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Tobias Necker, Martin Weissmann, Yvonne Ruckstuhl, Jeffrey Anderson, and Takemasa Miyoshi

Abstract

State-of-the-art ensemble prediction systems usually provide ensembles with only 20–250 members for estimating the uncertainty of the forecast and its spatial and spatiotemporal covariance. Given that the degrees of freedom of atmospheric models are several magnitudes higher, the estimates are therefore substantially affected by sampling errors. For error covariances, spurious correlations lead to random sampling errors, but also a systematic overestimation of the correlation. A common approach to mitigate the impact of sampling errors for data assimilation is to localize correlations. However, this is a challenging task given that physical correlations in the atmosphere can extend over long distances. Besides data assimilation, sampling errors pose an issue for the investigation of spatiotemporal correlations using ensemble sensitivity analysis. Our study evaluates a statistical approach for correcting sampling errors. The applied sampling error correction is a lookup table–based approach and therefore computationally very efficient. We show that this approach substantially improves both the estimates of spatial correlations for data assimilation as well as spatiotemporal correlations for ensemble sensitivity analysis. The evaluation is performed using the first convective-scale 1000-member ensemble simulation for central Europe. Correlations of the 1000-member ensemble forecast serve as truth to assess the performance of the sampling error correction for smaller subsets of the full ensemble. The sampling error correction strongly reduced both random and systematic errors for all evaluated variables, ensemble sizes, and lead times.

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Joshua P. Hacker, Jeffrey L. Anderson, and Mariusz Pagowski

Abstract

Strategies to improve covariance estimates for ensemble-based assimilation of near-surface observations in atmospheric models are explored. It is known that localization of covariance estimates can improve conditioning of covariance matrices in the assimilation process by removing spurious elements and increasing the rank of the matrix. Vertical covariance localization is the focus of this work, and two basic approaches are compared: 1) a recently proposed hierarchical filter approach based on sampling theory and 2) a more commonly used fifth-order piecewise rational function. The hierarchical filter allows for dynamic estimates of localization functions and does not place any restrictions on their form. The rational function is optimized for every analysis time of day and for every possible observation and state variable combination. The methods are tested with a column model containing PBL and land surface parameterization schemes that are available in current mesoscale modeling systems. The results are expected to provide context for assimilation of near-surface observations in mesoscale models, which will benefit short-range mesoscale NWP applications. Results show that both the hierarchical and rational function approaches effectively improve covariance estimates from small ensembles. The hierarchical approach provides localization functions that are irregular and more closely related to PBL structure. Analysis of eigenvalue spectra show that both approaches improve the rank of the covariance matrices, but the amount of improvement is not always directly related to the assimilation performance. Results also show that specifying different localization functions for different observation and state variable combinations is more important than including time dependence.

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Jonathan Poterjoy, Ryan A. Sobash, and Jeffrey L. Anderson

Abstract

Particle filters (PFs) are Monte Carlo data assimilation techniques that operate with no parametric assumptions for prior and posterior errors. A data assimilation method introduced recently, called the local PF, approximates the PF solution within neighborhoods of observations, thus allowing for its use in high-dimensional systems. The current study explores the potential of the local PF for atmospheric data assimilation through cloud-permitting numerical experiments performed for an idealized squall line. Using only 100 ensemble members, experiments using the local PF to assimilate simulated radar measurements demonstrate that the method provides accurate analyses at a cost comparable to ensemble filters currently used in weather models. Comparisons between the local PF and an ensemble Kalman filter demonstrate benefits of the local PF for producing probabilistic analyses of non-Gaussian variables, such as hydrometeor mixing ratios. The local PF also provides more accurate forecasts than the ensemble Kalman filter, despite yielding higher posterior root-mean-square errors. A major advantage of the local PF comes from its ability to produce more physically consistent posterior members than the ensemble Kalman filter, which leads to fewer spurious model adjustments during forecasts. This manuscript presents the first successful application of the local PF in a weather prediction model and discusses implications for real applications where nonlinear measurement operators and nonlinear model processes limit the effectiveness of current Gaussian data assimilation techniques.

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Lili Lei, Jeffrey L. Anderson, and Glen S. Romine

Abstract

For ensemble-based data assimilation, localization is used to limit the impact of observations on physically distant state variables to reduce spurious error correlations caused by limited ensemble size. Traditionally, the localization value applied is spatially homogeneous. Yet there are potentially larger errors and different covariance length scales in precipitation systems, and that may justify the use of different localization functions for precipitating and nonprecipitating regions. Here this is examined using empirical localization functions (ELFs). Using output from an ensemble observing system simulation experiment (OSSE), ELFs provide estimates of horizontal and vertical localization for different observation types in regions with and without precipitation. For temperature and u- and υ-wind observations, the ELFs for precipitating regions are shown to have smaller horizontal localization scales than for nonprecipitating regions. However, the ELFs for precipitating regions generally have larger vertical localization scales than for nonprecipitating regions. The ELFs are smoothed and then applied in three additional OSSEs. Spatially homogeneous ELFs are found to improve performance relative to a commonly used localization function with compact support. When different ELFs are applied in precipitating and nonprecipitating regions, performance is further improved, but varying ELFs by observation type was not found to be as important. Imbalance in initial states caused by use of different localization functions is diagnosed by the domain-averaged surface pressure tendency. Forecasts from analyses with ELFs have smaller surface pressure tendencies than the standard localization, indicating improved initial balance with ELFs.

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Hui Liu, Jeffrey L. Anderson, Ying-Hwa Kuo, and Kevin Raeder

Abstract

The importance of multivariate forecast error correlations between specific humidity, temperature, and surface pressure in perfect model assimilations of Global Positioning System radio occultation (RO) refractivity data is examined using the Ensemble Adjustment Filter (EAF) and the NCAR global Community Atmospheric Model, version 3. The goal is to explore whether inclusion of the multivariate forecast error correlations in the background term of 3D and 4D variational data assimilation systems (3DVAR and 4DVAR, respectively) is likely to improve RO data assimilation in the troposphere. It is not possible to explicitly neglect multivariate forecast error correlations with the EAF because they are not used directly in the algorithm. Instead, the filter only makes use of the forecast error correlations between observed quantities (RO here) and model state variables. However, because the forecast error correlations for RO observations are dominated by correlations with a subset of state variable types in certain regions, the importance of multivariate forecast error correlations between state variables can be indirectly assessed. This is done by setting the forecast error correlations of RO observations and some state variables (e.g., temperature) to zero in a set of assimilation experiments. Comparing these experiments to a control in which all state variables are impacted by RO observations allows an indirect assessment of the importance of multivariate correlations between state variables not impacted by the observations and those that are impacted. Results suggest that proper specification of the multivariate forecast error correlations in 3DVAR and 4DVAR systems should improve the analysis of specific humidity, surface pressure, and temperature in the troposphere when assimilating RO data.

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Robert Pincus, Robert J. Patrick Hofmann, Jeffrey L. Anderson, Kevin Raeder, Nancy Collins, and Jeffrey S. Whitaker

Abstract

This paper explores the degree to which short-term forecasts with global models might be improved if clouds were fully included in a data assimilation system, so that observations of clouds affected all parts of the model state and cloud variables were adjusted during assimilation. The question is examined using a single ensemble data assimilation system coupled to two present-generation climate models with different treatments of clouds. “Perfect-model” experiments using synthetic observations, taken from a free run of the model used in subsequent assimilations, are used to circumvent complications associated with systematic model errors and observational challenges; these provide a rough upper bound on the utility of cloud observations with these models. A series of experiments is performed in which direct observations of the model’s cloud variables are added to the suite of observations being assimilated. In both models, observations of clouds reduce the 6-h forecast error, with much greater reductions in one model than in the other. Improvements are largest in regions where other observations are sparse. The two cloud schemes differ in their complexity and number of degrees of freedom; the model using the simpler scheme makes better use of the cloud observations because of the stronger correlations between cloud-related and dynamical variables (particularly temperature). This implies that the impact of real cloud observations will depend on both the strength of the instantaneous, linear relationships between clouds and other fields in the natural world, and how well each assimilating model’s cloud scheme represents those relationships.

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Michael K. Tippett, Jeffrey L. Anderson, Craig H. Bishop, Thomas M. Hamill, and Jeffrey S. Whitaker

Abstract

Ensemble data assimilation methods assimilate observations using state-space estimation methods and low-rank representations of forecast and analysis error covariances. A key element of such methods is the transformation of the forecast ensemble into an analysis ensemble with appropriate statistics. This transformation may be performed stochastically by treating observations as random variables, or deterministically by requiring that the updated analysis perturbations satisfy the Kalman filter analysis error covariance equation. Deterministic analysis ensemble updates are implementations of Kalman square root filters. The nonuniqueness of the deterministic transformation used in square root Kalman filters provides a framework to compare three recently proposed ensemble data assimilation methods.

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Soyoung Ha, Chris Snyder, William C. Skamarock, Jeffrey Anderson, and Nancy Collins

Abstract

A global atmospheric analysis and forecast system is constructed based on the atmospheric component of the Model for Prediction Across Scales (MPAS-A) and the Data Assimilation Research Testbed (DART) ensemble Kalman filter. The system is constructed using the unstructured MPAS-A Voronoi (nominally hexagonal) mesh and thus facilitates multiscale analysis and forecasting without the need for developing new covariance models at different scales. Cycling experiments with the assimilation of real observations show that the global ensemble system is robust and reliable throughout a one-month period for both quasi-uniform and variable-resolution meshes. The variable-mesh assimilation system consistently provides higher-quality analyses than those from the coarse uniform mesh, in addition to the benefits of the higher-resolution forecasts, which leads to substantial improvements in 5-day forecasts. Using the fractions skill score, the spatial scale for skillful precipitation forecasts is evaluated over the high-resolution area of the variable-resolution mesh. Skill decreases more rapidly at smaller scales, but the variable mesh consistently outperforms the coarse uniform mesh in precipitation forecasts at all times and thresholds. Use of incremental analysis updates (IAU) greatly decreases high-frequency noise overall and improves the quality of EnKF analyses, particularly in the tropics. Important aspects of the system design related to the unstructured Voronoi mesh are also investigated, including algorithms for handling the C-grid staggered horizontal velocities.

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Hui Liu, Jeffrey Anderson, Ying-Hwa Kuo, Chris Snyder, and Alain Caya

Abstract

A nonlocal quasi-phase radio occultation (RO) observation operator is evaluated in the assimilation of Challenging Minisatellite Payload (CHAMP) radio occultation refractivity using a Weather Research and Forecasting (WRF) ensemble data assimilation system at 50-km resolution. The nonlocal operator calculates the quasi phase through integration of the model refractivity along the observed ray paths. As a comparison, a local refractivity operator that calculates the model refractivity at the observed ray perigee points is also evaluated. The assimilation is done over North America during January 2003 in two different situations: in conjunction with dense, high-quality radiosonde observations and with only satellite cloud drift wind observations. Analyses of temperature and water vapor with the RO refractivity assimilated using the local and nonlocal operator are verified against nearby withheld radiosonde observations. The bias and RMS errors of the analyses of water vapor and temperature using the nonlocal operator are significantly reduced compared with those using the local operator in the troposphere when the only additional observations are satellite cloud drift winds. The reduction of the bias and RMS errors is reduced when radiosonde observations are assimilated.

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Nedjeljka Žagar, Jeffrey Anderson, Nancy Collins, Timothy Hoar, Kevin Raeder, Lili Lei, and Joseph Tribbia

Abstract

Global data assimilation systems for numerical weather prediction (NWP) are characterized by significant uncertainties in tropical analysis fields. Furthermore, the largest spread of global ensemble forecasts in the short range on all scales is in the tropics. The presented results suggest that these properties hold even in the perfect-model framework and the ensemble Kalman filter data assimilation with a globally homogeneous network of wind and temperature profiles. The reasons for this are discussed by using the modal analysis, which provides information about the scale dependency of analysis and forecast uncertainties and information about the efficiency of data assimilation to reduce the prior uncertainties in the balanced and inertio-gravity dynamics.

The scale-dependent representation of variance reduction of the prior ensemble by the data assimilation shows that the peak efficiency of data assimilation is on the synoptic scales in the midlatitudes that are associated with quasigeostrophic dynamics. In contrast, the variance associated with the inertia–gravity modes is less successfully reduced on all scales. A smaller information content of observations on planetary scales with respect to the synoptic scales is discussed in relation to the large-scale tropical uncertainties that current data assimilation methodologies do not address successfully. In addition, it is shown that a smaller reduction of the large-scale uncertainties in the prior state for NWP in the tropics than in the midlatitudes is influenced by the applied radius for the covariance localization.

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