The Modern-Era Retrospective Analysis for Research and Applications (MERRA) was undertaken by NASA’s Global Modeling and Assimilation Office with two primary objectives: to place observations from NASA’s Earth Observing System satellites into a climate context and to improve upon the hydrologic cycle represented in earlier generations of reanalyses. Focusing on the satellite era, from 1979 to the present, MERRA has achieved its goals with significant improvements in precipitation and water vapor climatology. Here, a brief overview of the system and some aspects of its performance, including quality assessment diagnostics from innovation and residual statistics, is given.By comparing MERRA with other updated reanalyses [the interim version of the next ECMWF Re-Analysis (ERA-Interim) and the Climate Forecast System Reanalysis (CFSR)], advances made in this new generation of reanalyses, as well as remaining deficiencies, are identified. Although there is little difference between the new reanalyses i...

MERRA: NASA’s Modern-Era Retrospective Analysis for Research and Applications

The NCEP Climate Forecast System Reanalysis (CFSR) was completed for the 31-yr period from 1979 to 2009, in January 2010. The CFSR was designed and executed as a global, high-resolution coupled atmosphere–ocean–land surface–sea ice system to provide the best estimate of the state of these coupled domains over this period. The current CFSR will be extended as an operational, real-time product into the future. New features of the CFSR include 1) coupling of the atmosphere and ocean during the generation of the 6-h guess field, 2) an interactive sea ice model, and 3) assimilation of satellite radiances by the Gridpoint Statistical Interpolation (GSI) scheme over the entire period. The CFSR global atmosphere resolution is ~38 km (T382) with 64 levels extending from the surface to 0.26 hPa. The global ocean's latitudinal spacing is 0.25° at the equator, extending to a global 0.5° beyond the tropics, with 40 levels to a depth of 4737 m. The global land surface model has four soil levels and the global sea ice m...

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The NCEP Climate Forecast System Reanalysis

The Twentieth Century Reanalysis (20CR) project is an international effort to produce a comprehensive global atmospheric circulation dataset spanning the twentieth century, assimilating only surface pressure reports and using observed monthly sea-surface temperature and sea-ice distributions as boundary conditions. It is chiefly motivated by a need to provide an observational dataset with quantified uncertainties for validations of climate model simulations of the twentieth century on all time-scales, with emphasis on the statistics of daily weather. It uses an Ensemble Kalman Filter data assimilation method with background ‘first guess’ fields supplied by an ensemble of forecasts from a global numerical weather prediction model. This directly yields a global analysis every 6 hours as the most likely state of the atmosphere, and also an uncertainty estimate of that analysis.

The 20CR dataset provides the first estimates of global tropospheric variability, and of the dataset's time-varying quality, from 1871 to the present at 6-hourly temporal and 2° spatial resolutions. Intercomparisons with independent radiosonde data indicate that the reanalyses are generally of high quality. The quality in the extratropical Northern Hemisphere throughout the century is similar to that of current three-day operational NWP forecasts. Intercomparisons over the second half-century of these surface-based reanalyses with other reanalyses that also make use of upper-air and satellite data are equally encouraging.

It is anticipated that the 20CR dataset will be a valuable resource to the climate research community for both model validations and diagnostic studies. Some surprising results are already evident. For instance, the long-term trends of indices representing the North Atlantic Oscillation, the tropical Pacific Walker Circulation, and the Pacific–North American pattern are weak or non-existent over the full period of record. The long-term trends of zonally averaged precipitation minus evaporation also differ in character from those in climate model simulations of the twentieth century. Copyright © 2011 Royal Meteorological Society and Crown Copyright.

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The Twentieth Century Reanalysis Project

A theory for estimating the probability distribution of the state of a model given a set of observations exists. This nonlinear filtering theory unifies the data assimilation and ensemble generation problem that have been key foci of prediction and predictability research for numerical weather and ocean prediction applications. A new algorithm, referred to as an ensemble adjustment Kalman filter, and the more traditional implementation of the ensemble Kalman filter in which “perturbed observations” are used, are derived as Monte Carlo approximations to the nonlinear filter. Both ensemble Kalman filter methods produce assimilations with small ensemble mean errors while providing reasonable measures of uncertainty in the assimilated variables. The ensemble methods can assimilate observations with a nonlinear relation to model state variables and can also use observations to estimate the value of imprecisely known model parameters. These ensemble filter methods are shown to have significant advantag...

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An Ensemble Adjustment Kalman Filter for Data Assimilation

The ensemble Kalman filter (EnKF) is a data assimilation scheme based on the traditional Kalman filter update equation. An ensemble of forecasts are used to estimate the background-error covariances needed to compute the Kalman gain. It is known that if the same observations and the same gain are used to update each member of the ensemble, the ensemble will systematically underestimate analysis-error covariances. This will cause a degradation of subsequent analyses and may lead to filter divergence. For large ensembles, it is known that this problem can be alleviated by treating the observations as random variables, adding random perturbations to them with the correct statistics. Two important consequences of sampling error in the estimate of analysis-error covariances in the EnKF are discussed here. The first results from the analysis-error covariance being a nonlinear function of the backgrounderror covariance in the Kalman filter. Due to this nonlinearity, analysis-error covariance estimates may be negatively biased, even if the ensemble background-error covariance estimates are unbiased. This problem must be dealt with in any Kalman filter‐based ensemble data assimilation scheme. A second consequence of sampling error is particular to schemes like the EnKF that use perturbed observations. While this procedure gives asymptotically correct analysis-error covariance estimates for large ensembles, the addition of perturbed observations adds an additional source of sampling error related to the estimation of the observation-error covariances. In addition to reducing the accuracy of the analysis-error covariance estimate, this extra source of sampling error increases the probability that the analysis-error covariance will be underestimated. Because of this, ensemble data assimilation methods that use perturbed observations are expected to be less accurate than those which do not. Several ensemble filter formulations have recently been proposed that do not require perturbed observations. This study examines a particularly simple implementation called the ensemble square root filter, or EnSRF. The EnSRF uses the traditional Kalman gain for updating the ensemble mean but uses a ‘‘reduced’’ Kalman gain to update deviations from the ensemble mean. There is no additional computational cost incurred by the EnSRF relative to the EnKF when the observations have independent errors and are processed one at a time. Using a hierarchy of perfect model assimilation experiments, it is demonstrated that the elimination of the sampling error associated with the perturbed observations makes the EnSRF more accurate than the EnKF for the same ensemble size.

Ensemble Data Assimilation without Perturbed Observations

This article focuses on the construction, directly in physical space, of simply parametrized covariance functions for data-assimilation applications. A self-contained, rigorous mathematical summary of relevant topics from correlation theory is provided as a foundation for this construction. Covariance and correlation functions are defined, and common notions of homogeneity and isotropy are clarified. Classical results are stated, and proven where instructive. Included are smoothness properties relevant to multivariate statistical-analysis algorithms where wind/wind and wind/mass correlation models are obtained by differentiating the correlation model of a mass variable. the Convolution Theorem is introduced as the primary tool used to construct classes of covariance and cross-covariance functions on three-dimensional Euclidean space R3. Among these are classes of compactly supported functions that restrict to covariance and cross-covariance functions on the unit sphere S2, and that vanish identically on subsets of positive measure on S2. It is shown that these covariance and cross-covariance functions on S2, referred to as being space-limited, cannot be obtained using truncated spectral expansions. Compactly supported and space-limited covariance functions determine sparse covariance matrices when evaluated on a grid, thereby easing computational burdens in atmospheric data-analysis algorithms.

Convolution integrals leading to practical examples of compactly supported covariance and cross-covariance functions on R3 are reduced and evaluated. More specifically, suppose that gi and gj are radially symmetric functions defined on R3 such that gi(x) = 0 for |x| > di and gj(x) = 0 for |xv > dj, O di + dj and |x - y| > 2di, respectively, Additional covariance functions on R3 are constructed using convolutions over the real numbers R, rather than R3. Families of compactly supported approximants to standard second- and third-order autoregressive functions are constructed as illustrative examples. Compactly supported covariance functions of the form C(x,y) := Co(|x - y|), x,y ∈ R3, where the functions Co(r) for r ∈ R are 5th-order piecewise rational functions, are also constructed. These functions are used to develop space-limited product covariance functions B(x, y) C(x, y), x, y ∈ S2, approximating given covariance functions B(x, y) supported on all of S2 × S2.

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Construction of correlation functions in two and three dimensions

This article focuses on construction, directly in physical space, of three-dimensional covariance functions parametrized by a length-field, and on an application of these functions to improve the representation of the Quasi-Biennial Oscillation (QBO) in the Goddard Earth Observing System, Version 4 (GEOS-4) data assimilation system The covariance functions are obtained by fusing collections of auto-covariance functions having different constant length-scales with their associated cross-covariance functions This construction yields covariance functions with length-scales that can vary arbitrarily over any finite partition of the spatial domain A simple, and also motivating application of these functions is to the case where the length-scale varies in the vertical direction only The class of covariance functions with variable length-fields constructed in this article will be called multi-level to associate them with this application The multi-level covariance functions extend well-known single-level covariance functions depending only on a constant length-scale Generalizations of the familiar first-and third-order autoregressive covariances in three dimensions are given, providing multi-level covariances with zero and four continuous derivatives at zero separation, respectively Multi-level piecewise rational covariances with two continuous derivatives at zero separation are also provided Multi-level power-law covariances are constructed with continuous derivatives of all orders Additional multi-level covariance functions are constructed using the Schur product of single- and multi-level covariance functions A multi-variate, multi-level power-law covariance with a large troposphere-to-stratosphere length-field gradient is employed to reproduce the QBO from sparse radiosonde wind observations in the tropical lower stratosphere This covariance model is described along with details of the assimilation experiments The new covariance model is shown to represent the vertical wind shear associated with the QBO much more effectively than the multi-variate, multi-level covariance model in the baseline GEOS-4 system Copyright © 2006 Royal Meteorological Society

Gregory Gaspari

Papers

Construction of correlation functions in two and three dimensions

Construction and application of covariance functions with variable length‐fields