Atmospheric

2.3 Noise

2.3.2 Detector Noise

2.3.2.3 Atmospheric

Ground-based observations at submillimeter and millimeter wavelengths are strongly affected by water vapor (and to a lesser extent oxygen and ozone) in the atmosphere. Atmospheric absorption results in significant attenuation of astronomical signal as shown in Figure 2.14. Atmospheric emission increases background loading, which increases photon noise and degrades detector responsivity. Finally, fluctuations in atmo- spheric emission results in long-timescale noise that prevents the recovery of signal at large spatial scales.

This sensitivity to the amount and variability of atmospheric water vapor has driven the location of large (sub)millimeter telescopes to dry and stable (and often inhospitable) sites such as Antartica, the Atacama Desert in Chile, and the summit of Mauna Kea in Hawaii.

The majority of atmospheric water vapor is located in the troposphere at an effective heighthavg∼1 km.

The water vapor is poorly mixed with the other dry components of the atmosphere because the temperature of the atmosphere is near its condensation point [130]. This nonuniform distribution of water vapor is the dominant cause of fluctuations in atmospheric emission, which we will henceforth refer to as atmospheric noise.

Atmospheric noise above Mauna Kea has been studied at 143 and 268 GHz by Sayers et al. [37] using data collected with Bolocam. They find that the atmospheric noise is consistent with a Kolmogorov-Taylor (K-T) thin screen model [131]. This model assumes that there is a thin turbulent layer of thickness∆h

Figure 2.14: The atmospheric transmission on Mauna Kea when looking at zenith. The blue, green, and red curves correspond to the historical 25th, 50th, and 75th percentiles for the column depth of precipitable water vaporCPWon Mauna Kea. The curves were calculated using the ATM software [129].

at some heighthavgthat moves horizontally across the sky with an angular velocity w, and that within the layer the atmosphere behaves according to the K-T model of turbulence [132, 133, 134]. In the K-T model turbulent energy is constantly injected into the atmosphere at large scales by processes such as wind shear and convection. The energy cascades down to smaller scales through an inertial mechanism, producing a hierarchy of eddies. Eventually it reaches a small enough scale that it can be dissipated by viscous effects.

For a three-dimensional volume the model predicts that between the injection scaleL and the dissipative scaleη there will be a power spectrum of fluctuations due to turbulence that scales as|q|^−11/3, whereqis the three-dimensional spatial frequency. It can be shown (e.g., [131, 135, 136]) that this gives rise to a power spectral density of fluctuations in the brightness temperature of the sky that is given by

S_δTsky(α) =







B²_3D|α|^−11/3 _2∆hsine^havg |α| η B²_2D|α|^−8/3 L |α| _2∆hsin^havg _e,

(2.185)

whereα= [α_x,α_y] = [q_x,q_y]×_sine^havg is the angular wave number on the sky in units of 1/radians, B2D,3D are normalizations in units of mK rad^−5/6, ande is the elevation. The transition from the three-dimensional

to two-dimensional regime occurs at a physical scale of approximately 2∆h, at which point the projected power spectral density is expected to transition from a−11/3 to−8/3 power law. The normalizations will depend on several parameters — the average heighth_avgand thickness∆hof the turbulent layer, the elevation angle e, the observing wavelength νmm, and the column depth of precipitable water vaporCPW — so that

B2D,3D≡B2D,3D(havg,∆h,e,νmm,CPW). In the three-dimensional regime the exact dependence on the elevation,

height, and thickness can be derived

B²_3D(h_avg,∆h,e,νmm,CPW) =B²_3D(νmm,CPW)h^5/3_avg (sine)^−8/3 . (2.186) In the two-dimensional regime the dependence on these parameters is much more complicated and should be calculated numerically.

Sayers et al. [37] examined the cross-PSD between pairs of bolometers to constrain the angular wind speed and found a median value of|w| '30 arcmin/sec. Assuminghavg=1 km this corresponds to a physical value of'10 m/sec, which is reasonable. We note that this is much faster than the maximum scan speed that can be achieved at the CSO of 4 arcmin/sec. Sayers et al. [37] then employed random simulations of the sky in order to fit the timestream correlation between bolometers pairs as a function of pair separation to Equation (2.185). They attempted to constrainhavgand the normalization and exponent of the power-law. The data was insensitive tohavgbut was able to constrain the exponent to−3.3±1.1, consistent with the−11/3 scaling predicted for K-T fluctuations in the three-dimensional regime. They then sethavg=1 km and fixed the exponent at−11/3 to constrain the median amplitude of the fluctuations

B²_3D(143 GHz,1.68 mm) =280 mK²rad^−5/3,

B²_3D(268 GHz,1.68 mm) =4000 mK²rad^−5/3. (2.187) Note that the choice ofhavg=1 km is based on radiosonde measurements of the water vapor profile above Mauna Kea [137].

The fluctuations in the brightness temperature of the sky characterized by Equation (2.185) are caused by changes in the emissivity of the sky due to fluctuations in the column depth of precipitable water vapor. That is to say

S_δTsky(α) = ∂Tsky

∂ τ 2

∂ τ

∂CPW

2

S_δCPW(α), (2.188)

whereτ(νmm,CPW)is the atmospheric opacity. It is reasonable to assume that the size of the fluctuations in the precipitable water vapor will scale with the amount of precipitable water vapor, orδCPW∝CPW so that

S_δCPW(α)∝CPW². In this case

B²_3D(νmm,CPW)∝

Tatme^−τ ∂ τ

∂CPW

CPW

2

, (2.189)

where we have used the equationTsky= (1−e^−τ)T_atm for the sky temperature at zenith since the dependence on elevation angle has already been addressed. Sayers et al. [37] found that Equation (2.189) explained the dominant trend in the value ofB²_3DwithCPWand observing band. However, there was significant scatter on the order of 100−300% about this trend, suggesting that the assumptionδCPW∝CPWonly holds in a statistical sense, and that the value ofCPW at the time of an observation should only be used as a rough proxy for the amplitude of the atmospheric noise.

We are primarily interested in how the atmospheric noise will manifest in the timestreams of the MUSIC detectors. SinceS(α)is azimuthally symmetric and since the wind speed is much greater than the scan speed of the telescope, we should have that

ν≈ |α||w|, (2.190)

whereν is the temporal frequency. This assumes that the wind velocity is constant over the course of a scan. If we neglect beam smoothing and windowing effects we can directly insert Equation (2.190) into Equation (2.185) to obtain an expression forS_δTsky(ν). The atmospheric noise will be transduced via changes in quasi-particle density, just like astronomical signal, so we can express the power spectral density as

S_δnatm qp (ν) =

∂nqp

∂Popt

2

∂Popt

∂Tsky

2

B²_atm ν

|w|

−b_atm

=

τ_qp^effηphηopt(1−fspill)∆νmm

V∆

2

k²_BB²_atm ν

|w|

−b_atm

, (2.191)

where 8/3≤batm≤11/3 andB2D≤Batm≤B3D depending on the value ofνrelative to _2∆hsine^havg|w|. The partial derivatives were evaluated using Equations (2.7), (2.8), and (2.25). To summarize, the atmospheric noise will appear in the quasi-particle direction and will have a power spectral density that is a power-law in frequency with a steep spectral index between 8/3 and 11/3. The amplitude of the noise will on average scale with the column depth of precipitable water vaporCPW.

We expect the atmospheric noise recorded by each detector to be highly correlated with the atmospheric noise recorded by every other detector for the following reasons. The separation between the near and far field of our instrument is given by the Fraunhofer distance

d_f=2D²

λ ^{, (2.192)}

whereDis the diameter of the telescope andλ is the wavelength of the radiation. MUSIC uses aD=9 m diameter illumination of the CSO primary mirror [138], resulting ind_f = [80, 120, 155, 285]km for the

four bands. The vast majority of atmospheric water vapor resides in the troposphere between 0 and 10 km, and therefore is well within the near field of our instrument. In the near field, the beam patterns of individual detectors are well approximated by the primary illumination pattern, which is approximately a 9 m diameter top-hat. Assumingh_avg=1 km, the 14 arcmin field of view translates to a physical size of approximately 4 m at the height of the turbulent layer. This means that even detectors on opposite sides of the focal plane will have significant beam overlap at this height and at any given time the detectors will see many of the same atmospheric fluctuations. Therefore, the atmospheric noise present in the timestreams of different detectors will be highly correlated. One can perform a correlation analysis to remove this common signal [139, 140, 141]. Note that the atmospheric noise will be coherent across observing bands and spectral information can be used to improve its removal [142, 143, 144].

The atmospheric noise drops off quickly with frequency. Sayers et al. [37] found that for Bolocam the atmospheric noise is negligible compared to photon noise at frequenciesν_&0.5 Hz. Likewise for MUSIC the atmospheric noise is subdominant atν_&0.5−1.0 Hz, as we will show in Section 3.3.5. The long-timescales at which the atmospheric noise is of consequence correspond to fluctuations in atmospheric emission at large spatial scales. Again, assuming a heighthavg=1 km and using the median wind speed of 10 m/sec found by Sayers et al. [37],ν_.0.5 Hz corresponds to physical scales&20 m or angular scales&70 arcmin. Hence, the atmospheric fluctuations that we are actually sensitive to occur on scales that are much larger than the size of the focal plane, and therefore should be slowly varying over the focal plane. This further supports our assertion that the atmospheric noise will be highly correlated between detectors.