## 1. Introduction

[2] Forecasts of future seasonal streamflows are potentially valuable to a range of water managers and users, including irrigators, urban and rural water supply authorities, environmental managers, and hydroelectricity generators. Such forecasts can inform planning and management decisions to maximize returns on investments and available water resources and to ensure security of supply [e.g., *Chiew et al.*, 2003; *Plummer et al.*, 2009]. For many river and storage systems, a joint forecast of streamflows at multiple sites that accounts for intersite correlations is needed for managing water resources at a system scale. *Regonda et al.* [2006], *Westra et al.* [2007], and *Bracken et al.* [2010] provide some of the methods for joint forecasting of streamflows at multiple sites.

[3] In a previous paper, we developed a Bayesian joint probability (BJP) modeling approach for seasonal forecasting of streamflows at multiple sites [*Wang et al.*, 2009]. Specifically, the BJP approach provides probabilistic forecasts of streamflow volume totals over a forecast period, say the next 3 months, at multiple sites in the form of ensembles. The approach uses a Yeo-Johnson transformed multivariate normal distribution to model the joint distribution of future streamflows and their predictors such as antecedent streamflows, El Niño–Southern Oscillation indices and other climate indicators. The model parameters and their uncertainties are inferred from historical data using a Bayesian method. The parameters are then used for producing joint probabilistic forecasts of streamflows at multiple sites for future events. The BJP modeling approach is completed with a method for selecting predictors from a large number of candidate predictors based on pseudo Bayes factors calculated from cross-validation predictive densities (D.E. Robertson and Q. J. Wang, A Bayesian approach to predictor selection for seasonal streamflow forecasting, submitted to *Journal of Hydrometeorology*, 2010), and with a suite of methods and tools for verification of probabilistic forecasts [*Wang et al.*, 2009].

[4] The BJP modeling approach has been adopted by the Australian Bureau of Meteorology for seasonal streamflow forecasting in Australia [*Plummer et al.*, 2009]. One of the limitations of the BJP modeling approach as presented by *Wang et al.* [2009] is that it does not deal with zero flows, which occur on many streams in Australia (and in arid and semiarid regions elsewhere in the world). Zero flows may occur in dry seasons occasionally on perennial streams and frequently on intermittent streams. Zero flows can be the dominant state of many ephemeral streams. This study extends the BJP modeling approach so that it is applicable to seasonal flow forecasting for streams with zero value occurrences. The aim is to produce forecasts that give probabilities of zero flows and probability distributions of above zero flows.

[5] Antecedent streamflows are often useful for representing the initial catchment conditions and thus for serving as predictors of future streamflows. Zero values may occur with either antecedent streamflows or future streamflows. In a multisite problem, zero flows may occur in any of the streams being considered. Thus, in the BJP modeling setting, zero flows may occur in any combination of predictors and predictands and of sites. Mathematically, this leads to a mixed discrete-continuous multivariate probability distribution, which is extremely difficult to formulate and manipulate.

[6] Methods for stochastic generation and for downscaling of daily rainfall at multiple sites provide useful reference for modeling mixed discrete-continuous multivariate probability distributions. The most commonly used approach is to model rainfall occurrence and amounts separately [e.g., *Wilks*, 1998; *Charles et al.*, 1999; *Srikanthan and Pegram*, 2009]. *Bardossy and Plate* [1992] introduced an approach that directly transforms the discrete-continuous multivariate probability distributions to a continuous multivariate normal distribution. The transformation includes an allocation of the cumulative probabilities in the negative space of the transformed variables to the zero value points of the original rainfall variables. The approach does not require modeling rainfall occurrence and amounts separately and thus significantly simplifies the problem. A similar approach was used by *Frost et al.* [2007] in a context of stochastic generation of annual multisite hydrological data using a multivariate autoregressive lag-1 model. The approach uses a continuous multivariate distribution as the underlying distribution and then lumps the probability mass in the subspace below the zero thresholds of the variables to the zero value point of the variables. The lumping is done through a numerical integration.

[7] In this study we adopt essentially the same approach as *Bardossy and Plate* [1992] and *Frost et al.* [2007] as it is well suited to the extension of our original BJP modeling approach. However, we cast the problem to one of censored data. We treat the zero flows as censored data having unknown precise values but known to be below or equal to zero. In this way, the variables both before and after the Yeo-Johnson transform (see section 2) are considered to follow continuous multivariate distributions. The treatment allows us to greatly simplify the mathematical expressions required, and importantly, to deal with zero predictor values as well as zero predictand values. In the work of *Frost et al.* [2007], the zero threshold problem was considered only for variables at the current time step (equivalent to predictands), not for the variables at the lag-1 time step (equivalent to predictors) on which the variables at the current time step are conditioned.

[8] This paper presents the extended BJP modeling approach applicable to streams with zero value occurrences and demonstrates its use through a test application. In addition, improvements made to a number of detailed techniques in the work of *Wang et al.* [2009], including reparameterization, prior specification, and Markov chain Monte Carlo (MCMC) sampling, are also presented. A new skill score based on the root mean square error in probability (RMSEP) is introduced as an alternative to the linear error in probability space (LEPS). The paper is organized as follows. Model formulation is given in the next section. Model parameter inference in section 3 includes reparameterization, derivation of posterior distribution of parameters, specification of prior distribution of parameters, and MCMC sampling of the posterior distribution of parameters. Section 4 details the method for using the model to produce probabilistic forecasting, including the augmentation of censored predictor data. Section 5 includes a number of statistical measures and graphical methods for both overall and detailed verifications. Section 6 deals with model checking. A test application to forecasting streamflows at three river gauges in the Burdekin River catchment in northern Queensland, Australia, is given in section 7 to demonstrate the working of the extended BJP modeling approach. Section 8 completes the paper with a summary and conclusions. The paper also has three appendices to provide additional technical details.