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- Author or Editor: Gerald L. Browning x
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Abstract
Mathematical issues arising when applying four-dimensional variational (4DVAR) data assimilation to limited-area problems are studied. The derivation of the adjoint system for the initial-boundary value problem for a general hyperbolic system using the standard variational approach requires that the inflow adjoint variables at an open boundary be zero. However, in general, these “natural” boundary conditions will lead to a different solution than that provided by the global assimilation problem. The impact of using natural boundary conditions when there are errors (on the boundary) in the initial guess on the assimilated initial conditions is discussed.
A proof of the uniqueness of the solution for both forward and adjoint equations in the presence of open boundaries at each iteration of the minimization procedure is provided, along with an assessment of the convergence of numerical solutions.
Numerical experiments with a simple advection equation support the theoretical analyses. Numerical results show that if observational data are perfect, 4DVAR data assimilation using a limited-area model can produce a reasonable initial condition. However, if there are errors in the observational data at the open boundaries and if natural boundary conditions are assumed, boundary errors can contaminate the assimilated solutions.
Abstract
Mathematical issues arising when applying four-dimensional variational (4DVAR) data assimilation to limited-area problems are studied. The derivation of the adjoint system for the initial-boundary value problem for a general hyperbolic system using the standard variational approach requires that the inflow adjoint variables at an open boundary be zero. However, in general, these “natural” boundary conditions will lead to a different solution than that provided by the global assimilation problem. The impact of using natural boundary conditions when there are errors (on the boundary) in the initial guess on the assimilated initial conditions is discussed.
A proof of the uniqueness of the solution for both forward and adjoint equations in the presence of open boundaries at each iteration of the minimization procedure is provided, along with an assessment of the convergence of numerical solutions.
Numerical experiments with a simple advection equation support the theoretical analyses. Numerical results show that if observational data are perfect, 4DVAR data assimilation using a limited-area model can produce a reasonable initial condition. However, if there are errors in the observational data at the open boundaries and if natural boundary conditions are assumed, boundary errors can contaminate the assimilated solutions.
Abstract
The impact of observational errors on objective analyses is investigated with mathematical analyses, analytical examples, and real data experiments. Cases with observational errors at one or more stations are considered. It is found that in the presence of observational errors, the analysis error in an objective analysis scheme generally consists of two parts: the signal fitting error and the noise contamination error. Although every objective analysis scheme has its own procedure(s) to control the two errors, the procedures to suppress the noise contamination error in one and two dimensions are shown to be relatively ineffective. It is shown that the extension of an objective analysis method to more dimensions significantly reduces the noise contamination.
Based on these results, higher dimensional versions of the least squares polynomial fitting (LSPF) methods and the Barnes scheme are examined. In both analytic and real data experiments, the 3D and 4D LSPF methods and the 3D Barnes scheme show an enhanced ability to filter observational noise.
Abstract
The impact of observational errors on objective analyses is investigated with mathematical analyses, analytical examples, and real data experiments. Cases with observational errors at one or more stations are considered. It is found that in the presence of observational errors, the analysis error in an objective analysis scheme generally consists of two parts: the signal fitting error and the noise contamination error. Although every objective analysis scheme has its own procedure(s) to control the two errors, the procedures to suppress the noise contamination error in one and two dimensions are shown to be relatively ineffective. It is shown that the extension of an objective analysis method to more dimensions significantly reduces the noise contamination.
Based on these results, higher dimensional versions of the least squares polynomial fitting (LSPF) methods and the Barnes scheme are examined. In both analytic and real data experiments, the 3D and 4D LSPF methods and the 3D Barnes scheme show an enhanced ability to filter observational noise.
Abstract
The impact of observational and model errors on four-dimensional variational (4DVAR) data assimilation is analyzed for a general dynamical system. Numerical experiments with both the barotropic vorticity equation and the shallow water system are conducted. It is shown from the analysis and the numerical experiments that when there are random errors in observations or in model parameterizations, the 4DVAR assimilation method can suppress these errors; however, when the errors are systematic or biased, the 4DVAR assimilation method tends to either converge to the erroneous observations or introduce the model error into the data analysis, or both.
For a multiple-timescale fluid dynamical system, such as the shallow water equations with fluid depth corresponding to the external mode, the skewness in the system can amplify the errors, especially in the fast variable (e.g., the geopotential or height field).
Forecasts using the assimilated initial condition with an imperfect model indicate that the forecasts may or may not be improved, depending upon the nature of the model and observational errors, and the length of the assimilation and forecast periods.
Abstract
The impact of observational and model errors on four-dimensional variational (4DVAR) data assimilation is analyzed for a general dynamical system. Numerical experiments with both the barotropic vorticity equation and the shallow water system are conducted. It is shown from the analysis and the numerical experiments that when there are random errors in observations or in model parameterizations, the 4DVAR assimilation method can suppress these errors; however, when the errors are systematic or biased, the 4DVAR assimilation method tends to either converge to the erroneous observations or introduce the model error into the data analysis, or both.
For a multiple-timescale fluid dynamical system, such as the shallow water equations with fluid depth corresponding to the external mode, the skewness in the system can amplify the errors, especially in the fast variable (e.g., the geopotential or height field).
Forecasts using the assimilated initial condition with an imperfect model indicate that the forecasts may or may not be improved, depending upon the nature of the model and observational errors, and the length of the assimilation and forecast periods.
Abstract
The impact of discontinuous model forcing on the initial conditions obtained from 4DVAR data assimilation is studied with mathematic analyses, idealized numerical examples, and more realistic meteorological cases. The results show that a discontinuity in a parameterization, like a model bias, can introduce a systematic error in the assimilated initial fields. However, the most detrimental effect of a model discontinuity is the retention of roughness in the assimilated initial fields, although in some cases the 4DVAR procedure provides some smoothing effect. The obvious consequences of this roughness is that it will introduce spurious modes in the ensuing forecast, and derivatives of the assimilated initial data will be unrealistically large, which can lead to large errors in data analysis. The smoothing effect on the initial conditions with the addition of artificial diffusion to the constraining model is also studied. Possible solutions to the problem of 4DVAR data assimilation with discontinuous model forcing are discussed.
Abstract
The impact of discontinuous model forcing on the initial conditions obtained from 4DVAR data assimilation is studied with mathematic analyses, idealized numerical examples, and more realistic meteorological cases. The results show that a discontinuity in a parameterization, like a model bias, can introduce a systematic error in the assimilated initial fields. However, the most detrimental effect of a model discontinuity is the retention of roughness in the assimilated initial fields, although in some cases the 4DVAR procedure provides some smoothing effect. The obvious consequences of this roughness is that it will introduce spurious modes in the ensuing forecast, and derivatives of the assimilated initial data will be unrealistically large, which can lead to large errors in data analysis. The smoothing effect on the initial conditions with the addition of artificial diffusion to the constraining model is also studied. Possible solutions to the problem of 4DVAR data assimilation with discontinuous model forcing are discussed.
Abstract
The Limited-Area Model (LAM) developed at the National Center for Atmospheric Research for use in conjunction with the NCAR Global Circulation Model is described including details of the lateral boundary conditions. One set of experiments is described for which a 2½° glow simulation provides the correct or control data against which 2½° LAM forecasts are compared. Three cases are considered in which the LAM inflow boundary values are provided by the 2½° global forecast, a 5° global forecast, or are held fixed equal to the initial values. Forecasts produced by the LAM with finer gods (up to ⅝°) are also shown. Although based on simulated data, the results indicate that the Limited-Area Model shows good potential for short-range forecasting.
Abstract
The Limited-Area Model (LAM) developed at the National Center for Atmospheric Research for use in conjunction with the NCAR Global Circulation Model is described including details of the lateral boundary conditions. One set of experiments is described for which a 2½° glow simulation provides the correct or control data against which 2½° LAM forecasts are compared. Three cases are considered in which the LAM inflow boundary values are provided by the 2½° global forecast, a 5° global forecast, or are held fixed equal to the initial values. Forecasts produced by the LAM with finer gods (up to ⅝°) are also shown. Although based on simulated data, the results indicate that the Limited-Area Model shows good potential for short-range forecasting.
Abstract
The accuracy of finite-diffrerence approximations to the shallow water equations on a sphere is examined for flow cases having an analytic solution. Approximations over grids with the longitudinal grid increment (Δλ) increasing near the poles such that the distance between grid points is nearly constant have large errors near the poles. These large polar errors are caused by the large longitudinal grid increment used in the approximations and are reduced by using a grid with Δλ constant. The normally severe limit on the time step caused by the small distance between grid points near the pole can be relaxed by removing the short-wave-length, fast-moving waves by Fourier analysis. With our test case, which contains only large scales, this filtering method produced a solution which is almost identical to that obtained over the uniform grid using a small time step. In comparing second- and fourth-order schemes applied to the above test case, we find that the fourth-order schemes offer more improvement per computer time than second-order Themes with mesh reduction.
Abstract
The accuracy of finite-diffrerence approximations to the shallow water equations on a sphere is examined for flow cases having an analytic solution. Approximations over grids with the longitudinal grid increment (Δλ) increasing near the poles such that the distance between grid points is nearly constant have large errors near the poles. These large polar errors are caused by the large longitudinal grid increment used in the approximations and are reduced by using a grid with Δλ constant. The normally severe limit on the time step caused by the small distance between grid points near the pole can be relaxed by removing the short-wave-length, fast-moving waves by Fourier analysis. With our test case, which contains only large scales, this filtering method produced a solution which is almost identical to that obtained over the uniform grid using a small time step. In comparing second- and fourth-order schemes applied to the above test case, we find that the fourth-order schemes offer more improvement per computer time than second-order Themes with mesh reduction.
Abstract
Three-dimensional variational data assimilation (3DVAR) analysis is an important method used at many operational and research institutes in meteorology, for example, the National Centers for Environmental Prediction (NCEP) and the European Centre for Medium-Range Weather Forecasts (ECMWF). In 3DVAR analysis, different forms of cost functions and constraints (e.g., geostrophic balance) have been used. However, the impacts of these different forms of cost functions, covariances, and constraints on the 3DVAR solutions have not been completely analyzed due to their complexity. Using the Fourier analysis where the Fourier transformation is applicable, the impacts of different forms of cost functions and some commonly used physical constraints are demonstrated. In the particular case of geostrophic balance as the constraint, the large-scale motion of a 3DVAR analysis could be in geostrophic balance, but the mesoscale solution may be nearly unchanged if the penalty terms and the forms of J b (terms related to the background field in 3DVAR cost functions) and J o (related to the observation field) are chosen properly. This conclusion shows that the penalization of geostrophic imbalance can be used for mesoscale data assimilation without serious damage to the mesoscale features. More important for constructing a 3DVAR system, this paper also demonstrates that some formulations of J b can produce physically unexpected solutions. The theory is illustrated using numerical experiments.
Abstract
Three-dimensional variational data assimilation (3DVAR) analysis is an important method used at many operational and research institutes in meteorology, for example, the National Centers for Environmental Prediction (NCEP) and the European Centre for Medium-Range Weather Forecasts (ECMWF). In 3DVAR analysis, different forms of cost functions and constraints (e.g., geostrophic balance) have been used. However, the impacts of these different forms of cost functions, covariances, and constraints on the 3DVAR solutions have not been completely analyzed due to their complexity. Using the Fourier analysis where the Fourier transformation is applicable, the impacts of different forms of cost functions and some commonly used physical constraints are demonstrated. In the particular case of geostrophic balance as the constraint, the large-scale motion of a 3DVAR analysis could be in geostrophic balance, but the mesoscale solution may be nearly unchanged if the penalty terms and the forms of J b (terms related to the background field in 3DVAR cost functions) and J o (related to the observation field) are chosen properly. This conclusion shows that the penalization of geostrophic imbalance can be used for mesoscale data assimilation without serious damage to the mesoscale features. More important for constructing a 3DVAR system, this paper also demonstrates that some formulations of J b can produce physically unexpected solutions. The theory is illustrated using numerical experiments.
Abstract
The convergence of spectral model numerical solutions of the global shallow-water equations is examined as a function of the time step and the spectral truncation. The contributions to the errors due to the spatial and temporal discretizations are separately identified and compared. Numerical convergence experiments are performed with the inviscid equations from smooth (Rossby-Haurwitz wave) and observed (R45 atmospheric analysis) initial conditions, and also with the diffusive shallow-water equations. Results are compared with the forced inviscid shallow-water equations case studied by Browning et at. Reduction of the time discretization error by the removal of fast waves from the solution using initialization is shown. The effects of forcing and diffusion on the convergence are discussed. Time truncation errors are found to dominate when a feature is large scale and well resolved; spatial truncation errors dominate-for small-scale features and also for large scales after the small scales have affected them. Possible implications of these results for global atmospheric modeling are discussed.
Abstract
The convergence of spectral model numerical solutions of the global shallow-water equations is examined as a function of the time step and the spectral truncation. The contributions to the errors due to the spatial and temporal discretizations are separately identified and compared. Numerical convergence experiments are performed with the inviscid equations from smooth (Rossby-Haurwitz wave) and observed (R45 atmospheric analysis) initial conditions, and also with the diffusive shallow-water equations. Results are compared with the forced inviscid shallow-water equations case studied by Browning et at. Reduction of the time discretization error by the removal of fast waves from the solution using initialization is shown. The effects of forcing and diffusion on the convergence are discussed. Time truncation errors are found to dominate when a feature is large scale and well resolved; spatial truncation errors dominate-for small-scale features and also for large scales after the small scales have affected them. Possible implications of these results for global atmospheric modeling are discussed.
In August–September 2010, NASA, NOAA, and the National Science Foundation (NSF) conducted separate but closely coordinated hurricane field campaigns, bringing to bear a combined seven aircraft with both new and mature observing technologies. NASA's Genesis and Rapid Intensification Processes (GRIP) experiment, the subject of this article, along with NOAA's Intensity Forecasting Experiment (IFEX) and NSF's Pre-Depression Investigation of Cloud-Systems in the Tropics (PREDICT) experiment, obtained unprecedented observations of the formation and intensification of tropical cyclones. The major goal of GRIP was to better understand the physical processes that control hurricane formation and intensity change, specifically the relative roles of environmental and inner-core processes. A key focus of GRIP was the application of new technologies to address this important scientific goal, including the first ever use of the unmanned Global Hawk aircraft for hurricane science operations. NASA and NOAA conducted coordinated flights to thoroughly sample the rapid intensification (RI) of Hurricanes Earl and Karl. The tri-agency aircraft teamed up to perform coordinated flights for the genesis of Hurricane Karl and Tropical Storm Matthew and the nonredevelopment of the remnants of Tropical Storm Gaston. The combined GRIP– IFEX–PREDICT datasets, along with remote sensing data from a variety of satellite platforms [Geostationary Operational Environmental Satellite (GOES), Tropical Rainfall Measuring Mission (TRMM), Aqua, Terra, CloudSat, and Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO)], will contribute to advancing understanding of hurricane formation and intensification. This article summarizes the GRIP experiment, the missions flown, and some preliminary findings.
In August–September 2010, NASA, NOAA, and the National Science Foundation (NSF) conducted separate but closely coordinated hurricane field campaigns, bringing to bear a combined seven aircraft with both new and mature observing technologies. NASA's Genesis and Rapid Intensification Processes (GRIP) experiment, the subject of this article, along with NOAA's Intensity Forecasting Experiment (IFEX) and NSF's Pre-Depression Investigation of Cloud-Systems in the Tropics (PREDICT) experiment, obtained unprecedented observations of the formation and intensification of tropical cyclones. The major goal of GRIP was to better understand the physical processes that control hurricane formation and intensity change, specifically the relative roles of environmental and inner-core processes. A key focus of GRIP was the application of new technologies to address this important scientific goal, including the first ever use of the unmanned Global Hawk aircraft for hurricane science operations. NASA and NOAA conducted coordinated flights to thoroughly sample the rapid intensification (RI) of Hurricanes Earl and Karl. The tri-agency aircraft teamed up to perform coordinated flights for the genesis of Hurricane Karl and Tropical Storm Matthew and the nonredevelopment of the remnants of Tropical Storm Gaston. The combined GRIP– IFEX–PREDICT datasets, along with remote sensing data from a variety of satellite platforms [Geostationary Operational Environmental Satellite (GOES), Tropical Rainfall Measuring Mission (TRMM), Aqua, Terra, CloudSat, and Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO)], will contribute to advancing understanding of hurricane formation and intensification. This article summarizes the GRIP experiment, the missions flown, and some preliminary findings.
