4.1 Introduction
There are two approaches for an aircraft user namely non precision approach and precision approach. In non precision approach there is only horizontal guidance and no vertical guidance.
For non precision approach the user can use GPS signal (Enge et. al.,1996). In precision approach there are both horizontal guidance and vertical guidances. As per ICAO, for precision approach the GPS signal is not recommended, instead ICAO recommends the WAAS/LAAS. For this, the WAAS/LAAS must meet stringent requirements with respect to Required Navigation Performance (RNP) parameters. The SBAS uses the vector corrections, where as GBAS uses the scalar corrections. Precision approach is divided into three categories based on decision height (DH) and Runway Visual Range (RVR) (Enge et. al., 1996). The basic requirement for avionics navigation is that its position error should be very small. Most of the times, this requirement is not guaranteed. The user computes the protection levels in both horizontal and vertical direction by using satellite geometry and the range error. These Protection Levels (PL’s) are compared with alert limits to perform particular operation (surface landing, takeoff, etc) (Clark et. al., 2006).4.2 Protection Levels
The left over errors between airborne receiver and reference receivers are called residual decorrelation errors. The bounds on the errors that are caused due to thermal noise, multipath, nominal ionospheric variations and nominal troposphere variations are known as protection levels (Rife, Pullen. et. al., 2006). There are two types of protection levels, which gives the user about the integrity.
4.2.1 Horizontal Protection Level
The Horizontal Protection Level is the radius of a circle in the horizontal plane (the plane tangent to the WGS-84 ellipsoid), with its center being at the true position (Fig 1), (Nikiforov, 2002). It is based upon the error estimates provided by WAAS (WAAS Performance Analysis Report, Report #14, October 2005).
4.2.2 Vertical Protection Level (VPL)
The Vertical Protection Level is half of the length of a segment on the vertical axis (perpendicular to the horizontal plane of WGS-84 ellipsoid), with its center being at the true position (Fig 2) (Nikiforov, 2002). It is based upon the error estimates provided by WAAS (WAAS Performance Analysis Report, Report #14, October, 2005).
Fig 4.1 Horizontal protection Level Fig 4.2 Vertical Protection Level
4.3 Significance of VPL in avionics navigation
In order to meet integrity, an aircraft should satisfies HPL and VPL over the exposure window. The exposure window shows the duration of approach and landing operations. The duration of exposure window determines the prior probability that a hazardous anomaly occurs during the approach or landing operation. This exposure window is 150 seconds for Cat I and for Cat III the exposure window is 15 seconds (vertical direction) and 30 seconds (horizontal direction) (Rife and Phelts, 2008). The duration of exposure window in vertical direction is less and the integrity violation occurs if vertical position errors exceed the VPL for longer than time to alert during the exposure window.
The geometry distribution of the satellite constellation is poorer in vertical direction causing large vertical error than the horizontal error (Rife and Phelts, 2008). The vertical position error (VPE) is a multiplication of Vertical Dilute of Precision (VDOP) and User Range Error (URE) (Leva, 1994).
Where
HPE is Horizontal position error
HDOP is Horizontal dilution of precision
From Eq 1 it is clear that VPE is directly proportional to VDOP and URE. The VDOP value is always higher than HDOP indicating that VPE is greater than HPE. It is because the receivers get the signal from the satellites which are above the receiver. The HPE do not suffer because the receiver gets the signal from all sides. The VPE suffer because all satellites are above the receiver and none below the receiver. The blockage of the satellites below a user’s horizon leads to asymmetry, causing large error in vertical direction. (Misra et. al., 1999).
4.4 Estimation of protection level
The airborne receiver calculates three most dominant VPL expressions namely, fault free reference receiver (), single reference receiver fault () and ephemeris fault () (Rife and Phelts, 2008). The VPL is taken as maximum of these three expressions. The is considered, because it is greater than the and.
4.4.1 Estimation of
The Vertical protection level for fault free reference receiver () is calculated as (Murphy, 2000)
Where
is projection of the vertical component for the ith ranging source and is calculated from projection matrix also known as S matrix
is ith element of first row of the projection matrix S
is ith element of the third row of S
N is number of ranging sources used in the position solution
i is ranging source index
is fault free missed detection
is total error variance of individual satellite in view
For least square position estimation, the projection matrix is calculated as (DeCleene, 2000).
Where
S is Projection matrix
G is observation matrix
W is weighted matrix
The observation matrix is estimated in up-east-north (UEN) reference frame using line of sight (LOS) elevation (El) and LOS azimuth (Az). The formulae for evaluating observation matrix is (Grewal et. al., 2007 ).
The weighted matrix is defined as a function of error variance of each satellite in view and is evaluated as
Where
n is number of satellites in view
The equation to estimate total variance is
Where
is ground error variance for the ith ranging source
is residual tropospheric delay for the ith ranging source
is variance of the aircraft contribution to the corrected pseudorange error for ith ranging source. The total aircraft contribution includes the receiver contribution and the standard allowance for aircraft multipath
is residual ionospheric delay for the ith ranging source
Similar procedure can be followed for the estimation of
4.4.2 Estimation
The Horizontal protection level for fault free reference receiver () is calculated as (Murphy, 2000),
Where
=is the projection of the horizontal component for the ith ranging source
is the ith element of second row of S.
4.5 Parameters required for estimation of VPL
The following basic parameters are required for the estimation of VPL. These are briefly discussed in this section.
4.5.1 Tail area probability
Tail area probability is the area under the standard normal curve to the right or left of the threshold value of the test statistics for one tailed test. For two tailed test it is the combined area under the standard normal curve to the right of threshold and to the left of threshold. A test statistic is a function of the sample measurement upon which the statistical decision will be based. The shaded region of Fig 3 shows the one sided tail area probability and Fig 4 shows the two sided tail area probability. Tail area probability for different receivers and different categories are shown in Table 1.
Fig 4.3 One sided tail area probability Fig 4.4 Two sided tail area probability
In LAAS tail area probability depends on category and no of reference receivers.
Table 4.1 Tail area probability (Rowson, et al., 1998)
M is no of reference receivers
4.5.2 Geometry of satellite
The accurate position of the user is dependent on two parameters namely the Pseudo- Random Noise (PRN) code ranging precise and geometry of satellite in view (James Rankin, 1994). To describe the effect of geometry on position, the Dilution of Precisions
(DOP’s) parameters are used. Generally DOP’s are categorized into four namely Geometrical Dilution of Precision (GDOP), Position Dilution of Precision (PDOP), Vertical Dilution of Precision (VDOP) and Horizontal Dilution of Precision (HDOP).
The GDOP is defined as relationship between geometry of GPS satellites and the user (Jianping, et al, 1998). The Geometric Dilution of Precision (GDOP) is the value that indicates the effectiveness of the GPS satellite geometry on the positional accuracy of the user. The characteristics of GDOP is that the smaller the value is the more precise the position given by GPS satellite to the user. This characteristic is valid when the PRN code ranging precise is constant (Yang Yong, et al, 2004).
Fig 4.5 Illustration of Satellite Geometry
In order to calculate GDOP by user there should be a minimum of four satellites in view to the user. The typical values of DOP’s are as follows GDOP = 1.73, PDOP = 1.63, HDOP = 1.15, VDOP = 1.15 and TDOP = 0.58 (James, 2000).
4.5.3 Elevation and azimuth angle
The troposphere and ionospheric errors are calculated with respect to elevation angle. In order to calculate geometry matrix the user need elevation and azimuth angles.
a) Elevation angle
The elevation angle of an object in the sky (in the illustration below Figure 4.2 the object is the satellite) is the angle between the plane that forms the horizon and an imaginary line to the object. When the object is on the horizon the elevation angle is nearly 0°. When the object is directly overhead, or at "zenith," the elevation angle is 90°.
b) Azimuth angle
The satellite is vertically projected on the earth surface and the angle from that point to true north in anti clockwise direction is known as azimuth angle. It ranges from 0 to 360 degrees.
Figure 4.6 Elevation and azimuth angle of GPS satellite
4.5.4 Error variance
The various residual errors required in the estimation of VPL is discussed in detail in this section.
4.5.4.1 Residual troposphere pseudo-range error
In LAAS the troposphere error is dominant error compared to ionosphere error. This error is not completely cancelled by using differential method. The remaining error after differential correction is known as residual error. This residual error exists due to presence of difference in height between LGF and user. This residual error is considered for estimation of protection levels. The parameters required for evaluating this residual errors are sent by ground processing facility (LGF in LAAS ) to the user via a VHF link (Gary et. al., 2000).
4.5.4.2 Residual Ionosphere pseudo-range error
In LAAS this error in not much concern, because most of the ionospheric error between the user and LGF are common. These error are cancelled by using differential method. Still some residual error exists due to presence of the temporal and spatial decorrelation, which causes an ionosphere residual error (Gary et. al., 2000).
4.5.4.3 Airborne receiver pseudo-range error
The overall airborne RMS (Root mean square) pseudo-range error contains two components. These are noise and multipath. It is evaluated as (Murphy, 2000).
a) Receiver noise
The noise in the receiver is caused due to random behaviour of thermal noise. This noise is uncorrelated between ground reference receivers and airborne receiver. This error is dominant in LAAS (Rife and Phelts, 2008).
Where
is the elevation angle and are constants that depends on Airborne Accuracy Designators (AAD). Its values are shown in Table 2.
Table 4.2 Receiver noise modelling parameters (Rowson et. al., 1998)
b) Multipath
Multipath error will occur when the user receives the GPS signal through different paths. These paths can be direct LOS signal and through reflected signal of object surrounding the user (El-Rabbany, 2002 ). It cannot be corrected by using LAAS differential correction, because multipath error is uncorrelated between user and reference receivers. It can be modelled with respect to elevation angle (Gary et. al., 2000).
4.5.4.4 Ground receiver pseudo-range error
The ground reference receivers consider the errors from all contributing sources and calculate the error corrections. These corrections are calculated by averaging individual corrections obtained from reference receivers (Pervan et al., 2005). These corrections are denoted as. These errors are assumed to be zero mean Gaussian distribution (N (0,), but they never follow in real situations (Misra et. al., 2006). Even if it follows at core (till), the tails never follows Gaussian distribution. Therefore, tails of actual error distribution is heavier than Gaussian distribution (Braff and Shively, 2005).
The values are broadcast to the user by LGF for each satellite in view. The user receives these values as an input, to compute the protection levels (Lee, et al., 2006). The is calculated with respect to elevation angle of satellite (Eq. 12) (Rowson et. al., 1998).
Where
is elevation angle and are constants that depend on elevation angle and Ground Accuracy Designators (GAD) (Table 3)
The GAD is dependent on parameters such as elevation angle and constants values. These are defined by accuracy classes namely A, B and C (Matteo et. al., 2004). The accuracy of C is high compared to B and A (Gary et. al., 2000).
4.6 Results
Station 2 location is considered as airborne receiver location which is static. On 01 January 2009, t Stat RINEX file had downloaded from SOPAC which is maintained jointly by NASA and JPL. The VPL is drawn for two categories Cat I and Cat II&III for the Station 2 data as shown in Fig 4.7 and Fig 4.8 respectively.
Figure 4.7 Cat I plotted Graph VPL v/s time
Figure 4.8 Cat II&III plotted Graph VPL v/s time
On x axis the time (seconds) and on y axis the VPL (meters) is considered, the graph is drawn. The used formulae for estimation of VPL are eq 4.3. The variable is calculated with respect to elevation angle as given by RTCA Document (Stephen, et al., 1998). The graphs are drawn with an aid of MATLAB.
6 Conclusions
In this chapter, the significance of VPL is presented by comparing it with the VPE and HPE. The VPL estimation for fault free reference receivers along with necessary formulas is explained. To estimate VPL, understanding of tail probability and error variance is necessary. The residual troposphere pseudo-range error, residual ionosphere pseudo-range error, airborne receiver pseudo-range error and ground receiver pseudo range error are discussed. During precision approach, satisfying VPL is more critical than HPL. This is because HAL is more compared to VAL. The VPL graph is plotted by considering Station 2 data.
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