Scale-Dependent Inflation Algorithms for Ensemble Kalman Filters

This paper considers implications of different forms of the ensemble transformation in the ensemble square root filters (ESRFs) for the performance of ESRF-based data assimilation systems. It highlights the importance of using mean-preserving solutions for the ensemble transform matrix (ETM). The paper shows that an arbitrary mean-preserving ETM can be represented as a product of the symmetric solution and an orthonormal mean-preserving matrix. The paper also introduces a new flavor of ESRF, referred to as ESRF with mean-preserving random rotations. To investigate the performance of different solutions for the ETM in ESRFs, experiments with two small models are conducted. In these experiments, the performances of two mean-preserving solutions, two non-mean-preserving solutions, and a traditional ensemble Kalman filter with perturbed observations are compared. The experiments show a significantly better performance of the mean-preserving solutions for the ETM in ESRFs compared to non-mean-preserving solutions. They also show that applying the mean-preserving random rotations prevents the buildup of ensemble outliers in ESRF-based data assimilation systems.

Using radar observations, the performances of the ensemble square root filter (EnSRF) and an indirect three-dimensional variational (3DVar) data assimilation method were compared for a mesoscale convective system (MCS) that occurred in the Front Range of the Rocky Mountains, Colorado (USA). The results showed that the root mean square innovations (RMSIs) of EnSRF were lower than 3DVar for radar reflectivity and radial velocity and that the spread of EnSRF was generally consistent with its RMSIs. EnSRF substantially improved the analysis of the MCS compared with an experiment without radar data assimilation, and it produced a slight but noticeable improvement over 3DVar in terms of both coverage and intensity. Forecast results initiated from the final analysis revealed that EnSRF generally produced the best prediction of the MCS, with improved quantitative reflectivity and precipitation forecast skills. EnSRF also demonstrated better performance than 3DVar in the prediction of neighborhood probability for reflectivity at thresholds of 20 and 35 dBZ, which better matched the observed radar reflectivity in terms of both shape and extension. Additionally, the humidity, temperature, and wind fields were also improved by EnSRF; the largest error reduction was found in the water vapor field near the surface and at upper levels.

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A new multiscale, ensemble-based data assimilation (DA) method, multiscale local gain form ensemble transform Kalman filter (MLGETKF), is introduced. MLGETKF allows simultaneous update of multiple scales for both the ensemble mean and perturbations through assimilating all observations at once. MLGETKF performs DA in independent local volumes, which lends the algorithm a high degree of computational scalability. The multiscale analysis is enabled through the rapid creation of many pseudoensemble perturbations via a multiscale ensemble modulation procedure. The Kalman gain that is used to update the raw background ensemble mean and perturbations is based on this modulated ensemble, which intrinsically includes multiscale model space localization. Experiments with a noncycled statistical model show that the full background covariance estimated by MLGETKF more accurately resembles the shape of the true covariance than a scale-unaware localization. The mean analysis from the best-performing MLGETKF is statistically significantly more accurate than the best-performing scale-unaware LGETKF. The accuracy of the MLGETKF analysis is more sensitive to small-scale band localization radius than large-scale band. MLGETKF is further examined in a cycling DA context with a surface quasigeostrophic model. The root-mean-square potential temperature analysis error of the best-performing MLGETKF is 17.2% lower than that of the best-performing LGETKF. MLGETKF reduces analysis errors measured in kinetic energy spectra space by 30%–80% relative to LGETKF with the largest improvement at large scales. MLGETKF deterministic and ensemble mean forecasts are more accurate than LGETKF for full and large scales up to 5–6-day lead time and for small scales up to 3–4-day lead time, gaining ~12 h–1 day of predictability.

A hybrid parallel scheme for the ensemble square root filter (EnSRF) suitable for parallel assimilation of multiscale observations, including those from dense observational networks such as those of radar, is developed based on the domain decomposition strategy. The scheme handles internode communication through a message passing interface (MPI) and the communication within shared-memory nodes via Open Multiprocessing (OpenMP) threads. It also supports pure MPI and pure OpenMP modes. The parallel framework can accommodate high-volume remote-sensed radar (or satellite) observations as well as conventional observations that usually have larger covariance localization radii. The performance of the parallel algorithm has been tested with simulated and real radar data. The parallel program shows good scalability in pure MPI and hybrid MPI–OpenMP modes, while pure OpenMP runs exhibit limited scalability on a symmetric shared-memory system. It is found that in MPI mode, better parallel performance is achieved with domain decomposition configurations in which the leading dimension of the state variable arrays is larger, because this configuration allows for more efficient memory access. Given a fixed amount of computing resources, the hybrid parallel mode is preferred to pure MPI mode on supercomputers with nodes containing shared-memory cores. The overall performance is also affected by factors such as the cache size, memory bandwidth, and the networking topology. Tests with a real data case with a large number of radars confirm that the parallel data assimilation can be done on a multicore supercomputer with a significant speedup compared to the serial data assimilation algorithm.

The huge computational expense has been a main challenge while applying the sigma-point unscented Kalman filter (SPUKF) to a high-dimensional system. This study focuses on this issue and presents two methods to construct a reduced-rank sigma-point unscented Kalman filter (RRSPUKF). Both techniques employ the truncated singular value decomposition (TSVD) to factorize the covariance matrix and reduce its rank through truncation. The reduced-rank square root matrix is used to select the most important sigma points that can retain the main statistical features of the original sigma points. In the first technique, TSVD is applied on the covariance matrix constructed in the data space [RRSPUKF(D)], whereas in the second technique TSVD is applied on the covariance matrix constructed in the ensemble space [RRSPUKF(E)]. The two methods are applied to a realistic El Niño–Southern Oscillation (ENSO) prediction model [Lamont-Doherty Earth Observatory model, version 5 (LDEO5)] to assimilate the sea surface temperature (SST) anomalies. The results show that both the methods are more computationally efficient than the full-rank SPUKF, in spite of losing some estimation accuracy. When the truncation reaches a trade-off between cost expense and estimation accuracy, both methods are able to analyze the phase and intensity of all major ENSO events from 1971 to 2001 with comparable estimation accuracy. Furthermore, the RRSPUKF is compared against ensemble square root filter (EnSRF), showing that the overall analysis skill of RRSPUKF and EnSRF are comparable to each other, but the former is more robust than the latter.

Remote sensing data play a critical role in improving numerical weather prediction (NWP). However, the physical principles of radiation dictate that data voids frequently exist in physical space (e.g., subcloud area for satellite infrared radiance or no-precipitation region for radar reflectivity). Such data gaps impair the accuracy of initial conditions derived from data assimilation (DA), which has a negative impact on NWP. We use the barotropic vorticity equation to demonstrate the potential of deep learning augmented data assimilation (DDA), which involves reconstructing spatially complete pseudo-observation fields from incomplete observations and using them for DA. By training a convolutional autoencoder (CAE) with a long simulation at a coarse “forecast” resolution (T63), we obtained a deep learning approximation of the “reconstruction operator,” which maps spatially incomplete observations to a model state with full spatial coverage and resolution. The CAE was applied to an incomplete streamfunction observation (∼30% missing) from a high-resolution benchmark simulation and demonstrated satisfactory reconstruction performance, even when only very sparse (1/16 of T63 grid density) observations were used as input. When only spatially incomplete observations are used, the analysis fields obtained from ensemble square root filter (EnSRF) assimilation exhibit significant error. However, in DDA, when EnSRF takes in the combined data from the incomplete observations and CAE reconstruction, analysis error reduces significantly. Such gains are more pronounced with sparse observation and small ensemble size because the DDA analysis is much less sensitive to observation density and ensemble size than the conventional DA analysis, which is based solely on incomplete observations. Significance Statement Data assimilation plays a critical role in improving the skills of modern numerical weather prediction by establishing accurate initial conditions. However, unobservable regions are common in observation data, particularly those derived from remote sensing. The nonlinear relationship between data from observable regions and the physical state of unobservable regions may impede DA efficiency. As a result, we propose that deep learning be used to improve data assimilation in such cases by reconstructing a spatially complete first guess of the physical state with deep learning and then applying data assimilation to the reconstructed field. Such deep learning augmentation is found effective in improving the accuracy of data assimilation, especially for sparse observation and small ensemble size.

The National Center for Environmental Predictions (NCEP) has produced an ensemble meteorological reforecast product by using a fixed version of Global Forecast System (GFS) ensemble prediction system since 1 January 1979. The 15‐member ensemble product, with a global coverage at a 2.5° × 2.5° spatial resolution and a 14‐day lead time, has been used successfully by the River Forecast Centers of the National Weather Service (NWS) to produce basin scale precipitation and temperature ensemble forecasts in the US for several years now. This study evaluates the predictive skill of post‐processed ensemble forecasts based on GFS precipitation reforecast in China's Huai river basin. The evaluation is carried out in 15 sub‐areas of the Huai river basin and covers the 1/1/1981–31/12/2003 period. The Ensemble Pre‐Processing system version 3 (EPP3), developed at NWS, is used to develop joint probability distributions between forecasted ensemble mean precipitation and corresponding observations and to generate individual ensemble members that preserve space–time correlation of the observed precipitation data. Several statistical verification measures are used to quantify the goodness of fit between post‐processed (i.e. EPP3 processed) ensemble mean and observation and to assess the ensemble spread. Results indicate that the post‐processed forecasts have meaningful predictive skill for the first few days for ensemble daily precipitation forecasts. Predictive skill of ensemble forecasts of cumulative precipitation for lead times up to 14 days are significant. The forecast skill is highly dependent on seasonality, with relatively lower skills seen for wet summer season, when convective storm patterns dominate, as compared with other seasons. The predictive skill of the post‐processed ensemble precipitation is much better than the raw forecasts and the climatological ensemble forecasts. The results from this study suggest that the NCEP's GFS reforecasts can be a valuable resource for places other than the US. Copyright © 2012 John Wiley & Sons, Ltd.

In this study, a deterministic algorithm named Ensemble Square Root Filter (EnSRF), an algorithm significantly improved from the Ensemble Kalman Filter (EnKF), was used to integrate remotely sensed information (ASD spectral data, HJ-1 A/B CCD and Landsat-5 TM data) with a wheat (Triticum aestivum L.) growth model (WheatGrow). The analyzed values of model variables, leaf area index (LAI) and leaf nitrogen accumulation (LNA), were calculated based on EnSRF without perturbed measurements. Independent datasets were used to test EnSRF and the root mean square error (RMSE) values were 0.81 and 0.82 g m-2, with relative error (RE) values of 0.15 and 0.13, for LAI and LNA, respectively. RMSE values for LAI and LNA were 1.39 and 1.70 g m-2, respectively (RE, 0.28 and 0.34) based on EnKF, 1.17 and 1.80 g m-2 (RE, 0.24 and 0.35), respectively, based on the WheatGrow model alone, and 0.97 and 1.25 g m-2 (RE, 0.21 and 0.24), respectively, based on the remote sensing models. These results indicated that the LAI and LNA values based on EnSRF matched the measured values well compared with the EnKF, WheatGrow and remote sensing models. In addition, the predicted results are consistent with the temporal and spatial distribution of winter wheat growth status and grain yields in the study area, with RE values of less than 0.2 and 0.1 for LAI and LNA, respectively. These results provide an important approach for simulating winter wheat growth status based on combining remote sensing and crop growth models.

The initial condition accuracy is a major concern for tropical cyclone (TC) numerical forecast. The ensemble-based data assimilation techniques have shown great promise to initialize TC forecast. In addition to initial condition uncertainty, representing model errors (e.g. physics deficiencies) is another important issue in an ensemble forecasting system. To improve TC prediction from both deterministic and probabilistic standpoints, a Typhoon Ensemble Data Assimilation and Prediction System (TEDAPS) using an ensemble-based data assimilation scheme and a multi-physics approach based on Weather Research and Forecasting (WRF) model, has been developed in Shanghai Typhoon Institute and running realtime since 2015. Performance of TEDAPS in the prediction of track, intensity and associated disaster has been evaluated for the Western North Pacific TCs in the years of 2015–2018, and compared against the NCEP GEFS. TEDAPS produces markedly better intensity forecast by effectively reducing the weak biases and therefore the degree of underdispersion compared to GEFS. The errors of TEDAPS track forecasts are comparative with (slightly worse than) those of GEFS at longer (shorter) forecast leads. TEDAPS ensemble-mean exhibits advantage over deterministic forecast in track forecasts at long lead times, whereas this superiority is limited to typhoon or weaker TCs in intensity forecasts due to systematical underestimation. Four case-studies for three landfalling cyclones and one recurving cyclone demonstrate the capacities of TEDAPS in predicting some challenging TCs, as well as in capturing the forecast uncertainty and the potential threat from TC-associated hazards.

Bias correction of ensemble precipitation forecasts in the improvement of summer streamflow prediction skill

Forecasters routinely calibrate their confidence in model forecasts. Ensembles inherently estimate forecast confidence but are often underdispersive, and ensemble spread does not strongly correlate with ensemble-mean error. The misalignment between ensemble spread and skill motivates new methods for “forecasting forecast skill” so that forecasters can better utilize ensemble guidance. We have trained logistic regression and random forest models to predict the skill of composite reflectivity forecasts from the NSSL Warn-on-Forecast System (WoFS), a 3-km ensemble that generates rapidly updating forecast guidance for 0–6-h lead times. The forecast skill predictions are valid at 1-, 2-, or 3-h lead times within localized regions determined by the observed storm locations at analysis time. We use WoFS analysis and forecast output and NSSL Multi-Radar/Multi-Sensor composite reflectivity for 106 cases from the 2017 to 2021 NOAA Hazardous Weather Testbed Spring Forecasting Experiments. We frame the prediction task as a multiclassification problem, where the forecast skill labels are determined by averaging the extended fraction skill scores (eFSSs) for several reflectivity thresholds and verification neighborhoods and then converting to one of three classes based on where the average eFSS ranks within the entire dataset: POOR (bottom 20%), FAIR (middle 60%), or GOOD (top 20%). Initial machine learning (ML) models are trained on 323 predictors; reducing to 10 or 15 predictors in the final models only modestly reduces skill. The final models substantially outperform carefully developed persistence- and spread-based models and are reasonably explainable. The results suggest that ML can be a valuable tool for guiding user confidence in convection-allowing (and larger-scale) ensemble forecasts. Significance Statement Some numerical weather prediction (NWP) forecasts are more likely to verify than others. Forecasters often recognize situations where NWP output should be trusted more or less than usual, but objective methods for “forecasting forecast skill” are notably lacking for thunderstorm-scale models. Better estimates of forecast skill can benefit society through more accurate forecasts of high-impact weather. Machine learning (ML) provides a powerful framework for relating forecast skill to the characteristics of model forecasts and available observations over many previous cases. ML models can leverage these relationships to predict forecast skill for new cases in real time. We demonstrate the effectiveness of this approach to forecasting forecast skill using a cutting-edge thunderstorm prediction system and logistic regression and random forest models. Based on this success, we recommend the adoption of similar ML-based methods for other prediction models.

Top-down atmospheric inversion infers surface-atmosphere fluxes from spatially distributed observations of atmospheric compositions, which is a vital means for quantifying large-scale anthropogenic and natural emissions. In this study, we developed a Regional multi-Air Pollutant Assimilation System (RAPAS v1.0) based on the Weather Research and Forecasting/Community Multiscale Air Quality Modeling System (WRF/CMAQ) model, the three-dimensional variational (3DVAR) algorithm and the ensemble square root filter (EnSRF) algorithm. It is capable to simultaneously assimilate spatially distributed hourly in-situ measurements of CO, SO2, NO2, PM2.5 and PM10 concentrations to quantitatively optimize gridded emissions of CO, SO2, NOx, primary PM2.5 (PPM2.5) and coarse PM10 (PMC) on regional scale. RAPAS includes two subsystems, initial field assimilation (IA) subsystem and emission inversion (EI) subsystem, which are used to generate a "perfect" chemical initial condition (IC), and conduct inversions of anthropogenic emissions, respectively. A "two-step" inversion scheme is adopted in the EI subsystem in its each data assimilation (DA) window, in which the emission is inferred in the first step, and then, it is input into the CMAQ model to simulate the initial field of the next window, meanwhile, it is also transferred to the next window as the prior emission. The chemical IC is optimized through the IA subsystem, and the original emission inventory is only used in the first DA window. Besides, a "super-observation" approach is implemented based on optimal estimation theory to decrease the computational costs and observation error correlations and reduce the influence of representativeness errors. With this system, we estimated the emissions of CO, SO2, NOx, PPM2.5 and PMC in December 2016 over China using the corresponding nationwide surface observations. The 2016 Multi-resolution Emission Inventory for China (MEIC 2016) was used as the prior emission. The system was run from 26 November to 31 December, in which the IA subsystem was run in the first 5 days, and the EI subsystem was run in the following days. The optimized ICs at the first 5 days and the posterior emissions in December were evaluated against the assimilated and independent observations. Results showed that the root mean squared error (RMSE) decreased by 50.0–73.2%, and the correlation coefficient (CORR) increased to 0.78–0.92 for the five species compared to the simulations without 3DVAR. Additionally, the RMSE decreased by 40.1–56.3 %, and the CORR increased to 0.69–0.87 compared to the simulations without optimized emissions. For the whole mainland China, the uncertainties were reduced by 44.4 %, 45.0 %, 34.3 %, 51.8 % and 56.1 % for CO, SO2, NOx, PPM2.5 and PMC, respectively. Overall, compared to the prior emission (MEIC 2016), the posterior emissions increased by 129 %, 20 %, 5 %, and 95 % for CO, SO2, NOx and PPM2.5, respectively, indicating that there was significant underestimation in the MEIC inventory. The posterior PMC emissions, including anthropogenic and natural dust contributions, increased by 1045 %. A series of sensitivity tests were conducted with different inversion processes, prior emissions, prior uncertainties, and observation errors. Results showed that the "two-step" scheme clearly outperformed the simultaneous assimilation of ICs and emissions ("one-step" scheme), and the system is rather robust in estimating the emissions using the nationwide surface observations over China. Our study offers a useful tool for accurately quantifying multi-species anthropogenic emissions at large scales and near-real time.

We investigate the ability of hydrological multimodel ensemble predictions to enhance the skill of streamflow forecasts at short‐ to medium‐range timescales. To generate the multimodel ensembles, we implement a new statistical postprocessor, namely, quantile regression‐Bayesian model averaging (QR‐BMA). Quantile regression‐Bayesian model averaging uses quantile regression to bias correct the ensemble streamflow forecasts from the individual models and Bayesian model averaging to optimally combine their probability density functions. Additionally, we use an information‐theoretic measure, namely, conditional mutual information, to quantify the skill enhancements from the multimodel forecasts. We generate ensemble streamflow forecasts at lead times from 1 to 7 days using three hydrological models: (i) Antecedent Precipitation Index‐Continuous, (ii) Hydrology Laboratory‐Research Distributed Hydrologic Model, and (iii) Weather Research and Forecasting Hydrological modeling system. As forcing to the hydrological models, we use weather ensemble forecasts from the National Centers for Environmental Prediction 11‐member Global Ensemble Forecast System Reforecast version 2. The forecasting experiments are performed for four nested basins of the North Branch Susquehanna River, USA. We find that after bias correcting the streamflow forecasts from each model, their skill performance becomes comparable. We find that the multimodel ensemble forecasts have higher skill than the best single‐model forecasts. Furthermore, the skill enhancements obtained by the multimodel ensemble forecasts are found to be dominated by model diversity, rather than by increased ensemble size alone. This result, obtained using conditional mutual information, indicates that each hydrological model contributes additional information to enhance forecast skill. Overall, our results highlight benefits of hydrological multimodel forecasting for improving streamflow predictions.

Enhancing model parameterization with linearly constrained deep generative network for ensemble-based history matching