Atmospheric weighting functions and surface partial derivatives for remote sensing of scattering planetary atmospheres in thermal spectral region: general adjoint approach

doi:10.1016/j.jqsrt.2004.08.003

Journal of Quantitative Spectroscopy and Radiative Transfer

Volume 92, Issue 3, 15 May 2005, Pages 351-371

https://doi.org/10.1016/j.jqsrt.2004.08.003 Get rights and content

Abstract

An approach to formulation of inversion algorithms for remote sensing in the thermal spectral region in the case of a scattering planetary atmosphere, based on the adjoint equation of radiative transfer (Ustinov (JQSRT 68 (2001) 195; JQSRT 73 (2002) 29); referred to as Papers 1 and 2, respectively, in the main text), is applied to the general case of retrievals of atmospheric and surface parameters for the scattering atmosphere with nadir viewing geometry. Analytic expressions for corresponding weighting functions for atmospheric parameters and partial derivatives for surface parameters are derived. The case of pure atmospheric absorption with a scattering underlying surface is considered and convergence to results obtained for the non-scattering atmospheres (Ustinov (JQSRT 74 (2002) 683), referred to as Paper 3 in the main text) is demonstrated.

Introduction

In the previous two papers [1], [2], hereafter referred to as Papers 1,2, we applied an approach involving the adjoint radiative transfer (RT) equation to the analysis and evaluation of weighting functions for atmospheric profiles of temperature and mixing ratio of gaseous constituents. The analysis was done under a simplifying assumption that the atmospheric absorption has no temperature dependence, and no attempts were made to evaluate the sensitivity of observed radiances to parameters of the underlying surface.

In general, there are a number of atmospheric and surface parameters that are, directly, or indirectly, involved in radiative transfer. The observed radiances are, to some extent, sensitive to them. In another recent paper [3], hereafter referred to as Paper 3, we used an approach based on direct linearization to obtain corresponding sensitivities: the weighting functions for atmospheric parameters, and partial derivatives for surface parameters in the case of a blackbody (non-scattering) atmosphere with a scattering underlying surface. The intermediate weighting functions, or sensitivities, with respect to the radiative atmospheric parameters, Planck function and total absorption coefficient, turned out to be useful tools for evaluation of various atmospheric weighting functions. In a similar way, the partial derivatives, or sensitivities, to the surface parameters temperature, emissivity, and surface pressure, were obtained.

In the present paper we apply the adjoint approach to obtain the general results, analogous to those of Paper 3, in the case of scattering planetary atmospheres. In addition to the Planck function and total extinction (absorption plus scattering) coefficient, the atmospheric scattering adds one more radiative parameter, the total phase function of atmospheric scattering. As we see below, the corresponding intermediate weighting function provides a basis for evaluation of specific atmospheric parameters responsible for atmospheric scattering. Also, we evaluate corresponding surface partial derivatives.

Our plan is as follows. In the next section, we summarize the results obtained in Paper 3 for the blackbody atmospheres. In Section 3, we obtain analogous general results for scattering atmospheres using an ansatz employed in Paper 1, where the outgoing radiation was directly obtained from the solution of the radiative transfer (RT) equation at the top of the atmosphere (TOA). In Section 4, similar results are obtained based on an ansatz employed in Paper 2, where the outgoing radiation was obtained using integration of the source function of RT equation. In Sections 5 and 6 we discuss the obtained results. The concluding remarks are presented in Section 7.

Section snippets

Atmospheric weighting functions and surface partials for non-scattering planetary atmospheres

The results obtained in Paper 3 can be briefly summarized as follows. Let z be a geometric coordinate measured from TOA vertically down into the atmosphere; at TOA $z = 0$ and at the surface, $z = z_{0}$ . Let $μ$ be a cosine of the viewing angle, measured between nadir vertical and the direction of observation. The monochromatic radiances observed at TOA of a non-scattering planetary atmosphere in thermal IR can be written in the form $R (μ) = t (z_{0}, μ) ε B_{s} + \int_{z_{0}}^{0} B (z) d t (z, μ) .$ Here $t (z, μ)$ is an atmospheric transmission

Adjoint sensitivity analysis of radiances obtained directly from the solution of the forward RT equation

The general plan of action in this and the next sections is the same as in Papers 1,2. First, we specify the forward RT problem in the form of a linear operator equation. Then, based on a specific representation of observable radiances from the solution of the forward RT problem, we specify the observables weighting function. Then, we formulate the corresponding adjoint RT problem, again, in the form of a linear operator equation. Further on, using the forward and adjoint solutions, we obtain

Adjoint sensitivity analysis of radiances obtained using integration of the source function

Integration of the source function for computations of radiances from the solution of the RT equation was suggested by Kourganoff [5] in order to avoid the angular oscillations of the radiances computed using the discrete-ordinate method or spherical harmonics method. It turned out that the results for weighting functions and partial derivatives obtained using this approach for scattering atmospheres have a straightforward relation to results obtained for non-scattering atmospheres (Paper 3)

Discussion: inter-relation between obtained results and their convergence to the blackbody case

There is a straightforward inter-relation between the results obtained in Sections 3 and 4. Also, the expressions of Sections 3 and 4 converge to the corresponding expressions of Paper 3 summarized in Section 2 when the scattering becomes negligible.⁴ In this section we substantiate both these statements.

The

Discussion: applications of obtained results

In Paper 3 we considered four radiative parameters (two atmospheric: Planck function and absorption coefficient, and two surface ones: Planck function and emissivity) through which all physical parameters of the atmosphere and surface manifest themselves in the observed radiances. The corresponding four types of sensitivities to radiative parameters were analyzed there:

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Atmospheric Planck weighting function $K^{(B)} (z, μ)$ ;
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Atmospheric AC weighting function $K^{(κ)} (z, μ)$ ;
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Surface Planck partial derivative $\partial R$

Conclusion

The analytical framework presented above forms a theoretical basis for algorithms for reduction of observations of scattering planetary atmospheres in the thermal spectral region. The set of a few atmospheric weighting functions and surface partial derivatives for radiative parameters provides a “gate” to atmospheric weighting functions and surface partial derivatives for all physical parameters that impact the observed radiances and, consequently, can be retrieved from these observations,

Acknowledgements

The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology under a contract with the National Aeronautics and Space Administration, through the Mars Data Analysis Program.

References (6)

E.A. Ustinov
Adjoint sensitivity analysis of radiative transfer equationtemperature and gas mixing ratio weighting functions for remote sensing of scattering atmospheres in thermal IR
JQSRT
(2001)
E.A. Ustinov
Adjoint sensitivity analysis of radiative transfer equation: 2. Applications to retrievals of temperature in scattering atmospheres in thermal IR
JQSRT
(2002)
E.A. Ustinov
Analytic evaluation of the weighting functions for remote sensing of blackbody planetary atmospheresa general linearization approach
JQSRT
(2002)

There are more references available in the full text version of this article.

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