## A moderate-spectral-resolution transmittance model based on fitting the line-by-line calculation

Optics Express, Vol. 15, Issue 13, pp. 8360-8370 (2007)

http://dx.doi.org/10.1364/OE.15.008360

Acrobat PDF (900 KB)

### Abstract

A fast narrowband transmittance model, referred to as the Fast Fitting Transmittance Model (FFTM), is developed based on rigorous line-by-line (LBL) calculations. Specifically, monochromatic transmittances are first computed from a LBL model in a spectral region from 1 to 25000 cm-1 for various pressures and temperatures ranging from 0.05 hPa to 1100 hPa and from 200 K to 320 K, respectively. Subsequently, the monochromatic transmittances are averaged over a spectral interval of 1 cm^{-1} to obtain narrowband transmittances that are then fitted to various values of absorber amount. A database of fitting coefficients is then created that can be used to compute narrowband transmittances for an arbitrary atmospheric profile. To apply the FFTM to an inhomogeneous atmosphere, the Curtis-Godson (C-G) approximation is employed to obtain the weighted effective coefficients. The present method is validated against the LBLRTM and also compared with the high-spectral-resolution measurements acquired by the Atmospheric Infrared Sounder (AIRS) and High-resolution Interferometer Sounder (HIS). With a spectral resolution of 1 cm^{-1} and a wide spectral coverage, the FFTM offers a unique combination of numerical efficiency and considerable accuracy for computing moderate- to high-spectral-resolution transmittances involved in radiative transfer simulations and remote sensing applications.

© 2007 Optical Society of America

## 1. Introduction

1. S. A. Clough and M. J. Iacono, “Line-by-line calculations of atmospheric fluxes and cooling rates: Part II: Application to carbon dioxide, ozone, methane, nitrous oxide, and the halocarbons,” J. Geophys. Res. **100**, 16519–16535(1995). [CrossRef]

2. D. P. Kratz, G. M. Mlynczak, and C. J. Mertens, *et al.*, “An inter-comparison of far-infrared line-by-line radiative transfer models,” J. Quant. Spectrosc. Radiat. Transf. **90**, 323–341(2005). [CrossRef]

^{-1}, referred to as the Fast Fitting Transmittance Model (FFTM). This model is developed to efficiently compute atmospheric transmittance for wavelengths in the visible through far-infrared region, temperatures from 200 to 320 K, pressures from 0.05 hPa to 1100 hPa, and absorber amount over 7 orders. The present model is essentially based on several coefficients fitted from the LBL calculation. A database of fitting coefficients is then created that can be used to compute narrowband transmittances for an arbitrary atmospheric profile. The Curtis-Godson (C-G) approximation is used to obtain the weighted effective coefficients for an inhomogeneous atmosphere. The FFTM is validated against the LBL model and also compared to the observations from satellite-borne and airborne high-spectral-resolution infrared sensors.

## 2. Fast fitting transmittance model

^{-1}so that the combined transmittance is simply the product of the transmittances associated with individual species. This is a reasonable approximation for narrow band transmittance computations at most wavenumbers. In some spectral region, owing to the cross-absorption, the simple multiplication rule may degrade the accuracy of the results. This is a problem worthy of further studies. The continuum contributions of various gases can be considered separately because the continuum absorption varies smoothly with wavenumber.

*t*, pressure

*p*and absorber amount

*u*within a wavenumber interval of Δ

*ν*can be obtained via the following expression:

*k*is the absorption coefficient.

_{ν}*T̅*)) and the original monochromatic optical depth for water vapor in a small wavenumber range from 3500 to 3520 cm

_{ν}^{-1}. The monochromatic optical depths in this paper are computed from the Line-By-Line Radiative Transfer Model (LBLRTM) [1

1. S. A. Clough and M. J. Iacono, “Line-by-line calculations of atmospheric fluxes and cooling rates: Part II: Application to carbon dioxide, ozone, methane, nitrous oxide, and the halocarbons,” J. Geophys. Res. **100**, 16519–16535(1995). [CrossRef]

11. S. A. Clough, M. W. Shephard, E. J. Mlawer, J. S. Delamere, M. J. Iacono, K. Cady-Pereira, S. Boukabara, and P. D. Brown, “Atmospheric radiative transfer modeling: a summary of the AER codes,” J. Quant. Spectrosc. Radiat. Transf. **91**, 233–244 (2005). [CrossRef]

12. L. S. Rothmana, D. A. Jacquemarta, and A. Barbeb, *et al.*, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectrosc. Radiat. Transfer. **96**,139–204 (2005). [CrossRef]

^{-1}. Thus, we use a spectral resolution of Δ

*ν*=1 cm

^{-1}in this study. The mean spectral optical depth is computed over an interval of 1 cm

^{-1}in this paper. The degraded optical depth (red line in Fig. 1) with a spectral resolution of 1 cm

^{-1}is much smoother than its monochromatic counterpart.

_{2}absorption coefficient within one-wavenumber spectral interval centered at 2336.0 cm

^{-1}for a pressure of 0.1 hPa and a temperature of 200 K. Even within 1 cm

^{-1}, there are tens to hundreds of absorption lines, and the absorption coefficient varies by several orders (up to 10 orders). Thus, the mean transmittance within 1 cm

^{-1}no longer obeys Beer’s law.

*p*), temperature (

*t*), and absorber amount (

*u*). Because the mean spectrally absorption coefficient within bandwidth Δ

*ν*is a smooth function of absorber amount for a specific temperature

*t*and a pressure

*p*, we assume that the mean transmittance and mean absorption coefficient within a spectral width of 1 cm

^{-1}at wavenumber

*ν*be expressed as follows:

*k̄*is defined as:

_{ν}*C*are computed by non-linear regression of LBLRTM spectrally averaged transmittances versus absorber amount values that were allowed to vary by seven orders. In practice, M=4 is sufficient. The error introduced by omitting the contributions of M larger than 4 is smaller than 0.01% for most cases. The present fitting computation is carried out for 9 temperatures (200, 215, 230, 245, 260, 275, 290, 305 and 320 k), 9 pressures ranging from 0.05 hPa to 1100 hPa, and wavenumbers from 1 to 25000 cm

_{i}^{-1}with a resolution of 1cm

^{-1}. In terms of temperature and pressure, most realistic atmospheres are within the ranges considered in the present study. A database of the fitting coefficients

*C*for the above mentioned temperatures and pressures is obtained for seven main atmospheric absorption molecular species (H

_{i}_{2}O, CO

_{2}, O

_{3}, CO, N

_{2}O, CH

_{4}, and O

_{2}).

_{2}at 2383.0 cm

^{-1}as functions of temperature and pressure. Because the variations of the fitting coefficients

*C*versus temperature and pressure are smooth at all wavenumbers, the coefficients for arbitrary temperatures and pressures can be interpolated from the values corresponding to the aforementioned 9 temperatures and 9 pressures. Note that

_{i}*C*is close to zero. Thus, it is not necessary to use M larger than 4 in Eq. (3).

_{4}_{2}transmittances computed from the LBLRTM and FFTM methods with an absorber amount over 7 orders at various pressures within 1 cm

^{-1}at a wavenumber of 2336 cm

^{-1}as shown in Fig. 2. Figure 5 shows H

_{2}O transmittances at 5030 cm

^{-1}for various pressures, temperatures and absorber amounts. It is evident from Figs. 4 and 5 that the Eqs. (2) and (3) are suitable for describing the mean atmospheric transmittance, as the results from the FFTM are quite satisfactory over a large range of absorber amount up to 7 orders in comparison with LBLRTM. The fitting method is computationally efficient because the absorption is parameterized as a function of the absorber amount in Eq. (3).

13. Q. Fu and K. N. Liou, “A three-parameter approximation for radiative transfer nonhomogeneous atmosphere: application to the O3 9.6 μm band,” J. Geophys. Res. **97**, 13051–13058 (1992). [CrossRef]

14. L. S. Bernstein, A. Berk, and P. K. Acharya, *et al.*, “Very narrow band model calculations of atmospheric flux and cooling rates,” J. Atmos. Sci. **53**, 2887–2904 (1996). [CrossRef]

*k̅*is defined by Eq. (3) which is a function of temperature

*t*, pressure

*p*and absorber amount

*u*.

*U*is integrated absorber amount, which can be expressed as follows:

_{3}9.6 μm band transmission calculation. A number of three-parameter CG approaches have been developed to remedy this problem [13

13. Q. Fu and K. N. Liou, “A three-parameter approximation for radiative transfer nonhomogeneous atmosphere: application to the O3 9.6 μm band,” J. Geophys. Res. **97**, 13051–13058 (1992). [CrossRef]

14. L. S. Bernstein, A. Berk, and P. K. Acharya, *et al.*, “Very narrow band model calculations of atmospheric flux and cooling rates,” J. Atmos. Sci. **53**, 2887–2904 (1996). [CrossRef]

_{3}9.6 μm band could result in relatively large errors in the FFTM calculation in this study. To overcome this shortcoming, we introduce a new parameter

*f*for the O

_{3}9.6 μm band from 990 to 1070 cm

^{-1}as follows:

*U*is the vertical column ozone amount, and

_{0}*U*is ozone amount along the optical path.

*f*is obtained on the basis of comparing the results from Eq. (5) with the corresponding LBLRTM results at various values of

*U*/

*U*spanning from 0.2 to 5. In this study, a lookup table of the correction coefficient

_{0}*f*is developed for practical computations. Figure 6 shows the comparisons between the FFTM and LBLRTM computations after the aforementioned correction for the O

_{3}9.6 μm band for various paths and model atmospheres. Evidently, the FFTM results agree well with the LBLRTM solutions after the corrections are made.

11. S. A. Clough, M. W. Shephard, E. J. Mlawer, J. S. Delamere, M. J. Iacono, K. Cady-Pereira, S. Boukabara, and P. D. Brown, “Atmospheric radiative transfer modeling: a summary of the AER codes,” J. Quant. Spectrosc. Radiat. Transf. **91**, 233–244 (2005). [CrossRef]

15. D. C. Tobin, F. A. Best, P. D. Brown, S. A. Clough, R. G. Dedecker, R. G. Ellingson, R. K. Garcia, H. B. Howell, R. O. Knuteson, E. J. Mlawer, H. E. Revefrcomb, J. F. Short, P. F. W. van Delst, and V. P. Walden, “Downwelling spectral radiance observations at the SHEBA ice station: water vapor continuum measurements from 17 -26 μm,”J. Geophys. Res. **104**, 2081–2092 (1999). [CrossRef]

*T*. Specifically, for a homogenous atmosphere with a temperature

_{cn}*t*, a pressure

*p*, and an amount of each absorbing species (i.e.,

*u*,

_{1}*u*,

_{2}_{,}…

*u*), the total transmittance is given as follows:

_{k}*N*is the number of absorbing species, and

*U*,

_{1}*U*,

_{2}_{,}…

*U*are the integrated absorbing amounts of the

_{k}*N*absorbing species.

## 3. Validations

### 3.1 Validation against LBLRTM

^{-8}to more than 100. The fitting results agree with the LBLRTM quite well at all the wavenumbers.

^{-1}is displayed in the right panel of Fig. 10, where the differences between the two models are also shown with an offset of 1.1. The root-mean-square (rms) difference between the FFTM and LBLRTM results is 0.009 in the right panel of Fig. 10, indicating that the FFTM is applicable to non-homogeneous paths on the basis of the CG approximation.

_{2}O, CO

_{2}, O

_{3}, CO, N

_{2}O, CH

_{4}, and O

_{2}. A database of fitted coefficients for the spectral resolution of 1 cm

^{-1}is obtained for each gas at 9 discrete temperatures and 9 discrete pressures. The transmittance for each gas can be efficiently computed based on Eqs. (2) - (3) or Eqs. (5) - (7). The total transmittance at each wavenumber is the multiplication of the transmittances of all the gases and the continuum absorption.

^{-1}(8-10 μm) is shown in the right panel of Fig. 11. Evidently, the three models essentially agree with each other. The differences between the FFTM and LBLRTM are shown in the right panel of Fig. 11. The rms error of FFTM is approximately 0.012 in this diagram, less than the differences between LBLRTM and MODTRAN.

### 3.2 Comparisons with observations

16. J. A. Curry, P. V. Hobbs, and M. D. King, *et al.*, “FIRE arctic clouds experiment,” Bull. Amer. Meteorol. Soc. **81**, 5–29 (2000). [CrossRef]

^{-1}). The measurement data is averaged to match the 1 cm

^{-1}spectral resolution of the FFTM.

17. H. L. Wei, P. Yang, J. Li, B. A. Baum, H. L. Huang, S. Platnick, Y. X. Hu, and L. L. Strow, “Retrieval of ice cloud optical thickness from Atmospheric Infrared Sounder (AIRS) measurements,” IEEE Trans. Geosci. Remote Sensing. **42**, 2254–2267 (2004). [CrossRef]

^{-1}or lower). Given the specific instrumental response function of a sensor with a moderate- or high-spectral resolution, the present method can be applied to the simulation of the radiances observed by the sensor.

## 4. Summary

^{-1}for wavenumbers from 1 to 25000 cm

^{-1}. This model fits the mean transmittance for various values of absorber amount at 9 temperatures and 9 pressures based on the results computed from the LBLRTM. A database of the fitting coefficients was developed. An analytical expression based on a number of fitting coefficients is given for efficiently computing atmospheric transmittance. The CG approximation is used to extend the FFTM for transmittance computations for a non-homogenous atmosphere. The FFTM has been applied to 7 main absorbing gases in the atmosphere. In most cases, the FFTM results agree well with the LBLRTM counterparts. Furthermore, it was also shown that the simulations based on the FFTM agree well with the measurements acquired by the satellite-borne AIRS and airborne HIS instruments. The present model can be used to accurately and efficiently compute the infrared transmittances with a moderate- or high-spectral-resolution (1 cm

^{-1}or lower).

## Acknowledgments

## References and links

1. | S. A. Clough and M. J. Iacono, “Line-by-line calculations of atmospheric fluxes and cooling rates: Part II: Application to carbon dioxide, ozone, methane, nitrous oxide, and the halocarbons,” J. Geophys. Res. |

2. | D. P. Kratz, G. M. Mlynczak, and C. J. Mertens, |

3. | L. L. Strow, S. E. Hannon, S. Souza-Machado, H. E. Motteler, and D. C. Tobin, “An overview of the AIRS radiative transfer model,” IEEE Trans. Geosci. Remote Sens. |

4. | L. Moy, D. C. Tobin, P. Delst, and H. Woolf, “Clear sky forward model development for GIFTS,” (2004), http://ams.confex.com/ams/pdfpapers/71971. pdf. |

5. | L. M. McMillin, T. J. Kleespies, and L. J. Crone, “Atmospheric transmittance of an absorbing gas. 5. Improvements to the OPTRAN approach,” Appl. Opt. |

6. | R. W. Sunders, M. Matricardi, and P. Brunel, “An improved fast radiative transfer model for assimilation of satellite radiance observations,” QJRMS. |

7. | Q. Fu and K.N. Liou, “On the correlated K-distribution method for radiative transfer in non-homogeneous atmospheres,” J. Atmos. Sci. |

8. | D. P. Kratz and F. G. Rose, “Accounting for molecular absorption within the spectral range of the CERES window channel,” J.Quant. Spectrosc. Radiat. Transf. |

9. | J. L. Moncet, G. Uymin, and H. E. Snell, “Atmospheric radiance modeling using the optimal spectral sampling (OSS) method,” Proc. of SPIE |

10. | X. Liu, W. L. Smith, D. K. Zhou, and A. Larar, “Principal component-based radiative transfer model for hyperspectral sensors: theoretical concept,” Appl. Opt. |

11. | S. A. Clough, M. W. Shephard, E. J. Mlawer, J. S. Delamere, M. J. Iacono, K. Cady-Pereira, S. Boukabara, and P. D. Brown, “Atmospheric radiative transfer modeling: a summary of the AER codes,” J. Quant. Spectrosc. Radiat. Transf. |

12. | L. S. Rothmana, D. A. Jacquemarta, and A. Barbeb, |

13. | Q. Fu and K. N. Liou, “A three-parameter approximation for radiative transfer nonhomogeneous atmosphere: application to the O3 9.6 μm band,” J. Geophys. Res. |

14. | L. S. Bernstein, A. Berk, and P. K. Acharya, |

15. | D. C. Tobin, F. A. Best, P. D. Brown, S. A. Clough, R. G. Dedecker, R. G. Ellingson, R. K. Garcia, H. B. Howell, R. O. Knuteson, E. J. Mlawer, H. E. Revefrcomb, J. F. Short, P. F. W. van Delst, and V. P. Walden, “Downwelling spectral radiance observations at the SHEBA ice station: water vapor continuum measurements from 17 -26 μm,”J. Geophys. Res. |

16. | J. A. Curry, P. V. Hobbs, and M. D. King, |

17. | H. L. Wei, P. Yang, J. Li, B. A. Baum, H. L. Huang, S. Platnick, Y. X. Hu, and L. L. Strow, “Retrieval of ice cloud optical thickness from Atmospheric Infrared Sounder (AIRS) measurements,” IEEE Trans. Geosci. Remote Sensing. |

**OCIS Codes**

(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics

(010.1320) Atmospheric and oceanic optics : Atmospheric transmittance

(300.1030) Spectroscopy : Absorption

(300.6320) Spectroscopy : Spectroscopy, high-resolution

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: March 15, 2007

Revised Manuscript: May 18, 2007

Manuscript Accepted: May 30, 2007

Published: June 19, 2007

**Citation**

Heli Wei, Xiuhong Chen, Ruizhong Rao, Yingjian Wang, and Ping Yang, "A moderate-spectral-resolution transmittance model based on fitting the line-by-line calculation," Opt. Express **15**, 8360-8370 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-13-8360

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### References

- S. A. Clough and M. J. Iacono, "Line-by-line calculations of atmospheric fluxes and cooling rates: Part II: Application to carbon dioxide, ozone, methane, nitrous oxide, and the halocarbons," J. Geophys. Res. 100, 16519-16535 (1995). [CrossRef]
- D. P. Kratz, G. M. Mlynczak, C. J. Mertens, et al., "An inter-comparison of far-infrared line-by-line radiative transfer models," J. Quant. Spectrosc. Radiat. Transf. 90, 323-341 (2005). [CrossRef]
- L. L. Strow, S. E. Hannon, S. Souza-Machado, H. E. Motteler, and D. C. Tobin, "An overview of the AIRS radiative transfer model," IEEE Trans. Geosci. Remote Sens. 41, 303-313 (2003). [CrossRef]
- L. Moy, D. C. Tobin, P. Delst, and H. Woolf, "Clear sky forward model development for GIFTS," (2004), http://ams.confex.com/ams/pdfpapers/71971>pdf.
- L. M. McMillin, T. J. Kleespies, and L. J. Crone, "Atmospheric transmittance of an absorbing gas. 5. Improvements to the OPTRAN approach," Appl. Opt. 34, 8396-8399 (1995). [CrossRef] [PubMed]
- R. W. Sunders, M. Matricardi, and P. Brunel, "An improved fast radiative transfer model for assimilation of satellite radiance observations," Q. J. R. Meterol. Soc. 125, 1407-1425 (1999).
- Q. Fu and K. N. Liou, "On the correlated K-distribution method for radiative transfer in non-homogeneous atmospheres," J. Atmos. Sci. 49, 2139-2156 (1992). [CrossRef]
- D. P. Kratz and F. G. Rose, "Accounting for molecular absorption within the spectral range of the CERES window channel," J. Quant. Spectrosc. Radiat. Transf. 61, 83-95 (1999). [CrossRef]
- J. L. Moncet, G. Uymin, and H. E. Snell, "Atmospheric radiance modeling using the optimal spectral sampling (OSS) method," Proc. SPIE 5425, 368-374 (2004). [CrossRef]
- X. Liu, W. L. Smith, D. K. Zhou, and A. Larar, "Principal component-based radiative transfer model for hyperspectral sensors: theoretical concept," Appl. Opt. 45, 201-209 (2006). [CrossRef] [PubMed]
- S. A. Clough, M. W. Shephard, E. J. Mlawer, J. S. Delamere, M. J. Iacono, K. Cady-Pereira, S. Boukabara, and P. D. Brown, "Atmospheric radiative transfer modeling: a summary of the AER codes," J. Quant. Spectrosc. Radiat. Transf. 91, 233-244 (2005). [CrossRef]
- L. S. Rothmana, D. A. Jacquemarta, A. Barbeb, et al., "The HITRAN 2004 molecular spectroscopic database," J. Quant. Spectrosc. Radiat. Transfer. 96, 139-204 (2005). [CrossRef]
- Q. Fu and K. N. Liou, "A three-parameter approximation for radiative transfer nonhomogeneous atmosphere: application to the O3 9.6 μm band," J. Geophys. Res. 97, 13051-13058 (1992). [CrossRef]
- L. S. Bernstein, A. Berk, P. K. Acharya, et al., "Very narrow band model calculations of atmospheric flux and cooling rates," J. Atmos. Sci. 53, 2887-2904 (1996). [CrossRef]
- D. C. Tobin, F. A. Best, P. D. Brown, S. A. Clough, R. G. Dedecker, R. G. Ellingson, R. K. Garcia, H. B. Howell, R. O. Knuteson, E. J. Mlawer, H. E. Revefrcomb, J. F. Short, P. F. W. van Delst, V. P. Walden, "Downwelling spectral radiance observations at the SHEBA ice station: water vapor continuum measurements from 17 -26 μm,"J. Geophys. Res. 104, 2081-2092 (1999). [CrossRef]
- J. A. Curry, P. V. Hobbs, M. D. King, et al., "FIRE arctic clouds experiment," Bull. Amer. Meteorol. Soc. 81, 5-29 (2000). [CrossRef]
- H. L. Wei, P. Yang, J. Li, B. A. Baum, H. L. Huang, S. Platnick, Y. X. Hu, and L. L. Strow, "Retrieval of ice cloud optical thickness from Atmospheric Infrared Sounder (AIRS) measurements," IEEE Trans. Geosci. Remote Sens. 42, 2254-2267 (2004). [CrossRef]

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