J. Hydrol. Hydromech., 58, 2010, 3, 188–200
DOI: 10.2478/v10098-010-0018-9

INVERSE MODELING OF VADOSE ZONE FLOW PROCESSESUSING SQUARED ε-INSENSITIVITY LOSS FUNCTION
NAVIN K. C. TWARAKAVI1*), HIROTAKA SAITO2),
JIRKA ŠIMŮNEK3), M. TH. VAN GENUCHTEN4)
1*)

Corresponding Author Agronomy and Soils Department, Auburn University, Auburn, AL-36832; Mailto: twarakavi@auburn.edu
Department of Ecoregion Science, Tokyo University of Agriculture and Technology, Fuchu, Tokyo.
3)
Department of Environmental Sciences, University of California Riverside, Riverside, CA.
4)
Department of Mechanical Engineering, COPPE/LTTC, Federal University of Rio de Janeiro, UFRJ, Rio de Janeiro, RJ CEP
21945-970, Brazil.
2)

An accurate representation of reality in numerical variably-saturated flow models requires reliable
estimates of necessary model parameters. Inverse modeling seeks to estimate parameters such as thesaturated and residual water contents, the saturated hydraulic conductivity, the shape parameters of the soil
hydraulic functions, using easily attainable observations of actual or cumulative water fluxes, pressure
heads, water contents, and concentrations. The inverse procedure usually combines the nonlinear leastsquares-based (SSQ) parameter optimization method with a numerical solution of the variably-saturated
flow and transport equations. The SSQ-based inverse method is however sensitive to outliers. A novel
Squared ε-Insensitive Loss Function (SILF) approach is introduced in this study. The SILF approach isinspired by the ε-insensitive loss function proposed by Vapnik (1995). The objective function used in the
SILF approach is similar to the least-squares objective function, except that it penalizes only for errors
greater than a certain predefined acceptable error term ε. The SILF approach shows an improved
performance over the SSQ approach in estimating the soil hydraulic parameters. Apart from providing
robust estimates of the soil hydraulic parameters, the SILF approach also gives an approximation of therelative measurement error during sampling.
KEY WORDS: Vadose Zone, Inverse Problem, Insensitivity Loss Function.
Navin K. C. Twarakavi, Hirotaka Saito, Jirka Šimůnek, M. Th. Van Genuchten: INVERZNÉMODELOVANIE PRÚDENIA VODY VO VODOU NENASÝTENEJ PÔDE S POUŽITÍM
NECITLIVOSTNEJ STRATOVEJ FUNKCIE ε. J. Hydrol. Hydromech., 58, 2010, 3; 33 lit., 5 obr., 3 tab.
Presná reprezentácia skutočností v numerických modeloch prúdenia vo vodou nenasýtenej pôde vyžadujespoľahlivé určenie potrebných parametrov modelu. Inverzným modelovaním sa snažíme o určenie takých
parametrov, ako sú reziduálna vlhkosť pôdy, nasýtená hydraulická vodivosť, tvarové parametre
hydraulických funkcií pôdy, využijúc ľahko realizovateľné pozorovania momentálnych alebo
kumulatívnych tokov vody, tlakových výšok, vlhkostí pôdy a koncentrácií rozpustených látok. Inverzná
procedúra obyčajne kombinuje nelineárnu optimalizáciu parametrov založenú na metóde najmenšíchštvorcov (SSQ) s numerickým riešením transportných rovníc vo vodou nenasýtenej pôde. Táto metóda(SSQ) je však citlivá na náhodné chyby. Nová, necitlivostná stratová funkcia s necitlivosťou ε(SILF),
použitá v tejto štúdii, bola inšpirovaná návrhom publikovaným Vapnikom (1995).
Optimalizovaná funkcia použitá v prístupe SILF je podobná tej, ktorá sa používa v metóde najmenšíchštvorcov s tou výnimkou, že táto penalizuje len chyby väčšie ako je určitá preddefinovaná akceptovateľná
chyba ε. Pri určovaní hydraulických parametrov pôdy táto metóda SILF preukázala svoje prednosti pred
prístupom SSQ. Okrem toho, že metóda SILF dáva robustné odhady hydraulických parametrov pôdy,umožňuje tiež aproximáciu relatívnych chýb merania počas odberu vzoriek.
KĽÚČOVÉ SLOVÁ: vodou nenasýtená pôda, inverzný problém, necitlivostná stratová funkcia.

188

Inverse modeling of vadose zone flow processes using squared ε-insensitivity loss function

Introduction
Experimental determination of soil hydraulicproperties relating the soil water content, the capillary pressure head, and the hydraulic conductivity
can often be a time-consuming and work-intensive
process (Klute and Dirksen, 1986; Dirksen, 1991).
As a result, detailed characterization of soil hydraulic functions required for accurate model predictions of water flow in the heterogeneous vadosezone is seldom done. On the other hand, information about soil-water pressure heads and moisture
contents (also referred to as state variables) can beobtained readily and economically using instruments such as tensiometers, neutron probes, and/ortime domain reflectometry (TDR). The availabilityof state information, such as moisture contents,pressure heads and fluxes, in field-scale vadosezone studies is therefore much more common thanknowledge of soil hydraulic properties. As a result,much research effort has been devoted to estimating
soil hydraulic parameters using this so-called state
information. The utilization of state information(i.e., water content and pressure head data) to infer
model parameters is the quintessence of a class ofmodeling techniques referred to as “inverse problems”.
In the past two decades, a number of inverse
modeling studies for estimating soil hydraulic parameters in the vadose zone has been published(e.g., Kool et al. 1985; Kool and Parker 1987;
Eching and Hopmans 1993; van Dam et al. 1994;
Šimůnek and van Genuchten 1996, Hopmans et al.,1997, Vrugt et al., 2008). The inverse-problembased parameter estimation technique involves theindirect estimation of soil hydraulic functions using
repeated numerical solutions of the governing flow
equation for variable-saturated water flow, such as
the Richards equation (Richards, 1931; Jury et al.,1991), with different parameter sets, and comparing
computed results with experimental observations.
The Richards equation is first solved numericallyusing the parameterized soil hydraulic functions
with initial estimates for their parameters. These
parameters are then optimized by minimization of
the objective function that contains the sum ofsquared deviations between observed and predictedstate variables, using repeated numerical simulations of the flow process (Hopmans et al., 2002).
This iterative inversion of the flow equation in numerical models is in contrast to direct inversion
techniques used in analytical solutions (Yeh, 1986).

To summarize, numerical inverse estimation techniques involve the coupling of a numerical modelof flow with a parameter optimization algorithm
such as the Marquardt-Levenberg method(Marquardt, 1963) or various global optimization
methods (e.g., Vrugt et al., 2008).
Let ỹ = {ỹ1, ỹ2,…, ỹq} denotes the vector of observed state variables of size q (such as water contents, pressure heads, fluxes) at various locations
and times and Y(δ) = {Y1(δ), Y2(δ),…, Yq(δ)} denotes the corresponding model predictions using aparameter set δ = {δ 1, δ 2,…, δ s} of size s such thatδ ∈Θ ⊆ ℜs ,(1)

where ℜ s represents the s-dimensional Euclidean
space and Θ denotes the feasible parameter spacethat is predefined in the model using reasonable
upper and lower bounds for each parameter (Vrugt
et al., 2005). The prediction efficiency of the inverse model with a given parameter set δ is represented by the residual vector E(δ), which is linear/non-linear transformation of differences between the observed data, ỹ, and the model predictions, Y(δ):
E (δ ) = γ (Y (δ ), ~y ) = {e1 (δ ), e2 (δ ),....eq (δ )} ,(2)

where γ(.) is a user-defined linear/non-linear transformation function. In an inverse problem, one
estimates the “best parameter set” by minimizing
some kind of a lumped estimate (referred to as theobjective function) from the residual vector E(δ)
such that model predictions are as close as possible
to corresponding observed values (Vrugt et al.,2005).
The choice of the objective function is important
for a reliable estimation of model parameters. Oneof the commonly used objective functions is the
Sum of Squared Residuals (SSQ) which is based on
non-linear least squares. Essentially, the SSQ objective function is the sum of residuals wherein theresidual is the quadratic transformation of the difference between the observed data and model predictions:q

SSQ( E (δ )) = ∑ Wi [Yi (δ ) − yi ] ,
i =1

2

(3)

where Wi is a weight vector (i = 1..q) that assigns a
degree of importance to each residual. Popularly
used inverse formulations, such as those used in the
HYDRUS suite of softwares (Šimůnek et al., 2005,
189

N. K. C. Twarakavi, H. Saito, J. Šimůnek, M. Th. Van Genuchten2008), employ the weighted SSQ as the objectivefunction to estimate the best parameter set.
Non-linear least squares-based objective functions infer parameters by minimizing the sum of theweighted squared differences between the observed
state variables and those calculated by a mathematical model. A detailed literature review reveals
that non-linear least-squares-based objective functions (such as SSQ) have been the most widely used
objective functions for parameter optimization inhydrological sciences (Hopmans et al., 2002).
However, Finsterle and Najita (1998) have questioned the robustness of the least-squares criterion
because of the tendency of outliers in the measurements to strongly influence the outcome of the inverse modeling. Finsterle and Najita (1998) compared other robust estimators to the standard leastsquares objective function. It was observed thatrobust estimators decreased relative weights ofoutliers and thus made the estimated parameters
less affected by the presence of non-gaussian andgaussian errors (Šimůnek and Hopmans, 2002).
While significant attention has been given to addressing outliers in data sets during optimization,measurement errors that are a result of detection
limits and sensitivity of a particular instrument havebeen assumed to be negligible and ignored in thepast. In this paper, we propose a new approach tohandle measurement errors in datasets in inverse
problems by using a new objective function called
the Squared ε-Insensitive Loss Function (SILF)
proposed by Vapnik (1995). The SILF function isdefined as follows:q

SILF ( E (δ ), ε ) = ∑ Wi li2 (δ )
i =1

(4)

where
⎪⎧ Y (δ ) − yi − εli (δ ) = ⎨ i⎪⎩ 0

Yi (δ ) − yi > εYi (δ ) − yi ≤ εwhere ε is an optimally selected parameter called
the acceptable error. Fig. 1 compares the SILF and
SSQ objective functions. The SILF objective function is very similar to the SSQ objective function
except that residuals smaller than ε are considered
to be insignificant. Similarly to SSQ, a quadratictransformation is used for residuals greater then ε(Vapnik, 1995). One may observe that the SILF
objective function attains the form of the SSQ objective function for the acceptable error equal tozero (ε = 0). The performance of SILF objectivefunctions has been compared to traditional objec190

tive functions (SSQ, Absolute Mean) in previous
studies (e.g., Hastie et al., 2001), in which SILF
was shown to improve the fit between predicted
and observed data. Previous studies have shown an
improved fit of observed data when the SILF objective with a non-zero acceptable error was used,indicating that the SSQ objective function may notnecessarily lead to the best possible parameter estimate (e.g., Kwok and Tsang, 2003). In this paper,
we analyze the robustness of the SILF approach forinverse modeling of vadose zone flow processes.

Fig. 1. Significance of residuals to the value of SSQ and SILF
objective functions.
Obr. 1. Význam reziduálov vzhľadom k hodnotám funkcií SSQ
a SILF.

TheoryGoverning equations and numerical modeling
Variably-saturated flow in soils is a highly
nonlinear and dynamic phenomenon. Commonlyused numerical models for variably-saturated waterflow in the vadose zone typically employ the classical Richards equation (Richards, 1931). For aone-dimensional scenario, the Richards equation is
described mathematically as follows:
∂ ⎡
∂h
∂θ
⎤
= ⎢ K ( h) − K ( h) ⎥ − S ,∂z∂t ∂z ⎣⎦

(5)

where θ is the volumetric water content [–], h – thesoil water pressure head [L], t – time [T], z – thedistance from reference datum [L], K(h) – the unsaturated hydraulic conductivity as a function of hor θ [L T-1], and S is the sink term usually accounting for the root water uptake.
A high nonlinearity in the variably-saturated water flow equation exists due to the dependence ofthe hydraulic conductivity, K(h), and the watercontent, θ(h), on the pressure head, h. Conse-

Inverse modeling of vadose zone flow processes using squared ε-insensitivity loss function

quently, modeling of variably-saturated water flow
involves solving Eq. (5) along with a nonlinear
model that characterizes relationships between thewater content, pressure head, and unsaturated hydraulic conductivity. The Gardner exponential
model (Gardner, 1958), the Brooks and Coreymodel (Brooks and Corey, 1964) and the van
Genuchten-Mualem model (van Genuchten, 1980)
are among the most widely used approaches to represent the dependence of the hydraulic conductivity
and the water content on the capillary pressure
head. Readers are referred to Leij et al. (1997) for a
detailed discussion on the most commonly used soil
hydraulic models. Of all available soil hydraulicmodels, the van Genuchten-Mualem model (vanGenuchten, 1980) is perhaps the most widely usedmodel to characterize the hydraulic conductivity
and water content dependence on the capillary pressure head. The van Genuchten-Mualem model is
described as follows:θs −θr⎧⎪θ r +n m
θ (h) = ⎨1 + αh⎪θ⎩ s[

S e ( h) =

h<0

]

h≥0

θ ( h) − θ r
θs −θrl

K (h) = K s S e [1 − (1 − S em = 1 − 1 / n,

(6a)

(6b)
1/ m m 2

n > 1,

) ]

(6c)
(6d)

where θ(h) is the volumetric water content at thepressure head h [–], θs and θr – the saturated andresidual volumetric water contents [–], respectively;Se(h) – the degree of saturation at the pressure headh [–], Ks – the saturated hydraulic conductivity [LT-1], α (L-1) and n [–] are van Genuchten’s shapeparameters, and l is a tortuosity or pore connectivity
parameter estimated by Mualem (1976) to be 0.5.
Eqs. (5) and (6) have been used with or without
additional modifications to model variablysaturated water flow and have been successfully
extended into two and three dimensions (e.g.,
Pruess et al., 2004 or Šimůnek et al., 2006).
SILF inverse formulationŠimůnek and Hopmans (2002) and Hopmans etal. (2002) provide a detailed description of the SSQ
approach for estimating soil hydraulic parameters
using inverse modeling. For an inverse modelingproblem involving multiple measurement sets ofsize qi, the SSQ-based objective function φ used in
HYDRUS-1D (Šimůnek et al., 2005) is mathematically described as
p

qi

i =1

j =1

2

ϕ (Y (δ ), y , δ ) = ∑ ui ∑ vi , j ⎡⎣Yi , j (δ ) − yi , j ⎤⎦ ,(7)

where p is the number of measurement sets, ui – theweight corresponding to each measurement set, vi,j– the weight corresponding to the jth data point of
the ith measurement set, δ – the parameter set (asdescribed earlier in (1)), ỹi,j – the jth data point of theith measurement set and Yi,j(δ) is the corresponding
model prediction.
In the HYDRUS software, the MarquardtLevenberg algorithm (Marquardt, 1963) is used to
optimize the objective function. The MarquardtLevenberg algorithm combines the Newton andsteepest descent methods, and provides confidence
intervals for the optimized parameters. It should be
noted that the Marquardt-Levenberg algorithm is alocal minimization approach seeking a local minimum in the vicinity of the initial estimate of optimized parameters, requiring reasonable initial estimates of parameters. There are other local andglobal optimization approaches that could be also
considered to solve the optimization problem.
Readers are referred to the above referenced literature (e.g., Šimůnek and Hopmans, 2002; Hopmans
et al., 2002; Vrugt et al., 2005, 2008) for more details. The proposed SILF methodology essentiallyuses the same solution framework as SSQ. The only
difference between the SSQ and SILF based approaches is the definition of the objective function.
To ensure the robustness of the objective function in estimating parameter values during inverse
modeling, a good understanding of the nature ofmeasurement errors in collected datasets is needed(e.g., Roth et al., 1990; Šimůnek and van Genuchten, 1996). Most statistical methods, such as SSQbased inverse modeling, are based on the assumption of homoscedastic and Gaussian errors. Gaussian errors have a similar variance that is independent of the ‘true’ value of a particular variable.
However, one often encounters data when observing natural processes wherein the error variance isdirectly related to the ‘true’ variable value. Sucherrors are referred to as heteroscedastic errors. Homoscedastic errors may be considered as a special
case of heteroscedastic errors with a heteroscedastic
variance of zero. Considering the error distributionof typically observed soil data and inspired by the
SILF objective function, we propose for inverse
191

N. K. C. Twarakavi, H. Saito, J. Šimůnek, M. Th. Van Genuchten

modeling of vadose zone processes the SILF-based
objective function τ that uses relative errors insteadof absolute errors as follows:
p

qi

i =1

j =1

τ (Y (δ ), y , δ ) = ∑ ui ∑ vi, j ⎡⎣li , j (δ ) ⎤⎦

2

where
⎧
⎪ Yi, j (δ ) − yi, j − ε yi , j⎪⎪
li, j (δ ) = ⎨
⎪
⎪ 0
⎪⎩

Yi, j (δ ) − yi, jyi , jYi, j (δ ) − yi, jyi , j

>ε

≤ε
(8)

where symbols represent parameters, observations
and model predictions as described earlier in Eqs.
(3) through (6).
The parameter set, δ, to be estimated may include any combination of soil hydraulic parameters,such as θs, θr, Ks, α, n and l. The observed data set
can include observed water contents, θ(t), pressure
heads, h(t) and/or instantaneous or cumulative
boundary fluxes, q(t) (e.g., infiltration or outflow
data) at different times and locations, or retention,h(θ), or hydraulic conductivity, K(h), data. One
may observe that the SILF-based objective function, τ, attains the form of the weighted Root Mean
Squared Error (RMSE) approach (which is linearly
related to the SSQ-based objective function φ defined by Eq. (7)) when the acceptable error ε isequal to zero. The SSQ-based and SILF-based inverse models thus estimate the same parameter sets
when the acceptable error is zero. Similarly, whenobserved dataset involves only homoscedastic errors, the acceptable error ε may be multiplied by
one instead of the observed value ỹi,j.
In the SILF-based objective function τ (Eq. (8)),
the insensitivity of the error is considered on a relative basis. The error is considered to be significant
only if the ratio of the absolute residual ei(δ) to theobserved value is greater than the acceptable error,
ε. In scenarios wherein the soil measurements suffer from heteroscedastic errors, the SILF-based
objective function may be expected to show an
improvement in fitting the observed data.
The performance of the SILF objective function
is strongly dependent on the choice of ε (Kecman,
2001, Hastie et al., 2001). For example, one cannot
expect reliable estimates of parameters if ε is larger
than the range of the observed data. On the contrary, when the acceptable error ε is zero (same as

192

SSQ), the objective function may overfit measured
data by fitting measurement errors (Hastie et al.,2001). The value of ε must therefore be selected sothat it reflects the statistical distribution of errors in
the measured data in some way. Several researchers
have pointed out a direct linear dependence of the
optimal value of ε on the noise in the observed data
(Kwok and Tsang, 2003; Smola et al., 1998). The
optimal ε has been suggested to be between 0.6 and1.0 times the variance of the noise in the observed
data set. One may thus choose the optimal value ofε when the variance of noise in the observed data is
known. For the SILF objective function describedin (8), one may use an acceptable error that is 0.6–
–1.0 times the relative sensitivity of the measuring
instrument.
In many cases, the variance of the observed data
is not known. It may then be of interest to estimate
the noise in the observed data for quality assurance
and quality control purposes during the parameter
optimization process. In such case we suggest thefollowing approach: (a) vary the acceptable error, ε,from zero to a certain maximum value at equalintervals, (b) perform inverse modeling using the
SILF approach, (c) estimate the Root Mean
Squared Error (RMSE) between the observed andpredicted data, and (d) choose the optimal value of
ε that corresponds to the minimum RMSE.
The SILF formula was implemented in the inverse modeling framework of the HYDRUS-1D
software package (Šimůnek et al., 2005). No additional changes of the model were needed since therest of the inverse modeling framework for the
SILF approach is the same as for the SSQ approach. We then analyzed the performance of the
SILF approach by considering two sets of casestudies. In the first case, the soil hydraulic parameters were estimated using inverse modeling for ahypothetical numerical one-step outflow experiment, in which the observed data sets were createdby perturbing true values (model-generated) withheteroscedastic noises of varying levels. In the second case, we compared the performance of the
SILF and SSQ approaches for several tutorial examples supplied with the HYDRUS-1D software.
Case studies

Case Study 1: Numerical analysis of a one-stepoutflow experiment
In the first case study, we consider the inverse
solution of a one-step outflow experiment. The

Inverse modeling of vadose zone flow processes using squared ε-insensitivity loss function

experiment is setup based on the problem presented
by Kool et al. (1985). The soil profile, consisting of
a 3.95-cm long soil sample and a 0.57-cm thickceramic plate, is discretized using 50 nodes, five of
which represent the ceramic plate. Only a few discretization nodes were needed in the ceramic plate
since the plate remains saturated during the entire
experiment, thus causing the flow process in theplate to be linear. Outflow is initiated using a pressure head of –10 m imposed on the lower boundary.
The direct problem was first solved using the following soil hydraulic parameters: θr = 0.187, θs == 0.388, α = 0.042 cm-1, n = 1.535, Ks = 5.4 cm h-1,
Ks (of the ceramic plate) = 0.3 cm h-1, and l = 0.5, to
calculate cumulative bottom fluxes at selected time
steps that were assumed to represent the‘true’/unperturbed observed data set yi for the inverse problem. Perturbed data sets of varying noiselevels were then created as follows:yi* (ξ ) = yi [1 ± ξ *ψ (0,1)]

i = 1..q(8)

where q is the size of the observed data set (here q= 9), ξ – a perturbation level that describes themagnitude of noise, ψ(0,1) is a normally distributed
random number, and yp (ξ ) is the perturbed observed data as a function of ξ. Fig. 2 shows theunperturbed (true) and perturbed cumulative bottom
fluxes (ξ = 0.01, 0.03, 0.05 and 0.07) used in the
numerical experiment. The goal of the inverse
problem is to estimate soil hydraulic parameters, θr,α, and n, from observed cumulative bottom
fluxes y* (ξ ) . Similarly as Kool et al. (1987), we
assumed that θs and Ks were measured independently and were thus known. Both SSQ and SILF
approaches were used to estimate the soil hydraulicparameters using perturbed data sets. We used thesame initial parameter estimates for the SSQ andSILF approaches to allow for a reasonable comparison of the inverse modeling results. The obtained
soil hydraulic parameters were then compared totheir true values.
The following key objectives were addressed inthis analysis: (1) How well do the SILF and SSQ
approaches fit the perturbed and underlying true
observation data set? (2) How close to the ‘true’
parameter set are the soil hydraulic parameters estimated from perturbed observations using the SILF
and SSQ approaches? And finally, (3) does the
optimal acceptable error used in the SILF approachprovide an information about the noise in observations?

Fig. 2. Unperturbed (true) and perturbed observed data sets fordifferent perturbation levels (ξ = 0.01, 0.03, 0.05, and 0.07)
used in the numerical experiment.
Obr. 2. Pozorované údaje (nevyhladené reálne) a vyhladenéúdaje (s chybami merania a spracovania) pre rozdielne úrovne
spracovania (ξ = 0,01; 0,03; 0,05 a 0,07), ktoré boli použité
v numerických experimentoch.

The SILF and SSQ approaches were used to estimate the soil hydraulic parameters using the data
sets perturbed with different levels of noise. As
described earlier, the choice of the acceptable error
is crucial to the efficiency of the SILF approach. Inorder to estimate the optimal value of the acceptable error, a thorough search was performed as
described earlier. The optimal value of the acceptable error minimizes the Root Mean Squared Error(RMSEp) between the perturbed data and model
predictionsq

RMSE p =∑ (Y (δ ) − yp )
i =1

i

i

q

2

,

(9)

where Yi (δ ) is the model-predicted data that is
calculated using the parameter set δ estimated fromthe perturbed observed data set. As an example,
Fig. 3 shows the variation of the SILF-based function, τ, and the RMSEp (between predicted values
and observed data with the perturbed level ξ = 0.03)
for inverse solutions with different acceptable errors. It is interesting to observe that the solution
obtained using the SSQ approach does not produce
the minimal RMSEp even though the SSQ objectivefunction is designed to minimize RMSEp. The
RMSEp is minimized with an acceptable error ε == 0.031 for the perturbed data set of ξ = 0.03. Similar analysis carried on all perturbed data sets shows193

N. K. C. Twarakavi, H. Saito, J. Šimůnek, M. Th. Van Genuchtenq

RMSEup =∑ (Y (δ ) − y )
i =1

i

i

q

2

.

(10)

0.012

0.01

RMSEp
SILF

Error (in cm)

0.008

0.006

0.004

0.002
Optimal ε
0
0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Acceptable Error, εFig. 3. SILF and RMSEp for the perturbed data set (ξ = 0.03) inthe SILF approach as a function of the acceptable error. The
star symbol indicates RMSEp for the SSQ approach. The error
indicates the optimal estimate of the acceptable error.
Obr. 3. SILF a RMSEp pre súbor vyhladených údajov (ξ =0.03) pre SILF ako funkcia akceptovateľnej chyby. Hviezdička
označuje RMSEp pre SSQ. Chyba označuje optimálnu prijateľnú chybu.

194

0.08
0.07
0.06

Acceptable error, εa nearly 1:1 relationship between the optimal acceptable error and the perturbation level ξ of thedata set. Fig. 4 shows a plot between the optimal
acceptable error ε estimated with the SILF approach and the perturbation level ξ of the perturbed
data set. The observed linear trend between thesetwo variables is consistent with the observations
made in earlier applications of the SILF approachelsewhere (Smola et al., 1998). Similarly to Smolaet al. (1998), a strong linear relationship exists alsoin our analysis (ε = 0.95ξ; R2 = 0.97). This result isvery useful as it indicates that measurement errors,
as a result of instrument sensitivity during vadosezone sampling, can be estimated indirectly using
inverse modeling.
One of the biggest strengths of the SILF approach is the robustness in capturing the underlying
functional relationship despite of errors in data sets
(Hastie et al., 2001; Vapnik, 1995). Parameter sets
estimated using different perturbed data sets wereused to calculate cumulative outflow fluxes, whichwere then compared to the corresponding perturbed(RMSEp) and unperturbed (RMSEup, Eq. (10)) observations.

0.05
0.04
0.03

ε =0.95ξ

0.02

R = 0.97

2

0.01
0
0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Perturbation level, ξFig. 4. Relationship between the estimated optimal acceptableerror and the perturbation level. Note that the strong linear
dependency is similar to that suggested by Smola et al. (1998).
Obr. 4. Vzťah medzi určenou optimálnou prijateľnou chybou
a úrovňou vyhladenia chýb. Všimnite si, že lineárna závislosť
je podobná tej, ktorú navrhol Smola et al. (1998).

Tab. 1 shows estimates of RMSEup and RMSEp
for cumulative outflow fluxes using soil hydraulicparameter sets estimated using inverse analysis ofdifferent perturbed data sets. An obvious observation in Tab. 1 (as previously seen in Fig. 3) is that
the SILF approach seems to have smaller RMSE
values than SSQ approach for all perturbation levels. It can therefore be concluded that the SILF
approach is an improvement over SSQ for inverse
modeling purposes. Another interesting observationin Tab. 1 is the relative magnitude of RMSEup and
RMSEp. At low perturbation levels, the RMSEup issmaller than the RMSEp indicating that both approaches (SSQ and SILF) seem to capture the underlying hydrological process well in spite of thenoise in the data set. However, when perturbationlevels are high enough (ξ = 0.07) for the noise in
the data set to override the hydrological information, both SSQ and SILF (to a lesser extent as indicated by the smaller RMSEp and RMSEup) approaches seem to lose their generalization ability
and fail to capture the underlying functional relationship, resulting in RMSEup being greater than
RMSEp. Tab. 2 compares the soil hydraulic parameters estimated using the SILF and SSQ approaches from different data sets to their ‘true’ values. Parameters estimated using the SILF approach
are in general closer to the ‘true values’ than thoseestimated using the SSQ approach.

Inverse modeling of vadose zone flow processes using squared ε-insensitivity loss function

T a b l e 1. Root Mean Squared Errors (RMSE) between observed bottom water fluxes (perturbed and unperturbed) and those
calculated with parameters estimated using inverse analysis of the onestep outflow experiment.
T a b u ľ k a 1. RMSE medzi tokmi vody pozorovanými cez dolnú plochu vzorky pôdy (merané a vyhladené), ako aj vypočítané
s parametrami určenými inverznou analýzou jednostupňového výtokového experimentu.

Perturbationlevel, ξ
0.00
0.01
0.03
0.05
0.07

RMSEp [×10-3 cm]
SILF
SSQ approach
approach
0.00
0.00
3.90
3.89
11.20
9.30
20.40
20.30
18.52
18.39

Acceptable error,
ε
0.00
0.008
0.030
0.040
0.072

RMSEup [×10-3 cm]
SSQ
SILF
approachapproach
0.00
0.00
3.56
3.3
4.85
2.96
15.32
14.85
20.43
18.85

T a b l e 2. Soil hydraulic parameters (θr, α, and n) estimated from the unperturbed and perturbed data sets using the SSQ andSILF approaches. The values in parentheses are the estimates of 95% confidence intervals.
T a b u ľ k a 2. Hydraulické parametre pôdy (θr, α, a n) určené z nameraných a vyhladených hodnôt pomocou procedúr SSQ
a SILF. Hodnoty v zátvorkách sú odhady s 95% pravdepodobnosťou výskytu.
Inverse approach anddata set used

Acceptableerror εSSQ/SILF (ξ = 0.00)

0.00

SSQ (ξ = 0.01)–

SILF (ξ = 0.01)

0.008

SSQ (ξ = 0.03)–

SILF (ξ = 0.03)

0.030

SSQ (ξ = 0.05)–

SILF (ξ = 0.05)

0.040

SSQ (ξ = 0.07)–

SILF (ξ = 0.07)

0.072

θr

α

n

0.187
(0.186, 0.188)
0.182(0.171, 0.191)
0.182(0.174, 0.191)
0.169(0.140, 0.197)
0.179(0.149, 0.191
0.178(0.118, 0.238)
0.187(0.128, 0.214)
0.119(0.052, 0.186)
0.120(0.072, 0.185)

0.042(0.041, 0.043)
0.040(0.034, 0.045)
0.040(0.036, 0.043)
0.034(0.020, 0.047)
0.039(0.024, 0.043)
0.053(0.011, 0.081)
0.046(0.018, 0.061)
0.021(0.012, 0.0304)
0.022(0.014, 0.0334)

1.535(1.534, 1.536)
1.490(1.428, 1.551)
1.510(1.432, 1.544)
1.424(1.324, 1.522)
1.474(1.354, 1.531)
1.630(1.069, 2.001)
1.535(1.169, 1.893)
1.299(1.19, 1.403)
1.299(1.22, 1.39)

It may also be observed in Tab. 2 that parameters
estimated using the SILF approach tend to havesmaller confidence intervals than those estimatedusing the SSQ approach. The robustness of theinverse problem may be analyzed by plotting response surfaces (Hopmans et al., 2002). The response surface is obtained by solving the flow
equation, with the appropriate boundary and initialconditions, for many possible combinations of a
selected pair of parameters within their predetermined ranges, while keeping the other parametersconstant. This type of analysis should also be done
before the experiment is conducted to investigate
the well-posedness of the inverse problem (Hopmans et al., 2002). Fig. 5 presents one set of response surfaces showing the variation of the SSQ(SILF with the acceptable error of zero) and SILF

(for the optimal non-zero acceptable error) objective functions for the α and n parameters for the
inverse analysis with the perturbed data set (ξ == 0.03). Even though θr is a parameter of interest in
this exercise, we choose to discuss the response
surface for α and n parameters because the residual
water content tends to be relatively insensitive tothe inverse modeling procedure. For estimation of
the response surface, we assumed θr to be the optimized value of 0.187. The response surface of the
SILF objective function with the optimal acceptableerror indicates an improvement in inverse modeling. The greater slope of the response surface
around the minimum of the SILF objective function
implies smaller confidence intervals of optimized
parameters.
195

N. K. C. Twarakavi, H. Saito, J. Šimůnek, M. Th. Van Genuchten

1.9

1.9

(a)

1.8

1.7

1.6

1.6

1.5

1.5

n

n

1.7

1.4

1.4

1.3

1.3

1.2

1.2

1.1
0.01

(b)

1.8

0.02

0.03

0.04

0.05

α (1/cm)

-2

0.06

-1.8

0.07

0.08

0.09

1.1
0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

α (1/cm)

-1.6

-1.4

-1.2

Log(SSQ/ SILF)

-1

-0.8

-0.6

Fig. 5. Response surfaces indicating the sensitivity of the objective functions [(a) SSQ and (b) SILF] to the α an n [–] parameters
for the inverse analysis of the onestep outflow experiment using observed data perturbed with ξ = 0.03. The star symbol indicatesthe location of the ‘true’ parameter set and θr = 0.187.
Obr. 5. Povrchy, indikujúce citlivosť optimalizovaných funkcií [(a) SSQ a (b) SILF] k parametrom α a n [–] pre inverznú analýzu
jednokrokového výtokového experimentu s využitím vyhladených údajov s ξ = 0,03. Hviezdička označuje miesto „skutočného”
súboru parametrov a θr = 0,187.

Case Study 2: Tutorial examples from HYDRUS-1D
Previous studies (such as Toorman et al., 1992,
van Dam et al., 1992) have observed inverse modeling of one-step outflow experiments when the OF
is composed exclusively of cumulative outflow data
results in parameter estimates that are often unreliable and non-unique. In order to further verify if the
SILF approaches universally improves the inverse
modeling results, we performed a second case study
that involved the analysis of the performance of the
SSQ and SILF approaches for a variety of realworld examples. For this purpose we consider thefollowing examples that are provided as tutorials of
inverse modeling and distributed with the HYDRUS-1D software package (http://www.pcprogress.com/en/Default.aspx?hydrus-1d): (a) horizontal flow and redistribution (Vachaud, 1968), (b)horizontal infiltration (Vachaud, 1968), (c) upward
infiltration (Wierenga, personal communication),(d) drainage in a sand column (Vachaud, 1968), (e)196multistep outflow experiment (Hopmans, personalcommunication), and (f) transient flow involving
hysteresis (Vogel et al., 1996). Tab. 3 lists the typeof observed data and estimated parameters in eachof these examples, as well as the RMSE estimatesfor predictions using the SSQ and SILF approaches.
In each of these case studies that cover a variety ofreal-world vadose zone flow experiments, the SILF
approach exhibits an improvement in parameter
estimation (lower RMSE). Even though the improvement is not significant for some cases, thepresence (even if, insignificant) of an improvement
over SSQ approach indicates the need for further
exploration in to the efficacy of the SILF approach.
We would like to note that the SILF approach isinteresting for two reasons: (a) presence of an improvement in parameters over the SSQ approach aswell as (b) estimation of an acceptable error thatprovides an insight to the accuracy of the experiments.

Inverse modeling of vadose zone flow processes using squared ε-insensitivity loss function
T a b l e 3. Performance of the SSQ and SILF approaches in estimating the soil hydraulic parameters for the selected HYDRUS1D tutorial problems.
T a b u ľ k a 3. Výsledok aplikácie procedúr SSQ a SILF na určenie hydraulických parametrov pôdy pre vybraný príklad modelu
HYDRUS-1D; 1 – horizontálne prúdenie a redistribúcia; 2 – horizontálna infiltrácia, 3 – infiltrácia vertikálne nahor, 4 – drenáž
pieskovej vzorky pôdy, 5 – viackroková výtoková metóda, 6 – neustálené prúdenie s hysterézou.
Inverse model
Horizontal flow
and redistribution1)
Horizontal infiltration2)
Upward Infiltration3)

Observed data
[units]Estimated parameters

RMSE in
SSQ approach

θ(t) [–]θs, θr, α, n, Ks, l

0.0261SILF approach
Acceptableerror, ε0.01

θ(t) [–]θs, θr, α, n, Ks, l

0.04120.05

0.0396h(θ) [cm],θ(t) [–]
h(t) [cm]q(t) [cm s-1]
θ(t) [–]

θr, α, n, Ks, l

3.73780.01

3.73360.01440.15

0.01407.62980.15

7.52214.60590.06

4.6043θr, α, n, Ks, l
Drainage in a sandcolumn4)h(t) [cm]θs, θr, α, n, Ks, lMultistep outflow
q(t) [cm s-1]
experiment 5)h(t) [cm]α, αw*, nTransient flow
θ(t) [–]
involving hysteresis 6)*
shape parameter for the wetting branch of the retention curve

Summary and conclusions

Previous studies (such as Finsterle and Najita,
1998) have concentrated on robust estimators thatwere designed for tackling outliers in the data sets.
Another small, yet significant, source of error thatneeds to be addressed during inverse modeling is
the effect of measurement errors that are a result of
instrument sensitivity and detection limits. We hypothesized that consideration of measurement errors in the inverse modeling process is bound tofurther improve the reliability of parameter estimation.
In this paper we evaluated an inverse analysis ofvadose zone flow processes to estimate soil hydraulic parameters wherein the measurement errors
were considered. The proposed method, called the
SILF approach, is based on the objective functionthat uses the squared ε-insensitive loss function
proposed by Vapnik (1995) that sums up quadratically transformed estimates of residuals whose absolute values are above a pre-defined thresholdlevel (called the acceptable error). The SILF approach thus decreases the cumulative impact ofsmaller measurement errors by not penalizing themin the objective function. The performance of the
SILF approach was tested on a variety of case studies and compared to the standard non-linear leastsquare approach (SSQ), in which the acceptableerror is always set to zero. The SILF approach

RMSE
0.0257shows an improvement over the traditionally used
SSQ approach and provides more robust estimatesof soil hydraulic parameters.
One of the key requirements of the SILF approach is an apriori estimate of the acceptable errorε. As shown in this paper, when information regarding the acceptable error is not available, it can be
estimated (although it is time consuming) simultaneously with the soil hydraulic parameters using athorough, iterative, search-based technique. Apartfrom improved estimates of the soil hydraulic parameters over SSQ, the SILF approach can thus
provide information about the level of noise in observed data sets (the optimal acceptable error) that
may be used for quality control purposes at a laterstage. The estimated optimal acceptable error also
provides an approximation of the relative measurement and sampling errors caused during datacollection stage. For inverse models based on realworld experiments, we believe that the value of
acceptable error to be used can depend on the instrumentation or measurement errors involved. In
cases where such data is not available, the acceptable error has to be selected on a trial-and-error
basis.
There are many different types of errors associated with measurements that should be taken into
account. Even though the currently proposed approach lumps all kinds of errors together, estimation of hydraulic parameters may improve if these197

N. K. C. Twarakavi, H. Saito, J. Šimůnek, M. Th. Van Genuchten

errors can be separated in terms of their sources.
This is one of the areas of future research. The
SILF approach is expected to be useful in otherhydrological inverse modeling/parameter estimation problems, especially when apparent measurement and/or sampling errors are expected in observations and when the number of empirical model
parameters is large so that the model has the flexibility to over-fit the data. Application of the SILF
approach to other hydrological problems is also a
topic of our future research interest.
List of symbols
E(δ)
h
K(h)

– residual vector [–],
– soil water pressure head [L],– unsaturated hydraulic conductivity as a function ofh [L T-1],– saturated hydraulic conductivity [L T-1],
Ksl– tortuosity or pore connectivity parameter [–],n– van Genuchten’s shape parameter [–],
p
– number of measurement sets,sℜ– s-dimensional Euclidean space [–],
S
– sink term [L T-1],– degree of saturation at the pressure head h [–],Se(h)t– time [T],– weight corresponding to each measurement set,ui
Wi
– weight vector [–],
γ(.)– linear/non-linear transformation function [–],– perturbed observed data as a function of ξ,y * (ξ )z– distance from reference datum [L],α
– van Genuchten’s shape parameter [L-1],
ε– acceptable error [–],
Θ– feasible parameter space [–],
θ– volumetric water content [–],
θ(h)– volumetric water content at the pressure head h [–],– saturated volumetric water contents [–],θs
θr
– residual volumetric water contents [–],
– weight corresponding to the jth data point of the ithvi,jmeasurement set,
– jth data point of the ith measurement set,ỹi,jξ
– perturbation level that describes the magnitude
of noise,ψ(0,1) – normally distributed random number,Y(δ) = {Y1(δ), Y2(δ),…, Yq(δ)} – model predictions using aparameter set δ [–],
– vector of observed stateỹ ={ỹ1, ỹ2,…, ỹq}variables of size q [–],– parameter set of size s[–].
δ = {δ 1, δ 2,…, δ s}
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Received 27 April 2009
Accepted 23 June 2010

INVERZNÉ MODELOVANIE PRÚDENIA VODYVO VODOU NENASÝTENEJ PÔDEPOUŽITÍM NECITLIVOSTNEJ
STRATOVEJ FUNKCIE εNavin K. C. Twarakavi, Hirotaka Saito,
Jirka Šimůnek, M. Th. Van Genuchten
Predchádzajúce štúdie (ako napr. Finsterle a Najita,
1998) sa sústredili na robustné odhady eliminácie chýb v

databázach. Ostatné sú malé, ale stále významné chyby,
ktoré by mali byť vylúčené počas inverzného modelovania, ako je vplyv chýb merania, ktoré sú dôsledkom
citlivosti prístrojov a rozsahu merania. Predpokladalisme, že uváženie chýb merania počas inverzného modelovania je spojené s ďalším zvýšením spoľahlivosti určenia parametrov.
V tejto práci sa hodnotí inverzná analýza prúdenia
vody v nenasýtenej zóne pôdy, s uvážením chýb merania. Navrhnutá metóda nazvaná SILF je založená napoužití kvadratickej, necitlivostnej stratovej funkcie snecitlivosťou ε, navrhnutej Vapnikom (1995), sumujúcej
kvadraticky transformované odhady reziduálov, ktorých
absolútne hodnoty sú vyššie, ako preddefinovaný limit(nazývaný tiež akceptovateľnou chybou).
Metóda SILF znižuje sumárny vplyv malých chýbmerania tak, že ich nepenalizuje v optimalizačnej funkcii. SILF bol testovaný na viacerých prípadových štúdiách a výsledok bol porovnaný so štandardnou nelineárnou metódou najmenších štvorcov (SSQ), v ktorej jeprijateľná chyba vždy nastavená na nulu. Tento prístup
(SILF) je zlepšením tradične používaného prístupu SSQ
a výsledkom sú robustnejšie odhady hydraulických parametrov pôdy.
Jedným z kľúčových požiadaviek prístupu SILF je apriori určenie prijateľnej chyby ε. Ako je to ukázané v
tomto článku, ak nie je dostupná informácia o prijateľnej
chybe, potom táto chyba môže byť určená (hoci je točasovo náročné) simultánne s hydraulickými parametrami pôdy pomocou iteratívnej techniky. Okrem zlepšených odhadov hydraulických parametrov pôdy v porovnaní s SSQ, prístup SILF dáva informácie o úrovni šumuv meraných hodnotách – optimálna akceptovateľná chyba, ktorá môže byť využitá na neskoršiu verifikáciu.
Optimálna akceptovateľná chyba dáva tiež aproximáciu
relatívnych chýb merania a vzorkovania, počas zberuúdajov. Sme presvedčení, že pre inverzné modelovanie,založené na meraniach, akceptovateľná chyba závisí odvlastností prístrojov a od chýb merania. V prípade neexistencie takýchto údajov, prijateľná chyba môže byť
vybraná postupom pokus – chyba.
Existuje množstvo chýb v dôsledku meraní, ktoré musia byť vzaté do úvahy. Hoci bežne používaný postupzdružuje všetky druhy chýb, určenie hydraulických parametrov sa môže zlepšiť, ak tieto chyby môžu byť separované podľa ich zdrojov. Toto je jednou z oblastí budúceho výskumu. Procedúra SILF bude pravdepodobneužitočná aj v iných metódach inverzného modelovania,
resp. určovania parametrov, predovšetkým v prípadoch,keď sa očakávajú chyby pri meraní a vzorkovaní a keď
je počet parametrov modelu väčší. Použitie procedúry
SILF v iných hydrologických problémoch je predmetomďalšieho výskumu.
Zoznam symbolovE(δ)
h– reziduálny vektor [–],– tlaková výška pôdnej vody [L],199

N. K. C. Twarakavi, H. Saito, J. Šimůnek, M. Th. Van Genuchten
K(h)
Ks
l
n
p
ℜs
S
Se(h)
t
ui
Wi
γ(.)
y * (ξ )
z

α
ε

200

– nenasýtená hydraulická vodivosť ako funkcia h[L T-1],– nasýtená hydraulická vodivosť [L T-1],– tortuozita, alebo parameter konektivity pórov [–],– van Genuchtenov tvarový parameter [–],– počet sérií meraní,– s-rozmerný Euklidovský priestor [–],
– odberový člen [L T-1],– nasýtenosť pri tlakovej výške h [–],– čas [T],– váha zodpovedajúca každej sérii merania,– váhový vektor [–],– lineárna/nelineárna transformačná funkcia [–],– vyhladené pozorované údaje, ako funkcie ξ,– vzdialenosť od referenčnej úrovne [L],– van Genuchtenov tvarový parameter [L-1],– akceptovateľná chyba [–],

Θ
θ
θ(h)
θs
θr
vi,j
ỹi,j
ξ

– využiteľný priestor parametrov [–],
– objemová vlhkosť [–],
– objemová vlhkosť pri tlakovej výške h [–],
– nasýtená objemová vlhkosť [–],– reziduálna objemová vlhkosť [–],– váha zodpovedajúca údaju jth, sérii merania ith,– údaj jth v sérii merania ith,– úroveň vyhladenia, ktorá je charakterizovaná úrovňou šumu,ψ(0,1) – normálne rozdelenie náhodných čísel,– vektor pozorovanej stavovej
ỹ ={ỹ1, ỹ2,…, ỹq}premennej veľkosti q [–],
Y(δ)={Y1(δ), Y2(δ),…, Yq(δ)} – modelová predpoveď pomocou
série parametrov δ [–],– súbor parametrov veľkosti s [–].
δ ={δ 1, δ 2,…, δ s}