On the coarse-graining of hydrologic processes with increasing scales

The transformation of hydrologic processes from nonstationarity to stationarity with increasing scales is investigated. The fundamental mechanism of this transformation is the coarse-graining of hydrologic processes by aliasing and averaging operations. These operations are investigated by means of spectral relationships and hydrologic conservation equations. It is shown that hydrologic conservation equations at large scales may still be parsimonious due to the coarse-graining of hydrologic processes with increasing scales. When a larger scale process is formed by averaging a smaller scale process, the high frequency components of the smaller scale process are eliminated by the averaging operation, rendering the average hydrologic conservation equations to be quite simple in form. It is also shown that for the ensemble average form of a hydrologic conservation equation to be equivalent to its volume-average form, the parameter functions of that conservation equation at the immediately smaller scale must be ergodic. (C) 1999 Elsevier Science B.V. All rights reserved.