Groundwater Research, Rosbjerg et al. (eds) © 2000 Balkema, Rotterdam, ISBN 90 5809 133 3
Inverse Modeling of Unsaturated Flow Combined with Stochastic
Simulation Using Empirical Orthogonal Functions (EOF)
N.-O. Kitterød
University of Oslo, Department of Geophysics, Norway
S. Finsterle
University of California, Lawrence Berkeley National Laboratory, Berkeley, California, USA
Keywords: inverse modeling, unsaturated flow, EOF simulation
ABSTRACT: A Bayesian maximum likelihood method is used to derive optimal parameters for an
unsaturated flow problem. Liquid saturation data can be acquired with high spatial density at low
cost by indirect methods, making saturation a potentially useful primary observation type for inverse modeling. However, limited sensitivity of saturation to unsaturated flow parameters makes it
necessary to also include a priori information into the inversion procedure. The quality of the
estimated parameter set is expressed through a covariance matrix. Impacts of parameter uncertainties are evaluated by conditional stochastic simulations, in which not only the most likely
parameters are reproduced, but also the cross-correlation structure between the parameters.
Ignoring parameter cross-correlations in the simulation procedure lead to Monte Carlo realizations
with unlikely parameter combinations. In this paper, we present a conditional simulation method
which utilizes the orthogonal functions derived directly from the estimated covariance matrix.
1 MOTIVATION
In this study we examine the possibility of using liquid saturation data as primary observations to
estimate unsaturated hydraulic properties. A numerical model is developed based on the Richards
equation and the van Genuchten constitutive relations. The model is calibrated against liquid
saturation data using the maximum likelihood inversion method (Carrera and Neuman, 1986)
implemented in iTOUGH2 (Finsterle, 1999). This approach is potentially powerful because of the
relative ease with which liquid saturation can be mapped using geophysical and geostatistical
methods such as Ground Penetrating Radar (Hubbard et al., 1997) or - as done in this project - a
combination of ordinary kriging and high-resolution spatial mapping by the neutron scattering
method (Kitterød et al. 1997). These cost-effective methods provide spatially continuous images
that also contain important information about the geological architecture. Because liquid saturation
is not sufficiently sensitive to all unsaturated flow parameters, it is necessary to include prior
information about these parameters to constrain the inversion. The uncertainties in the estimated
parameters is evaluated and propagated through a prediction model to assess the uncertainties in the
simulated flow fields. Correlations among the estimated parameters must be accounted for to
reduce the impact of unlikely parameter combinations on the error analysis. Unlikely parameter
combinations are excluded directly during the Monte Carlo simulations using a method known as
Empirical Orthogonal Functions (EOF) or Karhunen-Loève simulation (Braud & Obled, 1991,
Kitterød and Gottschalk, 1997). This method has been used in the past to generate spatially
correlated property fields, but is equally applicable to uncertainty propagation analyses involving
correlated parameters, as discussed below.
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Groundwater Research, Rosbjerg et al. (eds) © 2000 Balkema, Rotterdam, ISBN 90 5809 133 3
2 CONDITIONAL PARAMETER SIMULATION
The generation of multiple parameter sets for subsequent Monte Carlo simulations starts with an
optimal estimate p, E[p]= p*, where p = pk, k = 1,…,n, and n is number of estimated parameters.
The covariance matrix of the estimated parameters, Cpp = E{(p+E[p])(p-E[p])}, can be expressed
as (Carrera and Neuman, 1986):
(
C pp = s 02 J T C −ZZ1 J
)
−1
(1)
where s02 is the estimated error variance, J is the Jacobian sensitivity matrix, and Czz is the a priori
matrix of measurement errors. Conditional realizations ξ of the stochastic vector p (denoted ξ p)
can be generated by applying the Karhunen-Loève theorem (Davenport & Root, 1958):
k = 1,..., n
(2)
ξ p = ξ Φkβk
where β k is the eigenvector for the kth parameter derived from:
C pp β Tk = µ k β k
k = 1,..., n
(3)
The stochastic coefficient ξ Φ is drawn from a Gaussian probability density function with pk* as
mean and the eigenvalue µk equal to the variance of pk, i.e., ξΦk ∈ N(pk*,µk). The expansion is often
called double orthogonal because
β k β Tj = D
k , j = 1,..., n
(4)
where D has entries dij = 0 if i ≠ j, and
E{ξ Φ k ξ Φ j } = δ kj µ k
k , j = 1,..., n
(5)
where δij = 0 if i ≠ j, and 1 if i = j.
3 RESULTS
The simulation algorithm is applied to a real case study. The results demonstrate that EOFsimulation narrows the realization outcome compared to standard Monte Carlo simulations where
the cross-correlation structure is ignored.
References:
Braud, I. & Obled, C. 1991. On the use of Empirical Orthogonal Function (EOF) analysis in the
simualtion of random fields. Stoch. Hydrol and Hydraul. 5, 125-134
Davenport, W.B. & Root, W.L. 1958. An introduction to the theory of random signalas and noise,
McGraw-Hill, New York
Finsterle, S. 1999. iTOUGH2 user’s guide. Earth Sciences Division, Lawrence Berkeley National Lab,
University of California, LBNL-40040
Carrera, J. & Neuman, S.P. 1986. Estimation of Aquifer parameters under transient and steady state
conditions: 1. Maximum likelihood method incororating prior information. Water Res. Res.22(2), 199-210
Hubbard, S.S., Peterson J.E. jr., Majer, E.L., Zawislanski, P.T. & Williams, K.H. 1997. Estimation of
permeable pathways and water content using tomographic radar data. The leading edge Nov, 1623-1628
Kitterød, N.-O. & Gottschalk, L. 1997. Simulation of normal distributed fields by Karhunen-Loève
expansion in combination with kriging. Stoch. Hydrol and Hydraul. 11, 459-482
Kitterød, N.-O. Langsholt, E., Wong, W.K. & Gottschalk, L. 1997. Geostatistic interpolation of soil
moisture. Nordic Hydrology 28(4/5), 307-328
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