In a recently introduced parameterization for the soil water
retention curve (SWRC) with a sigmoid wet branch and a logarithmic dry
branch, the matric potential at the junction point of the sigmoid and the
logarithmic branch (

Recently, de Rooij et al. (2021) proposed a closed-form expression for the
SWRC (soil water retention curve) with a distinct air-entry value, like the SWRC proposed by Ippisch et
al. (2006), a sigmoid shape in the intermediate range according to van
Genuchten (1980), and a logarithmic dry branch terminating at a finite
matric potential

Erroneous measurements in the dry range can lead to unrealistically low
values of the matric potential at oven dryness,

This note presents an alternative to de Rooij et al.'s (2021) model in which

In the testing phase, it was found that a commonly used convergence criterion used in parameter optimization did not necessarily give the best parameter values if the objective function was challenging. The second objective is therefore to present a parameter fitting algorithm that employs multiple convergence criteria, and optionally explores the parameter space prior to the fitting operation to reduce the search area during fitting. The corresponding open source code for fitting the improved SWRC is provided.

De Rooij et al. (2021) introduced a unimodal model for the SWRC by combining
those of Rossi and Nimmo (1994) and Ippisch et al. (2006), dubbed “RIA”.

By requiring the derivatives of the sigmoidal and logarithmic branches to
match at

The logarithm of shape factor

It therefore appears from Fig. 1 that plausible combinations of

A transect of Fig. 1 for

The limits imposed on the matric potential at the junction point

De Rooij et al. (2021) already fitted Eq. (

In the limit as

If Eq. (

Rossi and Nimmo (1994) fitted

The logarithm of the absolute value of the matric potential at the
junction point (

Fitted parameter values and the root mean square error (RMSE) of the best fits for 21 soils. The corresponding values of the derived parameter

For completeness, the multimodal version of Eq. (

The SWRC has five fitting parameters:

The parameters were fitted using the SCE algorithm (Duan et al., 1992, 1993), implemented in Fortran in a code that accompanies this paper. The most important features of the code are summarized here. The code itself, further details of the code and the algorithm, and a user manual can be downloaded (de Rooij, 2022). For each case, the code performs three optimization runs by minimizing the objective function: the root mean square error (RMSE) of the differences between fitted and observed volumetric water contents, weighted according to the error standard deviations of the observed matric potentials and corresponding water contents provided on input, as detailed by de Rooij (2022).

Ten convergence criteria are evaluated. Criteria 1 and 2 take into account
the results of the last few shuffles. The number of shuffles considered is
twice the number of fitting parameters or an internally set number (5),
whichever is larger.

In the best fits from the last set of shuffles, the range of a parameter exceeds neither the absolute nor the relative user-specified tolerance.

In the best fits from the last set of shuffles, the range of the objective function does not exceed its absolute user-specified tolerance.

The parameter range in the final complexes does not exceed the maximum internally set permissible value.

The volume of the hypercube enveloping the final complexes does not exceed the maximum internally set permissible value.

The parameter range in the most successful complex (minus the point with the highest RMSE) does not exceed the internally set maximum permissible value.

The volume of the hypercube enveloping the most successful complex (minus the point with the highest RMSE) does not exceed the internally set maximum permissible value.

A parameter does not exceed both the absolute and the relative user-specified tolerance in the final complexes.

A parameter does not exceed both the absolute and the relative user-specified tolerance in the most successful complex (without the point with the highest RMSE).

The change of the objective function between consecutive shuffles does not exceed the user-specified tolerance.

The root mean square error of the fit does not exceed a user-specified tolerance.

The continuity correction factor

Soil water retention data and fitted curves for soils of classes
A1 and A2 of Twarakavi et al. (2010). Curves fitted with

As Fig. 5, for soils of classes A3 and A4 of Twarakavi et al. (2010).

The algorithm generates large numbers of sets of fitted parameter values. A
random sample of these is used to determine the correlation matrix of the
parameters. The best fit, its RMSE, and its correlation matrix are reported
by the code for each of the three runs, and the run with the overall lowest
RMSE is identified. The code returns a table of the fitted curve based on
the best run, and reports the correction factor

Normally, the first complexes of each run are filled with randomly selected points in the valid regions of the parameter space. Optionally, only the complexes of run 3 are filled with randomly selected points, while the first complexes of run 1 are filled with the map points with the lowest RMSE. The first complexes of run 2 then contain these map points perturbed by adding random noise to the parameter values.

For each fitting parameter, a maximum and minimum value need to be provided. If these values are equal, the parameter is treated as a fixed value, and the dimensionality of the parameter space is reduced accordingly. The number of complexes is two (for eight of fewer fitting parameters) or four (see Duan et al., 1994). If this leads to frequent convergence at local minima, the number of complexes can be set to twice the number of fitting parameters. The number of individuals in a complex and the number of evolution steps are twice the number of fitting parameters plus one. The number of individuals in the subcomplexes is the number of fitting parameter plus one. The number of offspring in each evolution step is one. These settings are all in accordance with Duan et al. (1994).

When the user chooses to use the map of regularly spaced points in the parameter space to set the initial guesses of the first two runs, the code adapts the parameter ranges based on their ranges among the map points selected to fill the initial complexes.

The permitted parameter range for

No more than four convergence criteria were allowed to fail for convergence
to be achieved. If convergence was not achieved, up to 20 000 evaluations of
the objective function evaluations would be performed. The actual number
could be slightly higher because it was checked each time a shuffle had been
completed. A map was not created, and therefore all three optimization runs
started with random parameter combinations filling the complexes. The
maximum allowed relative change in the RMSE between consecutive shuffles was 10

The internally set relative tolerances for parameter variations for all complexes combined and for the most successful complex were both 0.01. The internally
set required maximum size of the hypercube enveloping the range of fitted
parameter values (again for all complexes combined and the most successful one), scaled by the volume of the hypercube defined by the minimum and
maximum allowed parameter values, equals 0.01

As Fig. 5, for soils of classes B2 and B4 of Twarakavi et al. (2010).

As Fig. 5, for soils of classes C2 and C4 of Twarakavi et al. (2010).

In the tables that follow, the soil texture classification according to Twarakavi et al. (2010) is provided. The advantage of this classification over the conventional USDA soil texture classification is that it better recognizes differences in hydraulic behavior between its textural classes. This makes it more relevant for soil water flow studies for which the SWRCs are typically used.

Table 1 shows the fitted parameter values of the best fits, the resulting
RMSE, and the corresponding value of

Fuentes et al. (1991) showed that for values of

The range of values of

Only the optimizations for soils 1142 and 2104 converged, with convergence criteria 4, 6, 8, 9, and 10 satisfied for all parameters for soil 1142, and criteria 4 through 10 for soil 2104. None of the correlation coefficients of the parameter pairs for either soil exceeded 0.31. The other optimizations ran until the maximum number of objective function evaluations was exceeded. For soil 1120, criteria 9 and 10 were met for all parameters. For the remaining soils, criteria 1, 2, and 9 were satisfied in all cases. For 14 soils, criterion 10 was met as well. For soil 3250, criterion 8 was also satisfied. The lack of convergence forced the code to keep exploring the parameter space, leading to a large proportion of randomly selected points because the reflection and contraction points determined by the SCE algorithm did not improve the fit. If the majority of points is randomly selected, there is no correlation between the parameters, and the correlation matrix does not provide any information.

In all cases, the fitted parameter values for the runs with

The reduced weights assigned to data points with

The fraction of adsorbed water increases when moving from sands (Figs. 5 and
6) through loams (Figs. 6 and 7) to clays (Fig. 8). Because the separation
between capillary and adsorbed water is abrupt and binary at

Most soils (2126 and 1142 in Fig. 5; 1120, 1121, 1143, 2110, and 2132 in Fig. 6; 1142 and 2126 in Fig. 7; 1122, 1180, and 1182 in Fig. 8) have observed saturated water contents that seem to be too large compared to the other data points. The causes (e.g., macropores or air inclusion) are not known. Data points at saturation were assumed to be very accurate and therefore had a high weight, which the plots reflect. It sometimes results in relatively low (more negative) air-entry values in coarse soils (Figs. 5 and 6 and Table 1, most notably soil 1142).

All B2 soils (silt loams) and two out of three B4 soils (both silty clay
loams) have high values of

This paper focuses on mineral soils with unimodal SWRCs. If an extension to multimodality is desired, Eqs. (12)–(16) provide a starting point. Further testing on soils with specific characteristics, such as volcanic or organic soils, may be worthwhile. In the case of organic soils, the risk of irreversible changes will require some caution when measuring points on the SWRC.

The Fortran code for fitting the SWRC parameters and post-processing and its user manual are available from the Zenodo repository (

The UNSODA database with the soil data can be downloaded at

The author is a member of the editorial board of

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

I thank the reviewers for their constructive comments.

The article processing charges for this open-access publication were covered by the Helmholtz Centre for Environmental Research – UFZ.

This paper was edited by Bob Su and reviewed by five anonymous referees.