Hydrologic uncertainty processor for probabilistic stage transition forecasting

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Abstract

The hydrologic uncertainty processor (HUP) is a component of the Bayesian forecasting system which produces a short-term probabilistic stage transition forecast (PSTF) based on a probabilistic quantitative precipitation forecast (PQPF). The PSTF specifies a sequence of families of predictive one-step transition density functions whose product gives the predictive joint density function of the actual river stage process {H1,…,HN}. A multivariate HUP is needed to quantify hydrologic uncertainty—the aggregate of all uncertainties arising from sources other than those quantified by the PQPF.

A Bayesian formulation of the multivariate HUP for a PSTF system is presented. The multivariate HUP supplies a family of posterior joint density functions of the actual river stage process {H1,…,HN}, conditional on a realization of the model river stage process {S1,…,SN} output from a deterministic hydrologic model, and an observation of the initial river stage H0. A posterior joint density function is factorized into posterior one-step transition density functions, each of which being obtained via Bayes theorem from a likelihood function and a prior one-step transition density function. To implement the HUP, a meta-Gaussian model is employed. The model is tested on data for daily river stage processes at four forecast points closing headwater basins of sizes 484, 1430, 1859, and 2372 km2, and located in the Eastern United States. The working of the HUP is illustrated, and general characteristics of the hydrologic uncertainty are inferred. The hydrologic uncertainty is significant and imposes a limit on the predictability of river stage transitions, which is currently about 48 h, given a 24-h PQPF.

Introduction

The Bayesian forecasting system (BFS) is a general theoretical framework for probabilistic forecasting via any deterministic hydrologic model (Krzysztofowicz, 1999). Within this framework, the first prototype system was developed (Krzysztofowicz, 2002) to produce a short-term probabilistic river stage forecast (PRSF) based on a probabilistic quantitative precipitation forecast (PQPF). For each time tn (n=1,…,N), the PRSF specifies a predictive density function of the river stage Hn to be observed at time tn. The second prototype system is now being developed to produce a probabilistic stage transition forecast (PSTF). The predictand is a time series of river stages H1,…,HN to be observed at times t1,…,tN. The PSTF specifies a sequence of families of predictive one-step transition density functions whose product gives the predictive joint density function of (H1,…,HN). Thereby the PSTF characterizes the stochastic dependence among the river stages H1,…,HN. This information is needed by decision systems such as flood response systems and reservoir control systems.

The BFS has three structural components, one of which is the hydrologic uncertainty processor (HUP). The purpose of the HUP is to quantify hydrologic uncertainty—the aggregate of all uncertainties arising from sources other than those quantified by the PQPF. An earlier article (Maranzano and Krzysztofowicz, 2004) identified the empirical dependence structures of the likelihood functions and the prior density functions from which the multivariate HUP is built. This article presents the HUP itself.

Section 2 gives a methodological background. Section 3 presents a Bayesian formulation of the HUP. Section 4 develops a meta-Gaussian model. Section 5 illustrates the model output for a single forecast point. Section 6 compares the likelihood parameter values for the HUPs at four forecast points and infers general characteristics of the hydrologic uncertainty.

Section snippets

Forecast points and hydrologic models

The results reported throughout the article are for four forecast points, whose characteristics are listed in Table 1. Samples of the actual river stages were extracted from archives of the US National Weather Service (NWS). Samples of the model river stages were produced via simulation performed at the NWS Ohio River Forecast Center using archived real-time input data and the operational forecast system (OFS) whose description can be found in Hudlow, 1988, Fread et al., 1995. The OFS is a

Concept

Let W denote the basin average precipitation amount to be accumulated during the 24-h period beginning at 1200 UTC on day n=0. Let V denote an indicator of precipitation occurrence, with V=0⇔W=0 and V=1⇔W>0. Each distribution in the HUP is conditioned on event V=v, where v∈{0,1}. Then, the event probability P(V=v), which is specified by the PQPF, enters the integrator of the BFS in order to mix two predictive one-step transition distributions, one conditional on zero precipitation (W=0) and

Modeling strategy

To implement the multivariate HUP, a meta-Gaussian model is formulated. This formulation is a generalization of the meta-Gaussian models developed earlier in connection with the univariate HUPs (Krzysztofowicz and Kelly, 2000, Krzysztofowicz and Herr, 2001). For this reason, the derivations are omitted and only the modeling strategy and the final expressions are presented.

The modeling strategy is to transform each variate, Hn or Sn, into a normally distributed variate, Wn or Xn, and assume

Input parameters

When N=3, the multivariate, precipitation-dependent, meta-Gaussian HUP for a season is specified by 78 parameters: (i) parameters {(αnv,βnv,γnv): v=0,1; n=0,1,2,3} of the marginal prior distributions Γnv of the actual river stages, (ii) parameters {(ᾱnv,β̄nv,γ̄nv): v=0,1; n=1,2,3} of the marginal initial distributions Λ̄nv of the model river stages, (iii) parameters {cnv: v=0,1; n=1,2,3} of the family of the prior density functions, and (iv) parameters {(anv,bnv,dnv,env,σnv): v=0,1; n=1,2,3}

Measure of informativeness

The simulation experiment performed at the NWS Ohio River Forecast Center enabled us to estimate 48 families of the likelihood functions (4 forecast points, 3 lead times, 2 precipitation events, 2 seasons). While not large, this set of cases offers the first opportunity to compare the likelihood parameters of the HUPs and to search for associations that might reveal some general characteristics of the hydrologic uncertainty.

Table 5, Table 6, Table 7, Table 8 report the likelihood parameter

Purpose of the multivariate HUP

A Bayesian formulation of the multivariate HUP for a PSTF system has been presented. The purpose of the HUP is to quantify the aggregate of all uncertainties about the actual river stage process, conditional on an estimate of that process output from a deterministic hydrologic model fed with a perfect forecast of the basin average precipitation amount (here for a 24-h period). The multivariate HUP generalizes the univariate HUP developed earlier in that it quantifies the stochastic dependence

Acknowledgements

Thomas E. Adams of the Ohio River Forecast Center, National Weather Service, performed all simulations. His skill and cooperation are greatly appreciated.

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