PATENT DOCUMENT

Abstract:
A method of calculating position data for an airborne aircraft using a GPS-based airborne navigation system includes the processing of a position component of a relative state function by fusing a plurality of different types of measurement data available in the GPS-based system into a weighted least squares algorithm to determine an appropriate covariance matrix for the plurality of different types of measurement data.

Full Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This applications claims the benefit under 35 U.S.C. Section 119(e), of Provisional Application No. 60/822,195, filed Aug. 11, 2006, the disclosure of which is incorporated herein by reference in its entirety. 
    
    
     FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
     Not Applicable 
     BACKGROUND 
     1. Field of the Invention 
     The present invention relates to Global Positioning System (GPS)-based Relative Navigation Systems, and more particularly to a method for processing the relative state of the position component of an airborne navigation system. 
     2. Background of the Invention 
     A need has been found for providing a rapidly deployable, adverse weather, adverse terrain, day-night, survivable, and mobile precision approach and landing capability that supports the principles of forward presence, crisis response and mobility. The capability should enable military forces to land on any suitable surface world wide (land and sea), under both peace time and hostile conditions, with ceiling and/or visibility the limiting factor. 
     The Joint Precision Approach and Landing System (JPALS) is an integral part of such a strategic system. JPALS is a differential GPS that provides a rapidly deployable, mobile, adverse weather, adverse terrain, day-night, and survivable precision approach and landing capability. The JPALS allows an aircraft to land on any suitable land or sea-based surface worldwide, while minimizing the impact to airfield operations, because of low ceiling or poor visibility. This approach and landing system provides the capability of performing conventional and special operations from a fixed-base, shipboard, and austere environments under a wide range of meteorological and terrain conditions. 
     One of the primary functions of sea-based JPALS is to determine the relative state (position, velocity and acceleration) of an airborne vehicle with respect to its assigned ship. This function is called “Relative State Function” or “Relative Navigation” (“RelNav”). Existing civil and military systems do not satisfy JPALS requirements because they have a number of shortcomings that limit joint operations. The multiplicity of systems, in and of itself, hinders inter-Service, civil, and allied operations. 
     Among the limitations of conventional systems, especially as applied to calculating position data of an aircraft closing in on a target, is that in order to calculate an aircraft&#39;s position, numerous measurements must taken. Certain measurement types have better accuracy but require more processing time to achieve results. Traditionally, four measurements of the same type are needed to be used in a solution; otherwise optimal solutions can not be made available. Conventional methods that use a weighted least squares position solution use only a single measurement type. Furthermore, conventional methods have been less than satisfactory when relative position estimation is needed between two moving vehicles (e.g., between an aircraft and a ship). 
     SUMMARY OF THE INVENTION 
     As used herein, the terms “invention” and “present invention” are to be understood as encompassing the invention described herein in its various embodiments and aspects, as well as any equivalents that may suggest themselves to those skilled in the pertinent arts. 
     The present invention provides a novel and effective system and method for a GPS-based airborne navigation system to process the position component of the Relative State Function by fusing different types measurement data, i.e., data representing several different measured parameters. More specifically, the present invention provides an improved method of calculating the position data of aircraft closing in on target locations. 
     The present invention enhances the speed and accuracy of such position calculations. This is accomplished by the fusing some or all of the different types of measurement data available in a GPS-based system into a weighted least squares algorithm to determine the appropriate covariance matrix, given the different measurement types. This method allows optimal accuracy in the position solution by including all satellites, and their best available measurements. Moreover, the invention can operate and provide satisfactory results whenever relative position estimation is needed between two vehicles (e.g., an aircraft and a ship). 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is an illustration of the overall set up of a relative navigation system, of the type in which the system and method of the present invention may be employed; 
         FIG. 2  is an architectural block diagram of an airborne relative state system that includes an airborne RelNav computer system and interfaces; 
         FIG. 3  is a flow diagram showing major functional/processing blocks of an airborne RelNav system including the RelNav computer system and interfaces; and 
         FIG. 4  is a flow chart identifying process steps to calculate position data in accordance with present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The following definitions are provided as they are typically (but not exclusively) used in GPS-based navigation systems implementing the various adaptive aspects of the present invention. 
     The GPS is a satellite-based navigation system having a constellation of 24 Earth orbiting satellites. These satellites are approximately uniformly dispersed around six circular orbits having four satellites in each orbit. Each GPS satellite transmits at two frequencies: L 1  (1575.42 MHz) and L 2  (1227.60 MHz). The L 1  frequency has two different spread-spectrum codes modulated on it: a coarse acquisition (CA) code, and a Y code. The Y code is also modulated onto the L 2  carrier. 
     Earth centered, Earth fixed (ECEF) is a Cartesian coordinate system used for GPS. It represents positions as X, Y, and Z coordinates in meters. The point (0,0,0) denotes the center of the earth, hence the name Earth-Centered. 
     World Geodetic System of 1984 (WGS 84) a geodetic reference systems used by GPS and developed by the U.S. Defense Mapping Agency. 
     Double Difference (technique) is a measurement method using two receivers. Differences are formed at each measurement epoch between two satellites on each receiver and then between the same two receivers. 
     To better understand the invention, an overall system description is given, and the function and operation of the present invention will be described in specific detail. 
       FIG. 1  shows an overall set up for a relative navigation system  100  between an aircraft  102  and a ship  106 . Both the aircraft  102  and the ship  106  receive L 1  and L 2  GPS measurements from a GPS satellite constellation  104 . The ship  106  processes its measurement data to generate its state (position, velocity) and estimated wide lane phase ambiguities. Ship measurement data are generated from as many as 4 GPS antennas. The computed ship state (position, velocity) and estimated wide lane phase ambiguities are broadcast, using radio frequencies, for use by all aircraft within broadcast range of the ship. 
     The aircraft  102  combines its own measurements with the ship measurements, state, and ambiguities received over the broadcast, to produce a relative vector solution between its own GPS antenna and the ship&#39;s reference point. It also determines the quality of the relative vector solution. 
     Referring to  FIG. 2 , an Airborne Relative State System  110  includes an Airborne RelNav computer system  116  that computes the relative position and velocity vector between the ship  106  and the aircraft  102 . It also computes the uncertainty of the computed position and velocity vectors. The computed relative state and its uncertainty are inputted to a guidance and control (G&amp;C) system  118  for guidance and control of the aircraft  102 . 
     To assure system modularity, the interfaces in the airborne relative state system  110  are well defined. Five such interfaces are shown in  FIGS. 2 and 3 : 
     Aircraft Avionic System  117  and its associated avionics interface (AI)  117 A; 
     Airborne GPS receiver  119  and its associated GPS receiver interface (AGRI)  119 A; 
     Airborne Data Link  112  and its associated data link interface (DLI)  112 A; 
     Configuration data  114  and its associated configuration date interface (CDI)  114 A; and 
     Airborne Guidance and Control system  118  and its associated guidance and control interface (GCI)  118 A. 
     The Relative State Function has been partitioned into several sub-functions, as illustrated in  FIG. 3 . The algorithms to implement each sub-function are not described; only the sub-functions performed are identified. A detailed description of functionality and algorithmic implementation is provided only for the Precision Relative State module (PRS)  137 , since its architecture and operation embody the present invention. 
     The functional representation of the RelNav computer system  116  in  FIG. 3  identifies nine functional blocks along with three sub-blocks. A Measurement Management and Validation (MMV) function  129  receives data from Airborne GPS receivers  119  via the AGRI  119 A, Configuration Data  114  via the CDI  114 A, and Ship Data Uncompressed (DU)  125  via the DLI  112 A. The DU  125  converts ship position and velocity data to WGS-84 ECEF coordinates. The MMV function  129  validates Pseudo Range (PR) and Carrier Phase (CP) data (“Measurement Validation”  127 ), synchronizes the use of ephemeral data between ship and aircraft (“Sync. of Ephemeral Data”  127   a ), propagates ship position and velocity data to aircraft time (“Propagation of Ship Data . . . ”  126 ), synchronizes aircraft measurements to ship time (“Sync. of Aircraft Data . . . ”  128 ), and computes the sigma value of each aircraft GPS measurement (“Compute Sigma”  128   a ). 
     A Ship Troposphere Correction (STC) function  131  applies tropospheric corrections to ship GPS measurements. An Aircraft Troposphere Correction function (ATC)  132  applies tropospheric corrections to aircraft GPS measurements. An Absolute Position and Velocity (APV) function  130  computes the aircraft&#39;s absolute position, the positions of satellites from the ephemeral data, and the aircraft&#39;s absolute velocity solution. A Relative Measurements (PM) function  133  calculates Double-Difference (DD) measurements (L 1  PR, L 2  PR, L 1  CP, L 2  CP, WL CP, NL PR) at ship time, and calculates variance components (multipath and noise, ionosphere corrections, troposphere corrections) of DD measurements from Single-Difference (SD) measurements (L 1  PR, L 2  PR, L 1  CP, L 2  CP, WL CP, NL PR) and variances at ship time. The satellite highest in elevation is chosen as a reference, and the PM function  133  calculates covariance matrices components (multipath and noise, ionosphere corrections, troposphere corrections) of the DD measurements. Since ship measurements may come from four separate antennas, four separate solutions may be required. 
     A Wide Lane Ambiguities (WLA) function  134  determines WL float ambiguities and covariance (Ship UD, Aircraft UD, SD, and DD). It determines the probability of correctly fixing WL ambiguities, and it also determines the Discrimination Ratio of integer ambiguities, validates integer ambiguities, and fixes WL ambiguities. An L 1 /L 2  Ambiguities (L 1 L 2 A) function  135  is performed assuming WL fixed ambiguities have been resolved successfully. It determines L 1  and L 2  float ambiguities and covariance, determines the probability of correctly fixing L 1  and L 2  ambiguities, determines the Discrimination Ratio (DR) of resolved L 1  and L 2  integer ambiguities, validates integer ambiguities, and fixes L 1  and L 2  ambiguities. A Basic Relative State (BRS) function  136  computes a Basic Relative State (position and velocity) at aircraft time and computes a Covariance of Basic Relative State (position and velocity). 
     The G &amp; C Interface (GCI)  118 A provides precision relative state outputs to Guidance and Control functions, and it receives input from the Precision Relative State module (PRS)  137 . The Avionics Interface (AI)  117 A provides Basic Relative State outputs to the aircraft avionics system. The PRS module  137  determines a Precision Relative State only if an aircraft&#39;s distance to the ship is less than 10 nautical miles A satellite should only be considered in a particular solution if its measurements have been consecutively available for a configurable amount of time. The initial interval is set at 30 seconds or 60 samples. 
     A functional block diagram for the Precision Relative State (PRS) module  137  is shown in  FIG. 4 , illustrating the different calculations performed to arrive at a Precision Relative State solution. A first computing module  138  computes code range and phase range values from Measurements and Covariance. DD NL Code Range Observables are calculated by Equations (1), (2) and (3): 
     
       
         
           
             
               
                 
                   
                       
                   
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             where R 1  and R 2  are DD code ranges 
           
         
         DD Phase Range Observables are calculated by Equations (4) to (9): 
         L 1  Float:
 
Φ 1     —     flo =Φ 1 −λ 1   N   1     —     flo 
 
 C   Φ     1     =C   DD     —     L     1—     m,e   +C   DD     —     L     1—     iono   +C   DD     —     L     1—     tropo 
 
 C   Φ     1—     flo    =C   Φ     1     +λ   1    2    C    N      1—     flo    (4)
 
         L 2  Float:
 
Φ 2     —     flo =Φ 2 −λ 2   N   2     —     flo 
 
 C   Φ     1     =C   DD     —     L     2—     m,e   +C   DD     —     L     2—     iono   +C   DD     —     L     2—     tropo 
 
 C   101     2—flo      =C   101      2      +λ   2    2    C    N      2—flo      (5)
 
         L 1  Fixed:
 
Φ 1     —     fix =Φ 1 −λ 1   N   1     —     fix 
 
 C    101      1—fix      =C    101      1      (6)
 
         L 2  Fixed:
 
Φ 2     —     fix =Φ 2 −λ 2   N   2     —     fix 
 
 C    101      2 —fix      =C    101    2    (7)
 
       
    
     
       
         
           
             
               
                 
                   
                       
                   
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         WL Fixed:
 
Φ wl     —     fix =Φ wl −λ wl   N   wl     —     fix 
 
 C    101      wl—fix      =C    101      wl      (9)
 
       
    
     After the calculations are performed by the first computing module  138 , first and second functional modules  139  and  140  select measurement data for further processing. The best data are selected based on measurement variance from each of the double difference pair, using the above-defined methods. The orders are: L 1  fixed, L 2  fixed, L 1  float, L 2  float, Wide Lane Fixed, Wide Lane Float, and Narrow Lane Code. This way every visible satellite is used in the solution, weighted only by its own relative uncertainty. 
     A third functional module  141  assembles data for a covariance matrix element calculation. The covariance matrix to be used in the solution is more difficult to generate than the other solutions and is done element by element. Diagonal elements for this solution are simply taken from the diagonal elements of the double difference covariance matrix and the measurement type is determined using the equations above. For examples if measurement two were a wide land fixed measurement, the M(2, 2) element would be taken from the WL fixed covariance matrix and placed in the hybrid covariance matrix in the same spot. 
     A fourth functional module  142  performs the covariance matrix element calculation. Off diagonal elements are determined by the relationships between each measurement type. The simplest way to provide the information is to show a complete example and provide the algorithms. For this example the hybrid solution includes each type of measurement in descending order as follows, and each of these measurements is the first measurement in its respective solution type. 
     L 1 _fixed 
     L 2 _fixed 
     L 1 _float 
     L 2 _float 
     WL_fixed 
     WL_float 
     NL_Code 
     The covariance matrix is shown by Equation (10): 
     
       
         
           
             
               
                 
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     where the elements of matrix calculated by Equations (11) to (32: 
     
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             λ 
                             2 
                             2 
                           
                           
                             λ 
                             1 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   b 
                   = 
                   
                     
                       σ 
                       
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           m 
                         
                         , 
                         
                           e 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           SV 
                         
                       
                       2 
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         L 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         iono 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   c 
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               λ 
                               1 
                             
                             
                               
                                 λ 
                                 2 
                               
                               - 
                               
                                 λ 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               λ 
                               2 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 1 
                               
                             
                           
                           
                             λ 
                             1 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   d 
                   = 
                   
                     
                       
                         - 
                         
                           ( 
                           
                             
                               λ 
                               2 
                             
                             
                               
                                 λ 
                                 2 
                               
                               - 
                               
                                 λ 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 2 
                                 2 
                               
                             
                             + 
                             
                               
                                 λ 
                                 2 
                               
                               ⁢ 
                               
                                 λ 
                                 1 
                               
                             
                           
                           
                             λ 
                             1 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   e 
                   = 
                   
                     
                       
                         ( 
                         
                           
                             λ 
                             WL 
                           
                           
                             λ 
                             1 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             λ 
                             2 
                           
                           
                             λ 
                             1 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   f 
                   = 
                   
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             λ 
                             2 
                           
                           
                             λ 
                             1 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   g 
                   = 
                   
                     
                       σ 
                       
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           m 
                         
                         , 
                         
                           e 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           SV 
                         
                       
                       2 
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             λ 
                             2 
                             4 
                           
                           
                             λ 
                             1 
                             4 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   h 
                   = 
                   
                     
                       
                         ( 
                         
                           
                             λ 
                             1 
                           
                           
                             
                               λ 
                               2 
                             
                             - 
                             
                               λ 
                               1 
                             
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               λ 
                               2 
                               3 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 1 
                               
                               ⁢ 
                               
                                 λ 
                                 2 
                                 2 
                               
                             
                           
                           
                             λ 
                             1 
                             3 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
             
               
                 
                   i 
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               λ 
                               2 
                             
                             
                               
                                 λ 
                                 2 
                               
                               - 
                               
                                 λ 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 2 
                                 4 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 2 
                                 3 
                               
                               ⁢ 
                               
                                 λ 
                                 1 
                               
                             
                           
                           
                             λ 
                             1 
                             4 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   j 
                   = 
                   
                     
                       
                         - 
                         
                           ( 
                           
                             
                               λ 
                               WL 
                             
                             
                               λ 
                               2 
                             
                           
                           ) 
                         
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             λ 
                             2 
                             3 
                           
                           
                             λ 
                             1 
                             3 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   k 
                   = 
                   
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             λ 
                             2 
                             3 
                           
                           
                             λ 
                             1 
                             3 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   l 
                   = 
                   
                     
                       
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                 λ 
                                 1 
                               
                               
                                 
                                   λ 
                                   2 
                                 
                                 - 
                                 
                                   λ 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             
                               λ 
                               1 
                             
                             
                               
                                 λ 
                                 2 
                               
                               - 
                               
                                 λ 
                                 1 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       
                         
                           ( 
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   λ 
                                   1 
                                 
                               
                               + 
                               
                                 λ 
                                 2 
                               
                             
                             
                               λ 
                               1 
                             
                           
                           ) 
                         
                         2 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
             
               
                 
                   m 
                   = 
                   
                     
                       
                         - 
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                 λ 
                                 1 
                               
                               
                                 
                                   λ 
                                   2 
                                 
                                 - 
                                 
                                   λ 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             λ 
                             2 
                           
                           
                             
                               λ 
                               2 
                             
                             - 
                             
                               λ 
                               1 
                             
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               λ 
                               2 
                             
                             
                               
                                 λ 
                                 2 
                               
                               - 
                               
                                 λ 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             λ 
                             1 
                           
                           
                             
                               λ 
                               2 
                             
                             - 
                             
                               λ 
                               1 
                             
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 2 
                                 3 
                               
                             
                             + 
                             
                               5 
                               ⁢ 
                               
                                 λ 
                                 2 
                                 2 
                               
                               ⁢ 
                               
                                 λ 
                                 1 
                               
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 1 
                                 2 
                               
                               ⁢ 
                               
                                 λ 
                                 2 
                               
                             
                           
                           
                             λ 
                             1 
                             3 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   n 
                   = 
                   
                     
                       
                         - 
                         
                           ( 
                           
                             
                               λ 
                               WL 
                             
                             
                               λ 
                               2 
                             
                           
                           ) 
                         
                       
                       * 
                       
                         ( 
                         
                           
                             λ 
                             1 
                           
                           
                             
                               λ 
                               2 
                             
                             - 
                             
                               λ 
                               1 
                             
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             λ 
                             WL 
                           
                           
                             λ 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         ( 
                         
                           1 
                           - 
                           
                             
                               λ 
                               1 
                             
                             
                               
                                 λ 
                                 2 
                               
                               - 
                               
                                 λ 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         
                             
                         
                         ⁢ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           tropo 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                       
                       2 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 1 
                               
                               ⁢ 
                               
                                 λ 
                                 2 
                               
                             
                             + 
                             
                               λ 
                               2 
                               2 
                             
                           
                           
                             λ 
                             1 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           
                               
                           
                           ⁢ 
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             iono 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             sv 
                           
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
             
               
                 
                   o 
                   = 
                   
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 1 
                               
                               ⁢ 
                               
                                 λ 
                                 2 
                               
                             
                             + 
                             
                               λ 
                               2 
                               2 
                             
                           
                           
                             λ 
                             1 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   p 
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               λ 
                               2 
                             
                             
                               
                                 λ 
                                 2 
                               
                               - 
                               
                                 λ 
                                 1 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             1 
                             + 
                             
                               
                                 λ 
                                 2 
                               
                               
                                 
                                   λ 
                                   2 
                                 
                                 - 
                                 
                                   λ 
                                   1 
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 2 
                                 2 
                               
                             
                             + 
                             
                               
                                 λ 
                                 2 
                               
                               ⁢ 
                               
                                 λ 
                                 1 
                               
                             
                           
                           
                             λ 
                             1 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   q 
                   = 
                   
                     
                       
                         - 
                         
                           ( 
                           
                             
                               λ 
                               WL 
                             
                             
                               λ 
                               1 
                             
                           
                           ) 
                         
                       
                       * 
                       
                         ( 
                         
                           
                             λ 
                             2 
                           
                           
                             
                               λ 
                               2 
                             
                             - 
                             
                               λ 
                               1 
                             
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             λ 
                             WL 
                           
                           
                             λ 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         ( 
                         
                           1 
                           + 
                           
                             
                               λ 
                               2 
                             
                             
                               
                                 λ 
                                 2 
                               
                               - 
                               
                                 λ 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 2 
                                 3 
                               
                             
                             + 
                             
                               
                                 λ 
                                 2 
                                 2 
                               
                               ⁢ 
                               
                                 λ 
                                 1 
                               
                             
                           
                           
                             λ 
                             1 
                             3 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
             
               
                 
                   r 
                   = 
                   
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     - 
                     
                       
                         ( 
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 λ 
                                 2 
                                 3 
                               
                             
                             + 
                             
                               
                                 λ 
                                 2 
                                 2 
                               
                               ⁢ 
                               
                                 λ 
                                 1 
                               
                             
                           
                           
                             λ 
                             1 
                             3 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   s 
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               λ 
                               WL 
                             
                             
                               λ 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             
                               λ 
                               WL 
                             
                             
                               λ 
                               1 
                             
                           
                           ) 
                         
                         2 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             m 
                           
                           , 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             λ 
                             2 
                             2 
                           
                           
                             λ 
                             1 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   t 
                   = 
                   
                     
                       
                         ( 
                         
                           
                             λ 
                             2 
                             2 
                           
                           
                             λ 
                             1 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
             
               
                 
                   u 
                   = 
                   
                     s 
                     + 
                     
                       
                         λ 
                         WL 
                         2 
                       
                       * 
                       
                         C 
                         
                           mw 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           SD 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           fl 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
             
               
                 
                   v 
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               λ 
                               NL 
                             
                             
                               λ 
                               1 
                             
                           
                           ) 
                         
                         2 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             M 
                           
                           , 
                           
                             E 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             
                               λ 
                               NL 
                             
                             
                               λ 
                               2 
                             
                           
                           ) 
                         
                         2 
                       
                       * 
                       
                         σ 
                         
                           
                             AB 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             L 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             M 
                           
                           , 
                           
                             E 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             key 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             _ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             SV 
                           
                         
                         2 
                       
                     
                     + 
                     
                       σ 
                       
                         AB 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         tropo 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         key 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         _ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         sv 
                       
                       2 
                     
                     + 
                     
                       
                         ( 
                         
                           
                             λ 
                             2 
                             2 
                           
                           
                             λ 
                             1 
                             2 
                           
                         
                         ) 
                       
                       * 
                       
                         σ 
                         
                           AB 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           L 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           iono 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           key 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           _ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           sv 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     The cases not covered in the example are the relationships between common solutions. In these cases the following applies: 
     L 1 _fixed to L 1 _fixed is equivalent to b; 
     L 2 _fixed to L 2 _fixed is equivalent to g; 
     L 1 _float to L 1 _float is equivalent to l; 
     L 2  float to L 2 _float is equivalent to p; 
     WL_fixed to WL fixed is equivalent to s; 
     WL_float to WL_float is equivalent to u; and 
     NL code to NL code is equivalent to v: 
     A fifth functional module  143  calculates a Relative Position Solution using the Weighted Least Squares Method. The measurements should include every satellite, using the most accurate (smallest variance) measurement available for that satellite. The calculations executed by different steps of the WLS algorithm are shown below. The initialization is performed once at the start of a filter. All other steps are iterated (2 or 3 times) until a solution converges. 
     Step 1 initializes the baseline vector b (at start of the filter) by Equation (33)
 
x k   − =0   (33)
 
     Step 2 calculates Measurements and Covariance by Equations (34), (35) and (36):
 
Z=Z Hybrid   (34)
 
C z =Σ Hybrid    (35)
 
W=C z   −1    (36)
 
     Step 3 calculates LOS (Line of Sight) vectors from Ship and Aircraft to Satellite j by Equations (37)-(42): 
     In the equations, satellite position {x s ,y s ,z s } is from ephemeris and ship position {x A ,y A ,z A } is from an absolute position solution. 
     LOS Vectors: 
     
       
         
           
             
               
                 
                   
                     
                       
                         Ship 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           A 
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         to 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         satellite 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         j 
                       
                     
                     
                       
                         
                           as 
                           ⁢ 
                           
                             ( 
                             j 
                             ) 
                           
                         
                         = 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       x 
                                       s 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     x 
                                     A 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       y 
                                       s 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     y 
                                     A 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       z 
                                       s 
                                     
                                     ⁡ 
                                     
                                       ( 
                                       j 
                                       ) 
                                     
                                   
                                   - 
                                   
                                     z 
                                     A 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   
                     
                       
                         Aircraft 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           B 
                           ) 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         to 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         satellite 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         j 
                       
                     
                     
                       
                         
                           
                             
                               
                                 bs 
                                 ⁡ 
                                 
                                   ( 
                                   j 
                                   ) 
                                 
                               
                               = 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           
                                             x 
                                             s 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             j 
                                             ) 
                                           
                                         
                                         - 
                                         
                                           x 
                                           B 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             y 
                                             s 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             j 
                                             ) 
                                           
                                         
                                         - 
                                         
                                           y 
                                           B 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             z 
                                             s 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             j 
                                             ) 
                                           
                                         
                                         - 
                                         
                                           z 
                                           B 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                         
                           
                             
                               = 
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         
                                           
                                             x 
                                             s 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             j 
                                             ) 
                                           
                                         
                                         - 
                                         
                                           ( 
                                           
                                             
                                               x 
                                               A 
                                             
                                             + 
                                             
                                               
                                                 b 
                                                 ^ 
                                               
                                               x 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             y 
                                             s 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             j 
                                             ) 
                                           
                                         
                                         - 
                                         
                                           ( 
                                           
                                             
                                               y 
                                               A 
                                             
                                             + 
                                             
                                               
                                                 b 
                                                 ^ 
                                               
                                               y 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             z 
                                             s 
                                           
                                           ⁡ 
                                           
                                             ( 
                                             j 
                                             ) 
                                           
                                         
                                         - 
                                         
                                           ( 
                                           
                                             
                                               z 
                                               A 
                                             
                                             + 
                                             
                                               
                                                 b 
                                                 ^ 
                                               
                                               z 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   
                     
                       
                         ( 
                         37 
                         ) 
                       
                     
                   
                   
                     
                       
                           
                       
                     
                   
                   
                     
                       
                           
                       
                     
                   
                   
                     
                       
                           
                       
                     
                   
                   
                     
                       
                         ( 
                         38 
                         ) 
                       
                     
                   
                 
               
             
           
         
       
         
         
           
             Magnitudes of LOS vectors:
 
 as _mag( j )=∥ as ( j )∥; and   (39)
 
 bs _mag( j )=∥ bs ( j )∥  (40)
 
Predicted SD Measurement (as−bs)
 
 dab ( j )= as _mag( j )− bs _mag( j );   (41)
 
Predicted DD Measurements
 
           
         
       
    
                     Z   k   -     =     [             dab   ⁡     (   1   )       -     dab   ⁡     (   2   )                     dab   ⁡     (   1   )       -     dab   ⁡     (   3   )                 …               dab   ⁡     (   1   )       -     dab   ⁡     (   K   )               ]             (   42   )               
using satellite  1  (highest) as reference
 
     Step 4 performs Linearization by evaluating Equations (43)-(45): 
     Unit LOS Vectors from Aircraft to Satellite j: 
                     [             1   x     ⁢     (   j   )                   1   y     ⁢     (   j   )                   1   z     ⁢     (   j   )             ]     =         bs   ⁡     (   j   )       /   bs_mag     ⁢     (   j   )               (   43   )               
Differences of Unit Vectors
   e   x (1 ,j )=1 x (1)−1 x ( j )   e   y (1 ,j )=1 y (1)−1 y ( j )   e   z (1 ,j )=1 z (1)−1 z ( j )   (44) 
H Matrix (Matrix Formed by Differenced Unit Vectors)
 
     
       
         
           
             
               
                 
                   H 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             
                               e 
                               x 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 2 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               e 
                               y 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 2 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               e 
                               z 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 2 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               e 
                               x 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 3 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               e 
                               y 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 3 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               e 
                               z 
                             
                             ⁡ 
                             
                               ( 
                               
                                 1 
                                 , 
                                 3 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           … 
                         
                         
                           … 
                         
                         
                           … 
                         
                       
                       
                         
                           … 
                         
                         
                           … 
                         
                         
                           … 
                         
                       
                       
                         
                           … 
                         
                         
                           … 
                         
                         
                           … 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     Steps 5 computes the Weighted Least Square Relative Position solution by Equations (46)-(49): 
     Filter Observable:
 
Δ Z=[Z   k   −Z   k   − ]  (46)
 
WLS Solution and Solution Covariance:
 
Δ{circumflex over (x)}=[H T WH] −1 H T W ΔZ   (47)
 
 {circumflex over (x)}   k   +   ={circumflex over (x)}   k   −   +Δ{circumflex over (x)}   (48)
 
Σ {circumflex over (x)} =[H T W H] −1    (49)
 
     The solution is arrived at by transforming each Relative Position Solution and its Covariance from WGS-84 to NED coordinates (north-east-down coordinates at ship reference point). 
     The final output from the process includes the following data:
         Hybrid Relative Position NED   Covariance of Hybrid Relative Position