*θ*being SZA, and

To minimize bias in solar absorption, Cronin (2014) points out that SZA should be chosen to most closely match the spatial- or time-mean planetary albedo, and he found “the absorption-weighted zenith angle is usually between the daytime-weighted and insolation-weighted zenith angles but much closer to the insolation-weighted zenith angle” (p. 2994).

Should the averaged SZA be determined by minimizing the bias in planetary albedo? And is the insolation-weighted-mean SZA more accurate than the daytime-mean SZA? We do not agree with either point as explained in detail below.

We denote the solar upward flux (or reflected flux) at the top of the atmosphere (TOA), downward flux (or transmitted flux) at surface and atmospheric absorption as *z* and

Figure 1a shows the relative errors by using *k* distribution scheme for gaseous transmission with O_{3}, H_{2}O, O_{2}, CH_{4}, and CO_{2} included in the solar spectrum range. The surface albedo varies from 0.1 to 0.6, covering most of the surface albedos of Earth. It is found that the relative error of

*n*-node Gaussian quadrature, the integration of moment

*l*is evaluated by

*x*,

*n*= 1), for the zero moment (

*l*= 0),

*l*= 1),

In Fig. 1b the benchmark results are calculated based on (5). The relative error of

Cronin (2014) has proposed an effective solar constant. According to Cronin, for

Figure 2 is the same as Fig. 1, but we replace

As in Fig. 1, but using the effective solar constant.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

As in Fig. 1, but using the effective solar constant.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

As in Fig. 1, but using the effective solar constant.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

Cronin (2014) has proposed a concept of absorption-weighted SZA,

Cronin (2014) shows that

The concept of

*δ*is the declination of the sun,

*φ*is the local latitude, and

*h*is the local solar time. The local solar time is from 0 to 24 h or converted to hour angle as 24 h = 2

*π*. The declination is accurately estimated by using the parameters of Earth’s orbit:

*N*is the day of the year beginning with

*h*; thus, the diurnal-mean cosine of SZA is

*μ*, which is equal to

The errors of the (top) upward solar flux at TOA, (middle) downward solar flux at the surface, and (bottom) atmospheric solar absorption. The benchmark results are calculated following (10). Results based on (left) diurnal-mean SZA of (9) and (right) diurnal insolation-weighted SZA of (11).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

The errors of the (top) upward solar flux at TOA, (middle) downward solar flux at the surface, and (bottom) atmospheric solar absorption. The benchmark results are calculated following (10). Results based on (left) diurnal-mean SZA of (9) and (right) diurnal insolation-weighted SZA of (11).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

The errors of the (top) upward solar flux at TOA, (middle) downward solar flux at the surface, and (bottom) atmospheric solar absorption. The benchmark results are calculated following (10). Results based on (left) diurnal-mean SZA of (9) and (right) diurnal insolation-weighted SZA of (11).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

By using ^{−2} in almost all the domains. The errors of ^{−2}, especially in the summer season. However, in this region the downward flux at the surface is generally over 500 W m^{−2}, the relative errors are only about 2%. For ^{−2} in the tropics and during the summer season in high-latitude regions.

The small error in solar flux and atmospheric solar absorption indicates that the daytime-mean SZA can characterize the solar insolation distribution in the atmosphere, as

By using the insolation-weighted mean, large errors occur as shown in the right column of Fig. 3. For ^{−2} in most of the regions. For ^{−2}!

*φ*on day

*N*,

*φ*. In Fig. 4, the annual-mean meridional distributions of

(a) The annual- and seasonal-averaged latitudinal distributions of diurnal-mean SZA from (13); the solid line is the result of North et al. (1981). (b) As in (a), but for diurnal insolation-weighted SZA. Seasons are June–August (JJA) and December–February (DJF).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

(a) The annual- and seasonal-averaged latitudinal distributions of diurnal-mean SZA from (13); the solid line is the result of North et al. (1981). (b) As in (a), but for diurnal insolation-weighted SZA. Seasons are June–August (JJA) and December–February (DJF).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

(a) The annual- and seasonal-averaged latitudinal distributions of diurnal-mean SZA from (13); the solid line is the result of North et al. (1981). (b) As in (a), but for diurnal insolation-weighted SZA. Seasons are June–August (JJA) and December–February (DJF).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

Figure 4a shows the comparison of the annual-mean SZA values. The parameterization proposed by North et al. (1981) is very close to the benchmark result, with relative errors of about 4% in the midlatitude region. In Ballinger et al. (2015), the annual-mean meridional distribution of SZA is parameterized as well. That parameterization strongly overestimates SZA in the higher latitudes.

Equation (13) is a general result based on an analytical formula of (9). By (13) the monthly or seasonal-mean SZA can be easily obtained, and this will be very useful for simplified climate models (Ballinger et al. 2015).

Similar to

*N*.

Figure 5 shows the global- and regional-averaged daytime-mean SZA. The results generally depend on the chosen latitude region and day number. However the global averaged *μ* changes from 1 to 0. This variation range of *μ* is the same as (1); thus, the integral in (1) represents an averaged SZA along any radius. Therefore, (1) is an averaged SZA on global scale, whereas (15) is an averaged SZA from a global integral over local daytime-mean SZA. It is interesting to find that the two different approaches lead to the same result. The globally averaged SZA is a constant regardless of the day number and Earth’s axial tilt angle.

The global- and regional-averaged diurnal-mean SZA. The tropics cover 0°–15°N, the midlatitudes cover 15°–45°N, and the subarctic covers 45°–60°N.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

The global- and regional-averaged diurnal-mean SZA. The tropics cover 0°–15°N, the midlatitudes cover 15°–45°N, and the subarctic covers 45°–60°N.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

The global- and regional-averaged diurnal-mean SZA. The tropics cover 0°–15°N, the midlatitudes cover 15°–45°N, and the subarctic covers 45°–60°N.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

Our calculations are based on the clear-sky condition. There is no physical meaning to study the averaged SZA over cloudy sky since clouds change from time to time. This is why the averaged SZA is usually applied to a clear sky, especially in the stratosphere, which is cloud free (Hogan and Hirahara 2016). However, if we assume that clouds remain the same in shape and location over an integral time period as done in Cronin (2014), the result of cloudy sky is very similar to that of clear sky as shown above.

In summary, the averaged SZA cannot be determined by minimizing the bias in planetary albedo. The accuracy of the insolation-weighted-mean SZA in planetary albedo is caused by the cancellation of two positive errors. On both a global scale and a latitude-dependent local scale, it is misleading to say that the insolation-weighted-mean SZA is more accurate than the daytime-mean SZA. The choice of daytime-mean SZA or insolation-weighted-mean SZA depends on the averaging process, as (3) or (5) for the global scale, and (10) or (12) for the latitude-dependent local scale. For radiation variables, the weighting process should follow (3) or (10) because the solar insolation has been built into the radiative transfer calculations, and the daytime-mean SZA should be used. However, if the solar insolation is weighted to a physical variable as in (5) or (12), the insolation-weighted-mean SZA should be used.

## Acknowledgments

The author thanks Dr. H. Barker and anonymous reviewers for their help comments and Professor P. Yang for his editorial efforts.

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