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Several studies have demonstrated the ability of long short-term memory (LSTM) machine-learning-based modeling to outperform traditional spatially lumped process-based modeling approaches for streamflow prediction. However, due mainly to the structural complexity of the LSTM network (which includes gating operations and sequential processing of the data), difficulties can arise when interpreting the internal processes and weights in the model. Here, we propose and test a modification of LSTM architecture that is calibrated in a manner that is analogous to a hydrological system. Our architecture, called “HydroLSTM”, simulates the sequential updating of the Markovian storage while the gating operation has access to historical information. Specifically, we modify how data are fed to the new representation to facilitate simultaneous access to past lagged inputs and consolidated information, which explicitly acknowledges the importance of trends and patterns in the data. We compare the performance of the HydroLSTM and LSTM architectures using data from 10 hydro-climatically varied catchments. We further examine how the new architecture exploits the information in lagged inputs, for 588 catchments across the USA. The HydroLSTM-based models require fewer cell states to obtain similar performance to their LSTM-based counterparts. Further, the weight patterns associated with lagged input variables are interpretable and consistent with regional hydroclimatic characteristics (snowmelt-dominated, recent rainfall-dominated, and historical rainfall-dominated). These findings illustrate how the hydrological interpretability of LSTM-based models can be enhanced by appropriate architectural modifications that are physically and conceptually consistent with our understanding of the system. 

We confirm that energy dissipation weighting provides the most accurate approach to determining the effective hydraulic conductivity (Keff) of a binary K grid. A deep learning algorithm (UNET) can infer Keff with extremely high accuracy (R2 > 0.99). The UNET architecture could be trained to infer the energy dissipation weighting pattern from an image of the K distribution, although it was less accurate for cases with highly localized structures that controlled flow. Furthermore, the UNET architecture learned to infer the energy dissipation weighting even if it was not trained directly on this information. However, the weights were represented within the UNET in a way that was not immediately interpretable by a human user. This reiterates the idea that even if ML/DL algorithms are trained to make some hydrologic predictions accurately, they must be designed and trained to provide each user-required output if their results are to be used to improve our understanding of hydrologic systems. 

We develop a simple Quantile Spacing (QS) method for accurate probabilistic estimation of one-dimensional entropy from equiprobable random samples, and compare it with the popular Bin-Counting (BC) and Kernel Density (KD) methods. In contrast to BC, which uses equal-width bins with varying probability mass, the QS method uses estimates of the quantiles that divide the support of the data generating probability density function (pdf) into equal-probability-mass intervals. And, whereas BC and KD each require optimal tuning of a hyper-parameter whose value varies with sample size and shape of the pdf, QS only requires specification of the number of quantiles to be used. Results indicate, for the class of distributions tested, that the optimal number of quantiles is a fixed fraction of the sample size (empirically determined to be ~0.25–0.35), and that this value is relatively insensitive to distributional form or sample size. This provides a clear advantage over BC and KD since hyper-parameter tuning is not required. Further, unlike KD, there is no need to select an appropriate kernel-type, and so QS is applicable to pdfs of arbitrary shape, including those with discontinuous slope and/or magnitude. Bootstrapping is used to approximate the sampling variability distribution of the resulting entropy estimate, and is shown to accurately reflect the true uncertainty. For the four distributional forms studied (Gaussian, Log-Normal, Exponential and Bimodal Gaussian Mixture), expected estimation bias is less than 1% and uncertainty is low even for samples of as few as 100 data points; in contrast, for KD the small sample bias can be as large as −10% and for BC as large as −50%. We speculate that estimating quantile locations, rather than bin-probabilities, results in more efficient use of the information in the data to approximate the underlying shape of an unknown data generating pdf.