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The paper extends the Representative Elementary Watershed (REW) theory for cold regions through explicit treatment of energy balance equations to include associated processes and process descriptions. A new definition of REW is presented which subdivides the REW into six surface sub-regions and two subsurface sub-regions. Vegetation, snow, soil ice, and glacier ice are included in the system so that such phenomena as evaporation/transpiration, melting, freezing, and thawing can be modeled in a physically reasonable way. The sub-stream-network is separated from other sub-regions so that the sub-REW-scale runoff routing function can be modeled explicitly. The final system of 24 ordinary differential equations (ODEs) can meet the requirements of most hydrological modeling applications, and the formulation procedure is re-arranged so that further inclusion of sub-regions and substances could be done more easily. The number of unknowns is more than the number of equations, which leads to the indeterminate system. Complementary equations are provided based on geometric relationships and constitutive relationships that represent geomorphological and hydrological characteristics of a watershed. Reggiani et al. (1999, 2000, 2001) and Lee et al. (2005b) have previously proposed sets of closure relationships for unknown mass and momentum exchange fluxes. Tian (2006) has applied Lee's procedures and formulas and Monte Carlo simulation method, and has come up with a determinate system based on the equations, though precluding energy balance ones, proposed in this paper. The additional geometric and constitutive relationships required to close the new set of balance equations will be pursued in a subsequent paper.