Modeling of groundwater heads based on second-order difference time series models

Historical groundwater heads at a confined aquifer in southwest Florida show a nonstationary long-term (multi-year) fluctuation. Stochastic modeling of these data is a main topic here. Ahn and Salas (Ahn, H., Salas, J.D., 1997. Groundwater head sampling based on stochastic analysis. Water Resour. Res. 33(12), 2769–2780) introduced an approach to build time series models of nonstationary data at different time intervals based on an observed time series sampled at a reference interval. The model utilized in their study was a first-order difference autoregressive integrated moving average model. However, some groundwater head data may also be fitted adequately by a second-order difference time series model. Thus, this study derived variance and autocovariance equations for the second-order difference time series model at various time intervals as a function of the parameters of the referenced model. The derived equations are useful for building a time series model at arbitrary time intervals. Unlike the first-order difference models, the variance and auto-covariance equations here are fully derivable, making the second-order difference models more convenient than the first-order difference models. The modeling procedure with the derived equations was tested through example problems of: (1) filling in gaps in time series; and (2) sampling frequency design. The results showed that the second-order difference model in some cases produces lower interpolation error than that of the first-order difference model.