10.4122/1.1000000297
Janssen, Gijs
Gijs
Janssen
gijs.janssen@wur.nl
Valstar, Johan
Johan
Valstar
johan.valstar@tno.nl
Van Der Zee, Sjoerd
Sjoerd
Van Der Zee
sjoerd.vanderzee@wur.nl
Janssen, Gijs
Gijs
Janssen
gijs.janssen@wur.nl
Measurement Network Design for Minimizing the Uncertainty in Breakthrough Predictions
XVI International Conference on Computational Methods in Water Resources
2006
2006
We are developing a first-order measurement network optimization algorithm that can
find the optimal combination and configuration of hydraulic conductivity, head and
travel time measurements. The method is based on a representer-based inverse method
for groundwater flow and transport applications (Valstar et al., Water Resources
Research 2004). In the representer approach, unknown variables are linearly expanded
in finite series which depend on unknown functions called representers. Each
representer quantifies the influence of a given measurement on the estimate of a
particular (state or static) variable. This procedure replaces the original inverse
problem by an equivalent problem where the number of independent unknowns is reduced
to the number of measurements. The representers can be shown to be equivalent to the
linearized cross-covariance between the measurement and the (state or static)
variable for which the representer is defined. As such, the representers can be used
to estimate the prior covariances of certain goal variables, if a (pseudo)
measurement for this variable is defined. Also, the representers can be used for a
first-order approximation of the posterior covariances of the goal variables if a
certain measurement set is assumed (because these covariances are functions of the
prior variances and the cross-covariances between all measurements and the cross-
covariances between the measurements and the goal variables).
In our study, we are interested in minimizing the uncertainty in the prediction of
contaminant breakthrough through confining layers. The uncertainty in breakthrough
is a convolution between the contaminant arrival time and contaminant arrival
location probability at the bottom of the confining layer. So the contaminant
arrival time and the contaminant arrival location are our goal variables, and we
derived representer definitions for advectively transported particles accordingly.
The posterior covariance of the goal variables obtained by the representer method
are based on a normal distribution. Using Monte Carlo calculation it turned out that
arrival times are nearly log-normally distributed. Therefore we choose the logarithm
of the arrival time rather than the arrival time itself as the goal variable. The
derivation of the travel time representers now requires an additional linearization.
However, the uncertainty estimates (both prior and posterior) of the arrival times
as given by the thus derived travel time representers turned out to approximate
Monte Carlo results very well , even for large variances of the hydraulic
conductivity field.
Once all representers that describe the relationships between a chosen set of
potential measurements (with their locations) and the goal variables are known, the
posterior covariances of the goal variables and therefore also the posterior
breakthrough prediction uncertainty can in principle be calculated for every
possible measurement set. Because the number of possible measurement combinations is
usually excessively large, we use a Genetic Algorithm for an efficient search for a
(close to) optimal measurement network within the available configuration space.
The novelty of this work lies primarily in the incorporation of travel time
measurements (for example Tritium/3He measurements) into a measurement network
optimization algorithm, which to our knowledge has never been reported before. It
requires an additional linearization in the derivation of the travel time
representers, which is also new. Furthermore, our approach of evaluating the
performance of a measurement network to the convolution of arrival time and arrival
location probability is a new approach. Within the applied inverse method this
requires the derivation of particle position representers, which has not been
published before.