A new vector radiative transfer solution using Padé approximants

doi:10.1016/j.jqsrt.2013.01.012

Elsevier

Journal of Quantitative Spectroscopy and Radiative Transfer

Volume 119, April 2013, Pages 12-22

Journal of Quantitative Spectroscopy and Radiative Transfer

https://doi.org/10.1016/j.jqsrt.2013年01月01日2 Get rights and content

Abstract

This paper solves the multi-layer multiple-scattering plane-parallel vector radiative transfer problem for solar and thermal sources. Efficient calculation of the Stokes vector for optically thin media is made possible by the method herein called VPARTM—"Vector Padé approximation radiative transfer method". The algorithm calculates the matrix exponential using the Padé approximation, with an accuracy prescribed by the user, from which the global reflection and transmission operators are obtained. In a multi-layer atmosphere, these operators can be combined using the adding method. Computational gain is enhanced by exploiting the symmetry inherent in the exponential solution and then exploiting the structure of this matrix to halve the order of the matrix solution. Computational efficiency is explored through a range of model configurations and the method is applied for two example cases, one for a range of wavelengths typically used to retrieve aerosol properties and one for an instrument similar to the GOSAT and OCO-2

0.76 - μ m O_{2}

A-band spectrometers.

Highlights

► We present a multiple-scattering plane-parallel vector radiative transfer solution. ► The solution is based on the Pade approximation to the matrix exponential. ► Timing results indicate that our solution is efficient especially for thin media. ► Numerical results show that our solution is accurate to

10^{- 4}

% compared to VLIDORT.

Introduction

As modern remote sensing instruments have improved in sensitivity, vector radiative transfer modeling has become an increasingly important component in retrieval algorithms. It has been shown that in some cases the errors incurred from using the scalar approximation (neglecting polarization) are considerable [1], [2], [3]; this can lead to significant errors in the retrieval of atmospheric and surface parameters. Vector radiative transfer models are also an important aspect in polarization correction algorithms for instruments sensitive to polarization [4], [5]. Polarization measurements have also been shown to provide important information in some retrieval scenarios. When combined with observations at multiple view angles, polarization can reveal information about particle size, shape, and/or refractive index that otherwise would be difficult to retrieve from intensity only measurements [6], [7], [8]. It has also been demonstrated that high resolution spectropolarimetry can reveal aerosol profile information [9], [10] while observations of polarized radiances are useful in separating atmospheric and surface reflectance contributions [11].

Unfortunately vector radiative transfer has an order of up to four times greater than that of scalar radiative transfer and therefore is computationally slower. When a large number of radiative transfer calculations are required, such as with high spectral resolution measurements, an efficient radiative transfer model is needed, especially when processing data in real-time applications..

The modern treatment of polarized radiative transfer is based on the work of Chandrasekhar [12] who formulated polarization in terms of the Stokes vector. An important contribution was provided by Kus ̆c ̆er and Ribaric ̆ [13] who formulated the phase matrix decomposition in azimuth angle using the addition theorem of generalized spherical functions. This was later more conveniently reformulated by Siewert [14], [15] which is the method used in most vector radiative transfer models today. With this formulation, extension of scalar radiative transfer codes to solve for the Stokes vector is relatively straightforward. Several vector methods have been developed including the discrete ordinate method [16], [17], matrix operator methods based on doubling and adding [18], the spherical harmonics method [19], the F_N method [20], and successive orders of scattering [21], [22].

In a previous paper [23], hereafter referred to as MG2010, we presented a scalar radiative transfer method based on the Padé approximation to the exponential of a matrix. It was found to be accurate and significantly faster compared to the discrete ordinate method, especially for optically thin media (optical thickness x less than one). In this study we extend this method to solve the vector radiative transfer equation and refer to this extension as the "Vector Padé approximation radiative transfer method" (VPARTM). In Section 2 a description of the radiative transfer formulation used in this paper is given beginning with the presentation of the basic equations in Section 2.1 followed by the matrix formulation and the general solution for the Stokes vector in Section 2.2. In Section 2.3 we discuss the solution methodology touching primarily on the differences required to extend the scalar solution in MG2010 to the vector case. In Section 3 some details and features of our implementation of VPARTM are discussed. Timing comparisons relative to a reference model for several model configurations are given in Section 4 while example results from two realistic applications are presented in Section 5 including accuracy and timing comparisons relative to a reference model. Finally, some concluding remarks are given in Section 6.

An implementation of VPARTM is available as one of several radiative transfer methods within the X Radiative Transfer Model (XRTM) software. XRTM is licensed under the GNU General Public License Version 3 and is available at http://reef.atmos.colostate.edu/∼gregm/xrtm.

Section snippets

Basic equations

The vector radiative transfer equation [12] for a single plane-parallel homogeneous layer is given by

μ \frac{d I (τ, μ, φ)}{d τ} = I (τ, μ, φ) - \frac{ω}{4 π} \int_{0}^{2 π} \int_{- 1}^{1} P (μ, φ, μ', φ') I (τ, μ', φ') d μ' d φ' - Σ (τ, μ, φ),

where

I (τ, μ, φ)

is the Stokes vector

[I, Q, U, V]^{T}

at optical depth

τ \in [0, x]

, x is optical thickness of the layer,

μ \in [- 1, 1]

is the cosine of the polar angle of propagation (measured from the upward normal or the negative

τ - direction

),

φ \in [0, 2 π]

is the azimuthal angle of propagation (measured clockwise when looking upward),

P (μ, φ, μ',

Implementation details

Both the scalar and vector implementations are distributed together and share as much code as possible. As a result, the same features that we listed in MG2010 for PARTM are available for VPARTM including a choice of (standard) Gauss–Legendre or double Gauss–Legendre quadrature, Wiscombe delta-M scaling [27], the Nakajima–Tanaka TMS correction [28], and the pseudo-spherical approximation [29]. For the surface boundary a generalized BRDF implementation including several common BRDF kernels is

Timing comparisons

In this section we present timing comparisons relative to the Vector Linearized Discrete Ordinate Radiative Transfer (VLIDORT) model version 2.3 [17]. VLIDORT is a general purpose plane parallel vector radiative transfer model appropriate for multiple scattering atmospheres with solar and thermal sources and is capable of computing both the vector radiance and its derivatives with respect to atmospheric and surface parameters. VLIDORT is an accurate and efficient implementation and is widely

Numerical results

In this section we have chosen two practical examples to demonstrate the use of VPARTM, one for a range of wavelengths typically used to retrieve aerosol properties and one for an instrument similar to the

0.76 - μ m

O₂ A-band spectrometers on the Greenhouse gases Observing SATellite (GOSAT, http://www.gosat.nies.go.jp/index_e.html) and the Orbital Carbon Observatory 2 (OCO-2) [36]. Results are presented for the intensity I and for the degree of polarization given by

P (0, μ, φ) = \sqrt{Q^{2} (0, μ, φ) + U^{2} (0, μ, φ) + V^{2} (}

Concluding remarks

In this paper we have presented a vector radiative transfer solution for a multi-layer, multiple-scattering, plane-parallel atmosphere with solar scattering and thermal emission sources and a lower surface boundary characterized by a BRDF. The solution method, based on the Padé approximation, is an extension of the scalar solution outlined in MG2010 to the vector case. It is shown that given the expansion of the phase matrix in terms of generalized spherical functions and a couple of additional

Acknowledgments

This research was supported under NASA Grants 1280999 and NNG04E35G. The authors would like to thank Denis O'Brien and Tommy Taylor for supplying the GOSAT/OCO test atmosphere.

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