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Transp Porous Med (2013) 98:485–504
DOI 10.1007/s11242-013-0155-9
Estimability Analysis and Optimisation of Soil Hydraulic
Parameters from Field Lysimeter Data
V. V. N g o ·M. A. Latifi ·M.-O. Simonnot
Received: 24 August 2012 / Accepted: 19 March 2013 / Published online: 9 April 2013
© Springer Science+Business Media Dordrecht 2013
Abstract Modelling the water-flow in the vadose zonerequires accurate hydraulic parame-
ters to be obtained at the relevant scale. Weighable lysimeters enable us to monitor hydraulic
data at an intermediate scale between lab and field scales and they can be used to optimise
these parameters. Parameter optimisation using inverse methods may be limited by the non-
uniqueness of the solution. In this contribution, an estimability method has been used to
assess the estimability of the van Genuchten–Mualem parameters, to evaluate the informa-
tion content of the data collected from a bare field lysimeter and to optimise the estimable
model parameters. Daily data were monitored from a 2 m3lysimeter, filled with the soil of a
former coking plant: pressure heads and water contents were measured at three depths (50,
100, 150 cm), cumulative boundary water fluxes. Water-flow was represented using the one-
dimensional single-porosity model implemented in HYDRUS-1D code. The estimability of
the 5 van Genuchten–Mualem hydraulic parameters and the information content of different
data were evaluated by sequentially calculating a sensitivity coefficient matrix. Optimisation
was achieved by the Levenberg-Marquardt algorithm. The estimability analysis revealed that
estimability of the soil hydraulic parameters, based on the combination of daily pressure
heads and water contents, was higher than those based on these data separately. In case of
2.4 being considered as a cut-off criterion for this study, all the parameters were considered
estimable from daily data in the decreasing order: θs,n,Ks,α,θ
r. Hydraulic parameters
were optimised in four scenarios: θsand nwere estimated with reliability while α, Ksand
θrwere uncertain. However, the narrow variations in measured data restricted parameter
optimisation.
V. V. N g o ·M. A. Latifi ·M.-O. Simonnot (B
)
Laboratoire Réactions et Génie des Procédés, Université de Lorraine-CNRS, 1 Rue Grandville,
BP 20451, 54001 Nancy Cedex, France
e-mail: Marie-Odile.Simonnot@univ-lorraine.fr
V. V. N g o
Laboratoire d’Hydrologie et de Géochimie de Strasbourg, Université de Strasbourg/EOST-CNRS,
1 Rue Blessig, 67084 Strasbourg Cedex, France
123
486 V. V. N g o e t a l .
Keywords Estimability analysis ·HYDRUS-1D ·Lysimeter ·Parameter optimisation ·
Single-porosity model
1 Introduction
Modelling of water-flow and solute transport in the vadose zone requires obtaining accurate
hydraulic parameters at the relevant scale. In the past, many authors have studied water-
flow and solute transport at the lysimeter scale, which is considered as an intermediate scale
between laboratory and field scales (Hermsmeyer et al. 2002a;Ingwersen et al. 2006;Luster
et al. 2008;Majdoub et al. 2001;Schoen et al. 1999;Seyfarth and Reth 2008). Lysimeters are
most often equipped with sensors, e.g. TDR probes and tensiometers, for monitoring water
contents and pressure heads. Weather data measurement and weighable lysimeters enable an
accurate controlling of fluxes at upper and lower boundaries (Durner et al. 2008;Meissner
et al. 2007).
Hydraulic parameters can be directly measured from soil samples at the bench scale. The
results obtained are quick and accurate, but differ from the one obtained at the field scale.
Therefore, inverse estimation methods have been widely used (e.g. Eching and Hopmans
1993;Si and Kachanoski 2000;Kohne et al. 2006;Mertens et al. 2006). Given the high
reliability of data obtained from weighable lysimeters, coupling flow models and optimizing
algorithms for determining hydraulic parameters at the lysimeter scale offers a great predictive
potential (Olyphant 2003). The determination of these parameters at this scale by inverse
simulation has been reported (Abbaspour et al. 1999;Sonnleitner et al. 2003;Kelleners et al.
2005;Wegehenkel et al. 2008;Durner et al. 2008). However, few studies have been published
on their reliable estimation at the lysimeter scale. Appropriate methods for assessing potential
applicability of an inverse estimation technique based on the available data are still being
intensively discussed in the scientific community.
Problems regarding the non-uniqueness of optimised parameters may occur (Simunek et al.
2001;Vrugt et al. 2001) due to various reasons, e.g. low sensitivities of optimised parameters
to the investigated criteria (Ines and Droogers 2002), mutual parameter dependency and high
measurement noise (Vrugt et al. 2001) or local algorithm optimisation (Kelleners et al. 2005).
These were met, for instance, in evaporation experiments performed on bare lysimeters, due
to the limited range in the experimental pressure head and water content data (Schwarzel et al.
2006). Soil moisture and pressure head variations were too small to produce a well-defined
global minimum in the objective function. The non-uniqueness problem can be partly solved
by including additional and more accurate measurements in the objective function (Russo
et al. 1991;Eching and Hopmans 1993;Van Dam et al. 1994). However, the information
content of experimental data appears to be much more important than the amount of data used
for model calibration (Gupta et al. 1998). Focus on subsets, where the model is most sensitive
to parameter changes, can also improve the quality of inverse solution (Vrugt et al. 2001).
Weighable lysimeters enable us to collect data regarding boundary water fluxes and soil
hydraulic variables (pressure head and water content at different depths) with a chosen time-
step. We used an estimability technique (Yao et al. 2003) that had been successfully applied
in chemical engineering, in order to assess the information content of our data and improve
parameter estimation (e.g. Jayasankar et al. 2009;Kou et al. 2005). This method provides
information about model parameter estimability from various output predictions, thereby
enabling us to account for correlations among parameters. It is based on the combination of
dynamic models and data as well as response variables available at different sampling rates
throughout each experimental run (Kou et al. 2005).
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Estimability Analysis and Optimisation of Soil Hydraulic Parameters 487
To begin with, in this method, we used the most simple single-porosity model (SPM) to
represent water-flow in the bare field lysimeter. SPM assumes that the soil is homogeneous
with the flow being described by the Richards equation; it involves the five parameters of the
van Genuchten–Mualem model. The numerical computer code, HYDRUS-1D, was used to
solve transport equations and optimise parameters. Firstly, the information content of different
daily data (pressure heads, water contents and cumulative water fluxes) was evaluated. Thus,
the more useful data would be chosen to optimise the set of estimable parameters in different
scenarios.
2 Material and Methods
2.1 Experimental System
The experimental system was a weighable lysimeter vessel (Meissner et al. 2000) described
in a previous contribution (Michel 2009). This stainless steel vessel (section: 1 m2, height:
200 cm) belonged to a fourfold lysimeter station implemented in the experimental site of the
GISFI (French Scientific Interest Group on Industrial Wastelands, www.gisfi.fr). In February
2008, it was filled with a 15-cm layer of sand at the bottom and 3 tons of a contaminated soil
from the upper horizon of a former coking plant (Homécourt, North-East part of France) (Sir-
guey et al. 2008). Large amounts of this contaminated soil were excavated, homogenised and
passed through a 4-cm sieve. The soil volume was 1.85 m3and the volumetric water content
0.28 cm3cm−3(Michel 2009). The soil sample was composed of 13.0 wt% clay, 19.2 wt%
silt and 67.8 wt% sand (Gujisaite 2008). The concentration of the 16 USEPA polycyclic aro-
matic hydrocarbons (PAHs) was 6.8gkg
−1(Michel 2009). The lysimeter surface consisted
of bare soil exposed to natural atmospheric conditions. Drainage water was collected from
the outlet at the bottom of the lysimeter.
The lysimeter was equipped with three sensors (TDR probe, tensiometer, suction probe) at
three depths: 50, 100 and 150 cm below ground surface (bgs) and with temperature sensors.
All these sensors were set up horizontally. Hourly monitoring of all the data (lysimeter weight,
volumetric water content, pressure head, bottom water flux, soil temperature) were done by
a data logger. Weather data (air temperature, relative humidity, wind speed and direction)
were also monitored. Hourly evaporation was calculated by Penmann–Monteith’s method
(Allen et al. 2000). Daily data were taken from hourly data at a 24-h interval (for example,
values recorded at 12 a.m. each day). The evaporation flux was derived from the changes of
lysimeter weight for 1 day interval. Water percolation and precipitation were also summed
up for 1 day interval.
2.2 Model Description
The one-dimensional SPM implemented in HYDRUS-1D software version 4.10 was used for
the simulations. Variably saturated water-flow was described by Richards’ equation (Simunek
et al. 2008):
∂θ (h)
∂t=∂
∂zK(h)∂h
∂z+1 (1)
in which θis the volumetric water content [L3L−3], his the pressure head [L], tis time [T],
zis the vertical coordinate [L], Kis the hydraulic conductivity function [LT−1].
123
488 V. V. N g o e t a l .
Tab le 1 Description and initial values of the 5 SPM parameters for contaminated soil in the lysimeter vessel
Parameter Unit Parameter description Initial value
θrcm3cm−3Residual water content 0.046
θscm3cm−3Saturated water content 0.41
αcm−1Shape factor in the soil water retention curve 0.03
n– Hydraulic property shape factor 1.49
Kscm d−1Saturated hydraulic conductivity 2.89
To solve this equation, we used van Genuchten–Mualem’s functions to express the rela-
tionships between θ,hand K(van Genuchten 1980):
Se=θ(h)−θr
θs−θr
=1
1+|αh|nmh<0(2)
Se=1h≥0(3)
m=1−1/nn>1(4)
K(Se)=KSSl
e1−1−S1/m
em2
,(5)
where Seis the effective water content [−], θrthe residual water content, θsthe saturated
water content, KSthe saturated hydraulic conductivity [LT−1]; α[L−1], n[−]andm[−],
are the empirical parameters. The parameter lof van Genuchten’s model in Eq. 5was set to
0.5 (Mualem 1976). Thus, there are five model parameters: θr,θ
s,α,nand Ks.
2.3 Model Parameters: Initial and Boundary Conditions
The initial parameter values are listed in Table 1. Water retention and hydraulic conductivity
parameters were estimated by Rosetta software (Schaap et al. 2001), which has been used
in different studies (e.g. Hermsmeyer et al. 2002b;Sander and Gerke 2007). This software
uses five pedotransfer functions for hierarchical estimation of the hydraulic parameters. The
input data of the Rosetta software comprised the soil composition, i.e. clay, silt and sand,
together with the bulk density of the contaminated soil. Preliminary calculations showed that
the hydraulic parameters of the sand layer had no significant influence on the results; hence,
this layer was neglected.
The initial condition was set by using the water content given by the TDR probes at initial
time.
θ(z,t)=θ(z)at t=0(6)
Since we measured water content at only three depths, the measured values of water
content at 50, 100 and 150 cm depths at the beginning of the experiment were consequently
set up for three soil layers 0–60, 60–120 and 120–185 cm, respectively.
At the soil surface, the flow-boundary condition was described by the “atmospheric bound-
ary condition with surface run off”. This boundary condition accounts for the input flux, such
123
Estimability Analysis and Optimisation of Soil Hydraulic Parameters 489
as precipitation and output one like evaporation. The equation for the upper boundary con-
dition is given as follows:
−K∂h
∂z−1=qT(t)at z=0,(7)
where qTis the net infiltration rate [LT−1]. The net infiltration was evaluated by subtracting
the precipitation to the evaporation for one day interval.
The 15 cm sand layer played the role of an infiltration layer; hence, the lower boundary
condition was specified as a seepage face. This boundary condition assumes that there is
no flow, q(t)=0, when the pressure head is negative. However, when the lower soil layer
becomes saturated, a zero pressure head is imposed at the lower boundary (Abdou and Flury
2004;Kasteel et al. 2007;Simunek et al. 2008):
h(z,t)=0atz=L(8)
2.4 Estimability Analysis
Estimability analysis requires the calculation of the normalised sensitivity coefficients of
the output predictions (Si)(e.g. pressure head or water content at a given depth, cumulative
evaporation at the soil surface, and cumulative percolation flux) to parameters ( pj). Each
sensitivity coefficient (Ii,j)was calculated from the approximation of the partial derivative
of Siwith respect to pjby centred finite difference:
Ii,j=∂Si
∂pj
≈S+
i−S−
i
2pj(9)
Ii,jwas then normalised (Hamby 1994):
Ii,jnorm =pj
SiS+
i−S−
i
2pj(10)
Siwas a predicted output obtained by direct simulation with the initial estimate of parameter
pj(Table 1). S+
iwas calculated at 1.1pjand S−
iat 0.9pjas pj=0.2pj. This means
that for each parameter pjthree direct simulations were run to calculate Ii,jnorm . All output
predictions Si,S+
i,andS−
ihave to correspond to data points.
Sensitivity analysis is necessary but not sufficient, as parameter correlation may hamper
parameter estimability (Larsbo and Jarvis 2006) and have an effect on the non-uniqueness
of optimised parameters. Therefore, it is important to carry out analysis, which represents
a measure of whether a given data set contains enough information to estimate parameters
from it (Jayasankar et al. 2009). Estimability is defined as the ability to accurately compute
parameters from a given data set and experimental conditions (Yao et al. 2003).
Parameter estimability depends not only on model structure and parameter sensitivity but
also on the possible correlations between parameters. In our case, estimability was evalu-
ated and a subset of estimable parameters of the SPM model was established based on the
method proposed by Yao et al. (2003). This method, which had been successfully applied in
chemical engineering (Jayasankar et al. 2009;Kou et al. 2005), enabled us to quantify the
effect of each model parameter on model predictions and helped account for correlations.
It compiles dynamic models and dynamic data, as well as response variables available at
different sampling rates throughout each experimental run (Kou et al. 2005). It was applied
as follows:
123
490 V. V. N g o e t a l .
For the output prediction Si, the matrix Ziwas built from the normalised sensitivity
coefficients calculated at each data point (t1,t2,...,t248 ):
Zi=⎡
⎢
⎢
⎣
Ii,1normt=t1···Ii,5normt=t1
Ii,1normt=t2···Ii,5normt=t2
···························
Ii,1normt=t248 ···Ii,5normt=t248
⎤
⎥
⎥
⎦
(11)
The number of columns is the number of parameters and the number of rows is the number
of daily data points (from t1to t248 at a regular interval).
The subset of estimable parameters was identified in 8 steps (Yao et al. 2003):
1. The magnitude of each column (quadratic sum of the elements) was calculated:
Mj=t=t248
t=t1Ii,jnorm2.(12)
2. The column with the highest magnitude represented the most estimable parameter p1.
3. The corresponding column was marked as Xi,L(L=1 for the first iteration).
4. Zi,L, prediction of the full sensitivity matrix Zi, was calculated using the subset of
columns Xi,Las:
Zi,L=Xi,LXT
i,LXi,L−1XT
i,LZi.(13)
5. The residual matrix Ri,Lwas built as:
Ri,L=Zi−Zi,L.(14)
6. The quadratic sum of the residuals was calculated in each column of Ri,L.Thecolumn
with the highest magnitude corresponded to the next estimable parameter (among the
remaining ones) having the largest effects on investigated data and not correlated with
the effects of the already selected parameters.
7. The corresponding column in Ziwas selected and included in the Xi,L. The new matrix
was called Xi,L+1.
8. The iteration counter was incremented and steps 4–7 were repeated until the last parameter
was treated.
After the determination of p1(steps 1, 2), we had to check if p1was correlated with the other
parameters. Orthogonalisation was used to rank individual parameter influences on the inves-
tigated data: the influence of p1on the other parameters had to be adjusted for any possible
correlation between the column of p1in Ziand the others, so that the “net influence” of each
of the remaining parameters could be assessed. The original columns of Ziwere regressed
on the column associated with p1. This resulted in a matrix having the same dimensions as
Zi, in which the column with respect to p1was composed of zeros (steps 3–5). The column
having the highest magnitude was then determined to the second parameter p2having the
second strongest “net influence” on the investigated data. Orthogonalisation was conducted
until the last model parameter was treated. Therefore, the use of the estimability analysis
method of Yao et al . (2003) allows us not only to rank the estimability order of parameters
but also to evaluate possible correlations among them, for a given experimental condition.
Yao et al . (2003) used 0.04 as a cut-off criterion for considering whether a model parameter
was estimable or not. The cut-off value shows that sufficient information would be contained
for estimating a parameter if a 10 % change in its initial value implied a 2 % change in
123
Estimability Analysis and Optimisation of Soil Hydraulic Parameters 491
output predictions. The choice of a cut-off criterion is somewhat arbitrary (Yao et al. 2003).
However, the approach seems to be reasonable for this work. Therefore, the approach of Yao
et al. (2003) was used herein to estimate the cut-off value, with a value of 2.4 being obtained
and considered as a cut-off criterion to conclude whether a parameter is estimable.
2.5 Parameter Identification
Parameter identification procedure was based on the minimisation of the differences between
experimental and simulated values. The objective function was (Simunek et al. 2008):
φ(b,q,p)=
mq
j=1
vj
nqj
i=1
wi,jq∗
j(x,ti)−qj(x,ti,b)2
+
mp
j=1
v
j
npj
i=1
w
i,jp∗
j(θi)−pj(θi,b)2
,(15)
where the first term on the right-hand side represents the deviations between the measured (q∗
j)
and the calculated (qj)space-time variables (e.g. pressure heads, water contents, cumulative
flux across the lysimeter boundary); mqis the number of sets of measurements; nqj is the
number of measurements in a particular measurement set; q∗
j(x,ti)represents a measurement
at time tifor the jth measurement set at location x; qj(x,ti,b) are the corresponding model
predictions for the vector of optimised parameters b(θr,θ
s,α,n,Ks);vjand wi,jare the
weights associated with a particular measurement set or point, respectively. The second term
of Eq. 15 represents the differences between the independently measured (p∗
j)and predicted
(pj)soil properties, such as retention or conductivity data at particular pressure heads;
mp,npj,p∗
j,pj(θi,b), v
j,w
ij have meanings similar to the first term, but are related to
soil hydraulic properties.
The weighting coefficient vjand v
jequalise the contribution of the different types of data
to the objective function Φ. These coefficients have the same equation form; hence, we only
write the one for vjherein (Simunek et al. 2008):
vj=1
nqjσ2
qj
(16)
in which σ2
qj is the measurement variance.
These coefficients allow a normalisation of the objective function with respect to the
variances of measurement type j.
The weight, wi,j, reflects uncertainty of measurement data (Simunek and Hopmans 2002).
In this work, the weight of each type of data was set up according to the uncertainty at lysimeter
scale evaluated by Fank (2006). The error measurements were 0.001 cm for the cumulative
outflow, 0.5 cm for the pressure heads and 0.01cm3cm−3for the water contents (Durner et al.
2008). We set wi,jequal to 0.001 for the daily cumulative bottom flux and daily cumulative
evaporation, 0.5 for the pressure heads, 0.01 for the water contents. w
i,jwas set equal to
0.001 or 0.1.
The Levenberg-Marquardt optimisation algorithm implemented in Hydrus-1D code was
used to minimise the objective function (Eq. 15) and to calculate confidence intervals for the
parameters (Simunek et al. 2008). Compared with global optimisation methods (Abbaspour
et al. 2001;Lambot et al. 2002;Mertens et al. 2004;Durner et al. 2008), this algorithm is
123
492 V. V. N g o e t a l .
based on a local optimisation method alone. The Hydrus-1D code provides useful statistical
information regarding the optimised parameters, e.g. their mutual correlation and confidence
intervals.
2.6 Model Performance Analysis
Model performance was evaluated by three statistical analysis criteria: the mean absolute
error, the root mean square error and the index of agreement. The mean absolute error (MAE)
describes the difference between observations (Oi)and simulated values (Si)as follows:
MAE =1
248
248
i=1
|Oi−Si|,(17)
where 248 is the number of daily observations.
The root mean square error (RMSE) and the index of agreement (IA) (Willmott 1981)are
given by the following equations:
RMSE =248
i=1(Oi−Si)2
N−1(18)
IA =1−248
i=1(Oi−Si)2
248
i=1Si−_
O+Oi−_
O2,(19)
where _
Ois the average of observations.
IA is ranged from 0 to 1.0, the higher IA, the better the agreement between experimental
and calculated values.
However, we mainly used the modified index of agreement in order to reduce the effect
of square terms (MIA) (Willmott et al. 1985):
MIA =1−248
i=1|Oi−Si|
248
i=1Si−_
O+Oi−_
O(20)
Using MIA is more interesting than using IA due to the elimination of the effect of squares
terms. Moreover, MIA is a more reliable statistical measure than the coefficient of determi-
nation, R2, which is less sensitive to extreme values (Legates and McCabe 1999).
2.7 Background Data
In case of estimability analysis of model parameters based on daily data, simulations were
done for a period of 248 day (February 15th to October 19th 2008). Over the entire moni-
toring period, the weighed lysimeter recorded 21.4 cm of net percolation and 39.6 cm of net
evaporation. Data of precipitation, cumulative percolation and cumulative evaporation for
the entire experimental period are presented in Fig. 1.
In case of the reference, 496 pressure head values measured at 100 and 150 cm depths
and 496 water content values measured at 50 and 100 cm bgs were included in the objective
function (the tensiometer at 50 cm and the TDR probe at 150 cm bgs were not working
during this period). Small changes were observed in pressure heads (between −1.7 and
123
Estimability Analysis and Optimisation of Soil Hydraulic Parameters 493
0
1
2
3
4
0
10
20
30
40
0 50 100 150 200 250
Precipitation (cm/day)
Cumulative output flow (cm)
Time (days)
Precipitation
Cumulative evaporation
Cumulative percolation
Fig. 1 Precipitation at the field site experiment, cumulative evaporation from the soil surface of the lysimeter,
and cumulative percolation across the bottom of the lysimeter
−25.7 cm at 100 cm bgs; between −4.1 and −42.3 cm at 150 cm bgs) and water contents
(0.25–0.29 cm3cm−3at 50 cm bgs and 0.27–0.31 cm3cm−3at 100 cm bgs).
3Results
3.1 Estimability Analysis
The results of parameter estimability are presented in Table 2. Each sub-table in Table 2
corresponds to one type of daily output prediction, i.e. pressure heads, water contents, cumu-
lative percolation and cumulative evaporation. The first row of each sub-table in Table 2
represents the magnitude of each column in the matrix Zicorresponding to each parame-
ter, prior to adjusting the correlations between parameters. From the second to the last row,
the magnitudes of each column were listed, removing the influences of the most estimable
parameter on the remaining ones. In case of each variable, it was seen that the estimability
decreased from the left to the right.
With respect to daily pressure head, five parameters were estimable with the decreas-
ing order: θs,n,Ks,αand θr. The set of estimable parameters using daily water contents
remained 3 out of the 5 parameters; one parameter, n, could be estimated from daily cumula-
tive bottom flux and evaporation. Thus, we can infer that (i) pressure head data were the most
interesting ones for parameter identification and (ii) cumulative bottom flux and evaporation
flux contained poor information. With respect to parameter optimisation, daily evapotranspi-
ration was found to be better than weekly evapotranspiration (Ines and Droogers 2002).
Parameter estimability is improved by increasing the number of different experimental
data (Yao et al. 2003). Hydraulic parameter estimability increased when pressure heads and
water contents were simultaneously included in the objective function (Simunek and Van
Genuchten 1997). Hence, we combined daily pressure heads and water contents (Table 3).
This resulted in a slight decrease in the corresponding magnitude of each parameter in the
sensitivity matrix, but the estimability order was the same as the previous one, namely:
θs,n,Ks,α,θ
r. On the whole, we found that parameter θrwas generally less sensitive
when compared to the others.
Possible correlations between parameters were looked into by computing the net influence
of the more estimable parameter(s) on the remaining one(s) from the magnitude of each
column of Zi:
123
494 V. V. N g o e t a l .
Tab le 2 Estimability order of the 5 SPM parameters as a function of daily measurements in the lysimeter: (a) pressure heads at three depths, (b) water contents at three depths,
(c) cumulative bottom flux and (d) cumulative evaporation
(a) Pressure head at 3 depths (b) Water contents at 3 depths (c) Cumulative bottom flux (d) Cumulative evaporation
θsnK
sαθ
rθsnK
sαθ
rnK
sθsαθ
rnθsKsθrα
23,071 6,813 4,010 764 8.85 584 251 4.39 0.16 0.39 24.2 13.0 0.53 0.07 0.01 4.96 4.09 0.16 0.07 0.48
06,266 3,957 653 8.82 0 41.6 2.49 0.16 0.18 0 1.37 0.16 0.03 0.01 0 0.64 0.05 0.03 0.07
003,899 540 8.53 0 0 2.41 0.16 0.04 0 0 0.09 0.01 0.01 0 0 0.03 0.03 0.01
000536 7.85 0000.160.01 0000.010 0000.010.003
0 0 006.16 00000.01 00000 00000.003
The bolded numbers are the magnitude of columns in the sensitivity matrix corresponding to the hydraulic parameters that are considered estimable
123
Estimability Analysis and Optimisation of Soil Hydraulic Parameters 495
Tab le 3 Estimability order of
the 5 SPM parameters when daily
pressure heads and water
contents at three depths in the
lysimeter were combined
θsnK
sαθ
r
23,655 7,065 4,014 765 9.23
06,421 3,959 656 9.18
003,903 541 8.79
0 0 0 537 8.11
0 0 0 0 6.40
Tab le 4 Net influence (%) of the
more estimable parameter(s) on
the remaining one(s) when daily
pressure heads and water
contents were combined
θsnK
sαθ
r
θs– 9.12 1.38 14.19 0.61
n–1.4217.59 4.15
Ks–0.63 7.82
α–20.99
θr–
nI =100 Rn+1−Rn
Rn,(21)
where Rn,Rn+1comprise the row order in Table 3for evaluating the influence between
different parameters based on the combination of daily pressure heads and water contents. If
nI =0, there is no correlation; the correlation is negligible between 0 and 5; low between 5
and 20; medium between 20 and 50; high between 50 and 75; and very high when above 75.
The influences of the more estimable parameter(s) on the remaining one(s) are listed in
Tabl e 4. These influences ranged from low to negligible. Hence, the estimation of the 5 SPM
parameters by numerical inversion, using the combination of daily pressure heads and water
contents, is expected to succeed.
Based on this estimability analysis, daily pressure heads are more pertinent for optimisa-
tion procedures than daily water contents. The best situation was obtained with a combination
of daily pressure heads and water contents. Nevertheless, in any case, daily pressure heads
and water contents were more reliable than daily cumulative bottom flux and evaporation, as
already observed (Ines and Droogers 2002;Vrugt et al. 2001).
As mentioned earlier, the cumulative bottom flux content and cumulative evaporation had
very poor information content for parameter optimisation. However, Durner et al. (2008)
suggested that the last value of cumulative bottom flux and cumulative evaporation should be
included in the objective function, in order to ensure meeting the overall mass balance of the
system. Hence, in the next section, this suggestion will be considered for the optimisation.
3.2 Optimisation Result
This section presents the optimisation results for four scenarios that differ in the way the data
were included in the objective function:
•the first scenario, SPM1, reference case, used the combination of daily pressure heads at
100 and 150 cm bgs and daily water contents at 50 and 100 cm bgs;
123
496 V. V. N g o e t a l .
Tab le 5 Soil hydraulic parameters, with corresponding confidence intervals, estimated from daily data using
HYDRUS-1D software for four scenarios
θrθsαnK
sR2
SPM1 0.2464 ±0.0029 0.3768 ±0.0094 0.4191 ±0.0563 1.68 ±0.09 464.09 ±153.98 0.9093
SPM2 0.2459 ±0.0028 0.3820 ±0.0109 0.4466 ±0.0691 1.68 ±0.08 568.01 ±238.53 0.9092
SPM3 0.2494 ±0.0026 0.4164 ±0.0146 0.5255 ±0.0823 1.78 ±0.10 1460.30 ±513.11 0.9092
SPM4 0.2449 ±0.0035 0.3873 ±0.0097 0.5268 ±0.1013 1.63 ±0.09 903.55 ±388.47 0.9088
•the second scenario, SPM2, used the same data as SPM1 plus the last value of cumulative
bottom flux and evaporation; the weight of these cumulative flux was set to 0.001;
•the third scenario, SPM3, used the same data as SPM1 plus 10 water retention values,
θ(h), at 100 cm bgs: one value corresponded to the highest pressure head value; one to
the lowest pressure head value and the 8 other values to 8 first daily data; the weights of
these 10 values were supposed to be 0.001;
•The fourth scenario, SPM4, used the same data as SPM3 with the weight of the 10 water
retention values equal to 0.1.
The estimated parameters of the four scenarios and their confidence intervals obtained
by numerical inversion are summarised in Table 5. Comparison of daily data and simulated
values are shown in Fig. 2for scenario SPM1 and Fig. 3for scenarios SPM2, SPM3 and
SPM 4. Figure 4exhibits the comparison between the optimised water retention curves for all
four scenarios—the measured ones that were determined for the soil sampled from the same
contaminated site, Gujisaite (2008), and the one estimated by using the Rosetta software.
One must be aware that the measured soil water retention curve done by Gujisaite (2008)
was obtained using the very fine and homogeneous contaminated soil, due to the diameter
size of the experimental device. Therefore, the water retention curve measured in the lab is
not considered as a reference curve to compare with the estimated ones herein.
In the reference case SPM1, the optimised θrvalue of 0.246 was very close to the lowest
water content at 50 cm bgs. The optimised value of parameter θrwas much higher than the
one estimated by the Rosetta software (Table 1), but a little lower than the one measured in
the lab (Fig. 4). This is why we have doubts on the accuracy of this optimised value despite its
narrow confidence interval, according to HYDRUS-1D code, knowing that θrwas the least
estimable parameter (Table 3).
The optimised θsvalue of 0.377 was higher than the highest water content at 100 cm bgs.
The optimised value of parameter θswas relatively lower than the one measured in the lab
(Fig. 4). Unfortunately, we did not have the measured water contents at 150 cm bgs that
could be higher than the ones at 100 cm bgs. The small confidence interval of this parameter
reflected the reliable estimation. The overestimation of θs,which often occurs in field-scale
studies, was attributed to air entrapment and/or irregular flow (Abbasi et al. 2004). In contrast,
Ramos et al. (2006) estimated the hydraulic parameters from tension disk infiltrometer data,
without finding an overestimation of θs.
Parameter nwas estimated with reliability, as shown by the relatively small confidence
interval. The optimised Ksvalue of 464.09 cm day−1seemed to be very high, having a large
confidence interval that revealed a high uncertainty. This poor result was partly explained by
the difficulty in Ksoptimisation due to soil spatial and temporal variability at the field scale
(Ramos et al. 2006). A rather high uncertainty was also obtained for α, which was attributed
123
Estimability Analysis and Optimisation of Soil Hydraulic Parameters 497
-50
-40
-30
-20
-10
0
0 50 100 150 200 250
Pressure head (cm)
Observed
Simulated_SPM1
-50
-40
-30
-20
-10
0
0 50 100 150 200 250
Pressure head (cm)
Observed
Simulated_SPM1
0.20
0.25
0.30
0.35
0.40
0 50 100 150 200 250
Water content (-)
Observed
Simulated_SPM1
0.20
0.25
0.30
0.35
0.40
0 50 100 150 200 250
Water content (-)
Observed
Simulated_SPM1
(c)
(d)
Time (days)
(b)
(a)
Fig. 2 Daily data and simulated values when daily pressure heads and water contents were simultaneously
included in the objective function (SPM1): apressure heads at 100 cm depth, bpressure heads at 150 cm
depth, cwater contents at 50 cm depth and dwater content at 100 cm depth
123
498 V. V. N g o e t a l .
-50
-40
-30
-20
-10
0
0 50 100 150 200 250
Pressure head (cm)
Observed
Simulated_SPM2
0.20
0.25
0.30
0.35
0.40
0 50 100 150 200 250
Water content (-)
Observed
Simulated_SPM2
-50
-40
-30
-20
-10
0
0 50 100 150 200 250
Pressure head (cm)
Observed
Simulated_SPM2
0.20
0.25
0.30
0.35
0.40
0 50 100 150 200 250
Water content (-)
Observed
Simulated_SPM2
-50
-40
-30
-20
-10
0
0 50 100 150 200 250
Pressure head (cm)
Observed
Simulated_SPM3
0.20
0.25
0.30
0.35
0.40
0 50 100 150 200 250
Water content (-)
Observed
Simulated_SPM3
-50
-40
-30
-20
-10
0
0 50 100 150 200 250
Pressure head (cm)
Observed
Simulated_SPM3
0.20
0.25
0.30
0.35
0.40
0 50 100 150 200 250
Water content (-)
Observed
Simulated_SPM3
-50
-40
-30
-20
-10
0
0 50 100 150 200 250
Pressure head (cm)
Observed
Simulated_SPM4
0.20
0.25
0.30
0.35
0.40
0 50 100 150 200 250
Water content (-)
Observed
Simulated_SPM4
-50
-40
-30
-20
-10
0
0 50 100 150 200 250
Pressure head (cm)
Observed
Simulated_SPM4
0.20
0.25
0.30
0.35
0.40
0 50 100 150 200 250
Water content (-)
Observed
Simulated_SPM4
(a) SPM2
(b) SPM3
(c) SPM4
Time (days) Time (days)
Fig. 3 Daily data and simulated values for 3 scenarios: aSPM2 using the data as in SPM1 plus the last value
of cumulative bottom flux and evaporation, bSPM3 using the data as in SPM1 plus 10 water retention values
and their weights were supposed to be 0.001 and cSPM4 using the data as in SPM3, the weight of 10 water
retention values were supposed to be 0.1
123
Estimability Analysis and Optimisation of Soil Hydraulic Parameters 499
0
20
40
60
80
100
120
0.2 0.3 0.4 0.5
Pressure heads (-cm)
Water contents (cm
3
cm
-3
)
Lab data
RETC
Rosetta softfware
SPM1
SPM2
SPM3
SPM4
Fig. 4 Comparison between water retention curves obtained by fitting the lab data with RETC code, those
optimised using the Hydrus-1D software for four scenarios, and the one estimated by using the Rosetta software
Tab le 6 Mean absolute error (MAE), root mean square error (RMSE), modified index of agreement (MIA)
between observed data and simulated values for four scenarios shown in Figs. 2and 3
MAE RMSE MIA
SPM1 SPM2 SPM3 SPM4 SPM1 SPM2 SPM3 SPM4 SPM1 SPM2 SPM3 SPM4
h100 3.71 3.69 3.69 3.71 5.27 5.23 5.26 5.23 0.52 0.52 0.52 0.53
h150 3.48 3.52 3.51 3.52 4.70 4.73 4.70 4.74 0.46 0.46 0.45 0.46
θ50 0.007 0.007 0.007 0.007 0.009 0.009 0.009 0.009 0.36 0.36 0.36 0.36
θ100 0.005 0.005 0.006 0.005 0.007 0.007 0.007 0.007 0.55 0.55 0.54 0.55
to the narrow range of daily pressure heads and water contents that created difficulty in
accurate determination of Ksand α.
The statistical result for observed and simulated pressure heads at 100 and 150 cm bgs
and water contents at 50 and 100 cm bgs are shown in Table 6. A better fit was obtained for
pressure heads at 150 cm than those at 100 cm bgs, as indicated by lower values of MAE and
RMSE and higher value of MIA for pressure heads at 150 cm bgs. However, RMSE values
of 5.23 and 4.70 cm for pressure heads at 100 and 150 cm bgs, respectively, were relatively
larger than the pure measurement error of 0.5 cm. Visual analysis of Fig. 2showed a wide
difference between observed and calculated pressure heads at two depths for the 15 first days.
This was assigned to uncertainty in pressure head data on account of lysimeter instability
in the initial period. Hence, during the summer period (days 125–190), the model could not
predict some pressure head data at 100 cm bgs, which decreased suddenly. However, the
simulated pressure heads at 150 cm bgs are very consistent with the measured ones for this
summer period. Hence, we believe that it was due to data errors and uncertainty rather than
to inaccuracy in the model.
Similarly, the agreement between observed and simulated water contents was better at 100
than at 50 cm bgs. Hence, the RMSE values 0.009 and 0.007 cm3cm−3for water contents,
123
500 V. V. N g o e t a l .
at 50 and 100 cm bgs, respectively, were very low—lower than the pure measurement error
(0.01 cm3cm−3). RMSE values for pressure heads were generally much higher than for
water contents: since pressure heads were measured in small soil volumes, measurement
uncertainties were larger than in case of water content that was measured in larger soil
volumes (Jacques et al. 2002). We noted that the measured soil volumes corresponding to
the pressure heads and water contents depend on the cup size of the tensiometers and the
type of the TDR probe (Ferré et al. 1998). In our study, we estimated that the measured soil
volumes varied in the range of 192–769 and 14–113 cm3for TDR probe and tensiometer,
respectively.
Previous contributions dealt with the estimation of hydraulic parameters from TDR and
tensiometer measurements. Jacques et al (2002) found RMSE values between 0.038 and
0.125 cm3cm−3for water contents,when optimising hydraulic parameters from water-flow
and solute transport at the bench scale. In another study devoted to the validation of a soil
water balance model from soil water content and pressure head data at the field scale, RMSE
values were between 0.010 and 0.035 cm3cm−3(Heidmann et al. 2000) or between 0.03 and
0.130 cm3cm−3(Wegehenkel 2005). Recently, Durner et al. (2008) also used a global optimi-
sation algorithm to optimise hydraulic parameters from TDR and tensiometer measurements
at the lysimeter scale under natural atmospheric conditions. They observed that the RMSE
value for the simulated and measured water content ranged from 0.009 to 0.20 cm3cm−3and
the one for simulated and measured pressure heads ranged from 0.67 to 31.82 cm.
The results of Scenario SPM2 were similar to that of SPM1 (Tables 5,6). The inclusion of
the last values of cumulative bottom flux and evaporation into the objective function did not
improve optimisation quality. The results of Scenario SPM3 were similar to that of SPM1
for θrand αbut showed higher values of θsand nand, above all, of Ks(Table 5). However,
the results of the statistical analysis were not modified (Table 6). Again, the addition of
some water retention values in the objective function did not increase information content,
as expected. Similar results were obtained in case of SPM4.
In conclusion, it is not necessary to include cumulative outflow (cumulative bottom flux
and evaporation) and initial values of hydraulic properties into the objective function. Daily
pressure heads and water contents were sufficient to successfully estimate parameters, such
as θsand n.
4 Discussion
4.1 Estimability of Soil Hydraulic Parameters
Among all the measurements, the highest estimabilities were obtained with daily pressure
heads. Similar results had been reported by Rocha et al. (2006), who investigated the effects
of various soil hydraulic properties on subsurface water-flow below furrows, during two
successive irrigation events, to analyse the effect of spatial variations in the initial soil water
content within the soil profile. The combination of daily pressure heads and water contents
also increased the estimability and all five hydraulic parameters were expected to be estimable.
The estimability order was θs,n,Ks,α,θ
r. Estimability of parameter θrwas the lowest,
based on the combination of daily pressure heads and water contents. This is due to the narrow
variations in measured pressure heads and water contents. The equation of the sensitivity
coefficient (Eq. 10) shows that the larger the variation ranges of the output prediction, the
higher the sensitivity coefficient. Thus, the information content of the investigated output
prediction strongly depends on its variation range.
123
Estimability Analysis and Optimisation of Soil Hydraulic Parameters 501
4.2 Optimisation
Following the estimability analysis, the optimisation of five parameters based primarily on
the combination of daily pressure heads and water contents in the objective function, were
carried out in different scenarios. The optimisation of parameters θsand nwas reliable.
However, the estimation of parameters αand KScontinued to remain inaccurate. The opti-
mised values of parameter θrfor all scenarios were generally much closer to the lowest
value of water content at the corresponding depths in the lysimeter. Furthermore, the opti-
mised values of parameter θrwere too high compared to its initial estimate determined by
the Rosetta software, but lower than the ones measured in the lab. It was believed that
the uncertainty of optimisation results was due to the restricted variations in measured
data. This result was well in keeping with the conclusions of Schwarzel et al. (2006),
who found that a bare field lysimeter could give rise to narrow pressure heads and water
contents (Schwarzel et al. 2006).
Our findings were also consistent with the result obtained by Kelleners et al. (2005), who
also determined soil hydraulic parameters at the lysimeter scale with the inverse method
from different experimental data. To solve this problem, it was suggested that at least one
independent measurement of θ(h)at a pressure head sufficiently lower than the lowest h0
should be included in the objective function (Schwartz and Evett 2003). In fact, Ramos et al.
(2006) showed that inclusion of two θ(h)at pressure heads of −100 and −15,850 cm allowed
an excellent fit of the measured data of tension infiltrometer disk. However, in case of the
field lysimeter experiment, it is likely to be impossible to get such a large range of data. We
can, however, increase the range of measured data to some extent, by collecting soil water
content from a point just close to the soil surface, as this layer is liable to become very dry
during summer. Thus, it offers more opportunities to successfully optimise the soil hydraulic
properties parameters.
As introduced in the Sect. 1, there are some major reasons that cause the non-uniqueness
issue for the determination of soil hydraulic parameters by inverse estimation method, such
as limited information of measurement data, strong correlation between different parame-
ters and local optimisation algorithm. By applying the estimability method of Yao et al.
(2003), information content of all daily data and correlation between van Genuchten–Mualem
parameters were evaluated systematically before the optimisation procedure. This helped
us to focus mainly on the available data containing most of the information content, i.e.
pressure heads followed by water contents, in this case, at the given experimental condi-
tions, thereby leading to better optimisation of the soil hydraulic parameters. Overall, the
issue of non-uniqueness could not be solved completely in this work. We believed this
was mainly due to the fact that a limited range of measurement data restricted the para-
meter optimisation. Moreover, the application of the local optimisation algorithm, such as
the Levenberg-Marquardt algorithm, also limited the accuracy of optimised parameters. In
addition, the use of the SPM model might also not be very suitable for this case, as the
preferential flow might exist within the soil of this large bare field lysimeter. Neverthe-
less, despite the doubts regarding the applicability of the Richards’ equation and SPM for
this soil with very high content of pollutant, the results of the statistical analysis revealed
a good agreement between measured and simulated daily pressure heads and water con-
tents, as indicated by the very low values of RMSE criterion. The application of the global
optimisation algorithm along with the use of the bimodal soil–water characteristic func-
tion is expected to better describe soil water dynamic within such a highly contaminated
soil.
123
502 V. V. N g o e t a l .
5 Conclusions
If 2.4 were considered as a cut-off criterion for this study, the set of estimable parameters
contained five parameters, based on daily pressure heads only and 3 out of the 5 parameters
based on daily water contents. However, according to daily cumulative bottom flux or evap-
oration, only parameter nwas estimable. Hence, daily measured pressure heads and water
contents should be included simultaneously in the objective function. This is because, on the
one hand, the combination also increased the parameter estimability and, on the other hand,
the combination did not result in any high correlation between different parameters.
When daily pressure heads and water contents were simultaneously included in the objec-
tive function, θsand nwere successfully estimated with narrowconfidence intervals. However,
uncertainties on optimised αand Kswere high, with large confidence intervals. This could
be explained by the fact that the bare field lysimeter produced narrow pressure heads and
water contents, which made accurate estimation of these parameters difficult. Inclusion of
the last values of daily cumulative bottom flux and evaporation or ten values of hydraulic
properties θ(h)did not help improve the optimisation procedure.
As shown in this work, estimability technique offers a useful method for estimating a
priori the reliability of the estimation for different soil hydraulic parameters and for pro-
viding information contents of different types of experimental data. This approach should
be generalised to help us focus on data having highest information contents. It could, thus,
enable us to improve the quality of the determination of soil hydraulic parameters.
Acknowledgments The authors thank the French Scientific Interest Group on Industrial Wastelands (www.
gisfi.fr), financially supported by the French government, the “Région Lorraine” and the “Conseil Général de
Meurthe et Moselle”. They also thank Noële Raoult and Dr. Julien Michel for data collection. The authors
highly appreciate the helpful and constructive comments of the four anonymous reviewers.
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