inverse modelling in estimating soil hydraulic functions ... · the study area. the procedure...

Inverse modelling in estimating soil hydraulic functions: a Genetic Algorithm approach

49

Hydrology and Earth System Sciences, 6(1), 49–65 (2002) © EGS

Inverse modelling in estimating soil hydraulic functions: a GeneticAlgorithm approach

Amor V.M. Ines1 and Peter Droogers1,a

1 International Water Management Institute P.O. Box 2075 Colombo, Sri Lanka1,a Water Engineering and Management Program, School of Civil Engineering,Asian Institute of Technology P.O. Box 4 Klong Luang 12120 Pathumthani, Thailand

Email for corresponding author: [email protected]

AbstractThe practical application of simulation models in the field is sometimes hindered by the difficulty of deriving the soil hydraulic properties ofthe study area. The procedure so-called inverse modelling has been investigated in many studies to address the problem where most of thestudies were limited to hypothetical soil profile and soil core samples in the laboratory. Often, the numerical approach called forward-backward simulation is employed to generate synthetic data then added with random errors to mimic the real-world condition. Inversemodelling is used to backtrack the expected values of the parameters. This study explored the potential of a Genetic Algorithm (GA) toestimate inversely the soil hydraulic functions in the unsaturated zone. Lysimeter data from a wheat experiment in India were used in theanalysis. Two cases were considered: (1) a numerical case where the forward-backward approach was employed and (2) the experimentalcase where the real data from the lysimeter experiment were used. Concurrently, the use of soil water, evapotranspiration (ET) and thecombination of both were investigated as criteria in the inverse modelling. Results showed that using soil water as a criterion provides moreaccurate parameter estimates than using ET. However, from a practical point of view, ET is more attractive as it can be obtained withreasonable accuracy on a regional scale from remote sensing observations. The experimental study proved that the forward-backward approachdoes not take into account the effects of model errors. The formulation of the problem is found to be critical for a successful parameterestimation. The sensitivity of parameters to the objective function and their zone of influence in the soil column are major determinants in thesolution. Generally, their effects sometimes lead to non-uniqueness in the solution but to some extent are partly handled by GA. Overall, itwas concluded that the GA approach is promising to the inverse problem in the unsaturated zone.

Keywords: Genetic Algorithm, inverse modelling, Mualem-Van Genuchten parameters, unsaturated zone, evapotranspiration, soil water

IntroductionIn developing improved water management alternatives,physically-based simulation models could be helpful. Thesesimulation models can be used as tools to understand thecurrent system better since they provide information overan unrestricted spatial and temporal resolution. Models arealso strong in providing information about processes thatare difficult to measure in the field such as capillary rise,soil evaporation and crop transpiration. Most important istheir ability to do scenario analysis as they can integrateeasily the impacts of any changes to the system. However,an obstacle to their practical application in the field is thedifficulty of deriving the soil hydraulic functions θ(h) and

K(h) of the study sites, where θ is the soil water, h is thehydraulic head and K is the unsaturated hydraulicconductivity. This seems to be of minor concern inconventional thinking but inevitably the veracity of the inputdata will be reflected in the end. According to Xevi et al.(1996) proper evaluation of the water balance in theunsaturated zone depends strongly on the appropriatecharacterisation of the soil hydraulic functions. Therefore,proper definition of the soil hydraulic parameters under fieldconditions should be taken into consideration.

Direct measurement in the laboratory using soil coresamples is the classic way to determine the soil hydraulicfunctions (Van Genuchten et al., 1991). Although this is

Amor V.M. Ines and Peter Droogers

50

relatively easy in principle, in practice it is difficult and time-consuming. To obtain sets of soil water at different hydraulicheads also needs specialised apparatus; likewise forconductivity values. The major concern with this approachis the question of whether the parameters derived in a soilcore sample with pre-defined boundary conditions arerepresentative of the field situation (Kool and Parker, 1988;Van Dam, 2000). A broad range of field methods was alsodeveloped to determine the soil hydraulic functions(Dirksen, 1991).

Using pedo-transfer functions is another approach, wheresoil texture, bulk density and organic matter content are usedto estimate soil hydraulic parameters (e.g. Vereecken et al.,1989, 1990; Wösten et al., 1998; Droogers et al., 2000;Antonopolous, 2000). These are very useful functions iflimited information is available about the soil. However,Wösten et al. (1998) stressed the need for further researchto develop a more robust and efficient methodology thatcould estimate soil hydraulic parameters practically for real-world applications.

The potential of a heuristic approach to estimate theMualem-Van Genuchten parameters has been investigatedrecently. Schaap et al. (1998) studied 19 Hierarchical NeuralNetwork (HNN) models (based on the input data) and foundthat the accuracy of the prediction improved when moreinput data were used. They concluded that the developedHNN models performed better than the published pedo-transfer functions in predicting soil water and hydraulicconductivity parameters. The use of the HNN modelsdeveloped is claimed to be attractive because of theconsiderable flexibility toward available input data.

The inverse modelling approach is also applied where themeasured soil hydraulic data are used as fitting criteria toestimate the soil hydraulic parameters by inverting thegoverning equation of the soil water movement. The inverseapproach has been implemented extensively in groundwaterstudies (e.g. Yeh, 1986). The obvious reason is that directmeasurement of aquifer characteristics is far more difficultthan in the vadoze zone. Using observed pressure headsand measured discharges, the transmissivity, specific yield,storage coefficient and other aquifer parameters could bedetermined by the inverse solution of the groundwater flowequation. Yeh (1986) reviewed the common techniques tosolve the inverse problem in the saturated zone. The reviewshowed that the problem could be solved using either a direct(equation error criterion) or an indirect approach (outputerror criterion). The indirect approach is more attractivebecause it does not require spatial and temporal data atregular intervals, hence avoiding the interpolation of datafrom sparse observations which could increase the noise inthe dataset. Gradient-dependent search algorithms are

widely used in this case. Non-linear regression is alsoadopted, e.g. MODFLOWP (Hill, 1992), the ModGA_Pused a Genetic Algorithm (Zheng, 1997).

Kool and Parker (1988) solved the inverse problem inunsaturated transient flows using the indirect approach withthe Levenberg-Marquardt algorithm. The Richards’ equationwas solved inversely during infiltration and redistributionin a hypothetical soil profile to determine the soil hydraulicfunctions simultaneously. The soil water and hydraulic headwere used as fitting criteria. Based on their works (Kool etal., 1985; Kool and Parker, 1988), several studies wereconducted to investigate the inverse problem in unsaturatedflows further. The method has been applied with reasonablesuccess in the laboratory using outflow experiments suchas the one-step (e.g. Van Dam et al., 1992) and multi-stepoutflow approach (e.g. Van Dam et al. 1994; Zurmühl andDurner, 1998) where the measured outflow and soil water(in some cases) were used as fitting criteria in estimatingthe soil hydraulic parameters. In contrast, Šimùnek et al.(1998) and Romano and Santini (1999) among othersapproached the inverse problem using the evaporationmethod (which is the opposite of the outflow experiments)where the soil water and hydraulic heads were usually usedas criteria. In general, most of the literature about the inverseapproach always emphasised the ill-posed nature of thesolution. This is caused mainly by (1) the non-uniquenessproblem and (2) instability in the solution. The non-uniqueness problem occurs when the parameters under studyhave low sensitivity to the criteria being investigated. Thiscould happen in the solution because at some point in thesearch space, the parameter(s) may not be sensitive anylonger. This problem is also caused by local optima and is athreat to gradient-dependent search algorithms. Theinstability problem, on the other hand, is caused by the highsensitivity of the parameters (Yeh, 1986; Kool and Parker,1988; Van Dam et al., 1992).

The inverse modelling approach seems to be a verypromising way to estimate soil hydraulic functions in theunsaturated zone where most plant activities areconcentrated. However, the fitting criteria used in previousstudies are mostly the soil water, hydraulic head and bottomflux. Soil water can be monitored easily in the field butbottom fluxes are more difficult to measure. Hydraulic headscan be measured using tensiometers. It seems that broaderapplication of the methodology in the field depends stronglyon the practicality of the criteria being used in terms ofspatial and temporal dimensions. Evapotranspiration couldbe explored in this case because: (1) it can be estimated ormeasured easily (Kite and Droogers, 2000) and (2) it has anadvantage in larger scale applications (e.g. an irrigationsystem) using remote sensing data e.g. SEBAL


51

(Bastiaanssen et al., 1998). In a recent study (Jhorar et al.,2002), the use of evapotranspiration as a criterion wasinvestigated in inverse modelling using SWAP (Van Damet al., 1997) and a parameter estimation package, PEST(Watermark Computing, 1994). The actualevapotranspiration and actual transpiration were used ascriteria, separately. The benchmark values of the criteriawere established using forward simulation by assumingsome base values of the Mualem-Van Genuchten parametersand then backward simulations were done to obtain theparameters again. To emulate the real-world situation,random errors were included in the benchmark values. Inreality, actual transpiration is difficult to obtain separatelyfrom evapotranspiration. Therefore, evapotranspirationalone is more realistic to explore as a criterion in inversemodelling. On this basis, robust search algorithms areneeded to investigate this option further. A heuristic yetpowerful search technique like Genetic Algorithms couldbe suitable in this case.

In summary, the objectives of this paper are: (1) to explorethe inverse problem in the unsaturated zone usingevapotranspiration, soil water and the combination of both,as fitting criteria, numerically, (2) the same, but usingmeasured evapotranspiration and soil water from lysimeterdata, and (3) to investigate the potential of GeneticAlgorithms in the solution of the inverse problem and howthe technique handles the issues of non-uniqueness andinstability.

MethodologySIMULATION MODEL

The Soil Water Atmosphere and Plant model generallyknown as SWAP (Van Dam et al., 1997) is a physicallybased, detailed agro-hydrological model that simulates therelationships of the soil, water, weather and plants. The coreof the model is the Richards’ equation (Eqn. 1) where thetransport of soil water is modelled by combining Darcy’slaw and the law of continuity:

( )( )

( )hSz

zhhK

thhC

t−

∂

+

∂∂∂

=∂∂=

∂∂

1θ (1)

where θ is the soil water content (cm3 cm-3), h the hydraulichead (cm), C the water capacity (cm-1), t the time (d), z thevertical distance taken positive upward (cm), K theunsaturated hydraulic conductivity (cm d-1) and S is the sinkterm (d-1). SWAP models the soil water movement byconsidering the spatial and temporal differences of the soilwater potentials in the soil profile. The governing equation

is solved numerically with the implicit scheme of Belmanset al. (1983), which can be applied effectively in saturatedand unsaturated conditions.

In SWAP, the soil hydraulic functions are described bythe analytical functions of Van Genuchten (1980) andMualem (1976) for the soil water retention (Eqn. 2) andhydraulic conductivity (Eqn. 3) referred hereafter as MVG:

( ) [ ]mnressat

resh

hα

θθθθ+

−+=1 (2)

( )2

1

11

−−=

m

meesat SSKhK λ

(3)

where θres is the residual soil water content (cm3 cm-3), θsat

the saturated soil water content (cm3 cm-3), α (cm-1), n (–),m (–) and λ (–) are empirical shape parameters and Ksat isthe saturated hydraulic conductivity (cm d-1). Equations 4aand 4b define the relative saturation, Se (–) and parameterm (–):

( )ressat

rese

hSθθθθ

−−

= (4a)

nm 11−= (4b)

SWAP does not only simulate the quantity of soil water butalso the quality and considers the effect of heat on the fateof solutes. Hysteresis, water repellency, soil swelling andshrinkage can be also assumed to affect soil water and solutetransport. For this study these options were not employed.

The water balance is solved by considering two boundaryconditions, the top and bottom boundaries. These boundariescan be either flux or head controlled. The Penman-Monteithequation is used to estimate evapotranspiration. SWAP usesthe leaf area index (LAI) or soil cover fraction (SC) tocalculate the potential transpiration and evaporation of apartly covered soil. The model separates firstly the potentialtranspiration (Tp) and potential evaporation (Ep) thensubsequently calculates the reduction of Tp in a morephysically based approach (Feddes et al., 1978; Maas andHoffman, 1977). The compounded effect of salt and water/oxygen stress to the actual transpiration (Ta) is consideredmultiplicative. In the case of wet soil, the soil evaporationis determined by the atmospheric demand and is equal toEp. However, as the soil dries, the soil hydraulic conductivityalso decreases and the evaporation is decreased to actualevaporation (Ea). The maximum evaporation (Emax) iscalculated using Darcy’s law at the top boundary, then Ea is


52

computed as the minimum of Ep and Emax. Alternatively, thesoil evaporation can be estimated using the empiricalequations of Black et al. (1969) or Boesten and Stroosnijder(1986). In general, the actual evaporation is calculated asthe minimum of Ep, Emax and the Ea using the empiricalfunctions.

The surface runoff is calculated as the ratio of thedifference between the depth of ponding water and themaximum sill (embankment) height to the resistance of thesoil to surface runoff. The surface detention is accountedfor in the resistance term. Field drainage can be simulatedusing the Hooghoudt and Ernst equations in homogenousand heterogeneous soil profiles. Drainage systems can bemodelled as a single or multi-level system. Bottom flux iscalculated according to the bottom boundary conditionadopted in the model.

A simple crop model and detailed crop model WOFOSTcan simulate crop growth in SWAP. The simple crop modelis based on the linear production function of Doorenbosand Kassam (1979). WOFOST (Supit et al., 1994) is ageneric crop model, capable of simulating the growth anddevelopment of most crops. In this study, the detailed cropmodel was used.

Several water management scenarios can be modelled inSWAP. Irrigation scheduling can be considered as fixed timeor according to a number of criteria. A combination ofirrigation prescription and scheduling is also possible. Thescheduling criteria define the timing and depth of irrigationin the growth process.

GENETIC ALGORITHMS

Genetic Algorithms (GAs) are mathematical models ofnatural genetics where the power of nature to develop,destroy, improve and annihilate life is abstracted and usedto solve complex optimisation problems. Holland (1975)developed this powerful technique and it has been appliedin various fields of science. GA is termed a global optimum-seeking algorithm (Zheng, 1997). The algorithm works bymimicking the mechanisms of natural selection to exploredecision search space for optimal solutions (Goldberg,1989).

Goldberg (1989) identified four unique attributes of GA(binary) among the more traditional optimization methods:(1) GA works with a coding of the parameter set (string),not with the parameters themselves, (2) GA searches froma population of points, not a single point, (3) GA usesobjective function information, not derivatives or otherauxiliary knowledge and (4) GA uses probabilistic transitionrules, not deterministic rules.

GA consists of three basic operators: the selection,

crossover and mutation (Cieniawski et al., 1995). First, aninitial set of individuals (strings) is generated. Thispopulation is a representative set of solutions to the problemunder investigation. Each individual is evaluated on itsperformance with respect to some fitness function whichrepresents the environment. Using this measure, theindividual competes in a selection process where the fittestsurvives and is selected to enter the mating pool; the lesser-fit individual dies. The selected individuals (parents) areassigned a mate randomly. Genetic information is exchangedbetween the two parents by crossover to form offspring.The parents are then killed and replaced in the populationby the offspring to keep the population size stable.Reproduction between the individuals takes place with aprobability of crossover. If a random number generated isless than the probability of crossover, crossover happens,otherwise not, and the parents enter into the new population.GAs are very aggressive search techniques; they tend toconverge quickly to a local optimum if the only geneticoperators used are selection and crossover. The reason isthat GA eliminates rapidly those individuals with poormeasures until all the individuals in the population areidentical. Without a fresh influx of new genetic materials,the solution stops there. To maintain diversity, some of thegenes are subjected to mutation to keep the population frompremature convergence (Goldberg, 1989; Cieniawski et al.,1995). Selection, crossover and mutation are repeated formany generations, with the expectation of producing thebest individual(s) that could represent the optimal or nearoptimal solution to the problem under study.

In this study, a modified-µGA was developed and used inthe inverse modelling. This is a slight modification of thesecurGA (small-elitist-creep-uniform-restarting GA) ofCarroll (in Yang et al., 1998). The µGA technique(Krishnakumar, 1989; Caroll, 1996, 1998) was extended toenable creep mutation to take place in the solution(securGA). In complex problems, however, more individualsmight be needed to arrive at the best solution. In thesecurGA, the innate nature of convergence of a µGA is stillbeing used, i.e. 95% uniformity of the genes whichsuppressed the restarting power of a µGA in largerpopulations. For this reason, a 90% convergence is used inthe modified-µGA.

SWAP-GA LINKAGE

Figure 1 shows the linkage between SWAP and GA. Thefitness function serves as an environment for the thrivingpopulation in each generation. The fitter individuals(represented by a high value of fitness) that survived thetest of the environment tend to reproduce and produce


53

Fig. 1. The SWAP-GA linkage for the inverse modeling


54

offspring that are expected to express better genetic traitsin the next generation, to cope up with the environmentaldemands. In this study, the fitness function used is expressedas follows:

(5)

where ξ is a weighting factor (–), θSWAP and θOBS are the soilwater content simulated and observed, respectively(cm3 cm-3), ETaSWAP and ETaOBS the actual evapotranspirationsimulated and observed, respectively (cm d-1), T is an indexof year, i is for day, j is for soil compartment, k representsan individual in the GA, N and M are indices of the numberof observations of θ and ETa and fitness is the measure ofan individual.

The θ and ETa are the criteria in the parameter estimationand their contributions to the fitness function is determinedby the weighting factor ξ. The observed and simulated soilwater and ET values used in evaluating the fitness functionare normalized using the observed minimum and maximumvalues of θ and ETa.

The Mualem-Van Genuchten (MVG) parameters in theSWAP-GA linkage are represented as binaries (0s and 1s),similar to the standard GA approach of multi-parameterrepresentation (see Fig. 2), where P stands for parameter.The real value of a parameter is defined as:

(6)

where Cmax is the maximum value of the parameter and Cmin

the minimum value, a represents alleles or bit value (0 or1), i is an index of bit position (referred as gene), L is lengthof the (sub) string, and j, the index for parameter.

LYSIMETER

Lysimeter data (Tyagi et al., 2000) from a wheat (Triticumaestivum) experiment in India (Karnal, Haryana) during the1991–1992 rabi season were used in the study. The lysimeteris a weighing type, with a surface area of 4 m2 and depth of2 m. Daily changes in lysimeter weight were recorded in adata logger and retrieved regularly. These weight changeswere subsequently converted to equivalent actual ET.Accurate daily ET data could be generated from thisexperiment except when there is excessive rainfall orirrigation application. Weather data were monitored usinga manual and an automatic weather station adjacent to thelysimeter site. In addition, data on crop, soil and watermanagement practices were collected and recorded duringthe duration of the experiment. Percolation losses werecollected from the sump located at the bottom of thelysimeter.

In the model set up, the soil profile was divided into twolayers: 0-60 cm (31% sand, 50% silt, 19% clay) and60–200 cm (29% sand, 49% silt, 22% clay). Soil watermonitoring was done mostly in the first layer at four depths(0–15, 15–30, 30–45 and 45–60 cm) using a Time DomainReflectometer (TDR). Soil water measurements were takenfrequently at irregular intervals and a total of 31 observationswere available. In the simulations, the soil column wasassumed to be well drained.

Using the data on soil texture, bulk density and organicmatter content, the estimates of the MVG parameters werederived with pedo-transfer functions (PTF) (Wösten et al.,1998; Droogers, 1999). These estimates (see Table 1) wereused in the numerical study as parameter base values wherethe performance of GA to handle the non-uniqueness andinstability issues was tested.

SENSITIVITY ANALYSES

The sensitivity of a parameter to the fitness functiondetermines the success of the estimation of the parameterinvolved. A straightforward method was applied by firstsetting the MVG parameters at an initial value as close as

+

−=

∑∑∑T i j

TijOBSSWAPN

kfitness1

1)(θθ

ξ

( )

−−+

∑∑T i

TiOBSaSWAPa ETETM1

11 ξ

( ) ( ) ( ) ( )[ ]jCjCa

jCjCj

j

L

L

i

ii

minmax11

1

min 2

2−+= −

=

−∑

Fig. 2. Parameter representation (8-parameter) in GA with the inverse problem


55

Tabl

e 1.

Sum

mar

y of

resu

lts in

inve

rse

mod

elin

g us

ing

a G

enet

ic A

lgor

ithm

.

Aver

age

Erro

rbR

2 , %

Pred

icte

d pa

ram

eter

val

ues

Crit

eria

Fitn

ess

θET

aθ

ETa

Soil

laye

r 1 (0

-60

cm)

Soil

laye

r 2 (6

0-20

0 cm

)

Num

eric

al c

ase

αn

Ksa

tθ sa

tcα

nK

sat

θ sat

Bas

e va

lues

a0.

0210

01.

2080

15.2

80.

384

0.01

800

1.13

7010

.19

0.36

5D

aily

dat

a4

para

met

ers

ETa

1435

.96.

9E-3

3.9E

-499

.22

99.9

90.

0201

01.

2168

--

0.02

753

1.16

78-

-θ

1458

6.0

1.1E

-51.

7E-5

99.9

999

.99

0.02

099

1.20

81-

-0.

0180

11.

1370

--

θ+ET

a26

65.9

3.0E

-31.

1E-4

99.9

399

.99

0.02

023

1.20

69-

-0.

0204

91.

1468

--

8 pa

ram

eter

sET

a33

75.9

1.3E

-11.

7E-4

97.9

399

.99

0.01

678

1.14

7922

.28

0.58

70.

0231

81.

1517

10.6

30.

355

θ28

4.1

5.5E

-48.

6E-4

99.9

799

.99

0.02

779

1.18

8128

.09

0.39

10.

0328

91.

1080

34.5

50.

372

θ+ET

a15

22.1

5.3E

-21.

9E-4

99.3

099

.99

0.02

587

1.15

8036

.87

0.44

90.

0450

91.

1173

39.3

30.

340

Wee

kly

Dat

a4

para

met

ers

ETa

1500

.61.

3E-2

3.1E

-499

.34

99.9

90.

0166

91.

1957

--

0.02

721

1.16

76-

-θ

301.

64.

6E-4

3.4E

-499

.95

99.9

90.

0212

51.

2033

--

0.01

679

1.13

72-

-θ+

ETa

1980

.22.

3E-2

1.2E

-495

.88

99.9

90.

0166

81.

1920

--

0.03

695

1.21

45-

-8

para

met

ers

ETa

6252

.41.

9E-2

7.4E

-599

.19

99.9

90.

0339

61.

1609

52.4

50.

364

0.03

517

1.09

2145

.55

0.34

7θ

61.2

2.3E

-33.

4E-3

97.2

099

.98

0.02

462

1.19

8721

.04

0.39

20.

0417

21.

1496

35.1

40.

403

θ+ET

a78

3.8

8.1E

-22.

9E-4

91.7

899

.99

0.01

666

1.14

5129

.68

0.43

00.

0568

31.

1709

26.7

50.

529

Exp

erim

enta

l cas

e

Lysi

met

er d

ata

4 pa

ram

eter

sET

a8.

41.

7E-1

7.4E

-254

.67

77.7

30.

0452

11.

6285

--

0.05

778

1.25

33-

-θ

8.2

2.6E

-27.

5E-2

66.1

377

.15

0.00

782

1.79

56-

-0.

0128

31.

0813

--

θ+ET

a8.

32.

6E-2

7.5E

-267

.20

77.1

80.

0075

81.

7976

--

0.01

089

1.08

13-

-8

para

met

ers

ETa

8.5

7.0E

-27.

4E-2

0.4

177

.93

0.01

703

2.15

621.

250.

575

0.04

317

1.17

070.

870.

461

θ8.

72.

4E-2

7.7E

-262

.37

77.7

90.

0058

52.

1493

54.4

20.

407

0.01

983

1.08

114.

130.

323

θ+ET

a8.

12.

8E-2

7.4E

-257

.02

77.5

20.

0131

61.

7818

2.55

0.56

80.

0433

01.

4522

4.66

0.58

7

a fr

om p

edo-

trans

fer f

unct

ion

(Wös

ten

et a

l., 1

998;

Dro

oger

s, 19

99);

θ res =

0.0

00 fo

r bot

h la

yers

; λ =

2.5

23 a

nd –

3.99

7 fo

r lay

er 1

and

2 re

spec

tivel

y.b θ

(cm

3 cm

-3);

ETa(c

m d

-1)

c in

the

expe

rimen

tal c

ase,

pre

defin

ed θ

sat h

as a

val

ue o

f 0.4

5 in

bot

h so

il la

yers

, bas

ed o

n av

aila

ble

soil

wat

er d

ata.


56

possible to the expected value and then running the SWAPmodel. Subsequently, MVG parameters are changed fromtheir initial values by a certain percentage and model outputis compared to the output from the initial runs. A parameterchange resulting in an insignificant change in output of themodel indicates that the parameter considered is lessimportant in the analysis. Moreover, such a parameter isvery unlikely to be estimated successfully, as it is notaffecting the fitness function used in the inverse modelling.The initial values of the parameters were set to the valuesobtained from the PTF as described above.

All six MVG parameters were subjected to this sensitivityanalysis. Actual ET was selected as a criterion because it isincluded in the fitness function and ET can be consideredas an integrated output from the complex soil-water-plant-atmosphere processes. A second criterion used was the fluxat the bottom of the soil profile. The bottom flux also reflectsthe soil-water-plant interactions and can be considered asan integrated representation of the soil water status in theentire profile, which is the other component of the fitnessfunction. Only the MVG parameters at the first soil layerwere set as variables during the sensitivity analysis.

PARAMETER ESTIMATION CASES

Two main cases were considered to estimate the MVGparameters: (1) whether generated observations were used(the numerical case) and (2) whether real-life observationsfrom the lysimeter experiment were used (the experimentalcase).

To evaluate the performance of GA, model-generatedoutput was used first instead of actual field measurements,i.e. the numerical case. Such an approach is common in manystudies, often extended with the incorporation of randomerrors in the generated “observations”. The MVG parametersas estimated from the PTF were considered for the numericalcase as the real parameter values. The SWAP model wasrun with these parameters and the generated output wasassumed in the inverse modelling as the “measured” values.The results from the sensitivity analyses, as presented later,were used to determine which parameters have to beestimated. However, it is well known that the inclusion oftoo many parameters in inverse modelling tends to result innon-uniqueness; therefore, the number of parameters forthe first test was limited to two per soil layer (4-parameterproblem) and in the second stage, four parameters per soillayer were considered (8-parameter problem).

In addition to these 4- and 8-parameter problems, twoother cases were considered. First, the daily model outputwas used as the “measured” values and second, the dailymodel outputs were aggregated into weekly values. The

latter can be seen as a first step towards the experimentalcase where daily measurements are usually scarce, especiallyfor soil water data.

The fitness function was constructed in such a way thatdifferent weights could be given to ET and soil water. Forthis study three options were considered: (1) only ET (ETa)was used in the fitness function (ξ = 0), (2) only soil water(θ) was used (ξ = 1) and (3) equal weights to soil water andET (ξ = 0.5), a multi-objective situation (θ + ETa). Obviously,additional values between 0 and 1 could be used but werenot considered in this study.

For the experimental case, actual ET data as well asmeasured soil water contents were used in the fitnessfunction. As in the numerical case, two different sets ofparameters were estimated per soil layer: (1) α and n and(2) α, n, Ksat and θsat. Obviously, the situation is differentfrom the numerical case as the exact parameter values areunknown for the lysimeter experiment.

To summarize, 18 different cases were considered, basedon the combinations of generated or measured data, two orfour MVG parameters per soil layer, daily or weekly dataand weights on ET and soil water (see Table 1).

Results and conclusionsSENSITIVITY ANALYSES

A summary of the output generated by SWAP using theparameter values derived from PTF is presented in Fig. 3.The amounts of rainfall and irrigation were derived fromthe recorded daily changes of lysimeter weight. These werecompared to the measured rainfall and irrigation during theexperiment; the lysimeter was able to produce highlycomparable values. The high sensitivity of the lysimeterallows the accurate measurements of the crop ET and thusincreases the level of confidence in the analysis.

SWAP is able to partition the actual soil evaporation andplant transpiration. In the figure, the interception by thecanopy during a rainfall event is incorporated in theevaporation. The crops were well watered during theexperiment, no significant water stress is evident along thecrop growth; only the minimal water stress experienced bythe crops before the third irrigation (day 22) is apparent.The abrupt reduction in relative transpiration is mainly dueto oxygen stress when the soil is saturated during anirrigation or rainfall. At some stages the crop could notrecover immediately from oxygen deficiency, which can benoticed after the fourth irrigation followed by the succeedingrainfall events. This increase in water input is evident in thesubsequently increased bottom flux during and after theseperiods. In the figure, the high quantity of bottom flux at


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Fig. 3. Daily output of the SWAP model for the base case using pedo-transfer functions

the onset of simulation is due mainly to the initial conditionused in the soil profile. All soil compartments wereinitialized at field capacity (h = –100 cm).

Before the start of the inverse modelling, the sensitivityof parameters was assessed to define the most applicableparameters to be optimised. The situation in Fig. 3 was usedas a base scenario where the sum of the daily actual ET(sumETa) and bottom flux (sumQbot) were used as sensitivity

criteria. The results of the sensitivity analysis (Fig. 4a andb) indicate clearly that θres and λ are non-sensitive parametersand can be ignored in further analyses: a change in one ofthese parameters does not affect the model outputsubstantially. The other four parameters are sensitive inchanging the model output; therefore, these could beincluded in the inverse modelling. Parameter n could betested only for values higher than the original 100% as n


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has a lower limit of 1 in the MVG equations. The sensitivityof Ksat was clearly limited to lower values, indicating thatthe parameter estimation by inverse modelling will yieldreliable results only when the predicted Ksat values alongthe search are lower than the expected value. Interestingly,the θsat mainly affects the bottom flux while ET is only partlyaffected. The results of the sensitivity analyses were usedto select the number of parameters to be estimated asexplained before, as shown in Table 1.

GENETIC ALGORITHM

An example of a typical parameter estimation using GA isshown in Fig. 5a and b. The figures present the GA solutionsof the 4-parameter problem using daily data in the numericalcase with the three criteria being used (ETa, θ and θ + ETa).The maximum and the average fitness show the typicalbehaviour of GA in comparison to other parameterestimation techniques. A maximum fitness value in ageneration corresponds to the measure of the best individualin that instantaneous population for any of the fittingcriterion mentioned above. The average value correspondsto the average fitness of the population. Fig. 5a shows thatan elitist GA is not gradually moving to an optimum, but astepwise improvement can be observed. After finding a newmaximum fitness value, the corresponding parametercombination (individual) is tagged as elite and carries thisidentity along the generations until a new elite is found inthe search. The GA continues the process of selection,crossover and mutation (creep) to search for new parametercombination that gives the best possible value of the fitnessfunction until the end of the generations. In the numerical

Fig. 4 (a) Sensitivity analysis of the soil hydraulic parameters to ET. (b) Sensitivity analysis of the soil hydraulic parameters to bottom flux.

Fig. 5. (a) Maximum fitness values in a generation with themodified-µGA solution (daily, 4-parameter). ETa indicates ET aloneas criterion, θ, soil water alone and θ + ETa, the combination.(b) Average fitness values in a generation with the modified-µGAsolution (daily, 4-parameter).


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case, the maximum number of generations was limited to1000 while in the experimental case it was 2000. The longertimeline in the experimental case was intended to producethe best possible solution with the real-world condition.Further, the average fitness (Fig. 5b) shows how the GAexploited the micro population to explore the search spaceunder each fitting criterion. When the average fitness is highand nearly touching the maximum fitness line, the populationis composed of highly measured individuals and is nearconvergence. On the other hand, when the value is low, someor even the majority of the population are lowly measuredindividuals (as compared to the elite). This contrast,however, is necessary in the search because diversity meansmore genetic traits to be introduced in the generations, whichis induced by the mutation and population restartmechanism. The crossover operation will exchange thesegenetic traits with the selected individuals.

The scope of the study is limited to the use of GA in theinverse problem. As such, the computational efficiency andaccuracy of results were not compared with other methods.The computational efficiency, however, can be gauged bythe GA parameters used in the search such as the number ofindividuals in a population and maximum number ofgenerations. With the present computing capabilities, doingthe GA processes (selection, crossover and mutation) aloneis not as computationally intensive as compared with thetime the integrated model SWAP takes to do the simulationsfor the evaluation of individual fitness. As observed, whenthe parameter combination is extreme, the model takes along time to finish the simulation. As regards accuracy, theresults to be discussed show that, in theory, GA is able tomatch the solution with high accuracy. In the study, the 4-and 8-parameter problems had a population size of 20 and30, respectively. A 0.5 probability of crossover was used inall cases and the probability of creep mutation was taken asthe reciprocal of the population size. In all cases, a randomnumber seed of –1000 was used in the search.

NUMERICAL CASE

The results of the 12 cases considered in testing theperformance of GA for the generated “observations” aregiven in Table 1. Considering the ability to match the“measured” and simulated values, the GA performance wasthought to be excellent, with R2 in all cases higher than 91%and in most cases even higher than 99%. The average errorin ET was always lower than 0.003 cm d-1. However, averagedeviations in soil water contents for the whole soil profilevaried between 0.13 and almost 0 cm3 cm-3. In this study,the soil profile is discretised into 33 compartments, i.e. 20and 13 compartments for the first and second soil layers,respectively.

Figure 6a and b shows the reduction in the average errorsbetween the “observed” and simulated ET and soil watercontents during the GA search for the 4-parameter problemusing daily data. In the case where only the soil water (θ)was included in the fitness function, the average errorbetween the “observed” and simulated soil water contentswas reduced rapidly during the first 400 generations.Interestingly, simultaneous reduction of the average errorin ET was also substantial, particularly during the first 100generations, then eventually decreasing with the soil wateralong the remaining generations. For the cases where onlyET or soil water and ET (θ +ETa) were included in the fitnessfunction, the reduction in the average errors in soil waterand ET are still very satisfactory: 0.007 and 0.003 cm3

cm-3, and 0.004 and 0.001 mm d-1, respectively. It isinteresting that optimising the fitness function using the soilwater contents only also improved the ET solution.

Fig. 6.(a) Average error of soil water (q) in a generation with amodified-µGA solution (daily, 4-parameter). ETa indicates ET aloneas criterion, θ, soil water alone and θ + ETa, the combination.(b) Average error of actual evapotranspiration (ETa) in a generationwith a modified-µGA solution (daily, 4-parameter).


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The ability of the GA to reproduce exactly the initialparameters gave a somewhat mixed result but some cleartrends could be observed. First of all, parameters of thesecond soil layer were more difficult to estimate than thosein the topsoil layer. Second, α and n were, in general, betterestimated than Ksat and θsat. Third, including daily values inthe fitness function gave better results than using weeklyvalues. Finally, the inclusion of ET or soil water content ora combination of both in the fitness function gave somediverse results; there is a tendency that using soil water onlyis the best option, although in one case including ET onlywas better (daily, 8-parameter).

As expected, using only soil water in the fitness functionwas best in all cases to predict the soil water (with respectto average error). However, this was not the case for ET.Sometimes ET was predicted better by including only ETin the fitness function (daily and weekly, 8-parameter),sometimes only with soil water (daily, 4-parameter) andsometimes the combination (weekly, 4-parameter). Apossible explanation for this counterintuitive result that ETin the fitness function does not improve the ET estimatesmight be the lack of any dry period in the simulation. Jhoraret al. (2002) have concluded already that such a period isessential in a successful inverse modelling approach basedon ET. The inclusion of ET in the fitness function meansthat less weight is given to θ, making the inverse modellingmore difficult.

As a conclusion: daily observations are better than weekly,four parameters are easier to estimate than eight and thecase for whether to use soil water, ET or a combination, isvague but with an inclination towards soil water.

It is clear from the results that the non-unique nature ofthe inverse problem is to some extent partly handled by GA.The first and second observations in the preceding discussioncould give some insight to this statement. The parameter nwas estimated best in all cases, both in the topsoil andsubsoil. This is attributed to its high sensitivity to both ofthe criteria included in the fitness function. For the otherparameters, there are regions in the search space where theyare not sensitive to either of the two components of thefitness function. GAs are very powerful in highly perturbedfunctions but cannot respond well to relatively flat surfaces.The algorithm is hard to narrow down for all the parameterpossibilities when there is no strong response from the fitnessfunction. The insensitivity factor in non-uniqueness is thenan innate aspect of the problem that even robust searchalgorithms like GA cannot handle. The possible solution tothis is that, when setting up the problem, narrow down asmuch as possible the parameter search space in the vicinityof the expected value.

Aside from the sensitivity of a parameter, the zone of

influence could also affect the solution which could explainwhy some parameters are more difficult to estimate in thesecond layer. Kool and Parker (1988) noticed this in theirstudy: the best sampling point of the hydraulic head duringinfiltration is in the vicinity of the infiltration front. For thecase of ET, the processes in the top 5–10 cm of soil couldbe described by the soil evaporation while below, until oneenters the regions influenced by root water extraction, it isaccounted for by plant transpiration. Probably, the definedbottom boundary condition could have influenced theinverse modelling process also.

Figure 7a–d shows the soil water simulations using thederived MVG parameters from ET as a criterion for the 4-and 8-parameter problems, daily and weekly cases. The caseof the 4-parameter problem (Fig. 7a and b) shows apromising fit with the base values. The minor discrepanciesin the predicted parameter values compared to the originals(see Table 1) did not cause significant deviations betweenthe simulated and “measured” soil water, both in the firstand second soil layers. When the frequency of data collectionis on a weekly basis, the solution is still very reasonable.On the other hand, Fig. 7c and d (daily and weekly, 8-parameter) show a different pattern, especially with the dailybasis in the first soil layer. Although the predicted parametervalues are reasonably better than the weekly basis (see Table1), the soil water situation is overestimated significantly.The reason for this is the high value of θsat predicted by theinverse modelling. Based on the sensitivity analysis, θsat haslow sensitivity to ET. Interestingly, a better estimate of θsatwas obtained using the weekly data.

EXPERIMENTAL CASE

Results of the inverse modelling using the lysimeter dataare also presented in Table 1. As the real parameter valuesare not known, as was the case in the numerical study, theonly criteria to test the parameter estimation by GA are theobserved soil water and ET values. The observed uniquenessin some parameters indicates the robustness of the GA.Overall, the value of the fitness function is low and thecomparison between observed and simulated values showsa big discrepancy between model and measurements.Average errors between observed and simulated ET areabout 0.7 mm d-1 for all cases. Errors in soil water contentsvary between 0.02 and 0.17 cm3 cm-3.

Scatter diagrams of soil water contents for the best (4-parameter, θ + ETa) and the worst (8-parameter, ETa) casesare shown in Fig. 8a and b. The range in observed soil watercontents is limited to wetter conditions (Fig. 8a), so no actualwater stress has occurred in the crops. According to Jhoraret al. (2002), a dry period is of paramount importance for a


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Fig. 7. (a) Simulated soil water as compared to the base values using ETa as search criterion (daily, 4-parameter). (b) Simulated soil water ascompared to the base values using ETa as search criterion (weekly, 4-parameter). (c) Simulated soil water as compared to the base valuesusing ETa as search criterion (daily, 8-parameter). (d) Simulated soil water as compared to the base values using ETa as search criterion(weekly, 8-parameter).


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Fig. 8. (a) Scatter diagrams of the best observed and simulated soilwater contents after a GA solution in the experimental study (4-parameter, θ + ETa ). (b) Scatter diagrams of the worst observed andsimulated soil water contents after a GA solution in the experimentalstudy (8-parameter, ETa )

successful parameter estimation, which is lacking in thiscase. Ranges in observed ET are from 0.3 to 6.5 mm d-1,where the lower values do not indicate water stress but areobservations at the emergence stage of the crop or after arainfall event or low evaporative demand (see Figs. 10 and3). For the worst case (Fig. 8b), the simulated and observedET values still match reasonably well (Table 1). However,the soil water contents show a big discrepancy betweensimulated and observed values. This shows that the soilwater was not matched properly during the search process;the lower cluster suggests that the soil water wasunderestimated and the upper cluster along the 45o linedepicts reasonably matched data (see Fig. 9b). But theaverage error suggests that the prediction improvedcompared to the case of the 4-parameter problem where ETwas used as the criterion (see Table 1). This result reveals

the danger of using only one statistical test criterion in thesolutions.

Figure 9a and b show the results of the soil watersimulations at the upper soil layer using the derived MVGparameters from the 4- and 8-parameter problems, with thethree search criteria being used. In the figures, it is obviousthat soil water (θ) as criterion is adequate to define the soilhydraulic parameters because of its direct relationship withthe parameters. The case of ET (Fig. 9a), however, isinteresting to explore. In theory, for a 4-parameter problem

Fig. 9a. Simulated and measured soil water using the GA derivedMVG parameters in the experimental study (4-parameter). ETaindicates ET alone as criterion, θ, soil water alone and θ + ETa, thecombination.


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with daily or weekly data, the use of ET as search criterioncan define the soil hydraulic parameters adequately. This istrue provided that the other (known) parameters are definedproperly. Figure 9a shows that the simulated and measuredsoil water contents using θ as criterion have reasonableagreement with each other, meaning that the other knownsoil hydraulic parameters defined in this case are quiteappropriate. Therefore, the prediction of the soil water usingET as search criterion was influenced by other factor(s) inthe process. Along this line, it is speculated that the

uncertainty involved in ET prediction is the main cause ofthis discrepancy. As observed in Table 1, the R2 of ET didnot change significantly in all cases.

Figure 10 shows the observed ET from the lysimeter(ETlys) and predicted potential ET (ETpot) from the SWAPmodel. The effect of model and data errors to ETpot can beobserved in the figure: there are times in the season whenETpot values are lower than the ETlys and the unusual oppositetrend in some days is apparent.

DiscussionMany studies presented on inverse modelling in the vadozezone are limited to the so-called forward-backwardsimulation approach, referred to here as the numerical case.Generated output of a model, with or without artificialrandom errors, is used as “observed” values in the objectivefunction to estimate the parameters. In most cases resultswere satisfactory but Jhorar et al. (2002) concluded thatparameters were difficult to assess and emphasis should beput on the prediction of the terms considered in the objectivefunction. In this study, the numerical case showed that theGA was able to represent the terms included in the fitnessfunction very well and parameters could be estimatedreasonably well, especially if only four parameters wereincluded. It should be taken into account that in this study,periods with water stress are completely lacking, which isconsidered essential to produce reliable results (Feddes etal., 1993; Van Dam, 2000; Jhorar et al., 2002). This leadsto the conclusion that the GA approach as used in this studyis a powerful tool in inverse modelling.

The cases where the actual lysimeter data were usedshowed a different picture. Parameter estimations are lesssuccessful and the ability of the model to produce similar

Fig. 9b. Simulated and measured soil water using the GA derivedMVG parameters in the experimental study (8-parameter). ETaindicates ET alone as criterion, θ, soil water alone and

Fig. 10. Comparison of the ETlys and calculated ETpot by SWAP.


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ET and soil water values as observed was worse than thenumerical case, although the overall performance can bedescribed as reasonable. In reality, better prediction of theactual ET may require parameterisation of the sensitiveparameter(s) that can influence the estimate of ET from thesimulation model significantly. At the point where GA couldnot improve the solution any more, the problem could havebeen controlled already by the model and data errors.

The different performances of the numerical andexperimental case indicate clearly the danger of focusingonly on the forward-backward approach. Such a numericalapproach takes no account of the simplifications,assumptions or errors included in any simulation model.Even the inclusion of a generated random error term in thesimulated “observations”, as seen as the ultimate test in manystudies, does not overcome this problem. This can berevealed only by the use of real field data.

In addition, GA as a tool is very promising for the inverseproblem in the unsaturated zone. Aside from its advantageof partially handling the causes of non-uniqueness and beingfavourable to highly perturbed functions, the setting up ofparameters to be investigated can conveniently be arrangedin a series of strings; no complex inversion matrix isrequired. The concepts of micro and restarting populationminimised the problem of the too much redundant fitnessfunction evaluations in a conventional GA and thusimproved the computational time. The features of securGAand 90% convergence criterion improved the search for asolution. From other tests, however, convergence criterialower that 90% did not improve the search; this could beexplained by the reduced opportunity of the good genes inan individual being shared and expressed in a generation.For the case of 90% convergence, this disadvantage iscompromised by the increased frequency of the fresh influxof new genetic materials. However, a comparison of GAwith other methods is highly recommended.

Furthermore, despite the weakness of ET as a criterion inthe inverse problem, its use for the practical application atregional scale is appealing because ET is so far the mostpractical and reliable hydrological component that can bederived from satellite imagery (Bastiaanssen et al., 1998).Setting up the inverse problem appropriately could probablycircumvent its weak points. Further research is needed toaddress this uncertainty.

AcknowledgementThis research has been made possible through the PhDResearch Grant awarded to the first author by theInternational Water Management Institute (IWMI),Colombo, Sri Lanka. Dr. David Carroll of CU Aerospace,

Urbana, IL, USA is acknowledged for his discussions onGenetic Algorithms. The authors are also very grateful toDr. N.K. Tyagi, Director of the Central Soil Salinity ResearchInstitute (CSSRI), Karnal, Haryana, India for permitting theuse of their lysimeter data for this study. The ResearchCommittee at Asian Institute of Technology (AIT), Bangkok,Thailand namely: Prof. Ashim Das Gupta, Assoc. Prof.Rainer Loof, Dr. Roberto Clemente and Dr. Kyoshi Hondaare greatly acknowledged for their continued support.

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inverse modelling in estimating soil hydraulic functions ... · the study area. the procedure...

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