Scale Issues in Remote Sensing: A Review on Analysis, Processing and Modeling <s> Wavelet transform method (WTM) <s> The wavelet variance provides a scale-based decomposition of the process variance for a time series or a random field and has been used to analyze various multiscale processes. Examples of such processes include atmospheric pressure, deviations in time as kept by atomic clocks, soil properties in agricultural plots, snow fields in the polar regions and brightness temperature maps of South Pacific clouds. In practice, data collected in the form of a time series or a random field often suffer from contamination that is unrelated to the process of interest. This paper introduces a scale-based contamination model and describes robust estimation of the wavelet variance that can guard against such contamination. A new M-estimation procedure that works for both time series and random fields is proposed, and its large sample theory is deduced. As an example, the robust procedure is applied to cloud data obtained from a satellite. <s> BIB001 </s> Scale Issues in Remote Sensing: A Review on Analysis, Processing and Modeling <s> Wavelet transform method (WTM) <s> Abstract Ecological experiments are usually conducted on small scales, but the ecological and environmental issues are usually on large scales. Hence, there is a clear need of scaling. Namely, when we deal with patterns and processes on larger scales, a special connection needs to be established on the small scales that we are familiar with. Here we presented a wavelet analysis method that could build relationships between spatial distribution patterns on different scales. With this method, we also studied how spatial heterogeneity and distribution patterns changed with the scale. We investigated the distribution and the habitat of C. ewersmanniana in two plots (200 m × 200 m; the distance between these two plots is 15 km) at Mosuowan desert. The results demonstrated that spatial heterogeneity and distribution patterns were incorporated into larger scales when the wavelet scale varied from one (5 m) to four (20 m). However, if the wavelet scale was above five (25 m), the spatial distribution patterns varied placidly, the oscillation frequency of landforms stabilized at 110 m, and the dynamic quantity period of C. ewersmanniana stabilized at 115–125 m. We also identified signal mutation points with wavelet analysis and verified the heterogeneity degree of local space with position variance. We found that position variance decomposed the distribution patterns on large scales into small sampling plots, and the position with the largest variance also had the strongest heterogeneity. In a word, the wavelet analysis method could scale-up spatial distribution patterns and habitat heterogeneity. With this method and other methods derived from this one, such as wavelet scale, wavelet variance, position variance and extremely direct-viewing graphs, wavelet analysis could be widely applied in solving the scaling problem in ecological and environmental studies. <s> BIB002
The wavelet transform method, which is a relatively new mathematical technique, has already been widely used in various disciplines. It uses a localized function in time or space. The wavelet's size can be adjusted and shifted to analyze a data set. Thus, we can investigate features of interest in the data set at an appropriate scale, for example, broad features at a large scale and fine features at a small scale. With the help of the wavelet transform method, we can find where changes in a data set take place and simultaneously measure how large these changes are. Percival BIB001 suggested that the wavelet variance calculated by the wavelet transform method is a natural tool to investigate the spatial scales of variability in remote sensing data. It can be used as an indicator to quantify the length scale of land surface. Through the analysis of simulating images, the hypothesis is confirmed that the dominant length scales in the landscape may correspond to the scale with the highest wavelet variance  . The best wavelet to identify length scales is the Haar wavelet. The results indicate that the wavelet transform method is also a suitable tool to analyze the heterogeneity of land surface and infer the optimum scale under which the main variability presented in the image may be lost. Apparently, GVM and WTM are different methods. One starts from the space domain, while the other starts from the frequency domain. However, these two methods will produce the same conclusion through several data simulations. It may be due to the fact that the cumulative variance and covariance are equal to the variance and covariance within the specific scale in the case of the Haar wavelet. Wavelet analysis would be a good tool to study multi-scale relationships of spatial pattern and heterogeneity BIB002 . However, the manner of WTM is also dependent on the mother wavelet. WTM's potential for using other wavelets, such as Daubechies, Coiflet and so no, needs to be further explored.