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- Author or Editor: Jidong Gao x
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Abstract
Generalized adjoint with modified discretization and generalized coarse-grain adjoint are derived for a vector system of equations that contains parameterized on/off switches. With vector examples, it is shown that the conventional adjoint minimization may have a convergence problem in multidimensional space. The problem can be solved by the generalized adjoint with modified discretization or by the generalized coarse-grain adjoint without modifying the traditional discretization in the forward model.
Abstract
Generalized adjoint with modified discretization and generalized coarse-grain adjoint are derived for a vector system of equations that contains parameterized on/off switches. With vector examples, it is shown that the conventional adjoint minimization may have a convergence problem in multidimensional space. The problem can be solved by the generalized adjoint with modified discretization or by the generalized coarse-grain adjoint without modifying the traditional discretization in the forward model.
Abstract
A new efficient dual-resolution (DR) data assimilation algorithm is developed based on the ensemble Kalman filter (EnKF) method and tested using simulated radar radial velocity data for a supercell storm. Radar observations are assimilated on both high-resolution and lower-resolution grids using the EnKF algorithm with flow-dependent background error covariances estimated from the lower-resolution ensemble. It is shown that the flow-dependent and dynamically evolved background error covariances thus estimated are effective in producing quality analyses on the high-resolution grid.
The DR method has the advantage of being able to significantly reduce the computational cost of the EnKF analysis. In the system, the lower-resolution ensemble provides the flow-dependent background error covariance, while the single-high-resolution forecast and analysis provides the benefit of higher resolution, which is important for resolving the internal structures of thunderstorms. The relative smoothness of the covariance obtained from the lower 4-km-resolution ensemble does not appear to significantly degrade the quality of analysis. This is because the cross covariance among different variables is of first-order importance for “retrieving” unobserved variables from the radar radial velocity data.
For the DR analysis, an ensemble size of 40 appears to be a reasonable choice with the use of a 4-km horizontal resolution in the ensemble and a 1-km resolution in the high-resolution analysis. Several sensitivity tests show that the DR EnKF system is quite robust to different observation errors. A 4-km thinned data resolution is a compromise that is acceptable under the constraint of real-time applications. A data density of 8 km leads to a significant degradation in the analysis.
Abstract
A new efficient dual-resolution (DR) data assimilation algorithm is developed based on the ensemble Kalman filter (EnKF) method and tested using simulated radar radial velocity data for a supercell storm. Radar observations are assimilated on both high-resolution and lower-resolution grids using the EnKF algorithm with flow-dependent background error covariances estimated from the lower-resolution ensemble. It is shown that the flow-dependent and dynamically evolved background error covariances thus estimated are effective in producing quality analyses on the high-resolution grid.
The DR method has the advantage of being able to significantly reduce the computational cost of the EnKF analysis. In the system, the lower-resolution ensemble provides the flow-dependent background error covariance, while the single-high-resolution forecast and analysis provides the benefit of higher resolution, which is important for resolving the internal structures of thunderstorms. The relative smoothness of the covariance obtained from the lower 4-km-resolution ensemble does not appear to significantly degrade the quality of analysis. This is because the cross covariance among different variables is of first-order importance for “retrieving” unobserved variables from the radar radial velocity data.
For the DR analysis, an ensemble size of 40 appears to be a reasonable choice with the use of a 4-km horizontal resolution in the ensemble and a 1-km resolution in the high-resolution analysis. Several sensitivity tests show that the DR EnKF system is quite robust to different observation errors. A 4-km thinned data resolution is a compromise that is acceptable under the constraint of real-time applications. A data density of 8 km leads to a significant degradation in the analysis.
Abstract
The impact of assimilating radar reflectivity and radial velocity data with an intermittent, cycled three-dimensional variational assimilation (3DVAR) system is explored using an idealized thunderstorm case and a real data case on 8 May 2003. A new forward operator for radar reflectivity is developed that uses a background temperature field provided by a numerical weather prediction model for automatic hydrometeor classification. Three types of experiments are performed on both the idealized and real data cases. The first experiment uses radial velocity data only, the second experiment uses both radial velocity and reflectivity data without hydrometeor classification, and the final experiment uses both radial velocity and reflectivity data with hydrometeor classification. All experiments advance the analysis state to the next observation time using a numerical model prediction, which is then used as the background for the next analysis. Results from both the idealized and real data cases show that, assimilating only radial velocity data, the model can reconstruct the supercell thunderstorm after several cycles, but the development of precipitation is delayed because of the well-known spinup problem. The spinup problem is reduced dramatically when assimilating reflectivity without hydrometeor classification. The analyses are further improved using the new reflectivity formulation with hydrometeor classification. This study represents a successful first effort in variational convective-scale data assimilation to partition hydrometeors using a background temperature field from a numerical weather prediction model.
Abstract
The impact of assimilating radar reflectivity and radial velocity data with an intermittent, cycled three-dimensional variational assimilation (3DVAR) system is explored using an idealized thunderstorm case and a real data case on 8 May 2003. A new forward operator for radar reflectivity is developed that uses a background temperature field provided by a numerical weather prediction model for automatic hydrometeor classification. Three types of experiments are performed on both the idealized and real data cases. The first experiment uses radial velocity data only, the second experiment uses both radial velocity and reflectivity data without hydrometeor classification, and the final experiment uses both radial velocity and reflectivity data with hydrometeor classification. All experiments advance the analysis state to the next observation time using a numerical model prediction, which is then used as the background for the next analysis. Results from both the idealized and real data cases show that, assimilating only radial velocity data, the model can reconstruct the supercell thunderstorm after several cycles, but the development of precipitation is delayed because of the well-known spinup problem. The spinup problem is reduced dramatically when assimilating reflectivity without hydrometeor classification. The analyses are further improved using the new reflectivity formulation with hydrometeor classification. This study represents a successful first effort in variational convective-scale data assimilation to partition hydrometeors using a background temperature field from a numerical weather prediction model.
Abstract
A hybrid three-dimensional ensemble–variational data assimilation (3DEnVAR) algorithm is developed based on the 3D variational data assimilation (3DVAR) and ensemble Kalman filter (EnKF) programs with the Advanced Regional Prediction System (ARPS). The method uses the extended control variable approach to combine the static and ensemble-derived flow-dependent forecast error covariances. The method is applied to the assimilation of simulated data from two radars for a supercell storm. Some sensitivity experiments are performed to answer questions about how flow-dependent covariance estimated from the forecast ensemble can be best used in the hybrid 3DEnVAR scheme. When the ensemble size is relatively small (with 5 or 10 ensemble members), it is found that experiments with a weaker weighting value for the ensemble covariance leads to better analysis results. Even when severe sampling errors exist, introducing ensemble-estimated covariances into the variational method still benefits the analysis. For reasonably large ensemble sizes (50–100 members), a stronger relative weighting (>0.8) for the ensemble covariance leads to better analyses from the hybrid 3DEnVAR. In addition, the sensitivity experiments also indicate that the best results are obtained when the number of the augmented control variables is a function of three spatial dimensions and ensemble members, and is the same for all analysis variables.
Abstract
A hybrid three-dimensional ensemble–variational data assimilation (3DEnVAR) algorithm is developed based on the 3D variational data assimilation (3DVAR) and ensemble Kalman filter (EnKF) programs with the Advanced Regional Prediction System (ARPS). The method uses the extended control variable approach to combine the static and ensemble-derived flow-dependent forecast error covariances. The method is applied to the assimilation of simulated data from two radars for a supercell storm. Some sensitivity experiments are performed to answer questions about how flow-dependent covariance estimated from the forecast ensemble can be best used in the hybrid 3DEnVAR scheme. When the ensemble size is relatively small (with 5 or 10 ensemble members), it is found that experiments with a weaker weighting value for the ensemble covariance leads to better analysis results. Even when severe sampling errors exist, introducing ensemble-estimated covariances into the variational method still benefits the analysis. For reasonably large ensemble sizes (50–100 members), a stronger relative weighting (>0.8) for the ensemble covariance leads to better analyses from the hybrid 3DEnVAR. In addition, the sensitivity experiments also indicate that the best results are obtained when the number of the augmented control variables is a function of three spatial dimensions and ensemble members, and is the same for all analysis variables.
Abstract
The assimilation of operational Doppler radar observations into convection-resolving numerical weather prediction models for very short-range forecasting represents a significant scientific and technological challenge. Numerical experiments over the past few years indicate that convective-scale forecasts are sensitive to the details of the data assimilation methodology, the quality of the radar data, the parameterized microphysics, and the storm environment. In this study, the importance of horizontal environmental variability to very short-range (0–1 h) convective-scale ensemble forecasts initialized using Doppler radar observations is investigated for the 4–5 May 2007 Greensburg, Kansas, tornadic thunderstorm event. Radar observations of reflectivity and radial velocity from the operational Doppler radar network at 0230 UTC 5 May 2007, during the time of the first large tornado, are assimilated into each ensemble member using a three-dimensional variational data assimilation system (3DVAR) developed at the Center for Analysis and Prediction of Storms (CAPS). Very short-range forecasts are made using the nonhydrostatic Advanced Regional Prediction System (ARPS) model from each ensemble member and the results are compared with the observations. Explicit three-dimensional environmental variability information is provided to the convective-scale ensemble using analyses from a 30-km mesoscale ensemble data assimilation system. Comparisons between convective-scale ensembles with initial conditions produced by 3DVAR using 1) background fields that are horizontally homogeneous but vertically inhomogeneous (i.e., have different vertical environmental profiles) and 2) background fields that are horizontally and vertically inhomogeneous are undertaken. Results show that the ensemble with horizontally and vertically inhomogeneous background fields provides improved predictions of thunderstorm structure, mesocyclone track, and low-level circulation track than the ensemble with horizontally homogeneous background fields. This suggests that knowledge of horizontal environmental variability is important to successful convective-scale ensemble predictions and needs to be included in real-data experiments.
Abstract
The assimilation of operational Doppler radar observations into convection-resolving numerical weather prediction models for very short-range forecasting represents a significant scientific and technological challenge. Numerical experiments over the past few years indicate that convective-scale forecasts are sensitive to the details of the data assimilation methodology, the quality of the radar data, the parameterized microphysics, and the storm environment. In this study, the importance of horizontal environmental variability to very short-range (0–1 h) convective-scale ensemble forecasts initialized using Doppler radar observations is investigated for the 4–5 May 2007 Greensburg, Kansas, tornadic thunderstorm event. Radar observations of reflectivity and radial velocity from the operational Doppler radar network at 0230 UTC 5 May 2007, during the time of the first large tornado, are assimilated into each ensemble member using a three-dimensional variational data assimilation system (3DVAR) developed at the Center for Analysis and Prediction of Storms (CAPS). Very short-range forecasts are made using the nonhydrostatic Advanced Regional Prediction System (ARPS) model from each ensemble member and the results are compared with the observations. Explicit three-dimensional environmental variability information is provided to the convective-scale ensemble using analyses from a 30-km mesoscale ensemble data assimilation system. Comparisons between convective-scale ensembles with initial conditions produced by 3DVAR using 1) background fields that are horizontally homogeneous but vertically inhomogeneous (i.e., have different vertical environmental profiles) and 2) background fields that are horizontally and vertically inhomogeneous are undertaken. Results show that the ensemble with horizontally and vertically inhomogeneous background fields provides improved predictions of thunderstorm structure, mesocyclone track, and low-level circulation track than the ensemble with horizontally homogeneous background fields. This suggests that knowledge of horizontal environmental variability is important to successful convective-scale ensemble predictions and needs to be included in real-data experiments.
Abstract
A variational scheme for the analysis of scalar variables is developed and compared to two-pass and three-pass versions of the Barnes analysis scheme. The variational scheme, appropriate for diagnostic studies, is similar to a previously developed variational method in that scalar gradient “observations”—derived directly from the scalar observations—are used in addition to the scalar observations themselves. The current scheme is different in that the cost function does not require analyses of the scalar field and its gradient; it simply requires scalar and gradient observations at their native locations. For the evaluation, randomly selected model gridpoint data are chosen to serve as pseudo-observations for the analysis schemes. By choosing appropriate model gridpoint data to serve as pseudo-observations, artificial data networks can be generated so as to mimic the spatial characteristics of real observational networks.
Results indicate that the proposed variational scheme is superior to both two-pass and three-pass Barnes schemes, increasingly so as the observations become more irregularly spaced. This is true even when the gradient information is not allowed to affect the variational analyses. When the observations are relatively sparse and irregularly distributed, further improvements in the variational analyses occur when the gradient information is properly included within the analysis scheme.
Abstract
A variational scheme for the analysis of scalar variables is developed and compared to two-pass and three-pass versions of the Barnes analysis scheme. The variational scheme, appropriate for diagnostic studies, is similar to a previously developed variational method in that scalar gradient “observations”—derived directly from the scalar observations—are used in addition to the scalar observations themselves. The current scheme is different in that the cost function does not require analyses of the scalar field and its gradient; it simply requires scalar and gradient observations at their native locations. For the evaluation, randomly selected model gridpoint data are chosen to serve as pseudo-observations for the analysis schemes. By choosing appropriate model gridpoint data to serve as pseudo-observations, artificial data networks can be generated so as to mimic the spatial characteristics of real observational networks.
Results indicate that the proposed variational scheme is superior to both two-pass and three-pass Barnes schemes, increasingly so as the observations become more irregularly spaced. This is true even when the gradient information is not allowed to affect the variational analyses. When the observations are relatively sparse and irregularly distributed, further improvements in the variational analyses occur when the gradient information is properly included within the analysis scheme.
Abstract
Velocity folding, or aliasing, is one of most significant impediments to the use of radial winds from Doppler weather radar. In this note, a variational algorithm is developed in which dealiasing is performed using wind gradient information. The key to the proposed method is that, by operating on gradients of velocity rather than on the velocity itself, aliasing ambiguities are readily identified and eliminated. The viability of the method is demonstrated by applying it to Weather Surveillance Radar-1988 Doppler (WSR-88D) observations from a winter-weather event and a tornadic supercell storm.
Abstract
Velocity folding, or aliasing, is one of most significant impediments to the use of radial winds from Doppler weather radar. In this note, a variational algorithm is developed in which dealiasing is performed using wind gradient information. The key to the proposed method is that, by operating on gradients of velocity rather than on the velocity itself, aliasing ambiguities are readily identified and eliminated. The viability of the method is demonstrated by applying it to Weather Surveillance Radar-1988 Doppler (WSR-88D) observations from a winter-weather event and a tornadic supercell storm.
Abstract
When on/off switches are triggered at discrete time levels by a threshold condition in a traditionally discretized model, the model solution is not continuously dependent on the initial state and this causes problems in tangent linearization and adjoint computations. It is shown in this paper that the problems can be avoided by introducing coarse-grain tangent linearization and adjoint without modifying the traditional discretization, although the coarse-grain gradient check can be performed only for finite perturbations.
Abstract
When on/off switches are triggered at discrete time levels by a threshold condition in a traditionally discretized model, the model solution is not continuously dependent on the initial state and this causes problems in tangent linearization and adjoint computations. It is shown in this paper that the problems can be avoided by introducing coarse-grain tangent linearization and adjoint without modifying the traditional discretization, although the coarse-grain gradient check can be performed only for finite perturbations.
Abstract
A diagnostic pressure equation is incorporated into a storm-scale three-dimensional variational data assimilation (3DVAR) system in the form of a weak constraint in addition to a mass continuity equation constraint (MCEC). The goal of this diagnostic pressure equation constraint (DPEC) is to couple different model variables to help build a more dynamic consistent analysis, and therefore improve the data assimilation results and subsequent forecasts. Observational System Simulation Experiments (OSSEs) are first performed to examine the impact of the pressure equation constraint on storm-scale radar data assimilation using an idealized tornadic thunderstorm simulation. The impact of MCEC is also investigated relative to that of DPEC. It is shown that DPEC can improve the data assimilation results slightly after a given period of data assimilation. Including both DPEC and MCEC yields the best data assimilation results. Sensitivity tests show that MCEC is not very sensitive to the choice of its weighting coefficients in the cost function, while DPEC is more sensitive and its weight should be carefully chosen. The updated 3DVAR system with DPEC is further applied to the 5 May 2007 Greensburg, Kansas, tornadic supercell storm case assimilating real radar data. It is shown that the use of DPEC can speed up the spinup of precipitation during the intermittent data assimilation process and also improve the follow-on forecast in terms of the general evolution of storm cells and mesocyclone rotation near the time of observed tornado.
Abstract
A diagnostic pressure equation is incorporated into a storm-scale three-dimensional variational data assimilation (3DVAR) system in the form of a weak constraint in addition to a mass continuity equation constraint (MCEC). The goal of this diagnostic pressure equation constraint (DPEC) is to couple different model variables to help build a more dynamic consistent analysis, and therefore improve the data assimilation results and subsequent forecasts. Observational System Simulation Experiments (OSSEs) are first performed to examine the impact of the pressure equation constraint on storm-scale radar data assimilation using an idealized tornadic thunderstorm simulation. The impact of MCEC is also investigated relative to that of DPEC. It is shown that DPEC can improve the data assimilation results slightly after a given period of data assimilation. Including both DPEC and MCEC yields the best data assimilation results. Sensitivity tests show that MCEC is not very sensitive to the choice of its weighting coefficients in the cost function, while DPEC is more sensitive and its weight should be carefully chosen. The updated 3DVAR system with DPEC is further applied to the 5 May 2007 Greensburg, Kansas, tornadic supercell storm case assimilating real radar data. It is shown that the use of DPEC can speed up the spinup of precipitation during the intermittent data assimilation process and also improve the follow-on forecast in terms of the general evolution of storm cells and mesocyclone rotation near the time of observed tornado.
Abstract
The utility of the anelastic vertical vorticity equation in a weak-constraint (least squares error) variational dual-Doppler wind analysis procedure is explored. The analysis winds are obtained by minimizing a cost function accounting for the discrepancies between observed and analyzed radial winds, errors in the mass conservation equation, errors in the anelastic vertical vorticity equation, and spatial smoothness constraints. By using Taylor’s frozen-turbulence hypothesis to shift analysis winds to observation points, discrepancies between radially projected analysis winds and radial wind observations can be calculated at the actual times and locations the data are acquired. The frozen-turbulence hypothesis is also used to evaluate the local derivative term in the vorticity equation. Tests of the analysis procedure are performed with analytical pseudo-observations of an array of translating and temporally decaying counterrotating updrafts and downdrafts generated from a Beltrami flow solution of the Navier–Stokes equations. The experiments explore the value added to the analysis by the vorticity equation constraint in the common scenario of substantial missing low-level data (radial wind observations at heights beneath 1.5 km are withheld from the analysis). Experiments focus on the sensitivity of the most sensitive analysis variable—the vertical velocity component—to values of the weighting coefficients, volume scan period, number of volume scans, and errors in the estimated frozen-turbulence pattern-translation components. Although the vorticity equation constraint is found to add value to many of these analyses, the analysis can become significantly degraded if estimates of the pattern-translation components are largely in error or if the frozen-turbulence hypothesis itself breaks down. However, tests also suggest that these negative impacts can be mitigated if data are available in a rapid-scan mode.
Abstract
The utility of the anelastic vertical vorticity equation in a weak-constraint (least squares error) variational dual-Doppler wind analysis procedure is explored. The analysis winds are obtained by minimizing a cost function accounting for the discrepancies between observed and analyzed radial winds, errors in the mass conservation equation, errors in the anelastic vertical vorticity equation, and spatial smoothness constraints. By using Taylor’s frozen-turbulence hypothesis to shift analysis winds to observation points, discrepancies between radially projected analysis winds and radial wind observations can be calculated at the actual times and locations the data are acquired. The frozen-turbulence hypothesis is also used to evaluate the local derivative term in the vorticity equation. Tests of the analysis procedure are performed with analytical pseudo-observations of an array of translating and temporally decaying counterrotating updrafts and downdrafts generated from a Beltrami flow solution of the Navier–Stokes equations. The experiments explore the value added to the analysis by the vorticity equation constraint in the common scenario of substantial missing low-level data (radial wind observations at heights beneath 1.5 km are withheld from the analysis). Experiments focus on the sensitivity of the most sensitive analysis variable—the vertical velocity component—to values of the weighting coefficients, volume scan period, number of volume scans, and errors in the estimated frozen-turbulence pattern-translation components. Although the vorticity equation constraint is found to add value to many of these analyses, the analysis can become significantly degraded if estimates of the pattern-translation components are largely in error or if the frozen-turbulence hypothesis itself breaks down. However, tests also suggest that these negative impacts can be mitigated if data are available in a rapid-scan mode.
