## 2 Instruments and Methods
### Measurements of the Atmosphere Optical Depth by the \"Sky Dip\" Method  
#### 2.2.1 Basics of the \"Sky Dip\" Method  
From the measurements at two zenith angles (\(\theta_{1}\) and \(\theta_{2}\)), the optical depth of the atmosphere in zenith in the first approximation can be obtained as [16]  
\[\tau=\frac{1}{\sec\theta_{2}-\sec\theta_{1}}\ln\frac{u_{0}-u_{1}}{u_{0}-u_{2}}\,. \tag{6}\]Here, \(u_{1}\) and \(u_{2}\) are the receiver responses at the zenith angles \(\theta_{1}\) and \(\theta_{2}\), respectively, while \(u_{0}\) is the response for the input emission with the brightness temperature \(T_{0}\). Most frequently, a reference angle near the horizon, where the optical depth should be high, is used for this purpose. Some variants of this method, which use measurements at 3 zenith angles and a reference area near the horizon, are considered in [30]. From measurements of the sky brightness in a range of zenith angles, the optical depth can be derived by fitting the measurement results with a function corresponding to Equation (4) (which can be also expressed in the logarithmic form), assuming a constant mean atmosphere temperature. In this case, a reference signal is needed anyway. Some examples of such devices are the 225 GHz tipping radiometers at the site of the 30 m IRAM radio telescope (D.L. John, private communication) and at the Large Millimeter Telescope (LMT) site [31]. 2.2 A Dual-Band Radiometer for Measuring the Atmospheric Opacity Developed at the Institute of Applied Physics RAS  
About 10 years ago, a dual-band radiometer MIAP-2 for measuring the atmospheric opacity was developed and manufactured at the Institute of Applied Physics of the Russian Academy of Sciences and GYCOM company [32]. The radiometer operates in the 3 mm and 2 mm atmospheric windows. At the front-end of the 3 mm channel, a broad-band millimeter-wave amplifier is used with the specified frequency range 84-99 GHz. In fact, it has no rejection of lower frequencies, which leads to contamination of the measurements by the strong contribution of the molecular oxygen absorption band near 60 GHz. The problem was partly fixed by the installation of a waveguide filter with the cross-section of \(1\times 2\) mm\({}^{2}\), which has a cutoff frequency of \(\sim\)75 GHz [33]. Nevertheless, the contribution of the low frequencies remains significant, and its consequences are discussed below. The front-end of the 2 mm channel includes the local oscillator on a Gunn diode with a built-in frequency doubler, a balanced mixer on Schottky-barrier diodes (SBDs), an intermediate-frequency amplifier (IFA) and a detector for the range of 4-8 GHz. The local oscillator frequency is 140 GHz. The channel detects the emissions in both lower and upper sidebands, which are 132-136 GHz and 144-148 GHz, respectively. At the front-ends of both channels, the modulator-calibrator is used. This device is described in detail in [34]. It is based on chains of series-parallel-connected SBDs, placed into the standard cross-section waveguide. It can serve as a modulator of the input signal or as a source of a calibration signal. Depending on the current, it can be in 3 states: (1) open (transmitting the input signal with low losses), (2) locked with the equivalent brightness temperature of 155-180 K (\"cold\" calibration level) and (3) locked with the equivalent brightness temperature of \(\sim\)300 K (\"warm\" calibration level). These features enable both sky dip measurements and absolute measurements of the sky brightness temperature. Both channels are equipped with the lens antennas, which are misphased conically shaped feed horns with a bent fracture and lenses. The half-power beam width of the antennas (with lenses) in both bands is about 2.5\({}^{\circ}\). The elevation scan is provided by the rotating common mirror in front of both horns, oriented at the angle of 45\({}^{\circ}\), and a mirror drive system, which is based on a stepper motor. The zenith angle range is from 0\({}^{\circ}\) to 90\({}^{\circ}\), and the step is 0.7\({}^{\circ}\).