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  1. 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. 
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    Free, publicly-accessible full text available July 11, 2024
  2. Abstract

    Unsteady transit time distribution (TTD) theory is a promising new approach for merging hydrologic and water quality models at the catchment scale. A major obstacle to widespread adoption of the theory, however, has been the specification of the StorAge Selection (SAS) function, which describes how the selection of water for outflow is biased by age. In this paper we hypothesize that some unsteady hydrologic systems of practical interest can be described, to first‐order, by a “shifted‐uniform” SAS that falls along a continuum between plug flow sampling (for which only the oldest water in storage is sampled for outflow) and uniform sampling (for which water in storage is sampled randomly for outflow). For this choice of SAS function, explicit formulae are derived for the evolving: (a) age distribution of water in storage; (b) age distribution of water in outflow; and (c) breakthrough concentration of a conservative solute under either continuous or impulsive addition. Model predictions conform closely to chloride and deuterium breakthrough curves measured previously in a sloping lysimeter subject to periodic wetting, although refinements of the model are needed to account for the reconfiguration of flow paths at high storage levels (the so‐called inverse storage effect). The analytical results derived in this paper should lower the barrier to applying TTD theory in practice, ease the computational demands associated with simulating solute transport through complex hydrologic systems, and provide physical insights that might not be apparent from traditional numerical solutions of the governing equations.

     
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  3. Abstract

    Spatially integrated transport models have been applied widely to model hydrologic transport. However, we lack simple and process‐based theoretical tools to predict the transport closures—transit time distributions (TTDs) and StorAge Selection (SAS) functions. This limits our ability to infer characteristics of hydrologic systems from tracer observations and to make first‐order estimates of SAS functions in catchments where no tracer data is available. Here we present a theoretical framework linking TTDs and SAS functions to hydraulic groundwater theory at the hillslope scale. For hillslopes where the saturated hydraulic conductivity declines exponentially with depth, analytical solutions for the closures are derived that can be used as hypotheses to test against data. In the simplest form, the hillslope SAS function resembles a uniform or exponential distribution (corresponding to flow pathways in the saturated zone) offset from zero by the storage in the unsaturated zone that does not contribute to discharge. The framework is validated against nine idealized virtual hillslopes constructed using a 2‐D Richards equation‐based model, and against data from tracer experiments in two artificial hillslopes. Modeled internal age, life expectancy, and transit time structures reproduce theoretical predictions. The experimental data also support the theory, though further work is needed to account for the effects of time‐variability. The shape and tailing of TTDs and their power spectra are discussed. The theoretical framework yields several dimensionless numbers that can be used to classify hillslope scale flow and transport dynamics and suggests distinct water age structures for high or low Hillslope number.

     
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  4. Abstract

    The hydrologic dynamics and geomorphic evolution of watersheds are intimately coupled—runoff generation and water storage are controlled by topography and properties of the surface and subsurface, while also affecting the evolution of those properties over geologic time. However, the large disparity between their timescales has made it difficult to examine interdependent controls on emergent hydrogeomorphic properties, such as hillslope length, drainage density, and extent of surface saturation. In this study, we develop a new model coupling hydrology and landscape evolution to explore how runoff generation affects long‐term catchment evolution, and analyze numerical results using a nondimensional scaling framework. We focus on hydrologic processes dominating in humid climates where storm runoff primarily arises from shallow subsurface flow and from precipitation on saturated areas. The model solves hydraulic groundwater equations to predict the water‐table elevation given prescribed, constant groundwater recharge. Water in excess of the subsurface capacity for transport becomes overland flow, which generates shear stress on the surface and may detach and transport sediment. This affects the landscape form that in turn affects runoff generation. We show that (a) four dimensionless parameters describe the possible steady state landscapes that coevolve under steady recharge; (b) hillslope length increases with increasing transmissivity relative to the recharge rate; (c) three topographic metrics—steepness index, Laplacian curvature, and topographic index—together provide a basis for interpreting landscapes that have coevolved with runoff generated via shallow subsurface flow. Finally we discuss the possibilities and limitations for quantitative comparisons between the model results and real landscapes.

     
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  5. Abstract

    Spatially integrated water transport dynamics at the hillslope scale have rarely been observed directly, and underlying physical mechanisms of those dynamics are poorly understood. We present time‐variable transit time distributions and StorAge Selection (SAS) functions for a 28 days tracer experiment conducted at the Landscape Evolution Observatory, Biosphere 2, the University of Arizona, AZ, USA. The observed form of the SAS functions is concave, meaning that older water in the hillslope was preferentially discharged than younger water. The concavity is, in part, explained by the relative importance of advective and diffusive water dynamics and by the geomorphologic structure of the hillslopes. A simple numerical examination illustrates that, for straight plan‐shaped hillslopes, the saturated zone SAS function is concave when the hillslope Péclet (Pe) number is large (and thus when the advective water dynamics are more pronounced). We also investigated the effect of hillslope planform geometry on the saturated zone SAS function using a model and found that the more convergent the plan shape is, the more concave the SAS function is. A numerical examination indicates that the unsaturated zone SAS function is concave for straight and convergent hillslopes when the soil thickness is uniform. The concavity of those subcomponent SAS functions signifies that the hillslope scale SAS function is concave for straight or convergent plan shape hillslopes when the hillslope Pe number is high.

     
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  6. Abstract

    The advance of a chemical weathering front into the bedrock of a hillslope is often limited by the rate weathering products that can be carried away, maintaining chemical disequilibrium. If the weathering front is within the saturated zone, groundwater flow downslope may affect the rate of transport and weathering—however, weathering also modifies the rock permeability and the subsurface potential gradient that drives lateral groundwater flow. This feedback may help explain why there tends to be neither “runaway weathering” to great depth nor exposed bedrock covering much of the earth and may provide a mechanism for weathering front advance to keep pace with incision of adjacent streams into bedrock. This is the second of a two‐part paper exploring the coevolution of bedrock weathering and lateral flow in hillslopes using a simple low‐dimensional model based on hydraulic groundwater theory. Here, we show how a simplified kinetic model of 1‐D rock weathering can be extended to consider lateral flow in a 2‐D hillslope. Exact and approximate analytical solutions for the location and thickness of weathering within the hillslope are obtained for a number of cases. A location for the weathering front can be found such that lateral flow is able to export weathering products at the rate required to keep pace with stream incision at steady state. Three pathways of solute export are identified: “diffusing up,” where solutes diffuse up and away from the weathering front into the laterally flowing aquifer; “draining down,” where solutes are advected primarily downward into the unweathered bedrock; and “draining along,” where solutes travel laterally within the weathering zone. For each pathway, a different subsurface topography and overall relief of unweathered bedrock within the hillslope is needed to remove solutes at steady state. The relief each pathway requires depends on the rate of stream incision raised to a different power, such that at a given incision rate, one pathway requires minimal relief and, therefore, likely determines the steady‐state hillslope profile.

     
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  7. Abstract

    This is the first of a two‐part paper exploring the coevolution of bedrock weathering and lateral flow in hillslopes using a simple low‐dimensional model based on hydraulic groundwater theory (also known as Dupuit or Boussinesq theory). Here, we examine the effect of lateral flow on the downward fluxes of water and solutes through perched groundwater at steady state. We derive analytical expressions describing the decline in the downward flux rate with depth. Using these, we obtain analytical expressions for water age in a number of cases. The results show that when the permeability field is homogeneous, the spatial structure of water age depends qualitatively on a single dimensionless number, Hi. This number captures the relative contributions to the lateral hydraulic potential gradient of the relief of the lower‐most impermeable boundary (which may be below the weathering front within permeable or incipiently weathered bedrock) and the water table. A “scaled lateral symmetry” exists when Hi is low: age varies primarily in the vertical dimension, and variations in the horizontal dimensionxalmost disappear when the vertical dimensionzis expressed as a fractionz/H(x) of the laterally flowing system thicknessH(x). Taking advantage of this symmetry, we show how the lateral dimension of the advection–diffusion‐reaction equation can be collapsed, yielding a 1‐D vertical equation in which the advective flux downward declines with depth. The equation holds even when the permeability field is not homogeneous, as long as the variations in permeability have the same scaled lateral symmetry structure. This new 1‐D approximation is used in the accompanying paper to extend chemical weathering models derived for 1‐D columns to hillslope domains.

     
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  8. Abstract. A comprehensive set of measurements and calculated metricsdescribing physical, chemical, and biological conditions in the rivercorridor is presented. These data were collected in a catchment-wide,synoptic campaign in the H. J. Andrews ExperimentalForest (Cascade Mountains, Oregon, USA) in summer 2016 during low-dischargeconditions. Extensive characterization of 62 sites including surface water,hyporheic water, and streambed sediment was conducted spanning 1st- through5th-order reaches in the river network. The objective of the sample designand data acquisition was to generate a novel data set to support scaling ofriver corridor processes across varying flows and morphologic forms presentin a river network. The data are available at https://doi.org/10.4211/hs.f4484e0703f743c696c2e1f209abb842 (Ward, 2019). 
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  9. Abstract. Although most field and modeling studies of river corridorexchange have been conducted at scales ranging from tens to hundreds of meters,results of these studies are used to predict their ecological andhydrological influences at the scale of river networks. Further complicatingprediction, exchanges are expected to vary with hydrologic forcing and thelocal geomorphic setting. While we desire predictive power, we lack acomplete spatiotemporal relationship relating discharge to the variation ingeologic setting and hydrologic forcing that is expected across a riverbasin. Indeed, the conceptual model of Wondzell (2011) predicts systematicvariation in river corridor exchange as a function of (1) variation inbaseflow over time at a fixed location, (2) variation in discharge withlocation in the river network, and (3) local geomorphic setting. To testthis conceptual model we conducted more than 60 solute tracer studiesincluding a synoptic campaign in the 5th-order river network of the H. J. Andrews Experimental Forest (Oregon, USA) and replicate-in-time experimentsin four watersheds. We interpret the data using a series of metricsdescribing river corridor exchange and solute transport, testing forconsistent direction and magnitude of relationships relating these metricsto discharge and local geomorphic setting. We confirmed systematic decreasein river corridor exchange space through the river networks, from headwatersto the larger main stem. However, we did not find systematic variation withchanges in discharge through time or with local geomorphic setting. Whileinterpretation of our results is complicated by problems with the analyticalmethods, the results are sufficiently robust for us to conclude that space-for-timeand time-for-space substitutions are not appropriate in our study system.Finally, we suggest two strategies that will improve the interpretability oftracer test results and help the hyporheic community develop robust datasets that will enable comparisons across multiple sites and/or dischargeconditions. 
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