## 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

It is critical to accurately calculate the absorption of radiation by molecules in the atmosphere in the study of the energy budget in the earth-atmosphere system, which is also essential to the implementation of various remote sensing techniques. The line-by-line (LBL) models [1] are the most accurate radiative transfer models for accounting for molecular absorption. Most recently, Kratz et al. [2] demonstrated that the LBL method is also quite accurate in the far-infrared spectral region. However, it is impractical to use the LBL method for many applications because of its significant demand on computational resources.

To minimize the computational effort, several moderate- to high-spectral- resolution fast models have been developed primarily for satellite-based remote sensing applications, such as the Stand-Alone Radiative Transfer Algorithm (SARTA) for the Atmospheric Infrared Sounder (AIRS) [3], the Pressure Layer Optical Depth (PLOD) for the Geostationary Imaging FTS (GIFTS) [4], the Optical Path Transmittance (OPTRAN) algorithm [5], and the Radiative Transfer TIROS Operational Vertical Sounder (RTTOV) [6]. These kinds of models are essentially regression-based approaches. Other types of fast models, such as the correlated-*k* distribution (CKD) method [7,8] and the Optimal Spectral Sampling (OSS) method [9], use a few representative monochromatic transmittances to derive narrowband or moderate- to high-spectral-resolution transmittance. These approaches compute the radiances at the top of the atmosphere (TOA) by using statistically selected TOA monochromatic radiances, and thus multiple scattering can be incorporated into these models in a straightforward way due to their pseudo-monochromatic feature. Most recently, Liu et al. [10] developed a novel method known as the Principle Component-based Radiative Transfer Model (PCRTM) for applications to hyperspectral remote sensing. Instead of predicating channel effective layer optical depths as in the aforementioned models, the PCRTM computes the Principle Component (PC) scores, which have much smaller dimensions as compared to the number of channels. This optimization leads to a significant decrease of computational effort without losing numerical accuracy.

This paper describes a new moderate-spectral-resolution transmittance model with a spectral resolution of 1 cm^{-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

In the present model, narrowband layer transmittance and layer-to-space or layer-to-layer transmittance are computed. We assume a random correlation between molecular species within a spectral width of 1 cm^{-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.

The mean transmittance over a homogenous path for a given temperature *t*, pressure *p* and absorber amount *u* within a wavenumber interval of Δ*ν* can be obtained via the following expression:

where *k _{ν}* is the absorption coefficient.

Figure 1 shows the mean spectral optical depth (defined as -log(*T̅ _{ν}*)) and the original monochromatic optical depth for water vapor in a small wavenumber range from 3500 to 3520 cm

^{-1}. The monochromatic optical depths in this paper are computed from the Line-By-Line Radiative Transfer Model (LBLRTM) [1, 11] using HITRAN2004 [12] molecular database. The monochromatic optical depth shows rapid variations with wavenumber. For most practical applications, the spectral band width in visible to infrared region is wider than 1 cm

^{-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.

To illustrate the variation of atmospheric absorption with wavenumber, Fig. 2 shows the CO_{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.

The mean transmittance for a specific atmospheric gas species in a homogeneous path is a function of pressure (*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:

where the mean absorption coefficient *k̄ _{ν}* is defined as:

The coefficients *C _{i}* 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

^{-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}).

Figures 3(a) and 3(b) show examples of the fitting coefficients for CO_{2} at 2383.0 cm^{-1} as functions of temperature and pressure. Because the variations of the fitting coefficients *C _{i}* 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

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

_{4}Figure 4 shows the CO_{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).

The Curtis-Godson (C-G) approximation method has been widely used to compute the transmittance for an inhomogeneous atmosphere for narrowband models [13, 14]. In this paper the C-G approximation for a non-homogeneous path is implemented by using effective fitting coefficients. The effective fitting coefficients are defined as follows:

where the mean absorption coefficient *k̅* is defined by Eq. (3) which is a function of temperature *t*, pressure *p* and absorber amount *u*.

The mean effective transmittance for a slant path can be written as

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

We also find that the following effective fitting coefficients can be as accurate as those defined by Eq. (4):

It is well known that the two-parameter CG approximation is inadequate for the O_{3} 9.6 μm band transmission calculation. A number of three-parameter CG approaches have been developed to remedy this problem [13, 14]. We also find that the application of the CG approximation of Eqs. (5) - (7) directly to the O_{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:

where *U _{0}* is the vertical column ozone amount, and

*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.

The continuum absorption of water vapor and other gases is computed using the new continuum model MT-CKD_1.0 [11] that has been updated with newly experimental data [15]. The MT-CKD continuum model includes the continuum absorption of water vapor, nitrogen, oxygen, carbon dioxide, and ozone. Additionally, the optical depth associated with Rayleigh scattering is also taken into account.

The total transmittance is the product of the individual molecular species transmittances and the contributions of the continuum absorptions *T _{cn}*. Specifically, for a homogenous atmosphere with a temperature

*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}The total transmittance for an inhomogeneous atmosphere is:

where *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

In this section, we first validate FFTM by comparing the FFTM and LBLRTM results for homogeneous paths and non-homogeneous slant paths. Then, we compare the results with some airborne or satellite-borne high-spectral infrared observations to further illustrate the accuracy of the FFTM.

#### 3.1 Validation against LBLRTM

Figure 7 shows the mean spectral optical depth for water vapor for a homogeneous path with t=250 k, p=1100 hPa, and u=3271 atm-cm. The solid line in Fig. 7 indicates the results computed from the LBLRTM algorithm, and the dashed line indicates the results from the FFTM. The mean optical depth of water vapor varies with a large range spanning from less than 10^{-8} to more than 100. The fitting results agree with the LBLRTM quite well at all the wavenumbers.

Figure 8 shows the comparison of the mean spectral transmittances computed from the LBLRTM and fitting methods for water vapor. A homogeneous path is assumed with t=200 K, p=1100 hPa, and u=100 atm-cm. The differences between FFTM and LBLRTM are shown in the upper part of the diagram with an offset of 1.05. The variance at each wavenumber is the root-mean-square (rms) difference between the two methods for 50 different absorbing amounts. Figure 9 shows the same comparisons but for a low pressure of 0.1 hPa. From Figs. 8 and 9, it is evident that the proposed FFTM is very accurate for homogeneous paths.

The results for a slant path by using CG approximation in Eqs. (5) - (7) are compared with the rigorous LBLRTM counterparts in the paragraph. Figure 10 shows the mean vertical spectral transmittance for water vapor from vertical path of 100 km to 0 km altitude using the AFGL mid-latitude summer atmospheric model with a 50 layer atmosphere. The effective coefficients and integrated absorber amount are computed from Eqs. (5) - (7). In Fig. 10, the solid lines indicate the LBLRTM results whereas the dash lines indicate the FFTM results. For clarity, a zoom-in plot for a wavenumber range of 600-800 cm^{-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.

We have applied the FFTM to 7 main absorbing gases in the atmosphere: H_{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.

To validate the results of the present method in the case of non-homogenous slant paths, we have compared the results obtained from the FFTM, LBLRTM and Modtran4.0. Figure 11 shows the atmospheric molecular transmittances computed from the FFTM, LBLRTM, and Modtran4.0 for a vertical path from 0 to 100 km based on the AFGL Mid-latitude summer atmospheric model. For clarity, the result for a wavenumber ranging from 1000 to 1250 cm^{-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.

The computational demand of the FFTM is small in comparison with LBLRTM. FFTM takes 0.97 second to compute the horizontal atmospheric spectral transmittance for seven major atmospheric gases including continuum absorption for wavenumbers from 1 to 25000 cm-1 on a 3.2 G Intel Pentium4 personal computer, while LBLRTM computations would take up to several minutes for low pressure cases.

#### 3.2 Comparisons with observations

To show the FFTM computation ability of moderate-to-high spectral resolution atmospheric transmittance or radiance, we compare the FFTM to AIRS and HIS observations in this subsection.

During the First ISCCP Regional Experiment Arctic Cloud Experiment (FIRE-ACE) [16], upwelling spectral radiances were observed by the high-resolution interferometer sounder (HIS) instrument onboard the ER-2 aircraft at an altitude of 20 km. A HIS brightness temperature spectrum at 22:18:01 UTC on May 22, 1998, a clear sky case, is used for comparison with the FFTM-based simulation. The radiances (brightness temperature) are computed from the FFTM with the profiles of temperature, pressure and relative humidity obtained from a nearby rawinsonde. The AFGL subarctic winter ozone profile, with a scaling factor, is used in the FFTM calculation. Figure 12 shows the comparison between the HIS observation and FFTM calculation for the HIS band 1 and band 2 (600–1700 cm^{-1}). The measurement data is averaged to match the 1 cm^{-1} spectral resolution of the FFTM.

We have further compared AIRS radiances with their counterparts calculated from the FFTM. The AIRS spectrum was taken from a clear sky overpass of Aqua over the United States on 6 September, 2002. The AIRS granule image was previously shown in [17]. The profiles of water vapor, temperature, and ozone, and the surface pressure and temperature, are from the ECMWF model. Figure 13 shows the observed AIRS spectral radiances and the calculated spectral brightness temperatures with FFTM.

Evidently from Figs. 11 and 12, the FFTM simulations generally agree with satellite-borne AIRS and airborne HIS observations. The overall consistency between FFTM and the airborne or satellite-borne observations shows that FFTM is applicable to the interpretation of the moderate- to high-spectral observations.

It should be pointed out that the FFTM is developed for general applications with a moderate spectral resolution (1 cm^{-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

We have developed a new moderate-spectral-resolution transmittance model, namely, FFTM, with a spectral resolution of 1 cm^{-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

Heli Wei acknowledges support by the President Foundation of Hefei Institutes of Physical Science and the Creative Foundation of Chinese Academy of Sciences (KGCX2-SW-413). Ping Yang acknowledges support from the National Science Foundation of the United States (ATMO-0239605).

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