A practical step-by-step guide to wavelet analysis is given, with examples taken from time series of the El Nino-Southern Oscillation (ENSO). The guide includes a comparison to the windowed Fourier transform, the choice of an appropriate wavelet basis function, edge effects due to finite-length time series, and the relationship between wavelet scale and Fourier frequency. New statistical significance tests for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise processes and using these to establish significance levels and confidence intervals. It is shown that smoothing in time or scale can be used to increase the confidence of the wavelet spectrum. Empirical formulas are given for the effect of smoothing on significance levels and confidence intervals. Extensions to wavelet analysis such as filtering, the power Hovmoller, cross-wavelet spectra, and coherence are described. The statistical significance tests are used to give a quantitative measure of changes in ENSO variance on interdecadal timescales. Using new datasets that extend back to 1871, the Nino3 sea surface temperature and the Southern Oscillation index show significantly higher power during 1880-1920 and 1960-90, and lower power during 1920-60, as well as a possible 15-yr modulation of variance. The power Hovmoller of sea level pressure shows significant variations in 2-8-yr wavelet power in both longitude and time.

† The contribution of R. J. Allan was written in the course of his employment at the Met Office, UK, and is published with the permission of the Controller of HMSO and the Queen's Printer for Scotland. ‡ The contributions of these authors were prepared as part of their official duties as US Federal Government employees.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. 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.

Variability of the Pacific decadal oscillation (PDO), on both interannual and decadal timescales, is well modeled as the sum of direct forcing by El Niño-Southern Oscillation (ENSO), the ''reemergence'' of North Pacific sea surface temperature anomalies in subsequent winters, and white noise atmospheric forcing. This simple model may be taken as a null hypothesis for the PDO, and may also be relevant for other climate integrators that have been previously related to the PDO.

Historical reanalyses that span more than a century are needed for a wide range of studies, from understanding large‐scale climate trends to diagnosing the impacts of individual historical extreme weather events. The Twentieth Century Reanalysis (20CR) Project is an effort to fill this need. It is supported by the National Oceanic and Atmospheric Administration (NOAA), the Cooperative Institute for Research in Environmental Sciences (CIRES), and the U.S. Department of Energy (DOE), and is facilitated by collaboration with the international Atmospheric Circulation Reconstructions over the Earth initiative. 20CR is the first ensemble of sub‐daily global atmospheric conditions spanning over 100 years. This provides a best estimate of the weather at any given place and time as well as an estimate of its confidence and uncertainty. While extremely useful, version 2c of this dataset (20CRv2c) has several significant issues, including inaccurate estimates of confidence and a global sea level pressure bias in the mid‐19th century. These and other issues can reduce its effectiveness for studies at many spatial and temporal scales. Therefore, the 20CR system underwent a series of developments to generate a significant new version of the reanalysis. The version 3 system (NOAA‐CIRES‐DOE 20CRv3) uses upgraded data assimilation methods including an adaptive inflation algorithm; has a newer, higher‐resolution forecast model that specifies dry air mass; and assimilates a larger set of pressure observations. These changes have improved the ensemble‐based estimates of confidence, removed spin‐up effects in the precipitation fields, and diminished the sea‐level pressure bias. Other improvements include more accurate representations of storm intensity, smaller errors, and large‐scale reductions in model bias. The 20CRv3 system is comprehensively reviewed, focusing on the aspects that have ameliorated issues in 20CRv2c. Despite the many improvements, some challenges remain, including a systematic bias in tropical precipitation and time‐varying biases in southern high‐latitude pressure fields.

The climate research community uses atmospheric reanalysis data sets to understand a wide range of processes and variability in the atmosphere, yet different reanalyses may give very different results for the same diagnostics. The Stratosphere–troposphere Processes And their Role in Climate (SPARC) Reanalysis Intercomparison Project (S-RIP) is a coordinated activity to compare reanalysis data sets using a variety of key diagnostics. The objectives of this project are to identify differences among reanalyses and understand their underlying causes, to provide guidance on appropriate usage of various reanalysis products in scientific studies, particularly those of relevance to SPARC, and to contribute to future improvements in the reanalysis products by establishing collaborative links between reanalysis centres and data users. The project focuses predominantly on differences among reanalyses, although studies that include operational analyses and studies comparing reanalyses with observations are also included when appropriate. The emphasis is on diagnostics of the upper troposphere, stratosphere, and lower mesosphere. This paper summarizes the motivation and goals of the S-RIP activity and extensively reviews key technical aspects of the reanalysis data sets that are the focus of this activity. The special issue “The SPARC Reanalysis Intercomparison Project (S-RIP)” in this journal serves to collect research with relevance to the S-RIP in preparation for the publication of the planned two (interim and full) S-RIP reports

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Studies using idealized ensemble data assimilation systems have shown that flow-dependent background-error covariances are most beneficial when the observing network is sparse. The computational cost of recently proposed ensemble data assimilation algorithms is directly proportional to the number of observations being assimilated. Therefore, ensemble-based data assimilation should both be more computationally feasible and provide the greatest benefit over current operational schemes in situations when observations are sparse. Reanalysis before the radiosonde era (pre-1931) is just such a situation.The feasibility of reanalysis before radiosondes using an ensemble square root filter (EnSRF) is examined. Real surface pressure observations for 2001 are used, subsampled to resemble the density of observations we estimate to be available for 1915. Analysis errors are defined relative to a three-dimensional variational data assimilation (3DVAR) analysis using several orders of magnitude more observations, both at the surface and aloft. We find that the EnSRF is computationally tractable and considerably more accurate than other candidate analysis schemes that use static background-error covariance estimates. We conclude that a Northern Hemisphere reanalysis of the middle and lower troposphere during the first half of the twentieth century is feasible using only surface pressure observations. Expected Northern Hemisphere analysis errors at 500 hPa for the 1915 observation network are similar to current 2.5-day forecast errors.

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