Measurement Network Design for Minimizing the Uncertainty in Breakthrough Predictions
Creators: name: Janssen, Gijs
name: Valstar, Johan
name: Van Der Zee, Sjoerd
name: Janssen, Gijs
Publisher: name: XVI International Conference on Computational Methods in Water Resources
Year: 2006

description: 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.