Process-based modelling offers interpretability and physical consistency in many domains of geosciences but struggles to leverage large datasets efficiently. Machine-learning methods, especially deep networks, have strong predictive skills yet are unable to answer specific scientific questions. In this Perspective, we explore differentiable modelling as a pathway to dissolve the perceived barrier between process-based modelling and machine learning in the geosciences and demonstrate its potential with examples from hydrological modelling. ‘Differentiable’ refers to accurately and efficiently calculating gradients with respect to model variables or parameters, enabling the discovery of high-dimensional unknown relationships. Differentiable modelling involves connecting (flexible amounts of) prior physical knowledge to neural networks, pushing the boundary of physics-informed machine learning. It offers better interpretability, generalizability, and extrapolation capabilities than purely data-driven machine learning, achieving a similar level of accuracy while requiring less training data. Additionally, the performance and efficiency of differentiable models scale well with increasing data volumes. Under data-scarce scenarios, differentiable models have outperformed machine-learning models in producing short-term dynamics and decadal-scale trends owing to the imposed physical constraints. Differentiable modelling approaches are primed to enable geoscientists to ask questions, test hypotheses, and discover unrecognized physical relationships. Future work should address computational challenges, reduce uncertainty, and verify the physical significance of outputs. Differentiable modelling is an approach that flexibly integrates the learning capability of machine learning with the interpretability of process-based models. This Perspective highlights the potential of differentiable modelling to improve the representation of processes, parameter estimation, and predictive accuracy in the geosciences.

Sensitivity analysis (SA) is en route to becoming an integral part of mathematical modeling. The tremendous potential benefits of SA are, however, yet to be fully realized, both for advancing mechanistic and data-driven modeling of human and natural systems, and in support of decision making. In this perspective paper, a multidisciplinary group of researchers and practitioners revisit the current status of SA, and outline research challenges in regard to both theoretical frameworks and their applications to solve real-world problems. Six areas are discussed that warrant further attention, including (1) structuring and standardizing SA as a discipline, (2) realizing the untapped potential of SA for systems modeling, (3) addressing the computational burden of SA, (4) progressing SA in the context of machine learning, (5) clarifying the relationship and role of SA to uncertainty quantification, and (6) evolving the use of SA in support of decision making. An outlook for the future of SA is provided that underlines how SA must underpin a wide variety of activities to better serve science and society. • Sensitivity analysis (SA) should be promoted as an independent discipline. • Several grand challenges hinder full realization of the benefits of SA. • The potential of SA for systems modeling & machine learning is untapped. • New prospects exist for SA to support uncertainty quantification & decision making. • Coordination rather than consensus is key to cross-fertilize new ideas.

• Time-variant variogram analysis reveals significant event scale parameter importance variability. • The type of modeling objectives used influences parameter importance. • Input rainfall intensity and hyetograph shape affects parameters importance. • VARS is an effective and robust approach for identifying key modeling parameters. Runoff and sediment yield predictions using rainfall-runoff modeling systems play a significant role in developing sustainable rangeland and water resource management strategies. To characterize the behavior and predictive uncertainty of the KINEROS2 physically-based distributed hydrologic model, we assessed model parameters importance at the event-scale for small nested semi-arid subwatersheds in southeastern Arizona using the Variogram Analysis of Response Surfaces (VARS) methodology. A two-pronged approach using time-aggregate and time-variant parameter importance analysis was adopted to improve understanding of the control and behavior of models. The time-aggregate analysis looks at several signature responses, including runoff volume, sediment yield, peak runoff, runoff duration, time to peak, lag time, and recession duration, to investigate the influence of parameter and input on the model predictions. The time-variant analysis looks at the dynamical influence of parameters on the simulation of flow and sediment rates at every simulation time step using the different forcing inputs. This investigation was able to address Simpson’s paradox-type issues where the analysis across the different objective functions and full data set vs. its subsets (i.e., different events and/or time steps) could yield inconsistent and potentially misleading results. The results indicated the uncertainties in the flow responses are primarily due to the saturated hydraulic conductivity, the Manning’s coefficient, the soil capillary coefficient, and the cohesion in sediment and flow-related responses. The level of influence of K2 parameters depends on the type of the model response surface, the rainfall, and the watershed size.

Abstract Many applications of global sensitivity analysis (GSA) do not adequately account for the dynamical nature of earth and environmental systems models. Gupta and Razavi (2018) highlight this fact and develop a sensitivity analysis framework from first principles, based on the sensitivity information contained in trajectories of partial derivatives of the dynamical model responses with respect to controlling factors. Here, we extend and generalize that framework to accommodate any GSA philosophy, including derivative-based approaches (such as Morris and DELSA), direct-response-based approaches (such as the variance-based Sobol’, distribution-based PAWN, and higher-moment-based methods), and unifying variogram-based approaches (such as VARS). The framework is implemented within the VARS-TOOL software toolbox and demonstrated using the HBV-SASK model applied to the Oldman Watershed, Canada. This enables a comprehensive multi-variate investigation of the influence of parameters and forcings on different modeled state variables and responses, without the need for observational data regarding those responses.

Abstract. Calibration is an essential step for improving the accuracy of simulations generated using hydrologic models. A key modeling decision is selecting the performance metric to be optimized. It has been common to use squared error performance metrics, or normalized variants such as Nash–Sutcliffe efficiency (NSE), based on the idea that their squared-error nature will emphasize the estimates of high flows. However, we conclude that NSE-based model calibrations actually result in poor reproduction of high-flow events, such as the annual peak flows that are used for flood frequency estimation. Using three different types of performance metrics, we calibrate two hydrological models at a daily step, the Variable Infiltration Capacity (VIC) model and the mesoscale Hydrologic Model (mHM), and evaluate their ability to simulate high-flow events for 492 basins throughout the contiguous United States. The metrics investigated are (1) NSE, (2) Kling–Gupta efficiency (KGE) and its variants, and (3) annual peak flow bias (APFB), where the latter is an application-specific metric that focuses on annual peak flows. As expected, the APFB metric produces the best annual peak flow estimates; however, performance on other high-flow-related metrics is poor. In contrast, the use of NSE results in annual peak flow estimates that are more than 20 % worse, primarily due to the tendency of NSE to underestimate observed flow variability. On the other hand, the use of KGE results in annual peak flow estimates that are better than from NSE, owing to improved flow time series metrics (mean and variance), with only a slight degradation in performance with respect to other related metrics, particularly when a non-standard weighting of the components of KGE is used. Stochastically generated ensemble simulations based on model residuals show the ability to improve the high-flow metrics, regardless of the deterministic performances. However, we emphasize that improving the fidelity of streamflow dynamics from deterministically calibrated models is still important, as it may improve high-flow metrics (for the right reasons). Overall, this work highlights the need for a deeper understanding of performance metric behavior and design in relation to the desired goals of model calibration.

Abstract VARS-TOOL is a software toolbox for sensitivity and uncertainty analysis. Developed primarily around the “Variogram Analysis of Response Surfaces” framework, VARS-TOOL adopts a multi-method approach that enables simultaneous generation of a range of sensitivity indices, including ones based on derivative, variance, and variogram concepts, from a single sample. Other special features of VARS-TOOL include (1) novel tools for time-varying and time-aggregate sensitivity analysis of dynamical systems models, (2) highly efficient sampling techniques, such as Progressive Latin Hypercube Sampling (PLHS), that maximize robustness and rapid convergence to stable sensitivity estimates, (3) factor grouping for dealing with high-dimensional problems, (4) visualization for monitoring stability and convergence, (5) model emulation for handling model crashes, and (6) an interface that allows working with any model in any programming language and operating system. As a test bed for training and research, VARS-TOOL provides a set of mathematical test functions and the (dynamical) HBV-SASK hydrologic model.

Abstract Dynamical earth and environmental systems models are typically computationally intensive and highly parameterized with many uncertain parameters. Together, these characteristics severely limit the applicability of Global Sensitivity Analysis (GSA) to high-dimensional models because very large numbers of model runs are typically required to achieve convergence and provide a robust assessment. Paradoxically, only 30 percent of GSA applications in the environmental modelling literature have investigated models with more than 20 parameters, suggesting that GSA is under-utilized on problems for which it should prove most useful. We develop a novel grouping strategy, based on bootstrap-based clustering, that enables efficient application of GSA to high-dimensional models. We also provide a new measure of robustness that assesses GSA stability and convergence. For two models, having 50 and 111 parameters, we show that grouping-enabled GSA provides results that are highly robust to sampling variability, while converging with a much smaller number of model runs.