The Cloud–Aerosol Lidar with Orthogonal Polarization (CALIOP) is an instrument aboard the polar-orbiting CALIPSO satellite . The CALIPSO satellite was part of the A-train satellite constellation until September 2018, when it was placed in a different orbit together with Cloudsat . CALIOP measures the attenuated backscatter at the 532 and 1064 nm wavelengths, which are disseminated as the CALIOP L1B product. Additional processing involving the identification of cloud and aerosol layers, as well as extinction retrieval, is performed to create the CALIOP L2 products . Both the CALIOP L1B and L2 products are used for collocation with GOES-16 ABI imagery.

The CALIPSO satellite carries two other instruments: the Wide Field Camera (WFC) and the Imaging Infrared Radiometer (IIR). The IIR instrument measures infrared radiances at three central wavelengths: 8.6, 10.6, and 12.0 µm . The imagery from the 10.6 µm band and its difference with the 12.0 µm band are used during the collocation process for visualization and manual inspection (see Figs. S4 and S5 in the Supplement).

The Advanced Baseline Imager (ABI) is a passive imaging radiometer that is carried aboard the GOES-R satellite series . The ABI features 16 imaging bands, 10 of which are in the infrared part of the electromagnetic spectrum. The nadir resolution of the ABI infrared bands is 2 km, and imagery is available every 5 min for the contiguous United States (CONUS) and every 10–15 min for the full disk. The contrail detection algorithm by uses orthographically projected GOES-16 ABI imagery, centered on CONUS. The boundaries of the domain where contrail detections are available are shown in Fig. .

The ERA5 reanalysis is used to provide wind data for the advection performed during the collocation process and comparisons to flight data. Geopotential data are used for conversion between geometric altitude and pressure altitude. The land–sea mask is used as an input for the altitude estimation models.

Contrails have previously been located in CALIOP data by use of contrail detections in Aqua MODIS imagery . At the time, the Aqua satellite was in the same orbit as CALIPSO such that the time difference between overpasses by the two satellites was around 75 s. Collocation of contrails detected on imagery of the GOES-16 ABI is different in that the two satellites do not share an orbit. The collocation is performed in two steps. The first “coarse” step considers whether contrails are detected within a 50 km distance of the ground track of CALIPSO (see Sect. S1 of the Supplement for more details). The second “fine” step accounts for advection of contrails between the CALIPSO overpass time and the GOES-16 ABI capture time, as well as parallax displacement. The collocation is performed for the years 2018 to 2022: the resulting dataset features 3267 collocated contrails.

An example collocation is shown in Fig. , which displays the CALIOP L1B attenuated backscatter and the corresponding IIR brightness temperature difference (BTD) imagery for a segment of an overpass on 8 August 2018. The attenuated backscatter is filtered to remove noise using the methods described in Sect. S1. The GOES-16 image shows that the CALIPSO satellite passes over four detected contrails, which are indicated using colored boundaries, with their corresponding cross-sections in the CALIOP L1B denoted by bounding boxes (and numbers). The approximate locations of these collocated contrails are also shown in the IIR image in Fig. . The first contrail encountered in Fig.  appears to be a thinner and potentially younger contrail embedded in another cloud and shows up as a narrow region of increased backscatter (white rather than gray) in the CALIOP L1B data. Contrail 2 has a larger vertical extent (> 1 km) when compared to the other contrail cross-sections found in this segment, and it also appears to be wider than the other contrails when viewed in the GOES-16 image. This particular contrail may actually be two different contrails given that the structure in bounding box 2 in the CALIOP L1B data shows two backscatter maxima, but these are not discernible as such at the resolution of the GOES-16 image (∼ 2 km) or the IIR image (1 km). The increased width of this contrail in the CALIOP data is partially caused by the angle between the CALIPSO ground track and the contrail. When correcting for this angle, however, contrail 2 is still found to be wider than contrails 3 and 4. The final two collocated contrails (3 and 4 in Fig. ) are of similar vertical and horizontal extent, both featuring a horizontally displaced fall streak as a result of vertical wind shear. The GOES-16 ABI image shows several cirrus clouds (such as the one between contrails 2 and 3) that could be contrails that are no longer recognized as such by the detection algorithm and which also seem to correspond to areas of increased backscatter (in gray) in the CALIOP L1B data. This highlights that, if a different contrail detection algorithm is used, the collocation process could be repeated to find more and/or different contrails.

In addition to locating contrails in CALIOP L1B data, we collocate 5 km cirrus layers – as identified in the CALIOP L2 product – with GOES-16 ABI data. The collocation process is nearly identical to that for contrails, except that the contrail detections described in Sect.  are no longer involved. This “cirrus dataset” is created to develop cirrus altitude estimation algorithms similar to that presented in , but for data from GOES-16 ABI rather than Meteosat SEVIRI. These algorithms are then assessed on their ability to estimate contrail altitudes and compared to algorithms developed using contrail data specifically. The possibility exists that contrails are recognized as cirrus layers by the CALIOP processing algorithms and thus appear in the CALIOP L2 5 km cloud layer product. These layers are removed from the cirrus dataset using the dataset of collocated contrails. More details on the collocation processes for both the contrail and cirrus datasets are given in the Supplement (Sects. S1 and S2).

Parallax displacement refers to the apparent displacement of clouds on satellite imagery that results from the incorrect geolocation of pixels containing the cloud in question. The effect is particularly noticeable at low satellite viewing angles and can amount to a displacement of the order of 10 km over the contiguous United States when viewed by the GOES-16 ABI. As such, neglecting this effect during the collocation procedure could lead to erroneous results. We correct the parallax effect using Vicente's method .

As part of the collocation procedure described in Sect. , we account for the advection of contrails by wind. The same advection process is also utilized when comparing flight data to contrail observations. This advection is done by means of the ERA5 data described in Sect. . The ERA5 wind data are interpolated by means of a bicubic interpolation in longitude and latitude , a quadratic polynomial interpolation in pressure, and a linear interpolation in time. The advection process is advanced in time by use of the fourth-order adaptive Runge–Kutta method from implemented in the GNU Scientific Library . The vertical position of the contrails or flight segments is affected by the vertical wind speeds from ERA5, but no ice particle sedimentation processes are accounted for.

The data from the CALIOP instrument use the altitude above mean sea level (a.m.s.l.​​​​​​​) as the vertical ordinate, thus being a geometric altitude. In contrast, most numerical weather prediction and ADS-B data utilize pressure (or pressure altitude) as vertical coordinates. Therefore, the process of advecting contrails and flight segments, as well as the comparison of collocated contrail altitudes with flight data, requires the conversion between geometric and pressure altitude. This conversion is performed by use of ERA5 geopotential altitude data, which also utilize the altitude above mean sea level as the datum. We assume that the geopotential altitude is equal to the geometric altitude: this leads to errors at higher altitudes as the gravitational field of the Earth weakens. Using the fact that the gravitational acceleration is proportional to the inverse square of the distance from the Earth's center, we estimate that this assumption leads to errors of ∼ 10 m when applied to objects around 10 km altitude.

Utilizing this assumption, we find the pressure altitude corresponding to a given object's geometric altitude as follows. We first interpolate the ERA5 geopotential data to the time and horizontal location of the object at multiple pressure levels. We then estimate the pressure at which the geopotential altitude of ERA5 equals the geometric altitude of the object. This results in an estimated pressure for the object. Finally, to convert this pressure to the pressure altitude, we use the US standard atmosphere . The process of converting a pressure (altitude) to a geometric altitude follows these steps in reverse.

Humidity and temperature profiles measured by radiosondes in the GCOS Reference Upper-Air Network (GRUAN) are used to compare the altitudes of regions conducive to persistent contrail formation with the altitudes found in the collocated CALIOP contrail dataset . The profiles supplied by GRUAN have already been corrected for radiosonde limitations such as the radiation dry bias and temperature-dependent lag.

We use data from the Lamont (SGP) site as this is located within the region for which contrail detections are available, and it also has more than 100 Vaisala RS92 sonde launches per year for the period between 2009 and 2018 . For the profiles corresponding to these launches, we compute the relative humidity with respect to ice by use of the approximations from . This relative humidity with respect to ice is used to ascertain whether ice supersaturation (ISS) is present, which is combined with the Schmidt–Appleman criterion (SAC) to evaluate whether persistent contrails may form. The SAC is computed by use of the approximation from , using the parameters shown in Table S2 of the Supplement.

Automatic Dependent Surveillance-Broadcast (ADS-B) data are used for comparing observed and predicted contrail altitudes with flight altitudes. We use a mixture of data from the OpenSky database and FlightAware (for times in 2023). The OpenSky data are used for comparison with the contrail altitude estimation performance on the test set in Sect. , whereas the FlightAware data are used for analyzing contrail detections and altitude estimates for a full day of data in Sect. . The OpenSky network does not have complete coverage throughout the domain for which we have contrail detections, and we thus use these data only in areas of the domain where full coverage is available.

When comparing flight and contrail altitudes, we account for the motion of the contrail between the time of formation and the time we observe it in a satellite image by advecting flight segments with ERA5 wind data as described in Sect. . We advect flight data from the 2 h leading up to the observation of a particular contrail. This choice of advecting only the past 2 h of flight data is motivated by the results from , who show that lifetimes of contrails observed in GOES-16 ABI imagery tend to be shorter than 2 h. Only advected flight segments that are sufficiently close – as determined by an estimate of ERA5 wind uncertainty and the duration of the advection process – to the contrail are considered in the comparison. The altitudes of the flight segments that are sufficiently close to the contrail are used to create a distribution of the distance flown at every 10th flight level. Further details of this process can be found in Sect. S9.

We develop both contrail and cirrus altitude estimation techniques using machine learning algorithms. Specifically, we use neural networks to “learn” a mapping between the input variables (e.g., GOES-16 ABI infrared brightness temperatures) and the contrail/cirrus top altitude. Neural networks are capable of learning nonlinear relationships from data and have been successfully applied to remote sensing retrieval problems .

We broadly distinguish between two neural network models: pixel-by-pixel models and image-level models. Pixel-by-pixel models, such as those developed by , operate independently on each individual pixel of the input satellite image. Image-level models consider (part of) the image simultaneously. Although the pixel-by-pixel option may lead to models containing fewer parameters and better interpretability of the results, image-level models have been shown to perform better for certain retrieval tasks . For cloud altitude estimation, it could be that viewing a satellite image (rather than a single pixel) allows a machine learning model to identify cloud overlap, a situation that is known to lead to less accurate predictions by some approaches , and incorporate this information into the resulting prediction. For the pixel-by-pixel models, we consider multilayer perceptrons (MLPs) with varying numbers and sizes of hidden layers, as well as activation functions. For the image-level model, the convolutional neural network (CNN) from is used. More details on the architecture of these neural networks can be found in the Supplement in Sect. S6.

The pixel-by-pixel models obtain the relevant inputs for a given collocated GOES-16 ABI pixel and are trained to output the CALIOP cloud-top altitude that was collocated with the relevant pixel. For the image-level models, we extract patches of 32 × 32 px that contain one or more collocated GOES-16 ABI pixels and present these to the CNN, similar to . The CNN predicts the cloud-top altitude for all pixels in the input image, but only the pixels in the output image which have CALIOP collocations are used during training: an example can be found in Fig. S11. When using the CNN for contrail altitude predictions after training is finished, the resulting output (for the whole image) is only considered for pixels where contrails are detected.

We use all 10 of the GOES-16 ABI infrared channels as input, as well as the latitude, the satellite viewing zenith angle, and an ERA5 land–sea mask. Temporal information is represented similar to by two additional inputs: sin⁡2πDOY365 and cos⁡2πDOY365, where DOY is the day of the year. The pixel-by-pixel models also receive the regional maximum brightness temperature for all 10 of the ABI channels, also suggested by . This regional maximum is the maximum in a 29 × 29 px patch centered on the collocated ABI pixel: this approximately corresponds to the region size used for SEVIRI by . The image-level models also receive a cloud mask as input. For the cirrus dataset, these cloud masks are obtained from the GOES-R ABI cloud-top phase (ABI-L2-ACTP) product . In this mask, the pixels for which the product indicates that the cloud-top phase is ice and the data quality flag is “overall good quality” are set to 1, whereas other pixels take the value of 0. For the contrail dataset, the outputs of the contrail detection algorithm are used as cloud masks. The CNN for contrail altitude estimation is trained by fine-tuning the CNN trained for cirrus altitude estimation on the contrail dataset. This avoids overfitting the CNN, which has significantly more parameters than the MLP models, to the relatively small contrail dataset. The fine-tuning process starts with the CNN trained on cirrus data. Then, all parameters except for those in the last two layers of the CNN are fixed to the values obtained by training on cirrus data. The parameter values of the last two network layers are optimized by training on the contrail dataset.

With the exception of the ERA5 land–sea mask and some of the inputs to the ACTP processing algorithm , all inputs to the neural networks used for altitude estimation are derived from observed satellite radiances. Previous machine learning approaches for cirrus altitude estimation have made use of numerical weather prediction (NWP) data such as surface temperature . Motivated by the potential of using the resulting contrail altitude estimates for the comparison to (and potentially assimilation with) NWP data, we choose not to include these inputs such that the resulting altitude estimates represent an independent piece of information.

For training of the neural networks, 80 % of the available data (for both the cirrus and contrail dataset) are used, and the remaining 20 % are split evenly between a validation and test set. The validation set is used during training to monitor overfitting, and the test set is used to evaluate the resulting models. The randomization is performed by grouping collocations by their CALIPSO overpass and then selecting the overpasses for the training, validation, and test sets at random. This avoids placing two or more nearby (or even neighboring) collocated pixels in the training and test sets, which could lead to an unrealistic evaluation of the algorithm's performance on unseen data.

In order to model predictive uncertainty, we use quantile regression neural networks . Rather than having a neural network output a single estimate of the cloud-top altitude, we output an estimate for several quantiles of the cumulative distribution function. These models are trained using the quantile loss function from . A known issue with quantile regression neural networks is the phenomenon of quantile crossing, where the “quantiles” predicted by the network are not monotonically increasing. We address this in a way similar to by fitting an isotonic regression to the neural network outputs. The frequency of occurrence and magnitude of these quantile crossings will be discussed in the Results section.

The dataset consists of 3267 contrails collocated between CALIOP L1 and GOES-16 ABI data for the years 2018 to 2022. The location of all contrails is shown in Fig. . The mean contrail top altitude decreases with increasing latitude, as was found by . The variance of this contrail top altitude distribution is found to increase with latitude. The location of the tropopause is known to decrease, on average, with increasing latitude. Given that ice supersaturation, a necessary condition for contrail persistence, is mostly found below the tropopause , this is a likely explanation for the observed decrease in the contrail top altitude with increasing latitude. Further analysis of the location of the collocated contrails relative to the local tropopause can be found in Sect. S5. We find that the majority (86.6 %) of collocated contrails are located below the local tropopause.

In Fig.  we show the seasonal distribution of the contrail top altitudes within the dataset, accompanied by vertical distributions for persistent contrail formation conditions and flight traffic. The data for persistent contrail formation are obtained from GRUAN profiles from the SGP site (location shown in the inset in Fig. ). The pressure altitude, which is used as the vertical coordinate in Fig. , is defined as the altitude in the International Standard Atmosphere corresponding to the pressure at which the contrail top is found. Since aircraft use pressure altitudes for vertical navigation, some seasonal variation in geometric contrail top altitudes (as measured by CALIOP) is expected to be caused by changes in temperature and thus the geometric altitude of certain pressure levels. To exclude this seasonal variation in the geometric altitude of a particular pressure level, Fig.  uses the pressure altitude as its vertical ordinate. Finally, the flight altitude distributions in Fig.  result from 1000 randomly sampled hours of ADS-B data in the years 2018 and 2019. Given the different locations and times at which the data in the three different plots are collected, Fig.  serves mostly to compare seasonal trends in the altitude of flights, regions conducive to persistent contrail formation, and observed contrail top altitudes.

On average, contrails are located at a pressure altitude of 11.1 km during winter (DJF) and 11.8 km during summer (JJA). This 700 m seasonal difference in the mean contrail location is statistically significant using Welch's t test at p = 0.01. In terms of geometric altitude, the difference between winter (11.3 km mean) and summer (12.2 km mean) is larger at 0.9 km than the pressure altitude difference at the same level of statistical significance. The vertical distribution of regions satisfying SAC and ISS (middle plot in Fig. ) also features higher altitudes during summer and lower altitudes in winter. Such seasonal changes in the vertical distribution of ice supersaturation have been reported by other studies that use radiosondes, also finding that ice supersaturation is on average located at higher altitudes during Northern Hemisphere spring–summer than in Northern Hemisphere winter. The contrail top altitudes in Fig.  are generally found to be higher than the regions that are ice-supersaturated and satisfy the Schmidt–Appleman criterion. Given the limited spatial coverage of the GRUAN data compared to the contrail top altitudes (whose spatial distribution is shown in Fig. ) as well as the different times at which these data were captured, this discrepancy is likely due to sampling effects.

The flight altitude distributions in Fig.  show that flight distance peaks at flight levels 340–360 (10.5 to 11 km). Seasonal changes in the weighted means of these distributions are all smaller than 100 m. Thus, the vertical variation of the observed contrail top altitude distributions cannot be explained by seasonal patterns in flight altitude alone: consideration of the seasonal variation in the location of persistent contrail formation conditions is necessary. This seasonal variation in the vertical location of persistent contrail-forming conditions is likely due to seasonal patterns in tropopause pressure, as ISS regions are known to be located closely below the tropopause .

The single GRUAN site for which data are shown in Fig.  does not provide full coverage of the spatial domain in which contrails are collocated (see Fig. ), nor does it have a significant number of sonde launches for the period 2018–2022 (31 versus 4486 launches between 2009 and 2017). Although statistics on the vertical distribution of ISS are known to vary with location and time , analyzed 15 years of aircraft in situ measurements and did not find any long-term trends in the occurrence of ISS besides seasonal variations. Similarly, we find that for the 9 years between 2009 and 2018 the seasonal trends in the altitude of regions satisfying both the SAC and ISS measured at the SGP site are qualitatively consistent (see Fig. S8).

This section evaluates the performance of the four machine learning approaches to contrail altitude estimation described in Sect.  by comparing the predicted altitudes to those measured by CALIOP for a subset of data not used during training (i.e., the test set). The evaluation of the predicted probability distributions for contrail top altitude is the subject of Sect. : this section focuses only on the means of these distributions.

The performance of the four different altitude estimation models on the contrail test dataset is shown in Fig. . These four models differ in the dataset(s) they have been trained (cirrus, contrails, or both) on and whether they consider pixels individually (MLP) or not (CNN). The cirrus altitude estimation models (Cirrus MLP and Cirrus CNN) are included in this comparison in order to evaluate their suitability for contrail altitude estimation, as well as the benefits of training an altitude estimation model on contrail data rather than cirrus data only. The two panels on the left in Fig.  show the mean errors as a function of altitude as a percentage of the ground-truth altitude. For all altitudes, the two MLP models show larger absolute errors than the two CNN models. The two MLP models also show larger relative errors for low (< 10 km) and high (> 14 km) contrails. The mean error (rather than the mean absolute error) plot in Fig.  illustrates that the MLP models have the tendency to overpredict the altitude for “low contrails” and underpredict the altitude for “high contrails”. The mean errors for the two CNN models are more uniform with altitude. The cirrus CNN achieves lower errors for the lower and upper ends of the altitude range considered here (8–15 km) than the contrail CNN, but the opposite is true for altitudes between 10 and 12 km.

The scatterplots on the right in Fig.  illustrate the spread between the ground truth (CALIOP altitudes) and the model estimates. We see that three out of four models achieve a root mean square error (RMSE) smaller than 1 km, with the exception being the Cirrus MLP (which operates pixel by pixel). This particular model, which is trained using cirrus data only, performs worse on the contrail test set (RMSE of 2.87 km) than on a dataset of unseen cirrus data points (RMSE of 1.05 km, shown in Sect. S7). This Cirrus MLP model is similar in design and performance to that introduced by , for which a detailed comparison can be found in Sect. S7.

The RMSE of a model can also be affected by the dataset it is evaluated on. For example, a model that always overpredicts by 10 % would have a higher RMSE on a dataset that has more variance. For this reason, we also show the coefficient of determination R2 in the scatterplots in Fig. , which takes such dependency on the underlying data into account. For the Cirrus MLP evaluated on the contrail test set, the R2 is -0.39. This can be interpreted as the model being worse than using the mean of the evaluation data as a predictor. However, when evaluated on an unseen cirrus data point we find an R2 of 0.77, indicating that the Cirrus MLP does have “skill” for the prediction of cirrus altitude estimation. We thus see that the performance of the Cirrus MLP – a cirrus altitude estimation algorithm that operates on a pixel-by-pixel basis – on the task of contrail altitude estimation is lower than what one would expect based on an evaluation with cirrus data alone. Nonetheless, the results of the other three algorithms indicate that this “skill gap” can be resolved by a choice of architecture, training data, or a combination of these.

For example, the Cirrus CNN algorithm which operates on multiple pixels simultaneously and was trained with cirrus data only is found to predict contrail altitudes with skill (R2 of 0.76). Its skill, as measured by R2, is, however, higher on a set of unseen cirrus data points (0.86) than it is for contrails. These results suggest that choosing a CNN model architecture over an MLP is beneficial for the performance on both cirrus and contrail data. Moreover, the Cirrus CNN is found to perform better in the task of contrail altitude estimation than the Contrail MLP: an algorithm that was trained using contrail data.

Referring to Fig. , the best-performing algorithm is found to be the Contrail CNN. This algorithm corresponds to the Cirrus CNN model that was fine-tuned on the contrail dataset. It combines the benefits of the choice of architecture (CNN over MLP) and the exposure to contrail data to achieve the lowest RMSE of all four algorithms at 570 m. We therefore utilize the contrail CNN for the remainder of this work.

The analysis in the previous section compared the mean contrail top altitudes estimated by the neural networks with the corresponding measurements by CALIOP. The best-performing algorithm, the CNN trained first on cirrus and then fine-tuned on contrail data (red in Fig. ), outputs 13 different quantile values that can be utilized to construct a probability distribution for the contrail top altitude. The estimated quantiles are 0.025, 0.05, 0.1, 0.2, …, 0.9, 0.95, and 0.975. This section discusses and quantifies the quality of the distributions constructed using these quantile predictions.

We compare the probability distributions from the CNN with two “baseline” models: one based on the CALIOP dataset distribution (a climatology, in a sense) and one based on the CNN mean and the error distribution on the test dataset. The first of these models represents the situation a priori, without any altitude estimation algorithm, where all that is available are the altitude data of the collocated contrails. The second model does include an altitude estimation algorithm (as it uses the mean from the CNN output) but without predictive uncertainty, which depends on the algorithm inputs: i.e., it is constant. An example of the probability density functions (PDFs) for the three models, and the corresponding contrail altitude found by CALIOP, is shown in the left panel in Fig. . More examples can be found in Fig. S14.

One desired property of predictive probability distributions is that they are calibrated. This means that when the model says an event should occur with probability 0.2, the event actually occurs 20 % of the time. We assess this, for each predicted quantile, by computing the number of times the CALIOP contrail top altitude is actually below that altitude. As an example, for the quantile value predicted for the 50 % quantile (i.e., the median), we would expect 50 % of the CALIOP data points to be below that value if the model is calibrated. By quantifying the fraction of contrails that are below the altitude predicted for each of the quantiles, the curves shown in the right panel of Fig.  are the result. The closer a particular curve is to the dashed black line, the better the calibration of the model. Figure  shows that the best-calibrated model is the one based on the CALIOP dataset distribution, which is expected given that it represents the distribution of the data directly. The CNN tends to be overconfident for most predicted probabilities, with the largest deviations occurring for predictions between probability 0.4 and 0.9. For example, when the CNN predicts that 60 % of the contrails should be below a particular altitude, Fig.  indicates that only 50 % of contrails will actually be found below this altitude. This overconfidence can be corrected by means of post-processing: we will show in the next section that the 95 % confidence intervals derived from the CNN predictions are calibrated, so we do not perform this correction here. Lastly, Fig.  shows that the error statistics model is overconfident for the majority of predicted probabilities and is the least-calibrated model considered here.

If a predictive probability distribution is calibrated, we would also like it to be sharp. That is, most of its probability mass should be close to the actual observed value (i.e., the CALIOP top altitude). To assess this, we use the continuous ranked probability score (CRPS) . For a given probability distribution F(y)=P(Y≤y) and observed value x, the CRPS is given by the integral 1CRPS(F,x)=∫-∞∞F(y)-I(y>x)2dy, where I is the indicator function. Note that CRPS(F,x)≥0, and CRPS(F,x)=0 if F(y)=I(y>x), which corresponds to a situation where all probability mass is located at the measurement location x. Lower values of the CRPS correspond to “sharper” probability distributions centered on the measurement x and are thus desirable. The mean CRPS values obtained for the three models are shown in Fig. . The CRPS values of the CNN (0.31) and the error statistics model (0.33) are both about 2 times as low as that of the dataset distribution model (0.65), indicating that the distributions predicted by these models are sharper when compared to the measurements. The improvement of the CNN CRPS with respect to that of the error statistics model is not as large: the CNN CRPS is about 6 % lower. In Sect. S8 we further analyze the size of the 95 % confidence intervals of the three models and find that the CNN confidence intervals are smaller than that from the error statistics model for over half the test set data points. Thus, the input-dependent predictive uncertainty of the CNN does provide additional value when compared to the constant uncertainty approach from the error statistics model.

The issue of quantile crossing was introduced in the Methods section and occurs when the quantile values that the neural network outputs are not monotonically increasing, and therefore do not define a valid probability distribution. In the test set, we found seven (out of 1344 total GOES-16 ABI pixels) occurrences, corresponding to a rate of 0.52 %. The amount by which the quantiles cross is a distance of 38 m on average with a maximum of 69 m: these values represent 0.1 % deviations from the values for which predictions are made (of the order of 10 km). These quantile crossings are corrected by means of an isotonic regression: the effect on the mean of the distributions is of the order of tens of micrometers.

This section further evaluates the performance of the contrail altitude estimation algorithm by comparison with flight data. Although interactions with wing tip vortices, wind-driven ascent and descent, radiative heating and cooling, and ice particle sedimentation can play a role, the observed contrail's vertical location is expected to be influenced most strongly by the altitude of the flight that formed it. Thus, the altitudes of flights near an observed contrail can be used to assess the correctness and plausibility of a contrail altitude estimate. Before any such comparison, flight tracks are advected using ERA5 wind data to account for contrail movement after formation.

Only in locations and during times of low air traffic is it plausible to visually compare observed contrails and their estimated altitudes with advected flight tracks. An example of such a situation is shown in Fig. , depicting a scene off the east coast of the United States at 04:00 local time in the morning. Flight data from the 2 h leading up to the image were advected to the satellite image observation time. The flight pressure altitudes are converted to geometric altitudes by use of ERA5 geopotential data. We also apply a parallax correction to “simulate” where the flight tracks would have been visible when viewed by the GOES-16 ABI, which has the effect of further aligning some of the advected flight tracks with the detected contrails. Figure e and f show the altitudes of the advected flight tracks and those estimated for the contrails. For the contrail top altitudes, the mean of the distribution estimated by the CNN is shown. The flights best aligning with the three detected contrails were cruising at flight levels 380 and 390, which correspond to geometric altitudes between 11.7 and 12.1 km. The mean of the estimated contrail top altitude of all detected contrail pixels is 11.9 km.

The comparison of flights to contrails is more complicated in areas of higher traffic density and makes analyses such as that presented in Fig.  infeasible given currently available tools. For these cases, we compare estimated contrail top altitudes to the distribution of distance flown rather than individual flight tracks and their altitudes, as shown in Fig. . The rationale for using these distributions of distance flown is that, in the absence of knowledge on the location of ice-supersaturated regions, they can be used to form an “a priori” confidence interval of the vertical location of the contrail. Figure  shows that all but four contrails (i.e., 7.5 %) are found within the 95 % confidence interval predicted by the model, which shows that these intervals are close to calibrated. Furthermore, the intervals that capture 95 % of the distance flown (not shown) are larger than the 95 % confidence intervals for the contrail top altitude distribution for all these data points: using the contrail top altitude distributions from the model narrows down the 95 % confidence interval of contrail altitude from 3.3 to 2.2 km, on average.