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_IC_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.

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 P₈ 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_IC_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.

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| > 1°C are summarized in Table 3.

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.