢ 
                         
                           
                             q 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                             
                               ⅆ 
                               x 
                             
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.52 
                   ) 
                 
               
             
           
         
       
     
     It follows now from (0.52), (0.1), and (0.7) that 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         
                           q 
                           L 
                           ′ 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                   ≥ 
                   
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         y 
                       
                       ⁢ 
                       
                         
                           p 
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           y 
                         
                       
                     
                     + 
                     
                       
                         ∫ 
                         
                           - 
                           y 
                         
                         ∞ 
                       
                       ⁢ 
                       
                         
                           p 
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           x 
                         
                       
                     
                   
                   ≥ 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         p 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ⅆ 
                           y 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.53 
                   ) 
                 
               
             
           
         
       
     
     For y≧0 it follows from first part of (0.5), and (0.8) that 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             y 
                           
                           ⁢ 
                           
                             
                               
                                 
                                   q 
                                   ′ 
                                 
                                 L 
                               
                               ⁡ 
                               
                                 ( 
                                 x 
                                 ) 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ⅆ 
                               x 
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               0 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     q 
                                     ′ 
                                   
                                   L 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                           + 
                           
                             
                               ∫ 
                               0 
                               y 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     q 
                                     ′ 
                                   
                                   L 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             + 
                             0 
                           
                           ≥ 
                           1 
                           ≥ 
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               y 
                             
                             ⁢ 
                             
                               
                                 p 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   ⅆ 
                                   y 
                                 
                                 . 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.54 
                   ) 
                 
               
             
           
         
       
     
     So it has been shown that with the definition given in (0.50) and provided that (0.1) hold true the following condition holds true for all y 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         
                           
                             q 
                             ′ 
                           
                           L 
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                   ≥ 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         p 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         y 
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.55 
                   ) 
                 
               
             
           
         
       
     
     We now compute for y≧0 and observing that q is symmetric, 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               y 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     q 
                                     ′ 
                                   
                                   R 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                         = 
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               y 
                               ∞ 
                             
                             ⁢ 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 q 
                                 ⁢ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               
                                 - 
                                 ∞ 
                               
                               
                                 - 
                                 y 
                               
                             
                             ⁢ 
                             
                               
                                 q 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                           - 
                           
                             
                               ∫ 
                               y 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 q 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.56 
                   ) 
                 
               
             
           
         
       
     
     It follows from (0.56), (0.1), and (0.7) 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               y 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     q 
                                     ′ 
                                   
                                   R 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             1 
                             - 
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 
                                   - 
                                   y 
                                 
                               
                               ⁢ 
                               
                                 
                                   q 
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                             - 
                             
                               
                                 ∫ 
                                 y 
                                 ∞ 
                               
                               ⁢ 
                               
                                 
                                   q 
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                           
                           ≤ 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             1 
                             - 
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 
                                   - 
                                   y 
                                 
                               
                               ⁢ 
                               
                                 
                                   p 
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                             - 
                             
                               
                                 ∫ 
                                 y 
                                 ∞ 
                               
                               ⁢ 
                               
                                 
                                   p 
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                           
                           ≤ 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               y 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 p 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.57 
                   ) 
                 
               
             
           
         
       
     
     For y≦0 if follows from (0.5), (0.6), and (0.8) 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               y 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     q 
                                     ′ 
                                   
                                   R 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     q 
                                     ′ 
                                   
                                   R 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           1 
                           - 
                           
                             
                               ∫ 
                               0 
                               ∞ 
                             
                             ⁢ 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 q 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ⅆ 
                                 x 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           0 
                           ≤ 
                           
                             1 
                             - 
                             
                               
                                 ∫ 
                                 y 
                                 ∞ 
                               
                               ⁢ 
                               
                                 
                                   p 
                                   ⁡ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ⅆ 
                                   x 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.58 
                   ) 
                 
               
             
           
         
       
     
     So we have shown with (0.57) and (0.58) that for any y the following holds true: 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       - 
                       
                         
                           ∫ 
                           y 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             
                               
                                 q 
                                 ′ 
                               
                               R 
                             
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                     
                     ≤ 
                     
                       1 
                       - 
                       
                         
                           ∫ 
                           y 
                           ∞ 
                         
                         ⁢ 
                         
                           
                             p 
                             ⁡ 
                             
                               ( 
                               x 
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ⅆ 
                             x 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       y 
                     
                     ⁢ 
                     
                       
                         p 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.59 
                   ) 
                 
               
             
           
         
       
     
     Combining (0.55) and (0.59) we finally get 
                       ∫     -   ∞     y     ⁢           q   ′     L     ⁡     (   x   )       ⁢           ⁢     ⅆ   x         ≥       ∫     -   ∞     y     ⁢       p   ⁡     (   x   )       ⁢           ⁢     ⅆ   y         ≥     1   -       ∫   y   ∞     ⁢           q   ′     R     ⁡     (   x   )       ⁢           ⁢     ⅆ   x                   (   0.60   )               
for any y.
 
     Assuming again as above m satellites and on each range two different types of contributions of range errors, say and ε i  and ε m+i  on range iε{1, . . . , m}, e.g. one due to the system in space (errors from orbit, satellite, clock, etc.) and one due to the local effects at the receiver location (errors from atmosphere, receiver noise, multipath, etc.). Let n=2·m. Then it is possible to prove that probability densities p i  are paired overbounded by the functions q L,i  and q R,i  defined by equations (0.14) and (0.15) with 
                         σ     L   ,   i       =       σ     R   ,   i       =       σ   i     =         SISA   i     ⁢           ⁢   for   ⁢           ⁢   all   ⁢           ⁢   i     =   1           ,   …   ⁢           ,     m   ⁢           ⁢   and       ⁢           ⁢     
     ⁢           ⁢         σ     L   ,   i       =       σ     R   ,   i       =       σ   i     =         σ     u   ,     i   -   m         ⁢           ⁢   for   ⁢           ⁢   all   ⁢           ⁢   i     =     m   +   1             ,   …   ⁢           ,   n             (   0.61   )                   -     b     L   ,   i         =       b     R   ,   i       =         b   i     &gt;     0   ⁢           ⁢   for   ⁢           ⁢   all   ⁢           ⁢   i       =   1         ,   …   ⁢           ,   n           (   0.62   )                   K     L   ,   i       =       K     R   ,   i       =       K   i     =         2   ·       (     1   +     erf   (       b   i         σ   i     ⁢     2         )       )       -   1         ⁢           ⁢   for   ⁢           ⁢   all   ⁢           ⁢   i     =   1           ,   …   ⁢           ,   n           (   0.63   )               
for the fault free case.
 
     Taking these definitions into account and remembering equality −erƒ(x)=erƒ(−x), equation (0.17) simplifies to 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       
                         w 
                         , 
                         int 
                       
                     
                     ⁡ 
                     
                       ( 
                       AL 
                       ) 
                     
                   
                   ≤ 
                   
                     
                       
                         2 
                         n 
                       
                       
                         
                           ∏ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             + 
                             
                               erf 
                               ( 
                               
                                 
                                   b 
                                   i 
                                 
                                 
                                   
                                     σ 
                                     i 
                                   
                                   ⁢ 
                                   
                                     2 
                                   
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                     · 
                     
                       ( 
                       
                         1 
                         - 
                         
                           erf 
                           ⁡ 
                           
                             ( 
                             
                               
                                 AL 
                                 - 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     n 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     
                                        
                                       
                                         M 
                                         
                                           w 
                                           , 
                                           i 
                                         
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       b 
                                       i 
                                     
                                   
                                 
                               
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       n 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           
                                             M 
                                             
                                               w 
                                               , 
                                               i 
                                             
                                           
                                           ⁢ 
                                           
                                             σ 
                                             i 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   0.64 
                   ) 
                 
               
             
           
         
       
     
     Observing again M i =M m+i  for all i=1, . . . m because these factors depend only on the geometry given by the m satellites and the receiver location but not on the special type of error contribution, this formular can be written as 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       
                         w 
                         , 
                         int 
                       
                     
                     ⁡ 
                     
                       ( 
                       AL 
                       ) 
                     
                   
                   ≤ 
                   
                     
                       
                         4 
                         m 
                       
                       
                         
                           ∏ 
                           
                             i 
                             = 
                             1 
                           
                           m 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 erf 
                                 ( 
                                 
                                   
                                     b 
                                     i 
                                   
                                   
                                     
                                       SISA 
                                       i 
                                     
                                     ⁢ 
                                     
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                           · 
                           
                             
                               ∏ 
                               
                                 i 
                                 = 
                                 1 
                               
                               m 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   erf 
                                   ( 
                                   
                                     
                                       b 
                                       
                                         m 
                                         + 
                                         i 
                                       
                                     
                                     
                                       
                                         σ 
                                         
                                           u 
                                           , 
                                           i 
                                         
                                       
                                       ⁢ 
                                       
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     · 
                     
                       ( 
                       
                         1 
                         - 
                         
                           erf 
                           ( 
                           
                             
                               AL 
                               - 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   m 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                      
                                     
                                       M 
                                       
                                         w 
                                         , 
                                         i 
                                       
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         b 
                                         i 
                                       
                                       + 
                                       
                                         b 
                                         
                                           m 
                                           + 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     m 
                                   
                                   ⁢ 
                                   
                                     
                                       M 
                                       
                                         w 
                                         , 
                                         i 
                                       
                                       2 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           SISA 
                                           i 
                                           2 
                                         
                                         + 
                                         
                                           σ 
                                           
                                             u 
                                             , 
                                             i 
                                           
                                           2 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     For the faulty case, satellite jε{1, . . . , m} is faulty, we get 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             σ 
                             
                               L 
                               , 
                               i 
                             
                           
                           = 
                           
                             
                               σ 
                               
                                 R 
                                 , 
                                 i 
                               
                             
                             = 
                             
                               
                                 σ 
                                 i 
                               
                               = 
                               
                                 
                                   
                                     SISA 
                                     i 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   for 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   all 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   i 
                                 
                                 = 
                                 1 
                               
                             
                           
                         
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         
                           
                             m 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             with 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             i 
                           
                           ≠ 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             and 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               σ 
                               
                                 L 
                                 , 
                                 i 
                               
                             
                           
                           = 
                           
                             
                               σ 
                               
                                 R 
                                 , 
                                 i 
                               
                             
                             = 
                             
                               
                                 σ 
                                 i 
                               
                               = 
                               
                                 
                                   
                                     σ 
                                     
                                       u 
                                       , 
                                       
                                         i 
                                         - 
                                         m 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   for 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   all 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   i 
                                 
                                 = 
                                 
                                   m 
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                         , 
                       
                     
                   
                   
                     
                       
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         
                           
                             n 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             and 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               σ 
                               
                                 L 
                                 , 
                                 j 
                               
                             
                           
                           = 
                           
                             
                               σ 
                               
                                 R 
                                 , 
                                 j 
                               
                             
                             = 
                             
                               
                                 σ 
                                 j 
                               
                               = 
                               
                                 SISMA 
                                 j 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.65 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           
                             - 
                             
                               b 
                               
                                 L 
                                 , 
                                 i 
                               
                             
                           
                           = 
                           
                             
                               b 
                               
                                 R 
                                 , 
                                 i 
                               
                             
                             = 
                             
                               
                                 
                                   b 
                                   i 
                                 
                                 &gt; 
                                 
                                   0 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   for 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   all 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   i 
                                 
                               
                               = 
                               1 
                             
                           
                         
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         
                           
                             n 
                             ⁢ 
                             
                               
                                   
                               
                               ⁢ 
                               
                                   
                               
                             
                             ⁢ 
                             with 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             i 
                           
                           ≠ 
                           j 
                         
                       
                     
                   
                   
                     
                       
                         
                           and 
                           ⁢ 
                           
                               
                           
                           - 
                           
                             b 
                             
                               L 
                               , 
                               j 
                             
                           
                         
                         = 
                         
                           
                             b 
                             
                               R 
                               , 
                               j 
                             
                           
                           = 
                           
                             
                               b 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                               
                             
                             + 
                             
                               TH 
                               j 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     0.65 
                     ′ 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           K 
                           
                             L 
                             , 
                             i 
                           
                         
                         = 
                         
                           
                             K 
                             
                               R 
                               , 
                               i 
                             
                           
                           = 
                           
                             
                               K 
                               i 
                             
                             = 
                             
                               2 
                               · 
                               
                                 
                                   ( 
                                   
                                     1 
                                     + 
                                     
                                       erf 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             b 
                                             i 
                                           
                                           
                                             
                                               σ 
                                               i 
                                             
                                             ⁢ 
                                             
                                               2 
                                             
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           
                             for 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             all 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             i 
                           
                           = 
                           1 
                         
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         n 
                         , 
                         
                           
                             i 
                             . 
                             e 
                             . 
                             
                                 
                             
                             ⁢ 
                             inclusive 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           j 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   0.66 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       p 
                       
                         w 
                         , 
                         int 
                       
                     
                     ⁡ 
                     
                       ( 
                       AL 
                       ) 
                     
                   
                   ≤ 
                   
                     
                       
                         2 
                         n 
                       
                       
                         
                           ∏ 
                           
                             i 
                             = 
                             1 
                           
                           n 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             + 
                             
                               erf 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     b 
                                     i 
                                   
                                   
                                     
                                       σ 
                                       i 
                                     
                                     ⁢ 
                                     
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                     · 
                     
                       ( 
                       
                         1 
                         - 
                         
                           erf 
                           ⁡ 
                           
                             ( 
                             
                               
                                 AL 
                                 - 
                                 
                                   
                                      
                                     
                                       M 
                                       
                                         w 
                                         , 
                                         j 
                                       
                                     
                                      
                                   
                                   ⁢ 
                                   
                                     TH 
                                     j 
                                   
                                 
                                 - 
                                 
                                   
                                     ∑ 
                                     
                                       i 
                                       = 
                                       1 
                                     
                                     n 
                                   
                                   ⁢ 
                                   
                                     
                                        
                                       
                                         M 
                                         
                                           w 
                                           , 
                                           i 
                                         
                                       
                                        
                                     
                                     ⁢ 
                                     
                                       b 
                                       i 
                                     
                                   
                                 
                               
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       n 
                                     
                                     ⁢ 
                                     
                                       
                                         ( 
                                         
                                           
                                             M 
                                             
                                               w 
                                               , 
                                               i 
                                             
                                           
                                           ⁢ 
                                           
                                             σ 
                                             
                                               i 
                                               ⁢ 
                                               
                                                   
                                               
                                             
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   0.67 
                   ) 
                 
               
             
           
         
       
     
     As shown before several times (e.g. see inequalities (0.44) and (0.45)) this formular can be written using the original symbols. This will be omitted here. 
     More worthy of mention is the fact that different choices of b i  lead to different values for p w,int (AL). The optimal choice, that gives smallest p w,int (AL), depends on the actual satellite/user geometry. Therefore the user has to determine the optimal choice of b i  by himself. 
     Process for Mapping of ICAO Overbounding to Poem 
     As a third embodiment the process for mapping of ICAO overbounding to POEM is described. 
     As already stated ICAO overbounding implies Galileo overbounding. Therefore the method of mapping Galileo overbounding to POEM described before is also applicable for ICAO overbounding. 
     The question arising is whether it is possible to get a factor smaller than 2 as necessary when mapping Galileo overbounding to POEM by definition of q L/R ≡2q for mapping without bias and q′ L (x)≡θ(−x)2q(x) and q′ R (x)≡θ(x)2q(x) for mapping with bias. 
     A positive function q L  is part of POEM if inequality 
                 ∫     -   ∞     y     ⁢         q   L     ⁡     (   x   )       ⁢     ⅆ   x         ≥       ∫     -   ∞     y     ⁢       p   ⁡     (   x   )       ⁢     ⅆ   x               
is fulfilled for all y. For negative y we have the same inequality for q itself. But for positive values of y we know nothing but
 
                 ∫     -   ∞     y     ⁢       p   ⁡     (   x   )       ⁢     ⅆ   x         ≤   1.         
Therefore the only condition we can use is
 
                 ∫     -   ∞     y     ⁢         q   L     ⁡     (   x   )       ⁢     ⅆ   x         ≥   1         
for all positive values of y. For reasons of continuity the same is valid for y=0. Using the Gaussian type of q L  we calculate
 
                 ∫     -   ∞     y     ⁢         q   L     ⁡     (   x   )       ⁢     ⅆ   x         =         K   L     2     ·       (     1   +     erf   ⁡     (     y       σ   L     ⁢     2         )         )     .             
Inserting y=0 leads to
 
                 1   ≤       ∫     -   ∞     0     ⁢         q   L     ⁡     (   x   )       ⁢     ⅆ   x           =           K   L     2     ·     (     1   +     erf   ⁡     (     0       σ   L     ⁢     2         )         )       =         K   L     2     ⇒       K   L     ≥   2           ,         
which answers the above question: we can not get a smaller factor than 2 in the definition of q L .
 
     A positive function q R  is part of POEM if inequality 
               1   -     K   R     +       ∫     -   ∞     y     ⁢         q   R     ⁡     (   x   )       ⁢     ⅆ   x           ≤       ∫     -   ∞     y     ⁢       p   ⁡     (   x   )       ⁢     ⅆ   x               
is fulfilled for all y. For positive y we have the same inequality for q itself (because K R =1 for q R ≡q). But for negative values of y we know nothing but
 
             0   ≤       ∫     -   ∞     y     ⁢       p   ⁡     (   x   )       ⁢       ⅆ   x     .               
Therefore the only condition that can be used is
 
               1   -     K   R     +       ∫     -   ∞     y     ⁢         q   R     ⁡     (   x   )       ⁢     ⅆ   x           ≤   0         
for all negative values of y. For reasons of continuity the same is valid for y=0. Using now the Gaussian type of q R , calculating the integral and inserting y=0 leads to
 
                     1   -               ⁢     K             ⁢   R                   ⁢   2         =       ⁢     1   -     K             ⁢   R       +                 ⁢     K             ⁢   R                   ⁢   2       ·     (     1   +     erf   ⁡     (     0             ⁢       σ   R     ⁢           ⁢     2           )         )                     =       ⁢         1   -     K             ⁢   R       +       ∫     -   ∞               ⁢   0       ⁢     q             ⁢   R       ⁢     (   x   )     ⁢     ⅆ   x         ≤   0     ⇒       K             ⁢   R       ≥   2                   
which answers the above question: we can not get a smaller factor than 2 in the definition of q R .
 
     The invention further discloses an apparatus that is configured to execute the methods described above. Such an apparatus could be part of one of the GNSS, e.g. a satellite of the GNSS or some ground systems or the receiver of a user. 
     Although the invention has been described for GNSS and the overbounding concepts Galileo Overbounding, ICAO Overbounding and Paired Overbounding with Excess Mass. However, it is to be understood by those skilled in the art that the invention is neither limited to GNSS nor to Galileo Overbounding, ICAO Overbounding and Paired Overbounding with Excess Mass, respectively. 
     An exemplary method in accordance with the present invention is illustrated in  FIG. 1 . Specifically, overbounds (q i (x)) of at least one error (ε i ) in a range domain is determined (step  110 ). The overbounds (q i (x)) of errors (ε i ) in the range domain are separately multiplied once by a first function θ(−x)·2 and another time by a second function θ(x)·2, wherein θ(x) is defined as follows: θ(x)=1 for x≧0 and θ(x)=0 for x&lt;0, to obtain a pair of overbounds (q iL (x) and q iR (x)) of errors (ε i ) in the range domain (step  120 ). An upper bound for the integrity risk at an alert limit (p w,int (AL)) in a position domain is determined using the pair of overbounds (q iL (x) and q iR (x)) of errors (ε i ) in the range domain (step  130 ). 
     For further clarifications the following is disclosed: 
     Prerequisites: 
     
       
         
           
             
               
                 
                   
                     
                       
                         q 
                         ∼ 
                       
                       L 
                     
                     := 
                     
                       2 
                       ⁢ 
                       
                         χ 
                         
                           ( 
                           
                             
                               - 
                               ∞ 
                             
                             , 
                             0 
                           
                           ] 
                         
                       
                       ⁢ 
                       q 
                     
                   
                   , 
                   
                     q 
                     := 
                     
                       
                         1 
                         
                           
                             2 
                             ⁢ 
                             πσ 
                           
                         
                       
                       ⁢ 
                       
                         exp 
                         ⁡ 
                         
                           ( 
                           
                             
                               - 
                               
                                 1 
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   · 
                                   σ 
                                 
                                 ) 
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     q 
                     L 
                   
                   := 
                   
                     
                       
                         K 
                         
                           
                             
                               2 
                               ⁢ 
                               π 
                             
                           
                           ⁢ 
                           σ 
                         
                       
                       ⁢ 
                       
                         exp 
                         ⁡ 
                         
                           ( 
                           
                             
                               - 
                               
                                 1 
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                   
                                     · 
                                     
                                       - 
                                       μ 
                                     
                                   
                                   σ 
                                 
                                 ) 
                               
                               2 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       with 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       μ 
                     
                     &lt; 
                     0 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     Which conditions have to be stipulated on K and μ, that it holds: 
     Let z≧0: It holds 
                       F       q   ∼     L       ≤     F     q   L         ⁢     
     ⁢         F       q   ∼     L       ⁡     (   z   )       =       1   ⁢     ≤   !     ⁢       F     q   L       ⁡     (   0   )       ⁢     ≤     monotonic   ⁢           ⁢   increasing       ⁢       F     q   L       ⁡     (   z   )         =         K   2     ⁢     (     1   +     erf   ⁡     (       z   -   μ         2     ⁢   σ       )         )       ⁢     
     ⇒     1   ⁢     ≤   !     ⁢       K   2     ⁢       (     1   +     erf   (       -   μ         2     ⁢   σ       )       )     .                       (   1   )               
Then it holds F {tilde over (q)}     L   (z)≦F q     L   (z) for z≧0.
 
     Set 
               K   :=     2     (     1   +     erf   (       -   μ         2     ⁢   σ       )       )         ,         
i.e. the smallest K will be used.
 
     Let z&lt;0: It is 
                   F       q   ∼     L       ⁡     (   z   )       =         ∫     -   ∞     z     ⁢     2   ⁢       x     (       -   ∞     ,   0     ]       ⁡     (   x   )       ⁢     q   ⁡     (   x   )       ⁢     ⅆ   x         =       2   ⁢       ∫     -   ∞     z     ⁢       q   ⁡     (   x   )       ⁢           ⁢     ⅆ   x           =       2   ⁢       F   q     ⁡     (   z   )         =       2   ·     1   2     ·     (     1   +     erf   (     z       2     ⁢   σ       )       )       =     1   +     erf   ⁡     (     z       2     ⁢   σ       )                   ,         
i.e. (1) is equivalent to
 
     
       
         
           
             
               1 
               + 
               
                 erf 
                 ⁡ 
                 
                   ( 
                   
                     z 
                     
                       
                         2 
                       
                       ⁢ 
                       σ 
                     
                   
                   ) 
                 
               
             
             ⁢ 
             
               ≤ 
               ! 
             
             ⁢ 
             
               
                 K 
                 2 
               
               ⁢ 
               
                 
                   ( 
                   
                     1 
                     + 
                     
                       erf 
                       ⁡ 
                       
                         ( 
                         
                           
                             z 
                             - 
                             μ 
                           
                           
                             
                               2 
                             
                             ⁢ 
                             σ 
                           
                         
                         ) 
                       
                     
                   
                   ) 
                 
                 . 
               
             
           
         
       
     
     With the K from above this will be: 
     
       
         
           
             
               1 
               + 
               
                 erf 
                 ⁡ 
                 
                   ( 
                   
                     z 
                     
                       
                         2 
                       
                       ⁢ 
                       σ 
                     
                   
                   ) 
                 
               
             
             ⁢ 
             
               ≤ 
               ! 
             
             ⁢ 
             
               
                 
                   1 
                   + 
                   
                     erf 
                     ⁡ 
                     
                       ( 
                       
                         
                           z 
                           - 
                           μ 
                         
                         
                           
                             2 
                           
                           ⁢ 
                           σ 
                         
                       
                       ) 
                     
                   
                 
                 
                   1 
                   + 
                   
                     
                       erf 
                       ⁡ 
                       
                         ( 
                         
                           
                             - 
                             μ 
                           
                           
                             
                               2 
                             
                             ⁢ 
                             σ 
                           
                         
                         ) 
                       
                     
                     
                       ︸ 
                       
                         
                           ≥ 
                           0 
                         
                         , 
                         
                             
                         
                         ⁢ 
                         
                           
                             for 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             μ 
                           
                           &lt; 
                           0 
                         
                       
                     
                   
                 
               
               . 
             
           
         
       
     
     Abbreviate: {tilde over (z)}:=z/(√{square root over (2)}σ), {tilde over (μ)}:=μ/(√{square root over (2)}σ) and multiply with (the positive) denominator of the right side, then (1) is equivalent to 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               1 
                               + 
                               
                                 erf 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     z 
                                     ∼ 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 erf 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     - 
                                     
                                       μ 
                                       ∼ 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ≤ 
                           ! 
                         
                         ⁢ 
                         
                           1 
                           + 
                           
                             
                               erf 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     z 
                                     ∼ 
                                   
                                   - 
                                   
                                     μ 
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                                
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         ⇔ 
                         
                           
                             
                               erf 
                               ⁡ 
                               
                                 ( 
                                 
                                   z 
                                   ∼ 
                                 
                                 ) 
                               
                             
                             + 
                             
                               erf 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     - 
                                     
                                       μ 
                                       ∼ 
                                     
                                   
                                   
                                     ︷ 
                                     
                                       = 
                                       
                                          
                                         
                                           μ 
                                           ∼ 
                                         
                                          
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 erf 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     z 
                                     ∼ 
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 erf 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     - 
                                     
                                       μ 
                                       ∼ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ⁢ 
                           
                             ≤ 
                             ! 
                           
                           ⁢ 
                           
                             
                               erf 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     z 
                                     ∼ 
                                   
                                   - 
                                   
                                     μ 
                                     ∼ 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                                
                               
                                 · 
                                 
                                   
                                     
                                       π 
                                     
                                     2 
                                   
                                   . 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           ⇔ 
                           
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             := 
                             
                               exp 
                               ⁡ 
                               
                                 ( 
                                 
                                   - 
                                   
                                     t 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                           
                             
                               
                                 ∫ 
                                 0 
                                 
                                   z 
                                   ∼ 
                                 
                               
                               ⁢ 
                               f 
                             
                             + 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ∫ 
                                 0 
                                 
                                    
                                   
                                     μ 
                                     ∼ 
                                   
                                    
                                 
                               
                               ⁢ 
                               f 
                             
                             + 
                             
                               
                                 erf 
                                 ( 
                                 
                                   z 
                                   ∼ 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   
                                      
                                     
                                       μ 
                                       ∼ 
                                     
                                      
                                   
                                 
                                 ⁢ 
                                 f 
                               
                             
                           
                           ⁢ 
                           
                             ≤ 
                             ! 
                           
                           ⁢ 
                           
                             
                               ∫ 
                               0 
                               
                                 
                                   z 
                                   ∼ 
                                 
                                 - 
                                 
                                   μ 
                                   ∼ 
                                 
                               
                             
                             ⁢ 
                             f 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Distinction of cases: 1. {tilde over (μ)}≦{tilde over (z)}&lt;0 and 2. {tilde over (z)}&lt;{tilde over (μ)}. 
     So let {tilde over (μ)}≦{tilde over (z)}&lt;0. Consider because of 
                       erf   ⁡     (     z   ∼     )         ︸     &lt;   0         ⁢         ∫   0          μ   ∼            ⁢   f       ︸     &gt;   0           &lt;   0     :         ∫   0     z   ∼       ⁢   f     +           ⁢       ∫   0          μ   ∼            ⁢   f         ⁢     =         z   ∼     &lt;   0     ,     f   ⁢           ⁢   symmetric   ⁢           ⁢   at   ⁢           ⁢   0         ⁢           ∫          z   ∼          0     ⁢   f     +       ∫   0          μ   ∼            ⁢   f       =       ∫          z   ∼                 μ   ∼            ⁢     f   .               
Notice |{tilde over (μ)}|≧|{tilde over (z)}|. (*)
 
     Furthermore, because of {tilde over (μ)}≦{tilde over (z)}: {tilde over (z)}−{tilde over (μ)}=|{tilde over (z)}−{tilde over (μ)}|, therefore 
     
       
         
           
             
               
                 ∫ 
                 0 
                 
                   
                     z 
                     ∼ 
                   
                   - 
                   
                     μ 
                     ∼ 
                   
                 
               
               ⁢ 
               f 
             
             = 
             
               
                 ∫ 
                 0 
                 
                    
                   
                     
                       z 
                       ∼ 
                     
                     - 
                     
                       μ 
                       ∼ 
                     
                   
                    
                 
               
               ⁢ 
               
                 f 
                 . 
               
             
           
         
       
     
     Now it holds |{tilde over (μ)}|−|{tilde over (z)}|=∥{tilde over (μ)}|−|{tilde over (z)}˜≦|{tilde over (z)}−{tilde over (μ)}| and f is monotone decreasing for positive arguments, therefore it holds 
     
       
         
           
             
               
                 ∫ 
                 
                    
                   
                     z 
                     ∼ 
                   
                    
                 
                 
                    
                   
                     μ 
                     ∼ 
                   
                    
                 
               
               ⁢ 
               f 
             
             ≤ 
             
               
                 ∫ 
                 0 
                 
                    
                   
                     
                       z 
                       ∼ 
                     
                     - 
                     
                       μ 
                       ∼ 
                     
                   
                    
                 
               
               ⁢ 
               f 
             
           
         
       
     
     (see  FIG. 2 ) 
     Altogether it has been shown: 
                       ∫   0     z   ~       ⁢   f     +       ∫   0          μ   ~            ⁢   f     +     erf   ⁢           ⁢       (     z   ~     )     ·       ∫   0          μ   ~            ⁢   f           &lt;         ∫   0     z   ~       ⁢   f     +       ∫   0          μ   ~            ⁢   f         =           ∫          z   ~                 μ   ~            ⁢   f     ≤       ∫   0            z   ~     -     μ   ~              ⁢   f       =       ∫   0       z   ~     -     μ   ~         ⁢   f         ,         
that is (2) for μ≦z&lt;0.
 
     Now let {tilde over (z)}&lt;{tilde over (μ)}, i.e. {tilde over (z)}−{tilde over (μ)}&lt;0. 
     As above (*), it follows 
     
       
         
           
             
               
                 
                   ∫ 
                   0 
                   
                     z 
                     ~ 
                   
                 
                 ⁢ 
                 f 
               
               + 
               
                 
                   ∫ 
                   0 
                   
                      
                     
                       μ 
                       ~ 
                     
                      
                   
                 
                 ⁢ 
                 f 
               
             
             = 
             
               
                 
                   ∫ 
                   
                      
                     
                       z 
                       ~ 
                     
                      
                   
                   
                      
                     
                       μ 
                       ~ 
                     
                      
                   
                 
                 ⁢ 
                 f 
               
               = 
               
                 - 
                 
                   
                     ∫ 
                     
                        
                       
                         μ 
                         ~ 
                       
                        
                     
                     
                        
                       
                         z 
                         ~ 
                       
                        
                     
                   
                   ⁢ 
                   
                     f 
                     . 
                   
                 
               
             
           
         
       
     
     Analogue it holds because of the symmetry of f for the right side of (2): 
     
       
         
           
             
               
                 ∫ 
                 0 
                 
                   
                     z 
                     ~ 
                   
                   - 
                   
                     μ 
                     ~ 
                   
                 
               
               ⁢ 
               f 
             
             = 
             
               
                 - 
                 
                   
                     ∫ 
                     
                       
                         z 
                         ~ 
                       
                       - 
                       
                         μ 
                         ~ 
                       
                     
                     0 
                   
                   ⁢ 
                   f 
                 
               
               = 
               
                 
                   - 
                   
                     
                       ∫ 
                       0 
                       
                         - 
                         
                           ( 
                           
                             
                               z 
                               ~ 
                             
                             - 
                             
                               μ 
                               ~ 
                             
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     f 
                   
                 
                 = 
                 
                   - 
                   
                     
                       ∫ 
                       0 
                       
                         
                           μ 
                           ~ 
                         
                         - 
                         
                           z 
                           ~ 
                         
                       
                     
                     ⁢ 
                     
                       f 
                       . 
                     
                   
                 
               
             
           
         
       
     
     Therefore, (2) is equivalent to: 
     
       
         
           
             
               
                 
                   
                     
                       
                         - 
                         
                           
                             ∫ 
                             
                                
                               
                                 μ 
                                 ~ 
                               
                                
                             
                             
                                
                               
                                 z 
                                 ~ 
                               
                                
                             
                           
                           ⁢ 
                           f 
                         
                       
                       + 
                       
                         
                           erf 
                           ⁡ 
                           
                             ( 
                             
                               z 
                               ~ 
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             
                                
                               
                                 μ 
                                 ~ 
                               
                                
                             
                           
                           ⁢ 
                           f 
                         
                       
                     
                     ⁢ 
                     
                       ≤ 
                       ! 
                     
                     ⁢ 
                     
                       - 
                       
                         
                           ∫ 
                           0 
                           
                             
                               μ 
                               ~ 
                             
                             - 
                             
                               z 
                               ~ 
                             
                           
                         
                         ⁢ 
                         
                           f 
                           . 
                         
                       
                     
                   
                   ⁢ 
                   
                     ⇔ 
                     
                       
                         erf 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             z 
                             ~ 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           - 
                           erf 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                              
                             
                               z 
                               ~ 
                             
                              
                           
                           ) 
                         
                       
                     
                   
                   ⁢ 
                   
                     
                       
                         ∫ 
                         0 
                         
                           
                             μ 
                             ~ 
                           
                           - 
                           
                             z 
                             ~ 
                           
                         
                       
                       ⁢ 
                       f 
                     
                     ≤ 
                     
                       
                         
                           ∫ 
                           
                              
                             
                               μ 
                               ~ 
                             
                              
                           
                           
                              
                             
                               z 
                               ~ 
                             
                              
                           
                         
                         ⁢ 
                         f 
                       
                       + 
                       
                         erf 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                              
                             
                               z 
                               ~ 
                             
                              
                           
                           ) 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             
                                
                               
                                 μ 
                                 ~ 
                               
                                
                             
                           
                           ⁢ 
                           
                             f 
                             . 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Now showing (3): 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               ∫ 
                               
                                  
                                 
                                   μ 
                                   ~ 
                                 
                                  
                               
                               
                                  
                                 
                                   z 
                                   ~ 
                                 
                                  
                               
                             
                             ⁢ 
                             f 
                           
                           + 
                           
                             erf 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                    
                                   
                                     z 
                                     ~ 
                                   
                                    
                                 
                                 ) 
                               
                               · 
                               
                                 
                                   ∫ 
                                   0 
                                   
                                      
                                     
                                       μ 
                                       ~ 
                                     
                                      
                                   
                                 
                                 ⁢ 
                                 f 
                               
                             
                           
                         
                         = 
                           
                         ⁢ 
                         
                           
                             
                               ∫ 
                               
                                  
                                 
                                   μ 
                                   ~ 
                                 
                                  
                               
                               
                                  
                                 
                                   z 
                                   ~ 
                                 
                                  
                               
                             
                             ⁢ 
                             f 
                           
                           + 
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   erfc 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                        
                                       
                                         z 
                                         ~ 
                                       
                                        
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             · 
                             
                               
                                 ∫ 
                                 0 
                                 
                                    
                                   
                                     μ 
                                     ~ 
                                   
                                    
                                 
                               
                               ⁢ 
                               f 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 ∫ 
                                 0 
                                 
                                    
                                   
                                     z 
                                     ~ 
                                   
                                    
                                 
                               
                               ⁢ 
                               f 
                             
                             - 
                             
                               erfc 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                      
                                     
                                       z 
                                       ~ 
                                     
                                      
                                   
                                   ) 
                                 
                                 · 
                                 
                                   
                                     ∫ 
                                     0 
                                     
                                        
                                       
                                         μ 
                                         ~ 
                                       
                                        
                                     
                                   
                                   ⁢ 
                                   f 
                                 
                               
                             
                           
                           = 
                           … 
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                           
                             
                               
                                 
                                   μ 
                                   ~ 
                                 
                                 - 
                                 
                                   z 
                                   ~ 
                                 
                               
                               &lt; 
                               
                                  
                                 
                                   z 
                                   ~ 
                                 
                                  
                               
                             
                             , 
                             
                               
                                 for 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   μ 
                                   ~ 
                                 
                               
                               &lt; 
                               0 
                             
                           
                         
                         ⁢ 
                           
                         ⁢ 
                         
                           
                             
                               
                                 ∫ 
                                 0 
                                 
                                   
                                     μ 
                                     ~ 
                                   
                                   - 
                                   
                                     z 
                                     ~ 
                                   
                                 
                               
                               ⁢ 
                               f 
                             
                             + 
                             
                               
                                 ∫ 
                                 
                                   
                                     μ 
                                     ~ 
                                   
                                   - 
                                   
                                     z 
                                     ~ 
                                   
                                 
                                 
                                    
                                   
                                     z 
                                     ~ 
                                   
                                    
                                 
                               
                               ⁢ 
                               f 
                             
                             - 
                             
                               
                                 
                                   erfc 
                                   ⁡ 
                                   
                                     ( 
                                     
                                        
                                       
                                         z 
                                         ~ 
                                       
                                        
                                     
                                     ) 
                                   
                                 
                                 
                                   ︷ 
                                   
                                     &gt; 
                                     0 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   ∫ 
                                   0 
                                   
                                      
                                     
                                       μ 
                                       ~ 
                                     
                                      
                                   
                                 
                                 ⁢ 
                                 f 
                               
                             
                           
                           ≥ 
                           … 
                         
                       
                     
                   
                   
                     
                       
                         
                           ≥ 
                           
                             0 
                             ≤ 
                             f 
                             ≤ 
                             1 
                           
                         
                         ⁢ 
                           
                         ⁢ 
                         
                           
                             
                               ∫ 
                               0 
                               
                                 
                                   μ 
                                   ~ 
                                 
                                 - 
                                 
                                   z 
                                   ~ 
                                 
                               
                             
                             ⁢ 
                             f 
                           
                           + 
                           
                             
                               min 
                               
                                 x 
                                 ∈ 
                                 
                                   [ 
                                   
                                     
                                       
                                         μ 
                                         ~ 
                                       
                                       - 
                                       
                                         z 
                                         ~ 
                                       
                                     
                                     , 
                                     
                                        
                                       
                                         z 
                                         ~ 
                                       
                                        
                                     
                                   
                                   ] 
                                 
                               
                             
                             ⁢ 
                             
                               
                                 f 
                                 ⁡ 
                                 
                                   ( 
                                   x 
                                   ) 
                                 
                               
                               · 
                               
                                 
                                   ( 
                                   
                                     
                                        
                                       
                                         z 
                                         ~ 
                                       
                                        
                                     
                                     - 
                                     
                                       ( 
                                       
                                         
                                           μ 
                                           ~ 
                                         
                                         - 
                                         
                                           z 
                                           ~ 
                                         
                                       
                                       ) 
                                     
                                   
                                   ) 
                                 
                                 
                                   ︷ 
                                   
                                     = 
                                     
                                       
                                         
                                           - 
                                           
                                             z 
                                             ~ 
                                           
                                         
                                         - 
                                         
                                           μ 
                                           ~ 
                                         
                                         + 
                                         
                                           z 
                                           ~ 
                                         
                                       
                                       = 
                                       
                                          
                                         
                                           μ 
                                           ~ 
                                         
                                          
                                       
                                     
                                   
                                 
                               
                             
                           
                           - 
                           
                             
                               erfc 
                               ⁡ 
                               
                                 ( 
                                 
                                    
                                   
                                     z 
                                     ~ 
                                   
                                    
                                 
                                 ) 
                               
                             
                             · 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               
                                 
                                   max 
                                   
                                     x 
                                     ∈ 
                                     
                                       [ 
                                       
                                         0 
                                         , 
                                         
                                            
                                           
                                             μ 
                                             ~ 
                                           
                                            
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   f 
                                   ⁢ 
                                   
                                     ( 
                                     x 
                                     ) 
                                   
                                 
                               
                               
                                 ︷ 
                                 
                                   = 
                                   1 
                                 
                               
                             
                             · 
                             
                               ( 
                               
                                 
                                    
                                   
                                     μ 
                                     ~ 
                                   
                                    
                                 
                                 - 
                                 0 
                               
                               ) 
                             
                           
                           = 
                         
                       
                     
                   
                   
                     
                       
                         
                           = 
                           
                             
                               
                                 
                                   
                                     f 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     is 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     monotone 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     decreasing 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                 
                               
                             
                             
                               
                                 
                                   for 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   positive 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   arguments 
                                 
                               
                             
                           
                         
                         ⁢ 
                           
                         ⁢ 
                         
                           
                             
                               ∫ 
                               0 
                               
                                 
                                   μ 
                                   ~ 
                                 
                                 - 
                                 
                                   z 
                                   ~ 
                                 
                               
                             
                             ⁢ 
                             f 
                           
                           + 
                           
                             
                               f 
                               ⁡ 
                               
                                 ( 
                                 
                                    
                                   
                                     z 
                                     ~ 
                                   
                                    
                                 
                                 ) 
                               
                             
                             · 
                             
                                
                               
                                 μ 
                                 ~ 
                               
                                
                             
                           
                           - 
                           
                             erfc 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 
                                    
                                   
                                     z 
                                     ~ 
                                   
                                    
                                 
                                 ) 
                               
                               · 
                               
                                  
                                 
                                   μ 
                                   ~ 
                                 
                                  
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Now it holds 
     
       
         
           
             
               erfc 
               ⁢ 
               
                   
               
               ⁢ 
               
                 ( 
                 x 
                 ) 
               
             
             ≤ 
             
               
                 f 
                 ⁡ 
                 
                   ( 
                   x 
                   ) 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               for 
               ⁢ 
               
                   
               
               ⁢ 
               x 
             
             &gt; 
             0. 
           
         
       
       
         
           
             
               
                 
                   ⌈ 
                   
                     
                       erfc 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     = 
                       
                     ⁢ 
                     
                       
                         
                           2 
                           
                             π 
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             x 
                             ∞ 
                           
                           ⁢ 
                           
                             
                               exp 
                               ⁡ 
                               
                                 ( 
                                 
                                   - 
                                   
                                     t 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                       
                       ⁢ 
                       
                         = 
                         
                           
                             
                                 
                               `` 
                             
                             ⁢ 
                             du 
                           
                           = 
                           
                             dt 
                             ″ 
                           
                         
                         
                           u 
                           := 
                           
                             t 
                             - 
                             x 
                           
                         
                       
                       ⁢ 
                       
                         
                           2 
                           
                             π 
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             0 
                             ∞ 
                           
                           ⁢ 
                           
                             
                               exp 
                               ⁡ 
                               
                                 ( 
                                 
                                   - 
                                   
                                     
                                       ( 
                                       
                                         u 
                                         + 
                                         x 
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               u 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       2 
                       
                         π 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                       ⁢ 
                       
                         
                           
                             exp 
                             ⁡ 
                             
                               ( 
                               
                                 - 
                                 
                                   u 
                                   2 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             exp 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     - 
                                     2 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   ux 
                                 
                                 - 
                                 
                                   x 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           u 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       exp 
                       ⁡ 
                       
                         ( 
                         
                           - 
                           
                             x 
                             2 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       2 
                       
                         π 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                       ⁢ 
                       
                         
                           
                             exp 
                             ⁡ 
                             
                               ( 
                               
                                 - 
                                 
                                   u 
                                   2 
                                 
                               
                               ) 
                             
                           
                           
                             ︸ 
                             
                               &gt; 
                               0 
                             
                           
                         
                         ⁢ 
                         
                           
                             exp 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   - 
                                   2 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 ux 
                               
                               ) 
                             
                           
                           
                             ︸ 
                             
                               
                                 0 
                                 &lt; 
                                 … 
                                 ⁢ 
                                 
                                     
                                 
                                 ≤ 
                                 1 
                               
                               , 
                               
                                 
                                   for 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   x 
                                 
                                 &gt; 
                                 0 
                               
                               , 
                               
                                 u 
                                 ≥ 
                                 0 
                               
                             
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           u 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   ≤ 
                     
                   ⁢ 
                   
                     
                       exp 
                       ⁡ 
                       
                         ( 
                         
                           - 
                           
                             x 
                             2 
                           
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       2 
                       
                         π 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                       ⁢ 
                       
                         
                           exp 
                           ⁡ 
                           
                             ( 
                             
                               - 
                               
                                 u 
                                 2 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           u 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                     ⁢ 
                     
                       
                         exp 
                         ⁡ 
                         
                           ( 
                           
                             - 
                             
                               x 
                               2 
                             
                           
                           ) 
                         
                       
                       ≡ 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                     
                   
                   ⌋ 
                 
               
             
           
         
       
     
     Therefore, it holds because of (4) 
     
       
         
           
             
               
                 
                   
                     
                       ∫ 
                       
                          
                         
                           μ 
                           ~ 
                         
                          
                       
                       
                          
                         
                           z 
                           ~ 
                         
                          
                       
                     
                     ⁢ 
                     f 
                   
                   + 
                   
                     
                       erf 
                       ⁡ 
                       
                         ( 
                         
                            
                           
                             z 
                             ~ 
                           
                            
                         
                         ) 
                       
                     
                     · 
                     
                       
                         ∫ 
                         0 
                         
                            
                           
                             μ 
                             ~ 
                           
                            
                         
                       
                       ⁢ 
                       f 
                     
                   
                 
                 ≥ 
                 
                   
                     
                       ∫ 
                       0 
                       
                         
                           μ 
                           ~ 
                         
                         - 
                         
                           z 
                           ~ 
                         
                       
                     
                     ⁢ 
                     f 
                   
                   + 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         
                            
                           
                             z 
                             ~ 
                           
                            
                         
                         ) 
                       
                     
                     · 
                     
                        
                       
                         μ 
                         ~ 
                       
                        
                     
                   
                   - 
                   
                     
                       f 
                       ⁡ 
                       
                         ( 
                         
                            
                           
                             z 
                             ~ 
                           
                            
                         
                         ) 
                       
                     
                     · 
                     
                        
                       
                         μ 
                         ~ 
                       
                        
                     
                   
                 
               
               = 
               
                 
                   ∫ 
                   0 
                   
                     
                       μ 
                       ~ 
                     
                     - 
                     
                       z 
                       ~ 
                     
                   
                 
                 ⁢ 
                 f 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 that 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 is 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   3 
                   ) 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 for 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 z 
               
               &lt; 
               
                 μ 
                 . 
               
             
           
         
       
     
     Because (3)           (2)         (1), everything has been shown.
     Add on for faulty case: Satellite j faulty! Shift TH j . 
     Referring to  FIG. 3 , the overbounding condition without shift by TH j  fulfilled, i.e. 
     
       
         
           
             q 
             := 
             
               
                 1 
                 
                   
                     
                       2 
                       ⁢ 
                       π 
                     
                   
                   ⁢ 
                   σ 
                 
               
               ⁢ 
               
                 exp 
                 ⁡ 
                 
                   ( 
                   
                     
                       - 
                       
                         1 
                         2 
                       
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           . 
                           σ 
                         
                         ) 
                       
                       2 
                     
                   
                   ) 
                 
               
             
           
         
       
     
                   q   ~       L   ,   S       :=     2   ⁢     χ     (       -   ∞     ,     -   TH       ]       ⁢     q   ⁡     (     ·     +   TH       )           ,     
     ⁢         q   ~       R   ,   S       :=     2   ⁢     χ     (     0   ,   ∝     )       ⁢     q   ⁡     (     ·     -   TH       )           ,     
     ⁢       q     L   ,   S       :=         K         2   ⁢   π       ⁢   σ       ⁢     exp   ⁡     (       -     1   2       ⁢       (         ·     +   TH       -     μ   L       σ     )     2       )       ⁢           ⁢   with   ⁢           ⁢     μ   L       &lt;   0                     q     R   ,   S       :=         K         2   ⁢   π       ⁢   σ       ⁢     exp   ⁡     (       -     1   2       ⁢       (         ·     -   TH       -     μ   R       σ     )     2       )       ⁢           ⁢   with   ⁢           ⁢     μ   R       &gt;   0                     F       q   ~       L   ,   S         ≤       F     q     L   ,   S         ⁢     :     ⁢       F       q   ~       L   ,   S         ⁡     (   z   )           =           F       q   ~     L       ⁡     (     z   +   TH     )       ≤       F     q   L       ⁡     (     z   +   TH     )         =       F     q     L   ,   S         ⁡     (   z   )               
with K L  as above!
 
     
       
         
           
             
               ( 
               
                 
                   Let 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   z 
                 
                 ≥ 
                 
                   
                     - 
                     TH 
                   
                   ⁢ 
                   
                     : 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   It 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   holds 
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         
                           
                             
                               F 
                               
                                 
                                   q 
                                   ~ 
                                 
                                 
                                   L 
                                   , 
                                   S 
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               z 
                               ) 
                             
                           
                           = 
                           
                             1 
                             ⁢ 
                             
                               ≤ 
                               ! 
                             
                             ⁢ 
                             
                               
                                 F 
                                 
                                   q 
                                   
                                     L 
                                     , 
                                     S 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   - 
                                   TH 
                                 
                                 ) 
                               
                             
                             ≤ 
                             
                               
                                 F 
                                 
                                   q 
                                   
                                     L 
                                     , 
                                     S 
                                   
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 z 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               F 
                               
                                 q 
                                 L 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 z 
                                 + 
                                 TH 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                           
                             
                               K 
                               2 
                             
                             ⁢ 
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   erf 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         z 
                                         - 
                                         
                                           ( 
                                           
                                             
                                               μ 
                                               L 
                                             
                                             - 
                                             TH 
                                           
                                           ) 
                                         
                                       
                                       
                                         
                                           2 
                                         
                                         ⁢ 
                                         σ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   ok 
                 
               
               ) 
             
             ⇒ 
             
               1 
               ⁢ 
               
                 ≤ 
                 ! 
               
               ⁢ 
               
                 
                   K 
                   2 
                 
                 ⁢ 
                 
                   ( 
                   
                     1 
                     + 
                     
                       erf 
                       ⁡ 
                       
                         ( 
                         
                           
                             - 
                             
                               μ 
                               L 
                             
                           
                           
                             
                               2 
                               ⁢ 
                               σ 
                             
                           
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     Claim: 
     
       
         
           
             
               
                 F 
                 
                   
                     q 
                     ~ 
                   
                   
                     R 
                     , 
                     S 
                   
                 
                 * 
               
               ≤ 
               
                 F 
                 
                   p 
                   ⁡ 
                   
                     ( 
                     
                       - 
                       TH 
                     
                     ) 
                   
                 
               
             
             , 
             
               
                 
                   F 
                   
                     p 
                     ⁡ 
                     
                       ( 
                       
                         + 
                         TH 
                       
                       ) 
                     
                   
                 
                 ≤ 
                 
                   
                     F 
                     
                       
                         q 
                         ~ 
                       
                       
                         L 
                         , 
                         S 
                       
                     
                   
                   ⁢ 
                   
                     : 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       F 
                       
                         
                           q 
                           ~ 
                         
                         
                           L 
                           , 
                           S 
                         
                       
                     
                     ⁡ 
                     
                       ( 
                       z 
                       ) 
                     
                   
                 
               
               = 
               
                 
                   2 
                   ⁢ 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       z 
                     
                     ⁢ 
                     
                       
                         
                           χ 
                           
                             ( 
                             
                               
                                 - 
                                 ∞ 
                               
                               , 
                               
                                 - 
                                 TH 
                               
                             
                             ] 
                           
                         
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       ⁢ 
                       
                         q 
                         ⁡ 
                         
                           ( 
                           
                             x 
                             + 
                             TH 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         x 
                       
                     
                   
                 
                 ⁢ 
                 
                   = 
                   
                     y 
                     := 
                     
                       x 
                       + 
                       TH 
                     
                   
                 
                 ⁢ 
                 
                   
                     2 
                     ⁢ 
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         
                           z 
                           + 
                           TH 
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               
                                 χ 
                                 
                                   ( 
                                   
                                     
                                       - 
                                       ∞ 
                                     
                                     , 
                                     
                                       - 
                                       TH 
                                     
                                   
                                   ] 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   y 
                                   - 
                                   TH 
                                 
                                 ) 
                               
                             
                             ︸ 
                           
                           
                             = 
                             
                               
                                 χ 
                                 
                                   ( 
                                   
                                     
                                       - 
                                       ∞ 
                                     
                                     , 
                                     0 
                                   
                                   ] 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 y 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                           q 
                           ⁡ 
                           
                             ( 
                             y 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           y 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         … 
                       
                     
                   
                   = 
                   
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         
                           z 
                           + 
                           TH 
                         
                       
                       ⁢ 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             χ 
                             
                               ( 
                               
                                 
                                   - 
                                   ∞ 
                                 
                                 , 
                                 0 
                               
                               ] 
                             
                           
                           ⁡ 
                           
                             ( 
                             y 
                             ) 
                           
                         
                         ⁢ 
                         
                           q 
                           ⁡ 
                           
                             ( 
                             y 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           y 
                         
                       
                     
                     = 
                     
                       
                         F 
                         
                           
                             q 
                             ~ 
                           
                           L 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           z 
                           + 
                           TH 
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
     Therefore, it holds 
     
       
         
           
             
               
                 
                   F 
                   
                     p 
                     ⁡ 
                     
                       ( 
                       
                         + 
                         TH 
                       
                       ) 
                     
                   
                 
                 ⁡ 
                 
                   ( 
                   z 
                   ) 
                 
               
               = 
               
                 
                   
                     ∫ 
                     
                       - 
                       ∞ 
                     
                     z 
                   
                   ⁢ 
                   
                     
                       p 
                       ⁡ 
                       
                         ( 
                         
                           x 
                           + 
                           TH 
                         
                         ) 
                       
                     
                     ⁢ 
                     
                       ⅆ 
                       x 
                     
                   
                 
                 = 
                 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       
                         z 
                         + 
                         TH 
                       
                     
                     ⁢ 
                     
                       
                         p 
                         ⁡ 
                         
                           ( 
                           y 
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         y 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           F 
                           p 
                         
                         ⁡ 
                         
                           ( 
                           
                             z 
                             + 
                             TH 
                           
                           ) 
                         
                       
                       ≤ 
                       
                         
                           F 
                           
                             
                               q 
                               ~ 
                             
                             L 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             z 
                             + 
                             TH 
                           
                           ) 
                         
                       
                       ≤ 
                       
                         
                           
                             F 
                             
                               
                                 q 
                                 ~ 
                               
                               
                                 L 
                                 , 
                                 S 
                               
                             
                           
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                         ⁢ 
                         
                           
                             F 
                             
                               
                                 q 
                                 ~ 
                               
                               
                                 R 
                                 , 
                                 S 
                               
                             
                             * 
                           
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         1 
                         - 
                         
                           
                             
                               K 
                               R 
                             
                             ︸ 
                           
                           
                             = 
                             1 
                           
                         
                         + 
                         
                           
                             F 
                             
                               
                                 q 
                                 ~ 
                               
                               
                                 R 
                                 , 
                                 S 
                               
                             
                           
                           ⁡ 
                           
                             ( 
                             z 
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         = 
                         
                           as 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           above 
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               F 
                               
                                 
                                   q 
                                   ~ 
                                 
                                 R 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 z 
                                 - 
                                 TH 
                               
                               ) 
                             
                           
                           ≤ 
                           
                             
                               F 
                               p 
                             
                             ⁡ 
                             
                               ( 
                               
                                 z 
                                 - 
                                 TH 
                               
                               ) 
                             
                           
                         
                         = 
                         
                           
                             
                               F 
                               
                                 p 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     - 
                                     TH 
                                   
                                   ) 
                                 
                               
                             
                             ⁡ 
                             
                               ( 
                               z 
                               ) 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ok 
                         
                       
                     
                   
                 
               
             
             ⁢ 
             
                 
             
           
         
       
     
     The foregoing disclosure has been set forth merely to illustrate the invention and is not intended to be limiting. Since modifications of the disclosed embodiments incorporating the spirit and substance of the invention may occur to persons skilled in the art, the invention should be construed to include everything within the scope of the appended claims and equivalents thereof.