Shorelines exhibit long-range dependence (LRD) and have been shown in some environments to be described in the wavenumber domain by a power law characteristic of scale-independence. Recent evidence suggests that the geomorphology of barrier islands can, however, exhibit scale dependence as a result of systematic variations of the underlying framework geology. The LRD of framework geology, which influences island geomorphology and its response to storms and sea level rise, has not been previously examined. Electromagnetic induction (EMI) surveys conducted along Padre Island National Seashore (PAIS), Texas, USA, reveal that the EMI apparent conductivity σa signal and, by inference, the framework geology exhibits LRD at scales up to 101 to 102 km. Our study demonstrates the utility of describing EMI σa and LiDAR spatial series by a fractional auto-regressive integrated moving average process that specifically models LRD. This method offers a robust and compact way for quantifying the geological variations along a barrier island shoreline using three parameters (p,d,q). We discuss how ARIMA (0,d,0) models that use a single parameter d provide a quantitative measure for determining free and forced barrier island evolutionary behavior across different scales. Statistical analyses at regional, intermediate, and local scales suggest that the geologic framework within an area of paleo-channels exhibits a first order control on dune height. The exchange of sediment amongst nearshore, beach and dune in areas outside this region are scale-independent, implying that barrier islands like PAIS exhibit a combination of free and forced behaviors that affect the response of the island to sea level rise.