A Variational Formalism for the Radiative Transfer Equation: Prelude to Model 3

The MODEL III variational data assimilation model is the third of four general assimilation models designed to blend weather data measured from space based platforms in the meteorological data mainstream in a way that maximizes the information content of the satellite data. Because there are many different observation locations and there are many instruments with different measurement error characteristics, it is also necessary to require that the blending be done to maximize the information content of the data and simultaneously to retain a dynamically consistent and reasonably accurate description of the state of the atmosphere. This is ideally a variational problem for which the data receive relative weights that are inversely proportional to measurement error and are adjusted to satisfy a set of dynamical equations that govern atmospheric processes. The advantage of MODEL III over the previous two models is that radiance, the atmospheric variable measured by satellite, becomes a dependent variable. In the previous versions, mean layer temperatures that had been retrieved from the radiances by some method, were included in the assimilation by substituting them in place of the rawinsonde temperatures. Now both rawinsonde temperatures and satellite radiances are included independently in the assimilation

A Variational Assimilation Method for Satellite and Conventional Data: Model 2 (version 1)

The Model II variational data assimilation model is the second of the four variational models designed to blend diverse meteorological data into a dynamically constrained data set. Model II differs from Model I in that it includes the thermodynamic equation as the fifth dynamical constraint. Thus, Model II includes all five of the primative equations that govern atmospheric flow for a dry atmosphere

A Variational Formalism for the Radiative Transfer Equation and a Geostrophic, Hydrostatic Atmosphere: Prelude to Model 3

The second step in development of MODEL III is summarized. It combines the four radiative transfer equations of the first step with the equations for a geostrophic and hydrostatic atmosphere. This step is intended to bring radiance into a three dimensional balance with wind, height, and temperature. The use of the geostrophic approximation in place of the full set of primitive equations allows for an easier evaluation of how the inclusion of the radiative transfer equation increases the complexity of the variational equations. Seven different variational formulations were developed for geostrophic, hydrostatic, and radiative transfer equations. The first derivation was too complex to yield solutions that were physically meaningful. For the remaining six derivations, the variational method gave the same physical interpretation (the observed brightness temperatures could provide no meaningful input to a geostrophic, hydrostatic balance) at least through the problem solving methodology used in these studies. The variational method is presented and the Euler-Lagrange equations rederived for the geostrophic, hydrostatic, and radiative transfer equations

Variational objective analysis for cyclone studies

Significant accomplishments during 1987 to 1988 are summarized with regard to each of the major project components. Model 1 requires satisfaction of two nonlinear horizontal momentum equations, the integrated continuity equation, and the hydrostatic equation. Model 2 requires satisfaction of model 1 plus the thermodynamic equation for a dry atmosphere. Model 3 requires satisfaction of model 2 plus the radiative transfer equation. Model 4 requires satisfaction of model 3 plus a moisture conservation equation and a parameterization for moist processes

Modification of a variational objective analysis model for new equations for pressure gradient and vertical velocity in the lower troposphere and for spatial resolution and accuracy of satellite data

Since late 1982 NASA has supported research to develop a numerical variational model for the diagnostic assimilation of conventional and space-based meteorological data. In order to analyze the model components, four variational models are defined dividing the problem naturally according to increasing complexity. The first of these variational models (MODEL I), the subject of this report, contains the two nonlinear horizontal momentum equations, the integrated continuity equation, and the hydrostatic equation. This report summarizes the results of research (1) to improve the way the large nonmeteorological parts of the pressure gradient force are partitioned between the two terms of the pressure gradient force terms of the horizontal momentum equations, (2) to generalize the integrated continuity equation to account for variable pressure thickness over elevated terrain, and (3) to introduce horizontal variation in the precision modulus weights for the observations

Application of satellite data in variational analysis for global cyclonic systems

The goal of the research is a variational data assimilation method that incorporates as dynamical constraints, the primitive equations for a moist, convectively unstable atmosphere and the radiative transfer equation. Variables to be adjusted include the three-dimensional vector wind, height, temperature, and moisture from rawinsonde data, and cloud-wind vectors, moisture, and radiance from satellite data. In order to facilitate thorough analysis of each of the model components, four variational models that divide the problem naturally according to increasing complexity were defined. The research performed during the second year fall into four areas: sensitivity studies involving Model 1; evaluation of Model 2; reformation of Model 1 for greater compatibility with Model 2; development of Model 3 (radiative transfer equation); and making the model more responsive to the observations

A Variational Assimilation Method for Satellite and Conventional Data: a Revised Basic Model 2B

A variational objective analysis technique that modifies observations of temperature, height, and wind on the cyclone scale to satisfy the five 'primitive' model forecast equations is presented. This analysis method overcomes all of the problems that hindered previous versions, such as over-determination, time consistency, solution method, and constraint decoupling. A preliminary evaluation of the method shows that it converges rapidly, the divergent part of the wind is strongly coupled in the solution, fields of height and temperature are well-preserved, and derivative quantities such as vorticity and divergence are improved. Problem areas are systematic increases in the horizontal velocity components, and large magnitudes of the local tendencies of the horizontal velocity components. The preliminary evaluation makes note of these problems but detailed evaluations required to determine the origin of these problems await future research

A multivariate variational objective analysis-assimilation method. Part 1: Development of the basic model

The variational method of undetermined multipliers is used to derive a multivariate model for objective analysis. The model is intended for the assimilation of 3-D fields of rawinsonde height, temperature and wind, and mean level temperature observed by satellite into a dynamically consistent data set. Relative measurement errors are taken into account. The dynamic equations are the two nonlinear horizontal momentum equations, the hydrostatic equation, and an integrated continuity equation. The model Euler-Lagrange equations are eleven linear and/or nonlinear partial differential and/or algebraic equations. A cyclical solution sequence is described. Other model features include a nonlinear terrain-following vertical coordinate that eliminates truncation error in the pressure gradient terms of the horizontal momentum equations and easily accommodates satellite observed mean layer temperatures in the middle and upper troposphere. A projection of the pressure gradient onto equivalent pressure surfaces removes most of the adverse impacts of the lower coordinate surface on the variational adjustment

Modification of a Successive Corrections Objective Analysis for Improved Derivative Calculations

The use of objectively analyzed fields of meteorological data for complex diagnostic studies and for the initialization of numerical prediction models places the requirements upon the objective method that derivatives of the gridded fields be accurate and free from interpolation error. A modification of an objective analysis developed by Barnes provides improvements in analyses of both the field and its derivatives. Theoretical comparisons, comparisons between analyses of analytical monochromatic waves, and comparisons between analyses of actual weather data are used to show the potential of the new method. The new method restores more of the amplitudes of desired wavelengths while simultaneously filtering more of the amplitudes of undesired wavelengths. These results also hold for the first and second derivatives calculated from the gridded fields. Greatest improvements were for the Laplacian of the height field; the new method reduced the variance of undesirable very short wavelengths by 72 pct. Other improvements were found in the divergence of the gridded wind field and near the boundaries of the field data

On the Concept of Varying Influence Radii for a Successive Corrections Objective Analysis

There has been a long standing concept by those who use successive corrections objective analysis that the way to obtain the most accurate objective analysis is first, to analyze for the long wavelengths and then to build in the details of the shorter wavelengths by successively decreasing the influence of the more distant observations upon the interpolated values. Using the Barnes method, the filter characteristics were compared for families of response curves that pass through a common point at a reference wavelength. It was found that the filter cutoff is a maximum if the filter parameters that determine the influence of observations are unchanged for both the initial and corrections passes. This information was used to define and test the following hypothesis. If accuracy is defined by how well the method retains desired wavelengths and removes undesired wavelengths, then the Barnes method gives the most accurate analyses if the filter parameter on the initial and corrections passes are the same. This hypothesis does not follow the usual conceptual approach to successive corrections analysis