Abstract
The Mualem-Van Genuchten equations have become very popular in recent decades. Problems were encountered fitting the equations’ parameters through sets of data measured in the laboratory: parameters were found which yielded results that were not monotonic increasing or decreasing. Due to the interaction between the soil moisture retention and the hydraulic conductivity relationship, some data sets yield a fit that seems not to be optimal. So the search for alternatives started. We ended with the cubical spline approximation of the soil physical characteristics. Software was developed to fit the spline-based curves to sets of measured data. Five different objective functions are tested and their results are compared for four different data sets. It is shown that the well-known least-square approximation does not always perform best. The distance between the measured points and the fitted curve, as can be evaluated numerically in a simple way, appears to yield good fits when applied as a criterion in the optimization procedure. Despite an increase in computational effort, this method is recommended over the least square method.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahuja L.R., Green R.E. and Chong S.-K. (1980). A simplified functions approach for determining soil hydraulic conductivities and water characteristics in situ. Water Resour. Res. 16(5): 947–953
Ahuja L.R., Ross J.D., Bruce R.R. and Cassel D.K. (1988). Determining unsaturated hydraulic conductivity from tensiometric data alone. Soil Sci. Soc. Am. J. 52(1): 27–34
Alexander L. and Skaggs R.W. (1987). Predicting unsaturated hydraulic conductivity from texture. J. Irrig. Drain. Eng. 113(2): 184–192
Baker K.R. (2006). Optimization Modeling with Spreadsheets. Thompson Brooks/Cole, Belmont, CA
Bitterlich S., Durner W., Iden S.C. and Knabner P. (2004). Inverse estimation of the unsaturated soil hydraulic properties from column outflow experiments using free-form parameterizations. Vadose Zone J. 3: 971–981
Bloemen G.W. (1980). Calculation of hydraulic conductivities from texture and organic matter content. Z. Pflanz. Bodenkunde 143(5): 581–605
Boels D., Veerman G.J., Wit K.E. and Gils J.B.H.M. (1978). Theory and systems of automatic determination of soil moisture characteristics and unsaturated hydraulic conductivities. Soil Sci. 126(4): 191–199
Bouma J. and Denning J.L. (1972). Field measurement of unsaturated hydraulic conductivity by infiltration through gypsum crusts. Soil Sci. Soc. Am. Proc. 36: 846–847
Bouma J., Hille D.I., Hole F.D. and Amerman C.R. (1971). Field measurement of unsaturated hydraulic conductivity by infiltration through artificial crusts. Soil Sci. Soc. Am. Proc. 35: 362–364
Bresler E., Russo D. and Miller R.D. (1978). Rapid estimate of unsaturated hydraulic conductivity function. Soil Sci. Soc. Am. J. 42(1): 170–172
Bruce R.R. (1972). Hydraulic conductivity evaluation of the soil profile from soil water retention relations. Soil Sci. Soc. Am. Proc. 36(4): 555–561
Bruckler L., Renault P. and Aries F. (1987). Laboratory estimation of apparent soil thermal conductivity using a numerical approach. Soil Sci. 143(6): 387–397
Cesari L. (1983). Optimization – Theory and Applications. Springer Verlag, New York, USA
Elsenbeer H. (2001). Pedotransfer functions is hydrology. J. Hydrol. 251: 121–122
Erh K.T. (1972). Application of spline functions to soil science. Soil Sci. 114: 333–338
Fuentes C., Haverkamp R. and Parlange J. (1992). Parameter constraints on closed-form soil water relationships. J. Hydrol. 134(1–4): 117–142
Fuentes C., Haverkamp R., Parlange J.-Y., Brutsaert W., Zayani K. and Vachaud G. (1991). Constraints on parameters in three soil-water capillary retention equations. Transport Porous Media 6(4): 445–449
Gill P.E., Murray W. and Wright M.H. (1981). Practical Optimization. Academic Press, London, UK
Halbertsma, J.M., Veerman G.J.: A new calculation procedure and simple set-up for the evaporation method to determine soil hydraulic functions. Report 88, DLO-Staring Centrum, Wageningen (1994)
Hampton D.R. et al.: A method to fit soil hydraulic curves in modelsof subsurface flow in unsaturated soils. In: Gambolati, et al. (eds) Computational methods in subsurface hydrology. Computational mechanics Publ, Boston MA, pp 175–180
Hansen S., Jensen H.E. and Nielsen N.E. (1990). DAISY: Soil Plant Atmosphere System Model. Copenhagen, Denmark
Heinen, M.: Dynamics of water and nutrients in closed, recirculating cropping systems in glasshouse horticulture. With special attention to lettuce grown in irrigated sand beds. Ph.D. Thesis, Wageningen Agricultural University (1997)
Jhorar R.K., Bastiaansen W.G.M., Fedddes R.A. and Dam J.C. (2001). Inversely estimation soil hydraulic functions using evapotranspiration fluxes. J. Hydrol. 258(1–4): 198–213
Kastanek F.J. and Nielsen D.R. (2001). Description of soil water characteristics using cubic spline interpolation. Soil Sci. Soc. Am. J. 65: 279–283
Kool, J.B., Parker, J.C.: Estimating soil hydraulic properties from transient flow experiments: SFIT user’s guide. Report, Electric Power Research Institute, Palo Alto, California (1987)
Kool J.B. and Parker J.C. (1988). Analysis of the inverse problem for transient unsaturated flow. Water Resour. Res. 24(6): 817–830
Kool J.B., Parker J.C. and Genuchten M.Th. (1985). Determining soil hydraulic properties from one-step outflow experiments by parameter estimation: I. Theory and numerical studies. Soil Sci. Soc. Am. J. 49(6): 1348–1354
Kool J.B., Parker J.C. and Genuchten M.Th. (1987). Parameter estimation for unsaturated flow and transport models – a review. J. Hydrol. 91: 255–293
Leij F.J., Russell W.B. and Lesch S.M. (1997). Closed-form expressions for water retention and conductivity data. Ground Water 35(5): 848–858
Leong E.C. and Rahardjo H. (1997). Permeability functions for unsaturated soils. J. Geotech. Geoenviron. Eng. 123(12): 1118–1126
Luenberger, D.G.: Introduction to Linear and Non-linear Programming. Addison-Wesley Publishing Company (1973)
Marquardt D.W. (1963). An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Math. 11(2): 431–441
Mertens, J.: Parameter estimation strategies in unsaturated zone modelling. Ph.D. Thesis, Leuven Catholic University (2003)
Metselaar, K.: ‘Auditing predictive models : a case study in crop growth’. Ph.D. thesis, Wageningen Agricultural University (1999)
Minasny B., McBratney A.B. and Bristow K.L. (1999). Comparison of different approaches to the development of pedotransfer functions for water-retention curves. Geoderma 93(3–4): 225–253
Nelder J.A. and Mead R. (1965). A simplex method for function minimization. Comput. J. 7: 308–313
Nemes, A.: Multi-scale hydraulic transfer functions for Hungarian soils. Ph.D. Thesis, Wageningen Agricultural University (2003)
Nemes A., Schaap M.G., Leij F.J. and Wösten J.H.M. (2001). Description of the unsaturated soil hydraulic database UNSODA Version 2.0. J. Hydrol. 251(3–4): 151–162
Press W.H., Flannery B.P., Teukolsky S.A. and Vetterling W.T. (1986). Numerical Recipes: The Art of Scientific Computing. Cambridge University Press, Cambridge, UK
Price W.L. (1979). A controlled random search procedure for global optimization. Comput. J. 4: 367–370
Prunty L. and Casey F.X.M. (2002). Soil water retention curve description using a flexible smooth function. Vadose Zone J. 1: 179–185
Russo D., Zaidel J. and Laufer A. (1998). Numerical analysis of flow and transport in a three-dimensional partially saturated heterogeneous soil. Water Resour. Res. 34(8): 1451–1468
Rust B.W. and Burrus W.R. (1972). Mathematical Programming and the Numerical Solution of Linear Equations. American Elsevier Publishing Company Inc., New York, USA
Schaap M.G. and Leij F.J. (1998). Using neural networks to predict soil water retention and soil hydraulic conductivity. Soil Till. Res. 47(1/2): 37–42
Schaap M.G., Leij F.J. and Genuchten M.Th. (2001). ROSETTA: a computer program for estimating soil hydraulic parameters with hierarchical pedotransfer functions. J. Hydrol. 251: 163–176
Schuh W.M. and Bauder J.W. (1986). Effect of soil properties on hydraulic conductivity-moisture relationships. Soil Sci. Soc. Am. J. 50(4): 848–855
Simunek, J., Sejna, M., Van Genuchten, M.Th.: The HYDRUS-1D software package for simulating the one-dimensional movement of water, heat and multiple solutes in variably-saturated media, version 2.0. Technical Report IGWMC-TPS-70, Int. Ground Water Model. Cent., Colo. Sch. of Mines, Golden (1998)
Spriet J.A. and Vansteenkiste G.C. (1982). Computer-aided Modeling and Simulation. Academic Press, London, UK
Stephens D.B. and Rehfeldt K.R. (1985). Evaluation of closed-form analytical models to calculate conductivity in a fine sand. Soil Sci. Soc. Am. J. 49(1): 12–19
Stolte, J.: Manual for soil physical measurements: version 3. Technical Document 37, DLO-Staring Centrum, Wageningen (1997)
Stolte, J., Veerman, G.J.: Description of the evaporation method, crust method and hanging water column method to determine the hydraulic conductivity and water retention characteristics in Dutch. Rapport 74, DLO-Staring Centrum, Wageningen (1990)
Stolte, J., Wesseling, J.G., Verzandvoordt-Van Dijck, S.: Soil physical characteristics made available for a large group of users. in preparation (2007)
Stolte, J., Wesseling, J.G., Wösten, J.H.M.: Pedotransfer functions for hydraulic and thermal properties of soil and the tool HERCULES. Report 126, DLO-Staring Centrum, Wageningen (1996)
Stoof, C., Wesseling, J.G., Ritsema, C.J.: Effect of texture on hydrological behaviour of putting green rootzone mixes. Agron. J. (2007, submitted)
Tarantola A. (1987). Inverse Problem Theory. Methods for Data Fitting and Model Parameter Estimation. Elsevier Science Publishers B.V., Amsterdam, The Netherlands
Booltink H.W.G., Bouma J. and Alphen B.J. (2001). Combining pedotransfer functions with physical measurements to improve the estimation of soil hydraulic properties. Geoderma 103(1/2): 133–147
Hendriks Th.H.B. and Beek P. (1983). Optimization Techniques: Principles and Applications (in Dutch). Bohn, Sheltema en Holkema, Utrecht, The Netherlands
Van Dam, J.C., Huygen, J., Wessling, J.G., Feddes, R., Kabat, P., Van Walsum, P.E.V., Groenendijk, P., Van Diepen, C.A.: Theory of SWAP version 2.0: simulation of water flow, solute transport and plant growth in the Soil-Water-Atmosphere-Plant environment. Technical Document 45, DLO-Staring Centrum, Wageningen (1997a)
Van Dam, J.C., Huygen, J., Wesseling, J.G., Feddes, R.A., Kabat, P., Van Walsum, P.E.V., Groenendijk, P.: SWAP Users Manual, Version 2.0. Simulation of flow and transport processes in the Soil-Water-Air-Plant environment. Report 71, Dept. of Water Resources, Wag. Agr. University, Wageningen, The Netherlands (1997b)
Van Dam, J.C., Stricker, J.N.M., Droogers, P.: From one-step to multi-step: determination of soil hydraulic functions by outflow experiments. Report 7, Dept. of Water Resources, Wag. Agr. University, Wageningen, The Netherlands (1990)
Genuchten M.Th. (1980). A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44(3): 892–898
Van Genuchten, M.Th., Leij, F.J., Yates, S.R.: The RETC code for quantifying the hydraulic functions of unsaturated soils. Technical report, EPA, Ada (1991)
Vogel, T.: SWM 2 – numerical model of two-dimensional flow in a variably saturated porous medium. Research Report 87, Research report/Agricultural University, Department of Hydraulics and Catchment Hydrology (1997)
Vrugt J.A., Bouten W. and Gupta H.V. (2002). Toward improved identifiability of hydrologic model parameters: the information content of experimental data. Water Resour. Res. 38(12): 48/1–13
Wind, G.P.: Capillary conductivity data estimated by a simple method. Proc. unesco/iash, Water in the unsaturated zone. Symp. 1966. Wageningen, The Netherlands (1966)
Wösten, J.H.M.: Description of the soil moisture retention and hydraulic conductivity characteristics from the Staringseries with analytical functions (in Dutch). Rapport 2019, STIBOKA Wageningen (1987)
Wösten, J.H.M., Lilly, A., Nemes, A., LeBas, C.: Using existing soil data to derive hydraulic parameters for simulation models in environmental studies and in land use planning. Report 156, DLOSC (1998)
Wösten J.H.M., Pachepsky Ya.A. and Rawls W.J. (2001a). Pedotransfer functions: bridging the gap between available basic soil data and missing soil hydraulic characteristics. J. Hydrol. 251: 123–150
Wösten, J.H.M., Veerman, G.J., DeGroot, W.J.M., Stolte, J.: Soil moisture retention and hydraulic conductivity characteristics of top- and sub-soils in The Netherlands: the Staringseries (in Dutch). Rapport, Alterra, Wageningen (2001b)
Yeh W.W.-G. (1986). Review of parameter identification procedures in groundwater hydrology: the inverse problem. Water Resour. Res. 22(2): 95–108
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Wesseling, J.G., Ritsema, C.J., Stolte, J. et al. Describing the soil physical characteristics of soil samples with cubical splines. Transp Porous Med 71, 289–309 (2008). https://doi.org/10.1007/s11242-007-9126-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-007-9126-3