Development of an extended Schnute model for more physically realistic representations of soil water retention and moisture capacity curves

Abstract Commonly used soil water retention, θ(h), and moisture capacity, C(h), functions implicitly assume that (i) the θ(h) data curve is sigmoid-shaped with an inflection and (ii) the C(h) data curve has a value of zero at soil saturation. Desorption measurements on intact soils indicate, however, that the θ(h) data curve is frequently convex-monotonic in shape with no inflection, and C(h) at saturation is often a finite negative value rather than zero. As these model-data mismatches may cause substantial error in simulation or prediction of near-saturated soil hydraulic properties and water flow, a new “Extended Schnute” θ(h)–C(h) function was proposed that can provide θ(h) curve shapes and saturated C(h) values which are consistent with θ(h) and C(h) measurements. The new function and/or its nested Schnute sub-model provided high-quality and physically realistic fits to desorption data collected from intact cores of coarse sand, loamy sand, loam, clay loam, sandy clay loam, clay, and organic clay soils; and it out-performed or equalled the three-parameter van Genuchten θ(h)–C(h) function for every data-set. The new function also provided accurate and physically realistic representations of θ(h) and C(h) data from structured soils containing macropores and strongly graded pore size distributions. It was concluded that the Extended Schnute model is capable of providing accurate and physically realistic representations for a wide range of θ(h) and C(h) data, and it was further recommended that this model be considered over other models when measurements indicate that θ(h) is convex-monotonic in shape and/or C(h) is not zero at soil saturation.


Introduction
The amount of water within the drainable (interconnected) pore space of undisturbed, rigid to semi-rigid soil is usually expressed as water volume per unit bulk soil volume, and referred to as "pore water content" or "soil water content", θ [L 3 L −3 ].The corresponding energy status of that water is conveniently expressed on a per unit water weight basis, and is referred to as "pore water tension head," h [L] (h ≥ 0).The equilibrium θ versus h relationship defines the soil water retention curve, θ (h), and it describes how a soil retains water against energy gradients resulting from gravity, capillarity, solutes, temperature and overburden pressure (Kutilek and Nielsen 1994).The corresponding slope of the retention curve (i.e., dθ /dh versus h) defines the soil moisture capacity curve, C(h), and it describes the volume of water sorbed or released by a soil per unit change in tension head.The θ (h) and C(h) curves are considered fundamental properties which are necessary for characterizing the storage and transmission of water, air and solutes in soil (e.g., Or and Wraith 2002;Assouline and Or 2013;Reynolds and Drury 2022).
Because of their fundamental nature, θ (h) and C(h) determine or strongly influence many aspects of soil productivity, hydrology and environmental impact, such as crop growth, soil aeration, rainfall infiltration and drainage, surface runoff and erosion, greenhouse gas emissions, groundwater recharge, and surface and groundwater pollution.For example, Assouline (2021) reviewed how θ (h) and C(h) play prominent roles in determining saturated and unsaturated soil permeability to water and air; drainage time to field capacity and field capacity water content; root zone plantavailable water and air capacities; rainfall infiltration capacity and time to ponding; and evaporation rates of soil water.Qiu et al. (2023) demonstrated how relative plant-available water and air derived from θ (h) curves are important regulators of soil microbial respiration, the Birch effect, and mineralization of carbon and nitrogen compounds from soil organic matter.Reynolds (2018Reynolds ( , 2019)), Assouline and Or (2014) and others have shown how θ (h) relationships and associated parameters can be incorporated into analytic expressions describing soil water flow at field capacity; and how these analytic expressions may in turn be used to optimize air, water and nutrient dynamics during crop production.Nimmo (2013) reviewed how C(h) can be used to infer and characterize soil pore size and aggregate size distributions, as well as the connections between pore/aggregate size distributions and the transmission of water, gases and solutes.Reynolds et al. (2008Reynolds et al. ( , 2009) ) and others have further shown how pore size distributions and θ (h) and C(h) curves can be used to infer soil health or quality.
The θ (h) curve is typically comprised of discrete (h, θ ) data points obtained from desorption measurements taken in the field or from laboratory soil samples (e.g., Or and Wraith 2002).As θ (h) data are usually sparse, statistically noisy and unevenly spaced with h extending over several orders of magnitude, this "data curve" is almost always replaced by a fitted θ (h) function that is smooth, differentiable, and continuous throughout the h-domain of interest.Smoothness and continuity ensures good behaviour and tractability in analytic and numerical simulation analyses, and differentiability ensures that the corresponding C(h) function is easily calculated and internally consistent with the θ (h) function (Or and Wraith 2002;Assouline and Or 2013).
Due to the extreme physical and hydraulic complexity of soils, fitted θ (h) and C(h) functions are always empirical or quasi-empirical, and many empirical-parametric forms have been proposed over the past seven decades (e.g., Gardner 1956;Brooks and Corey 1964;Brutsaert 1966;Visser 1966;Campbell 1974;Haverkamp et al. 1977;van Genuchten 1980;Tani 1982;Russo 1988;Kosugi 1994Kosugi , 1996;;Fredlund and Xing 1994;Assouline et al. 1998;Groenevelt and Grant 2004;Dexter et al. 2008;Peters 2013;Rudiyantoa et al. 2020;Reynolds et al. 2021).Unfortunately, all commonly used θ (h) and C(h) functions have implicit assumptions and characteristics that are not always compatible with actual θ (h) and C(h) data.Reynolds and Drury (2022) showed, for example, that although θ (h) and C(h) data curves are predominantly convex-monotonic in shape with a finite nonzero C(h)-axis intercept, the widely used θ (h) and C(h) functions of van Genuchten (1980), Assouline et al. (1998), and Groenevelt and Grant (2004) assume that θ (h) has an inflection, and that C(h) is bell-shaped with zero intercept.In addition, the more flexible Weibull-based θ (h)-C(h) function recently proposed by Reynolds and Drury (2022) allows a finite, nonzero C(h)-axis intercept only for the special case of an exponential θ (h) function.These incompatibilities and limitations have been shown to force substantial mismatches between near-saturated θ (h) and C(h) data and the fitted functions (Reynolds and Drury 2022); to produce considerable error in the magnitude and shape of soil water diffusivity and hydraulic conductivity relationships (e.g., Vogel et al. 2001;Reynolds and Drury 2022); and to cause instability and poor convergence in numerical simulations of unsaturated soil water flow (e.g., Vogel et al. 2001).By adding a fictious "airentry" tension head to the van Genuchten θ (h)-C(h) function, Vogel at al. (2001) and Ippisch et al. (2006) were able to avoid C(h) = 0 at h = 0, and thereby enhance the performance of both the van Genuchten-Mualem hydraulic conductivity function and the HYDRUS numerical model.Although this tactic usually improves overall numerical simulation of infiltration and drainage (see, e.g., figs.6-8 in Vogel et al. 2001; Table 1.Special-case models contained within the Schnute biological growth model (eq.3) (adapted from Schnute 1981).Of the myriad of empirical functions available for fitting experimental Y versus X data, the Schnute (1981) function or "model" is perhaps unique in that it is extremely general and flexible, yet still simple and easy to apply.The model can simulate sigmoid or monotonic Y(X) curves; it can assume or mimic no less than 10 divergent and widely applied functions (Table 1); and of particular importance for this study, it's derivative function (i.e., dY/dX vs. X) has a flexible, finite and nonzero intercept on the dY/dX axis.In addition, iterative model-data fits tend to converge easily and quickly (Lei and Zhang 2006), and fitting parameter estimates are statistically stable (Schnute 1981;Bredenkamp and Gregoire 1988;Yuancai et al. 2001).Hence, the objective of this study was to determine if the Schnute (1981) model could be adapted and extended to provide accurate and physically realistic representations for a wide range of θ (h) and C(h) data.We start by recasting the Schnute model in terms of θ (h) and C(h) data and parameters (Section 2.1); then extend the Schnute model (Extended Schnute) to further increase flexibility (Section 2.2); and then apply several θ (h) data-sets and model-data fit metrics (Section 2.3) to assess Schnute and Extended Schnute model performance relative to a Weibull θ (h) model (Reynolds and Drury 2022) and the van Genuchten θ (h) model (van Genuchten 1980) (Section 3).

Recasting the Schnute model in terms of θ(h) and C(h)
The Schnute model (Schnute 1981) was originally developed to describe biological growth, and it is based on the premise that growth acceleration, dZ dt , relative to current growth rate, dG dt , is a linear function of growth rate relative to current biological size, G(t), that is,: where t [T] is time, a [T −1 ] is an unrestricted empirical scale constant, b [-] is an unrestricted empirical shape constant, and Z (t), is given by: where G(t) is any measure of biological size through time (e.g., length, diameter, circumference, weight, volume, etc.).The general solution of eqs. 1 and 2 is (Schnute 1981): where G 1 = G(t 1 ), G 2 = G(t 2 ), t 1 and t 2 are, respectively, the initial and final times in the G vs. t data-set (t 2 > t 1 ), and Q [−] is a constant given by: The a and b constants are usually determined by curvefitting eq. 3 to G vs. t data.
Equations 1-4 can be recast in terms of soil water using: , and α [L −1 ] and β [−] are, respectively, empirical scale and shape constants which are obtained by curve-fitting eq.7 to θ vs. h data.Equation 7 constitutes the "Schnute" θ (h) function; and the corresponding "Schnute" C(h) function is therefore given by: At present, α and β in eqs.7-9 are purely empirical with no formal mechanistic linkage to soil physical or hydraulic properties.
In eqs.7-9, perhaps the most convenient and physically relevant assignments of θ 1 , θ 2 , h 1 , and h 2 are: where θ S [L 3 L −3 ] and h E [L] are the saturated soil water content and air-entry or water-entry tension head, respectively, and θ WP [L 3 L −3 ] and h WP [L] are the plant permanent wilting point soil water content and tension head, respectively.Note also that residual soil water content, θ R [L 3 L −3 ] (i.e., θ at very large h), is readily estimated from eq. 7 by setting h = ∞ to obtain: where E [−] is defined by eq. 8. Hence, θ R is not curve-fitted, and it is guaranteed to be physically realistic (i.e., θ R > 0; Assouline and Or 2013;Reynolds and Drury 2022) for finite β and physically realistic θ 1 and θ 2 values.Note as well that eq. 9 combined with eqs. 10 and 11 condenses to: and therefore saturated moisture capacity can take on any value except zero or infinity since α, β, θ WP , and θ S must all be finite and nonzero.This feature is particularly useful because it allows considerable flexibility in the value of the Schnute moisture capacity curve at saturation (eq.13) and the angle at which the Schnute retention curve (eq.7) intersects the θ -axis, while most other models are extremely limited in this regard.As alluded to in Section 1, the van Genuchten (1980), Assouline et al. (1998) and Groenevelt and Grant (2004) models all assume a saturated moisture capacity of zero and a retention curve intersection angle of 90 • , while the adapted Weibull function of Reynolds and Drury (2022) allows only three options at saturation: that is, dθ dh hE = 0 and θ (h) intersection angle = 90 • (same as above); dθ dh hE = −∞ and θ (h) intersection angle = 0 (not physically realistic); dθ dh hE and θ (h) intersection angle both finite but based on an assumed exponential θ (h) curve.Another consequence of eqs.7-13 is that the Schnute θ (h) and C(h) curves are always convexmonotonic in shape; and hence, θ (h) has no inflection and C(h) has no peak.

Extension of the Schnute model to improve flexibility
Reynolds and Drury (2022) showed for intact soil that saturated moisture capacity is commonly finite and negative (i.e., C(h) < 0 at h ≤ h E ), but values of zero (i.e., C(h) = 0 at h ≤ h E ) do occur occasionally.To allow for possible C(h) = 0 at h ≤ h E , an "Extended" Schnute retention function was created by adding a third curve-fitting parameter, γ [−], to eq. 7: which in turn produces an Extended Schnute moisture capacity function: where and E is replaced by T in eq.12. Hence, when a curvefit of eq. 14 to θ vs. h data produces γ > 1, the (h − h 1 ) (γ -1)  term in eq.15 → 0 as h → h 1 thereby causing C(h) = 0 for h ≤ h 1 .Note that setting γ = 1 in eqs.14 and 15 recreates the original Schnute functions (eqs.7 and 9), while setting β = 1 and T = 0 produces two fitting parameter Weibull functions (i.e., α and γ fitted) which are simplified versions of the three-parameter Weibull functions (α, γ , θ R fitted) described in Reynolds et al. (2021) and Reynolds and Drury (2022).The Schnute and two-parameter Weibull models are consequently special cases "nested" within the Extended Schnute model (see Section 3.7.1 for related discussion).
As discussed in Reynolds and Drury (2022) and elsewhere, the retention curve inflection (h I , θ I ) is an important and often-used point for characterizing soil physical and hydraulic properties.The inflection locates both the maximum slope of the retention curve and the maximum value or "peak" of the capacity curve.The inflection is usually obtained by equating the retention curve curvature (second derivative) to zero, then solving for h to obtain h I , and then back-substituting h I into the retention curve equation to obtain θ I (see e.g., Reynolds et al. 2021 for details).The Extended Schnute function curvature, d 2 θ /dh 2 , can be written as: and equating to zero produces: where θ (h I ) refers to the inflection water content (θ I ) at the inflection tension head (h I ).Analytic solution of eq.18 for h I seems problematic, however numerical solution via the Solver add-on in Excel is straightforward.Note also that since the Weibull (β = 1, T = 0) and Schnute (γ = 1) functions are nested within the Extended Schnute model, then the Weibull version of eq.18 is given by: which readily simplifies to: while the Schnute version of eq.18 simplifies to: Equation 21 can be solved for h I by analytical expansion (e.g., eq.27 in Schnute 1981), or by direct numerical solution using Solver .
It should perhaps also be mentioned that since the retention curve inflection corresponds to the moisture capacity peak, h I can also be determined by using Solver to numerically locate the maximum value of the moisture capacity function within the h-domain of interest (e.g., between h E and h WP ).For the Extended Schnute model (and all nested or special case sub-models), this involves applying the Solver "Max" algorithm with eq. 15 as the objective function; setting h as the iteration variable; and then using the visual peak of the fitted moisture capacity curve to select an initial guess value for h I .This approach may prove useful when numerical solutions of complicated curvature equations (e.g., eq.18) are prone to inaccuracy or non-convergence.The method may also help verify that numerical solution of the curvature equations (eqs. 18, 19, and 21) found the global minimum within the desired h-domain, rather than a local minimum or a value outside the h-domain.

Assessing model performance
The Extended Schnute model (eqs.14-16) was assessed against the Schnute, Weibull, and van Genuchten (1980) models using four performance metrics, two hypothetical θ (h) data-sets, and seven measured θ (h) data-sets.The Schnute and Weibull models (eqs.14-16 with γ = 1 and β = 1, respectively) were selected because they are nested within the Extended Schnute model.The van Genuchten function (Section 2.3.1) was selected because it is by far the most widely applied θ (h)-C(h) model.Performance metrics (Section 2.3.2) included adjusted coefficient of determination (R A 2 ), normalized standard error of regression (SER N ), partial F-test (F ES-X ), and ranked probability (P X ).The data-sets (Section 2.3.3)included hypothetical θ (h) data based on the natural exponential function, hypothetical θ (h) data with an inflection, and θ (h) desorption measurements obtained from literature and intact cores collected from natural field soils ranging in texture from coarse sand to organic clay.The models were assessed for their ability to fit both retention data and the corresponding moisture capacity data.
It should also be noted that the focus of this study is on applying θ (h) models within the plant-available or "capillary" soil water range (i.e., θ WP ≤ θ ≤ θ S ).Modifications such as those described by Peters (2013) and Rudiyantoa et al. (2020) may be required to apply some or all of the models to drier soil conditions.

van Genuchten model
The classic three-fitting-parameter van Genuchten (1980) equations for water retention, θ (h), moisture capacity, C(h), and inflection point tension head, h I , can be written as: and where c [L −1 ] and n [−] are empirical scale and shape constants, respectively, and all other parameters are as previously defined.Note that the van Genuchten moisture capacity function (eq.23) is restricted to zero at soil saturation (i.e., C(h) = 0 at h = 0) because of the h-term in the numerator is raised to a power greater than zero.This limitation also applies to many other soil water models, including those developed by Assouline et al. (1998) and Groenevelt and Grant (2004).
The van Genuchten model is typically applied by least squares fitting eq. ( 22) to θ (h) data using c, n, and θ R as adjustable curve-fitting parameters.Some practitioners constrain the fitting algorithm to θ R ≥ 0 because negative residual soil water content is physically unrealistic (e.g., Greminger et al. 1985;van Genuchten et al. 1991;Reynolds and Drury 2022).Others maintain that θ R has no physical meaning in eqs.22 and 23 (i.e., it is just an empirical fitting parameter), and should therefore be allowed to take on any value required (including large negative values) to achieve the best model-data fit (e.g., van Genuchten and Nielsen 1985).The θ R ≥ 0 constraint was used in this study; however the impacts of allowing θ R < 0 are briefly described in Section 3.7.2.

Model performance metrics
As mentioned above, accuracy and plausibility of the various models were quantified using R A 2 , SER N , F ES-X , and P X .The R A 2 and SER N metrics are primarily measures of goodness of model-data fit; F ES-X determines if the Extended Schnute model-data fit is significantly improved relative to the fits of its simpler sub-models (in this case, Schnute and Weibull); and P X is a relative "likelihood" ranking of candidate models (in this case, Extended Schnute, Schnute, Weibull, and van Genuchten) considering both model parsimony and goodness of model-data fit.These metrics were calculated using: where and ; 0 ≤ P S ≤ 100% (30) is the regression total sum of squares, θ P i and θ M i are, respectively, the modelpredicted and measured θ values at each h-value, θ M i is the mean of the measured θ values, N is the number of data points (measurements), and L is the number of model fitting parameters (excluding intercept).In eqs.27 and 28, SSE X is the regression sum of squared errors of the nested model in question (i. where SSE X is the regression sum of squared errors for the fitted model in question, and N and L are as defined above.From a purely statistical perspective, the overall "best" model for representing the retention and capacity curve data is the one that yields the largest R A 2 and P X values, and the smallest SER N and AIC C values.Further detail on calculation and interpretation of the above metrics is given in Burnham and Anderson (2002), Banks andJoyner (2017), andReynolds et al. (2021).

Soil water retention and moisture capacity data-sets
As mentioned above, the test data-sets included hypothetical θ (h) data based on the natural exponential function, contrived hypothetical θ (h) data with an inflection, and measured θ (h) data obtained from literature and intact soil cores collected from field soils ranging in texture from coarse sand to organic clay.
The natural exponential function θ (h) data (Fig. 1a) were obtained from Reynolds and Drury (2022), and were generated using: where k = 0.0025 cm −1 and 0 ≤ h ≤ 5000 cm.By definition, the natural exponential function and its first derivative (slope) have no inflection (e.g., Nekola et al. 2008;Reynolds and Drury 2022).The contrived hypothetical θ (h) data, on the other hand, was designed to yield an inflection (and capacity curve peak) at h I = 75 cm (Fig. 2b).These two data-sets generate the envelope of retention curve and capacity curve shapes commonly observed in intact soils; that is, (i) retention and capacity curves both convex-monotonic with no inflection or peak, respectively (Figs. 1a, 1b), and (ii) sigmoid-shaped retention curve with a well-defined inflection (Fig. 2a), and a bellshaped capacity curve with a well-defined peak (Fig. 2b).
The measured θ (h) desorption data were acquired from literature and from intact (undisturbed) soil cores; and they included coarse sand (from Table 1 in Grant et al. 2010), loamy sand, loam, clay loam, sandy clay loam, clay, and organic clay textures.The cores (10 cm diameter by 10 cm long, 0-10 cm depth range, and 5-10 replicates per texture) were processed using tension table, tension plate, and pressure plate extractor methods (chapter 71 in Carter and Gregorich 2008), and the (h, θ ) data points were measured at h = 0, 5, 10, 30, 50, 100, 225, 350, and 15 000 cm.The corresponding moisture capacity data-sets were estimated using simple two-point finite difference approximations of retention curve slope (Reynolds et al. 2021;Reynolds and Drury 2022): and N is the number of measured (h i , θ i ) data pairs in the retention curve data-set with (h 0 , θ 0 ) = (h E , θ S ).As eq.35 is only a first-order approximation of slope, model fits to moisture capacity data will generally be poorer than model fits to retention data.

Model fitting to data
The Extended Schnute, Schnute, Weibull and van Genuchten retention curves were fitted to the soil water Table 2. Parameter values and associated metrics for model fits to calculated water retention data, (h) vs. h (eq.34), and moisture capacity data, −d /dh vs. h (eq.35), from a hypothetical natural exponential function (Fig. 1a and 1b).Fitted models included Extended Schnute, ES (eq.14); Schnute, S (eq.14, γ = 1); two-parameter Weibull, W (eq. 14, β = 1); and three-parameter van Genuchten,vG (eq. 22).Fitted parameters included α, β, and γ for ES; α and β for S; α and γ for W; and R , c, and n for vG.retention data using iterative nonlinear least squares regression applied via the Solver add-on in Excel .The curve fitting parameters varied by model and included: α, β, and γ for Extended Schnute; α and β for Schnute (γ = 1); α and γ for Weibull (β = 1); and c, n, and θ R for van Genuchten with θ R constrained to ≥ 0. Initial parameter guess values were obtained by informal trial-and-error matching of model prediction to retention data, and several initial values were used to ensure that consistent global solutions were obtained.Selected Solver options included generalized reduced gradient numerical iteration, slope calculation using central derivatives, automatic scaling, and a 10 −10 convergence criterion.The moisture capacity functions (eqs.15 and 23) were not fitted because the moisture capacity data-sets were derived using eq.35, which makes the capacity data both approximate and not independent from the retention data.It should also be mentioned that the Extended Schnute model is flexible with respect to fitting parameters, that is, any number and combination of α, β, γ , θ 1 , θ 2 , h 1 , and h 2 can be selected to make optimal use of available information.

Model fits to hypothetical natural exponential (h) data
The Extended Schnute, Schnute, and two-parameter Weibull models all produced excellent fits to the natural exponential water retention [ (h)] and moisture capacity [C(h) = −d /dh] data, as evidenced by close visual correspondence between model and data (Figs.1a, 1b) and excellent model-data fit metrics (e.g., R A 2 = 1.0000;SER N ≤ 0.016%; Table 2).This was not unexpected, as γ = 1 (which was specified for Schnute, and a curve-fitting result for Extended Schnute and Weibull) creates an exponential function in the Extended Schnute, Schnute and two-parameter Weibull models (Table 1, Section 2.2, see also Reynolds et al. 2021).The nearly equal SER N and AIC C values among the three models (Table 2) indicate statistically equivalent fits, but the Extended Schnute model was substantially less preferred than Schnute or Weibull (partial F-tests highly non-significant, low P ES relative to P S and P W ; Table 2) because of the parsimony penalty associated with Extended Schnute having three fitting parameters (α, β, and γ ) as opposed to Schnute and Weibull having just two fitting parameters (α and β for Schnute; α and γ for Weibull).Note as well that Schnute and Weibull have equal P X and AIC C values because for this dataset they are effectively the same model, that is, an exponential function with two fitting parameters.
The van Genuchten model also fits the (h) retention curve data reasonably well (R A 2 = 0.9881; SER N = 5.22%; Table 2), but it overestimates the data for 10 ≤ h ≤ 100 cm and for h > 400 cm (Fig. 1a).The overestimate appears to be the result of the model's required inflection, which occurred at h I = 72.7 cm; and this likely caused the very poor model fit (Fig. 1b) to the wet-end moisture capacity data (R A 2 = −2.0663;SER N = 80.86%;Table 2), as well as nil ranked probability (P vG ≈ 0; Table 2).As a consequence, the van Genuchten model was not competitive with the Extended Schnute, Schnute, Table 3. Parameter values and associated metrics for model fits to retention curve data, θ (h) vs. h, and moisture capacity data, −dθ /dh vs. h (eq.35), from a hypothetical retention curve with an inflection at (h I , θ I ) = (75 cm, 0.34 cm 3 cm −3 ) (Fig. 2a and 2b).Fitted models included Extended Schnute, ES (eq.14); Schnute, S (eq.14, γ = 1); two-parameter Weibull, W (eq. 14, β = 1); and three-parameter van Genuchten,vG (eq. 22).Fitted parameters included α, β, and γ for ES; α and β for S; α and γ for W; and θ R , c, and n for vG. is specified saturated volumetric water content at tension head, h = h S = 0 cm.† θ R is calculated or fitted residual volumetric water content at tension head, h → ∞. ‡ Calculated using eqs.27 and 28, where F ES-S compares the Extended Schnute and Schnute models, and F ES-W compares the Extended Schnute and twoparameter Weibull models.# Calculated using eqs.29-32,where subscripts ES, S, W, and vG represent, respectively, the Extended Schnute, Schnute, two-parameter Weibull, and threeparameter van Genuchten models.
and Weibull models for simulating a natural exponential function.
Note in Fig. 1a that the van Genuchten inflection point (h I = 72.7 cm, I = 0.85) is obscured and does not correspond to the tension head at the (apparent) maximum retention curve slope (at h = 400 cm).As discussed in Reynolds and Drury (2022), this occurs because log-transformation of the h-axis regressively "stretches" small-h data and progressively "compresses" large-h data.Small-h stretching often obscures the true inflection (which applies only to a linear h-axis); and the opposing processes of small-h stretching and large-h compression often produces a "pivot" point where scale stretching and compression cancel each other, thereby creating an "apparent" inflection (Fig. 1a).In addition, the coordinate of the pivot point depends on the type of h-axis transformation (e.g., log 10 h, log e h, h 1/2 , etc.), and is therefore simply an artefact of nonlinear scale transformation (Reynolds and Drury 2022).
Notwithstanding the above, the pivot point in this particular example (Fig. 1a) is related to both the exponential function scale constant and model fitting parameters through the relationship: where h P (cm) is the pivot point tension head, k = 0.0025 cm −1 is the specified scale constant of the natural exponential function (eq.34), W −1 (cm) is the inverse of the fitted Weibull scale constant (i.e., the inverse of the Weibull α value), and β/α is the ratio of the fitted scale and shape parameters of the Schnute and Extended Schnute models (Table 2).
Unfortunately, eq.36 does not hold for all data-model combinations, but general expressions for h P are given in Appendix A.

Model fits to hypothetical data with an inflection
The Extended Schnute and two-parameter Weibull models achieved comparable fits to the hypothetical inflection data (Fig. 2; Table 3), with the fits being "excellent" for the retention curves (R A 2 ≥ 0.9996, SER N ≤ 1.46%), and "good" for the moisture capacity curves (R A 2 ≥ 0.9613, SER N = 11.63%-13.07%).The two models also predicted the coordinate of the data inflection (h I = 75 cm, θ I = 0.34 cm 3 cm −3 ) with only 3.6% discrepancy (i.e., h I = 72.3cm, θ I = 0.34 cm 3 cm −3 for both Extended Schnute and two-parameter Weibull).Given the closeness of the above metrics, it was not surprising that the parsimony penalty for Extended Schnute caused a lower ranked probability relative to Weibull (P ES = 6.87% vs. P W = 93.13%)as well as a non-significant partial F-test (F ES-W = 0.9141).As discussed above, log 10 transformation of the h-axis obscured the Extended Schnute, Weibull, and data-set inflections (located by the triangle in Fig. 2a), and it produced an artefact pivot point (located by the square in Fig. 2a) which is offset from the true inflections.These results confirm that the Extended Schnute and two-parameter Weibull models can accurately represent retention and moisture capacity data that contain an inflection.
The Schnute and van Genuchten fits to the data-set were not competitive with the Extended Schnute and Weibull fits  3).The overall Schnute fit was relatively poor (e.g., retention curve SER N = 17.69%,F ES-S < 0.0001, P S = 0) -and apparently a consequence of its inability to simulate an inflection, as evidenced both visually in Fig. 2b by a very poor model-data fit, and quantitatively in Table 3 by extremely poor fit metrics for moisture capacity (R A 2 = −0.0897,SER N = 69.31%).Although the retention curve and inflection coordinate were reasonably predicted by van Genuchten (e.g., retention curve R A 2 = 0.9950, retention curve SER N = 5.15%, h I = 64.6 cm, θ I = 0.36 cm 3 cm −3 ), overestimation of the retention curve in the 25 cm ≤ h ≤ 50 cm range and underestimation for h > 300 cm (Fig. 2a) apparently caused nil ranked probability (P vG = 0.00%), a physically unrealistic θ R = 0 value, a substantial overestimate of the capacity curve peak (Fig. 2b), and a poor model-data fit for moisture capacity (R A 2 = 0.6587, SER N = 38.79%)(Table 3).

Model fits to an intact coarse sand soil
All four models provide visually plausible fits to the coarse sand retention curve (Fig. 3a), as well as good retention curve fit metrics (Table 4).However, the correspond-ing moisture capacity fits (Fig. 3b) reveal substantial and physically important model-data mismatches for the Schnute and two-parameter Weibull models; and as a result, their moisture capacity fit metrics were poor (R A 2 = 0.4690 and 0.4953; SER N = 113.93%and 111.07%), partial F-tests were highly significant (F ES-S = 0.0052, F ES-W = 0.0016), and ranked model probabilities were negligible (P S = 0.50%, P W = 0.06%) (Table 4).Although the moisture capacity data (Fig. 3b) strongly suggests a zero Y-axis intercept (i.e., −dθ /dh = 0 at h = 0), the Schnute model predicts an intercept of −0.025 cm −1 , and the two-parameter Weibull model predicts an intercept of negative infinity (fitted Weibull γ < 1, Table 4).The Schnute result for moisture capacity was expected, as it assumes no inflection (Section 2.1), but the Weibull result was surprising because this model can recognize and provide accurate fits to inflected (sigmoidal) data, as shown in Fig. 2. Given that the three-parameter Weibull model (i.e., θ R , α, and γ fitted) of Reynolds and Drury (2022) also gave an infinite moisture capacity intercept and a similar fit as the two-parameter Weibull (data not shown), it appears that the Weibull model was over-influenced by retention curve data that were not adjacent to the intercept.At any rate, Table 4 and Fig. 3b show that the Schnute and Weibull models were not competitive from both statistical and physical perspectives, despite having good visual fits and reasonably good fit metrics for the retention curve.
In contrast, the Extended Schnute and van Genuchten models provided excellent visual fits to both the retention and moisture capacity curves (Fig. 3a and 3b); they gave similar and physically plausible estimates of the inflection point (h I = 3.9 cm, θ I = 0.33 cm 3 cm −3 for Extended Schnute; h I = 4.0 cm, θ I = 0.33 cm 3 cm −3 for van Genuchten); and they produced good retention curve fit metrics with Extended Schnute being slightly better (e.g., R A 2 = 0.9985 vs. 0.9981, SER N = 4.54% vs. 5.17%, θ R = 0.012 cm 3 cm −3 vs. 0.010 cm 3 cm −3 , Table 4).In addition, the Extended Schnute and van Genuchten fit metrics for moisture capacity were much better than those for Schnute and Weibull, with those for Extended Schnute being marginally better than the van Genuchten values (Table 4).Hence, both Extended Schnute and van Genuchten provided similar high-quality representations of the coarse sand soil data, although the Extended Schnute fit was sufficiently better fit to produce a larger ranked probability (P S = 75.83%)than van Genuchten (P vG = 23.61%).

Model fits to an intact loamy sand soil
The Extended Schnute, Schnute, and van Genuchten models all tracked the loamy sand retention data very well, while two-parameter Weibull deviated systematically (Fig. 4a).As a result, the fit metrics for the Weibull retention curve were consistently the poorest of the four models (Table 5), with the model yielding by far the lowest ranked probability (P W = 0.04%) and a highly significant partial F-test (F ES-W = 0.0027).The retention curve fit metrics for Extended Schnute, Schnute, and van Genuchten (Table 5) suggest that van Genuchten provided that best overall fit (largest R A 2 and P vG ; lowest SER N and AIC C ), with Extended Schnute and Table 4. Parameter values and associated metrics for model fits to retention curve data, θ (h) vs. h, and moisture capacity data, −dθ /dh vs. h (eq.35), from an intact coarse sand soil (Fig. 3a and 3b).Fitted models included Extended Schnute, ES (eq.14); Schnute, S (eq.14, γ = 1); two-parameter Weibull, W (eq. 14, β = 1); and three-parameter van Genuchten,vG (Eq. 22).Fitted parameters included α, β, and γ for ES; α and β for S; α and γ for W; and θ R , c, and n for vG.= 0 cm.† θ R is calculated or fitted residual volumetric water content at tension head, h → ∞. ‡ Calculated using eqs.27 and 28, where F ES-S compares the Extended Schnute and Schnute models, and F ES-W compares the Extended Schnute and twoparameter Weibull models.# Calculated using eqs.29-32, where subscripts ES, S, W, and vG represent, respectively, the Extended Schnute, Schnute, two-parameter Weibull, and threeparameter van Genuchten models.
Schnute being effectively tied for second-best due to mixed but similar results (i.e., R A 2 slightly larger and SER N slightly smaller for Extended Schnute versus Schnute; AIC C slightly lower and P X substantially larger for Schnute versus Extended Schnute; non-significant F ES-S = 0.1378).
The corresponding moisture capacity results paint a different picture, however.In this case, all four models tracked the moisture capacity data reasonably well (Fig. 4b), and the model-data fit metrics followed the pattern: Weibull slightly better than Schnute; Schnute better than Extended Schnute; and Extended Schnute equivalent to van Genuchten (Table 5).Note also that Extended Schnute and van Genuchten both predicted capacity-axis intercepts of zero with near-zero inflections which were distant from the pivot point (i.e., h I = 3.63 cm, θ I = 0.54 cm 3 cm −3 for Extended Schnute; h I = 3.92 cm, θ I = 0.54 cm 3 cm −3 for van Genuchten; h P = 38.24cm, θ P = 0.36 cm 3 cm −3 ; Fig. 4), while Weibull predicted an infinite intercept (fitted γ < 1, Table 5) and Schnute predicted a finite intercept of −0.0145 cm −1 .The moisture capacity data, on the other hand, strongly suggests a finite, nonzero intercept (Fig. 4b); hence the second most probable Schnute model (P S = 27.58%,Table 5) rather than the most probable van Genuchten model (P vG = 54.62%,Table 5) seems best from the perspective of physical plausibility and realism.

Model fits to an intact loam soil
The retention curve data for intact loam soil (Fig. 5a) had a similar shape and relative pivot point position (h P = 75.67cm, θ P = 0.32 cm 3 cm −3 ) to that of the loamy sand soil (Fig. 4a), but the loam moisture capacity data (Fig. 5b) produced a more uniform convex-monotonic shape than the loamy sand data (Fig. 4b).All four models provided visually plausible fits to the water retention and moisture capacity data of the intact loam (Fig. 5a and 5b).Model ranking according to the fit metrics was Extended Schnute > Weibull > Schnute > van Genuchten for the retention curve, but Weibull > Extended Schnute > Schnute > van Genuchten for moisture capacity (Table 6).The change in ranking between retention and moisture capacity appears to be due mainly to Weibull fitting the first three moisture capacity points (i.e., h = 2.55, 7.5, and 20 cm) better than the other models (Fig. 5b).
Although Extended Schnute and two-parameter Weibull were the most probable and second-most probable models, respectively (P ES = 53.65%,P W = 27.06%),they both seem inappropriate for this data-set because they predict negative infinity for the moisture capacity intercept (i.e., fitted γ < 1, Table 6) which is physically unrealistic.The van Genuchten model also seems inappropriate because it gives the poorest overall fit with negligible ranked probability (Table 6); it predicts a physically unrealistic θ R = 0 value (Table 6); and it predicts an inflection (located at h I = 1.97 cm, θ I = 0.43 cm 3 cm −3 ) even though the moisture capacity data strongly suggests that no inflection exists (Fig. 5b).The statistically third-place Schnute model, on the other hand, seems both physically viable and appropriate because it combines a nonzero intercept on the moisture capacity axis (i.e., C(h = 0) = -0.0057cm −1 ) with model-data fits that are both visually and numerically plausible (Figs.5a, 5b; SER N = 1.67%,P S = 19.04%,Table 6).The retention and moisture capacity data curves for the intact clay loam, sandy clay loam, clay and organic clay soils (data not shown) were similar in overall shape to the loam data curves (Fig. 5a and 5b).For these fine textured soils, the Extended Schnute, Weibull, and van Genuchten models were consistently non-competitive relative to Schnute.Specifically, only the Schnute model provided physically realistic fits to the strongly convex-monotonic moisture capacity data (i.e., it predicted finite, nonzero saturated moisture capacity values, Table 7), and this model was always by far the most probable (i.e., P S ≥ 74.19%,Table 7).In contrast, the two-parameter Weibull model consistently predicted physically unrealistic (infinite) moisture capacity intercepts (fitted Weibull γ < 1), and it generated partial Ftests that were always significant (F ES-W < 0.05) along with ranked probabilities that were always negligible (P W ≤ 0.09%) (Table 7).The van Genuchten model consistently predicted a physically unrealistic θ R = 0 value, it assumes a moisture capacity intercept of zero and predicts h I > 0 which are unrealistic for these data, and its ranked probability was always nil (P vG ≤ 0.02%) (Table 7).The Extended Schnute model was always substantially less preferred than Schnute (P ES ≤ 25.72% vs. P S ≥ 74.19%), and the corresponding partial F-tests were always strongly non-significant (F ES-S >> 0.05) (Table 7).
The above results likely reflect the hydraulics of soils containing macropores and a strongly graded (polydisperse) pore size distribution.Macropores (e.g., shrinkage cracks, worm holes, and root channels) prevent a zero intercept on the moisture capacity axis by causing immediate soil drainage from saturation (i.e., air-entry and water-entry occurs at h = 0); and strong grading of pore sizes discourages retention curve inflections and capacity curve peaks (see e.g., fig.4.19 in Kutilek and Nielsen 1994 for a conceptual illustration).To further illustrate, the pore size distribution, dθ /dR vs. R, of the intact loam soil (Section 3.5) was calculated using capillary rise theory and the relationships: where R [L] is effective pore radius (equivalent cylinder), σ [MT −2 ] is air-water interfacial surface tension, τ [−] is water contact angle with the pore wall (τ = 0 • assumed), ρ [ML −3 ] is water density, g [LT −2 ] is gravitational acceleration, and dθ /dR [L −1 ] is pore size frequency (Nimmo 2013).The pore size frequency vs. pore radius relationship (which was derived from the moisture capacity "data" in Fig. 5b) implies that size frequency increases monotonically and at an increasing rate with decreasing pore radius (Fig. 6).Regression analysis confirmed this behaviour, yielding an approximate power function relationship, dθ /dR = 0.02R −1.156 for 0.0002 ≤ R ≤ 0.6 mm (R 2 = 0.9882), which has a negative monotonic slope (i.e., d 2 θ /dR 2 = −0.02312R−2.156 ) and a nonzero curvature (i.e., d 3 θ /dR 3 = 0.0498R −3.156 ) which both increase rapidly with decreasing R (Fig. 6).The presence of a strongly graded pore size distribution with macropores is further supported by the fact that macroporosity (h ≤ 10 cm, R ≥ 0.15 mm) comprised 10.2% of the loam soil's total measured porosity (total porosity = 0.440 cm 3 cm −3 ), while drainage porosity (0 ≤ h ≤ 100 cm, R ≥ 0.015 mm) comprised 31.9% of total porosity, storage porosity for plant-available water (15 000 ≤ h < 100 cm, 0.0001 ≤ R < 0.015 mm) comprised 38.0% of total porosity, and residual porosity (h > 15 000 cm, R < 0.0001 mm) comprised 30.1% of total porosity.The pore size distributions and porosity relationships of the intact clay loam, sandy clay loam, clay, and organic clay soils were all similar to those of the intact loam.
Setting (θ 1 , h 1 ) = (θ S , h S ) and (θ 2 , h 2 ) = (θ WP , h WP ) in the Schnute and Extended Schnute models affords the dual ad-vantages of fitting the full range of crop-relevant retention data (i.e., θ for h S ≤ h ≤ h WP ), and providing "exact" fits to two of the most important (and most commonly measured) points on the soil water retention curve (i.e., θ = θ S at h = h S = 0, and θ = θ WP at h = h WP = 15 000 cm).This tactic additionally reduces the number of curve-fitting parameters to just two or three (i.e., α and β for Schnute; α, β, and γ for Extended Schnute), which in turn reduces the number of (h, θ ) data points required to obtain model fits and parameter values that are accurate, statistically stable and unique.The θ (h) models of Assouline et al. (1998) and Groenevelt and Grant (2004) (as described in Grant et al. 2010) can also be "forced" through one or more θ (h) data points.

van Genuchten θ(h)-C(h) fits using unconstrained θ R
As mentioned in Section 2.3.1, the van Genuchten θ (h)-C(h) model (eqs.22 and 23) was fitted using the constraint, θ R ≥ 0 (e.g., Tables 2-7).As there is some debate over whether θ R should be constrained or not, eq.22 was re-fitted using unconstrained θ R to determine its effects on θ R value and model-data fit metrics.The resulting θ R values ranged from no change for the coarse sand and loamy sand soils (unconstrained θ R = 0.010 and 0.066, respectively) to slight negative values for the hypothetical inflection data and loam soil (unconstrained θ R = −0.023and − 0.029, respectively) and to large negative values for the natural exponential data and the clay, organic clay, clay loam, and sandy clay loam soils (unconstrained θ R = −0.153,−0.496, −18.91, −22.06, and − 32.41, respectively).Despite large changes in some θ R values, rank-ing of model-data fits (as determined by R A 2 , SER N , AIC C , and P X ) either remained exactly the same as for constrained θ R (this occurred for the natural exponential, hypothetical inflection, coarse sand, loamy sand, loam, and sandy clay loam datasets), or van Genuchten replaced Weibull as the third best fitting model while Schnute and Extended Schnute remained as best fitting and second best fitting, respectively (this occurred for the clay loam, clay, and organic clay soil datasets).Hence, allowing unconstrained θ R produced a minor improvement in the ranking of the van Genuchten fit relative to Weibull in three out of nine cases, but had no effect on the ranking of the Schnute and Extended Schnute models.

Assessing model fits to θ(h) and C(h) data
As exemplified in Figs.1a-5a, retention curve data and fitted models often appear to have (i) an intersection angle of 90 • on the water content axis (which is equivalent to a saturated moisture capacity value of zero) and (ii) visually good model-data fits that seem plausible.Figures 1b-5b show, however, that (i) moisture capacity is often not zero at saturation, and hence the intersection angle on the water content axis is often not 90 • and (ii) model fits to near-saturated moisture capacity data are frequently poor and/or physically unrealistic despite good fits to the corresponding retention data.As discussed above and in Reynolds and Drury (2022), these contradictions and inaccuracies stem largely from (i) h-axis nonlinearity due to log 10 transformation (which causes regressive stretching of small h data and progressive compression of large h data) and (ii) the implicit assumption by popular retention curve models that the water content axis is intersected at 90 • .These problems can often be circumvented, however, by fitting the Schnute or Extended Schnute models which allow flexible intersection angles on the water content axis; and also by basing assessments of model accuracy and physical realism on the ability of the model to fit both retention curve data and near-saturated moisture capacity data.

Summary and conclusions
Commonly used θ (h)-C(h) models, such as those of van Genuchten (1980), Assouline et al. (1998), and Groenevelt and Grant (2004), implicitly assume that θ (h) has an inflection and intersects the θ -axis at a fixed angle of 90 • , and that C(h) is bell-shaped with a value of 0 at soil saturation.Measurements of water release from intact soil suggests, however, that θ (h) and C(h) are often convex-monotonic in shape, with θ (h) intersecting the θ -axis at a variety of angles ≤ 90 • , and saturated C(h) is finite and ≤ 0. As this model-measurement contradiction can potentially cause substantial error in simulation of near-saturated soil hydraulic properties and water flow (Reynolds and Drury 2022), a new θ (h)-C(h) model derived from the Schnute (1981) biological growth function (eq. 3) was proposed that provides curve shapes, θ -axis intersection angles and saturated C(h) values which are flexible, physically plausible, and consistent with θ (h) and C(h) data.
Fig. 6.Pore size distribution for the intact loam soil (Section 3.5, Fig. 5a and 5b).Equivalent cylinder pore radius, R, was calculated using eq.37; pore size frequency, dθ /dR, was calculated using eq.38.Triangles are based on finite difference estimates of the intact loam moisture capacity data (Fig. 5b); solid line is the corresponding Schnute model which was fitted to the intact loam retention data (Fig. 5a); and dash-dot line is a power function regression based on moisture capacity data.Note that the pore radius axis is reversed to be physically consistent with the tension head axis in Fig. 5b. to generate θ (h) curves which can be convex-monotonic or sigmoid-shaped with θ -axis intersection angle ≤ 90 • , as well as C(h) curves that can be convex-monotonic or bell-shaped with C(h) finite and ≤ 0 at soil saturation.The new model is extremely general in that it can assume or approximate 17 important and widely applied models, including von Bertalanffy,Pütter No. 1,Pütter No. 2,Gompertz,Richards,Logistic,Linear,Quadratic,nth Power,Exponential,Schnute,twoparameter Weibull,Mitscherlich,Rayleigh,Lognormal,Normal,and Gumbel.The Extended Schnute model was assessed against the Schnute, two-parameter Weibull and three-parameter van Genuchten (eqs.22-24) models using four model-data fit metrics (Section 2.3.2), and nine data-sets (Section 2.3.3).The data-sets included hypothetical data with no inflection, and convex-monotonic θ (h) and C(h) (natural exponential function); hypothetical data with a known inflection, sigmoid θ (h), and bell-shaped C(h); and experimental data with various θ (h) and C(h) shapes which were obtained from intact cores collected in coarse sand, loamy sand, loam, clay loam, sandy clay loam, clay, and organic clay soils.
The Extended Schnute model (eqs.14-16) and/or it's nested  were able to provide highquality representations of all nine θ (h)-C(h) data-sets in terms of favourable model-data fit metrics, good visual fits to θ (h) and near-saturated C(h) data, and physically realistic predictions of the saturated C(h) value (Figs.1-5; Tables 2-7).The Schnute sub-model was also particularly adept at fitting θ (h) and C(h) data from structured soils with macropores and strongly graded pore size distributions (Table 7).In contrast, the Weibull fits were equivalent to the Extended Schnute and Schnute fits only for the natural exponential data (Fig. 1; Table 2) and hypothetical data with an inflection (Fig. 2; Table 3); and the Weibull model consistently returned unrealistic saturated C(h) values (i.e., γ < 1, C(h = 0) = −∞) for the intact soil core data (Figs.3-5; Tables 4-7).Similarly, the van Genuchten model (eqs.22-24) was competitive with Extended Schnute only for the coarse sand data (Fig. 3; Table 4), while all other van Genuchten fits had poorer fit metrics or returned physically unrealistic values for h I , saturated C(h), or θ R (Figs. 4-5, Tables 4-7).
It is concluded that the Extended Schnute model (Eqs.14-16) in combination with its nested Schnute sub-model (Eqs.7-9) is capable of providing accurate and physically realistic representations for a wide range of θ (h) and C(h) data--especially for near-saturated soil where hydraulic properties and water flow often change rapidly with changing h, and for structured soils containing macropores and a strongly graded pore size distribution.It is further recommended that the Extended Schnute model and its Schnute sub-model be considered over other commonly used models when θ (h) data curves appear convex-monotonic in shape with no evidence of an inflection; when C(h) data curves show no evidence of a peak; and when saturated C(h) is not zero.The substantial generality and flexibility of the Extended Schnute model may also make this function useful for characterizing many other types of agricultural data besides θ (h) and C(h).
figs.1-2 and 5-6 inIppisch et al. 2006), it does not remedy unrealistic hydraulic function behaviour at near-saturation.Hence, further development of more flexible and physically realistic θ (h)-C(h) functions is warranted and well justified.Of the myriad of empirical functions available for fitting experimental Y versus X data, theSchnute (1981) function or "model" is perhaps unique in that it is extremely general and flexible, yet still simple and easy to apply.The model can simulate sigmoid or monotonic Y(X) curves; it can assume or mimic no less than 10 divergent and widely applied functions (Table1); and of particular importance for this study, it's derivative function (i.e., dY/dX vs. X) has a flexible, finite and nonzero intercept on the dY/dX axis.In addition, iterative model-data fits tend to converge easily and quickly(Lei and Zhang 2006), and fitting parameter estimates are statistically stable(Schnute 1981;Bredenkamp and Gregoire 1988;Yuancai et al. 2001).Hence, the objective of this study was to determine if theSchnute (1981) model could be adapted and extended to provide accurate and physically realistic representations for a wide range of θ (h) and C(h) data.We start by recasting the Schnute model in terms of θ (h) and C(h) data and parameters (Section 2.1); then extend the Schnute model (Extended Schnute) to further increase flexibility (Section 2.2); and then apply several θ (h) data-sets and model-data fit metrics (Section 2.3) to assess Schnute and Extended Schnute model performance relative to a Weibull θ (h) model(Reynolds and Drury 2022) and the van Genuchten θ (h) model (van Genuchten 1980) (Section 3).
e., Schnute or Weibull), SSE ES is the regression sum of squared errors of the Extended Schnute model, K is the number of fitting parameters in the Extended Schnute model (K = 3), J is the number of fitting parameters in the nested model (J = 2 for Schnute and Weibull), and F ES-X is the partial F-test significance level, where F.DIST.RT is the Excel representation of the right-tailed F probability distribution.If F ES-S < 0.05, the Extended Schnute model fit is deemed better (at p < 0.05) than Schnute; and if F ES-W < 0.05, the Extended Schnute model fit is deemed better (at p < 0.05) than Weibull.As discussed in Reynolds et al. (2021), 0% ≤ SER N < 10% indicates excellent model prediction of the data; 10% ≤ SER N < 20% indicates good prediction; 20% ≤ SER N ≤ 30% indicates fair prediction; and SER N > 30% indicates poor prediction.From a combined goodness of modeldata fit and parsimony perspective, eqs.29-32 determine the relative degree to which the Extended Schnute (P ES ), Schnute (P S ), Weibull (P W ), or van Genuchten (P vG ) models represent the θ (h) data-set (P ES + P S + P W + P vG = 100%).The ω ES , ω S , ω W , and ω vG parameters are relative weights (ω ES + ω S + ω W + ω vG = 1) derived from the corrected Akaike Information Criterion (AIC C ) for each model (Banks and Joyner 2017):

Fig. 1 .
Fig.1.Fits of the Extended Schnute, Schnute, two-parameter Weibull, and three-parameter van Genuchten models to hypothetical data calculated using a natural exponential function: (a) water retention curve and (b) moisture capacity curve.Circles are calculated water content (eq.34); diamonds are finite difference estimates of moisture capacity (eq.35); triangle demarks the van Genuchten inflection; and square demarks retention curve pivot point on the log 10 h axis.

Fig. 2 .
Fig. 2. Fits of the Extended Schnute, Schnute, two-parameter Weibull, and three-parameter van Genuchten models to hypothetical retention curve data with an inflection at (h I , θ I ) = (75 cm, 0.34 cm 3 cm −3 ): (a) water retention curve and (b) moisture capacity curve.Circles are water content; diamonds are finite difference estimates of moisture capacity (eq.35); triangles demark model-predicted inflection points; and square demarks retention curve pivot point on the log 10 h axis.

Fig. 3 .
Fig. 3. Fits of the Extended Schnute, Schnute, two-parameter Weibull, and three-parameter van Genuchten models to data from an intact coarse sand soil: (a) water retention curve and (b) moisture capacity curve.Circles are measured water content; diamonds are finite difference estimates of moisture capacity (eq.35); triangles demark model-predicted inflection points; and square demarks retention curve pivot point on the log 10 h axis.

Fig. 4 .
Fig. 4. Fits of the Extended Schnute, Schnute, two-parameter Weibull, and three-parameter van Genuchten models to data from an intact loamy sand soil: (a) water retention curve and (b) moisture capacity curve.Circles are measured water content; diamonds are finite difference estimates of moisture capacity (eq.35); triangles demark model-predicted inflection points; and square demarks retention curve pivot point on the log 10 h axis.Each measured water content is the arithmetic mean of 5-10 replicates; standard error bars are too small to be visible.

Fig. 5 .
Fig. 5. Fits of the Extended Schnute, Schnute, two-parameter Weibull, and three-parameter van Genuchten models to data from an intact loam soil: (a) water retention curve and (b) moisture capacity curve.Circles are measured water content; diamonds are finite difference estimates of moisture capacity (eq.35); triangle demarks model-predicted inflection point; and square demarks retention curve pivot point on the log 10 h axis.Each measured water content is the arithmetic mean of 5-10 replicates; standard error bars are too small to be visible.
Calculated using eqs.27 and 28, where F ES-S compares the Extended Schnute and Schnute models, and F ES-W compares the Extended Schnute and twoparameter Weibull models.# Calculated using eqs.29-32, where subscripts ES, S, W, and vG represent, respectively, the Extended Schnute, Schnute, two-parameter Weibull, and threeparameter van Genuchten models.
S is normalized saturated volumetric water content at tension head, h = h S = 0 cm.† R is calculated or fitted normalized residual volumetric water content at tension head, h → ∞. ‡ θ S is measured saturated volumetric water content at tension head, h = h S = 0 cm.† θ R is calculated or fitted residual volumetric water content at tension head, h → ∞. * R is calculated or fitted residual volumetric water content at tension head, h → ∞. ‡ Calculated using eqs.27 and 28, where F ES-S compares the Extended Schnute and Schnute models, and F ES-W compares the Extended Schnute and two-parameter Weibull models.
Water Retention Curve, θ vs. h Measured saturated water content, θ S (−) * * θ S is measured saturated volumetric water content at tension head, h = h S = 0 cm.† θ Significance level of partial F-test comparing the fit of the Extended Schnute model to the fit of the nested Schnute model (F ES-S ) and the fit of the nested two-parameter Weibull model (F ES-W ) (Eqs.27 and 28).