Zeeman effect in atmospheric O 2 measured by ground-based microwave radiometry

Abstract. In this work we study the Zeeman effect on stratospheric O2 using ground-based microwave radiometer measurements. The interaction of the Earth magnetic field with the oxygen dipole leads to a splitting of O2 energy states, which polarizes the emission spectra. A special campaign was carried out in order to measure this effect in the oxygen emission line centered at 53.07 GHz. Both a fixed and a rotating mirror were incorporated into the TEMPERA (TEMPERature RAdiometer) in order to be able to measure under different observational angles. This new configuration allowed us to change the angle between the observational path and the Earth magnetic field direction. Moreover, a high-resolution spectrometer (1 kHz) was used in order to measure for the first time the polarization state of the radiation due to the Zeeman effect in the main isotopologue of oxygen from ground-based microwave measurements. The measured spectra showed a clear polarized signature when the observational angles were changed, evidencing the Zeeman effect in the oxygen molecule. In addition, simulations carried out with the Atmospheric Radiative Transfer Simulator (ARTS) allowed us to verify the microwave measurements showing a very good agreement between model and measurements. The results suggest some interesting new aspects for research of the upper atmosphere.


Introduction
The Zeeman effect is a phenomenon which occurs when an external magnetic field interacts with a molecule or an atom of total electron spin different from 0. Such an interaction will split an original energy level into several sublevels (Lenoir, 1967). In the atmosphere, oxygen is an abundant molecule which in its ground electronic state has a permanent magnetic dipole moment coming from two parallel electron spins. The interaction of the magnetic dipole moment with the Earth magnetic field leads to a Zeeman splitting of the O 2 rotational transitions. In this state, each rotational level with quantum number N is split into three levels of total quantum number J (J J ) following a Hund's coupling case (Pardo et al., 1995). This effect was studied by Gautier (1967) and Lenoir (1967Lenoir ( , 1968 in the 60 GHz band of the main isotopologue 16 O 2 . It is established, from these works, that the Earth's magnetic field splits the different Zeeman components over a range of a few megahertz around the center of each rotational line. The shape of each component is governed by a pressure broadening mechanism up to 60 km of altitude and by a Doppler mechanism above (Pardo et al., 1995).
Zeeman splitting of millimeter-wavelength emissions of oxygen molecules must be taken into account for altitudes above 45 km in the terrestrial atmosphere when modeling the radiative transfer of these molecules. Temperature soundings of the atmosphere at high altitudes are not possible without including this effect (Von Engeln et al., 1998;Von Engeln and Buehler, 2002;Stähli et al., 2013;Shvetsov et al., 2010). Observation of the Zeeman effect from ground-based measurements was first performed by Waters (1973) for atmospheric O 2 at 53 GHz. Pardo et al. (1995) were able to measure the Zeeman substructure for atmospheric 16 O 18 O at 233.95 GHz. For this rare isotopic species the relative abundance is much lower than for 16 O molecule, and its emission from upper atmospheric layers can be observed and the Zeeman substructure detected from the ground (Pardo et al., 1995). The main difficulty for observations of the Zeeman structure of 16 O molecule comes from its very broad tropospheric emission and the high opacity of low layers which eliminate any structure.

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The observation of this effect for 16 O has also been possible from satellite measurements. Hartmann et al. (1996) observed the Zeeman broadening of the oxygen emission line of the 9 + line in the 61.1509 ± 0.062 GHz frequency range using the Millimeter-Wave Atmospheric Sounder on the NASA space shuttle during the ATLAS missions. Comparison of satellite measurements and radiative transfer models including the Zeeman effect have also been addressed (Han et al., 2007(Han et al., , 2010Schwartz et al., 2006). Han et al. (2007) used spectral passband measurements from the Special Sensor Microwave Imager/Sounder (SSMIS) on board the Defense Meteorology Satellite Program F-16 satellite to measure the oxygen magnetic dipole transitions (7+, 9+, 15+, and 17+; Rosenkranz, 1993). These measurements were used to validate a fast model developed from the radiative transfer model of Rosenkranz and Staelin (1988). Moreover, the measurements were also used together with data from the Microwave Limb Sounder (MLS) on board the Aura spacecraft for assimilation in a numerical weather prediction (NWP) model (Hoppel et al., 2013). Schwartz et al. (2006) also reported a comparison of another radiative transfer model with measurements of the 118 GHz oxygen line from MLS.
In this work we present an experiment in which the Zeeman broadening of the oxygen emission line at 53.0669 GHz is observed and the polarization state of the radiation due to this effect is detected for the first time using a ground-based microwave radiometer.
The measurements were possible using a fast Fourier transform (FFT) spectrometer with 1 GHz of bandwidth to measure the whole oxygen emission line centered at 53.07 GHz and a narrow spectrometer (4 MHz) to measure the center of the line with a very high resolution (1 kHz). These measurements have been compared to a model which includes the Zeeman-splitting effect. The incorporation of this effect to the forward model will allow extension to the temperature retrievals beyond 50 km. This improvement in the forward model will be very useful for the assimilation of brightness temperatures in NWP models. It is also important to note that ground-based measurements of the atmosphere with good temporal resolution complement satellite measurements, which are temporally limited by their satellite's orbital parameters.
The paper is organized as follows: in Sect. 2, the instrumentation and the measurements are briefly outlined. The Zeeman effect theory and the modeling are presented in Sect. 3. Section 4 presents the results of this study. Firstly the simulations using a model are addressed and secondly the tropospheric correction performed to the radiometer measurements and the results obtained during this campaign are presented. Finally, the conclusions are given in Sect. 5.

Instrumentation and measurements
The TEMPERA (TEMPERature RAdiometer) radiometer is a microwave radiometer that provides temperature profiles from the ground to around 50 km (Stähli et al., 2013). This is the first microwave radiometer that measures temperature in the troposphere and stratosphere at the same time. The instrument is a heterodyne receiver at a frequency range of 51-57 GHz. Figure 1 shows a picture of TEMPERA, which is operated in a temperature-stabilized laboratory in the ExWi building of the University of Bern (Bern, Switzerland; 575 m above sea level; 46.95 • N, 7.44 • E). In this lab a styrofoam window allows views of the atmosphere over the zenith angle range from 30 to 70 • . The instrument mainly consists of three parts: the front end to collect and detect the microwave radiation and two back ends consisting of a filter bank and a digital FFT spectrometer for the spectral analysis. The radiation is directed into the corrugated horn antenna using an off-axis parabolic mirror. The antenna beam has a half power beam width (HPBW) of 4 • . The signal is then amplified and downconverted to an intermediate frequency for further spectral analysis. A noise diode in combination with an ambient hot load is used for calibration in each measurement cycle. The noise diode is calibrated regularly (about once a month) using a cold load (liquid nitrogen) and a hot load (ambient). The receiver noise temperature T N is in the range from 475 to 665 K. More details about the calibration with TEMPERA can be found in Stähli et al. (2013).
For tropospheric measurements the instrument uses a filter bank with four channels. By switching the local oscillator frequency with a synthesizer, it is possible to measure at 12 frequencies. In this way TEMPERA covers uniformly the range from 51 to 57 GHz at positions between the emission lines. Tropospheric retrievals are not addressed in this paper and more details about this measurement mode can be found in Stähli et al. (2013) and Navas-Guzmán et al. (2014).
The second back end is used for stratospheric measurements and contains a digital FFT spectrometer (Acqiris AC240) for the two emission lines centered at 52.5424 and 53.0669 GHz. The FFT spectrometer measures the two emission lines with a resolution of 30.5 kHz and a bandwidth of 960 MHz. The receiver noise temperature T N for the receiver-spectrometer combination is around 480 K. An overview of the technical specifications is given in Table 1.
An example of FFT measurements is shown in Fig. 2 (upper panel). This figure shows the brightness temperature on  16 January of 2012 for the oxygen emission line centered at 53.07 GHz. The red box indicates the influence of the Zeeman effect by the broadened line shape in the center with a kind of a plateau (round line shape around the line center: ±1 MHz). A second spectrometer was installed in TEMPERA in order to measure with a higher resolution the narrow spectral region where a broadening in the oxygen emission line is produced due to the Zeeman effect. This narrow-band software defined ratio (SDR) spectrometer consists of 4096 channels which cover a bandwidth of 4 MHz with a resolution of    1 kHz. An example of a monthly mean brightness temperature spectrum centered at 53.07 GHz measured with the SDR spectrometer is shown in Fig. 2 (lower panel).
Moreover, a set of two auxiliary mirrors was installed on the roof of the ExWi building in the University of Bern (Fig. 3). A rotating mirror allows one to observe the atmosphere under different azimuth angles and with a fixed elevation angle, while the fixed mirror directs the radiation from the rotating mirror into TEMPERA radiometer. The main goal of using these auxiliary mirrors is to measure the Zeeman-broadened oxygen line under different angles relative to the Earth's magnetic field.
A special campaign was carried out in autumn of 2013 in order to detect the Zeeman effect with TEMPERA. Three months of measurements (September-November 2013) were performed using these auxiliary mirrors and the SDR spectrometer. A special measurement cycle was designed for TEMPERA during this period. Periodic cycles of almost 5 min were performed. This whole cycle consisted of 13 subcycles, each one starting with a hot load calibration in combination with a noise diode for 10 s followed by other 10 s of atmospheric measurements in one azimuth direction. A total of 13 azimuth angles were scanned ranging from 71.5 to 191 • in steps of 10 • during the whole cycle. The elevation angle was fixed at 60 • during all the measurements since it was found as the angle at which the intensity of the emission lines was highest (Stähli et al., 2013).

Theory
The Zeeman effect (Zeeman, 1897) occurs because the spin of unpaired electrons couples to the external magnetic field, changing the internal energy of the molecule. A transition between two of these altered energy levels can change the frequency dependence of the absorption spectrum. The Zee-man energy change is calculated by where g depends on the line and molecule (see, e.g., Christensen and Veseth (1978) for molecular oxygen), µ 0 is the Bohr magneton, M(J M ) is the projection of J on the magnetic field, and H is the magnetic field vector. The quantum numbers necessary can be found in most databases, e.g., HI-TRAN (Rothman et al., 2013). There are 2J + 1 possible M for a state level (these are M = −J, −J + 1, · · ·, J − 1, J ), and M can only change by 0 or ±1. A transition without changing the value of M is called a π transition, and a transition with changing M is called a σ ± transition. The total line strength is not altered by the effect but will be distributed among the new lines. Each line "produced" by this procedure then undergoes the same broadening mechanisms (thermal and pressure) to create the absorption spectrum.
In addition to splitting the line, the change in energy level depends on the direction of the magnetic field and the propagation path of the radiation, which means that the absorption also depends on the polarization of the radiation. The main polarization occurs along the magnetic field in the plane perpendicular to the propagating radiation. If H is entirely in this plane, then the radiation will be linearly polarized along H for σ ± transitions and linearly polarized perpendicular to H for π transitions. If H is parallel/anti-parallel to the path of the propagating radiation, then the σ + and σ − transitions will circularly polarize the radiation in opposite ways, and π transitions do not affect the radiation at all. The polarizing effect will generally scale between the two cases above as a function of the angle that H forms with the direction of the propagating radiation.

Modeling
The first official release of the Atmospheric Radiative Transfer Simulator (ARTS) was by Buehler et al. (2005) as a flexible/modular code base for radiative transfer simulations. Since then, ARTS has been under continual development.
One key release is version 2.0 by Eriksson et al. (2011), which describes the ARTS scripting potential and a few of the modules. Presently, ARTS is at version 2.2; the latest version includes, among other new features, a module that calculates the Zeeman effect presented by Larsson et al. (2014).
In short, ARTS calculates each of the three polarization components individually before adding their absorption contributions to a Stokes vector propagation matrix. The polarization of the radiation is internally kept in a universal coordinate system defined by the sensor through all of the propagation. The line shape should return both its imaginary and real part to account for dispersion-caused polarization rotation. The input magnetic field is either static or three 3-Dgridded fields, one field for each coordinate: x, y, and z. This propagation matrix is then sent to the radiative transfer calculator, which solves the vector radiative transfer equation ( in, e.g., del Toro Ininiesta, 2003) where I is the Stokes vector, r is the path vector, K is the propagation matrix, and B is the Stokes version of Planck's function for blackbody radiation. For details on the ARTS Zeeman effect module see Larsson et al. (2014). The calculated relative Zeeman pattern for the line measured by TEMPERA can be found in Fig. 4. This otherwise singular line is split into 159 Zeeman lines, 53 for each polarization component. The plot has been renormalized for readability. The strongest split line accounts for less than 1.5 % of the original strength of the line, the maximum splitting from the central line is ∼ 27.99 kHz µT −1 , and the splitting between the individual lines is about 1.08 kHz µT −1 within a component. The last number is significantly small. The thermal broadening in the stratosphere is under normal conditions larger than the magnetic line splitting above Bern, so individual Zeeman lines cannot be discerned from the overall shape.

Brightness temperature simulations incorporating the Zeeman effect
Brightness temperature spectra have been simulated using the ARTS model which was described in the previous section. ARTS was set with all the information about instrumental aspects and location of TEMPERA in order to simulate the same measurement conditions. The brightness temperature was calculated for 13 azimuth angles (71.5 : 10 : 191.5 • ) and a fixed elevation angle (60 • ) and to simulate the atmospheric conditions of 15 October 2013 (Figs. 5 and 6). The altitude of the platform was set at 12 km in order to avoid any tropospheric effect in the spectra. On 15 October 2013, the total intensity of the magnetic field over Bern at the altitude of 50 km was 46 547 nT with a declination of 1 • 21 44 and an inclination of 62 • 46 16 (www.ngdc.noaa.gov/geomag/ magfield.shtml). Figure 5 shows the calculated brightness temperature spectra for a linear horizontal polarization of the oxygen emission line centered at 53.07 GHz in a range of 5 MHz. From these simulations we can observe that the spectra are almost identical for most of the frequency range plotted here and differences are only observed in the central part when the observational azimuth angle is changed. In the narrow central frequency range we can observe that both the shape and the intensity of the spectra changes for the different observations. For the higher azimuth angles the brightness temperature spectra are lower and the shape is flatter, while for lower angles the spectra are higher and the shape is less flat. The maximum difference in brightness temperature between the most intensive spectrum (91.5 • ) and the least intensive (191.5 • ) is 2.5 K. Figure 6 shows linear vertical polarization. We observe a similar pattern for linear vertical polarization as for linear horizontal polarization, with the peak strength of the signal changing mostly in the center of the line as a function of the azimuthal angle. However, the change is much smaller for linear vertical polarization, which only has a maximum difference of 1 K between the most and least intense spectra. Also, the change with azimuthal angle is inverted compared to linear horizontal polarization. For the linear vertical polarization the most intensive spectrum corresponds to the observational angle of 181.5 • while the least intensive corresponds to 91.5 • . This behavior is clearly associated with the polarized nature of the Zeeman effect, since the polarized state of the observed radiation changes when the angle between the propagation path and the direction of the Earth magnetic field is varied. It is also interesting to note from Figs. 5 and 6 that the differences between horizontal and vertical polarization are very small when close to the 181 • azimuth angle. This is in good agreement with theory, as this direction corresponds to measurements of radiation which has been propagated along the magnetic field towards TEMPERA. This parallel propagation results in minimal differences among linear polarizations.
The brightness temperature has also been simulated without considering the Zeeman effect in the ARTS model. These simulations correspond to the dashed lines shown in Figs. 5 and 6. We found that when the Zeeman module is not active there is no difference in the spectra for difference observational angles. Moreover, the spectrum presents higher brightness temperature values and it does not show any broadening in the center of the oxygen emission line.
In order to compare the simulated spectra from ARTS with the measurements, the effects of the different optical components of TEMPERA on the polarization state of the radiation, as well as the vertically polarized observing antenna, have to be considered. A full characterization of the polarization state of the radiation can be done by means of the Stokes vector, s, which is defined as where ε and µ are the electric and magnetic constants, respectively, < · > indicates time average, and E v and E h are the complex amplitudes for vertical and horizontal polarization. The first Stokes component (I ) is the total intensity, the second component (Q) is the difference between vertical and horizontal polarization, and the last two components, U and V , correspond to linear ±45 • and circular polarization, respectively. The Stokes components are converted to brightness temperature by inverting the Planck function (Eriksson et al., 2011); this new Stokes vector of brightness temperatures is denoted as s . The calculus of the measured brightness temperature (T p b ) considering the sensor polarization response can be expressed as (cf. Eriksson et al. (2011), Eq. 19) where p is a row vector of length 4 which describes the sensor polarization response. In the case of TEMPERA, whose antenna is vertically polarized, the vector p is [1100]. The rotation of the Stokes reference frame due to the reflection in the different mirrors and the rotation of the external mirror is considered using the transformation matrix L(χ), which allows one to obtain a consistent definition between the polarization directions for atmospheric radiation and sensor re- sponse. This matrix is defined as (Liou, 2002) The rotational angle (χ) has been calculated using the GRASP software package (www.ticra.com/products/ software/grasp). This software package allows design and analysis of complex reflector elements using physical optics, physical theory of diffraction and the method of moments. Figure 7 shows the setup of this simulation, where we can see the different TEMPERA components (horn antenna, parabolic mirror and the two auxiliary mirrors, the fixed and the rotating mirror) and the ray tracing of an electric field which is propagated from the antenna to the atmosphere. The calculated angle χ can be expressed as χ = ϕ +ϕ offset , where ϕ is the observational azimuth angle defined in our experiments (ϕ = 71.5 : 10 : 191.5 • ) and ϕ offset is 141.5 • . Once the sensor polarization response and the rotation of the polarized radiation have been characterized, the effective brightness temperature can be calculated as (Eriksson et al., 2006) Figure 8 shows the effective brightness temperature spectra calculated for the case simulated in ARTS (15 October 2013) in Figs. 5 and 6. For these spectra we can appreciate again the same pattern as in the ARTS simulations, with almost the same intensity on the wings of the oxygen emission line and some differences in the central frequencies when the azimuth angle is changed. The highest brightness temperature is found at 71.5 • , while the lowest is found at 191.5 • . The latter position corresponds to the maximum broadening found when the direction of observation is almost antiparallel to the direction south-north. The maximum difference between the most and the least intense spectra is 2.5 K. In order to study the difference in the broadening of each azimuth observational spectrum due to the Zeeman effect, we have calculated the ratio among each spectrum to the averaged spectrum from all the observational angles. Figure 9 shows these ratios for the different azimuth angles. Azimuthal behavior with a ratio below unity in the center of the line also has a ratio above unity in the wings, which means that the line experienced more than average broadening. The opposite is also true: azimuthal behavior with values above unity in the center means less than average broadening. The different ratios show a clear azimuth dependence, indicating that the highest broadening is found when the azimuth angle is 191.5 • while the smaller broadening is found at 71.5 • .

Tropospheric correction of SDR spectrometer
A ground-based microwave radiometer measures a superposition of emission and absorption of radiation at different altitudes. The received intensity at ground level can be expressed in the Rayleigh-Jeans limit (hυ kT ) as a function of the brightness temperature. In these conditions the radiative transfer equation is given by where T b is the brightness temperature at frequency υ, T 0 is the brightness temperature of the cosmic background radiation, T (z) is the physical temperature at height z, z 0 is the Earth surface, z 1 is the upper boundary in the atmosphere, α is the absorption coefficient, and τ is the opacity. The opacity is defined as The contribution of the troposphere to the brightness temperature measured with a microwave radiometer at ground level is very important and it could be very different depending on the observational direction or on the period of measurements. After oxygen, water vapor and liquid water (clouds) are the most important components in the atmosphere, the emissions of which have relevance in the microwave spectrum. It is very important to correct our measurements for any tropospheric effect in order to ensure that the changes observed in our measurements for different observational directions come from the stratosphere (Zeeman effect) and not from the troposphere.
Since the tropospheric portion of the pathlength provides a relatively spectrally flat signal the microwave radiative transfer equation can be rewritten as where T b (z trop ) is the brightness temperature as observed from the tropopause, τ is the tropospheric zenith opacity, and T trop is the effective temperature of the troposphere. From this equation the opacity can be calculated as Since the atmospheric opacity is dominated by the contribution from the troposphere, the stratospheric contribution is considered negligible and the cosmic background radiation, T bg , is in practice used instead of T b (z trop ) in Eq. (10). This means that the calculated τ actually is approximately the total atmospheric opacity and hence includes the minor contribution from altitudes above the troposphere (e.g., absorption by stratospheric O 2 and H 2 O) (Forkman et al., 2012).
T trop has been estimated using a linear model between the weighted tropospheric temperature and the ground temperature (Ingold et al., 1998). The weighted tropospheric temperature was calculated using radiosonde measurements. Radiosondes are launched twice a day at the aerological station of MeteoSwiss in Payerne (40 km west of Bern). One year of radiosonde data was used and the linear fit found between T trop at 53 GHz and the ground temperature T z 0 was T trop = 0.8159T z 0 + 47.21 K. The constant term T bg is independent of frequency and has a value of 2.7 K (Gush et al., 1990). The term T b (z 0 ) is measured using the wings of the oxygen emission line centered at 53.07 GHz for every azimuth angle. The simultaneous measurements performed with the FFT spectrometer allow us to measure in the wings of the oxygen emission line, where most of the contribution to the brightness temperature comes from the troposphere. In the frequency range of interest, the tropospheric attenuation increases with increasing frequency. In order to account for this, we determine the correction factor at each frequency using a linear fit between the frequency ranges highlighted in red in Fig. 10.
Once all the terms are calculated, the brightness temperature corrected for tropospheric effects can be obtained as It is interesting to note that for the correction presented in this section we have used the scalar radiative transfer equation, since this tropospheric correction is independent of polarization state. This assumption is valid if scattering can be neglected, which should hold in the absence of strong precipitation.

Stratospheric brightness temperature measurements
As already described in Sect. 2, a special campaign of microwave radiometer measurements has been performed for 3 months in autumn 2013 in the ExWi building of University of Bern. During this campaign, TEMPERA was set with a special configuration in order to be able to observe the Zeeman effect from ground-based measurements. Radiometer measurements in different azimuth angles (13 angles) were carried out in order to scan the atmosphere under different angles between the propagation path and the local Earth magnetic field. Figure 11 shows mean monthly brightness temperature spectra obtained for different azimuth angles in October 2013. All the measurements were corrected for tropospheric effects following the procedure described in the previous section. Figure 11a shows the whole range (4 MHz) measured with the SDR spectrometer. From this plot we observe that the mean spectra for the different azimuth angles show almost identical values outside of the narrow central region. However, differences in the intensity and Monthly Brightness temperature from SDR corrected of tropospheric effects (October 2013).
show a higher broadening for the highest azimuth angle for both, measurements and ions. In order to compare in a more quantitative way the measurements with the model we mpared the ratio between the maximum mean brightness temperature of each spectrum and n value for all the spectra at the central frequencies (range of ± 0.25 MHz). Equation 12 s explicitly the expression of these calculation: in the shape are observed in the very narrow range centered on 53.067 GHz. Figure 11b shows a zoom of the spectra in the central frequencies for some selected azimuth angles. We can observe that for higher azimuth angles the spectra show lower values of brightness temperature and flatter shapes in the central frequency range (±0.5 MHz), while higher brightness temperature and less flat shapes are observed in lower azimuth angles. These results are in good agreement with the simulations performed including the Zeeman effect with ARTS (Sect. 4.1). However, we can notice that there is an offset in the brightness temperature spectra from model (highest peak ∼ 64 K) and from measurements (highest peak ∼ 54 K). The offset could be due to an inappropriate consideration of the continuum absorption from secondary species (water vapor, ozone, etc.) in the forward model and the fact that the contribution from line mixing to the oxygen spectra is not modeled. Other reasons that could explain some differences could be related to the uncertainties of the tropospheric correction in the measurements and to the fact we are comparing different periods of measurements: 1 day for the simulations (15  Figure 12. Brightness temperature spectra from the SDR spectrometer (solid lines) and simulated from ARTS (dash lines) for two different observational angles.
October 2015) and 1 month of integrated measurements (October 2013). In any case, while the baseline offset affects the absolute difference between model and data, the shape of the center line (±2 MHz) is not altered. Thus the main conclusions of Zeeman polarization measurements and the ARTS module validation are solid. Figure 12 shows a direct comparison of the brightness temperature spectra from SDR measurements (solid lines) and from ARTS model (dashed lines) for two observational azimuth angles (91.5 and 181.5 • ). An offset correction has been applied to the simulated spectra in order to compare with the measurements. Although the absolute values are not exactly the same for the modeled and measured spectra in the center of the oxygen emission line we can clearly observe that the behavior of the spectra for the two azimuth angles are the same. The spectra show a higher broadening for the highest azimuth angle for both measurements and simulations. In order to compare the measurements with the model in a more quantitative way, we have compared the ratio of the maximum mean brightness temperature of each spectrum to the mean value for all the spectra at the central frequencies (range of ±0.25 MHz). Equation (12) indicates explicitly the expression of these calculations: where υ 1 and υ 2 indicate the frequency range which corresponds to an interval of 0.5 MHz centered at 53.067 GHz. ψ i is the observational azimuth angle for a specific position and n t is the total number of positions scanned by TEMPERA (13 positions). Figure 13 shows these ratios calculated with the ARTS model simulating the conditions of 15 October 2013 and the ones obtained from the mean monthly spectra (October 2013)  Figure 13. Ratios of maximum brightness temperature of each spectrum to the mean value for all the spectra at the central frequencies for TEMPERA radiometer (red points) and ARTS (blue points). measured by TEMPERA. We can observe that, in general, there is a very good agreement between the measurements and the model. Both simulations and measurements show higher ratio values (> 1), which indicate a smaller broadening regarding the averaged spectrum for the smallest azimuth angles and a larger broadening (ratios < 1) for the largest angles. The relative differences between both ratios are lower than 1 % for all the azimuth angles. We observe that the ratios for some azimuth angles are almost identical while some discrepancies are observed for other ones. The errors for the TEMPERA measurements have been estimated by evaluating the uncertainties associated with the different terms of the tropospheric correction (Eq. 11). The error bars shown in Fig. 13 have been calculated using error propagation theory and they presented values very similar (∼ 0.01) for all the observational angles. The errors associated with the simulations were obtained by evaluating the ratio of the simulated spectra plus Gaussian white noise. The calculated uncertainties presented values much smaller than the ones found for the measurements (maximum value of 4.6 × 10 −4 ). It is important to note that the differences found between measurements and simulations are within the measurement uncertainties. From this comparison we can conclude that the agreement between measurements and model is clear. These results show the polarized state of the radiation due to the Zeeman effect, which is revealed for a different broadening in the spectra when the angle of the Earth magnetic field and the observational path is changed.

Conclusions
This work presents an experiment in which the Zeeman broadening of the oxygen emission line at 53.0669 GHz is observed and the polarization state of the radiation due to this effect is detected for the first time using a ground-based microwave radiometer. A special campaign was carried out in order to detect this effect with the TEMPERA radiometer. The installation of a fixed and a rotating mirror in front of TEMPERA allowed us to measure under different angles between the observational path and the Earth's magnetic field direction. A total of 13 azimuth angles were scanned ranging from 71.5 to 191.5 • . In addition, the use of a narrow spectrometer (4 MHz) allowed us to measure the center of the oxygen emission line with a very high resolution of 1 kHz.
The brightness temperature spectra for the different azimuth angles were simulated using the ARTS model. This forward model applies a vector radiative transfer code which includes the Zeeman effect. ARTS was set up with all the information about instrumental aspects and location of TEM-PERA in order to simulate the same measurement conditions. These simulations showed almost identical spectra for most of the frequency range (4 MHz) and differences were only observed in the central part when the observational azimuth angle was changed. The spectra considering linear horizontal polarization showed lower values of brightness temperature and flatter shapes for the highest azimuth angles, while for lower angles the spectra showed higher values and the shapes were less flat. The maximum difference in brightness temperature between the most intensive spectrum (91.5 • ) and the least intensive (191.5 • ) was 2.5 K. For the linear vertical polarization the effect in the central frequencies was smaller, with a maximum difference of brightness temperature of 1 K between the most and the least intensive spectra; the azimuthal order was the inverse. These results are an evidence of the polarized nature associated with the Zeeman effect, which shows changes in the polarized state of the observed radiation when the angle between the propagation path and the direction of the Earth's magnetic field is varied.
In order to compare the ARTS simulations with the measurements the effects on the polarization state of the radiation due to the different optical components of TEMPERA radiometer were taken into account using the GRASP software package. The effective brightness temperature calculated after this correction showed that the most intense and least-broad spectrum was found at 71.5 • and the least intense and most-broad spectrum was found at 131.5 • . The maximum difference between both spectra was 2.3 K.
Similar behavior to the simulations was observed for the measured spectra from the TEMPERA radiometer. A direct comparison of the ratios of the maximum brightness temperature of each spectrum to the mean value for all the spectra at the central frequencies showed a very good agreement between the model and the measurements. Both simulations and measurements showed a smaller broadening for the smallest azimuth angles and a larger broadening for the largest angles. The small discrepancies found for some azimuth angles were always within of the measurement uncertainties.