This thesis presents a new method to estimate the Apparent Proper Motion (APM) vector and its uncertainty for supernova light echoes (LEs) and tests its usefulness in practice on LEs due to two old Galactic supernovae (SNe) - Cas A and Tycho. Ten instances of two-dimensional cross-correlation (2-D CC) of images containing light echoes at diferent epochs are employed to examine how well this new method works in practice. The images selected for this work originate from KPNO 4m Mosaic 1.1 images and were originally processed by the Pan-STARRS pipeline. All the APM estimates reported in this thesis are within 1sigma of estimates based on supernova distance and age provided reasonable inclinations are assumed. It was found that several factors tend to reduce the expected precision of this method and these include: 1) the existence of more than one LE feature for each epoch, 2) longer intervals between the two epochs lead to a bias, and 3) the existence of dust filaments at more than one depth along the line of sight. The results of three LE fields which were in common with the previous studies by Rest et al. in 2008 and 2011, were compared and a good agreement was found between them in difference-images with the same time interval. Since pixel values have a significant role in the introduced method, a control region is considered to eliminate the defect of the irrelevant residuals to the LE features. Hence, the introduced method was not straightforward. In addition, this method was not thoroughly manual independent, as the benefits of the visual measurement from the previous method reported by Rest et al. (2008) and (2011) were adopted for this method. However, compared to the previous manual technique, there were much less manual measurements were taken for the whole LE features in one frame. Considering all the challenges, the CC method is favourable as the APM vector uncertainty can be determined, which has not been achievable with previous method before. / Thesis / Master of Science (MSc)