## Introduction

The military importance of visible condensation trail (contrail) formation in the wake of aircraft became obvious during the strategic bombing campaigns of World War II. The war renewed interest in contrails and led to considerable research to explain further the physics of contrails and to develop forecast techniques for their occurrence. These efforts are thoroughly described by Schumann (1996a). Today, the basic approach used to forecast contrails still employs these fundamental concepts. For example, the U.S. Air Force Weather Agency (AFWA) contrail forecast algorithm is based on the method of Appleman (1953, 1957). Although traditionally credited to Appleman within the U.S. Air Force, this forecast technique builds upon the efforts of numerous researchers (Schumann 1996a).

### Brief review of Schmidt–Appleman theory

Most current contrail forecasting techniques follow the approach referred to as Schmidt–Appleman theory by Schumann (1996a). Schmidt–Appleman theory uses fundamental thermodynamic principles to calculate a theoretical critical temperature *T*_{c} at which the mixture of the engine exhaust and the ambient atmosphere achieves saturation with respect to water. Critical temperature at 100% relative humidity is a function of the environmental water saturation vapor pressure and the fuel combustion characteristics of the aircraft. Combustion of aviation fuel results in the addition of water vapor to the aircraft exhaust, increasing the relative humidity of the wake, while adding heat, decreasing the relative humidity of the wake. The ratio of moisture and heat added to the wake is known as the contrail factor and can be expressed in units of mixing ratio per degree. Published values for contrail factors vary between 0.0295 g kg^{−1} K^{−1} (Pilié and Jiusto 1958) and 0.049 g kg^{−1} K^{−1} (Peters 1993).

If the aircraft wake and the environment mix adiabatically and isobarically, the specific water mass (*m*) and enthalpy per mass (*h*) of the moist air are conserved (Schumann 1996a). If it is further assumed that heat and water mix similarly in the plume, and that there are not other sources of heat loss or addition, then it can be assumed that mixing occurs along a straight line on an *h–m* diagram (Schumann 1996a) in which the slope is equal to the contrail factor. Assuming also sufficiently low temperatures and no phase change of water substance occurs, the mixing will also follow along a straight mixing line on a vapor partial pressure versus temperature diagram (Schumann 1996a; Schrader 1997). The temperature at the point where the mixing line is tangent to the saturation vapor pressure curve is the critical temperature at 100% relative humidity and represents the warmest possible temperature for contrail formation. At this temperature, the derivative of the saturation vapor pressure curve with respect to temperature on a vapor partial pressure versus temperature diagram is the same as the slope of the mixing line, if the mixing line slope is expressed in units of Pa^{−1} K^{−1}. This temperature can be found iteratively (Schumann 1996a; Schrader 1997), graphically (Ferris 1996), or by using an explicit approximation of such iterative solutions (Schumann 1996a). The calculation of critical temperature for relative humidities lower than 100% is straightforward and is not discussed here (see, e.g., Schumann 1996a; Schrader 1997). When the environmental temperature is lower than the critical temperature calculated for the relative humidity and pressure at flight level, contrails are forecast to occur.

The correctness of these thermodynamic criteria for contrail formation is firmly established. Jensen et al. (1998) studied the ambient temperatures and humidities required for contrail formation using highly accurate in situ aircraft measurements and contrail observations. They found that visible contrails formed only when the exhaust plume was saturated with respect to water (within the accuracy of their measurements) and that saturation with respect to ice was not sufficient for contrail formation. This finding agrees with Appleman’s (1953) assumption that contrails will not form until the aircraft wake is saturated with respect to water. Jensen et al. (1998) also showed that the threshold temperature required for contrail formation agreed extremely well with the critical temperatures calculated using Schumann’s (1996a) algorithm, which is an application of Schmidt–Appleman theory discussed below. The results of Jensen et al. (1998) agree with the conclusions of Kärcher et al. (1998), who show that contrail formation temperatures can be predicted thermodynamically (using Schmidt–Appleman theory) provided the flight-level relative humidity, temperature, and fuel combustion characteristics for the aircraft are known. The fuel combustion characteristics determine the mixing line slope, using either the contrail factor or the propulsion efficiency as described below. These findings confirm that the Schmidt–Appleman theory is essentially correct and that the physical basis for contrail formation is well understood.

### Refinements to Schmidt–Appleman theory

There have been numerous efforts to develop further refinements to improve the accuracy of contrail forecast techniques (Peters 1993; Coleman 1996; Schumann 1996a; Schrader 1997). The following is a brief summary of some of these efforts.

Peters (1993) performed a validation study of the Appleman (1953, 1957) contrail forecast method using a large database of U.S. Air Force contrail observations. Peters found that the Appleman method correctly forecasted the occurrence of contrails in less than 30% of the cases below 12.19 km (40 000 ft). Peters attributed this result to the fact that modern high-bypass jet engines emitted less heat per mass of water vapor emitted. He proposed engine-specific corrections to the contrail factors for low-bypass, high-bypass, and nonbypass engines to account for these differences. This approach differed from that of Appleman (1953), who recommended the use of a single contrail factor for all jet aircraft based upon a study of typical jet engines in use at that time. Busen and Schumann (1995) also showed that the contrail factor varies as flight parameters change and that the use of a single contrail factor for all modern jet aircraft is not appropriate.

Coleman (1996) developed a slightly different approach to forecasting contrails. Rather than proceeding directly in the traditional way described above, he began by deriving a fundamental necessary condition for contrail formation and showed this fundamental condition is consistent with the Appleman (1953) method. He derived an explicit solution for critical temperature as a function of water vapor mixing ratio rather than relative humidity. This method was not used in the current study.

Schumann (1996a) presented a detailed rederivation of the calculation of critical temperature using the Schmidt–Appleman theory, which accounts for the fact that some of the heat produced by combustion is converted into kinetic energy of the motion of the wake of the aircraft. This reduces the heat in the plume and increases the ratio of water to heat in the plume, increasing the contrail factor and leading to higher critical temperatures and warmer threshold temperatures for contrail formation. Following this theory, the slope of the mixing line (or contrail factor) is estimated as a function of the propulsion efficiency *η,* which is a measure of the work performed against drag forces to propel the aircraft. The propulsion efficiency is a function of cruise speed, engine thrust, specific combustion heat, and fuel flow rate per engine. The relationship between the contrail factor CF and the propulsion efficiency *η* is CF = *E*_{I}*C*_{p}/(1 − *η*)*Q,* where *E*_{I} is the water vapor emission index (typically 1.25), *C*_{p} is the specific heat capacity of air (1004 J kg^{−1} K^{−1}), and *Q* is combustion heat (assumed to be 43 MJ kg^{−1}). As a consequence, CF takes its minimum for *η* = 0 of 0.0292 g kg^{−1} K^{−1} for typical fuels and is larger in all practical cases. Propulsion efficiency accounts for the same differences in engine performance noted by Peters and also accounts for the airspeed and drag of the aircraft (Schumann 1996a). As implemented in this study, the method applies to stagnant plumes and ignores the kinetic energy in the plume early in the plume’s life cycle (close to the engine exhaust where exhaust velocities are high relative to the environment). Schumann (1996a) also extended this theory to account for nonstagnant plumes.

Schrader (1997) explained the use of the Schmidt–Appleman theory and presented a method for the calculation of critical temperature (using the contrail factor to represent the slope of the mixing line) that corrected a few minor errors in the work of Peters (1993). Schrader’s (1997) algorithm also uses different contrail factors for high-bypass, nonbypass, and low-bypass engines to improve forecast accuracy. Although the contrail factor for an individual aircraft varies as flight parameters change, Schrader (1997) pointed out that one can use representative values for the contrail factor that account for the generic propulsion efficiencies of typical engine types when preparing contrail forecasts used for different types of aircraft at the same time. This procedure is currently followed by the AFWA, which produces separate contrail forecasts for high-bypass, nonbypass, and low-bypass engine types. Because propulsion efficiency is highest for high-bypass engines and lowest for nonbypass engines, high-bypass engines have the highest contrail factors, and nonbypass engines have the lowest contrail factors (Schrader 1997). The Schrader (1997) algorithm is essentially the same as the current AFWA algorithm. Some minor details of the AFWA implementation differ from the implementation of the Schrader (1997) algorithm used here (e.g., AFWA uses representative relative humidities rather than forecast relative humidities at pressure below 30 kPa, and the iteration method used is slightly different). The Schumann (1996a) and Schrader (1997) algorithms are slightly different applications of the same Schmidt–Appleman theory and vary primarily in the way the slope of the mixing line is estimated. In the application of the Schrader (1997) algorithm, the contrail factor (slope of the mixing line) is prescribed, whereas in the Schumann (1996a) method, the contrail factor is calculated from the propulsion efficiency and other parameters as described above.

### Forecasting contrails at low relative humidity

Hanson and Hanson (1995) proposed a change to the Schmidt–Appleman theory that they argued would produce more accurate contrail forecasts at low relative humidity. However, they noted their algorithm (hereinafter referred to as the Hanson and Hanson algorithm) cannot be used to calculate *T*_{c} at 0% relative humidity;Hanson and Hanson (1995) justified this through the statement that “such a critical temperature would be an extremely large magnitude negative number that would not be characteristic of the physical conditions under examination.” The implication is that contrails are unlikely to occur under extremely dry conditions.

Schumann (1996b) and Schrader et al. (1997) each showed that the thermodynamic theory proposed by Hanson and Hanson (1995) is incorrect. In particular, Schumann (1996b) and Schrader et al. (1997) pointed out that the Hanson and Hanson algorithm does not adequately account for water vapor contributed by the combustion of jet fuel because it forecasts a *T*_{c} for contrail formation that is independent of the contrail factor at 0% relative humidity. Schrader et al. (1997) also stated that considerable evidence exists (e.g., Peters 1993) that contrails are not uncommon in very dry atmospheres. The data collected in the present study further demonstrate this. Schrader et al. (1997) provided a limited study based on 13 aircraft observations, which showed that significant differences exist between the *T*_{c} produced by the Hanson and Hanson algorithm and that produced by the Appleman (1953) forecast method.

Subsequently, Hanson and Hanson (1998) attempted to validate their proposed method using a contrail database of 1014 observations collected by Air Weather Service (Bjornson 1992) and a database of 10 low relative humidity observations they collected near Anchorage, Alaska. They claimed good results (100% agreement for the Anchorage observations) and stated that “complete agreement is indicative of the accuracy that may be expected if the data sample were to be taken for other regions as well.” Investigation of the accuracy of this claim is the main purpose of the current study. Hanson and Hanson (1998) used the premise that contrails are expected to occur when the dewpoint temperature *T*_{d} (not environmental temperature *T*_{e}) is lower than the critical temperature. This methodology is not physically intuitive and differs from the basic Schmidt–Appleman theory described above. As noted by Hanson and Hanson (1998), the radiosonde database they used is also somewhat deficient for validation purposes because only synoptic radiosonde data were available for comparison. The radiosonde data in many cases were up to 2° in latitude or 6 h in time different from the aircraft observations.

In this paper, results are presented from a contrail forecast validation study that used an asynoptic radiosonde database and ground-based aircraft observations obtained within 3 h of radiosonde ascent. The results provide a statistically significant comparison of the Hanson and Hanson (1995), Schumann (1996a), and Schrader (1997) algorithms. This database was collected during conditions in which the relative humidity was less than 30% (as measured by the radiosonde), so it provides a good test of the claims made in Hanson and Hanson (1998) for improved accuracy under low relative humidity conditions. For direct comparison with the claims in Hanson and Hanson (1998), validation of the Hanson and Hanson algorithm was carried out by comparing *T*_{c} against *T*_{d} (although there is no physical justification for doing this) as well as against *T*_{e} in the usual way as prescribed by Schmidt–Appleman theory.

## Data collection

Vaisala RS80 (and a few Atmospheric Instrumentation Research GPS-700) radiosondes were used to measure profiles of relative humidity and temperature near Wright–Patterson Air Force Base (WPAFB), Ohio, on 32 different observation days in 1996 and 1997. Launches were made at various times of day, from early morning to early afternoon, and were coordinated to coincide with the aircraft observations made from WPAFB. The nearest synoptic upper-air data from Wilmington, Ohio, (about 42 km away) were not used, because of the space and time differences involved.

Radiosonde data were collected approximately every 1.5 s during balloon ascent, which provided a high-resolution database. The accuracy of the relative humidity measurements from these soundings is discussed below. After the balloon ascended to commercial flight levels, local air traffic was observed within 3 h of balloon ascent. Contrail and aircraft observations were collected using spotting telescopes or a video camera with 120× magnification. For each aircraft observed, the aircraft type, airline, time overhead (or nearest to viewing position), direction of flight, and contrail conditions were noted. Federal Aviation Administration (FAA) flight logs were used to determine which aircraft could be positively identified.

The FAA flight logs provided the time of entry into and exit from the local area, the aircraft type, the flight plan type (overflying, landing, or departing), and the flight altitude for each aircraft. Aircraft that were not positively identified were eliminated from the database. Also, aircraft that were landing or departing were not used since their flight profiles were changing during observation. This procedure was adequate to positively identify 64% of the aircraft observed in 1996, but the database was biased toward contrailing aircraft because they were much easier to observe.

To alleviate this bias and collect a larger database in 1997, the WPAFB air traffic control (ATC) Digital Bright Radar Indicator Tower Equipment (DBRITE) was used to aid in aircraft identification. The DBRITE provided instantaneous readout of the aircraft’s call sign, type, and flight level, allowing ample time for visual identification and contrail observation. Virtually 100% of aircraft were positively identified in 1997, including a large number of noncontrailing aircraft. The result was a larger (318 observations with relative humidity less than 30%) and more balanced dataset (164 contrailing and 154 noncontrailing aircraft). The observations are summarized in Table 1.

Because the aircraft altitudes from the 1996 observations were obtained using only FAA flight logs, the overall quality of the flight logs was estimated using the 1997 aircraft-reported altitudes. The results showed the FAA-reported altitudes were very accurate (over 98% of the flight log altitudes matched the aircraft-reported altitudes exactly). Therefore, both 1996 and 1997 databases were combined to increase the sample size. Separate analysis of the 1996 and 1997 databases (not discussed here) provided similar results to the results discussed below.

## Forecast and verification procedure

### Critical temperature forecasts

To examine the claims made in Hanson and Hanson (1998), only observations with flight-level relative humidity less than 30% were used. The flight-level pressures and relative humidities were used to prepare critical temperature forecasts (really nowcasts) for each observed aircraft using the Schumann (1996a) and Schrader (1997) algorithms (hereinafter referred to as the Schumann and Schrader algorithms, respectively) and the Hanson and Hanson algorithm. The reader is referred to the original articles for further details of the respective algorithms and their implementation. As noted by Schrader et al. (1997), the equation for *P*_{8} in Hanson and Hanson (1995) and Hanson and Hanson (1998) must be corrected to duplicate the results shown in Hanson and Hanson (1995).

An iterative procedure was used to produce a critical temperature for contrail formation for each observation. For the Hanson and Hanson (1995) and Schrader algorithms, a constant contrail factor of 0.039 g kg^{−1} K^{−1} was assumed for high-bypass engines, and a contrail factor of 0.034 g kg^{−1} K^{−1} was assumed for low-bypass engines. These contrail factors imply *η* = 0.251 for CF = 0.039 g kg^{−1} K^{−1} and *η* = 0.141 for CF = 0.034 g kg^{−1} K^{−1} if the factor (*E*_{I}*C*_{p})/*Q* takes the values specified above. These constant contrail factors were chosen because they were the same contrail factors used in the AFWA contrail forecast algorithm (I. Laracuente 1997, personal communication). The effect of error in this choice of contrail factor on the critical temperature forecasts is discussed below. While the contrail factor in reality varies as flight parameters change (Busen and Schumann 1995), the values used here are assumed to be representative of the engine type of the aircraft that were observed. Critical temperatures computed using the Schrader (1997) and Hanson and Hanson (1995) algorithms were checked against published values. Values of *T*_{c} calculated using our implementation compared closely to Schrader’s (1997) values (e.g., values at 25 kPa using a contrail factor of 0.034 g kg^{−1} K^{−1} matched published values to two decimal places). The maximum difference between calculated values of *T*_{c} and published values in Hanson and Hanson (1995) (at pressures equal to or below 30 kPa) was 0.03°C.

As described above, the Schumann algorithm uses propulsion efficiency *η* to account for the conversion of heat to kinetic energy in the aircraft wake in estimating the slope of the mixing line. Using this method, the slope of the mixing line (*G* in Schumann’s notation) is a function of the emission index for water vapor of the aircraft, the specific combustion heat of the fuel, the pressure at flight level, the specific heat at constant pressure, and the propulsion efficiency of the aircraft. In this study the assumed emission index for water was 1.25 kg kg^{−1} and the specific combustion heat used was 43 MJ kg^{−1}, which are characteristic of a Boeing 747 burning kerosene (Schumann 1996a). Propulsion efficiency itself is a function of engine thrust, true airspeed, specific combustion heat, and rate of fuel flow (Schumann 1996a).

Because these factors can vary even under cruise conditions, it is problematic to choose the proper propulsion efficiency for a particular aircraft. In this study, propulsion efficiencies for low-bypass engine types characteristic of common airframe and engine combinations under cruise conditions were assumed (Schumann 1998, personal communication). A wide variety of aircraft with high-bypass engines were observed. For these aircraft, a propulsion efficiency of 0.34 (0.35 for Boeing 747) was assumed, which is approximately in the midrange of expected high-bypass propulsion efficiencies that range from 0.3 to 0.4 (Schumann 1996a). Propulsion efficiencies assumed in the current study are shown in Table 2. The effect of errors in these assumed values of propulsion efficiency must be considered in interpretation of the results and is discussed below. Comparison with published values of *T*_{c} in Schumann (his Table 3) showed that our calculated values of *T*_{c} were consistently about 0.25 too cold. These differences most likely result from differences in the method used to compute vapor pressure and its derivatives. For our calculations, the Goff–Gratch approximation was used to calculate vapor pressure, while Schumann used the Sonntag (1994) formula. The difference in vapor pressure resulting from these formulas is about 2.75 % at −60°C. Also, the derivatives of vapor pressure were calculated analytically, while Schumann (1996a) used centered finite difference approximations, which could result in small differences in implementing Schumann’s equations for calculating critical temperature at relative humidities other than 100%. Other differences may be due to small variations in the details of implementing the algorithms. These small differences are much smaller than the probable error in calculated values of *T*_{c} − *T*_{e} discussed below and do not affect the validity of the following results.

## Results

Forecasts for flight levels at environmental temperatures (*T*_{e}) colder than the critical temperature (*T*_{c} − *T*_{e} > 0) were categorized as “contrail forecast,” while flight levels where *T*_{c} − *T*_{e} < 0 were assigned “no contrail forecast.” As mentioned above, separate Hanson and Hanson (1995) forecasts were also made using *T*_{c} − *T*_{d} > 0 as the contrail forecast condition for comparison with the results in Hanson and Hanson (1998). These categorical forecasts were then evaluated subjectively and objectively.

### Subjective evaluation

For subjective evaluation, the distribution of correct and incorrect forecasts was examined as a function of the calculated values of *T*_{c} − *T*_{e}. Using the Schmidt–Appleman theory, contrails are expected to occur when *T*_{c} − *T*_{e} > 0, and contrails are not expected to occur when *T*_{c} − *T*_{e} < 0 (assuming the observations and forecast algorithms are perfect). Any deviation from this distribution must be caused by errors in either the forecast values of *T*_{c} or the observed value of *T*_{e}, with which *T*_{c} is compared. Potential sources of errors in *T*_{c} and *T*_{e} are discussed below. If these sources of error are not sufficient to explain the distribution of *T*_{c} − *T*_{e} for the incorrect forecasts, then it may reasonably be suspected that the forecast algorithm itself is in error.

Figure 1 shows the distribution of contrail (Con) and no contrail (NoCon) cases versus forecast *T*_{c} − *T*_{e} in 1°C bins for each of the three algorithms. Figure 1a shows that contrails occurred even when the Hanson and Hanson algorithm predicted *T*_{c} − *T*_{e} smaller than −30°C. The critical temperatures produced by the Hanson and Hanson algorithm were considerably colder (about 12°C on average) than those produced by either of the other algorithms. Contrails occurred 130 times when the Hanson and Hanson algorithm predicted *T*_{c} − *T*_{e} smaller than −4°C. Figure 1a shows that the Hanson and Hanson algorithm introduces bias in the calculated value of *T*_{c}. Also, the distribution of *T*_{c} − *T*_{e} does not conform to the distribution expected from the Schmidt–Appleman theory for any reasonable estimate of assumed errors in *T*_{c} or *T*_{e}. Figures 1b and 1c show the Schumann and Schrader algorithms predicted negative values of *T*_{c} − *T*_{e} only 20% and 27% of the time when contrails occurred, respectively. Of these, only two predictions of *T*_{c} − *T*_{e} smaller than −4°C occurred for each algorithm when contrails were observed. Similarly, the Schumann and Schrader algorithms predicted *T*_{c} − *T*_{e} larger than 4°C only once each when a contrail was not observed.

Figure 2 more clearly shows the distribution of incorrect forecasts versus *T*_{c} − *T*_{e}. From Fig. 2a, substantial numbers of incorrect forecasts occurred across the range of negative values of *T*_{c} − *T*_{e} using the Hanson and Hanson algorithm. From Figs. 2b and 2c, the values of *T*_{c} − *T*_{e} are clustered near zero using the Schumann and Schrader algorithms. A test of normality for the distribution of *T*_{c} − *T*_{e} for the incorrect Schumann and Schrader forecasts strongly supports the hypothesis that *T*_{c} − *T*_{e} is normally distributed with a Wilk–Shapiro score (Conover 1980) of 0.98 in each case. The mean value of *T*_{c} − *T*_{e} for incorrect forecasts using the Schrader algorithm was −0.7°C, with a standard deviation of 2.3°C. Corresponding values for incorrect Schumann forecasts are 0.1° and 2.1°C. For the Hanson and Hanson algorithm, the mean value of *T*_{c} − *T*_{e} was −12.7°C with a standard deviation of 8.4°C.

Figures 2b and 2c show that only three incorrect forecasts occurred for each of the Schumann or Schrader algorithms when the absolute value of *T*_{c} − *T*_{e} is greater than 4°C. Assuming *T*_{c} − *T*_{e} for the incorrect forecasts is normally distributed and using the standard deviations given above, *T*_{c} − *T*_{e} for the incorrect forecasts is between −2.2° and 0.8°C and −1.3° and 1.5°C 50% of the time for the Schrader and Schumann algorithms, respectively. The observations in which at least one of the Schumann or Schrader algorithms was incorrect and *T*_{c} − *T*_{e}|

Conservative estimates of the errors in this study are sufficient to explain the observed distribution of *T*_{c} − *T*_{e} for the incorrect forecasts by the Schumann and Schrader algorithms but not for the Hanson and Hanson algorithm. The errors in *T*_{c} − *T*_{e} result from errors in the flight–level pressure and relative humidity, incorrect choice of propulsion efficiency for the Schumann algorithm, incorrect choice of contrail factor in the Schrader algorithm (which is equivalent to an incorrect choice of propulsion efficiency, emission index for water, and/or heat of combustion in the Schumann algorithm), and/or error in the environmental temperature at flight level used to calculate *T*_{c} − *T*_{e}. (Error may also be introduced in the Schumann algorithm because of incorrectly assumed values of the emission index for water and the specific combustion heat; these errors are not further considered here.)

Errors in the flight-level pressure may result from errors in altitude reporting and altimeter errors. By FAA regulation (Spence 1997), the error in reported altitude at 9144 m (30 000 ft) could be as much as 93 m (305 ft), which corresponds to a pressure error of about 400 Pa. For the 318 observations in this study, a 400-Pa pressure error introduces a mean change in *T*_{c} of about 0.2°C.

Errors in propulsion efficiency and contrail factor can be more significant. An error in propulsion efficiency of +0.05 (−0.05) results in a mean change in *T*_{c} of +0.7°C (−0.7°C) using the 318 observations of this study. The high-bypass propulsion efficiency used could also easily be in error by this amount. Reasonable assumed errors in the contrail factor are also significant; a change in contrail factor of +0.005 g kg^{−1} K^{−1} (−0.005 g kg^{−1} K^{−1}) introduces a mean change in *T*_{c} of about +1.2°C (−1.3°C) using our dataset with the Schrader algorithm.

Critical temperature increases monotonically as relative humidity increases but is less sensitive at lower relative humidity than at higher relative humidity (e.g., see Fig. 3 in Schrader 1997). According to Leiterer et al. (1997), the RS80 radiosondes used in this study underreport relative humidity by about 5% (at −40°C) to 15% (at −60°C). A conservative estimate of relative humidity measurement error of −7.5% in our dataset leads to a mean change in *T*_{c} of about 0.3°C and 0.1°C using the Schrader and Schumann algorithms, respectively. This small difference between the Schrader and Schumann algorithms was unexpected and may be due to the slightly different methods used to calculate critical temperatures at relative humidities less than 100%. The error introduced by nonrepresentative relative humidity measurements could be much larger. Using our full dataset, the average difference between critical temperatures calculated for a relative humidity of 1% and a relative humidity of 100% was 9.3 K using the Schumann (1996a) algorithm. Most of this difference (7.8 K) occurred between 50% and 100% relative humidity.

Another likely source of error in *T*_{c} − *T*_{e} is due to inaccuracy in *T*_{e} resulting from measurement error by the radiosonde and spatial and temporal differences between the measurement and the observation. A conservative estimate of the error in *T*_{e} at flight level is ±2°C, and larger deviations are not unreasonable.

To estimate the effect of these potential errors on the forecast algorithms, critical temperature forecasts for the entire dataset of 318 observations were recalculated using the worst-case combinations of the errors described above (−7.5% relative humidity bias, *T*_{e} of ±2°C, contrail factor error of ±0.005 g kg^{−1} K^{−1}, and propulsion efficiency error of ±0.05). These calculations showed the errors estimated above were sufficient to explain approximately 88% (59/67) of the incorrect forecasts by the Schumann algorithm and 94% (59/63) of the incorrect forecasts by the Schrader algorithm. Additional consideration of the possible error introduced by nonrepresentative relative humidity measurements is sufficient to explain all of the error in the Schumann and Schrader algorithms by itself.

Using the Hanson and Hanson algorithm, the estimated errors were sufficient to explain only 30% (43/151) of the incorrect forecasts (correctly comparing *T*_{c} with *T*_{e} rather than *T*_{d}). The additional error due to the nonrepresentative relative humidity measurements is not sufficient to explain the poor performance of the Hanson and Hanson forecasts, as shown by the large number of incorrect forecasts with *T*_{c} − *T*_{e} less than −10°C in Fig. 2a.

### Objective comparison

As mentioned above, it is impractical regularly to produce forecasts for large numbers of aircraft (as the AFWA does) without using representative estimates of the mixing line slope (Schrader 1997). For this reason, it is of interest to military forecasters (and their customers) to examine the skill of contrail forecast algorithms under the assumption that representative flight parameters can be used (as is done here) to estimate the slope of the mixing line. Although the results achieved using this assumption do not represent the maximum possible skill of the algorithms under ideal conditions, the results are of interest because they are suggestive of the quality of the support that military operators might expect (assuming inputs of the same accuracy could be provided to the forecast algorithms). For this reason, the performance of the algorithms was further quantified objectively using 2 × 2 contingency tables (described in the appendix).

Table 4 shows the results of the Schumann, Hanson and Hanson (1995), and Schrader algorithms for the 318 cases in which the relative humidity at flight level was less than 30%. The Schumann and Schrader algorithms achieved similar hit rates of 0.79 and 0.80, respectively. This similar result is expected because these two algorithms essentially differ only in the way in which the slope of the mixing line is estimated. The Hanson and Hanson algorithm achieved a hit rate of 0.53 when *T*_{c} was compared with *T*_{e}, and 0.62 when *T*_{c} was compared with *T*_{d} as in Hanson and Hanson (1998). The Hanson and Hanson (1995) algorithm produced a very small number of “yes” forecasts under dry conditions when *T*_{c} was compared with *T*_{e}; most of the correct forecasts occurred in conditions where contrails did not occur. This is reflected in the low critical success index and low false alarm rate (due to the low number of forecast events). As shown above, the critical temperatures produced by the Hanson and Hanson algorithm were very cold when compared with the critical temperatures produced by the Schumann and Schrader algorithms, so that few yes contrail forecasts were produced by the Hanson and Hanson algorithm. All contingency tables are statistically significant with *p* values <0.01.

The Hanson and Hanson algorithm can be expected to perform better when *T*_{d} is used as the basis of comparison, as in Hanson and Hanson (1998), although there is no physical justification for using this technique. In this study, when *T*_{c} was compared with *T*_{d}, the Hanson and Hansons algorithm achieved a higher hit rate but produced a larger number of false alarms. Table 4 shows the Hanson and Hanson algorithm incorrectly forecast contrails to occur (*T*_{d} − *T*_{c} < 0) for 120 of 154 cases in which contrails were not observed. As shown in Table 4, the observed *T*_{d} was colder than the Hanson and Hanson forecast *T*_{c} 163 times for the 164 events in which contrails were observed to occur, while *T*_{e} was colder than *T*_{c} for only 14 of the 164 events. This contributed to the higher hit rate for the Hanson and Hanson algorithm (0.62 versus 0.53) when *T*_{d} − *T*_{c} < 0 was used as the forecast criteria. For cases in which contrails do not occur, the Schmidt–Appleman theory predicts that *T*_{e} > *T*_{c}. Therefore, *T*_{d} may be warmer or colder than *T*_{c}, and less accurate forecasts will be produced with the Hanson and Hanson algorithm using *T*_{d} as the basis of comparison. This is also shown in Table 4. When contrails were not observed in this study, correct no contrail forecasts were made by the Hanson and Hanson algorithm in 153 of 154 events using *T*_{e} as the basis of comparison, and in only 34 of 154 events when *T*_{d} was compared with *T*_{c}.

The skill of each algorithm was estimated using the Heidke Skill Score (HSS) and Kuiper’s Skill Score (KSS) (described in the appendix). These results are also shown in Table 4. Both skill scores show the Hanson and Hanson algorithm was marginally skillful in comparison with the reference forecasts (also described in the appendix) when *T*_{c} was compared with *T*_{e}. The Schumann and Schrader algorithms both achieved more skillful forecasts (Table 4). Although the skill of the Hanson and Hanson (1995) algorithm was slightly better when *T*_{c} was compared with *T*_{d}, it still showed relatively low skill in comparison with the other two algorithms.

## Conclusions

These results support the findings of Schrader et al. (1997), who showed significant differences between the critical temperatures produced by the Hanson and Hanson algorithm and those produced by correct application of the Schmidt–Appleman theory. As shown here, these differences are significant at low relative humidity. These results are statistically significant and do not support the claims made in Hanson and Hanson (1998) for improved accuracy of their algorithm at low relative humidities. The Schumann (1996a) and Schrader (1997) algorithms are principally the same. Any differences in the results of the two algorithms are solely due to different input values, in particular for the contrail factor or the propulsion efficiency and the ratio of water vapor emissions per combustion heat of the fuels used, and (perhaps) small differences in the numerical implementation of the algorithms.

## Acknowledgments

We thank Steve Weaver, Mike Kapel, and John Polander of the 88th Weather Squadron at Wright–Patterson AFB who provided radiosondes and balloons and assisted in radiosonde launches. We also appreciate the assistance of Wright–Patterson Air Force Base Air Traffic Control personnel. Thanks also to the anonymous reviewer who suggested numerous improvements, and to Ulrich Schumann who provided many helpful comments and suggested the low-bypass propulsion efficiencies in Table 2.

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## APPENDIX

### Summary of Statistics Used in Contingency Tables

The forecasts were placed into one of four categories:*a* (contrail forecast and observed), *b* (contrail forecast but not observed), *c* (contrail not forecast but observed), and *d* (contrail not forecast and not observed). Based on these assignments, the following statistical measures of accuracy were calculated following Wilks (1995).

*Hit Rate.*The hit rate was calculated as (*a*+*d*)/*n,*where*n*is the total number of observations. This represents the percentage of forecasting opportunities in which the forecast method correctly predicted the observed event.*Critical Success Index (CSI).*The CSI only considers events in which contrails were forecast or observed. Therefore, it represents the fractional hit rate once the correct “no” forecasts are removed. Also known as Threat Score, the CSI was calculated as*a*/(*a*+*b*+*c*).*Probability of Detection (POD).*The POD is the number of events observed that were forecast, or*a*/(*a*+*c*).*False Alarm Rate (FAR).*This gives the proportion of events forecast that did not occur. FAR was computed using*b*/(*a*+*b*).*Bias Ratio.*This is a comparison of the average forecast with the average observation and is the ratio of yes forecasts to yes observations. The bias ratio was calculated using (*a*+*b*)/(*a*+*c*). A perfectly unbiased forecast has a bias ratio of one.

Forecast verification data in contingency tables are also commonly characterized using measures of relative accuracy or skill (Wilks 1995). The following measures of skill were calculated.

*Heidke Skill Score (HSS).*This score uses the hit rate as the basic measure of accuracy. As described by Wilks (1995), the reference accuracy measure used in the HSS is the hit rate achieved by a random forecast, where the marginal distributions of the randomly generated forecasts and observations are the same as the actual verification dataset. Perfect forecasts have an HSS of one, forecasts equivalent to the reference forecasts have an HSS of zero, and a negative HSS indicates forecasts that were less skillful than the random forecasts. Following Wilks (1995), the HSS is calculated as*Kuipers Skill Score (KSS).*The reference forecasts used in the KSS are random forecasts, which are constrained to be unbiased (Bjornson 1992; Wilks 1995). Therefore, the hypothetical reference forecasts have the same probability distribution as the climate data of the sample, and the probability of forecasting an event is considered to be the same as the probability of observing an event. Following Wilks (1995), the KSS is calculated as

For the measures of accuracy, bias, and skill to be meaningful, it must be shown there is an association between column classification (contrails observed) and row classifications (contrails forecast), using a test of independence. For this purpose, Pearson chi-square and *p* values were calculated for each contingency table. If the *p* value was less than 0.01, the notion of independence was rejected and the contrail condition observed was declared to be dependent on the forecast category. All of the results presented here had *p* values less than 0.01, so the results are considered to be statistically significant at the 1% level. Because the number of expected observations in some cells of the tables was very small, the Fisher–Irwin exact test (Conover 1980; Sachs 1984) was also used to ensure statistical significance. All contingency tables passed the Fisher–Irwin exact test, also at the 1% confidence level.

Number of incorrect forecasts vs forecast *T*_{c} − *T*_{e} for each algorithm: (a) Hanson and Hanson, (b) Schrader, and (c) Schumann. Hanson and Hanson forecasts compared critical temperature *T*_{c} with *T*_{e} following Schmidt–Appleman theory.

Citation: Journal of Applied Meteorology 39, 1; 10.1175/1520-0450(2000)039<0080:ACOECT>2.0.CO;2

Number of incorrect forecasts vs forecast *T*_{c} − *T*_{e} for each algorithm: (a) Hanson and Hanson, (b) Schrader, and (c) Schumann. Hanson and Hanson forecasts compared critical temperature *T*_{c} with *T*_{e} following Schmidt–Appleman theory.

Citation: Journal of Applied Meteorology 39, 1; 10.1175/1520-0450(2000)039<0080:ACOECT>2.0.CO;2

Number of incorrect forecasts vs forecast *T*_{c} − *T*_{e} for each algorithm: (a) Hanson and Hanson, (b) Schrader, and (c) Schumann. Hanson and Hanson forecasts compared critical temperature *T*_{c} with *T*_{e} following Schmidt–Appleman theory.

Citation: Journal of Applied Meteorology 39, 1; 10.1175/1520-0450(2000)039<0080:ACOECT>2.0.CO;2

Summary of all 1996 and 1997 daily observations. Only observations with relative humidity <30% were used for calculation of critical temperatures.

Assumed propulsion efficiencies and contrail factors used with Schumann and Schrader algorithms, respectively.

Ambient conditions at flight level and forecast values of |*T _{c}* −

*T*| where either Schumann or Schrader forecast was incorrect, with (|

_{e}*T*−

_{c}*T*| > 1°C). Contrail factor and propulsion efficiencies used were as given in Table 2.

_{e}Objective verification statistics for Schumann, Hanson and Hanson, and Schrader algorithms for cases with radiosonde measured flight-level relative humidity <30%. Total number of observations is 318 (164 contrail, 154 no contrail). Hanson and Hanson forecasts were made by comparing the critical temperature (*T _{c}*) with the environmental temperature (

*T*) and the dewpoint temperature (

_{e}*T*). Statistical measures are described in the appendix.*

_{d}