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Edward S. Epstein

Abstract

The problem of decision making in applied meteorology is approached from the point of view of decision theory and subjectivist statistics. The modern concept of “utility” is discussed, and optional rules for decision making based on the availability of a limited amount of meteorological data are presented and discussed. Bayes' theorem forms the basis for the statistical estimation of the frequencies of various alternative weather events. The method is applied to a single example for the purpose of illustration, but it is emphasized that the generality of these techniques is great and that they warrant further study.

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Edward S. Epstein

Abstract

The likelihood ratio of the data for a hypothesis of some change, relative to the hypothesis of no change, is a suitable statistical measure for the detection of climate change. Likelihood ratios calculated on the basis of Angell and Korshover's (1977) global mean temperature, updated through 1980, do not show convincing evidence of recent climate change. It is possible to calculate probabilities of obtaining future values of likelihood ratios, depending on the postulated future climate change. A modest but significant climate change, such as that expected to occur from an increase of atmospheric carbon dioxide, is likely to be detected from global mean surface temperatures within ten years. The joint behavior of the troposphere and stratosphere is more likely to discriminate between climate change and no change than are surface temperatures. In this case, a climate change that can be attributed to carbon dioxide increase should be detectable by 1986.

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Edward S. Epstein

Abstract

Appropriately defined goodness-or-fit statistics are shown to provide a reasonable and objective means to determine the optimum number of harmonies to represent an annual climatology. The method is described in terms of its application, with varying degrees of success, to 5-day temperature means, their standard deviations, and to 5-day means of daily maximum and minimum temperatures.

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Edward S. Epstein

Abstract

When the initial values, or the parameters, of prognostic equations are not known with certainty, there must also be errors in the solution. The initial conditions may be represented by an ensemble, each member of which is consistent with all available knowledge. The mean of this ensemble is a reasonable "best" solution to the prognostic equation. Following Gleeson, we have examined the behavior of the error in the forecast, as represented by the rms deviation of the ensemble members from their mean, for a few simple equations. We have further examined the time-dependent behavior of the ensemble mean, as opposed to the solution obtained by applying the prognostic equation to the original mean values. These are, in general, different. It is concluded that optimum procedures for forecasting, i.e., solving prognostic equations, require includingterms in the equations to represent the influence of the initial uncertainties. Since the nature of these uncertainties may also have profound influences on the error of the forecast, this aspect, too, must be taken into consideration.

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EDWARD S. EPSTEIN

Abstract

The meaning of probabilistic weather forecasts is discussed from the point of view of a subjectivist concept, of probability. The prior degree of belief of probabilities of the weather in question, for a given forecast statement, is expressed analytically as a beta function. Bayes' theorem is used to modify this degree of belief in the light of experience, producing a posterior degree of belief which is also in the form of a beta function. By establishing an arbitary criterion that one should always be able to assign at least as much belief to the probability interval implied by the forecast as to any other equivalent interval, a method of quality control for probability forecasts is developed. Appropriate tables are given to permit application of the method, and the implications of the method, for both forecaster and forecast user, are discussed.

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EDWARD S. EPSTEIN

Abstract

The relationship between point and area precipitation probabilities is examined on the basis of a simple model in which circular precipitation cells of uniform size are distributed at random over an area that is large compared to the forecast area. From knowledge of the cell size and the number of cells per unit area it is then possible to state both the point and area precipitation probabilities. Formulas and graphs of these relationships are shown. When the cells are large, point and area precipitation probabilities are almost equal, but they differ markedly when the cells are small. Joint and conditional probabilities of precipitation at two or more stations are also examined. An extension of the model is presented in which uncertainty regarding the density of cells is expressed as an elementary probability density, and the effects of this on the expected point and area precipitation probabilities are shown.

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EDWARD S. EPSTEIN

Abstract

Vertical velocities have been computed for the lower stratosphere for two independent winter periods, by employing a form of the adiabatic method. The regions studied were in both cases outside the polar vortex. The flow pattern was divided into stationary (long-wave) and transient (short-wave) components. The vertical velocity pattern associated with the stationary long wave is precisely that described by Kochanski [3]; i.e., the air rises in moving from warm troughs to cold ridges. The pattern associated with the short waves is more complicated. There is a maximum of warm, advection in the vicinity of short-wave ridges, and cold advection near troughs. Local temperature changes, however, very nearly compensate the advection, with the net result that in the mean the vertical velocities associated with short-wave patterns are small, but tend to be positive near ridges and negative near troughs. Superimposition of the short and long wave, however, can lead to any conceivable combination of signs of advection, local temperature change, vertical velocity, and position with respect to ridge or trough.

The single parameter which is most useful in specifying the vertical velocity is the temperature advection.

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Edward S. Epstein

Abstract

The details of the weather are not predictable beyond one to two weeks. At longer time ranges, averages of the weather over space and time can be usefully predicted only to the extent that the variations of the averages exceed the “noise” produced by the omnipresent but unpredictable transient weather. This margin of potential predictability is not large, but parts of it are being exploited in routinely issued monthly and seasonal forecasts. The format and utilization of these forecasts, the methods by which they are routinely produced, and prospects for improvements are discussed.

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Allan H. Murphy
and
Edward S. Epstein

Abstract

Attributes of the anomaly correlation coefficient, as a model verification measure, are investigated by exploiting a recently developed method of decomposing skill scores into other measures of performance. A mean square error skill score based on historical climatology is decomposed into terms involving the anomaly correlation coefficient, the conditional bias in the forecast, the unconditional bias in the forecast, and the difference between the mean historical and sample climatologies. This decomposition reveals that the square of the anomaly correlation coefficient should be interpreted as a measure of potential rather than actual skill.

The decomposition is applied to a small sample of geopotential height field forecasts, for lead times from one to ten days, produced by the medium range forecast (MRF) model. After about four days, the actual skill of the MRF forecasts (as measured by the “climatological skill score”) is considerably less than their potential skill (as measured by the anomaly correlation coefficient), due principally to the appearance of substantial conditional biases in the forecasts. These biases, and the corresponding loss of skill, represent the penalty associated with retaining “meteorological” features in the geopotential height field when such features are not predictable. Some implications of these results for the practice of model verification are discussed.

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Edward S. Epstein
and
Allan H. Murphy

Abstract

On most forecasting occasions forecasts are made for several successive periods, but decision-making models have traditionally neglected the impact of the potentially useful information contained in forecasts for periods beyond the initial period. The use and value of multiple-period forecasts are investigated here in the context of a recently developed dynamic model of the basic cost-loss ratio situation. We also extend previous studies of this model by examining the impacts—on forecast use and value—of assuming (i) that weather events in successive periods are dependent and (ii) that the forecasts of interest are expressed in probabilistic terms. In this regard, expressions are derived for the expected expenses associated with the use of climatological, imperfect (categorical or probabilistic) and perfect multiple-period forecasts under conditions of dependence and independence between events.

Numerical results are presented concerning expected expense and economic value, based on artificially generated forecasts that incorporate the effects of the decrease in forecast quality as lead time increases. Comparisons are made between multiple-period and single-period forecasts, between dependent and independent events, and between probabilistic and categorical forecasts. For some values of the relevant parameters (e.g., cost-loss ratio, climatological probability), the availability of information for longer lead times can substantially increase economic value. It appears, however, that (i) current imperfect forecasts achieve relatively little of this potential value and (ii) improvements in forecasts at longer lead times must be accompanied by improvements at the shortest lead times for these benefits to be realized. Dependence (i.e., persistence) between events generally reduces weather-related expected expenses, sometimes quite substantially, and consequently reduces forecast value. The results also demonstrate once again the potential economic benefits of expressing forecasts in a probabilistic rather than a categorical formal.

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