First order analysis of uncertainty in numerical models of groundwater flow part: 1. Mathematical development

Source abstract

In part 1 of this paper, first and second order analysis of uncertainty is applied to numerical models of groundwater flow. The models are cast in state‐space form, with boundary conditions and inputs that are functionally dependent, but statistically independent, of time. Using a compact matrix calculus notation, first and second order Taylor series expansions of the model equations are derived and used to estimate the mean and variance‐covariance properties of piezometric head predictions, given corresponding statistics for aquifer parameters: material properties, initial conditions, boundary conditions, and inputs. The mathematical results demonstrate that the prediction uncertainty is a function of the magnitude of the parameter uncertainty, and sensitivity of the predictions to the parameters. Furthermore, the first order estimate of the piezometric head is identical to the deterministic result. Part 2 of this paper, to be presented later, will illustrate these and other results through numerous applications of the methodology.