Deep learning, hydrological processes and the uniqueness of place

One of the things that we learn from the history of science is that, with some notable exceptions beloved of philosophers of science, knowledge and understanding progress over time. Looking back, we see that understanding of the natural world has (mostly) progressed. Sometimes alternative theories have awaited experimental confirmation; sometimes a new experimental technique has led to significant theoretical advances. We hope, of course, to see some of that progression, and to make a contribution to it, over the time scale of our own careers in science. It is therefore somewhat disconcerting to have something you wrote more than 30 years ago cited (in Nearing et al., 2020) as if the comments were relevant today. Things should have changed, even in hydrology. The context is that of the availability of the new techniques of machine learning and deep learning and their application to hydrological data. Nearing et al. (2020) suggest that in many respects not much has actually changed, since I wrote about the need for a new paradigm in hydrological modelling in 1987 (Beven, 1987). They go on to suggest that machine learning and deep learning can produce models that perform just as well, if not better, than conceptual models and process-based hydrological models, including for catchments treated as ungauged (see also Kratzert, Klotz, Shalev et al., 2019; Kratzert, Klotz, Herrnegger, et al., 2019). Should this be considered surprising? Not necessarily in the case of individual catchments—if there are consistent anomalies or epistemic uncertainties in catchment data that mean, for example, that water balance constraints are not well met, then a deep learning (DL) model can compensate for those anomalies in ways that a conceptual model, constrained by water balance cannot. If there are consistent anomalies between the conceptual structure of a hydrological model in a particular catchment and the nature of the hydrological processes in that catchment then again a DL model might well be able to capture that behaviour better than a deficient process description (although it is worth noting that DL models are also subject to choices in structure and multiple hidden parameters; that is what gives them flexibility in fitting the training data). Nearing et al. (2020) point out that there are techniques for incorporating conservation constraints into physically constrained DL models (see also Wang, Zhang, Chang, recession characteristics can vary dramatically over short distances and with scale in some regions, depending on changes in geology and soils (e.g., Oudin, Kay, Andréassian, Jencso Bergstrom, Jencso, and the study using the correlation and regression tree data analysis method of Fang Kratzert, Klotz, Shalev, et al., 2019). So DL has not solved the ungauged catchment problem, but it did show better performance than the two conceptual models that were compared (and that will be subject to similar problems of the input data and, from the choice of conceptual assumptions, the potential to have quite the wrong structure for specific catchments). Clearly there must be other issues at play here. Secondly, the sensitivity coefficients for the DL show distinct variations in behaviour across the catchments in the training data set. This reflects the range of hydrological responses across catchment characteristics and scales (although there is no need to specify classes using this methodology; Kratzert, Klotz, Herrnegger, et al., 2019 also show that the LSTM model performs somewhat better than a k-means clustering algorithm for several hydrological signatures). The essence of predicting an ungauged catchment is then to map that catchment into the DL model space. This is analogous to the mapping suggested by Beven (2000) as a way of assessing the uncertainty in parameterizations of a conceptual model (and then thinking about how that uncertainty might be further constrained). Such a mapping is necessarily based on the catchment characteristics that are available to the DL as indices. But most catchment indices are only surrogate variables and geology, for example, is one characteristic that is not very well reflected in indices that can be used for prediction (a base flow index cannot serve that purpose, since it will not be available a priori for ungauged catchments but must itself be predicted). In this respect the use of a DL approach might indeed be advantageous because the nature of the training data can compensate for the lack of direct hydrological relevance of such indices (and in doing so have more flexibility than, e.g., the multiple regressions to estimate the model parameters used in other more traditional regionalization approaches). This idea of mapping a catchment (gauged or ungauged) into the model space was originally raised by Beven (2000; 2001; 2002a; 2002b) in the context of uniqueness of place in hydrological simulation. It was extended in Beven and Freer (2001) in the application of a conceptual hydrological model. The idea was that, given our limited knowledge of catchment characteristics and understanding of catchment responses, it would be impossible to map that catchment to a single point in the model space (this could apply to both model structures and parameter sets). The uncertainty in model predictions does not then come from the model space itself (even if providing stochastic rather than deterministic outputs), it comes from the mapping of an area of reality into the model space given the available information. The concept then allows consideration of how that mapping might be made more precise, perhaps by applications of likelihoods, hypothesis testing and model rejection, and the collection of additional data. It is interesting to consider the application of a similar concept to DL models. Kratzert, Klotz, Brenner, Schulz, and Herrnegger (2018) give an example of these where a pre-trained LSTM model based on regional data (effectively defining a generalised model space) can be refined by more limited data for an individual basin; this gives better results than only training on the limited data for the catchment itself, even though the regional model might include catchments with a wide range of behaviours. In addition, as Nearing et al. (2020) suggest, DL models, given enough training data, can be trained to represent not only deterministic responses but also the parameters of the uncertainty in outputs (variances or quantiles). The DL model space might also therefore be deterministic or stochastic, but the range of outcomes is well defined given a set of inputs. The nature of that space will, however, be dependent on how the DL has been trained and what objective functions have been used. In predicting an ungauged catchment, with its unique characteristics, we then have the problem of mapping that catchment, for which we have limited knowledge, into that model space. We should surely not expect that the mapping will be to a precise point in that space, but that there will be some uncertainty associated with our lack of knowledge. This is one way of structuring our understanding of how that catchment might function when that mapping is applied in a thoughtful way. To some extent a stochastic DL should be able to reflect the variation seen for somewhat similar gauged catchments in the training data set, given a big enough sample, even though all those gauged catchments have their own uniqueness of place poorly reflected in characteristic indices. This will be the case for those ungauged catchments that are similar in the way defined by the training process (remembering that for all machine learning methods extrapolating beyond the range of the training data for nonlinear systems may not be well constrained). In this sense therefore DL models provide a model space constrained by the training data into which the characteristics of some new ungauged catchment can be mapped with the aim of predicting the response of that catchment without worrying about any process information. Nearing et al. (2020) posit that DL might be able to do this better than the hydrologist, though again information about the effect of geology on hydrological response might be particularly limited (for both!). More process information could, of course, be used if it was available in some form for all the training data catchments and the catchment of interest. In applying process-based models we have traditionally expected the right sort of response to be predicted by specifying the physical parameters of a catchment. That this has proven rather difficult should not actually be surprising, it was already underlying my call for a new paradigm in modelling in 1987. Since then, computational power has hugely increased, but knowledge of catchment processes and characteristics in most catchments has not (with some notable research catchment and critical zone observatory exceptions). The difficulty in predicting the future responses of catchment management strategies that might change those characteristics should therefore be even less surprising. It is evident that DL can then only help us in that when such changes are reflected in the indices used to control the pathways through the DL network (and when there are informative relevant cases in the training data). Question 1 points again to lack of knowledge. That might be lack of knowledge about inconsistencies in the data or lack of knowledge about how to represent processes. That DL models are not able to compensate for that lack of knowledge in some catchments suggests that the data issues might be the first thing to look at (see also Beven, 2019; Beven et al., 2019). Indeed, experience suggests that applying data learning methods can reveal anomalies of interest in a data set (e.g., Iorgulescu Freer, McMillan, McDonnell, Güntner, Uhlenbrook, Seibert, Hopp Jencso Tetzlaff, Birkel, Dick, Geris, Harman, 2019) might be more useful. They require considerable effort, but perhaps this effort would be justified if we really want to improve our predictive models and learn more by inference from DL approaches. It would be interesting, for example, to see a DL study of a catchment where the time scales of hydrographs and water table responses (and also residence times over time) are quite different to see what insights could be gained (see Beven, 2020; McDonnell Blair et al., 2019). Others will not agree of course. 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