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Richard H. Jones

Abstract

This note corrects two misprints and a degrees of freedom error in a recent paper on estimating the variance of time averages, and suggests an alternative method for estimating the spectral density near zero frequency on which the variance depends.

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Richard H. Jones

Abstract

The role of statistics in numerical weather prediction should be to aid in the handling of random errors and disturbances. The actual prediction is a problem in dynamics. When the initial conditions are observed with error, there is information in the past forecasts which could increase the accuracy of the numerical predictions. The techniques of control theory provide an optimal method for combining past forecasts with current observations. This paper demonstrates the method on simulated non-linear time series.

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Richard H. Jones

Abstract

A maximum likelihood estimation procedure is developed for estimating the spatial wavenumber spectrum and falloff slope for a stationary process on a circle when multiple realizations are available at the same unequally spaced locations. The complete model allows for observational error, an arbitrary number of free spectral lines, and the assumed form αk −β for higher wavenumbers, where k is the wavenumber and α and β are parameters to be estimated. The estimation technique involves iterative, nonlinear optimization; however, since maximum likelihood estimation is used, confidence limits can be obtained for the estimated parameters. A numerical example is given using 500 mb temperature at 15 stations located near 53N, and the estimated falloff rate (β) and 95% confidence limits are 2.03±0.20 when observational error is not included in the model. However, when observational error is included, the slope increases significantly to near 3. This increase is expected since observational error is similar to variance at high wavenumbers. The example demonstrates that care must he taken when interpreting the results of various estimation procedures, since the results are highly dependent on the assumed model. An objective method is used for selecting the appropriate model, and indicates that observational error should be included.

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Richard H. Jones

Abstract

A numerical-statistical forecasting method is presented in which statistical techniques art used on the errors of numerical predictions. This, in effect, uses earlier numerical predictions to obtain best estimates of the initial conditions. A technique is given whereby the method can be applied by estimating the necessary parameters, and it is shown that this procedure converges to the optimal estimate.

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Richard H. Jones

Abstract

Data analysis with unequally spaced observations is approached by assuming that a finite number of observations are available at known locations in space or time. No assumptions are made about the distribution of the station locations. Two types of observational error are considered, stationary (or homogeneous) and independent. Variance transfer functions are calculated which, when multiplied by the signal spectrum or error spectrum and integrated, give the contribution to the variance of estimates. Operations such as smoothing, interpolation, estimating derivatives, gradients of two-dimensional fields, divergence of two-dimensional vector fields, and spectrum estimates are considered. Aliasing takes the form of irregular side lobes, and it is concluded that the variance transfer functions should be plotted in most practical situations involving unequally spaced observations. Simulation studies are presented where the station locations have a random distribution, but the analysis does not use this information and is applicable to any distribution of stations.

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Richard H. Jones

Abstract

A method of linear prediction of stationary multivariate time series is discussed from the point of view of meteorological applications. Tests of significance are given and it is shown by examples that the method is practical even when the dimensionality of the series becomes quite large.

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Richard H. Jones

Abstract

The variance of a time average of a stationary time series depends on the spectral density near frequency zero rather than on the variance of the process. Equations are given for estimating the variance of a time average by fitting a low-order autoregression to the data. Details are given for selecting the order of the autoregression. An example is presented which uses an analysis of variance approach for testing for climatic trends, allowing for diurnal and annual variability and serial correlation.

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Richard H. Jones

Abstract

A method of estimating the spectral density of the nondeterministic component of a meteorological time series which is uncontaminated by the periodic mean variations is presented. This method does not require knowledge of the mean variations. The estimated spectrum is used to calculate the Wiener-Kolmogoroff prediction constants and to estimate the linear predictability of the series. Examples are given using meteorological and artificially generated time series.

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RICHARD H. JONES

Abstract

Equations are derived for the estimation of the parameters in a nonlinear model for the probability of more than two mutually exclusive and exhaustive events. The estimated probabilities are between zero and one and sum to one. The equations for least squares and maximum likelihood estimation are given, and it is pointed out that the maximum likelihood estimate has the form of weighted least squares with estimated weights giving more weight to low probability events.

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A. H. Woodcock and Richard H. Jones

Abstract

A recent study in Queensland, Australia, associates long-term downward trends in rain amount with the productivity of the sugarcane industry. The relationship is attributed to an increasing colloidal stability in the clouds caused by additional cloud condensation nuclei shown to be present in the smoke coming from local cane-harvesting fires.

As an additional test of the hypothesis, the rainfall records of several sugar-producing areas in Hawaii are examined where burning prior to harvesting is also practiced. Two physically similar leeward coastal areas were selected for comparison, one because it is downwind from a major cane-growing region and the other because it is not. The data suggest a downward trend in rainfall over periods of 30–60 years at both areas, but the trends are not statistically significant. However, the records for areas along the windward coastal regions of the two northwesternmost islands indicate an upward trend. It is concluded that factors other than cane-fire smoke are probably involved in any rainfall trends which may exist.

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