Scaling Soil Water Retention Curves using a Correlation Coefficient Maximization Approach

Transcription

1 Scaling Soil Water Retention Curves using a Correlation Coefficient aximization Approac LAKSHAN NANDAGIRI Department of Applied ecanics and Hydraulics, National Institute of Tecnology Karnataa, Suratal, India JAN DE LEEUW Department of Statistics, University of California, Los Angeles In contrast to existing similar-media scaling metods wic minimize sum of squared differences (SS) between a mean soil water retention curve and scaled soil water pressure data, we propose a new metod involving maximization of te correlation coefficient (R) between measured and estimated soil water pressure eads. Wit tis new criterion, multivariate statistical procedures are implemented resulting in an explicit non-iterative solution for te set of scale factors describing te spatial variability of measured soil water retention curves. Performance of te proposed metod was tested wit publised data of insitu soil water retention measurements made at a site in Nort Daota, USA. Scaling was successful as indicated by substantial reduction in SS and matced reported performances of existing iterative metods. For te dataset used, our algoritm provides te optimal solution in a non-iterative manner and introduces minimal distortions into original retention measurements. INTRODUCTION Regression-based functional normalization tecniques (Tillotson and Neilsen, 984) are simple scaling tools tat may be used to caracterize and quantify spatial variability of soil ydraulic properties. By providing information on te structure of spatial eterogeneity in soil water retention and ydraulic conductivity curves, scaling facilitates stocastic

2 simulation of soil water/solute flow at field and watersed scales (eg., Pec et al, 977; Sarma and Luxmoore, 979; Hopmans and Stricer, 989; Űnlu et.al., 990). Scaling metods use measured data sets of soil water pressure () versus soil moisture content (θ) and ydraulic conductivity (K) versus θ (or ) obtained from a number of locations in a eterogeneous landscape and yield two sets of scale factors (α) relating te soil water retention curve [(θ)] and te ydraulic conductivity curve [K(θ) or K()] at eac location to teir respective representative mean curves. Altoug similitude requirements demand tat te scale factor derived at a sampling location from te (θ) curve be identical to tat derived from te K(θ) curve, field soils rarely appear to beave as perfect similar media. To counter tis, Clausnitzer et. al., (99) proposed a novel metod of scaling bot curves simultaneously to yield a single set of scale factors. Neverteless, most earlier investigators ave observed: i) ig degree of correlation between te two sets of scale factors for a wide variety of soils (Warric et al., 977; Russo and Bresler, 980) ii) more effective scaling of K(θ) curves using scale factors derived from (θ) curves (Warric et al., 977) and iii) negligible errors in simulating soil water processes using scale factors derived from (θ) data alone (Aua et al., 984). Consequently, based on te assumption of approximate similar-media beavior, it is generally accepted tat a single set of scale factors derived from scaling soil water retention data alone may be sufficient to describe te spatial variability of bot ydraulic functions (Hopmans and Stricer, 989). Several similar-media metods (under te class of functional normalization tecniques) to scale soil water retention curves ave been proposed (e.g., Warric et al., 977; Russo and Bresler, 980; Rao et. al., 983; Aua et al., 984; Williams and Aua, 99; Daamen et. al., 99; Clausnitzer et al., 99), but all of tem are based on determining optimal scale factors troug some form of minimization of te sum of squared differences (SS) between a mean curve and scaled water retention data. As Hopmans (987) notes in a

3 comparative evaluation of several approaces, metods differ mostly in te manner in wic te mean (θ) curve is represented. In tis tecnical note, we present a new metod to scale soil water retention curves troug maximization of te correlation coefficient (R) between soil water pressure eads estimated from a mean curve and measured pressure eads. Altoug one may prove from statistical teory tat maximizing R is equivalent to minimizing SS, we felt it was wortwile investigating te former approac wit te obective of being able to tae advantage of analogous procedures adopted in multivariate statistical metods, notably canonical correlation. Accordingly, our algoritm yields an explicit non-iterative solution for te optimal scale factors and constitutes an improvement over existing metods, wic employ iterative solution procedures. In subsequent sections of tis note, we present te teory of regression-based similarmedia scaling and a description of te proposed R-maximization scaling procedure, and demonstrate its application using a dataset of soil water retention measurements made by earlier investigators in Nort Daota, USA. THEORY Similar-edia Scaling Scaling teory, based on te concept of similar media assumes tat, if (, θ) are measured values of soil water pressure (negative in unsaturated soils but treated as positive in tis paper) and corresponding soil moisture content at any location, ten is related to a mean soil water pressure (ĥ) corresponding to te same moisture content troug te scaling relationsip,

4 _ λ λ () α _ were λ and λ represent microscopic caracteristic lengts of te reference soil and te soil at te particular location respectively, and α is te scale factor wic is constant for te location for all values of moisture contents. However, owing to differences in pore structures of soils, use of effective saturation ( θ θ ) ( θ θ ) Θ (were subscripts s and r refer to saturation and residual moisture contents respectively), is preferred to te use of θ. Te r obective of all scaling metods is to derive values of α, one for eac sampling location, and tereby caracterize te spatial variability in (Θ) curves. An attractive coice for representing te soil water retention curve is te model proposed by van Genucten (980): s r Θ n n n + γ () wic may be re-cast as, n n n Θ γ (3) in wic and γ and n are fitting parameters. As a first step, Equation (3) is fitted to (,θ) observations made at eac location and optimal values of θ s and θ r obtained are used to transform observed θ values at eac location

5 to corresponding Θ values. Equation (3) is ten fitted to (,Θ) observations pooled from all sampling locations so as to derive optimal values of model parameters γ and n. Te optimized form of Equation (3) is ten representative of te first estimate of te mean soil water retention curve for all te sampled locations. Assuming tat te similar media scaling relationsip [Equation ()] is valid for all values at all locations, a set of scale factors may be derived by minimization of te sum of squared differences (SS) between te mean curve and scaled data. n SS i, α i, for i n ;,.. (4) i were i, is te i t soil water pressure ead measured at te t location, ĥ i, is te i t mean soil water pressure ead at te t location, n is te number of observations at te t location and is te total number of sampled locations. Analytical procedures for minimization of SS may ten be employed by setting, ( SS) α 0,... (5) subect to a normalizing constraint α (6) Tis procedure yields te desired solution of te set of scale factors, [for example, refer Equations (9) and (0) of Clausnitzer et. al., 99]. However, optimal scale factors

6 still need to be determined troug an iterative process in wic Equation (3) is fitted to scaled soil water pressures using scale factors obtained in te previous iteration. Te procedure is repeated until SS converges to a minimum or if convergence is not obtained, te smallest SS attained is assumed to yield te optimal set of scale factors. Proposed R-maximization Scaling Procedure Instead of defining SS [Equation (4)] and minimizing it using Equations (5) and (6), we wor wit te correlation coefficient (R) between te two variables ĥ and, defined in general terms as: R, cov, var() var() (7) For convenience, we use matrix notations and accordingly define to be a (n x ) vector containing n observations of soil water pressure eads and ĥ to be a (n x ) vector of n estimates of water pressure eads, bot corresponding to te same values of effective saturation Θ. Te n values are partitioned into groups (corresponding to sampling locations) eac wit n values suc tat n n. We set n n. Let and u represent te group means and group variances of, and ĥ and v te corresponding group means and group variances of ĥ. Let c represent te group covariances between and ĥ. Te total covariance for te n observations of and ĥ may be written as,

7 + c, cov (8) and te total variance of will be, ( ) + u var (9) Now, if all values of in group are multiplied by a weigt, te total covariance and variance get modified as, + c, cov (0) ( ) + u var () Te total variance [Equation (0)] may be represented in matrix form as, ' g cov, () were g ' is te transpose of a ( x ) vector g wose elements are given by,

8 + c g for, (3) Similarly, total variance [Equation ()] may be written as, ( ) A ' var (4) were A is a ( x ) symmetric matrix wose diagonal elements are defined by, ( ) q u a + for q (5) and off-diagonal elements by, for q (6) wit,. and q,. Using Equation (7) we may now write te correlation coefficient as a function of weigts as, ( ) { } ) ( var, cov, Var Var R A g ' ' (7) q q q a

9 Te obective is to determine te ( x ) vector tat maximizes R(ĥ, ). To ensure a unique solution, it is necessary to constrain var() in Equation (7) to unity wile maximizing cov(ĥ, )[orrison, 990]. Note tat var(ĥ) is constant and remains unaffected by. Accordingly, we set up te maximization problem,, 0 R (8) subect to te constraint, ' A (9) Introducing te Lagrangian multiplier (µ), te optimization problem defined by Equation (8) reduces to, g - µ A 0 (0) From Equation (0) we ave, µ A g () wic upon substitution into te constraint defined by Equation (9) yields,

10 ' A µ ' g A g () Tis implies, ' - µ g A g (3) Combining Equations () and (3), we get te expression of te desired optimal ( x ) vector tat maximizes te correlation coefficient as, A g (4) ' g A g and te optimal correlation coefficient as, R ' - g A g var (5) Any scalar operation on will not cange its optimality property. Hence, we apply te normalizing constraint given by Equation (6) troug te following operation [Equation (6)] to ensure tat our vector of weigts is comparable to scale factors α derived by oter investigators.

11 α (6) Te following steps may be adopted for implementing te proposed metod.. For eac group (location) determine, ĥ,, u, and c.. Use Equation (3) to setup te g vector and Equations (5) & (6) to setup te A matrix. 3. Compute g' and A Determine te optimal vector of weigts using Equation (4) and te optimal correlation coefficient using Equation (5). 5. Transform into equivalent vector of scale factors α using Equation (6). EXAPLE APPLICATION To demonstrate te applicability of te proposed R-maximization procedure for scaling water retention curves, we used measurements made by Scu et.al., (99) in Dicey County, Nort Daota, USA. Teir dataset comprises in-situ soil water retention measurements made using a paired neutron probe-tensiometers setup during profile internal drainage tests at a number of sites. For tis study, we selected water retention data (,θ) of te A p orizon (8 cm dept) at five sites (A, B, C, D, E) located in two topo- sequences separated by a distance of 8 m. In all, measurements pooled from te five locations yielded 30 data points, wic we felt was sufficient to demonstrate our procedure. Computations were performed in icrosoft Excel spreadseet software. We used te Solver Add-in to fit Equation (3) to retention data as per te metodology suggested by Wrait and Or (998) and a freeware add-in (atrix.3) to perform matrix operations. Te performance of te scaling metod was evaluated using te following criteria:

12 i) percentage reduction in SS [Equation (4)] prior to and after scaling ii) correlation coefficient (R US-DS ) between unscaled data and descaled data obtained by applying te appropriate scale factor to te final mean soil water retention curve and iii) root mean square error (RSE) between unscaled and descaled data. RESULTS AND DISCUSSION Figure sows te effect of scaling on measured soil water retention data. Unscaled values wic are scattered around a mean soil water retention curve, coalesce into a narrow band around te final curve. Table sows te set of optimal scale factors tat bring about tis transformation and Table sows van Genucten model parameters of te mean soil water retention curve prior to and after scaling. Performance criteria for te scaling metod are summarized in Table 3. Starting wit a large SS of 596 and a correlation coefficient (R) of between unscaled data and ĥ values, our algoritm attains te maximum possible value of R SS reduces to 3060 indicating an impressive reduction of 90.6%. Altoug we implemented an iterative procedure in an effort to furter maximize R, te result obtained at te end of te first iteration [Table 3] was te optimal one. In comparison, Warric et. al.,(977) report SS reduction of 54.5% at te end of te first iteration and maximum reduction of 85.5% after 0 iterations, wen applying a SS-minimization scaling metod to retention data of a Panoce soil. Using a similar SS-minimization metod to scale water retention data of a sandy loam soil, Clausnitzer et. al., (99) report SS reductions of 95.5% and 83.% for A and B soil orizons respectively, but te number of iterations employed is not reported. Our optimal solution also appears to introduce small distortions into te original data as indicated by R US-DS of and RSE of 4.3 cm, results wic are similar to tose reported for existing SS-minimization metods (Hopmans, 987). Figure provides a grapical comparison of original measurements and descaled values of.

13 CONCLUSIONS Wile existing scaling metods are based on minimization of sum of squared differences (SS) between a mean soil water retention curve and scaled ydraulic data, we propose a new algoritm based on maximization of correlation coefficient (R) between measured and estimated soil water pressure eads. Preliminary tests indicated satisfactory performance of te proposed metod wen applied to in-situ measured soil water retention measurements. For te data set used, te algoritm yields te optimal solution in a noniterative manner. However, more compreensive evaluation wit larger and more diverse data sets is needed to validate tese findings and also to determine te probability distribution of scale factors derived using te proposed metod. Since similar-media scaling tecniques are nowadays being incorporated into ydrologic models, an example being te Root Zone Water Quality odel (RZWQ) (Stars et. al., 003), a non-iterative metod of te type proposed in tis wor would be more amenable for integration into suc modeling systems. Acnowledgements. We tan William. Scu, Nort Daota Water Commission, for providing soils data and Jan Hopmans, University of California, Davis for words of encouragement. REFERENCES Aua, L.R., J.W.Nancy, and D.R.Nielsen, Scaling soil water properties and infiltration modelling, Soil Sci. Soc. Am.J., 48, , 984. Clausnitzer, V., J.W. Hopmans, and D.R.Nielsen, Simultaneous scaling of soil water retention and ydraulic conductivity curves, Water Resour. Res., 8, 9-3, 99.

14 Daamen, C.C., Zenua Xiao, and J.A.Robinson, Estimation of water retention function using scaling teory and soil pysical properties, Soil Sci. Soc. Am. J., 55, 8-3, 99. Hopmans, J.W., A comparison of various metods to scale soil ydraulic properties, J. Hydrol., 93, 4-56, 987. Hopmans, J.W., and J.N..Stricer, Stocastic analysis of soil water regime in a watersed, J. Hydrol., 05, 57-84, 989. atrix.3, ttp://digilander.liberto.it/foxes/matrix3.zip orrison, D.F., ultivariate statistical metods, cgraw-hill Inc., 495 p, 990. Pec, A.J., R.J. Luxmoore, and J.C. Stolzy, Effects of spatial variability of soil ydraulic properties in water budget modeling, Water Resour. Res., 3, , 977. Rao, P.S.C., R.E. Jessup, A.C. Hornsby, D.K. Cassel, and W.A. Pollens, Scaling soil microydrological properties of Laeland and Konowa soils using similar media concepts, Agric. Water anage., 6, 77-90, 983. Russo, D., and E. Bresler, Scaling soil ydraulic properties of a eterogeneous field, Soil Sci. Soc. Am. J., 44, , 980.

15 Scu, W.., R.L. Cline and.d. Sweeney, Unsaturated soil ydraulic properties and parameters for te Oaes area, Dicey County, Nort Daota, Water Resources Investigation No.8, Nort Daota State Water Commision, 99. Sarma,.L., and R.J. Luxmoore, Soil spatial variability and its consequences on simulated water balance, Water Resour. Res., 5, , 979. Stars, P.J., G.C. Heatman, L.R. Aua and Liwang a, Use of limited soil property data and modeling to estimate root zone soil water content. J. Hydrol., 7, 3-47, 003. Tillotson, P.., and D.R. Nielsen, Scale factors in soil science, Soil Sci. Soc. Am.J., 48, , 984. Ünlu, K., D.R. Nielsen, and J.W. Biggar, Stocastic analysis of unsaturated flow: Onedimensional onte Carlo simulations and comparisons wit spectral perturbation analysis and field observations, Water Resour. Res., 6, 07-8, 990. Van Genucten,.T., A closed form equation for predicting te ydraulic conductivity of unsaturated soils, Soil Sci. Sco. Am. J., 44, , 980. Warric, A.W., G.J. ullen and D.R. Nielsen, Scaling field measured soil ydraulic properties using a similar media concept, Water Resour. Res., 3, , 977. Williams, R.D., and L.R.Aua, Evaluation of similar-media scaling and a one-parameter model for estimating te soil water caracteristic, J. Soil Sci., 43, 37-48, 99.

16 Wrait, J.., and Dani Or, Nonlinear parameter estimation using spreadseet software. J. Nat. Resour. Life Sci. Educ., 7, 3-9, 998.

17 FIGURE CAPTIONS Figure : a) Original and b) scaled soil water retention data and fitted mean curves Figure : Comparison of original and descaled soil water pressure ead

18 TABLE : Optimal Scale Factors Site α A B C D E TABLE :Van Genucten odel [Equation (3)] parameters for mean soil water retention curve Parameter Unscaled Scaled γ (cm - ) n TABLE 3: Performance Criteria for Proposed Scaling etod Criteria Unscaled Scaled SS %SS reduction R R US-DS RSE 4.3

19 Soil Water Pressure Head (cm) a) Unscaled Effective Saturation Soil Water Pressure Head (cm) b) Scaled Effective Saturation Fig.. a) Original and b) scaled soil water retention data and fitted mean curves

20 00 observed (cm) : descaled (cm) Fig.. Comparison of original and descaled soil water pressure ead