the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Improving atmospheric path attenuation estimates for radio propagation applications by microwave radiometric profiling

### Ayham Alyosef

### Domenico Cimini

### Lorenzo Luini

### Carlo Riva

### Frank S. Marzano

### Marianna Biscarini

### Luca Milani

### Antonio Martellucci

### Sabrina Gentile

### Saverio T. Nilo

### Francesco Di Paola

### Ayman Alkhateeb

### Filomena Romano

Ground-based microwave radiometer (MWR) observations of downwelling brightness temperature
(*T*_{B}) are commonly used to estimate atmospheric attenuation at relative
transparent channels for radio propagation and telecommunication purposes. The atmospheric
attenuation is derived from *T*_{B} by inverting the radiative transfer equation with a
priori knowledge of the mean radiating temperature (*T*_{MR}). *T*_{MR} is usually
estimated by either time-variant site climatology (e.g., monthly average computed from atmospheric
thermodynamical profiles) or condition-variant estimation from surface meteorological
sensors. However, information on *T*_{MR} may also be extracted directly from MWR
measurements at channels other than those used to estimate atmospheric attenuation. This paper
proposes a novel approach to estimate *T*_{MR} in clear and cloudy sky from independent MWR
profiler measurements. A linear regression algorithm is trained with a simulated dataset obtained
by processing 1 year of radiosonde observations of atmospheric thermodynamic profiles. The
algorithm is trained to estimate *T*_{MR} at K- and V–W-band frequencies (22–31 and
72–82 GHz, respectively) from independent MWR observations at the V band
(54–58 GHz). The retrieval coefficients are then applied to a 1-year dataset of real
V-band observations, and the estimated *T*_{MR} at the K and V–W band is compared with
estimates from nearly colocated and simultaneous radiosondes. The proposed method provides
*T*_{MR} estimates in better agreement with radiosondes than a traditional method, with
32 %–38 % improvement depending on frequency. This maps into an expected improvement in
atmospheric attenuation of 10 %–20 % for K-band channels and ∼30 % for V–W-band
channels.

There is a continuous trend to use higher frequencies in the development of satellite communication (SatCom) as lower-frequency bands become saturated (e.g., Biscarini et al., 2017). Europe's current Earth observation programs with the Sentinel satellite constellation generate a daily data volume of terabytes, requiring new broadband links to access the data. In future interplanetary explorer missions, the need for high-throughput communications will also become more pressing due to a wider range of observed parameters and teleoperated landers or rovers to avoid data loss due to limited onboard memory or data compression (Jebril et al., 2007; Acosta et al., 2012). In remote areas on Earth, like Antarctica, it is of concern to forward scientific data via satellite to the research facilities (Bonifazi et al., 2002). All mentioned scientific applications have in common that the increase in data volume requires higher transmission capacities than those currently available. Current high-throughput SatCom systems operate at the X (8–12 GHz), Ku (12–18 GHz), K (18–26), and Ka (26–40 GHz) band, and presumably their next implementation will use Q (40–50 GHz) and V (50–75 GHz) bands, whereas the W band (75–110 GHz) appears to be the next natural evolution (Riva et al., 2014). Moving beyond the X and Ku bands to less congested higher frequencies increases the available bandwidth, allowing smaller equipment that consequently reduces the size of the satellite and launch vehicle (Cianca et al., 2011; Acosta et al., 2012; Emrick et al., 2014).

Ground-based microwave radiometer (MWR) observations of downwelling brightness temperature
(*T*_{B}) are commonly used to estimate atmospheric attenuation at relatively
transparent microwave channels for radio propagation and telecommunication purposes (e.g., Marzano et
al., 2006; Marzano, 2007; Biscarini et al., 2019). However, higher frequencies are characterized by
larger dynamics of atmospheric propagation effects, mainly because of higher atmospheric losses
(rain, clouds, and atmospheric gases). Planning of V- and W-band SatCom systems requires experimental
data to characterize these unexplored atmospheric radio channels (Mattioli et al., 2013; Riva et
al., 2014; Biscarini and Marzano, 2020). Radio wave propagation models can provide a reliable
estimate of atmospheric path attenuation but have typically been validated only for frequencies
up to 50 GHz (Riva et al., 2014). These models, recommended by the International
Telecommunication Union (ITU), are based on past experimental campaigns at K–Ka and Q bands, whereas
designing the Earth–satellite link budget at V and W bands would require satellite beacon data that
are currently not available. It is then essential to investigate the behavior of electromagnetic
waves in the V and W bands to improve existing models and validate them with independent measurements
(Biscarini et al., 2019).

In response to this need, a measurement campaign has been recently planned to characterize the V-
and W-band satellite atmospheric radio channel through ground-based microwave radiometric
observations. The core observatory is located at Politecnico di Milano (Milan, Italy), where a
four-channel MWR, including two V- and W-band channels at 72.5 and 82.5 GHz, respectively,
is operated. An independent MWR with a 14-channel temperature and humidity profiler is also operated
in Spino d'Adda, 25 km from Milan (Italy). Atmospheric path attenuation is derived from MWR
*T*_{B} observations by inverting the radiative transfer equation with a prior knowledge of
the mean radiating temperature (*T*_{MR}). A priori *T*_{MR} is usually obtained
from monthly average values computed from radiosondes (e.g., Martellucci, 2007), inferred
from surface meteorological sensors (e.g., Luini et al., 2018), or derived from radio propagation
models (e.g., Mattioli et al., 2013; Biscarini and Marzano, 2020). The uncertainty in
*T*_{MR} estimates contributes to the path attenuation uncertainty. To the aim of reducing
this uncertainty, in this work we propose an original approach increasing the accuracy of
*T*_{MR} estimates by exploiting independent MWR profiler measurements. This is a follow-up
of the work presented at the 11th International Symposium on Tropospheric Profiling (Cimini et al.,
2019). The paper is structured as follows: Sect. 2 describes the methodology, and Sect. 3
presents the available dataset; Sect. 4 presents the results and the obtained performance, and
Sect. 4 summarizes the results, providing hints for future work.

The atmospheric brightness temperature *T*_{B} (K), measured by a MWR at frequency *f*_{i}
and elevation angle *θ*, can be used to estimate the atmospheric total path attenuation
*A*_{MWR}(*f*_{i},*θ*) (dB) using the following expression
(e.g., Marzano, 2007; Ulaby and Long, 2014):

where *T*_{C} is the cosmic background temperature (usually set to 2.73 K in the
microwave and millimeter-wave range) and *T*_{MR}(*f*_{i},*θ*) is the mean
radiating temperature (in K), which is given by (e.g., Han and Westwater, 2000)

where *T*(*s*) and *α*(s) are the atmospheric physical temperature and absorption
coefficient along the path *s* and
$\mathit{\tau}\left(\mathrm{0},\mathrm{\infty}\right)={\int}_{\mathrm{0}}^{\mathrm{\infty}}\mathit{\alpha}\left(s\right)ds$ is the total atmospheric
opacity (Np) from the surface to the top of the atmosphere. As Eq. (2) suggests, the mean radiating
temperature represents the mean temperature along the optical path weighted by the atmospheric
transmission ${T}_{\mathrm{A}}={e}^{-\mathit{\tau}}$, i.e., the inverse of the atmospheric loss
*L*_{A}=*e*^{τ}. Note that Eqs. (1) and (2) are derived from the radiative transfer
equation for a non-scattering atmosphere (Schwarzschild's equation) and adopting the Rayleigh–Jeans
approximation (Janssen, 1993), which is commonly used in the microwave range to simplify Planck's law
with a linear relationship with temperature *T*;
${B}_{f}\left(T\right)\approx \mathrm{2}k\frac{{f}^{\mathrm{2}}}{{c}^{\mathrm{2}}}T$, where *k* and *c* are the Boltzmann and speed
of light constants, respectively. In these conditions, the atmospheric opacity can be written as

and thus the atmospheric total path attenuation, which is simply the atmospheric loss in decibels (dB) units,
can be rewritten in terms of *τ* as

Note that, as discussed in Han and Westwater (2000) and Janssen (1993), Eq. (1) is just an
approximation of the exact formulation. In the frequency range used here, this approximation is
valid within 2 % of the exact formulation, and thus it is adopted here for the sake of
simplicity. Moreover, atmospheric scenarios with rainfall and snowfall are excluded since multiple
scattering is not included in Eq. (1) and thus in this work (see Marzano et al., 2006; Biscarini and
Marzano, 2020). *T*_{MR} can be easily calculated from the atmospheric profiles of the physical
temperature and absorption coefficient through Eq. (2). In clear-sky conditions, radiosonde profiles
of temperature and humidity are sufficient to compute *T*_{MR}, while in the presence of clouds
assumptions must be made on the vertical distribution of condensed water (e.g., Salonen and Uppala,
1991).

Thus, the mean radiating temperature plays a role in mapping the brightness temperature to the
atmospheric opacity and then total path attenuation, and the operational estimate of atmospheric
attenuation from radiometric *T*_{B} observations requires some a priori knowledge of
*T*_{MR}. Traditionally, *T*_{MR} was treated as a constant determined
climatologically from a dataset of atmospheric profiles, usually radiosondes. This assumption
propagates uncertainty in the attenuation estimates through Eq. (1). However, as long as
*T*_{B} is relatively low, e.g., for zenith and low-frequency observations, the
*T*_{MR} uncertainty contribution to attenuation is rather small, and thus precise
knowledge of *T*_{MR} is not crucial.

On the other hand, with increasing *T*_{B} values, e.g., in the case of observations at lower
elevation angles and/or at relatively more opaque higher frequencies, accurate *T*_{MR}
estimates gain more importance. One consequence is that *T*_{MR} uncertainties cause
significant calibration errors when large air masses (i.e., pointing at a low elevation angle) are
used. For example, it has been demonstrated that using a *T*_{MR} climatological mean (with
9 K standard deviation based on a 13-year dataset) introduces up to 1.4 K
uncertainty in tipping curve calibration at K-band channels, exploiting elevation angles down to
∼15^{∘} (Han and Westwater, 2000).

Thus, methods are usually exploited to reduce *T*_{MR} uncertainties, especially when low-angle and/or high-frequency observations are involved. One simple method is to divide the
*T*_{MR} climatology into seasons, efficiently reducing the standard deviation of the
climatological mean. A slightly more sophisticated method exploits time interpolation of the
*T*_{MR} monthly mean (Martellucci, 2007). However, these methods do not consider the actual
meteorological conditions, which may significantly differ from the seasonal or monthly mean. In
order to consider the actual meteorological conditions, another method is predicting
*T*_{MR} from the surface air temperature using regression analysis. Surface-based
temperature measurements, along with *T*_{MR} calculated from radiosonde measurements, provide
the means to derive linear regression coefficients relating surface temperature to
*T*_{MR}. It has been shown that this method reduces the calibration uncertainty in K-band
channels by a factor of ∼3 (Han and Westwater, 2000). Other surface measurements, such as
pressure and humidity, may also be considered among the predictors in addition to temperature. This
last method, relating *T*_{MR} to surface pressure, temperature, and humidity (PTU)
measurements, likely represents the current best practice (Luini et al., 2018). Note that hereafter
relative humidity is used as the humidity variable.

However, the PTU method may be inaccurate in particular cases, i.e., when surface conditions are not
well correlated with upper air. One obvious case is the occurrence of strong temperature
inversions. To circumvent this problem, another method was suggested by Han and Westwater, (2000):
*T*_{MR} prediction could be improved by using boundary temperature profiles from an MWR
profiler or a radio acoustic sounding system, which accurately recovers boundary layer surface
temperature inversions (Martner et al., 1993). To our knowledge, this has not been
demonstrated yet.

Thus, this analysis builds on this suggestion and presents a method to derive *T*_{MR} from
combined surface measurements and MWR profiler observations, demonstrating the reduced uncertainty
with respect to the other methods introduced above.

The proposed method is demonstrated estimating *T*_{MR} at four channels in K and V–W bands
from surface measurements and independent MWR profiler observations. The dataset considered here
consists of experimental data collected in 2015–2016 at two sites involved within the ESA WRad
campaign. The MWR operated in Spino d'Adda is a humidity and temperature profiler (HATPRO)
manufactured by Radiometer Physics GmbH (RPG) measuring *T*_{B} at 14 channels from the K to
V band (22.24, 23.04, 23.84, 25.44, 26.24, 27.84, 31.4, 51.26, 52.28, 53.86, 54.94, 56.66, 57.3,
58.0 GHz). The MWR operated at Politecnico di Milano is a LWP-U72-82 manufactured by RPG
measuring *T*_{B} at four channels, two at the K band (23.84 and 31.4 GHz) and two between the
V and W bands (72.5 and 82.5 GHz). During the considered period, both MWRs constantly pointed
at ∼35^{∘} elevation towards the geostationary satellite Alphasat, collecting
one sample per second. Standard meteorological sensors are located near the two MWRs to provide the
environmental PTU measurements.

In addition, the dataset includes the atmospheric thermodynamical profiles measured by radiosondes
launched operationally twice a day from the Linate airport in Milan (∼5 km from
Politecnico di Milano). The two radiosondes per day are launched at 11:30 and 23:30 UTC. Radiosonde
profiles in the period from January 2015 to December 2016 have been collected for this
analysis. Atmospheric thermodynamical profiles from each radiosonde have been processed to compute
the simulated *T*_{MR} in clear and cloudy conditions using the Wave Propagation Laboratory
(WPL) radiative transfer code. This code was originally developed at the U.S. National Oceanic and
Atmospheric Administration (NOAA; Schroeder and Westwater, 1991), implementing the millimeter-wave
propagation model (MPM; Liebe, 1989), and has since been updated with refined spectroscopic
parameters (Rosenkranz, 2017), as described in Cimini et al. (2018) and references therein. The
cloud water content is modeled using the Teknillinen KorkeaKoulu (TKK) method (Salonen and Uppala, 1991; Luini et al., 2018).

The experimental implementation is pictured in Fig. 1. *T*_{B}, *T*_{MR}, and PTU
simulated from the 2-year dataset of radiosonde profiles are used in the training and test
phases. Synthetic noise, with zero mean and standard deviation equal to the expected instrument
accuracy, has been added to simulate the instrument uncertainty. In the training phase, a half-dataset (2016) is used to train two versions of a multivariate linear regression to estimate
*T*_{MR} from either PTU only or PTU and *T*_{B}. From the set of 14 HATPRO
channels available, we selected the five higher-frequency V-band channels (51.26, 52.28, 53.86,
54.94, 56.66, 57.3, 58.0 GHz). These channels are mostly sensitive to atmospheric
temperature and are less affected by hydrometeors than lower-frequency K-band channels, which makes
them more suited for the operational whole-sky estimate of *T*_{MR}. In the test phase, the
two versions of regression coefficients are used to estimate *T*_{MR} from either PTU only or
PTU and *T*_{B} from the remaining dataset (2015). The resulting *T*_{MR} values are then
compared with “true” values computed from simultaneous radiosondes. Finally, in the validation
phase, the two versions of regression coefficients are fed with real measurements, either from the PTU
sensor only or with the PTU sensor and five HATPRO V-band channels. The resulting *T*_{MR} values
are again compared with true radiosonde values and also applied to real LWP-U72-82 observations
to estimate atmospheric attenuation through Eq. (1).

In the validation phase, the multivariate regression trained with the simulated dataset from 2016 is
applied to real observations in 2015 and validated against *T*_{MR} computed from radiosonde
profiles. For the considered pointing angle (35^{∘} elevation), the cloud liquid water path
estimated from radiosondes reaches 2.8 mm for the training set, while the liquid water path
estimated from MWR observations within the validation set reaches 4.6 mm. The results from
the two versions of regression coefficients, one applicable to surface PTU measurements only and the
other applicable to PTU measurements and five V-band channels *T*_{B}, are compared
here. The implemented equation and coefficients for the multivariate regression are given in
Appendix A. The output dataset consists of *T*_{MR} and *A* at four frequencies
(23.84, 31.4, 72.5, and 82.5 GHz) retrieved at 1 min temporal resolution. One
example of 24 h time series is shown in Fig. 2. For all four considered frequencies, it is
evident that *T*_{MR} from PTU and *T*_{MR} from PTU and *T*_{B} follow a
similar diurnal cycle, decreasing up to 05:00, then rapidly increasing until noon, then remaining
stable for a few hours, and finally decreasing again after 17:00 UTC. However, there seems to be a
factor of ∼2 in the peak-to-peak variation; e.g., at 23.84 GHz, *T*_{MR}
peak-to-peak variation is ∼9 K for *T*_{MR} (PTU), while it is ∼4 K for
*T*_{MR} (PTU and *T*_{B}). *T*_{MR} computed from the two daily radiosondes,
representing our reference “truth”, seems to confirm that *T*_{MR} (PTU and
*T*_{B}) is correct in estimating a smaller variation. The statistical comparison from the
validation phase is reported in Figs. 2 and 3, considering a set of 638 radiosondes in 2015. From
this dataset, the *T*_{MR} climatological variations in Milan in clear and cloudy sky are
estimated to be ∼7.6–8.2 K, depending upon K- and V–W-band channels. Time colocation
with radiometric observations is achieved by averaging the estimated *T*_{MR} within
15 min of the radiosonde release time. All the considered statistical scores show that
*T*_{MR} (PTU and *T*_{B}) agrees better than *T*_{MR} (PTU) with the
reference radiosondes for all four considered frequency channels (two K and two V–W bands). In
particular, the average difference (AVG), the root mean square difference (RMSD), and the correlation
coefficient (COR) with respect to *T*_{MR} from radiosondes are reported in Table 1. Four
methods to estimate *T*_{MR} are reported in Table 1: seasonal climatology (monthly mean),
time-interpolated monthly mean, regression from PTU, and finally regression from PTU and
*T*_{B}. As one would expect, Table 1 indicates that condition-dependent methods (e.g., the
two regression types) outperform methods simply based on climatology. The only score that is better
for climatology methods is AVG, i.e., the average difference over 1 year. This is somewhat
expected, as the climatology methods minimize the annual mean difference by definition. Nonetheless,
the regression methods show modestly higher AVG values. Conversely, the regression methods show
substantially better RMSD and COR scores with respect to climatological methods, which confirms that
regression methods are preferable when accurate estimates of *T*_{MR} and atmospheric
attenuation are desired. Table 1 also clearly indicates that the regression based on PTU and
*T*_{B} outperforms the one based on PTU only. For the considered K- and V–W-band
frequencies, the improvement ranges between ∼0.2 and 0.8 K in average difference,
∼1.0 and 1.4 K in RMSD, and ∼4 % and 7 % in correlation. This quantitatively demonstrates
that the consideration of V-band channels within the regression brings in significant
information on *T*_{MR}, as originally foreseen by Han and Westwater (2000).

Given the radio propagation purposes, the question is whether the improvements in *T*_{MR}
estimation given in Table 1 bring significant improvements in atmospheric attenuation estimates. In
order to investigate this, we propagate *T*_{MR} and *T*_{B} uncertainty through
Eq. (1) to obtain the uncertainty of atmospheric attenuation. From Eqs. (3)–(4), the uncertainty in
atmospheric attenuation is simply related to the uncertainty in atmospheric opacity as

where

is the uncertainty in atmospheric opacity due to the uncertainty in *T*_{MR} and
*T*_{B}, i.e., *σ*_{TMR} and *σ*_{TB}. Thus, we compute the
uncertainty of atmospheric attenuation *σ*_{A} in the case that *T*_{MR} is estimated
from PTU with *T*_{B} and from PTU only by replacing *σ*_{TMR} in Eq. (6)
with the *T*_{MR} uncertainty in Table 1 and *σ*_{TB} with a
typical value for MWR *T*_{B} uncertainty, i.e., 0.5 K (e.g., Cimini et al.,
2003). The percentual improvement brought by the *T*_{MR} estimated with the proposed method
(A, based on PTU and *T*_{B}) over the conventional method (B, based on PTU only)
is quantified by

for both *T*_{MR} and *A*. Table 2 summarizes the percentual improvements for the four
considered frequencies in the K and W band. Thus, with respect to the conventional PTU method, the
proposed method on average improves the *T*_{MR} estimates by more than 32 %, and it is
expected to improve the *A* estimates by 10 %–20 % at K-band channels and
∼30 % at V–W-band channels. In terms of radio propagation measurements, the achieved
improvement level is rather modest (fraction of a decibel) in clear-sky conditions when
*T*_{B} and the atmospheric attenuation are low, but it becomes more and more important as
*T*_{B} and the attenuation increase (e.g., heavy clouds and precipitation) due to the
(*T*_{MR}−*T*_{B}) factor in the denominator of Eqs. (1) and (6).

To show an example of application, we select one day (31 December 2018) for which data from the
14-channel MWR in Spino d'Adda and the four-channel MWR at Politecnico di Milano are available,
together with the PTU readings. PTU and *T*_{B} at the five higher-frequency V-band
channels (51.26, 52.28, 53.86, 54.94, 56.66, 57.3, 58.0 GHz) of the 14-channel MWR are used
to compute *T*_{MR} at the frequencies of the four-channel MWR (23.84, 31.40, 72.50,
82.50 GHz). *T*_{MR} and the observed *T*_{B} at the four channels are used
to compute the attenuation. Results for both PTU only and for the PTU and *T*_{B} method are shown in
Fig. 5 (*T*_{MR}) and Fig. 6 (attenuation). Figures 5 and 6 also show *T*_{MR} and
attenuation computed from the radiosonde profiles (twice daily) and the model profiles (every
6 h) from the nearest grid point of the global analysis produced by the European Centre for
Medium-Range Weather Forecasts (ECMWF). The difference between the PTU and PTU with *T*_{B}
methods is evident between midnight and 08:00. As indicated by the radiosonde profile (not
shown), that night was characterized by a temperature inversion near the surface about 8 K
strong and 160 m deep. This causes the surface temperature (used in the PTU method) to
decouple from that of the upper air. Conversely, the PTU and *T*_{B} method brings in
information on lower-atmospheric temperature. The *T*_{MR} difference between the two methods is
4–6 K at 08:00, rapidly decreasing as the Sun warms up the surface and fading to
negligible values around noon.

A similar behavior is found in attenuation (Fig. 6), although the difference is less
striking. Attenuation from radiosondes and ECMWF profiles is mostly closer to that from the PTU and
*T*_{B} method. However, a proper validation would require a dataset with a larger dynamical
range and an independent reference valid in both clear and cloudy conditions. In fact, neither
radiosonde nor ECMWF profiles can be assumed as a reference in cloudy conditions due to the lack of
accurate cloud water content, which for radiosondes is modeled statistically (TKK method), while
for ECMWF it represents a larger scale than the local one. The collection of a reference dataset is
indeed the main objective of the WRad campaign through the application of Sun-tracking microwave
radiometry (Biscarini et al., 2019, and references therein).

In this paper we propose an approach to estimate *T*_{MR} from radiometric observations at the
V band (sensitive to atmospheric temperature) in addition to surface measurements of PTU, which
represents the current best practice. The approach was suggested in Han and Westwater (2000) but
never attempted to our knowledge. Here, we implement the suggested approach by applying multivariate
linear regression to radiometric and radiosonde observations collected in the Milan area
(Italy). Two independent microwave radiometers are considered, one atmospheric profiler operating at
14 channels in the K and V bands and one four-channel radiometer operating at two K-band channels and two between
V- and W-band channels. The implemented approach exploits five V-band channels of the microwave
profiler (namely at 53.86, 54.94, 56.66, 57.3, and 58.0 GHz) together with surface PTU
measurements to estimate *T*_{MR} at the K- and V–W-band frequencies of the four-channel
radiometer. The conventional method is also implemented, estimating *T*_{MR} at the
frequencies of the four-channel radiometer from PTU measurements only. Results from the proposed and
conventional methods are validated against *T*_{MR} from simultaneous radiosondes, showing
improvement in all channels and statistical scores (∼0.2–0.8 K in average difference,
∼1.0–1.4 K in RMSD, and ∼4 %–7 % in correlation, depending upon
frequency). This corresponds to a decrease in *T*_{MR} estimation uncertainty by 32 % to
38 %, depending upon frequency. The improvement in *T*_{MR} estimation is then mapped
into the improvement in attenuation estimates for radio propagation purposes by propagating typical
*T*_{MR} and *T*_{B} uncertainties into the atmospheric attenuation equation. This
results in expected improvements in atmospheric attenuation estimates of the order of
10 %–20 % at K-band channels and ∼30 % at V–W-band channels. Although this level
of improvement leads to modest change in absolute attenuation in clear sky (fraction of a
decibel), it becomes more and more important (a few decibels) with the increasing attenuation
typical of cloudy and rainy conditions. In summary, this paper demonstrates the validity of the Han and
Westwater (2000) idea, and it provides a quantitative assessment of the improvements brought by the
proposed method over the conventional PTU method for estimating *T*_{MR} and atmospheric
attenuation at the cost of higher observation complexity (two radiometers in a relatively small
area). This limitation may be overcome by the increasing availability of MWR profilers currently
deployed at several ground stations serving satellite telecommunication (e.g., ESA Tracking Network
in Cebreros, Malargüe, and New Norcia) as well as observatories devoted to atmospheric research and
operational weather forecast (Cimini et al., 2020). Concerning the radio propagation purposes,
future work will include the application of the proposed method to the dataset collected within the
ESA WRad campaign (August 2019–August 2021) to further validate the improvements in
atmospheric attenuation estimates in whole-sky conditions, eventually contributing to the future
assessment of the V–W-band link budget for Earth–satellite telecommunication.

Multivariate multiple linear regression (Bevington and Robinson, 2003) is used here to estimate
*T*_{MR} at four frequencies (23.8, 31.4, 72.5, 82.5 GHz). To clarify, note that the
term multivariate refers to statistical models that have more than one dependent or outcome variable
(predictands), while multiple (or multivariable) refers to statistical models that have more than
one independent or input variable (predictors) (e.g., Hidalgo and Goodman, 2013). Following Cimini
et al. (2006) and references therein, a general equation for the multivariate multiple linear
regression between $\widehat{\mathit{x}}$ (vector of predictands) and ** y** (vector of
predictors) is

where **D** is the matrix of linear regression coefficients, and *x*_{0},
*y*_{0}, and **C**_{xy} and **C**_{yy} are estimated from
the training set (a priori knowledge) as the mean values for ** x** and

**, the covariance matrix of simultaneous**

*y***and**

*x***, and the autocovariance matrix of**

*y***, respectively. In this work, the predictands $\widehat{\mathit{x}}$ are**

*y**T*

_{MR}at four frequencies. Thus, for any measured

*k*-dimension vector of predictors

*y*_{i}, the estimated

*T*

_{MR}for each channel

*j*is

In this study, two versions are implemented with different sets of predictors. The first version
considers three variables as predictors (*k*=3): air pressure, temperature, and relative humidity
(PTU) measured by standard meteorological sensors. The second version considers eight variables as
predictors (*k*=8): the three PTU readings and *T*_{B} at five V-band channels
(53.86, 54.94, 56.66, 57.3, 58.0 GHz). From the training set, we obtain the following values
for *x*_{0}, indicating the mean *T*_{MR} (K) at four frequencies:

While *x*_{0} is the same for the two versions of multivariate multiple linear regression,
both *y*_{0} and **D** depend on the number of predictors. For the first version
** y** contains the mean PTU measurements, i.e., a vector of three components, and

**D**is as in Table A.1:

For the second version, ** y** contains the PTU measurements and

*T*

_{B}at five V-band channels, i.e., a vector of eight components, and

**D**is as in Table A.2:

The underlying software code and data can be accessed upon request by emailing the corresponding author: domenico.cimini@imaa.cnr.it.

AyhA and DC conceived the study, processed the data, and wrote the paper. LL and CR provided the MWR data and insights on methodology for radio propagation applications. FSM led the WRad project and provided funds for the study. MB, LM, and AM contributed to the understanding of requirements. SG, STN, FDP, FR, and AymA contributed to data processing. All authors contributed to writing, reviewing, and editing the published version of the paper.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Tropospheric profiling (ISTP11) (AMT/ACP inter-journal SI)”. It is a result of the 11th edition of the International Symposium on Tropospheric Profiling (ISTP), Toulouse, France, 20–24 May 2019.

Support from COST – European Cooperation in Science and Technology (https://www.cost.eu, last access: 1 April 2021) – under action CA18235 “PROBE” is acknowledged.

This research has been supported by the European Space Agency (ESA) as part of the WRad project (ESA contract no. 4000125141/18/NL/AF).

This paper was edited by Paolo Di Girolamo and reviewed by Ed R. Westwater and one anonymous referee.

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- Abstract
- Introduction
- Methodology
- Dataset and implementation
- Results
- Conclusions
- Appendix A: Coefficients for multivariate multiple linear regression
- Code and data availability
- Author contributions
- Competing interests
- Special issue statement
- Acknowledgements
- Financial support
- Review statement
- References

- Abstract
- Introduction
- Methodology
- Dataset and implementation
- Results
- Conclusions
- Appendix A: Coefficients for multivariate multiple linear regression
- Code and data availability
- Author contributions
- Competing interests
- Special issue statement
- Acknowledgements
- Financial support
- Review statement
- References