Auxiliary Material for 2006WR005721 (Mudelsee: "Long memory of rivers from spatial aggregation") Table 1. Runoff records and ARFIMA(1,d,0) fit results. River/ Station Interval A d Phi D_AIC Residual tests Remarks Tributary (km**2) Rank Turn. L-B =============================================================================================================================== Weser Hann. Muenden 01/1857-04/2002 12,442 0.151 ñ 0.064 0.423 ñ 0.074 -0.033 + + + Weser Bodenwerder 01/1857-04/2002 15,924 0.230 ñ 0.054 0.356 ñ 0.064 -0.026 + + + Weser Vlotho 01/1857-04/2002 17,618 0.294 ñ 0.047 0.292 ñ 0.057 -0.020 + + + Weser Intschede 01/1857-04/2002 37,720 0.298 ñ 0.052 0.314 ñ 0.062 -0.024 + + + Elbe/Jizera Turice 11/1910-10/1990 2159 0.180 ñ 0.064 0.395 ñ 0.074 -0.039 + + + Note interval start Elbe Decin 11/1899-10/1990 51,104 0.235 ñ 0.062 0.357 ñ 0.072 -0.031 + + + Elbe Dresden 11/1899-10/1990 53,096 0.277 ñ 0.055 0.305 ñ 0.066 -0.024 + + + Elbe Barby 11/1899-10/1990 94,060 0.284 ñ 0.058 0.325 ñ 0.069 -0.029 + + + Elbe Wittenberge 11/1899-10/1990 123,532 0.291 ñ 0.060 0.365 ñ 0.070 -0.041 + + + Elbe Neu-Darchau 11/1899-10/1990 131,950 0.264 ñ 0.065 0.394 ñ 0.075 -0.045 + + + Rhine Domat/Ems 01/1920-12/2003 3229 -0.019 ñ 0.063 0.661 ñ 0.056 -0.101 + - - Rhine Rekingen 01/1920-12/2003 14,718 0.152 ñ 0.079 0.590 ñ 0.074 -0.113 + + + Rhine Cologne 01/1920-12/2003 144,232 0.246 ñ 0.076 0.367 ñ 0.088 -0.030 + + + Rhine Lobith 01/1920-12/2003 160,800 0.246 ñ 0.073 0.364 ñ 0.085 -0.030 + + + Colorado near Cisco 04/1934-09/2003 62,419 0.274 ñ 0.078 0.552 ñ 0.080 -0.140 + + + Colorado Lees Ferry 04/1934-09/2003 289,562 0.318 ñ 0.057 0.573 ñ 0.047 -0.248 + - - Colorado below Hoover 04/1934-09/2003 444,703 0.430 ñ 0.048 0.355 ñ 0.061 -0.086 + + - Dam Mississippi St. Paul 04/1906-09/2003 95,312 0.442 ñ 0.041 0.283 ñ 0.053 -0.053 + - + Mississippi Clinton 04/1906-09/2003 221,704 0.362 ñ 0.053 0.262 ñ 0.064 -0.022 + + + Mississippi Keokuk 04/1906-09/2003 308,210 0.339 ñ 0.061 0.339 ñ 0.073 -0.038 + + + Nile/ Mongalla 01/1912-12/1982 450,000 0.493 ñ 0.010 0.418 ñ 0.034 -0.182 + + - Monthly data White Nile Nile/ Malakal 01/1912-12/1982 1,080,000 0.484 ñ 0.022 0.386 ñ 0.042 -0.079 + - - Monthly data White Nile Nile/ Roseires Dam 01/1912-12/1982 210,000 -0.024 ñ 0.092 0.644 ñ 0.086 -0.075 + + - Monthly data Blue Nile Nile/ Sennar 01/1912-12/1982 230,000 0.145 ñ 0.076 0.409 ñ 0.089 -0.034 + - - Monthly data Blue Nile Nile Tamaniat 01/1912-12/1982 2,000,000 0.148 ñ 0.063 0.587 ñ 0.058 -0.139 + - - Monthly data Nile Hudeiba 01/1912-12/1982 2,010,000 0.095 ñ 0.071 0.625 ñ 0.063 -0.127 + - - Monthly data Nile Dongola 01/1912-12/1982 2,690,000 0.159 ñ 0.060 0.612 ñ 0.052 -0.174 + + - Monthly data Nile Aswan Dam 01/1871-12/1967 2,850,000 0.200 ñ 0.054 0.559 ñ 0.054 -0.125 + - - Monthly data, note interval start and end =============================================================================================================================== Notes (1) Time resolution is daily if not otherwise noted. (2) Interval dates are given as month/year. (3) A, size of catchment area [Bodo, 2001; Global Runoff Data Centre, 2006]; d, long-memory parameter; Phi, parameter of the autoregressive model component. Errors given are 1-sigma standard errors. (4) D_AIC, difference of AIC values. AIC is Akaike's [1973] information criterion, which measures fit quality. A lower AIC value indicates a superior fit quality. D_AIC is the AIC value for the fitted ARFIMA(1,d,0) model (with autoregressive component) minus the AIC value for the fitted ARFIMA(0,d,0) model (without autoregressive component). A negative D_AIC value indicates that the ARFIMA(1,d,0) model is a better description of the data than the ARFIMA(0,d,0) model. (5) Following residual tests [Brockwell and Davis, 1991] were performed (99% level) to check for uncorrelatedness and absence of trend: Rank, rank test; Turn., turning-point test; L-B, Ljung-Box test. A positive (negative) test result is indicated by a plus (minus) sign. (6) Blue Nile and Nile: Area values are rough estimates [Bodo, 2001]. References Akaike, H. (1973) Information theory and an extension of the maximum likelihood principle, in Second International Symposium on Information Theory, edited by B. N. Petrov and F. Csaki, pp. 267-281, Akademiai Kiado, Budapest. Bodo, B. A. (2001) Annotations for Monthly Discharge Data for World Rivers (excluding former Soviet Union), http://dss.ucar.edu/datasets/ds552.1/docs/docglob.pdf, National Center for Atmospheric Research, Boulder, Colo. Brockwell, P. J., and R. A. Davis (1991) Time Series: Theory and Methods, 2nd ed., 577 pp., Springer, New York. [Section 9.4 therein] Global Runoff Data Centre (2006), GRDC Station Catalogue, http://grdc.bafg.de/servlet/is/910, Federal Institute of Hydrology, Koblenz, Germany.