## 1. Introduction

[2] Trace gases such as carbon dioxide are now being measured from space using spectroscopic observations of reflected sunlight in the near infrared. The SCanning Imaging Absorption spectroMeter for Atmospheric CartograpHY (SCIAMACHY) has been measuring various trace gases such as CO_{2}, CO, CH_{4}, and O_{3} since March 2002 aboard the ENVISAT satellite [*Bovensmann et al.*, 1999]. The Greenhouse gases Observing SATellite (GOSAT) was launched in January 2009 and carries the Thermal And Near-infrared Sensor for carbon Observation Fourier Transform Spectrometer (TANSO-FTS), designed to make global observations of near-surface CO_{2} and CH_{4} concentrations [*Hamazaki et al.*, 2005]. The Orbiting Carbon Observatory (OCO) was to make similar measurements of carbon dioxide [*Crisp et al.*, 2004] but experienced a launch failure in February 2009. The primary objective of these missions is to identify and quantify the sources, where these gases are emitted, and sinks, where they are removed from the atmosphere. This is a particularly challenging remote sensing measurement because the presence of surface sources and sinks must be inferred from the small spatial and temporal variations that they produce in the background distribution. Simulations of CO_{2} fluxes with source-sink inversion models indicate that estimates of the column-averaged CO_{2} dry air mole fraction with accuracies of 1 ppm (out of the 388 ppm background) are needed on regional scales at monthly intervals to retrieve CO_{2} sources and sinks on these scales [*Miller et al.*, 2007; *Chevallier et al.*, 2007; *Baker et al.*, 2008]. This requires retrieval models with accuracies of a fraction of one percent.

[3] For OCO, GOSAT, and other similar sensors, model simulations of spectra are generally required for parameter retrievals, data assimilation, and also simple visualization and sensitivity studies. However, in atmospheres with even a modest amount of scattering, multiple-scattering calculations are required to obtain accurate estimates of visible and near-infrared instrumental radiances. These simulations can be time-consuming as they often require tens of thousands of monochromatic radiative transfer (RT) calculations for simulating a single instrumental spectrum. Remote sensing retrieval calculations are even more demanding because they require not just radiances, but often the radiance Jacobians as well, which specify the derivatives of the radiances with respect to each parameter of interest (e.g., trace gas abundance, aerosol optical properties and distribution, surface properties) for each monochromatic spectral point.

[4] Many multiple scattering codes in use today rely on the adding-doubling method [e.g., *van de Hulst*, 1980; *Hansen*, 1971] or the discrete ordinate method [e.g., *Stamnes et al.*, 1988]. Both methods compute the radiation coming along a discrete set of zenith angles or streams, and interpolate between these angles to find the radiation at other angles. A code is said to be run with 2*N*_{s} streams when *N*_{s} is the number of streams in either hemisphere. In nadir viewing mode, the computational burden is proportional to *N*_{s}^{3}, because the codes largely rely on multiplication and inversion of *N*_{s} × *N*_{s} matrices. For off-nadir viewing, an additional loop over *N*_{s} is required due to the additional azimuthal Fourier modes required [see, e.g., *Liou*, 2002], leading to an approximate *N*_{s}^{4} dependency. Therefore, using the smallest number of streams allowed for a given application is often highly desirable.

[5] The number of streams required to achieve a given accuracy varies with the amount and type of atmospheric scattering and absorption, as well as the surface properties. Generally speaking, the stronger the atmospheric scattering, the higher the number of streams required. As an example, Figure 1 shows the number of full-sphere streams required to achieve 0.1% accuracy in top-of-atmosphere (TOA) intensity for a wide variety of profiles and near-infrared wavelengths, versus a reference calculation with 256 streams. The profiles are a subset of the *Chevallier* [2001] set of 60-layer ECMWF profiles; see section 2 for details of the optical properties and radiative transfer. The solar zenith angle is variable, but the observation zenith angle is fixed at 0° The independent variable in Figure 1 is the integrated scattering optical depth from TOA to the level in the atmosphere where the cumulative gas absorption optical depth to the TOA equals 5.0; scattering below this point in the atmosphere has little effect on the TOA radiance, as virtually all radiation from below this point will be absorbed before reaching the TOA. The error bars represent ±1 standard deviation and represent the variability among different profiles. Note that these calculations were made with a doubling-adding code and employed both Delta-M scaling [*Wiscombe*, 1977] and the TMS (single-scattering) correction of *Nakajima and Tanaka* [1988]; these methods can substantially reduce the required number of streams to achieve a given accuracy for most cases. It is seen that for more significantly scattering profiles, a high *N*_{s} is often required to achieve this level of accuracy.

[6] To understand the implications of Figure 1, let us consider a typical optimal estimation approach to retrieving CO_{2} concentration from OCO or GOSAT measurements [e.g., *Bösch et al.*, 2006; *Oshchepkov et al.*, 2008; *Butz et al.*, 2009]. OCO and GOSAT measure roughly the same shortwave bands; Table 1 gives some details of the OCO bands. OCO and GOSAT were designed to yield CO_{2} estimates with random and systematic biases no larger than ∼0.3% [*Crisp et al.*, 2004; *Suto et al.*, 2008]. To obtain this, we suggest the corresponding radiative transfer errors to be in the neighborhood 0.1%. This is somewhat less than the expected instrument noise of OCO, and it was shown by *Hasekamp and Butz* [2008] that RT errors on the order of 0.1% lead to CO_{2} retrieval errors of 0.05–0.20 ppm, which constitutes an acceptable 20% or less of the 0.3% total error budget.

Band | Name | Spectral Range (cm^{−1}) | ILS FWHM^{a} (cm^{−1}) |
---|---|---|---|

- a
ILS FWHM, instrument line shape full-width at half-maximum.
| |||

1 | Oxygen A | 12,950–13,200 | 0.63 |

2 | WCO_{2} | 6170–6290 | 0.27 |

3 | SCO_{2} | 4805–4900 | 0.21 |

[7] To fully resolve the spectral lines, on the order of 40,000 spectral points are required for the simulation of all three OCO bands. The standard OCO retrieval algorithm was to be run for scattering optical depth of 0.3 or less; the GOSAT team will process similarly clear profiles. From Figure 1 it is seen that roughly 16 streams would generally be needed to reach the desired 0.1% accuracy level for this amount of scattering. Performing a typical line-by-line approach thus would require tens of thousands of independent, 16-stream calculations.

[8] Many methods have been put forth to reduce the computational expense of the line-by-line approach described above. Most strive to reduce the number of monochromatic calculations needed, or the complexity of the multiple scattering calculation needed at each spectral grid point. Traditional correlated *k* distribution methods [e.g., *Goody et al.*, 1989; *Lacis and Oinas*, 1991; *Fu and Liou*, 1992] are an example of the first class of methods, but can lead to errors of tens of percent even in modestly scattering atmospheres [*Duan et al.*, 2005]. However, several techniques have been proposed to reduce the computational time for multiple scattering calculations in the visible and near IR [*West et al.*, 1990; *Min and Harrison*, 2004; *Natraj et al.*, 2005; *Duan et al.*, 2005; *Hasekamp and Butz*, 2008; *Boesche et al.*, 2009]. Most of these techniques replace the line-by-line calculations at each spectral wavelength with a smaller set of representative calculations, typically binned by absorption optical depth or something similar, and reconstruct the full spectrum from this small set of calculations. Some of the techniques also take into account the vertical structure of the absorption in some fashion in order to reduce errors. Several of the techniques achieve errors on the order of 1–2%, while some are substantially better. *Natraj et al.* [2005] (hereafter N05) reports errors of less than 0.3% in the O_{2} A band, but only tests a single case with total aerosol optical depth (AOD) of 0.2. *Hasekamp and Butz* [2008] (hereafter H08) report errors of better than 0.13% in all three OCO bands for a similarly thin case (AOD = 0.3), and better than 0.5% for a more challenging high-cloud case. *Duan et al.* [2005] (hereafter D05) report errors better than 0.5% for all but the most challenging multilayered cloud systems. Most authors report results for nadir observations and modest solar zenith angles of 30°–40°.

[9] Of the studies described above, the work of H08, N05, and D05 achieve errors at or near the desired accuracy for OCO-type retrievals, but all have some drawbacks. H08 requires both radiances and Jacobians to be calculated, as it works using a first-order correction to the binned radiance involving the corresponding derivatives with respect to the atmospheric and surface optical properties. While the N05 technique does not require derivatives, it requires on the order of 1000 high-accuracy calculations in the oxygen A band alone, substantially more than the other approaches. The technique of D05 requires only a small number of bins and does not require radiance derivatives. They calculate the first order of scattering contribution to each spectral radiance exactly and model the multiple-scattering component as piecewise analytic functions in the absorption optical thickness and a variable related to the fraction of the absorption that occurs above scattering layers of cloud or aerosol. They apply their method for total intensity (polarization was not considered), and mostly consider simple one and two-layer cloud systems in their analysis; it is not clear if their analytic fits will generalize to situations with more complicated profiles of scattering or with polarizing atmospheric and/or surface properties.

[10] In this paper, we describe a simple, tunable method to accelerate visible and near-infrared multiple-scattering radiance calculations with a minimal loss in accuracy. This method, called Low-Streams Interpolation (LSI), can be used in conjunction with any scalar or vector multiple-scattering radiative transfer model, and does not require derivatives to be calculated. This method builds upon the work of several previous authors but differs in several important respects. It identifies a strong relationship between radiative transfer errors and two easy-to-calculate properties of the gas absorption, and utilizes this relationship to accelerate the RT calculations. The motivation for and description of the method are given in section 2. In section 3, results are shown depicting both the accuracy and computational efficiency of the method. Section 4 gives a summary of the results and compares and contrasts the technique to those of previous authors.