1. Introduction
Since the seminal work by Kaplan (1959), thermal IR radiation has been used extensively to infer the vertical structure of atmospheric temperature and greenhouse gases (most notably, water vapor, ozone, and CO2), as well as cloud properties such as cloud-top temperature and cloud optical depth. As far as IR cloud remote sensing is concerned, many retrieval algorithms have been implemented in actual operations, to name a few, split-window algorithms (e.g., Inoue 1985; McMillin 1975), CO2 slicing algorithm (e.g., Smith et al. 1970; Chahine 1974; McCleese and Wilson 1976), and optimal estimation algorithms (e.g., Wang 2016a,b).
Given the well-known importance of thermal contrast in IR remote sensing, an interesting question to ask is, If scattering is presented in such an isothermal atmosphere, how much could it affect the above statement? Will it make an observable difference at the top of the atmosphere? In reality, clouds can have single scattering albedo as high as 0.85–0.9 in certain IR window or microwindow regions, e.g., ice cloud at the far-IR dirty window around 400 cm−1 and mid-IR window around 1100 cm−1, and liquid clouds in the shortwave-IR window around 2700 cm−1 (Kuo et al. 2017). Even though, in reality, the entire atmosphere column cannot be isothermal, there are situations where the lower troposphere can be nearly isothermal. For example, the polar wintertime or the extrapolar regions during episodes of cold-air outbreaks tend to have nearly isothermal lower troposphere with persistent stratus clouds coverage at the same time. Thus, such a theoretical consideration of scattering in an isothermal atmosphere can be a good starting point to explore some real-world applications too.
The remaining sections are arranged as follows. Section 2 presents two-stream analytical solutions, as well as their physical interpretations, to the scattering in an isothermal atmosphere for a variety of boundary conditions. Section 3 further illustrates the problem with a Monte Carlo approach and photon budget analysis. Section 4 quantifies the effect of scattering with a more sophisticated radiative transfer model with realistic molecular spectroscopy and cloud optical properties. Further discussion and speculations are presented in section 5.
2. Two-stream analytical solutions
a. A trivial solution: Blackbody surfaces at both upper and lower boundaries
b. Blackbody bottom and space top
1) Pure absorption case ( )
2) Pure forwarding scattering ( )
3) Semi-infinite atmosphere (τ0 +∞)
4) Generic cases
Figure 1 shows the solutions to several combinations of
The two-stream solution for an isothermal atmosphere with an underlying blackbody surface of the same temperature and zero downward radiance at the top of the atmosphere. The (left) upwelling and (right) downwelling radiances are shown. The blackbody radiation at the surface and atmospheric temperature is 100 (arbitrary unit). Results from different combinations of single scattering albedos and asymmetry factors are shown in different colors as labeled.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0050.1
The radiance divergence, i.e., d(I↑ − I↓)/dτ, for the same four solid line cases and two dashed line cases shown in Fig. 1.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0050.1
Contrasting results between sections 2a and 2b, it is obvious that the boundary condition matters here for such deviation from blackbody behavior. In the real world, the lower troposphere can be close to isothermal in some weather conditions. Downwelling radiance at the upper boundary of such an isothermal layer is not zero, which is different from the idealized case above. However, scattering can still make a difference even if there is downwelling radiance at the upper boundary. The only exception is that the amount of downwelling radiance across the upper boundary is the amount of radiance emitted by a surface with the same temperature as the isothermal-layer temperature [i.e., the boundary condition described by Eq. (7) in section 2a].
3. Monte Carlo simulation and photon “budget” analysis
To further understand the contribution of scattering to TOA radiances, we carried out a two-stream Monte Carlo simulation so the trajectory of each photon can be traced and counted. In the simulation, photons can be emitted from the surface as well as any layers within the atmosphere, and they travel in either upward or downward directions. When traveling through the atmosphere, each photon has a chance to interact with particles within a layer. The probability is dictated by the optical thickness of the layers. For those which interact with the atmosphere, they could be absorbed or scattered, forward or backward, as determined by single scattering albedo and asymmetry factor. We trace each photon to its destination and count the number of photons that get absorbed by each layer in the atmosphere, reach the surface, or reach the TOA, respectively. To be consistent with the two-stream analytical solution in the previous section, the zenith angle (μ) along the upward and downward direction is set at cosμ = 1/1.66. The number of photons released from the surface is 106. The atmosphere is divided into 100 layers with equal optical thickness and the number of photons emitted from each layer depends on the emissivity of the layer (i.e., 106 photons are emitted if ε = 1). Compared to the theoretical derivation, the Monte Carlo method allows us to trace each photon and count the statistics. Table 1 shows such budget analyses for three cases in Figs. 1 and 2:
Photon budget analysis based on two-stream Monte Carlo simulation for three opaque cases. Photons are traced and counted for their emission origins (surface, upward or downward from the atmosphere) and their final destinations (reaching TOA, absorbed by the surface, and absorbed by the atmosphere). Cases 1 and 2 have identical extinction optical depth and cases 2 and 3 have identical absorptive optical depth.
Cases 1 and 2 have the same extinction optical depth but case 1 has no scattering. Cases 2 and 3 have the same absorptive optical depth (thus the same number of photons emitted by the atmosphere) but case 3 has no scattering. The results from Monte Carlo simulations agree well with the analytical solutions, as shown in Fig. 3. The differences between them are within 0.25 units throughout the atmosphere (inset in each panel).
The radiance fields for an isothermal atmosphere for the three cases investigated in Table 1. (left) Upwelling and (right) downwelling radiance are shown. The surface has the same temperature as the atmosphere, and its blackbody emission is 100 (arbitrary unit). Dash–dotted lines are results from a two-stream Monte Carlo simulation, and solid lines from the two-stream analytical solutions. They may not be visible since they overlap each other. The inset in each panel shows the radiance difference between the analytical and the Monte Carlo solutions.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0050.1
For the cases with the same extinction optical depth, total photons emitted by the atmosphere are reduced in the presence of scattering. As a result, although a higher percentage of atmospheric upward emission can reach the TOA in case 2 than in case 1 (10.4% vs 6.5%), case 1 still has more photons originating from the upward atmosphere emission and reaches the TOA than case 2 does. In case 2, only 0.9% of the total photons emitted downward are reflected to the TOA, which cannot change the contrast between case 1 and case 2.
For the cases with the same absorption optical depth, the number of photons emitted by the atmosphere is thus identical. Scattering in case 2 leads to ∼0.9% of upward photons emitted by the atmosphere being removed by reflection to the surface, but it also puts the same number of downward photons emitted by the atmosphere reflected to the TOA. Meanwhile, scattering leads to more atmospheric absorption of photons (88.7% in case 2 vs 87.5% in case 3). As a result, the total number of photons reaching the TOA is less in case 2 than in case 3. For these cases, photons emitted from the surface have little chance of reaching the TOA due to the large optical depth. Case 2 shows 10% of surface emission reflected back to the surface due to scattering. Accordingly, there is a reduction in the number of photons reaching the TOA (or absorbed by the atmosphere) compared to case 3. If we let g = 1 in case 2, the photon budget will be identical to that of case 3 (no more than 0.1% difference in any category due to Monte Carlo sampling), consistent with the analytical solution discussed in section 2b(2).
Table 2 shows three sets of nonopaque cases. The budget analyses depicted above can be applied to such nonopaque cases as well. In a nutshell, for the cases with the same extinction optical depth, scattering implies less emission from the atmosphere, which leads to a reduction of the radiances that can reach the TOA. Reflection of surface emission back to the surface further contributes to the decrease of TOA upwelling radiances. For the cases with the same absorptive optical depth, backscattering contributes to the reduction of upwelling radiance at the TOA in two ways: 1) reducing the number of photons that are emitted from the surface and reach the TOA by the combined effect of atmospheric reflection and absorption; 2) reducing the number of photons emitted from the atmosphere and reach the TOA by increased atmospheric absorption (a consequence of scattering photons backward instead of forwarded). Note that, in this isothermal configuration, the reflection of atmospheric upward and downward emissions offset each other and make no contribution to the change of TOA upwelling radiance (as well as the change of surface downwelling radiance). As far as the TOA upwelling radiance is concerned, the real role of backward scattering is to increase the chance of photons being absorbed within the atmosphere instead of reaching the boundaries (both top and bottom).
4. A more realistic simulation: 2016 East Asia cold wave
The above discussions are all based on highly idealized situations. To explore to what extent such isothermal scattering can affect the actual observations, we conduct more realistic simulations using actual sounding profiles of temperature and humidity and more sophisticated radiation transfer modeling, moderate resolution atmospheric transmission (MODTRAN) v5.2. MODTRAN is a radiative transfer code extensively used by the remote sensing community. Further details of it can be found in Anderson et al. (2007).
From 18 to 26 January 2016, East Asia was struck by a strong and broad cold-air outbreak. On 24 January, the Hong Kong Observatory at King’s Park (WMO station ID 45004) recorded its coldest temperature in 59 years. The cold wave left Hong Kong blanketed by low clouds over days, with the lower troposphere nearly isothermal. Figure 4 shows the sounding profiles of temperature and dewpoint temperature reported by the King’s Park station in Hong Kong at 1200 UTC 23 January 2016, as well as the true-color image, retrieved cloud optical thickness, and cloud-top pressure based on Aqua MODIS retrievals at 0600 UTC 23 January 2016, as summarized on NASA Worldview website (https://worldview.earthdata.nasa.gov/). It is clear from the temperature profile that the atmosphere is close to isothermal from the surface to ∼3 km above. By using such a sounding profile, our primary goal in this section is to explore whether it is possible to detect water clouds within the isothermal layer from the TOA radiance, instead of developing a fully operational retrieval algorithm. Therefore, to produce simulated radiance as close as possible to the observed MODIS radiance is not a focus here, as many other factors besides cloud scattering can contribute to the differences between simulated and observed radiances.
(left) The sounding profile obtained at 1200 UTC 23 Jan 2016 by the Hong Kong Observatory: air temperature in red and dewpoint temperature in green. The blue hatched area indicates the cloud region set in the simulation. The purple line indicates the temperature profile assumed in cases 3 and 4, i.e., isothermal from the surface to 700 hPa and then identical to the actual sounding profile. (right) MODIS (top right) true color image, (middle right) retrieved cloud optical thickness, and (bottom right) cloud-top pressure derived from Aqua MODIS observation at 0600 UTC 23 Jan 2016. The location of the Hong Kong Observatory is labeled with a black circle with a square in its center. All right panels are produced using NASA Global Imagery Browse Services (GIBS).
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0050.1
In our MODATRAN calculation, cloud base is set at 1000 m, which is close to the lifting condensation level (LCL) decided by the sounding profile. Cloud top is determined to be ∼600 hPa by the MODIS observations (Fig. 4). However, for the purpose of our study to study LW scattering in the isothermal layer and to avoid introducing the effect of scattering in a nonisothermal layer, the cloud top is deliberately set at 2000 m. MODIS-retrieved optical thickness is used as the cloud optical thickness in the calculation. All clouds here are liquid clouds, and the default liquid cloud properties in MODTRAN are used. Four sets of MODTRAN calculations are conducted:
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Clr noScat: Temperature and humidity profiles are based on the radiosonde observation. No clouds are specified in the model (i.e., clear sky). LW scattering is turned off.
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Cld noScat: Same as 1, but a cloud is put in the isothermal layer (i.e., cloudy sky). It can only emit and absorb thermal IR radiation. The cloud properties are specified as described above.
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Clr Scat: Same as 1, but LW scattering is enabled (essentially the same as case 1).
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Cld Scat: Same as 2, but LW scattering is enabled. Clouds can now scatter thermal IR radiation.
The remaining configurations for all cases are the same. All greenhouse gases except water vapor are assumed at the default values of midlatitude winter prescribed in MODTRAN. Simulations are then carried out at 2-cm−1 spectral resolution (FWHM).
Cloud radiative effect on TOA radiance can be derived by subtracting TOA radiance in cloudy-sky cases from that in clear-sky cases (i.e., case 2 − case 1 and case 4 − case 3). Figure 5a shows the simulated cloud radiative effect on TOA radiance when LW scattering is enabled (red line) or disabled (blue line). As discussed in previous sections, in the absence of LW scattering, upwelling radiance at the top of an isothermal layer is independent of the amount of absorption within that layer. Because the temperature profile in the real world is not perfectly isothermal, adding an absorbing cloud within the quasi-isothermal layer will result in a small difference in simulated TOA radiance (blue line). However, when LW scattering is included in the simulation, there is a large dip at the atmospheric window region (red line), indicating that the TOA radiance is reduced because of the added cloud layer. Figure 5b shows a case where the atmospheric temperature from surface to 3200 m is set constant. The blue line is now completely flat at zero, suggesting that the added cloud layer will make no impact on TOA radiance when cloud LW scattering is ignored. When the LW scattering physics is included, however, a difference in TOA radiance can be clearly seen (red line in Fig. 5b), with similar magnitude in the window region as its counterpart in Fig. 5a.
(a) Simulated TOA radiance difference between the cloudy-sky case and the clear-sky case when LW scattering is enabled (red) or disabled (blue) using the real sounding profile at Hong Kong site at 1200 UTC 23 Jan 2016. (b) As in (a), except that the temperature from the surface to 3200 m is homogenized to create a perfect isothermal layer.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0050.1
To see how our results can be applied in remote sensing problems, we convolved simulated radiance with the spectral response functions of MODIS channels 28–31, i.e., channels in the mid-IR window and at the edge of the H2O band, to produce nadir-view synthetic MODIS radiances. Figure 6 shows the differences in simulated brightness temperature (BT) due to clouds in the nearly isothermal layer at MODIS IR channels 27–36. To show the capability to retrieve cloud properties, we added two cases where cloud optical thickness is different. When LW scattering is ignored (Fig. 6a), the change in BT is within 0.25 K throughout the MODIS IR channels. The LW scattering in the nearly isothermal layer can cause a much larger BT difference, especially at MODIS band 29 (Fig. 6b). The BT dip at MODIS band 29 is approximately 1.25 K when cloud optical thickness is large (τvis = 10 and 60) and nearly 0.75 K when cloud optical thickness is small (τvis = 1). The BT decrease at MODIS band 30 is ∼0.5 K for all cases. The BT difference at other MODIS IR bands is much smaller. We also calculated the BT differences when the atmospheric temperature from surface to 3200 m is set to be constant. Figures 6c and 6d show that when a cloud is added to an isothermal layer, it is impossible to detect it based on TOA radiance without scattering being included. Only when LW scattering is included in our simulation can we see a reduced brightness temperature at MODIS bands 29–32. The amount of decrease depends on cloud properties such as cloud optical thickness (Fig. 6d). Such dependency enables us to retrieve cloud properties in this isothermal layer from the space.
Changes of the simulated brightness temperatures (BTs) over MODIS IR channels when a cloud layer is added to the nearly isothermal layer. (a) LW scattering is not included; (b) LW scattering is included; (c),(d) as in (a) and (b), except the atmospheric temperature in the nearly isothermal layer is set to be constant so that it becomes perfectly isothermal. MODIS bands 29–32 are labeled using red font. Different colors represent different visible optical thickness of the cloud layer.
Citation: Journal of the Atmospheric Sciences 80, 11; 10.1175/JAS-D-23-0050.1
5. Conclusions and discussion
Using an analytical two-stream solution and a Monte Carlo approach, we examine how the thermal IR scattering can affect the radiation field under a special circumstance, i.e., an isothermal atmosphere. Without scattering, the radiation field for an isothermal atmosphere with an underneath blackbody surface can be easily understood using Kirchhoff’s law: the upwelling radiance always follows the blackbody radiation regarding the actual values of optical properties in any atmospheric layer, and the downwelling radiance increases with the optical depth. With scattering, however, the upwelling radiance no longer follows Planck’s law. Instead, as long as the asymmetry factor is less than one, the TOA upwelling radiance will be smaller than the blackbody radiation. Compared to a nonscattering isothermal atmosphere with the same extinction optical depth, the presence of scattering implies a smaller absorption optical depth and, consequently, leads to a smaller emissivity (i.e., a larger transmissivity) for the atmosphere. This further leads to two opposite effects: a reduction of the total amount of atmospheric thermal emission and an increase in the amount of surface emission that can reach the TOA. The first effect outweighs the latter one and leads to a reduction of upwelling radiance at the TOA (note two effects would perfectly cancel each other if no scattering were presented and only a reduction of optical depth happened). Compared to a nonscattering isothermal atmosphere with the same absorption optical depth, backscattering increases the chance of photons (regardless of their origin) being absorbed within the atmosphere instead of reaching the boundaries (both top and bottom). Consequently, the TOA upwelling radiance also becomes smaller than the blackbody radiation. Using sounding profiles and more realistic and sophisticated radiative transfer modeling, we further show that such scattering effect of isothermal cloud can cause approximately −1.5-K brightness temperature difference for the nadir-view radiance of MODIS channel 29, a mid-IR window channel frequently used for cloud retrievals.
Thermal contrast (i.e., temperature gradient) is a prerequisite for infrared remote sensing in absence of scattering. However, we showed here that even in the absence of thermal contrast, scattering could affect the TOA radiance. Thus, it allows one to infer cloud properties with such scattering signals even when thermal contrast does not exist. Here we use an actual sounding profile in the subtropics under a severe cold-air outbreak event to illustrate how the cloud scattering within a nearly isothermal layer can affect the TOA radiance. Liquid cloud is used in our study. A nearly isothermal layer can be encountered in polar regions, where the clouds could be mixed phased or ice. Ice clouds have similar single scattering albedo and asymmetric factor as liquid clouds in the mid-IR window region and higher single scatter albedo in the far-IR dirty window (Kuo et al. 2017). Therefore, the same LW scattering mechanism could also be useful for polar cloud observations. Furthermore, scattering affects not only the scalar radiance field but also the polarization of the radiation (Coy et al. 2020; Shanks 2022). Thus, it can be expected that polarized IR measurements can further enhance cloud retrieval capability under such isothermal environments.
Acknowledgments.
We sincerely acknowledge three anonymous reviewers for their insightful suggestions on improving the quality and clarity of our article. XLH thanks Graeme Stephens for inspirational discussions, which partly motivated this study. Lead author CXF is supported by the NASA FINNEST program under Grant 80NSSC22K1433. Part of XLH’s effort is supported by NOAA Grant NA20NES4400007 with a subcontract from Texas A&M University.
Data availability statement.
We acknowledge the use of MODIS images and retrievals provided by services from NASA’s Global Imagery Browse Services (GIBS), part of NASA’s Earth Observing System Data and Information System (EOSDIS). They can be accessed at https://gibs.earthdata.nasa.gov/. The sounding data can be downloaded from the Department of Atmospheric Science at University of Wyoming (https://weather.uwyo.edu/upperair/sounding.html). MODTRAN is a radiative transfer simulation software that can be purchased from Spectral Sciences Inc. (http://modtran.spectral.com/).
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