The most important class of precipitation nowcasting models is based on the extrapolation of radar echoes along the background advection field. These models first estimate the advection field from a sequence of past radar images with methods such as variational echo tracking or optical-flow-based methods like the Lucas–Kanade method . In the classical case of the pure extrapolation nowcast, the most recently observed frame is simply extrapolated along the estimated advection field, often using a semi-Lagrangian scheme . Extrapolation nowcasting does not model the growth and decay of precipitation, so many extensions attempting to make up for that have been developed. One important method is Spectral Prognosis (S-PROG) by . S-PROG is based on the scale-dependence of the lifetime and evolution of features, decomposing the field into additive components corresponding to different spatial scales and evolving each of them separately using an autoregressive (AR2) model in Lagrangian (flow frame of reference) coordinates, enabling modeling the scale-dependent behavior of precipitation.

STEPS (Short-Term Ensemble Prediction System) by , is an influential ensemble nowcasting model based on S-PROG. In STEPS, stochastic perturbations are added to the motion field in order to model its uncertainty. Just like with S-PROG, the growth and decay of the precipitation field is modeled by decomposing it using a cascade of scales with the autoregressive model applied to each of these scales separately in Lagrangian coordinates. Unlike in S-PROG, stochastic noise is injected at each scale, concurrent with the AR modeling. Over time, various models have expanded upon STEPS; one recent example is LINDA (Lagrangian INtegro-Difference equation model with autoregression) , which uses an integro-difference equation model with rain cell detection and convolutions for modeling the loss of predictability at small scales. LINDA produces nowcasts that are particularly well-suited to the prediction of strong localized rainfall.

With significant recent advances in deep learning, the interest in its use for precipitation nowcasting has increased. One of the first deep-learning models to have been used explicitly for precipitation nowcasting is the convolutional LSTM (ConvLSTM) model , which combines the temporal prediction capacity of the long short-term memory (LSTM) neural networks with 3D convolutions modeling spatiotemporal features in one model for spatiotemporal nowcasting. ConvLSTM has later been improved by the TrajGRU model that replaces the heavy LSTM structure with a lighter GRU (gated recurrent unit) structure and is capable of learning an active location variant structure for the recurrent connections.

Apart from doing the temporal modeling using recurrent units, a popular approach has been to use fully convolutional neural networks, often two-dimensional, thus avoiding the modeling of explicit temporal dependencies. These networks have often been based on U-Net-type architectures, one early example of which is the model by , which predicts the exceedance of rainfall over three distinct intensity thresholds for a 1 h lead time. A more useful model is RainNet by . RainNet nowcasts rainfall continuously, one time step at a time, inserting the predicted frames back into the network in order to make multiple lead time predictions. Similarly, FureNET by nowcasts rainfall 1 h at a time using polarimetric input variables, in addition to observed rain rates, via multiple encoder branches and late fusion in the decoder of a residual U-Net architecture and brings improvement compared to using plain rain rates.

The principal problem of using the above deep-learning models for deterministic precipitation nowcasting is that of the increasing blurring of nowcasts with lead time. This is the natural consequence of attempting to minimize the pixel-wise forecasting error in the presence of uncertainties inherent to the task of predicting precipitation. Such loss functions thus behave in the same fashion as S-PROG and STEPS by explicitly filtering out scales through their loss of predictability. One way to resolve the problem is to use generative modeling, which is the approach taken by with their Deep Generative Model of Radar (DGMR). DGMR is an adversarially trained convolutional gated recurrent unit (ConvGRU)-based generative model capable of generating realistic time series of future radar observations that outperform both classical and deep-learning baseline models. In addition to deterministic nowcasts, DGMR is also capable of making ensemble-based probabilistic nowcasts.

Making probabilistic precipitation nowcasts using deep learning has been explored less often than deterministic nowcasts, despite the clear benefit of the probabilistic approach in operational use. In addition to DGMR, other existing probabilistic models are MetNet and its successor MetNet-2 . MetNet aggregates weather radar, satellite, and orographic information over a large area to predict a probability distribution of rain rate per pixel in one forward pass for a single lead time, with an architecture consisting of a spatial aggregator of inputs, a ConvLSTM spatial encoder, and a spatial decoder with axial attention. The model is shown to outperform the HRRR NWP model on an F1 metric for lead times up to 8 h. MetNet-2 improves upon its predecessor by adding the data assimilation context as an input and aggregating data over a larger area. This enables it to outperform, or at worst rival, HRRR and HREF (High-Resolution Ensemble Forecast) models in continuous ranked probability score (CRPS) and critical success index (CSI) metrics for lead times up to 12 h.

In addition to playing an important role in precipitation nowcasting, the importance of uncertainty quantification (UQ) has also been recognized in deep learning . In the field of machine learning, uncertainty in the predictions can be divided into two separate components: epistemic and aleatoric uncertainty. Epistemic uncertainty represents the lack of knowledge in the model, and it is reducible through improving the model or bringing in more training data. Aleatoric uncertainty, on the other hand, is inherent to the input data, and no amount of additional training data or model improvement will reduce it. Aleatoric uncertainty that varies over the input data is said to be heteroscedastic; a constant uncertainty is called homoscedastic.

Many approaches to the quantification of uncertainty have been developed on the deep-learning side. One particularly important theme driving the development in this realm has been operational safety and countering overconfident predictions made by black box models overfitting the training data. Bayesian neural networks (BNNs) have emerged as a candidate for addressing that issue. They work by placing probability distributions over the weights, which are estimated via the means of Bayesian inference and yield a predictive distribution for data through their marginalization.

Although exact Bayesian inference is intractable for large neural networks, suitable approximations exist. These are commonly divided into Markov chain Monte Carlo (MCMC)- and variational inference (VI)-based methods . MCMC methods predict better weight distributions but are more computationally expensive and thus often reserved for small-scale problems where performance is key. VI, on the other hand, is more scalable and has been applied to larger neural networks. The idea behind variational inference is to approximate the true posterior of weights with a simpler analytic one (the variational posterior) and to estimate the variational posterior which is the closest to the true one. Thanks to advances by and subsequently with the Bayes by Backprop (BBB) algorithm, it is now possible to use mini-batch optimization for mean field VI (i.e., assuming fully factorizable variational posteriors) on large networks, opening up possibilities for the use of VI in problems such as precipitation nowcasting that require large amounts of input data and numerous model parameters.

Later, Monte Carlo Dropout techniques, among other variants, have been identified as being equivalent to approximate Bayesian inference due to losing some model expressivity but gaining ease of implementation. Based on this, have developed a technique for estimating the epistemic and heteroscedastic aleatoric variance components separately in deep-learning regression tasks. They estimate the epistemic uncertainty with the variance of predictions made via Monte Carlo Dropout and add a separate component to their network for predicting the aleatoric component. The predictions are modeled as having Gaussian likelihoods, with means equal to the prediction point estimates and variances equal to the aleatoric term described. These terms are then learned by minimizing a Gaussian negative log-likelihood loss function and taking them and observations as inputs. This approach has recently started to be applied to problems such as the segmentation of satellite images , remaining useful for life prognostics and long-term synoptic-scale precipitation forecasts .

We propose the Deep Ensemble-based Uncertainty Combining radar Echo nowcasting (DEUCE) model for probabilistic precipitation nowcasting. The idea of the model is to apply the aleatoric and epistemic decomposition of uncertainty by to a Bayesian convolutional neural network with mean field variational inference to produce a predictive distribution of radar reflectivity. This distribution is then sampled to generate ensemble nowcasts of weather radar echo images. The research questions that we will attempt to answer are the following:

Can we produce both powerful and reliable ensemble precipitation nowcasts using Bayesian neural networks with uncertainty decomposition? Specifically, is such a model competitive when compared to classical baseline models when assessed with a variety of quantitative probabilistic prediction skill metrics and based on a qualitative assessment?

A neural network fθ(x)=y^ is a universal function approximator which can be used for regression tasks, mapping an input tensor x to a predicted output tensor y^ and approximating a ground-truth output tensor y, using its learned network parameters θ that are represented as a list of tensors. In the case of radar-based precipitation nowcasting with neural networks, we approximate a function mapping the spatiotemporal time series of past radar observation images x=x1,x2,…,xLin, where Lin corresponds to the number of input time steps and to future radar observation images y=y1,y2,…,yLout, where Lout corresponds to the number of output time steps. In the DEUCE model, both x and y represent processed radar reflectivity data, and the network fθ(x)=y^,σ2 outputs a tuple of predicted reflectivity field time series y^=y^1,y^2,…,y^Lout, along with fields estimating the aleatoric uncertainties σ2=σ12,σ22,…,σLout2 corresponding to each of the pixels of y^.

For the task of precipitation nowcasting, the neural network has to be capable of outputting predictions for multiple lead times, i.e., discrete time steps in the future, corresponding to future radar observations. DEUCE achieves this using a variant of U-Net as its functional architecture, taking in a sequence of 12 radar reflectivity fields x, predicting y^,σ2 that corresponds to the nowcast for the next 12 time steps in a single forward pass.

A schematic representation of the main components of DEUCE and how they are connected is presented in Fig. . The architecture consists of a single encoder branch, extracting features from x at different spatial scales and semantic levels. The feature maps from these different scales are preserved for later use through skip connections. The (largest-scale) latent state produced by the encoder and the intermediate feature maps mediated by skip connections are then fed to two independent decoders: one outputting y^ and the other outputting log⁡σal2. Using separate decoders for the outputs is preferable over a single combined decoder to avoid the blending of adjacent features, which would be detrimental to the expressivity of the model.

The network contains two-dimensional (spatial) convolutions. These are represented by conv2d 3x3 and conv2d 1x1 labels denoting layers with filter sizes 3 and 1, respectively. Using 2D convolutions, temporal dependencies are only present implicitly. This approximation casts the nowcasting task as a simple image sequence-to-sequence translation problem, which reduces the computational resources needed compared to explicit modeling of the temporal aspect. The convolutional layers use partial convolutions in which missing values are masked, and only valid values are used to normalize the convolutions. Although we do not work with missing data, this design choice helps in providing better-quality predictions near image borders by reducing, e.g., various artifacts related to them. ReLU denotes the rectified linear unit activation function, concat is concatenation along the channel dimension, upsample is nearest-neighbor upsampling by a scale of 2, and Max Pooling is maximum pooling by a scale of 2.

Conventional neural networks are deterministic in their nature, meaning that they only ever yield the same output y^ for a given input x and parameters θ. Our goal is to produce a reliable estimate of the uncertainty associated with the approximation produced by the neural network. Because this approximation is merely a function of the input data and the functional model including parameters, considering the uncertainty in these sources separately should allow the approximation of the total predictive uncertainty in nowcasts.

Hence, epistemic uncertainty is modeled by placing probability distributions on functional model parameters θ, effectively turning the model stochastic. A Bayesian approach is taken in this regard, placing a prior distribution upon the weights and estimating the most likely posterior distribution given that prior and the training data. The estimation of the true posterior is an intractable task for a large-scale neural network, which is why variational inference (VI) is used to learn approximate posterior estimates for weights. VI limits the space of acceptable posterior distributions to a parameterized family, whose learned parameters replace the point estimates of classical neural network weights. Here, we aim to minimize the Kullback–Leibler (KL) divergence DKL between the true and variational posteriors, which is a measure of the similarity between two probability distributions. As such, the objective is stated as 1θ*=arg⁡min⁡θDKL[q(w∣θ)‖p(w∣D)]=arg⁡min⁡θ∫q(w∣θ)log⁡q(w∣θ)p(w∣D)dw, where θ denotes the variational posterior parameters, θ* the optimal parameters, w the sampled network weights, D=(x,y) the problem data, q(w∣θ) the variational posterior, and p(w∣D) the exact posterior of network weights. In practice, this is not directly solvable, so the optimization is accomplished through the maximization of an evidence lower-bound (ELBO) proxy objective. The objective is defined as 2ELBO(D,θ)=Eq(w∣θ)[log⁡p(w,D)]-Eq(w∣θ)[log⁡q(w∣θ)],3=Eq(w∣θ)[log⁡p(D∣w)]︷likelihood+Eq(w∣θ)[log⁡p(w)]︷prior-Eq(w∣θ)[log⁡q(w∣θ)]︷posterior︸=-DKL[q(w∣θ)‖p(w)],i.e., the complexity term, consisting of the log-likelihood, log-prior, and log-posteriors, with the last two terms commonly grouped together as the complexity term. Here Eq(w∣θ) denotes the expected value of the probability density of interest over the variational posteriors.

According to , in Bayesian neural networks and using mini-batch optimization, the ELBO objective, as stated in Eq. (), can be approximated as 4ELBOiπ(Di,θ)≈1N∑n=1N(log⁡p(Di∣wi,n)+πilog⁡p(wi,n)-πilog⁡q(wi,n∣θ)), which acts as an unbiased Monte Carlo estimator of the ELBO and is our final loss function. Here, the cost is calculated for each ith of the M mini-batches in an epoch drawing N Monte Carlo samples of the variational posteriors of the weights each time. πi denotes an arbitrary weighting of the complexity term, using the same rule as in in this work, which is πi=2M-i/(2M-1). This serves to make the regularization effect of the prior stronger earlier, allowing data to be more important later in the training. In DEUCE, the variational posterior distributions q are modeled as diagonal Gaussian distributions, and the Bayes By Backprop (BBB) algorithm, using the re-parameterization trick by , is employed for their optimization. The prior distribution p(w), on the contrary, is fixed as a hyperparameter and is identically and independently distributed for each parameter as a normal distribution with zero mean and a variance of 0.1. This allows us to potentially calculate the complexity cost in closed form , rather than with the Monte Carlo estimate of Eq. (), hence reducing the computational cost of training.

The likelihood cost of Eq. (), similar to , is modeled for the ith mini-batch and the nth Monte Carlo sample as the Gaussian log likelihood 5log⁡P(D∣w)=-1P∑p=1P12e-sp(yp-y^p)2+12sp, where the cost is averaged over p=1…P pixels of the Lout×W×H spatiotemporal time series s, y and y^. Lout refers to the length of the time series, W refers to the width of the images, and H refers to the height of the images. i and n indices of fields are omitted here for clarity. Here, y denotes the observed reflectivity fields, y^ denotes the predicted reflectivity fields using the nth weights sampled from the network, and s:=log⁡σ2 refers to the corresponding logarithm of the aleatoric variances predicted by the network with those weights. The logarithm of the aleatoric variances estimate is taken because optimizing using it is more computationally stable and was found to work better than simply using variance constrained to be positive with a ReLU output activation function, especially when dealing with variances approaching zero.

The procedure for producing the primary outputs of DEUCE to make probabilistic nowcasts is presented in Fig. . First, N raw network outputs are produced, which are stochastic pairs of reflectivity field sequences y^n and logarithmic aleatoric variance field sequences σn. Each nth of those sampled outputs draws different weights from the learned variational posterior distributions, which is reflected in the output distribution. The σn are converted to their non-logarithmic version σn2, and individual stochastic runs are stacked into a pair of raw ensembles y^,σ2. At this point, the epistemic uncertainty is embedded in y^, but the aleatoric uncertainty is separate and only present in σal2. Hence, in order to allow the combination of these uncertainties, y^ is divided into the prediction mean y^mean and the epistemic variance σep2 by taking the mean and variance over y^, respectively. Additionally, the aleatoric variance σ2 is summarized by taking its mean, denoted σal2. These three outputs, y^mean, σep2, and σal2, form the base from which probabilistic nowcasts are computed.

The total mean and uncertainty in the prediction can thus be estimated as 6y^mean=1N∑n=1Ny^n,σpred2≈1N∑n=1Ny^n2-(1N∑n=1Ny^n)2︷σep2+1N∑n=1Nσn2︷σal2, where σpred2 denotes the predictive variance. This means that the predictive variance can be estimated as the sum of the variance of the predicted reflectivity fields, which is the epistemic variance, and of the mean of the predicted aleatoric variance fields. These quantities are sufficient for making probabilistic nowcasts such as calculating exceedance probabilities for future radar reflectivity values, allowing us to model the predictive distribution of this reflectivity as normally distributed with mean y^mean and variance σpred2. This formulation is admissible, as reflectivity of precipitation (in dBZ units) is known to have a normal distribution which follows from the distribution of the precipitation rate being log-normal . Additionally, it is interesting to note that the relationship between the ensemble and the predictive distribution here is opposite to that of NWP, where perturbations to initial conditions lead to ensembles that themselves define the predictive distribution .

Nevertheless, some applications of probabilistic precipitation nowcasting – such as flood modeling – assume ensemble-based nowcasts, where each member of the ensemble represents a physically plausible precipitation scenario. One could of course randomly sample the predictive distribution to generate an ensemble, which would correctly approximate pixel-wise statistics, but the spatiotemporal structure of the fields would be lost. In an attempt to remedy this, we post-process outputs to generate ensemble members respecting the spatial covariance structure of the input field x as 7y^nens=y^mean+σpred2⊗ϵcorr,n, where y^nens=y^n,1ens,y^n,2ens,…,y^n,Loutens denotes the newly generated ensemble member, ⊗ denotes an element-wise multiplication broadcast over Lout frames, and ϵcorr,n is a correlated Gaussian random field of shape W×H. ϵcorr,n is generated to match the average spatial correlation structure of x using fast Fourier transform (FFT) filtering. The structure is obtained non-parametrically from the power spectrum of x . The technique is equivalent to that used to generate perturbation fields in STEPS . Even though this method accounts for the spatial structure of the precipitation time series, it is not capable of modeling its temporal structure, which is assumed constant. The ensembles produced in this way shall be denoted y^ens, which is in contrast to the raw predicted reflectivity fields denoted y^.

The dataset used for this work comes from the Finnish Meteorological Institute radar network. It consists of cropped lowest-altitude radar reflectivity composites chosen from rainy days during the summer period of the years 2019–2021. The dataset is identical to that used by , except for using longer time series. The composites are built from the two lowest-elevation angle scans interpolated into an 1×1 km Cartesian grid. The chosen area covers southern Finland, with the bottom-left corner at coordinates (59.01° N, 20.55° E) and the top-right corner at coordinates (63.62° N, 30.27° E). The spatial extent of this crop is 512×512 km, corresponding to 512×512 pixel square images, suitable for training a neural network. The composites are available with a temporal resolution of 5 min. The extent of the bounding box is additionally illustrated in Fig. , along with the coverage of Finnish Meteorological Institute radars. From this we see that the advantage of the crop is that it has a higher-density radar cover than its surroundings.

The data were selected on a day-by-day basis, selecting the 100 d with the most pixels having reflectivity values over 35 dBZ. The days were then divided into 6 h long blocks from which blocks with below 1 % of the pixels with reflectivity values over 20 dBZ were removed. These remaining blocks were then randomly split into training, validation, and verification datasets with a ratio of 6:1:1. The division into blocks was done in order to limit the number of successive time series present in different splits, as they exhibit high correlation, and not using any blocks would make the training, validation, and verification sets dependent, as the same events would be present in all of them. A time of 6 h was deemed a sufficient time for temporal correlations to mostly disappear. Last, 2 h long time series, corresponding to 24 images each, were then extracted from these blocks using a sliding window principle, with a stride of one, omitting those time series with missing data. The final training, validation, and verification datasets ended up containing 10 780, 1813, and 1666 time series, respectively.

The input time series were read from HDF5 files, stored there with an 8 bit scale-offset lossy compression scheme, ranging from -32 to 96 dBZ at a resolution of 0.5 dBZ. The images were then converted to floating point values, and a threshold of 8 dBZ was applied, replacing values below the threshold with -10 dBZ. This served as a simple way to remove non-meteorological targets and other clutter that could interfere with the training and prediction while maintaining most of the relevant precipitation echoes. Finally, the reflectivity values were normalized between zero and one. Computed predictions were converted back into reflectivity values by applying the inverse of the transformation before saving them, using the same scheme as with the input data.

The dataset reflectivity distribution is depicted in Fig. , with the -32 dBZ minimum value pixels left out of the histogram for clarity. The threshold is shown to divide the data into retained and discarded parts, and a Gaussian density is fitted to the part most likely to purely consist of precipitation, which exceeds 20 dBZ. Below the threshold, there seem to be multiple peaks in density, likely involving insects, birds, and miscellaneous clutter, as well as increasing noise the lower we go on the scale. While the highest-reflectivity density values seem to follow the Gaussian fit well, part of the density between 10 and 20 dBZ remains unexplained, and this range is likely to contain a mixture of precipitation and clutter. Still, this is not a major issue, since the most interesting precipitation to predict corresponds to reflectivity values well above 20 dBZ.

For the training of the network, the Adam optimizer was used with an initial learning rate of 1×10-4 and other parameters set to their PyTorch default values. The network was trained with that learning rate for 20 epochs, after which the learning rate was lowered to 1×10-5 for 8 more epochs, and finally further lowered to 1×10-6 for 1 final epoch. A validation epoch was carried out after each epoch in which equitable threat score (ETS) metrics were calculated for converted precipitation estimates (Sect. ) of predictions and summed over thresholds of 0.5, 1.0, 5.0, 10.0, 20.0, and 30.0 mm h-1, as well as each lead time. This validation score showed improvement over the whole training process.

The training procedure for DEUCE is presented in Fig.  for a single epoch. Both input sequence lengths Lin and output sequence lengths Lout were 12, corresponding to 1 h each. For the training and validation epochs, the batch size was set to two, and the number of produced Monte Carlo samples of posteriors N was set to two as well, which was the most that our GPU could fit during training. In order to increase the variance between the gradients of mini-batch members, Flipout re-parameterization was applied to the sampled weights, multiplying the random sampling coefficient of weights with a random sign matrix and effectively adding randomness inside batches for a low computational cost. The closed-form of the KL divergence between two Gaussian distributions was used for the calculations of the ELBO complexity term instead of Monte Carlo estimates in the final model training.

The input time series Xi was pre-processed first, as described in Sect. , and, in the case of training data, was then augmented by applying in succession a random horizontal flip, a random vertical flip, and a rotation by an angle randomly chosen between 0, 90, 180, and 270°. This was done to improve the variety in the training dataset and consequently improve the generalization performance of the trained network.

The performance of the DEUCE model is verified against the pySTEPS implementation of multiple extrapolation-based precipitation methods. The verification is divided into the qualitative inspection of ensembles produced in two case studies, into an analysis of DEUCE uncertainty composition, into the verification of the (probabilistic) performance of the whole ensemble, and into the verification of the (deterministic) performance of the ensemble mean, i.e., its fidelity in representing the true variation in the radar images. The four types of verification performed, along with the relevant DEUCE product, the baseline models used, and the evaluation criteria are summarized in Table .

In probabilistic verification experiments, N=48 ensemble members are used both for producing the raw outputs y^,σ2 and for drawing the post-processed ensemble y^ens, as well as for making the baseline ensemble model predictions. All of the predictions made for the verification of DEUCE are made until a 60 min lead time and thresholded at 8 dBZ, serving as an estimate for minimum observable precipitation. Four precipitation thresholds are considered where the verification involves evaluating the quality of a prediction exceeding a particular reflectivity value. Converted using the Z–R relationship presented in Sect. , these are 20 dBZ (≈0.5 mm h-1), 25 dBZ (≈1.3 mm h-1), 35 dBZ (≈5.7 mm h-1), and 45 dBZ (≈25.5 mm h-1), which correspond to very light, light, moderate, and heavy rain, respectively.

The results of the first case study in Figs. , , and suggest that DEUCE ensemble nowcasts are able to give reasonable uncertainty and exceedance probability estimates at multiple thresholds and lead times and that DEUCE nowcasts look similar to those given by STEPS, despite being less grainy. The results of the second case study are detailed in Appendix . Figure  shows an example of what individual ensemble members look like at different prediction lead times for the first case. The predictions all start out being quite similar, but they eventually diverge, driven by the increasing predictive uncertainty and different patterns of correlated noise. The ensemble members exhibit variety while preserving a moderate amount of realism that is nevertheless limited by the increasing smoothing of the predictive mean and variance fields with lead time.

The relative contribution of aleatoric and epistemic uncertainty for the first case study is presented in Fig. . We can see that most of the predictive uncertainty in fact comes from the aleatoric part. Epistemic uncertainty is of much smaller magnitude, and its contribution is further reduced when working in terms of variance in the calculation of predictive uncertainty. We can see that epistemic uncertainty does not extend as much away from the core of the predicted reflectivity as for aleatoric uncertainty, which reflects a small variance in the raw y^ ensemble. This overlap can be seen in probabilistic nowcasts as aleatoric uncertainty overshadowing the contribution of epistemic uncertainty.

A more detailed view of the contribution of aleatoric and epistemic components is provided in Fig. , with statistics over the whole verification dataset. A histogram of the uncertainties aggregated over all lead times and observed reflectivity values is shown to the left of Fig. . Epistemic uncertainty has a very narrow distribution, mostly between 0–5 dBZ, which means that its average value could not have varied much in different cases, lead times, and observed reflectivity values, pointing to a small model uncertainty response to these factors. Aleatoric uncertainty, on the other hand, has a long-tail distribution centered around 10 dBZ but going up to values over 30 dBZ, which means that we cannot exclude a dependence on these external factors.

Such dependencies are confirmed when inspecting the bar plots on the right of Fig. , where mean aleatoric uncertainty shows a clear dependence on prediction lead time and to some degree on observed reflectivity. Aleatoric uncertainty seems to clearly increase with lead time and to first slightly decrease before increasing again in relation to observed reflectivity. One possible explanation for this last observation is that reflectivity values below 20 dBZ often correspond to the edges of precipitation cells, which are difficult to predict, and that reflectivity values over 35 dBZ often correspond to heavy precipitation with a short lifetime and thus bad predictability. In between those, there are more predictable precipitation patterns, such as the interior of stratiform precipitation cells. Epistemic uncertainty, on the other hand, does not seem to show any particular dependence on prediction lead time, which might have something to do with the fact that the model predicts all lead times at once, making it possibly more difficult for the predictions to vary, depending on the lead time. There is, on the other hand, a slight increase in the epistemic uncertainty with observed reflectivity, which might be an accurate reflection of the relatively smaller amount of training data available for high observed reflectivity values.

The reliability diagrams and sharpness histograms for probabilistic nowcasts are depicted in Fig. . It can first be noted that, in general, DEUCE nowcasts are very close to the dashed black line, indicating a perfectly reliable forecast. Sometimes, this is similar to baseline models, but in some cases, such as a long lead time and a high threshold, DEUCE is closer to the diagonal than baselines. This is, however, not reflected in the ECE scores at 35 dBZ shown in Table , as smaller forecast probabilities are weighted much higher due to their sample count here, making STEPS the most reliable model at 35 dBZ when using this metric. An important pattern is that compared to baseline models, DEUCE is prone to slight under-forecasting of the exceedance probabilities. This is particularly the case for a short lead time (5 min), where the effect is the most pronounced. As lead times grow longer and thresholds get higher, nowcasting gets harder, and there is an overall tendency in all models, but particularly LINDA-P, to over-forecast threshold exceedance.

From the sharpness histogram, it is seen that the distribution of forecast probabilities is more or less uniform at low thresholds but more biased towards small exceedance probabilities at higher thresholds. These higher thresholds are where the difference between DEUCE and baselines is visible, as there is a considerably lower number of cases of high forecast probability in DEUCE than in baselines.

The rank histogram of nowcasts is shown in Fig. . It is apparent here that DEUCE is constantly slightly biased towards predicting reflectivity values that are too low and that the spread is large at short lead times but less significant later on. STEPS exhibits a very balanced flat histogram, but LINDA, on the other hand, has a U-shaped histogram characteristic of an ensemble breadth that is too small in general.

The results for the ROC area under the curve probabilistic nowcast metric are shown in Fig. . In this benchmark, DEUCE achieves the best results at all thresholds. We can notice that STEPS has good discriminative power at low thresholds but that it does not scale well to higher ones and that LINDA-P is not competitive at lower thresholds but excels as the threshold grows. Nevertheless, DEUCE manages to perform better than both in their skillful areas.

Last, the CRPS verification metric is depicted in Fig. . It can be seen that the lowest and best score is achieved by STEPS at all lead times. DEUCE comes second, slightly above STEPS, and LINDA-P lags far behind. Overall, with CRPS, it can be seen that DEUCE achieves adequate results of the order of baseline models.

From the quantitative probabilistic verification, it can be summarized that DEUCE achieves satisfactory and well-rounded performance. The model does not significantly lack in any category in particular and offers a good trade-off between forecast reliability and discriminatory power.

Here, we analyze the results of the comparison of the deterministic nowcast skill between DEUCE non-augmented mean predictions y^mean and baseline predictions. First, a depiction of the mean nowcasting error (ME) until a 60 min lead time is presented in Fig. . While extrapolation nowcasts have, on average, a ME slightly below zero, DEUCE is more strongly negatively biased, while LINDA-D is strongly positively biased.

Furthermore, the equitable threat score (ETS) results for reflectivity thresholds of 20, 25, and 35 dBZ are shown in Fig. . The ETS score of DEUCE is competitive for the 20 dBZ threshold, but at 25 dBZ, its progression at lead times longer than 30 min is already worse than the baseline. At 35 and 45 dBZ, DEUCE already performs worse than the baseline at any lead time examined. The reason for this weakness is the compound effect of intrinsic CNN prediction smoothing and the averaging of ensemble members. This smoothing effect is also visible in the radially averaged power spectral density (RAPSD) results for nowcasts presented in Fig. . Average RAPSD is computed for nowcasts at lead times of 5, 15, 30, and 60 min. It can clearly be seen that, compared to baselines, the fields predicted by DEUCE lose more power at small spatial scales and that this effect is heavily amplified at longer lead times, which again illustrates the above compound effect. However, this effect seems to be damped by augmenting the mean prediction with uncertainty-weighted correlated noise following the structure of the input field, especially at longer lead times.