The researchers developed a new model called spatial autoregressive fractionally integrated moving average (sp-ARFIMA) to study spatial dependence in data. By adding a fractional integration parameter to the model, they could show how the range of dependence can change across space. They compared this model to time-series models and other spatial models. They also created a method to estimate the model's parameters and tested it in simulations. Finally, they applied the model to study atmospheric fine particles, specifically aerosol optical thickness, which is important for weather, climate, and environmental science.