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Abstract
Particle filters are ensemble-based assimilation schemes that, unlike the ensemble Kalman filter, employ a fully nonlinear and non-Gaussian analysis step to compute the probability distribution function (pdf) of a system’s state conditioned on a set of observations. Evidence is provided that the ensemble size required for a successful particle filter scales exponentially with the problem size. For the simple example in which each component of the state vector is independent, Gaussian, and of unit variance and the observations are of each state component separately with independent, Gaussian errors, simulations indicate that the required ensemble size scales exponentially with the state dimension. In this example, the particle filter requires at least 1011 members when applied to a 200-dimensional state. Asymptotic results, following the work of Bengtsson, Bickel, and collaborators, are provided for two cases: one in which each prior state component is independent and identically distributed, and one in which both the prior pdf and the observation errors are Gaussian. The asymptotic theory reveals that, in both cases, the required ensemble size scales exponentially with the variance of the observation log likelihood rather than with the state dimension per se.
Abstract
Particle filters are ensemble-based assimilation schemes that, unlike the ensemble Kalman filter, employ a fully nonlinear and non-Gaussian analysis step to compute the probability distribution function (pdf) of a system’s state conditioned on a set of observations. Evidence is provided that the ensemble size required for a successful particle filter scales exponentially with the problem size. For the simple example in which each component of the state vector is independent, Gaussian, and of unit variance and the observations are of each state component separately with independent, Gaussian errors, simulations indicate that the required ensemble size scales exponentially with the state dimension. In this example, the particle filter requires at least 1011 members when applied to a 200-dimensional state. Asymptotic results, following the work of Bengtsson, Bickel, and collaborators, are provided for two cases: one in which each prior state component is independent and identically distributed, and one in which both the prior pdf and the observation errors are Gaussian. The asymptotic theory reveals that, in both cases, the required ensemble size scales exponentially with the variance of the observation log likelihood rather than with the state dimension per se.
Abstract
Tropical cyclone formation over the eastern Pacific during 2005 and 2006 was examined using primarily global operational analyses from the National Centers for Environmental Prediction. This paper represents a “vortex view” of genesis, adding to previous work on tropical cyclone formation associated with tropical waves. Between 1 July and 30 September during 2005 and 2006, vortices at 900 hPa were tracked and vortex-following diagnostic quantities were computed. Vortices were more abundant during periods of an enhanced “Hadley” circulation with monsoon westerlies around 10°N in the lower troposphere. This zonally confined Hadley circulation was significantly stronger during the genesis of developing vortices. Developing vortices were stronger at the outset, with a deeper potential vorticity maximum, compared to nondeveloping vortices. This implies that developing disturbances were selected early on by favorable synoptic-scale features.
The characteristic time-mean reversal of the meridional gradient of absolute vorticity in the lower troposphere was found to nearly vanish when the aggregate contribution of strong vortices was removed from the time-mean vorticity. This finding implies that it is difficult to unambiguously attribute development to a preexisting enhancement of vorticity on the synoptic scale. The time-mean enhancement of cyclonic vorticity primarily results from the accumulated effect of vortices. It is suggested that horizontal deformation in the background state helps distinguish developing vortices from nondevelopers, and also biases the latitude of development poleward of the climatological ITCZ axis.
Abstract
Tropical cyclone formation over the eastern Pacific during 2005 and 2006 was examined using primarily global operational analyses from the National Centers for Environmental Prediction. This paper represents a “vortex view” of genesis, adding to previous work on tropical cyclone formation associated with tropical waves. Between 1 July and 30 September during 2005 and 2006, vortices at 900 hPa were tracked and vortex-following diagnostic quantities were computed. Vortices were more abundant during periods of an enhanced “Hadley” circulation with monsoon westerlies around 10°N in the lower troposphere. This zonally confined Hadley circulation was significantly stronger during the genesis of developing vortices. Developing vortices were stronger at the outset, with a deeper potential vorticity maximum, compared to nondeveloping vortices. This implies that developing disturbances were selected early on by favorable synoptic-scale features.
The characteristic time-mean reversal of the meridional gradient of absolute vorticity in the lower troposphere was found to nearly vanish when the aggregate contribution of strong vortices was removed from the time-mean vorticity. This finding implies that it is difficult to unambiguously attribute development to a preexisting enhancement of vorticity on the synoptic scale. The time-mean enhancement of cyclonic vorticity primarily results from the accumulated effect of vortices. It is suggested that horizontal deformation in the background state helps distinguish developing vortices from nondevelopers, and also biases the latitude of development poleward of the climatological ITCZ axis.
Abstract
The quality of convective-scale ensemble forecasts, initialized from analysis ensembles obtained through the assimilation of radar observations using an ensemble Kalman filter (EnKF), is investigated for cases whose behaviors span supercellular, linear, and multicellular organization. This work is the companion to , which focused on the quality of analyses during the 60-min analysis period. Here, the focus is on 30-min ensemble forecasts initialized at the end of that period. As in , the Weather Research and Forecasting (WRF) model is employed as a simplified cloud model at 2-km horizontal grid spacing. Various observation-space and state-space verification metrics, computed both for ensemble means and individual ensemble members, are employed to assess the quality of ensemble forecasts comparatively across cases. While the cases exhibit noticeable differences in predictability, the forecast skill in each case, as measured by various metrics, decays on a time scale of tens of minutes. The ensemble spread also increases rapidly but significant outlier members or clustering among members are not encountered. Forecast quality is seen to be influenced to varying degrees by the respective initial soundings. While radar data assimilation is able to partially mitigate some of the negative effects in some situations, the supercell case, in particular, remains difficult to predict even after 60 min of data assimilation.
Abstract
The quality of convective-scale ensemble forecasts, initialized from analysis ensembles obtained through the assimilation of radar observations using an ensemble Kalman filter (EnKF), is investigated for cases whose behaviors span supercellular, linear, and multicellular organization. This work is the companion to , which focused on the quality of analyses during the 60-min analysis period. Here, the focus is on 30-min ensemble forecasts initialized at the end of that period. As in , the Weather Research and Forecasting (WRF) model is employed as a simplified cloud model at 2-km horizontal grid spacing. Various observation-space and state-space verification metrics, computed both for ensemble means and individual ensemble members, are employed to assess the quality of ensemble forecasts comparatively across cases. While the cases exhibit noticeable differences in predictability, the forecast skill in each case, as measured by various metrics, decays on a time scale of tens of minutes. The ensemble spread also increases rapidly but significant outlier members or clustering among members are not encountered. Forecast quality is seen to be influenced to varying degrees by the respective initial soundings. While radar data assimilation is able to partially mitigate some of the negative effects in some situations, the supercell case, in particular, remains difficult to predict even after 60 min of data assimilation.
Abstract
Estimates of three components of an uncertainty budget for a coastal ocean model in a wind-forced regime are made based on numerical simulations. The budget components behave differently in the shelf regime, inshore of the 200-m isobath, and the slope-interior regime, between the 200-m isobath and a fixed longitude (126°W) that is roughly 150 km offshore. The first of the three budget components is an estimate of the uncertainty in the ocean state given only a known history of wind stress forcing, with errors in the wind forcing estimated from differences between operational analyses. It is found that, over the continental shelf, the response to wind forcing is sufficiently strong and deterministic that significant skill in estimating shelf circulation can be achieved with knowledge only of the wind forcing, and no ocean data, for wind fields with these estimated errors. The second involves initial condition error and its influence on uncertainty, including both error growth with time from well-known initial conditions and error decay with time from poorly known initial conditions but with well-known wind forcing. The third component is that of boundary condition error and its influence on the interior solutions, including the dependence of that influence on the specific location along the boundary of the boundary condition error. Boundary condition errors with amplitude comparable to the root-mean-square variability at the boundary lead eventually to errors equal to the root-mean-square variability in the slope-interior regime, and somewhat smaller errors in the shelf regime. Covariance estimates based on differences of the wind-forced solutions from the ensemble mean are not dramatically different from those based on the full fields, and do not show strong state dependence.
Abstract
Estimates of three components of an uncertainty budget for a coastal ocean model in a wind-forced regime are made based on numerical simulations. The budget components behave differently in the shelf regime, inshore of the 200-m isobath, and the slope-interior regime, between the 200-m isobath and a fixed longitude (126°W) that is roughly 150 km offshore. The first of the three budget components is an estimate of the uncertainty in the ocean state given only a known history of wind stress forcing, with errors in the wind forcing estimated from differences between operational analyses. It is found that, over the continental shelf, the response to wind forcing is sufficiently strong and deterministic that significant skill in estimating shelf circulation can be achieved with knowledge only of the wind forcing, and no ocean data, for wind fields with these estimated errors. The second involves initial condition error and its influence on uncertainty, including both error growth with time from well-known initial conditions and error decay with time from poorly known initial conditions but with well-known wind forcing. The third component is that of boundary condition error and its influence on the interior solutions, including the dependence of that influence on the specific location along the boundary of the boundary condition error. Boundary condition errors with amplitude comparable to the root-mean-square variability at the boundary lead eventually to errors equal to the root-mean-square variability in the slope-interior regime, and somewhat smaller errors in the shelf regime. Covariance estimates based on differences of the wind-forced solutions from the ensemble mean are not dramatically different from those based on the full fields, and do not show strong state dependence.
Abstract
The effectiveness of the ensemble Kalman filter (EnKF) for assimilating radar observations at convective scales is investigated for cases whose behaviors span supercellular, linear, and multicellular organization. The parallel EnKF algorithm of the Data Assimilation Research Testbed (DART) is used for data assimilation, while the Weather Research and Forecasting (WRF) Model is employed as a simplified cloud model at 2-km horizontal grid spacing. In each case, reflectivity and radial velocity measurements are utilized from a single Weather Surveillance Radar-1988 Doppler (WSR-88D) within the U.S. operational network. Observations are assimilated every 2 min for a duration of 60 min and correction of folded radial velocities occurs within the EnKF. Initial ensemble uncertainty includes random perturbations to the horizontal wind components of the initial environmental sounding. The EnKF performs effectively and with robust results across all the cases. Over the first 18–30 min of assimilation, the rms and domain-averaged prior fits to observations in each case improve significantly from their initial levels, reaching comparable values of 3–6 m s−1 and 7–10 dBZ. Representation of mesoscale uncertainty, albeit in the simplest form of initial sounding perturbations, is a critical part of the assimilation system, as it increases ensemble spread and improves filter performance. In addition, assimilation of “no precipitation” observations (i.e., reflectivity observations with values small enough to indicate the absence of precipitation) serves to suppress spurious convection in ensemble members. At the same time, it is clear that the assimilation is far from optimal, as the ensemble spread is consistently smaller than what would be expected from the innovation statistics and the assumed observation-error variance.
Abstract
The effectiveness of the ensemble Kalman filter (EnKF) for assimilating radar observations at convective scales is investigated for cases whose behaviors span supercellular, linear, and multicellular organization. The parallel EnKF algorithm of the Data Assimilation Research Testbed (DART) is used for data assimilation, while the Weather Research and Forecasting (WRF) Model is employed as a simplified cloud model at 2-km horizontal grid spacing. In each case, reflectivity and radial velocity measurements are utilized from a single Weather Surveillance Radar-1988 Doppler (WSR-88D) within the U.S. operational network. Observations are assimilated every 2 min for a duration of 60 min and correction of folded radial velocities occurs within the EnKF. Initial ensemble uncertainty includes random perturbations to the horizontal wind components of the initial environmental sounding. The EnKF performs effectively and with robust results across all the cases. Over the first 18–30 min of assimilation, the rms and domain-averaged prior fits to observations in each case improve significantly from their initial levels, reaching comparable values of 3–6 m s−1 and 7–10 dBZ. Representation of mesoscale uncertainty, albeit in the simplest form of initial sounding perturbations, is a critical part of the assimilation system, as it increases ensemble spread and improves filter performance. In addition, assimilation of “no precipitation” observations (i.e., reflectivity observations with values small enough to indicate the absence of precipitation) serves to suppress spurious convection in ensemble members. At the same time, it is clear that the assimilation is far from optimal, as the ensemble spread is consistently smaller than what would be expected from the innovation statistics and the assumed observation-error variance.
Abstract
The predictability of coastal ocean circulation over the central Oregon shelf, a region of strong wind-driven currents and variable topography, is studied using ensembles of 50-day primitive equation ocean model simulations with realistic topography, simplified lateral boundary conditions, and forcing from both idealized and observed wind time series representative of the summer upwelling season. The main focus is on the balance, relevant to practical predictability, between deterministic response to known or well-predicted forcing, uncertainty in initial conditions, and sensitivity to instabilities and topographic interactions. Large ensemble and single-simulation variances are found downstream of topographic features, associated with transitions between along-isobath and cross-isobath flow, which are in turn related both to the time-integrated amplitude of upwelling-favorable wind forcing and to the formation of small-scale eddies. Simulated predictability experiments are conducted and model forecasts are verified by standard statistics including anomaly correlation coefficient, and root-mean-square error. A new variant of relative entropy, the forecast relative entropy, is introduced to quantify the predictive information content in the forecast ensemble, relative to the initial ensemble. The results suggest that, even under conditions of relatively weak wind forcing, the deterministic response is stronger than instability growth over the 3–7-day forecast intervals considered here. Consequently, important elements of the coastal circulation should be accessible to predictive, dynamical forecasts on the nominal 7-day predictability time scale of the atmospheric forcing, provided that sufficiently accurate initializations are available. These results on predictability are consistent with inferences drawn from recent modeling studies of coastal ocean circulation along the central Oregon shelf, and should have general validity for other, similar regions.
Abstract
The predictability of coastal ocean circulation over the central Oregon shelf, a region of strong wind-driven currents and variable topography, is studied using ensembles of 50-day primitive equation ocean model simulations with realistic topography, simplified lateral boundary conditions, and forcing from both idealized and observed wind time series representative of the summer upwelling season. The main focus is on the balance, relevant to practical predictability, between deterministic response to known or well-predicted forcing, uncertainty in initial conditions, and sensitivity to instabilities and topographic interactions. Large ensemble and single-simulation variances are found downstream of topographic features, associated with transitions between along-isobath and cross-isobath flow, which are in turn related both to the time-integrated amplitude of upwelling-favorable wind forcing and to the formation of small-scale eddies. Simulated predictability experiments are conducted and model forecasts are verified by standard statistics including anomaly correlation coefficient, and root-mean-square error. A new variant of relative entropy, the forecast relative entropy, is introduced to quantify the predictive information content in the forecast ensemble, relative to the initial ensemble. The results suggest that, even under conditions of relatively weak wind forcing, the deterministic response is stronger than instability growth over the 3–7-day forecast intervals considered here. Consequently, important elements of the coastal circulation should be accessible to predictive, dynamical forecasts on the nominal 7-day predictability time scale of the atmospheric forcing, provided that sufficiently accurate initializations are available. These results on predictability are consistent with inferences drawn from recent modeling studies of coastal ocean circulation along the central Oregon shelf, and should have general validity for other, similar regions.
Abstract
Hybrid ensemble–three-dimensional variational analysis schemes incorporate flow-dependent, ensemble-estimated background-error covariances into the three-dimensional variational data assimilation (3DVAR) framework. Typically the 3DVAR background-error covariance estimate is assumed to be stationary, nearly homogeneous, and isotropic. A hybrid scheme can be achieved by 1) directly replacing the background-error covariance term in the cost function by a linear combination of the original background-error covariance with the ensemble covariance or 2) through augmenting the state vector with another set of control variables preconditioned upon the square root of the ensemble covariance. These differently proposed hybrid schemes are proven to be equivalent. The latter framework may be a simpler way to incorporate ensemble information into operational 3DVAR schemes, where the preconditioning is performed with respect to the background term.
Abstract
Hybrid ensemble–three-dimensional variational analysis schemes incorporate flow-dependent, ensemble-estimated background-error covariances into the three-dimensional variational data assimilation (3DVAR) framework. Typically the 3DVAR background-error covariance estimate is assumed to be stationary, nearly homogeneous, and isotropic. A hybrid scheme can be achieved by 1) directly replacing the background-error covariance term in the cost function by a linear combination of the original background-error covariance with the ensemble covariance or 2) through augmenting the state vector with another set of control variables preconditioned upon the square root of the ensemble covariance. These differently proposed hybrid schemes are proven to be equivalent. The latter framework may be a simpler way to incorporate ensemble information into operational 3DVAR schemes, where the preconditioning is performed with respect to the background term.
Abstract
Accurate predictions in numerical weather models depend on the ability to accurately represent physical processes across a wide range of scales. This paper evaluates the utility of model time tendencies, averaged over many forecasts at a given lead time, to diagnose systematic forecast biases in the Advanced Research version of the Weather Research and Forecasting (WRF) Model during the 2010 North Atlantic hurricane season using continuously cycled ensemble data assimilation (DA). Erroneously strong low-level heating originates from the planetary boundary layer parameterization as a consequence of using fixed sea surface temperatures, impacting the upward surface sensible heat fluxes. Warm temperature bias is observed with a magnitude 
This study is the first to diagnose systematic forecast bias in a limited-area mesoscale model using its forecast tendencies. Unlike global models where relatively fewer time steps typically encompass a DA cycling period, averaging all short-term forecast tendencies can require potentially large data. It is shown that 30-min averaging intervals can sufficiently represent the systematic model bias in this modeling configuration when initializing forecasts from an ensemble member that is generated using a DA system with an identical model configuration. However, the number of time steps before model error begins to dominate initial condition (IC) errors may vary between modeling configurations. Model and IC error are indistinguishable in short-term forecasts when initialized from the ensemble mean, a global analysis from a different model, and an ensemble member using a different parameterization.
Abstract
Accurate predictions in numerical weather models depend on the ability to accurately represent physical processes across a wide range of scales. This paper evaluates the utility of model time tendencies, averaged over many forecasts at a given lead time, to diagnose systematic forecast biases in the Advanced Research version of the Weather Research and Forecasting (WRF) Model during the 2010 North Atlantic hurricane season using continuously cycled ensemble data assimilation (DA). Erroneously strong low-level heating originates from the planetary boundary layer parameterization as a consequence of using fixed sea surface temperatures, impacting the upward surface sensible heat fluxes. Warm temperature bias is observed with a magnitude
This study is the first to diagnose systematic forecast bias in a limited-area mesoscale model using its forecast tendencies. Unlike global models where relatively fewer time steps typically encompass a DA cycling period, averaging all short-term forecast tendencies can require potentially large data. It is shown that 30-min averaging intervals can sufficiently represent the systematic model bias in this modeling configuration when initializing forecasts from an ensemble member that is generated using a DA system with an identical model configuration. However, the number of time steps before model error begins to dominate initial condition (IC) errors may vary between modeling configurations. Model and IC error are indistinguishable in short-term forecasts when initialized from the ensemble mean, a global analysis from a different model, and an ensemble member using a different parameterization.
Abstract
Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophic theory also represents a leading-order theory in the sense that it is derivable from the primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophic theory, and the centrality of potential vorticity, the authors have recently developed a systematic asymptotic framework from which balanced, next-order corrections in Rossby number can be obtained. The approach is illustrated here through numerical solutions pertaining to unstable waves on baroclinic jets. The numerical solutions using the full primitive equations compare well with numerical solutions to our equations with accuracy one order beyond quasigeostrophic theory; in particular, the inherent asymmetry between cyclones and anticyclones is captured. Explanations of the latter and the associated asymmetry of the warm and cold fronts are given using simple extensions of quasigeostrophic– potential-vorticity thinking to next order.
Abstract
Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophic theory also represents a leading-order theory in the sense that it is derivable from the primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophic theory, and the centrality of potential vorticity, the authors have recently developed a systematic asymptotic framework from which balanced, next-order corrections in Rossby number can be obtained. The approach is illustrated here through numerical solutions pertaining to unstable waves on baroclinic jets. The numerical solutions using the full primitive equations compare well with numerical solutions to our equations with accuracy one order beyond quasigeostrophic theory; in particular, the inherent asymmetry between cyclones and anticyclones is captured. Explanations of the latter and the associated asymmetry of the warm and cold fronts are given using simple extensions of quasigeostrophic– potential-vorticity thinking to next order.
Abstract
Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophy also represents a leading-order theory in the sense that it is derivable from the full primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophy, and the centrality of potential vorticity, a systematic asymptotic framework is developed from which balanced, next-order corrections in Rossby number are obtained. The simplicity of the approach is illustrated by explicit construction of the next-order corrections to a finite-amplitude Eady edge wave.
Abstract
Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophy also represents a leading-order theory in the sense that it is derivable from the full primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophy, and the centrality of potential vorticity, a systematic asymptotic framework is developed from which balanced, next-order corrections in Rossby number are obtained. The simplicity of the approach is illustrated by explicit construction of the next-order corrections to a finite-amplitude Eady edge wave.
