Forecast Error Covariance Research Articles

On‐line machine‐learning forecast uncertainty estimation for sequential data assimilation

AbstractQuantifying forecast uncertainty is a key aspect of state‐of‐the‐art numerical weather prediction and data assimilation systems. Ensemble‐based data assimilation systems incorporate state‐dependent uncertainty quantification based on multiple model integrations. However, this approach is demanding in terms of computations and development. In this work, a machine‐learning method is presented based on convolutional neural networks that estimates the state‐dependent forecast uncertainty represented by the forecast error covariance matrix using a single dynamical model integration. This is achieved by the use of a loss function that takes into account the fact that the forecast errors are heteroscedastic. The performance of this approach is examined within a hybrid data assimilation method that combines a Kalman‐like analysis update and the machine‐learning‐based estimation of a state‐dependent forecast error covariance matrix. Observing system simulation experiments are conducted using the Lorenz'96 model as a proof‐of‐concept. The promising results show that the machine‐learning method is able to predict precise values of the forecast covariance matrix in relatively high‐dimensional states. Moreover, the hybrid data assimilation method shows similar performance to the ensemble Kalman filter, outperforming it when the ensembles are relatively small.

Optimal reconciliation of hierarchical wind energy forecasts utilizing temporal correlation

Independent wind energy forecasts of a wind farm at different time horizons have limited accuracy, and they show disagreement despite relating to the same wind farm. The limited forecast accuracy is attributable to the insufficient information at a particular time horizon of the wind energy time series, whereas applying distinct forecasting methods to several time series of non-identical patterns at different time scales causes disagreement among forecasts. Mutual disagreement among less accurate forecasts negatively impacts the decision-making capabilities in associated power systems activities at distinct time scales. The configuration of time series expressing different time horizons at different levels of a non-overlapped hierarchically aggregated framework manifests a temporal hierarchy. Forecast combination through reconciliation of time series forecasts drawn at different hierarchical levels of temporal hierarchy using any state-of-the-art method facilitates the sharing of diverse information across the hierarchy; consequently, accuracy and mutual agreement of forecasts improve. Such benefits may be further enhanced by embedding intra- and inter-level forecast error correlations in the forecast reconciliation process. However, the forecast error covariance matrix of temporal hierarchy becomes a complex high-dimensional structure while accommodating intra- and inter-level correlations. Estimating such a matrix is challenging since the high-dimensional structure severely impedes the identifiability of model parameters. Besides, in the hierarchical forecast reconciliation process, the number of predictor variables is generally higher than the number of samples. This condition gives rise to a singular covariance matrix, making it non-invertible, and thus obstructs its parameter estimation. This work employs the MinT(shrinkage) covariance matrix estimator that considers all correlations and shrinks the non-diagonal components of the matrix toward zero to avert the complexity and, therefore, the non-identifiability. Additionally, the shrinkage parameter λ of MinT(shrinkage) conveniently obtains the invertible matrix. The case study validates that while incorporating the intra- and inter-level forecast error correlations, MinT(shrinkage) provides competitively accurate and mutually agreed forecasts over other reconciliation methods.

Representing uncertainty in limited-area data assimilating ocean models

A limited-area ocean prediction system is developed to acquire forecast error covariances related to uncertainty in atmospheric forcing and turbulent mixing using perturbed model parameters within an Ensemble Kalman Filter (EnKF). The system performs sequential data assimilation delivering realistic ocean state estimation and forecasts. It is initialised to observations using the EnKF, the hybrid-EnKF and Ensemble Optimal Interpolation (EnOI). It has higher resolution than the parent global model, includes tides and assimilates altimetric sea level anomaly to constrain the offshore mesoscale circulation. Dynamic ensemble spread introduced by parameter uncertainty shows agreement with error estimates obtained from forecast innovation statistics. Hybrid EnKF offers improvements to both EnKF and EnOI, having smaller analysis increments and forecast errors. The ensemble mean of the hybrid-EnKF is more accurate than any member due to the introduction of parameterised error which behaves as additive inflation in the hybrid EnKF system. This indicates the parameterisation acts as a non-linear filter for model forecast error growth. To demonstrate consistency, experiments are carried out for various regions that differ in their oceanographic situations. The hybrid EnKF data assimilation system enables practical use of dynamic ensembles that capture the flow dependent errors which are mixed with static or climatological covariances for situations where model error is systematically under-estimated.

Cross-temporal probabilistic forecast reconciliation: Methodological and practical issues

Forecast reconciliation is a post-forecasting process that involves transforming a set of incoherent forecasts into coherent forecasts which satisfy a given set of linear constraints for a multivariate time series. In this paper, we extend the current state-of-the-art cross-sectional probabilistic forecast reconciliation approach to encompass a cross-temporal framework, where temporal constraints are also applied. Our proposed methodology employs both parametric Gaussian and non-parametric bootstrap approaches to draw samples from an incoherent cross-temporal distribution. To improve the estimation of the forecast error covariance matrix, we propose using multi-step residuals, especially in the time dimension where the usual one-step residuals fail. To address high-dimensionality issues, we present four alternatives for the covariance matrix, where we exploit the two-fold nature (cross-sectional and temporal) of the cross-temporal structure, and introduce the idea of overlapping residuals. We assess the effectiveness of the proposed cross-temporal reconciliation approaches through a simulation study that investigates their theoretical and empirical properties and two forecasting experiments, using the Australian GDP and the Australian Tourism Demand datasets. For both applications, the optimal cross-temporal reconciliation approaches significantly outperform the incoherent base forecasts in terms of the continuous ranked probability score and the energy score. Overall, the results highlight the potential of the proposed methods to improve the accuracy of probabilistic forecasts and to address the challenge of integrating disparate scenarios while coherently taking into account short-term operational, medium-term tactical, and long-term strategic planning.

The Hydro-ABC model (Version 2.0): a simplified convective-scale model with moist dynamics

Abstract. The prediction of convection (in terms of position, timing, and strength) is important to achieve for high-resolution weather forecasting. This problem requires not only good convective-scale models, but also data assimilation systems that give initial conditions which neither improperly hinder nor improperly hasten convection in the ensuing forecasts. Solving this problem is difficult and expensive using operational-scale numerical weather prediction systems, and so a simplified model of convective-scale flow is under development (called the “ABC model”). This paper extends the existing ABC model of dry convective-scale flow to include mixing ratios of vapour and condensate phases of water. The revised model is called “Hydro-ABC”. Hydro-ABC includes transport of the vapour and condensate mixing ratios within a dynamical core, and it transitions between these two phases via a micro-physics scheme. A saturated mixing ratio is derived from model quantities, which helps determine whether evaporation or condensation happens. Latent heat is exchanged with the buoyancy variable (ABC's potential-temperature-like variable) in such a way to conserve total energy, where total energy is the sum of dry energy and latent heat. The model equations are designed to conserve the domain-total mass, water, and energy. An example numerical model integration is performed and analysed, which shows the development of a realistic looking anvil cloud and excitation of inertio-gravity and acoustic modes over a wide range of frequencies. This behaviour means that Hydro-ABC is a sufficiently challenging model which will allow experimentation with innovative data assimilation strategies in future work. An ensemble of Hydro-ABC integrations is performed in order to study the possible forecast error covariance statistics (knowledge of which is necessary for data assimilation). These show patterns that are dependent on the presence of convective activity (at any model's vertical column), thus giving a taste of flow-dependent error statistics. Candidate indicators/harbingers of convection are also evaluated (namely relative humidity, hydrostatic imbalance, horizontal divergence, convective available potential energy, convective inhibition, vertical wind, and the condensate mixing ratio), some of which appear to be reliable diagnostics concerning the presence of convection. These diagnostics will be useful in the selection of the relevant forecast error covariance statistics when data assimilation for Hydro-ABC is developed.

Predicting power ramps from joint distributions of future wind speeds

Abstract. Power ramps are sudden changes in turbine power and must be accurately predicted to minimize costly imbalances in the electrical grid. Doing so requires reliable wind speed forecasts, which can be obtained from ensembles of physical numerical weather prediction (NWP) models through statistical postprocessing. Since the probability of a ramp event depends jointly on the wind speed distributions forecasted at multiple future times, these postprocessing methods must not only correct each individual forecast but also estimate the temporal dependencies among them. Typically though, crucial dependencies are adopted directly from the raw ensemble, and the postprocessed forecast is limited to the tens of members computationally feasible for an NWP model. We extend statistical postprocessing to include temporal dependencies using novel multivariate Gaussian regression models that forecast 24-dimensional distributions of next-day hourly wind speeds at three offshore wind farms. The continuous joint distribution forecast is postprocessed from an NWP ensemble using flexible generalized additive models for the components of its mean vector μ and for parameters defining the forecast error covariance matrix Σ. Modeling these parameters on predictors which characterize the empirical joint distribution of the NWP ensemble allows forecasts for each hour and their temporal dependencies to be adjusted in one step. Wind speed ensembles of any size can be simulated from the postprocessed joint distribution and transformed into power for computing high-resolution ramp predictions that outperform state-of-the-art reference methods.

Super-resolution data assimilation

Increasing model resolution can improve the performance of a data assimilation system because it reduces model error, the system can more optimally use high-resolution observations, and with an ensemble data assimilation method the forecast error covariances are improved. However, increasing the resolution scales with a cubical increase of the computational costs. A method that can more effectively improve performance is introduced here. The novel approach called “Super-resolution data assimilation” (SRDA) is inspired from super-resolution image processing techniques and brought to the data assimilation context. Starting from a low-resolution forecast, a neural network (NN) emulates the fields to high-resolution, assimilates high-resolution observations, and scales it back up to the original resolution for running the next model step. The SRDA is tested with a quasi-geostrophic model in an idealized twin experiment for configurations where the model resolution is twice and four times lower than the reference solution from which pseudo-observations are extracted. The assimilation is performed with an Ensemble Kalman Filter. We show that SRDA outperforms both the low-resolution data assimilation approach and a version of SRDA with cubic spline interpolation instead of NN. The NN’s ability to anticipate the systematic differences between low- and high-resolution model dynamics explains the enhanced performance, in particular by correcting the difference of propagation speed of eddies. With a 25-member ensemble at low resolution, the SRDA computational overhead is 55% and the errors reduce by 40%, making the performance very close to that of the high-resolution system (52% of error reduction) that increases the cost by 800%. The reliability of the ensemble system is not degraded by SRDA.

Integrating Recurrent Neural Networks With Data Assimilation for Scalable Data‐Driven State Estimation

AbstractData assimilation (DA) is integrated with machine learning in order to perform entirely data‐driven online state estimation. To achieve this, recurrent neural networks (RNNs) are implemented as pretrained surrogate models to replace key components of the DA cycle in numerical weather prediction (NWP), including the conventional numerical forecast model, the forecast error covariance matrix, and the tangent linear and adjoint models. It is shown how these RNNs can be initialized using DA methods to directly update the hidden/reservoir state with observations of the target system. The results indicate that these techniques can be applied to estimate the state of a system for the repeated initialization of short‐term forecasts, even in the absence of a traditional numerical forecast model. Further, it is demonstrated how these integrated RNN‐DA methods can scale to higher dimensions by applying domain localization and parallelization, providing a path for practical applications in NWP.

Casting vector time series: algorithms for forecasting, imputation, and signal extraction

Recursive algorithms, based upon the nested structure of Toeplitz covariance matrices arising from stationary processes, are presented for the efficient computation of multi-step ahead forecast error covariances for nonstationary vector time series. Further, we discuss time reversal to forecast the past, and a filtering algorithm for imputation of missing values. These quantities are required to quantify multi-step ahead forecast error and signal extraction error. An information filter is presented, which provides imputations for arbitrary patterns of missing values (such as ragged edge patterns occurring in mixed frequency data). The methods are applied to multivariate retail data exhibiting trend dynamics and seasonality.

The Maximum Likelihood Ensemble Filter with State Space Localization

AbstractA new method for ensemble data assimilation that incorporates state space covariance localization, global numerical optimization, and implied Bayesian inference is presented. The method is referred to as the MLEF with state space localization (MLEF-SSL) due to its similarity with the maximum likelihood ensemble filter (MLEF). One of the novelties introduced in MLEF-SSL is the calculation of a reduced-rank localized forecast error covariance using random projection. The Hessian preconditioning is accomplished via Cholesky decomposition of the Hessian matrix, accompanied with solving triangular system of equations instead of directly inverting matrices. For ensemble update, the MLEF-SSL system employs resampling of posterior perturbations. The MLEF-SSL was applied to Lorenz model II and compared to ensemble Kalman filter with state space localization and to MLEF with observation space localization. The observations include linear and nonlinear observation operators, each applied to integrated and point observations. Results indicate improved performance of MLEF-SSL, particularly in assimilation of integrated nonlinear observations. Resampling of posterior perturbations for an ensemble update also indicates a satisfactory performance. Additional experiments were conducted to examine the sensitivity of the method to the rank of random matrix and to compare it to truncated eigenvectors of the localization matrix. The two methods are comparable in application to low-dimensional Lorenz model, except that the new method outperforms the truncated eigenvector method in case of severe rank reduction. The random basis method is simple to implement and may be more promising for realistic high-dimensional applications.

Assessing the performance of an ocean observing, analysis and forecast System for the Mid-Atlantic Bight using array modes

The efficacy of an ocean observing, analysis, and forecasting system for the Mid-Atlantic Bight and the Gulf of Maine is explored using the concept of array modes. The analysis-forecast system is based on a triply nested configuration of the Regional Ocean Modeling System (ROMS) in conjunction with 4-dimensional variational (4D-Var) data assimilation. The array modes identify the degrees of freedom (df) of the signal and of the noise resolved by the observations, and are used here to quantify the extent to which the existing network of platforms and instruments are able to observe the ocean across different dynamical regimes ranging from quasi-geostrophic through the mesoscale and down to the sub-mesoscale. The ocean observing system includes the U.S. National Science Foundation’s Ocean Observatories Initiative Pioneer Array. In general, it is found that the df of the signal are largely associated with in situ observations from the Pioneer Array. On the other hand, a combination of satellite remote sensing and in situ observations potentially contribute to the df of the noise associated with uncertainties in the measurements. The array modes also provide information about the reduction in the expected analysis and forecast error covariance due to assimilating the observations. Here too observations from the Pioneer Array are found to significantly influence the veracity of the analyses and forecasts, and the circulation is instrumental in propagating observational information to other parts of the model domain. An approach is presented in which the array modes are used to quantify the impact of data assimilation on the expected forecast error covariance of forecasts initialized from the 4D-Var ocean state estimates. The advantage of this approach over others in common use is that it is independent of forecast error norm and circumvents the need for generating potentially large and costly ensembles.

Refinement of CrIS channel selection for global data assimilation and its impact on the global weather forecast

AbstractTo maximize the contribution of the Cross-track Infrared Sounder (CrIS) measurements to the global weather forecasting, we attempt to choose the CrIS channels to be assimilated in the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS). From pre-selected 431 CrIS channels, 207 channels are newly selected using a one-dimensional variational (1D-Var) approach where the channel score index (CSI) is used as a figure of merit. Newly selected 207 channels consist of 85 temperature, 49 water vapor, and 73 surface channels, respectively. In addition, to examine how the channels are selected if the forecast error covariance is differently defined depending on the latitudinal regions (i.e., Northern and Southern Hemispheres, and tropics), the same selection process is carried out repeatedly using three regional forecast error covariances. From three regional channel sets, two-channel sets are made for the global data assimilation. One channel set is made with 134 channels overlapped between three regional channel sets. Another channel set consists of 277 channels that is the sum of three regional channel sets. In the global trial experiments, the global CrIS 207 channels have a significant positive forecast impact in terms of the improvement of GFS global forecasting, as compared with the forecasts with the operational 100 channels as well as the overlapped 134 and the union 277 channel sets. The improved forecast is mainly due to the additional temperature/water vapor channels of the global CrIS 207 channels that are selected optimally based on the global forecast error of operational GFS.

Empirical determination of the covariance of forecast errors: An empirical justification and reformulation of hybrid covariance models

AbstractDuring the last decade, the replacement of static climatological forecast error covariance models with hybrid error covariance models that linearly combine localised ensemble covariances with static climatological error covariances has led to significant forecast improvements at several major forecasting centres. Here, a deeper understanding of why the hybrid's superficially ad hoc mix of ensemble‐based and climatological covariances yields such significant improvements is pursued. In practice, ensemble covariances are not equal to the true flow‐dependent forecast error covariance matrix. Here, the relationship between actual forecast error covariance and the corresponding ensemble covariance is empirically demonstrated. Using a simplified global circulation model and the local ensemble transform Kalman filter (LETKF), the covariance of the set of actual forecast errors corresponding to ensemble covariances close to a fixed target value is computed. By doing this for differing target values, an estimate of the actual forecast error covariance as a function of ensemble covariance is obtained. A demonstration that the hybrid is a much better approximation to this estimate than either the static climatological covariance or the localised ensemble covariance is given. The empirical estimate has two features that current hybrid error covariance models fail to represent: (i) The weight given to the static covariance matrix is an increasing function of the horizontal separation distance of the covarying model variables, and (ii) for small ensemble sizes and ensemble covariances near zero but negative, the actual forecast error covariance is a decreasing function of increasing ensemble covariance. While the first finding has been anticipated by other authors, the second finding has not been anticipated, as far as the authors are aware. Here, (ii) is hypothesised to be a consequence of spurious sample correlations and variances associated with reduced ensembles. Consistent with this hypothesis, the non‐monotonicity of this relationship is almost eliminated by quadrupling the ensemble size.

Assimilation of lidar planetary boundary layer height observations

Abstract. Lidar backscatter and wind retrievals of the planetary boundary layer height (PBLH) are assimilated into 22-hourly forecasts from the NASA Unified – Weather and Research Forecast (NU-WRF) model during the Plains Elevated Convection at Night (PECAN) campaign on 11 July 2015 in Greensburg, Kansas, using error statistics collected from the model profiles to compute the necessary covariance matrices. Two separate forecast runs using different PBL physics schemes were employed, and comparisons with six independent radiosonde profiles were made for each run. Both of the forecast runs accurately predicted the PBLH and the state variable profiles within the planetary boundary layer during the early morning, and the assimilation had a small impact during this time. In the late afternoon, the forecast runs showed decreased accuracy as the convective boundary layer developed. However, assimilation of the Doppler lidar PBLH observations was found to improve the temperature and V-velocity profiles relative to independent radiosonde profiles. Water vapor was overcorrected, leading to increased differences with independent data. Errors in the U velocity were made slightly larger. The computed forecast error covariances between the PBLH and state variables were found to rise in the late afternoon, leading to the larger improvements in the afternoon. This work represents the first effort to assimilate PBLH into forecast states using ensemble methods.

On the behavior of ocean analysis and forecast error covariance in the presence of baroclinic instability

The properties of the expected analysis and forecast error covariance matrices are explored using a novel method based on the tangent linearization and adjoint of a 4-dimensional variational (4D-Var) data assimilation system. The method is applied to the mesoscale circulation that develops in the presence of a baroclinically unstable mid-latitude ocean temperature front using a series of paternal twin experiments that employ both strong and weak constraint 4D-Var. Adopting the traditional view of Empirical Orthogonal Functions (EOFs) of a covariance matrix as the semi-major axes of a multi-dimensional hyper-ellipsoid, variations in the volume of the analysis and forecast error hyper-ellipsoids are explored which provides information about the flow of probability through state–space. The complementary variations in the expected total variance of the covariance matrix are also investigated. Two different kinds of behavior are identified that are associated with either the demise or growth of baroclinic instabilities. In both cases, the volume of the hyper-ellipsoid decreases during the 4D-Var analysis cycle. During the subsequent forecasts, the volume of the forecast error hyper-ellipsoid initially continues to collapse under both scenarios. During this time, the hyper-ellipsoid becomes increasingly elongated along some of the semi-major axes as forecast errors grow in preferential directions. Growth in these directions is controlled by the most unstable error modes, and projection of forecast error on to the precursors of these modes has been shown previously to be characterized by upscale energy transfer and non-normal processes. For the case of the growing wave, the forecast error hyper-ellipsoid continues to collapse through to the end of the forecast period. However, for the decaying wave, the hyper-ellipsoid may undergo expansion at longer forecast lead times.

An EnOI‐Based Data Assimilation System With DART for a High‐Resolution Version of the CESM2 Ocean Component

AbstractAn ensemble optimal interpolation (EnOI) data assimilation system for a high‐resolution (0.1° horizontal) version of the Community Earth System Model Version 2 (CESM2) ocean component is presented. For this purpose, a new version of the Data Assimilation Research Testbed (DART Manhattan) that enables large‐state data assimilation by distributing state vector information across multiple processors at high resolution is used. The EnOI scheme uses a static (but seasonally varying) 84‐member ensemble of precomputed perturbations to approximate samples from the forecast error covariance and utilizes a single model integration to estimate the forecast mean. Satellite altimetry and sea surface temperature observations along with in situ temperature and salinity observations are assimilated. This new data assimilation framework is then used to produce a global high‐resolution retrospective analysis for the 2005–2016 period. Not surprisingly, the assimilation is shown to generally improve the time‐mean ocean state estimate relative to an identically forced ocean model simulation where no observations are ingested. However, diminished improvements are found in undersampled regions. Lack of adequate salinity observations in the upper ocean actually results in deterioration of salinity there. The EnOI scheme is found to provide a practical and cost‐effective alternative to the use of an ensemble of forecasts.

Dynamical Structures of Cross-Domain Forecast Error Covariance of a Simulated Tropical Cyclone in a Convection-Permitting Coupled Atmosphere–Ocean Model

AbstractUnderstanding the dynamics of the flow-dependent forecast error covariance across the air–sea interface is beneficial toward revealing the potential influences of strongly coupled data assimilation on tropical cyclone (TC) initialization in coupled models, and the fundamental dynamics associated with TC air–sea interactions. A 200-member ensemble of convection-permitting forecasts from a coupled atmosphere–ocean regional model is used to investigate the forecast error covariance across the oceanic and atmospheric domains during the rapid intensification of Hurricane Florence (2018). Forecast uncertainties in both atmospheric and oceanic domains, from an Eulerian perspective, increase with forecast lead time, mainly from TC displacement errors. In a storm-relative framework, the ensemble forecast uncertainties in both domains are predominantly caused by differences in the simulated storm intensity and structure. The largest ensemble spread in the atmospheric pressure, temperature, and wind fields can be found within the TC inner-core region. Alternatively, the largest ensemble spread in the upper-ocean currents and temperature fields are located along the cold wake behind the storm. Cross-domain ensemble correlations between simulated atmospheric (oceanic) observations and oceanic (atmospheric) state variables in the storm-relative coordinates are highly anisotropic, variable dependent, and ultimately driven by the dynamics of TC air–sea interactions. Meaningful and dynamically consistent cross-domain ensemble correlations suggest that it is possible to use atmospheric and oceanic observations to simultaneously update state variables associated with the coupled ocean–atmosphere prediction of TCs using strongly coupled data assimilation. Sensitivity experiments demonstrate that at least 60–80 ensemble members are required to represent physically consistent cross-domain correlations and minimize sampling errors.

Divergence of the Ensemble Transform Kalman Filter (LETKF) by Nonlocal Observations

Ensemble Kalman filters are powerful tools to merge model dynamics and observation data. For large system models, they are known to diverge due to subsampling errors at small ensemble size and thus possible spurious correlations in forecast error covariances. The Local Ensemble Transform Kalman filter (LETKF) remedies these disadvantages by localisation in observation space. However, its application to nonlocal observations is still under debate since it is still not clear how to optimally localize nonlocal observations. The present work studies intermittent divergence of filter innovations and shows that it increases forecast errors. Nonlocal observations enhance such innovation divergence under certain conditions, whereas similar localisation radius and sensitivity function width of nonlocal observations minimizes the divergence rate. The analysis of the LETKF reveals inconsistencies in the assimilation of observed and unobserved model grid points which may yield detrimental effects. These inconsistencies inter alia indicate that the localisation radius should be larger than the sensitivity function width if spatially synchronised system activity is expected. Moreover, the shift of observation power from observed to unobserved grid points hypothesised in the context of catastrophic filter divergence is supported for intermittent innovation divergence. Further possible mechanisms yielding such innovation divergence are ensemble member alignment and a novel covariation between background perturbations in location and observation space.

Climate Model State Estimation Using Variants of EnKF Coupled Data Assimilation

Abstract Data assimilation (DA) experiments are performed to assess impacts of observations in climate model state estimation through the cross-domain ocean–atmosphere forecast error covariances (cross covariances). Specifically, we explore strongly and weakly coupled DA variants using the Climate Analysis Forecast Ensemble (CAFE) system. This comprises 96 ensemble members of the Geophysical Fluid Dynamics Laboratory (GFDL) CM2.1 climate model assimilating observational data from the ocean, atmosphere, and sea ice realms with the ensemble Kalman filter (EnKF). Sequences of atmospheric synoptic time-scale coupled forecasts (7 days) are carried out with model consistent initialization. Unassimilated forward-independent observations are used to quantify forecast innovation error-growth rates. The results show benefit for the slow components of the atmosphere and ocean subsurface when strongly coupling ocean observations to the atmosphere. In the present system, projecting fast atmospheric observations onto the ocean subsurface through the cross covariances benefits the oceanic and atmospheric near-surface layers; however, this leads to deterioration in the ocean subsurface. Particular variants of coupled DA are able to constrain the ocean and atmosphere. The forecasts initialized with these variants have predictability at intraseasonal time scales. Errors associated with the dominant intraseasonal mode of variability, the Madden–Julian oscillation (MJO), are decomposed into normal mode functions. Consistent with recent studies showing large MJO events are concurrent with rapid error growth associated with nonlinear interactions, we find a clear relationship between the strength of a given MJO event and the related forecast innovations. Our results demonstrate consistent system behavior in relation to capturing real-world disturbances that affect climate predictability.

Evaluating Ensemble Kalman Filter Analyses of Severe Hailstorms on 8 May 2017 in Colorado: Effects of State Variable Updating and Multimoment Microphysics Schemes on State Variable Cross Covariances

Abstract Ensemble Kalman filter (EnKF) analyses of the storms associated with the 8 May 2017 Colorado severe hail event using either the Milbrandt and Yau (MY) or the NSSL double-moment bulk microphysics scheme in the forecast model are evaluated. With each scheme, two experiments are conducted in which the reflectivity (Z) observations update in addition to dynamic and thermodynamic variables: 1) only the hydrometeor mixing ratios or 2) all microphysical variables. With fewer microphysical variables directly constrained by the Z observations, only updating hydrometeor mixing ratios causes the forecast error covariance structure to become unreliable, and results in larger errors in the analysis. Experiments that update all microphysical variables produce analyses with the lowest Z root-mean-square innovations; however, comparing the estimated hail size against hydrometeor classification algorithm output suggests that further constraint from observations is needed to more accurately estimate surface hail size. Ensemble correlation analyses are performed to determine the impact of hail growth assumptions in the MY and NSSL schemes on the forecast error covariance between microphysical and thermodynamic variables. In the MY scheme, Z is negatively correlated with updraft intensity because the strong updrafts produce abundant small hail aloft. The NSSL scheme predicts the growth of large hail aloft; consequently, Z is positively correlated with storm updraft intensity and hail state variables. Hail production processes are also shown to alter the background error covariance for liquid and frozen hydrometeor species. Results in this study suggest that EnKF analyses are sensitive to the choice of MP scheme (e.g., the treatment of hail growth processes).