0 10 20 30
Lag (in Ax units)
Figure 9 Comparison of correlation function for monthly rainfall (from Kang and Ramirez46). See ftp site for color image.
scale environment. Being able to parameterize the scaling characteristics of precipitation as a function of such variables is a prerequisite for implementing of down-scaling methodologies based on random cascades. Perica and Foufoula-Georgiou77 introduced a spatial downscaling scheme that is able to statistically reproduce the spatial heterogeneity of observed precipitation fields at subgrid scales while being conditioned on large-scale averages and physical properties. They computed multi-scale standardized fluctuations using an orthogonal Haar wavelet decomposition and found that, at least for the range of scales of their analysis, these fluctuations exhibited normality and simple scaling. They also found that the scale-independent parameter H characterizing the simple scaling behavior of the standardized fluctuations was strongly dependent on the convective instability of the prestorm environment, namely on the convective available potential energy. The utility of the model in reproducing the small-scale statistical variability of precipitation as well as the fraction of area covered by rain at all subgrid scales was demonstrated,77 and the relationship between H and the convective available potential energy of the prestorm
From Kang and Ramirez.46
environment established.77 On the other hand, the relationship between the /S-log-normal random cascade model parameters and the mean of the large-scale precipitation intensity was also observed and established.46'75
Most downscaling methodologies proposed in the literature only deal with the spatial variability of the precipitation field. The temporal evolution of the fields is usually described independently of the spatial downscaling, so that these schemes do not properly account for the temporal correlation structure, i.e., persistence, of the precipitation fields at subgrid scales. Recently, the linkage between the spatial and temporal scaling of precipitation fields has been explicitly addressed.12'75'113'114 Over and Gupta75 propose a model for space-time description of rainfall distributions based on multiplicative random cascades with independent weights in space, which are time varying according to an imposed structure. Carsteanu and Foufoula-Georgiou12 argue that space and time variations of rainfall are necessarily connected. They postulate and experimentally verify a Taylor-like hypothesis stating that the power law variation for the moments is the same in time and space. Venugopal et al.114 found that for spatial scales of 2 to 30 km and for temporal scales of 10 min to several hours, the evolution of precipitation remained statistically invariant under a transformation of the type t ~ L~, where z is a so-called dynamic scaling exponent. That is, they found that the space-time organization of rainfall fields is scale-invariant and that its characteristics can be obtained by a simple renormaliza-tion of the space and time coordinates as implied by the t ~ Lz transformation. They used the above results to develop a space—time precipitation downscaling scheme that is capable of preserving not only the spatial correlation of precipitation but also the temporal correlation at subgrid scales.
Finally, Seed et al." have modeled the space-time behavior of radar precipitation using a multiplicative bounded (multifractal) cascade, each level of which was linked to the same level at the next time step via a different ARMA( 1,1) model. Also Pegram and Clothier,76 developed the so-called string-of-beads model in which power-law filtering of Gaussian random fields in space and time is used to capture the correlation structure of the rainfall process. Two autoregressive models, one at the image scale, the other at the pixel scale, drive the string-of-beads model. The spatial power-law filtering then ensures that the generated fields scale correctly in space and time.
1. Aksoy, H., and M. Bayazit, A model for daily flows of intermittent streams, Hydrol. Proc., 14( 10), 1725-1744, 2000.
2. Anthes, R. A., and T. T. Warner, Development of hydrodynamic models suitable for air pollution and other mesometeorological studies, Monthly Weather Rev., 106, 1045-1078, 1978.
3. Bartolini, R, and J. D. Salas, Modeling of streamflow processes at different time scales, Water Resonr. Res., 29(8), 2573-2588, 1993.
4. Black, T. J., The new NMC mesoscale eta model: Description and forecast examples, Weather Forecast., 9, 265-278, 1994.
5. Box, G. E. P., and G. M. Jenkins, Time Series Analysis Forecasting and Control, Holden-Day, San Francisco, 1976.
6. Bras, R. L., and I. Rodriguez-Iturbe, Random Functions and Hydrology, Addison-Wesley, Reading, MA, 1985.
7. Buishand, T. A., Some remarks on the use of daily rainfall models, J. Hydrol., 36, 295308, 1977.
8. Burian, S. J., S. R. Durrans, S. Tomic, R. L. Pimentel, and C. N. Wai, Rainfall disaggregation using artificial neural networks, ASCE J. Hydrol. Eng., 5(3), 299-307, 2000.
9. Burlando, P., and R. Rosso, Comment on "Parameter estimation and sensitivity analysis for the modified Bartlett-Lewis Rectangular pulses model of rainfall by Islam et al., J. Geophys. Res., vol. 95, no. D3, 1990, p. 2093-2100," J. Geophys. Res., 96(D5), 93919395, 1991.
10. Burlando, P., and R. Rosso, Stochastic models of temporal rainfall: Reproducibility, estimation and prediction of extreme events, in J. Marco Segura, R. Harboe, and J. D. Salas (Eds.), Stochastic Hydrology and its Use in Water Resources Systems Simulation and Optimization, Kluwer Academic Publishers, The Netherlands, 1993, pp. 137-173.
11. Cadavid, L. G., J. D. Salas, and D. C. Boes, Disaggregation of short-term precipitation records, in Water Resources Papers, Vol. 106, Colorado State University, Fort Collins, CO, 1992.
12. Carsteanu, A., and E. Foufoula-Georgiou, Assessing dependence among weights in a multiplicative cascade model of temporal rainfall, J Geophy. Res., I()I(D2126, 363-26, 370, 1996.
13. Chang, T. J., M. L. Kavvas, and J. W. Delleur, Daily precipitation modeling by discrete autoregressive moving average processes, Water Resour. Res., 20, 565-580, 1984.
14. Chebaane, M., J. D. Salas, and D. C. Boes, Product periodic autoregressive processes for modeling intermittent monthly streamflows, Water Resour. Res., 52(5), 1513-1518, 1995.
15. Chin, E. H., Modeling daily precipitation occurrence process with Markov chain, Water Resour. Res., 13(6), 949-956, 1977.
16. Claps, P., F. Rossi, and C. Vitale, Conceptual-stochastic modeling of seasonal runoff using autoregressive moving average models and different scales of aggregation, Water Resour. Res., 29(8), 2545-2559, 1993.
17. Cowpertwait, P. S. P., and P. E. O'Connell, A regionalized Neyman-Scott model of rainfall with convective and stratiform cells, Hydrol. Earth Syst. Sci., 1, 71-80, 1997.
18. Delleur, J. W., and M. L. Kavvas, Stochastic models for monthly rainfall forecasting and synthetic generation, J. Appl. Meteor., 17, 1528-1536, 1978.
19. Dudhia, J., A nonhydrostatic version of the Penn State/NCAR mesoscale model: Validation tests and the simulation of an Atlantic cyclone and cold front, Monthly Weather Rev., 121, 1493-1513, 1993.
20. Eagleson, P., Climate, soil and vegetation, 2, The distribution of annual precipitation derived from observed storm sequences, Water Resour. Res., 14, 713-721, 1978.
21. Eltahir, E. A. B., A feedback mechanism in annual rainfall in Central Sudan, J. Hydrol, 110, 323-334, 1989.
22. Entekhabi, D., I. Rodriguez-Iturbe, and P. S. Eagleson, Probabilistic representation of the temporal rainfall process by a modified Neyman-Scott rectangular pulse model: Parameter estimation validation, Water Resour. Res., 25(2), 295-302, 1989.
23. Entekhabi, D., and P. S. Eagleson, Land surface hydrology parameterization for atmospheric general circulation models including subgrid scale spatial variability, J. Climate, 2(8), 816-831, 1989.
24. Epstein, D., and J. A. Ramirez, Spatial disaggregation for studies of climatic hydrologic sensitivity, ASCE J. Hydr. Div., 720(12), 1449-1467, 1994.
25. Evora, N. D., and J. R. Rousselle, Hybrid stochastic model for daily flows simulation in semiarid climates, ASCE J. Hydrol, 5(1), 33^12, 2000.
26. Ewen, J., G. Parkin, and P. E. O'Connell, SHETRAN: Distributed River Basin flow and transport modeling system, ASCE J. Hydrol. Eng., 5(3), 250-258, 2000.
27. Fernandez, B., and J. D. Salas, Periodic gamma autoregressive processes for operational hydrology, Water Resour. Res., 22(10), 1385-1396, 1986.
28. Fernandez, B., and J. D. Salas, Gamma-autoregressive models for streamflow simulation, J. Hydr. Eng. ASCE, 116(11), 1403-1414, 1990.
29. Fiering, M. B., and B. B. Jackson, Synthetic Streamflows, Water Resources Monograph 1, American Geophysical Union (AGU), Washington, DC, 1971.
30. Foufoula-Georgiou, E., and P. Guttorp, Compatibility of continuous rainfall occurrence models with discrete rainfall observations, Water Resour. Res., 22, 1316-1322, 1986.
31. Foufoula-Georgiou, E., and D. P. Lettenmaier, A Markov renewal model of rainfall occurrences, Water Resour. Res., 23(5), 875-884, 1987.
32. Foufoula-Georgiou, E., and W. Krajewski, Recent advances in rainfall modelling, estimation and forecasting, Rev. Geophys., Suppl., 1125-1137, July 1995.
33. Giorgi, F., and L. O. Mearns, Approaches to the simulation of regional climate change: A review, Rev. Geophys., 29(2), 191-216, 1991.
34. Giorgi, F., M. R. Marinucci, and G. T. Bates, Development of a second-generation regional climate model (RegCM2). Part I: Boundary-layer and radiative transfer processes, Monthly Weather Rev., 121, 2794-2813, 1993a.
35. Giorgi, F., M. R. Marinucci, and G. T. Bates, Development of a second-generation regional climate model (RegCM2). Part II: Convective processes and assimilation of lateral boundary conditions, Monthly Weather Rev., 121, 2814-2832, 1993b.
36. Grygier, J. C., and J. R. Stedinger, Condensed disaggregation procedures and conservation corrections for stochastic hydrology, Water Resour. Res., 24, 1574-1584, 1988.
37. Gupta, V K., and E. Waymire, Multiscaling properties of spatial rainfall and river flow distributions, J. Geoph. Res., 95(D3), 1999-2009, 1990.
38. Gupta, V K., and E. Waymire, A statistical analysis of mesoscale rainfall as a random cascade, J. Appl. Meteor, 12(2), 251-267, 1993.
39. Guttorp, P., Stochastic Modeling of Scientific Data, Chapman Hall, London, 1995.
40. Gyasi-Agyei, Y., and G. R. Willgoose, A hybrid model for point rainfall modeling, Water Resour. Res., 33(1), 1699-1706, 1997.
41. Hershenhorn, J., and D. A. Woolhiser, Disaggregation of daily rainfall, J. Hydrol., 95, 299-322, 1987.
42. Hipel, K. W., and A. I. McLeod, Time Series Modeling of Water Resources and Environmental Systems, Elsevier, Amsterdam, 1994.
43. Hirsch, R. M., Synthetic hydrology and water supply reliability, Water Resour. Res., 15(6), 1603-1615, 1979.
44. Hosking, J. R. M., Fractional differencing, Biometrika, 68, 165-176, 1981.
45. Intergovernmental Panel on Climate Change (IPCC), Summary for Policymakers, report of Working Group I of the IPCC, available on-line, http://www.ipcc.ch/, 2001.
46. Kang, B., and J. A. Ramirez, Comparative study of the statistical features of random cascade models for spatial rainfall downscaling, in J. A. Ramirez (Ed.), Proc. AGU Hydrol. Days 2001, Hydrology Days Publications, Fort Collins, CO, 2001, pp. 151-164.
47. Karl, T. R., W. C. Wang, M. E. Schlesinger, R. W. Knight, and D. Portman, A method of relating general circulation model simulated climate to the observed local climate. Part I: Seasonal statistics. J. Climate, 3, 1053-1079, 1990.
48. Katz, R. W., On some criteria for estimating the order of a Markov chain, Technometrics, 25(3), 243-249, 1981.
49. Katz, R. W., and M. B. Parlange, Generalizations of chain-dependent processes: Application to hourly precipitation, Water Resour. Res., 31, 1331-1341, 1995.
50. Kavvas, M. L., L. J. Cote, and J. W. Delleur, Time resolution of the hydrologic time series models, J. Hydrol., 32, 347-361, 1977.
51. Kavvas, M. L., and J. W. Delleur, A stochastic cluster model of daily rainfall sequences, Water Resour. Res., 17(4), 1151-1160, 1981.
52. Kelman, J, A stochastic model for daily streamflow, J. Hydrol., 47, 235-249, 1980.
53. Koch, R. W., A stochastic streamflow model based on physical principles, Water Resour. Res., 21(4), 545-553, 1985.
54. Koepsell, R. W., and J. B. Valdes, Multidimensional rainfall parameter estimation from a sparse network, ASCE J. Hydr. Eng., 117(1), 832-850, 1991.
55. Krajewski, W. F., and J. A. Smith, Sampling properties of parameter estimators for a storm field rainfall model, Water Resour. Res., 25(9), 2067-2075, 1989.
56. Lane, W. L., Applied Stochastic Techniques (Last Computer Package), User Manual, Division of Planning Tech. Services, Bureau of Reclamation, Denver, CO, 1979.
57. Lane, W. L., Corrected parameters estimates for disaggregation schemes, in V P. Singh (Ed.), Statistical Analysis of Rainfall and Runoff, Water Resources Publications (WRP), Littleton, CO, 1982.
58. Lanza, L. G., A conditional simulation Model of intermittent rain fields, Hydrol. Earth Sys. Sci. 4( 1), 173-183,2000.
59. Leavesley, G. H., R. W. Lichty, B. M. Troutman, and L. G. Saindon, Precipitation-Runoff-Modelling-System—User's Manual, USGS Water Resour. Invest. Report, U.S. Geological Survey, 83-4238, 1983.
60. Le Cam, L. A., A stochastic description of precipitation, in J. Newman (Ed.), Proc. IV Berkeley Symp. on Math., Statis. & Prob., University of Calif. Press, Berkeley, 1961, pp. 165-186.
61. Lettenmaier, D. P., and S. J. Burges, Operational assessment of hydrologic models of long-term persistence, Water Resour. Res., 13(1), 113-124, 1977.
62. Loucks, D. P., J. R., Stedinger, and D. Haith, Water Resources Systems Planning and Analysis, Prentice Hall, Englewood Gifts, NJ, 1981.
63. Lovejoy, S., and B. B. Mandelbrot, Fractal properties of rain and a fractal model, Tellus, 37A, 209-232, 1985.
64. Mandelbrot, B. B., and J. R. Wallis, Computer experiments with fractional Gaussian noises: Part 1, Averages and variances, Water Resour. Res., 5(1), 228-241, 1969.
65. Matalas, N. C., Mathematical assessment of synthetic hydrology, Water Resour. Res., 3(4), 937-945, 1967.
66. McKerchar, A. I. and J. W. Delleur, Application of seasonal parametric stochastic models to monthly flow data, Water Resour. Res., 10, 246-255, 1974.
67. Mellor, D., The modified turning bands (MTB) model for space-time rainfall. I. Model definition and properties, J. Hydrol, 175(l^t), 113-127, 1996.
68. Murrone, F., F. Rossi, and P. Claps, Conceptually-based shot noise modeling of stream-flows at short time interval, Stochast Hydrol. Hydraul., 11(6), 483-510, 1997.
69. Neyman, J., and E. L. Scott, Statistical approach to problems of cosmology, J. R. Stat. Soc. Ser. B, 20(1), l^t3, 1958.
70. Obeysekera, J. T. B., and J. D. Salas, Modeling of aggregated hydrologic series, J. Hydrol., 86, 197-219, 1986.
71. Obeysekera, J. T. B., G. Tabios, and J. D. Salas, On parameter estimation of temporal rainfall models, Water Resour. Res., 25(10), 1837-1850, 1987.
72. O'Connell, P. E., A simple stochastic modeling of Hurst's law, in 1971 Warsaw Symp. in Mathematical Models in Hydrology, International Association of Hydrologic Sciences, Pub. vol. 100, No. 1, 1974, pp. 169-187.
73. O'Connell, P. E. Stochastic modeling of long-term persistence in streamflow sequences, Ph.D. dissertation, Imperial College of Science and Technology, University of London, England, 1974.
74. Ormsbee, L. E., Rainfall disaggregation model for continuous hydrologic modeling, ASCEJ. Hydraul. Eng., 115(94), 507-525, 1989.
75. Over, T. M., and V J. Gupta, A space-time theory of mesoscale rainfall using random cascades, J. Geophys. Res., 101(D21), 26319-26331, 1996.
76. Pegram, G. G. S., and A. N. Clothier, High resolution space-time modeling of rainfall: The string of beads model, WRC Report no. 752/1/99, report to the Water Research Commission, Pretoria, South Africa, 1999.
77. Perica, S. E., and E. Foufoula-Georgiou, Model for multiscale disaggregation of spatial rainfall based on coupling meteorological and scaling descriptions, J. Geophys. Res. Atmos. 101(D21), 26347-26361, 1996.
78. Pielke, R. A., and R. Avissar, Influence of landscape structure on local and regional climate, Landscape Ecol., 4, 133-155, 1990.
79. Pielke, R. A., W. R. Cotton, R. L. Walko, C. J. Tremback, M. E. Nicholls, M. D. Moran, D. A. Wesley, T. J. Lee, and J. H. Copeland, A comprehensive meteorological modeling system—RAMS, Meteor. Atmos. Phys., 49, 69-91, 1992.
80. Pielke, Sr., R. A., Overlooked issues in the U.S. national climate and IPCC assessments, Preprints, in 11th Symp. on Global Change Studies, 80th AMS Annual Meeting, Long Beach, CA, January 9-14, 2000, pp. 32-35.
81. Pielke, Sr., R. A., and L. Guenni, Vulnerability assessment of water resources to changing environmental conditions, 1GBP Newslett., 39, 21-23, 1999.
82. Ramirez, J. A., and R. L. Bras, Conditional distributions of Neyman-Scott models for storm arrivals and their use in irrigation control, Water Resour. Res., 21, 317-330, 1985.
83. Ramirez, J. A., and S. Senarath, A statistical-Dynamical parameterization of canopy interception and land surface-atmosphere interactions, J. Climate, 13, 4050-4063, 2000.
84. Rasmussen, P. F., J. D. Salas, L. Fagherazzi, J. C. Rassam, and B. Bobee, Estimation and validation of contemporaneous PARMA models for streamflow simulation, Water Resour. Res., 32(10), 3151-3160, 1996.
85. Richardson, C. W., and D. A. Wright, WGEN: A Model for Generating Daily Weather Variables, U.S. Department of Agriculture, Agriculture Research Service, ARS-8, August, 1984.
86. Rodriguez- Iturbe, I., V K. Gupta, and E. Waymire, Scale considerations in the modeling of temporal rainfall, Water Resour. Res., 20(11), 1611-1619, 1984.
87. Rodriguez-Iturbe, I., D. R. Cox, and V Isham, Some models for rainfall based on stochastic point processes, Proc. R. Soc. Lond. Ser. A, 410, 269-288, 1987.
88. Rodriguez-Iturbe, I., B. Febres de Power, and J. B. Valdes, Rectangular pulses point process models for rainfall: Analysis of empirical data, J. Geophys. Res., 92(D8), 96459656
89. Rodriguez-Iturbe, I., B. Febres de Power, M. B. Sharifi, and K. Georgakakos, Chaos in rainfall, Water Resour. Res., 25(1), 1667-1675, 1989.
90. Roldan, J., and D. A. Woolhiser, Stochastic daily precipitation models: 1. A comparison of occurrence processes, Water Resour. Res., 18(5), 1451-1459, 1982.
91. Salas, J. D., D. C. Boes, V Yevjevich, and G. G. S. Pegram, Hurst phenomenon as a pre-asymptotic behavior, J. Hydrol., 44(1), 1-15, 1979.
92. Salas, J. D., J. R. Delleur, V Yevjevich, and W. L. Lane, Applied Modeling of Hydrologic Time Series, Water Resources Publications, Littleton, CO, 1980.
93. Salas, J. D., and D. C. Boes, Shifting level modelling of hydrologic series, Adv. Water Resour., 3, 59-63, 1980.
94. Salas, J. D., and M. Chebaane, Stochastic modeling of monthly flows in streams of arid regions, in Proc. Intern. Symp. HY&IR Div. ASCE, San Diego, CA, 1990, pp. 749-755.
95. Salas, J. D., and M. W. Abdelmohsen, Determining streamflow drought statistics by stochastic simulation, in Proc. U.S.-PRC Bilateral Symp. on Droughts and Arid-Region Hydrology, Tucson, AZ, September 16-20, 1991, U.S. Geological Survey Open-File Report No. 91-244, 1991.
96. Salas, J. D., and J. T. B. Obeysekera, Conceptual basis of seasonal streamflow time series models, ASCE J. Hydraul. Eng., 118(9,), 1186-1194, 1992.
97. Salas, J. D., Analysis and modeling of hydrologic time series, in D. R. Maidment (Ed.) Handbook of Hydrology, McGraw-Hill Book, New York, 1993, Chapter 19.
98. Santos, E., and J. D. Salas, Stepwise disaggregation scheme for synthetic hydrology, ASCE J. Hydr. Eng., 118(5), 765-784, 1992.
99. Seed, AW, R Srikanthan, and M. Menabde, A space and time model for design storm rainfall, J. Geophy. Res., accepted for publication.
100. Sharma, A., D. G. Tarboton, and U. Lall, Streamflow simulation: A nonparametric approach, Water Resour. Res., 33(2), 291-308, 1997.
101. She, Z. S., and E. C. Waymire, Quantized energy cascade and log-Poisson statistics in fully developed turbulence, Phys. Rev. Lett., 74(2), 1995.
102. Shukla, J., J. Anderson, D. Baumherner, C. Brankovic, Y. Chang, E. Kalnay, L. Marx, T. Palmer, D. Paolino, J. Ploshay, S. Schubert, D. Straus, M. Suarez, and J. Tribbia, Dynamical seasonal prediction, Bull. Am. Meteor. Soc, 81, 2593-2606, 2000.
103. Smith, J. A., and A. F. Karr, A point process model of Summer season rainfall occurrences, Water Resour. Res., 19(1), 95-103, 1983.
104. Smith, J. A., and W. F. Krajewski, Statistical modeling of space-time rainfall using radar and rain gage observations, Water Resour. Res., 25(10), 1893-1900, 1987.
105. Stedinger, J. R., and R. M. Vogel, Disaggregation procedures for generating serially correlated flow vectors, Water Resour. Res., 20(1), 47-56, 1984.
106. Stedinger, J. R., D. P. Lettenmaier, and R. M. Vogel, Multisite ARMA(1,1) and disaggregation models for annual streamflow generation, Water Resour. Res., 21, 497509, 1985a.
107. Stedinger, J. R., D. Pei, and T. A. Cohn, A condensed disaggregation model for incorporating parameter uncertainty into monthly reservoir simulations, Water Resour. Res., 21(5), 665-675, 1985b.
108. Tarboton, D. G., A. Sharma, and U. Lall, Disaggregation procedures for stochastic hydrology based on nonparametric density estimation, Water Resour. Res., 34, 107-119, 1998.
109. Tessier, Y., S. Lovejoy, and D. Schertzer, Universal multifractals: Theory and observations for rain and clouds, J. Appl. Meteorol., 32(2), 223-250, 1993.
110. Treiber, B., and E. J. Plate, A stochastic model for the simulation of daily flows, Hydrol. Sci. Bull., 22(1), 175-192, 1977.
111. Valencia, D. R., and J. C. Schaake, Jr., Disaggregation processes in stochastic hydrology, Water Resour. Res., 9(3), 580-585, 1973.
112. Vecchia, A., J. T. B. Obeysekera, J. D. Salas, and D. C. Boes, Aggregation and estimation for low-order periodic ARMA models, Water Resour. Res., 19(5), 1297-1306, 1983.
113. Venugopal, V, and E. Foufoula-Georgiou, Energy decomposition of rainfall in the time-frequency-scale domain using wavelet packets, J. Hydrol., 187, 3-27, 1996.
114. Venugopal, V, E. Foufoula-Georgiou, and V Sapozhnikov, Evidence of dynamic scaling in space-time rainfall, J. Geophys. Res., 104(D24), 31599-31610, 1999.
115. Walko, R. L., L. Band, J. Baron, T. G. Kittel, R. Lammers, T. J. Lee, D. Ojima, R. A. Pielke, Sr., C. Taylor, C. Tague, C. J. Tremback, and P. L. Vidale, Coupled atmosphere-biophysics-hydrology models for environmental modeling, J. Appl. Meteor., 39, 931-944, 2000.
116. Waymire, E., V. K. Gupta, and I. Rodriguez-Iturbe, A spectral theory of rainfall intensity at the meso-/? scale, Water Resour. Res., 20(10), 1453-1465, 1984.
117. Wigley, T. M. L., P. D. Jones, K. R. Briffa, and G. Smith, Obtaining sub-grid-scale information from coarse resolution general circulation model output, J. Geophys. Res., 95(D2), 1943-1953, 1990.
118. Wilks, D. S., Statistical Methods in the Atmospheric Sciences, Academic, San Diego, C A, 1995.
119. Woolhiser, D. A., and H. B. Osborn, A stochastic model of dimensionless thunderstorm rainfall, Water Resour. Res., 21(4), 511-522, 1985.
120. Yevjevich, V, Stochastic Processes in Hydrology, Water Resouces Publications, Littleton, CO, 1972.
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