Problems of Scale and Regionalization in River Basins The occurrence of the hydrologic cycle covers a very wide range of space and time scales, and involves physical, chemical, and ecological processes. Therefore, in modeling and making predictions, one is required to understand how various properties and measurements behave under a change of scale. At the spatial and temporal scales of interest in river basins, space–time variability and fluctuations are displayed in input, output, and storage elements of the components of the hydrologic cycle and in their interactions. This variability is part of the physics and in this sense is different from the measurement noise. We can use the term physical-statistical or statistical-dynamical to describe such systems. An understanding of the physics of these systems in the presence of variability and fluctuations through mathematical notions of randomness has been and continues to be one of the central challenges of hydrology and constitutes the main theme of this chapter. A river basin contains a channel network, as shown in Figure 4.1, and systems of hills on both sides of the channels in the network. Rainfall and/or snowmelt are transformed into runoff, and sediments are eroded over hills, and these in turn are fed into a channel network for their journey toward an ocean. The hydrologic cycle on a hillside involves transformation of rainfall to surface runoff, infiltration through the near-surface unsaturated soils, and evapotranspiration from the soil surface into the atmosphere. The infiltrated water goes into recharging the soil moisture in the unsaturated zone, the aquifers, and some of it also appears as subsurface runoff in a channel network.