Appendix B: BIBLIOGRAPHY  No Title  Single Doppler Radar Retrieval

Appendix A: Discussion of Uncertainty Elements

These sections provide additional detail on the elements of the uncertainty analysis presented in Chapter 2. Each is numbered to correspond to the tables in that chapter.

1.  Operation of the dead-weight standard. Items 1)-4) pertain to calibration of the 1501 pressure transducer that is used as a transfer standard to calibrate the transducers in the aircraft. The calibration of the transfer standard is performed by comparison to a Bell and Howell dead-weight standard, for which the manufacturer claims a 3-$\sigma$ inaccuracy limit of 0.015% of full scale. This leads to an error in the calibration of the 1501 transducer of 0.1 mb. Because the transfer standard is then used to calibrate the transducers used in the aircraft, any inaccuracy in the calibration of the transfer standard acts as a bias when the airborne sensors are calibrated, so only a bias limit is shown in Table 1a for item 1. The dead-weight standard is recertified every five years by the Bureau of Naval Weapons, and the history of these recertifications supports this estimate of the bias limit. Other factors affecting the bias include the correction for gravity at the calibration site, variation in temperature of the unit, etc., but these corrections are minor in comparison to the above major source of bias.

2.  Calibration of the dead-weight standard.  The dead-weight sensor may be read with a resolution of 0.01 mb. Such a resolution contributes insignificantly to the precision of the measurement of pressure, or to the bias in the calibration of the transfer standard. 3.  Repeatability of the 1501 transfer standard.  The transfer standard used is a Rosemount 1501MD2/1501C01 pressure transducer. Its stated inaccuracy limit includes effects of non-linearity, hysteresis, temperature variations, calibration at Rosemount, and variability in the response of the transducer. Rosemount quotes a 3-$\sigma$ static error limit of 0.026% of full scale (or a 2-$\sigma$ limit of about 0.2 mb) for this transducer. However, the temperature range for this specification is -55 to +71$^\circ$C, and other ranges of conditions are much broader than used in the laboratory calibration, so we expect the unit to perform better in the limited range used (about 15-25$^\circ$C).  Furthermore, most of the above effects enter when the unit is used to calibrate the airborne transducers, and will be included below. The additional contribution applicable to the calibration of the transfer standard is only the repeatability of that transducer. We have therefore used half of the static error limit, or 0.1 mb, in the table.^14.1

4.  Height uncertainty in 1501 calibration.  The uncertainty introduced by differences in height of the pressure sensors is negligible because both units are located next to each other at the same elevation.

5.  Inaccuracy of 1501 transfer standard. The remaining entries in Table 1a apply to the calibration of the transducers in the aircraft by comparison to the transfer standard. The dynamic inaccuracy of the transfer standard indicates the limit expected for the errors caused by vibration, acceleration, overpressure, or power supply variation. In the calibration laboratory, conditions are such that the unit is not subjected to vibrations or accelerations, and the pressure and temperature are stable, so we have used 50% of the manufacturer's limit for this error contribution. This is probably an overestimate because accelerations and vibration are the main source of dynamic inaccuracy, but some contribution is included for possible sensitivity to power supply variations. The contributions have been divided between random errors and bias because there are aspects of each in the variations in power supply voltage. 6.  Stability of the 1501 transfer standard. The stability of the transfer standard is specified as 0.025% of full scale per year. As the unit is recalibrated at least each six months, the associated bias limit is estimated to be about 0.1 mb. The calibration history of the transfer standard supports this limit. 7.  Resolution of the 1501 transfer standard. The transfer standard produces a digital output with a resolution of 0.03 mb. The 2-$\sigma$ precision associated with such digitization is about 0.02 mb, as argued under item 3). When the transfer standard is used to calibrate the airborne transducers, this digitization again introduces a bias. 8.  Leaks in the lines during calibration. Careful checks for line leaks are performed during calibration by checking the ability of the system to maintain a constant pressure for a period of about 10 min, so the effect of leaks is either negligible or is corrected before the calibration is continued. The table entry is an indicator of this possible source of error, which we believe has been removed. 9.  Height uncertainty during calibration. During calibration of the aircraft sensor, the aircraft and the laboratory are separated horizontally by about 100 ft and vertically by less than 3 ft. A 3 ft separation would cause a difference of about 0.1 mb in the pressure sensed by the two units. A correction is applied, reducing the bias limit to about 0.04 mb representing uncertainty in the height difference and in the temperature in the pressure lines (affecting the static pressure gradient). 10.  Curve fit inaccuracy. The resulting calibration is represented by a quadratic expression relating the measured voltage to the pressure. This quadratic expression is fitted to the calibration data points, and this overconstrained fit shows that the pressure can be represented by the quadratic expression over the total pressure range of the transducer to a (root-mean-square) inaccuracy of less than 0.05 mb. A bias limit for the fit is therefore 0.1 mb. 11.  Airborne data system digitization. This and the next three items apply only to p_W, the pressure measured by the 1201 transducer attached to the wing pitot-static ports. The calibration of this sensor is performed by recording the output from the aircraft pressure transducer through the aircraft data system. A separate analysis of this system (documented in the hardware manual, Walther et al. 19xx), led to estimated uncertainties for the data system of 0.0024% (bias) and 0.011% (precision). When converted to pressure for a 0-1050 mb transducer, these limits lead to the values in the table.

12.  Static inaccuracy, hysteresis, 1201 transducer. The specification for the hysteresis of the 1201 transducer is 0.02%, so the bias limit used is 0.21 mb. This enters as a bias because the pressure change during calibration is always in the same direction; this term might be minimized by different procedures.^14.2

13.  Static inaccuracy, repeatability, 1201 transducer. The repeatability of the 1201 transducer affects the validity of the calibration, and is assumed to be a random error. The manufacturer's specification (3-$\sigma$) is 0.02%, so we estimate 2S=0.14 mb. 14.  Static inaccuracy, voltage variations, 1201 transducer. The calibration is performed with the unit installed in the aircraft, so the performance of the 1201 transducer is dependent on the stability of the aircraft power supply at 28 VDC.  This voltage may vary by $\pm$1V, or $\pm$4%. The transducer is specified as stable to 0.002% for 1% changes in supply voltage, so the associated error is 0.08 mb. We interpret the $\pm$1V variation as a 2-$\sigma$ limit, and so use 0.08 mb as a corresponding limit. This is listed as a random variation in the sense that repeated calibrations would be subject to different voltages, but could appear as a bias if the voltage used during calibration systematically differs from that in flight.

15.  Repeatability of the transfer standard. This and the next entry apply to the measurement p_F, measured by the Rosemount Model 1501 transducer connected to the static ports located on the fuselage. There is no entry for digitization by the airborne data system, because the transducer itself produces a digital output. As discussed under item 3), repeatability of the p_F pressure transducer contributes to the uncertainty in the calibration of the pressure tranducer, and as before [cf. item 3)] we have used 50% of the specification because of the favorable conditions that apply during the calibration. 16.  Dynamic inaccuracy of the 1501 transducer. As in item 5), only 50% of the specified uncertainty attributable to dynamic inaccuracy of the 1501 transducer is included in the table, because the primary source of this inaccuracy, vibration and acceleration, is negligible during the calibrations. However, the calibrations are performed while the airborne transducer is installed in the aircraft, and so are subject to voltage variations that contribute to the specified dynamic inaccuracy. The specified range of input voltages is $\pm$4 V, and this is much more variation than occurs during ground calibrations, so reduction of the specified tolerance should be justified. This estimate is probably still too high; dynamic inaccuracy most likely introduces a negligible contribution to the uncertainty during calibration.

17.  Net calibration error, 1201 transducer. The calibration as described in Table 1a leads to a net bias error for the 1201 transducer (obtained by adding the bias uncertainties in quadrature) of 0.30 mb and a net precision of 0.19 mb. When the calibration is used to interpret measurements from the pressure transducers, however, both these errors introduce biases, so they have been summed in quadrature and a net bias error of 0.36 mb is used to characterize the inaccuracy of the calibrations.

18.  Dependence oof 1201 transducer on temperature. The specified temperature range for the transducer is -55 to +75$^\circ$C.  Over this range, Rosemount has specified the inaccuracy as 0.3% (3-$\sigma$ limit), and it appears that the temperature dependence is the dominant contributor to this uncertainty estimate. However, our normal operation spans only part of the specified temperature range; normal flight conditions vary over about $\pm30$C or only about half of the specified temperature range. We have therefore reduced the estimated inaccuracy by 50% (which should be conservative because the extreme inaccuracy probably occurs at the extreme limits of the temperature range). This is listed as a bias because the inaccuracy is probably a consistent function of temperature and could not be reduced by repeated measurements if the temperature remained the same.

19.  Dependence of 1201 transducer on temperature change. The conditions under which the primary uncertainty analysis presented here is valid require straight-and-level flight. Some entries in the table, however, include estimates for other conditions; they are marked with the symbol ``.'' In the case of descents, the temperature change causes significant errors in the output of the 1201 transducer. This has been studied in a temperature chamber, where it was found that errors of 3 mb could be produced by temperature changes of about 2$^\circ$C/min, such as would be encountered in a 1000 ft/min descent. Typical temperature changes encountered during level flight are usually only 0.2$^\circ$C/min or less, and because of thermal lags and variable atmospheric structures will at least partially have the character of random errors. We have therefore used an error estimate of 0.2 mb for the random and also the bias error associated with temperature changes of the transducer, for level flight, but have used a bias limit of 3.0 mb for 1000 ft/min descents or climbs.^14.3

20.  Dynamic inaccuracy, vibration. Rosemount estimates that the effect of acceleration on the 1201 transducer is 0.02% for a 1-g (9.8 m s^-2) acceleration. Studies of the accelerations which aircraft experience show that 1-g accelerations are rare and are confined to the vertical. We have aligned the 1201 transducer along the longitudinal axis of the aircraft where accelerations are normally <0.1 g, and we treat the resulting error as a random error because the horizontal accelerations are not expected to be correlated with pressure in a consistent way. We use 2S=(0.02%/g)x0.1gx1050 mb = 0.2 mb.

21.  Dynamic inaccuracy, noise. When held at a constant pressure, the noise output from the 1201 transducer is about 5 mV peak-to-peak for frequencies less than 1000 Hz. If this is interpreted as corresponding to a standard deviation of $\pm$1.8 mV, the 2-$\sigma$ error limit would be (3.6 mV/(10 V)=3.6x10^-2% or 0.36 mb. However, the output of the transducer is filtered by a Butterworth four-pole analog filter with a cut-off frequency that is usually 1 Hz, so this filtering should reduce the noise to negligible levels. Our experience during calibrations is that the signal remains constant within the digitization resolution of the data system, and plots of the variance spectra from measurements in flight show no sign of a contribution from noise. The quantization noise from the data system is 0.04 mb, and the above evidence indicates that the noise from the transducer is less than this, so we have included a random-error contribution of 0.04 mb to represent an upper limit to the noise produced by the transducer.

22.  Line leaks. The lines connecting the pressure transducer to the static ports on the pitot-static tube are checked for leaks via a static pressure check as described under item 8). This item is included in the table as an indication of this potential source of error, but the contribution should normally be negligible.

23.  Time lag. The response time of the 1201 transducer is about 15 ms. The pressure at the flight altitude might change at the rate of about 0.5 mb/s, but seldom changes faster than this, so the measurement might lag by 0.5x15x10^-3 mb or <0.01 mb. This is negligible in comparison to other tabulated errors. Another time lag is introduced by the line connecting the pressure transducer and the static ports, but this line is only about 1 m long and the expected time delay from such a line is only a few milliseconds (Brown et al. 1983), so this contribution is similarly negligible.

24.  Airborne data system. The effects of the airborne data system entered the calibration as discussed under item 11), but they enter again during the measurement phase. Including this contribution twice is conservative, because some portion of the errors will be similar from calibration to measurement phase and will therefore cancel. However, the uncertainty contributions are slightly different in the measurement phase because the digitization error no longer contributes to the bias during the measurement phase. Justification for these entries is provided under item 11).

25.  Aerodynamic effects. As the aircraft yaw, pitch, and roll change, there are influences on the static pressure sensed by the (wing-mounted) static source. These have been estimated from maneuvers flown with a trailing cone. One effect, that of roll, can be corrected: the sensed pressure will change with the altitude of the sensor relative to the fuselage of the aircraft, according to:

$$(dp/d\tau) = \rho g (W/2) {\rm sin}(\Phi)$$ (14.1)

 

where $\rho$ is the air density, g is the acceleration due to gravity, W is the full wingspan of the aircraft, and $\Phi$ is the roll angle (taken positive for the right wing down). For roll angles of $\pm$5, this gives a 2-$\sigma$ limit of 0.07 mb; however, a correction for this effect can be applied to the measurements, and that correction leaves a negligible residual error. Maneuvers in which the yaw angle ($\beta$) varied showed a variation in measured pressure of (dp/d$\beta$)=0.1 mb/; as the yaw variation is typically less than 0.5, the estimated contribution is 2$\sigma$=0.1 mb under moderately turbulent conditions (such that the root-mean-square lateral wind speed relative to the aircraft is about 1 m/s). A similar contribution is included for vertical wind fluctuations that cause changes in the angle-of-attack. The bias estimate of 0.2 mb corresponds to the error expected from an angle of attack (or yaw) systematically different from zero, and arises because the normal angle of attack of the King Air is about 2. [check on this: orientation of tubel vs ac axis?]

26.  Static defect. A trailing-cone calibration is used to determine the static defect caused by aerodynamic effects of the aircraft in flight. For p_W, these effects might arise from distortion of the airflow in the vicinity of the wing-tip and associated effects on the static pressure there. Tests of the static defect are discussed in Appendix A; these indicate that the tolerance associated with the static defect is consistent with that estimated in the preceding item, so no additional contribution is included here.

27.  Truncation during data processing. All calculations are performed on computers using at least 32-bit (and in most cases 60-bit) floating point words, so the effects of truncation during calculation are negligible in comparison to the effects of the initial 14-bit digitization by the airborne data system.

28.  Calibration uncertainty, 1501 transducer. The calibration as described in Table 1a leads to a net bias error for the 1501 transducer (obtained by adding the bias uncertainties in quadrature) of 0.24 mb, and a precision limit of 0.07 mb. When the calibration is used to interpret measurements from the pressure transducers, however, both these errors introduce biases, so they have been summed in quadrature and a net bias error of 0.25 mb is used to characterize the inaccuracy of the calibrations.

29.  Digitization noise. The digitization error introduced by the digital transducer affects the measurement in the same was as it affects the calibration, as discussed earlier in items 7) and 15).

30.  Static inaccuracy. Rosemount quotes a static error limit of 0.026% of full scale, or about 0.3 mb, for this transducer. However, this is a 3-$\sigma$limit, and includes the effects of calibration which we have treated separately. Contributions from linearity, hysteresis, and temperature dependence of the transducer response enter this error limit. In our application, non-linearity is represented (at least partially) by the non-linear fits to the calibration data, so effects of non-linearity should be minimized. Also, the transducer is installed in the cabin where the temperature remains about 20$^\circ$C$\pm$10$^\circ$C, and so varies over a much smaller range than the specified operating range of -55 to +71$^\circ$C. It therefore seems reasonable to reduce this error contribution to about 50% of the quoted value.^14.4

31.  Dynamic inaccuracy, acceleration and vibration. The most important contributor to dynamic inaccuracy is vibration and acceleration. The sensor is specified to provide the quoted accuracy when operating under 10 g varying accelerations and 6 g continuous accelerations. These limits are more than twice the accelerations that the research aircraft are designed to withstand, and accelerations exceeding 1 g are seldom experienced, so use of this error limit to characterize the dynamic inaccuracy should be conservative. (The specifications do not separate this effect from others including power supply variation, but those are negligible contributors in comparison to this major source of error.) The contribution is listed as a random error because sustained accelerations are seldom experienced; instead, varying accelerations and vibrations are more common, and they are probably randomly associated with pressure and so have the character of a random error.

32.  Dynamic inaccuracy, voltage variations. The contributions of variations in the power supply (in this case, the primary aircraft 28 V supply) are usually within the range of $\pm$1 V, while the instrument is specified to meet error tolerances for voltage variations of $\pm$4 V, so we assume that the error contribution from this source is negligible. In any case, it is included in the estimate for the preceding errors because that estimate is actually the total dynamic error, including the effect of variations in the voltage of the power supply.

33.  Long-term stability. The long-term stability of the transducer is specified as 0.025%/y, or about 0.27 mb/y. However, the transducers are recalibrated before and after each project, or typically within 1 mo of any research use, so this tolerance has been reduced to about 10% of the quoted value or 0.03 mb. The periodic calibrations themselves have an uncertainty substantially larger than this, so this low tolerance cannot be documented, but calibrating the sensors so close to any use should reduce the effects of drift to negligible levels.

34.  Response time. The quoted response time for the pressure sensor is 75 ms to indicate a 63% response to a step change. Under normal conditions, the pressure at the flight level seldom changes faster than 0.5 mb/s, so the measurement might lag by as much as 0.075x0.5$\approx$0.04 mb.^14.5 An additional lag is introduced because the pressure lines are up to 5 m long, and so can introduce a time lag of about 20 ms. The combined effects of these two sources are that an error of about 0.05 mb might result. During rapid climbs or descents (of up to 2000 ft/min or about 10 m/s), an error of about 0.1 mb could occur. This is listed in the tables as a bias because it would normally always be in the direction of a lag in measured pressure behind the correct pressure.

35.  Line leaks. The lines connecting the pressure transducer to the static ports on the fuselage are checked for leaks via a static pressure check as described under item 8). This item is included in the table as an indication of this potential source of error, but the contribution should normally be negligible.

36.  Static defect. A trailing-cone calibration is used to determine the static defect caused by aerodynamic effects of the aircraft in flight. For p_F, these effects can arise because the static pressure at the locations of the pressure ports on the fuselage might differ, for some flight conditions, from the true static pressure. This source of error is discussed in detail in Appendix A, where justification for the tabulated numbers is presented. See also item 38) below, representing the effects of uncertainties in the parameters needed to make the correction for this effect.

37.  Truncation during data processing. All calculations are performed on computers using at least 32-bit (and in most cases 60-bit) floating point words, so the effects of truncation during calculation are negligible in comparison to the effects of the initial 14-bit digitization by the airborne data system.

38.  Effects of airspeed and angle of attack. The correction for the static defect is dependent on the Mach number and the angle of attack:

$$\Delta p = d_1 + d_2 M + d_3 \alpha,$$ (14.2)

 

where $\Delta$p is the pressure correction, M is the Mach number, and $\alpha$ is the angle of attack. Typical values for the coefficients are d₁=-3.33 mb, d₂=3.94 mb, and d₃=0.225mb/. Airspeed is generally measured with an inaccuracy of about 0.5 m/s, so the uncertainty in M is typically 0.0015. Similarly, $\alpha$ is determined with an inaccuracy of less than 0.1. These uncertainties lead to respective uncertainties in $\Delta$p of 0.006 mb and 0.02 mb, so uncertainties in M and $\alpha$ introduce negligible uncertainties in the results.
 
 

Summary  

For straight-and-level flight, the estimated uncertainty limits for the pressures p_W and p_F are about 1.1 and 2.1 mb, respectively, as listed in Table 1d. During rapid climbs or descents, the uncertainty in p_W increases to about 3.2 mb. The uncertainty in p_W is dominated by the biases that arise from temperature variations, while the largest contribution to uncertainty in p_F arises from uncertainty in the correction for the static defect. Improvements in both sensors might be made; e.g., the transducer for p_W could be maintained at constant temperature, and further study of the static defect (or perhaps use of a different sensing port exhibiting a smaller static defect) could reduce the uncertainty in p_F.

Despite the smaller uncertainty associated with p_W, p_F is usually used for calculations such as true airspeed because there is a substantial lateral offset between the sensors for p_W (on the wing tip) and those for wind gusts (on the nose of the aircraft). It has not been determined how different these measurements are, however, or if there are pressure variations large enough to necessitate use of p_F instead of p_W for this reason.

These estimates apply when the aircraft is in relatively steady flight, not in sharp turns or rapid climbs and descents. It is thought that these estimates should apply when the wind variance (at scales to which the aircraft does not respond, or typically <300 m in the vertical wind and <2000 m in the horizontal wind) is <1 m²s^-2.

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