Effects of Spatial Variability on Annual Average Water Balance

Transcription

1 WATER RESOURCES RESEARCH, VOL. 23, NO. 11, PAGES , NOVEMBER 1987 Effects f Spatial Variability n Annual Average Water Balance P. C. D. M!LLY Department f Civil Engineering, Water Resurces Prgram, Princetn University, Princetn, New Jersey P.S. EAGLESON Department f Civil Engineering, Massachusetts Institute f Technlgy, Cambridge Spatial variability f sil and vegetatin causes spatial variability f the water balance. Fr an area in which the water balance is nt affected by lateral water flw, the frequency distributins f strm surface runff, evaptranspiratin, and drainage t grundwater are derivable frm distributins f sil hydraulic parameters by means f a pint water balance mdel and lcal applicatin f the vegetal equilibrium hypthesis. Means and variances f the cmpnents f the budget can be fund by Mnte Carl simulatin r by apprximate lcal expansins. Fr a fixed set f mean sil parameters, sil spatial variability may induce significant changes in the areal mean water balance, particularly if strm surface runff ccurs. Variability f the pre size distributin index and permeability has a much larger effect than that f effective prsity n the means and variances f water balance variables. The imprtance f the pre size distributin index implies that the micrscpic similarity assumptin may underestimate the effects f sil spatial variability. In general, the presence f sil variability reduces the sensitivity f water balance t mean prperties. Fr small levels f sil variability, there exists a unique equivalent hmgeneus sil type that reprduces the budget cmpnents and the mean sil misture saturatin f an inhmgeneus area. INTRODUCTION The lcal hydrlgic respnse f the land surface t atmspheric frcing (precipitatin, slar and atmspheric radiatin, etc.) is determined by the prperties f the surface and by the frcing. Bth f these factrs may vary significantly at spatial scales smaller than the scale f practical hydrlgic analyses. Since analyses f small-scale physical prcesses in catchments suggest that the lcal respnse is a nnlinear functin f these variables, we infer that the prblem f spatial integratin t the catchment scale is nt trivial. It is nt at all clear, fr example, under what cnditins a ne-dimensinal mdel f the sil bundary layer, grunded in sil water hydrdynamics, can represent the average behavir f a spatially variable natural catchment. This prblem f lcal variability has been ne f the majr impediments t the successful applicatin f Darcy scale physical thery t explain hydrlgic respnse at catchment scales. Variability f atmspheric frcing is mst apparent in the hetergeneity f rainfall fields. Individual strms have significant internal spatial and tempral structure and are generally in mtin relative t the grund. The result is that rainfall intensity histries, and even aggregate quantities such as ttal strm depth, are smetimes variable at scales smaller than thse f the catchment f interest. With increasingly large averaging perids (i.e., many strms), the spatial crrelatin naturally increases. Hwever, the shrt-term hydrlgic respnse is nt necessarily determined uniquely by such lngtime averages, whse unifrmity is therefre nt especially helpful in the analysis f single events. On the ther hand, the time average water balance can be parameterized in terms f rainfall statistics that are quite unifrm spatially. Variability f the land surface (itself, the sil texture and Cpyright 1987 by the American Gephysical Unin. Paper number 7W /87/007W structure, the vegetatin type and density, and gemrphlgical parameters), is the secnd majr surce f variability f hydrlgic fluxes. Sil prperties affect the ability f the sil surface bundary layer t transmit water int r frm the sil. The vegetatin alternately enhances r hinders the transmissin prcess. Gemrphlgical factrs f imprtance include thse inducing lateral subsurface flws, which cmplicate (and ften gvern) the direct runff generatin prcess. The spatial variability f the surface, unlike that f the frcing, is a factr that cannt be eliminated by averaging in time; surface prperties f untilled sils are s persistent in time as t be effectively static. In this paper we examine the effect f spatial variability f sil and vegetatin n water balance. In particular, we are cncerned with the extent t which their areal variatins affect the tempral-spatial averages f the majr hydrlgic fluxes. We are als interested in the questin f the existence f an equivalent hmgeneus sil capable f reprducing the average behavir f an inhmgeneus area. Theretical analyses f water balance n spatially variable sils have been described by Peck et al. [1977] and by Sharma and Luxmre [1979]. They emplyed dynamic simulatin mdels in analyses f the water balance f a particular catchment n a mnthly time scale, accunting fr strage changes. Sil spatial variability was represented using the cncept f micrscpic scale similarity. Sharma and Luxmre demnstrated the imprtance f sil variability and nted the difficulty f making generalizatins in light f the dependence f results n the sil-plant-weather cmbinatin. They bserved that their neglect f variatins f vegetatin prperties assciated with sil variability was unrealistic. Our treatment fllws the basic apprach first laid dwn by Peck et al. [1977-1, but ffers several new insights, primarily as a result f ur different apprach t the pint water balance. By restricting ur attentin t the annual average water balance, we are able t emply the results f Eaglesn [1978a, b, 2135

2 2136 MILLY AND EAGLESON: EFFECTS OF SPATIAL VARIABILITY ON ANNUAL AVERAGE WATER BALANCE c, d, e, f, g], which cncisely express the majr cmpnents f the water balance as functins f sil and atmspheric parameters and which accunt fr the dependence f equilibrium vegetatin densities n sil prperties. The cmputatinal simplicity f the water balance mdel allws us t btain detailed results regarding the sensitivity f average hydrlgic fluxes t the variance f sil parameters fr a wide range f sil types and climatic cnditins. It als permits us t examine the effect f departures frm micrscpic similarity in the analysis. Just as imprtant, the parametric frugality f the mdel allws us t infer frm thse results sme relatively general cnclusins regarding the effects f sil spatial variability. In what fllws the term "water balance" will refer t the annual average water balance, which will be further averaged here in space. FRAMEWORK Climate Parameters d, average rate f ptential evapratin; mt mean time between strms; mtr mean strm duratin; m mean duratin f rainy seasn; mpa mean annual rainfall; c parameter f gamma distributin f strm depth' a mean annual air temperature. Other Parameters h 0 surface retentin capacity; Z effective depth t water table; k v ptential transpiratin efficiency; M vegetal canpy density. Given these parameters, the water balance mdel yields expected values f the fllwing variables' We suppse that an inhmgeneus area A may be repre- S average sil misture saturatin; sented, t first rder, as a battery f parallel, independent, ETA annual evaptranspiratin; ne-dimensinal sil clumns, each described by a deterministic mdel yielding utputs as functins f lcal inputs and RsA annual surface runff; Rg A annual recharge t grundwater. parameters. In rder t derive the areal average water balance, Furthermre, invcatin f the vegetal equilibrium hypthesis we integrate the pint water balance ver the regin A, using [Eaglesn, 1978f-] allws us t slve the water balance withthe jint density functin f the sil prperties as a weighting ut specifying M;in that case, the water balance mdel als functin. Thus if y is any utput dependent n the sil prperprvides the value f M as an utput. Accrding t the equities t with jint frequency distributin f(t) in A, then the librium hypthesis, natural systems f vegetatin evlve areal average (y), defined by tward a canpy density that maximizes the average sil misture saturatin. dx (1) With respect climate, nly certain lng-term statistics are prescribed; the mdel implicitly perfrms a time average f is given by the dynamics [Eaglesn, 1978a, b, c, d, e, f, g]. In cntrast, cnstant sil parameters are specified, with the implicatin that the mdel is based n a physical cnceptin f hmge- (Y) =, g( )f( ) d (2) neity within the mdeled area. Direct applicatin f the mdel t a hetergeneus area implicitly requires an assumptin that in which x is the spatial crdinate, g( ) is the utput func- "effective" sil parameters may be defined. This assumptin is tin relating t t y, and fl is the set f all t fund in A. examined in a later sectin f this paper. (Ntatin fllws the main text f this paper.) One f the critical assumptins underlying the ne- In principle, the areal integratin shuld include nt nly dimensinal water balance mdel used here is that surface the sil parameters, but als the inputs, such as rainfall and runff is generated nly by the infiltratin-excess mechanism. ptential evapratin rate. While these may be highly variable There is n prvisin fr runff prduced at riparian seepage in space at any time, it is nly climatic statistics that enter the faces. Our cnclusins are therefre limited t catchments fr water balance mdel we use t represent g( ). These statistics which this assumptin is valid r t thse prtins f catchexhibit cnsiderably greater crrelatin in space than the in- ments in which the water table des nt apprach the surface. stantane s values, but, f curse, may als have significant spatial variability. We are therefre ignring such temprally JOINT DENSITY OF SOIL PARAMETERS cnstant, spatially variable effects as rgraphic influences n In rder t prceed within the prpsed framewrk, we rainfall statistics, tpgraphic influences n net radiatin, and require a statistical descriptin f the distributins and crres frth. latins f the three independent sil parameters n e, k(1), and m, which we dente cllectively by. The nature f the jint POINT WATER BALANCE We emply the statistical-dynamic water balance mdel f Eaglesn a, b, c, d, e, f, g] t explain, t first rder, the transfrmatin f lcal hydrlgic inputs and surface prperties t hydrlgic fluxes. The input parameters t the mdel are defined as fllws. Sil Parameters variability f these parameters has nt been well-defined and is an area f nging research. The marginal distributins f sme hydraulic parameters have been studied, either directly r indirectly, by a number f investigatrs. The analyses are mainly f tw basic types. In the first, data are cllected frm numerus surces f wide gegraphic rigin and gruped by sil textural classificatin. The resulting statistics characterize variability within a texture class ver the set f all sils. Such ne effective prsity f sil; analyses have been reprted by Clapp and Hrnberger [19783, k(1) sil permeability; by Brakensiek et al. [1981], and by Rawls et al. [1982]. The m pre size distributin index f sil. secnd type f analysis uses a large set f data frm a single

3 MILLY AND EAGLESON: EFFECTS OF SPATIAL VARIABILITY ON ANNUAL AVERAGE WATER BALANCE 2137 lcatin, field, r watershed t develp the densities, and therefre seems mre relevant t the current analysis. Warrick and Nielsen [1980] prvide a review f the earlier wrk in this area, and numerus wrks have been published subsequently, particularly in the sil science literature. It is rather well-established that the hydraulic cnductivity, and hence the intrinsic permeability, is ften lgnrmally distributed in the field. Effective prsity is sufficiently wellrepresented by the nrmal distributin. The parameter m, which is the expnent in the Brks-Crey [Brks and Crey, 1964] equatin fr the sil misture characteristic, was fund t be lgnrmal by Brakensiek et al. [1981] within a textural class, but its lcal variability within a field r a catchment has nt received much attentin. Our investigatins f available sil misture retentin data supprt the use f the lgnrmal distributin at the lcal scale. On the basis f the freging, we adpt the multivariate nrmal distributin t describe sil variability in this wrk. We treat he, In [k(1)], and In (m) as the three variates, dented by IO)il [1 ne ] 1 [ln(m)_] m = r 2 = n [k(1) (3) Then the jint density functin f m is f(t) = (2zr)3/2[Q[ /2 exp [-«(t -- III)TQ - (0}-- IIl)] (4) (T (0.6) BYERS AND STEPHENS (1983) x TRANSECT (I.2)... ANDERSON AND CASSEL (1986) (I. I) A HORIZON COELHO (1974) 300 mm DEPTH (0.8) AHUJA et l. (1984) O- 150mm DEPTH (0.9) NIELSEN et l. (1973) (I.8) 305mm LOAMY SAND DEPTH "] _ (2'0)(2 )7) SANDY LOAM [ LOAM,,,, (2.1). CLAY LOAM m (2.._5_) SlLTY CLAY [ ua (2.1) / SILTY CLAY LOAM) (2.5) ALL TEXTURES -2, IO -,5 tn[k(i),mm 2] Fig. 1. Fitted lgnrmal frequency distributins f k(1). Parenthetical numbers at left are values f a. in which (5) Q = EE(t - p)(t - p)r] (6) a a a2p 2 '10'3p 1 g3 lp31 g392p32 32 The vectr g is the mean value f, the matrix Q is the cvariance f, a is the standard deviatin f w, and Pu is the crrelatin f w and w. Fr cnvenience, we scale the standard deviatins by a parameter a, creases with the hrizntal dimensin, but there are bviusly ther imprtant factrs determining the sil variability. The data prvided by Brakensiek et al. [1981] can be prcessed t yield an estimate f a equal t 2.5 fr the entire United States f America. The hypthetical extreme case f an area having a bimdal sil distributin with permeabilities differing by a factr f 1000 yields a value f abut 3.5. Fr areas f interest fr the water balance prblem, it seems reasnable t fcus n values f ain the range frm abut WATER BALANCE SENSITIVITY TO THE SOIL VARIANCE ffi = ffo i= 1, 2, 3 (8) Mnte Carl Analysis f the Means The evaluatin f (2) is cmplicated by the fact that g( ) is We set f2 equal t unity, s a is equivalent t the standard nt knwn as an explicit functin; a numerical integratin deviatin f the lgarithm f permeability. We fix the values scheme is required. Since t is three-dimensinal and the ff and f3 as cnstants fr the entire investigatin. Based n evaluatin f g( ) is nt simple, a straightfrward disur interpretatin f sme f the limited available data, we cretizatin methd is cmputatinally unattractive. We intake f t be 0.05 and f3 t be 0.4. These shuld be cnsidered stead emply a Mnte Carl technique fr evaluatin f (2). as representative values with n universal significance. The We use a randm sampling scheme t generate numerus parameter a is used as an indicatr f the level f variability equally prbable values f t. With a set f N t i values s in the sil parameters. generated, we apprximate (2) accrding t It is helpful t cnsider the range f values that a may assume. Typical fitted frequency distributins f the lgarithm f permeability are pltted in Figure 1 fr representative textural classes and fr selected field sites; Table 1 summarizes the experimental findings f sme investigatrs. Fr sites Fr all results reprted in this paper, we used N equal t 500, whse characteristic hrizntal dimensin is 1000 m r less, the value f a typically ranges frm 0.5 t 1.0, but may becme even larger. Fr instance, Nielsen et al. [1973] fund which was fund thrugh numerus sensitivity analyses t be sufficient fr accurate estimatin f the means. We als examined the variances f the utputs, estimated accrding t the smaller values n a relatively hmgeneus field, while 1 N Andersn and Cassel [1986] fund values as high as 2.6, which O'y i 1 [ ](O)/) they speculated may have been caused by the presence f tree rt channels in the lwer sil hrizns. In general, a in- We have chsen the tw climatic data sets used by Eaglesn 1 N <y> =, a(t,) (9) -- <y>]2 (10)

4 2138 MILLY AND EAGLESON: EFFECTS OF SPATIAL VARIABILITY ON ANNUAL AVERAGE WATER BALANCE TABLE 1. Representative Values f S, the Standard Deviatin fln [k(1)] Characteristic Hrizntal Length f Site, Estimated Surce f Data m Sil a Byers and Stephens [1983] 15 fluival sand Cassel [1983] 60 Nrflk lamy sand Bresler et al. [1984] 90 Hamra Red Mediterranean Ahuja et al. [1984] 300 three silt lams Andersn and Cassel [1986] 500 Prtsmuth sandy lam Celh [1974] 1000 Pima clay lam Nielsen et al. [1973] 1500 Panche sil series Brakensiek et al. [1981] 5 x USDA texture classes Brakensiek et al. [1981] 5 x 106 all textures cmbined 2.5 [1978f]; the data fr Clintn, Massachusetts, and fr Santa mean value f In [k(1)], with n e and In (m) remaining fixed. Paula, Califrnia, are summarized in Table 2. Fr the sil parameters, the mean values!s are assigned the values given by Eaglesn ['1978e] as typical fr each f fur textural classes f sil; these are given in Table 3. We cnsider the case where there is n crss-crrelatin amng the sil parameters. We cnsider nly the case when the surface retentin capacity is negligible and the water table is s deep that there is n capillary rise t the surface. We take k v t be unity, and we Figure 4 shws, fr the Clintn climate, the sensitivity f the water balance t In [k(1)] fr a hmgeneus sil and fr a sil with a - equal t 6. Nt nly is the sensitivity f the balance t the mean generally reduced by inhmgeneity, but additinally the curves fr the inhmgeneus case are remarkably linear ver the range f pssible means. Several similar calculatins revealed that the effect f averaging is similar fr In (m) and fr the ther sil-climate cmbinatins. emply the vegetal equilibrium hypthesis t evaluate M. Figure 2 displays the slutins fr mean sil water satu- Mnte Carl Analysis f the Variances ratin s f the fur sils at bth lcatins. The parameter s We calculated the variances f the varius water balance is pltted as a functin f variability f sil type. One striking feature f these plts is the relatively small dependence f s utputs accrding t (10). In ur discussin f the results, we shall fcus n the cases f sandy lam and clay lam sils n the sil variance. The secnd remarkable feature is the with the Clintn climate; the first is qualitatively representative f all cases with sandy lam r silt lam sils, while the secnd is representative f all cases with clay lam r clay sils. small sensitivity t climate in cmparisn with the larger sensitivity t sil type. This is apparently explained by the fact that sil parameters such as permeability vary ver a wider range than the climatic parameters such as precipitatin and ptential evapratin. Figure 3 shws the majr water balance cmpnents fr each sil-climate cmbinatin as functins f sil variability. All pltted quantities are nrmalized by average precipitatin. As is seen in Figure 2, the effect f sil type dminates that f climate. In cntrast t Figure 2, significant sensitivity t a is visible. In general, it may be said that variability tends t equalize the magnitudes f the three cmpnents. Fr example, surface runff appears in the case f sandy lam and silt lam as a 2 grws frm 0 t 6. In cntrast, it decreases frm a very large fractin in the case f clay. Whereas nly the clay and the clay lam sils yield strm surface runff in the hmgeneus case, all inhmgeneus sils prduce it. A cnsequence f the behavir nted abve is that there is less sensitivity f the water balance t the average sil type when sil variability is cnsidered. We illustrate this mre directly by cnsidering ur clay lam sil and varying the TABLE 2. Climatic Parameters Figure 5 shws variances f s as functins f a - fr the tw cases. The relatins are rughly linear. In all cases examined, the standard deviatin f s was between 0.03 and 0.1 when 6 2 was unity. Figure 6 shws variances f the nrmalized cmpnents f the water balance as functins f a 2. In the case f the sandy lam, variances are quite small fr a 2 less than unity. This seems t be assciated with the virtual absence f surface runff in this regin; in all f ur results, flux variances are disprprtinately small when the surface runff cmpnent is very small. This suggests that much f the variability in fluxes induced by sil variance is cntrlled by strm surface runff. In the case f the clay lam sil, variances are much larger and apprach maximum values asympttically. The rder f magnitude f the maxima can be estimated by cnsidering the hypthetical case where variability f a particular flux is s great that the flux is unifrmly distributed between zer and ne. In that situatin, the variance wuld be abut This is cnsistent with Figure 6 and ther calculatins nt reprted here. Parameter Clintn Santa Paula,,, mm s-1 mtb, S mr, S mpa, mm mt, S K 1.74 x x x x 10? x x x x 10? TABLE 3. Mean Sil Parameters Clay Silt Sandy Clay Lam Lam Lam n e k(1), mm x x x x 10-7 m

5 . MILLY AND EAGLESON: EFFECTS OF SPATIAL VARIABILITY ON ANNUAL AVERAGE WATER BALANCE <S > 0.6- O,4- CLAY CLAY LOAM SILT LOAM 08 2 =6 0.2 ø'' SANDYLOAM CLINTON SANTA PAULA Fig. 2. Areal means f s as functins f %2 fr varius sils and climates. Apprximate Analysis f the Means It is difficult t infer frm applicatin f (9) the manner in which individual sil parameters affect the average water balance. Furthermre, the effects f crrelatin are nt easily discerned. In this sectin we prduce an alternative expressin fr (y) that shws the interactin f parameters explicitly, albeit nly apprximately. Assuming that sil variability is small, we develp secnd-rder apprximatins fr the mean values f the utput variables f the water balance. We expand the utput y in a Taylr series arund the mean pa- rameter set y = g( ) = g( ) + ( - )' -d g( ) ( ) Taking the expected value, and nececting terms fr which n is 3 r greater, we find (Y) = g( ) + i=t, % Qu ( 2) E 0.2 /!E[Rga ]/mpa k(i),mm 2 Fig. 4. Water balance as functin f mean In [k(1)] fr the clay lam sil with Clintn climate. r (Y) = g(p) + a 2,63-2 f r _2 f r32 f3 2 +,,0,02 f,f, 2 + 0, ] f2f3p23)] (13) + &2& in which all partial derivatives are taken at m equal t p. This equatin tells us, t a first apprximatin, that the effect f sil spatial variability f any input parameter n any utput variable is directly prprtinal bth t the variance f the parameter and t the secnd derivative f the utput with SANDY LOAM () SILT LOAM CLINTON SANTA PAULA CLINTON - SANTA PAULA I.O 0.8 C LAY LOAM I, ( c ) _ It E [ETa ]/mp A CLAY (d) 0.6 O.4 II E[Rsa ]/mpa CLINTON SANTA PAULA I 0.2 -, E[R a]/mpa I Fig. 3. Areal averagexpected water balance cmpnents as functins f a 2 fr variu sils and climates.

6 2140 MILLY AND EAGLESON' EFFECTS OF SPATIAL VARIABILITY ON ANNUAL AVERAGE WATER BALANCE s% 0.02 SANDY LOAM CLAY LOAM TABLE 4. Sensitivity f Mean Evaptranspiratin t Sil Variability 1 02 g. 1 02g 1 02g 2 0(_D22 2 0c032 Sil Type 2 &0 x 2 fx 2 f22 --' f32 Clintn Sandy lam Silt lam Clay lam Clay O.OI I I I I Santa Paula Sandy lam Silt lam Clay lam Clay Fig. 5. Variance f s as functin f a 2 fr tw sils with Clintn climate. respect t the parameter. Effects may als arise due t the interactin f tw parameters when they are crrelated with each ther Fr each Of the cmbinatins f sil and climate already discussed, and fr each f the majr utput variables, we have cmputed the terms (32g/3i3j)f f. Table 4 summarizes the results fr evaptranspiratin. These numbers are representative f results fr all ther utputs, which are therefre nt presented here. Each term in Table 4 represents the change in the mean utput resulting frm a particular term in (13) when a 0 is unity. In assessing the imprtance f a given term, it is helpful t recall that each f the utputs cnsidered is dimensinless and is cnfined t the interval [0, 1]. The relative imprtance f the different terms in (13) varies frm case t case, but sme generalizatins can be made. Let us cnsider first the terms assciated directly with the variances f the three parameters. Fr the cases in which n sur c O.lO SANDY LOAM E [ETA ]/mpa _ E [RsAi/mpA [] E i i I CLAY LOAM OOA A I ( 2 I i 4 i I. 6 Fig. 6. Variances f nrmalized water balance fluxes as functins f a 2 fr tw sils with Clintn climate. - A A Sil Type &x 0092 fzf2 Owz flf Clintn Sandy lam Silt lam Clay lam Clay Santa Paula Sandy lam Silt lam Clay lam Clay Nte {fx, r 2, c03} = {ne, In [k(1)], In (m)}. face runff is generated in the hmgeneus situatin, the effect f spatial variability f any f these parameters is minimal. This is cnsistent with ur earlier bservatin that strm surface runff is an imprtant surce f variability. Fr the sils f finer texture, variability f m is mst effective in changing the utputs, and variability f k(1) als has a significant effect. The effect f variability f ti e is negligible. The imprtance f the terms that cntain the crss derivatives will depend n the magnitude f the crrelatin between parameters, but generally appears t be minimal. The table entries represent the maximal effect, which ccurs when tw parameters are perfectly crrelated. In any case, the tie- k(1) term is cnsistently small. The tie- m term is generally smewhat larger, yielding abslute changes f the utputs ptentially (fr IP 31-1) as high as 3% fr the situatins in which surface runff ccurs. The,largest crss-derivative term is the m - k(1) term. The apprximatin implied by (12) crrespnds t straight lines that are tangent t the curves in Figure 3 at zer variance; this crrespndence was verified numerically in the curse f ur wrk as a check n the cnsistency f the calculatins. The range f validity f the apprximatin (12) is nt, hwever, readily apparent frm Figure 3 fr all cases, since the plts d nt reslve sme f the changes in slpe fr small variance. In summary, this apprximate analysis suggests that sil spatial variability has a significant effect n the areal average water balance nly fr cases in which surface runff is generated. In thse cases, the utput is mst strngly affected by variability f m, next by that f k(1), and nly slightly by variability f tie' Crss-crrelatin f parameters is unimpr- tant.

7 MILLY AND EAGLESON' EFFECTS OF SPATIAL VARIABILITY ON ANNUAL AVERAGE WATER BALANCE 2141 Apprximate Analysis f the Variances As in the case f the mean, we may use (11) t derive an apprximate result fr the variance that ffers insights absent frm the Mnte Carl apprach. Fr sufficiently small values f a 2 we find r i= = 3 &øi & Qu (14) by gk( ), then the requirement fr equivalence is <yk> = gt,( e) k = 1, 2,.'-, K (17) If e satisfies (17), then we may say that is an equivalent parameter set with respect t the K utputs. Several questins naturally arise cncerning the existence and uniqueness f, as well as its functinal dependence n f( ) and ther parameters. These questins are mst directly addressed by viewing (17) as a system f equatins defining %. Direct, exact slutin f (17) fr any useful set f gn des nt appear feasible, given the cmplex nature f the equatins. In this sectin we cnsider nly an apprximate case, the case where a 0 is small. We apprximate the left side f (17) using (12), and represent the right side f (17) using (11), in bth cases replacing g by gn, 1 c32g Q0 = g,(p) = j= in which the partial derivatives are evaluated at equal t It. We see, t a first apprximatin, that the variance f any water balance utput variable is directly prprtinal t a0 e. = k = 1, 2,..., K (18) We emplyed (15) t estimate the relative cntributins f each f the three sil parameters t variance f water balance Fr a sufficiently small, will be clse enugh t It that we cmpnents. The findings parallel the bservatins made in may drp all but the leading term in the infinite sum. This discussin f Table 4. In brief, variance f In (m) cntributes yields a new system f equatins fr e the mst t utput variance, and variance f In [k(1)] cntributes the bulk f the remainder. 1 02g, Qij The range f validity f (15) is variable; departures frm i=1 j=l prprtinality in the relatin between a0 e and ay e are indica- tive f failure f (15). Referring t Figure 6, we see that the apprximatin is nt bad fr the balance cmpnents in the case f sandy lam when a02 is less than unity, mainly be- Ocøic% (O,)e It). g (19) k= 1,2,...,K where derivatives are evaluated using the mean sil. This is a set f K linear equatins fr the three unknwn cmpnents cause there is n surface runff and hence n variability f f ) e. In the case where K is 3 and the matrix Og/3 is fluxes. Hwever, in the case f clay lam, the range f validity nnsingular, there exists a uniquequivalent sil parameter vectr is much smaller. Fr the variance f s, the situatin is similar. Fr sandy lam, (15) is accurate up t a e near 2, as can be me -- It -[- Qij (20) seen in Figure 5. Less apparent in Figure 5 is the inadequacy f (15) fr the clay lam' the slpe is quite small at the rigin, but increases several times ver near the rigin. ON THE EXISTENCE OF AN EQUIVALENT HOMOGENEOUS SOIL TYPE We have shwn that the areal average water balance depends n spatial variability f sil parameters. In applicatins f the water balance mdel, hwever, it is usually desirable t wrk with a single hmgeneusil type. In such a situatin, it is ften argued that the single ne-dimensinal clumn may capture the essential dynamics and that the sil type used is an "effective" sil type, functinally dependent upn the distributin f actual sil types within the mdeled area. In this sectin we prpse a precise definitin f an equivalent hm- geneus sil and explre the cnditins fr its existence in the case f lw sil variance. The general idea f an equivalent hmgeneus sil is that it shuld yield the same respnse as the spatially variable ensemble f sils. If we take the measure f respnse t be the areally averaged utput y, then we require that the equivalent sil type have sil parameters.} e satisfying (16) Typically, we are cncerned with a set f K utputs yn (k - 1, 2,..-, K). If the crrespnding utput functins are dented If / is singular r if K is 1 r 2 then there is an infinite set f equivalent parameter sets. If K is reater than 3 and t ere is n redundancy in (l ), then there can be n equivalent param- eter set. The value f K equal t 3 seems t be especially relevant here. Of the three nrmalized water balance fluxes (evaptranspiratin, surface runff, grundwater recharge), nly tw f them yield independent cnditins n e since their sum is precipitatin, which is independent f. A third independent utput is the mean sil misture saturatin. Equatin (20) can therefre be used, fr a 0 small, t cnstruct the equivalent hmgeneus sil with respect t the three water balance fluxes and the mean sil misture saturatin. We nte that the equivalent sil depends nt nly n the density functin fr the sil parameters, but als n the climate parameters. SUMMARY AND CONCLUSIONS Our study explres sme effects f sil spatial variability n annual average water balance. We use idealized dynamic mdel, and ur findings shuld be interpreted in light f the simplifying assumptins made. The mst imprtant f these are the fllwing. 1. The spatially variable land area behaves as a battery f parallel hmgeneus sil clumns withut dynamic interac-

8 2142 MILLY AND EAGLESON' EFFECTS OF SPATIAL VARIABILITY ON ANNUAL AVERAGE WATER BALANCE tin. Excess infiltratin capacity at ne lcatin cannt cm- M vegetal canpy density. pensate fr a deficiency elsewhere. m pre size distributin index f sil. 2. The spatial variability f climate is negligible within the mpa mean annual precipitatin. mdeled area. 3. The annual average water balance at a pint in th e area mtb mean time between strms. mr, mean strm duratin. is described by the mdel f Eaglesn [1978a, b, c, d, e, f, g], m, mean duratin f wet seasn. the majr assumptins f which are summarized by Eaglesn N number f samples in Mnte Carl cmputatins. [1978a]. n e effective prsity f sil. 4. The vegetal equilibrium hypthesis f Eaglesn [1978f] Q Cvariance f t. applies. 5. The effective prsity, the lgarithm f permeability, Rg A annual recharge t grundwater. RsA annual surface runff. and the lgarithm f the pre size distributin irldex f the s time average sil misture saturatin. sil fllw a jint nrmal distributin. 'a mean annual temperature. 6. The water table is deep, the surface retentin capacity is x lcatin vectr in A. negligible, and the ptential transpiratin efficiency is unity. Keeping in mind these limitatins f the study, we have the fllwing cnclusins. y a set f water balance variables. y any water balance variable. yk kth cmpnent f y. 1. The areal mean f the time average sil misture s is Z depth t water table. insensitive i the variance f the sil hydraulic parameters, parameter f gamma distributin f strm depth. despite its strng dependence n the mean sil parameters. is mean f t. 2. The average divisin f precipitatin amng surface Pij crss crrelatin f ci and runff, grundwater recharge, and evhptranspiratin may be a standard deviatin f In [k(1)]. significantly affected by the variance f sil prperties. The a standard deviatin f significance f the effect depends upn the mean sil parame- as standardeviatin f s. ters. Sils f high average permeability exhibit a slight in- ay standardeviatin f y. crease in surface runff due t sil variance, while thse f lw fl set f all t values fund in A. average permeability exhibit decreased surface runff and in- t sil parameter vectr {tie, In [k(1)], In (m)}. creasedrainage t grundwater. As a general rule, increased t ith sample value f t used in Mnte Carl runs. variance f sil type tends t equalize the magnitudes f the c ith cmpnent f t. three majr water balance cmpnents. ( ) areal average n A. 3. A crllary f 2 is that the sensitivity f the water bal- El- ] expected value at a pint in A. ance t the mean sil prperties is markedly reduced by the presence f spatial variability. Acknwledgments. This wrk was supprted by the Natinal Science Fundatin under grants ATM , ATM , and 4. When the sil hydraulic prperties are parameterized in CEE terms f effective prsity, permeability, and pre size distributin index, variability f the first has a negligibleffect n the average water balance relative t variability f the ther tw. REFERENCES Ahuja, L. R., J. W. Naney, and D. R. Nielsen, Scaling sil water prperties and infiltratin mdeling, Sil $ci. $c. Am. J., 48(5), 5. The significant effect f the pre size distributin index , n the means and variances suggests that departures frm Andersn, S. H., and D. K. Cassel, Statistical and autregressive analysis f sil physical prperties f Prtsmuth sandy lam, Sil micrscpic similarity in the sil are imprtant. $ci. $c. Am. J., 50(5), , , The bulk f the variance f water balance variables is Brakensiek, D. L., R. L. Engleman, and W. J. RawIs, Variatin within explained by variance f permeability and pre size distri- texture classes f sil water prperties, Trans. Am. $c. Agric. Eng., butin index. 24(2), , Bresler, E., G. Dagan, R. J. Wagenet, and A. Laufer, Statistical analy- 7. Fr sufficiently small variance f the sil parameters, sis f salinity and texture effects n spatial variability f sil hythere exists a unique equivalent hmgeneus sil type that draulic cnductivity, Sil $ci. $c. Am. J., 48(1), 16-25, yields the same mean sil misture saturatin and the same Brks, R. H., and A. T. Crey, Hydraulic prperties f prus mean water balance cmpnents as an inhmgeneus sil. media, Hydrl. Pap. 3, Cl. State Univ., Frt Cllins, Byers, E., and D. B. Stephens, Statistical and stchastic analyses f NOTATION hydrauli cnductivity and particle-size in a fluvial sand, Sil Sci. A mdeled area. $c. Am. J., 47(6), , Cassel, D. K., Spatial and tempral variability f sil physical prper- ErA annual evaptranspiratin. ties fllwing tillage f Nrflk lamy sand, Sil $ci. $c. Am. J., 47(2), , average rate f ptential evapratin. Clapp, R. B., and G. M. Hrnberger, Empirical equatins lbr sme f( ) jint density f t. sil hydraulic prperties, Water Resur. Res., 14(4), , f/ rati f ai t a. Celh, M. A., Spatial variability f water related sil physical g( ) functin transfrming t t y. prperties, Ph.D. dissertatin, Dep. f Sils, Water, and Eng., Univ. g( ) functin transfrming t t y. f Ariz., Tucsn, Eaglesn, P.S., Climate, sil, and vegetatin, 1, Intrductin t water gk( ) functin transfrming t t Ykbalance dynamics, Water Resur. Res., 14(5), , 1978a. h sil surface retentin capacity. Eaglesn, P.S., Climate, sil, and vegetatin, 2, The distributin f K number f elements in y. annual precipitatin derived frm bserved strm sequences, Water k(1) sil permeability. Resur. Res., 14(5), , 1978b. k v ptential transpiratin efficiency. Eaglesn, P.S., Climate, sil, and vegetatin, 3, A simplified mdel f

9 MILLY AND EAGLESON: EFFECTS OF SPATIAL VARIABILITY ON ANNUAL AVERAGE WATER BALANCE 2143 sil misture mvement in the liquid phase, Water Resur. Res., water prperties, Trans. Am. Sc. Agric. Eng., 25(2), , 14(5), , 1978c Eaglesn, P.S., Climate, sil, and vegetatin, 4, The expected value f Sharma, M. L., and R. J. Luxmre, Sil spatial variability and its annual evaptranspiratin, Water Resur. Res., 14(5), , cnsequences n simulated water balance, Water Resur. Res., 1978d. 15(6), , Eaglesn, P.S., Climate, sil, and vegetatin, 5, A derived distributin Warrick, A. W., and D. R. Nielsen, Spatial variability f sil physical f strm surface runff, Water Resur. Res., 14(5), , 1978e. prperties in the field, edited by D. Hillel, in Applicatins f Sil Eaglesn, P.S., Climate, sil, and vegetatin, 6, Dynamics f the Physics, Academic, Orland, Fla., annual water balance, Water Resur. Res., 14(5), , 1978f Eaglesn, P.S., Climate, sil, and vegetatin, 7, A derived distributin P.S. Eaglesn, Department f Civil Engineering, Massachusetts f annual water yield, Water Resur. Res., 14(5), , 1978g. Institute f Technlgy, Cambridge, MA Nielsen, D. R., J. W. Biggar, and K. T. Erh, Spatial variability f P. C. D. Milly, Water Resurces Prgram, Department f Civil field-measured sil-water prperties, Hilgardia, 42(7), , Engineering, Princetn University, Princetn, NJ Peck, A. J., R. J. Luxmre, and J. L. Stlzy, Effects f spatial variability f sil hydraulic prperties in water budget mdeling, Water Resur. Res., 3(2), , Rawls, W. J., D. L. Brakensiek, and K. E. Saxtn, Estimatin f sil (Received May 1, 1987; revised August 3, 1987; accepted August 4, 1987.)