Statistical downscaling methods are vital tools in translating global and regional climate model output to actionable guidance for climate impact studies. General circulation models (GCMs) and regional climate models (RCMs) are used to provide projections of future climate scenarios; however, coarse resolution and systematic biases result in unrealistic behaviour, particularly for extreme events . In recognition of these limitations, downscaling is routinely performed to correct raw GCM and RCM outputs. This is achieved either by dynamical downscaling, running a nested high-resolution simulation or via statistical methods. Comparisons of statistical and dynamical downscaling suggest that neither group of methods is clearly superior ; however, in practice computationally cheaper statistical methods are widely used.

Major classes of statistical downscaling methods are model output statistics (MOS) and perfect prognosis (PP; ). MOS methods explicitly adjust the simulated distribution of a given variable to the observed distribution using variations of quantile mapping . Though these methods are widely applied in impact studies, they struggle to downscale extreme values and artificially alter trends . In contrast, in PP downscaling, the aim is to learn a transfer function f such that 1y^=f(x,Z), where y^ is the downscaled prediction of a given climate variable whose true value is y at location x and Z is a set of predictors from the climate model . This is based on the assumption that while sub-grid-scale and parameterised processes are poorly represented in GCMs, the large-scale flow is generally better resolved .

Multiple different models have been trialled for parameterising f. Traditional statistical methods used for this purpose include multiple linear regression , generalised linear models and analogue techniques . More recently, there has been considerable interest in applying advances in machine learning to this problem, including relevance vector machines , artificial neural networks , auto-encoders , recurrent neural networks , generative adversarial networks and convolutional neural networks . These models are trained in a supervised framework by learning a mapping from low-resolution predictors to downscaled values at a particular set of locations for which observations are available. Unsupervised downscaling using normalising flows has also been proposed .

Limitations remain in these models. In many climate applications it is desirable to make projections that are both (i) consistent over multiple locations and (ii) specific to an arbitrary locality. The problem of multi-site downscaling has been widely studied, with two classes of approaches emerging. Traditional methods take analogues or principal components of the coarse-resolution field as predictors. The spatial dependence is then explicitly modelled for a given set of sites, using observations at those locations to train the model . More recent work has sought to leverage advances in machine learning, for example convolutional neural networks (CNNs), for feature extraction . These methods take in a grid of low-resolution predictors and output downscaled predictions either on a fixed grid or at a pre-determined list of sites. The question naturally arises as to how we can generate predictions at new locations at test time. Models trained in one location can be applied in another using transfer learning . In this case, however, the output predictions are still at the resolution or list of sites determined at training time (i.e. a CNN model trained on 0.1∘ resolution will output 0.1∘ resolution predictions, regardless of where it is applied). To make predictions on a grid with a different resolution or at a new set of locations requires interpolation of model predictions or taking the closest location.

In this study we propose a new approach to statistical downscaling using a convolutional conditional neural process model (convCNP; ), a state-of-the-art probabilistic machine learning method combining ideas from Gaussian processes (GPs) and deep neural networks. This model learns a mapping between a gridded set of low-resolution predictors and a continuous stochastic process over longitude and latitude representing the downscaled prediction of the required variable. In contrast to previous work where discrete predictions are made at a list of locations determined at training time, the stochastic process output from the convCNP can be queried at any location where a prediction is required. Although to our knowledge this is the first application of such a model in downscaling, similar work has demonstrated the advantages of learning a mapping from discrete input data to continuous prediction fields in modelling idealised fluid flow .

The specific aims of this study are as follows.

Develop a new statistical model for downscaling GCM output capable of generating a stochastic process as a prediction that can be queried at an arbitrary site.

Section 2 outlines the development of the downscaling model and presents the experimental setup used to address aims 2–4. Section 3 compares the performance of the statistical model to an ensemble of baselines. Sections 4 and 5 explore model performance at unseen locations and the impact of including local topographic data. Finally, Sect. 6 presents a discussion of these results and suggestions for further applications.

We first outline the development of the statistical downscaling model, followed by a description of three validation experiments.

The convCNP model outperforms all VALUE baselines on median mean absolute error (MAE) and Spearman correlation for both maximum temperature and precipitation. Comparisons of convCNP model performance at the 86 VALUE stations to each model in the VALUE baseline ensemble are shown in Fig. 4. The low MAE and high Spearman correlation indicate that the model performs well at capturing day-to-day variability.

For maximum temperature, the mean bias is larger than baseline models at many stations, with interquartile range -0.02 to 0.08 ∘C. This is a direct consequence of training a global model as opposed to individual models to each station which will trivially correct the mean . Though larger than baseline models, this error is still small for a majority of stations. Similarly for precipitation, though mean biases are larger than many of the VALUE models, the interquartile range is just -0.07 to 0.12 mm. For precipitation, the bias in convCNP relative wet day frequency (R01) and mean wet day precipitation (SDII) are comparable to the best models in the VALUE ensemble (not shown).

When downscaling GCM output for impact studies, it is of particular importance to accurately reproduce extreme events . In line with previous work comparing the VALUE baselines , an extreme event is defined to be a value greater than the 98th percentile of observations. Comparisons of biases in the 98th percentile of maximum temperature and precipitation are shown in Fig. 5. The convCNP performs similarly to the best baselines, with a median bias of -0.02 ∘C for temperature and -2.04 mm for precipitation across the VALUE stations. R10 biases are comparable to baselines, with a median bias of just -0.003 mm.

The convCNP model outperforms the GP-baseline at unseen stations. Results for MAE, Spearman correlation and mean bias are shown in Fig. 6. For maximum temperature, the convCNP model gives small improvements over the baseline model, with Spearman correlations of 0.99 (0.98) and MAE of 1.19 ∘C (1.35 ∘C) for the convCNP (GP baseline). Importantly, large outliers (> 10 ∘C) in the baseline MAE are not observed in the convCNP predictions. Figure 7 shows the spatial distribution of MAE for the convCNP and GP-baseline together with the difference in MAE between the two models. This demonstrates that stations with high MAE in the convCNP model are primarily concentrated in the complex topography of the European Alps. The GP-baseline model displays large MAE not only in the Alps but also at other locations, for example in Spain and France. The convCNP improves predictions at 82 out of the 86 stations.

Repeating this analysis for precipitation, the convCNP model gives substantial improvement over the baseline for MAE and Spearman correlation. Spearman correlations are 0.57 (0.20) and MAE 2.10 mm (2.71 mm) for convCNP (GP-baseline). In contrast to maximum temperature, there is no clear link between topography and MAE, though again convCNP predictions have large MAE for multiple stations located in the Alps. The convCNP model improves on baseline predictions at 80 out of 86 stations.

Comparisons between models for extreme metrics are shown in Fig. 8. For maximum temperature, the convCNP has slightly lower absolute 98th percentile bias than the baseline. For precipitation, errors are substantially lower, with median absolute 98th percentile bias of 4.90 mm for convCNP compared to 22.92 mm for GP-baseline. The spatial distributions of 98th percentile bias for maximum temperature and precipitation predictions together with the difference in absolute bias are shown in Fig. 9. For maximum temperature, the convCNP does not improve on the baseline at all stations. The GP-baseline exhibits uniformly positive biases, while the convCNP model has both positive and negative biases. Improvements are seen through central and eastern Europe, while the convCNP performs comparatively poorly in southern Europe and the British Isles. For precipitation, predictions have low biases across much of Europe for the convCNP, with the exception of in the complex terrain of the Alps. GP-baseline biases are negative throughout the domain. For this case, convCNP predictions have lower bias at 84 of the 86 validation stations.

A limitation to the analysis of the standard climate metrics in Table 2 is that these only assess certain aspects of the predicted distribution. To assess the calibration of the models, we next examine the probability integral transform (PIT) values. The PIT value for a given prediction is defined as the cumulative density function (CDF) of the distribution predicted by the convCNP model evaluated at the true observed value. These values can be used to determine whether the model is calibrated by evaluating the PIT for every model prediction at the true observation, and plotting their distribution. If the model is properly calibrated, it is both necessary and sufficient for this distribution to be uniform . PIT distributions for maximum temperature and wet-day precipitation are shown in Fig. 10. For temperature, the model is well calibrated overall, although the predicted distributions are often too narrow, as demonstrated by the peaks around zero and one indicating that the observed value falls outside the predicted normal distribution. Calibration of the precipitation model is poorer overall. The peak in PIT mass around zero indicates that this model often over-predicts rainfall accumulation. Performance varies between individual stations for both temperature and precipitation, with examples of PIT distributions for both well-calibrated and poorly calibrated stations shown in Fig. 10.

Results of the topography ablation experiment are shown in Fig. 11 (mean metrics) and Fig. 12 (extreme metrics). These figures compare the performance on each metric between the convCNP model with all topographic predictors and convCNP models trained with no topography, elevation and elevation difference only, and mTPI only.

For maximum temperature, inclusion of topographic information improves MAE, mean bias and Spearman correlation. Models including only mTPI or no topographic predictors have a number of stations with very large MAE, exceeding 10 ∘C at several stations. Unsurprisingly, these stations are found to be located in areas of complex topography in the Alps (not shown). Including elevation both decreases the median MAE and corrects errors at these outliers, with further improvement observed with mTPI added. A similar pattern is seen for mean bias. More modest improvements are seen for precipitation, though inclusion of topographic data does result in slightly improved performance.

For maximum temperature, inclusion of topographic data results in reduced 98th percentile bias. This is primarily as a result of including elevation and elevation difference data, with limited benefit derived from the inclusion of mTPI. In contrast, for precipitation, models with topographic correction perform worse than the elevation agnostic model for both 98th percentile and R10 biases. This reduced performance for precipitation may result from overfitting.

This study demonstrated the successful application of convCNPs to statistical downscaling of temperature and precipitation. The convCNP model performs well compared to strong baselines from the VALUE ensemble on both mean and extreme metrics. For both variables the convCNP model outperforms an interpolation-based baseline. Inclusion of sub-grid-scale topographic information is shown to improve model performance for mean and extreme metrics for maximum temperature and mean metrics for precipitation. The convCNP model has a significant advantage over these baselines in that the output prediction is a continuous function, allowing predictions to be made at an arbitrary (longitude, latitude, elevation) location. Although only temperature and precipitation are considered in this study, the model is easily applied to any climate variable with available station observations, for example wind speed.

Several areas remain for future work, both within the convCNP model and in comparison to other downscaling methods. In the convCNP predictions, representation of certain metrics, notably precipitation extremes requires further improvement, particularly in areas with complex topography. The topography ablation experiments demonstrate that the convCNP P98 bias increases in regions with complex topography. Dynamically, this is likely due to local flow effects such as Föhn winds , which depend on the incident angle of the background flow. A possible explanation for this is that the MLP is insufficient to model these effects. Further experimentation with adding a second CNN to capture the sub-grid-scale processes and possibly conditioning predictions of this model on local flow is left as a topic for future research. Another avenue for improving model performance would be to change the distribution predicted by the convCNP. Model calibration results presented in Sect. 4 indicate that the temperature downscaling model could be improved using a distribution with heavier tails. Precipitation model calibration requires improvement, with the model frequently under-predicting wet-day accumulations. A possible explanation for this is that the left-hand tail of the gamma distribution decays rapidly. For cases where the mode of the predicted distribution is greater than zero, small observed accumulations are heavily penalised. Previous work has acknowledged that the Bernoulli–gamma distribution used in this study is not realistic for all sites and suggested that representation of precipitation extremes can be improved using a Bernoulli–gamma–generalised Pareto distribution . Future work will explore improving the calibration of the downscaling models using mixture distributions and normalising flows to improve the calibration of the model. A further possibility for extending the convCNP model would be to explicitly incorporate time by building recurrence into the model .

Future work will also focus on developing a standardised framework to compare the convCNP model to a variety of deep-learning baselines, building on the work of . Although some studies have indicated that in certain cases deep-learning models offer little advantage over widely used statistical methods such as those included in the VALUE ensemble , others suggest that deep-learning methods offer improved performance . Further work is required both to rigorously compare the convCNP model to other machine learning models for downscaling and to generate a standardised intercomparison of models more broadly. Examination of a larger set of metrics, particularly for precipitation, would also be beneficial.

The final aspect to consider is extending these promising results downscaling reanalysis data to apply to future climate simulations from GCMs. An in depth analysis of the convCNP model performance on seasonal and annual metrics would be beneficial in informing application to impact scenarios. A limitation in all PP downscaling techniques is that applying a transfer function trained on reanalysis data to a GCM makes the assumption that the predictors included in the context set are realistically simulated in the GCM . Future work will aim to address this issue through training a convCNP model directly on RCM or GCM hindcasts available through projects such as EURO-CORDEX .