Particle filters are ensemble-based assimilation schemes that, unlike the ensemble Kalman filter, employ a fully nonlinear and non-Gaussian analysis step to compute the probability distribution function (pdf) of a system’s state conditioned on a set of observations. Evidence is provided that the ensemble size required for a successful particle filter scales exponentially with the problem size. For the simple example in which each component of the state vector is independent, Gaussian and of unit variance, and the observations are of each state component separately with independent, Gaussian errors, simulations indicate that the required ensemble size scales exponentially with the state dimension. In this example, the particle filter requires at least 1011 members when applied to a 200-dimensional state. Asymptotic results, following the work of Bengtsson, Bickel and collaborators, are provided for two cases: one in which each prior state component is independent and identically distributed, and one in which both the prior pdf and the observation errors are Gaussian. The asymptotic theory reveals that, in both cases, the required ensemble size scales exponentially with the variance of the observation log-likelihood, rather than with the state dimension per se. 2

There is increasing consensus in the hydrologic literature that an appropriate framework for streamflow forecasting and simulation should include explicit recognition of forcing, parameter and model structural er-ror. This paper presents a novel Markov Chain Monte Carlo (MCMC) sam-pler, entitled DiffeRential Evolution Adaptive Metropolis (DREAM), that is especially designed to efficiently estimate the posterior probability den-sity function of hydrologic model parameters in complex, high-dimensional sampling problems. This MCMC scheme adaptively updates the scale and orientation of the proposal distribution during sampling, and maintains de-tailed balance and ergodicity. It is then demonstrated how DREAM can be used to analyze forcing data error during watershed model calibration us-ing a 5-parameter rainfall- runoff model with streamflow data from two dif-ferent catchments. Explicit treatment of precipitation error during hydro-logic model calibration not only results in prediction uncertainty bounds that are more appropriate, but also significantly alters the posterior distribution

In data assimilation for a system which evolves in time, one combines past and current observations with a model of the dynamics of the system, in order to improve the simulation of the system as well as any future predictions about it. From a statistical point of view, this process can be regarded as estimating many random variables, which are related both spatially and temporally: given observations of some of these variables, typically corresponding to times past, we require estimates of several others, typically corresponding to future times. Graphical models have emerged as an effective formalism for assisting in these types of inference tasks, particularly for large numbers of random variables. Graphical models provide a means of representing dependency structure among the variables, and can provide both intuition and efficiency in estimation and other inference computations. We provide an overview and introduction to graphical models, and describe how they can be used to represent statistical dependency and how the resulting structure can be used to organize computation. The relation between statistical

Operational flood forecasting requires that accurate estimates of the uncertainty associated with model-generated streamflow forecasts be provided along with the probable flow levels. This paper demonstrates a stochastic ensemble implementation of the Sacramento model used routinely by the National Weather Service for deterministic streamflow forecasting. The approach, the simultaneous optimization and data assimilation method (SODA), uses an ensemble Kalman filter (EnKF) for recursive state estimation al-lowing for treatment of streamflow data error, model structural error, and parameter uncertainty, while enabling implementation of the Sacramento model without major modification to its current structural form. Model parameters are estimated in batch using the shuffled complex evolution metropolis stochastic-ensemble optimization approach (SCEM-UA). The SODA approach was implemented using parallel com-puting to handle the increased computational requirements. Studies using data from the Leaf River, Mis-sissippi, indicate that forecast performance improvements on the order of 30 % to 50 % can be realized even with a suboptimal implementation of the filter. Further, the SODA parameter estimates appear to be less biased, which may increase the prospects for finding useful regionalization relationships. 1. Introduction and

www.hydrol-earth-syst-sci.net/18/1289/2014/

model averaging and data assimilation

Hydrologic data assimilation with a hillslope-scale-resolving model and L band radar observations: Synthetic experiments with the ensemble Kalman filter

hydro-climatic projection and urban development effects

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into a hydrologic model