Hydrology and Earth System Sciences www.hydrol-earth-syst-sci.net 1027-5606 1607-7938 10 4 2006 10.5194/hess-10-469-2006 http://www.hydrol-earth-syst-sci.net/10/469/2006/ http://www.hydrol-earth-syst-sci.net/10/469/2006/hess-10-469-2006.html http://www.hydrol-earth-syst-sci.net/10/469/2006/hess-10-469-2006.pdf 469 484 2006-07-03 Mapping mean and variance of runoff in a river basin L. Gottschalk lars.gottschalk@geo.uio.no I. Krasovskaia E. Leblois E. Sauquet Department of Geosciences, University of Oslo, P.O.~Box~1022 Blindern, 0315 Oslo, Norway Cemagref, 3 bis quai Chauveau, CP 220, 69336 Lyon Cedex 09, France The study presents an approach to represent the two first order moments of temporal runoff variability as a function of catchment area and aggregation time interval, and to map them in space. The problem is divided into two steps. First, the first order moment (the long term value) is analysed and mapped applying an interpolation procedure for river runoff. In a second step a simple random model for the river runoff process is proposed for the instantaneous point runoff normalised with respect to the long term mean. From this model analytical expressions for the time-space variance-covariance of the inflow to the river network are developed, which then is used to predict how the second order moment varies along rivers from headwaters to the mouth. The observation data are handled by a hydrological information system, which allows to display the results either in the form of area dependence of moments along the river branches to the basin outlet or as a map of the variation of the moments across the basin. The findings are demonstrated by the example of the Moselle drainage basin (French part). Beldring, S., Roald, L. A. and Voksø, A.: Avrenningskart for Norge (Runoff map of Norway, in Norwegian), Norwegian Water and Energy Directorate Report, 2, 2002, Oslo, Norway, 2002. Gandin, L. S. and Kagan, P. L.: Statisticheskie metody dlya interpretatsii meteorologicheskykh dannykh. (Statistical methods for interpretation of meteorological data, in Russian), Gidrometeoizdat, Leningrad, 1976. Gergov, G. I.: Zakonomernosti v izmenenii modulya stoka (Regularities in the variation of specific runoff, in Russian ), Meteorologiya i Gidrologiya 8, 75–81, 1972. Ghosh, B.: Random distances within a rectangle and between two rectangles, Bull. Calcutta Math. Soc., 43, 1951. Gottschalk, L.: Correlation and covariance of runoff, Stochastic Hydrology and Hydraulics, 7, 85–101, 1993a. Gottschalk, L.: Interpolation of runoff applying objective methods, Stochastic Hydrology and Hydraulics, 7, 269–281, 1993b. Gottschalk, L.: Methods for analysing variability, in: Encyclopaedia of Hydrological Sciences, edited by: Anderson, M. G. and McDonnell, J. J., John Wiley & Sons, 1, 6, 95–122, 2005. Gupta, V. K. and Waymire, E.: Multiscaling properties of spatial rainfall and river flow distribution, J. Geophys. Res., 95(D3), 1999–2009, 1990. Krasovskaia, I. and Gottschalk, L.: Stability of river flow regimes, Nordic Hydrology, 23, 137–154, 1992. Leblois, E. and Sauquet, E.: Grid elevation models in hydrology – Part~1: Principles and a literature review; Part~2: HydroDem, User's manual, Cemagref, Technical Notes, Lyon, 80 pp., 2000. Matérn, B.: Spatial Variation, Meddelanden fr\aan Statens Skogsforskiningsinstitut, 49(5), 1960. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T.: Numerical recipes in Pascal, Cambridge University Press, N.Y., 1992. Sauquet, E.: Une cartographie des écoulements annuels et mensuels d'un grand basin versant structurée par la topologie du réseau hydrographique, Thèse – Institut National Polytechnique de Grenoble 1, 356 pp., 2000. Sauquet, E., Gottschalk, L., and Leblois, E.: Mapping average annual runoff: A hierarchical approach applying a stochastic interpolation scheme, Hydrol. Sci. J., 45(6), 799–815, 2000. Singh, V. P.: Hydrological systems: Rainfall-Runoff Modeling, Prentice Hall, Englewood Cliffs, New Jersey, 480 pp., 1988. Tukey, J. W.: Bias and confidence in not-quite large samples, (Abstract), Ann. Math. Stat., 29, 614, 1961. Vanmarcke, E.: Random Fields. Third printing, MIT press, Cambridge, Mass., 382 pp., 1988.