stochastic analysis of soil water monitoring for drip irrigation management in heterogeneous soils

Stochastic Analysis of Soil Water Monitoring for Drip IrrigationManagement in Heterogeneous Soils

Dani Or*

ABSTRACTMany drip irrigation management schemes rely on frequent moni-

toring of soil water content and matric potential, using various sensors(e.g., tensiometers or time domain reflectometry probes). The soilwater information is used either for irrigation scheduling or for ad-justing schedules based on evapotranspiration measurements. Spatialvariations in soil properties induce variations in wetting patterns aboutthe drippers, which complicate the acquisition and interpretation ofinformation on soil water status. The objective of this study was toquantify the effects of mild spatial variation in soil hydraulic propertieson wetting patterns and the consequences on soil water sensor place-ment and interpretation. The soil hydraulic properties (saturated hy-draulic conductivity, Ks, and the exponent a of the exponential unsatu-rated hydraulic conductivity function) were considered as randomspace functions. Small perturbation expansions were applied to analyti-cal solutions of steady state flow from point sources. The resultinganalytical expressions relate the variability of Ks and a to the expectedvariability in soil water pressure head (h) and relative saturation (S)about a point source. Comparisons of the analytical predictions withMonte Carlo simulations (for surface and buried sources) resulted inexcellent agreement for coefficients of variation of ln(A"s) and a < 0.3.The expressions may be used as first approximations to define regionswith small uncertainty that may be most suitable for soil water monitor-ing and control in drip-irrigated fields. Another application is fordetermining the minimum number of sensors needed to obtain esti-mates with a prescribed estimation error. The derived spatial covari-ance functions of 5 and h may be used to identify regions in the fieldwith different wetting patterns or simply to interpolate measurementsto unmeasured portions of the field.

FREQUENT MONITORING of soil water content and soilwater pressure head is at the core of many drip

irrigation management schemes. Soil water informationobtained by various sensors (e.g., tensiometers, gypsumblocks, or TDR probes) is often used for schedulingirrigation or for adjusting irrigation schedules based onevapotranspiration measurements. The properties ofmost soils vary from one place to another; such spatialvariations in soil hydraulic properties affect water distri-bution patterns about drippers at different locations inthe field. These variations in wetted volumes and inspatial distributions of soil water pressure head and watercontent present a problem for sensor placement relativeto the drippers or the crop, and complicate interpretationsbased on soil water information.

Aspects of anisotropy, and the effects of deterministicvariations in soil properties with depth on water flowfrom point and line sources were analyzed theoreticallyby Philip (1972) and Philip and Forrester (1975), andexperimentally by Merrill et al. (1978) and Dirksen(1978). These studies illustrate (and quantify) the impactof heterogeneous soil properties in significantly modi-fying the flow regimes and the distributions of flow

Department of Plants, Soils, and Biometeorology, Utah State Univ.,Logan, UT 84322^820. Received 14 Apr. 1994. "Corresponding author([email protected]).

Published in Soil Sci. Soc. Am. J. 59:1222-1233 (1995).

attributes compared with flow in idealized homogeneoussoil. The applicability of such analyses to practical prob-lems of natural variability is limited because (i) theyprovide a description of mean behavior only, and (ii)most soil properties vary in an irregular and erraticfashion, which is more suitable for statistical descriptionand analysis. The alternative is a stochastic approachthat considers and characterizes spatial variations in soilproperties as random space functions. This approach hasbeen useful in many studies (Yeh et al., 1985; Hopmanset al., 1988; Rubin and Or, 1993) and has provided ameans for analytical solutions to otherwise intractableflow problems.

This study is designed to provide an improved frame-work for collecting and interpreting soil water informa-tion needed for management of drip-irrigated fields inwhich spatial variations in soil hydraulic properties arerelatively small. The main reasons for limiting the discus-sion to conditions of mild variability are (i) many agricul-tural fields are parcels of land selected on the basisof homogeneity in soil conditions (within plant rootingzones) and are cultivated to further homogenize the soilconditions (at top layer) and (ii) these conditions areamenable to first-order statistical analysis, which yieldsanalytical solutions.

The specific objectives of this study were to: (i) quan-tify the effects of spatial variation in soil hydraulic proper-ties on the variability of soil water pressure head andwater content patterns about a dripper (point source),and (ii) use the results for providing guidelines for soilwater sensor placement and interpretation under condi-tions of mild soil variability.

The theoretical part of the study is based on the applica-tion of first-order small-perturbation expansions to ana-lytical solutions for steady water flow from point sources.These analyses provided analytical expressions relatingthe (permanent) variability of soil hydraulic propertiesto variability of wetting patterns (in terms of soil waterpressure head and water content) about a dripper source.The analytical solutions were then used to propose guide-

, lines for sensor placement given (i) soil properties andthe extent of the soil variability, (ii) irrigation systemparameters (source strength and position), and (iii) irriga-tion management objectives and constraints.

MATHEMATICAL DEVELOPMENTSSteady Flow from Point Sources

Philip (1968,1971) and Raats (1971) considered multidimen-sional steady infiltration from various source geometries, pointor line sources, buried or placed on the soil surface. Thegoverning flow equation is a combination of Darcy's law withconservation of mass (continuity) in an unsaturated homoge-

Abbreviations: TDR, time domain reflectometry; MC, Monte-Carlo; CV,coefficient of variation.

1222

OR: SOIL WATER MONITORING STOCHASTIC ANALYSIS FOR DRIP IRRIGATION MANAGEMENT 1223

neous porous medium:

dK(h)dz [1]

where z is the vertical coordinate, positive downward, h is thesoil water pressure head (negative), and K(h) is the unsaturatedhydraulic conductivity. Analytical solutions to Eq. [1] may beobtained by introducing a transformation variable known as thematric flux potential (p (Gardner, 1958; Raats, 1971) defined as

h [2]and expressing K(h) as proposed by Gardner (1958):

K(h) = *<e°* [3]where a is the slope of dln[ K(h)] /dh or the rate of reduction inhydraulic conductivity with h, and Ks is the saturated hydraulicconductivity. Substitution of Eq. [2] and [3] into Eq. [1] yieldsa quasilinear steady flow equation (in one variable, q>):

[4]

Equation [4] was solved analytically for various boundaryconditions and flow geometries. In this study, we focusedon solutions for soil water pressure head and water contentdistributions about point sources (representing a single dripper)placed either on the soil surface or buried.

The solution for the steady state distribution of (p about apoint source buried in an infinite medium (denoted as <pB),subject to the following boundary condition is

= 0 [5]where r is the radial distance from the source; and with aknown source strength q (L3 T~}) at z = 0, r = 0, is givenas (Philip, 1968)

4n(r2 l/2 exp[a/2[z - (r2 [6]

Similarly, a solution for the distribution of matric fluxpotential about a point source located on the surface (cps) ofa semiinfinite porous medium was obtained by Raats (1971):

= 2q>Bage"

471•Ei{-a/2[z z2)1/2]j [7]

where cpB is the matric flux potential distribution around aburied point source given in Eq. [6], and Ei is the exponential

S CO

(e~*/x)dx). The boundary condi-tion (Eq. [5]) and a source strength q placed at z = 0 (surface)and r = 0 were assumed in the derivation of Eq. [7].

The distribution of h may be obtained from <p (surface orburied) using the following transformation:

h(r,z) = ln[aq>(r,z)/A'J[8]

An example of the steady-state distribution of soil water pres-sure head about buried (Eq. [6]) and surface (Eq. [7]) pointsources is depicted in Fig. 1, using mean values of the parame-ters in Table 1.

The estimation of soil water content (9) distribution abouta point source requires a retention model describing the rela-tionship between h and 9. One such model based on the

Radial Distance (cm)

-340 30 20 10 -10 -20 -30 -40

Soilwater

pressurehead

Ocm

-10

-20

-30

-40

-50

-60-70-80

-90-100-110

40 30 20 10 -10 -20 -30 -40

Fig. 1. Steady state distribution of soil water pressure head from (a)buried and (b) surface point sources.

unsaturated hydraulic conductivity in Eq. [3] was proposedby Russo (1988). The retention model was derived by usingthe exponential hydraulic conductivity model (Eq. [3]) coupledwith Mualem's (1976) expression for relating K(h) to 9 — h,

Table 1. Statistical parameters for Monte-Carlo simulation study.

MeanSTDtCorrelation^:

scale, I, mq, cm3 h-'

a

Prescribed

0.050.01

6.0500

Generatedi

0.0510.0103

5.8

y =Prescribed

0.4050.1

20.0

ln(/QGenerated

, _ ,>

0.3990.103

20.4

t STD = standard deviation.| For exponential covariance function.

1224 SOIL SCI. SOC. AM. J., VOL. 59, SEPTEMBER-OCTOBER 1995

resulting in the following S(h) relationship:

S(h) = [easa*(l - [9]where S(h) is the relative saturation defined as S(h) = (6 -6r)/(6s — 6r), 6 is soil water content, 6r and 0S are residualand saturated water contents, respectively, and u, is a parameterthat accounts for tortuosity and connectivity between pores.While |i should in general be regarded as an unknown parame-ter, for reasons of parsimony and simplicity, we followedMualem (1976) and van Genuchten (1980) in selecting a con-stant value of n = 0.5 in our subsequent applications. Themain advantage of the retention model in Eq. [9] over otherexpressions is that the estimation of soil water content is basedon previously mentioned parameters (a and Ks) only. Note,however, that 9r and 9S (or effective porosity) must be knownto convert effective saturation to volumetric water content.These additional random variables will not be discussed here,and results will be presented in terms of effective saturation.

The application of steady-state solutions to approximate thedistribution of flow attributes under drip irrigation systems isjustified by the high frequency of drip irrigation, and therelatively small volumes of wetted soil. Steady-state approxi-mations were evaluated by Bresler (1978) and Merrill et al.(1978), who have shown that they provide reasonable estimatesof the conditions for significant parts of the flow domain andduring part of the time. Rawlins and Raats (1975) pointed outthat when irrigation is very frequent, pulses of h and S aredamped out a short distance away from the source. Moreover,even when irrigations are not very frequent, steady-state condi-tions develop rapidly near the source (Merrill et al., 1978).

Under ideal conditions of homogeneous and uniform soils,the local (steady-state) distribution of h and 6 (or 5) may bedescribed completely by Eq. [6] to [9]. Such information hasbeen used for design purposes (Bresler, 1978) or may be usedfor selection of sensor locations relative to the dripper (basedon irrigation management objectives). In reality, however,even in fields with soils that are considered to be homogeneousand uniform, the distribution of h and 0 is uncertain due touncertainty in soil hydraulic properties. This uncertainty compli-cates the placement of a soil water sensor relative to the dripper.In addition, several sensors are often used concurrently to provideinformation from different locations in the field. The possibilityof different sensor readings further confounds the problem ofmonitoring and interpreting soil water status.

We developed expressions for considering effects attribut-

able to spatial variability in soil hydraulic properties indepen-dently from deterministic effects due to flow geometry (i.e.,the local distribution of h and S about a point source).

Models of Variability in Soil Hydraulic PropertiesFlow Cells and Local Coordinates

In most drip-irrigated fields, the spacing between dripperlines and between drippers along the lines is fixed. This grid-like arrangement of drippers and the symmetrical geometry offlow about each of the drippers create hydraulically independentflow cells that are isolated from one another by vertical stream-lines at their boundaries (i.e. , no-flow boundaries) . Consideringthe relatively small soil volume that is actively wetted by adripper (<1 m3), we assume that the soil hydraulic propertieswithin each of these flow cells are constant mainly becauseresolving variabilities at such a small scale in the field is notpractical or feasible by present methods.

It is difficult to determine the minimal spacing betweendrippers (dmm) for which point source approximation remainsvalid. A possible approximate criterion may be based on soilparameters (Ks and a), dripper flow rate (q), and Wooding's(1968) analysis for steady flow from a shallow and circularsurface pond. Adopting the notion of a saturated pond formingaround a dripper (i.e., ignoring the point nature of the sourcefor a moment), then clearly, when neighboring ponds merge,a dripper line may be considered as a continuous line source.Hence, the minimal dripper spacing for a point source approxi-mation to hold should be larger than the pond's saturated radius,rs (Bresler, 1978): dmm < [4/(a2n2) + ql(nKs)]m - 21 (an).Typical dripper spacing is larger than rs, and in many cases,it is in the range of 0.5 to 1 m.

At the field scale, we consider a situation where the maindirection of variability in soil hydraulic properties (a and Ks)is in the horizontal plane. The drip-irrigated field is assumedto be comprised of homogeneous flow cells (emanating fromthe drippers), each with its own soil properties (a and Ks).The definition sketch in Fig. 2 depicts a single flow cell anda plane view of the spatial distribution of various wettingpatterns (from surface point sources) affected by spatial varia-tions in soil properties.

A similar model was proposed by Dagan and Bresler (1979)for one-dimensional flow and was analyzed recently by Rubinand Or (1993). The justification for this type of model is forfields whose vertical scale of heterogeneity is small comparedwith the horizontal one. Such conditions may exist when thefield's top layer is homogeneous and the vertical extent of theflow domain (i.e., wetting) is much smaller compared withthe lateral extent of a drip-irrigated field. This analysis appliesalso to situations where the drippers are not regularly distributed(e.g., landscaping), as long as the flow cells are hydraulicallyindependent and the wetted soil volumes are relatively small.

Spatial Variability of Soil Properties and Flow AttributesThe soil spatial variability is defined through the spatial

variability of the parameters a and Y = ln(Ks). These parame-ters are expressed as random space functions, each comprisedof an expected value and a random fluctuation:

a = ma + a' <a> = /naY=mY+Y'

<a'> = 0= 0

Fig. 2. A definition sketch for the field-scale (x-y coordinates) lateraldistribution of flow cells with different wetting patterns and a singleflow cell.

where angle brackets denote the expected value operator. Al-though sufficient field data are available to justify the selectionof lognormal distribution for Ks, information on natural distri-butions of a is limited. White and Sully (1992) show evidence


suggesting a lognormal distribution for a but acknowledgethat their data is insufficient to make a clear determination.Messing and Jarvis (1993) found that the large values of a(responsible for the large tail in the distribution) may beattributed to the presence of a bimodal pore system. Nearsaturation, the large pore system dominates the flow withlarge a values, and as the soil becomes unsaturated, different(smaller) a values were obtained. In other words, inferencesof a and its distribution based on near-saturated conditionsmay be biased by the presence of exceedingly large values(White and Sully, 1992). In the absence of sufficient fielddata, we adopted a normal distribution for a in this study formathematical convenience and because the study is limited tosmall variabilities for which the moments of the normal andlognormal distributions are practically the same. Nevertheless,the results of the analysis are not strongly dependent on thetype of distribution, and if a is lognormally distributed, then thearithmetic moments of a may be substituted by the lognormalmoments: /na = exp(mao + 0.Salic) and ai = ml[exp(OOG) — 1], where mac and o^G are the expectation and varianceof the normal variable ln(a).

Two-point covariances and cross-covariances are specifiedas products of the respective fluctuations:

Ca(X,X') = «X '(*)<*'(*')>

CYa(x,x') = [H]and are assumed to be stationary, i.e., functions of the separa-tion distances k = \x — x'\ only. Here, and subsequently,x = (x,y) denotes a cartesian coordinate in the horizontal plane(field coordinate), and p(z,r) = (z1 + r2)"2 denotes a localradial coordinate about the point source (Fig. 2). Within eachflow cell (located at x,y), the source is located at p = 0 (z =0, r = 0).

Small Perturbation AnalysisThe basis for the small-perturbation analysis is the assump-

tion that the variabilities of a and Y are small. Mild soilvariability is defined by the following condition:

[12]

Substitution of Eq. [10] into Eq. [8] and expansion of theanalytical solution into a Taylor series (about wa and m r), andretaining zero- and first-order terms only, yields a first-orderapproximation for h as h(p) = <h(p)> + h'(p). The expectedvalue of h, </i(p)>, is given as

ma[13]

where <cp> is mean (p obtained from Eq. [6] or [7] dependingon the source geometry, with ma and e"1"" replacing a and Ks,respectively. The expected value </i(p)> refers to h at a fixedp (a fixed position relative to source) averaged across thehorizontal plane (i.e., the entire field). A random fluctuationh'(p) is expressed as a weighted sum of random fluctuationsin a and Y obtained by subtracting <h(p)> from the seriesapproximation of h(p):

where AY = <dh/dY> = - l//na and Aa(p) = <dh/da>. Fora buried point source, Aa(p) is given by

maz - 2ma<h(p)>1ml

[15]

where the superscript B denotes a buried source. For a surfacepoint source, /4a(p) is given as

(2 + maz)

where the superscript S denotes surface source and Ei =Ei[— ma(z + p)/2]. Note that AY is identical for surface andburied point sources.

The spatial covariance of h, C/,, is determined as the expectedvalue of the product of h' (Eq. [14]) evaluated at two points(jc, p) and (*', p'):

Aa(x,p)Aa(x',p')Ca(x;x') + Ar(x)Ar(x')Cr(xix')+ Car(x[x')[Aa(x,p)Ar(x') + AY(x)Aa(x',p')] [17]

Note that (i) G correlates h at any position relative to thedripper using the horizontally correlated properties a and Y,and (ii) for x = x' and p = p', Ch reduces to the point varianceof h, C*(0) = oi

A similar procedure applies for 5, i.e., a small-perturbationexpansion and retaining zero- and first-order terms only. Theexpected value of 5 is

<S(h)> = [e°-s««<*> (1 - 0.5ma <A>)]°-8 [18]and the spatial covariance of S is given by

D(p)D(p') imlCh(x,p;x',p') + ma [<h(x,p)>Cah(x'-x,p)+ <h(x',p')>Cha(x',p';x)]+ <h(x,p)Xh(x',p'>Ca(x;x')} [19]

with

S'Cc.p) = S(x,p) - <S(x,p)>Cah(x';x,p) = Aa(p)Ca(x';x) + AYCaY(x';x)

= -0.4inn<A(p)><5(p)>2-ma<h(p)>

h'(p) = Aa(p)a'+ AYY' [14]

Summarizing the above results, small-perturbation analyticalsolutions to flow from point sources have been developed forthe expected values and covariances of soil water pressure headand effective saturation. The moments of the flow variables (hand S) are expressed in terms of the moments of a and Ks—both are measurable soil hydraulic properties that characterizethe soil spatial variability. Once the soil variability is character-ized (in terms of a and Ks), estimates of variability in irrigationdecision variables (h and S) may be obtained and assist indesigning efficient sensor deployment.

METHODSThe analytical expressions for the moments of h and S (Eq.

[17] and [19]) were evaluated by means of a MC study. Themethodology of the MC study involves (i) generating random


fields of o and Y with prescribed statistical and spatial proper-ties, (ii) solving analytically the flow equation for the appro-priate source position (surface or buried) at each point in space,where each dripper may be characterized by different pairs ofa and Y, (iii) computing the statistics of the resulting flowvariables h and S by averaging across the entire random fieldto obtain the expected values and moments for each pointrelative to the source, and finally (iv) comparing the "experi-mental" moments with analytically predicted moments, usingthe expressions developed in this study where the only inputsare the prescribed statistics used for the generation of therandom fields.

The methodology and the random field generation are de-tailed in Rubin and Or (1993); thus, only relevant informationwill be repeated here. The soil properties a and Y were modeledas first-order autoregressive (AR[1J) processes (Box and Jen-kins, 1976; Sharp, 1982) with specified means m*, varianceso,?, and integral scales // with / = 1,2 referring to a and Y,respectively. The univariate AR(1) model for a stochasticprocess / (/ = 1,2) is given by

k,i = a\,idk-\,i [20]where a\j is the autoregression parameter, e*,, is a zero-mean,normally distributed random fluctuation, and dkj (k = ...—2, —1,0,1,2,..) is a sequence of equally spaced data withthe Markov property for the conditional probabilities, i.e.,Prob(rfk|<&-i,<ii-2,...) = ¥ro\)(dk\dk-\). The covariance of the/th AR(1) process is given by

The relation between the autoregression parameter (a\j) andthe integral scale (/,) of an exponential covariance is simply

-1 [22]

Each random field was comprised of 10000 elements; thisfield size provided consistent and stable results in agreementwith the prescribed statistics (Rubin and Or, 1993). A compari-son between prescribed and generated autocorrelation functionsof a and Y is depicted in Fig. 3 (this realization was used forsubsequent MC comparisons). To keep the present analysessimple, we considered a zero cross-correlation between aand Y in all cases. However, Eq. [17] and [19] account for

Autocorrelation - Realization #2la =6 m; IY=20 m; Exponential Model

0 5 10 15 20 25 30 35 40___ Lag(m)_____|O Gen-a~ a Gen-Y —Theoretical |

Fig. 3. A comparison of theoretical (prescribed) and generated auto-correlation functions of exponential slope (a) and log saturatedconductivity (Y; I = integral pairs).

cross-correlation between a and Y, which is likely to occurin natural soils (i.e., a sandy soil is likely to be characterizedby both large Ks and large a). A summary of the parametersused for the simulations is given in Table 1.

Finally, MC simulations were performed to identify somepractical limits of applicability of proposed analytical expres-sions. This was done by gradually increasing the variances ofa and Ks (or their CVs) and comparing MC results withanalytical predictions.

RESULTS AND DISCUSSIONComparisons of Mean and Variance Profiles

of h and 5In this section, we show comparisons between analyti-

cal predictions and MC results of mean flow attributesand their variances sampled along two vertical profilesat radial distances of 0 and 30 cm from the source. Thesource strength in all comparisons was q = 500 cm3

h~', and the mean soil properties were a = 0.05 cm"1

and Ks = 1.5 cm IT1 (Table 1).The assumption of stationarity in the lateral direction

(x-y coordinates) of a and Y makes the variances of hand 5 functions of the p coordinate only. The variancesare given by Eq. [17] and [19] for x = x' and p = p',as O2(p) = C(x,p;x,p). The variations with depth of themean and standard deviation of h and 5 (i.e., <h[p]>,0h[p], <S[p]>, and Os[p]) at two radial distances froma point source are depicted in Fig. 4 and 5, for subsurfaceand surface sources, respectively.

The agreement between the MC results (symbols) andthe analytically predicted variances (lines) for the twosource positions and the two flow attributes is very good.Figure 4 depicts the distribution of the moments of hand 5 about a point source that is buried at a depth of40 cm below the soil surface. The variances vanish atthe source (i.e., o/,-*0 and Os~*0, as h~*Q and S-*!) forboth sources (Fig. 4 and 5), as expected under saturatedconditions.

The vertical distribution of the soil water pressurehead along r = 0 about a buried source (Fig. 4) isnot symmetrical, and <h> decreases (becomes morenegative) more rapidly in the upward direction than inthe downward direction due to the effect of gravity.Mean effective saturation, <S>, follows the generaltrends shown by <h>.

The vertical distribution of the standard deviation ofh, OA(P) in Fig. 4a and 5a reflects the dependency ofCA(P) on </z(p)>, which can be verified theoretically byinspection of Eq. [15] and [16]. Experimental evidenceof a consistent dependency of o/,(p) on <h> was reportedby Hendrickx and Wierenga (1990) based on data col-lected in (surface) drip-irrigated fields.

The behavior of ag(p) in Fig. 4b and 5b is quitedifferent from the monotonous and well-behaved shapesof <5> or o/,(p). The standard deviation of 5 along r =0 increases rapidly at the vicinity of the source (in anasymmetric fashion for the buried source in Fig. 4b), andthen decreases gradually to lower values (less variability)away from the source. This peculiar behavior is a result

OR: SOIL WATER MONITORING STOCHASTIC ANALYSIS FOR DRIP IRRIGATION MANAGEMENT

-40-

1227

a (cm)h

0 -20 -40 -60 -80 -100 -120

<h> (cm)

-20-

oIM*

£"5.<D

——— T - (0 cm)- - - T - (30 cm)

• MC - (0 cm)• MC - (30 cm)

0.05 0.04 0.03 0.02 0.01 0.00 0.2

a . ( - )0.6 0.8 1.0

( - )

Fig. 4. A comparison between analytical predictions (lines) and Monte Carlo (MC) simulations (symbols) of the vertical distributions of (a) meansoil water pressure head (<h>) and standard deviation of soil water pressure head, (04), and (b) mean relative saturation (<S>) and standarddeviation of relative saturation (as), at radial distances of 0 and 30 cm from a buried point source (T = theoretical prediction; STD = standarddeviation).

of the highly nonlinear relationships between h and 5.A similar phenomenon was observed for one-dimensionalunsaturated flow to a water table by Rubin and Or (1993),and the following explanation is proposed. The variabilityof S (as inferred from h using Eq. [9]) is influenced bytwo factors: (i) by the value of h, and (ii) by the magnitudeof OH- The largest variations in S due to h are expectedat h values where the slope dS/d/z is largest, which was

found to be at h = h" as given by (Rubin and Or, 1993):

A*=-|m_^] =-— [23]cr a

The second factor determining as, is the magnitude ofah\k=h*. When h = h*, even small variations in h aremapped (or "stretched") by the retention curve to inducelarge variations in 5. For example, h" in Fig. 4 and 5


- - - - T - (30 cm)• MC - (0 cm)• MC - (30 cm)

10025 20

100

15

a (cm)h

-20 -40 -60 -80 -100 -120

<h> (cm)

• - T - (30 cm)• MC - (0 cm)• MC - (30 cm)

0.05 0.8 1.0

Fig. 5. A comparison between analytical predictions (lines) and Monte Carlo (MC) simulations (symbols) of the vertical distributions of (a) meansoil water pressure head (<h>} and standard deviation of h (a*) and (b) mean relative saturation (<5>) and standard deviation of S (<r.s) fora surface point source (T = theoretical distribution; STD = standard deviation).

is about —45 cm, and the largest values of as occur atthe approximate depths where <h> = —45 cm (e.g.,compare the positions of <h> — —45 cm in Fig. 4awith the positions of maximum as in Fig. 4b).

Limits of the Small-perturbation ApproximationThe crucial assumption in the analysis is that of mild

soil variability. We wish to identify some practical limitsfor the applicability of the analytical approximations by

comparisons with a range of variances of a and Y. Themean values of a and Y and the flow rate q used in allcomparisons are shown in Table 1. The range of variabil-ity was obtained by varying the CV of soil parameters(e.g., CVa = Oa/Wa, and CVKi = aY). Figure 6 depictsa comparison of profiles of <h> and o/, at two radialdistances from a buried source for two values of CV(the same value of CV was used for both, a and Y).From the results in Fig. 6, it is evident that the analytical

OR. SOIL WATER MONITORING STOCHASTIC ANALYSIS FOR DRIP IRRIGATION MANAGEMENT 1229

T - (0 cm)T - (30 cm)MC - (0 cm)MC - (30 cm)

10

cr (cm)h

-20 -40 -60 -80 -100 -120

<h> (cm)Fig. 6. A comparison between analytical predictions (lines) and Monte Carlo (MC) simulations (symbols) of the vertical distributions of mean

soil water pressure head (<A>) and standard deviation of h (ah) at radial distances of 0 and 30 cm from a buried point source with (a) CV =0.3, and CV = 0.4 (T = theoretical prediction; STD = standard deviation).

approximation applies for variabilities up to CV = 0.3(Fig. 6a), and increasing the variability to CV = 0.4resulted in large discrepancies between the analyticalapproximation and MC results (representing simulatedfield variability). Additional simulation results (notshown) showed that the extent of discrepancy increaseswith the increase in variability and that the analyticalapproximation is more sensitive to variability in a thanto variability in Y (Rubin and Or, 1993). Based on thelimited tests, we conclude that the analytical approxima-tions provide reasonable predictions for CVs smallerthan 0.3. A few examples of reported variabilities forboth a and Xs are given in Table 2. The data indicatethat the approximations may be applicable only for the

data of Jensen and Mantoglou (1992). However, thelimited data on the variabilities of a and Ks in relativelyhomogeneous agricultural fields make it difficult to assessthe applicability of the proposed approach at present.

Spatial Covariances of h and SFor purposes of spatial interpolation and identification

of areas within an irrigated field with different wettingpatterns (i.e., different distributions of h and 5), thespatial covariance functions of h and 5 must be specified.These covariance functions are dependent on the spatialvariability of the soil properties (a and Y), as can beseen in Eq. [ 17] and [19]. Examples of the autocorrelation


Table 2. Ranges of a and Ks and their field variability in different soils.

SourceSoil type

(# of measurements) Layer <a> a.„„-!

Smettem and Clothier(1989)t

White and

Jensen and(1992)5

Sully (1992)

Mantoglou

Manawatu fine sandyloam (n = 9)

Bungendore loamysand (n = 40)

Jyndevad coarse sand(n = 24/layer)

SurfaceSubsurface

Top

MediumBottom

0.084 0.040.161

1.14

0.2540.2840.02

0.195

0.6

0.0490.1290.025

cvt

0.471.20.56

0.19§

0.461.24

<K,>

h-'0.220.322.21

8.29

11.5998.94

0

0.0.0.

4.

12.55.

KS

13896

,54

15,8

CV

0.450.930.41

0.55

1.050.56

t CV = coefficient of variation.t Original data reported sorptivity (S) which was converted to a; aa was estimated from measured a, and <TKs by first-order analysis.§ A typical value for the entire range or pressure heads measured in the field.1 Data were obtained from three soil depths (layers) and four different soil water pressure heads (ft) values, data in table are for ft = - 1 cm.

functions C/,(/t)/C*(0) and Cs(fc)/Cs(0) (denoted for brev-ity as CH and C) for subsurface and surface sources aredepicted in Fig. 7a to 7d (symbols mark MC results andlines mark theoretical predictions). The autocorrelationfunctions of a and Fare also shown in Fig. 7 (parametersare given in Table 1), denoted as Ca and CY. A sketchshowing the positions for which the autocorrelation func-tions were evaluated (and the positions of the pointsources) is given at the bottom of Fig. 7.

The agreement between MC simulation results andtheoretical predictions in Fig. 7 is very good; in mostcases, the prediction lines are indistinguishable from thesymbols. Note the close resemblance between Ch andCa for all the positions depicted in Fig. 7. The reasonfor the dominance of the integral scale of Ca (/„) on thespatial structure of CH (or the integral scale Ih) stemsfrom the large differences in the magnitudes of Aa andAy-the weight functions for computing CH in Eq. [17].It can be shown that in most cases, Aa»Ay. Thus, theweight given to /„ in the computation of Ch (Eq. [17])is much larger than that given to IY.

The spatial structure of Cs is more complicated thanthat of CH, as can be seen in Fig. 7. This is a result ofcompeting influences of the two different integral scalesof Ca and CY (Ia and IY) on the integral scale of Cs (Is)-To illustrate the interplay of the integral scales, wesimplify Cs (Eq. [19]) and neglect some indexing nota-tion, to obtain

Cs = D2 + Ca (Aama + <h»2] [24]From Eq. [24], it is clear that when Aaa = -<h>, thestructure of Cs is determined exclusively by the integralscale of Y, Iy. The analysis can be carried a step furtherby expanding and simplifying the terms in A%a = —<h>for a buried source. The result is an expression thatdescribes the region about a buried point source whereIY completely dominates Is:

[25]

The derivation of a similar expression for a surfacesource is somewhat more complicated.

Summarizing, the MC simulation results show thatthe spatial covariances of h and 5 are accurately predictedby the proposed analytical expressions (Eq. L17] and[19]) for CVs of 0.2 and 0.25 for a and ln(ATs), respec-

tively. The integral scale of h is dominated by that ofa, which characterizes the unsaturated behavior of theporous medium. While /« dominates the structure of Csin most of the flow domain, there are regions where thespatial structure of Cy (i.e., Iy) takes over and controls/s. The usefulness of Ch and Cs lies in the need forspatial interpolation of sparse information to estimatethe conditions in unmeasured portions of the field (e.g.,using geostatistical methods such as kriging).

Applications for Sensor Placementand Interpretation

Soil Water Pressure HeadWhen soil water pressure head is used as the decision

variable for drip irrigation scheduling, the behavior ofmean h and its variance should be considered at twoscales of observation: (i) the local, or the dripper scale,and (ii) the field scale. At the dripper scale, it may bepossible to take advantage of the dependency of o/, on</z> in establishing thresholds and prospective measure-ment locations. A sensible selection criterion for sensordeployment at the local scale should consider zones withlow OH and high <h> (less negative). The potentialbenefits are not only in the reduction of measurementuncertainty, but also the attainment of steady-state flowconditions is likely to occur earlier during an irrigationevent.

An important deciding factor for sensor placementis a particular threshold value of h. Some irrigationscheduling schemes are designed to maintain a targetrange of soil water pressure heads in the wetted soilvolume by using a prescribed threshold value of h at acertain location (or locations) to turn irrigation on oroff. The definition of a proper threshold requires agro-nomic and economic considerations that are beyond thescope of this study. Due to the complexity of a properdefinition of a threshold, most available recommenda-tions are empirical (Taylor and Ashcroft, 1972; Caryand Fisher, 1983; Phene and Howell, 1984). This studyprovides means for explicit consideration of informa-tional uncertainty as part of threshold and sensor place-ment selection.

A main concern at the field scale is determining theminimum number of soil water pressure head sensorsneeded to obtain an estimate of h for the field with a


Buried - ACF of h and S @ r=0 cmDCh (z=-40)OCh (z=5)VCs(z=-40)XCs(2=5)

-0.240

Surface - ACF of h and S @ r=0 cm

-0.2

Buried - ACF of h and S @ r=30 cm Surface - ACF of h and S @ r=30 cm

-40 -30 -20-10 0 10 20 30 40

40

SO-40 -30 -20 -10 0 10 20 30 40

Fig. 7. Theoretical predictions (lines) and Monte Carlo (MC) simulations (symbols) of the autocorrelation functions of soil water pressure head(h) and relative saturation (5): (a and b) buried source, at two depths (5 and - 40 cm) and two radial distances of (a) 0 and (b) 30 cm; and(c and d) surface source, two depths (5 and 50 cm) and two radial distances of (c) 0 and (d) 30 cm from the source. A sketch with theevaluation positions appears beneath (b) (subsurface) and (d) (surface) (C = spatial covariance).

prescribed estimation error. This problem was studiedby Hendrickx and Wierenga (1990), who determinedthe relationships between h and ch experimentally. Wepropose to use the analytically derived estimates of a/,(at a particular position relative to the dripper) to deter-mine the number of tensiometers (or other pressure headsensors) assuming independent sampling locations. Theassumption of spatial independence is justified because,in practice, only a few sensors are affordable, and thesefew sensors are likely to be distributed at separationdistances that are larger than the integral scale of h, Ih.The number of sensors (n) needed to estimate h to be

within ±d units of the mean <h> for (1 - a)100% ofthe time may be determined by (Warrick and Nielsen,1980)

n =d2 [26]

where Za/2 is the normalized deviate (i.e., Zai2 = (h —<h>)/Oh), and 1 — a is the cumulative probability suchthat Prob(Za/2 < Z) = 1 - a (for a = 0.05, Za/2 =1.96 » 2). Given an acceptable error (or confidenceinterval) and a sensor location, the number of sensorsn may be determined as follows: (i) use Eq. [17] to


estimate ah based on mean soil properties and theirvariability and source placement and strength; and (ii)use Eq. [26] with oh to estimate n.

All estimates of uncertainty in this study should beconsidered nonconservative, since they are based on thefollowing assumptions: (i) steady state flow conditions,where transient conditions prevail for long periods andare likely to introduce larger variability; and (ii) smallvariability, and thus the use of first-order analysis, ne-glecting higher order terms.

Soil Water Content (Saturation) MeasurementsRepeated and nondestructive soil water monitoring in

drip-irrigated fields was limited in the past by constraintsassociated with the neutron probe, i.e., the exclusion ofnear soil surface measurements, and averaging of largesoil volumes influenced by the measurement. Recently,with the introduction of the TDR technique (Topp et al.,1980), the prospects for repeated soil water contentmonitoring in well-defined soil volumes seem promising.The main advantage of soil water content monitoring isthat the information may be readily used to estimateamounts of irrigation water needed.

The nonlinear relationships between S and h inducean irregular spatial distribution of 05. This means thatsmall changes in soil texture between dripper locations(i.e., relatively sandy vs. clayey spots) may result inrelatively large differences in water content at a certainrange of pressure heads (comparisons are among differentdripper locations but at a fixed position relative to thedripper). Figure 8 depicts the spatial distribution of asabout buried and surface point sources in a soil character-ized by the parameters in Table 1. The area with thelargest 05 coincides (partially) with the equipotential ofh' (Eq. [23]) denoted as the hatched area in Fig. 8a (fora buried source). The cross-hatched area in Fig. 8aencompassing h" ± average ah (i.e., -45 + 15 cm) isgiven for qualitative evaluation only, because the spatialdistribution of 05 is also affected ("stretched") by theshape of Oh (compare with Fig. 4). Note also the existenceof a region above and to the side of the sources (darkarea), where the variability in S is minimal. In general,the area under the dripper appears to be the most suscepti-ble to spatial errors in soil water content measurements(i.e., water contents vary from the wettest region passingthrough h* on the retention curve). To avoid the regionswith high 05, we propose to first identify the positionof h* equipotential about the dripper and then excludethis region from consideration for soil water contentsensor placement (e.g., TDR). The low os region (dark)seems ideal for soil water monitoring; it is easily accessi-ble and sufficiently close to the source (to attain steadystate). Its exact position, however, should be determinedfor each combination of soil properties, source place-ment, and source strength.

Most soil water measurements are not point measure-ments but rather involve some spatial averaging (e.g.,linear averaging using TDR probes or volume averagingusing a neutron probe). Such spatial averaging mayintroduce uncertainty into the measurement. For exam-

Radial Distance (cm)

STD S

! 0.030

j 0.025

I 0.020

50 40 30 20 10 0 -10 -20 -30 -40 -50 H|0.oi5

JO.010

10.005

' 0.000

10Fig. 8. Distribution of standard deviation of relative saturation, 5,

(<TS) about (a) buried point source, and (b) surface point sourcefor a soil with parameters as in Table 1. The cross-hatched areain (a) encompasses h' ± a/, = —45 + 15 cm (a/, = standarddeviation of soil water pressure head, //).

pie, considering TDR probes, the soil water content ismeasured along a linear distance determined by the lengthof the TDR probes, a path that may combine watercontents of various levels of uncertainty (Fig. 8). Onecan use the analytical expressions developed in this studyto account for this additional source of uncertainty byintegration of the averaging path considering its positionand orientation relative to the source and using informa-tion on mean soil water content and its moments, whichcan be derived analytically.

SUMMARY AND CONCLUSIONSAnalytical solutions to steady-state flow from surface

and buried point sources were expanded to obtain analyti-cal expressions for the first two moments of soil watercontent and pressure head. These moments are expressed


in terms of the spatial moments of two soil propertiescharacterizing the soil variability, Ks and a.

The regular spacing between drippers in irrigated fieldsand the axial flow symmetry about a dripper createhydraulically independent flow cells. While the soil prop-erties within each flow cell were assumed uniform, thefield spatial variability was expressed by variable Ks anda among flow cells. The extent of variability was assumedto be small to enable the application of a first-orderperturbation expansion.

The new analytical expressions provided predictionsof the moments of h and S, which were in good agreementwith MC simulation results (for CVs < 0.3). The nonlin-ear relationships between h and S result in a localizedincrease in 05 in regions where dS/d/z is largest. Whilethe spatial structure of h was dominated by that of a(70), the spatial structure of S was a result of competingelfects between the integral scales of a and of Y =ln(*s).

The spatial covariance functions are needed for inter-polation of h and S to unmeasured locations in the field.Most practical applications, however, require informa-tion on the point variance of the attribute of interest (hor 5). The new analytical expressions provide a meansfor considering various aspects of sensor placement rela-tive to the dripper to reduce the level of uncertainty dueto spatial variations in soil properties.

Finally, though the theoretical approach adopted hereis based on many simplifying assumptions, it providesa first approximation for rational selection of sensorplacement and interpretation in heterogeneous soils. Thisshould be viewed as a first step toward unveiling thecomplex interaction between soil heterogeneity and flowattributes, while preserving h-S nonlinearity and multidi-mensional flow geometry. The ultimate test of the pro-posed approach is in the field, and a series of such testsare now underway.

ACKNOWLEDGMENTSThis research was supported in part by Grant IS-2131-92RC

from the U.S.-Israel Binational Research and DevelopmentFund (BARD). The partial support of the Utah AgriculturalExperimental Station (UAES) is gratefully acknowledged. Ap-proved as UAES Journal Paper 4709.

stochastic analysis of soil water monitoring for drip irrigation management in heterogeneous soils

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