Linear Form of the Radiative Transfer Equation Revisited

A linear form of the radiative transfer equation for the simultaneous retrieval of atmospheric temperature and absorbing gas profiles and surface temperature from radiance spectra was derived in the classic paper by Smith et al. (1991). The derivation started with the RTE difference between the true and initial radiance spectra. It is shown that there exists a dual representation of this RTE difference, and the resulting dual linear form of the RTE appears more general than the original linear form. Introduction The spectrum of radiance exiting the earth-atmosphere system, as described by the radiative transfer equation (RTE), is a nonlinear functional of atmospheric temperature and absorbing gas profiles as well as surface temperature and emissivities. The physical retrieval of these meteorological variables from the RTE is a mathematically ill-posed problem. With the advance in infrared sensor technology from the multispectral sounders (e.g. HIRS, GOES, MODIS) with tens of bands to the so-called hyperspectral sounders (e.g. HIS, NAST-I, AIRS, IASI, CrIS, GIFTS) with thousands of channels, the physical retrieval has evolved from a mathematically underdetermined problem into an overdetermined one as respect to a typical fast RTM with hundreds of atmospheric variables. The advance in infrared sensor technology may also affect NWP data assimilation. For multispectral sounders, assimilation of tens of radiance values instead of hundreds of retrieved meteorological variables per field of view (FOV) in a global or regional 3D/4D-VAR assimilation system has not only shown improved assimilation accuracy but also reduce assimilation time. For hyperspectral sounders, assimilation of hundreds of retrieved meteorological variables instead of thousands of radiance values per FOV may be a foreseeable direction in the future to greatly reduce the assimilation problem size and thus assimilation time if the 3D/4D-VAR system would make the most of full hyperspectral spectra. For physical retrieval or data assimilation formulated as an optimal variational problem involving radiance observations from a multispectral or hyperspectral sounder, an effective and efficient search direction toward the minimum of the associated cost function relies on the guidance of the so-called Jacobians, which are the Frêchet derivatives of radiances with respect to atmospheric and surface variables in a linear form of the RTE or RTM. Ideally an exact linear form with exact analytical Jacobians is the most desired for accurate and efficient physical retrieval and data assimilation. When such a linear form is not analytically derivable, the next desirable candidate is the tangent linear and adjoint models based on algorithmic differentiation (AD), a technique that uses chain rules to numerically evaluate derivatives of composite functions specified by the computer program of the RTM. This process achieves the same numerical accuracy as the exact analytical Jacobians, but the latter is several times faster in computation. A last resort to compute the Jacobians is the approximation of the Frêchet derivatives by the “brute-force” finite differences for each variable at each pressure level, but this approach is much less accurate and too time-consuming for practical usage in physical retrieval and data assimilation. Deriving a better linear form of the RTE with more accurate Jacobians for the simultaneous physical retrieval of atmospheric temperature and absorbing gas profiles and surface temperature from radiance spectra has been a continued endeavor since the early 70’s. In the classic paper by Smith et al. (1991), a new approximate linear form of the RTE was presented. This type of monochromatically-approximate linear form has been still used in the physical retrieval by some authors, and the paper inspired Huang et al. (2002) to accomplish the derivation of the exact linear form with correct analytical Jacobians for the widely-used McMillin-Fleming-Eyre-Woolf type fast radiative transfer models in satellite remote sensing. Examples of this type of fast radiative transfer models are the RTTOV and RT-IASI models (e.g. Eyre, 1991, Saunders et al., 1999; Matricardi et al., 2001). Currently, the RTTOV and RT-IASI models carry the tangent linear and adjoint codes instead of the more computationally-efficient exact analytical Jacobian subroutine for the inverse problems. The quality of analytical Jacobians is crucial for physical retrieval and variational data assimilation in an efficient, reliable and accurate fashion. By numerically comparing the linear form with exact analytical Jacobians with the linear form with monochromaticallyapproximate analytical Jacobians for the 19 HIRS bands under the 0.1% perturbation of the 1976 U.S. Standard Atmosphere, Huang et al. showed the latter is lack of accuracy (worse than the “brute-force” finite differences) and even yields the wrong signs for some HIRS bands, implying wrong search directions for finding the optimal inverse solution. Nevertheless, the classical derivation of the linear form by Smith et al. (1991) represents a historic landmark on the simultaneous physical retrieval, signifying the last major endeavor for deriving approximate analytical Jacobians as well as a key inspiration for later successful derivation of exact analytical Jacobians. The derivation of the linear form by Smith et al. (1991) is of pedagogical interest. Without resorting to the concepts in calculus of variations, Smith et al. started with the finite difference between the RTEs for the true and initial radiance spectra to derive the linear form. In this paper we prove that there exists a dual representation of this RTE difference, and show some interesting outcomes from the resulting dual linear form. The rest of the paper is organized as follows. The section below summarizes the derivation of the linear form by Smith et al. and the section following details the derivation of its dual linear form. Linear Form by Smith et al. (1991) The true spectrum of monochromatic radiance exiting the earth-atmosphere system is 0 ( ) ( ) ( ) ( ) S P v v s v s v v R B p p B p d p τ = − ∫ τ , (1) where Bv is the Planck radiance with subscript v denoting spectral frequency, τ v p ( ) the total transmittances of the atmosphere above atmospheric pressure level p. The subscript s denotes the earth's surface. The radiance spectrum corresponding to an assumed initial temperature and absorbing gas profiles is 0 0 0 0 0 0 ( ) ( ) ( ) ( ) s v v s v s v v p R B p p B p d p τ = − ∫ τ , (2) where a superscript 0 denotes the initial quantity. With the following linear perturbation definitions: δ R R R v v v ≡ − 0 , δ B p B p B p v v v ( ) ( ) ( ) ≡ − 0 , (3) 0 ( ) ( ) ( ), v v v p p δ τ τ τ ≡ − p the difference between the true and initial radiance spectra is [ ] 0 0 0 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). s s v v s v s v s v s v v v v p p R B p p B p p B p d p B p d p δ δ τ δ τ δτ δ τ = + − − ∫ ∫ (4) Performing integration by parts on the first integral term in Eq. (4), the equation becomes 0 0 0 0 ( ) ( ) ( ) ( ) ( ) ( ). (5) s s v v s v s v v v v p p R B p p B p d p p dB p δ δ τ δ τ δ τ = − + ∫ ∫ Using the linear approximations to the Planck radiance perturbations: 0 0 ( ( )) ( ) ( ) ( ) ( ), (6) ( ) v v v B T p B p T p p T p T p ∂ δ δ β δ ∂ = ≡ d B p p d T p v v ( ) ( ) ( ) = β 0 , (7) and the difference of monochromatic transmittances in terms of the difference of the absorbing gas profiles: 0 0 0 0 1 1 ln ( ) ( ) ( ) ln ( ) ( ) ( ) , (8) ( ) i i N N v v v v v i i i i d p p p p p U p d U p τ δ τ τ δ τ τ δ = = = ≈ ∑ ∑