Jon E. Ahlquist

An Ensemble Forecasting Primer

An ensemble forecast is a collection (an ensemble) of forecasts that all verify at the same time. These forecasts are regarded as possible scenarios given the uncertainty associated with forecasting. With such an ensemble, one can address issues that go beyond simply estimating the best forecast. These include estimation of the probability of various events and estimation of the confidence that can be associated with a forecast. Global ensemble forecasts out to 10 days have been computed at both the U.S. and European central forecasting centers since December 1992. Since 1995, the United States has computed experimental regional ensemble forecasts focusing on smaller-scale forecast uncertainties out to 2 days. The authors address challenges associated with ensemble forecasting such as 1) formulating an ensemble, 2) choosing the number of forecasts in an ensemble, 3) extracting information from an ensemble of forecasts, 4) displaying information from an ensemble of forecasts, and 5) interpreting ensemble forecasts. Two synopticscale examples of ensemble forecasting from the winter of 1995/96 are also shown.

Improved Skill for the Anomaly Correlation of Geopotential Heights at 500 hPa

This paper addresses the anomaly correlation of the 500-hPa geopotential heights from a suite of global multimodels and from a model-weighted ensemble mean called the superensemble. This procedure follows a number of current studies on weather and seasonal climate forecasting that are being pursued. This study includes a slightly different procedure from that used in other current experimental forecasts for other variables. Here a superensemble for the „ 2 of the geopotential based on the daily forecasts of the geopotential fields at the 500hPa level is constructed. The geopotential of the superensemble is recovered from the solution of the Poisson equation. This procedure appears to improve the skill for those scales where the variance of the geopotential is large and contributes to a marked improvement in the skill of the anomaly correlation. Especially large improvements over the Southern Hemisphere are noted. Consistent day-6 forecast skill above 0.80 is achieved on a day to day basis. The superensemble skills are higher than those of the best model and the ensemble mean. For days 1‐6 the percent improvement in anomaly correlations of the superensemble over the best model are 0.3, 0.8, 2.25, 4.75, 8.6, and 14.6, respectively, for the Northern Hemisphere. The corresponding numbers for the Southern Hemisphere are 1.12, 1.66, 2.69, 4.48, 7.11, and 12.17. Major improvement of anomaly correlation skills is realized by the superensemble at days 5 and 6 of forecasts. The collective regional strengths of the member models, which is reflected in the proposed superensemble, provide a useful consensus product that may be useful for future operational guidance.

Climatology of Normal Mode Rossby Waves

Abstract Three years of twice-daily NMC global operational analyses were projected onto normal mode Rossby waves to produce a climatology of these waves. For zonal wavenumbers. 1 through 4, annual average geopotential amplitudes at 50 kPa are about 5 gpm for the gravest symmetric meridional mode, and 10 and 20 gpm for the next two meridional model although the amplitude for a given time and latitude can greatly exceed the average. Seasonal average amplitudes differ by less than ±25% from the annual average. The modes’ frequencies drift during the course of a year, but this variation is not correlated with season. Autocorrelations of Rossby wave time series become negligible for lags greater than approximately ten days, which is of the order of the wave period. For all ten modes examined, geopotential fluctuations exist in both Northern and Southern Hemispheres.

Use of Differentiable and Nondifferentiable Optimization Algorithms for Variational Data Assimilation with Discontinuous Cost Functions

Abstract Cost functions formulated in four-dimensional variational data assimilation (4DVAR) are nonsmooth in the presence of discontinuous physical processes (i.e., the presence of “on–off” switches in NWP models). The adjoint model integration produces values of subgradients, instead of gradients, of these cost functions with respect to the model’s control variables at discontinuous points. Minimization of these cost functions using conventional differentiable optimization algorithms may encounter difficulties. In this paper an idealized discontinuous model and an actual shallow convection parameterization are used, both including on–off switches, to illustrate the performances of differentiable and nondifferentiable optimization algorithms. It was found that (i) the differentiable optimization, such as the limited memory quasi-Newton (L-BFGS) algorithm, may still work well for minimizing a nondifferentiable cost function, especially when the changes made in the forecast model at switching points to the...

A Study of Baroclinic Wave Behavior over Bottom Topography Using Complex Principal Component Analysis of Experimental Data

Abstract Complex principal component analysis is applied to data from three laboratory experiments of flow over two-wave sinusoidal bottom topography in a thermally driven, rotating annulus of fluid. The experiments are conducted at the same imposed temperature contrast (ΔT) and at three different rotation rates (Ω). In each case, the intensity of the wave activity is maximum downstream of the two topographic ridges. The analysis, however, reveals a fundamental difference in the behavior of the waves at lower rotation rates than at the highest rotation rate. At the lower Ωs, the baroclinic waves travel over the topographic ridges with diminished intensity and amplify on the other side of each ridge, with the result that the flows downstream of the two ridges are coherent. At the largest Ω, at which the Rossby number, Ro, is very small and the friction parameter, r = E½/Ro (where E is proportional to the Ekman number), is rather large, the waves downstream of each ridge are decoupled from those downstream...

Examination of Numerical Results from Tangent Linear and Adjoint of Discontinuous Nonlinear Models

Abstract The forward model solution and its functional (e.g., the cost function in 4DVAR) are discontinuous with respect to the models control variables if the model contains discontinuous physical processes that occur during the assimilation window. In such a case, the tangent linear model (the first-order approximation of a finite perturbation) is unable to represent the sharp jumps of the nonlinear model solution. Also, the first-order approximation provided by the adjoint model is unable to represent a finite perturbation of the cost function when the introduced perturbation in the control variables crosses discontinuous points. Using an idealized simple model and the Arakawa–Schubert cumulus parameterization scheme, the authors examined the behavior of a cost function and its gradient obtained by the adjoint model with discontinuous model physics. Numerical results show that a cost function involving discontinuous physical processes is zeroth-order discontinuous, but piecewise differentiable. The ma...

Free software and information via computer network

Through “anonymous ftp” accounts accessible over the Internet computer network, one can obtain much free quality software, documentation, and information relevant to meteorology. A program named archie can locate sites that offer desired software. This article lists some of the more useful sites and give suggestions for distributing software. Bitnet users have LISTSERV sources for software and information as well as connections to Internet.

Application of fuzzy implication to probe nonsymmetric relations: Part I

Abstract Nine fuzzy implication operators were studied to determine their ability to identify nonsymmetric relations. The fuzzy degrees of the antecedent and consequent propositions of each implication were evaluated using the cumulative distribution function, which has the property that fuzzy degrees are uniformly distributed between 0 and 1 and hence are as fuzzy as possible. Under these conditions, average implications from the nine operators are confined to small subintervals of [0, 1]. For eight out of nine implication operators, the average implication largely duplicates information from the correlation coefficient. These points are explained theoretically. Confidence limits for fuzzy implication are also discussed.

Geographic Uncertainty in Environmental Security

This volume contains 17 papers presented at the NATO Advanced Research Workshop on Fuzziness and Uncertainty held in Kiev, Ukraine, 28 June to 1 July 2006. Eleven of the papers deal with fuzzy set concepts, while the other six (papers 5, 7, 13, 14, 15, and 16) are not fuzzy. A reader with no prior exposure to fuzzy set theory would benefit from having an introductory text at hand, but the papers are accessible to a wide audience. In general, the papers deal with broad issues of classification and uncertainty in geographic information.

Calendars and Software

Abstract Meteorologists need to be able to manipulate arbitrary dates in the past, present, and future. Here calendar rules for both the Julian (old) and Gregorian (modern) calendars are reviewed. The author describes free software available online to perform calendrical conversions involving (day, month, year), (day of the year, year), and Julian day number for both the Julian and Gregorian calendars.