This paper reviews the methods which may be used to estimate the state of the atmosphere, i.e., the distribution of temperature and composition, from measurements of emitted thermal radiation such as are made by remote sounding instruments on satellites. The principles of estimation theory are applied to a linear version of the problem, and it is shown that many of the apparently different methods to be found in the literature are particular cases of the same general method. As an aid to understanding, the optimum linear solution is described in terms of the geometry of n dimensions, with n = 3 for illustration. In generalizing the approach to the nonlinear problem there are two stages: (1) finding any member of the infinite family of possible solutions, which may be done by any convenient iterative method, and (2) finding the optimum solution by satisfying appropriate constraints.

Retrieval of atmospheric temperature and composition from remote measurements of thermal radiation

In this paper we study a family of gradient descent algorithms to approximate the regression function from reproducing kernel Hilbert spaces (RKHSs), the family being characterized by a polynomial decreasing rate of step sizes (or learning rate) By solving a bias-variance trade-off we obtain an early stopping rule and some
probabilistic upper bounds for the convergence of the algorithms We also discuss the implication of these results in the context of classification where some fast convergence rates can be achieved for plug-in classifiers Some connections are addressed with Boosting, Landweber iterations, and the online learning algorithms as stochastic approximations of the gradient descent method

http://web.mit.edu/lrosasco/www/publications/earlystop.pdf

On Early Stopping in Gradient Descent Learning

Numerical solution of ill-posed problems is often accomplished by discretization (projection onto a finite dimensional subspace) followed by regularization. If the discrete problem has high dimension, though, typically we compute an approximate solution by projecting the discrete problem onto an even smaller dimensional space, via iterative methods based on Krylov subspaces. In this work we present a common framework for efficient algorithms that regularize after this second projection rather than before it. We show that determining regularization parameters based on the final projected problem rather than on the original discretization has firmer justification and often involves less computational expense. We prove some results on the approximate equivalence of this approach to other forms of regularization, and we present numerical examples.

/pdf/choosing-regularization-parameters-in-iterative-methods-for-2682bw2ejk.pdf

Choosing Regularization Parameters in Iterative Methods for Ill-Posed Problems

A computationally fast line-by-line method for the determination of atmospheric absorption is described. This method is based on the creation of an Automatized Atmospheric Absorption Atlas (4A) covering all possible plausible atmospheric conditions (temperature, mixing ratios of absorbing gases, zenith angle). It is applied to synthetic computations of atmospheric transmittances and radiant energies associated with three types of satellite observations: radiometric measurements made by HIRS/2 (High-Resolution Infrared Sounder) on TIROS-N; infrared images taken from the geostationary satellite METEOSAT; interferometric experiment IRIS (Infrared Interferometer Spectrometer) on VOYAGER (NASA's mission to Jupiter, Saturn and possibly Uranus). For all three experiments, comparisons were made with real observations and are presented associated with radiosonde data for the first. Concerning computation times, a gain of a factor varying between 15 and 40 is obtained when using the 4A line-by-line method ...

A Fast Line-by-Line Method for Atmospheric Absorption Computations: The Automatized Atmospheric Absorption Atlas

. This paper gives an update of the RTTOV (Radiative Transfer for TOVS) fast radiative transfer model, which is widely used in the satellite retrieval and data assimilation communities. RTTOV is a fast radiative transfer model for simulating top-of-atmosphere radiances from passive visible, infrared and microwave downward-viewing satellite radiometers. In addition to the forward model, it also optionally computes the tangent linear, adjoint and Jacobian matrix providing changes in radiances for profile variable perturbations assuming a linear relationship about a given atmospheric state. This makes it a useful tool for developing physical retrievals from satellite radiances, for direct radiance assimilation in NWP models, for simulating future instruments, and for training or teaching with a graphical user interface. An overview of the RTTOV model is given, highlighting the updates and increased capability of the latest versions, and it gives some examples of its current performance when compared with more accurate line-by-line radiative transfer models and a few selected observations. The improvement over the original version of the model released in 1999 is demonstrated.

/pdf/an-update-on-the-rttov-fast-radiative-transfer-model-5g5nf9xd03.pdf

An update on the RTTOV fast radiative transfer model (currently at version 12)

At a given pressure level the atmospheric transmittance for an absorbing gas with a constant mixing ratio varies only with the temperature profile of the atmosphere. A simple transmittance model based on temperature differences is derived for monochromatic radiation. This model then is extended to the more important case of polychromatic radiation through the use of scaling approximations. The resulting algorithm for calculating transmittances for an arbitrary temperature profile is simple to use, accurate, and computationally fast because only arithmetic operations are required. In fact, resulting transmittances agreed with line-by-line calculations to within 0.0031 for the cases tried. Details for calculating the expansion coefficients are provided.

Atmospheric transmittance of an absorbing gas: a computationally fast and accurate transmittance model for absorbing gases with constant mixing ratios in inhomogeneous atmospheres.

Models exist which allow the calculation of atmospheric transmittance at a given zenith angle for an absorbing gas with a constant mixing ratio. However, many applications require transmittances at several zenith angles. A simple, fast, and accurate model for calculating the angular dependence is given. This model is computationally fast because only the four arithmetic operations are used. Details for calculating the expansion coefficients are provided. When this technique is combined with a procedure for calculating transmittances at a fixed angle, it is possible to calculate transmittances for slant paths at arbitrary zenith angles and temperature profiles, provided the mixing ratio is constant. This technique was evaluated with a method capable of calculating transmittances at zero zenith angle with an accuracy of 0.0031. For zenith angles ranging from 0° to 40°, transmittances agree with line-by-line calculations to within 0.0038.

Atmospheric transmittance of an absorbing gas. 2: A computationally fast and accurate transmittance model for slant paths at different zenith angles.

Equivalence of regularization and truncated iteration in the solution of III-posed image reconstruction problems

The method of real-time retrieval of atmospheric temperature profiles from Nimbus IV Satellite Infrared Spectrometer observations currently used in dynamical weather analysis-forecast operation is described. Each vertical temperature profile is determined by its deviation from a “guess” profile. The deviation is expressed as a linear combination of differences between the measured radiances and those computed from the guess profile. The coefficients are estimated, by matrix inversion, from the weighting functions (i.e., derivatives of atmospheric transmittance functions), which are regularized by the ratio of the expected variance of the measurement errors to the expected variance of the errors in the guess profile. The deviations are iterated until the variance of the radiance residuals is less than the expected variance of the measurement errors. For weather analysis-forecast operation the dynamical forecast is used as the first guess; therefore, the calculated profiles should differ from the f...

Retrieval of Atmospheric Temperature Profiles from Satellite Measurements for Dynamical Forecasting

Computed tomography is a medical diagnostic technique in which x-ray transmission measurements at numerous angles through the human body are processed by computer to produce cross-sectional pictures of the body. A modification of this technique, using emitted infrared or microwave radiation instead of transmitted x-ray radiation, can be applied to satellite radiance measurements taken along the orbital track at various angles. Cross sections of the vertical atmospheric temperature structure (or the gaseous constituent density structure) are retrieved from the collection of radiance measurements taken at various angles and frequencies. The advantage of this technique over conventional remote sensing methods is the additional information acquired by viewing a given point in the atmosphere at several angles as well as at several frequencies. The physical and geometric concepts involved are discussed along with the mathematical formulation of the problem and the practical aspects of applying the tech...

Henry E. Fleming

Papers

Atmospheric transmittance of an absorbing gas: a computationally fast and accurate transmittance model for absorbing gases with constant mixing ratios in inhomogeneous atmospheres.

Atmospheric transmittance of an absorbing gas. 2: A computationally fast and accurate transmittance model for slant paths at different zenith angles.

Equivalence of regularization and truncated iteration in the solution of III-posed image reconstruction problems

Retrieval of Atmospheric Temperature Profiles from Satellite Measurements for Dynamical Forecasting

Satellite Remote Sensing by the Technique of Computed Tomography