A key challenge in climate science is to quantify the forced response in impact-relevant variables such as precipitation against the background of internal variability, both in models and observations. Dynamical adjustment techniques aim to remove unforced variability from a target variable by identifying patterns associated with circulation, thus effectively acting as a filter for dynamically induced variability. The forced contributions are interpreted as the variation that is unexplained by circulation. However, dynamical adjustment of precipitation at local scales remains challenging because of large natural variability and the complex, nonlinear relationship between precipitation and circulation particularly in heterogeneous terrain. Building on variational autoencoders, we introduce a novel statistical model – the Latent Linear Adjustment Autoencoder (LLAAE) – that enables estimation of the contribution of a coarse-scale atmospheric circulation proxy to daily precipitation at high resolution and in a spatially coherent manner. To predict circulation-induced precipitation, the Latent Linear Adjustment Autoencoder combines a linear component, which models the relationship between circulation and the latent space of an autoencoder, with the autoencoder's nonlinear decoder. The combination is achieved by imposing an additional penalty in the cost function that encourages linearity between the circulation field and the autoencoder's latent space, hence leveraging robustness advantages of linear models as well as the flexibility of deep neural networks. We show that our model predicts realistic daily winter precipitation fields at high resolution based on a 50-member ensemble of the Canadian Regional Climate Model at 12 km resolution over Europe, capturing, for instance, key orographic features and geographical gradients. Using the Latent Linear Adjustment Autoencoder to remove the dynamic component of precipitation variability, forced thermodynamic components are expected to remain in the residual, which enables the uncovering of forced precipitation patterns of change from just a few ensemble members. We extend this to quantify the forced pattern of change conditional on specific circulation regimes. Future applications could include, for instance, weather generators emulating climate model simulations of regional precipitation, detection and attribution at subcontinental scales, or statistical downscaling and transfer learning between models and observations to exploit the typically much larger sample size in models compared to observations.
Precipitation is a key climate variable that is highly relevant for impacts such as floods or meteorological drought. Precipitation simulations at high resolution
Dynamical adjustment techniques have been developed to separate forced and internal variability via a co-interpretation of target variables such as temperature or precipitation using circulation information: a circulation proxy (such as a sea-level pressure pattern) is used to estimate the circulation-induced (dynamic) contribution to temperature or precipitation variability. For example, dynamical adjustment of precipitation has revealed that the spatial pattern and amplitude of observed residual (predominant thermodynamic) precipitation trends at the scale of the entire Northern Hemisphere mid- and high-latitude land areas are in good agreement with the expected anthropogenically forced trends from model simulations
Techniques for dynamical adjustment have relied largely on linear regression
In this work, we leverage recent advances in machine learning to propose a novel statistical model – the Latent Linear Adjustment Autoencoder (LLAAE) – suitable for dynamical adjustment (and potentially further applications) in daily, high-resolution precipitation fields. In recent years, deep learning techniques have gained in popularity in machine learning due to large improvements in neural network architectures, optimization algorithms as well as computing power and frameworks. Among the class of deep generative models, the introduction of variational autoencoders (VAEs)
Specifically, during training, the Latent Linear Adjustment Autoencoder encodes daily precipitation fields into a low-dimensional latent space and subsequently decodes them for reconstruction. In addition, we formulate the objective function such that the latent space can be regressed linearly on the circulation proxy. For dynamical adjustment, we use the estimate of the latent space based on circulation, which is then decoded for predicting daily precipitation fields at high spatial resolution. In other words, the final model is nonlinear, consisting of a linear part and a nonlinear part, where the latter is a deep neural network. It enables prediction of the portion of the precipitation field that can be explained by circulation (i.e. the dynamic component of precipitation). Moreover, several further climate science applications of the Latent Linear Adjustment Autoencoder are conceivable, such as for example weather generators emulating regional climate model simulations, detection and attribution at subcontinental scales, or statistical downscaling, and are discussed further below.
In summary, the objectives of this paper are the following:
We introduce a novel statistical model – the Latent Linear Adjustment Autoencoder – as a versatile technique for applications in climate science, particularly for making better use of high-resolution climate simulations by estimating circulation-induced (dynamic) precipitation at high resolution from coarse-scale circulation information. We illustrate the Latent Linear Adjustment Autoencoder by applying it to dynamical adjustment of daily high-resolution precipitation from simulations over central Europe. More specifically, the LLAAE will be used to separate forced precipitation trends from internal variability.
Following
Let
Illustration of a standard autoencoder model: the spatial fields
We build on variational autoencoders
We extend the standard VAE model to make it suitable for dynamical adjustment by adding a linear component
In more detail, we consider the following objective to train the encoder
Illustration of the Latent Linear Adjustment Autoencoder after training: the input features
After training the components
The spatial field is modelled jointly in our approach – the optimization is performed over the whole spatial field at once – in contrast to
To evaluate our statistical model, we use the Canadian Regional Climate Model Large Ensemble (CRCM5-LE,
This approach yields 50 approximately independent realizations of the climate system
We focus on precipitation as the target climate variable throughout the main text. A subset of results for temperature can be found in Appendix
SLP is regridded to a spatial resolution of 1
We use RCM simulation data from 1955 to 2100 to allow for 5 years of spinup. We train our model using daily data from December to February (DJF) from nine ensemble members (“kba”, “kbc”, “kbe”, “kbg”, “kbi”, “kbk”, “kbm”, “kbq”, “kbs”). The results in the main text are based on training data that comprise the years 1955–2070. In Appendix
Examples of (i) original precipitation fields (left column), (ii) reconstructions (centre column), and (iii) predictions (right column). The examples are chosen such that the
To illustrate the spatial coherence of our approach, we show five example target (daily) data points of
We evaluate the extent to which the forced response of precipitation can be uncovered with a small number of ensemble members using dynamical adjustment
We begin by showing a selection of reconstructed precipitation fields
MSE (based on precipitation data in mm d
The proposed model yields spatially coherent predictions and explains a large proportion of the variance of
Proportion of variance explained (
In this subsection, we evaluate our predictions of the circulation-induced component in the framework of dynamical adjustment
Dynamical adjustment of the full domain (land grid cells only) using autoencoders; (left) seasonal precipitation totals simulated by three members of the high-resolution RCM in black (top: “kbb”, middle: “kct”, bottom: “kcu”), the predicted (“circulation-induced”) component for three “holdout” ensemble members (blue), and the forced response (average across all 50 members, red); (right) residuals from the prediction (black dots) and the forced response (red).
The effect of dynamical adjustment can be seen in Fig.
Dynamical adjustment of 50-year winter precipitation trends (2020–2069); (left column) forced precipitation response (2020–2069 linear winter precipitation trends averaged over all 50 ensemble members); (middle column) linear 50-year winter precipitation trends in three randomly selected dynamically adjusted ensemble members (top: “kbb”, middle: “kct”, bottom: “kcu”); (right column) linear 50-year winter precipitation trends in the original ensemble members (top: “kbb”, middle: “kct”, bottom: “kcu”); (bottom row) scatter plot contour lines of 50-year precipitation trends in the forced response against 50-year precipitation trends over land in dynamically adjusted (middle) and originally simulated ensemble members (right).
RMSE of 50-year trends, calculated by averaging
It is more challenging, however, to identify and evaluate the forced precipitation response at the local scale of individual grid points. To this end, Fig.
However, large variability in individual ensemble members is superimposed on the signal of forced change (Fig.
Figure
While dynamical adjustment of long-term trends of temperature and precipitation has become a standard tool for the detection of forced thermodynamic trends
Thus, we assess to what extent the forced precipitation response can be uncovered under specific circulation conditions from a small number of ensemble members. We create composites of the dominant mode of atmospheric winter circulation over Europe as diagnosed by EOF analysis over the historical period (1955–2020) in the RCM simulations.
The first EOF of the coarse-resolution SLP field is shown in Fig.
Dynamical adjustment of 50-year winter precipitation trends (2020–2069) under the “EOF1+” regime (25 % of all days that project strongest on the first EOF, i.e. those that show a strong westerly flow).
We now generate composites of “EOF1+” and “EOF1-” regimes by isolating days that exceed the 75th percentile (“EOF1+”) and those that fall below the 25th percentile (“EOF1-”) in terms of the first principal component
(Fig.
For the 50-year forced precipitation trend on “EOF1+” winter days (obtained by averaging across all ensemble members), there is a more pronounced precipitation increase on the western slopes of the Alps and in most parts of the domain north and west of the Alps (Fig.
Raw simulations for sets of three holdout members show variable 50-year (2020–2069) precipitation trends under the “EOF1+” regime (Fig.
Dynamical adjustment of 50-year winter precipitation trends (2020–2069) under the “EOF1-” regime (25 % of all days that project weakest on the first EOF).
Forced precipitation trends for 2020–2069 under “EOF1-” conditions differ from “EOF1+” conditions due to a change in the synoptic situation: the forced spatial pattern has generally weaker precipitation changes (due to overall drier conditions during “EOF1-”), and precipitation increases are confined towards southeastern Europe (Fig.
The application of dynamical adjustment to composites of specific circulation regimes raises the question as to whether the Latent Linear Adjustment Autoencoder may be applicable to understanding the dynamical component in extreme precipitation events. While the LLAAE may be able to fill an important gap in reconstructing the dynamical component of daily precipitation fields, possibly including days with extreme precipitation (at least, the component proportional to surface pressure), it exhibits a tendency to smooth predicted precipitation fields (Fig.
Overall, we conclude that dynamical adjustment enables approximating the forced response from high-resolution simulations with only a few ensemble members. This is possible for both long-term trends in seasonal precipitation totals as well as for trends under more specific circulation regimes. The improvement for the “EOF1+” and “EOF1-” circulation regimes can be evaluated from Fig.
One of the main uncertainties in dynamical adjustment is the question of whether and how to detrend the climate data (circulation fields and/or precipitation) prior to dynamical adjustment. This is somewhat subjective and often discussed as an inherent uncertainty in the literature (see, e.g.
Another important question is how much training data are necessary to achieve the presented results. One may argue that it is computationally cheaper to estimate the forced response using a – say – nine-ensemble-member mean, instead of training the LLAAE based on simulations from nine ensemble members (as done in this work). Indeed, as machine learning algorithms are known to require rather large amounts of training data, “proving” the case of LLAAE dynamical adjustment in a large ensemble may not be as straightforward. While we have shown that dynamical adjustment based on LLAAEs reduces the number of ensemble members required to identify a proxy of the forced response for local-scale 50-year winter precipitation trends substantially (Fig.
Beyond the computational aspects, however, we anticipate the ultimate applications of LLAAE-based dynamical adjustment not on a large ensemble (where the forced response is typically approximated with the ensemble average,
In principle, there are alternative approaches for statistical learning in the context of dynamical adjustment and also alternative options to employ deep neural networks. For instance, one could extend the method of
Furthermore, one may wonder why the autoencoder is needed in the architecture of the LLAAE if the encoder is discarded when predicting dynamic precipitation from SLP. Using the autoencoder for estimation allows us to link SLP EOFs as input with the 2-D precipitation fields as output. Removing the intermediate stage of the autoencoder would constitute a challenging estimation problem as the autoencoder helps to estimate the decoder. We are not aware of alternative machine learning (ML) algorithms for this input/output combination and the LLAAE is novel in this regard.
In this work, we have first introduced the Latent Linear Adjustment Autoencoder, which combines a linear model with the nonlinear decoder of a variational autoencoder. By combining a linear model, which takes a circulation proxy as input, with the expressive nonlinear (deep neural network) decoder, it can be easily trained and allows for jointly modelling the dynamically induced high-resolution spatial field of the climate variable of interest. The main methodological novelty is that we add a linear model to the variational autoencoder and include an additional penalty term in the loss function that encourages linearity between the circulation proxy and the latent space. This leverages the advantages of a linear relationship between circulation variables and latent space variables, hence enhancing robustness, while also benefiting from the advantages of deep neural networks (i.e. flexibility in modelling nonlinearities, such as those that occur in high-resolution orographic precipitation). Future work targeting climate applications could explore the robust transfer of LLAAEs between different climate models, reanalyses data, or observations by using ideas from transfer learning or distributional robustness
Second, as the main application, we have tested the applicability of the Latent Linear Adjustment Autoencoder to dynamical adjustment of high-resolution precipitation based on daily data at regional scales. Based on a circulation proxy, the Latent Linear Adjustment Autoencoder predicts dynamic (circulation-induced) precipitation at high resolution. An estimate of the forced precipitation response can then be separated from internal variability, leaving a higher signal-to-noise ratio compared to raw multidecadal trends. With only one or two ensemble members, root mean squared errors are roughly halved compared to raw trends when estimating the forced response (see Fig.
Further use cases of the Latent Linear Adjustment Autoencoder may include further applications of dynamical adjustment, including transfer learning across different high-resolution simulations such as EURO-CORDEX models
Lastly, a further broad application of the Latent Linear Adjustment Autoencoder within climate science may lie in the area of model emulation
Overall, the Latent Linear Adjustment Autoencoder may prove a versatile tool for climate and atmospheric science, specifically for modelling relationships between large-scale predictors and local and nonlinear precipitation at high resolution.
In this section, we detail the architecture used for the encoder and decoder of the proposed model. Additionally, we report the most important hyperparameters. All further details can be found in the accompanying code; see the “Code and data availability” section below for details.
For the encoder and the decoder, we use three convolutional layers and one residual layer
As discussed in the main text, one of the main uncertainties in dynamical adjustment is how to ensure that the statistical model does not fit a thermodynamic, forced signal and hence only models the dynamic internal variability. Fitting a forced signal can potentially be mitigated by (i) an appropriate choice of the training period and (ii) suitable pre-processing of the data. Furthermore, another important question is how much training data are necessary to achieve the presented results, even though we do not see the ultimate use case of LLAAEs to be used for dynamical adjustment in large ensembles (also see the discussion in Sect.
Training 1955–2020: mean-squared error (MSE, based on precipitation data in mm d
Training 1955–2020: proportion of variance explained (
Dynamical adjustment of the full domain (land grid cells only) but training only based on 1955–2020.
We train on the shorter period from 1955–2020 (as opposed to 1955–2070), using the same nine ensemble members as described in the main text. This corresponds approximately to a
Dynamical adjustment of the full domain (land grid cells only) but training only based on 1955–2020 and with precipitation detrended instead of SLP before model training and dynamical adjustment.
The question of whether and how to detrend prior to dynamical adjustment is open, somewhat subjective, and often discussed as an inherent subjective choice and uncertainty in dynamical adjustment papers (see, e.g. Precipitation change cannot be modelled by a single additive mean change across the whole distribution. For instance, precipitation change is known to increase the variance of the precipitation distribution There may be some dynamically induced changes in precipitation, but it would be hard to evaluate this without any additional simulations where dynamical effects and thermodynamical effects could be separated.
Overall, we conclude that our simple SLP detrending (without detrending precipitation) is a useful approach for introducing LLAAEs as a versatile tool for dynamical adjustment, as demonstrated by the fact that the residuals of individual ensemble members after dynamical adjustment match the ensemble-mean trend of precipitation very well (e.g. Fig.
The following results are based on temperature anomalies. In Fig.
Figure
Temperature anomalies. Examples of (i) original temperature fields
Temperature anomalies. MSE for each grid cell for the temperature predictions.
Temperature anomalies. Proportion of variance explained (
The Latent Linear Adjustment Autoencoder model is free and open source. It is distributed under the MIT software license which allows unrestricted use. The source code is available at the following GitHub repository:
CH and NM conceptualized the Latent Linear Adjustment Autoencoder. CH, SS, and NM conceptualized the climate applications with support from AP and FL. CH, SS, and NM developed the methodology. CH did the formal analysis. CH and SS did the investigation and visualization. CH and SS wrote the original draft. CH, SS, AP, and NM reviewed and edited it.
The authors declare that they have no conflict of interest.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank Raul Wood and Martin Leduc for providing the climate model simulations. The production of ClimEx was funded within the ClimEx project by the Bavarian State Ministry for the Environment and Consumer Protection. The CRCM5 was developed by the ESCER centre of Université du Québec à Montréal (UQAM;
This material is based in part upon work supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation (NSF) under cooperative agreement no. 1947282, and by the Regional and Global Model Analysis (RGMA) component of the Earth and Environmental System Modeling Program of the US Department of Energy's Office of Biological & Environmental Research (BER) via NSF IA 1844590. Sebastian Sippel acknowledges funding provided by the Swiss Data Science Centre within the project “Data Science-informed attribution of changes in the Hydrological cycle” (DASH, ID C17-01). Flavio Lehner has been supported by the Swiss National Science Foundation (grant no. PZ00P2_174128).
This paper was edited by Gerd A. Folberth and reviewed by Ségolène Berthou and one anonymous referee.