Abstract

A large number of quantities have to be measured and processed to determine the atmospheric-state variables, which are the actual measurands, from aircraft-based measurements. A great part of the dependencies between these quantities depends on the aerodynamic state of the aircraft. Aircraft-based meteorological measurements, hence, require in-flight calibration. Most operators of research aircraft perform some kind of calibration, but the schemes used and the degree they are documented greatly vary. The flight maneuvers and calculation methods required, however, are published in a number of partly overlapping and partly contradictory publications. Some methods are only presented as a minor issue in publications mainly focused on atmospheric processes and are therefore hard to find. For an aircraft user, it is hence challenging to either perform or verify a calibration because of missing comprehensive guidance. This lack was stated on occasion of the in-flight calibration of the German research aircraft Polar5 carried out for the field experiment Investigation of Katabatic Winds and Polynyas during Summer (IKAPOS). In the present paper, a comprehensive review of the existing literature on this field and a practical guide to the wind calibration of a research aircraft to be used for turbulent flux measurements are given.

1. Introduction

a. Motivation

Aircraft-based in situ measurements are among the most valuable tools for the investigation of physical processes in the atmosphere. Although aircraft have always been used as meteorological sensor platforms (Moninger et al. 2003), and the equations to calculate wind vector and other quantities are well known (e.g., Tjernström and Friehe 1991; Lenschow 1986), obtaining good atmospheric measurements is still a complex and challenging task. Typically, 15 individual quantities have to be measured to obtain the three-dimensional (3D) wind vector (e.g., Metzger et al. 2011).

The design of the sensing probes is a science of its own (for examples, see Spyers-Duran and Baumgardner 1983; Crawford and Dobosy 1992; Haman et al. 2001; Spiess et al. 2007; Wang and Geerts 2009), integrating sensors requires engineering skills (e.g., AEEC 2001), and is subject to strict regulations (FAA 1968, 1995). While the actual sensors can be calibrated in a laboratory, the corrections needed due to flow distortion by the aircraft body require an in-flight calibration of each instrumented aircraft (e.g., Lenschow 1976; Brown 1988; Tjernström and Friehe 1991; Bange and Roth 1999; Williams and Marcotte 2000).

Most aircraft available for atmospheric research are operated by instrumentation engineers and scientists who have great expertise in airborne measurements. Hence, usually the operators ensure a proper calibration (Lenschow et al. 2007). However, in some cases the required expertise is not present—for example, leased commercial aircraft or multipurpose aircraft that are only occasionally instrumented—and a satisfying in-flight calibration may be left to the user. Planning and performance of such a calibration, however, exceed the expertise of many users.

Lenschow (1986) gives a comprehensive recommendation for suitable calibration methods. But this book was written before the public availability of the Navigation Satellite Timing and Ranging (NAVSTAR) system (Parkinson and Gilbert 1983)—often referred to as “the” global positioning system (GPS)—and fully digital data acquisition. Hence, later advances (such as, e.g., Crawford et al. 1993; Tjernström and Samuelsson 1995; Matejka and Lewis 1997; Khelif et al. 1999; Kalogiros and Wang 2002) are not incorporated.

The intention of the present paper is, hence, to give aircraft data users a practical guide to a state-of-the art in-flight calibration. The recommendations given are developed for the German research aircraft Polar5 on occasion of the experiment Investigation of Katabatic Winds and Polynyas during Summer (IKAPOS; Heinemann et al. 2011), but they can be easily transferred to any other aircraft of similar weight, ranging from single-engine light aircraft (e.g., Crawford and Dobosy 1992) over twin-engine aircraft (e.g., Kalogiros and Wang 2002) to quad-engine utility aircraft (e.g., Khelif et al. 1999) and twin-engine business jet (Tjernström and Friehe 1991).

b. IKAPOS

IKAPOS was performed in June 2010, using the German polar aircraft “Polar5” of the Alfred Wegener Institute (AWI), which was based at Qaanaaq (northwest Greenland). The investigations comprised studies of the summertime katabatic wind system in the coastal area of north and northwest Greenland, and of atmosphere/sea ice/ocean exchange processes over the North Water (NOW) polynya (see Heinemann et al. 2011). Polar5 is a Basler BT-67 (Fig. 1) that consists of a modified Douglas DC-3 airframe retrofitted with turboprop engines (Herber et al. 2008). The main modifications comprise a slight stretching of the fuselage, reinforced structure, and redesigned outer wing leading edge and wing tip.

Fig. 1.

Polar5 fitted with the nose boom carrying the turbulence sensors. (inset) Detail of the nose boom: the tip to the left is the five-hole probe, the housed sensors are (top to bottom) Lyman-α hygrometer, inlet for dewpoint mirror through a deicable Rosemount housing, slow temperature sensor (TE) in deicable Rosemount housing, and fast temperature sensor (TRvF) in non-deicable Rosemount housing (see section 2f for details).

Fig. 1.

Polar5 fitted with the nose boom carrying the turbulence sensors. (inset) Detail of the nose boom: the tip to the left is the five-hole probe, the housed sensors are (top to bottom) Lyman-α hygrometer, inlet for dewpoint mirror through a deicable Rosemount housing, slow temperature sensor (TE) in deicable Rosemount housing, and fast temperature sensor (TRvF) in non-deicable Rosemount housing (see section 2f for details).

Polar5 has a permanent “basic” meteorological instrumentation and was carrying its optional nose boom and various radiation sensors during IKAPOS. The sensors involved in calculating energy fluxes are listed in Table 1.

Table 1.

Polar5 turbulence instrumentation.

Polar5 turbulence instrumentation.
Polar5 turbulence instrumentation.

All sensing elements delivering analog output as well as the analog-to-digital converters (ADCs) had been calibrated prior to the experiment. For sensors with calibrations expected to be stable, manufacturer-supplied calibration coefficients are used, such as for the inertial navigation system (INS), GPS, and the altimeters.

Default data output of the Polar5 data acquisition system contains temperature values only with a resolution of about 0.025 K. This turned out to be too coarse, for example, to investigate the sensor inertia (section 2f). Extracting analog-to-digital converter readings from a backup of the onboard database, however, allows calculation of calibrated sensor readings with almost five significant digits.

As known from Polar5’s predecessor, Polar2, the data acquisition system tends to record occasional spikes. It appears that they are caused by uncorrected data communication errors, since spikes even occur in fully digitally recorded quantities such as INS position (Th. Garbrecht 2002, personal communication). Although it was commonly believed that these were caused by very high frequency (VHF) radio, the spikes persisted after the use of VHF was discontinued. To identify spikes, a simple method is used that removes all values that deviate more than a threshold value from high-pass-filtered time series. To determine the threshold, the normalized probability density distribution of each high-pass-filtered measurement time series is compared to a Gaussian error function (Drüe 2001). For IKAPOS, a value of 3.5 standard deviations was determined as a suitable threshold.

Upon delivery, the manufacturer of the nose boom, MessWERK GmbH, performed an extensive verification of wind measurements (Cremer 2008). This report, however, gives the empirical relationships determined using formulations that differ from the equations commonly used (e.g., by Lenschow 1986; Bögel and Baumann 1991; Khelif et al. 1999; Strunin and Hiyama 2004) or by Vörsmann et al. (1989), which is a somewhat expanded version of Vörsmann (1990). Hence, the results are difficult to compare to these other calibration methods. Furthermore, the local angle-of-attack corrections are determined by quasi-static maneuvers that do not allow for time shifts between the time series (Bögel and Baumann 1991), which might be not optimal for turbulence measurements.

2. In-flight calibration

In general, “calibration” denotes setting up a conversion between the sensor reading (measured quantity) and the sought quantity (the measurand; JCGM 2012), but most authors use in-flight calibrations to yield correction functions for preexisting sensor calibrations determined in a wind tunnel or by the sensor manufacturer (Vörsmann 1990; Crawford et al. 1993; Bange and Roth 1999; Kalogiros and Wang 2002; Metzger et al. 2011). The usual exception are studies that describe the development of new probes (e.g., Crawford and Dobosy 1992; Haman et al. 2001) or use the aircraft radome as a flow-angle sensor (e.g., Brown et al. 1983; Tjernström and Friehe 1991; Khelif et al. 1999).

The measurements usually calibrated in flight are angle of attack (α), sideslip angle (β), static pressure (ps) offset and defect (also called position error; Brown 1988), airspeed, and temperature probe recovery factor (Williams and Marcotte 2000; Drüe 2001; Kalogiros and Wang 2002). If sensors are mounted only occasionally, it has also turned out to be essential to determine the current sensor alignment for each single experiment (Vörsmann 1990; Freese and Kottmeier 1998; Drüe 2001). In case the sensors have to be removed for transfer flights or if the schedule for the aircraft is rather tight, an in-flight calibration under favorable atmospheric conditions in the area of the experiment is required. At the beginning of the IKAPOS experiment, a calibration flight was performed on 12 June 2010, west of Qaanaaq (Fig. 2).

Fig. 2.

Flight path of the calibration flight performed for IKAPOS on 12 Jun 2010. The black line represents the 3D view, and the gray line is its projection to the surface. The white square marks Qaanaaq airport. Elevation data are taken from the Geoscience Laser Altimeter System (GLAS) dataset (Greenland above 78°N; DiMarzio et al. 2007) and the Global Land One-Kilometer Base Elevation (GLOBE) database (elsewhere; Hastings et al. 1999).

Fig. 2.

Flight path of the calibration flight performed for IKAPOS on 12 Jun 2010. The black line represents the 3D view, and the gray line is its projection to the surface. The white square marks Qaanaaq airport. Elevation data are taken from the Geoscience Laser Altimeter System (GLAS) dataset (Greenland above 78°N; DiMarzio et al. 2007) and the Global Land One-Kilometer Base Elevation (GLOBE) database (elsewhere; Hastings et al. 1999).

The basic idea of all maneuvers used for in-flight calibration is to vary only one quantity in a way such that the variation can be measured at least in two ways, of which one does not need calibration. The uncertainty of the calibration coefficients or of the calibrated measured values (see Axford 1968) may be assessed in a number of ways: many authors use propagation of sensor uncertainties (e.g., Tjernström and Friehe 1991; Crawford and Dobosy 1992; Garman et al. 2006), and the analysis of specific flight maneuvers (e.g., Tjernström and Friehe 1991; Williams and Marcotte 2000; Kalogiros and Wang 2002), but also the spread of multiple calibrations can be used (e.g., Spyers-Duran and Baumgardner 1983; Vörsmann 1990; Bögel and Baumann 1991; Drüe 2001). We chose to use the last method because it can be difficult for a mere user of an aircraft to gather all the information to calculate proper propagation of uncertainty for all sensors and because extra flight maneuvers are usually expensive for the user.

a. Time lags

Different processing speeds of the individual sensor systems can cause time lags between the recorded time series. Analog measurements are associated with a time stamp upon A/D conversion. Digital data receive a time stamp when being sent—directly or via the Aeronautical Radio Incorporated (ARINC) bus (AEEC 2001)—to the data acquisition system. Both kinds of processing require a certain processing time.

Internal lags of navigation data may arise from the internal data processing: INS integrates acceleration-to-yield speed and position, while GPS derives position-to-yield speed. Since both calculations are done at finite time intervals using backward differences, the results may be shifted in time too.

These time shifts are usually on the order of or below 1 s. Such lags seem irrelevant for navigation purposes and are hence often neglected in aircraft sensor development (de Mendonça et al. 2007). In the case of scientific data collection, however, they may lead to erroneous application of fast-varying corrections (Bögel and Baumann 1991) or inhibit accurate calculation of turbulent fluxes (Kristensen et al. 1997).

For the present study, the lags of all measurements of the same quantity were determined. To assess the lags between measurements of different quantities, the lags of measurements from the same group (i.e., from the same A/D converter or generated by one digital system) are assumed to be identical. The relative lag between each pair of groups was then determined by choosing a pair of quantities, one from each group, for which highly correlated values are expected—for example, radar height and static pressure (over water or sufficiently flat terrain). To determine the relative lag between the measurements of two quantities, a flight segment was chosen, on which both are expected to cover a sufficiently wide range of values. Mean difference and covariance of both are calculated for a reasonable range of time lags (see Fig. 3 for an example). Depending on the physical relation, the time lag corresponding to the minimum difference, the minimum covariance, or the maximum covariance was chosen (Drüe 2001); see Table 2.

Fig. 3.

Example: RMS difference (m) and (cross) correlation vs time shift, for radar height (HradMPm) and barometric altitude (HICAOmMP) for a series of ascents and descents. The maximum covariance (and hence maximum correlation) is found at a shift of 0.29 s. Note how well the maximum and minimum are defined.

Fig. 3.

Example: RMS difference (m) and (cross) correlation vs time shift, for radar height (HradMPm) and barometric altitude (HICAOmMP) for a series of ascents and descents. The maximum covariance (and hence maximum correlation) is found at a shift of 0.29 s. Note how well the maximum and minimum are defined.

Table 2.

Lags of all sensor groups relative to INS (vertical) acceleration.

Lags of all sensor groups relative to INS (vertical) acceleration.
Lags of all sensor groups relative to INS (vertical) acceleration.

The internal lags between GPS and INS outputs can be determined by comparing derived position change, integrated accelerations, and ground speed in the same way (Tjernström and Samuelsson 1995). Unfortunately, it turned out that the resolution of the position data as recorded by Polar5 was too coarse for this procedure and steplike structures dominate the time series, even after applying a low-pass filter. However, internal lags of the INS could be determined in the course of the local flow-angle calibration (see section 2d) and are included in Table 2.

Since all lags are relative, the choice of a time reference—that is, zero lag—is arbitrary. We chose to give the lags in Table 2 relative to INS vertical acceleration because in our experience, this measurement is most crucial for retrieving the vertical wind variation. Some readers might argue that some INS systems tend to exhibit unsteady internal lags, but such variations were not observed in the course of the repeated local flow-angle calibrations (see section 2d).

b. Static pressure

The measurement of the static pressure ps on an aircraft may be influenced by instrument deficits and by distortion of the flow field. Static pressure for navigation is usually measured at ports on each side of the fuselage. The deviation of the indicated (i.e., actually measured) pressure psi from the actual static pressure ps mainly depends on the location of the static pressure ports and the airspeed (Gracey 1958). This difference is usually referred to as static (pressure) defect (Khelif et al. 1999) or position error (Brown 1988). The only way to obtain a complete correction of the static defect is by comparison to an undisturbed pressure measurement taken on a trailing cone towed behind the aircraft (Brown 1988).

Since altitude information used for navigation is deduced from the measured static pressure, strict airworthiness regulations—that is, Society of Automotive Engineers (SAE) standard AS942A—ensure that the uncertainty altitude is, at least for modern passenger aircraft types (ICAO 2002), less than 15 m, requiring the uncertainty of static pressure to be less than 0.4 hPa. This tolerance requires that the correction of the position error has to be included in the altimeter calibration inside the air data computer (ADC; FAA 1988). Researchers using the static pressure from the aircraft altimeter, hence, usually can use the absolute value without corrections (e.g., Lenschow 1986; Bögel and Baumann 1991). Any modification of the aircraft body, however, especially in places ahead of the static pressure ports (nose boom, probes, inlets, and similar) can affect a previous calibration of the static pressure and may require new trailing cone flights.

To verify the calibration of ps (or in case trailing cone flights are not possible), some authors use tower flyby maneuvers (Sethuraman et al. 1979; Tjernström and Friehe 1991; Metzger et al. 2011), which is also the preferred method for certification of small aircraft (FAA 2011). Other authors use a missed approach (aircraft descends to a runway but does not touch down; Williams and Marcotte 2000), takeoff runs (aircraft accelerates, rolling on a runway until lift is generated but does not take off; Crawford and Dobosy 1992; Kalogiros and Wang 2002), or a low pass (aircraft flies a few meters at a constant height over the full length of a runway; Vörsmann 1990) maneuver. Takeoff runs do have the disadvantage of possible disturbances by the ground effect (Stengel 2004), while a low pass allows for checking the vertical wind offset, an additional benefit.

If the static pressure is measured on a five-hole probe such as the Rosemount 858 used on Polar5, then the airspeed dependency of the static defect must be calibrated, either together with dynamic pressures qc, as described in the next section (section 2c; Vörsmann 1990; Bange and Roth 1999; Kalogiros and Wang 2002), or by a trailing cone flight in an accelerating or decelerating maneuver (Brown 1988).

In the case of Polar5, the barometric altitude calculated from the ps nose boom five-hole probe was within ±15 m around the ADC altitude, and the difference between ps measured at touchdown and the surface pressure was below the surface pressure accuracy of 1 hPa. Hence, no further offset correction was applied.

Note that the uncertainty in the height (vertical distance to the ground) calculated from altitude (vertical distance to sea level) determined from ps and the terrain elevation is often of a similar order or even much larger than the uncertainty caused by the measurement of ps, because the horizontal variation of the surface pressure and the air temperature profile below the aircraft are usually not known. Hence, it is advisable to determine height from the aircraft radar altimeter or from an additional laser altimeter, for example, for flux measurements near the ground (e.g., Drüe and Heinemann 2007; van den Kroonenberg et al. 2012).

c. True airspeed

As for the static port, the pressure at the side holes of a flow-angle sensor—such as Rosemount 858 (Rosemount 1988) or the best aircraft turbulence (BAT) probe (Garman et al. 2006)—is not equal to the static pressure, since the airflow is partly decelerated or the pressure field is distorted by the aircraft body and propulsion. The static defect is generally regarded as a function of the dynamic pressure that might be defined as a polynomial (Brown 1988), but it is mostly approximated by a linear function (Williams and Marcotte 2000; Cremer 2008). If the indicated pressure values are already corrected for their instrument offset, then the difference may be expressed as an empirical factor (Lenschow 1972). It has been called “pressure coefficient” (by Nacass 1992), but we call it “pressure partitioning factor” E (after Vörsmann 1990). It relates indicated pressures to the undisturbed values via

 
formula
 
formula

where ps is static pressure, qc is dynamic pressure, and an i appended to the subscript denotes indicated values. In our case, psi and qci were already corrected for their instrument offset, which was determined in a laboratory on the ground.

If dynamic pressure (and flow angle) is not measured on a nose boom but at the radome (i.e., the aircraft nose), then impact pressure and static pressure correction have to be determined, including an additional dependency on the flow angles α and β (Tjernström and Friehe 1991), because an aircraft body is usually much less rotationally symmetric than a nose boom.

The usual procedure to determine E is to perform a straight flight at constant (pressure) altitude with increasing airspeed from close to stall speed to maximum measurement speed. If the wind field is uniform, then linear regression of true airspeed (TAS) versus ground speed yields E (Lenschow 1986). It should be noted that qc has to be calculated for a compressible flow (Lenschow 1986; Kalogiros and Wang 2002).

Many authors just use the terms acceleration and deceleration, but as Williams and Marcotte (2000) and Vörsmann (1990) point out, it is important to change speed gradually. The reason is that the process of changing throttle or the blade angle of the propellers results in significant deviations in the pressure measurements (e.g., ±0.5 hPa in Kalogiros and Wang 2002). These variations decay with time and usually become insignificant after less than a minute (Vörsmann 1990).

Another popular method used for this purpose is the “racetrack” maneuver (Williams and Marcotte 2000; Metzger et al. 2011)—that is, two parallel flight tracks connected by two 180° turns (see Fig. 4b)—which is performed at a number of different constant (indicated) airspeeds (Williams and Marcotte 2000; FAA 2008).

Fig. 4.

Schematic view of selected flight maneuvers: (a) wind squares, (b) racetrack maneuver, (c) eights, and (d) procedure turn.

Fig. 4.

Schematic view of selected flight maneuvers: (a) wind squares, (b) racetrack maneuver, (c) eights, and (d) procedure turn.

Since such a maneuver was not performed during IKAPOS, a more simple method (also used by Telford and Wagner 1974; Telford et al. 1977; Vörsmann 1990) had to be employed: in the course of the calibration flight, two full circles were flown at an altitude of 3000 m, well above the boundary layer, where turbulence was minimal. From these data, the horizontal wind vector was calculated. Then E was varied until the amplitude in both the eastward and northward wind components are minimized (Vörsmann et al. 1989); see Fig. 5. This procedure was performed separately for both circles, which yields

 
formula
Fig. 5.

Dependence of the amplitude of the eastward wind component during a full circle on the pressure partitioning factor E [see (1) and (2)].

Fig. 5.

Dependence of the amplitude of the eastward wind component during a full circle on the pressure partitioning factor E [see (1) and (2)].

The correction formula for pressure partitioning used by the manufacturer of the nose boom (Cremer 2008) is equivalent to

 
formula
 
formula

where K0,p, K1,p, K0,q, and K1,q are empirical coefficients. Both equations allow for an offset in the pressure measurements, in contrast to (1) and (2). But because the variation of TAS is only a few tens of a percent, they do not allow a sufficiently precise extrapolation to zero. Although (4) and (5) might represent better a nonlinear behavior of the sensors, we prefer to determine the zero offsets on the ground and to correct the sensor readings before calculating the pressure values.

The actual representation in Cremer (2008) contains an additional term that is a function of α, β, and ps, apparently because it is assumed that the measured TAS value actually represents the component of the TAS vector along the probe axis. But as Lenschow (1986) points out, the usual assumption is that the measured TAS value equals the magnitude υTAS of the vector (see also Stickney et al. 1994), which we use in our data processing scheme, too. Although the difference is usually negligible (Williams and Marcotte 2000), both assumptions are equivalent if they are properly reflected in the wind calculation.

Comparing (4) to (1) yields

 
formula

Although this expression depends on airspeed, the variation of E is only of the order of 10−5 (or 0.01% relative change) over the full speed performance range of typical propeller-driven aircraft. The values K0,p = 0.27 hPa and K1,p = 1.063 supplied by the manufacturer hence correspond to E = 0.08, which is quite different from our value (E = 0.1969). In turn, if applied to the full-circle maneuvers from IKAPOS, then the manufacturer value yields a variation of the horizontal wind components that is clearly larger than the variation obtained by using our value. This demonstrates the need for a new calibration before every experiment.

Since the differences correspond to a change in the static pressure offset of only 0.25 hPa, a replacement of the pilot’s altimeter or wear of its static port (Brown 1988) in the meantime could explain the change in E.

d. Flow angles

Similar to the measurement of TAS, pressure field deformation around the aircraft causes the local flow angles αL and βL at the tip of the flow-angle sensor to be different from the flow angles α and β of the general airflow around the aircraft (Lenschow 1986). The correction needs to be determined in flight because it depends on the flight dynamic state of the aircraft, in particular airspeed, throttle, and aerodynamic configuration (setting of flaps, rudder, elevator, etc.; Garman et al. 2008). However, measurements are usually made in clean configuration (retracted flaps, gear, etc.; see Stengel 2004); hence, such influences (Drüe 2011) may be neglected.

Several authors introduce an interdependence term in the calibrations of α and β. In our and other authors’ experience (Kalogiros and Wang 2002), however, such interdependences are small as long as α and β remain small.

Either vanes are used as a flow-angle sensor (e.g., Axford 1968; Grant and Hignett 1998) or pressure differences on a forward-facing hemispheric structure are converted to the flow angle (e.g., Bange and Roth 1999; Lenschow et al. 2007). In the case of a five-hole probe—like Rosemount 858 or the BAT probe—the local flow angles are determined from the measured pressure differences using the manufacturer-supplied (Rosemount 1988) wind tunnel calibration (Bennett 1975; Crawford and Dobosy 1992; Garman et al. 2006; Metzger et al. 2011), or by a calibration device integrated into a wing-mounted five-hole probe (which allows for tilting the probe while maintaining α and β; Wood et al. 1997). Should pressure ports at a radome be used, the results of theoretical calculations (Nacass 1992) or indicated pressure differences (Brown et al. 1983; Tjernström and Friehe 1991) are used.

1) Angle of attack

The most usual procedure to determine the conversion between αL and the (vertical) angle of attack α is a straight and level flight with a stepwise increase of TAS (Lenschow 1986; Tjernström and Friehe 1991; Crawford and Dobosy 1992; Williams and Marcotte 2000; Kalogiros and Wang 2002). Fewer authors use racetrack maneuvers (Williams and Marcotte 2000), starlike patterns (Garman et al. 2006; van den Kroonenberg et al. 2008), or wind squares—that is, box patterns at a constant altitude with four 90° turns and short straights in between—flown at a range of airspeeds kept constant throughout the pattern (Fig. 4a; Cremer 2008; French et al. 2007; Metzger et al. 2011). Although the latter maneuvers favor the assumption of constant horizontal wind, the need to wait at least 30 s after each turn for the aerodynamic state to settle (Bögel and Baumann 1991) causes a much longer overall trajectory, generating higher costs.

Since the angle of attack α can be assumed to equal the aircraft pitch angle θ in steady flight and zero vertical wind (Williams and Marcotte 2000), a linear regression between the local angle of attack αL and the free stream angle of attack α = θ then yields the coefficients of the linear equation

 
formula

Some authors only determine the offset of α from these maneuvers and prefer to perform slow oscillations of the pitch angle, also called (slow) pitching maneuver, to determine sensitivity Cα in (7) (Brown et al. 1983; Vörsmann 1990; Bögel and Baumann 1991) or to check the robustness of the calibration of α against vertical speed or acceleration (Lenschow 1986; Williams and Marcotte 2000).

Another common approach is to calculate the vertical wind throughout a pitching maneuver and to minimize the vertical wind variance by variation of coefficients α0 and Cα (e.g., French et al. 2007; Petersen and Renfrew 2009). For IKAPOS, we use a modified form of the approach presented by Bögel and Baumann (1991). For this method, the aircraft has to perform a series of fast pitch angle variations of ±10° in an area where zero vertical wind can be expected.

The usual assumption is that the vertical wind is zero well above the atmospheric boundary layer (ABL) in calm synoptic conditions (e.g., Lenschow 1986; Bögel and Baumann 1991). This condition has been found to be not satisfied in areas above or in the lee of mountain ridges or islands (Corby 1954), near deep convection (Stull 1976), or above areas from which katabatic wind systems originate (Klein and Heinemann 2002). Another approach used by van den Kroonenberg et al. (2008) and Metzger et al. (2011), because of insufficient ceiling (highest altitude at which it can sustain level flight) of their aerial vehicles, are horizontal flight segments (“ABL runs”) at constant height in the lower ABL over flat terrain. Although the variance of the vertical wind σw is large, its mean is zero, if the flight segment is long enough.

With w = 0, the vertical component of the TAS vector (wTAS) equals the vertical speed of the aircraft wac. Recalling the approximation wTASυTAS sin(θα) illustrates well that a pure pitching oscillation causes periodic variations in both vertical speeds (Bögel and Baumann 1991). Aircraft speed wac is calculated primarily from the INS vertical acceleration, as described in Drüe (2001), following the method of Matejka and Lewis (1997). To allow for internal lags of data processing, a time shift dtINS of the INS vertical acceleration is introduced. Finally, Powell’s method (Press et al. 1986) is used to minimize the RMS of the difference (wacwTAS) by synchronous variation of α0, Cα, and dtINS. However, as Tjernström and Friehe (1991) and Bögel and Baumann (1991) point out, it is important to use the full wind calculation, including all corrections, in this optimization.

During IKAPOS, four series of oscillations were performed, which yield

 
formula
 
formula
 
formula

The vertical wind before and after application of the calibration according to (8)(10) is shown in Fig. 6.

Fig. 6.

Example of a fast pitching maneuver: Vertical wind (right axis) before (dashed, using manufacturer coefficients) and after calibration (solid) vs time of flight. Pitch angle (left axis) is shown as dashed–dotted line.

Fig. 6.

Example of a fast pitching maneuver: Vertical wind (right axis) before (dashed, using manufacturer coefficients) and after calibration (solid) vs time of flight. Pitch angle (left axis) is shown as dashed–dotted line.

The nose boom manufacturer calibration of α (Cremer 2008) uses

 
formula

with K0,α = −1.15° and K1,α = 0.087.

Other literature (e.g., Lenschow 1986) uses a formulation equivalent to our (7):

 
formula

where K1 = 0.079 is a manufacturer-supplied value from wind tunnel calibration of the Rosemount 858’s five-hole probe, valid for around 130 kt (1 kt = 0.51 m s−1; typical measurement speed).

The values determined by Cremer (2008) hence correspond to Cα = 0.908° hPa−1, which differs by about 5% from our value (Cα = 0.960° hPa−1 ± 0.021° hPa−1). The smaller value of Cremer (2008) presumably reflects the neglected influence of time shift between INS and the pressure sensors.

The value K0,α = −1.15° differs significantly from our value (α0 = −2.06°). A difference of almost 1° seems surprising at first glance, since the nose boom mounting is rather solid, which makes it an unlikely source of 1° misalignment. However, it has to be considered that such offset includes mechanical tolerances in the nose boom and in the INS mounting, as well as pressure sensor temporal behavior and aging.

The nose boom sensor head has a mounting base of 210 mm and the tolerance for a regular 4-mm washer is 0.2 mm (ISO 4759, class A); the resulting reproducibility of the nose boom orientation is on the order of 0.2°. The same applies to the INS inertial reference system from the Honeywell Laseref V family, which has an outside dimension of 6.5 in. (16 cm), allowing a mean orientation tolerance of 0.1° (Honeywell 2004). The pressure transducer offsets were determined in a laboratory calibration by the manufacturer of the nose boom and turned out to be on the order of 0.05 hPa (at least more than 40 min after powering them on). Assuming a differential pressure of 5 hPa, this uncertainty roughly equals 0.5°. Hence, the sum of all such uncertainties may sum up to a mission-to-mission change of the offset of α on the order of almost 1°.

2) Sideslip

In the literature, three main procedures to determine the conversion between the horizontal pressure difference at the tip of the five-hole probe into the horizontal flow angle β are found, which all assume a linear relationship,

 
formula

Most authors use a reverse-heading maneuver (also known as go-and-return maneuver) on a straight trajectory at constant altitude (e.g., Lenschow 1986; Wood et al. 1997; Williams and Marcotte 2000). As Lenschow (1986) points out, it is important to perform a procedure turn (Fig. 4d; FAA 2008) in between to stay in the same air volume. The difference between the two heading angles in each direction then corresponds to twice the offset of β (Tjernström and Friehe 1991).

Other authors use full circles (Lenschow 1986; Lenschow et al. 1999) or wind squares (Cremer 2008) and adjust the coefficients β0 and Cβ by minimizing the variance of the horizontal wind. According to Bögel and Baumann (1991), calculations are facilitated by flying close to one of the four cardinal headings (0, 90, 180, or 270) because deviations are visible as variations of one wind component at a time, while other authors recommend flying boxes with two sides parallel to the mean wind direction to minimize the influence of the drift angle variations (Vörsmann 1990).

The third approach is to perform one or more slow yawing oscillations (also called sideslip maneuver). If they are carried out slowly with a period of 10 s or more, then the procedure is similar to that used for α (Lenschow 1986; Tjernström and Friehe 1991; Williams and Marcotte 2000; Kalogiros and Wang 2002; Garman et al. 2006). Applying fast yawing oscillations and minimizing the difference between the indicated airspeed and ground speed (Bögel and Baumann 1991), however, allows for a time shift between pressure and attitude measurements and drift during the maneuver (Williams and Marcotte 2000).

During the IKAPOS calibration flight, the wind field in the area turned out to be rather inhomogeneous (Fig. 7). Hence, none of the above-mentioned techniques could be applied and a split technique was used: The sensitivity Cβ and time shift dtINS were determined from fast yawing maneuvers. For this procedure, both prescribing offset β0 as zero and including β0 in the variation process yielded almost the same results (Fig. 8).

Fig. 7.

Horizontal wind measured on a horizontal box pattern flown during of the calibration flight over Smith Sound (Fig. 2). Land surfaces are shaded gray. Note the inhomogeneity of the wind field. Each straight segment is about 45 km long (12-min flight).

Fig. 7.

Horizontal wind measured on a horizontal box pattern flown during of the calibration flight over Smith Sound (Fig. 2). Land surfaces are shaded gray. Note the inhomogeneity of the wind field. Each straight segment is about 45 km long (12-min flight).

Fig. 8.

Horizontal wind (right axis) assuming zero offset (thin line) and after offset calibration (thick line). The eastward component is plotted as solid lines, and the northward component is plotted as dashed lines. Yaw angle (left axis) is shown by crosses.

Fig. 8.

Horizontal wind (right axis) assuming zero offset (thin line) and after offset calibration (thick line). The eastward component is plotted as solid lines, and the northward component is plotted as dashed lines. Yaw angle (left axis) is shown by crosses.

Offset β0 is determined from two perpendicular reverse-heading maneuvers; see Fig. 9. To check the plausibility of this result, the same procedure was applied to the corners of the box pattern shown in Fig. 7, which yielded a similar result (β0 = 0.97) but with much greater uncertainty, because the inhomogeneity of the wind field can make the actual wind direction vary between the box sides much more than during a reverse-heading maneuver.

Fig. 9.

Horizontal wind components during a reverse-heading maneuver. Representation as in Fig. 8, but true heading (left axis), which is shown as plus signs.

Fig. 9.

Horizontal wind components during a reverse-heading maneuver. Representation as in Fig. 8, but true heading (left axis), which is shown as plus signs.

To summarize, the split technique yields

 
formula
 
formula
 
formula

Analogous to the previous section, the nose boom manufacturer (Cremer 2008) used

 
formula

whereas most authors prefer Lenschow (e.g., Lenschow 1986), who used

 
formula

Assuming a typical measurement speed of 130 kt, the values K0,β = −0.6°, K1,β = 0.088°, and K2,β = 0.025° reported by Cremer (2008) correspond to β0 = −0.15° and Cβ = 0.898°.

The difference of offset β0 between Cremer (2008) and this study is on the same order as for α0. The magnitude of Cβ is the same for Cremer (2008) and this study. The sign, however, is opposite, which is hard to explain by any uncertainties. It appears more likely that during the last assembly of the nose boom sensor, the hoses connecting the left and right holes of the five-hole probe to the respective pressure sensors were interchanged.

e. Angular rates

The calibration report by Cremer (2008) states that no significant correlation between (uncorrected) vertical wind and pitch rate could be found, although wind was not corrected for the probe movement because of aircraft body rotations. As the report states, this is surprising. A possible reason is that this correlation did not allow for time shifts between nose boom and INS measurements.

A distance of 5 m between the mounting position of the INS inertial reference system and the five-hole probe provides a large cantilever (Brown et al. 1983). Therefore, any rotation of the aircraft body causes quite a large movement of the flow-angle probe. Some authors argue that this contribution to the sensor motion is negligible, for example, because typical angular rates cause rather small speeds of the nose boom tip. But when looking at the distribution of angular rates, we found that the contribution from rotations of the body should not be omitted.

During the calibration flight the pitching rate exceeded 0.54° s−1, corresponding to about 0.05 m s−1 vertical speed of the probe in 10% of the time. Values for the other axes are similar. In consequence, for 10% of all data points, the error of the wind measurement caused by neglecting the rotation of the body exceeds 0.1 m s−1, which is quite significant compared to all other contributions.

Allowing a time lag between INS and flow-angle measurements (see section 1) revealed a slight dependency—a correlation coefficient of 0.55—between the local angle of attack αL and the INS pitching rate in the present study. The vertical wind was hence corrected for the vertical movement of the flow-angle sensor, calculated from the INS angular rates and the three-dimensional cantilever between the INS and flow-angle sensor positions (Vörsmann et al. 1989). It should be noted that this correction is already included in the full wind calculation used in section 1.

f. Fast temperature sensor

The nose boom of Polar5 carries two Rosemount 102 temperature sensors, one of them in a Rosemount 102 deiced housing (model 102-EJ2BB), called TE, and the other one in a Rosemount 102 nondeiced housing (model 102-E4AL), called TRvF. The idea of both is to supplement measurements of a more robust sensor by a faster sensor.

1) Recovery factor

Air passing the sensor location inside the housing is (incompletely) decelerated, which causes compression of the air inside the housing and, in consequence, a rise in temperature by adiabatic heating δTi. To convert the measured (“indicated”) temperature into true (“static”) temperature, the recovery factor r of the sensor housing must be known. Then as in Lenschow (1986),

 
formula

where δTi is the temperature increase; Ti and Ts are indicated and static temperatures, respectively; υ0 is the mean speed of the airflow, and cp is the specific heat capacity of air. It is often assumed that the flow speed at the sensor location is equal to the true airspeed, υ0 = υTAS. As a consequence it is possible to use the recovery factor as provided by the manufacturer or in a wind tunnel (e.g., Vörsmann 1990; Crawford and Dobosy 1992).

In the case of Rosemount, the manufacturer supplies a recovery correction value η that can be related to the recovery factor r via . For the target airspeed of 65 m s−1 during measurements, we used ηTE = 0.000 30 or r(TE) = 0.961 for sensor TE and ηTRvF = 0.000 70 or r(TRvF) = 0.908 for sensor TRvF (for Mach 0.2, see Stickney et al. 1994).

Since, the local flow velocity at the probe position is likely to be slightly reduced relative to the overall aircraft airspeed (Williams and Marcotte 2000), it might be useful to determine the recovery factor for a particular combination of housing and mounting position. Lenschow (1986) suggested varying the aircraft speed on a straight and level flight segment. A linear regression of versus Ti then yields r as slope and Ts as intercept. We performed two acceleration maneuvers at the beginning and end of two reverse-heading maneuvers, which ensures that they are performed in the same air volume. Averaging the fitted slopes yields then

 
formula

which does not conflict with the manufacturer value, but it does not add great value because of the large uncertainty.

As Williams and Marcotte (2000) point out, this method requires assumptions about static and airspeed that are often fairly crude appropriations. They recommend the method by Leise and Masters (1993), that is, to pass a racetrack pattern twice with different airspeeds. Then r can be calculated from indicated temperature and INS- or GPS-measured ground speed, which has much greater accuracy.

2) Response time

The inertia of a temperature sensor in a housing usually behaves like a so-called two-component system (Rodi and Spyers-Duran 1972). Such a system has two response times [see Inverarity (2000) for a comprehensive review]. On a short time scale (e.g., <1 s) it behaves like having a response time τ2, while on a longer time scale (e.g., >1 s) it behaves like having a response time τ1. For the non-deicable Rosemount 102-E2AL housing (Stickney et al. 1994), Spyers-Duran and Baumgardner (1983) found τ1 = 0.17 s and τ2 = 45 ms [values from other sources are listed in Inverarity (2000)].

If these values are assumed valid for TRvF, then the time constant of another sensor can be determined by fitting the phase spectrum of the two time series recorded by both sensors. Figure 10 shows coherence, phase lag, and calculated response time difference versus frequency. Below 1 Hz, the coherence is close to one and the mean response time difference is −1.15 s. Assuming the above-mentioned value for τ1 for TRvF yields

 
formula

between about 2 and 20 Hz, the coherence drops to values below 0.2 and the phase lag becomes poorly defined. Above 20 Hz, however, coherence increases to 0.8 and the phase lag exhibits less scatter (see error bars in Fig. 10). Such behavior could be caused either by both sensors exhibiting identical inertia in this range or by one senor being too slow to yield anything but random noise (de Sá et al. 2002). Earlier experience with the TE sensor, however, had shown that it has a faster response time τ2(TE) that is similar to τ1(TRvF) (Drüe 2001). As a consequence, TRvF can be used as the exclusive temperature sensor with TE as a backup, for example, in case of icing.

Fig. 10.

(top) Coherence, (middle) phase difference, and (bottom) response time difference vs frequency for the temperature indicated by sensor TRvF vs sensor TE (see text).

Fig. 10.

(top) Coherence, (middle) phase difference, and (bottom) response time difference vs frequency for the temperature indicated by sensor TRvF vs sensor TE (see text).

g. Pyranometer misalignment

As discussed with the flow-angle sensors in section 1, the alignment of the hemispheric radiation sensors mounted on top and below the fuselage of Polar5 can change slightly with every installation. Hence, minimal angular changes are expected to occur from mission to mission.

The misalignment for a mission can be determined by in-flight calibration, using the modified method of Freese and Kottmeier (1998), as described by Drüe (2001). This method applies a low-pass filter to the data and then determines global radiation Kdown by linear regression of the indicated global radiation Kdowni versus the dimensionless incidence coefficient c = cos(ζ)/cos(Θ), where Θ represents the solar zenith angle and ζ is the angle between the pyranometer zenith and the sun. The mechanical offset angles are then varied until the variance of the corrected Kdown is minimized.

This procedure was applied to two full circles performed during the calibration flight (see Fig. 11), which yields

 
formula
 
formula
Fig. 11.

Solar radiation recorded during two full circles flown as part of the IKAPOS calibration flight. Uncorrected (dashed) and corrected (solid) (top) shortwave downward radiation flux and (bottom) solar zenith angle seen from the pyranometer (solid) and in the geodetic system (dashed).

Fig. 11.

Solar radiation recorded during two full circles flown as part of the IKAPOS calibration flight. Uncorrected (dashed) and corrected (solid) (top) shortwave downward radiation flux and (bottom) solar zenith angle seen from the pyranometer (solid) and in the geodetic system (dashed).

h. Check of calibration

To determine if a new calibration is needed or to assess the uncertainty of the coefficients, it is useful to check a present calibration. This is usually done by performing additional maneuvers. Analyzing power spectra of calculated quantities allows for identifying remaining noise, artifacts, and other forms of inadequate data processing. Finally, calculating uncertainties of the final atmospheric quantities such as temperature and wind components allows for quantifying the success of the calibration.

1) Additional maneuvers

Several authors recommend simple maneuvers either to add some redundancy to the actual calibration maneuvers or to enable an independent verification of calibration results. As a reasonable rule of thumb, French et al. (2007) state that none of the wind speed components should vary more than 10% of the parallel aircraft speed component during such a check. Petersen and Renfrew (2009) even recommend repeating such patterns several times during a longer experiment.

The authors recommend reverse-heading maneuvers or L-shaped flight segments to verify TAS, sideslip (Lenschow 1986), and temperature sensor recovery factors (Lenschow and Pennell 1974). To check flow-angle offsets, full circles or eights (two opposite full circles, see Fig. 4c) at a range of roll angles (Lenschow 1986; Williams and Marcotte 2000), or ABL runs (van den Kroonenberg et al. 2008; Metzger et al. 2011; see section 1) are recommended.

Several authors use comparison flights using two or more aircraft (Lenschow and Pennell 1974; Lilly et al. 1982; Spiess et al. 2007) for the same purpose. Among these Lilly et al. (1982) particularly recommend L-shaped formation flights. Such procedures are of special importance, if measurements taken by different aircraft are composed to one dataset. For safety, however, the aircraft have to maintain a certain separation, which makes it hard to distinguish between instrumental differences and atmospheric gradients (Drüe et al. 2010).

2) Spectra

Figure 12 shows power spectra from a flight section of a measurement flight carried out on 15 June 2010 in fairly stable conditions over ocean at approximately 50-m height. The flight direction was to the west, almost perpendicular to the average (northerly) wind direction. The spectra are smoothed over equal logarithmic intervals of frequency and multiplied by frequency. Both spectra show a reasonably defined inertial subrange with the expected −⅔ slope above 2 Hz. At lower frequencies, contamination by aircraft motions is visible. Such contamination often appears as a peak corresponding to one of the autopilot control loops. The small peak near 9 Hz in the vertical wind component, hence, most probably corresponds to the autopilot vertical speed control loop. Above 15 Hz, the spectra deviate from the slope, because of excessive (vertical wind) or scarce (along wind) antialiasing filtering. As also reported by Kalogiros and Wang (2002), the spectrum of the along-wind component decreases less, or increases in the present case, because of high-frequency noise of the TAS raw data. Although the spectra are averaged over rather small (1/20 decade) bins, no spikes are visible at multiples of 50 or 60, which would indicate a ripple in the power supply or persisting timing issues in digital data transmission (both found in Drüe and Heinemann 2007). The flight was performed in a stable boundary layer, but the spectral density values are much higher than in a strongly stable boundary layer (Drüe and Heinemann 2007), because of the rather high wind speed of 18 m s−1. Because of a typical airspeed of around 70 m s−1, the scaled maximum in the along-wind spectrum is located at a scaled frequency (fz/Ua) of around 1.0, which is close to the values found by Kalogiros and Wang (2002).

Fig. 12.

Power spectra scaled by frequency [f S(f)] of (top) temperature, the horizontal wind components (second) parallel and (third) perpendicular to the flight direction, and (bottom) vertical wind at 50-m height above ocean. A dashed −⅔ slope is plotted for comparison. The data shown are high-pass filtered with a 600-s cutoff period.

Fig. 12.

Power spectra scaled by frequency [f S(f)] of (top) temperature, the horizontal wind components (second) parallel and (third) perpendicular to the flight direction, and (bottom) vertical wind at 50-m height above ocean. A dashed −⅔ slope is plotted for comparison. The data shown are high-pass filtered with a 600-s cutoff period.

3) Uncertainty of the measured atmospheric quantities

The methods to estimate the uncertainty of the measured results that are used in the literature are repeated measurements (e.g., Cremer 2008), propagation of uncertainty [also known as Gaussian error propagation (GEP); e.g., Drüe (2001)], a differential error analysis (DEA; Tjernström and Friehe 1991), or a Monte Carlo simulation (Buzorius et al. 2006). Although theoretically identical, GEP and DEA differ in practice. Using GEP usually means calculating uncertainties of each intermediate variable and propagating it to the next result of each individual formula, where it is used. DEA, in contrast, means evaluating the total differential of the full equation describing a measurand. Because of the complexity of the differentials involved, GEP is tempting to omit terms considered “minor” and DEA is rarely found in literature, probably because it is practically impossible using pen and paper.

Because of this complexity, one might argue that omitting terms in most steps of GEP leads to a potential underestimation of the combined uncertainty. Therefore, we have additionally performed a DEA of the full equations (not shown here because of excessive space requirements) using the open-source computer algebra system “Maxima” (Joyner 2006). The calculations are performed using typical values for a low-level straight flight (taken from a constant level flight over Baffin Bay on 15 June 2010). As a result, Maxima yields the value of each partial derivative to each individual input quantity. The contribution of each input quantity to the standard measurement uncertainty is this gradient times the uncertainty of the input quantity. The standard uncertainty of the measured value of each measurand is then calculated by the law of propagation (defined in JCGM 2008).

All quantities not being natural constants are regarded as input quantities: measured quantities, calibration values of sensors and ADCs, and aircraft dimensions. The instrument precisions were determined from the latest laboratory calibration of each sensor, alternatively with the manufacturer-supplied specifications. In our case, laboratory calibrations of the sensing elements yielded the following precisions: temperature (TE, TRvF): ±1 × 10−3 K, static pressure sensor: ≤0.1 hPa, differential pressure sensors: ≤0.001 hPa, INS pitch and roll angles: 0.1°, and INS heading: 0.4°. Since the exact type of INS used in the Polar5 avionics is not known to us, we assume the precisions of the attitude angles to be identical to the rather similar Honeywell avionics suite on the Dassault Falcon research aircraft of Deutsches Zentrum für Luft- und Raumfahrt (DLR): pitch/roll: 0.1° and heading: 0.4°. The precisions of the ADCs by means of the standard uncertainty of the digitized value are determined from the manufacturer’s specifications: 0.05 K, in the case of temperature; and 0.13% of the reading plus 60 ppm of the full (electric) range, in the case of the pressure sensors. Using these values, we obtain the uncertainties given in Table 3.

Table 3.

Calculated standard measurement uncertainties (fourth column) of the atmospheric measurands (first column). Values in parentheses in the fourth column represent values estimated by “manual” GEP. Second and third columns give contribution input quantities and their respective contributions. Note that third and fourth column contain absolute values in the respective units indicated in the first column. For clarity, contributions of less than 10% of the total value are omitted. Sensing element calibration (SEC), pressure partitioning factor (PPF) E, differential pressure sensor (DPS), and static pressure sensor (SPS).

Calculated standard measurement uncertainties (fourth column) of the atmospheric measurands (first column). Values in parentheses in the fourth column represent values estimated by “manual” GEP. Second and third columns give contribution input quantities and their respective contributions. Note that third and fourth column contain absolute values in the respective units indicated in the first column. For clarity, contributions of less than 10% of the total value are omitted. Sensing element calibration (SEC), pressure partitioning factor (PPF) E, differential pressure sensor (DPS), and static pressure sensor (SPS).
Calculated standard measurement uncertainties (fourth column) of the atmospheric measurands (first column). Values in parentheses in the fourth column represent values estimated by “manual” GEP. Second and third columns give contribution input quantities and their respective contributions. Note that third and fourth column contain absolute values in the respective units indicated in the first column. For clarity, contributions of less than 10% of the total value are omitted. Sensing element calibration (SEC), pressure partitioning factor (PPF) E, differential pressure sensor (DPS), and static pressure sensor (SPS).

The values for the standard uncertainty given in Table 3 are similar or lower than other precisions presented in literature (e.g., Vörsmann 1990; Metzger et al. 2011). Tjernström and Friehe (1991) report slightly smaller values for the uncertainty of wind components, which is apparently caused by their attitude reference system having errors of about compared to the INS of Polar5. On the other hand, they estimate their calculated temperature of having a greater uncertainty than ours, probably because of a disadvantageous positioning of the temperature sensor in front of the windscreen.

In the last column of Table 3, the results of a typical “manual” GEP are shown for comparison. It is remarkable that the GEP result is not always smaller than the result of the full-scale DEA. It appears justified to use GEP at least as a quick method to get a reasonable estimation of the uncertainties.

3. Conclusions

We summarize our results as a recommendation that allows for the determination of the best calibration, even if we did not perform all maneuvers in the experiment IKAPOS, which is used to exemplify most procedures. Based on the procedures we performed, as described, and based on the review of all literature we could find, the maneuvers recommended below have been shown to provide adequate in-flight calibration for turbulence measurements, at least for the type of investigations performed by the Polar5 for the IKAPOS field campaign. Although investigators on other aircraft and/or different configurations may want to add or repeat calibration maneuvers for further validation, we believe that this set is the best choice for an economic state-of-the-art in-flight calibration, in many cases.

The aim of the recommended set of maneuvers is to yield a good calibration without any need to add patterns for “better” values. Although we mentioned several times the goal to minimize the calibration flight duration, we used this criterion only to make our selection among equally suited choices. We hence see no need to change the recommendations for the case that unlimited flight time is available. Some maneuvers, however, that may provide some more redundancy but are not essentially needed are marked as optional.

All sensors should be well calibrated in the laboratory before the start of the experiment. Data acquisition should be set up to record data with sufficient resolution. Even if the accuracy of the absolute value recorded does not require the finest possible resolution, the calibration procedures may benefit from the additional information on fluctuations of the measured values, for example, the calculations presented in section 2g.

To determine the static pressure offset, no dangerous maneuvers close to the ground—such as tower flybys—are required, nowadays, if the aircraft altimeter complies with reduced vertical separation minimum (RVSM) requirements for pilot altimeters (±~1,5 hPa; ICAO 2002). If the compliance is ensured even after doing all scientific modifications to the aircraft body, then comparing the aircraft altimeter to the scientific static pressure sensor yields a sufficient estimation of the static pressure offset. To double-check the offset, a low pass appears to be optimal, since it also allows for a check of the vertical wind calibration.

True airspeed (via the pressure partitioning) should be calibrated by straight and level flights at increasing and decreasing speeds. It is essential to change speed in steps rather than continuously. Racetrack maneuvers seem likewise adequate but require at least roughly twice the flight time.

The (vertical) angle of attack is best calibrated by fast pitching maneuvers, minimizing vertical wind variance by variation of offset and sensitivity (α0, Cα) and the sensor time lags. Likewise, sideslip sensitivity Cβ and sensor time lags are calibrated by fast yawing maneuvers. For calibration of offset β0, a reverse-heading maneuver seems most adequate.

Ideally, full circles flown at two different roll angles are the best choice to determine the radiation sensor misalignment.

Finally, a calibration flight should contain enough redundancy to 1) make the results robust against failure of a single maneuver, 2) allow checking the results on independent data, and 3) yield an estimate of the uncertainty of the results achieved.

Hence, we recommend the following flight program:

  • speed steps: straight flight at constant altitude; increase TAS in steps, approximately five steps around typical measurement speed; procedure turn; decelerate TAS in steps again;

  • square at 0, 90, 180, and 270 heading on each side;

  • straight flight (3–5 min);

  • fast pitch, roll, and yaw oscillations (each ±10°);

  • straight flight (1–3 min);

  • eights (one left and one right circles) at 10° and 20° roll angle;

  • two (cross or L shaped) reverse-heading maneuvers;

  • (optional) acceleration maneuver: straight flight at constant altitude, decelerate to close to stall speed, slowly accelerate to maximum cruise speed, slowly decelerate, procedure turn, slowly accelerate to maximum cruise speed, slowly to close to stall speed; and

  • (optional) low pass above a flat surface (runway).

Acknowledgments

IKAPOS is supported by Deutsche Forschungsgemeinschaft (DFG) under Grant HE2740/9. The aircraft program was funded by the Alfred Wegener Institute (AWI), Bremerhaven, Germany. We thank anonymous reviewer 4 for the discerning comments, which helped to improve the manuscript.

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