Determining Soil Hydraulic Properties from One-step Outflow Experiments
by Parameter Estimation: I. Theory and Numerical Studies1
J. B. KOOL, J. C. PARKER, AND M. TH. VAN GENUCHTEN2
ABSTRACTThe numerical feasibility of determining water retention and hydraulic conductivity functions simultaneously from one-step pressure
outflow experiments on soil cores by a parameter estimation method
is evaluated. Soil hydraulic properties are assumed to be represented
by van Genuchten's closed-form expressions involving three unknown parameters: residual moisture content 6, and coefficients aand n. These parameters are evaluated by nonlinear least-squares
fitting of predicted to observed cumulative outflow with time. Numerical experiments were performed for two hypothetical soils toevaluate limitations of the method imposed by constraints of uniqueness and sensitivity to error. Results indicate that an accurate solution of the parameter identification problem may be obtained when(i) input data include cumulative outflow volumes with time corresponding to at least half of the final outflow and additionally the
final outflow volume; (ii) final cumulative outflow corresponds to a
sufficiently large fraction (e.g., >0.5) of the total water between
saturated and residual water contents; (iii) experimental error in
outflow measurements is low; and (iv) initial parameter estimates
are reasonably close to their true values.
Additional Index Words: unsaturated hydraulic conductivity measurement, water retention measurement, transient flow, inverse problem, optimization.
Kool, J.B., J.C. Parker, and M.Th. van Genuchten. 1985. Determining soil hydraulic properties from one-step outflow experimentsby parameter estimation: I. Theory and numerical studies. Soil Sci.
Soc. Am. J. 49:1348-1354.

HE INCREASING REALIZATION that knowledge ofTspatial variability of soil hydraulic properties isof paramount importance for the prediction of fieldscale flow and transport processes has greatly accentuated the need for more efficient means of determining these properties. Recently, interest has arisen in
the feasibility of simultaneously determining water retention and hydraulic conductivity functions fromtransient flow data by parameter estimation methods(Zachman et al., 1981; 1982; Dane and Hruska, 1983;
Hprnung, 1983). Since transient flow data can be relatively quickly obtained, much of the tedious and time
consuming nature of more conventional methods ofdetermining hydraulic properties may be avoided. The
parameter estimation approach is based on the assumption that relationships between volumetric watercontent 6, hydraulic conductivity K, and pressure head
h are described by known mathematical expressions
with a small number of parameters. (In this paper we
use the term pressure head to refer to the components
of total potential attributable to matric and hydrostatic components but excluding nonatmospheric gas
pressure contributions.) The problem of determining
6(h) and K(h) thus becomes a problem of determining
values of the initially unknown parameters. Experimentally, the procedures involve measurement of some
flux-controlled attribute(s) during transient flow. The
flow process is then simulated numerically using
guessed initial values of the unknown parameters.
Simulations are repeated with improved parameter estimates until simulated and observed results match.
Assuming that sufficiently accurate expressions for6(h) and K(h) are used, the main requirement for theparameter estimation problem is that the input datacontain sufficient information to define a unique solution. Hprnung (1983) considered a hypothetical experiment in which a vertical column, initially at hydrostatic equilibrium with zero matric potential at thelower boundary is subjected to a constant downward
flux at the upper surface. Two unknown parameters
in a five parameter model for the soil hydraulic prop1
Contribution from the Dep. of Agronomy, Virginia Polytechnic
Institute and State Univ., Blacksburg, VA 24061 and U.S. Salinity
Laboratory, 4500 Glenwood Dr., Riverside, CA 92501. Received
202 Nov. 1984. Approved 6 June 1985.
Research Associate and Associate Professor, Virginia Polytechnic Institute and State Univ., Blacksburg, VA 24061, and Research
Soil Scientist, U. S. Salinity Laboratory, respectively.

KOOL ET AL.: SOIL HYDRAULIC PROPERTIES FROM OUTFLOW EXPERIMENTS BY PARAMETER ESTIMATION: I.

erties were determined from column outflow rates. It
was concluded that under these experimental conditions outflow rates alone do not give sufficient information to satisfy uniqueness. Only when additional
information was included in the input data could theuniqueness problem be resolved. Hornung used thefinal steady state pressure head at a fixed point inside
the column as additional input. Zachman et al. (1981)
considered a similar problem involving gravity drainage from an initially saturated column of sand andinvestigated the usefulness of various types of measurements for estimating the coefficients in a four parameter model with two coefficients assumed known.
They concluded that cumulative drainage vs. timeyielded the best results while offering the additional
advantage that the data are easily obtained. No problems involving uniqueness were reported.
We wish to investigate the feasibility of using a parameter estimation approach to obtain 6(h) and K(h)from measurements of cumulative outflow with timefrom initially saturated soil in a pressure desorption
cell following a step change in gas pressure. Experimental details and results for undisturbed soil cores
will be presented in a companion paper (Parker et al.,1985). Here, we present the theoretical background of
the method and investigate numerical constraints ofthe technique by evaluating solution uniqueness andsensitivity to errors in input data for two hypotheticalsoils.

Hydraulic Model
The experimental procedure which we consider involves
the measurement of cumulative outflow with time from asoil core at high initial water content, subjected to an instantaneous increment in pneumatic pressure at the top witha saturated porous plate in place at the bottom. We requirea solution of Richards' equation which may be written forthe one-dimensional case with vertical distance x taken positive downward and with pneumatic potentials translated to
the lower boundary condition for notational convenience:"[1]

where C = d6/dh is the water capacity. The appropriate
initial and boundary conditions are

h = h0(x)t = 0, Q<x<L

dh/dx = 1

t > 0, x = 0
t > 0, x = L

[2a]
[2b]
[2c]

where x = 0 is taken at the top of the core, x = L is thebottom of the porous plate, hL is the pressure head at thebottom of the porous plate, and ha = &p/pg where Ap is thegauge gas pressure applied to the core, g is gravitational acceleration, and p is the density of water. For the problem
under consideration, we solve Eq. [1] for the two-layer system of soil column and porous plate. Since the porous plateremains saturated, hydraulic properties of the plate in thepressure cell will be independent of h with C = 0 and K =Kr~ 0(x,t)] dx[3]

where A is the core area perpendicular to flow. The integral
in Eq. [3] is evaluated using the composite trapezoidal rule.
We assume soil hydraulic properties are described by van
Genuchten's model (van Genuchten, 1978a; 1980):
1
[4]

h >0
[5]

and
6 = (0 - 0,)/(0, - Or)[6]
where 6, is the saturated water content (6 at h — 0); 0, is the"residual" water content; K, is the saturated hydraulic conductivity; and a and n are empirical parameters. Expressions
for K(h) and C(h) follow from Eq. [4] through [6]. Of thefive parameters Kn 6S, dn a, and n in these expressions, thefirst two have clear physical significance. The residual watercontent is defined nominally as the water content at which
K—.0 and h—» — oo. Literally, this condition is only met if 6r= 0. In practice, Eq. [4] through [6] are not applied at very
low pressure heads (e.g., h < — 150m) and 8r must be regarded as a strictly empirical coefficient which yields the best
fit to the data over the desired range in 6. The parameters
a and n are inversely related to the air-entry tension andwidth of the pore size distribution, respectively (van Genuchten, 1978a). From our own data and that in the literature, we observe that for desorption, a generally ranges from

0.5 to 5.0 m"1, while n usually varies from 1.1 to about 3.5

THEORYh = hL - h

Q(t) = A1349

The solution of Eq. [1] and [2] was obtained by a modified
version of the Galerkin finite element model of van Genuchten (1978b). Cumulative outflow Q(t) was subsequently
calculated as(van Genuchten, 1980; van Genuchten and Nielsen, 1985).
For a, the lowest value reported is 0.15 m"' for a heavy clay
soil, while for n the upper limit is about 10 for materials
with extremely narrow pore size distributions. High valuesof a and n generally correspond to sandy soils while finetextured soils have lower values.
In the parameter estimation problem we assume that 6S,
Ks, and Kf have been measured independently. Values of 8na, and n are sought by numerical inversion of the flow problem.
Parameter Estimation Procedure
The experimental procedure results in a set of cumulative
outflow measurements Q at specific times /, (/ = 1,2,.. .N).
These Q(t,) are employed as input data for the numericalinversion problem. Let Q(b,t,) be the numerically calculated
values of outflow corresponding to a trial vector of parameter values \b\ where [b\ is the three-dimensional vector [a,n, 6T}. The problem we pose is to find an optimum combination of parameters {b°} that minimizes the objective function:[8]

where w, is a weighting function taken as unity in the simulations considered here. To determine {b°} we employ an
optimization algorithm based on Marquardt's maximum
neighborhood method (Marquardt, 1963). This method represents an optimum combination of the method of steepest
descent and the Gauss-Newton method, and is widely used
for nonlinear least-squares optimization (Beck and Arnold,
1977).
We wish to investigate the adequacy of cumulative outflow volumes [Q(ti),.. .Q(tN)] observed at times t:,.. .,tN todefine unique solutions to the inverse problem. It is antic-

1350

SOIL SCI. SOC. AM. J., VOL. 49, 1985

30

Table 1. Assumed van Genuchten model parameters for
hypothetical soils in numerical experiments.

Q.

Soils

hMO.
3

Sandy loam
Clay loam

O
.-•20

K,
0.47
0.45

0.17
0.24

8.7 x 10-"
6.9 X 10-'

1.00
0.67

2.00
1.395

couples the nonlinear least-squares analysis with the numerical solution of the flow problem are given by Kool etal. (1985). Machine readable copies of the program are available from the authors on request.

U)
>

ID 10

hM.

0

1

2

3

4
T I M E Ir"7)

B

6

•

Fig. 1. Cumulative outflow with time from sandy loam for h" = 1
and 10 m. indicates outflow at equilibrium Q(tm).
12

3
O

eH
O

hMOn

u

ID
U

0

1

2

3

4

5

6

T I M E kr"*lFig. 2. Cumulative outflow with time from lay loam for h" = 1 m
and 10 m. indicates outflow at equilibrium £>(O- 3m« = 19.2
mL.

ipated that the probability of nonuniqueness will increaseas Q(tN) diminishes relative to the equilibrium outflow volume Q(O> where t^, is the time to effectively reach equilibrium. The probability of nonuniqueness is also expected toincrease as Q(tN) diminishes relative to the total drainable
volume of water in the column Qmax that corresponds to adecrease in water content from Bs to 6r For a given pressure
increment this suggests the desirability of collecting data overthe entire outflow process 0<t<tai. On the other hand, minimizing tN allows a reduction in experimental effort andcomputer time. As a compromise between these two requirements we employ transient data for the first hours ofoutflow supplemented by the total volume of outflow atequilibrium 2(O- This means that experiments must becontinued to equilibrium. However, the frequency of observation after the first hours of the experiment may be diminished. Also, since h(x,t^) is known from the imposedboundary conditions, the corresponding moisture content
distribution can be obtained directly from Eq. [4], thus allowing Q(O to be calculated with Eq. [3] without the needto solve the flow equation to large times.
Further details on the numerical procedures and documentation of the FORTRAN IV program ONESTEP which

SOLUTION UNIQUENESS
The parameter identification procedure is examined
for two hypothetical soils with contrasting hydraulicproperties. Van Genuchten model parameters corresponding approximately to those of a sandy loam anda clay loam soil were chosen (Table 1). For brevity,
we will refer to the two hypothetical soils by thesenames. Outflow data for the two soils (Fig. 1 and 2)
were generated with the numerical solution assuming
54-mm diam by 40-mm long soil cores and 5.7-mm
thick porous plates with KP=L39 X 10~8 m s~'. Initial conditions were hydrostatic with /z 0 =0 at the vertical center of the core and outflow levels were taken
to give hi, = 20 mm. Simulations were performed forpneumatic heads h" of 1.0 and 10.0 m, and for outflow
volumes collected over different time periods. Parameter estimation runs were carried out using different
initial (a, n, 0r) values chosen to represent the range
in parameter values that can be expected for mostnatural soils. The iterative optimization procedure wascontinued until the relative change in each parameter
became <1%. Results are summarized in Table 2
which gives initial and final (a, n, 0r) values and thedegree of divergence between predicted and "observed" outflow. The latter is expressed as the proportion of variance of predicted outflow that is not
attributable to its linear regression on observed outflow (Snedecor and Cochran, 1967) represented by thequantity 1 — r2 where r is the correlation coefficient.
The number of times the flow equation was solved
reflects the speed of convergence.
Sandy Loam
We first consider numerical inversion of the ha =1m outflow data for the sandy loam soil using (2(0values for 0<*<3 h and t = tm. At t = 3 h G(0/(2(O> 0.99 (Fig. 1). Clearly, the results in Table 2 (Examples A1-A7) indicate that the input data in thiscase are insufficient to define a unique solution to theparameter estimation problem. Final parameter values exhibit marked heterogeneity and fits to the outflow data
are rather unconvincing as evidenced by large
1 — r2 values, especially for cases Al, A2, A4, andA5, indicating that the solutions converge to local,
rather than global minima. A comparison of predicted
retention curves for the seven sets of final parametervalues (Fig. 3) shows that over the observed range inpressure heads (0 to — 1 m) the curves cluster into twogroups with cases Al, A2, A4, and A5 along a spurious
path and the curves for the other cases very close to
the correct curve. Note that all curves yield essentiallyKOOL ET AL.: SOIL HYDRAULIC PROPERTIES FROM OUTFLOW EXPERIMENTS BY PARAMETER ESTIMATION: I.

1351

Table 2. Solutions of parameter estimation problem.

Case

Al
A2
A3
A4
A5
A6
A7

Bl

Final estimates

Initial estimatesha

Outflow
times

a

n

m

h

nr'

-

1
1
1
1
1
1
1

0-3, too
0-3, to.
0-3, too
0-3, too
0-3, to,,0-3, too
0-3, too

5.00
0.50
1.00
5.00
2.50
0.50
5.00

10
10

0-6,
0-6,

too
too

10

0-6,

too

10

»r

a

n

»r

I -r'

No. of trial
solutionsm"'_

m 3 m' J

X 10«

1.10
1.10
3.00
1.50
2.50
3.00
2.75

msm"3
Sandy loam
0.150
0.225
0.075
0.150
0.150
0.005
0.225

3.11
3.51
0.83
4.66
3.86
0.77
1.18

1.159
1.183
1.945
1.260
1.201
1.936
2.439

0.001
0.081
0.080
0.214
0.127
0.035
0.260

2600
2900
70
5100
3100
300
340

66

too
to.
too
too

5.00
0.50
1.00
5.00
2.50
0.50
5.00

1.10
1.10
3.00
1.50
2.50
3.00
2.75

0.150
0.225
0.075
0.150
0.150
0.005
0.225

1.00
1.03
0.99
1.00
1.00
1.02
1.35

2.002
1.940
1.985
1.997
2.011
2.001
3.647

0.170
0.166
0.169
0.170
0.171
0.170
0.194

0.08
7.8
1.0
0.15
0.74
0.20
1300

29
25
21
27
43
17
44

0.300
0.200
0.250
0.150
0.150
0.300
0.200
0.250
0.150
0.150
0.300
0.200
0.250
0.150
0.150

0.83
0.60
0.59
0.39
0.56

1.477

160
20
50
560
410

16
21
43
51
47

0.68
0.64
0.59
0.75
0.78

1.391
1.374
1.337
1.458
1.487

0.67
0.63
0.65
0.70
0.68

1.392
1.362
1.382
1.421
1.403

0.298
0.207
0.199
0.018
0.172
0.240
0.229
0.206
0.264
0.273
0.238
0.223
0.234
0.251
0.244

B2
B3
B4
B5
B6
B7

10
10
10

0-6,
0-6,
0-6,
0-6,

Cl
C2
C3
C4
C5

1
1
1
1
1

0-6,
0-6,
0-6,
0-6,
0-6,

too
t,,,
to.
too
too

0.50
1.00
0.50
2.00
2.50

1.50
1.40
1.20
1.60
2.00

Dl
D2
D3
D4
D5

10

10

0-6,
0-6,
0-6,
0-6,
0-6,

too
too
too
too
too

0.50
1.00
0.50
2.00
2.50

1.50
1.40
1.20
1.60
2.00

El
E2
E3
E4
E5

10
10
10
10
10

0-12, fa,
0-12, to.
0-12, ^
0-12, too
0-12, t,»

0.59
1.00
0.50
2.00
2.50

1.50
1.40
1.20
1.60
2.00

41
21
41
41
30
48

Clay loam

10
10
10

the same water content at — h = h" = 1 m corresponding to <2(O- At lower pressure heads, the curves
diverge erratically leading to large errors in 6(h) predictions at potentials much beyond the range encountered experimentally.
This nonuniqueness for the sandy loam with h" =1 m may be attributed to the fact that experimental
data span such a narrow range of soil water contents:<2(O/(2max = 0.29. By contrast, for h" = 10m Q(tJ/finm = 0.89. Parameter estimation trials for ha = 10
m outflow data with 0</<6 h and tm were carried out
using the same starting values as before. At t = 6 h
the soil is quite close to equilibrium with <2(0/(2(O= 0.97. Unlike the h" = 1m experiments, we now
obtain excellent agreement between predicted and observed outflow and between predicted and actual parameter values (cases B, Table 2) in all cases except
B7 which used initial values (5.0, 2.75, 0.225). The
corresponding solution (1.35, 3.647, 0.194) differs
considerably from the correct set and clearly does not
represent an absolute minimum on the response surface since 1 — r2 is large compared to the other solutions. However, in absolute terms the deviations arestill quite small and could go unnoticed if we were
dealing with real soils, as opposed to the hypotheticalsoils considered here, where agreement between calculated and observed results will always be less thanperfect.
The results for sandy loam above indicate that evenfor the larger pressure increment, the outflow data do
not provide sufficient information to uniquely deter-

1.367
1.371
1.297
1.343

8.6
0.52
2.6
3.3
5.5

21
25
34
17
40

0.34

21
21
25
36
102

4.2

0.90
2.2
0.27

mine the three unknown parameters. On the positiveside, however, it may be noted that only one case of
nonuniqueness was observed. Furthermore, the solution (1.35, 3.647, 0.194) represents hydraulic properties which may be questioned on physical grounds since
they are more likely for a sand than for a sandy loamo;sFig. 3. Predicted retention curves for cases Al to 7 of Table 2 for h?
= 1 m outflow experiments on sandy loam soil. Curve 0 represents
correct 9(h) relationship.

1352

SOIL SOC. SCI. AM. J., VOL. 49, 1985
I0 3

Table 3. Sensitivity of parameter estimation problem solution
to random error in input data for sandy loam.t
I(P =• 0.02)

10Z

II (p == 0.05)

HI(p = 0.15)

Rep a

Repb

Rep a

Repb

Rep a

RepbJ

0.96
1.957
0.168
850
17

1.01
2.115
0.178
860
21

0.88
1.748
0.146

0.90
1.745
0.145

0.78
1.934
0.172

1.84
4.913
0.190

7000

2000

8800

25

27

44

0

Final aFinal nFinal flrliJ 10'No. trials

UCC• 10 m. Units of a in m"1,
t Outflow data
Qp (t) for t 0 to 6 h and tm
6r in m' m'3, n dimensionless.
} Last selected parameter values. Execution stopped because of persistentoscillations in the solution of direct problem.

J

1 - r'( x 10s)

29

a.
I
10"

o.o

0.1

0.2
0.3
6 feV)

0.4

0.5

Fig. 4. Predicted retention curves for cases Cl to S of Table 2 for
h° = 1 m outflow experiments on clay loam soil. Curve 0 represents correct 9(h) relationship.

soil. Finally, we observe that good results were obtained in all cases where initial parameter estimates
were reasonably close to the true values. This suggeststhat for the sandy loam, nonuniqueness would not be
a serious problem in practice.
Clay Loam
The sandy loam example showed that initial parameter estimates can influence the results of the parameter estimation procedure. Fortunately, the range inpossible parameter values for a given soil is rather
narrow and reasonably close first estimates can be obtained from the texture of a sample. It is thus justified
to use only initial parameter values that represent reasonable guesses in solving the inversion problem. Values were so chosen for the hypothetical clay loam soil.
We consider first the h" = 1m outflow experiments
employingQ(t) for 0<a<6 h and tm. At t = 6 h Q(t)/G(O = 0.70. For five cases with different initial values, final parameter estimates yield reasonably goodfits to observed outflow data (cases C, Table 2).

Whereas the final parameter estimates exhibited someheterogeneity, essentially identical 0(/z) predictions
were obtained (Fig. 4) over the experimental range inh from 0 to — 1 m. For h < — 1 m, predictions diverge
markedly, reflecting nonuniqueness associated awith thenarrow range in water contents during the h = 1m
outflow experiment for which (2(O/Qmax = 0.12.
For the h" — 10 m experiment on the clay loam<2(O/(2max increases to 0.53. Initially we employ as
input to the inversion problem the outflow volumes
Q(t) for 0<?<6 h and tm as well as the same combination of initial parameter values as used for the h"= 1 m experiments. Solutions for all cases result ingood agreement with measured outflow data, (cases
D, Table 2). Also, variations in fitted parameters areless than were found for the h" = 1 m case. However,differences in fitted 0, values still cause predicted d(h)and K(h) curves to diverge at pressure heads < —10

m. Note that Q(tjQ(t^ for this soil was only 0.35 whenusing input data with outflow to t = 6 h. The input
data thus cover a much smaller part of the outflow
process than was the case with the sandy loam soil.
Therefore, the inversion problem was solved again forthe same initial parameter values using outflow data
up to =
12 h, and also including (?(p. After 12 h Q(t)/G(O 0.44. This leads to solutions (cases E, Table2) that are very close to the correct parameter values
and all closely approximate the "true" hydraulic properties of the soil. Except for case E5, speed of convex
gence is generally quite good.
SENSITIVITY TO EXPERIMENTAL ERROR
We turn now to an investigation of the effects ofrandom error (noise) in the input data on the parameter estimation problem solution. We utilize the hypothetical outflow data for the sandy loam soil, uponwhich we overlay random errors using:+2p(R- 0.5)][9]QP(ti) =where Qp(t,) denotes the outflow with added random
error at times tl•. = 0 to 6 h and also includes <2(O>Qo(t,) represents the exact outflow, R is a random
number between 0 and 1, and p is the relative error.
To simulate measurement error of 2, 5, and 15%, we
successively set p to 0.02 (case I), 0.05 (case II) and
0.15 (case III), respectively. For the solution of theinversion problem we use initial parameter values(2.50, 1.50, 0.150). These values represent "best guess"estimates. When these initial values were used withthe exact outflow data, the solution quickly converged
to exactly the correct parameter values, requiring theflow equation to be solved 17 times. At each noiselevel we generated two series of outflow data denoted
as a and b, each consisting of 10 data points. Twoseries were used to get some indication about the variation that might occur at a given level of error. Table
3 shows that the inversion problem is sensitive to errors in input data. The effect on the parameter estimates is small but noticeable at the 2% level, greaterat the 5% level, and quite dramatic at the 15% error
level. For case Ilia, errors in the parameters are relatively small but deviations between predicted outflow and input data remainquite high as can be seenfrom the value of 1 — r2. For case Illb, the solution
failed to converge at all and program execution wasinterrupted because the selected parameter values led
to persistent oscillations in the solution of the flow
equation. For the chosen time and space discretiza-

KOOL ET AL.: SOIL HYDRAULIC PROPERTIES FROM OUTFLOW EXPERIMENTS BY PARAMETER ESTIMATION: I.
Table 4. Solutions of parameters estimation problem for sandy
loam with various degrees of error in input Ks values, t
A

KS/KS
1.25

(x 10°)

No. of trial
solutions

90

33

»r

1.18
1.956
0.171

90

40

1 -r"

Final parameter valueat
a

n
0.75

1.50

a

0.83

n

2.046

»r

0.169

a

1.31
1.919
0.172

80

31

0.62
2.144
0.164

580

30

1.53
1.675
0.150

130

27

n
»r

0.50

a
n
»r

2.0

a

n
»r

t Outflow data Q (t) for t = 0 to 6 h and tx. Units of a in m"1, 0r in m3 m"=
n dimensionless.

tion, numerical difficulties were encountered for large
values of n when 0(k) and K(h) tend towards step functions.
In addition to experimental errors in outflow volumes, significant errors are also possible in the input
values for Ks; such errors may be due to analyticalinaccuracies or to the sensitivity ofKs to small changes
in soil structure during testing. To evaluate the effects
of uncertainty in Ks we carried out parameter estimation analyses for h" = 10 m outflow data for the
sandy loam soil assuming error-free outflow measurement using initial parameter values of (0.025, 1.5, 0.15)
and employing erroneous input Ks values. Results are
given in Table 4 for KS/KS varying between 0.5 and
2.0, where Ks is the input value and Ks is the true
value. For errors of ± 25% in Ks, effects on final parameter estimates are relatively minor. However, using input Ks values that are either half or twice as large
as the correct value results in serious errors in parameter estimates.
CONCLUSIONS
The determination of soil hydraulic properties byparameter estimation allows for considerable freedom
in choosing experimental boundary conditions andmeasurements to be used as input data. It has been
our approach to choose boundary conditions andmeasurements in such a way that the experimental
procedure can be kept simple, yet applicable to a variety of soils, rather than search for a method that is
optimum in terms of uniqueness and sensitivity todata variability. Results of Zachman et al. (1981) suggest that cumulative drainage data may be an attractive choice from either point of view. Although ourinvestigation of solution uniqueness is by no means
exhaustive, results for two hypothetical soils suggestthat nonuniqueness, while of some concern, need not
be a serious problem. Good results can be obtainedfor one-step pressure outflow experiments using cumulative outflow with time as input data if the pressure increment is selected to yield a relatively low final
reduced water content and if input data include a reasonably large portion of the transient flow process.

1353

Using cumulative outflow Q(t) to at least about 50%
of (2(O and including as well the value Q(tm) appear
to be generally sufficient for accurate results.
Initial parameter values should be reasonably close
to their actual values to reduce chances of finding anerroneous solution and to enhance the speed of convergence. An approximate range for a and « has already been given. Using"average" initial values, forinstance a = 2.50 nr1, n = 1.75 and 6r = 0.150,
should give good results for most medium textured
soils. These initial values can be suitably adjusted forsoils with different particle size distributions. As acheck on the results, it should be verified that a solution of the identification problem corresponds toreasonable 6(h) and K(h) for the particular soil. In caseof uncertainty, we recommend that the inversion process be repeated with different initial estimates.
Our results show that solutions of the parameter
estimation problems are sensitive to errors in measured data. This is in contrast to findings by Zachman
et al. (1982), who studied a gravity drainage experiment and found that relative errors in input data wereapparently dampened in the solution process. Our
simulations suggest that parameter estimates are moresensitive to errors in outflow measurements than toerrors in input saturated conductivity values. On theother hand, uncertainty in unsaturated conductivity
due to experimental error and/or variations in actual
saturated conductivity during testing may, in practicalcircumstances, be considerably larger than errors in
observed outflow. Also, we evaluated both sources oferror independently, while in practice they will becompounded. All this emphasizes the need for accurate experimental measurements. Convergence speedof the optimization routine is variable and sensitive
to small changes in input data and/or initial parametervalues (reflecting the irregularity of the response surface). The average number of times the flow equation
had to be solved for all cases discussed in this study
was 34. This corresponds to an average CPU-time ofjust below 1 min on the IBM 3081 computer used. In
one case (case E5, Table 2) the solution was found toconverge very slowly. In practice, excessively longcomputer runs are avoided by setting an upper limit
to the number of function evaluations allowed. If thesolution fails to converge in the allowed number of
evaluations, we suggest that the solution process be
started anew with different initial parameter values.

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SOIL SCI. SOC. AM. J., VOL. 49, 1985

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van Genuchten, M.Th. 1978b. Numerical solutions of the one-dimensional saturated-unsaturated flow equation. Res. Rep. 78-WR09, Dep. of Civil Engineering, Princeton, NJ.

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