Hidden Markov Models (HMM) can be used for downscaling daily rainfall occurrences and amounts from GCM simulations. The HMM fits a model to observed rainfall records by introducing a small number of discrete rainfall states. These states allow a diagnostic interpretation of observed rainfall variability in terms of a few rainfall patterns. The states are ‘hidden’ from the observer, i.e., they are not directly observable. The time sequence of which state is active on each day follows a Markov chain. Thus, the state which is active ‘today’ depends only on the state which was active ‘yesterday’ according to transition probabilities. The HMM can be thought of as a cluster analysis of multi-site rainfall observations, with an explicit (Markov chain) time dependence. In its ability to make simulations, it also bears similarities to existing stochastic weather generators.
The HMM also allows you to simulate rainfall at each of the station locations, such that key statistical properties (eg. rainfall probabilities, dry/wet spell lengths) of the simulated rainfall match those of the observed rainfall records. This can be useful for generating large numbers of synthetic realizations of rainfall for input into statistical analysis, or input into a crop simulation model, for example.
The HMM also provides a basis for downscaling GCM simulations to the station scale, or calibrating estimates of observed rainfall. Downscaling is accomplished using a nonhomogeneous HMM (NHMM) in which predictors are incorporated into the HMM. The predictors modulate the transitions between the states over time. The transition matrix is no longer homogeneous in time, hence the name. Missing observations are allowed, and are treated in a consistent probabilistic manner. Additional options are available in MVNHMM (and HMMTool) for modeling spatial dependencies between stations. Seasonal forecast downscaling applications of this model are described in Robertson et al. (2004, 2006, 2009). A climate change downscaling application for crop modeling is described by Chen et al. (2013).
Two innovations are introduced in the new Bayesian NHMM: firstly that parameter uncertainties are explicitly modeled, and secondly that the exogenous covariates can influence the rainfall distributions at the station level (through a GLM component), as well as through the transition probabilities between the states. This second attribute adds flexibility that may be particularly useful for downscaling of GCM climate change projections.
Chen, C., A. M. Greene, A. W. Robertson, and W. Baethgen, 2013: Scenario Development for Estimating Potential Climate Change Impacts on Crop Production in the North China Plain. Int. J. Climatology, DOI: 10.1002/joc.3648.
Greene, A. M., Robertson, A. W. and Kirshner, S. (2008), Analysis of Indian monsoon daily rainfall on subseasonal to multidecadal time-scales using a hidden Markov model. Q.J.R. Meteorol. Soc., 134: 875–887. doi: 10.1002/qj.254
Greene, A. M., Robertson, A. W., Smyth, P. and Triglia, S. (2011), Downscaling projections of Indian monsoon rainfall using a non-homogeneous hidden Markov model. Q.J.R. Meteorol. Soc., 137: 347–359. doi: 10.1002/qj.788
Robertson, A. W., S. Kirshner, and P. Smyth, 2004: Downscaling of daily rainfall occurrence over Northeast Brazil using a Hidden Markov Model. J. Climate, 17, 4407-4424.
Robertson, A. W., S. Kirshner, P. Smyth, S. P. Charles, and B. C. Bates, 2006: Subseasonal-to-Interdecadal Variability of the Australian Monsoon Over North Queensland. Quart. J. Royal Meteor. Soc., 132, 519-542.
Robertson, A. W. , V. Moron, and Y. Swarinoto, 2009: Seasonal predictability of daily rainfall statistics over Indramayu district, Indonesia. Int. J. Climatology, 29, 1449-1462.