One-Dimensional Problem Formulation Clause Samples

One-Dimensional Problem Formulation. The first part of this research is concerned with the feasibility of estimating NAPL source locations and compositions from dissolved-phase measurements. The feasibility question is important since the processes involved are non-linear and the sensitivity of the estimation to the measurements is unknown. Estimation of residual saturation from dissolved concentration measurements can succeed only if Sn is sufficiently sensitive to the measurements. Otherwise the estimation problem will be ill-posed (i.e., the solution may be non-unique or very sensitive to small measurement errors). The success of the estimation is also affected by the degree of non-linearity of the forward model. Our estimation method is based on a series of local linearizations of the forward model in the neighborhood of the current parameter estimates. If the forward model is very non-linear, even a local linearization may produce a poor parameter estimate. In order to examine the feasibility of our estimation objective we constructed a simplified one-dimensional (1-D) problem. This 1-D problem is also useful for exploring the relative effectiveness of varying sampling strategies. While a multi- dimensional or distributed source problem would be too computationally expensive to run for multiple scenarios, the 1-D problem is very amenable to exploratory analysis. In our 1-D formulation, the residual NAPL mixture is considered to be a point source of dissolved phase contamination at an uncertain location. We treat the source as a time-varying concentration boundary condition for a ground-water transport model. The source concentrations are calculated from the dissolution of a NAPL mass. The unknown parameters are the initial masses of each constituent, or chemical, in the NAPL mixture and the location of the source. Measurements of dissolved concentration are used to estimate these parameters. The remainder of this section describes the mathematical formulation of the state equations, the measurement equation, and the estimation algorithm as well as the simulated problem, which represents the “true” set of states estimated in Section 4.