The parametrization of supersaturation w.r.t ice in the UTLS is based on the assumption that ice crystals are formed in most cases by the homogeneous nucleation process when the temperature is lower than 235 K. We introduce a saturation ratio coefficient k which is a function of the average temperature in the grid-box T¯ and represents the ratio of saturation to be reached to have homogeneous nucleation processes activated in the grid-box. For temperatures colder than 235 K, the saturation ratio coefficient is set such as 1k=Ccalib⋅f(T¯),f(T)=2.583-T/207.8K where f(T) represents the homogeneous nucleation threshold as a function of T see e.g., and Ccalib is a calibration coefficient used for the tuning of the model. From 235 K, a linear return is made back to k=1 at 250 K (k=1 for T¯≥250 K). This implies that the range of temperature where supersaturation is allowed in our study is set for temperatures colder than 250 K collocating with the pure ice phase limit within ARPEGE microphysical scheme, this range being compatible with persistent contrail applications. In higher temperature conditions where mixed-phase or pure liquid conditions can exist, the ARPEGE cloud scheme is not directly impacted by the modification.

The second step is to write the local thermodynamic adjustment equation to include the saturation ratio coefficient in the classical scheme. The following assumptions are used:

when ice supersaturation conditions are allowed (i.e. k>1), condensate can exist within a grid-box for any point where the saturation ratio k is locally exceeded.

The difference between the modified and the original ARPEGE scheme relies in the introduction of variable k in Eq. () to define the nucleation threshold, and the following consists in reworking the original cloud scheme equations with k>1. To calculate the mean condensed content and the mean cloud cover in the grid-box, a statistical description of the distance to supersaturation qt-kqsat in the grid-box is used. We introduce the notation x¯ (resp. x′) to represent the grid-box mean value (resp. the local perturbation) of a variable x. Following the cloud distribution concepts developed by and , and assuming that pressure fluctuations are negligible in the grid-box , the local difference between qt and kqsat(T,p) can be expressed by 3qt-kqsat(T,p)=Qc,k+s, where Qc,k characterizes the average distance to supersaturation (k>1) resp. saturation (k=1) in the grid-box, such as 4Qc,k=aL,kq¯t-kq¯sat(T¯L,p¯), and s represents the local deviation around the value of Qc,k and can be described by 5s=aL,kq′t-kbLTL′, with L being the latent heat of condensation, cp the specific heat capacity of air at constant pressure, and TL=T-(L/cp)⋅qc the liquid water temperature which is conserved through the condensation process. The factors aL,k and bL account for changes in qsat due to latent heating and are given by 6aL,k=(1+kbLL/cp)-1,7bL=εLq¯sat(T¯L,p¯)/RT¯L2, where R is the dry air specific gas constant and ε is the ratio of dry air constant and vapor constant.

The local deviation s is statistically modeled by a centered G-distribution with a standard deviation σs,k such as s∼G[0,σs,k]. This implies that qt-kqsat follows a G-distribution with a mean equal to Qc,k and a standard deviation σs,k which gives 8qt-kqsat∼G[Qc,k,σs,k].

The mean cloud fraction C+ and mean cloud condensate content q¯c+ after adjustment can be expressed in relation to the centered and reduced probability distribution G[0,1](t) (see calculations in Appendix ), such as 9C+=∫-Qc,k/σs,k+∞G[0,1](t)dt,10q¯c+=σs,1∫-Qc,k/σs,k+∞G[0,1](t)t+Qc,1σs,1dt. It is important to note that steps from Eqs. () to () do not depend on the probability distribution that is chosen, hence the methodology can be applied to cloud schemes based on the same type of statistical distribution concepts.

The cloud scheme used in ARPEGE assumes a symmetric triangular probability function for the setting of the s distribution. Applying this methodology in our context, the probability density function is represented by a symmetric triangular distribution with a finite support, its lower and upper limits spanning from -6σs,k to 6σs,k, as represented in Fig. . The cloud fraction is non-zero when -Qc,k<6σs,k indicating the presence of condensed water when the critical point corresponding to -Qc,k=6σs,k is obtained. This equality can be used to define the critical relative humidity threshold U¯c,k=q¯v/q¯sat(T¯,p¯). Evaluating Qc,k at this critical point where q¯t=q¯v and T¯L=T¯ leads to 11σs,k=aL,k6q¯sat(k-U¯c,k). This expression exhibits a direct relation between the width of the distribution σs,k and the critical relative humidity U¯c,k. This allows the cloud scheme closure by the definition of a critical relative humidity threshold, as the s distribution is fully determined by its standard deviation. Defining αk=Qc,k/σs,k6, the integration of Eqs. () and () gives an explicit algorithm (see Algorithm ) for the computation of C+ and q¯c+ (see Appendix  for calculations). Setting k≡1 returns to original Smith cloud scheme.

The final closure of the system of equations can be given by an arbitrary definition of Uc,k for all values of k. However, some simple limiting cases in the s formulation given in Eq. () can be exhibited and allows to restrict the closure to the choice of Uc,1: Closure 1

– If predominance of humidity variability term over temperature variability term is assumed in the s formula i.e. qt′≫kbLTL′, then σs,k/σs,1 is equal to aL,k/aL,1 and closure is given by: 13U¯c,k=U¯c,1+k-1.

The operational version of the ARPEGE model runs 4 times a day with a stretched native grid of 5 km above Europe and 25 km over Oceania and with a 105 level vertical grid. In the following we will denote “ARP-new” the ARPEGE model with the modified cloud scheme, and “ARP-op” the version of the model with the operational (non-modified) cloud scheme. We also perform the verification of RH_ice provided by the high resolution (HRES) configuration of the IFS model operated by ECMWF cy48r1. It has a mean native resolution of approximately 9 km worldwide and 137 vertical levels. ARP-op and IFS forecasts start from their own respective operational analysis. The ARP-new forecast starts from the analysis of ARP-op.

For all NWP models, runs of 00:00 and 12:00 Z are used, and hourly time steps from T+6 to T+17 are considered for output lead terms. Each output is interpolated on a regular 0.25° lat/lon horizontal grid. The vertical output grid is composed of 21 vertical pressure levels corresponding to flight levels ranging from FL250 to FL450 with intervals of 1000 ft (304.8 m).

IAGOS is a European non-profit association involving research organizations, universities and weather services from Germany, France and the U.K, which provides in situ commercial aircraft observations in relation with air quality and climate change e.g.. Ambient air temperature and relative humidity measurements are one of the main IAGOS-core data products. They are measured by a platinum resistance sensor (Pt100) and a capacitive sensor (Humicap-H, Vaisala, Finland), respectively, which are combined into the so-called ICH sensor (IAGOS Capacitive Hygrometer ). The accuracy of the instruments used for RH measurements is considered approximately 5 % with typical 1–3 min instrument response time (≃15–50 km of flight if a cruising speed of 250 m s⁻¹ is considered) e.g.. Aircrafts equipped with IAGOS sensors operate at a cruising altitude between 9 and 13 km, providing a comprehensive in situ dataset of UTLS humidity on a global scale, although northern extra-tropical and, in particular, Europe and Northern Atlantic flight paths are more frequently sampled.

The IAGOS observational dataset and NWP models have very different resolutions. Indeed, the horizontal outputs of the model data is 0.25° (20–25 km at mid-latitudes), while the output frequency of the observations is 4 s along the flight track, resulting in a spatial resolution of approximately 1 km between adjacent observed points, considering regular cruise speeds. In order to have comparable resolutions for the needs of calibration and verification, the IAGOS dataset is smoothed and under-sampled. A temporal interval of 100 s with a centered mean-filter window is applied to obtain an under-sampled resolution of ∼ 25 km. Nearest point interpolation in time and space dimensions is then used to assign each observation to the NWP outputs in closest proximity.

The RH_ice histogram with the associated frequency bias for RHice>100 % (Fig. a) and the Mean Absolute Error (MAE) (Fig. b) for different observed humidity categories are calculated for the verification dataset and for three models: ARP-new, ARP-op and IFS. Model results are compared with filtered and unfiltered IAGOS observations, the latter being used to compute the distribution plots of direct model errors (Fig. c).

Figure a shows that ARP-new permits supersaturation, exhibiting a close alignment with the observed RH_ice histogram, although it does not allow supersaturation exceeding 115 %, as underlined in the calibration study (Sect. ). The frequency bias of ARP-new for RH_ice events greater than 100 % is neutral. ARP-new offers enhanced reliability in comparison to ARP-op, which shows a peak at 100 %, standing out from the rest of the curve and followed by a precipitous drop beyond 100 %, indicating that supersaturation is not forecast. Moreover, ARP-new gets to remove the over-representation of the 70–100 % RH_ice categories of ARP-op. Concerning IFS, as shown in previous studies e.g., the results show a pronounced peak at 100 % and an under-estimation for higher RH_ice values evidenced by the frequency bias (0.81) on RH_ice events greater than 100 %. IFS has a significant shortcoming when confronted with values exceeding 110 %. We note that there is no significant difference between filtered and unfiltered observation histograms, showing that applying a 100 s mean-filter does not alter the properties represented in the original IAGOS dataset.

While the histogram gives a general idea of the distribution of the data, the MAE analyzes each observed RH_ice point and its corresponding forecast as to how close they are in value and provides an average of the pointwise model's accuracy. Therefore, the conclusions of both are complementary. In Fig. b, ARP-new shows lower MAE values than ARP-op for RH_ice categories above 95 %, while in lower categories the MAE is just slightly higher. This suggests that most of the values removed from the over-representation of the 70 %–100 % categories in ARP-op, have been correctly adjusted to represent supersaturated values in ARP-new. For categories ranging from 95 % to 105 %, the IFS model shows the lowest MAE values across all models, however, exhibiting a steady increase for categories above 105 %. In fact, categories around 100 % are significantly over-forecast by IFS, ensuring a lower amount of error for this specific categories of observations. In return, there is a strong MAE increase for under-predicted categories over 105 %. A similar trend is observed in ARP-new, however, for higher RH_ice, around 115 %, i.e. precisely when the histogram reveals a shortcoming. As a result, ARP-new exhibits the lowest MAE for higher supersaturated values.

In the analysis of the direct model error, Fig. c shows that ARP-new has a median bias centered on 0, slightly better than the median bias of ARP-op and IFS. However, IFS shows a narrower interquartile range (11.06 %), compared to (13.87 %) for ARP-new. This indicates that, after bias correction, IFS has a slightly better overall accuracy in representing humidity.

In Sect. , the RH_ice threshold has been set at 100 % in order to identify ISSRs in both forecasts and observations. However, there is a high interest in studying the sensitivity of scores on ISSR threshold definition for both forecast and observations:

Regarding forecasts, varying the RH_ice threshold provides an overview of the NWP model performance range and should be considered for a complete evaluation of NWP model discriminatory skills. Since models are not perfectly calibrated to observations, there may be forecast threshold values that offer better scores than the 100 % threshold regarding the end-user needs (see e.g. who used a threshold lower than 100 % for ISSR detection in ERA5 models, and and who used post-processing recalibration of the RH_ice parameter).

In order to perform this sensitivity study, a robust approach in the context of imbalanced datasets is to use a graphical tool called the Precision-Recall curve . The Precision-Recall curve is drawn by calculating the recall (i.e. hit rate) and precision (i.e. 1-FAR) scores at several RH_ice thresholds for the forecast. In general terms, the curve closest to the top/right corner corresponds to the best discrimination capabilities, where precision and hit rate (recall) are optimal, i.e., equal to 1. This graphical tool is roughly equivalent to the widely used Receive Operation Characteristic (ROC) curve, but the Precision-Recall curve is better used in the context of rare events .

In Fig. , we show the Precision-Recall curves computed with no neighborhood tolerance. We represent four distinct plots, each corresponding to a different RH_ice threshold for the observations: 95 %, 100 %, 105 % and 115 %. We include ARP-op in this plot because, although it does not permit supersaturation, it may be capable of detecting ISSRs with a threshold of less than 100 %. For each curve, we calculate the Average Precision (AP), which is defined as the area under the Precision-Recall curve. The AP offers a concise overview of the Precision-Recall curve, facilitating direct comparisons between models and their scores across varying RH_ice ranges. Moreover, we represent the diagonal line where neutral frequency bias is obtained.

The results obtained with the observation threshold of 100 % (Fig. b) indicate that all three models have close APs, with an advantage for IFS. In addition, IFS exhibits the highest precision and recall scores close to the diagonal where neutral frequency bias is obtained. A comparison of the APs obtained with the 95 % threshold (Fig. a) reveals an improvement for all models with IFS being significantly better in the diagonal region. At higher humidity ranges, with the threshold of 105 % and 115 % (Fig. c and d), the APs are significantly lower for all the models, with ARP-new and ARP-op now performing the best. When comparing APs obtained for relative humidity above 100 % versus 115 %, a decrease of approximately 60 % in scores is revealed for all models.

The comparison of ARP-new model with ARP-op model shows that both models have a similar ability to discriminate ISSRs. For example, forecast threshold of 90 % in the non-modified ARPEGE corresponds to broadly the same HR and precision values as a threshold of 100 % with the modified cloud scheme. We conclude that the modified scheme strongly enhanced the reliability of the ARPEGE forecast of ISSR (as shown by Fig. ), while being neutral on its inherent discrimination skills.

A comparison of the monotony of the Precision-Recall curves reveals that ARP-new and ARP-op exhibit divergent behavior compared to IFS. The ARPEGE curves are quasi monotonic, indicating that an increase in the decision threshold of the forecast is accompanied by an increase in precision. In contrast, IFS does not follow the same behavior. From a certain threshold close to 98 %, an increase in the decision threshold results in a decline in both precision and recall. This penalizes the global AP score for IFS, while it exhibits its maximum potential of ISSR discrimination for thresholds close to 98 %.

In addition to providing an overview of the performance of each model, the Precision-Recall curves can also be used to adjust the decision threshold in line with specific preferences regarding non-detections and false alarms. For example, should a pointwise HR of 0.6 be requested, we can ascertain the RH_ice threshold that should be selected in each model to discriminate ISSRs and the pointwise precision that will be achieved. If we fix the observation threshold at 100 % and aim for a HR of 0.6, we will have to choose RH_ice = 97 % as the threshold for the ARP-new forecast, giving a precision of 0.49. Similarly, we would have to choose a threshold of 87 % for ARP-op, with a precision of 0.47, and 97 % for IFS, with 0.55 precision.

Finally, it should be noted that the Precision-Recall curve can also be calculated with a neighborhood tolerance, following the methodology presented in Sect.  to calculate the HR and FAR.

In this section, we investigate a transatlantic IAGOS flight from 23 September 2022 between Frankfurt (Germany) and Washington D.C. (USA). A comparison is made between the observed and forecast ISSRs using IAGOS measurements and NWP data from both ARP-new, ARP-op and IFS. Figure  shows RH_ice along the IAGOS flight path. Take-off and landing are not included in this graph, as flight levels below FL250 are not of interest for this study.

There are two distinct areas where the values are greater than 100 %, shown in orange and red (Fig. a). The first is over the Atlantic Ocean (region A), and the second is the vicinity of Newfoundland, Canada (region B). In Fig. b, the time series show the temporal evolution of RH_ice, with ISSR events highlighted in bottom plots to facilitate comparison between observations and forecasts. In this case, the ARP-new model has a very good agreement with the occurrence of ISSR events in regions A and B, particularly in region B. In the region A, RH_ice reveals that the ARP-new curve is slightly below the ISSR threshold while the observations start being supersaturated, yet leading to “miss” events when, in fact, the global view of the episode is well predicted. This demonstrates the value of the neighborhood approach as a means of correctly interpreting scores. ARP-op, as it does not predict RH_ice values higher than 100 %, does not detect the ISSRs. However, it follows closely the time evolution of observations, even though it remains below the observations for RHice>100 %. With regard to IFS, while the plot comparing the ISSRs indicates that it did not fully predict the episode, the relative humidity evolution reveals a pattern very closely aligned with the observations, but with a deficiency in reaching higher supersaturated values, confirming the IFS under-prediction of high values of RH_ice, as shown in Fig. . In the case of IFS, introducing a neighborhood tolerance in the verification process allows to counterbalance this under-representation of ISSR and exhibit the potential of detection capabilities of this model.

Figure provides a zoomed spatial view of regions A and B. ISSR observed events in the flight trajectory are compared to ARP-new forecast at corresponding flight levels, and the presence of multiple persistent contrails is shown on the Cirrus Reflectance Product from Suomi NPP/VIIRS at times close to the flight path. This example highlights the good agreement between observations and forecast and confirms the potential of what could be made using NWP model forecasts to predict environments favorable to persistent contrail triggering.

A modified cloud scheme for ARPEGE NWP global model is presented in this work to enhance the representation of relative humidity w.r.t ice, and in particular supersaturation, which is a necessary condition for contrail persistence. The modeling is based on a generalization of the Smith cloud scheme currently used in the operational ARPEGE. This modification can be implemented without any major modification and does not imply supplementary computational effort. A notable point is that the methodology developed in this article reworks the statistical concepts while incorporating ISSR parametrization, which allows for extensions to other atmospheric models using a similar framework. This is the case, for example, of the ICE3 microphysical scheme implemented in the French regional NWP model AROME , which uses a turbulence-based Gaussian distribution for the representation of clouds, and which could benefit from these developments.

Some common assumptions are shared with the parameterization implemented in the IFS model , in particular the use of a temperature function based on to describe the ice nucleation threshold, and the assumption that once this threshold is locally exceeded, local adjustment is instantly obtained back to saturation in the cloud. However, there are major differences between the two models in terms of their respective microphysical frameworks. For example, cloud fraction is diagnostic in ARPEGE and prognostic in IFS. These different frameworks also implies differences in the representation of subgrid variability: IFS uses a dedicated distribution combining uniform (out-cloud) and Dirac (in-cloud) distributions, compatible with its prognostic cloud fraction parametrization, while ARPEGE uses a triangular distribution. The ARPEGE model also differs from other GCMs that allow supersaturation, such as ECHAM or CAM see e.g.. These models are used for climate prediction and include supersaturation w.r.t ice, generally associated with a parametrization of contrail cirrus. In both cases, the triggering of homogeneous nucleation is proposed in a similar manner, mainly driven by a temperature function, but the thermodynamic in-cloud adjustment to saturation is relaxed, within a more complex framework of a 2-moment microphysical scheme.

In the new scheme, some assumptions are made regarding the introduction of supersaturation in ARPEGE cloud scheme, most of them having already been discussed in and . For example, in the modified scheme, the adjustment in the cloud to ice saturation is assumed to be instantaneous, which is probably a major limitation of the physical description of ice representation in our context and an explanation of the cut-off on the highest supersaturation observed in the RH_ice distribution histogram (Fig. a). Indeed, it has been shown that RH_ice decreases with a relaxation time that can exceed several time steps, to finally reach a few percent above saturation, thus allowing local in-cloud supersaturation w.r.t ice. Taking into account pre-existing ice and a better description of the physics of the vapor deposition should then improve the representation of the highest values of ISSR. proposed a cooling-cloud formation-phase relaxation process as an alternative to instantly adjusting to the equilibrium value. This modification requires the addition of new prognostic variables describing the history of in-cloud supersaturation, which adds further complexity, but could result in a better description of RH_ice distribution for the highest values. Regarding cirrus cloud formation, homogeneous nucleation is considered dominant compared to heterogeneous nucleation at the altitudes and temperatures of interest for the contrail application. It can be justified by the presence of relatively pristine environments with few ice nucleating particles see e.g.. Thus, a threshold with temperature dependence issued from is used to represent the maximum local value of supersaturation which can be obtained before homogeneous nucleation processes are activated. highlight that for slow updrafts the amount of generated ice crystals could however become large enough to reduce the supersaturation limit. This could lead to incorporate the vertical velocity in the expression of this coefficient. Regarding the use of microphysical 2-moment schemes, we acknowledge that cloud formation processes can be better represented in NWP models involving more detailed physical processes . For GCM climate models where contrail clouds are parametrized e.g., this type of scheme has proven useful for estimating the physical and optical properties of contrails, such as optical length see e.g.. However, there is currently no consensus that obtaining a satisfactory representation of RH_ice in the UTLS necessarily requires the introduction of a second moment into the microphysical scheme, and the balance between computational costs and benefits in terms of improved forecasts must be carefully evaluated before implementing such a complex scheme in an operational NWP system.

We decided to focus this study on the Europe and North Atlantic domain due to the high density of aircraft in this area, which provides a large number of available observations . However, we believe that this work could be extended to tropical regions. In this case, it could be calibrated accordingly, for example by modifying the Ccalib coefficient (see calibration procedure, Sect. ), given that tropical climatology differs from mid-latitude regions and that the occurrence of ISSR depends on factors such as dominant deep convection . Similarly, North America and Asia are large continental areas with large mountain ranges, and the distribution of processes leading to ISSR in the atmosphere (e.g. warm conveyor belt, convection) is likely to differ from European/North Atlantic climatology, which could alter the results.

The final steps towards operationalizing the modified scheme are to verify the impact on the general parameters of the global NWP model (temperature, wind, etc.) and to adjust all the parameterizations accordingly. Indeed, the introduction of supersaturation implies a reduction of the cloud cover and cloud ice contents in the altitudes which are impacted by the modification. This change will have a direct impact on temperature through changes in radiative and latent heat transfers, with an expected cooling effect, mainly in the upper troposphere (see Appendix ). The intensity of these impacts, their consequences for the dynamics, and whether they benefit or not for ARPEGE will be assessed in a dedicated forthcoming study.

In this work, particular attention has been paid to the definition of appropriate verification metrics. In addition to traditional verification methods on humidity variable (e.g. Frequency histograms, Mean Absolute Error), the statistical results emphasize the importance of adopting a neighborhood approach when evaluating the capacity of a model to discriminate ISSRs. Introducing a neighborhood tolerance allows to link the quality of the forecast to a spatial scale, as well as addressing several scoring issues (e.g. double penalty, observation uncertainty, interpolation on gridded data). The trajectory-neighborhood method is intuitive, simple to implement and results can be interpreted in the frame of a flight trajectory. However, it does not include forecast information on the adjacent vertical levels, nor of points to the right or left of the flight trajectory. Consequently, the method compromises between maintaining symmetry in dealing with forecasts and observations and ensuring straightforward interpretation with the flight trajectory, while disregarding the potential to consider forecasts of ISSRs that may be offset to one side of the flight trajectory.

The verification metrics HR, FAR and Fβ-score with the use of a neighborhood tolerance, give easily interpretable information on spatial scales at which a certain degree of correspondence between observed and forecast events is achieved, but this methodology can hide biases and should be completed by its estimation to have a complete overview of the performances of the forecasts. The FSS score is less directly interpretable, but provides a summary of all the factors that can assess NWP model quality (including true negatives in the verification). The Recall-Precision plot has shown to be a valuable tool to measure the sensitivity of different NWP models to RH_ice decision threshold, and provide a full NWP model inter-comparison outcome. Further verification metrics could also be explored by methods based on neighborhood-populating contingency tables e.g., each method having their qualities and drawbacks regarding for example inconsistent event definitions or keeping symmetry between hit and false alarm calculations. Other advanced spatial verification techniques, such as object-based methods could provide further information on NWP model quality, for example by targeting most significant ISSR features. However, implementing this kind of methodology can present some challenges regarding mathematical data processing to obtain geometrically coherent ISSR features.

In future works, other observational dataset could be useful for further verification of the NWP models. IAGOS dataset is open-accessible and quality-checked, relevant for humidity measurements in the UTLS. However, there is an heterogeneity in sampling due to the operational flight paths of participating airlines . The highest measurement density is found across the North Atlantic flight tracks, North-Eastern America and Europe at cruise altitudes (200 to 250 hPa). In addition, the sampling is biased by avoiding aviation hazards (e.g. deep convective clouds and the outflow of such clouds, turbulent regions) and by route optimization according to the jet stream. Radiosonde observations which mainly sample the atmosphere above continental surfaces represent a valuable complementary database in world regions or weather situations where IAGOS data are less represented. This observation source also makes it possible to assess humidity at levels above or below typical cruising altitudes see e.g.. Also, contrails tracked using satellite imagery, ground-based cameras and lidars , along with flight and ground-based contrail observations COOP Program represent an additional data source. However, verification is less direct, as both contrail formation (Schmidt-Appleman) and its persistence (ISSR) must be considered together.

This study shows that ARP-new has an enhanced agreement with RH_ice observations from IAGOS measurements in the UTLS when compared to ARP-op. Results of Sect.  show that it corrects the dry bias of RH_ice above 70 % and overcomes the limit at 100 % that the operational model exhibits. However, it over-predicts slightly occurrences of RH_ice in the range of 105 % to 115 % and under-predicts values above 115 %. As a confirmation of previous studies, IFS exhibits also a capability to produce supersaturation w.r.t ice, but with an overestimation of values very close to 100 %, a general underestimation of ISSRs, and a shortcoming in representing values above 105 %.

In terms of model performance to detect ISSR events, ARP-new and IFS models show close skills as shown by the FSS score verification in Sect. . For ARP-new, HR and FAR results indicate that when ISSR was observed in IAGOS, an ISSR was forecast in a 150 km upstream or downstream the trajectory (i.e. 10 min of flight) in ∼ 80 % of the cases. Symmetrically when an ISSR was forecast, ISSR event could be observed in ∼ 70 % of the cases. This indicates a rather good spatial correspondence between forecasts and observations. For IFS, the scores are even better but must be tempered because of the IFS dry bias which is artificially corrected when neighborhood tolerance is applied. These statistical results are well illustrated by the case study presented in Sect. . On this case, the correspondence between forecast and observation patterns is exhibited for both ARPEGE and IFS models on RH_ice spatio-temporal evolution. On the one hand, ARP-new corresponds closely to observations in the ISSR zones. On the other hand, ARP-op and, to a lesser extent, IFS show underestimation of the highest RH_ice values in accordance with their statistical dry bias on ISSR.

The sensitivity study driven on the RH_ice threshold provides additional information. In particular, it allows a comparison on NWP model skills for detecting ISSR events, relaxing the constraint on the choice of RH_ice decision thresholds in the forecast. This comparison shows that ARP-new outperforms IFS in predicting the highest supersaturated events and that, conversely, IFS outperforms ARP-new when slight undersaturation thresholds are considered. In addition, the range of HR and FAR pairs that can be obtained by each model with adjusted thresholds is measured. It shows that even without allowing supersaturation, the ARP-op has also skill for discriminating ISSRs, similar to ARP-new if decision thresholds lower than 100 % are applied, which is a major conclusion of this work. This suggests that post-processing recalibration of RH_ice outputs can be a workable strategy to use non-allowing supersaturation NWP forecast such as ARP-op. Finally, the study shows that IFS and ARP-new recall-precision curves have a shortcoming on precision when decision thresholds greater than a certain amount (98 % for IFS, resp. 110 % for ARP-new) are applied in the forecast. Indeed, while higher thresholds should lead to greater success ratio of ISSR forecasts, the results show a stagnation of precision beyond these thresholds. This is particularly true for IFS leading to the conclusion that applying thresholds greater than 98 % to detect ISSR is de facto sub-optimal. We believe that this behavior should be further analyzed to understand its cause, probably related to the saturation adjustment process.

It is important to note that the comparison between different NWP models which is conducted in this study is not able to separate the specific contribution of the cloud and microphysical schemes in ARPEGE or IFS regarding ISSR forecast. In a schematic view, the quality of a NWP model for predicting supersaturation w.r.t ice is the result of the quality in representing several processes across various scales, among them: at the largest scales, the global circulation, which induces the positioning of high water content areas in the right geographical location, and at a lower scale the microphysical processes that allows to estimate the right amount of supersaturation reached in these areas. Comparing separately cloud schemes within the same NWP model could be done in dedicated research work but will represent a certain amount of implementation and calibration work. A first, lighter approach, which could be used to minimize the impact of global circulation on scores, would be to base forecasts on the same analysis at initial time.

Moving forward to end-user verification metrics is sketched out in this work by proposing the use of spatial metrics in the verification to take into account the spatial scales of the operational context. This may involve the inclusion of a margin of uncertainty when planning avoidance strategies. For instance, according to Fig. , including a 150 km margin to avoid an ISSR forecast helps guarantee that ∼ 80 % of the ISSRs that actually occurred will be avoided, and that ∼ 70 % of the flight paths will be appropriately rerouted, otherwise an ISSR would have been crossed. The introduction of the Fβ-score is also a user-oriented approach as it allows to give differential weight in scoring to detection or precision capabilities of the forecast. Indeed, precision has a great importance in a contrail avoidance operation where supplementary CO2 emissions are emitted.

Assessment of quality regarding contrail application requires a further step beyond this study, involving traffic and use of climate metrics with associated CO2 and non-CO2 contributions. Some additional degrees of freedom in the use of NWP models in the operational context can also be obtained which opens optimization possibilities. For example, the sensitivity study on RH_ice decision threshold presented in this work can be used as a base for future optimization work which would tune this threshold according to climate impact metrics and operational constraints. In the same vein, the use of a multi-model approach or/and ensemble forecasts could also prove very useful for this type of trade-off problem by tuning a probability threshold on persistent contrails occurrence conditions.