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Roman Krzysztofowicz

Abstract

A probabilistic quantitative precipitation forecast (PQPF) is prepared judgmentally by a meteorologist based on a guidance PQPF. The predictand of a judgmental PQPF is the spatially averaged precipitation amount. The predictand of a guidance PQPF produced by a statistical model is the point precipitation amount. Therefore, a procedure is needed for point-to-area rescaling of the PQPF. Theoretically based equations for rescaling are presented. The equations incorporate two predictive parameters, which characterize the precipitation field being forecast: the quotient of the area covered by a precipitation cell to the area of averaging (cell/area quotient), and the degree of certainty about the precipitation pattern (pattern certainty factor). Both parameters can be judgmentally quantified by the meteorologist during PQPF preparation. The same parameters can be entered into an inverse procedure for area-to-point rescaling of the judgmental PQPF.

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Roman Krzysztofowicz

Abstract

Consider an event definable in terms of two subevents as, for example, the occurrence of precipitation within a 24-h period is definable in terms of the occurrence of precipitation within each of the 12-h subperiods. A complete forecast must specify three probabilities; these may be marginal probabilities, one for the period and two for subperiods. Theoretical relations between these probabilities are investigated and solutions are presented to three problems encountered in operational forecasting: (i) guaranteeing that the marginal probabilities jointly obey the laws of probability, (ii) structuring admissible procedures for adjusting the initial (guidance) probabilities by forecasters, and (iii) formulating optimal estimators of the probability for period in terms of the probabilities for subperiods.

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Roman Krzysztofowicz

Abstract

From the theory of sufficient comparisons of experiments, a measure of skill is derived for categorical forecasts of continuous predictands. Called Bayesian correlation wore (BCS), the measure is specified in terms of three parameters of a normal-linear statistical model that combines information from two sources: a prior (climatological) record of the predictand and a verification record of forecasts. Three properties characterize the BCS: (i) It is meaningful for comparing alternative forecasts of the same predictand, as well as forecasts of different predictands, though in a limited sense; (ii) it is interpretable as correlation between the forecast and the predictand; and, most significantly, (iii) it orders alternative forecast systems consistently with their ex ante economic values to rational users (those who make decisions by maximizing the expected utility of outcomes under the posterior distribution of the predictand). Thus, by maximizing the BCS, forecasters can assure a utilitarian society of the maximum potential economic benefits of their forecast.

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Roman Krzysztofowicz
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Roman Krzysztofowicz

Under the auspices of the Cooperative Program for Operational Meteorology, Education and Training, pilot research has been conducted from 1991 to 1996 on operational methods for producing Probabilistic Quantitative Precipitation Forecasts (PQPFs) and Probabilistic River Stage Forecasts (PRSFs). The first integrated forecasts were produced using operational data and models of the National Weather Service on 19 October 1996. The PQPF was for a 24-h period and the PRSFs were for lead times of 1, 2, and 3 days. This article discusses these pilot forecasts, their interpretation, and their advantages to forecasters and end users with regard to communicating uncertainties and making decisions.

All those whose duty it is to issue regular daily forecasts know that there are times when they feel very confident and other times when they are doubtful as to the coming weather. It seems to me that the condition of confidence or otherwise forms a very important part of the prediction, and ought to find expression.

—W. E. Cooke (1906)

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Roman Krzysztofowicz and Thomas A. Pomroy

Abstract

Disaggregative invariance refers to stochastic independence between the total precipitation amount and its temporal disaggregation. This property is investigated herein for areal average and point precipitation amounts accumulated over a 24-h period and disaggregated into four 6-h subperiods. Statistical analyses of precipitation records from 1948 to 1993 offer convincing empirical evidence against the disaggregative invariance and in favor of the conditional disaggregative invariance, which arises when the total amount and its temporal disaggregation are conditioned on the timing of precipitation within the diurnal cycle.

The property of conditional disaggregative invariance allows the modeler or the forecaster to decompose the problem of quantitative precipitation forecasting into three tasks: (i) forecasting the precipitation timing; (ii) forecasting the total amount, conditional on timing; and (iii) forecasting the temporal disaggregation, conditional on timing. Tasks (ii) and (iii) can be performed independently of one another, and this offers a formidable advantage for applications.

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Ashley A. Sigrest and Roman Krzysztofowicz

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The predictand of a probabilistic quantitative precipitation forecast (PQPF) may be either a point precipitation amount or a spatially averaged precipitation (SAP) amount. At the current state of the art, it is the SAP amount (the volume of water accumulated over an area during a period) that is most predictable. This case study compares the climatic PQPFs of the two predictands within a river basin in the Appalachians, then highlights similarities and distinctions of which the forecasters should be aware. Empirical relations reveal whether or not a given statistic of the point precipitation amount is (i) locally invariant, that is, does not vary appreciably within some area so that a single estimate (e.g., a spatial average) can approximate the statistic at every point within the area, and (ii) amenable to averaging, that is, can be averaged over some area to obtain an approximation to the statistic of the SAP amount. The study also illustrates the effect of elevation on the statistics of point precipitation and highlights seasonal differences. The conclusions point to a need for local climatic guidance to help forecasters in calibrating PQPFs.

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Roman Krzysztofowicz and Ashley A. Sigrest

Abstract

From 1 August 1990 to 31 July 1995, the Weather Service Forecast Office in Pittsburgh prepared 6159 probabilistic quantitative precipitation forecasts. Forecasts were made twice a day for 24-h periods beginning at 0000 and 1200 UTC for two river basins. This is the first in a series of articles devoted to a comprehensive verification of these forecasts. The property verified herein is calibration: a match between forecast probabilities and empirical frequencies of events.

Monthly time series of calibration statistics are analyzed to infer (i) trends in calibration over time, (ii) the forecasters’ skill in quantifying uncertainty, (iii) the adaptability of forecasters’ judgments to nonstationarities of the predictand, (iv) the possibility of reducing biases through dynamic recalibration, and (v) the potential for improving calibration through individualized training.

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Roman Krzysztofowicz and W. Britt Evans

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A sequence of meteorological predictands of one kind (e.g., temperature) forms a discrete-time, continuous-state stochastic process, which typically is nonstationary and periodic (because of seasonality). Three contributions to the field of probabilistic forecasting of such processes are reported. First, a meta-Gaussian Markov model of the stochastic process is formulated, which provides a climatic probabilistic forecast with the lead time of l days in the form of a (prior) l-step transition distribution function. A measure of the temporal dependence of the process is the autocorrelation coefficient (which is nonstationary). Second, a Bayesian processor of forecast (BPF) is formulated, which fuses the climatic probabilistic forecast with an operational deterministic forecast produced by any system (e.g., a numerical weather prediction model, a human forecaster, a statistical postprocessor). A measure of the predictive performance of the system is the informativeness score (which may be nonstationary). The BPF outputs a probabilistic forecast in the form of a (posterior) l-step transition distribution function, which quantifies the uncertainty about the predictand that remains, given the antecedent observation and the deterministic forecast. The working of the Markov BPF is explained on probabilistic forecasts obtained from the official deterministic forecasts of the daily maximum temperature issued by the U.S. National Weather Service with the lead times of 1, 4, and 7 days. Third, a numerical experiment demonstrates how the degree of posterior uncertainty varies with the informativeness of the deterministic forecast and the autocorrelation of the predictand series. It is concluded that, depending upon the level of informativeness, the Markov BPF is a contender for operational implementation when a rank autocorrelation coefficient is between 0.3 and 0.6, and is the preferred processor when a rank autocorrelation coefficient exceeds 0.6. Thus, the climatic autocorrelation can play a significant role in quantifying, and ultimately in reducing, the meteorological forecast uncertainty.

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Roman Krzysztofowicz and W. Britt Evans

Abstract

The Bayesian processor of forecast (BPF) is developed for a continuous predictand. Its purpose is to process a deterministic forecast (a point estimate of the predictand) into a probabilistic forecast (a distribution function, a density function, and a quantile function). The quantification of uncertainty is accomplished via Bayes theorem by extracting and fusing two kinds of information from two different sources: (i) a long sample of the predictand from the National Climatic Data Center, and (ii) a short sample of the official National Weather Service forecast from the National Digital Forecast Database. The official forecast is deterministic and hence deficient: it contains no information about uncertainty. The BPF remedies this deficiency by outputting the complete and well-calibrated characterization of uncertainty needed by decision makers and information providers. The BPF comes furnished with (i) the meta-Gaussian model, which fits meteorological data well as it allows all forms of marginal distribution functions, and nonlinear and heteroscedastic dependence structures, and (ii) the statistical procedures for estimation of parameters from asymmetric samples and for coping with nonstationarities in the predictand and the forecast due to the annual cycle and the lead time. A comprehensive illustration of the BPF is reported for forecasts of the daily maximum temperature issued with lead times of 1, 4, and 7 days for three stations in two seasons (cool and warm).

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