3 The State of the Aircraft
The primary sources of information on the location and motion of the aircraft are inertial navigation systems and global positioning systems. Both are described in this section, and combined results that merge the best features of each into composite variables for location and motion are also discussed. Useful references for material in this section are Lenschow (1972) and RAF Bulletin 23.
3.1 Inertial Reference Systems
An Inertial Navigation System (INS) or Inertial Reference Unit (IRU) provides measurements of aircraft position, velocity relative to the Earth, acceleration and attitude or orientation. The IRU provides basic measurements of acceleration and angular rotation rate, while the INS integrates those measurements to track the position, altitude, velocity, and orientation of the aircraft. For the GV, the system is a Honeywell Laseref IV HG2001 GD03 Inertial Reference System; for the C-130, it is a Honeywell Model HG1095-AC03 Laseref V SM Inertial Reference System. These systems are described on the EOL web site, at this URL. Data from the IRS come via a serial digital bit stream (the ARINC digital bus) to the Aircraft Data System (ADS). Because there is some delay in transmission and recording of these variables, adjustments for this delay are made when the measurements are merged into the processed data files, as documented in the NetCDF header files and as discussed in Section 2.3. Typical delays are about 80 ms for variables including ACINS, PITCH, ROLL, and THDG.
Some variables are recorded only in the original “raw” data files and are not usually included in final archived data files; these are discussed at the end of this subsection. See also the discussion in Section 10 for information on results from inertial systems that were used prior to installation of the present Honeywell systems.
An Inertial Navigation System “aligns” while the aircraft is stationary by measurement of the variations in its reference frame caused by the rotation of the Earth. Small inaccuracy in that alignment leads to a “Schuler oscillation” that produces oscillatory errors in position and other measurements, with a period \(\tau_{Sch}\) of about 84 minutes (\(\tau_{Sch}=2\pi\sqrt{R_{E}/g}\)). Position errors of less than \(1.0\,\)n mi/h are within normal operating specifications. See Section 3.4 for discussion of additional variables, similar to the following, for which corrections are made for these errors via reference to data from a Global Positioning System.
Some projects have used smaller Systron Donner C-MIGITS Inertial Navigation Systems with GPS coupling, usually in connection with special instruments like a wing-mounted wind-sensing system. For these units, variable names usually begin with the letter C but otherwise have names matching the following variables (e.g., CLAT). GPS coupling via a Kalman filter is incorporated in the measurements from these units.
Uncertainties associated with measurements from the IRS are discussed in a Technical Note, available at this URL. See page 7 of that document and the tables on pages 41 and 49.
3.1.1 Standard Variables
Latitude (º): LAT
The aircraft latitude or angular distance north of the equator in an Earth reference frame. Positive values are north of the equator; negative values are south. The resolution is 0.00017\(^{\circ}\) and the accuracy is reported by the manufacturer to be 0.164\(^{\circ}\) after 6 h of flight. Values are provided by the INS at a frequency of 10 Hz.
Longitude (º): LON
The aircraft longitude or angular distance east of the prime meridian in an Earth reference frame. Positive values are east of the prime meridian; negative values are west. The resolution is 0.00017\(^{\circ}\) and the accuracy is reported by the manufacturer to be 0.164\(^{\circ}\) after 6 h of flight. Values are provided by the INS at a frequency of 10 Hz.
Aircraft True Heading (\(^{\circ}\)): THDG
The azimuthal angle between the center-line of the aircraft (pointing ahead, toward the nose) and a line of meridian. This azimuthal angle is measured in a polar coordinate system oriented relative to the Earth with polar axis upward and azimuthal angle measured relative to true north. The heading thus indicates the orientation of the aircraft, not necessarily the direction in which the aircraft is traveling. The resolution is 0.00017\(^{\circ}\) and the uncertainty is quoted by the manufacturer as 0.2\(^{\circ}\) after 6 h of flight. Values are provided by the INS at a frequency of 25 Hz. “True” distinguishes the heading from the magnetic heading, the heading that would be measured by a magnetic compass. For more information on the coordinate system used, see RAF Bulletin 23.
Aircraft Pitch Attitude Angle (\(^{\circ}\)): PITCH
The angle between the center-line of the aircraft (pointing ahead, toward the nose) and the horizontal plane in a reference frame relative to the Earth with polar axis upward. Positive values correspond to the nose of the aircraft pointing above the horizon. The resolution is 0.00017\(^{\circ}\) and the uncertainty is quoted by the manufacturer as 0.05\(^{\circ}\) after 6 h of flight. Values are provided by the INS at a frequency of 50 Hz.
Aircraft Roll Attitude Angle (\(^{\circ}\)): ROLL
The angle of rotation about the longitudinal axis of the aircraft required to bring the lateral axis (along the wings) to the horizontal plane. Positive angles indicate that the starboard (right) wing is down (i.e., a clockwise rotation has occurred from level when facing forward in the aircraft). The resolution is 0.00017\(^{\circ}\) and the uncertainty is quoted by the manufacturer as 0.05\(^{\circ}\) after 6 h of flight. Values are provided by the INS at a frequency of 50 Hz.
Aircraft Vertical Acceleration (m s-2): ACINS
The acceleration upward (relative to the Earth) as measured by an inertial reference unit. With INSs now in use, the internal drift that arises when this measurement is integrated to get aircraft vertical speed and then altitude is removed by the INS via pressure damping through reference to the pressure altitude.6 Positive values are upward. The sample rate is 50 Hz and the resolution is 0.0024 m s\(^{-2}\).
Computed Aircraft Vertical Velocity (m/s): VSPD
The upward velocity of the aircraft, or rate-of-climb relative to the Earth, as measured by the INS. VSPD is determined within the INS by integration of the vertical acceleration, with damping based on measured pressure to correct for accumulated errors in the integration of acceleration. The sample rate is 50 Hz with a resolution of 0.00016 m/s. The Honeywell Laseref INS employs a baro-inertial loop, similar to that described below for WP3 and the Litton LTN-51, to update the value of the acceleration. This variable is also filtered within the INS so that there is little variance with frequency higher than 0.1 Hz.
Aircraft Rate of Climb (m/s): ROC (new 2017)
The rate of climb or upward speed of the aircraft, as measured by the INS with correction so as to represent the derivative of the geometric height. This variable is calculated by integration of the variable ACINS and then addition of the low-pass-filtered difference between that integral and the climb rate determined from the hydrostatic equation. The result retains the high-frequency response from the INS while matching the low-frequency average value determined from the hydrostatic equation, and so represents change in geometric height. This memo contains additional background information on this variable.
\[\begin{align}
g(z,\lambda)=g_{e}\left(\frac{1+g_{1}\sin^{2}(\lambda)}{(1-g_{2}\sin^{2}\lambda)^{1/2}}\right)(1-(k_{1}-k_{2}\sin^{2}(\lambda))z+k_{3}z^{2})
\tag{3.1}
\end{align}\]
where \(g_{e}=9.780327\) m s\(^{-2}\), \(g_{1}=0.00193185\), \(g_{2}=0.00669438\), and
{\(k_{1},k_{2},k_{3}\)} = {3.15704\(\times 10^{-7}\mathrm{m}^{-1}\),
2.10269\(\times 10^{-9}\mathrm{m}^{-1}\), 7.37452\(\times 10^{-14}\mathrm{m}^{-2}\)}
\(g\) = acceleration of gravity \(^{(a)}\) at latitude \(\lambda\) and altitude \(z\) above the WGS-84 geoid
\(R_{d}{}^{\dagger}\) = gas constant for dry air
\(T_{K}\) = absolute temperature = (ATX + 273.15)
\(a\) = ACINS = upward acceleration as measured by the INS [m s-2]
\(p\) = PSXC = measured ambient pressure [hPa]
\(\Delta p\) = difference between current and last value of PSXC
\(\Delta t\) = time between samples (1/\(f\) where \(f\) is the sample
frequency)
\(F_{L}\) = low-pass Butterworth filter (cf. p. ).
1. From consecutive measurements of pressure, estimate the rate of climb
from the hydrostatic equation:
\[\begin{equation}
w_{p}=-\frac{R_{d}T_{k}}{gp}\frac{\Delta p}{\Delta t}
\tag{3.2}
\end{equation}\]
2. Add the current measurement of acceleration to the cumulative sum:
\[\begin{equation}
w_{p}^{*}\leftarrow w_{p}^{*}+a\Delta t
\tag{3.3}
\end{equation}\]
3. Define ROC as the sum of \(w_{p}^{*}\) and the low-pass filtered value
of (\(w_{p}-w_{p}^{*}\)):
\[\begin{align}
\mathrm{ROC} & =w_{p}^{*}+F_{L}(w_{p}-w_{p}^{*})
\tag{3.4}
\end{align}\]
Inertial Altitude (m): ALT
The altitude of the aircraft as provided by an INS, with pressure damping applied within the INS to the integrated aircraft vertical velocity to avoid the accumulation of errors. The value therefore is updated to the pressure altitude, not the geometric altitude, and should be regarded as a measurement of pressure altitude that has short-term variations as provided by the INS. The sample rate is 25 Hz with a resolution of 0.038 m. In some projects ALT also referred to the altitude from the avionics GPS system; the preferred and current variable name for that is ALT_G.
Aircraft Ground Speed (m/s): GSPD
The ground speed of the aircraft as provided by an INS. The resolution is 0.0020 m/s, and the INS provides this measurement at a frequency of 10 Hz. Formerly GSF. Update to GSPD occurred in 2014.
Aircraft Ground Speed East Component (m/s): VEW
The east-directed component of ground speed as provided by an INS. The resolution is 0.0020 m/s, and the INS provides this measurement at a frequency of 10 Hz.
Aircraft Ground Speed North Component (m/s): VNS
The north-directed component of ground speed as provided by an INS. The resolution is 0.0020 m/s, and the INS provides this measurement at a frequency of 10 Hz.
Distance East/North of a Reference (km): DEI, DNI
Distance east or north of a project-dependent reference point. These are derived outputs obtained by subtracting a fixed reference position from the current position. The values are determined from measurements of latitude and longitude and converted from degrees to distance in a rectilinear coordinate system. The reference position can be either the starting location of the flight or a user-defined reference point (e.g., the location of a project radar). The accuracy of these values is dependent on the accuracy of the source of latitude and longitude measurements (see LAT and LON), and the calculations are only appropriate for short distances because they do not take into account the spherical geometry of the Earth.
LON\(_{ref}\) = reference longitude (º)
LAT\(_{ref}\) = reference latitude (º)
\(C_{deg2km}=\) conversion factor, degrees latitude to km
\(\equiv\) 111.12 km / \(^{\circ}\)
\[\begin{equation} \begin{split} \mathrm{DEI} = & \mathrm{C_{deg2km}}(\mathrm{\{LON\}}-\{\mathrm{LON}_{ref}\})\cos(\mathrm{\{LAT\}}) \notag \\ \mathrm{DNI} = & \mathrm{C_{deg2km}}(\mathrm{\{LAT\}}-\{\mathrm{LAT}_{ref}\}) \end{split} \tag{3.5} \end{equation}\]
Radial Azimuth/Distance from Fixed Reference FXAZIM, FXDIST
Azimuth and distance from a project-dependent reference point. The units of the azimuthal angle are degrees (relative to true north) and the distance is in kilometers. These are calculated by rectangular-to-polar conversion of DEI and DNI, described in the preceding paragraph.
3.1.2 Additional Special-Use Variables
The following INS and IRU variables are not normally included in archived data files, but their values are recorded by the ADS and can be obtained from the original “raw” data files:
Raw Lateral Body Acceleration (m/s2): BLATA
The raw output from the IRU lateral accelerometer. Positive values are toward the starboard, normal to the aircraft center line. The sample rate is 50 Hz with a resolution of 0.0024 m s-2.
Raw Longitudinal Body Acceleration (m/s2): BLONA
The raw output from the IRU longitudinal accelerometer. Positive values are in the direction of the nose of the aircraft and parallel to the aircraft center line. The sample rate is 50 Hz with a resolution of 0.0024 m s-2.
Raw Normal Body Acceleration (m/s2): BNORMA
The raw output from the IRU vertical accelerometer. Positive values are upward in the reference frame of the aircraft, normal to the aircraft center line and lateral axis. The sample rate is 50 Hz with a resolution of 0.0024 m s-2.
Raw Body Pitch Rate (º/s): BPITCHR
The raw output of the IRU pitch rate gyro. Positive values indicate the nose moving upward and refer to rotation about the aircraft’s lateral axis. The sample rate is 50 Hz with a resolution of 0.0039º/s.
3.2 Global Positioning Systems
Primary GPS variables specifying the position and velocity of the aircraft (strictly speaking, of the GPS antenna) are provided by GPS receivers, currently a NovAtel Model OEM-628 receiver. Prior to c. 2014 a NovAtel OEM4 was on the C-130 and a OEMV on the GV. See this link for a description of these systems. The coordinate system used for all GPS measurements is the World Geodetic System WGS-84.7 The uncertainty of the horizontal position measurements for the Novatel OEM628 receiver currently in use is specified by NovAtel to be typically 1.2 m RMS, 0.6 m, or 0.1 m for single-point L1/L2 measurements, WAAS mode, or OmniSTAR / TerraStar mode, respectively. This specification is subject to ionospheric and tropospheric conditions, satellite geometry, interference, etc. Vertical uncertainty is approximately twice the horizontal uncertainty. Because variables are stored as 4-byte single-precision floating point numbers, the inherent storage precision can limit the precision of the recorded position to about 1 m, depending on latitude. The full-precision data is available from the original aircraft data files upon request. The accuracy of velocity measurements is 0.03 m/s RMS. Prior to November 2014 all variables provided by the NovAtel GPS receivers were recorded at 5 Hz. Between then and May 2021 the output rate was 10 Hz, and since June 2021 the output rate is 20 Hz.
Prior to January 2014, latitude and longitude were recorded from the NMEA GPGGA log with a resolution of 0.0001 arcminutes (1.6x10-6 degrees), while altitude was recorded with 0.01 m resolution. Speed-over-ground was recorded from the NMEA GPRMC log with resolution of 0.001 knots (5x10-4 m/s). Starting in January 2014 two Novatel specific logs (named BESTPOS and BESTVEL) have also been recorded to preserve more significant digits in the measurements. The BESTPOS log has a horizontal position resolution of 10−11 degrees and altitude resolution of 10-4 m and also reports the estimated uncertainty (in meters) of the horizontal position and altitude. The BESTVEL log reports the horizontal speed-over-ground and the vertical speed with 1x10-4 m/s resolution.
Until April 2022, the speed data (GGSPD and GGVSPD) reported by the Novatel BESTVEL log used the default calculation method, such that the speed calculation was done using measured Doppler shift of the satellite signals in the single-point and WAAS modes, but calculated using the position difference between successive samples in the OmniSTAR/TerraStar mode. In some circumstances a peculiarity in the position difference method could cause spikes to appear in the vertical speed data. This occurred when the altitude, instead of smoothly changing, would ‘jump’ a few centimeters more than expected between two samples, resulting in a single sample spike of as much as a meter/second in the velocity data. When using the GPS speed data for wind calculations these spikes would appear there as well. Since April 2022 the receivers are set to use Doppler shift velocity calculations for the data in the BESTVEL log for all positioning modes thereby eliminating this artifact.
Prior to 2012, some of the following variables were also available from alternate Garmin GPS16 receivers, for which the variable name is qualified by the name of that unit; e.g., GGLAT_GMN for GGLAT as measured by a Garmin GPS unit. In addition, some of the measurements from the GPS units that are part of the aircraft avionics systems are recorded; these are denoted by a suffix “_G” or “_A”. The accuracy of the position measurements for that unit was stated to be 25 meters (horizontal) and 35 meters (vertical) under “steady-state conditions.”8 Likewise, velocity measurements are within 0.2 m/s for all axes. Measurement resolution is that of 4-byte IEEE format (about 6 significant digits). All variables were provided by the Trimble receivers at 1 Hz.
For some uses, a special correction is needed for variables GGVEW, GGVNS, and GGVSPD, which measure the motion at the GPS antenna relative to the Earth. The conventional wind calculation addresses the difference between the motion at the radome (where the relative wind is measured) and the INS (where variables VEW, VNS, VSPD are measured) arising from rotation of the aircraft. However, if GGVSPD is used instead of VSPD for vertical wind or GGVEW and GGVNS are used (perhaps via the complementary filter) for the horizontal wind, an additional correction is needed for the displacement between the GPS antenna and the INS receiver. A correction for aircraft rotation is therefore applied to GGVSPD, as described below.
3.2.1 Standard Variables
GPS Latitude (º): GGLAT, LAT_G; also formerly GLAT
The aircraft latitude measured by a global positioning system. Positive values are north of the equator; negative values are south. These variables are recorded in netCDF files as single-precision values. GGLAT is provided by the data-system GPS; LAT_G and LATF_G are from the avionics system GPS. LATF_G is a fine-resolution measurement that requires special processing.GPS Longitude (º): GGLON, LON_G; also formerly GLON
The aircraft longitude measured by a global positioning system. Positive values are east of the prime meridian; negative are west. GGLON is provided by the (or a) data-system GPS; LON_G and LONF_G are from the avionics system GPS. LONF_G is a fine-resolution measurement that requires special processing.
GPS Ground Speed (m/s): GGSPD, GSPD_G
The aircraft ground speed measured by a global positioning system. GGSPD originates from a data-system GPS; GSF_G originates from an avionics-system GPS.
GPS Ground Speed Vector East Component (m/s): GGVEW, VEW_G
The eastward component of ground speed measured by a global positioning system. GGVEW originates from a data-system GPS; VEW_G originates from an avionics-system GPS. In the case of GGVEW, when this is used in the calculation of horizontal wind, the following correction would be needed:
\[\begin{equation}
\mathrm{\{GGVEWA\}}=\mathrm{\{GGVEW\}-L_G}\dot{\psi}\frac{\cos\psi}{\cos\phi}
\tag{3.6}
\end{equation}\]
where GGVEWA is the corrected value used in the wind calculation, , and \(\psi\) and \(\phi\) are respectively the heading and roll angles, LG = − 4.30 m for the GV, and \(\dot{\psi}\) is the rate-of-change of heading (in radians). The variable BYAWR transmitted from the INS gives the rate-of-change of heading after conversion from \(^{\circ}\thinspace s^{-1}\) to radians s − 1. This correction is not applied in normal processing because the use of the complementary filter, discussed in Sect. 3.4, makes it of negligible importance. More information is contained in this memo.
GPS Ground Speed Vector North Component (m/s): GGVNS, VNS_G
The northward component of ground speed as measured by a global positioning system. GGVNS originates from a data-system GPS; VNS_G originates from an avionics-system GPS. In the case of GGVNS, when this is used in the calculation of horizontal wind, the following correction would be needed:
\[\begin{equation}
\mathrm{\{GGVNSA\}} = \mathrm{\{GGVNS\}}+L_{G}\dot{\psi}\frac{\sin\psi}{\cos\phi}
\tag{3.7}
\end{equation}\]
where GGVNSA is the corrected value used in the wind calculation, \(\psi\) and \(\phi\) are respectively the heading and roll angles, LG = − 4.30 m for the GV, and \(\dot{\psi}\) is the rate-of-change of heading (in radians). The variable BYAWR transmitted from the INS gives the rate-of-change of heading ψ̇ after conversion from \(^{\circ}\thinspace s^{-1}\) to radians s − 1. This correction is not applied in normal processing because the use of the complementary filter, discussed in Sect. 3.4, makes it of negligible importance, as discussed in the note referenced for GGVEW.
GPS-Measured Aircraft Vertical Velocity (m/s): GGVSPD; also (obsolete) VSPD_G and GVZI
The aircraft vertical velocity provided by a GPS unit. Positive values are upward. When GGVSPD is used in the calculation of vertical wind, the following correction (omitted before 2017) is applied: \[\begin{equation} \mathrm{\{GGVSPDA\}} = \mathrm{\{GGVSPD\}}-L_{G}\dot{\theta} \tag{3.8} \end{equation}\] where LG = − 4.30 m for the GV and \(\dot{\theta}\) is the rate-of-change of the pitch angle, corresponding to the IRU variable BPITCHR*π/180. The variable GGVSPDA is used internally but not recorded in the data archives. See this memo for additional justification.
GPS Altitude (m MSL): GGALT, GALT_A
The aircraft altitude measured by a global positioning system. The measurement is with respect to the geoid as represented internally by the GPS receiver and is determined by adding the adjustment –GGEOIDHT to the direct measurement relative to the ellipsoidal Earth model of the GPS, which is defined by WGS-84. Positive values are above the reference surface. GGALT originates from a data-system GPS; GALT_A originates from an avionics-system GPS. See the discussion of height at the beginning of this subsection and the variable GGEOIDHT below for interpretation of these GPS-based measurements.
GPS Geopotential Altitude (m): GEOPTH
The aircraft geopotential altitude above mean sea level. If g(z, λ) is the acceleration of gravity as represented by the formula in the Table of Constants in Section 1.3, then the formula used for calculation of GEOPTH is obtained by integrating that formula from the reference surface for MSL (the geoid, Δ above the WGS84 reference ellipse) to the geometric altitude H, which is H + Δ above the reference ellipse. The result is normally close (within about 0.5 m) to that obtained with Δ = 0. There are additional details in this memo.
\(H\) = aircraft altitude above mean sea level, [m] (GGALT)
\(\lambda\) = latitude (GGLAT) converted to radians
\(Z(H,\ \lambda)\) = aircraft geopotential height [m] (GEOPTH)
\(g_{0}\) = constant acceleration of gravity as defined for the International
Standard Atmosphere
\(g_{e},\,g_{1},\,g_{2},\,k_{i}\) as defined in the Table of Constants
in Sect. 1.3
\(\Delta\)= height of the geoid above the WGS-84 reference ellipse
(GGEOIDHT)
\[\begin{equation} \mathrm{\{GEOPHT\}}=Z(H,\lambda)=\frac{1}{g_{0}}\Biggl\{ g_{e}\left(\frac{1+g_{1}\sin^{2}\lambda}{(1-g_{2}\sin^{2}\lambda)^{1/2}}\right) \notag \\ \times \left(H-\frac{1}{2}\left((H+\Delta)^{2}-\Delta^{2}\right)(k_{1}-k_{2}\sin^{2}\lambda)+\frac{1}{3}\left((H+\Delta)^{3}-\Delta^{3}\right)k_{3}\right)\Biggr\} \tag{3.9} \end{equation}\]
GPS Aircraft Track Angle (º): GGTRK, TKAT_G
The direction of the aircraft track (degrees clockwise from true north) as measured by a data-system global positioning system (GGTRK) or an avionics-system GPS (TKAT_G).
GPS Height of the Geoid (m): GGEOIDHT
Height of geoid, approximating mean sea level, above the WGS-84 ellipsoid. Also commonly called the geoid undulation The height above mean sea level is found by subtracting this value from the height above the WGS-84 reference ellipse as provided by GPS-based measurements.The NovAtel OEM628 receivers use the EGM96 model which provides geoid height on a 0.5° x 0.5° grid and 0.1 m height resolution.
GPS Satellites Tracked: GGNSAT
The number of satellites tracked by a GPS receiver. The receiver may not use all tracked satellites when calculating the position and velocities.
GPS Quality Flag: GGQUAL
GPS quality flag:
| GGQUAL | description |
|---|---|
| 0 | Invalid |
| 1 | Valid but without quality enhancement. Approximately 1.2 m RMS horizontal accuracy. |
| 2 | Receiving OmniSTAR/TerraStar corrections but not fully converged to the OmniSTAR/Terrastar position accuracy specification. Horizontal accuracy will be between 1.2 m and 0.1 m RMS. |
| 5 | Fully locked-in OmniSTAR XP or TerraStar C, usually starting after about 20 minutes of tracking the GPS satellites and receiving the OmniSTAR or TerraStar data feed. This mode provides 0.1 m RMS or better horizontal position accuracy. This is described in some documents as differential GPS. |
| 9 | Measurement enhanced by the Satellite-Based Augmentation System, a means of improving GPS accuracy and integrity by broadcasting from geostationary satellites wide area corrections for GPS satellite orbits and ionospheric delays. In the US, this uses the Wide-Area Augmentation System or WAAS. This is described in some data files as a differential-GPS measurement. Horizontal accuracy is approximately 0.6 m RMS. |
GPS Status: GGSTATUS, GSTAT_G, GSTAT
The status of the GPS receiver. A value of 1 indicates that the receiver is operating normally; a value of 0 indicates a warning regarding data quality. GGSTATUS indicates the status of the data-system GPS; GSTAT_G indicates the status of the avionics-system GPS. The obsolete variable GSTAT, formerly used for the same purpose, has the reverse meaning: A value of 0 indicates normal operation and any other code indicates a malfunction or warning regarding poor data accuracy.
Altitude, Latitude, and Longitude Standard Deviation (m): GGALTSD, GGLATSD, GGLONSD
The estimated standard deviation in meters (1-sigma) of the altitude, latitude, and longitude measurements. These values are reported by the BESTPOS log in use since 2014.
Differential Age (s): GGDAGE
The time since the last OmniSTAR/TerraStar corrections data was received. The corrections data typically update approximately every 15 seconds, and the estimated position uncertainty increases during the time between updates. Once the age exceeds the timeout (300 s by default) the receiver will exit OmniSTAR/TerraStar mode.
Horizontal Dilution of Precision: GGHDOP
A measure of the impact the spatial geometry of the observed satellites has on the calculated horizontal position uncertainty. Values less then 2 are considered good. This is superceded by the reported measurement standard deviation recorded in the GGALTSD, GGLATSD, and GGLONSD variables.
3.3 Other Measurements of Aircraft Altitude
Geometric Radar Altitude (Extended Range) APN-159 (APN-159) (m): HGME
The distance to the surface below the aircraft, measured by a radar altimeter. There are two outputs from an APN-159 radar altimeter, one with coarse resolution (CHGME) and one with fine resolution (HGME). Both raw outputs cycle through the range 0-360 degrees, where one cycle corresponds to 4,000 feet for HGME and to 100,000 feet for CHGME. To resolve the ambiguity arising from these cycles, 4,000-foot increments are added to HGME to maintain agreement with CHGME. This preserves the fine resolution of HGME (1.86 m) throughout the altitude range of the APN-159.
Geometric Radar Altitude (Extended Range) APN-232 (m): HGM232
Altitude above the ground as measured by an APN-232 radar altimeter.
Height Above Terrain (m): ALTG
The aircraft altitude above the Earth’s surface as represented by the next variable. If GGALT is the altitude above mean sea level, ALTG=GGALT−SFC.
Height of the Earth’s Surface (m MSL): SFC
The altitude of the Earth’s surface at a location directly below the aircraft. The data source is the Shuttle Radar Topography Mission of 2000. The height estimate is described in this memo.
ISA Pressure Altitude (m): PALT
The geopotential altitude in the International Standard Atmosphere where the pressure is equal to the reference barometric (ambient) pressure (PSXC).9
The pressure altitude is best interpreted as a variable equivalent to the measured pressure, not as a geometric altitude. In the following description of the algorithm, some constants (identified by the symbol ‡) are specified as part of the ISA and so should not be “improved” to more modern values such as those given in the table of constants in Section 1.3 (e.g., R0‡).10
A note at this link describes the pressure altitude in more detail and documents the change that was made in November 2010.
\(T_{0}^{\ddagger}\)= 288.15 K, reference temperature
\(\lambda_{a}^{\ddagger}\) =
-0.0065 \(^{\circ}\)C per geopotential meter = the lapse rate for the troposphere\(^{\ddagger}\)
\(p\) = measured static (ambient) pressure, hPa, usually from PSXC
\(p_{0}^{\ddagger}\) = 1013.25 hPa, reference pressure for PALT=0 \(^{\ddagger}\)
\(M_{d}^{\ddagger}\) = 28.9644 kg/kmol = molecular weight of dry
air, ISA definition \(^{\ddagger}\)
\(g^{\ddagger}\) = 9.80665 m s\(^{-2}\), acceleration of gravity \(^{\ddagger}\)
\(R_{0}^{\ddagger}\) = universal gas constant, defined\(^{\ddagger}\)
as 8.31432\(\times10^{3}\) J kmol-1 K-1
\(z_{T}^{\ddagger}\) = altitude of the ISA tropopause = 11,000 m \(^{\ddagger}\)
\(x=-R_{0}^{\ddagger}\lambda_{a}^{\ddagger}/(M_{d}^{\ddagger}g^{\ddagger})\)
\(\approx\) 0.1902632 (dimensionless)
For pressure > 226.3206 hPa (equivalent to a pressure altitude < \(z_{T}\)):
\[\begin{equation}
\mathrm{PALT}=-\left(\frac{T_{0}^{\ddagger}}{\lambda^{\ddagger}}\right)\left(1-\left(\frac{p}{p_{0}^{\ddagger}}\right)^{x}\right)
\tag{3.10}
\end{equation}\]
otherwise, if \(T_{T}\)
and \(p_{T}\)
are respectively the temperature and pressure at the altitude \(z_{T}\):
\[T_{T}=T_{0}+\lambda^{\ddagger}z_{T}^{\ddagger}=216.65\,\mathrm{K}\]
\[p_{T}=p_{0}^{\ddagger}\Bigl(\frac{T_{0}^{\ddagger}}{T_{T}}\Bigr)^{\frac{g^{\ddagger}M_{d}^{\ddagger}}{\lambda^{\ddagger}R_{0}^{\ddagger}}}=226.3206\,\mathrm{hPa}\]
\[\begin{equation}
\mathrm{PALT}=z_{T}^{\ddagger}+\frac{R_{0}^{\ddagger}T_{T}}{g^{\ddagger}M_{d}^{\ddagger}}\ln\left(\frac{p_{T}}{p}\right)
\tag{3.11}
\end{equation}\]
which, after conversion from natural to base-10 logarithm, is coded
to be equivalent to the following:
## transition pressure at the assumed ISA tropopause:
#define ISAP1 226.3206
## reference pressure for standard atmosphere:
#define ISAP0 1013.25
if (psxc > ISAP1) {
palt = 44330.77 * (1.0 - pow(psxc / ISAP0, 0.1902632))
} else {
palt = 11000.0 + 14602.12 * log10(ISAP1 / psxc)
}
3.4 Combining IRS and GPS Measurements
Measurements from the global positioning and inertial navigation systems are combined to produce new variables that take advantage of the strengths of each, so that the resulting variables have the long-term stability of the GPS and the short-term resolution of the INS. This section describes some variables that result from this blending of variables. These corrected variables are usually the best available when the GPS and IRS are both functioning.
One can determine if the GPS is functioning by examining the GPS status variables described in the previous section or by looking for spikes or “flat-lines” in the data. If the GPS data are missing for a short time (a few seconds to a minute), accuracy is not affected. However, longer dropouts will result in uncertainties degrading toward those of the INS. Without the GPS or another ground reference, the IRS error cannot be determined empirically, and one should assume that it is within the manufacturer’s specification (1 nautical mile of error per hour of flight, 90% CEP). When the GPS is active, RAF estimates that the correction algorithm produces a position error on order of 1.5 meters due to the GPS system. Due to the nature of the algorithm, the error will increase from about 1.5 meters to the INS specification in about one-half hour after GPS information is lost.
GPS-Corrected Inertial Ground Speed Vector (m/s): VEWC, VNSC
These variables result from combining GPS and INS output of the east and north components of ground speed from a complementary-filter algorithm. Positive values are toward the east and north, respectively. The smooth, high-resolution, continuous measurements from the inertial navigation system, {VNS, VEW}, which can slowly accumulate errors over time, are combined with the measurements from the GPS, {GVNS, GVEW}, which have good long-term stability, via an approach based on a complementary filter. A low-pass filter, FL({GVNS, GVEW}), is applied to the GPS measurements of groundspeed, which are assumed to be valid for frequencies at or lower than the cutoff frequency fc of the filter. Then the complementary high-pass filter, denoted (1 − FL)({VNS, VEW}), is applied to the IRS measurements of groundspeed, which are assumed valid for frequencies at or higher than fc. Ideally, the transition frequency would be selected where the GPS errors (increasing with frequency) are equal to the IRS errors (decreasing with frequency).
The procedure is use now is documented in the Technical Note on Wind Uncertainty, beginning on p. 125. It is a three-pole Butterworth low-pass filter, originally coded following the algorithm described in Bosic, S. M., 1980: Digital and Kalman filtering : An Introduction to Discrete-Time Filtering and Optimum Linear Estimation, p. 49. As described in this memo, it has been revised (2014) to use coefficients generated by the R routine “butter().” The digital filter used is recursive, not centered, to permit calculation during a single pass through the data. If the cutoff frequency lies where both the GPS and INS measurements are almost the same, then the detailed characteristics of the filter (e.g., phase shift) in the transition region do not matter because the complementary filters have canceling effects when applied to the same signal. The transition frequency fc was chosen to be (1/600) Hz. The Butterworth filter was chosen because it provides flat response away from the transition.11
CONSTANTS (dependent on time constant \(\tau\)):(a)
\(a=\frac{2\pi}{\tau}\),
\(a_{2}\)=\(a\ e^{-a/2}(\cos(a\sqrt{\frac{3}{2}})+\sqrt{\frac{1}{3}}\sin(a\sqrt{\frac{3}{2}}))\),
\(a_{3}\)=2\(e^{-a/2}\)\(\cos(a\sqrt{\frac{3}{2}})\), \(a_{4}\)=\(e^{-a}\)
__________
(a) For processing prior to the 2014, the factor \(\sqrt{\frac{3}{2}}\)
was erroneously \(\frac{\sqrt{3}}{2}\).
// input x = unfiltered signal
// output returned is low-pass-filtered input
// tau determines the cutoff
// zf[] saves values for recursion
zf[2] = -a * x + a2 * zf[5] + a3 * zf[3] - a4 * zf[4];
zf[1] = a{*}x + a4{*}zf[1];
zf[4] = zf[3];
zf[3] = zf[2];
zf[5] = x;
return(zf[1] + zf[2]); The net result then is the sum of these two filtered signals, calculated as described in the following boxes:
VEW = IRS-measured east component
of the aircraft ground speed
VNS = IRS-measured north component of the aircraft ground speed
GGVEW = GPS-measured east component of the aircraft ground speed
GGVNS = GPS-measured north component of the aircraft ground speed
\(F_{L}()\) = three-pole Butterworth low-pass recursive digital filter
\[\begin{align}\begin{split} \{\mathrm{VNSC}\} & = F_{L}(\mathrm{\{GGVNS\})}+(1-F_{L})(\{\mathrm{VNS\}})\\ \{\mathrm{VEWC}\} & = F_{L}(\mathrm{\{GGVEW\})}+(1-F_{L})(\{\mathrm{VEW\}}) \end{split}\tag{3.12} \end{align}\]
This result is used as long as the GPS signals are continuous and flagged as being valid. When that is not the case, some means is needed to avoid sudden discontinuities in velocity (and hence wind speed), which would introduce spurious effects into variance spectra and other properties dependent on a continuously valid measurement of wind. To extrapolate measurements through periods when the GPS signals are lost (as sometimes occurs, for example, in turns) a fit is determined to the difference between the best-estimate variables {VNSC,VEWC} and the IRS variables {VNS,VEW} for the period before GPS reception was lost, and that fit is used to extrapolate through periods when GPS reception is not available. The procedure is as described below.
1. If GPS reception has never been valid earlier in the flight, use the INS values without correction.
2. Whenever both GPS and INS are good, update the low-pass-filtered estimate of the difference between them. This is added to the INS measurement to obtain the corrected variable. Also update a least-squares fit to the difference between the GPS and INS groundspeeds, for each component. The errors are assumed to result primarily from a Schuler oscillation, so the three-term fit is of the form \(\Delta=a_1+a_2\sin(\Omega_{Sch}t)+a_3\cos(\Omega_{Sch}t)\), where \(\Omega_{Sch}\) is the angular frequency of the Schuler oscillation (taken to be \(2\pi/5067\) s, and \(t\) is the time since the start of the flight. A separate fit is used for each component of the velocity and each component of the position (discussed below under LATC and LONC). The fit matrix used to determine these coefficients is updated each time step but the accumulated fit factors decay exponentially with a 30-min decay constant, so the terms used to determine the fit are exponentially weighted over the period of valid data with a time constant that decays exponentially into the past with a characteristic time of 30 min. This is long enough to determine a significant portion of the Schuler oscillation but short enough to emphasize recent measurements of the correction.
3. When GPS data become invalid, if sufficient data (spanning 30 min) have been accumulated, invert the accumulated fit matrices to determine the coefficients {\(a_1,a_2,a_3\)} and then use the formula for \(\Delta\) in the preceding step to extrapolate the correction to the IRS measurements while the GPS measurements remain invalid. Doing so immediately would introduce a discontinuity in {VNSC,VEWC}, however, so the correction \(\Delta\) is introduced smoothly by adjusting {VNSC, VEWC} as follows: If \(dvy\) is the adjustment added to the INS measurement, adjust it according to
\(dvy^\prime=\eta\ dvy+(1-\eta)\Delta\) where \(dvy^\prime\) is the sequentially adjusted correction and \(\eta=0.995\) s\(^{-1}\) is chosen to give a decaying transition with a time constant of about 5.5 min. This has the potential to introduce some artificial variance at this scale and so should be considered in cases where variance spectra are analyzed in detail, but it has much less influence on such spectra than a discontinuous transition would. Ideally, the current fit and the last filtered discrepancy (VNSC0\(-\)GVNS0 should be about equal, so transitioning between them should not introduce a significant change.
4. To avoid transients that would result from switching abruptly to the complementary-filter solution when the GPS measurements again become valid, the correction factors (e.g., \(dvy\)) are also updated smoothly toward the complementary-filter solution, using for example \(dvy^\prime=\eta\ dvy+(1-\eta)F_L(v_y^{GPS}-v_y^{IRS})\) where \(F_L\) is the low-pass filter and \(v_y\) is the northward component of aircraft groundspeed.
GPS-Corrected Inertial Latitude and Longitude (º): LATC, LONC
Combined GPS and IRS output of latitude and longitude. Positive values are north and east, respectively.These variables are the best estimate of position, obtained by the following approach:
LAT = latitude measured by the IRS
LON = longitude measured by the IRS
GGLAT = latitude measured by the GPS
GGLON = longitude measured by the GPS
VNSC = aircraft ground speed, north component, corrected
VEWC = aircraft ground speed, east component, corrected
1. Initialize the corrected position at the IRS position at the start of the flight or after
any large change (>5\(^\circ\)) in the IRS position.
2. Integrate forward from that position using the aircraft groundspeed with components
{VNSC,VEWC}. Note that in the absence of GPS information this will introduce
long-term errors because it does not account for the Earth’s spherical geometry. It
provides good short-term accuracy, but the GPS updating in the next step is needed to
compensate for the difference between a rectilinear frame and the Earth’s spherical
coordinate frame and provides a smooth yet accurate track.
3. Use an exponential adjustment to the GPS position, with time constant that is typi-
cally about 100 s.(a)
4. To handle periods when the GPS becomes invalid, use an approach analogous to that
for groundspeed, whereby a Schuler-oscillation fit to the difference between the GPS
and IRS measurements is accumulated and used to extrapolate through periods when
the GPS is invalid.
__________
(a) specifically, LATC += η(GLAT-LATC) with η = 2π/(600 s)
For earlier projects using the Litton LTN-51 INS, this is a direct measurement without adjustment for changes in gravity during flight and without pressure-damping. Previous use employed a baro-inertial loop to compensate for drift in the integrated measurement. See the discussion of WP3 below.↩︎
There are four measures of height or altitude discussed in this technical note, height relative to the WGS-84 reference surface, height relative to mean sea level, geopotential height and pressure height. The WGS-84 height (measured by GPS instruments) is height relative to a reference system in which zero is defined by a specified reference ellipsoid representing the shape of the Earth. This is not defined to be a level surface in the sense of being a gravitational equipotential surface. The geoid, however, is such a surface that approximates mean sea level (MSL) and height above mean sea level is measured relative to the geoid. The geoid is more structured than the WGS-84 reference ellipsoid and departs significantly from it, often by several 10s of meters. Even the geoid does not represent mean sea level exactly because local mean sea level can be influenced by variations in water density, mean wind, or ocean circulation, but the geoid is usually the reference used for measurements labeled “MSL” except when fine-scale local effects must be considered. There is a variable included below, GGEOIDHT, that provides a measure of the difference. Modern GPS receivers typically incorporate a model geoid and report altitude as height above the geoid. The Novatel receivers use the EGM96 geoid (https://cddis.nasa.gov/926/egm96/egm96.html). Geopotential height is the height above mean sea level that would give the geopotential, or gravitational potential energy per unit mass, of the actual parcel if that mass were raised against standard gravity (not varying, e.g., with latitude or height) to that altitude. For the purpose of this definition, standard gravity is defined to be 9.80665 m s − 2. Finally, pressure altitude, defined in detail below, is the altitude in the ISA Standard Atmosphere where the pressure matches a specified value; it is not a geometric coordinate but rather a measure of pressure.↩︎
The GPS signals at one time suffered from “selective availability,” a US DOD term for a perturbed signal that degraded GPS absolute accuracy to 100 meters. This was especially noticeable in the altitude measurement, so GALT normally was not useful. As of 1 May 2000, selective availability was deactivated to allow everyone to obtain better position measurements. See the Interagency GPS Executive Board web site for more information on selective availability and GPS measurements prior to 2000.↩︎
See “U.S. Standard Atmosphere, 1976”, NASA-TM-A-74335, available for download at this URL.↩︎
Prior to and including some projects in 2010, processing used slightly different coefficients: for aircraft other than the GV, T0/λ was represented by -43308.83, the reference pressure p0 was taken to be 1013.246, and the exponent x was represented numerically by 0.190284. For the GV, the value of T0/λ was taken to be 44308.0, the transition pressure pT was 226.1551 hPa, x = 0.190284, and coefficient \(\frac{R_{0}^{\prime}T_{T}}{gM_{d}}\) was taken to be 6340.70 m instead of 6341.620 m as obtained below. The difference between these older values and the ones recommended below is everywhere less than 10 m and so is small compared to the expected uncertainty in pressure measurements, because 1 hPa change in pressure leads to a change in pressure altitude that varies from about 8–40 m over the altitude range of the GV.↩︎
For historical reasons, the details of the now obsolete filter as originally coded and used for many years are described here. For the current version with coefficients, see the memo referenced above.↩︎