Application of an error statistics estimation method to the PSAS forecast error covariance model

2006 ◽  
Vol 23 (1) ◽  
pp. 33-44
Author(s):  
Runhua Yang ◽  
Jing Guo ◽  
Lars Peter Riishøjgaard
Keyword(s):  
Forecast Error ◽  
Estimation Method ◽  
Covariance Model ◽  
Error Statistics ◽  
Error Covariance

How is the Balance of a Forecast Ensemble Affected by Adaptive and Nonadaptive Localization Schemes?

2015 ◽  
Vol 143 (9) ◽  
pp. 3680-3699 ◽  
Author(s):  
Ross N. Bannister
Keyword(s):  
Forecast Error ◽  
Covariance Matrices ◽  
Error Statistics ◽  
Error Covariance ◽  
Localization Scheme ◽  
Adaptive Schemes ◽  
Convective Scale ◽  
Free Parameters ◽  
Hydrostatic Balance ◽  
First Time

Abstract This paper investigates the effect on balance of a number of Schur product–type localization schemes that have been designed with the primary function of reducing spurious far-field correlations in forecast error statistics. The localization schemes studied comprise a nonadaptive scheme (where the moderation matrix is decomposed in a spectral basis), and two adaptive schemes: a simplified version of Smoothed Ensemble Correlations Raised to a Power (SENCORP) and Ensemble Correlations Raised to a Power (ECO-RAP). The paper shows, the author believes for the first time, how the degree of balance (geostrophic and hydrostatic) implied by the error covariance matrices localized by these schemes can be diagnosed. Here it is considered that an effective localization scheme is one that reduces spurious correlations adequately, but also minimizes disruption of balance (where the “correct” degree of balance or imbalance is assumed to be possessed by the unlocalized ensemble). By varying free parameters that describe each scheme (e.g., the degree of truncation in the schemes that use the spectral basis, the “order” of each scheme, and the degree of ensemble smoothing), it is found that a particular configuration of the ECO-RAP scheme is best suited to the convective-scale system studied. According to the diagnostics this ECO-RAP configuration still weakens geostrophic and hydrostatic balance, but overall this is less so than for other schemes.

scholarly journals Efficient Adaptive Error Parameterizations for Square Root or Ensemble Kalman Filters: Application to the Control of Ocean Mesoscale Signals

2010 ◽  
Vol 138 (3) ◽  
pp. 932-950 ◽  
Author(s):  
Jean-Michel Brankart ◽  
Emmanuel Cosme ◽  
Charles-Emmanuel Testut ◽  
Pierre Brasseur ◽  
Jacques Verron
Keyword(s):  
Kalman Filter ◽  
Covariance Matrix ◽  
Error Estimates ◽  
Forecast Error ◽  
Observation Error ◽  
Square Root ◽  
Error Statistics ◽  
Error Covariance Matrix ◽  
Error Covariance ◽  
Optimal Estimates

Abstract In Kalman filter applications, an adaptive parameterization of the error statistics is often necessary to avoid filter divergence, and prevent error estimates from becoming grossly inconsistent with the real error. With the classic formulation of the Kalman filter observational update, optimal estimates of general adaptive parameters can only be obtained at a numerical cost that is several times larger than the cost of the state observational update. In this paper, it is shown that there exists a few types of important parameters for which optimal estimates can be computed at a negligible numerical cost, as soon as the computation is performed using a transformed algorithm that works in the reduced control space defined by the square root or ensemble representation of the forecast error covariance matrix. The set of parameters that can be efficiently controlled includes scaling factors for the forecast error covariance matrix, scaling factors for the observation error covariance matrix, or even a scaling factor for the observation error correlation length scale. As an application, the resulting adaptive filter is used to estimate the time evolution of ocean mesoscale signals using observations of the ocean dynamic topography. To check the behavior of the adaptive mechanism, this is done in the context of idealized experiments, in which model error and observation error statistics are known. This ideal framework is particularly appropriate to explore the ill-conditioned situations (inadequate prior assumptions or uncontrollability of the parameters) in which adaptivity can be misleading. Overall, the experiments show that, if used correctly, the efficient optimal adaptive algorithm proposed in this paper introduces useful supplementary degrees of freedom in the estimation problem, and that the direct control of these statistical parameters by the observations increases the robustness of the error estimates and thus the optimality of the resulting Kalman filter.

A Hybrid Background Error Covariance Model for Assimilating Glider Data into a Coastal Ocean Model

2011 ◽  
Vol 139 (6) ◽  
pp. 1879-1890 ◽  
Author(s):  
Max Yaremchuk ◽  
Dmitri Nechaev ◽  
Chudong Pan
Keyword(s):  
Null Space ◽  
Coastal Ocean ◽  
Ocean Model ◽  
Covariance Operator ◽  
Covariance Model ◽  
Error Statistics ◽  
Background Error ◽  
Background Error Covariance ◽  
Error Covariance ◽  
Coastal Ocean Model

Abstract A hybrid background error covariance (BEC) model for three-dimensional variational data assimilation of glider data into the Navy Coastal Ocean Model (NCOM) is introduced. Similar to existing atmospheric hybrid BEC models, the proposed model combines low-rank ensemble covariances with the heuristic Gaussian-shaped covariances to estimate forecast error statistics. The distinctive features of the proposed BEC model are the following: (i) formulation in terms of inverse error covariances, (ii) adaptive determination of the rank m of with information criterion based on the innovation error statistics, (iii) restriction of the heuristic covariance operator to the null space of , and (iv) definition of the BEC magnitudes through separate analyses of the innovation error statistics in the state space and the null space of . The BEC model is validated by assimilation experiments with simulated and real data obtained during a glider survey of the Monterey Bay in August 2003. It is shown that the proposed hybrid scheme substantially improves the forecast skill of the heuristic covariance model.

scholarly journals A Study of Rain Forecast Error Structure Based on Radar Observations over a Continental-Scale Spatial Domain

2016 ◽  
Vol 144 (8) ◽  
pp. 2871-2887
Author(s):  
Majid Fekri ◽  
M. K. Yau
Keyword(s):  
Forecast Error ◽  
Error Variance ◽  
Precipitation Forecast ◽  
Local Error ◽  
Error Statistics ◽  
Continental Scale ◽  
Error Covariance ◽  
Hourly Rainfall ◽  
Dependent Error

Abstract This study examines the univariate error covariances of hourly rainfall accumulations using two different NWP models and a mosaic of radar reflectivity over a continental-scale domain. The study focuses on two main areas. The focus of the first part of the paper is on the ensemble-based and the innovation-based error variance and correlation estimations. An ensemble of forecasts and a set of observations provide the basis for estimating the errors in two different ways. The results indicate that both ensemble- and innovation-based methods lead to comparable variance estimations, while the local error correlation estimates have larger differences due to the sensitivity of calculations to the gradient of the variance field. The second part of the paper uses innovations for identifying the errors. The focus of this part is on a prognostic method for estimating the error statistics from the background based on the Bayesian inference technique. The case study shows that the predictive model produces a similar result regarding the magnitude and the dispersion of variance in comparison with the innovation and ensemble-based variances. This study represents a step toward estimating local error variances and local error correlations to construct a nonhomogeneous and precipitation-dependent error covariance matrix of rainfall. These results will be used in a future paper in the design of a 2D-VAR Assimilation Method for Blending Extrapolated Radars (AMBER) with NWP precipitation forecast to form a precipitation nowcasting model.

scholarly journals A Nonvariational Consistent Hybrid Ensemble Filter

2015 ◽  
Vol 143 (12) ◽  
pp. 5073-5090 ◽  
Author(s):  
Craig H. Bishop ◽  
Bo Huang ◽  
Xuguang Wang
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Forecast Error ◽  
Model Space ◽  
High Rank ◽  
Covariance Model ◽  
Ensemble Data Assimilation ◽  
Error Covariance ◽  
Ensemble Data ◽  
Ensemble Transform Kalman Filter

Abstract A consistent hybrid ensemble filter (CHEF) for using hybrid forecast error covariance matrices that linearly combine aspects of both climatological and flow-dependent matrices within a nonvariational ensemble data assimilation scheme is described. The CHEF accommodates the ensemble data assimilation enhancements of (i) model space ensemble covariance localization for satellite data assimilation and (ii) Hodyss’s method for improving accuracy using ensemble skewness. Like the local ensemble transform Kalman filter (LETKF), the CHEF is computationally scalable because it updates local patches of the atmosphere independently of others. Like the sequential ensemble Kalman filter (EnKF), it serially assimilates batches of observations and uses perturbed observations to create ensembles of analyses. It differs from the deterministic (no perturbed observations) ensemble square root filter (ESRF) and the EnKF in that (i) its analysis correction is unaffected by the order in which observations are assimilated even when localization is required, (ii) it uses accurate high-rank solutions for the posterior error covariance matrix to serially assimilate observations, and (iii) it accommodates high-rank hybrid error covariance models. Experiments were performed to assess the effect on CHEF and ESRF analysis accuracy of these differences. In the case where both the CHEF and the ESRF used tuned localized ensemble covariances for the forecast error covariance model, the CHEF’s advantage over the ESRF increased with observational density. In the case where the CHEF used a hybrid error covariance model but the ESRF did not, the CHEF had a substantial advantage for all observational densities.

scholarly journals Four-Dimensional Data Assimilation and Balanced Dynamics

2006 ◽  
Vol 63 (7) ◽  
pp. 1840-1858 ◽  
Author(s):  
Lisa J. Neef ◽  
Saroja M. Polavarapu ◽  
Theodore G. Shepherd
Keyword(s):  
Data Assimilation ◽  
Degrees Of Freedom ◽  
Forecast Error ◽  
Wave Mode ◽  
Slow Manifold ◽  
Error Statistics ◽  
Ensemble Kalman Filters ◽  
Error Covariance ◽  
Balanced Dynamics ◽  
Four Dimensional Data Assimilation

Abstract The problem of spurious excitation of gravity waves in the context of four-dimensional data assimilation is investigated using a simple model of balanced dynamics. The model admits a chaotic vortical mode coupled to a comparatively fast gravity wave mode, and can be initialized such that the model evolves on a so-called slow manifold, where the fast motion is suppressed. Identical twin assimilation experiments are performed, comparing the extended and ensemble Kalman filters (EKF and EnKF, respectively). The EKF uses a tangent linear model (TLM) to estimate the evolution of forecast error statistics in time, whereas the EnKF uses the statistics of an ensemble of nonlinear model integrations. Specifically, the case is examined where the true state is balanced, but observation errors project onto all degrees of freedom, including the fast modes. It is shown that the EKF and EnKF will assimilate observations in a balanced way only if certain assumptions hold, and that, outside of ideal cases (i.e., with very frequent observations), dynamical balance can easily be lost in the assimilation. For the EKF, the repeated adjustment of the covariances by the assimilation of observations can easily unbalance the TLM, and destroy the assumptions on which balanced assimilation rests. It is shown that an important factor is the choice of initial forecast error covariance matrix. A balance-constrained EKF is described and compared to the standard EKF, and shown to offer significant improvement for observation frequencies where balance in the standard EKF is lost. The EnKF is advantageous in that balance in the error covariances relies only on a balanced forecast ensemble, and that the analysis step is an ensemble-mean operation. Numerical experiments show that the EnKF may be preferable to the EKF in terms of balance, though its validity is limited by ensemble size. It is also found that overobserving can lead to a more unbalanced forecast ensemble and thus to an unbalanced analysis.

scholarly journals On the Sensitivity Equations of Four-Dimensional Variational (4D-Var) Data Assimilation

2008 ◽  
Vol 136 (8) ◽  
pp. 3050-3065 ◽  
Author(s):  
Dacian N. Daescu
Keyword(s):  
Data Assimilation ◽  
Forecast Error ◽  
Computational Effort ◽  
Water Model ◽  
Error Statistics ◽  
Background Error ◽  
Practical Applications ◽  
Error Covariance ◽  
Sensitivity Equations ◽  
The Impact

Abstract The equations of the forecast sensitivity to observations and to the background estimate in a four-dimensional variational data assimilation system (4D-Var DAS) are derived from the first-order optimality condition in unconstrained minimization. Estimation of the impact of uncertainties in the specification of the error statistics is considered by evaluating the sensitivity to the observation and background error covariance matrices. The information provided by the error covariance sensitivity analysis is used to identify the input components for which improved estimates of the statistical properties of the errors are of most benefit to the analysis and forecast. A close relationship is established between the sensitivities within each input pair data/error covariance such that once the observation and background sensitivities are available the evaluation of the sensitivity to the specification of the corresponding error statistics requires little additional computational effort. The relevance of the 4D-Var sensitivity equations to assess the data impact in practical applications is discussed. Computational issues are addressed and idealized 4D-Var experiments are set up with a finite-volume shallow-water model to illustrate the theoretical concepts. Time-dependent observation sensitivity and potential applications to improve the model forecast are presented. Guidance provided by the sensitivity fields is used to adjust a 4D-Var DAS to achieve forecast error reduction through assimilation of supplementary data and through an accurate specification of a few of the background error variances.

scholarly journals Multivariate Error Covariance Estimates by Monte Carlo Simulation for Assimilation Studies in the Pacific Ocean

2005 ◽  
Vol 133 (8) ◽  
pp. 2310-2334 ◽  
Author(s):  
Anna Borovikov ◽  
Michele M. Rienecker ◽  
Christian L. Keppenne ◽  
Gregory C. Johnson
Keyword(s):  
Monte Carlo ◽  
Data Assimilation ◽  
Forecast Error ◽  
Independent Set ◽  
Velocity Fields ◽  
Optimal Interpolation ◽  
Model Errors ◽  
Error Statistics ◽  
Practical Applications ◽  
Monte Carlo Techniques

Abstract One of the most difficult aspects of ocean-state estimation is the prescription of the model forecast error covariances. The paucity of ocean observations limits our ability to estimate the covariance structures from model–observation differences. In most practical applications, simple covariances are usually prescribed. Rarely are cross covariances between different model variables used. Here a comparison is made between a univariate optimal interpolation (UOI) scheme and a multivariate OI algorithm (MvOI) in the assimilation of ocean temperature profiles. In the UOI case only temperature is updated using a Gaussian covariance function. In the MvOI, salinity, zonal, and meridional velocities as well as temperature are updated using an empirically estimated multivariate covariance matrix. Earlier studies have shown that a univariate OI has a detrimental effect on the salinity and velocity fields of the model. Apparently, in a sequential framework it is important to analyze temperature and salinity together. For the MvOI an estimate of the forecast error statistics is made by Monte Carlo techniques from an ensemble of model forecasts. An important advantage of using an ensemble of ocean states is that it provides a natural way to estimate cross covariances between the fields of different physical variables constituting the model-state vector, at the same time incorporating the model’s dynamical and thermodynamical constraints as well as the effects of physical boundaries. Only temperature observations from the Tropical Atmosphere–Ocean array have been assimilated in this study. To investigate the efficacy of the multivariate scheme, two data assimilation experiments are validated with a large independent set of recently published subsurface observations of salinity, zonal velocity, and temperature. For reference, a control run with no data assimilation is used to check how the data assimilation affects systematic model errors. While the performance of the UOI and MvOI is similar with respect to the temperature field, the salinity and velocity fields are greatly improved when the multivariate correction is used, as is evident from the analyses of the rms differences between these fields and independent observations. The MvOI assimilation is found to improve upon the control run in generating water masses with properties close to the observed, while the UOI fails to maintain the temperature and salinity structure.

Theoretical Application of Ensemble Kalman Filter to Adsorptive Solute Cr(VI) Transfer from Soil into Surface Runoff

2014 ◽  
Vol 919-921 ◽  
pp. 1257-1261
Author(s):  
Chao Qun Tan ◽  
Ju Xiu Tong ◽  
Bill X. Hu ◽  
Jin Zhong Yang
Keyword(s):  
Kalman Filter ◽  
Data Assimilation ◽  
Ensemble Kalman Filter ◽  
Surface Runoff ◽  
Amplification Factor ◽  
Forecast Error ◽  
Assimilation Effect ◽  
Solute Transfer ◽  
Error Covariance ◽  
Theoretical Application

This paper mainly discusses some details when applying data assimilation method via an ensemble Kalman filter (EnKF) to improve prediction of adsorptive solute Cr(VI) transfer from soil into runoff. Based on this work, we could make better use of our theoretical model to predict adsorptive solute transfer from soil into surface runoff in practice. The results show that the ensemble number of 100 is reasonable, considering assimilation effect and efficiency after selecting its number from 25 to 225 at an interval of 25. While the initial ensemble value makes little difference to data assimilation (DA) results. Besides, DA results could be improved by multiplying an amplification factor to forecast error covariance matrix due to underestimation of forecast error.

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