ALBERT

Description: Interpolation of spatial data has been regarded in many different forms, varying from deterministic to stochastic, parametric to nonparametric, and purely data-driven to geostatistical methods. In this study, we propose a nonparametric interpolator, which combines information theory with probability aggregation methods in a geostatistical framework for the stochastic estimation of unsampled points. Histogram via entropy reduction (HER) predicts conditional distributions based on empirical probabilities, relaxing parameterizations and, therefore, avoiding the risk of adding information not present in data. By construction, it provides a proper framework for uncertainty estimation since it accounts for both spatial configuration and data values, while allowing one to introduce or infer properties of the field through the aggregation method. We investigate the framework using synthetically generated data sets and demonstrate its efficacy in ascertaining the underlying field with varying sample densities and data properties. HER shows a comparable performance to popular benchmark models, with the additional advantage of higher generality. The novel method brings a new perspective of spatial interpolation and uncertainty analysis to geostatistics and statistical learning, using the lens of information theory.

Description: In this paper we propose adaptive clustering as a new method for reducing the computational efforts of distributed modelling. It consists of identifying similar-acting model elements during runtime, clustering them, running the model for just a few representatives per cluster, and mapping their results to the remaining model elements in the cluster. Key requirements for the application of adaptive clustering are the existence of (i) many model elements with (ii) comparable structural and functional properties and (iii) only weak interaction (e.g. hill slopes, subcatchments, or surface grid elements in hydrological and land surface models). The clustering of model elements must not only consider their time-invariant structural and functional properties but also their current state and forcing, as all these aspects influence their current functioning. Joining model elements into clusters is therefore a continuous task during model execution rather than a one-time exercise that can be done beforehand. Adaptive clustering takes this into account by continuously checking the clustering and re-clustering when necessary. We explain the steps of adaptive clustering and provide a proof of concept at the example of a distributed, conceptual hydrological model fit to the Attert basin in Luxembourg. The clustering is done based on normalised and binned transformations of model element states and fluxes. Analysing a 5-year time series of these transformed states and fluxes revealed that many model elements act very similarly, and the degree of similarity varies strongly with time, indicating the potential for adaptive clustering to save computation time. Compared to a standard, full-resolution model run used as a virtual reality “truth”, adaptive clustering indeed reduced computation time by 75 %, while modelling quality, expressed as the Nash–Sutcliffe efficiency of subcatchment runoff, declined from 1 to 0.84. Based on this proof-of-concept application, we believe that adaptive clustering is a promising tool for reducing the computation time of distributed models. Being adaptive, it integrates and enhances existing methods of static grouping of model elements, such as lumping or grouped response units (GRUs). It is compatible with existing dynamical methods such as adaptive time stepping or adaptive gridding and, unlike the latter, does not require adjacency of the model elements to be joined. As a welcome side effect, adaptive clustering can be used for system analysis; in our case, analysing the space–time patterns of clustered model elements confirmed that the hydrological functioning of the Attert catchment is mainly controlled by the spatial patterns of geology and precipitation.

Description: When constructing discrete (binned) distributions from samples of a data set, applications exist where it is desirable to assure that all bins of the sample distribution have nonzero probability. For example, if the sample distribution is part of a predictive model for which we require returning a response for the entire codomain, or if we use Kullback–Leibler divergence to measure the (dis-)agreement of the sample distribution and the original distribution of the variable, which, in the described case, is inconveniently infinite. Several sample-based distribution estimators exist which assure nonzero bin probability, such as adding one counter to each zero-probability bin of the sample histogram, adding a small probability to the sample pdf, smoothing methods such as Kernel-density smoothing, or Bayesian approaches based on the Dirichlet and Multinomial distribution. Here, we suggest and test an approach based on the Clopper–Pearson method, which makes use of the binominal distribution. Based on the sample distribution, confidence intervals for bin-occupation probability are calculated. The mean of each confidence interval is a strictly positive estimator of the true bin-occupation probability and is convergent with increasing sample size. For small samples, it converges towards a uniform distribution, i.e., the method effectively applies a maximum entropy approach. We apply this nonzero method and four alternative sample-based distribution estimators to a range of typical distributions (uniform, Dirac, normal, multimodal, and irregular) and measure the effect with Kullback–Leibler divergence. While the performance of each method strongly depends on the distribution type it is applied to, on average, and especially for small sample sizes, the nonzero, the simple “add one counter”, and the Bayesian Dirichlet-multinomial model show very similar behavior and perform best. We conclude that, when estimating distributions without an a priori idea of their shape, applying one of these methods is favorable.

Description: Second Workshop on Information Theory and the Earth Sciences; Santander, Spain, 16–19 May 2018

Description: This article introduces an improvement in the Series Distance (SD) approach for the improved discrimination and visualization of timing and magnitude uncertainties in streamflow simulations. SD emulates visual hydrograph comparison by distinguishing periods of low flow and periods of rise and recession in hydrological events. Within these periods, it determines the distance of two hydrographs not between points of equal time but between points that are hydrologically similar. The improvement comprises an automated procedure to emulate visual pattern matching, i.e. the determination of an optimal level of generalization when comparing two hydrographs, a scaled error model which is better applicable across large discharge ranges than its non-scaled counterpart, and "error dressing", a concept to construct uncertainty ranges around deterministic simulations or forecasts. Error dressing includes an approach to sample empirical error distributions by increasing variance contribution, which can be extended from standard one-dimensional distributions to the two-dimensional distributions of combined time and magnitude errors provided by SD. In a case study we apply both the SD concept and a benchmark model (BM) based on standard magnitude errors to a 6-year time series of observations and simulations from a small alpine catchment. Time–magnitude error characteristics for low flow and rising and falling limbs of events were substantially different. Their separate treatment within SD therefore preserves useful information which can be used for differentiated model diagnostics, and which is not contained in standard criteria like the Nash–Sutcliffe efficiency. Construction of uncertainty ranges based on the magnitude of errors of the BM approach and the combined time and magnitude errors of the SD approach revealed that the BM-derived ranges were visually narrower and statistically superior to the SD ranges. This suggests that the combined use of time and magnitude errors to construct uncertainty envelopes implies a trade-off between the added value of explicitly considering timing errors and the associated, inevitable time-spreading effect which inflates the related uncertainty ranges. Which effect dominates depends on the characteristics of timing errors in the hydrographs at hand. Our findings confirm that Series Distance is an elaborated concept for the comparison of simulated and observed streamflow time series which can be used for detailed hydrological analysis and model diagnostics and to inform us about uncertainties related to hydrological predictions.

Description: In this study we propose and demonstrate a data-driven approach in an “information-theoretic” framework to quantitatively estimate precipitation. In this context, predictive relations are expressed by empirical discrete probability distributions directly derived from data instead of fitting and applying deterministic functions, as is standard operational practice. Applying a probabilistic relation has the benefit of providing joint statements about rain rate and the related estimation uncertainty. The information-theoretic framework furthermore allows for the integration of any kind of data considered useful and explicitly considers the uncertain nature of quantitative precipitation estimation (QPE). With this framework we investigate the information gains and losses associated with various data and practices typically applied in QPE. To this end, we conduct six experiments using 4 years of data from six laser optical disdrometers, two micro rain radars (MRRs), regular rain gauges, weather radar reflectivity and other operationally available meteorological data from existing stations. Each experiment addresses a typical question related to QPE. First, we measure the information about ground rainfall contained in various operationally available predictors. Here weather radar proves to be the single most important source of information, which can be further improved when distinguishing radar reflectivity–ground rainfall relationships (Z–R relations) by season and prevailing synoptic circulation pattern. Second, we investigate the effect of data sample size on QPE uncertainty using different data-based predictive models. This shows that the combination of reflectivity and month of the year as a two-predictor model is the best trade-off between robustness of the model and information gain. Third, we investigate the information content in spatial position by learning and applying site-specific Z–R relations. The related information gains are only moderate; specifically, they are lower than when distinguishing Z–R relations according to time of the year or synoptic circulation pattern. Fourth, we measure the information loss when fitting and using a deterministic Z–R relation, as is standard practice in operational radar-based QPE applying, e.g., the standard Marshall–Palmer relation, instead of using the empirical relation derived directly from the data. It shows that while the deterministic function captures the overall shape of the empirical relation quite well, it introduces an additional 60 % uncertainty when estimating rain rate. Fifth, we investigate how much information is gained along the radar observation path, starting with reflectivity measured by radar at height, continuing with the reflectivity measured by a MRR along a vertical profile in the atmosphere and ending with the reflectivity observed by a disdrometer directly at the ground. The results reveal that considerable additional information is gained by using observations from lower elevations due to the avoidance of information losses caused by ongoing microphysical precipitation processes from cloud height to ground. This emphasizes both the importance of vertical corrections for accurate QPE and of the required MRR observations. In the sixth experiment we evaluate the information content of radar data only, rain gauge data only and a combination of both as a function of the distance between the target and predictor rain gauge. The results show that station-only QPE outperforms radar-only QPE up to a distance of 7 to 8 km from the nearest station and that radar–gauge QPE performs best, even compared with radar-based models applying season or circulation pattern.

Description: Surface topography is an important source of information about the functioning and form of a hydrological landscape. Because of its key role in explaining hydrological processes and structures, and also because of its wide availability at good resolution in the form of digital elevation models (DEMs), it is frequently used to inform hydrological analyses. Not surprisingly, several hydrological indices and models have been proposed for linking geomorphic properties of a landscape with its hydrological functioning; a widely used example is the “height above the nearest drainage” (HAND) index. From an energy-centered perspective HAND reflects the gravitational potential energy of a given unit mass of water located on a hillslope, with the reference level set to the elevation of the nearest corresponding river. Given that potential energy differences are the main drivers for runoff generation, HAND distributions provide important proxies to explain runoff generation in catchments. However, as expressed by the second law of thermodynamics, the driver of a flux explains only one aspect of the runoff generation mechanism, with the driving potential of every flux being depleted via entropy production and dissipative energy loss. In fact, such losses dominate when rainfall becomes runoff, and only a tiny portion of the driving potential energy is actually transformed into the kinetic energy of streamflow. In recognition of this, we derive a topographic index called reduced dissipation per unit length index (rDUNE) by reinterpreting and enhancing HAND following a straightforward thermodynamic argumentation. We compare rDUNE with HAND, and with the frequently used topographic wetness index (TWI), and show that rDUNE provides stronger discrimination of catchments into groups that are similar with respect to their dominant runoff processes. Our analysis indicates that accounting for both the driver and resistance aspects of flux generation provides a promising approach for linking the architecture of a system with its functioning and is hence an appropriate basis for developing similarity indices in hydrology.