## 1. Introduction

*F*contributed by the outgoing radiances

*I*emerging from the top of the atmosphere (TOA) at all possible angles. Their relationship can be expressed as Eq. (1), assuming azimuthal homogeneity:

*θ*is the zenith angle,

*ϕ*is the azimuth angle,

*π*:

With the advancement of measurement technology as well as the improvement of radiation codes, spectrally decomposed radiation has been increasingly used for weather and climate monitoring and model validation (Huang et al. 2007; Feldman et al. 2015; Huang 2013; Bani Shahabadi et al. 2016). When simulating spectrally decomposed OLR fluxes from an atmospheric profile, an adequate radiance-to-irradiance conversion scheme is required. The DFA method with a constant diffusivity angle is often used in such applications to reduce computing costs (e.g., Huang et al. 2013; Pan and Huang 2018). However, the DFA method, although widely used for the spectrally integrated OLR, is not validated for the spectrally decomposed irradiance fluxes. As shown in the following sections, using constant-angle DFA across the longwave spectrum (100–2800 cm^{−1}) would lead to considerable biases. Other authors have discussed relevant problems. For instance, Mehta and Susskind (1999) used empirically determined diffusivity angles for different spectral bands that are independent of atmospheric condition. Zhao and Shi (2013) proposed a bridge function-based scheme to calculate the diffusivity angle according to the optical depth. In general, these methods were based on the consideration of angular integration of radiance in an isolated atmospheric layer and thus do not account the geometry of the line of sight, which in reality is complicated, for instance, by the curvature of Earth’s surface.

In this paper, based on our physical interpretation of the DFA, we propose a method for determining the diffusivity angle at every frequency according to specific atmospheric conditions. This method treats the window band (weak atmospheric absorption) and gas absorption bands (strong atmospheric absorption) differently, which as shown below is necessary according to the physical interpretation of the DFA method. We also provide a scheme that corrects the bias due to the path geometry. The paper is structured as follows: in section 2, we present our physical interpretation of the DFA method; in section 3, we adjust the scheme to account for the effect of spherical Earth surface geometry; the performance and implication of this method are demonstrated in section 4.

## 2. Physical model

*θ*at spectral frequency of

*υ*is (Goody and Yung 1989)

*τ*is the nadir optical depth from TOA to a vertical level, and

*υ*. The first term of Eq. (3) corresponds to the atmospheric contribution

*τ*. According to Planck function, the blackbody emission

*τ*, given Eq. (4):

^{−1}(in a CO

_{2}absorption band). In this comparison, the temperature profile

*W*is the so-called weighting function, which is the derivative of transmission function with respect to altitude

*z*:

*W*is only a function of

*τ*, in the limb-view case, the same set of

*μ*:

This solution

The physical derivation above also indicates under what condition the DFA as given by Eq. (15) would fail: when the ELDM does not apply. Most notably, Eq. (12) assumes the surface contribution *τ* approaches zero (Li 2000; Zhao and Shi 2013). In the following section, we will use a different approach to find an appropriate diffusivity angle in the window band.

## 3. Window band model

Because of its sensitivity to the surface climate, the radiance in the window is especially important for climate monitoring. As addressed in the previous section, the ELDM poorly represents the window band. In this section, we focus on finding an approach better suited for the weak atmospheric absorption feature in the window band.

In the window band, the surface emission is a primary contributor to the TOA flux. With an elevated viewing angle, which creates a longer optical path, the atmosphere becomes more opaque, leading to more contribution from the atmosphere and less from the surface. Thus both atmospheric and surface effects are nonnegligible.

*α*and

^{−1}. As shown in Fig. 1c, this approximation [Eq. (18)] generally yields error within 5% in the window band compared to the truth [

*θ*and the total optical depth

*μ*in a Taylor series expansion at

To validate the above result [Eq. (24)], we set a simple radiative transfer model that accurately computes the source function of each atmospheric layer and vertically integrates it according to Eq. (3). The directional radiance is obtained at each 0.01° between 0° and 89.99°, from which the irradiance flux is integrated. We then match the directional radiance and irradiance to find the

Diffusivity angle considering (a) plane-parallel and (b) realistic path geometry. In (a), the approximation [Eq. (24)] and truth are shown as solid and dashed lines, respectively. In (b), approximation [Eq. (29)] and truth are shown as solid and dashed lines, while different atmospheric lapse rates (K km^{−1}) are indicated by color. The dotted line is the constant diffusivity angle at 52.66°.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0246.1

Diffusivity angle considering (a) plane-parallel and (b) realistic path geometry. In (a), the approximation [Eq. (24)] and truth are shown as solid and dashed lines, respectively. In (b), approximation [Eq. (29)] and truth are shown as solid and dashed lines, while different atmospheric lapse rates (K km^{−1}) are indicated by color. The dotted line is the constant diffusivity angle at 52.66°.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0246.1

Diffusivity angle considering (a) plane-parallel and (b) realistic path geometry. In (a), the approximation [Eq. (24)] and truth are shown as solid and dashed lines, respectively. In (b), approximation [Eq. (29)] and truth are shown as solid and dashed lines, while different atmospheric lapse rates (K km^{−1}) are indicated by color. The dotted line is the constant diffusivity angle at 52.66°.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0246.1

## 4. Spherical correction

In the previous discussion, a plane-parallel geometry is assumed, so that the zenith angle *θ* from the surface level to the TOA is constant along the line of sight. As shown in Fig. 3, in reality, the effective zenith angle *z* and becomes larger farther away from the observer, resulting in an extended geometry path compared to plane-parallel geometry, because of the curvature of Earth’s surface.

Path zenith angle affected by Earth surface curvature. *H* is the observer height, and *z* is the emission-layer height. *θ* and *H* and the emission-layer *z*, respectively.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0246.1

Path zenith angle affected by Earth surface curvature. *H* is the observer height, and *z* is the emission-layer height. *θ* and *H* and the emission-layer *z*, respectively.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0246.1

Path zenith angle affected by Earth surface curvature. *H* is the observer height, and *z* is the emission-layer height. *θ* and *H* and the emission-layer *z*, respectively.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0246.1

*z*:

*τ*–

*z*relationship described in Eq. (5), we have

In contrast to previous results, where the atmospheric condition is not relevant in the solution, now *α* becomes an important factor influencing the outgoing irradiance when the total optical depth is small. This result illustrates that, aside from optical properties, the choice of diffusivity angle is also dependent on atmospheric conditions, such as surface temperature and lapse rate, which determine *α*. This is a factor ignored in previous studies.

Following a similar simulation test described in previous section, Fig. 2b reveals that, with the effect of the surface curvature, the choice of diffusivity angle decreases with an increasing Γ, accounting for the effect of increasing *α*; similar dependence can be found on the wavenumber *υ* and

## 5. Validation tests and discussion

In this section, we perform a series of tests with different atmospheric profiles to verify the accuracy of our revised DFA algorithms, in comparison with the conventional constant diffusivity-angle approach.

In these tests, we calculate the optical depths using a radiative transfer model MODTRAN 5.2, from 100 to 2800 cm^{−1} at 0.1-cm^{−1} resolution, and then apply a path geometry model that we develop to generate the directional radiance. The path geometry model here can implement either plane-parallel geometry or the realistic path that combines refraction and spherical Earth surface. The realistic path geometry model has been validated against the geometry package of MODTRAN. Note that the realistic path geometry option considers refraction, although its impact is insignificant compared to the spherical geometry, which we focus on here. We angularly integrate the directional radiances from 0° to 90° zenith angle at a 0.1° interval following Eq. (2), at each wavenumber, to generate the reference spectral flux (truth).

In addition to the U.S. standard profile (McClatchey et al. 1972), errors averaged in five bands with different atmospheric profiles are shown in Table 1. Besides the two DFA approaches, we also include a double Gaussian quadrature (2GQ) method (Li 2000) in the comparison. The result (Table 1) shows that the 2GQ method generally achieves better accuracy than the constant-angle DFA method (equivalently a 1GQ method) at the price of doubled computation. However, there is a similar pattern of overestimation in the window band to the constant-angle DFA method when 2GQ is used, resulting from the neglect of the surface curvature effect. As demonstrated in Figs. 4c and 4d, the surface curvature effect may lead to larger than 1 K of irradiance differences, and it is the major source of error in this case. Failure to account for this effect leads to up to 1.5 W m^{−2} overestimation with the constant-angle DFA approach. To overcome this problem, we propose a similar correction scheme to the Gaussian quadrature (GQ) method in Eq. (A4) in the appendix and generally reduce the error to 0.55 W m^{−2} as presented in Table 2, although 2GQ shows little improvement compared to the constant-angle DFA method after this correction.

Errors (W m^{−2}) in five spectral bands (cm^{−1}) caused by a constant diffusivity angle

Absolute error of irradiance flux [expressed by blackbody brightness temperature (K)] induced by constant diffusivity angle (red, *y* axis on the right. These results are smoothed using a median filter of width = 10 cm^{−1}.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0246.1

Absolute error of irradiance flux [expressed by blackbody brightness temperature (K)] induced by constant diffusivity angle (red, *y* axis on the right. These results are smoothed using a median filter of width = 10 cm^{−1}.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0246.1

Absolute error of irradiance flux [expressed by blackbody brightness temperature (K)] induced by constant diffusivity angle (red, *y* axis on the right. These results are smoothed using a median filter of width = 10 cm^{−1}.

Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-18-0246.1

As presented in Fig. 4 and Table 1, the spectrally dependent DFA given in this study approximates the truth very well throughout the spectra. It improves the constant-angle DFA approach, and its accuracy is comparable with 2GQ with a plane-parallel geometry in window bands. Compared to the GQ methods, it achieves a noticeable improvement with errors being within 0.50 W m^{−2} in all atmospheric conditions even when a realistic geometry is considered. However, a few caveats should be noted. From the derivation above, it is clear that the solution [Eqs. (29) and (30)] relies on the condition that a constant lapse rate prevails in the vertical range over which the weighting function spans. It means that the DFA approach may incur more uncertainty at the frequencies where the weighting function peaks near an abrupt change in lapse rates, such as around the tropopause or near a surface temperature inversion layer. The same limitation also exists in the correction of the surface geometry effect in Eq. (28). Moreover, the analytical solutions derived here depend on the *τ*–*z* relation given by Eq. (5). This relation may be obscured by clouds, especially in a partially cloudy sky. In the case of an overcast, thick cloud layer, for example, occurring at the tropopause level, because the relation largely retains above the cloud, we find our approach yields errors within 0.5 K throughout the middle infrared spectrum.

In conclusion, this study reinvestigates the diffusivity angle approach. We find a physical explanation of why the conventional constant-angle DFA works in the absorption bands and an analytical solution to improve the accuracy in the window bands. This solution takes account of both optical and atmospheric properties while keeping the simplicity of the expression.

## Acknowledgments

We thank Jiangnan Li and two anonymous reviewers for their comments. This work is supported by a grant from the Discovery Program of the Natural Sciences and Engineering Research Council of Canada (Grant RGPIN-2019-04511). The source code of this diffusivity angle approach can be downloaded online (https://huanggroup.wordpress.com/2018/10/02/diffusivity-angle-script/).

## APPENDIX

### Spherical Correction of Gaussian Quadrature Method

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