EsƟmaƟon of Unsaturated Flow Parameters Using
GPR Tomography and Groundwater Table Data
O
R
M. Bagher Farmani,* Nils-O o Ki erød, and Henk Keers
Flow parameters in the vadose zone of an ice-contact delta near Oslo’s Gardermoen Airport were es mated by inverse flow
modeling combining me-lapse ground-penetra ng radar (GPR) travel- me (TT) tomography and groundwater (GW) table
data. Natural water infiltra on from the 2006 snowmelt was used as the upper boundary condi on in forward flow simulaons. The lower boundary condi on was the ou lux of water from the vadose zone derived by me deriva ve of the GW
table. The inverse flow modeling was done using different combina ons of me-lapse GPR TT tomography and me series of
the level of the GW table. The purpose was to assess the improvements in parameter es ma ons and the informa on value
of the different data sets. Flow parameters es mated by condi oning only on me-lapse GPR TT tomography capture the
development of the wet front but fail to simulate the GW table fluctua ons. Inverse flow modeling condi oned on only the
GW table did not simulate the wet front correctly but decreased the objec ve func on be er than condi oning on me-lapse
GPR data. If the inversion is condi oned on both me-lapse GPR data and the GW table fluctua ons, the final es mates of the
flow parameters are close to the es mates from the inversion condi oned on only GW table data. This result was obtained
because the moving GW table was monitored con nuously in me while the GPR data were sampled only three mes during
the infiltra on event; thus, observa ons of the GW table had higher weights in the objec ve func on than did observa ons
derived from GPR tomograms. Finally, we did forward flow modeling with the es mated parameter sets and compared the
flow paths with an independent tracer experiment performed at the field site in 1999. The results showed that anisotropy in
the intrinsic permeability was an important parameter that should be considered when simula ng the flow paths. However,
volumetric soil water content distribu on is not strongly related to the anisotropy of intrinsic permeability. Therefore,
anisotropy cannot be correctly es mated by inverse flow modeling condi oned on volumetric soil water content only.
A
: EM, electromagnetic; GPR, ground-penetrating radar; LM, Levenberg–Marquardt; SCE, shuffled complex evolution; 2D,
two-dimensional; 3D, three-dimensional.
E
flow parameters is of fundamental importance for the modeling and understanding of hydrological
processes in the subsurface and is therefore required for optimal
management of water resources. Flow parameters can be determined in a laboratory using small-scale samples and/or using in
situ field tests. Even though these traditional methods provide a
reasonably accurate estimation of flow parameters of soil samples,
they cannot automatically be used in flow modeling because
soil samples are usually available at a different scale than what is
required for flow modeling. Inverse flow modeling conditioned
on flow measurements provides an alternative method to estimate
flow parameters.
The use of geophysical data in flow inversions has increased
in recent years because of the extensive spatial coverage offered by
M.B. Farmani, Dep. of Geosciences, Univ. of Oslo, 1022 Blindern, 0315
Oslo, Norway; N-O. Ki erød, Norwegian Ins tute for Agricultural and
Environmental Research, Frederik A. Dahls vei 20, 1432 Ås, Norway; H.
Keers, Schlumberger Cambridge Research, Madingley Rd., Cambridge CB3
0EL, UK. Received 8 Nov. 2007. *Corresponding author
(m.b.farmani@geo.uio.no).
Vadose Zone J. 7:1239–1252
doi:10.2136/vzj2007.0169
© Soil Science Society of America
677 S. Segoe Rd. Madison, WI 53711 USA.
All rights reserved. No part of this periodical may be reproduced or
transmi ed in any form or by any means, electronic or mechanical,
including photocopying, recording, or any informa on storage and
retrieval system, without permission in wri ng from the publisher.
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
1239
geophysical methods and their ability to sample the subsurface in
a minimally invasive manner at the same scale as the flow model.
Of the different geophysical methods, GPR and resistivity methods are the ones most used in inverse flow modeling. Lambot et al.
(2004) combined electromagnetic inversion of GPR signals with
inverse flow modeling to estimate the flow parameters of a type of
sand in laboratory conditions. Kitterød and Finsterle (2004) used
surface GPR reflection profiles to define the geometry of the flow
model and measurements of water saturation in flow inversion.
The use of different types of geophysical data or both
geophysical and hydrological data to condition inverse flow
modeling has received considerable attention and has shown
to be promising. Lambot et al. (2006) proposed an integrated
electromagnetic–hydrodynamic inverse modeling approach to
identify field-scale unsaturated soil hydraulic properties and electric profiles from off-ground time-lapse GPR data. Linde et al.
(2006a) demonstrated that hydrogeological parameters can be
better characterized using joint inversion of cross-well electrical resistivity and GPR travel time data rather than individual
inversions. Kowalsky et al. (2005) estimated the soil flow parameters as well as the petrophysical parameters with the joint use of
time-lapse GPR travel times and neutron meter data. Binley et al.
(2002) estimated saturated hydraulic conductivity of Sherwood
sandstone by comparing the flow model results with the GPR and
resistivity images. Hyndman et al. (1994) combined synthetic
seismic and tracer data to estimate the geological structure, the
effective hydraulic conductivity, and the seismic velocities of geological zones using a zonation algorithm. Hyndman and Gorelick
(1996) developed the work done by Hyndman et al. (1994) and
used cross-well seismic, hydraulic and tracer data to estimate the
three-dimensional zonation of an aquifer along with the hydraulic
properties as well as the seismic velocities for these zones. Linde
et al. (2006b) did inverse flow modeling of tracer test data using
GPR tomographic constraints. A review of different techniques
to estimate the hydrological parameters using geophysical data
was given by Hubbard and Rubin (2000).
Even though GPR tomographic images can usually be used
to estimate geological structure, caution should be exercised in
estimating the geometry of the unsaturated flow model using
GPR tomographic images alone, if capillary or permeability
barriers exist in the vadose zone (Farmani et al., 2008b). To
overcome this challenge, Farmani et al. (2008b) suggested optimizing geological structure, in addition to the flow parameters,
as an integrated part of inverse flow modeling. They presented
a methodology to estimate the flow parameters and calibrate
the geological structure conditioned by time-lapse soil water
content estimates derived from the GPR tomograms. Farmani
et al. (2008b) defined the initial geometry of their flow model
using all available data including knowledge of the local geology,
well cores, and the GPR tomograms. The geological structure
is defined using different sets of control points. The positions
of these control points can be optimized during the inversion.
They introduced an objective function to minimize the differences
between the soil water content estimates and the flow model by
using weights derived from ray tracing in the objective function.
The weights were larger in areas with no artifacts and lower in
areas affected by artifacts.
In this study, we applied the methodology described above
to datasets collected during the snowmelt event in 2006 at a field
site in Norway close to Oslo’s Gardermoen airport (Fig. 1). The
methodology is developed further by using two different datasets
in the optimization procedure. We use the optimum geological
structure suggested by Farmani et al. (2008b) for our field site and
evaluate the improvements in parameter estimations by combining GPR and groundwater table data. Data acquisition is usually
expensive, and the extra cost should result in improved estimations.
We performed different inversions with and without the GPR and
groundwater table observations to address the improvement from
using different combinations of data. In comparison with the work
done by Farmani et al. (2008b) the GPR data has a denser dataacquisition geometry in the present study, and soil water content
is estimated in two perpendicular cross-sections rather than one
cross-section as used by Farmani et al. (2008b).
The ultimate test of the optimization procedure is crossvalidation of the estimated parameters against independent data
not used in the inverse modeling. The purpose of this study is
to compare the performance of different parameter sets derived
from different observations against an independent tracer test.
The inversion results are used in forward modeling and compared
with an independent tracer experiment performed at the field site
in 1999 (Alfnes et al., 2004). The forward simulations are run
with the same infiltration history used in the tracer experiment,
and the computed water movement in the vadose zone is compared with the observed water movement as reported by Alfnes
et al. (2004). Instead of using the tracer experiments directly as
observations for estimation of flow parameters in inverse modeling, we argue that estimation of flow parameters should be
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
1240
F . 1. Loca on of the Moreppen field site near Oslo’s Gardermoen
airport, Norway (top). The loca ons of the four ground-penetra ng
radar (GPR) wells (k14, k16, k18, and f22), and the groundwater well
(gw) used in this study are shown in the bo om half of the figure.
conditioned on easily acquired observations, while expensive
observations, such as field tracer experiments, should be used for
cross-validation. Using this approach the methodology can be
applied to other areas where estimations of travel time and travel
paths are of crucial importance for groundwater safety.
In this paper, we first outline the theory of nonlinear ray tracing and estimation of soil moisture content from GPR tomograms.
We then briefly present equations for forward flow simulations
and the method we use for inverse modeling. In the next section
we apply the methodology to field data acquired at an ice-contact
delta near Oslo’s Gardermoen airport (Norway). In the following
section the inversion results are cross-validated using results from
the tracer experiment undertaken by Alfnes et al. (2004). Finally,
we highlight some important results in the discussion.
Methodology
GPR Travel-Time Tomography
Electromagnetic (EM) waves generated by GPR antennas
used in this study are in the frequency band where ray theory is
valid (e.g., Kline and Kay, 1965; Vasco et al., 1997). According
to this theory, energy propagates along ray paths. Hence, ray
tracing is used to find the curved ray paths and to compute the
corresponding travel times (see Keers et al. [2000] and Farmani
et al. [2008a] for more details). In travel-time tomography, the
travel time residuals (the differences between the computed and
observed travel times), δTi, are related to the velocity variation
δvk (e.g., Nolet, 1987) by
δ Ti = ∑ −
k
l ik
δ vk
vk2
where lik is the length of ray i through velocity cell k and the starting velocity in cell k is denoted by vk. In this paper, the velocity
values are given on square grids with a grid size of 10 cm by 10
cm. In our study, Eq. [1] is extended to account for small errors
in the source and receiver locations (Keers et al., 2000) and stabilized by adding two terms to minimize a combination of velocity
variation (damping) and velocity gradients (smoothing):
⎛ δ T ⎞⎟ ⎛ L
L R L S ⎞⎟ ⎛⎜ δ V ⎞⎟
⎜⎜ ⎟ ⎜⎜
⎟⎟ ⎜
⎟⎟
⎜⎜ 0 ⎟⎟⎟ = ⎜⎜ λ I 0
0S ⎟⎟ ⎜⎜⎜ δ VR ⎟⎟
R
⎜⎜ ⎟⎟ ⎜⎜ 1
⎟⎟ ⎜
⎟
⎜⎝ 0 ⎠⎟⎟ ⎝⎜ λ 2D 0R 0S ⎠⎟⎟ ⎝⎜ δ VS ⎠⎟⎟⎟
[2]
L
where L is the coefficient matrix consisting of the parameters in
Eq. [2]; LR and LS are matrices containing 0s and 1s depending
on whether the source/receiver is active; δVR and δVS represent
the source and receiver statics, which are time shifts associated with
small errors in the source and receiver locations; λ1 is the damping factor; λ2 is the smoothing factor; I is the identity matrix; D is
the smoothing operator; and 0R and 0S are zero matrices. In this
study the damping and smoothing factors are kept constant for all
inversions. The matrix L is a sparse matrix, and, therefore, it can be
solved efficiently using the sparse equations and least squares (LSQR)
algorithm (Paige and Saunders, 1982). We use this algorithm to solve
Eq. [2] iteratively. Velocities obtained for one iteration are used as
the starting velocities for the next iteration. Further details on the
tomography method are given in Farmani et al. (2008a).
The final tomogram gives a spatial velocity distribution of
the vadose zone. We apply the conventional assumption that the
EM-velocity of the soil, v, is described by
c
[3]
v≈ a
ε
[4]
= −5.75 × 10−2, b = 3.09 × 10−2, c = −7.44 × 10−4,
9.634 × 10−6. According to Topp et al. (1980), the
where a
and d =
uncertainty in the values of θ in Eq. [4] is about σTopp = 0.0089.
Applicability of Topp’s model for sandy soil has been tested by
Ponizovsky et al. (1999). In addition, Farmani et al. (2008a)
cross-validated the applicability of this model for our research
field site. For low-loss materials εa ≈ ε and, therefore, θ can be
determined from the velocity using Eq. [3] and [4].
Forward Flow Modeling
Water flow in a heterogeneous variably saturated porous
medium is modeled by Richards’ equation (Richards, 1931; Jury
et al., 1991; Comsol Multiphysics, 2007):
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
θ = θ r + S e (θ s − θ r )
[6]
n ⎞−m
⎛
⎜⎜
α p ⎟⎟
⎟⎟
S e = ⎜⎜1 +
ρ f g ⎠⎟⎟
⎜⎝
1
1
αm
C=
(θ s − θ r ) S e m 1 − S e m
1− m
(
[7]
)
m
1241
[8]
m ⎤2
⎡
1
k r = S e L ⎢1− 1- S e m ⎥
[9]
⎢⎣
⎥⎦
Here, θr is the residual water content and θs is the saturated water
content; and α, m, n, and L are parameters that characterize the
porous medium. In this paper we set Qs and Ss to zero, L = 0.5,
and m = 1 – 1/n. A unique solution of Eq. [5] requires boundary
conditions. It also requires initial conditions for the transient problems. The boundary conditions used in this study are as follows:
(
)
p = p0
where ca is the velocity of an EM wave through the air and ε
is the relative dielectric permittivity of the soil. For a range of
sediments (from clay to sandy loam), Topp et al. (1980) found a
general experimental relationship between volumetric soil water
content and apparent permittivity. They also introduced separate
relationships for different types of soils. Since the vadose zone at
our research field site consists mainly of sand, Topp’s model for
sandy loam is used in this study:
θ = a + bε + cε 2 + d ε 3
⎡ k
⎤
∂p
+ ∇. ⎢− s kr ∇ (p + ρ f gz )⎥ = Q s
[5]
⎢⎣ η
⎥⎦
∂t
where pressure p is the dependent variable; C is the specific
capacity [C = (∂Se/∂p)] expressed in Eq. [8]; Se is the effective
saturation; Ss is the storage coefficient related to the compression
and expansion of the pore space and the water; ks is the intrinsic
or absolute permeability; η is the fluid viscosity; kr is the relative permeability; ρf is the density of water; g is the gravitational
acceleration; z is the vertical coordinate (positive upward); and
Qs represents a source or a sink.
Richards’ equation is highly nonlinear because p and kr vary
as a function of volumetric water content θ. For these parameters
constitutive relations are needed. In this study we use Mualem’s
(1976) and van Genuchten’s (1980) relations:
[C + S e S s ]
[1]
[10]
n⋅
ks
kr ∇ (p + ρ f gz )= 0
η
[11]
n⋅
ks
kr ∇ (p + ρ f gz )= N 0
η
[12]
⎡ ks
⎤
ks
n ⋅ ⎢ 1 kr1∇ (p1 + ρ f gz )− 2 kr2 ∇ (p2 + ρ f gz )⎥ = 0
[13]
⎢ η
⎥
η
⎣
⎦
where n is the vector normal to the boundary. Equations [10–13]
are used to define the known distribution of pressure at boundary,
impervious boundary, inward or outward flux at boundary, and
continuity across the boundary, respectively. We solve Richards’
equation using the finite element code COMSOL3.3a (Comsol
Multiphysics, 2007).
Inverse Flow Modeling
The volumetric soil water content estimated using GPR
travel-time tomography and Topp’s model, θ, gives the soil water
content distribution at various times; these are considered to be
the observed soil water content for the inverse flow modeling. The
groundwater table observations are another set of observed data
used in the inverse flow modeling in this study. Given a combination of flow parameters and a geological structure, the flow model
predicts the soil water content distribution at the same times as the
observed soil water content. In addition, the flow model predicts
the groundwater table at the times of observations. By minimizing
the difference between the observed and computed soil water content and groundwater table, we can invert for the flow parameters.
The geometry of the flow model is typically built using various data,
such as knowledge of local geology, measurements on soil samples,
and seismic and GPR data. Farmani et al. (2008b) showed that
the geometry can be modified during the inverse flow modeling.
However, because the field site in this study is the same as the field
site they modified the geometry for, we use their suggested geometry as a fixed geometry during the inverse flow modeling.
Because forward flow modeling is time consuming, the flow
model consists of a relatively small number of distinct geological units, in this paper referred to as subdomains. After solving
Richards’ equation for the flow model, the saturations obtained
for the finite elements are mapped to the square velocity cells to
compute the difference between the observed and computed soil
water contents. The objective function to be minimized is the
sum of the squared weighted residuals for all data sets:
nt
N
F = A∑ ∑ σ -2jk* [θ*jk − θ jk ( p1 , p2 ,..., p M )]2
j =1 k =1
ntt
where p1,..., pM are the flow parameters that are optimized; θ*jk
and θjk are respectively the observed and the computed volumetric soil water content at a given cell k and survey time j; N
is the number of velocity cells; nt is the number of surveys; gw*t
and gwt are respectively the observed and computed groundwater
table levels at the time t; ntt is the number of groundwater table
observations; A and B are the parameters weighting the importance
of each data set used in the objective function; σ−gw2* is the variance
of the groundwater table observations; and σ -2*
jk is the weight for the
soil water content estimate.
Parameters A and B can be estimated by a global sensitivity
analysis or Monte Carlo simulation for each kind of observation
separately and, then, a regression analysis of the objective function. An easier approach, which we applied in this study, is to
estimate A and B by computing an average of some realizations
of the objective function for each kind of observation separately
and, subsequently, calculating A and B as scaling factors. We
used 1000 realizations in this study and, then, calculated A and
B in such a way that each data set has approximately the same
contribution to the value of the objective function. The function
σ -jk2* is defined as
-2
σ -2*
jk = W jk σ Topp
[15]
where σ2Topp is the error variance of Topp’s
model; Wjk is the
r
weight derived from the ray coverage; ∑ lijk is the total length of
i =1
the rays passing through the kth cell in the jth survey; r is the
total number of ray paths; and χ is a small number that is used
as a lower bound in case there are some cells with no or very
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
∂F
∂ pi
i = 1,..., M
[16]
This way, the impact of the flow parameters on the objective
function can be assessed. Note that even though the unit of Eq.
[16] varies for each type of parameters, the results of different
inversions can still be compared with each other for the same
type of parameters.
Descrip on of the Field Site
t =1
W jk
Si =
Field Example
[14]
2
*
*
+ Bσ -2
gw ∑ [gw t − gw t ( p1 , p 2 ,..., p M )]
⎛
⎞⎟
r
⎜⎜
⎟⎟
⎜⎜ ∑ lijk
⎟⎟
i
1
=
= max ⎜⎜
,χ ⎟⎟⎟
r
⎜⎜
⎛
⎞
⎜⎜ l ⎟⎟ ⎟⎟⎟
∑
ijk ⎟
⎜⎜⎜ max
⎟
⎜
⎝ k ⎝ i=1 ⎠⎟ ⎠⎟
little ray coverage in the medium. In this study χ = 0.05. The
objective function is minimized iteratively using the shuffled
complex evolution (SCE) global search algorithm (Duan et al.,
1992). Farmani et al. (2008b) showed that the SCE algorithm
and Levenberg–Marquardt (LM) algorithm give very similar
results for these specific kinds of problems when applied to synthetic data sets. Therefore, the LM algorithm can be used as a
fast alternative. In addition, Farmani et al. (2008b) showed that
these algorithms are not very sensitive to moderate perturbations
in the initial parameters.
After the optimization, the sensitivity of the objective function
to the optimized flow parameters can be estimated by computing
The inverse flow modeling described in the previous section was applied to field data obtained at Moreppen, near Oslo’s
Gardermoen airport in Norway (Fig. 1). Moreppen has been the
subject of numerous studies related to sedimentological, hydrological, geophysical and geochemical processes in the saturated and
unsaturated zone (see, for example, Aagaard et al., 1996; Langsholt
et al., 1996; Tuttle and Aagaard, 1996; French and Binley, 2004;
Farmani et al., 2008a). Moreppen is a part of the Gardermoen
delta, an ice-contact delta formed after the last ice age about 9500
yr ago, with an area of approximately 80 km2. The delta is vertically
divided into three main sedimentary structures: bottomset, foreset,
and topset. The vadose zone at Moreppen is about 4 to 5 m thick.
The vadose zone contains the topset unit (roughly the upper two
meters) as well as the upper part of the foreset unit.
The Gardermoen aquifer is a superposition of two delta lobes
that have a semi-axial symmetry around the paleo-glacier portals.
The groundwater divide forms, therefore, a semicircular shape, and
because the permeable layer of the aquifer is thicker at the proximal
side of the delta, the groundwater divide is closer to the distal part
of the delta (Jørgensen and Østmo, 1990). The groundwater table
varies throughout the year, in relation to the amount of precipitation, with the highest levels during the snowmelt period. Moreppen
is located close to the groundwater divide.
The foreset at Moreppen consists of about 95% fine sand. The
remainder of the structure contains coarse sand and gravel as well as
some fine lenses of sandy silt. It was deposited in a shallow marine
environment and progrades in a southwesterly direction (dip directions vary between 180 and 240 degrees) with dips between 15 and
30 degrees. The topset was deposited in a fluvial environment. It
consists mainly of pebbly sand. The bedding in the topset is subhorizontal (Tuttle et al., 1996; Tuttle and Aagaard, 1996).
Observa ons
The GPR time lapse data were collected from March to May
2006. During this time the snow that had fallen during the winter
of 2005–2006 started to melt. The water infiltration rate, based on
1242
the measurements at Moreppen and a meteorological station nearby,
the objective function in the inverse flow modeling, are shown
is shown in Fig. 2.
in Fig. 3d, 3e, and 3f. Note that reasonable continuity of the
Moreppen contains a number of PVC-cased wells (wells k14,
k16, k18, and f22; see Fig. 1). The distances between k14
and k16, k16 and k18, and k16 and f22 are 2.4 m, 2.5 m,
and 2.3 m, respectively. Multi-offset cross-well GPR data
were collected from wells k14 and k16, wells k16 and k18,
and wells k16 and f22 on 18 March (just before the snow
started to melt), 21 April (during the snowmelt) and 6 May
(after the snowmelt). The groundwater well in Fig. 1 was
used to monitor the fluctuation of the groundwater table
from 1 January to 31 May.
All nine GPR datasets were collected using a stepfrequency radar provided by the Norwegian Geotechnical
Institute (Kong and By, 1995). Start and stop frequencies
for data acquisition were 50 and 900 MHz, respectively,
with 199 evenly stepped frequencies. The cross-well GPR
data were acquired from the depth of 0.8 m to a depth of
4.2 m in cross-wells k14–k16 and k16–k18 and 3.9 m in
cross-well k16–f22 with 0.1 m spacing between the source
antennas and between the receiver antennas. Travel times
from source–receiver pairs with offset angles higher than 45°
were not used in the tomographic inversion. First arrival
travel-time tomography was applied to all these datasets.
The volumetric soil water content distribution was found F . 2. Infiltra on history at the Moreppen research site from 1 Jan. 2006
using Eq. [3–4], and is shown in Fig. 3a, 3b and 3c. The to 31 May 2006. Circles indicate the days on which the ground-penetra ng
radar (GPR) surveys were acquired.
weights derived from tomograms, σ -2*
jk , which are used in
F . 3. Volumetric soil water content (in percent) es mated using curved ray ground-penetra ng radar (GPR) travel- me tomography, at
Moreppen on (a) 18 Mar. 2006, (b) 21 Apr. 2006, and (c) 6 May 2006. Corresponding spa al weights (Eq. [15]) for volumetric soil water content used in the objec ve func on (Eq. [14]) for the same dates are illustrated in (d), (e), and (f).
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
1243
volumetric soil water content across well k16 is an independent
quality check of the tomograms.
As expected, the soil water content in the vadose zone
increased from the first survey toward the second and third surveys. An interesting phenomenon, which is clearly observed in the
soil water content estimates at the second survey on 21 Apr. 2006
(Fig. 3b), is the wet front located at the depth of approximately
3.2 m. The continuous infiltration started 19 d before the day of
the second survey (Fig. 2). In other words, the wet front was at
the depth of 3.2 m after 19 d. This observation correlates very
well with the observation of Alfnes et al. (2004), who performed
a tracer test at Moreppen to estimate the flow parameters in the
vadose zone. They reported that the maximum mass observed
at the depth of 3.1 m at Moreppen was 17 d after the infiltration started.
In this study, the tomographic inversion procedure is similar
to that of Farmani et al. (2008a), applied to the data collected
at Moreppen during the snowmelt in 2005. They showed that
their volumetric soil water content estimates were consistent with
various quality checks, such as the equivalent estimates from a calibrated neutron meter sampled from an observation tube nearby,
water balance computations, and knowledge of the local geology
(Farmani et al., 2008a).
In addition to the cross-well GPR and groundwater table
data, air temperature, soil temperature, and precipitation data in
2006 were available. Air temperature and precipitation data were
used to compute the infiltration rate (Fig. 2). Soil temperature
is necessary for estimation of water viscosity, which is used in
flow modeling. Furthermore, a zero-offset surface GPR reflection
profile was acquired in May 2006 (see Fig. 4), and core samples of
the wells were also available. These data were used to determine
the initial geological structure of the flow model.
Parameteriza on of Flow Model
The geological (soil) structure of the flow model is similar
to that of Farmani et al. (2008b), estimated for the Moreppen
research field site (Fig. 5a). They derived the initial geological
structure between wells k14 to k18 from the surface GPR reflection profiles, the tomograms during the snowmelt event in 2005,
the core samples, and knowledge of the local geology/sedimentology. They then optimized the initial geological structure during
the inverse flow modeling. For more details on the geological
model see Farmani et al. (2008a, b). Figure 4 shows the match
between their geological model and the surface GPR reflection
profiles along the wells k14 to k18. Note that no data processing
has been done on the surface reflection profiles except the amplitude recovery. Still, the match between the geological structure
and the surface GPR reflection profile is reasonably good. This is
also evident along wells k16 to f22 (Fig. 4). The geological model
between wells k14 and k18 was projected to the plane k16–f22
using the general apparent dip seen in the surface GPR reflection
profile in this plane. The forward flow modeling is performed in
two dimensions (2D) along the main dip direction of the foreset
layers. The strike and true dip of the foreset dipping layers was
computed using apparent dips shown in Fig. 4. By performing
forward flow modeling along the main dip direction, the outof-plane flow is minimized. The meshes used to solve Richards’
equation are also shown in Fig. 5a. We use 1758 finite elements
to solve Richards’ equation.
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
1244
F . 4. Surface ground-penetra ng radar (GPR) reflec on profiles
between the four wells (k14, k16, k18, and f22) at Moreppen
acquired in May 2006.
The infiltration history at Moreppen during the snowmelt
in 2006 (Fig. 2) depicts a spatial average of the infiltration into
the monitoring area. The spatial distribution of infiltration was
never completely uniform because of the small-scale topography
and shade from the trees located on the south side of the field
site. However, French and Binley (2004) indicate that spatial
variability of the flow intensity is smoothed out as water percolates in the vadose zone. In addition, they discovered that the
spatial heterogeneity in the infiltration rate decreases during the
melting period. Since our tomograms are from a depth of 0.8 m
onward and our second and third surveys were performed much
later than the beginning of the snowmelt, we neglect the spatial
heterogeneity in the infiltration. The groundwater table showed a
fluctuation pattern correlated to the infiltration during the period
of modeling (blue line in Fig. 6). The level of the groundwater
table first declined steadily from a depth of 4.6 m to a depth of
4.98 m over a period of 98 d, then rose sharply to a depth of 4.19
m in 45 d, and afterward stayed constant during the last 7 d of
the modeling.
At the beginning of each realization during the inverse flow
modeling, the flow model is first run for a steady-state condition with a daily infiltration of 0.5 mm. The steady-state flow
modeling is done to estimate the initial pressure state. An average
infiltration rate of 0.5 mm is assumed for the long winter period
before the snowmelt started. However, there is no continuous
infiltration during the winter. The infiltration totally depends on
the temperature. For example, before the snowmelt at the end
of winter in 2006, there was infiltration of much more than 0.5
mm d–1 on some days, whereas there was no infiltration during
the rest of the time (Fig. 2). Based on the previous work done at
the field site, it can be concluded that the steady-state infiltration
F . 5. (a) Geometry of the flow
model. Gray
triangles show
the meshing used
to solve Richards’ equa on.
Numbers show
the indices of the
various subdomains. Volumetric
soil water content
(in percent) on
18 Mar. 2006,
21 Apr. 2006
and 6 May 2006,
es mated using
inverse flow modeling (b), (c), and
(d) condi oned
on groundpenetra ng radar
(GPR) volumetric
soil water content es mates
(Fig. 3a, 3b, and
3c), (e), (f), and
(g) condi oned
on groundwater
table, (h), (i), and
(j) condi oned on
both GPR volumetric soil water
content es mates
and groundwater
table, (k), (l), and
(m) condi oned
on groundwater
table using anisotropic intrinsic
permeability,
a higher upper
boundary for the
range of variaon in intrinsic
permeability and
a lower boundary
for the range of
varia on in the
van Genuchten
parameter α.
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
1245
which is defined using Eq. [12]. Interior boundaries of the
model are defined using Eq. [13].
Inverse Flow Modeling Using GPR and Time Series
of Groundwater Table
For the inverse flow modeling the residual water content
of the individual subdomains is taken from previous laboratory and field measurements and fixed (Table 1) (Pedersen,
1994; Kitterød et al., 1997). The water viscosity in each
subdomain is estimated using the average soil temperature
during the period of infiltration and is fixed. The intrinsic
permeability, porosity, and the van Genuchten parameters n
and α are determined by the inversion. The initial guess of
the flow parameters, which are optimized, and the a priori
search space were derived by interpreting the available data
and previous studies done at the field site (Table 1) (Pedersen,
1994; Kitterød et al., 1997; Kitterød and Finsterle, 2004).
The inverse flow modeling is performed four times. First,
it is conditioned on GPR volumetric soil water content estiF . 6. Observed and es mated groundwater table at Moreppen from 20
mates (A = 1 and B = 0 in Eq. [14]). Next, it is conditioned
Jan. 2006 to 31 May 2006.
on the groundwater table fluctuations (A = 0 and B = 1000).
For this, we set B equal to 1000 to make its contribution to
has no significant impact on the inversion result if observations
the value of the objective function approximately the same order
are started some days after the real infiltration starts (Kitterød and
as the contribution of the GPR volumetric soil water content
Finsterle, 2004). The first GPR volumetric soil water content estiestimates. Because the number of groundwater table observamates were obtained 77 d after starting the modeling. Therefore,
tions, 148 observations, is much less than the number of GPR
the steady-state infiltration has no influence on the GPR voluvolumetric soil water content estimates, 7353 estimates, B needs
metric soil water content estimates. However, groundwater table
to be high. In addition, the groundwater table at the start time
observations are from the first day of modeling. To eliminate the
of the model is fixed to the observed groundwater and, therefore,
influence of steady state infiltration on the simulated groundwater
the difference between the observed and simulated groundwater
table, we use groundwater table data from Day 20 of the period
tables is generally less than the difference between the observed
of modeling.
and simulated volumetric soil water content. Third, it is condiSince the groundwater table is used to condition the inverse
tioned on both GPR volumetric soil water content estimates and
flow modeling, the lower boundary of the flow model should be
groundwater table (A = 1 and B = 1000). Fourth, it is conditioned
deep enough to cover the fluctuation of the groundwater table
on the groundwater table by assuming an anisotropic intrinsic
during the period of modeling. We set the lower boundary of the
permeability in the foreset, a larger search space for the intrinsic
flow model at a depth of 5.3 m. Since Moreppen is close to the
permeability, the van Genuchten parameter α, and a lower steadygroundwater divide, the groundwater flow at Moreppen is mainly
state outflow estimated by groundwater table data from Day 1
vertical. To estimate the outflow of the saturated zone, we use
to 88 (Δh/Δt = –ΔR/Δt = 3.9θs mm d–1). Based on previous
the groundwater table data (Fig. 6). Changes in the groundwater
work done at Moreppen by Alfnes et al. (2004), the anisotropy
table are a function of changes in inflow ΔG and changes in
direction is in the dip direction of the foreset unit (subdomains
outflow ΔR: Δh = ΔG − ΔR. During Days 14 to 88, inflow to
3, 4, and 5). In the fourth inversion, we fix the direction of the
saturated zone was negligible: ΔG/Δt = 0 (see Fig. 2). We use
anisotropy in the foreset in the dip direction, but estimate the
the drop in the groundwater table from Day 50 to Day 80 to calanisotropic ratio (ksh/ksv) during the inversion.
culate steady state outflow: Δh/Δt = −ΔR/Δt = 4.34θs mm d–1.
The objective function is minimized using the SCE algoBecause the porosity of the soil layers is unknown, the outflux
rithm, which is a global minimization method. Each realization
from the model is modified based on the porosity values before
of the flow model took approximately 500 s on a PC Pentium 4
starting each realization. We assume that ΔR/Δt is constant
with a CPU of 3 GHz and 3 GB of RAM. In our experience, the
during the infiltration event. In each realization, for the steady
SCE algorithm uses roughly 200 realizations in each iteration.
state flow model, the groundwater table is fixed at the observed
Therefore, each iteration of the inverse modeling takes more than
depth of the groundwater table at Day 1, 1 Jan. 2006, which was
1 d on a normal PC. We stopped the inversion after 6 iterations.
4.6 m. This is done by using Eq. [10] for the lower boundary of
The optimized flow parameters for each inversion are presented
the model where p0 = 0.7. Then, for the transient flow model,
in Table 1. The estimated volumetric soil water content along
the outflux from the lower boundary of the model is fixed using
the wells by the flow inversions are shown in Fig. 5b to 5m. The
Eq. [12], whereas the groundwater table can fluctuate freely. The
observed groundwater table and groundwater water table estisides of the model are assumed to be impervious boundaries (Eq.
mated by the flow inversion are shown in Fig. 6.
[11]). The sides are far enough from the part of interest of the
The value of the objective function decreased by 33% in
model and, therefore, they have no impact on the results. The
Inversion 1, by 75% in Inversion 2, by 56% in Inversion 3, and
upper boundary has of course an inward flux due to the snowmelt,
by 88% in Inversion 4; in all cases after 6 iterations (Fig. 7). The
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
1246
T
1. Fixed, ini al, and op mized flow parameters for the natural snowmelt event at Moreppen condi oned on ground-penetra ng
radar (GPR) water content es mates, condi oned on groundwater table, condi oned on both GPR water content es mates and groundwater table, and condi oned on groundwater table using anisotropy intrinsic permeability. Values in parentheses are the ranges of varia ons.
Subdomains are labeled in Fig. 5a.
Dom.
θ r†
η
Ksh/ksv
α
n
ks (log10)
θs
Ksh/ksv
α
Inv. no. 1
1
2
3
4
5
0.05
0.05
0.05
0.05
0.05
Fixed
1.71 × 10 –3
1.67 × 10 –3
1.61 × 10 –3
1.61 × 10 –3
1.61 × 10 –3
1
1
1
1
1
15 (10–30)
15 (10–30)
15 (10–30)
15 (10–30)
15 (10–30)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
Ini al guess
−10 (−10 to −12)
−10 (−10 to −12)
−10 (−10 to −12)
−10 (−10 to −12)
−10 (−10 to −12)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
–
–
–
–
–
Op mized by GPR es mates
18.86 2.13 −11.24
0.3
16.12 2.20 −10.70
0.3
14.93 2.15 −10.67
0.28
18.63 1.96 −11.00
0.28
14.93 2.15 −10.60
0.28
Inv. no. 2
1
2
3
4
5
0.05
0.05
0.05
0.05
0.05
Fixed
1.71 × 10 –3
1.67 × 10 –3
1.61 × 10 –3
1.61 × 10 –3
1.61 × 10 –3
1
1
1
1
1
15 (10–30)
15 (10–30)
15 (10–30)
15 (10–30)
15 (10–30)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
Ini al guess
−10 (−10 to −12)
−10 (−10 to −12)
−10 (−10 to −12)
−10 (-10--12)
−10 (−10 to −12)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
–
–
–
–
–
Op
18.63
15.98
18.48
17.09
17.63
Inv. no. 3
1
2
3
4
5
0.05
0.05
0.05
0.05
0.05
Fixed
1.71 × 10 –3
1.67 × 10 –3
1.61 × 10 –3
1.61 × 10 –3
1.61 × 10 –3
1
1
1
1
1
15 (10–30)
15 (10–30)
15 (10–30)
15 (10–30)
15 (10–30)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
Ini al guess
−10 (−10 to −12)
−10 (−10 to −12)
−10 (−10 to −12)
−10 (−10 to −12)
−10 (−10 to −12)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
–
–
–
–
–
Op mized by GPR & groundwater table
22.10 2.19 −10.78
0.29
–
19.26 2.17 −10.51
0.29
–
15.75 2.34 −10.03
0.35
–
16.81 2.39 −10.69
0.35
–
19.86 2.37 −10.25
0.35
–
Inv. no. 4
1
2
3
4
5
0.05
0.05
0.05
0.05
0.05
Fixed
1.71 × 10 –3
1.67 × 10 –3
1.61 × 10 –3
1.61 × 10 –3
1.61 × 10 –3
1
1
–
–
–
15 (5–30)
15 (5–30)
15 (5–30)
15 (5–30)
15 (5–30)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
2 (1.5–2.5)
Ini al guess
−10 (−9 to −12)
−10 (−9 to −12)
−10 (−9 to −12)
−10 (−9 to −12)
−10 (−9 to −12)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
0.30 (0.25–0.35)
Op
–
13.50
–
14.54
1(1–10) 8.80
1(1–10) 5.28
1(1–10) 14.63
n
ks (log10)
θs
Ksh/ksv
–
–
–
–
–
mized by groundwater table
2.13 −10.40
0.27
–
1.96 −10.55
0.27
–
2.33 −10.02
0.35
–
2.18 −10.78
0.35
–
2.13 −11.03
0.35
–
mized by groundwater table
2.18 −9.30
0.30
–
2.11 −9.20
0.31
–
2.41 −9.12
0.35 1.05
1.85 −10.67
0.35 1.05
2.15 −10.92
0.35 1.05
† θ s is the maximum water content, θ r is the residual water content, η (kg m−1 s−1) is the water viscosity, α (m−1) and n are the van Genuchten parameters,
ks (m2) is intrinsic permeability, and ksh/ksv is anisotropic ra o.
main reduction in value of the objective function occurred after
the first iteration in Inversions 2, 3, and 4. After the second iteration, the objective function decreased slightly in all inversions.
Because the computation is expensive on a standard currentday PC, the inversion can be stopped, in principle, after the
second iteration.
The sensitivity of the objective function to the optimized flow
parameters for all inversions is presented in Fig. 8. In Inversion
1, the objective function is generally more sensitive to the small
changes in the value of the flow parameters in comparison to the
other inversions. In Inversion 2, the sensitivity of the objective
function decreases considerably in comparison with Inversion 1,
especially for parameters n and α. In Inversion 3, the sensitivities
of the intrinsic permeability decrease. The rest of the parameters
in this inversion show the same pattern of sensitivity as Inversion
2. In Inversion 4, the sensitivity to the value of α for Subdomains
3 and 4 considerably increases. Note that the value of α in these
subdomains is smaller than the estimates of α for the other subdomains in this inversion and also smaller than all subdomains
in the other inversions (Table 1). Therefore, the sensitivity analysis shows that if α is small, then the depth of the groundwater
table is highly sensitive to this parameter. Anisotropy in intrinsic
permeability was not shown to have larger impact on the groundwater table than the other parameters.
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
1247
Forward SimulaƟons of Independent Tracer Test
The question we want to address is: how well are our parameter estimates able to mimic the main features of an independent
tracer experiment? For this purpose we use a tracer experiment
undertaken at the Moreppen research field site in 1999 (Søvik et
al., 2002; Alfnes et al., 2004). In the tracer experiment, an area of
3 by 6 m, just south of a trench penetrating the unsaturated zone,
was used for the experiment. This trench is located approximately
25 m north-east of well k14 in Fig. 1. From the trench, 25 suction
cups were installed horizontally into the ground to avoid vertical
flow along the sampling tubes. The tracer experiment started with
a background wetting of the field by constant irrigation of 30 mm
d–1 in 7 d. During the experiment, the net background infiltration was increased to 43 mm d–1. A tracer solution with 1000
mg L–1 of Br was applied at a rate of 25 L every other hour for a
period of 3 d, forming a line source perpendicular to the trench
wall. This line forms a point source in the local horizontal coordinate of 1 m in all figures in Fig. 9. Note that the tracer experiment
has been done some distance from our modeling area. However,
the general soil structure observed in the trench is very similar to
our model. The most important layer in the tracer experiment is
perhaps a thin low-permeability layer in the foreset, shown with
black tilted lines in Fig. 9a and 9b. We chose Day 5 and Day 16
from the tracer experiment and show the concentration of Br in
these 2 d with red circles in Fig. 9a and 9b, respectively. The size
of the circles is proportional to the Br concentration. The blue
asterisks show the suction cups that have no Br concentration.
F . 7. Misfit reduc on of the objec ve func on during the inversion.
F . 8. Sensi vi es (Eq. [16]) of the op mized flow parameters. The symbol
on the x axis corresponds to the parameter, and the number corresponds to
the subdomain. Note that the unit of the various parameters differs.
is not relevant in this exercise, we fix the groundwater table
at the bottom of our models during the period of modeling.
In addition, we use a random intrinsic permeability for each
subdomain. The random intrinsic permeability is calculated
by assuming a lognormal distribution with a mean equal to
our estimate from the inversions and a standard deviation
equal to 0.1 for all subdomains. The final flow models from
Inversions 1, 2 and 3 are run twice, the second time by
using the anisotropic intrinsic permeability with the ratios
suggested by Alfnes et al. (2004). The flow paths are computed by particle tracking. The flow path after 5 and 16 d
are shown with yellow lines in Fig. 9c to 9n. The final flow
model from Inversion 4 is only run using the estimated flow
parameters and flow path is shown in Fig. 9o. The background color in Fig. 9 is the volumetric soil water content.
From Fig. 9, it is clear that if the anisotropy is not taken
into account, the flow model fails to simulate the true flow
path. In addition, Fig. 9 shows that anisotropy does not have
a large influence on the soil water content distribution. This
can be seen by comparing the soil water content distribution
in each two isotropic and anisotropic models (e.g., Fig. 9c
and 9e). Therefore, when conditioning the flow model on
soil water content distribution, it is hard to estimate the
anisotropy. This seems to be even more challenging when
the infiltration is more or less uniform and on a much larger
scale, such as with a snowmelt event. Even though Fig. 9o
is able to simulate the true flow path, it fails to simulate the
true travel time of water. It is clear that the intrinsic permeability in this model has been underestimated.
The horizontal and vertical traveling distances 5 and
16 d after the beginning of the infiltration are shown in Fig.
10a and 10b, respectively. In these figures, we only illustrate
the traveling distance for flow models found by considering
the anisotropic intrinsic permeability. In addition, the traveling distance for tritiated water (HTO), which was used as
another tracer in the same tracer experiment, is also shown in
Fig. 10. The difference between the Br traveling distance and
the HTO traveling distance is an indication that uncertainty
exists in the observations. If we assume that the anisotropy
ratios offered by Alfnes et al. (2004) are correct, then the
best models for Moreppen are Models 2 and 3, including
the anisotropy ratios offered by Alfnes et al. (2004). Model
1 overestimates the travel time of water because of an underestimation of porosity in the foreset.
Discussion
The concentration of Br over time shows the travel time of water
and the flow path in the vadose zone. It seems that the flow
path in the foreset is highly controlled by the low-permeability
layer. Alfnes et al. (2004) noticed that to simulate the observed
flow path, anisotropy in the intrinsic permeability in the vadose
zone should be taken into account. They suggested an anisotropic intrinsic permeability with a horizontal-to-vertical anisotropy
ratio of 4 and 10 for the topset layers (respectively Subdomains 1
and 2 in Fig. 5a) and an anisotropy ratio of 10 in the dip direction for all foreset layers.
We use the same infiltration as was used in the tracer experiment and run forward simulations with the flow parameters as
estimated from the flow inversion. Because the groundwater table
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
1248
The final estimates of the flow parameters conditioned on
GPR volumetric soil water content estimates, Inversion 1 in Table
1, are able to simulate the main pattern of the volumetric soil
water content distribution in the vadose zone: The vadose zone
gets wetter in time due to the natural infiltration of snowmelt.
In addition, the wet front observed using GPR tomography at
the time of the second survey at a depth of approximately 3.2 m
(Fig. 3b) is correctly simulated by the final flow estimates (Fig.
5c). However, the volumetric soil water content in subdomains
4 and 5 is underestimated by the flow model, especially at the
time of the third survey (Fig. 5d). The capillary rise at the time
of the third survey is also much weaker in the flow model than
the GPR estimates. The mismatch between the flow model and
GPR estimates for these areas of the model was expected
since we assign much lower weights to the GPR estimates
at these areas in the objective function than at the other
areas. Because EM ray coverage in these areas is very poor
(Fig. 3d, 3e and 3f ), the GPR estimates are not as reliable
as at the other areas. Farmani et al. (2008b) showed that
major tomographic artifacts exist in regions with poor ray
coverage. Therefore, this mismatch cannot be considered to be a disadvantage, but rather the advantage of
the method that can extract more reliable data from
the tomograms to condition the inverse flow modeling on it.
When the inverse flow modeling is conditioned
on GPR volumetric soil water content estimates, the
groundwater table fluctuation estimated by the final flow
parameters differs quite a bit from the observed fluctuation (Fig. 6). The objective function, Eq. [14], consists
of two terms: F = FGPR + Fgw, where FGPR and Fgw are
called the GPR term and the groundwater term, respectively. For Inversion 1, the objective function value for
the final estimates of the flow parameters is F = FGPR =
16,900. However, if we also consider the groundwater
term, Fgw, in the objective function, then the value of the
objective function increases to 64,700. This means that
Fgw = 47,800, which is quite large. The observed groundwater table can be considered as independent data to
cross-validate the results of the Inversion 1. However, the
computed groundwater table is not reasonably matched
with the observed groundwater table. The groundwater
term in the objective function for Inversion 1 is 3.32
times larger than the groundwater term in the objective
function for Inversion 2.
Based on the groundwater table observations at
Moreppen, the groundwater table started to rise from
Day 98 in the period of the modeling. However, the flow
model suggests that the groundwater table starts to rise
from Day 116. The main reason for this mismatch is
probably that the flow model simulates the wet front at
a depth of 3.2 m at the time of the second survey (Day
111). This means that water cannot reach the groundwater at Day 98. The infiltration slows down in Inversion 1
by assigning smaller values to the intrinsic permeability
of all subdomains in comparison to the other inversions
where groundwater table data are used. This is clearly
seen in Table 1 for the estimated intrinsic permeability
for all subdomains. There are two possible explanations
for how the groundwater table starts to rise before the
F . 9. Flow path and travel me of water based on the infiltra on used in a
wet front reaches the groundwater. The first possibility is tracer experiment at Moreppen in 1999. Concentra on of Br 5 d (a) and 16 d
the existence of the preferential flow paths in the vadose (b) a er the trace experiment was started. Flow paths for the final flow modzone. This was also one of the conclusions of Alfnes et els respec vely from Inversions 1, 2, 3 a er 5 d (c, g, k) and 16 d (d, h, l). Flow
al. (2004) and Farmani et al. (2008a) at Moreppen. The paths for the final flow models respec vely from Inversions 1, 2, 3 using anisoother possibility would be a more complicated geometry tropic intrinsic permeability suggested by Alfnes et al. (2004) a er 5 d (e, i, m)
and 16 d (f, j, n). Flow paths for the final flow model from Inversion 4 a er 5 d
than what has been used in the flow model.
(o). Yellow lines show the flow path, and background of the figures c to o is the
The final estimates of the flow parameters condi- volumetric soil water content distribu on.
tioned on the groundwater table, Inversion 2 in Table
1, are able to simulate the groundwater table fluctuation
time shift for the deepest groundwater table, from Day 98 in
reasonably well. The final depth of the groundwater table is estithe observations to Day 107 in the simulation. This shift can
mated correctly. However, there are two important mismatches
be compensated by either introducing preferential flow paths in
between observations and estimates. The first mismatch is the
the model, extending the upper boundary of the search space for
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
1249
F . 10. (a) Horizontal and (b) ver cal traveling distance in me for
tracers bromide and tri ated water (HTO) in the tracer experiment
(observed) and for all flow models by assuming an anisotropic intrinsic
permeability (simulated) 5 and 16 d a er the start of the infiltraon. The ver cal depth of 5 m means that the front head has already
reached the groundwater.
intrinsic permeability (ks > 1e-10), or introducing an anisotropic
intrinsic permeability. However, based on the cross-validation
exercise, the intrinsic permeability cannot be larger (Fig. 9o).
The second mismatch is the slower increase in the depth of the
groundwater table for simulation.
For Inversion 2, the GPR volumetric soil water content estimates can be considered as independent data to cross-validate the
results of the inversion. The objective function value for the final
estimates of the flow parameters is F = Fgw = 14,400. However,
if we also consider the GPR term, FGPR, in the objective function for Inversion 2, then the value of the objective function
increases to 41,100. This means that FGPR = 26,700. Therefore,
the GPR term in the objective function for Inversion 2 is only
1.58 times larger than the GPR term in the objective function
for Inversion 1. This means that conditioning the inverse flow
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
1250
modeling to the groundwater table fluctuation is probably more
robust than conditioning the inverse flow modeling to the GPR
volumetric soil water content estimates, when few volumetric
soil water content estimates are available in time. However, note
that the clear existence of the wet front at a depth of 3.2 m in
the second GPR survey (Fig. 3b) is missed in the flow model (Fig.
5f ) because of the faster movement of the wet front. This is due
to the assignment of lower values of the intrinsic permeability in
comparison to Inversion 1.
In Inversion 3, where the inverse flow modeling is conditioned to both the GPR volumetric soil water content estimates
and the groundwater table fluctuations, the final estimates of
the flow parameters simulate the groundwater table better than
does the GPR volumetric soil water content distribution alone.
The value of the objective function is F = FGPR + Fgw = 21,700
+ 15,400 = 37,100. The groundwater term is only 1000 larger
than in Inversion 2, whereas the GPR term is 4800 larger than in
Inversion 1. This was expected because much more groundwater
table data were available in time than were the GPR data. Because
of this, the sensitivity pattern for Inversion 3 is very similar to
the sensitivity pattern for Inversion 2.
In Inversion 4, we attempted to improve the inversion results
of Inversion 2 by introducing anisotropy in the intrinsic permeability for the foreset layers, using a larger upper boundary in
the search space for intrinsic permeability and a smaller lower
boundary in the search space for the van Genuchten parameter α.
With this new model we were able to decrease the objective function by more than 50%, which is a considerable improvement.
The objective function value for the final estimates of the flow
parameters is F = Fgw = 7000. If we also consider the GPR term,
FGPR, in the objective function then the value of the objective
function increases to 53,300. This means that FGPR = 46,300.
The GPR term is larger for Inversion 4 than for Inversion 2. This
means that even though we simulated the groundwater table
better by modifying the model, the simulation of volumetric soil
water content distribution became worse in comparison to the
Inversion 2. Moreover, a comparison with the tracer test revealed
that the estimated intrinsic permeability for the subdomains from
this inversion does not seem to be correct.
However, any conclusion should be taken with caution
at this stage. The tracer test data also contains some sources of
uncertainty. For example, the total mass recovery for Br was
reported to be 70% by Alfnes et al. (2004). In addition, tritiated
water observations show discrepancies with the Br observations
(Fig. 10) (Søvik et al., 2002). In general, it seems that all of our
models overestimate the water velocity in the topset. More studies need to be done to be able to efficiently condition the inverse
flow modeling on both GPR volumetric soil water content estimates and the groundwater table. For example, it is interesting
to see how the final estimates of the flow parameters change
with changing parameters A and B in Eq. [14]; or how much
better we can estimate the groundwater fluctuation using variable
groundwater outflow in time. Also, it is interesting to estimate
the three-dimensional (3D) geological structure on a larger scale
than the observations to see if any information is lost when the
modeling is done in 2D. If the geological structure varies spatially, for example by a change in the thickness and dip of the
layers or discontinuity of the layers, the flow modeling should
be performed in 3D. Unsaturated flow in a heterogeneous and
anisotropic geological formation is of course a 3D problem, and
in the beginning of this project we performed some preliminary
simulations in 3D. Inverse flow modeling in 3D requires extensive computer resources, and the geological complexity had to be
simplified to achieve feasible simulations. Thus, there is a tradeoff between adequate model of geology in 2D and too-coarse
numerical mesh in 3D. In our case, the main geological variability
is kept by describing the structures in the principal directions
along strike and dip. It is not unlikely that small-scale variability
in 3D will increase focusing and defocusing effects in the unsaturated zone. If this small-scale variability is taken into account,
the estimation uncertainty may be reduced further. This task,
however, is postponed until more effective numerical algorithms
for flow computations are available, together with more powerful
hardware. A full 3D simulation sounds to be feasible on normal
PCs in the near future.
Conclusions
In this study, we used the method presented by Farmani et
al. (2008b) to estimate the flow parameters in the vadose zone
of an ice-contact delta near Oslo’s Gardermoen airport (Norway)
by combining cross-well GPR travel-time tomography (obtained
in 2006), a soil-physics relationship, groundwater table data, and
inverse flow modeling.
If the inverse flow modeling is conditioned on time-lapse
GPR volumetric soil water content estimates, assigning the
weights to the GPR water content estimates could reduce the
influence of tomographic artifacts on the final estimates of the
flow parameters. However, while the wet front observed at a depth
of 3.2 m in the second GPR survey was satisfactorily simulated
by the inverse flow modeling, the observed groundwater table
showed a considerable mismatch with the simulated groundwater
table. The inverse flow modeling conditioned on groundwater
table data was able to simulate the GPR volumetric soil water
content estimates better than the inverse flow modeling conditioned on GPR volumetric soil water content estimates simulated
the groundwater table. However, the inverse flow modeling conditioned on groundwater table could not simulate the wet front
at the depth of 3.2 m at the time of the second GPR survey.
When the inverse flow modeling was conditioned on both timelapse volumetric soil water content estimates obtained from the
GPR data and the groundwater table data, the final estimates
of the flow parameters were very similar to the estimates from
the inverse flow modeling conditioned on only the groundwater table. This was because GPR data were available only for
three days whereas groundwater table data were daily available.
Simulation of the independent tracer test showed that anisotropy is an important parameter in simulating the true flow path
at the scale of our problem.
A
The authors thank Professor Per Aagaard at the Department of
Geosciences, University of Oslo, for financial and scientific support.
Also, many thanks to Dr. Fan Nian Kong of the Norwegian Geotechnical Institute, for advice and for letting us use NGI’s step-frequency
radar; to the COMSOL support team; and to staff at the Bioforsk Soil
and Environment.
www.vadosezonejournal.org · Vol. 7, No. 4, November 2008
1251
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