uncertainty in estimation of soil hydraulic parameters by inverse modeling: example lysimeter...

Uncertainty in Estimation of Soil Hydraulic Parameters by Inverse Modeling:Example Lysimeter Experiments

Karim C. Abbaspour, Marion A. Sonnleitner,* and Rainer Schulin

ABSTRACTAn increasingly attractive alternative to the direct measurement

of soil hydraulic properties is the use of inverse procedures. We investi-gated the consequences of using different variables or combinationsof variables from among pressure head, water content, and cumulativeoutflow on the estimation of hydraulic parameters by inverse model-ing. We also looked at a new multiplicative formulation of the objec-tive function which does not require weights for different variables.The inverse study combined a global optimization procedure, Sequen-tial Uncertainty Fitting (SUFI), with a numerical solution of the one-dimensional variably saturated flow equation. We analyzed multistepdrainage experiments with controlled boundary conditions on twolarge lysimeters. Estimated hydraulic parameters based on differentobjective functions were all different from each other; however, asignificance test of simulation results based on these parameters re-vealed that most of the parameter sets produced simulation resultsthat were statistically the same. Notwithstanding the significance test,ranking of the performances of the fitted parameters on the basis ofthe mean square error (MSE) statistic revealed that they were highlyconditional with respect to the variables and the mathematical formu-lation of the objective function. To obtain statistically unconditionalsets of parameters, we introduce and discuss the concept of "parameterconditioning" instead of "parameter fitting". Parameter conditioningidentifies a parameter domain such that when propagated in a stochas-tic simulation, all or most of the measured points of a variable arewithin the 95% confidence interval of the Bayesian distribution ofthat variable.

DETERMINATION OF SOIL HYDRAULIC PROPERTIES is animportant step in studies of soil water behavior

under drainage and soil infiltration conditions. Success-ful applications of sophisticated numerical modelslargely depend on knowledge of the soil water retentionand flow curves. Although there exist many laboratoryand field methods for the estimation of the unsaturatedsoil hydraulic parameters (Green et al., 1986; Klute andDirksen, 1986; Klute, 1986), most methods still remainrelatively tedious, expensive, time-consuming, and mayinclude large errors for most practical applications.

An increasingly attractive alternative to the directmeasurement of soil hydraulic properties is the use ofan inverse procedure. In this procedure, measurementsof primary model variables such as water content, pres-sure head, discharge, or concentration are used to obtainmodel inputs such as parameters of the retention curveand saturated hydraulic conductivities. Inverse estimationtechniques usually involve the coupling of a numericalmodel of flow with a parameter optimization algorithmsuch as the Levenberg-Marquardt method (Marquardt,1963). For one-step outflow experiments, Kool et al.(1985) and Kool and Parker (1987) presented computer

Swiss Federal Institute of Technology, Dep. of Soil Protection, Gra-benstrasse 3, 8952 Schlieren, Switzerland. Received 25 Mar. 1998.*Corresponding author ([email protected]).

Published in Soil Sci. Soc. Am. J. 63:501-509 (1999).

models applicable to laboratory columns. To overcomethe non-uniqueness problem, multistep approacheswere introduced by van Dam et al. (1994) and Echingand Hopmans (1993). Also, the need for additional wa-ter content data at very low values of the pressure headwas reported by van Dam et al. (1992) and Bohne etal. (1993), while the need for more pressure head mea-surements within the soil profile was suggested by Toor-man et al. (1992) and Eching and Hopmans (1993).

Santini et al. (1995) used evaporation experiments(Gardner and Miklich, 1962; Wind, 1968; Wendroth etal., 1993) for inverse parameter estimation, while Simu-nek and van Genuchten (1996) coupled inverse parame-ter estimation and tension disk infiltrometer data. Mostof the above studies were conducted on laboratory soilcolumns or with numerically generated data.

In our study, we used measured water content, pres-sure head, and cumulative discharge from two largelysimeters to obtain the soil hydraulic parameters bythe SUFI program of Abbaspour et al. (1997), which isbriefly discussed in the next section. In performing in-verse modeling, an important question is how much andwhat type of fitting data should be used, and how well-defined an optimized parameter set can possibly be.From an experimental point of view, this informationis very important because some variables are more diffi-cult to measure, depending on the experiment, thanothers. For example, in laboratory soil columns andlysimeters cumulative discharge data are much easierto collect than pressure head or water content data.Tensiometers and time domain reflectometry (TDK)probes are expensive and time-consuming to install andmonitor; however, in a plot study or a larger scale fieldstudy, collection of discharge may be a very difficult orimpossible task and it may be much easier to measurepressure head or water content data.

We investigated the consequences of using differentvariables or combinations of variables from among pres-sure head, water content, and cumulative outflow onthe parameter estimation. We also looked at a newmultiplicative formulation of the objective function thatdoes not require weights for different variables. Further-more, we introduce the concept of "parameter condi-tioning" as opposed to "parameter fitting" and arguethat parameter conditioning is more suitable for soilhydrological analysis.

THEORYSequential Uncertainty Fitting (SUFI) Inverse

Simulation ProgramSome important features of the SUFI program are that,

unlike least square optimization algorithms using steepest de-

Abbreviations: ESS, exhaustive stratified sampling; MSE, meansquare error; RSS, random stratified sampling; SUFI, sequential un-certainty fitting; TDK, time domain reflectometry.

501

502 SOIL SCI. SOC. AM. J., VOL. 63, MAY-JUNE 1999

scent, it does not suffer from stability or convergence problemsbecause it is always performed in a direct mode. Also, SUFIcontains safeguards to prevent falling into local minima (Ab-baspour et al., 1997). This feature does not often exist in leastsquare minimization algorithms.

The generic global optimization program SUFI can be de-scribed as follows. Let g(x): D —> R be a continuous andbounded function, and x be an n-dimensional state vector.Also, let g(a) be the objective function, R be real numbers,and D be the search space x. The objective is to find the statevector x that minimizes g(x). Without loss of generality, wecan take D as the hyperparallelepiped D = {x,\^~ < *, <&1"; i = 1, 2, . . . , n\ where £,~ and £*" denote, respectively,

the lower and upper bounds of the state variable xt. The upperand lower bounds are based on prior information.

The optimization problem proceeds by discretization of thesearch space D, by dividing the interval [£f, &] of each param-eter AC,- into a number, say mh of strata. If each component ofthe state vector x can take on a discrete set of values *,•(/,•),where jt = 1, 2,. . ., m,, clearly there would be M = WiW2 • • •mn permutations. Elements of the space P = {z\, . . ., ZM]constitute the discretized space of input parameters whereZi = (xi(i), x2(l), • • •, *«(!)} and ^(1) is the first moment ofthe first stratum of the first variable.

The next step in optimization consists of sampling from thespace of the input parameters, P. The SUFI program offers twopossibilities for sampling the parameter space P, exhaustivestratified sampling (ESS), which includes all possible combina-tions of parameters, and random stratified sampling (RSS),where RSS is a randomly selected subset of ESS. The choiceof a sampling strategy depends on the speed of simulationprograms, and the size of M.

An objective function is usually expressed as some measureof the difference between measured and simulated variables.If ESS sampling was selected, then the optimization wouldproceed by calculating the objective function for all the ele-ments of P and the results would be used to calculate theBayesian distribution of the objective function (Benjamin andCornell, 1970). In RSS sampling, the Bayesian distributionwould be based on a random subsample of P.

The aim of SUFI is then to minimize the objective function,or more precisely in our case, to reduce the uncertainty in theBayesian distribution of the objective function by a searchalgorithm described in detail by Abbaspour et al. (1997). Inshort, the simulation program is executed for each possible

3gt/5

Time, or SpaceFig. 1. A conceptualization of "parameter conditioning" vs. "parame-

ter fitting". Regions depicted by Curves a, b, and c show differentdegrees of conditioning on the measured data. Curve d is a "fitted",or "highly conditioned" curve. Symbols represent measured valuesof the variable.

combination of the parameters in the space of P, and subse-quently, the parameter strata are scored on the basis of thesmallness of the objective function. After scoring, the parame-ters are updated by eliminating the strata with the largestscores. The above steps are repeated again with new parameterdomains until the global minimum is reached and no furtherimprovements can be made to the objective function.

Conditional vs. Fitted ParametersIn a demonstration example in Fig. 1, the reduction in the

95% confidence interval of the Bayesian distribution of anarbitrary variable is illustrated for a few iterations. Initially,based on a prior uncertainty domain of the parameters, the95% confidence interval is very large (Curves a in Fig. 1)indicating large parameter uncertainties. As iterations pro-ceed, the 95% confidence interval decreases as depicted byCurves b, and c, and the parameters become more and moreconditioned on the measurements of the variable being used.In SUFI, we can continue the iterations until the upper andthe lower limits of the 95% confidence interval coincide witha single curve, Curve d. This condition is generally referredto as "fitting" and Curve d is the "fitted curve", or in our caseit could also be referred to as a "highly conditioned" curve.The fitted curve is produced by a set of single-valued parame-ters, but, as is also evident in Fig. 1, fitted curves often missmost of the measurements. We believe that a simulationshould be able to reproduce measured data points at the mea-surement times or locations, a situation referred to as "respect-ing" or "honoring" the measurements in a geostatistical con-cept (Deutsch and Journel, 1992). Furthermore, a simulationthat honors the measurements is referred to as a "conditional"simulation, which in environmental studies has tremendousadvantages over an "unconditional" simulation whenever aMonte Carlo type algorithm is invoked.

Borrowing the phrase "conditional" from geostatistics, wemaintain that conditional estimation of parameters, ratherthan fitted estimates, is more realistic for environmental stud-ies. However, unlike a geostatistical estimation, in parameterestimation it would be impossible to obtain a set of parametersthat would produce a simulation honoring every measuredpoint at the location and time of the measurement, simplybecause of measurement errors, heterogeneity, and inade-quate models. So, in what we refer to as parameter condition-ing, the objective is to obtain parameter distributions such that,when propagated stochastically through a simulation program,the 95 % confidence interval of the Bayesian distribution of thesimulated variable would contain all, or most of the measuredpoints. In Fig. 1 this is shown by the shaded region of Curves c.

MATERIALS AND METHODSLysimeter Experiment

We studied two representative lysimeters, each with a dif-ferent soil type, in order to determine soil hydraulic parame-ters for an experimental facility consisting of 36 identical lysim-eters filled with those two soil types. Each lysimeter had anarea of 3 m2 and was 1.5 m deep. The bottom 50 cm consistedof a drainage package of sand and fine gravel. In the firstlysimeter, the 1-m soil profile consisted of a single layer ofcalcareous loamy sand, and in the second one, a two-layer soilsystem consisting of a 40-cm layer of humus-rich acidic sandyloam (top soil material) on top of a 60-cm layer of morecompacted acidic sandy loam (subsoil material). Both soilsoriginated from typical Swiss forest sites. All lysimeters withthe same soil type were filled in the same way with the homoge-nized soil material and were planted with cloned beech (Fagus

ABBASPOUR ET AL.: UNCERTAINTY IN PARAMETERS OBTAINED BY INVERSE MODELING 503

sylvatlca L.) and spruce [Picea abies (L.) Karsten] saplings.Lysimeters were allowed to settle for 2 yr before the start ofthe drainage experiment.

The drainage experiment was conducted in March whenevaporation was negligible but the soil was no longer frozen.To control the water input into the lysimeters a tent was builtto protect them from rain. The lysimeters were initially slowlysaturated from below for a period of 16 d until the water tableappeared at the soil surface. Water table height could be readin the transparent tubes installed against a meter stick.

Lowering of the water table was conducted in steps of 5 cm.The excess water after each lowering was pumped from thetubes until no further drainage could be observed. Duringeach step of the lowering, drainage rate was measured as afunction of time, and water content and water potential weremeasured at the end. Pressure head was measured with verti-cally installed tensiometers at the 25-, 50-, and 75-cm depthswith three instruments at each depth. The average of the threetensiometers was used to'represent the pressure head at eachdepth. Water content was measured with 25-cm-long TDRprobes. Probes were vertically installed, also in triplicates, withtheir centers at depths of 25, 50, and 75 cm, and horizontallyinstalled at the 25-cm depth.

Laboratory ExperimentSoil cores of 5-cm diameter were removed to a depth of

95 cm from 16 other lysimeters of the experimental facilitycontaining the same soils as the ones used in the drainageexperiments. In total we had eight cores from each of the twosoil types used in the drainage experiment. The sampling wasperformed using a metal cylinder of 20 cm in length and 5 cmin diameter that was screwed onto a metal stick. This devicewas pushed into the soil in steps of 16 cm down to the finaldepth. To minimize disturbance, only the middle 5 cm of thecores were used for analysis. From the loamy sand we had 22samples, from the top of the sandy loam we had 5 samples,and from the bottom of the sandy loam we had 18 samples,which were still intact after preparation.

Saturated hydraulic conductivity was measured first on allof the samples using a constant head method as described byKlute and Dirksen (1986). The drainage branch of the waterretention curve was measured afterwards on the same sampleson a sand and clay bed with a hanging water column in stepsfrom 1 to 345 cm. A pressure plate apparatus was used athigher suctions to drain soil samples of 2.5-cm height. Thefollowing suction steps were performed with all the samples:1, 5, 10, 20, 50, 100, 345, 2000, and 15 000 cm of water. TheRETC program (van Genuchten et al., 1991) was used to fita retention curve to the measured data as shown in Fig. 2.

Governing Flow EquationsWe assumed a one-dimensional isothermal Darcian flow in

a variably saturated rigid porous medium, expressed by thefollowing form of the Richards equation:

[i]where 9 is the volumetric water content (cm3 cm""3), h is thepressure head (cm), K is the hydraulic conductivity (cm h"1),z is a vertical coordinate (cm) positive downward, and t is time(h). Initial and boundary conditions applied to the drainageexperiments were as follows:

o.eo -i ***** Loamy sandxxxxx Top layer sandy loam•kirtrtrtt Bottom layer sandy loam

o.io10 100 1000 10000Pressure head (cm)

Fig. 2. Retention curves based on laboratory measurements (sym-bols) and fitted curves (lines) calculated with RETC.

- 1 = 0

h(z,t) = z = L

[3]

[4]where h{ is the initial pressure head (cm), L is the coordinateof the bottom of the soil, and H0 is the pressure head (cm)imposed at the bottom of the soil. Equation [1], subject to theabove initial and boundary conditions, was solved numericallyusing the HYDRUS5 code (Vogel et al., 1996). To reconcilethe simulated water contents with the TDR probe measure-ments, we averaged the simulations over the 25-cm depth ofthe probes.

The unsaturated soil hydraulic properties in this paper wereassumed to be described by the following equations (van Gen-uchten, 1980):

es - er= es

K(9) = K&5 [1

h < 0 [5]

h > 0 [6]]2 [7]

h(z,t) = t = 0 [2]

where n and a are the van Genuchten parameters, Se is theeffective water content, m = 1 — 1/n, Or and 9S are the residualand saturated water contents (cm3 cm"3), respectively, and Ksis the saturated hydraulic conductivity (cm h"1).

Inverse ProcedureThe hydraulic characteristics defined in Eq. [5] to [7] contain

five unknown parameters, 9r, 6S, a, n, and Ks. We treated thevalues of the saturated water contents as known and set themequal to the values determined after saturating the lysimeters,and estimated the other four parameters.

All four parameters 6r, a, n, and Ks were assumed to beindependent and uniformly distributed. On the basis of ourexperience, prior estimates of uncertainty about each parame-ter were expressed as follows. For loamy sand, 9r = [0.0, 0.15],K, = [10.0,100.0], a = [0.01, 0.1], and n = [2, 4], and for bothlayers of sandy loam, 9r = [0.05, 0.15], Ks = [0.1, 50], a = [0.1,0.7], and n = [1.1,1.5]. For the second lysimeter (two-layeredsandy loam) we used two different inverse approaches. In thefirst approach, all eight parameters were estimated simultane-ously. In the second approach, the parameters for the top soilwere estimated first on the basis of the water content at the


25-cm depth alone, while the pressure head measurements atthe 25-cm depth were used as the bottom boundary condition;then, these parameters were treated as known and those forthe bottom soil were estimated on the basis of the pressurehead and water content of the bottom layer. Because estima-tion of too many parameters becomes computationally moredemanding and even impossible, the second approach, if suc-cessful, would make parameter estimation of multilayered soilprofiles more feasible.

Two different types of objective functions were used forminimization as follows:

0.60 i

2;=i ",-2 27=1 k=\

= n 2[8]

[9]1=1 7=1 t=l

where jt"1 is a measured variable, x? is a simulated variable, w,is the weight for variable i, tt is the number of measurementsover time for the variable i, s, is the number of measurementsover space for the variable i, and n is the number of variablesused in the objective function. For determining the weightswt for each variable, different methods are presented in theliterature, we used the following equations for xl (cumulativedischarge, q), x2 (pressure head, h), and xz (water content, 9)after Kool et al. (1987):

H>1 = 1

2 2 XTik

2 27=1 k=lsi <i

2

27=1 Jt=l

[10]

[11]

[12]

Since different choices of the weights make significant differ-ences to the estimated parameters (Abbaspour et al., 1997),the advantage of the formulation in Eq. [9] is that it eliminatesthe need for calculating the weights altogether.

For the loamy sand, we formulated 12 different objectivefunctions for inverse analysis based on: (q), (q2), (6), (h),

), (q,9,h), (9,6)* (qjt^, (Wf, and (

0.1010 100 1000Pressure head (cm)

10000

Fig. 3. Retention curves for loamy sand based on the 12 differentformulations of the objective function in Table 1. Symbols showthe measured values.

where in (q2) only one-third of the data in (q ) was used andg9 indicates formulation of the objective function as in Eq.[9]. For the two-layered sandy loam lysimeter, there were 18different formulations: (q), (8), (h), (<?,6), (q,h), (Q,h), (q,Q,h),

(6,/z )*""•, and (9,8^)™-, where FTL stands for thecase in which the parameters for the top layer were estimatedfirst and then fixed.

RESULTS AND DISCUSSIONThe experiment with uniform loamy sand resulted in

241 cumulative discharge data, 111 pressure head data(t2 = 37, s2 = 3), and 148 water content data (r3 = 37,s3 = 4). The pressure head values ranged from +61 to-42 cm, the water content values ranged from 0.53 to0.23, and the saturated water content for the loamy sandwas fixed at 0.53 (cm3 cm~3). Table 1 contains a list ofthe parameters fitted to different variable combinations

Table 1. Unsaturated soil hydraulic parameters obtained by inverse method using different combinations of measured variables in thegoal function. Example loamy sand.

Variables?

(<7)(92)(6)(A)(9,6)<<7,A)<e,A)<9,e,A)(9,e)«»(<?,*)"(Q,h)*>(.qflji)*Average and standard deviationLaboratory

Residual volumetricwater content

e,cm3 cm~3

0.13750.13750.13750.13750.13250.13750.13750.13750.13250.13750.10250.1375

0.1354 ± 0.00430.138 ± 0.0056

a

cm-'0.036250.037750.033250.069550.035750.040750.061250.039750.035750.036750.031750.03825

0.0413 ± 0.01120.036 ± 0.0036

n

2.5252.4753.0753.5252.5752.5251.8252.5752.6252.5252.9752.625

2.737 ± 0.30002.256 ± 0.1252

Saturatedhydraulic conductivity

K,cmh '73.5042.518.5062.5051.5015.2515.2515.5037.0057.5016.5021.50

35.4 ± 20.560.1 ± 0.011

t g9 indicates that parameters were determined using the goal function based on Eq. [9].


0.60 -i 0.60-1

0.0010 100 1000Pressure head (cm)10000 10 100 1000

Pressure head (cm)10000

Fig. 4. Retention curves for the top (a) and the bottom (b) layer of sandy loam based on the different formulations of the objective function.Symbols show the measured values.

and objective functions. The differences in the parame-ters were not very large, but with the exception of 0r,they were significantly different at the P = 0.05. Parame-ters obtained by the laboratory experiment are alsolisted in Table 1. The fitted retention curves for all 12cases are plotted in Fig. 3.

For the second lysimeter, we had 194 cumulative dis-charge data, 111 pressure head data (t2 = 37, s2 = 3),and 148 water content data (t3 = 37, s3 = 4). The pressurehead values ranged from +67 to -20 cm, the watercontent values ranged from 0.47 to 0.27 cm3 cm~3, andthe saturated water content for the top layer of sandyloam was fixed at 0.47 (cm3 cm~3), and for the bottomlayer at 0.42 (cm3 cm"3). Fitted parameters obtained forthe two-layered sandy loam soil (values not shown) alsoshowed significant differences at P = 0.05 with respectto different variables in the goal function. The plots ofthe retention curves for the top and the bottom layerof sandy loam are shown, respectively, in Fig. 4a and4b. For the top layer of the sandy loam in Fig. 4a, theapparent outlier is the retention curve based on (h),and for the bottom layer in Fig. 4b, the two outliers arebased on (h) and (A)™-.Table 2. Ranks of the mean square error of the simulation results

calculated for parameters obtained using different variables inthe inverse method. Example loamy sand.___________

Variables! Cumulative discharge q Pressure head h Water content 6

(9)(92)(6)(A)(9,8)<9,A)<O,A)(9,e,A)<9,e)"(aW(W(q,OJt)*>Laboratory

14

101338

1175296

12

————— cm -0*ooooooooooooooo

12951

103248

1176

13

OOOooooooooo0

cm8

101

1359

1163724

12

3 cm-'OOOooooooooooo00

t g9 indicates that parameters were determined using the goal functionin Eq. [9].

t O symbols indicate the results of the significance test. Those variableswith the same number of diamonds produced statistically similar simula-tion results.

For each estimated parameter set, the MSE betweenthe measured and simulated values for the three vari-ables q, h, and 6 was calculated. For ease of presentationand comparison, only the ranks are listed in Table 2and Table 3 for the loamy sand and the sandy loam,respectively. We also performed a significance test (Zar,1984), using the MSE values, to determine whether sim-ulations based on different parameter sets were differ-ent from each other at the P = 0.05.

Tables 2 and 3 also show the results of the significancetest with diamond-shaped symbols, where for example,simulation of the cumulative discharge for the loamysand in Table 2 resulted in three distinct parameterclasses: (h), laboratory, and all others.

Global Minimum and Response SurfacesSome interpretations of the above results could be

made as follows. Using the SUFI algorithm, we did notTable 3. Ranks of the mean square error of the simulation results

calculated for parameters obtained using different variables inthe goal function. Example sandy loam. ____________

Variables! Cumulative discharge q Pressure head h Water content u>

(q)(8)(A)(9,0)(9,A)(0,A)(9,e,A)(9,8)"(«,*)*(8,*)*(q,W(9)m

(A)™(8)"L

(9,e,A)m

(9,0)m

(9,A)m

<6,A)m

Laboratory

112194

13181532

1659

1711687

1014

——————— VI11 ———

Ot 8OOOOooooooooooooooooooo

141

10546

11739

152

121816171319

OOOOOoooooooooooooooooooooooooooooooooooo

cm3 cm 3

13 01

1912101829

1415118

1745673

16

OOOO0oooooooooooooooooo

t g9 indicates that parameters were determined using the the formulationof the goal function in Eq. [9]; FTL stands for the case where theparameters of the first layer were calculated first and then fixed.

t O symbols indicate the results of the significance test. Those variableswith the same number of diamonds produced statistically similar simula-tion results.


encounter a problem with parameter identification dueto lack of a clear minimum. Sometimes the uncertaintyin a parameter must be made very small before otherparameters begin to show signs of sensitivity to theobjective function. We looked at the (a-n), (a-Ks), and(n-Ks) response surfaces of the objective function forthe parameters of loamy sand based on (0) [only (a-n)shown in Fig. 5]. The (a-n) surface in Fig. 5a was plottedby dividing the range of each parameter into 40 discretepoints resulting in 1600 grid points. The global mini-mum, as indicated in Table 1 for (0) occurs at a =0.03325, n = 3.075, Ks = 18.50, and 9r = 0.1375 and hasan objective function value of 0.0083 as calculated byEq. [8]. The (a-n) surface in Fig. 5a, where Ks = 18.50,and 0r = 0.1375, shows the presence of many local min-ima. In this case, the parameter n becomes sensitiveonly within a narrow range of a values, i.e., 0.030 to0.036. For values of a <0.030, the objective function iscompletely insensitive to the parameter n as the contourlines are almost parallel to the n axis. If the grid usedto generate the response surface is not fine enough, thenthere is always a good chance that the response surfacecould give the misleading impression that there was noclear global minimum. To demonstrate this we plotteda 50 by 50 grid on a much smaller interval for (a-n)surface as shown in Fig. 5b. In this graph, a global mini-mum can easily be seen; however, due to the existenceof several local minima, many optimization programswould have difficulty detecting the global minimum.

Furthermore, it should be emphasized that in produc-ing Fig. 5, we knew the values of the other parameters(Ks and 0r) for the global minimum. Without this knowl-edge no conclusions could have really been reachedregarding the existence of unique values for n and a.Many interpretations regarding the existence of uniqueand identifiable parameters are based on the two-parameter response surfaces in the literature (e.g., Toor-man and Wierenga, 1992; Simunek and van Genuchten,

1996). It should generally be realized that for more thantwo parameters, no conclusions can be made in a two-parameter plot, unless the other parameters are eitherknown (usually not the case) or are also allowed to varya large number of times.

The plots of Fig. 5 indicate that sometimes a parame-ter must have a certain degree of accuracy before an-other parameter shows any signs of sensitivity. This car-ries significant implications for the commonly usedmethod of sensitivity analysis. Since all other parametersare usually fixed and the parameter of interest is system-atically varied, fixing the value of a in this case to any-thing outside the interval 0.030 to 0.036 would give thefalse impression that the objective function was not at allsensitive to n. Therefore, for a more accurate sensitivityanalysis, other parameters should also systematicallyvary.

Performance of Different Proceduresin Inverse Modeling

The results of the significance test in Tables 2 and 3indicated that the laboratory determined parameterswere not very suitable for simulations in a larger scaleas the MSE was almost the highest for both lysimeters.Also, we can conclude that in both lysimeters, simula-tions of most variables on the basis of parameters ob-tained by only (q) or (0) were statistically the same asthose obtained by all (q,Q,h). In fact for the loamy sand,with the exception of parameters based on pressurehead alone, any other variable or combination of vari-ables could have been used for identification of theparameters in an inverse model with statistically similarsimulation results. This conclusion can have far reachingconsequences for the design of experimental setups forthe purpose of parameter identification by inverse meth-ods. For example, water content or cumulative dischargedata alone in a multistep experiment would be sufficientfor parameter estimation in a lysimeter experiment.

4.00

3.50

3.00

2.50

3.50

3.30 -

0.03 0.04 0.05Alpha (cm )

0.08 ).030 0.038 0.0400.034 0.038 0.038Alpha (cm"1)

Fig. 5. Contours of the objective function based on water content (6) for the loamy sand soil, (a) coarse grid a-n plane, (b) finer grid a-n planebased on a smaller region of the parameters.


Furthermore, parameters obtained with the formula-tion of the objective function based on Eq. [9] performedthe same or better than the more traditional formulationbased on Eq. [8]. If further testing of this objectivefunction produces similar results, then the problem ofcalculating the weights for different variables in theobjective function could be eliminated. Also, parameterestimation by inverse analysis could be performed layerby layer in multilayered soils with good results if pres-sure head or water content data were obtained foreach layer.

Notwithstanding the results of the significance test,it can further be deduced from the results of Tables 2and 3 that to simulate discharge it is best to fit theparameters on (q) alone. To simulate pressure head oneobtains the best results upon fitting the parameters on(/z) alone, and to simulate water content, similarly oneshould obtain parameters only by fitting them on (9)alone. Since we are estimating fitting hydraulic parame-ters rather than soil properties, it is not unreasonableto suggest that we could use different parameter setsfor different simulation purposes. However, this wouldrequire that we should measure everything, a proposi-tion that may defeat the purpose of employing an in-verse simulation in the first place.

Obtaining Geostatistically Conditional andStatistically Unconditional Parameters

An important conclusion drawn from the results ofTables 2 and 3 is that parameters obtained by an inverseanalysis are always statistically conditional. The condi-tioning factors are the type, quantity, and quality of thevariables used in the objective function as well as themathematical formulation of the objective function.

100 -.

0.01100Time

Fig. 6. Graphs of the last iteration by SUFI of parameter conditioningfor the loamy sand soil using an objective function based on (q).The solid outer lines are the 95% confidence interval of the Bayes-ian distribution of the simulated cumulative discharge, the dashedline is the expected simulation, and the symbols are the mea-sured values.

Other conditioning factors that were not addressed here,but could be as important, include the model of hydrau-lic characteristics, simulation program, experimental setup, and boundary conditions. We emphasize that for agiven soil, the persistent idea that there exists a set of"unique" hydraulic parameters only to be discoveredby the right experimental setup and combination ofvariables, which can then serve as the soil hydraulicfinger print, is not valid. The behavior of the soil hydrau-lic parameters is more compatible with the notion ofrandom variables than single-valued soil properties.

From a statistical point of view, "conditioned" or"conditional" parameters are not very useful. Unfortu-nately geostatistics and classical statistics use the com-mon phrase "conditional" to refer to different things.From a geostatistical point of view, we should like tohave parameters that are "conditional", meaning simu-lations with these parameters honor the measured data;however, from a statistical point of view we should liketo have "unconditional" parameters, meaning generalpurpose parameters that can be used for all cases. Fittedparameters are highly "conditional", in the sense thatthey are best used for only the case for which theywere conditioned.

To obtain statistically unconditional parameters, weneed to integrate over all the conditioning factors, andthis would produce a domain of retention curves thatwould be the approximate outer boundary of all theretention curves in Fig. 3 and 4. The sought-after param-eters would then be expressed by an expected valueaccompanied by a standard deviation. An estimate ofthese statistics would be the averages and the standarddeviations given in Table 1 for the loamy sand; however,the method of obtaining these estimates is not verypractical as it requires estimation of parameters for alarge number of cases.

0.60 -i

0.101000010 too loqo

Pressure head (cm)Fig. 7. Water retention domain for the loamy sand soil obtained by

conditioning the hydraulic parameters on the measured cumulativedischarge. Symbols show the measured values.


0.66-1 0.66 -i

0.60 -

0.16

0.66 -i

0.16

Time (h60 100

Time (h

"nO.46 -

•<->0.40 -

I00.36uhi-gO.30

o.ee

50cm 0.64 -i

r]0.63

, 0.61 -

I

Time (hZOO 280

0.60

200 860

75cm

l6o 160Time (h)

Fig. 8. Results of the stochastic simulation of the water content for loamy sand determined on the basis of parameters obtained by conditioningon the measured cumulative discharge (a) at 25-cm depth vertically installed TDK (b) at 25-cin depth horizontally installed TDK, (c) at 50-cm depth, and (d) at 75-cm depth. Solid lines are the 95% confidence interval band of the simulated results, dashed line is the expected watercontent, and symbols are the measured values.

A different approach for obtaining statistically uncon-ditional, and yet geostatistically conditional, parametersis to perform "parameter conditioning" rather than fit-ting, as discussed above. Using only (q) as the condition-ing variable, we performed parameter conditioning forthe loamy sand. Figure 6 shows the last iteration wherethe 95% confidence interval of the distribution of thecumulative discharge honors all the measured points.The intervals for the parameters of the flow and reten-tion curves were 6r = [0.12, 0.14], a = [0.025, 0.055],n = [1.8,3.0], and Ks = [16,40]. The graph of the domainof the retention curves is shown in Fig. 7 along with themeasured values. The advantage of performing parame-ter conditioning instead of fitting is that any single vari-able could be used in the objective function to obtainthe parameters. Also, given some of the characteristicfeatures of environmental data such as model and pa-rameter uncertainty, heterogeneity, and measurementerrors, it is more reasonable to depict environmentalparameters probabilistically than as absolute values. Us-ing the conditional parameters based on (q), we simu-lated stochastically, using uniform distribution of pa-rameters within the conditional intervals, the water

content at the four measured locations by propagatingthe hydraulic parameter uncertainties with a MonteCarlo type simulation and calculated the 95% confi-dence interval of the Bayesian distribution of the watercontents. The results are shown in Fig. 8 for all fourlocations, where most of the water content data arehonored by the (g)-based parameters.

CONCLUSIONSWe estimated unsaturated hydraulic parameters by

coupling the SUFI inverse program with a numericalsolution of Richards' equation. Measurements of cumu-lative discharge, water content, and pressure head ob-tained by multistep drainage experiments in two largelysimeters were used as fitting and conditioning data.For the first lysimeter, which contained uniform loamysand, we formulated 12 different objective functions onthe basis of different combinations of the variables, andtwo different formulations of the objective functions.For the second lysimeter, which contained a two-layeredsandy loam system, we formulated 18 different objectivefunctions on the basis of different combinations of the


variables, different formulations of the objective func-tion, and different inverse approaches. We obtainedsimilar conclusions for both soils, i.e., estimated parame-ters were highly conditioned on the variables that wereincluded in the objective function, laboratory-deter-mined parameters did not perform well for simulationof lysimeter experiments, and objective functions basedon the multiplicative function [Eq. 9] performed betterthan the traditional summation formulation [Eq. 8]. Wealso observed that using the SUFI program parametersbased on only cumulative discharge or water contentproduced simulation results statistically the same asthose based on a combination of water content, pressurehead, and cumulative discharge. Finally, we introducedand discussed the concept of "parameter conditioning",proposed that in environmental studies parameter con-ditioning would be more appropriate than parameterfitting, and showed that parameters conditioned on asingle variable could provide statistically unconditionalparameters that would at the same time produce simula-tions honoring the measurements.

ACKNOWLEDGMENTSThe authors are grateful for the experimental help provided

by Werner Attinger, and editorial comments of Prof. Dr. Han-nes Fliihler and Dr. Thomas Gimmi.

uncertainty in estimation of soil hydraulic parameters by inverse modeling: example lysimeter...

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