A multidimensional fuzzy least-squares regression approach for estimating hydraulic gradients in unconfined aquifer formations and its application to the Gulf Coast aquifer in Goliad County, Texas

Abstract

Epistemic uncertainties arise during the estimation of hydraulic gradients in unconfined aquifers due to planar approximation of the water table as well as data gaps arising from factors such as instrument failures and site inaccessibility. A multidimensional fuzzy least-squares regression approach is proposed here to estimate hydraulic gradients in situations where epistemic uncertainty is present in the observed water table measurements. The hydraulic head at a well is treated as a normal (Gaussian) fuzzy variable characterized by a most likely value and a spread. This treatment results in hydraulic gradients being characterized as normal fuzzy numbers as well. The multidimensional fuzzy least-squares regression has an exact analytical form and as such can be implemented easily using matrix algebra methods. However, the method was noted to be sensitive to round-off and truncation errors when the epistemic uncertainties are small. A closeness index based on the cardinality of a fuzzy number is used to evaluate how well the regression model fits the fuzzy hydraulic head observations. A fuzzy Euclidian distance measure is used to compare two fuzzy numbers and to evaluate how fuzziness in the observed hydraulic heads affects the fuzziness in the estimated hydraulic gradients. The Euclidian distance measure is also used to ascertain the influence of each well on the fuzzy hydraulic gradient estimation. The fuzzy regression framework is illustrated by applying it to evaluate hydraulic gradients in the unconfined portion of the Gulf Coast aquifer in Goliad County, TX. The results from the case-study indicate that there is greater uncertainty associated with the estimation of the hydraulic gradients in the vertical (Z-axis) direction. The epistemic uncertainties in the hydraulic head data at the wells have a significant impact on the gradient estimates when they are of the same order of magnitude as the most likely values of the observed heads. The influence analysis indicated that 5 of the 13 wells in the network had a critical influence on at least one of the hydraulic gradients. Three wells along the northeastern section of the study area and bordering the Victoria County were noted to have the least influence on the regression estimates. The fuzzy regression framework along with the associated goodness-of-fit and influence measures provides a useful set of tools to characterize the uncertainties in the hydraulic heads and gradients arising from data gaps and planar water table approximation.