Wind measurements combine a measurement of relative wind with a measurement of aircraft motion to determine the air motion relative to the ground. The aircraft motion has long been measured by an IRS, and recently also by a GPS. These have complementary strengths: The IRS provides very good information on short-term motion but drifts with a characteristic period of more than an hour, while the GPS provides good absolute accuracy but sometimes is unable to receive the GPS signals and (except in differential-GPS mode) can have short-term errors that make short segments of the track look jagged. To take advantage of the strengths of each, a complementary-filter calculation was developed and implemented in the 1980s, but it was never documented publicly. This memo is partly an attempt to remedy that and partly a suggestion to make some minor changes to how it is implemented. This note is complementary to the information in the document on RAF Processing Algorithms, section 3.4, and contains additional detail as well as notes regarding implementation of changes.
To accomplish this, a low-pass filter, , is applied to the GPS measurements of groundspeed, {GVNS,GVEW}, which are assumed to be valid for frequencies at or lower than the cutoff frequency of the filter. Then the complementary high-pass filter, denoted ()(), is applied to the IRS measurements of groundspeed, {VNS,VEW}, which are assumed valid for frequencies at or higher than . Ideally, the transition frequency would be selected where the GPS errors (increasing with frequency) are equal to the IRS errors (decreasing with frequency). The filter used is a three-pole Butterworth lowpass filter, 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. 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 valid and are almost the same, then the detailed characteristics of the filter in the transition region (e.g., phase shift) do not matter because the complementary filters have cancelling effects when applied to the same signal. The transition frequency was chosen to be (1/600) Hz (but this value can be overridden via the “defaults” file). The Butterworth filter was chosen because it provides flat response away from the transition. The resulting variables for aircraft motion, {VNSC,VEWC}, are then each the sum of two filtered signals, calculated as described in the following box:
This is straightforward and effective when both sets of measurements (IRS and GPS) are available. The approach in use becomes more complicated when the GPS signals are lost, as sometimes happens in sharp turns. Then some means is needed to avoid sudden discontinuities in velocity (and hence windspeed), 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 measurements are not available, 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 described in section 3.4 of ProcessingAlgorithms.pdf.
The changes proposed in that section are as follows:
Note that it is not necessary to filter gvns and vns separately because only the difference between the filtered results is used.
The following provides more documentation of the fit procedure used to determine the Schuler oscillation. The errors are assumed to result primarily from this oscillation, so the three-term fit is of the form , where is the angular frequency of the Schuler oscillation (taken to be and 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 about 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. The procedures for accumulating the matrices for the fit are as follows: