a. Static pressure defect

The static pressure on commercial aircraft is measured from ports at proper locations on the fuselage that depend on the aircraft design. In our case, the static pressure ports were located on both sides of the fuselage, just in front of the cockpit (Fig. 1). The dynamic pressure P_d was estimated as the difference between the total pressure by a Pitot tube at the copilot side and the static pressure using a differential pressure transducer. This dynamic pressure was corrected for the compressibility effect similarly with the radome “total” pressure P_tr mentioned above. Well-designed Pitot probes show very small sensitivity to flow angles up to 20° and, thus, a zero position error of measured total pressure is usually assumed. The measured static pressure, however, deviates from that in the free airstream due to the flow distortion by the aircraft. This error is the position error or static pressure defect dP_s (defined as measured local static pressure P_sm minus free airstream static pressure P_s) and is usually estimated using a trailing cone system (Brown 1988; Khelif et al. 1999), so that the static pressure of the undisturbed air behind the aircraft is measured. Because such a system was not available, a different approach (Tjernström and Friehe 1991; Bögel and Baumann 1991) was used, which is based on the hydrostatic balance assumption in the free atmosphere above the boundary layer. We note that the trailing cone system limits the maneuvers that can be performed by some aircraft and is required to be at the same altitude with the aircraft, so that no altitude error is introduced in the estimation of static pressure defect. The dynamic pressure P_d is computed from the difference between total pressure and static pressure and, thus, contains a −dP_s error.

We used data from the speed runs and sideslip maneuvers to find the dependence of static pressure defect on dynamic pressure P_d, attack angle α, and sideslip angle β. Usually the angle dependence of static pressure defect, which is more significant at low airstream speed (relatively large flow angles) as in our case, is not considered. Bögel and Baumann (1991) found a significant angle dependence of static pressure defect for a 858AJ Rosemount probe mounted on a boom at the nose of a Falcon aircraft. In our approach, the static pressure during the maneuvers was corrected for altitude changes relative to the altitude of the first data point of each maneuver using the hydrostatic balance assumption. We used the corrected DGPS altitude measurement as described in the beginning of section 3. Then, the static pressure defect was computed as the difference of this height adjusted static pressure with its value at that first point of each maneuver. This procedure actually computes the static pressure defect minus its value at the first point of each maneuver leg. Thus, the constant term of the fit of the computed static pressure defect by a polynomial function of the flow parameters P_d, α, and β [see Eq. (2) below] cannot be determined in this way.

A first approximation of this term was found by requiring in-flight maneuvers data to match the static pressure defect estimates obtained during takeoff runs using as a reference the average value of static pressure before the aircraft starts to increase speed. This takeoff static pressure defect includes the effect of the proximity to the ground and the takeoff configuration of the flaps and, thus, it is different from the in-flight data. Using data just after takeoff, when the aircraft was at an altitude of two to three times the wing span and the hydrostatic balance assumption for height adjustment, we found that the proximity to the ground may increase the static pressure defect by about 0.4 hPa relative to the in-flight conditions. However, these data were of limited length with significant scatter. An accurate estimate of the constant term in the static pressure defect approximation was obtained from the reverse heading maneuvers by minimizing the differences between the wind components before and after the turn (see section 3d). The longitudinal wind component in the aircraft frame of reference is more sensitive to the correct removal of the static pressure defect, since it is directly connected to the airstream speed and, thus, dynamic pressure. This step was performed at the end of the calibration process described in the next paragraphs and the process was repeated using the new accurate constant term of the static pressure defect approximation.

Figure 2 shows the computed static pressure defect dP_s for the case of 17 July 1999 maneuvers as a function of dynamic pressure P_d. As it can be seen in Fig. 2, the dP_s corresponding to longitudinal acceleration (with respect to earth) of the aircraft is lower by about 0.5 hPa with respect to deceleration. This implies that the flow around the aircraft is different in these two cases, which has also been observed by Tjernström and Friehe (1991). As it is shown by Kalogiros and Wang (2002, manuscript submitted to J. Atmos. Oceanic Technol.) this was mainly the effect of the aircraft propellers' slipstream (tube of high speed) at the position of the static pressure sensor (see Fig. 1). Figure 2 also shows the static pressure defect, which deviates significantly from a second-order polynomial of P_d for large attack and sideslip angles at the points indicated by the arrows. The dependence of dP_s on P_d, α, and β was approximated in a least absolute error sense (robust estimation of the parameters of the fit to avoid data outliers) by the following equation:

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where

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on average during the maneuvers. The units of the angles and the pressures are radians and hectoPascals, respectively. The local flow angles at the radome used in the fit were computed using the Pitot measured P_d in Eqs. (1a) and (1b), because of the less noisy differential measurement of P_d compared to the postcalculation of P_d using P_tr and P_s in Eq. (1c).

During the speed run maneuvers, the positive acceleration values were in the range 0 to 1 m s⁻² (0.53 m s⁻² median value) and the negative ones were in the range −0.5 to −1.5 m s⁻² (−0.81 m s⁻² median value). During normal flight conditions (almost no acceleration from changes in engines' thrust) a linear interpolation of C₀ to zero acceleration (C₀ = −0.278), assuming that the values of C₀ reported just after Eq. (2) correspond to the above median acceleration values, was used. In Eq. (2) we note the quadratic (symmetric) dependence of dP_s on flow angles, which was evident from the time series of dP_s (especially during the sideslip maneuvers with large positive and negative angle values of sideslip angle).

Figure 3 shows the time series of dP_s measured against the one approximated by Eq. (2) for the speed run and sideslip maneuvers on 17 July 1999. The sudden increases of dP_s at its local maxima indicated by arrows during the speed run correspond to abrupt changes from acceleration to deceleration of the aircraft. Most of the time the error of the approximation is within ±0.2 hPa, which is quite low if one considers the validity of hydrostatic balance assumption and the accuracy of the DGPS altitude measurements.

b. Correction for the nonspherical shape of the radome

We performed a simple check to detect any possible deviation of the pressure distribution on the radome from the spherical model Eqs. (1a)–(1c). The pitot measured dynamic pressure P_d used in these equations was corrected for the static pressure defect as well as static pressure P_s. In the case of very small sideslip angle (speed run) or attack angle (sideslip maneuver) Eq. (1c) can be solved for the remaining flow angle. Then, the corresponding pressure difference ΔP_α or ΔP_β can be estimated from Eqs. (1a)–(1b). Equation (1c) is expected to hold for the near spherical radome because the true stagnation point is very close (flow angles up to 15°) to the design stagnation point (the center port) where “total” pressure P_tr is measured. The differential pressure ports are located at 33° from the center port and, thus, their measurements may depart significantly from Eqs. (1a)–(1b). Figures 4a and 4b show the estimated pressure differences for the attack and sideslip pressure ports against the measured ones for the speed run and the sideslip maneuver, respectively, on 22 July 1999. The deviation of the experimental data from the theoretical model is significant (more than 5%) and, thus, the pressure differences should be corrected before being used for the estimation of local flow angles with Eqs. (1a)–(1c). We note that the deviation is of opposite sign between attack angle and sideslip angle ports, implying a different shape of the radome on those two axes. However, errors in the laboratory calibration of the differential sensors may contribute to this behavior, too. Using the above test we found large scatter for the maneuver data on 19 July 1999 and thus, they were not used in the radome calibration below.

The measured ΔP_α in Fig. 4a has been actually corrected for a small near linear dependence on the acceleration of the aircraft with unknown origin (probably a problem of the corresponding sensor). This was concluded from the correlation of the original measured ΔP_α with the estimated one (similar to Fig. 4a), which showed a looplike behavior (even though with considerable scatter). Most of this looplike behavior and scatter could be removed if a dependence of measured ΔP_α on the aircraft acceleration during the speed run maneuver is assumed. If ΔP_α is not corrected for this dependence, the result is a looplike behavior in the calibration plot of the radome local attack angle against the estimated free airstream one (see Fig. 5a). This would suggest an unrealistic large dependence of the attack angle calibration and, thus, of the upwash/flow distortion effect on the acceleration of the aircraft. We note that a small dependence is actually expected mainly because of the suction effect of the propeller's slipstream at the nose of the aircraft as it is shown by Kalogiros and Wang (2002, manuscript submitted to J. Atmos. Oceanic Technol.).

c. Variance method of radome calibration

A commonly used technique to calibrate the local flow angles at the radome so that they correspond to true free airstream flow angles is to calculate the calibration factors that minimize the variance of the wind components (Tjernström and Friehe 1991; Khelif et al. 1999). The assumption made here is that errors in the estimation of the free air stream flow angles increase the wind variance. This method is easy to use and can be applied in the presence of turbulence in the boundary layer, but does not reveal any possible nonlinear dependences or problems in the data.

In the application of this method we assumed a linear dependence of free airstream flow angles to local flow angles α and β with no cross-dependence. The full equations for the estimation of wind components in the meteorological frame of reference were used. The calibration coefficients (slope and offset) were assumed to be independent of the Mach number, which is justified by the small range (0.15–0.25) of Mach number at which the Twin Otter flies. For faster aircraft that fly in a broader range of Mach number, these coefficients are found to be slowly varying functions of the Mach number (Brown et al. 1983; Tjernström and Friehe 1991). We computed the calibration coefficients that minimized the sum of the horizontal wind components variances and the second moment of vertical wind velocity. We used the second moment of vertical wind velocity instead of the variance so that the mean vertical wind velocity was also minimized. The calibration coefficients were computed using all the maneuver data for each experimental day. The following calibration equations were obtained (angles in rad):

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where α_f and β_f are the free airstream attack and sideslip flow angles, respectively. Using the data from 17 July and 22 July 1999 we found substantially different calibration coefficients (especially for the attack angle). The values of the coefficients in Eqs. (3a) and (3b) correspond to the latter day. The decision of using the results of this day was based on the analysis described in the next section.

d. Direct method of radome calibration

Next, we transformed the estimated wind components to airstream components in the aircraft reference system using the inverse transform of the one described by Lenschow (1986) for the calculation of the wind components in the meteorological frame of reference. The airstream velocity components in the aircraft system u_a, υ_a, and w_a are related to airstream flow angles through the following equations (Crawford and Dobosy 1992):

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The flow angles α_f and β_f are spherical coordinates similar to Eqs. (1a)–(1c). The corresponding wind components defined by Lenschow (1986) are somewhat different because in that case α_f is not a spherical coordinate. According to Eqs. (5a)–(5c), the free airstream flow angles can be estimated as β_f = tan⁻¹(υ_a/u_a) and α_f = tan⁻¹(cosβ_fw_a/u_a).

Figures 5a and 5b show the estimated free airstream flow angles α_f and β_f, respectively, against the corresponding local flow angles α and β at the radome for the maneuvers performed on 22 July 1999. Cases with absolute sideslip angle greater than 0.05 rad (about 3°) were not included in the calibration of attack angle in Fig. 5a. The attack angle results for the data from 17 July 1999 showed a nonlinear dependence at small angles resulting in an erroneous calibration offset and, thus, they were rejected from the radome calibration. The following calibration equations were obtained:

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The value of the calibration offset of the sideslip angle C_β was estimated to be 0.0062 using the method described above. Most of the dispersion around the solid lines especially for attack angle values above 0.15 rad (8.5°) is probably due to wind variations during the maneuvers. Alternatively, we can use the time derivative of the wind equations (Lenschow 1986) assuming zero vertical wind velocity, steady (in the small time period and space of each maneuver) wind components and small roll, pitch, and attack angles to find an estimate of the time derivative of the β_f angle during the sideslip maneuvers. The correlation between the time derivatives of β and β_f gave a similar result with Eq. (6b) for the slope coefficient but in our case the dispersion of the data points was larger than the one in Fig. 5b, probably because the time derivative amplified the noise in the data.

The calibration coefficients in Eqs. (6a)–(6b) are very close to the ones in Eqs. (3a)–(3b), which were estimated using the variance method, with the exception of the offset of the sideslip angle. A careful analysis revealed that an error of 0.5 m s⁻¹ in the horizontal wind components estimated using Eq. (4) could even change the sign of C_β. However, the slope coefficient was less sensitive to such errors. Thus, we used the reverse heading maneuver data with the slope coefficient given by Eq. (6b) to estimate C_β. The wind component that is more sensitive to the value of the calibration offset of the sideslip is the lateral one in the aircraft frame of reference, since it is directly connected to the sideslip angle. The value of the calibration offset of the sideslip angle found by minimizing the differences between the average estimated wind components in the meteorological frame of reference for the legs before and after the turn of the reverse heading maneuver was C_β = −0.0087. This value is very close to the value of the calibration offset of the sideslip angle given by Eq. (3b). In the data analysis presented in the next paragraphs the radome calibration given by Eqs. (6a) and (6b) with the above value of C_β was used.

The results of the procedure for the estimation of the above sideslip offset are shown in Fig. 6, where the wind varies along the maneuver legs but with similar horizontal variations between the forward and backward legs. Track distance is the distance from a common point of the two legs along the aircraft track, where negative distance means that the aircraft approaches that point. We note that the assumption that the wind components in the aircraft system should have the same magnitude but change sign (this can be seen in Fig. 6b) is often used in the reverse heading maneuver. However, this requires an exact reverse maneuver, which is not usually the case. We used the more general method of minimizing the change of the components of the wind in the meteorological frame of reference during the maneuver.

a. Maneuvers results

The reverse heading maneuver is a good test for the quality of the estimation of the horizontal components of the wind, which depend mainly on the accuracy of airstream speed U_a and sideslip angle. The corresponding average horizontal wind components for each of the two legs of the reverse maneuver shown in Fig. 6a agree with each other within 0.1 m s⁻¹. Figures 7a and 7b show the estimated wind longitudinal and lateral components of the horizontal wind with respect to the average heading direction of the aircraft against the corresponding airspeed components U_a cosα_f and U_a sinβ_f for the speed run and the sideslip maneuver, respectively, on 22 July 1999. The results satisfy adequately (about 5% root-mean-square error) the commonly used criterion of 10% contamination error for the quality of the performance of the system (Lenschow 1986).

Figure 8 shows time series of the estimated vertical wind velocity w and the vertical aircraft velocity w_p for the pitch maneuver on 22 July 1999. We use 20-Hz data to show the response of the system during possible fast variations of the vertical velocity of the aircraft. The pitch maneuvers are the most demanding ones for testing the contamination of the vertical component of the wind by the motion of the aircraft because of the large vertical velocity of the aircraft involved. Our results in Fig. 8 show very small correlation of the estimated vertical wind velocity with the vertical velocity of the aircraft for most of the time during the pitch maneuver. The deviation of the estimated vertical wind velocity from near zero values occurs at time periods of fast changes of the vertical velocity of the aircraft, when the airflow around the aircraft is highly unsteady. Figures 9a and 9b are scatter diagrams of w against w_p and the corresponding airspeed component U_a sinα_f. From these figures it can be concluded that the root-mean-square error of the vertical wind velocity, which is assumed to have near zero mean value in the air where the maneuvers took place, is about 5% of the vertical velocity of the aircraft and is not significantly correlated with it or the corresponding air stream component. However, we can see from Fig. 8 that the instantaneous error of the vertical wind velocity during abrupt changes of the vertical velocity of the aircraft can increase significantly.

b. Boundary layer results

In this section sample results from the DECS boundary layer data at about 0930 local time on 7 July 1999 (a day with limited stratocumulus cloud cover over the experimental area) are presented as a quality test of profile and turbulent wind statistics. The raw 100-Hz data were filtered at 10-Hz Nyquist frequency to avoid aliasing and resampled at 20 Hz. The 10-Hz DGPS data showed very little spectral energy above 2 Hz (not shown here) and, thus, it was plausible to linearly interpolate them to 20 Hz. Static pressure was low-pass filtered at 3 Hz to suppress the observed high frequency noise. A simple analysis of the equations (Lenschow 1986) used to derive the wind components shows that dynamic pressure variations dominate over the static pressure variations at high frequencies. Thus, this filtering does not affect significantly the high-frequency content of the wind estimates.

Figures 11a–d show profiles (20-point moving averages) of wind speed and direction, potential temperature, and water vapor specific humidity from a vertical spiral sounding (ascent followed by descent). A nearly well-mixed layer up to 230 m and a strong temperature inversion in the layer 450–650 m can be seen in these figures. The diameter of the spiral was 7 km and the aircraft horizontal velocity was higher by 10 to 20 m s⁻¹ during descent compared to ascent. Despite the change of airplane heading during the spiral motion and the difference in horizontal velocity of the aircraft between ascent and descent, the corresponding wind speed and direction profiles display the same layer structures especially above 450 m. Between 250 and 450 m, a difference in the boundary layer structure among ascent and descent is evident in the virtual temperature and water vapor mixing ratio profiles in addition to the wind profile.

Figure 12 shows the power spectra of the wind velocity components with respect to the sampling direction and the virtual temperature from a 30-km straight level leg at an altitude of about 36 m above the sea surface. The sampling direction (convection velocity) of a moving platform (aircraft) is the air stream direction. The airstream vector is the vector difference of wind speed and aircraft velocity. The average wind direction in the case of Fig. 12 is 335°, the aircraft path (track) direction is 87°, and sampling direction is 75°, almost perpendicular to the average wind direction. The spectra are averaged over segments of 1024 points and then smoothed over equal logarithmic intervals of frequency. The virtual potential temperature time series were quite noisy above 3 Hz and, thus, the corresponding spectra are shown only up to this frequency. All the power spectra show a well-defined isotropic inertial subrange with the expected −5/3 slope above 2 Hz and no evident contamination aircraft motions at low frequencies. At frequencies higher than 8 Hz, the drop of the power spectra is due to the antialiasing filter. The power spectrum of the longitudinal component falls off less rapidly because it is affected by high frequency (mainly above 10 Hz) noise of the raw data. The power spectrum of the lateral wind component (which is almost along the mean wind direction) shows significant low frequency–mesoscale energy in the frequency decade just before the inertial subrange as opposed to the vertical wind component. These spectral properties are consistent with the idea of elongation of eddies of the along wind component due to stretching by the mean wind shear (Nicholls and Readings 1981). Thus, when sampling is in the across wind direction, energy from small wavenumbers in the spectrum of the along wind component is redistributed to larger wavenumbers (higher frequencies).

Figure 13 shows the cospectra of momentum and heat fluxes for the same measurement leg with Fig. 12. The cospectra seem to approach the expected inertial subrange slope of −7/3 above 2–3 Hz (we note, however, that there is not sufficient bandwidth above these frequencies) and, thus, they adequately cover the significant energy-containing region even at this low altitude. The isotropic behavior in the inertial subrange requires just that the turbulent fluxes are zero or decreasing faster than the power spectra in this range (Kaimal and Finnigan 1994) as shown in Fig. 13.

Figure 14 shows power spectra ratios of vertical and lateral wind components to the power spectrum of the longitudinal wind component with respect to the sampling direction for the same flight leg with Fig. 12. In the sampling direction, frequency corresponds to the wavenumber component on this direction according to Taylor's hypothesis of frozen turbulence. Thus, the above ratios of the frequency power spectra of the velocity components in the inertial subrange should be 4/3 according to local isotropy (Kaimal and Finnigan 1994), which is a very good test of the quality of the radome calibration. Figure 14 shows that for the lateral wind velocity the ratio is very close to the isotropic value in the inertial subrange, while for the vertical wind velocity it is lower than unity. This behavior was repeated in almost all of the experimental days. Similar behavior of the vertical wind velocity spectrum is found in the results of other researchers using aircraft measurements over sea (Nicholls and Readings 1981; Tjernström and Friehe 1991) or over land (Lenschow et al. 1991). The drop of the power spectra ratios above 8 Hz is due to the antialiasing filter and the high frequency noise of the longitudinal component discussed above. Kalogiros and Wang (2002, manuscript submitted to J. Atmos. Oceanic Technol.) show that the observed lower than unity S_w/S_ux ratio in the inertial subrange is not an indication of deviation from local isotropy, but an effect related to aerodynamic response of the wing vortex and, thus, of the induced upwash at the nose of the aircraft. They propose, also, a method to correct the time series of the vertical wind velocity or its power and cross spectra with other parameters using in-flight fast acceleration measurements.