Measurements of Tropospheric Ice Clouds with a Ground-based CMB Polarization Experiment, POLARBEAR

Since the size of ice crystals is sufficiently smaller than the wavelength, the scattering of microwave radiation by ice crystals is described by Rayleigh scattering. The electric field ${{\boldsymbol{E}}}_{\mathrm{sc}}$ scattered by an ice crystal located at the origin is expressed as (e.g., Landau & Lifshitz 1960)

$$\begin{eqnarray}&&{{\boldsymbol{E}}}_{\mathrm{sc}}(r\hat{{\boldsymbol{n}}})=-\displaystyle \frac{{\omega }^{2}}{{c}^{2}r}\hat{{\boldsymbol{n}}}\times (\hat{{\boldsymbol{n}}}\times {\boldsymbol{P}}),\end{eqnarray} \tag{ 1 }$$

where r is the distance from the origin, $\hat{{\boldsymbol{n}}}$ is a unit vector toward the propagation direction, ω is the angular frequency of the wave, and c is the speed of light. In general, the electric dipole moment ${\boldsymbol{P}}$ is expressed as

$$\begin{eqnarray}&&{\boldsymbol{P}}=V{\boldsymbol{\alpha }}{{\boldsymbol{E}}}_{\mathrm{in}},\end{eqnarray} \tag{ 2 }$$

where V is the volume of the scatterer, ${{\boldsymbol{E}}}_{\mathrm{in}}$ is the incident electric field, and ${\boldsymbol{\alpha }}$ is the polarizability matrix, calculated as

$$\begin{eqnarray}&&{\boldsymbol{\alpha }}=\displaystyle \frac{\epsilon -1}{4\pi }{\left[{\bf{I}}+(\epsilon -1){\boldsymbol{\Delta }}\right]}^{-1},\end{eqnarray} \tag{ 3 }$$

where ${\bf{I}}$ is the identity matrix and ${\boldsymbol{\Delta }}$ is the depolarization factor, which is a positive definite symmetric matrix satisfying $\mathrm{Tr}({\boldsymbol{\Delta }})=1$ and depending on the shape and orientation of the scatterer. In the case of spheroids, ${\boldsymbol{\Delta }}$ is parameterized as a diagonal matrix $\mathrm{diag}\{(1-{{\rm{\Delta }}}_{z})/2,(1-{{\rm{\Delta }}}_{z})/2,{{\rm{\Delta }}}_{z}\}$, where Δ_z < 1/3, Δ_z = 1/3, and Δ_z > 1/3 correspond to the prolate (column), spherical, and oblate (plate) shapes, respectively. The relative permittivity of ice is about 3.15 for microwave radiation (Warren & Brandt 2008).

In the simple case of spherical particles, we can obtain the total cross section of the Rayleigh scattering σ_R as

$$\begin{eqnarray}&&{\sigma }_{{\rm{R}}}=\displaystyle \frac{8\pi }{3}\displaystyle \frac{{V}^{2}{\alpha }^{2}{\omega }^{4}}{{c}^{4}}\propto {D}_{e}^{6}{\omega }^{4},\end{eqnarray} \tag{ 4 }$$

where $\alpha =(3/4\pi )(\epsilon -1)/(\epsilon +2)$. We can see the well-known dependence on the size of the scatterer D_e and the frequency of the light ω. This strong dependence on the particle size and the variation of the size distribution in clouds cause huge uncertainty in the prediction of σ_R by orders of magnitude. At the observing frequency of Polarbear, ω/(2π) = 150 GHz, the cross section σ_R becomes ∼10⁻¹⁶ m² for D_e = 20 μm and ∼10⁻¹² m² for D_e = 100 μm. By assuming that all the particles in a cloud have the same size and using the typical number density of ice crystals, n ∼ 10⁴–10⁵ m⁻³, and the typical thickness of cirrus clouds, Δh ∼ 10³ m, we estimate the optical depth of the clouds as τ ∼ 10⁻⁹–10⁻⁴. Note that the estimate increases for 220 or 280 GHz due to the frequency dependence.

The calculation above has been substantially simplified by ignoring the size and shape distributions but does indicate that larger ice crystals in clouds are the main contributor to the scattering of microwave radiation and that the optical depth would be ≲10⁻³ at most.

However, scattering by clouds changes the direction of the thermal radiation from the ground and injects it into the line of sight. Since almost half of the solid angle as seen from the cloud is covered by the ground at ambient temperature, scattering of the ground emission at levels below 1% could cause an additional signal at the  ∼1 K level. The clouds are randomly distributed in the sky and are gradually varying and moving due to atmospheric turbulence and wind, which leads to low-frequency variations of the signal from the clouds. Therefore, clouds can become an important source of low-frequency noise (see Section 4 for further discussions). The cloud signal is significantly larger than the current detector noise level for CMB observations, which is lower than 1 mK over a few seconds of beam-crossing time, and thus can be detected instantaneously.

There are mainly two types of effects that polarize the light scattered by the ice crystals. The first is due to the curvature of the ground. The second is due to the horizontal alignment of ice crystals with plate or column shape. We explain the two effects in the following. Our estimate and observation suggest the latter is dominant. Other subsidiary effects are also discussed in Section 4.3.

The three-dimensional positional relations among the telescope, the clouds, and the ground produce polarization (Figure 1). As shown in Equations (1) and (2), the polarization of the scattered light is determined by its scattering angle and the polarization of the incident light. By taking spherical coordinates (θ, ϕ) centered at the clouds and aligning the z-axis to zenith as shown in Figure 2, the ground radiation can be expanded in spherical harmonics ${Y}_{l}^{m}(\theta ,\phi )$ with m = 0 because of the axial symmetry. If we assume that the ground is a blackbody with a uniform temperature T_g, the expansion coefficients ${a}_{l,0}$ are obtained as

$$\begin{eqnarray}&&{a}_{l,0}=2\pi {T}_{{\rm{g}}}{\int }_{0}^{\tfrac{\pi }{2}-\delta \theta }{Y}_{l}^{0}(\theta ,0)\sin \theta \,d\theta .\end{eqnarray} \tag{ 5 }$$

Here, δθ is the look-down angle of the horizon from the clouds, which is approximately $\sqrt{2h/R}$, with the altitude of the clouds' center above sea level h and the radius of the Earth R. In particular, the monopole (l = 0) and quadrupole (l = 2) components are calculated as

$$\begin{eqnarray}&&\displaystyle \frac{{a}_{\mathrm{0,0}}}{\sqrt{4\pi }{T}_{{\rm{g}}}}=\displaystyle \frac{1-\sin \delta \theta }{2}\quad \mathrm{and}\quad \displaystyle \frac{{a}_{\mathrm{2,0}}}{\sqrt{4\pi }{T}_{{\rm{g}}}}=\displaystyle \frac{\sqrt{5}}{4}\sin \delta \theta {\cos }^{2}\delta \theta ,\end{eqnarray} \tag{ 6 }$$

respectively. Again, if we assume spherical particles to deal with the scattering simply by the optical depth τ, the Stokes parameters of the scattered light are calculated as (Hu & White 1997)

$$\begin{eqnarray}&&\left(\begin{array}{c}I\\ Q\end{array}\right)\approx \displaystyle \frac{\tau {T}_{{\rm{g}}}}{\sqrt{4\pi }}\left(\begin{array}{c}1\\ 0\end{array}\right){a}_{\mathrm{0,0}}+\displaystyle \frac{\tau {T}_{{\rm{g}}}}{10}\left(\begin{array}{c}{Y}_{2}^{0}(\theta ,\phi )\\ -{\sqrt{6}}_{2}{Y}_{2}^{0}(\theta ,\phi )\end{array}\right){a}_{\mathrm{2,0}},\end{eqnarray} \tag{ 7 }$$

where ${Y}_{2}^{0}(\theta ,\phi )\equiv \sqrt{5/(16\pi )}(3{\cos }^{2}\theta -1)$, ${}_{2}{Y}_{2}^{0}(\theta ,\phi )\,\equiv \sqrt{15/(32\pi )}{\sin }^{2}\theta$ and the other Stokes parameters, U and V, are zero. Here, the polarizations are defined on the usual base vectors (${{\boldsymbol{e}}}_{\theta }$, ${{\boldsymbol{e}}}_{\phi }$), and the negative Stokes Q represents horizontal linear polarization. The polarization fraction p is the ratio of Q to I, thus

$$\begin{eqnarray}&&| p| \approx \displaystyle \frac{3}{4\sqrt{5}}\displaystyle \frac{{a}_{\mathrm{2,0}}}{{a}_{\mathrm{0,0}}}{\sin }^{2}\theta \approx \displaystyle \frac{3}{4}\sqrt{\displaystyle \frac{h}{2R}}{\sin }^{2}\theta .\end{eqnarray} \tag{ 8 }$$

Putting h = 10 km, R = 6400 km, and θ = 45° into Equation (8) results in $| p| \sim 1 \%$.

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Horizontally aligned ice crystals with column and plate shapes scatter horizontal electric fields more efficiently, and can therefore produce a larger polarized signal. We approximate the crystal shapes as spheroids and directly calculate Equations (1)–(3). In the case of column shape, we set the long axis horizontal but assume that its azimuth is random. The polarization fraction calculated from the model is shown in Figure 3. Here, all the crystals are assumed to have the same shape and to be horizontally aligned, i.e., the long axis of the spheroid is in the horizontal plane. Thus, these estimates give an upper limit, whereas real polarization fractions are expected to be smaller. However, the amplitude of the polarization fraction is considerably larger than the spherical case shown as the black line. This is because the horizontally aligned crystals have a tendency to scatter horizontally polarized light even without a quadrupolar anisotropy of the incident radiation field.

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Figure 4 shows an example of the bolometer timestreams from a four-hour observation on 2014 December 18. During the observation, the telescope is azimuthally scanning the sky back and forth in an azimuth range of 133°–156° at a constant elevation of 30°. The precipitable water vapor (PWV) increases from ∼0.8 mm to ∼1.6 mm during the time of this data set. The PWV is provided by the APEX experiment (Güsten et al. 2006) using a commercial LHATPRO microwave radiometer (Rose et al. 2005). The APEX PWV will be partially correlated with the PWV at the Polarbear site 6 km away.

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The intensity signal shown in the top panel is continuously fluctuating by ±3 K like a random walk, which is due to atmospheric turbulence. The Stokes Q in the middle panel, on the other hand, has negatively directed, burst-like structures, down by as much as ∼0.3 K relative to an offset A₀ ∼ 0.2 K. The Stokes U also shown in the middle panel has much smaller variation than Q, which means that the burst-like signal is horizontally polarized. This property agrees with the expectation for the cloud signal that we described in Section 2.2.

The bottom panel shows the slope of the simple linear regression between the intensity and Q polarization signals, i.e., the signed Q polarization fraction. The intensity signal is a combination of the clouds and atmosphere, whose contributions are both a few kelvin. However, the timescale of the polarized-burst signal is shorter than that of the atmosphere. Thus, in this calculation, we apply a simple high-pass filter by subtracting the baseline for each ∼50 s one-way scan and minimize the contribution of the atmosphere. In the absence of the bursts in Q, the polarization is ∼0.1%, which is consistent with the level expected solely due to instrumental intensity-to-polarization leakage and no atmospheric polarization (Takakura et al. 2017). On the other hand, the polarization fraction significantly increases to 5%–10% at the timings of the bursts. The polarization fraction of 10% at the elevation of 30° is larger than the estimate for the spherical case in Equation (8) but can be explained by horizontally aligned column or plate crystals as shown in Figure 3.

Figure 5 is another illustration of the same data: a map of the Q polarization data as a function of the azimuth and time accompanied with snapshots from the webcam. Each horizontal row of the map corresponds to each leftward or rightward scan, which takes 50 s. Around the time of 19:45:05, there is a structure within the row, which means that the variation of the polarized-burst signal is more rapid than the scan time. In the next two scans, the structure appears consistently but at different azimuths, which suggests that the source of the signal is moving within the scan area. The timescale of the motion is several minutes. The existence of the polarized-burst signals and their motion from right to left agree well with those of the clouds in the webcam. This result also supports the argument that the origin of the signal is a cloud. Note that the maximum elevation of the webcam FOV is 20° and it does not exactly cover the scanning elevation of 30°. However, the sizes of the clouds in the photos are sufficiently large to cover most of the sky, thus we suppose that the clouds would expand to the line of sight of the telescope.

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We have considered other possibilities to create the polarized-burst signal, but none of them can explain the data as well as the clouds. Sudden variations in responsivity may couple to the steady instrumental polarization A₀ and cause apparent variations in the Q timestream. However, this cannot explain the variation of Q(t) to negative values in Figure 4 because the responsivity of the TES detector does not change its sign. Besides, the 2f signal, which is another stable optical signal from the CRHWP, does not exhibit such variations. Variations in temperature of the primary mirror could change the instrumental polarization, but the polarization fraction of the effect is expected to be less than 0.1% (Takakura et al. 2017). The far sidelobe of the telescope may have larger polarization leakage and see the ground and another part of the sky. However, the spurious signal from the ground should stay at the same azimuth, while that from the sky should be gradual rather than burst-like fluctuations. Condensation and evaporation of water vapor on the primary mirror may also cause spurious polarization. We do not have concrete evidence to reject it, but the correlation of the polarized burst with the humidity is weaker than that with the cloud detection in Section 3.3.

We analyze all of the photos from the webcam and obtain the statistics of the clouds at the Polarbear site, in the Atacama Desert in Chile. Note again that the FOV of the webcam covers a small fraction of the sky and that the telescope points to a sky region outside the FOV in all the observations. The webcam images are not useful during the night. We also removed the pictures taken at dawn or dusk in the following analysis because the gradient of brightness in the sky increases the false detection rate.

The basic idea of the cloud detection algorithm is to find white regions in the picture. First, we mask the telescope, mountain, and ground, and split the sky into 29 patches as shown in Figure 6. For each patch, we calculate the average red–green–blue color $(R,G,B)$ among the pixels. Then, we convert the color into hue–saturation–value color $(H,S,V)$, specifically (Smith 1978)

$$\begin{eqnarray}&&S=\displaystyle \frac{\max (R,G,B)-\min (R,G,B)}{\max (R,G,B)},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&V=\max (R,G,B).\end{eqnarray} \tag{ 12 }$$

We set the thresholds to detect the clouds in each patch as S < 0.1 and V < 0.98, where the former condition rejects the blue sky and the latter cuts pixels saturated by the Sun. The performance of the cloud detection is checked by eye for pictures from several days chosen randomly. The algorithm often fails to detect faint clouds as in Figure 6 but rarely make false detections, which are occasionally caused by ghost images, i.e., images that appear at spurious positions due to multiple reflections.

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Figure 7 shows the rate of cloud detection for each patch, i.e., the number of shots with positive cloud detection in the patch divided by the total number of shots. Although there is a small gradient in elevation, possibly due to the difference in the area of the sky, there is no clear tendency for clouds to appear in any particular region of the FOV. The clouds may be created outside of the webcam FOV, but they are expected to persist sufficiently long enough that they will pass across the FOV.

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Figures 8 and 9 show annual and daytime variations of the cloud detection rate per shot. Here we flag a shot as "cloudy" when a cloud is detected in at least one patch according to the algorithm specified above. In Figure 8, we can see the significant increase in cloud detection rate around February, which is known as Altiplanic winter (see Erasmus & Van Staden 2001 for more details about the climate of northern Chile). During that season, the cloud detection rate seems to increase in the afternoon (Figure 9). It might be due to the lift of the atmosphere heated up by thermal conduction from the ground, which is also heated by sunlight. During July through December, on the other hand, the daytime trend is moderate.

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Figure 10 shows the correlation between the cloud detection rate per shot and the APEX PWV. There is a clear correlation between the cloud detection and PWV. This is an expected trend and provides additional support and validation to our cloud detection method.

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The overall cloud detection rate per shot is about 26%. Note that we use only the daytime photos and have no information during the night.

By using the polarized-burst signals in the bolometer data, we also detect clouds as shown in Figure 4. We use the Polarbear data from 2014 July to 2017 January. The cloud detection results are compared with the webcam study described in Section 3.2.

Since the typical size of the clouds should be larger than the FOV of Polarbear, 3°, the cloud signal is correlated among all the detectors. Thus, we can improve the sensitivity to detect the clouds by averaging the timestreams over all the detectors. Similar to Kusaka et al. (2014), we separate the averaged polarization timestreams, Q(t) and U(t), into two components using the method of principal component analysis (PCA) as

$$\begin{eqnarray}&&{X}_{1}(t)+{{iX}}_{2}(t)=\{Q(t)+{iU}(t)\}\,{e}^{-i\phi },\end{eqnarray} \tag{ 13 }$$

where the rotation ϕ is determined to maximize the variance of ${X}_{1}(t)$. Since the secondary component ${X}_{2}(t)$ is dominated by the detector white noise, the signal-to-noise ratio (S/N) of the cloud is calculated as

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}=\sqrt{\displaystyle \frac{\sigma {({X}_{1})}^{2}}{\sigma {({X}_{2})}^{2}}-1},\end{eqnarray} \tag{ 14 }$$

where σ denotes the standard deviation. The polarization angle of the signal ψ is obtained as

$$\begin{eqnarray}&&\psi =\displaystyle \frac{\phi }{2}.\end{eqnarray} \tag{ 15 }$$

Here, we assume that the signal is almost horizontal as shown in Figure 4, and constrain π/2 < ϕ < 3π/2, which cannot be determined by PCA due to degeneracy.

Figure 11 shows a histogram of the polarization angle of the polarized-burst signal ψ with S/N > 10 for each scan. If we have any instrumental noise sources other than the clouds, e.g., intensity leakage due to the detector nonlinearity, the polarized ground structure, and the HWP encoder error, they could appear at angles ψ ≁ 90°. Having only the single peak at ψ ∼ 90° is evidence that such extra noises are not significant and that all the polarized-burst signals are most likely coming from clouds. There are still possibilities of the responsivity variation due to electrical noise and the temperature variation of the primary mirror, but they cannot explain the coincidence with the webcam described below. Both the width of the peak, rms = 268, and the offset from ψ = 90° might be due to a systematic error of the cloud signal or the instrument, which is still under investigation (see more discussions in Section 4.3).

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Each of the CMB observations typically takes one hour and contains 40–70 left and right scans at a constant elevation. We calculate the S/N and polarization angle ψ of the cloud signal for each scan and take the values with the highest S/N as representative of the CMB observation. On the other hand, the detection of clouds by the webcam is determined by a detection in at least one of the ∼12 photos taken during the observation. Note that the CMB observations are performed at all times of the day, but the webcam can be used during the daytime only.

Table 1 shows the coincidence of the cloud detection in the bolometer data with that in the webcam. For the data with clouds in the webcam, the rate of polarized-burst detection significantly increases to 46.3% compared to 1.3% for the data without clouds. Assuming that the appearance of polarized bursts in the bolometer data and the appearance of clouds in the webcam each occur at their observed rates but are also independent, then the significance of observing such a strong covariance between them is estimated to be >24σ. Note again that the FOV of the webcam does not cover the sky regions of the CMB observations. That could be the reason for the deviation from a perfect separation. Also, the daytime rate of polarized bursts is 16.1%, which is smaller than ∼30%, the fraction of data with clouds detected by the webcam (Section 3.2). This is because CMB observations are not performed when PWV > 4 mm, or the webcam is more sensitive to clouds than bolometers in this analysis using the thresholds above. For the night data, we have no information on the clouds from the webcam, but polarized-burst signals are detected in 9.7% of the data.

Figure 12 shows the distribution of the S/N for each data set. Note that it shows the partial data with high S/N (Table 1). Again, it shows the clear difference between the data with and without clouds detected by the webcam. Besides, the S/N distribution for the night data is very similar to that for the daytime data with clouds. As explained in Section 2.2, the clouds scatter the thermal emission from the ground, which also takes place at night. Our results support the expectation that the cloud signal also exists during the night.

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CMB measurements require thousands of hours of observations with high-sensitivity detectors to measure the faint CMB signals. The cloud signal is just noise that lowers the quality of the observations.

One simple approach to reduce the impact of the clouds is to drop noisy data, but that would inevitably reduce the observation efficiency.²⁶ It is possible to detect cloud signals in the bolometer data to an extent as performed in Section 3.3. We find cloud-like polarized-burst signals in 16.1% (9.7%) of one-hour observations (Table 1) and in 5.2% (3.0%) of leftward or rightward scans during the daytime (night). One could use image analysis similar to Section 3.2, in which we detect clouds in 26% of all the daytime shots of the webcam. As shown in Figure 10, the fraction of data without clouds improves if we observe only in good PWV. But this decreases the total amount of data.

Even with data cuts, we expect residuals from faint clouds below the detection threshold. These residuals will be present at low frequency with a 1/f behavior and degrade the detector performance at large angular scales. Such residuals cannot be mitigated by polarization modulation techniques such as the CRHWP used in Polarbear.

If we detect the clouds using the bolometer data as performed in Section 3.3, the residual noise level will depend on the S/N threshold of the cloud detection. For example, to achieve an S/N of 10, the power of the cloud signal should be 100 times larger than that of the detector noise. In other words, residual 1/f noise below the threshold could naïvely degrade the sensitivity of the CMB angular power spectra by a factor of 100 in the worst case. Tightening the threshold will mitigate the contamination but also decrease the observation efficiency. Thus, optimization of the threshold is required to maximize performance.

The effect of clouds will depend on the telescope FOV and the detector beam size. Small-aperture telescopes with a large FOV have a high probability of seeing clouds. Large-aperture telescopes with small beams have a high instantaneous S/N of a cloud, because many detectors simultaneously observe the same cloud, which is larger than the beam size.

These studies could inform forecasting and optimization of future CMB experiments, such as CMB-S4 (Abazajian et al. 2016).

Systematic errors due to the residual cloud signal are also of concern for CMB measurements.

The cloud signal is horizontally polarized, i.e., Q < 0, and does not fluctuate symmetrically between positive and negative. This highly non-Gaussian fluctuation could lead to possible systematics in the map because many map-making algorithms assume Gaussianity for noise fluctuations. This systematics can be mitigated to some extent by parallactic angle rotation.

The cloud signal would affect foreground removal due to its distinct frequency dependence. By using maps at multiple frequencies, we separate the CMB and the other foregrounds that are stationary in the sky, i.e., not associated with any Earth or atmospheric motion, such as the Galactic dust and synchrotron emissions. Since the cloud signal has a Rayleigh scattering spectrum that is ∝ω⁶ (see Equation (4) with additional ω² from the spectrum of the ground emission), it would appear to rise in frequency, similar to a dust component (approximately ω^1.5), but much more steeply. On the other hand, the atmospheric motion would likely decorrelate with astrophysical foregrounds, making control of the clouds appropriate for a time domain analysis, rather than in maps.

Clouds staying in the same position may also cause systematic errors. However, there is no significant localized cloud feature appearing in Figure 7. In addition, any localized clouds would be fixed to ground features, such as mountains, but the rotation of the sky would change the relative positions and suppress systematic errors.

The daytime trend of the cloud detection rate shown in Figure 9 might cause a systematic difference in the additional noise from the residual cloud signals between the morning and afternoon observations. However, the yearly motion of the Earth gradually shifts the observing time of the CMB patch fixed on the sky. For year-long observations, the difference will be at least partially averaged.

Performing null tests sensitive to the clouds is necessary to validate the data. A split of rising versus setting can test for localized clouds, and a split of summer versus winter can test the impact of the diurnal variation as mentioned above. A split of low-PWV versus high-PWV can test the cloud rate because of their correlation as shown in Figure 10.

While ice clouds are a nuisance in CMB observations, the polarized signal from the clouds could be a useful calibrator for the absolute polarization angle. As explained in Section 2.2 and demonstrated in Figures 4 and 11, the signal is horizontally polarized mainly because the column or plate ice crystals are aligned horizontally by gravity. The center of cirrus clouds lie at an altitude of ∼10 km above sea level, so the distance from the telescope is sufficient to achieve a far-field measurement for Polarbear with the 2.5 m diameter aperture observing at 150 GHz. Furthermore, the clouds are diffuse objects, making beam systematics of less concern. These two properties are better than those for near-field calibrators, such as the sparse wire grid in front of the telescope (Tajima et al. 2012; Kusaka et al. 2018), which need a connection from the near-field measurement to the far-field beam that relies on the optics model. Although the spectrum of the cloud signal depends strongly on the observing frequency as ∝ω⁶, the polarization angle does not depend on the frequency. This addresses an uncertain polarization angle rotation feature of Tau A, a popular polarized celestial source. Cloud calibration would make it possible to operate the detectors with typical sky loading, as opposed to extra loading when observing a source near the ground, and it would not have uncertainty in extrapolating the pointing model of the telescope.

In Figure 11, the precision of the polarization angle calibration for each scan is only 27, but the uncertainty of the mean value can be reduced by accumulating statistics to 003, provided the errors are independent and Gaussian distributed. That is better than the statistical uncertainty of 016 from the polarization angle calibration from nulling the apparent correlation between the CMB E- and B-mode patterns from two years of Polarbear data (The POLARBEAR Collaboration et al. 2017). In addition, the cloud polarization is absolutely referenced to gravity, and it does not use assumptions about the symmetry properties of the CMB.

On the other hand, the cloud signal may have its own systematic errors. For example, wind and electrification may slightly tilt the ice crystals. The ground emission may not be uniform due to local features, e.g., deserts, mountains, lakes, snowfields, etc. The contribution of the Sun can become non-negligible. We have estimated the systematic error by splitting the data into subsets for various observation conditions: splits can be by year, day–night, scan direction, PWV, outside temperature, and wind speed. The median value for each subset has a variation of about 04. However, that value also includes the systematic error of the instrument such as variations in the time constant value of the detector during the observation and imperfection of the pointing model. Further investigation is necessary to separate them, but the possibility of having many measurements with various conditions demonstrates the potential usefulness of the cloud signal as a calibrator of polarization angle.

For future ground-based CMB experiments aimed at more precise measurement of CMB polarization, such as the CMB-S4, steps to mitigate the contamination of the cloud signal will be necessary.

One approach is remote sensing of clouds from the ground on site. In Section 3.1, we have demonstrated a simple cloud detection technique using the webcam for monitoring the telescope. Even with the limitation of its FOV, a significant coincidence of the cloud detections between the webcam and bolometers is observed as shown in Table 1 and Figure 12. This can be improved by using a whole-sky camera and a co-mounted infrared camera, which would be useful during the night (e.g., Suganuma et al. 2007). As already mentioned by Pietranera et al. (2007), the most informative but challenging method is polarized lidar (e.g., Lewis et al. 2016), which shoots a laser pulse into the sky, receives the scattered light, and characterizes the atmospheric properties along the line of sight including the shape, size distribution, and orientation of the ice crystals. These tools would enable reliable cloud detection and precise data selection. This would also help with understanding the clouds and reducing the systematic error in polarization angle.

Another approach might be to perform foreground separation in the time domain. The clouds are obviously the frontmost component of the foregrounds for CMB observations. The cloud signal has a frequency dependence markedly different from that of the CMB and other astrophysical foregrounds, i.e., approximately ω⁶ as opposed to ω⁻³ for synchrotron and ω^1.5 for dust. Therefore, it would be possible to separate the cloud signal in measurements with multi-frequency bands. Here, it is important to observe the same position at the same time among detectors with different frequency bands. The multi-chroic detector technique used in, e.g., the Simons Array experiment (Stebor et al. 2016) would be useful for that purpose.

Of course, satellite missions are the best solution to avoid the clouds. Balloon-borne experiments may see clouds in the mesosphere, but their impact would be small because the particles in mesospheric clouds are smaller than those in tropospheric clouds.