Soil Science Society of America Journal 65:279-283 (2001)
© 2001 Soil Science Society of America
DIVISION S-1-SOIL PHYSICS
Description of Soil Water Characteristics Using Cubic Spline Interpolation
Ferdinand J. Kastaneka and
Donald R. Nielsenb
a Institute for Hydraulics and Rural Water Management, Univ. of Agriculture, Forestry and Renewable Resources, Muthgasse 18, A-1190 Vienna, Austria
b Dep. of Land, Air and Water Resources, Univ. of California, Davis, CA 95616
Corresponding author (drnielsen{at}ucdavis.edu)
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ABSTRACT
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Solutions of differential equations describing soil water behavior frequently require a mathematical function for the soil water characteristics. Although spline interpolation provides a method to connect the data points, in many cases its use yields poor results. The mathematical representation of the soil water characteristic may also be shaped according to the imagination and judgment of the investigator. In such cases, some additional virtual data points are chosen through which the spline function passes exactly to yield a mathematical description of the entire curve consistent with the initially measured points. A computer program SWC-SPLINE written in Delphi 2 allows the investigator to easily mark and change the position of the assumed virtual data points interactively according to visual inspection.
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INTRODUCTION
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THE SOIL WATER CHARACTERISTIC describes the relationship between soil water content and the matric potential of water. Many equations describing the soil water characteristic appear in the literature. Most of them have been developed for soil water contents from water saturation to approximately those of the permanent wilting point of plants. Generally, at least two factors preclude an accurate description of the soil water characteristic. First, because a theoretical relationship has yet to be derived, descriptive equations for the soil water characteristic remain empirical. Second, experimental measurements are subject to observational and instrumental errors (Baveye and Sposito, 1985). Recently, Dourado-Neto et al. (1997) developed personal computer software for 12 analytical models appearing in the literature to describe the soil water characteristic. The most universally used model, published by van Genuchten (1980), is applicable to S-shaped relationships, while that of Brooks and Corey (1964) is applicable to C-shaped relationships. A special feature of both models is the introduction of the concept of the residual water content 
r, which is approached asymptotically by both models. For measurements manifesting S- and C-shaped curves, the models of van Genuchten and Brooks and Corey show excellent fit, but for many sets of measurements, their models as well as those of others reported by Dourado-Neto et al. (1997) yield questionable results.
Our objective here is to provide another method to describe the soil water characteristic that has several advantages over those presently being used in soil hydrology. We have used the proposed classical cubic spline interpolation function to describe some measured soil water characteristic curves presently available in the literature and illustrate its use with several data sets.
Without an analytical basis for drawing a smooth curve through a sequence of measured points, the visual or draw-by-eye method continues to be commonly practiced in soil science as well as in other disciplines. The smooth curve without an analytical equation representing the data is evaluated at selected points along its path. Values between these selected points are usually found by linear interpolation. Curve templets are sometimes used to not only smooth measured data but also to interpolate between measured values. With such visual techniques, boat builders use thin wooden or metal rods-splines—to shape the hull of a boat between its skeletal frames.
Piece-by-piece linear straight connections between two given neighboring points-a polynomial of order one—is the simplest alternative. Data outside the neighboring points under consideration have no effect on the interpolating linear relationship. Unfortunately, such linear interpolation between two or more pairs of measured points yields discontinuous derivatives at each junction of linear segments. At the other extreme, a polynomial of order (n - 1) can be used for interpolating between n measured points. Although this alternative passes through all of the measured points, it is usually not acceptable because it generally creates a nonrealistic, undulating curve between the measured points.
The most versatile method is that using polynomial functions of order three, which need four data points. For more than four data points an infinite number of polynomials of order three can be drawn for one interval between only two data points. Interpolating with natural cubic spline functions may help to solve this problem.
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MATERIALS AND METHODS
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Theory of Spline Interpolation
A cubic spline function interpolating between n data points with the independent data xi and the dependent data yi = f(xi) inside the closed interval [x1, xn] and with x1 < x2 < x3... < xn is composed piece-by-piece with cubic polynomials Pi between each subinterval [xi, xi+1]
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Equation [1] has the following first and second derivatives
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Interpolation with cubic-spline means that neighboring intervals have different coefficients for their interpolating third-order polynomials, but at their junctions their values as well as their respective first and second derivatives are equal to give smooth transitions
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Moreover,
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and for natural cubic splines,
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With this requirement the coefficients of all polynomials of order three may be calculated by solving a system of linear equations (Engeln-Muellerges and Reuter, 1986).
The Concept of Virtual Data Points
Erh (1972) recommended using a spline interpolation to describe the soil water characteristics by passing a smooth line through a set of measured data. He found reasonable results for some examples. However, for most data found in the literature, simple spline interpolation yields unrealistic nonmonotonic behavior, and for scattered points, a simple spline interpolation is altogether inapplicable.
Fritsch and Carlson (1980) developed an algorithm to guarantee monotonic behavior if the user does not require continuity of the second derivative. Their curve fitting routine tested for one data set by Hampton (1990) gave satisfactory results. Their method demands monotonic data assumed to be sufficiently accurate to warrant interpolation, not approximation. Hence, their method does not work for scattered data points or for data with observational and instrumentation errors.
A cubic spline can also be used to pass a smooth line through a set of data containing errors of measurement. In this case, the smooth line should approximate the measured data rather than pass exactly through each and every point. The measured data points give some idea about the shape of this relationship.
It is a reasonable assumption to choose some additional data points-virtual data points—according to the imagination of the user through which the cubic splines pass exactly to yield the proposed mathematical representation of the soil water characteristic. In most cases the first assumption of virtual data points does not show good results because of undulating behavior. Small changes in the position of the assumed data points or addition of more virtual data points may help overcome this problem. However, changing the position of one data point or adding more virtual data points may change the shape of the entire relationship because all points are interconnected. To facilitate the choice of the virtual data points the visual computer program SWC-SPLINE was written in Delphi 2 (Kastanek and Nielsen, 1999) so that the user can mark and change the position of the assumed data points interactively according to visual inspection.
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RESULTS
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Traditional Methods
The data published by Dane and Puckett (1992) show a distinct hump (assumed to be real and not an experimental artifact) that cannot be modeled with the equation of van Genuchten or that of Brooks and Corey (Fig. 1)
. For the calculation of the root-mean-square error in this and subsequent figures, the matric potential has been taken as the independent variable and the water content as the dependent variable. The data published by Stephens and Rehfeldt (1985), taken from Durner (1992), is a good example of a bimodal-shaped soil water characteristic (Fig. 2)
. The discrepancies between the measured data and the approximations by the equation of van Genuchten as well as that of Brooks and Corey are not trivial. Durner replaced the model of van Genuchten (1980) by a multi-van Genuchten model consisting of the superposition of two or more van Genuchten equations, each weighted to describe the measured set of data. For each of the van Genuchten equations the maximum water content o and the residual water content r are the same while other parameters are unique for each superpositioned equation. Although the results of this multi-equation model depend on the number of van Genuchten equations included, the functional properties of the basic van Genuchten model are retained-continuously differentiable, asymptotic to a zero slope towards the fine and large pores, and strongly monotonic over the entire moisture range. If the number of linearly imposed van Genuchten equations is small, the function is well behaved for interpolation purposes, reducing the noise in the measured data (Durner, 1994).
Spline and Virtual Data Points
When all measured data points are passed through and described by a simple spline curve, the root-mean-square error is zero. The simple spline method (Erh, 1972) shows no monotonic interpolation for the example of Dane and Puckett (1992) (Fig. 1) and for that of Stephens and Rehfeldt (1985) (Fig. 2). But adding two virtual points to the measurements shown in Fig. 1, a reasonable representation of the soil water characteristic by cubic spline interpolation is achieved (Fig. 3)
. Adding one virtual point to the measurements shown in Fig. 2 provides description of the measurements in the vicinity water saturation (Fig. 4)
. Replacing one measured point with a virtual point allows the soil water characteristic to be monotonic everywhere. Dane et al. (1994) used relationships similar to the soil water characteristic to describe the relationship between capillary pressure and water content when water is displaced by trichloroethylene instead of air. Their results (a good example of scattered data) show C-shaped curves and discontinuities at the displacement pressure point. For this case Dane and Jalbert (1999) suggested using the Brooks and Corey (1964) equation instead of that of van Genuchten (1980). Although it is not possible to describe discontinuous functions with cubic splines, it is possible to divide the course of the function into several intervals and use separate splines for each interval. For the relationships presented by Dane et al. (1994), the spline interpolation was only performed for the continuous part and a straight line was added between the displacement pressure point and zero matric potential head (Fig. 5) . Such a procedure is an extension of the proposed method with cubic splines. It should be noted that because the input data of water contents and matric potentials are represented pixel by pixel on the screen of the computer, their values may deviate slightly from their representation in the computer. Because a considerably large number of coefficients is required (Table 1), this method is recommended only if other methods do not yield satisfactory results, for example, for the data of Dane and Puckett (1992) or Stephens and Rehfeldt (1985).
We note that the method has advantages and disadvantages. The advantages are:- The concept applying virtual data points yields a curve that always fits the measured data points in an excellent manner.
- In contrast to robust, empirical functions the proposed method defines mathematical functions that include the sense and experience of the user for each set of data.
- There is almost no restriction on the shape of the relationship. S-shaped relationships as modeled by the equation of van Genuchten (1980) as well as C-shaped relationships as modeled by Brooks and Corey (1964) may be described equally by the method proposed.
- Spline functions are easy to differentiate and to integrate.
- There is no need to apply the concept of the residual water content or an asymptotical approach.
- Although discontinuities cannot be approximated by splines, the proposed method allows for the discontinuous shape of the soil water characteristic at the displacement pressure point.
- Extrapolation may be achieved either by continuation of the tangents at the end-points or by adding more visual data points outside the range under consideration. We warn that extrapolation is generally not desirable for every method including the one presented here.
The disadvantages are:
- The overall result is not a simple equation with a few parameters that can be handled with a pocket computer.
- For cubic spline interpolation at least four data point have to be known.
- Natural cubic spline functions for n data points consist of (n -1) cubic polynomial equations with 4(n - 1) coefficients. They are difficult to survey and only can be handled with a computer.
- Visual spline interpolation does not guarantee monotonic behavior. This may be achieved by proper choice of the virtual data points.
- Realizing that the coefficients of van Genuchten or Brooks and Corey equations are curve-shape coefficients (Durner, 1994), it is easy to understand their effect on the shape of a curve. In contrast, it is almost impossible to predict the consequences of changing one or more coefficients of the spline function on the shape of a curve.
- According to MATHCAD (1994), an inverse solution for a polynomial equation of order three exists. In contrast to the multi-van Genuchten-method, water content as well as matrix potential may be evaluated independently. Due to its clumsy length, the inverse equation with the water content as the independent variable is difficult to handle.
- The method proposed is only easily applied with a computer program that allows marking and changing the position of the assumed data points interactively.
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CONCLUSIONS
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Assuming virtual data points, the cubic spline method successfully described the soil water characteristic of all data sets available to the authors. The virtual spline method is a useful alternative to the traditional methods because it is the only method where the results remain on the judgment of a qualified personal decision and not on an automated procedure without personal innovation. On the other hand, the virtual spline method should not replace existing methods if they are adequate. Because of its greater flexibility, the virtual spline method should be used where other methods are not applicable. We conclude that the method provides an additional representation of the soil water characteristic curve.
A corresponding interactive computer program (SWC-SPLINE) was developed and may be downloaded free of charge from http://ihlww.boku.ac.at/.
Received for publication November 19, 1999.
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REFERENCES
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- Baveye, P., and G. Sposito. 1985. Macroscopic balance equations in soils and aquifers: The case of space- and time-dependent instrumental response. Water Resour. Res. 21:1116–1120.
- Brooks, R.H., and A.T. Corey. 1964. Hydraulic properties of porous media. Hydrol. Pap. 3, Colorado State Univ., Fort Collins.
- Dane, J.H., M. Oostrom, and B.C. Missildine. 1994. Determination of capillary pressure–saturation curves involving TCE, water and air for a sand and a sandy loam. EPA/600/R/94/005. Robert S. Kerr Environ. Res. Lab., U.S. EPA, Ada, OK.
- Dane, J.H., and W.E. Puckett, 1992. Field soil hydraulic properties based on physical and mineralogical information. p. 389– 403. In M.Th. van Genuchten et al. (ed.) Indirect methods for estimating the hydraulic properties of unsaturated soils. U.S. Salinity Lab., Riverside, CA.
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- Kastanek, F., and D.R. Nielsen. 1999. An interactive program to represent the soil water characteristics by cubic splines. Computer Software Demonstration. ASA, CSSA, and SSSA 1999 Annual Meetings, Salt Lake City, UT.
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TOP
ABSTRACT
INTRODUCTION
