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The two‐term infiltration equation I(t)=St+At$I(t) = S\sqrt {t} + A t$ is commonly used to determine the sorptivity, S$S$(LT−1/2)$({\text{LT}}^{-1/2})$, and product, A=cKs$A = c K_{\text{s}}$(LT−1)$({\text{LT}}^{-1})$, of the dimensionless multiple c$c$ and saturated soil hydraulic conductivity Ks$K_{\text{s}}$(LT−1)$({\text{LT}}^{-1})$ from cumulative vertical infiltration measurements I∼1,...,I∼n$\tilde{I}_{1},\ldots,\tilde{I}_{n}$ (L) at times t1,...,tn$t_{1},\ldots,t_{n}$ (T). This reduced form of the quasi‐analytical power series solution of Richardson's equation of Philip enjoys a solid physical underpinning but at the expense of a limited time validity. Using simulated infiltration data, Jaiswal et al. have shown this time validity to equal about 2.5 cm of cumulative infiltration. The goals of this work are twofold. First, we investigate the extent to which cumulative infiltration measurements larger than 2.5 cm bias the estimates of S$S$ and Ks$K_{\text{s}}$. Second, we investigate the impact of epistemic errors on the inferred time validities and parameters. Partial infiltration curves up to 2.5 cm of cumulative vertical infiltration improve substantially the agreement between actual and least squares estimates of S$S$ and Ks$K_{\text{s}}$. But this only holds if the data generating infiltration process follows Richardson's equation and experimental conditions satisfy assumptions of soil homogeneity and a uniform initial water content. Otherwise, autocorrelated cumulative infiltration residuals will bias the least squares estimates of S$S$ and Ks$K_{\text{s}}$. Our findings reiterate and reinvigorate earlier conclusions of Haverkamp et al. and show that epistemic errors deteriorate the physical significance of the coefficients of infiltration functions. As a result, the parameters of infiltration functions cannot simply be used in storm water and vadose zone flow models to forecast runoff and recharge at field and landscape scales unless these predictions are accompanied by realistic uncertainty bounds. We conclude that the time validity of Philip's two‐term equation is an elusive theoretical quantity with arbitrary physical meaning. Core Ideas: Structural model errors of infiltration functions bias the values of S$S$ and Ks$K_s$.Least squares estimates of Ks$K_s$ compare poorly to measured values.Epistemic uncertainty turns coefficients of infiltration functions into fitting parameters.The time validity of Philip's two‐term equation is an elusive theoretical quantity.Treatment of epistemic errors enhances usefulness of plot‐scale infiltration experiments for large‐scale prediction. [ABSTRACT FROM AUTHOR]

Distribution forecasts P over future quantities or events are routinely made in hydrology but usually traded for a (likelihood‐weighted) mean or median prediction to accommodate error measures or scoring functions such as the mean absolute error or mean squared error. Case in point is the so‐called KG efficiency (KGE) of Gupta et al. (2009, https://doi.org/10.1016/j.jhydrol.2009.08.003) and improvements thereof (Lamontagne et al., 2020, https://doi.org/10.1029/2020wr027101), which have rapidly gained popularity among hydrologists as alternative scoring functions to the commonly used Nash and Sutcliffe (1970, https://doi.org/10.1016/0022‐1694(70)90255‐6) efficiency, but are equally exclusive in how they quantify model performance using only single‐valued output of the quantities of interest. This point‐valued mapping necessarily implies a loss of information about model performance. This paper advocates the use of probabilistic watershed model training, evaluation and diagnostics. Distribution evaluation opens a mature literature on scoring rules whose strong statistical underpinning provides, as we will demonstrate, the theory, context and guidelines necessary for the development of robust information‐theoretically principled metrics for watershed signatures. These so‐called hydrograph functionals are scalar‐valued mappings of major behavioral watershed functions embodied in a strictly proper scoring rule. We discuss past developments that led to the current state‐of‐the‐art of distribution evaluation in hydrology and review scoring rules for dichotomous and categorical events, quantiles (intervals) and density forecasts. We are particularly concerned with elicitable functionals and scoring rule propriety, discuss the decomposition of scoring rules into a sharpness, reliability and entropy term and present diagnostically appealing strictly proper divergence scores of hydrograph functionals for flood frequency analysis, flow duration and recession curves. The usefulness and power of distribution‐based model evaluation and diagnostics by means of scoring rules is demonstrated on simple illustrative problems and discharge distributions simulated with watershed models using random sampling and Bayesian model averaging. The presented theory (a) enables a more complete evaluation of distribution forecasts, (b) offers a statistically principled means for watershed model training, evaluation, diagnostics and selection using hydrograph functionals and/or extreme events and (c) provides a universal framework for metric development of watershed signatures, promoting metric standardization and reproducibility. Plain Language Summary: The past decades have witnessed an unbridled growth in goodness‐of‐fit metrics of hydrologic models. These metrics may satisfy the needs of hydrologists but lack conforming theory and principles. This state of affairs (a) elicits improper model training and evaluation, (b) provokes and supports misguided inferences, (c) impedes statistically‐principled uncertainty quantification, metric standardization and development of universal model benchmarks and (d) obfuscates determination of whether the model has finished learning. What is more, most hydrologic model evaluation metrics in use today are rather exclusive in how they quantify model performance using only single‐valued simulated output of the quantities of interest. Predictive distributions derived from (quasi)‐Bayesian methods or ensembles are usually traded for a (likelihood‐weighted) mean or median prediction to accommodate error measures (scoring functions) such as the mean absolute error. This implies a large loss of information. This paper develops a distribution‐based approach to hydrologic model evaluation and diagnostics. Distribution evaluation opens the necessary theory and guidelines for development of robust information‐theoretically principled metrics of watershed signatures. These so‐called hydrograph functionals are scalar‐valued mappings of major behavioral watershed functions embodied in a strictly proper scoring rule. The hydrograph functionals offer a statistically principled means for hydrologic model evaluation, diagnostics and selection. Key Points: Scoring rules of hydrograph functionals provide an information‐theoretically principled means for watershed model training, evaluation, and diagnosticsWe present strictly proper (divergence) scores for flood frequency analysis, flow duration, and recession curvesPropriety and elicitability offer useful working paradigms for metric development of hydrograph functionals [ABSTRACT FROM AUTHOR]

In this Reply, we address the concerns of Knoben and Clark (2023, https://doi.org/10.1029/2022WR034294) or KC23 that "the assumptions needed to effectively use difference‐based variance estimation methods are not always met by hourly streamflow records." There should be little doubt that the assumptions of our difference‐based estimator will sometimes be violated in hourly streamflow records but the results from de Oliveira and Vrugt (2022, https://doi.org/10.1029/2022wr032263) and confirmed by the findings in our Reply show that such violations are sporadic enough not to affect much the error variance estimates. Snowmelt as pointed out by KC23 (https://doi.org/10.1029/2022WR034294) may not have received sufficient attention in our paper, yet their 365‐day record is simply not long enough to demonstrate bias of our discharge error variance estimates (and their dependence on flow level). This would require analysis of a much longer, multi‐year, streamflow record of a snowmelt‐driven watershed. The snowmelt catchment analyzed in dOV22 (https://doi.org/10.1029/2022wr032263) did not demonstrate bias in the discharge error variance estimates. We also provide additional clarification on the interpretation of the variance estimates obtained with the nonparametric estimator, and discuss the main issues in the test case presented in Knoben and Clark (2023, https://doi.org/10.1029/2022WR034294)). Key Points: We address the concerns of Knoben and Clark (2023) on the use of the nonparametric estimator with hourly discharge dataWe provide further clarification on the interpretation of the variance estimates obtained with the nonparametric estimator [ABSTRACT FROM AUTHOR]

This comment addresses the work of Wang and Li (DOI: 10.1007/s12293-009-0012-0, hereafter referred to as WL) published online on September 01, 2009 in a (special) issue of Memetic Computing. We respectfully beg to differ in opinion with WL and argue that the concept of multi-strategy evolutionary search used by WL is not particularly novel. Similar ideas have been presented elsewhere, and these publications date back to at least early 2007. For instance, in a series of papers published in PNAS (2007), SSSAJ (2008), WRR (2008), IEEE-TEVC (2009), JH (2010), Vrugt and coworkers have introduced AMALGAM, a multi-method (or ensemble) search approach to solve emerging single and multiple objective search and optimization problems. In the past few years, MATLAB, C++, Python, and R implementations of AMALGAM have been distributed extensively among researchers and practitioners in various fields of study.

Climatic drivers of floods have been widely used to improve nonstationary flood frequency analysis (FFA). However, the forecast ability of nonstationary FFA with out-of-sample prediction has not been comprehensively evaluated. We use 379 flood records from Brazil to assess the ability of process-informed nonstationary models for out-of-sample FFA using the generalized extreme value (GEV) distribution. Five drivers of floods are used as covariates: annual temperature, El Nino Southern Oscillation, annual rainfall, annual maximum rainfall, and annual maximum soil moisture content. Our results reveal that a nonstationary model is preferable when there is a significant correlation between flood and climate covariates in both the training period and full record. The rainfall-based covariates lead to better out-of-sample nonstationary FFA models. These findings highlight that using climate information as covariates in nonstationary FFA is a promising approach for estimating future floods and, hence, better infrastructure design, risk assessment and disaster preparedness. [ABSTRACT FROM AUTHOR]

Variance‐based analysis has emerged as method of choice for quantifying the sensitivity of the output, y, of a scalar‐valued square‐integrable function, f ∈ L2(χ $\boldsymbol{\upchi }$), to its d ≥ 1 input variables, x = {x1, ..., xd}, with support x∈χ⊆Rd $\mathbf{x}\in \boldsymbol{\upchi }\subseteq {\mathbb{R}}^{d}$. The prototype of this approach, Sobol's method is a generalization of the analysis of variance (ANOVA) to d > 2 independent input variables and decomposes y, as sum of elementary functions of zeroth‐, first‐, second‐, up to dth‐order. This independence assumption is mathematically convenient but may not be borne out of the causal or correlational relationships between the x's. This paper is concerned with variance‐based sensitivity analysis (SA) for correlated input variables, for example, multivariate dependencies in a posterior parameter distribution. We use high‐dimensional model representation (HDMR) of Li et al. (2010, https://doi.org/10.1021/jp9096919), Li and Rabitz (2012, https://doi.org/10.1007/s10910-011-9898-0) and replace Sobol's elementary functions with so‐called component functions with unknown expansion coefficients to disentangle the structural, correlative and total contribution of input factors. We contrast the default HDMR methodology with cubic B‐splines and sequential coefficient estimation against its successor, HDMRext of Li and Rabitz (2012, https://doi.org/10.1007/s10910-011-9898-0), which uses polynomial component functions with an extended orthonormalized basis. Benchmark experiments confirm that HDMR and HDMRext parse out the structural and correlative contributions of input factors to the model output and infer an optimal experimental design with parameter correlation. Our last study applies HDMRext to probabilistic SA of a watershed model. The multivariate posterior parameter distribution supports model emulation and yields sensitivity indices that pertain to measured discharge data. Key Points: We use high‐dimensional model representation for variance‐based sensitivity analysis with dependent input variablesWe use this method to disentangle the parameters' structural and correlative contributions across the posterior distributionCorrelation can profoundly affect input variable importance, optimal experimental design and physical underpinning of sensitivity indices [ABSTRACT FROM AUTHOR]

In this paper, we introduce a relatively simple data‐driven method for the representation of the uncertainty in daily discharge records. The proposed method relies only on hourly discharge data and takes advantage of a nonparametric difference‐based estimator in the characterization of random errors in discharge time series. We illustrate with corrupted streamflow data that the nonparametric estimator provides an accurate characterization of the nature (homoscedastic or heteroscedastic) and magnitude of these errors. In addition, we demonstrate the practical usefulness of the estimator using discharge time series of 500+ watersheds of the Catchment Attributes and MEteorology for Large‐sample Studies data set. This analysis reveals that the magnitude of errors of aleatory nature in the investigated discharge records is rather small (less than 3% for 80% of the records). We then combine the effect of random errors and measurement frequency into a daily variance estimate, which serves as input to a streamflow generation approach. This procedure produces replicates of the discharge record which portray accurately the assigned streamflow uncertainty, preserve key statistical properties of the discharge record and are hydrologically realistic. The proposed method facilitates Bayesian analysis and supports tasks such as model diagnostics, data assimilation, uncertainty quantification and regionalization. Key Points: We present a model‐agnostic method for the generation of replicates of discharge recordsThe nonparametric estimator provides an accurate estimate of random errors, which for the investigated discharge records are rather smallThe replicates portray accurately the assigned streamflow uncertainty and preserve statistical and hydrologic properties [ABSTRACT FROM AUTHOR]

The dual-probe heat-pulse (DPHP) method is useful for measuring soil thermal properties; however, the probes of a DPHP sensor can deflect when inserted into the soil. Theoretical analysis has shown that measurements of thermal conductivity (h) should be unaffected by deflection-induced changes in probe spacing. To verify this result, the conductivities of water, dry sand, and saturated sand were measured using DPHP sensors with probes subject to inward deflection, no deflection, and outward deflection. No error in X was detected when probes were deflected inward by an amount that caused a 14% reduction in probe spacing. Outward deflection (15% increase in spacing) caused error in X estimates, but the errors were small ([less than or equal to] 0.04 W [m.sup.-1][ K.sup.-1]) and likely to be of little consequence in most applications. We conclude that estimates of X obtained with the DPHP method are largely unaffected by changes in probe spacing caused by deflection. doi: 10.2136/sssaj2010.0016N

Inverse modeling has become increasingly popular for estimating effective hydraulic properties across a range of spatial scales. In recent years, many different algorithms have been developed to solve complex multiobjective optimization problems. In this study, we compared the efficiency of the Nondominated Sorting Genetic Algorithm (NSGA-II), the Multiobjective Shuffled Complex Evolution Metropolis algorithm (MOSCEM-UA), and AMALGAM, a multialgorithm genetically adaptive search method for multiobjective estimation of soil hydraulic parameters. In our analyses, we implemented the HYDRUS-1D model and used observed pressure head data at three different depths from the Spydia experimental field site in New Zealand. Our optimization problem was posed in a multiobjective context by simultaneously using three complementary RMSE criteria at each depth. We analyzed the trade-off between these criteria and the adherent Pareto uncertainty. The results demonstrate that all three algorithms were able to find a good approximation of the Pareto set of solutions, but differed in the rate of convergence to this distribution. Small differences in performance of the various algorithms were observed because of the relative high dimension of the optimization problem in combination with the presence of multiple local optimal solutions within the three-objective search space. The Pareto parameter sets yielded satisfactory results when simulating the transient tensiometric pressure at predetermined observation points in the investigated vadose zone profile. The overall best parameter set was found by AMALGAM with RMSE values of 0.14, 0.11, and 0.17 m at the 0.4-, 1.0-, and 2.6-m depths, respectively. In contrast, the fit errors were substantially higher at these respective depths, with RMSE values ranging from 0.87 to 1.49 m, when using soil hydraulic parameters derived from laboratory analysis of small vadose zone cores. Abbreviations: AMALGAM, a multialgorithm genetically adaptive search method; MOSCEM-UA, Multiobjective Shuffled Complex Evolution Metropolis algorithm; MVG, Mualem-van Genuchten; NSGA-II, Nondominated Sorting Genetic Algorithm; OI, Oruanui ignimbrite; P1 and P2, Palaeosols; SCE-UA, Shuffled Complex Evolution algorithm; TI, Taupo ignimbrite.

The sustainability of irrigated agriculture in many arid and semiarid areas of the world is at risk because of a combination of several interrelated factors, including lack of fresh water, lack of drainage, the presence of high water tables, and salinization of soil and groundwater resources. Nowhere in the United States are these issues more apparent than in the San Joaquin Valley of California. A solid understanding of salinization processes at regional spatial and decadal time scales is required to evaluate the sustainability of irrigated agriculture. A hydro-salinity model was developed to integrate subsurface hydrology with reactive salt transport for a 1,400-k[m.sup.2] study area in the San Joaquin Valley. The model was used to reconstruct historical changes in salt storage by irrigated agriculture over the past 60 years. We show that patterns in soil and groundwater salinity were caused by spatial variations in soil hydrology, the change from local groundwater to snowmelt water as the main irrigation water supply, and by occasional droughts. Gypsum dissolution was a critical component of the regional salt balance. Although results show that the total salt input and output were about equal for the past 20 years, the model also predicts salinization of the deeper aquifers, thereby questioning the sustainability of irrigated agriculture. regional hydrology | salinization | vadose zone

Soil water infiltration is central to hydrologic studies and lends itself for detailed experimentation and mathematical–physical modeling. The most rigorous of these approaches numerically solve Richards' equation, possibly coupled through a heterogeneous soil to a surface water routine and groundwater model. This approach is computationally expensive and prone to mass balance errors and overparameterization. Analytic solutions of the infiltration process obviate the need for specification of hydraulic functions and simplify computation and inverse determination of soil properties. This paper investigates the usefulness of Parlange's three‐parameter infiltration equation for forward and inverse modeling of vertical infiltration experiments. This quasi‐exact implicit solution of Richards' equation, also credited to Haverkamp and coworkers in recent literature, is valid for the entire infiltration event, matches cumulative infiltration data from different soils and its (super‐)parameters, S, Ks, and β, exhibit a solid mathematical–physical underpinning. Nonetheless, Parlange's equation has not entered mainstream use for infiltration simulation and data analysis in the absence of a robust, exact and efficient numerical solution. This paper builds upon the recent work of Jaiswal et al. and presents theory, algorithms, and source codes of two numerical procedures for forward and inverse modeling of Parlange's infiltration equation. We illustrate the procedures using measured infiltration data from the Soil Water Infiltration Global (SWIG) database. Our findings highlight the potential of Parlange's equation for infiltration modeling and hydraulic characterization of the soil sorptivity, S [L T−1/2], saturated hydraulic conductivity, Ks [L T−1], and unitless coefficient, β. Parlange's infiltration equation provides a powerful alternative to mathematically more convenient explicit infiltration equations that suffer physical underpinning and/or a limited time validity. Core Ideas: Methods for exact and robust forward and inverse modeling of Parlange's three‐parameter infiltration equation are presented.Methods allow for a rapid inverse estimation of S, Ks, and β from cumulative infiltration measurements.Time and infiltration form of Parlange's equation may provide contrasting parameter values.Most infiltration experiments are too short to warrant an accurate estimation of Ks and β. [ABSTRACT FROM AUTHOR]