Eugene A. Ustinov

Adjoint Sensitivity Analysis of Radiative Transfer Equation: Temperature and Gas Mixing Ratio Weighting Functions for Remote Sensing of Scattering Atmospheres in Thermal IR

Abstract Sensitivity analysis based on of the adjoint equation of radiative transfer is applied to the case of atmospheric remote sensing in the thermal spectral region with non-negligible atmospheric scattering. Analytic expressions for the weighting functions for retrievals of temperature and gas mixing ratio are derived. It is demonstrated that these expressions include the case of pure absorption as a particular case when single scattering albedo of atmospheric scattering can be neglected.

Adjoint sensitivity analysis of radiative transfer equation: 2. Applications to retrievals of temperature in scattering atmospheres in thermal IR ☆

Abstract An approach to formulation of inversion algorithms for thermal sounding in the case of scattering atmosphere based on the adjoint equation of radiative transfer (Ustinov, JQSRT 68 (2001) 195, referred to as Paper 1 in the main text) is applied to temperature retrievals in the scattering atmosphere for the nadir viewing geometry. Analytical expressions for the weighting functions involving the integration of the source function are derived. Temperature weighting functions for a simple model of the atmosphere with scattering are evaluated and convergence to the case of pure atmospheric absorption is demonstrated. The numerical experiments on temperature retrievals are carried out to demonstrate the validity of the expressions obtained.

Analytic evaluation of the weighting functions for remote sensing of blackbody planetary atmospheres: a general linearization approach

An analytic approach is proposed for the evaluation of weighting functions for remote sensing of a blackbody planetary atmosphere based on straightforward, general linearization. In the present paper, this approach is applied to the case of remote sensing with the nadir (down-looking) geometry. Expressions for weighting functions for various atmospheric parameters are derived. It is demonstrated that in a realistic case of temperature-dependent atmospheric absorption, an additional term appears in the expression for the temperature weighting function which contains the temperature derivative of the atmospheric absorption coefficient. The approach is applied to the case of a semi-infinite atmosphere and then, to the atmosphere of a finite optical depth with the underlying surface. In this, latter case, the expressions are also obtained for partial derivatives of observed radiances with respect to surface parameters: surface pressure, temperature and emissivity.

Three Approaches to Sensitivity Analysis of Models

There are three ways to implement the sensitivity analysis of quantitative models. The simplest finite-difference (FD) approach requires multiple re-runs of the forward model according to the number of model input parameters. Although this approach uses the forward model without any modifications and does not require any analytic work, its application results in a very computer-intensive algorithm, which may become a prohibitive factor in practical applications to models with a large number of input parameters. Two other approaches of sensitivity analysis—the linearization approach and adjoint approach—are substantially more computer-efficient and require just single runs of a corresponding model derived from the initial, baseline model. The general formulation and comparison of these approaches is presented in this chapter.

Applications of Sensitivity Analysis in Remote Sensing

In this chapter we consider the practical applications of sensitivity analysis in remote sensing. After a brief review of various types of sensitivities, we consider three main areas of applications: the error analyses of input and output parameters and the solution of inverse problems. The error analysis of output parameters with given errors of input parameters is most straightforward. The corresponding algorithm involves, essentially, only matrix multiplication. The error analysis of input parameters with given requirements to errors of output parameters becomes more complicated if the matrix of sensitivities cannot be inverted directly. The solution of inverse problems is the most sophisticated area of application of sensitivity analysis. Here, the simple forms of least squares method and of the method of statistical regularization are presented.

Sensitivity Analysis of Analytic Models: Linearization and Adjoint Approaches

In this chapter we apply the linearization and adjoint approaches of sensitivity analysis to three analytic models, for which we have obtained the solutions earlier, in Chap. 4, using the formalism of differential calculus (for discrete parameters) and variational calculus (for continuous parameters). We will reproduce these results using the approaches of sensitivity analysis described in general form in Chap. 3, which will provide a form of validation of these approaches.

Sensitivity Analysis of Numerical Models

In the previous chapter, the linearization and adjoint approaches were applied to analytic models, and the obtained results matched those obtained in Chap. 4 using methods of differential calculus and variational calculus. This provided, albeit not rigorous but hopefully still convincing validation of general formulations of the linearization and adjoint approaches formulated in Chap. 3. In this chapter, the linearization and adjoint approaches of sensitivity analysis will be applied to selected numerical models for which analytic solutions are not available.

Introduction: Remote Sensing and Sensitivity Analysis

By its basic definition, remote sensing is simply indirect measurement. In a vast variety of practical applications, we cannot measure quantities of interest directly, but we can measure some other quantities that are related to quantities of interest by some known relations. For example, targeted measurements of spectral radiances on top of the atmosphere in the thermal infrared spectral region provide a capability to measure atmospheric profiles of temperature and mixing ratios of atmospheric constituents. Another example: targeted measurements of position and velocity of a spacecraft orbiting a planet provide a capability to measure spherical harmonics of the gravity field of this planet.

Sensitivity Analysis of Models with Higher-Order Differential Equations

All models considered in previous chapters are based on differential equations of first order. There exists a wide variety of models however, that are based on higher-order differential equations, such as the Poisson equation or wave equation. While application of the linearization approach to forward problems with these equations poses no substantial problems application of the adjoint approach, which needs formulation of corresponding adjoint problems, becomes more and more sophisticated with increasing the order of equations. In a nutshell, one has to apply the Lagrange identity rule as many times, as the order of the equations dictates, and this procedure becomes increasingly complicated [see, e.g., (Marchuk 1995)]. In this chapter we present an alternative approach based on the standard techniques using the reduction of the higher-order differential equation to a system of differential equations of first order. Further on, this system is represented in the form of a matrix differential equation of first order complemented by corresponding matrix initial-value conditions (IVCs) and/or boundary conditions (BCs). We present the general principles of application of this matrix approach and the results of its application to a set of problems based on selected stationary and non-stationary equations of mathematical physics.