4 The State of the Atmosphere

4.1 Information on Instruments and Calibrations

The instruments used to collect the measurements that lead to the variables in this section are described on the EOL web site, in the “State Parameters” section at this URL. The data acquisition and processing for these variables and the calibration coefficients used where applicable are described.

4.2 Variable Names

Measurements of some meteorological state variables like pressure, temperature, and water vapor pressure may originate from multiple sensors mounted at various locations on an aircraft. To distinguish among similar measurements, many variable names incorporate an indication of where the measurement was made. In this document, locations in variable names are represented by “x”, where “x” may be one of the following:

Character	location
B	bottom (or bottom-most)
B	(obsolete) boom
F	fuselage
G	(obsolete) gust probe
R	radome
T	top (or top-most)
W	wing

In addition, a true letter ’X’ (not replaced by the above letters) may be appended to a measurement to indicate that it is the preferred choice among similar measurements and is therefore used to calculate derived variables that depend on the measured quantity. Other suffixes sometimes used to distinguish among measurements are these: ’D’ for a digital sensor; ’H’ for a heated (usually, anti-iced) sensor, ’L’ for port-side sensors, and ’R’ for starboard-side sensors.

4.3 Pressure

4.3.1 Standard Pressure Measurements

Static or Ambient Pressure (hPa): PSx, PSxC, PS_A, PSX, PSXC, PSFD, PSFRD, PSTF

The atmospheric pressure at the flight level of the aircraft, measured by a calibrated absolute (barometric) transducer at location x. PSx is the measured static or ambient pressure before correction, and it may be affected by local flow-field distortion. PS_A is the pressure measurement taken from the avionics system on the aircraft, processed via unknown algorithms in the avionics system that may smooth, correct, and perhaps delay the result. PSxC is PSx corrected for local flow-field distortion (see RAF Bulletin #21 and the discussion in this memo and this supplement), and PSXC is the preferred corrected measurement used for derived calculations. These measurements have been made using various sensors, so it is best to consult the project documentation for the transducer used. Recent measurements from both the C-130 and the GV have been made using a Paroscientific Model 1000 Digiquartz Transducer.

Corrections to the pressures have been determined by reference to some standard, including a “trailing cone” sensor, the pressure PS_A from the cockpit avionics system, or (since 2012) the Laser Air Motion Sensing System (LAMS). The latter correction is discussed in the memo referenced above, where corrections used prior to 2011 are also discussed. Beginning in 2012, the deduced corrections Δp to the measured pressures as functions of dynamic pressure q, angle of attack α,¹² and the Mach number M are described by the following equations and coefficients:

For the C-130,¹³ \[\begin{equation} \frac{\Delta p}{p_{m}}=d_{0}+\frac{q_{m}}{p_{m}}(d_{1}+d_{4}\frac{\alpha^{2}}{a_{r}^{2}})+d_{2}\frac{\alpha}{a_{r}}+d_{3}\thinspace M \tag{4.1} \end{equation}\]

where, for p_m = PSFD , q_m = dynamic pressure (QCF), α = ATTACK and a_r = 1^∘ (included to keep the equation and coefficients dimensionless), and {d₀, d₁, d₂, d₃, d₄} = {-4.389e-03, -2.966e-02, -6.831e-05, 2.672e-02, 2.4466e-03}. For PSFRD, the corresponding coefficients are {0.007372, 0.12774, −6.8776e-4, −0.02994, 0.001630}. The latter coefficients are significantly different from the coefficients for PSFD, but the static ports where PSFRD is measured are at a different location on the fuselage so different flow-distortion effects are expected.

For the GV,¹⁴ \[\begin{equation} \frac{\Delta p}{p}=a_{0}+a_{1}\frac{q}{p}+a_{2}M^{3}+a_{3}\frac{\alpha}{a_{r}} \tag{4.2} \end{equation}\]
where, for p = PSF, q = QCF, α = ATTACK, and a_r = 1^∘ (included to keep the equation and coefficients dimensionless) {a₀, a₁, a₂, a₃} = { − 0.012255, 0.075372,  − 0.087508, 0.002148}}.

In equations (4.1) and (4.2) the Mach number is calculated from the uncorrected measurements of p and q, using dry-air values for R, c_v and c_p, via
\[\begin{equation} M=\left\{ \left(\frac{2c_{v}}{R}\right)\left[\left(\frac{p+q}{p}\right)^{R/c_{p}}-1\right]\right\}^{1/2}\,\,\,. \tag{4.3} \end{equation}\]

The pressure PSTF is measured at the static-pressure port of a pitot-static tube mounted on the top of the fuselage. The correction to this pressure (leading to PSTFC) differs from the pressure corrections listed above in that it is based on an empirical fit to other measurements of static pressure. The airflow at that sensor is distorted by the fuselage, so that measurement is usually less reliable than other measures of static pressure. See this note for additional discussion of this measurement.

For additional information on these correction coefficients, see this note and Cooper et al. (2014).¹⁵

Dynamic Pressure (hPa): QCx, QCxC, QCX, QCXC, QCTF, QCTFC, QC_A

The pressure excess caused by bringing the airflow to rest relative to the aircraft. These quantities represent the difference between the total pressure p_t as measured at the inlet of a pitot tube or other forward-pointing port and the ambient pressure that would be present in the absence of motion through the air. The variable QC_A is provided by the avionic data system and is subject to an unknown delay and smoothing. The variables ending in “C” have been corrected for flow-distortion effects, mostly arising from errors in the measurement of static pressure. Since 2012, the corrections are based on measurements from the LAMS system as described for PSxC, and they have the same functional form as in (4.1) and (4.2) except that the correction applied to q is  − Δp with reversed sign because q = p_t − p_a and the error arises primarily from the error in p_a. The same correction is applied to QCR because it is also measured relative to the static pressure ports so errors in the pressure sensed at those ports affect QCR in the same way that QCF is affected. See the notes referenced in the preceding section, and also RAF Bulletin 21 for the corrections applied to earlier data files.¹⁶

A Rosemount Model 1221 differential pressure transducer is used for current measurements of dynamic pressure on the C-130, and a Honeywell PPT transducer is used on the GV. This measurement enters the calculation of true airspeed and Mach number and so is needed to calculate many derived variables.

In the case of QCRC from the GV, one additional correction is applied (beginning 2017). The uncorrected measurement QCR is affected by flow angles, while QCF is not (for modest angle of attack or sideslip), so an additional adjustment is needed. The needed correction can be found by using an empirical relationship matching QCR to QCF, which leads to the following equation:
\[\begin{equation} \mathrm{\{QCR\}}=b_0+b_1\mathrm{\{QCR\}}+b_2\mathrm{\{AKRD\}}^2+b_3\mathrm{\{SSRD\}}^2-\Delta p \tag{4.4} \end{equation}\]

where \(\Delta p\) is given by (4.2) and the coefficients are {b_0 − 3}={ − 0.5635, 0.9982, 0.0273, 0.0562}. Some justification for this correction is contained in this note. A similar correction is not made for measurements from the C-130 radome because they do not appear to be necessary, as discussed in that note.

The measurement QCTF is made at the dynamic-pressure port of a pitot-static sensor mounted on the top of the fuselage. Because this is located in a region of airflow distortion around the fuselage of the GV, special processing is required. The method proposed in this note is to calculate a corrected dynamic pressure from QCTFC=QCTF+PSTF-PSXC. For high-rate files, PSXC is low-pass-filtered with a cut-off frequency of 0.5 Hz to eliminate high-frequency noise in this measurement.

D-Value (m): DVALUE

The difference between geopotential altitude and pressure altitude (m). This variable is calculated from {GEOPHT}−{PALT} and, for appropriate flight segments, can be used to measure height gradients on a constant-pressure surface. Prior to 2018, this was calculated from {GGALT} – {PALT}.

Special Pressure Measurements (hPa): PSDPx, CAVP, PCAB, PS_VXL, PSURF

PSDPx and CAVP_x are measurements of the pressure in the housing of the dew-point sensors, as discussed in connection with DPxC. PCAB is a measurement of the pressure in the cabin of the aircraft. PS_VXL is the pressure measured by the VCSEL hygrometer. PSURF is the estimated surface pressure calculated from HGME (a radar-altimeter measurement of height), TVIR, PSXC, and MR using the thickness equation as shown in the box below. TVIR and MR are described later in this section, and HGME was described in Section 3.3. The average temperature for the layer is obtained by using HGME and assuming a dry-adiabatic lapse rate from the flight level to the surface. Because of this assumption, the result is only valid for flight in a well-mixed surface layer or in other conditions in which the temperature lapse rate matches the dry-adiabatic lapse rate.¹⁷

PSXC = ambient pressure [hPa]
HGME = (radar) altitude above the surface [m]
TVIR = virtual temperature [\(^{\circ}\mathrm{C}\)]
PSURF = estimated surface pressure [hPa]
\(g\) = acceleration of gravity\(^{\dagger}\)
\(R_{d}\) = gas constant for dry air\(^{\dagger}\)
\(c_{pd}\) = specific heat of dry air at constant pressure\(^{\dagger}\)

\[\begin{equation} T_{m}=(\mathrm{\{TVIR\}}+T_{0})+0.5\mathrm{\{HGM\}}\frac{g}{c_{pd}} \tag{4.5} \end{equation}\] \[\begin{equation} \mathrm{PSURF}=\mathrm{\{PSXC\}}\,\exp\left\{ \frac{g\,\{\mathrm{HGME}\}}{R_{d}T_{m}}\right\} \tag{4.6} \end{equation}\]

4.4 Temperature

Recovery Temperature (ºC): RTx, RTHx, RTHRx, RTX

The recovery temperature is the temperature sensed by a temperature probe that is exposed to the atmosphere. In flight, the temperature is heated above the ambient temperature because it senses the temperature of air near the sensor that has been heated adiabatically during compression as it is brought near the airspeed of the aircraft. These variables are the measurements of that recovery temperature from calibrated temperature sensors at location x, for processing prior to about 2012; more recently, the names are simply RTF# or RTH# where # is a number starting with 1.¹⁸ For Rosemount or HARCO temperature probes in current use, the recovery temperature is near the total temperature, but all probes must be corrected to obtain either true total temperature or true ambient temperature. In the standard output, the variable name also conveys the sensor type: RTF# or RTx is a measurement from a Rosemount Model 102 non-deiced temperature sensor, RT#H or RTHx is the measurement from a Rosemount Model 102 anti-iced (heated) temperature sensor, and RTH# or RTHx is the measurement from a HARCO heated sensor. Some past experiments also used a reverse-flow temperature housing and a fast-response “K” housing; the associated variable names for these probes were TTRF and TTKP.¹⁹

Ambient Temperature (^∘C): ATx, ATX, ATHx, ATxD

The temperature of the atmosphere at the location of the aircraft, as it would be measured by a sensor at rest relative to the air. {#ATX} The ’x’ in the name of the variable used for ambient temperature, ATx, conveys the same information regarding sensor type and location as the variable name used with total (recovery) temperature. See the discussion above regarding RTx. The ambient temperature (also known as the static air temperature) is calculated from the measured recovery temperature, which is increased above the ambient temperature by dynamic heating caused by the airspeed of the aircraft. The calculated temperature therefore depends on the recovery temperature RTx as well as the dynamic and ambient pressure, usually respectively QCXC and PSXC. The ambient and dynamic pressures are first corrected from the raw measurements QCX and PSX to obtain variables that account for deviations caused by airflow around the aircraft and/or position-dependent systematic errors, as discussed in the section describing PSxC. The following basic equations are developed on the basis of conservation of energy for a perfect gas undergoing an adiabatic compression.

This section combines discussion of the calculations of temperature and airspeed, to reflect the linkage between these derived measurements. To provide accuracy in the equations, this discussion considers effects of the humidity of the air on characteristics like the gas constant and the specific heats. Most archived data before 2012 used values for dry air, although a special variable TASHC has been used to represent the true airspeed in cases where the correction was significant. That variable is based on a good approximation to the results from the following equations; see the discussion of TASHC later in this section. TASHC is now considered an obsolete variable. New variables ATxD and TASxD have been introduced that neglect the humidity corrections and perform all calculations as if the humidity is negligible.


As discussed above, temperature sensors on aircraft that are exposed to the airflow do not measure the total temperature but rather the temperature of the air immediately in contact with the sensing element. This air will not have undergone an adiabatic deceleration completely to zero velocity and hence will have a temperature T_r somewhat less than the total temperature T_t that would require the air to reach zero velocity. T_r is the measured or “recovery” temperature., The ratio of the actual temperature difference attained to the temperature difference relative to the total temperature is defined to be the “recovery factor” \(\alpha_r\): \[\begin{equation} \alpha_{r}=\frac{T_{r}-T_{a}}{T_{t}-T_{a}} \tag{4.7} \end{equation}\] where \(T_a\) is the ambient air temperature. From conservation of energy:
\[\begin{equation} \frac{U_{a}^{2}}{2}+c_{p}^{\prime}T_{a}=\frac{U_{r}^{2}}{2}+c_{p}^{\prime}T_{r}=\frac{U_{t}^{2}}{2}+c_{p}^{\prime}T_{t} \tag{4.8} \end{equation}\] where primes on quantities like c_p^′, or (below) c_v^′ and R^′ denote properties of moist air, respectively the specific heat at constant pressure, specific heat at constant volume, and gas constant.

Moist-Air Considerations

Primes on the symbols denote that these values should be moist-air values, appropriately weighted averages of the dry-air and water-vapor contributions. The practice prior to 2014 was to use the dry-air values for specific heats and the gas constant, except as described in connection with TASHC below. Since 2014, calculations use the appropriate values for moist air, except that to avoid errors introduced by unrealistically high measurements of humidity the humidity correction was limited to be less than or equal to the equilibrium value at the measured temperature. The formulas used for the specific heats and gas constant of moist air in terms of the water vapor pressure \(e\), the specific heats for dry air (\(c_{pd}=\frac{7}{2}R_{0},\,c_{vd}=\frac{5}{2}R_{0}\)) and water vapor (\(c_{pw}=4R_{0},\,c_{vw}=3R_{0}\)), and the ratio of molecular weights (\(\epsilon=M_{W}/M_{d}\)) are those of Khelif et al. 1999:
\[\begin{equation} R^{\prime}=R_{d}/[1+(\epsilon-1)\frac{e}{p}] \end{equation}\] \[\begin{equation} c_{v}^{\prime} = \frac{(p-e)R^{\prime}}{pR_{d}}\frac{5R_{0}}{2M_{d}}+\frac{eR^{\prime}}{pR_{w}}\frac{3R_{0}}{M_{w}}=c_{vd}\frac{R^{\prime}}{R_{d}}\left(1+\frac{1}{5}\frac{e}{p}\right) \tag{4.9} \end{equation}\] \[\begin{equation} c_{p}^{\prime} = c_{pd}\frac{R^{\prime}}{R_{d}}\left(1+\frac{1}{7}\frac{e}{p}\right) \tag{4.10} \end{equation}\] \[\begin{equation} \gamma\,^{\prime} = \gamma_{d}\frac{1+\frac{1}{7}\frac{e}{p}}{1+\frac{1}{5}\frac{e}{p}} \tag{4.11} \end{equation}\]
See also the discussion of TASHC below and the reference there for Khelif et al. 1999.

In (4.8) {U_a, U_r, U_t} are respectively the aircraft true airspeed, the airspeed relative to the aircraft of the air in thermal contact with the sensor, and the airspeed of air relative to the aircraft when fully brought to the motion of the sensor (i.e., zero). Then, from (4.8) \[\begin{equation} T_{a}=T_{r}-\alpha_{r}\frac{U_{a}^{2}}{2c_{p}^{\prime}} \tag{4.12} \end{equation}\]

The temperature sensors used on RAF aircraft are designed to decelerate the air adiabatically to near zero velocity. Recovery factors determined from wind tunnel testing for the Rosemount sensors are approximately 0.97 (unheated model) and 0.98 (heated models).²⁰ These values have also been confirmed from flight maneuvers, often from “speed runs” where the aircraft is flown level through its speed range and the variation of recovery temperature with airspeed is used with (4.12), with the assumption that T_a remains constant, to determine the recovery factor. Data files and project reports normally document what recovery factor was used for calculating the true airspeed and ambient temperature for a particular project.

Because the values used in processing have varied, the project reports should be consulted to find what was used for particular projects. The Goodrich Technical Report 5755 documents wind-tunnel testing of the probes formerly made by Rosemount. Their plot showed that, for heated sensors, there is a significant variation with Mach number (M); cf their Eq. (38). The dependence in their plot is represented well by the following equations, where \(\alpha_r^{[h]}\) refers to heated probes and \(\alpha_r^{[u]}\) to unheated probes:
\[\begin{equation} \alpha_r^{[h]} = 0.988+0.053(\log_{10}M)+0.090(\log_{10}M)^2+0.091(\log_{10}M)^3 \tag{4.13} \end{equation}\] \[\begin{equation} \alpha_r^{[u]}=0.9959+0.0283(\log_{10}M)+0.0374(\log_{10}M)^2+0.0762(\log_{10}M)^3 \tag{4.14} \end{equation}\]

Some studies of the recovery factor are discussed further in this memo and this technical report.

The true airspeed \(U_a\) is used in (4.12) to calculate the ambient temperature \(T_a\). However, the ambient temperature is also needed to calculate the true airspeed. Therefore the constraints imposed on ambient temperature and true airspeed by the measurements of recovery temperature, total pressure (the pressure measured by a pitot tube pointed into the airstream and assumed to be that obtained when the incoming air is brought to rest relative to the aircraft), and ambient pressure must be used to solve simultaneously for the two unknowns, temperature and airspeed.

The relationship is conveniently derived by first calculating the dimensionless Mach number \(M\), which is the ratio of the airspeed to the speed of sound \(U_{s}=\sqrt{\gamma^{\prime}R^{\prime}T_{a}}\), where \(\gamma^\prime\) is the ratio of specific heats of (moist) air, \(c_p^\prime /c_v^\prime\) and \(R^\prime\) is the gas constant for moist air. The Mach number is a function of air temperature only and can be determined as follows:

a). Express energy conservation, as in (4.12), in the form
\[\begin{equation} d\left(\frac{U^{2}}{2}\right)+c_{p}^{\prime}dT=0\,\,\,\,. \tag{4.15} \end{equation}\]
where the total derivatives apply along a streamline as \(U\) changes from \(U_a\) to \(U_t=0\) and \(T\) changes from \(T_a\) to \(T_t\).
b). Use the perfect gas law to replace \(dT\) with \(\frac{pV}{nR}(\frac{dV}{V}+\frac{dp}{p})\) where \(V\) and \(p\) are the volume and pressure of a parcel of air. Then use the expression for adiabatic compression in the form \(pV^\gamma = \mathrm{constant}\) to replace the derivative \(\frac{dV}{V}\) with \(-\frac{1}{\gamma}\frac{dp}{p}\), leading to \(dT=\frac{R^{\prime}T}{c_{p}^{\prime}}\frac{dp}{p}\) or, after integration, \(T(p)=T_{a}\left(\frac{p}{p_{a}}\right)^{R^{\prime}/c_{p}^{\prime}}\). Using this expression for \(T\) in the formula for \(dT\) and then integrating both total derivatives in (4.15) along the streamline leads to
\[\begin{equation} \frac{U_{a}^{2}}{2}+c_{p}^{\prime}T_{a}=c_{p}^{\prime}T_{a}\left(\frac{p_{t}}{p_{a}}\right)^{\frac{R^{\prime}}{c_{p}^{\prime}}} \tag{4.16} \end{equation}\]
where \(p_t\) is the total pressure (i.e., PSXC+QCXC) and \(p_a\) is the ambient pressure (PSXC).
c). Use the above definition of the Mach number \(M\) (\(M=U_a/U_s\)) in the form \(U_a^2=\gamma^\prime M^2 R^\prime T_a\) to obtain:
\[\begin{equation} M^{2}=\left(\frac{2c_{v}^{\prime}}{R^{\prime}}\right)\left[\left(\frac{p_{t}}{p_{a}}\right)^{\frac{R^{\prime}}{c_{p}^{\prime}}}-1\right] \tag{4.17} \end{equation}\]
which is the same as (4.3). This equation shows that \(M\) can be found from measurements of \(p_t\) and \(p_a\) alone, except for the moist-air corrections.

d). Use the expression for ambient temperature in terms of recovery temperature and airspeed, (4.12), to obtain the temperature in terms of the Mach number and the recovery temperature:
\[\begin{align} T_{a} & = T_{r}-\alpha_{r}\frac{U_{a}^{2}}{2c_{p}^{\prime}}=T_{r}-\alpha_{r}\frac{M^{2}\gamma^{\prime}R^{\prime}T_{a}}{2c_{p}^{\prime}}\notag\\ & = \frac{T_{r}}{1+\dfrac{\alpha_{r}M^{2}R^{\prime}}{2c_{v}^{\prime}}} \tag{4.18} \end{align}\]


e). Express the true airspeed \(U_a\) as
\[\begin{equation} U_{a}=M\sqrt{\gamma\,^{\prime}R^{\prime}T_{a}} \tag{4.19} \end{equation}\]

Then the temperature is found as described in the following box:²¹

RTX = recovery temperature (\(T_{r})\)
QCxC = dynamic pressure, corrected (\(q_{a}\))
PSXC = ambient pressure, after airflow/location correction (\(p_{a}\))
MACHx = Mach number based on QCxC and PSXC; cf. (4.17)
MACHX = best Mach number, based on QCXC and PSXC
\(\alpha_{r}\) = recovery factor for the particular temperature sensor
\(R^{\prime}\), \(c_{v}^{\prime}\) and \(c_{p}^{\prime}\) as defined above and in the list of symbols

From (4.17), \[\begin{equation} \mathrm{MACHx}=\left\{ \left(\frac{2c_{v}^{\prime}}{R^{\prime}}\right)\left[\left(\frac{\mathrm{\{PSXC\}+\{QCxC\}}}{\mathrm{\{PSXC\}}}\right)^{\frac{R^{\prime}}{c_{p}^{\prime}}}-1\right]\right\} ^{1/2} \tag{4.20} \end{equation}\]
From (4.18), \[\begin{equation} \mathrm{ATx}=\frac{\mathrm{\left(\{RTx\}+T_{0}\right)}}{\left(1+\dfrac{\alpha_{r}\mathrm{(\{MACHX\})}^{2}R^{\prime}}{2c_{v}^{\prime}}\right)}-T_{0} \tag{4.21} \end{equation}\]

In-cloud Air Temperature, Radiometric (^∘C): AT_ITR

The radiometric ambient air temperature measured by the In-cloud Air Temperature Radiometer, which measures the radiometric temperature in the 4.3 μm CO₂ band.[AT_ITR] Its primary use is in water cloud when the standard thermometers are affected by wetting. In clear air the temperature is an average over an integrating range of up to 100s of meters away from the aircraft, whereas in clouds the integrating range is as little as 10 meters due to the IR opacity of water droplets. The calibration is by a polynomial fit of the internal reference temperature and measured radiance to the ATX temperature outside of clouds.>

4.5 Humidity

Dewpoint/Frostpoint (ºC): DP_DPx

The dew point or frost point is measured by an EG&G Model 137, a General Eastern Model 1011B or a Buck Model 1011C dew-point hygrometer. Below 0^∘C the instrument is assumed to be responding to the frost point, although occasionally in climbs there is a short transition near the freezing level before the condensate on the mirror of the instrument freezes and there may be a measurement error before the condensate freezes. The measurements are corrected for the housing pressure, the enhancement of the equilibrium vapor pressure arising from the total pressure (discussed below), and conversion from frost point if appropriate. The result is the temperature at which the equilibrium vapor pressure over a plane water surface in the absence of other gases would match the actual water-vapor pressure. Dew/frost-point hygrometers measure the equilibrium point in the presence of air, and the presence of air affects the measurement in a minor way that is represented by a small correction here named the “enhancement factor.” In the case where the dew-point or frost-point sensor is exposed to ambient air directly, the enhancement factor is defined so that the ambient water vapor pressure \(e_a\) is related to \(T_p\), the measured dew or frost point in the presence of air having total pressure \(p\), by \(e_a=f(p,T_P)e_s(T_p)\) where \(e_s(T_p)\) is the vapor pressure in equilibrium with ice or water at the dew or frost point \(T_p\) in the absence of air. Calculation of DP_D removes this dependence, so the vapor pressure obtained from \(e_s(DP\_DPx)\) will be that vapor pressure corresponding to equilibrium in the absence of air. In addition, if the measurement is below \(0^\circ\mathrm{C}\), it is assumed to be a measurement of frost point and a corresponding dew point is calculated from the measurement (also with correction for the influence of the total pressure on the measurement). Some changes were made to these calculations in 2011; for more information, see this memo.

An additional correction is needed in those cases where the pressure in the housing of the instrument (measured as PSDPx or CAVP_x) differs from the ambient pressure, because the changed pressure affects the partial pressure of water vapor in proportion to the change in total pressure and so changes the measured dew point from the desired quantity (that in the ambient air) to that in the housing. This is especially important in the case of the GV because the potential effect increases with airspeed. If the pressure in the housing is measured or otherwise known (e.g., from correlations with other measurements), then this correction can be introduced into the processing algorithm at the same time that the correction for the presence of dry air is introduced, and the enhancement factor should be evaluated at the pressure in the housing.

The relationship between water-vapor pressure and dew- or frost-point temperature is based on the Murphy and Koop²² (2005) equations.²³ They express the equilibrium vapor pressure as a function of frost point or dew point and at a total air pressure \(p\) via equations that are equivalent to the following:
\[\begin{equation} e_{s,i}(M_{FP})= b_{0}^{\prime}\exp(b_{1}\frac{(T_{0}-M_{FP})}{T_{0}M_{FP}}+b_{2}\ln(\frac{M_{FP}}{T_{0}})+b_{3}(M_{FP}-T_{0})) \tag{4.22} \end{equation}\] \[\begin{equation} e_{s,w}(M_{DP})=c_{0}\exp\left((\alpha-1)c_{6}+d_{2}(\frac{T_{0}-M_{DP}}{M_{DP}T_{0}})\right)+d_{3}\ln(\frac{M_{DP}}{T_{0}})+d_{4}(M_{DP}-T_{0}) \tag{4.23} \end{equation}\] \[\begin{equation} f(p, T_P)= 1 + p(f_1 + f_2T_P + f_3T_P^2) \tag{4.24} \end{equation}\] where \(e\) is the water vapor pressure, \(M_{FP}\) or \(M_{DP}\) is the mirror temperature at the frost or dew point, respectively, expressed in kelvin, \(T_0\) = 273.15 K, \(e_{s,i}(M_{FP})\) is the equilibrium vapor pressure over a plane ice surface at the temperature \(M_{FP}\), \(e_{s,w}(M_{DP})\) is the equilibrium vapor pressure over a plane water surface at the temperature \(M_{DP}\) (above or below \(T_0\)), and \(f(p,T_P)\) is the enhancement factor at total air pressure \(p\) and temperature \(T_P\), with \(T_P\) equal to \(M_{DP}−T_0\) when above \(T_0\) and \(M_{FP}−T_0\) when below \(0^\circ\)C.

The coefficients used in the above formulas are given in the following tables, with the additional definitions that \(\alpha_T=\tanh(c_5(T-T_x))\), \(T_X=218.8\) K, and \(d_i=c_i+\alpha_Tc_{i+5}\) for i = {2,3,4}:

coefficient value \(b_0^\prime\) \(6.11536\) hPa \(b_1\) \(-5723.265\) K \(b_2\) \(3.53068\) \(b_3\) \(-0.00728332\) K\(^{-1}\) \(f_1\) \(4.923\times 10^{-5}\) hPa\(^{-1}\) \(f_2\) \(-3.25\times 10^{-7}\)hPa\(^{-1}\)K\(^{-1}\) \(f_3\) \(5.84\times 10^{-10}\)hPa\(^{-1}\)K\(^{-2}\)

coefficient value \(c_0\) \(6.091886\) hPa \(c_1\) \(6.564725\) \(c_2\) \(-6763.22\) K \(c_3\) \(-4.210\) \(c_4\) \(0.000367\) K\(^{-1}\) \(c_5\) \(0.0415\) K\(^{-1}\) \(c_6\) \(-0.1525967\) \(c_7\) \(-1331.22\) K \(c_8\) \(-9.44523\) \(c_9\) \(0.014025\) K\(^{-1}\)

The vapor pressure in the instrument housing, \(e_h\), is related to the sensed dew or frost point according to equation (4.23) or (4.22), but further corrections must also be made for the enhancement factor and to account for possible difference between the pressure in the sensor housing (\(p_h\)) and the ambient pressure (\(p_a\)):
\[\begin{equation} e_{a}=f(p_{a},T_{p})e_{h}\frac{p_{a}}{p_{h}} \tag{4.25} \end{equation}\] Because processing to obtain the corrected dew point DP_DPx from the ambient vapor pressure \(e_a\) would require difficult inversion of the above formulas, interpolation is used instead. A table constructed from (4.23) and another constructed from (4.22), giving water vapor pressure as a function of frost point or dew point temperature in \(1^\circ\mathrm{C}\) increments from \(-100\) to \(+50^\circ\mathrm{C}\), is then used with three-point Lagrange interpolation (via a function described below as \(F_D(e)\)) to find the dew point temperature from the vapor pressure.²⁴

Tests of these interpolation formulas against high-accuracy numerical inversion of formulas (4.23) and (4.22) showed that the maximum error introduced by the interpolation formula was about \(0.004^\circ\mathrm{C}\) and the standard error about \(0.001^\circ\mathrm{C}\). This inversion then provides a corrected dew point that incorporates the effects of the enhancement factor as well as differences between the ambient pressure and that in the housing. The algorithm is documented in the box below.

\(M_{p}\) = MP_DPx from instrument x [\(^{\circ}\)C], or alternately
RHO = water vapor density measurement [\(\mathrm{g\ }\mathrm{m}^{-3}\)]; only one is used in any calculation
ATX = reference ambient temperature [\(^{\circ}C\)]
\(T_{K}\)=ATX+\(T_{0}\) \(^{\dagger}\) = ambient temperature [K]
\(p\) = PSXC = reference ambient pressure [hPa]
\(p_{h}\) = CAVP_x = pressure in instrument “x” housing [hPa]
\(e_{t}\) = intermediate vapor pressure used for calculation only
\(e\) = EWx = water vapor pressure from source x [hPa]
\(M_{w}\) = molecular weight of water\(^{\dagger}\)
\(R_{0}\) = universal gas constant\(^{\dagger}\)
\(f(p_{h},T_{p})\) = enhancement factor (cf. (4.24))
\(F_{d}(e)\) = interpolation formula giving dew point temperature from water vapor pressure

For dew/frost point hygrometers, producing the measurement DPx:
    if MIRRTMP_DPx < 0\(^\circ\)C:
        obtain \(e_{t}\) from (4.22) using \(T_{FP}\)=MIRRTMP_DPx + \(T_{0}\)
    else (i.e., MIRRTMP_DPx \(\geq 0^\circ\)C):
        obtain \(e_{t}\) from (4.23) using \(T_{DP}=\mathrm{MIRRTMP\_DPx}+T_{0}\)
    correct \(e_{t}\) for enhancement factor and internal pressure to get ambient vapor pressure \(e\):
\[\begin{equation} e=f(p_{h},T_{P})\,(\frac{p}{p_{h}})\,e_t \tag{4.26} \end{equation}\]     obtain DP_DPx by finding the dew point corresponding to the vapor pressure \(e\):
\[\begin{equation} \mathrm{\{DP_DPx\}} = F_{d}(e) \tag{4.27} \end{equation}\] - - - - - - - - - - - - - - - - - - - -
For other instruments producing measurements of vapor density (RHO [g m\(^{-3}\)]:^(a)
    find the water vapor pressure in units of hPa: \[\begin{equation} e = (\mathrm{\{RHO\}}\,R_{0}\,T_{K}\,/\,M_{w})\times 10^{-5} \tag{4.28} \end{equation}\]     find the equivalent dew point:
\[\begin{equation} \mathrm{\{DP_XXXC\}} = F_{d}(e) \tag{4.29} \end{equation}\] __________
^(a) prior to 2011 the following formula was used: \[Z=\frac{\ln((\mathrm{\{ATX\}}+273.15)\,\mathrm{\{RHO\}}}{1322.3}\]
\[\mathrm{\{DP\_XXX\}}=\frac{273.0\,Z}{(22.51-Z)}\]

For other instruments that measure vapor density, such as a Lyman-alpha or tunable diode laser hygrometers (including the Vertical Cavity Surface Emitting Laser (VCSEL) hygrometer), a similar conversion is made from vapor density to dew point, as described below.

Corrected Dew Point (^∘C): DPXC

The reference value for dew point and frost point. This variable is chosen to equal one of the DP_XXX measurements based on the judgement of the Project Managers as to which DP_XXX is most reliable.

Dew/Frost Point (^∘C): DPx, DP_x

The dew point or frost point measured directly by the chilled mirror dewpoint hygrometer without correction for possible pressure differences between the ambient air and the air pressure over the mirror.

Mirror Temperature (^∘C): MIRRTMP_DPx

The temperature of the chilled mirror reported by the chilled mirror hygrometer.

Dew Point Determined from the VCSEL Hygrometer (^∘C): DP_VXL

The dew point temperature determined from the measured water vapor density from the VCSEL hygrometer. The calculation is as described at the bottom of the box immediately above this paragraph (above the footnote). The water vapor density converted from a molecular density [molecules\(\,\)cm\(^{-3}\)] to a mass density [g\(\,\)m\(^{-3}\)] via²⁵ {CONCV_VXL} * \(2.9915\times 10^{17}\) is used for {RHO}. DP_VXL is given by DPxC on the last line of that algorithm box. See CONCV_VXL below.

Frost Point Temperature from the CR2 Cryogenic Hygrometer (^∘C): FP_CR2, MIRRORT_CR2

The mirror temperature in the CR2 cryogenic hygrometer, which is normally the frost point inside the measuring chamber of the instrument. The measurement is often suspect when the value is above about -15^∘C; the measurement is intended for use below this value. The CR2 is a cabin-mounted instrument, so the measured pressure (P_CR2) in the instrument must be used with the ambient pressure (PSXC) to convert the measurement to ambient humidity measures like DP_CR2 and EW_CR2.

Corrected Dew Point Temperature from the CR2 Cryogenic Hygrometer (^∘C): DP_CR2C

The dew point temperature corresponding to equilibrium at the ambient humidity, as determined by the CR2 hygrometer. The measurement of the mirror temperature inside the CR2, FP_CR2, is converted to a vapor pressure assuming equilibrium water vapor pressure relative to a plane ice surface at that temperature, and the resulting vapor pressure is converted to an ambient value via the assumption that the ratio of vapor pressure internal to the instrument to ambient vapor pressure is the same as the corresponding total pressure ratio. The resulting ambient vapor pressure (EW_CR2) is then converted to an equivalent ambient dew point. The steps are the same as those in the algorithm box for DPxC above, with these substitutions: FP_CR2 is used for DPx and P_CR2 for \(p_h\).

Uncorrected Water Vapor Number Density from the VCSEL Hygrometer (molecules cm^− 3): RAWCONC_VXL

The uncorrected water vapor number density reported by the VCSEL hygrometer. This is determined by comparing the measured absorption peak height against a reference spectrum generated using the HITRAN spectral parameters, the ambient temperature and the ambient pressure.²⁶

Corrected Water Vapor Concentration from the VCSEL Hygrometer (molecules cm^− 3): CONCV_VXL

The corrected water vapor number density produced by the VCSEL hygrometer, after minor corrections for ambient temperature, pressure, laser intensity and water vapor concentration. For more information on calibration and data processing for this instrument, see the instrument web page and additional documentation there.

Voltage Output from the UV Hygrometer (V): XSIGV_UVH

The voltage from a modern (as of 2012) version of the Lyman-alpha hygrometer, which provides a signal that represents water vapor density. The instrument also provides measurements of pressure and temperature inside the sensing cavity; they are, respectively, XCELLPRES_UVH and XCELLTEMP_UVH. See the discussion of EW_UVH below for the data-processing algorithm that uses this variable.

Water Vapor Number Density from the UV Hygrometer (molecules cm^− 3): CONCH_UVH

Water vapor number density (or concentration of molecules) measured by the UV Hygrometer. This is the direct measurement from the instrument. Its calculation relies on a bench calibration that fits the water vapor number density to the Beers-Lambert absorption law and corrects for output offsets and the effect of UV absorption by atmospheric constituents other than water vapor. See also the discussion of EW_UVH in the paragraph that immediately follows.

Water Vapor Pressure (hPa): EWx, EWX, EW_VXL, EW_UVH, EDPC (obsolete)

The ambient vapor pressure of water, also used in the calculation of several derived variables. It is often obtained from an instrument measuring dew point or water vapor density. In the case where it is derived from a measurement of dew point (DPx), a correction is applied for the enhancement factor that influences dew point or frost point measurements.²⁷ The formula for obtaining the ambient water vapor pressure as a function of dew point is given in the discussion of DPxC above, Eqs. (4.23) and (4.24), where the calculation of the variables EWx and EWX are also discussed. EWX (or previously EDPC) is the preferred variable that is selected from among the possibilities {EWx} for subsequent calculation of derived variables.

For the case where water vapor pressure is determined by the VCSEL hygrometer, EW_VXL is determined from CONCV_VXL: EW_VXL=Ck{CONCV_VXL}{ATX+273.15) where k is the Boltzmann constant and C = 10^− 4(cm/m)³(hPa/Pa) converts units to hPa.

In the case where the water vapor pressure is determined from the UV Hygrometer data, this variable is calculated using one of two methods:

1. Using the ideal gas law to convert the water vapor number density from the UV Hygrometer to water vapor pressure, using XCELLTEMP_UVH and XCELLPRES_UVH, the measured temperature and pressure in the absorption cell, via the equation:
   \[\begin{equation} \mathrm{\{EW\_UVH\}=C\,\{CONC\_UVH\}\notag \\ \times\frac{k\,(\mathrm{\{XCELLTEMP\_UVH\}+273.15)\,\mathrm{\{PSX\}}}}{\mathrm{{\{XCELLPRES\_UVH\}}}}} \tag{4.30} \end{equation}\]
   or
   
2. Through use of a polynomial fit with coefficients fitted to {EWX}:
   \[\begin{equation} \mathrm{\{EWX\}}=c_0 + c_1(\mathrm{\{XSIGV\_UVH\}}) + c_2(\mathrm{\{XSIGV\_UVH\}}^2) \tag{4.31} \end{equation}\]
   where {EWX} is a reference water vapor pressure provided by another instrument. This preserves the fast-response characteristics of the UV hygrometer while linking the absolute values to a baseline provided by a more stable instrument. This can be done on a flight-by-flight basis and largely eliminates drift.²⁸ See the project reports to determine which method was used for a particular project.

Relative Humidity (per cent or Pa/hPa): RHUM

The ratio of the water vapor pressure to the water vapor pressure in equilibrium over a plane liquid-water surface, scaled to express the result in units of per cent or Pa/hPa:

EWX = atmospheric water vapor pressure (hPa)
ATX = ambient air temperature [\(^{\circ}\mathrm{C}\)]
\(T_{0}=273.15\) K
\(e_{s.w}(\mathrm{ATX+T_{0}})\) = equilibrium water vapor pressure at dewpoint ATX (hPa)
          (see Eq. (4.23) for the formula used.)

\[\begin{equation} \mathrm{\{RHUM\}}=100\%\,\times\,\frac{\mathrm{\{EWX\}}}{e_{s,w}(\mathrm{\{ATX\}+T_{0}})} \tag{4.32} \end{equation}\]

To follow normal conventions, the change in equilibrium vapor pressure that arises from the enhancement factor is not included in the calculated relative humidity, even though the true relative humidity should include the enhancement factor as specified in (4.24) in the denominator of (4.32).

Relative Humidity with respect to Ice (per cent or Pa/hPa): RHUMI

The ratio of the water vapor pressure to the water vapor pressure in equilibrium over a plane ice surface, scaled to express the result in units of per cent or Pa/hPa:

EWX = atmospheric water vapor pressure (hPa)
ATX = ambient air temperature [\(^{\circ}\mathrm{C}\)]
\(T_{0}=273.15\) K
\(e_{s.i}(\mathrm{ATX+T_{0}})\) = equilibrium water vapor pressure at frostpoint ATX (hPa)
          (see Eq. (4.22) for the formula used.)

\[\begin{equation} \mathrm{\{RHUMI\}}=100\%\,\times\,\frac{\mathrm{\{EWX\}}}{e_{s,i}(\mathrm{\{ATX\}+T_{0}})} \tag{4.33} \end{equation}\]

To follow normal conventions, the change in equilibrium vapor pressure that arises from the enhancement factor is not included in the calculated relative humidity, even though the true relative humidity should include the enhancement factor as specified in (4.24) in the denominator of (4.33).

Absolute Humidity, Water Vapor Density (g/m³):RHOx

The water vapor density computed from various measurements of humidity as indicated by the ’x’ suffix, and conventionally expressed in units of g kg^− 1 or per mille. The calculation proceeds in different ways for different sensors. For sensors that measure a chilled-mirror temperature, the calculation is based on the equation of state for a perfect gas and uses the water vapor pressure determined by the instrument, as in the following box:

ATX = ambient temperature (\(^{\circ}\mathrm{C}\))
EWX = water vapor pressure, hPa
\(C_{mb2Pa}\)= conversion factor, hPa to Pa} = 100 Pa hPa\(^{-1}\) (conversion factor to MKS units)
\(C_{kg2g}=10^{3}\,\mathrm{g\,kg}^{-1}\) = (conversion factor to give final units of g\(\,\)m\(^{-3}\))
\(T_{0}\) = 273.15,K

\[\begin{equation} \mathrm{\{RHOx\}} = C_{kg2g}\frac{C_{mb2Pa}\mathrm{\{EWX\}}}{R_{w}\mathrm{(\{ATX\}+T_{0})}} \tag{4.34} \end{equation}\]

For instruments measuring the vapor pressure density (including the Lyman-alpha probes and the newer version called the UV hygrometer), the basic measurement from the instrument is the water vapor density, RHOUV or **** RHOLA, determined by applying calibration coefficients to the measured signals (XUVI or VLA). In addition, a slow update to a dew-point measurement is used to compensate for drift in the calibration. The processing used for early projects with the Lyman-alpha instruments is similar but more involved and won’t be documented here because the instruments are obsolete. See RAF Bulletin 9 for the processing previously used for archived measurements from the Lyman-alpha hygrometers.

Specific Humidity (g/kg): SPHUM

The mass of water vapor per unit mass of (moist) air, conventionally measured in units of g/kg or per mille.

PSXC = ambient total air pressure. hPa
EWX = ambient water vapor pressure, hPa
\(C_{kg2g}=10^{3}\,\)g\(\,\)kg\(^{-1}\) (conversion factor to give final units of g\(\,\)kg\(^{-1}\))
\(M_{w}=\)molecular weight of water\(^{\dagger}\)
\(M_{d}=\)molecular weight of dry air\(^{\dagger}\)

\[\begin{equation} \mathrm{\{SPHUM\}} = C_{kg2g}\frac{M_{w}}{M_{d}}(\mathrm{\frac{\{EWX\}}{\mathrm{\{PSXC\}-(1-\frac{M_{w}}{M_{d}})\{\mathrm{EWX}\}}}}) \tag{4.35} \end{equation}\]

Mixing Ratio (g/kg): MR, MRCR, MRLA, MRLA1, MRLH

The ratio of the mass of water to the mass of dry air in the same volume of air, conventionally expressed in units of g/kg or per mille. Mixing ratios may be calculated for the various instruments measuring humidity on the aircraft, and the variable names reflect the source: MR from the dewpoint hygrometers, MRCR from the cryogenic hygrometer, MRLA from the Lyman-alpha sensor, MRLA1 if there is a second Lyman-alpha sensor, MRLH from a tunable-diode laser hygrometer, and MRVXL from the VCSEL hygrometer (also a laser hygrometer). The example in the box below is for the case of the dewpoint hygrometers; others are analogous.

PSXC = ambient total air pressure. hPa
EWX = ambient water vapor pressure, hPa
\(C_{kg2g}=10^{3}\,\)g\(\,\)kg\(^{-1}\) (conversion factor to give final units of g\(\,\)kg\(^{-1}\))
\(M_{w}=\)molecular weight of water\(^{\dagger}\)
\(M_{d}=\)molecular weight of dry air\(^{\dagger}\)

\[\begin{equation} \mathrm{\{MR\}}=C_{kg2g}\frac{M_{w}}{M_{d}}\frac{\mathrm{\{EWX\}}}{(\mathrm{\{PSXC\}-\{EWX\})}} \tag{4.36} \end{equation}\]

4.6 Derived Thermodynamic Variables

Potential Temperature (K): THETA

The absolute temperature reached if a dry parcel at the measured pressure and temperature were to be compressed or expanded adiabatically to a pressure of 1000 hPa. It does not take into account the difference in specific heats caused by the presence of water vapor, and water vapor can change the exponent in the formula below enough to produce errors of 1 K or more.

ATX = ambient temperature, \(^{\circ}\)C
PSXC = ambient pressure (hPa)
\(p_{0}\) = reference pressure = 1000 hPa
\(R_{d}\) = gas constant for dry air\(^{\dagger}\)
\(c_{pd}\) = specific heat at constant pressure for dry air\(^{\dagger}\) \(T_0=273.15\,\mathrm{K}\)

\[\begin{equation} \mathrm{\{THETA\}}=\left(\mathrm{\{ATX\}}+T_{0}\right)\left(\frac{p_{0}}{\mathrm{\{PSXC\}}}\right)^{R_{d}/c_{pd}} \tag{4.37} \end{equation}\]

Pseudo-Adiabatic Equivalent Potential Temperature (K): THETAP, THETAE

The absolute temperature reached if a parcel of air were to be expanded pseudo-adiabatically (i.e., with immediate removal of all condensate) to a level where no water vapor remains, after which the dry parcel would be compressed to 1000 hPa. Beginning in 2011, pseudo-adiabatic equivalent potential temperature is calculated using the method developed by Davies-Jones (2009).²⁹ This is discussed in the memo available at this link. The following summarizes that study. The Davies-Jones formula is:
\[\begin{equation} \Theta_{P}=\Theta_{DL}\exp\{\frac{r(L_{0}^{*}-L_{1}^{*}(T_{L}-T_{0})+K_{2}r)}{c_{pd}T_{L}}\} \tag{4.38} \end{equation}\]
\[\begin{equation} \Theta_{DL}=T_{K}(\frac{p_{0}}{p_{d}})^{0.2854}\,(\frac{T_{k}}{T_{L}})^{0.28\times10^{-3}r} \tag{4.39} \end{equation}\]
where \(T_K\) is the absolute temperature (in kelvin) at the measurement level, \(p_d\) is the partial pressure of dry air at that level, \(p_0\) is the reference pressure (conventionally 1000 hPa), \(r\) is the (dimensionless) water vapor mixing ratio, \(c_p\) the specific heat of dry air, \(T_L\) the temperature at the lifted condensation level (in kelvin), and \(T_0=273.15\,\mathrm{K}\). The coefficients in this formula are \(L_0^* = 2.56313\times 10^6\mathrm{J\,kg^{-1}}\), \(L_1^* = 1754\,\mathrm{J\,kg^{-1}K^{-1}}\), and \(K_2 = 1.137\times 10^6\mathrm{J\,kg^{-1}}\). The asterisks on \(L_0^*\) and \(L_1^*\) indicate that these coefficients depart from the best estimate of the coefficients that give the latent heat of vaporization of water, but they have been adjusted to optimize the fit to values obtained by exact integration. Note that, unlike the formula discussed below that was used prior to 2011, the mixing ratio must be used in dimensionless form (i.e., kg/kg), not with units of g/kg. The following empirical formula, developed by Bolton (1980),³⁰ is used to calculate \(T_L\):
\[\begin{equation} T_{L}=\frac{\beta_{1}}{3.5\ln(T_{K}/\beta_{3})-\ln(\mathrm{e/\beta_{4}})+\beta_{5}}+\beta_{2} \tag{4.40} \end{equation}\]
where \(e\) is the water vapor pressure, \(\beta_1 = 2840\,\mathrm{K}\), \(\beta_2 = 55\,\mathrm{K}\), \(\beta_3 = 1\,\mathrm{K}\), \(\beta_4 = 1\,\mathrm{hPa}\), and \(\beta_5 = -4.805\). (\(\beta_3\) and \(\beta_4\) have been introduced into (4.40) only to ensure that arguments to logarithms are dimensionless and to specify the units that must be used to achieve that.)

\(T_K\) = ATX + \(T_0\) = ambient temperature [K]
\(e\) = EWX = water vapor pressure
\(p_d\) = PSXC - EWX = partial pressure of dry air [hPa]
\(p_{0}\) = reference pressure = 1000 hPa
\(r\) = MR = water vapor mixing ratio
\(R_{d}\) = gas constant for dry air\(^{\dagger}\)
\(c_{pd}\) = specific heat at constant pressure for dry air\(^{\dagger}\)
\(T_L\) = temperature at the lifted condensation level (LCL) [K]
\(L_0^*+L_1^*(T_L-T_0)\) = latent heat of vaporization at the LCL
\(L_0=2.56313 × 10^6\) J\(\,\)kg\(^{-1}\), \(L_1=1754\) J\(\,\)kg\(^{-1}\)K\(^{-1}\)
\(K_2=1.137 × 10^6\) J\(\,\)kg\(^{-1}\)
\(\beta_{1-5}\) = {2840 K, 55 K, 1 K, 1 hPa, −4.805}

\[\begin{equation} T_{L}=\frac{\beta_{1}}{3.5\ln(T_{K}/\beta_{3})-\ln(\mathrm{e/\beta_{4}})+\beta_{5}}+\beta_{2} \tag{4.41} \end{equation}\] \[\begin{equation} \Theta_{DL}=T_{K}(\frac{p_{0}}{p_{d}})^{0.2854}\,(\frac{T_{k}}{T_{L}})^{0.28\times10^{-3}r} \tag{4.42} \end{equation}\] \[\begin{equation} \Theta_{P}=\Theta_{DL}\exp\{\frac{r(L_{0}^{*}-L_{1}^{*}(T_{L}-T_{0})+K_{2}r)}{c_{pd}T_{L}}\} \tag{4.43} \end{equation}\]

Prior to 2011, the variable called the equivalent potential temperature³¹ and named THETAE in the output data files was that obtained using the method of Bolton (1980), which used the same formula to obtain the temperature at the lifted condensation level (\(T_L\)) and then used that temperature to find the value of potential temperature of dry air that would result if the parcel were lifted from that point until all water vapor condensed and was removed from the air parcel. The formulas used were as follows:

\(T_{L}\)= temperature at the lifted condensation level, K
\(T_0=273.15\,\mathrm{K}\) ATX = ambient temperature [\(^{\circ}\mathrm{C}\)]
EDPC = water vapor pressure [hPa] – now superceded by EWX
MR = mixing ratio [g/kg]
THETA = potential temperature [K]

\[\begin{equation} T_{L}=\frac{2840.}{3.5\ln(\mathrm{\{ATX\}+T_{0}})-\ln(\mathrm{\{EDPC\}})-4.805}+55 \tag{4.44} \end{equation}\] \[\begin{align} \mathrm{\{THETAE\}} = & \mathrm{\{THETA\}}\left(\frac{3.376}{T_{L}}-0.00254\right)\notag \\ & \times (\mathrm{\{MR\}})(1+0.00081(\{MR\})) \tag{4.45} \end{align}\]

Differences vs the new formula are usually minor but can be as much as 0.5 K.****

Virtual Temperature (ºC): TVIR

The temperature of dry air having the same pressure and density as the air being sampled. The virtual temperature thus adjusts for the buoyancy added by water vapor.

ATX = ambient temperature, \(^{\circ}\mathrm{C}\)
\(r\) = mixing ratio, dimensionless {[}kg/kg{]} = {MR}/(1000 g/kg)
\(T_{0}=273.15\),K

\[\begin{equation} \mathrm{TVIR}=(\mathrm{\{ATX\}}+T_{0})\left(\frac{1+\frac{M_{d}}{M_{w}}r}{1+r}\right)-T_{0} \tag{4.46} \end{equation}\]

Virtual Potential Temperature (K): THETAV

A potential temperature analogous to the conventional potential temperature except that it is based on virtual temperature instead of ambient temperature. Dry-adiabatic expansion or compression to the reference level (1000 hPa) is assumed. As for THETA, use of dry-air values for the gas constant and specific heat at constant pressure can lead to significant errors in humid conditions. For further information, see this note.

TVIR = virtual temperature [\(^{\circ}\mathrm{C}\)]
PSXC = ambient pressure [hPa]
\(R_{d}=\)gas constant for dry air\(^{\dagger}\)
\(c_{pd}=\)specific heat at constant pressure for dry air\(^{\dagger}\)
\(T_{0}=273.15\,\)K
\(p_{0}\) = reference pressure, conventionally 1000 hPa

\[\begin{equation} \mathrm{THETAV}=\left(\mathrm{\{TVIR\}}+T_{0}\right)\left(\frac{p_{o}}{\mathrm{\{PSXC\}}}\right)^{R_{d}/c_{pd}} \tag{4.47} \end{equation}\]

Wet-Equivalent Potential Temperature (K): THETAQ

The absolute temperature reached if a parcel of air were to be expanded adiabatically (i.e., retaining the condensed water in the liquid phase and accounting for the specific heat of that condensate) to a level where no water vapor remains, after which the condensate would be removed and the resulting dry parcel compressed to 1000 hPa. This variable was not included in data archives prior to 2012. Emanuel (1994) gives the following formula (his Eq. 4.5.11):
\[\begin{equation} \Theta_{q}=T(\frac{p_{0}}{p_{d}})^{\frac{R_{d}}{c_{pt}}}\exp\left\{ \frac{L_{v}r}{c_{pt}T}\right\} \left(\frac{e}{e_{s,w}(T)}\right)^{-rR_{w}/c_{pt}} \tag{4.48} \end{equation}\]
where \(\Theta_q\) is the wet-equivalent potential temperature, \(L_v\) the latent heat of vaporization, \(r\) the (dimensionless) water-vapor mixing ratio, \(c_{pt} = c_{pd}+r_tc_w\) with \(r_t\) the total-water mixing ratio including vapor and condensate, \(c_w\) the specific heat of liquid water, and other symbols are as used previously. See this memo for additional discussion of this variable, for values to use for the latent heat and specific heat, and in particular for analysis indicating that \(\Theta_Q\) evaluated with this formula can be expected to vary from the true adiabatic value by a few tenths kelvin (in a worst case, by about 1 K) because of variation in (and uncertainty in) the specific heat of supercooled water at low temperature. The details of the calculation are described in the following box. Note that this algorithm only uses the liquid water content as measured by a King probe, PLWCC; other similar calculations could be based on other measures of liquid water such as that from a cloud-droplet spectrometer.

\(e=\){EWX}\(*100\) = water vapor pressure [Pa]
ATX = ambient temperature (\(^{\circ}\mathrm{C}\))
\(r=\){MR}/1000. = mixing ratio (dimensionless)
\(p_{d}=\)({PSXC}\(-\){EWX})\(*100\) = ambient dry-air pressure [Pa]
\(p_{0}=\)reference pressure for potential temperature, 10\(^{5}\)Pa
\(\chi=\){PLWCC}/1000.=cloud liquid water content [kg\(\,\)m\(^{-3}\)]
\(R_{d}=\)gas constant for dry air\(^{\dagger}\)
\(\rho_{d}=\)density of dry air = \(\frac{p_{d}}{R_{d}(\{ATX\}+T_{0})}\)
\(c_{pd}=\)specific heat of dry air\(^{\dagger}\)
\(c_{w}=\)specific heat of liquid water\(^{\dagger}\)
\(L_{v}=L_{0}+L_{1}\mathrm{\{ATX\}}\) where \(L_{0}=2.501\times10^{6}\mathrm{J}\,\mathrm{kg^{-1}}\) and \(L_{1}=-2370\,\mathrm{J\,\mathrm{kg^{-1}\,\mathrm{K^{-1}}}}\)

\[\begin{equation} r_{t}=r+(\chi/\rho_{d}) \tag{4.49} \end{equation}\] \[\begin{equation} c_{pt}=c_{pd}+r_{t}c_{w} \tag{4.50} \end{equation}\]
If outside cloud or below 100% relative humidity, define
\[\begin{equation} F_{1}=\left(\frac{e}{e_{s,w}(T)}\right)^{-\frac{rR_{w}}{c_{pt}}} \tag{4.51} \end{equation}\]
otherwise set \(F_{1}=1\). Then
\[\begin{equation} T_{1}=\mathrm{(\{ATX\}}+T_{0})\left\{ \frac{p_{0}}{(\mathrm{\{PSXC\}}-\mathrm{\{EDPC\})}}\right\} ^{\frac{R_{d}}{c_{pt}}} \tag{4.52} \end{equation}\] \[\begin{equation} \mathrm{\{THETAQ\}}=T_{1}F_{1}\exp\left\{ \frac{L_{v}r}{c_{pt}(\{\mathrm{ATX\}}+T_{0})}\right\} \tag{4.53} \end{equation}\]

4.7 Wind

RAF Bulletin 23 documents the calculation of wind components, both with respect to the earth (UI, VI, WI, WS and WD) and with respect to the aircraft (UX and VY). In data processing, a separate function (GUSTO in GENPRO, gust.c in NIMBUS) is used to derive these wind components. That function uses the measurements from an Inertial Navigation System (INS) as well as aircraft true airspeed, aircraft angle of attack, and aircraft sideslip angle. The wind components calculated in GUSTO/gust.c are used to derive the wind direction (WD) and wind speed (WS). Additional variables UIC, VIC, WSC, WDC, UXC, and VYC are also calculated based on the variables VNSC, VEWC discussed in Section 3.4, which combine INS and GPS information to obtain improved measurements of the aircraft motion. Those are usually the highest-quality measurements of wind because the merged INS/GPS variables combine the high-frequency response of the INS with the long-term accuracy of the GPS.

There is an extensive discussion of the wind-sensing system and the uncertainties associated with measurements of wind in this Technical Note. The details contained therein and in Bulletin 23 will not be repeated here, so those documents should be consulted for additional information. There are two exceptions that are discussed in more detail here:

1. The calculation of vertical wind is described in more detail below for the variables WI and WIC.

2. Because measurements obtained by a GPS receiver are often used, the motion of the GPS receiving antenna relative to the IRU must be considered. Standard processing corrects for the motion of the gust system relative to the IRU arising from aircraft rotation, but a similar correction is needed because the GPS antenna is displaced from the IRU. The displacement is almost entirely along the longitudinal axis of the aircraft, so GPS-measured velocities like GGVNS, GGVEW, and GGVSPD (denoted here \(v_n\), \(v_e\), \(v_u\)) need correction as follows to give measurements that apply at the location of the IRU. Then these variables can be used in place of or to complement similar measurements from the IRU in the processing algorithms. The equations are:
   \[\begin{align}\begin{split} \delta v_{u} = & -L_{G}\dot{\theta}\notag \\ \delta v_{e} = & -L_{G}\dot{\psi}\,{\cos\psi}\notag \\ \delta v_{n} = & L_{G}\dot{\psi}\,{\sin\psi} \end{split} \tag{4.54} \end{align}\]
   where \(\theta\) and \(\psi\) respectively are the pitch and heading angles and \(L_G\) is the distance forward along the longitudinal axis from the IRU to the GPS antenna (\(−4.30\) m for the GV and \(-9.88\) m for the C-130 during and after 2015). The negative signs indicate that the GPS antennas are behind the IRUs. The dots over the attitude-angle symbols represent time derivatives, so for example \(\dot{\theta}\) is the rate of change of the pitch angle. All angles are expressed in radians. The correction terms should be added to the GPS-measured velocity components so that they represent the motion of the IRU relative to the Earth. This is done for the vertical wind, beginning in 2017, but for horizontal wind the complementary filter (discussed below) removes high-frequency fluctuations from the GPS-derived measurements so incorporation of these changes would have negligible effect. For more information, see this note.

The variables pertaining to the relative wind are described in the next subsection, and the variables characterizing the wind are then described briefly in the last subsection. Some additional detail is included in cases where procedures are not documented in that earlier bulletin.

4.7.1 Relative Wind

Wind is measured by adding two vectors, the measured air motion relative to the aircraft (called the relative wind) and the motion of the aircraft relative to the Earth. The following are the measurements used to determine the relative wind. The motion of the aircraft relative to the ground was discussed in Section 3.1, and the combination of these two vectors to measure the wind is described in RAF Bulletin 23.

RAF uses the radome gust-sensing technique³² to measure incidence angles of the relative wind (i.e., angles of attack and sideslip). The pressure difference between sensing ports above and below the center line of the radome is used, along with the dynamic pressure measured at a pitot tube and referenced to the static pressure source, to determine the angle of attack. The sideslip angle is determined similarly using the pressure ports on the starboard and port sides of the radome. A Rosemount Model 858AJ gust probe has occasionally been used for specialized measurements. The radome measurements are made by differential pressure sensors located in the nose area of the aircraft and connected to the radome by semi-rigid tubing.

Mach Number (dimensionless): MACHx, MACHX

The Mach Number that characterizes the flight speed. The Mach number is defined as the ratio of the flight speed (or the magnitude of the relative wind) to the speed of sound. See Eq. (4.17) for the equation used. Many relatively old archived data files have instead a variable XMACH2, which is the square of MACHx.

Aircraft True Airspeed (m/s): TASx, TASxD, TASX

The flight speed of the aircraft relative to the atmosphere. This derived measurement of the flight speed of the aircraft relative to the atmosphere is based on the Mach number calculated from both the dynamic pressure at location x and the static pressure. See the derivation for ATx . The different variables for TASx (TASF, TASR, etc) use different measurements of QCxC in the calculation of Mach number. The variable TASxD is the result of calculations for which the Mach number, air temperature, and true airspeed are determined for dry instead of humid air. See the discussion of ATX for an explanation of how humidity is handled in the calculation of true airspeed.

(see (4.17) and (4.18) for MACHx and ATX)
Note dependence of MACHx on choices for QCXC and PSXC
TASx depends on QCXC, PSXC, ATX
     where PSXC and ATX are the preferred choices
\(\gamma^{\prime}\), \(R^{\prime}\), and \(T_{0}\): See the List of Symbols

\[\begin{equation} \mathrm{TASx}=\mathrm{\{MACHx\}}\sqrt{\gamma^{\prime}R^{\prime}\mathrm{\,(\{ATX\}}+T_{0})} \tag{4.55} \end{equation}\]

Aircraft True Airspeed (Humidity Corrected) (m/s): TASHC

This derived measurement of true airspeed accounted for deviations of specific heats of moist air from those of dry air. See List, 1971, pp 295, 331-339, and Khelif, et al., 1999. This variable is no longer used because the standard calculation of TASX (documented in the preceding paragraph) now uses moist-air values of the specific heats and gas constant. The equation previously used for this variable, given by Khelif et al. 1999,³³ added a moisture correction to the true airspeed derived for dry air, as follows:

\(q\) = specific humidity (dimensionless) = SPHUM/1000.
       for SPHUM expressed in g/kg
\(c=0.000304\,\mathrm{kg\,g^{-1}}=0.304\) (dimensionless)

\[\begin{equation} \mathrm{\{TASHC\}} = \mathrm{\{TASX\}} (1.0 + c\,q) \tag{4.56} \end{equation}\]

Attack Angle Differential Pressure (mb): ADIFR

The pressure difference between the top and bottom pressure ports of a radome gust-sensing system. This measurement is used to determine the angle of attack; see AKRD below. **** Obsolete variable ADIF is a similar variable used for old gust-boom systems or for Rosemount Model 858AJ flow-angle sensors.

Sideslip Angle Differential Pressure (mb): BDIFR

The pressure difference between starboard and port pressure inlets of a radome gust-sensing system. This measurement is used to determine the sideslip angle; see SSRD below. Obsolete variable BDIF is a similar variable used for old gust-boom systems or for Rosemount Model 858AJ flow-angle sensors.

Attack Angle, Radome (º): AKRD

The angle of attack of the aircraft. This derived measurement represents the angle between the longitudinal axis of the aircraft and the component of the relative wind vector in the plane of port-starboard symmetry of the aircraft. The tangent of the angle of attack is the ratio of the vertical to longitudinal component of the relative wind. Positive values indicate flow moving upward (in the aircraft reference frame) relative to the longitudinal axis. The calculation is based on ADIFR and a measurement of dynamic pressure, and so is the measurement produced by a radome gust-sensing system. Empirical sensitivity coefficients for each aircraft, determined from special flight maneuvers, are used; see RAF Bulletin 23 and this Technical Note for more information. The sensitivity coefficients listed below have changed when the radomes were changed or refurbished, so the project documentation should be consulted for the values used in a particular project. For more information on the latest C-130 calibration, see this note.
Prior to 2017, the procedure was based on the following algorithm:

ADIFR = attack differential pressure, radome [hPa]
QCF = uncorrected dynamic pressure [hPa]
MACH = uncorrected Mach number based on QCF and PSF without humidity correction
\(e_{0},\,e{}_{1},\,e_{2}\) = sensitivity coefficients determined empirically; typically:
     {4.7532, 9.7908, 6.0781} for the C-130^(a)
     {4.605\(\,[^{\circ}]\), \(18.44\,[^{\circ}]\), \(6.75\,[^{\circ}]\)} for the GV
__________
^(a) Prior to Jan 2012, when the GV radome was changed: {5.516, 19.07, 2.08}

\[\begin{equation} \mathrm{\{AKRD\}}=e_{0}+\frac{\{\mathrm{ADIFR}\}}{\{\mathrm{QCF}\}}\left(e_{1}+e{}_{2}\mathrm{\{MACH\}}\right) \tag{4.57} \end{equation}\]

See also this memo.

Beginning in 2017, a different strategy was used, as documented in more detail in this memo. Two variables were used to represent the angle of attack, \(A\)={ADIFR}/{QCF} and \(q\)={QCF}. However, each was filtered into complementary low-pass and high-pass components, with the cutoff frequency at (1/600) Hz, and the separate components were used to represent the separate components of angle of attack according to the following formula:

ADIFR = attack differential pressure, radome [hPa]
QCF = uncorrected dynamic pressure [hPa]
\(A\) = (ADIFR/QCF) = \(A_{f}+A_{s}\) where \(A_{f}\) is the high-pass and \(A_{s}\) the low-pass component
\(e_{1},\,d_{0},\,d{}_{1},\,d_{2}\) = sensitivity coefficients determined empirically; typically, for the GV,
     \(e_{1}=21.481\,[^{\circ}]\)
     \(d_{1-3}\) = {\(4.5253\,[^{\circ}]\), \(19.9332\,[^{\circ}]\), \(-0.00196099\,[^{\circ}\mathrm{hPa}^{-1}]\)}

\[\begin{equation} \mathrm{\{AKRD\}}=d_{0}+d_{1}A_{s}+d_{2}\mathrm{\{QCF\}_{s}+}e_{1}A_{f} \tag{4.58} \end{equation}\]

****

Reference Attack Angle (º): ATTACK

The reference angle of attack used to calculate derived variables. This variable is the reference selected from other measurements of angle of attack in the data set. In most projects, it is equal to AKRD. It is used where attack angle is needed for other derived calculations (e.g., wind measurements).

Sideslip Angle (Differential Pressure) (º): SSRD

The angle of sideslip of the aircraft. This derived measurement represents the angle between the longitudinal axis of the aircraft and the projection of the relative wind onto the plane determined by the longitudinal and lateral axes. Positive values indicate airflow from the starboard side. This variable is derived from BDIFR and a dynamic pressure using a sensitivity function that has been determined empirically for each aircraft.

BDIFR = differential pressure between sideslip pressure ports, radome [hPa]
QCXC = dynamic pressure [hPa]
\(s_{0},\,s{}_{1}\) = empirical coefficients dependent on the aircraft and radome configuration
      = {-0.000983, (1/0.08189) \(^\circ\)} for the C-130
      = {-0.0025, (1/0.04727) \(^\circ\)} for the GV^(a)

\[\begin{equation} \mathrm{\{SSRD\}} = s_{1}(\frac{\mathrm{\{BDIFR\}}}{\{\mathrm{QCXC}\}}+s_{0}) \tag{4.59} \end{equation}\] __________
^(a) The technical note on wind uncertainty recommended using SSRD=\(e_{0}+e_{1}\){BDIFR}/{QCF} with \(e_{0}=0.008\) and \(e_{1}=22.302\). This has not yet been used in processing as of May 2022.

Reference Sideslip Angle (º): SSLIP

The reference sideslip angle used to calculate derived variables. This variable is the reference selected from other measurements of sideslip angle in the data set. In most projects, it is equal to SSRD. It is used where sideslip angle is needed for other derived calculations (e.g., wind measurements).

4.7.2 Wind Components and the Wind Vector

Wind Vector Components (m/s): UI, VI, WI

The three-dimensional wind vector with respect to the earth, as determined from the inertial reference systems. UI is the east-west component with positive values toward the east, VI is the north-south component with positive values toward the north, and WI is the vertical component with positive values toward the zenith.
The calculation of WI differs from the description in Bulletin 23 because the output from the inertial reference system is different for the modern units now in use. The vertical wind is the sum of the vertical gust component (represented approximately by TASX sin(ATTACK-PITCH)) and the motion of the aircraft as measured by VSPD (discussed in Section XXX). Bulletin 23 describes the historical calculation of the vertical motion of the aircraft via a barometric-inertial feedback loop, but equivalent calculations (including pressure damping to the pressure altitude) are incorporated into current IRS units so VSPD already is the product of such a calculation. To calculate WI, VSPD is therefore used in place of the obsolete variable WP3 that was discussed in Bulletin 23.
WIC should usually be used instead of WI because VSPD, entering WI, is updated to the pressure altitude and so can have false variations in baroclinic conditions. WIC uses GGVSPD (or in some cases older GPS-based rate-of-climb variables) in place of VSPD and so is more reliable.
****

Wind Speed and Direction (m/s and º): WS, WD

The magnitude and direction of the horizontal wind. These variables are obtained in a straightforward manner from UI and VI. The resulting wind direction is relative to true north and represents the direction from which the wind blows. That is the reason that 180^∘ appears in the following algorithm.

UI = easterly component of the horizontal wind
VI = northerly component of the horizontal wind
atan2 = 4-quadrant arc-tangent function producing output in radians from -\(\pi\) to \(\pi\)
\(C_{rd}\) = conversion factor, radians to degrees, = 180/\(\pi\) [units: \(^{\circ}\),/,radian]

\[\begin{align} \mathrm{WS} = & \sqrt{\mathrm{\{UI\}}^{2}+\{\mathrm{VI\}}^{2}}\tag{4.60}\\ \mathrm{WD} = & C_{rd}\mathrm{\,atan2(\{UI\},\,\{VI\})}+180^{\circ} \tag{4.61} \end{align}\]

Wind Vector Longitudinal and Lateral Components (m/s): UX and VY

The horizontal wind vector relative to the frame of reference attached to the aircraft. UX is parallel to the longitudinal axis and positive toward the nose. VY is along the lateral axis and normal to the longitudinal axis; positive is toward the port (or left) wing.

GPS-Corrected Wind Vector, East and North Components (m/s): UIC, VIC

The horizontal wind components respectively toward the east and toward the north. They are derived from measurements from an inertial reference unit (IRU) and a Global Positioning System (GPS), as described in the discussion of VEW and VNS above. They are calculated just as for UX and VY except that the GPS-corrected values for the aircraft groundspeed are used in place of the IRU-based values. They are considered “corrected” from the original measurements from the IRU or GPS, as described in Section 3.4.

Wind Vector, Vertical Component (m/s): WIC

The component of the wind in the vertical direction. This is the standard calculation of vertical wind, obtained from the difference between the measured vertical component of the relative wind and the vertical motion of the aircraft (usually GGVSPD in recent projects).**** This should be used in preference to WI if the latter is present; see the discussion of WP3 in Section 3.1. Positive values are toward the zenith.

GPS-Corrected Wind Speed and Direction (m/s and ^∘): WSC, WDC

The magnitude and direction of the wind vector, obtained by combining measurements from GPS and IRU units. These variables are obtained in a straightforward manner from UIC and VIC, using equations analogous to (4.60) and (4.61) but with UIC and VIC as input measurements. They are expected to be the preferred measurements of wind because they combine the best features of the IRU and GPS measurements.

GPS-Corrected Wind Vector, Longitudinal and Lateral Components (m/s): UXC, VYC

The longitudinal and lateral components of the three-dimensional wind, similar to UX and VY, but corrected by the complementary-filter algorithm that combines IRU and GPS measurements. See the discussion in Section 3.4. The components UXC and VYC are toward the front of the aircraft and toward the port (left) wing, respectively.

4.8 Special-Use Remote Sensors

The above variables are normally included in the archived netCDF files from projects, but there are a few remote sensors that provide additional state-parameter measurements in some projects. These include:

• Microwave Temperature Profiler (http://www.eol.ucar.edu/instruments/microwave-temperature-profiler) {MTP}) – remotely sensed temperature profiles
  
• Dropsonde System (https://www.eol.ucar.edu/observing_facilities/avaps-dropsonde-system) {AVAPS}) – profiles of temperature, humidity, and wind vs pressure.
  
• GPS-Occultation Sensor (http://www.eol.ucar.edu/instruments/gnss-instrument-system-multi-static-and-occultation-sensing) {GISMOS}) – atmospheric soundings of refractivity via GPS occultation.

The links provided connect to descriptions of these instruments on the EOL web site, and each provides a summary of how data are acquired and processed. These measurements are not normally part of the archived netCDF project files. Those interested in using these measurements should contact EOL data management (mailto:raf-dm@eol.ucar.edu) for access to the measurements and for information on how the measurements are processed.

---

12. A weakness is this form for the pressure correction is that occasionally the radome ports become plugged with ice and the measurement of angle of attack is not available. When the variable ATTACK representing angle of attack is invalid, the angle of attack is instead calculated from PITCH−VSPD/TASX, which approximates the angle of attack if the vertical wind is zero.↩︎

13. For C-130 measurements prior to 2012 but after September 2003, the correction applied to PSF was Δp = p + max ((3.29 + {QCX} * 0.0273),4.7915) using units of hPa. Prior to Sept 2003, the correction was Δp = max ((4.66 + 11.4405Δp_α/Δq_r), 1.113). For both PSFD and PSFRD, the correction prior to (2012?) was Δp = p + max ((3.29 + {QCX}  0.0273),4.7915). For GV measurements Aug 2006 to 2012, Δp= (-1.02 + 0.1565q) + q1(0.008 + q1(7.1979e-09q1 - 1.4072e-05). Before Aug 2006: Δp=(3.08 - 0.0894{PSF}) + {QCF}(-0.007474 + {QCF}4.0161e-06).↩︎

14. See this memo and this revisionfor details regarding implementation of this representation of Δp for the GV.↩︎

15. Atmos. Meas. Tech., 7, 3215-3231, 2014 doi:10.5194/amt-7-3215-2014.↩︎

16. C-130, prior to 2012: For QCFC: subtract max(4.66+11.4405 * \(\mathrm{\{ADIFR\}}\) / \(\mathrm{\{QCR\}}\), 1.113); For QCFRC prior to Sept 2003: same as for QCFC; after/including Sept 2003, subtract max(3.29+\(\mathrm{\{QCX\}}\)  0.0273, 4.7915); For QCRC: subtract max((3.29+\(\mathrm{\{QCX\}}\)  0.0273), 4.7915). GV Aug 2006 to 2012: For QCF, subtract (1.02+\(\mathrm{\{PSF\}}\)(0.215 - 0.04 * \(\mathrm{\{QCF\}}\)/1000.) + {QCF} * (\(-0.003266\) + \(\mathrm{\{QCF\}}\) * 1.613e-06))↩︎

17. The symbol ^† indicates that values are included in the table of constants in Sect. 1.3.↩︎

18. Prior to 2012, these variables were called “total temperature” and symbols starting with ’TT’ instead of ’RT’ were used. That name was misleading because these values are not true total-temperature measurements, for which the air would be at the same speed as the aircraft, but instead recovery-temperature measurements. The name has been changed to correct this mis-labeling, although this was a long-standing convention in past datasets.↩︎

19. See the related obsolete variables TTx, which are previously used names for these variables. The names were changed to clarify that the quantity represented is the recovery temperature, not the total temperature.↩︎

20. The recovery factor determined for the now-obsolete NCAR reverse-flow sensor was 0.6. The recovery factor for the now retired NCAR fast-response (K-probe) temperature sensor was 0.8.↩︎

21. A problem sometimes arises from use of the measured humidity, because that measurement might be obviously in error. For example, following descents the dew point determined from chilled-mirror hygrometers sometimes overshoots the correct value significantly, producing dew-point measurements well above the measured temperature. If such measurements are used, the result can produce a significant error in derived variables based on the humidity-corrected gas constant and specific heats. If the measurements are flagged as bad, there will be gaps in derived variables. To avoid these two errors, the corrections applied to the gas constant and specific heats are treated as follows: (i) The humidity correction is limited to not more than that given by the water-equilibrium humidity at the temperature ATXD, calculated using dry-air specific heats and gas constant. (ii) If the humidity from the primary sensor is flagged as a missing measurement (e.g., from a dew-point sensor), a secondary measurement is used (e.g., the VCSEL) in cases when the secondary sensor is almost always present in an experiment. (iii) As a backup, the variables TASxD and ATxD are always calculated omitting the humidity correction to the gas constant and the specific heats. These variables usually provide continuous measurements, although they will be offset from the humidity-corrected values. The offset indicates the magnitude of the correction when both are present, and one of the variables TASxD (ATxD) may be selected as TASX (ATX) in cases where missing values might cause a problem for derived variables.↩︎

22. Q. J. R. Meteorol. Soc. (2005), 131, pp. 1539–1565↩︎

23. Prior to 2010, the vapor pressure relationship used was the Goff-Gratch formula as given in the Smithsonian Tables (List, 1980).↩︎

24. prior to 2011 the conversion was made using the formula DPxC = 0.009109 + DPx(1.134055 + 0.001038DPx). For instruments producing measurements of vapor density (RHO), the previous Bulletin 9 section incorrectly gave the conversion formula as DPxC = 273.0Z/(22.51 − Z), a conversion that would apply to frost point, not dew point. However, the code in use shows that the conversion was instead 237.3Z/(17.27 − Z), where Z in both cases is Z = ln ((ATX + 273.15)RHO/1322.3).↩︎

25. The conversion factor is given by this formula:
    
    \[C^{\prime}=\frac{10^{6}\mathrm{cm}^{3}}{\mathrm{m}^{3}}\times\frac{M_{W}^{\dagger}}{N_{A}^{\dagger}}\]
    where N_A is the Avogadro constant, 6.022147 × 10²⁶ molecules kmol^− 1.↩︎

26. For details see Zondlo, M. A., M. E. Paige, S. M. Massick, and J. A. Silver, 2010: Vertical cavity laser hygrometer for the National Science Foundation Gulfstream-V aircraft. J. Geophys. Res., 115, D20309, doi:10.1029/2010JD014445.↩︎

27. prior to 2011, this variable was calculated using the Goff-Gratch formula. See the discussion of DPXC for more information on previous calculations.↩︎

28. For more details see Beaton, S. P. and M. Spowart, 2012: UV Absorption Hygrometer for Fast-Response Airborne Water Vapor Measurements. J. Atmos. Oceanic Technol., 29. DOI: 10.1175/JTECH-D-11-00141.1↩︎

29. Davies-Jones, R., 2009: On formulas for equivalent potential temperature. Mon. Wea. Review, 137, 3137–3148.↩︎

30. Bolton, D., 1980: The computation of equivalent potential temperature. Mon. Wea. Rev., 108, 1046–1053.↩︎

31. The AMS glossary defines equivalent potential temperature as applying to the adiabatic process, not the pseudo-adiabatic process; the name of this variable has therefore been changed.↩︎

32. Brown, E. N, C. A. Friehe, and D. H. Lenschow, 1983: Journal of Climate and Applied Meteorology, 22, 171–180↩︎

33. Khelif, D., S.P. Burns, and C.A. Friehe, 1999: Improved wind measurements on research aircraft. Journal of Atmospheric and Oceanic Technology, 16, 860–875.↩︎