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20 ALLUVIAL FAN FLOODING thus establishing a relationship between inundation depth above a point (such as the finished floor ofa house) and the recurrence interval. Figure 1-4a(iv) shows the cases for the main channel and the higher surface (labeled 2 on the cross section), which is often called the overbank area or the floodway jril/ge. The stage recurrence graph developed in this manner always has some uncertainty, but it is usually not presented. The riverine approach provides a clear method that allows us to communicate about flooding and to make reproducible calculations of its severity. Over time it has become widely accepted and its weaknesses seldom questioned. Figure 1-4b shows the analogous components of an analytical approach to alluvial fan flooding. First, the discharge recurrence relationship is estimated for the apex of the fan. Second, based on the knowledge that the perceived or historical flow path may not convey all of the water during a flood, it is assumed that the actual flow path has no greater chance of occupying the perceived channel than it does of straying to any location on the fan. This default assumption is shown in Figure 1-4b(ii). The relationship between depth and discharge is then determined using the method proposed by Dawdy (1979) (although a case can be made for altering this step by using the process-based knowledge described in Chapter 3 and an alternative solution to the conditional probability). Finally, the inundation depth and velocity are delineated based on the assumption that the entire fan surface is subject to flooding. The predicted degrees of flood hazard for the surfaces labeled I and 2 are identical because the procedure knows nothing of the differences between them. Real floods on alluvial fans are, of course, much more complex than this. An important implication of this approach to the prediction of flood risk on alluvial fans is illustrated in Figure 1-5, which portrays a flood-prone surface with three distinct elevations, labeled I, 2 (the main, recently occupied channel), and 3. In the traditional riverine flooding paradigm (Figure 1-5a) the historical channel is the main conveyor of the base flood and the computation of the water surface in "overbank" areas is based on the conveyance capacity of a single cross section that includes surface 2. This approach shows surface 3 as "wet" merely as a consequence of surface 2 being too small to convey the entire flood. Surface 1 is above the computed base flood elevation and could therefore be shown as outside of the 100-year floodplain on the FIRM. In such areas, we imply that the probability density function that describes flow path location within the lateral domain is narrow and strongly peaked, that is, that all of the flow behaves hydraulically as a single channel contained within a relatively narrow zone that does not shift during the event. Figure 1-5b portrays the case of alluvial fan flooding where, during the base flood event, the channel might separate into two branches upstream of the cross-section, allowing a flow path to develop that invades the higher surface I. After such a flow split occurs upstream, surface 1 may be flooded even during events smaller than the 100-year event depending on the specific behavior of a real sequence of floods. If the alluvial fan flooding paradigm is applied to the situation (Figure 1-5b), surface I is treated as a separate flow path, which it may well become during an actual flood. The corresponding probability density function for this situation shows three peaks indicating that each of surfaces I, 2, and 3 have a finite chance of conveying water during a flood. Based on this potential, multiple scenarios are analyzed that represent the possible distribution of flows on the three surfaces. From this information, a separate baseflood elevation is computed for each surface via the conditional probability equation. The FIRM in this case would show that surface I is indeed subject to flooding. (Note: The probability density functions in these figures are shown to illustrate the difference between the two flooding perspectives. Showing the