scholarly journals Background-error correlation length-scale estimates and their sampling statistics

2008 ◽  
Vol 134 (631) ◽  
pp. 497-508 ◽  
Author(s):  
O. Pannekoucke ◽  
L. Berre ◽  
G. Desroziers
Keyword(s):  
Correlation Length ◽  
Length Scale ◽  
Background Error ◽  
Error Correlation ◽  
Sampling Statistics

scholarly journals The Effect of Thinning and Superobservations in a Simple One-Dimensional Data Analysis with Mischaracterized Error

2018 ◽  
Vol 146 (4) ◽  
pp. 1181-1195 ◽  
Author(s):  
Ross N. Hoffman
Keyword(s):  
Correlation Length ◽  
Weighted Least Squares ◽  
Simple Problem ◽  
Length Scale ◽  
Observation Error ◽  
Analysis Method ◽  
Analysis Problem ◽  
Background Error ◽  
One Dimensional ◽  
Error Correlation

A one-dimensional (1D) analysis problem is defined and analyzed to explore the interaction of observation thinning or superobservation with observation errors that are correlated or systematic. The general formulation might be applied to a 1D analysis of radiance or radio occultation observations in order to develop a strategy for the use of such data in a full data assimilation system, but is applied here to a simple analysis problem with parameterized error covariances. Findings for the simple problem include the following. For a variational analysis method that includes an estimate of the full observation error covariances, the analysis is more sensitive to variations in the estimated background and observation error standard deviations than to variations in the corresponding correlation length scales. Furthermore, if everything else is fixed, the analysis error increases with decreasing true background error correlation length scale and with increasing true observation error correlation length scale. For a weighted least squares analysis method that assumes the observation errors are uncorrelated, best results are obtained for some degree of thinning and/or tuning of the weights. Without tuning, the best strategy is superobservation with a spacing approximately equal to the observation error correlation length scale.

scholarly journals Forecast Divergences of a Global Wave Model

2005 ◽  
Vol 133 (8) ◽  
pp. 2148-2162 ◽  
Author(s):  
Diana J. M. Greenslade ◽  
Ian R. Young
Keyword(s):  
Correlation Length ◽  
Surface Wind ◽  
Wave Model ◽  
Surface Wind Speed ◽  
Length Scale ◽  
Similar Distribution ◽  
Background Error ◽  
Latitudinal Dependence ◽  
Wave Models ◽  
Seasonal Signal

Abstract One of the main limitations to current wave data assimilation systems is the lack of an accurate representation of the structure of the background errors. One method that may be used to determine background errors is the “NMC method.” This method examines the forecast divergence component of the background error growth by considering differences between forecasts of different ranges valid at the same time. In this paper, the NMC method is applied to global forecasts of significant wave height (SWH) and surface wind speed (U10). It is found that the isotropic correlation length scale of the SWH forecast divergence (LSWH) has considerable geographical variability, with the longest scales just to the south of the equator in the eastern Pacific Ocean, and the shortest scales at high latitudes. The isotropic correlation length scale of the U10 forecast divergence (LU10) has a similar distribution with a stronger latitudinal dependence. It is found that both LSWH and LU10 increase as the forecast period increases. The increase in LSWH is partly due to LU10 also increasing. Another explanation is that errors in the analysis or the short-range SWH forecast propagate forward in time and disperse and their scale becomes larger. It is shown that the forecast divergence component of the background error is strongly anisotropic with the longest scales perpendicular to the likely direction of propagation of swell. In addition, in regions where the swell propagation is seasonal, the forecast divergence component of the background error shows a similar strong seasonal signal. It is suggested that the results of this study provide a lower bound to the description of the total background error in global wave models.

scholarly journals A new exponentially-decaying error correlation model for assimilating OCO-2 column-average CO<sub>2</sub> data, using a length scale computed from airborne lidar measurements

2021 ◽  
Author(s):  
David F. Baker ◽  
Emily Bell ◽  
Kenneth J. Davis ◽  
Joel F. Campbell ◽  
Bing Lin ◽  
...  
Keyword(s):  
Correlation Length ◽  
Error Model ◽  
Airborne Lidar ◽  
Length Scale ◽  
Correlation Model ◽  
Correlation Lengths ◽  
New Model ◽  
The Mean ◽  
Error Correlation ◽  
Along Track

Abstract. To check the accuracy of column-average dry air CO2 mole fractions (XCO2) retrieved from Orbiting Carbon Overvatory (OCO-2) data, a similar quantity has been measured from the Multi-functional Fiber Laser Lidar (MFLL) aboard aircraft flying underneath OCO-2 as part of the Atmospheric Carbon and Transport (ACT)-America flight campaigns. Here we do a lagged correlation analysis of these MFLL-OCO-2 column CO2 differences and find that their correlation spectrum falls off rapidly at along-track separation distances of under 10 km, with a correlation length scale of about 10 km, and less rapidly at longer separation distances, with a correlation length scale of about 20 km. The OCO-2 satellite takes many CO2 measurements with small (~3 km2) fields of view (FOVs) in a thin (<10 km wide) swath running parallel to its orbit: up to 24 separate FOVs may be obtained per second (across a ~6.75 km distance on the ground), though clouds, aerosols, and other factors cause considerable data dropout. Errors in the CO2 retrieval method have long been thought to be correlated at these fine scales, and methods to account for these when assimilating these data into top-down atmospheric CO2 flux inversions have been developed. A common approach has been to average the data at coarser scales (e.g., in 10-second-long bins) along-track, then assign an uncertainty to the averaged value that accounts for the error correlations. Here we outline the methods used up to now for computing these 10-second averages and their uncertainties, including the constant-correlation-with-distance error model currently being used to summarize the OCO-2 version 9 XCO2 retrievals as part of the OCO-2 flux inversion model intercomparison project. We then derive a new one-dimensional error model using correlations that decay exponentially with separation distance, apply this model to the OCO-2 data using the correlation length scales derived from the MFLL-OCO-2 differences, and compare the results (for both the average and its uncertainty) to those given by the current constant-correlation error model. To implement this new model, the data are averaged first across 2-second spans, to collapse the cross-track distribution of the real data onto the 1-D path assumed by the new model. A small percentage of the data that cause nonphysical negative averaging weights in the model are thrown out. The correlation lengths over the ocean, which the land-based MFLL data do not clarify, are assumed to be twice those over the land. The new correlation model gives 10-second XCO2 averages that are only a few tenths of a ppm different from the constant-correlation model. Over land, the uncertainties in the mean are also similar, suggesting that the +0.3 constant correlation coefficient currently used in the model there is accurate. Over the oceans, the twice-the-land correlation lengths that we assume here result in a significantly lower uncertainty on the mean than the +0.6 constant correlation currently gives – measurements similar to the MFLL ones are needed over the oceans to do better. Finally, we show how our 1-D exponential error correlation model may be used to account for correlations in those inversion methods that choose to assimilate each XCO2 retrieval individually, and to account for correlations between separate 10-second averages when these are assimilated instead.

scholarly journals Calculating an error correlation length scale from MFLL-OCO-2 column-average CO2 differences and using it to average OCO-2 data

2021 ◽  
Author(s):  
David F Baker ◽  
Emily Bell ◽  
Kenneth J Davis ◽  
Joel F Campbell ◽  
Bing Lin ◽  
...  
Keyword(s):  
Correlation Length ◽  
Length Scale ◽  
Error Correlation

scholarly journals Emulation of an ensemble Kalman filter algorithm on a flood wave propagation model

2013 ◽  
Vol 10 (6) ◽  
pp. 6963-7001
Author(s):  
S. Barthélémy ◽  
S. Ricci ◽  
O. Pannekoucke ◽  
O. Thual ◽  
P. O. Malaterre
Keyword(s):  
Correlation Length ◽  
Computational Cost ◽  
Error Variance ◽  
Observation Point ◽  
Propagation Model ◽  
Length Scale ◽  
Background Error ◽  
Background Error Covariance ◽  
Covariance Functions ◽  
Error Covariance

Abstract. This study describes the emulation of an Ensemble Kalman Filter (EnKF) algorithm on a 1-D flood wave propagation model. This model is forced at the upstream boundary with a random variable with gaussian statistics and a correlation function in time with gaussian shape. This allows for, in the case without assimilation, the analytical study of the covariance functions of the propagated signal anomaly. This study is validated numerically with an ensemble method. In the case with assimilation with one observation point, where synthetical observations are generated by adding an error to a true state, the dynamic of the background error covariance functions is not straightforward and a numerical approach using an EnKF algorithm is prefered. First, those numerical experiments show that both background error variance and correlation length scale are reduced at the observation point. This reduction of variance and correlation length scale is propagated downstream by the dynamics of the model. Then, it is shown that the application of a Best Linear Unbiased Estimator (BLUE) algorithm using the background error covariance matrix converged from the EnKF algorithm, provides the same results as the EnKF but with a cheaper computational cost, thus allowing for the use of data assimilation in the context of real time flood forecasting. Moreover it was demonstrated that the reduction of background error correlation length scale and variance at the observation point depends on the error observation statistics. This feature is quantified by abacus built from linear regressions over a limited set of EnKF experiments. These abacus that describe the background error variance and the correlation length scale in the neighboring of the observation point combined with analytical expressions that describe the background error variance and the correlation length scale away from the observation point provide parametrized models for the variance and the correlation length scale. Using this parametrized variance and correlation length scale with a diffusion operator makes it possible to model the converged background error covariance matrix from the EnKF without actually integrating the EnKF algorithm. This method was finally applied to a case with two different observation point with different error statistics. It was shown that the results of this emulated EnKF (EEnKF) in terms of background error variance, correlation length scale and analyzed water level is close to those of the EnKF but with a significantly reduced computational cost.

Estimation and Impact of Nonuniform Horizontal Correlation Length Scales for Global Ocean Physical Analyses

2014 ◽  
Vol 31 (10) ◽  
pp. 2330-2349 ◽  
Author(s):  
Andrea Storto ◽  
Simona Masina ◽  
Srdjan Dobricic
Keyword(s):  
Data Assimilation ◽  
Correlation Length ◽  
Global Ocean ◽  
Length Scale ◽  
Background Error ◽  
Length Scales ◽  
Skill Scores ◽  
Uniform Background ◽  
Analysis System ◽  
The Impact

Abstract Optimally modeling background-error horizontal correlations is crucial in ocean data assimilation. This paper investigates the impact of releasing the assumption of uniform background-error correlations in a global ocean variational analysis system. Spatially varying horizontal correlations are introduced in the recursive filter operator, which is used for modeling horizontal covariances in the Centro Euro-Mediterraneo sui Cambiamenti Climatici (CMCC) analysis system. The horizontal correlation length scales (HCLSs) were defined on the full three-dimensional model space and computed from both a dataset of monthly anomalies with respect to the monthly climatology and through the so-called National Meteorological Center (NMC) method. Different formulas for estimating the correlation length scale are also discussed and applied to the two forecast error datasets. The new formulation is tested within a 12-yr period (2000–11) in the ½° resolution system. The comparison with the data assimilation system using uniform background-error horizontal correlations indicates the superiority of the former, especially in eddy-dominated areas. Verification skill scores report a significant reduction of RMSE, and the use of nonuniform length scales improves the representation of the eddy kinetic energy at midlatitudes, suggesting that uniform, latitude, or Rossby radius-dependent formulations are insufficient to represent the geographical variations of the background-error correlations. Furthermore, a small tuning of the globally uniform value of the length scale was found to have a small impact on the analysis system. The use of either anomalies or NMC-derived correlation length scales also has a marginal effect with respect to the use of nonuniform HCLSs. On the other hand, the application of overestimated length scales has proved to be detrimental to the analysis system in all areas and for all parameters.

scholarly journals An atmospheric inversion over the city of Cape Town: sensitivity analyses

2018 ◽  
Author(s):  
Alecia Nickless ◽  
Peter J. Rayner ◽  
Robert J. Scholes ◽  
Francois Engelbrecht ◽  
Birgit Erni
Keyword(s):  
Spatial Distribution ◽  
Correlation Length ◽  
Prior Information ◽  
Fossil Fuel ◽  
Cape Town ◽  
Uncertainty Reduction ◽  
Sensitivity Analyses ◽  
Sensitivity Tests ◽  
Error Correlation ◽  
Biogenic Fluxes

Abstract. We present sixteen different sensitivity tests applied to the Cape Town atmospheric Bayesian inversion analysis from March 2012 until June 2013. The reference inversion made use of a fossil fuel inventory analysis and estimates of biogenic fluxes from CABLE (Community Atmosphere Biosphere Land Exchange model). Changing the prior information product and the assumptions behind the uncertainties in the biogenic fluxes had the largest impact on the inversion results in terms of the spatial distribution of the fluxes, the size of the aggregated fluxes, and the uncertainty reduction achieved. A carbon assessment product of natural carbon fluxes, used in place of CABLE, and the Open-source Data Inventory for Anthropogenic CO2 product, in place of the fossil fuel inventory, resulted in prior estimates that were more positive on average than the reference configuration. The use of different prior flux products to inform separate inversions provided better constraint on the posterior fluxes compared with a single inversion. For the Cape Town inversion we showed that, where our reference inversion had aggregated prior flux estimates that were made more positive by the inversion, suggesting that the CABLE was overestimating the amount of CO2 uptake by the biota, when the alternative prior information was used, fluxes were made more negative by the inversion. As the posterior estimates were tending towards the same point, we could deduce that the best estimate was located somewhere between these two posterior fluxes. We could therefore restrict the best posterior flux estimate to be bounded between the solutions of these separate inversions. The assumed error correlation length for NEE fluxes played a major role in the spatial distribution of the posterior fluxes and in the size of the aggregated flux estimates, where ignoring these correlations led to posterior estimates more similar to the priors compared with the reference inversion. Apart from changing the prior flux products, making changes to the error correlation length in the NEE fluxes resulted in the greatest contribution to variability in the aggregated flux estimates between different inversions. Those cases where the prior information or NEE error correlations were altered resulted in greater variability between the aggregated fluxes of different inversions compared with the uncertainty around the posterior fluxes of the reference inversion. Solving for four separate weekly inversions resulted in similar estimates for the weekly fluxes compared with the single monthly inversion, while reducing computation time by up to 75 %. Solving for a mean weekly flux within a monthly inversion did result in differences in the aggregated fluxes compared with the reference inversion, but these differences were mainly during periods with data gaps. The uncertainty reduction from this inversion was almost double that of the reference inversion (47.2 % versus 25.6 %). Taking advantage of more observations to solve for one flux, such as allowing the inversion to solve for separate slow and fast components of the fossil fuel and NEE fluxes, as well as taking advantage of expected error correlation between fluxes of homogeneous biota, would reduce the uncertainty around the posterior fluxes. The sensitivity tests demonstrate that going one step further and assigning a probability distribution to these parameters, for example in a hierarchical Bayes approach, would lead to more useful estimates of the posterior fluxes and their uncertainties.

Local Adjustment of the Background Error Correlation for Surface Analyses over Complex Terrain

2005 ◽  
Vol 20 (2) ◽  
pp. 149-160 ◽  
Author(s):  
David T. Myrick ◽  
John D. Horel ◽  
Steven M. Lazarus
Keyword(s):  
Surface Temperature ◽  
Correlation Matrix ◽  
Analytic Solution ◽  
Weather Conditions ◽  
Optimal Interpolation ◽  
Unbiased Estimation ◽  
Background Error ◽  
Advanced Regional Prediction System ◽  
Regional Prediction ◽  
Error Correlation

Abstract The terrain between grid points is used to modify locally the background error correlation matrix in an objective analysis system. This modification helps to reduce the influence across mountain barriers of corrections to the background field that are derived from surface observations. This change to the background error correlation matrix is tested using an analytic case of surface temperature that encapsulates the significant features of nocturnal radiation inversions in mountain basins, which can be difficult to analyze because of locally sharp gradients in temperature. Bratseth successive corrections, optimal interpolation, and three-dimensional variational approaches are shown to yield exactly the same surface temperature analysis. Adding the intervening terrain term to the Bratseth approach led to solutions that match more closely the specified analytic solution. In addition, the convergence of the Bratseth solutions to the best linear unbiased estimation of the analytic solution is faster. The intervening terrain term was evaluated in objective analyses over the western United States derived from a modified version of the Advanced Regional Prediction System Data Assimilation System. Local adjustment of the background error correlation matrix led to improved surface temperature analyses by limiting the influence of observations in mountain valleys that may differ from the weather conditions present in adjacent valleys.

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