The fractal dimension (D) of a surface can be viewed as a summary or average statistic for characterizing the geometric complexity of that surface. The D values are useful for measuring the geometric complexity of various land cover types. Existing fractal methods only calculate a single D value for representing the whole surface. However, the geometric complexity of a surface varies across patches and a single D value is insufficient to capture these detailed variations. Previous studies have calculated local D values using a moving window technique. The main purpose of this study is to compute local D values using an alternative way by incorporating the geographical weighting scheme within the original global fractal methods. Three original fractal methods are selected in this study: the Triangular Prism method, the Differential Box Counting method and the Fourier Power Spectral Density method. A Gaussian density kernel function is used for the local adaption purpose and various bandwidths are tested. The first part of this dissertation research explores and compares both of the global and local D values of these three methods using test images. The D value is computed for every single pixel across the image to show the surface complexity variation. In the second part of the dissertation, the main goal is to study two major U.S. cities located in two regions. New York City and Houston are compared using D values for both of spatial and temporal comparison. The results show that the geographical weighting scheme is suitable for calculating local D values but very sensitive to small bandwidths. New York City and Houston show similar global D results for both year of 2000 and 2016 indicating there were not much land cover changes during the study period
