APPENDIX A River Network Extraction from Large DEMs Excerpted

At the beginning of this work, it became clear that databases spanning a wide range of scales would be needed to properly test various hypotheses of scale-invariance for river networks. However, the question of what range of scales is required for these tests (and for the concept of scale-invariance to be meaningful) does not have a clear-cut answer. In the eld of critical phenomena, where scale-invariance is well-documented and better understood, it has been suggested that one should not claim scale-invariance for a system unless this invariance can be shown to hold over at least three decades in scale. This is because many functions other than power laws will plot as approximately straight lines on a log-log plot, and it is easier for this to happen when the range of scales is small, especially in the presence of noise, which is always present in geophysical data sets. If we adopt this \three decades" rule of thumb, then we nd that virtually all previous studies of hydrological systems have been too small to verify scale-invariance. In fact, some studies have claimed scale-invariance based on only four powers of two in scale. However, it is unfair to judge prior studies too harshly, because (1) geophysical data over such a wide range of scales is generally unavailable, and (2) the data volumes necessary to pass the three-decades test are enormous. Fortunately, digital terrain data is an exception to (1) because it is widely available at a variety of resolutions and across a broad range of scales. This data takes the form of rectangular arrays of average elevation values, which are called digital elevation models or simply, DEMs. Moreover, over the last decade computer algorithms have been developed for extracting drainage network patterns from DEMs, and advances in computer technology have made it 2 possible to store enormous quantities of data on a single machine. Coupled with software to automate the measurement of geometric and topologic quantities from the extracted networks, these facts open the way for hydrologists and ge-omorphologists to carry out comprehensive studies of enormous river systems. There is no doubt that access to such large quantities of data will lead to major advances in our understanding of these systems. Other methods exist for obtaining a digital representation of a drainage network pattern. For example, the blue lines and other linear features from topographic maps have been digitized for many areas in …