Abstract
Most precipitation data comes in the form of daily rainfall totals collected across a network of rain gauges. Research over the past several years on the statistical modeling of rainfall data has led to the development of models in which rain events are formed according to some stochastic process, and deposit rain over an area before they die. Fitting such models to daily data is difficult, however, because of the absence of direct observation of the rain events. In this paper, we argue that such a fitting procedure is possible within a Bayesian framework. The methodology relies heavily on Markov chain simulation algorithms to produce a reconstruction of the unseen process of rain events. As applications, we discuss the potential of such methodology in demonstrating changes in precipitation patterns as a result of actual or hypothesized changes in the global climate.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Reference
Best, D.J. & Fisher, N.I. (1979), Efficient simulation of the von Mises distribution. Applied Statistics 28, 152–157.
Bonnell, M. and Sumner, G. (1992), Autumn and winter daily precipitation events in Wales 1982–1983 to 1986–1987. International Journal of Climatology 12, 77–102.
Cox, D.R. and Isham, V. (1988), A simple spatial-temporal model of rainfall. Proc. Roy. Soc. Lond. A 415, 317–328.
Cox, D.R. & Isham, V. (1994), Stochastic models of precipitation. In Statistics for the Environment, Volume 2 ( V. Barnett & F. Turkman, editors), pp. 3–18. Chichester: John Wiley.
Cressie, N.A.C. (1991), Statistics for Spatial Data. Wiley, New York.
Devroye, L. (1986), Non-Uniform Random Variate Generation. Springer-Verlag, New York.
Green, P.J. (1995), Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711–732.
Hastings, W.K. (1970), Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109.
Henderson, K.G. and Robinson, P.J. (1994), Relationships between the Pacific/North American teleconnection patterns and precipitation events in the southeastern United States. International J. Climatology 14, 307–323.
Henderson-Sellers, A. and McGuffie, K. (1987), A Climate Modelling Primer. Wiley, New York.
Le Cam, L. (1961), A stochastic description of precipitation. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 3, ed. J. Neyman, Berkeley, CA, pp. 165–186
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953), Equation of state calculations by fast computing machines. Journal of Chemical Physics 21, 1087–1092.
Müller, P. (1991), A generic approach to posterior integration and Gibbs sampling. Preprint, Duke University.
Phelan, M.J. (1992), Aging functions and their nonparametric estimation in point process models of rainfall. Chapter 6 of Statistics in the Environmental and Earth Sciences, eds. A. Walden and P. Guttorp, Halsted Press, New York.
Phelan, M.J. and Goodall, C.R. (1990), An assessment of a generalized Waymire—Gupta—Rodriguez-Iturbe model for GARP Atlantic Tropical Experiment rainfall. J. Geophys. Res. 95, 7603–7616.
Ripley, B.D. (1981), Spatial Statistics. Wiley, New York.
Robinson, P.J. (1994), Precipitation regime changes over small watersheds. In Statistics for the Environment, Volume 2 ( V. Barnett & F. Turk-man, editors), pp. 43–59. Chichester: John Wiley.
Rodriguez-Iturbe, I., Cox, D.R. and Eagleson, P.S. (1986), Spatial modelling of total storm rainfall. Proc. Roy. Soc. Lond. A 403, 27–50.
Rodriguez-Iturbe, I., Cox, D.R. and Isham, V. (1987), Some models for rainfall based on stochastic point processes. Proc. Roy. Soc. Lond. A 410, 269–288.
Rodriguez-Iturbe, I., Cox, D.R. and Isham, V. (1988), A point process model for rainfall: further developments. Proc. Roy. Soc. Lond. A 417, 283–298.
Smith, A.F.M. and Roberts, G.O. (1993), Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J.R. Statist. Soc. B 55, 3–23.
Smith, R.L. (1994), Spatial modelling of rainfall data. In Statistics for the Environment, Volume 2 ( V. Barnett & F. Turkman, editors), pp. 19–41. Chichester: John Wiley.
Stern, R.D. and Coe, R. (1984), A model fitting analysis of daily rainfall data (with discussion). J.R. Statist. Soc. A 147, 1–34.
Taylor, G.I. (1938), The spectrum of turbulence. Proc. Roy. Soc. Lond. A 164, 476–490.
Tierney, L. (1994), Markov chains for exploring posterior distributions. Ann. Statist. 22, 1701–1728.
Waymire, E. and Gupta, V.K. (1981a), The mathematical structure of rainfall representations, 1: A review of the stochastic rainfall models. Water Resources Research 17, 1261–1272.
Waymire, E. and Gupta, V.K. (1981b), The mathematical structure of rainfall representations, 2: A review of the theory of point processes. Water Resources Research 17, 1273–1285.
Waymire, E. and Gupta, V.K. (1981c), The mathematical structure of rainfall representations, 3: Some applications of the point process theory to rainfall processes. Water Resources Research 17, 1287–1294.
Waymire, E., Gupta, V.K. and Rodriguez-Iturbe, I. (1984), Spectral theory of rainfall intensity at the meso-β scale. Water Resources Research 20, 1453–1465.
Woolhiser, D.A. (1992), Modeling daily precipitation — progress and problems. Chapter 5 of Statistics in the Environmental and Earth Sciences, eds. A. Walden and P. Guttorp, Halsted Press, New York.
Additional References
Bilonick, R. A. (1988). “Monthly hydrogen ion deposition maps for the Northeastern U.S. from July 1982 to September 1984,” Atmospheric Environment, 22, 1909–1924.
Breslow, N. E. and Clayton, D. G. (1993). “Approximate inference in generalized linear mixed models,” Journal of the American Statistical Association, 88, 9–25.
Breslow, N. E. and Lin, X. (1995). “Bias correction in generalized linear mixed models with a single component of dispersion,” Biometrika, 82, 81–91.
Diggle, P. J., Liang, K-Y, and Zeger, S. L. (1994). Analysis of Longitudinal Data, Oxford University Press: New York.
Drum, M. and McCullagh, P. (1993). “REML Estimation with exact co- variance in the logistic mixed model,” Biometrics, 49, 677–689.
Journel, A. G. (1983). “Non-parametric estimation of spatial distributions,” Mathematical Geology, 15, 445–468.
Karim, M. R. and Zeger, S. L. (1992). “Generalized linear models with random effects: salamander mating revisited,” Biometrics, 48, 63 1644.
McCulloch, C. E. (1994). “Maximum likelihood variance components estimation for binary data,” Journal of the American Statistical Association, 89, 330–335.
Schall, R. (1991). “Estimation in generalized linear models with random effects,” Biometrika, 40, 917–927.
Zeger, S. L. and Karim, M. R. (1991). “Generalized linear models with random effects: a Gibbs sampling approach,” Journal of the American Statistical Association, 86, 79–86.
References
Diggle, P.J., Moyeed, R. and Tawn, J.A. (1996), Model-based geostatistics. Technical Report, Lancaster University.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag New York, Inc.
About this paper
Cite this paper
Smith, R.L., Robinson, P.J. (1997). A Bayesian Approach to the Modeling of Spatial-Temporal Precipitation Data. In: Gatsonis, C., Hodges, J.S., Kass, R.E., McCulloch, R., Rossi, P., Singpurwalla, N.D. (eds) Case Studies in Bayesian Statistics. Lecture Notes in Statistics, vol 121. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2290-3_5
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2290-3_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-94990-1
Online ISBN: 978-1-4612-2290-3
eBook Packages: Springer Book Archive