# Seasonal and diurnal dynamics of radio noise for 8-20MHz poleward-oriented
mid-latitude radars

###### Abstract

Based on ray tracing in a smooth ionosphere described by the IRI-2016 model we
have infered the seasonal-diurnal dynamics of radio noise observed by four
mid-latitude HF radars. In the calculations, noise is assumed to propagate
from the radar dead zone boundary. Noise absorption along the ray path is
simulated from the IRI-2016 electron density, and from the molecular nitrogen
density and electron temperatures obtained from the NRLMSISE-00 model. Model
results are compared with experimental radar data, and good agreement between
the two is demonstrated. The model makes it possible to estimate the amount of
absorption in D- and E- layers under average undisturbed conditions. This is
important for the retrieval of long term variations in the electron density in
the lower ionosphere. The model also makes it feasible to interpret vertical
absorption in experimental data, thereby significantly expanding the
capability of HF radars to monitor the lower ionosphere.

Radio Science

Institute of Solar-Terrestrial Physics, SB RAS, Irkutsk, Russia Department of
Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Canada
Bradley Department of Electrical and Computer Engineering, Virginia
Polytechnic Institute and State University, Blacksburg, VA, USA Research
Institute in Astrophysics and Planetology (IRAP), Toulouse University, CNRS,
CNES, Toulouse, France

Oleg<EMAIL_ADDRESS>
Physics-based seasonal-daily noise level model built for pole-oriented HF
radars

Good agreement between modeled noise level and experimental observations

Model allows estimating vertical absorption from noise measurements

##  1 Introduction

Noise level studies have recently become an integral part of theoretical and
applied research on lower ionosphere dynamics during solar flares and coronal
mass ejections through the use of SuperDARN and similar high-frequency (HF)
radars [Berngardt . (2018), Bland . (2018), Berngardt . (2019), Bland .
(2019), Berngardt (2020)]. HF noise has traditionally been considered to come
from a mixture of several components - anthropogenic, cosmic, and atmospheric.

For radars with an equatorward field of view, it has been assumed that the
main contribution to observed HF noise originates from tropical lightning
discharges and that the signals from these discharges propagate over large
distances through the ionospheric waveguide [Pederick Cervera (2014)].
Modeling based on this notion has been validated reasonably well, with noise
levels predictions close to experimental observations.

In studies by [Ponomarenko . (2016)] and [Berngardt (2020)], seasonal-diurnal
variations of the noise level were measured by the mid-latitude SAS and EKB
radars at a fixed frequency. In this paper we refer to radars, including SAS,
as mid-latitude radars for having geographic latitudes lower than 60 degrees.
The fields of view of both radars point toward the polar regions and the
shapes of the seasonal-diurnal dependence as a function of local solar time
are similar. As illustrated in (Fig.1A), they take the form of an oval, the
contour of which is determined by the position of the solar terminator. This
stated, there is no obvious reason why the HF noise observed in mid-latitude
poleward pointing radars like SAS and EKB should be closely associated with
lightning. In addition, the ray paths emerging from equatorial sources are so
long that the noise should be absorbed almost completely.

As a result of the above considerations, [Berngardt . (2019)] suggested that
the observed noise pattern was not so much related to the source of noise in
poleward pointing mid-latitude HF radars as it was related to ionospheric
conditions. The authors proposed that in most cases the sources of noise
should be considered to be anthropogenic and that the noise pattern is
dictated by the border of the so-called ’dead zone’ often observed though
ionospheric ground echoes [Samson . (1990)]. The dead zone is an area around
the radar that is free of hop-propagating rays in terms the geometric optics
approach that can complete a path between a ground point and the radar, hence,
points within the dead zone are not sources of noise for the radar ( in terms
of wave optics approach, used in this paper, some weak signals can be detected
even in the dead zone ). Dead zones mark the boundary of a noise region where
ray paths are focused by ionospheric effects. Specifically, this means that
both the trajectory of propagation of noise signal and the intensity of the
noise focusing at the border of the dead zone depend on the height
distribution of the electron density along the ray path, which has clear and
strong diurnal and seasonal variations. Based on that notion, a model of the
noise pattern for poleward pointing radars at mid-latitudes would have to
properly account for the seasonal and daily variations of the noise level
through the proper use of ionospheric and atmospheric models, given that the
noise and its pattern both depend on the atmospheric and ionospheric densities
through absorption and ray path focusing/de-focusing, respectively.

Figure 1: A) Seasonal-diurnal pattern of average noise intensity at the EKB
radar [Berngardt (2020)]. The lines correspond to the position of solar
terminator at two different heights (at ground level - dashed line and at
300km height - dot-dashed line); B-C) Fields of view, in AACGMv2 [Shepherd
(2014)] geomagnetic coordinates, of the four mid-latitude radars involved in
the current study, for the Northern and Southern hemispheres; D) Noise
propagation and collecting scheme for the proposed model, illustrating the
focusing mechanism for enhancing the intensity, $I_{F}(x)$, around the
boundary of the dead zone.

Mathematical empirical approaches have been introduced to reproduce noise
patterns of the kind shown in Fig.1A, irrespective of the physical origin of
the patterns. One such approach uses the mean daily dynamics of noise level
over the previous 30 days [Bland . (2018)]. Another consists of using special
auto-regressive models [Berngardt (2020)]. While these models have their
usefulness, no physical model has apparently been built to this day to
reproduce the seasonal-daily dependence of the noise pattern. One clear
advantage of building such a physical model would be the ability to predict a
’quiet-day’ curve, namely, a curve of the expected noise level for ordinary
ionospheric conditions. Reproducing an observed quiet-day curve would make it
possible to estimate the vertical absorption level, allowing for the
extraction of the long-period variations associated with the D- and E-layers
of the ionosphere. In turn, building a physical model of noise level would
make it feasible to effectively use SuperDARN and similar radars for
estimating the electron density in the lower part of the ionosphere.

With the above in mind, we have developed a physics-based numerical model of
the noise at HF frequencies. We have applied the results of the model to the
seasonal-diurnal dynamics of noise level at four mid-latitude radars, namely,
the Canadian SAS SuperDARN radar (52.2N, 106.5W), the US BKS SuperDARN radar
(37.1N, 78.0W), the French KER SuperDARN radar (Southern French Territory,
49.2S, 70.1E) and the Russian EKB ISTP SB RAS radar (56.5 N, 58.5E). Each of
these radars has a field of view with an antenna beam pattern that points
nearly poleward (Fig.1B-C), meaning that the influence of tropical
thunderstorm centers on the noise production should be negligible. We describe
our model in Section 2. In Section 3 we present its results and compare with
observations while discussing the implications for the plasma densities in the
lower ionosphere. Our conclusions are presented in Section 4.

##  2 Description of the physics-based noise model

Our model is based on a mechanism proposed by [Berngardt . (2018), Berngardt .
(2019)]. It assumes that the noise level in polar-oriented radars operating in
the 8-20 MHz range is determined by a focusing of noise signals from various
anthropogenic sources, and that the main sources contributing to the noise are
located around the boundary of the so-called ’dead zone’. Near the dead zone,
the amplitude of propagating signals is amplified significantly due to spatial
focusing of radiowaves by the ionosphere [Tinin . (1992)].

Within the framework of this model, the following factors need to be
considered:

  1. 1.

the spatial distribution of the amplitudes of the sources of noise and the
noise emission direction pattern for each source;

  2. 2.

the ray path of the radio signals in an irregular ionosphere;

  3. 3.

the intensity produced by the focusing of radio signals on the boundary of the
dead zone;

  4. 4.

the absorption of the signal over the ray path as it passes through the D- and
E-regions;

  5. 5.

the antenna beam pattern of the radar receiver.

In general, each of the above characteristics has its own temporal and
frequency dependencies. This stated, in the following calculations, we assumed
that noise sources approximately have an isotropic directional pattern, equal
amplitudes, and are uniformly distributed over the Earth’s surface around the
radar. In the presence of intense localized anthropogenic sources (radio
stations, industrial sources, railways) or for radars facing thunderstorm
activity centers, this assumption has to break down. However, for radars
located at a significant distance from intense anthropogenic or thunderstorm
activity centers, this assumption should be acceptable.

To be specific, according to the interpretation suggested by [Berngardt .
(2018), Berngardt . (2019), Berngardt (2020)] noise propagates from a noise
source and focuses just beyond the boundary of the source’s dead zone, as
illustrated in Fig.1D. Within that framework the noise received by a radar is
a superposition (integral) of noise source contributions with weights defined
by dead zone multiplier $I_{F}$ calculated for the radar position. The
integration is made over the azimuth $\theta$, elevation $\alpha$, and range
$s$. It takes into account the antenna beam pattern, signal attenuation with
distance and signal attenuation due to absorption in the lower ionosphere.
This means that, to a first approximation, the noise intensity received by the
radar, ${I_{rcv}}$, can be obtained from the expression

${I_{rcv}}\approx\int_{\Omega,S}I_{src}\left[x(\alpha,\theta)\right]I_{F}\left[x(\alpha,\theta,s)-x_{0}(\alpha,\theta)\right]A_{t}(\alpha,\theta)G(\alpha,\theta)s^{-2}\cos(\alpha)d\alpha
d\theta ds$ (1)

where $I_{src}$ is the noise intensity distribution of the noise source at the
ground, $I_{F}$ is the dead zone focusing multiplier, $s$ is the distance from
the source to the receiver, and $s^{-2}$ is the attenuation in intensity due
to distance. Also, $G(\alpha,\theta)$ is the shape of the antenna beam
pattern, and $A_{t}(\alpha,\theta)$ is the absorption over the trajectory of a
ray. In these expressions $x$ is a phase path length along a ray that starts
at specific values of $\theta$ and $\alpha$, whereas $x_{0}$ is the phase path
length at the focusing point, namely, at the dead zone boundary. It should be
understood that $x_{0}$ does not depend on $s$ but that $x$ does depend on $s$
because the integration is over the distances of the various noise sources to
the receiver. In other words, the $\alpha$ and $\theta$ integrations are over
the receiver angles, while the integration over $s$ is actually over the
distance of the noise sources to the radar.

As a further approximation, we assume that the variations of the various
parameters are such that the noise intensity $I_{rcv}$ can be described
accurately enough with the simpler expression

${I_{rcv}}\approx I_{0}\overline{I_{F}}A_{t}G_{A}\overline{A_{R}}$ (2)

where $I_{0}$ is the intensity of ground noise sources and replaces the
$I_{src}$ if the latter can be assumed to be isotropic (later in our
simulations $I_{0}$ is used as a normalization constant, calculated over one
year). In addition, $\overline{I_{F}}$ is the integral over all distances to
the source of the focusing intensity at the boundary of the dead zone, whereas
$A_{t}$ becomes the absorption of the radio signal power over the whole ray
path, and $G_{A}$ is the beam pattern of the receiving antenna integrated over
$\theta$, i.e., over all azimuths. Also, $\overline{A_{R}}$ is the integrated
signal attenuation with respect to distance. Consistent with this definition,
the $\overline{a}$ symbol describes an averaging of quantity $a$ over the
range. So to build Eq.2 we replace average value of the product of the various
non-negative parameters in Eq.1 by the product of average values of the
parameters with a single ray path from the border of the radar dead zone.

Finally, taking into account that power attenuates with propagation trajectory
length $s$ as $s^{-2}$, we can estimate:

$\overline{A_{R}}\approx\int_{S}^{S_{max}}s^{-2}ds\approx S^{-1}$ (3)

This result is valid as long as $S<<S_{max}$ (maximal radar range).

At a fixed frequency, we can assume $I_{0}$ to be constant. This holds if we
can assume to first order that the various noise sources are not only
isotropic, but also homogeneous, incoherent, and equal in intensity. This is a
rather rough assumption, but, as shown below, it proves to be sufficient for
an adequate modeling of the noise pattern.

###  2.1 The propagation trajectory

Propagation trajectories were obtained with the use of wave optics based on a
calculation made at fixed pre-determined starting values of the elevation
angles within 0-90 degrees with 0.09 degree steps. For a given azimuthal
direction or radar beam we could describe the system in terms of spherical
coordinates for the radial distance $\rho$ and the elevation angle $\phi$. We
restricted the calculations to the high-frequency limit for the index of
refraction, $n$, and did not include absorption in the calculations, i.e., we
used the ordinary wave expression given by

$n^{2}=1-f_{e}^{2}/f_{0}^{2}$ (4)

where $f_{e}$ and $f_{0}$ are the plasma frequency and radar frequency,
respectively. For this situation the differential equations leading to phase
and group delays as a function of starting elevation angles can be described
by the following set of equations [Kravtsov Orlov (1983)]:

$\left\\{\begin{array}[]{l}\frac{\partial R_{\rho}}{\partial s}=P_{\rho}\\\
\frac{\partial R_{\phi}}{\partial s}=\frac{1}{R_{\rho}}P_{\phi}\\\
\frac{\partial P_{\rho}}{\partial s}=\frac{f_{e}}{f_{0}^{2}}\frac{\partial
f_{e}}{\partial R_{\rho}}+\frac{1}{R_{\rho}}P_{\phi}^{2}\\\ \frac{\partial
P_{\phi}}{\partial s}=\frac{1}{R_{\rho}}(\frac{f_{e}}{f_{0}^{2}}\frac{\partial
f_{e}}{\partial\phi}-P_{\rho}P_{\phi})\\\
\psi=\psi_{0}+\int\limits_{S_{0}}^{S}P^{2}ds\end{array}\right.$ (5)

where $\overrightarrow{R}$ is the radius vector describing the position of the
wave impulse at a particular point in the trajectory and where
$\overrightarrow{P}$ is the wave impulse, while $\psi$ is the phase path (or
eikonal). Integration was carried over the trajectory variable $s$ with
starting and end points at $S_{0}$ and $S$, and with $\psi_{0}$ being
associated with the initial value of the eikonal. In such calculations, the
regions where phase delay goes through an extremum (minimum) correspond to the
regions of focusing, that is to say, they describe the borders of the dead
zone for different propagation modes.

In the numerical calculations giving rise to our model, we assumed that the
radio wave propagated in a plane that was associated with a given azimuth over
a globe arc within the framework of the spherical Earth model. This stated,
our ionospheric model remained inhomogeneous in the propagation plane, namely,
over the distance to the source and the elevation angle. In other words the
ionospheric model remained a function of latitude and altitude. Specifically,
we used the IRI-2016 [Bilitza . (2017)] ionospheric model for this purpose. In
order to work with a smooth and continuous ionosphere along the trajectory, it
was necessary to assume that the signal propagation plane did not depend on
the starting elevation angle when seeking numerical solutions to the set of
differential equations (5). While correct for the middle(central) beams of the
radar antenna pattern, this was not strictly true for the extreme ones, since
the antenna beam pattern is actually described by a conical shape [Shepherd
(2017)]. But, as will be shown later, this rough approximation is sufficient
for carrying the model calculations forward with reasonable accuracy.

For the specific ray path calculations we used a grid with step sizes of 222
km (equivalent to $2^{o}$ in latitude at Earth’s surface) in distance and 3 km
in height in the propagation plane. For calculations between grid points, an
interpolation was performed using two-dimensional local quadratic B-splines to
obtain continuous values and smooth spatial derivatives. When later adding
absorption effects we used the the NRLMSISE-00 model for the neutral
atmosphere, from which we extracted the molecular nitrogen density and the
electron temperature interpolated between grid points in the same way.

###  2.2 Modeling the focusing near the dead zone boundary

####  2.2.1 Determining $\overline{I_{F}}$

In a two-dimensional situation such as the one used here, the focusing of the
radio waves emitted from a point source can be described analytically near the
dead zone, if the ionospheric peak can be approximated by a parabola. In that
case, if we neglect insignificant multipliers, the signal amplitude $A_{F}(x)$
over the phase path length $x$ has a well-known dependence given by [Tinin .
(1992)]:

$A_{F}(x)\sim\frac{1}{\Lambda}Ai(\frac{2^{2/3}}{\Lambda}[\overline{x}-x])$ (6)

where $Ai()$ is the Airy function describing the shape of the signal focusing
versus phase path length, and where

$\Lambda=\left(\frac{1}{2k^{2}}\frac{\partial^{2}\overline{x}}{\partial\alpha^{2}}\right)^{1/3}$
(7)

is a quantity inversely proportional to the size of the focusing area
associated with the second derivative of the phase path length over the
elevation angle $\alpha$. Note that Equation (6) does not take into account
the decay of the wave field with distance.

By integrating (6) over the range, and taking into account that noise sources
are independent, it therefore follows that the range-integrated power
$\overline{I_{F}}$, corresponding to the total power of the noise superposed
from various noise sources homogeneously placed at different ranges is given,
to a first approximation, by:

$\overline{I_{F}}\sim\frac{1}{\Lambda}\int Ai^{2}(\overline{\eta}-\eta)d\eta$
(8)

where the integration is made over the dimensionless quantity
$\eta={2^{2/3}x}/{\Lambda}$ and $\overline{\eta}>>1$.

This means that the focusing power $\overline{I_{F}}$ in (2) thus can be
approximated reasonably well with the simple expression:

$\overline{I_{F}}\sim\frac{1}{\Lambda}$ (9)

####  2.2.2 Determination of the dead-zone position

To find the propagation path of the focusing signal in an arbitrary
ionosphere, it is necessary in the context of wave optics to extract the
dependence of the phase delay $\psi$ on the elevation angle $\alpha$. We use
the fact that, at the focal points, the phase delay dependence on the
elevation angle is weakest, meaning that the derivative of the phase becomes
zero [Tinin . (1992)]. Since we considered many possible ray paths, the focal
points spread into a small area. Qualitatively, the size of such an area is a
measure of the number of rays arriving at the receiving point with the same
phase, thereby adding to the total amplitude of the received noise signal
through constructive interference of the rays. In terms of our model, this
size is inversely proportional to the cubic root of the second derivative of
the phase delay over the elevation angle (Eq.7) at the point where the first
derivative becomes zero (i.e. near the dead zone border), so that the power of
the focused signal is proportional to the size of this area [Kravtsov (1968)].

It is obvious that in a complex ionosphere there can be several focusing
regions (modes). To find the positions (elevation angles) of the groundscatter
modes, we analyzed all the possible propagation trajectories by calculating
the phase delay $\psi$ for each elevation angle $\alpha_{i}$. After
calculating the dependence of the phase delay on the elevation angle
$\psi(\alpha)$ the elevation angles $\alpha_{i}$ of the rays getting to the
dead zones were determined from the position where the phase delay reached its
minimal value:

$\alpha_{i}:\left.\frac{d\psi(\alpha)}{d\alpha}\right|_{\alpha=\alpha_{i}}=0$
(10)

We should add that the local dependence $\psi(\alpha)$ was fitted to a
parabola to make the evaluation of the second derivative of $\psi(\alpha)$
more easily to facilitate the evaluation of $\Lambda$ (Eqn 7).

We obtained a collection of angles $\alpha_{i}$ corresponding to different
groundscatter modes. In a regular, spherically layered, non-magnetized,
parabolic ionosphere, there is usually no more than one ground-scatter mode -
there is only one local minimum of the phase-elevation characteristic. In a
more complex but more realistic heterogeneous multi-layer ionosphere, there
can be several groundscatter modes, with the presence of a magnetic field
possibly adding extra possibilities. Just with the propagation of ordinary
rays, we found many instances of multiple ground-scatter modes, each with its
own phase, group delay and central elevation angle $\alpha$.

Examples of electron density profile at different local time(LT), as well as
phase and group delay for single-mode (black line) and multi-mode (red and
green lines) signals, calculated using the IRI-2016 model, are shown in
Fig.2A-C. Modeling was carried out for the EKB radar (56.5N, 58.5E), for the
northern direction of radio wave propagation (azimuth 0, beam 2), for June 1,
2013. It can be seen from the figure that several modes correspond to several
well-defined (E and F) layers in the electron density.

Figure 2:  A-C)-Examples of electron density profile (A), phase path (B) and
group path (C), calculated using the IRI-2016 model for the EKB radar for the
north-facing direction on 01 June, 2013; D) - model radar antenna beam
patterns in the elevation plane used in our various simulations: LPDA beam
model at 10 MHz for the EKB, SAS and KER radars [Berngardt . (2020), Arnold .
(2003)], and TTFD beam model at 14 MHz for the BKS radar [Sterne . (2011)].

When multi-mode noise trajectories were obtained, we chose the mode
(trajectory) that contributed the most to the noise signal intensity, after
having taken signal absorption and antenna pattern attenuation into account.
Following this, the modeled noise level was chosen to be the intensity of the
mode with maximum intensity after taking all factors into account, namely,
propagation, absorption, and antenna beam pattern.

###  2.3 Antenna pattern

Another important parameter of the model is the signal attenuation due to the
antenna pattern $G_{A}$. The attenuation can reach several tens of dB, and
therefore it is very important to take it into account at low and high
elevation angles. In this paper, we have dealt with the data from several
radars: EKB from ISTP SB RAS, and SAS, KER and BKS from SuperDARN. Every one
of these radars has a different type of antenna and a different $G_{A}$.

Taking into account that the azimuthal width of the antenna beam pattern is
defined by a phased array geometry and size considerations (that much is
similar for all the radars under consideration), the integrated antenna
multiplier $G_{A}$ has been taken to simply be proportional to the antenna
pattern in the elevation plane, that is to say:

$G_{A}(\alpha)=\int_{\Omega}G(\alpha,\Theta)d\Theta\approx
G(\alpha)\Delta\Theta$ (11)

where $\Delta\Theta$ is the azimuthal beam width of the phased array antenna
beam pattern and is the same for all the radars considered here. In addition,
$G(\alpha)$ is the antenna beam pattern of a single antenna in the elevation
plane.

Fig.2D shows the model antenna beam patterns of the various radars at 10-14
MHz frequency that were used for our modeling. It should be noted that the
main difference between the various antenna beam patterns comes from the fact
that the EKB, SAS and KER radars use Log-Periodic Dipole Array antennas (LPDA)
at the top of masts approximately 15 meters above the ground [Arnold . (2003),
Berngardt . (2020)], while the BKS radar uses Twin Terminated Folded Dipole
antennas (TTFD) at a lower height [Sterne . (2011)]. This significantly
increases the amplitude of the vertical lobe on the BKS radar and broadens its
elevation antenna pattern as compared to the EKB, SAS and KER radars.

###  2.4 Noise absorption over the trajectory

Another important parameter affecting the radio noise intensity is the
absorption of the signal over the propagation trajectory. The absorption per
unit length $L[dB]$ was calculated within the framework of a classical model
[Zawdie . (2017)] in which frequencies are considered to be high enough to
justify neglecting magnetic field effects:

$L[dB]=-\frac{20}{log(10)}\frac{e^{2}}{2\epsilon_{0}m_{e}c}\frac{n_{e}\nu_{e}}{\nu_{e}^{2}+\left(2\pi
f_{0}\right)^{2}}$ (12)

where, as before, $f_{0}$ is the operating frequency of the radar, $n_{e}$ is
the electron density, and $m_{e},c,\epsilon_{0}$ are the electron mass, the
speed of light in vacuum and the dielectric constant of vacuum, respectively
while $\nu_{e}$ is the effective electron collision frequency. Since $\nu_{e}$
is only effectual in the lower part of the ionosphere we considered it to be
equal to the electron collision frequency with molecular nitrogen
$\nu_{e,N_{2}}$. Using [Schunk Nagy (2000)], it was assumed to be given by:

$\nu_{e}\simeq\nu_{e,N_{2}}=2.33\cdot 10^{-11}\cdot n_{N_{2}}(1-1.21\cdot
10^{-4}T_{e})T_{e}$ (13)

where $n_{N_{2}}$ is the molecular nitrogen density and $T_{e}$ is the
electron temperature. The accuracy of this approximation has been discussed in
detail by [Zawdie . (2017)].

Figure 3:  Panel A: Comparison between the monthly average and median electron
densities at 75 km altitude extracted from the three different models listed
in the legend. Panels B to D: Modeling results for beam 2 of the EKB radar,
beam 2 (0∘ azimuth) for the year 2013, based on the standard IRI-Ne option of
the model IRI-2016. Panel B: modeled absorption profile per 1 km propagation
trajectory for May 2013. Panel C: same as for panel B but for November 2013.
Panel D: seasonal-daily variation of the maximum value of the absorption per 1
km of the radio signal trajectory in the ionosphere.

The total absorption of the radio wave $L_{int}[dB]$ was calculated as the
integral of the unit absorption over the propagation trajectory $S$, namely,
[Zawdie . (2017)]:

$L_{int}[dB]=\int_{S}L[dB]ds$ (14)

In the simulations, the absorption was taken into account only for the D- and
E-layers, namely, in the 60-120 km altitude range, where significant
contributions to the absorption were possible if $n_{e}$ was potentially large
enough and where the collision frequency could compete with $f_{0}$. For our
model simulations, we describe the absorption factor $A_{t}$ in Eqn (2) with

$A_{t}=\exp\left(\frac{ln(10)}{10}L_{int}[dB]\right)$ (15)

####  2.4.1 Weighting the factors that control the absorption

To evaluate the absorption of the radio signal from the above expression, we
used well-known empirical models, namely, the IRI-2016 model for a
determination of the electron density $n_{e}$, and the NRLMSISE-00 model for a
determination of the effective electron collision frequency based on the
density of molecular nitrogen $n_{N_{2}}$ and the electron temperature
$T_{e}$. For calculations between grid points, interpolation was performed
with two-dimensional local quadratic B-splines to obtain continuous values
required for trajectory calculations. Then, in the plane of the propagation
trajectory of the radio signal, the absorption was tabulated over a grid which
had 222 km range resolution and 5 km height resolution. In the simulations,
interpolation based on local quadratic B-splines was once again used to
account for a smooth spatial dependence.

One source of uncertainty in the absorption calculation comes from accounting
for the electron density in the D-layer, which is no easy task, to say the
least [Danilov . (2002), Bilitza Reinisch (2008)]. Even the standard Reference
ionospheric model, IRI-2016, includes several different models for electron
density at heights 65-110 km. To demonstrate the complicated task at hand,
Fig.3A shows a comparison of the monthly average level of electron density at
75 km height between three electron density models for the March 2017 (IRI,
FIRI and Danilov’s). The IRI-2016 model comes with the standard IRI-Ne D-layer
model. Other models of the D-layer come from [Danilov . (1995), McKinnell
Friedrich (2006)].

Fig.3A shows that there are clearly important differences in the D region
between the models. In particular, the nighttime electron density level could
be significant (as demonstrated, for example, in [Gomonov . (2019)]) and
should be taken into account when simulating the diurnal variation of the
absorption. Taking this into account leads us to conclude that the diurnal
variation of the absorption inferred from the ionospheric models is associated
not with the absolute value of the electron density, but with its variation
over its minimum (nighttime) value. This means that the absolute electron
density of the D-layer cannot be correctly determined through coherent scatter
radar absorption data using noise level observations without a very careful
analysis of the latter and without an absolute calibration of these
observations on a long term basis. A comparison between observations and
numerical prediction is done in the next subsection.

Before a detailed comparison we want to illustrate the effect of seasonal
variations on absorption, Figures 3B and 3C show plots of the unit absorption
per kilometer of the trajectory $L[dB]$ as a function of altitude that were
calculated for May and November of 2013 to the North of the EKB radar (azimuth
0 degrees, radar beam 2), using the standard IRI-2016 model to describe the
D-layer. The radiowave propagation path used for the calculation corresponds
to the trajectory of the groundscatter signal propagation, i.e. to the
trajectory from the dead zone boundary to the radar. The absorption values are
close, in the end, to the results obtained by [Zawdie . (2017)].

Figure 3D also shows the full seasonal-daily variation of the maximum value
$L_{max}[dB]$ of absorption per km of the trajectory in the ionosphere

$L_{max}[dB]=Max\left(-L[dB]\right)_{h=[60..120km]}$ (16)

The calculation is based on the standard IRI-2016 D-layer model for the EKB
radar site. As can be seen, the maximum absorption is observed in the summer
under daytime conditions. This is the result of the ionization of the lower
part of the ionosphere being greatest at that time of day and year. It
contrasts with winter and/or nighttime conditions when the lower ionosphere
should have only weak ionization if any at all, according to the empirical
models.

##  3 Results and discussion

###  3.1 Comparison between simulations and experimental observations

For the years 2013 and 2014, we have compared the observed noise level for our
four mid-latitude radars with calculations based on our noise level model
using three different D-layer models included into the international reference
ionospheric model IRI-2016: (1) the standard IRI-Ne [Bilitza (1997)] model,
(2) the FIRI model [McKinnell Friedrich (2006)] and (3) the Danilov model
[Danilov . (1995)]. The comparisons are shown in Fig.4.

Figure 4: Comparison between observations and simulations of noise level
seasonal-daily dependence for 2013 based on three different D-layer models,
namely, IRI-2016, FIRI, and Danilov. Panels A to D: EKB radar, 10MHz, at 0∘
azimuth with the top panel for observations and the next 3 panels for each
model. Panels E to H: same as panels A to D, but for the KER radar at 10 MHz
for a 180∘ azimuth. Panel I: SAS observations reported by [Ponomarenko .
(2016)]. Panels J to L: same as panels B to D, but for the SAS radar at 10 MHz
for a $-23^{\circ}$ azimuth. Panels M to P: same as panels A to D but for the
BKS radar at 14 MHz at $0^{\circ}$ azimuth.

A detailed comparison can be made with more confidence for the data associated
with the EKB and SAS radars, because the data from these radars came from
noise level observations at a fixed frequency close to 10 MHz on a 24 hours
per day basis. From the results displayed in Figs.4A to 4D, and 4I to 4L for
the EKB and SAS radars respectively, it can be seen that using standard IRI-Ne
D-layer model leads to a significant overestimation of noise absorption, while
the FIRI and Danilov models lead to better qualitative agreement between model
calculations and observations in regions away from the regions of low noise
prediction by the model (i.e., the uniformly dark central areas in the panels
that describe the computational results).

Data from the BKS and KER radars are harder to use for a comprehensive
analysis because they come from a variety of daytime and nighttime sounding
conditions. Nevertheless, the daytime data that were retrieved (for 14 MHz at
BKS, and for 10 MHz at KER) demonstrate a good agreement with our model,
particularly for the KER radar and especially when using Danilov’s model of
the D-layer.

As one can see, the KER radar (southern hemisphere) seasonal-daily noise has a
distribution that differs significantly from the northern hemisphere EKB, SAS
and BKS radars. This difference appears to come from the difference in
hemispheres that are to be associated with different seasons and
atmosphere/ionosphere conditions. Indeed our model shows a good correspondence
with the observations in both the northern and southern hemispheres, implying
that the model that we formulated based on Eqn. (2) has validity.

###  3.2 Relationship between propagation parameters and noise variations

For a qualitative analysis, it is useful to demonstrate the relationship
between the propagation parameters and the seasonal-diurnal variation of the
noise level. Fig.5 shows the simulated results of seasonal-diurnal variation
of various propagation parameters at the EKB radar. It can be seen from the
figure that:

  * •

the annular shape of the noise level (Fig.5E) is controlled mainly by the
total absorption over the propagation trajectory (Fig.5F): in other words, the
more absorption there is, the less noise is seen, with the two patterns
matching very well;

  * •

the total absorption over the trajectory, in turn, is related to the
trajectory elevation angle (which changes the propagation length of the signal
in the absorbing layer) - the higher the elevation angle, the shorter this
length and the higher the noise level (Fig.5B).

  * •

the total absorption is also related to the unit absorption in the regular D-
and E-layers - the stronger the absorption, the lower the noise level(Fig.5F
versus 3D). We should note that preliminary modeling has shown that the
absorption below 85 km altitude is comparable to the absorption above 85 km;

  * •

the attenuation of the noise level in the winter daytime is associated with
antenna pattern attenuation at high elevation angles(Fig.5D versus Fig.5B).

Figure 5: Simulated seasonal-diurnal variation of parameters associated with
the propagation of radio waves for the EKB radar in 2013 at 10 MHz. We used
standard IRI-Ne model for D-layer simulation.

###  3.3 Saturation effect and quiet day curve

As could be inferred from Fig.5 the central region embedding summer daytime
conditions seems to be associated with noticeably weaker noise levels than
expected from the model calculations. The predictions for weak noise levels
are based on strong absorption during sunlit conditions so that little noise
from ground sources should be expected to be found. We have carried out more
detailed comparisons to assess this situation. For the purpose at hands, we
focused on only two radars (EKB and SAS) because these were radars that used a
fixed sounding frequency (10 MHz) on a continuous basis throughout the day.
This allows for a numerical comparison between the model and observations:
Fig.6 shows some detailed examples of daily variations of the noise on the EKB
and SAS radars for different dates and their comparison with our proposed
model when we use the Danilov D-layer model for taking into account absorption
at lower altitudes, the IRI-2016 ionosphere model for taking into account
propagation effects, and NRLMSISE-00 neutral atmosphere model for calculating
the electron collision frequency.

As might have been expected from Fig.5, the various panels in Fig.6 show a
satisfactory agreement between the observations and the model at high noise
levels, but there is generally a strong disagreement when the model noise
level is predicted to be low. We see two possibilities that could explain this
situation. One is that there may well be a saturation effect associated with
analog and digital sensitivity levels of the radars. Alternatively, or even in
association with instrumental noise, cosmic noise could be a factor, as it can
sometimes be as high as 20 dB below anthropogenic noise, as shown in [ITU-R
P.372-14 (2020)](Figs.2,39). Cosmic noise comes from much higher elevation
angles and could make a significant contribution when anthropogenic noise is
not too strong because of absorption. We have not queried this question any
further as the focus of the present paper is anthropogenic sources from the
ground. Our model of these contributions produces the bright arc patterns in
Fig.4. The important message is that these arcs fit well with the notion of
noise from anthropogenic sources.

Figure 6: Comparison of model (red line) and measured daily noise variations
(black line) on radars EKB (beam 2, A-D) and SAS (beam 7, E-H), at 10MHz

To demonstrate that saturation effects are very real, we introduce Fig.7.
Fig.7A and B show the results of a statistical comparison between the model
and the observations using the EKB and KER radar data in the year 2013. The
color shows the number of cases when a particular combination of experimental
and model noise values was obtained.

Consider Fig. 7A first: clearly, when the observed noise level is less than 20
dB, there is no connection between the predicted noise level and the observed
one. However, when the observed noise level is greater than 20 dB there is a
linear relation between prediction and observations. From this it should be
clear that the measured EKB noise levels are saturated below 20 dB, thereby
having nothing to do with ionospheric absorption. By contrast, above 20 dB,
the effects of ionospheric absorption are seen. The clear implication is that
the model cannot be used below 20 dB in the EKB case because of noise
saturation effects affecting the observations.

Fig.7B shows that a similar effect takes place for the KER radar. In that
case, however, the saturation level is higher (32 or 33 dB), and the physical
information that can be extracted from noise variations is for noise levels
that are weaker by about 8 to 10 dB.

Fig.7C and D display similar information in a somewhat different way through
monthly statistics. For those panels, we kept track of the radar data that
could be suitably compared with a numerical interpretation of the observations
by the proposed model (as per the numbers inferred from panels A and C,
meaning a noise level above 20 dB for the EKB radar and above 32 or 33 dB for
the KER radar). This being the case, Fig.7C shows that the probability that
the noise level at EKB be 20 dB above the numbers expected from our
simulations is always above 60% at all times of the year, with a somewhat
higher value (closer to 80%) in June. The numbers are different for KER, as
seen from Fig.7D. We note that the statistics are too weak to give any
significance to the June peak in that case (a change from 32 dB to 33 dB would
put the peak well inside a region of minimum probability).

From our analysis of Fig.7A to D we may conclude that about a half of the
daily noise data at EKB is suitable for the numerical interpretation. These
periods where the simulations work reasonably well are usually associated with
a weak D-layer ionization level. They come mainly from nighttime and autumn-
winter months. This much is clear after we take another look at Fig.4. The
smaller size of the dataset in KER is caused by its smaller dynamical range
(Fig.7B), itself most likely caused by a lower level of distributed surface
anthropogenic sources around the radar (Fig.1C) , which is isolated in the
Southern Indian Ocean.

Figure 7: Comparison between model and experimental data for the EKB (A) and
KER (B) radars and monthly statistics of radar data available for adequate
numerical analysis within the framework of the proposed model: for EKB (C) and
KER (D) radars. Dashed line in (A,B) corresponds to linear dependence, the
color indicates the corresponding number of occurences of combinations of
experiment vs. model values. E) Vertical absorption at 10MHz calculated from
experimental EKB noise data using the suggested model.

With the above reservations concerning extra noise sources in mind, the
proposed model can be used to build a quiet day curve (the results of which
are presented through the red traces in Fig. 6). However, for the model to be
put to good use, we need to take into account the saturation effect of the
radar sensors at low noise levels. The saturation level clearly differs for
each specific radar (Fig.7A,B) , and its mechanism and dynamics should be
studied in the future. As shown above, this model depends on a number of
factors, namely, (1) the trajectory of propagation of the noise signals (or
ground-scatter signals), (2) the antenna pattern in the elevation plane of
propagaion, as well as (3) the mode of the received signal as it is affected
by propagation in a three-dimensionally inhomogeneous ionosphere. The model
used for the ‘quiet’ ionospheric D-layer is also significant in the
construction of the quiet day curve. The comparison of observations with model
simulations (Fig.4) shows that the use of the traditional IRI-2016 model leads
to significant overestimation of the noise level. The Danilov model of the
D-layer leads to better fits to observed noise levels than the other models,
at least for the EKB radar (Fig.4).

###  3.4 Implications for the modeling of ionospheric absorption

Based on the use of Eqn.(2) for the construction of our model, one can exclude
all propagation effects (antenna pattern attenuation $G_{A}$, radio wave
attenuation with distance $\overline{A_{R}}$, and focusing effects
$\overline{I_{F}}$ ) on the noise level variations $\overline{I_{N,rcv}}$ as
well as exclude attenuation variations in the elevation angles through
propagation for a trajectory through an angle $\alpha$ in the D-layer (valid
for a thin non-refracting absorption layer). In other words the intensity is
essentially ’fixed’ through the relation

$I_{N,fxd}[dB]=10\cdot\log_{10}\left(\frac{\overline{I_{N,rcv}}}{I_{0,N}\overline{I_{F}}G_{A}\overline{A_{R}}}\right)$
(17)

while the vertical absorption becomes

$A_{t,vert}[dB]=(I_{N,fxd}[dB]-Max(I_{N,fxd}[dB]))\cdot\sin(\alpha)$ (18)

where the maximum value is calculated over the year to calibrate absorption
when it reaches a minimum. As a result, one can obtain an estimate for the
equivalent vertical absorption $A_{t,vert}$ near the radar. Fig.7E shows the
seasonal-daily dynamics of the vertical absorption $A_{t,vert}$ near EKB based
on this approach to the analysis of the noise observations at 10 MHz. As one
can see, the vertical absorption goes through a maximum at summer daytime.
This corresponds well, qualitatively, to the D-layer absorption expected from
the IRI-2016 and NRLMSISE-00 models, as shown in Fig.3D.

##  4 Conclusion

In this paper, we have developed a numerical model of the seasonal-diurnal
dynamics of the radio noise at polar-oriented HF-radars. The model has focused
on four high-frequency mid-latitude coherent scatter radars - EKB from ISTP SB
RAS, and SAS, BKS and KER from SuperDARN. The numerical results have been
compared with observations.

The results from our simulations give credence to the model proposed earlier
in [Berngardt . (2018), Berngardt . (2019)] for interpreting the radio noise
level in terms of the spatial focusing of ground sources of anthropogenic
origin in the vicinity of the ‘dead zone’ propagation boundary. Using the
IRI-2016 and NRLMSISE-00 models, it has been shown here that seasonal-diurnal
variations of the noise level can be explained within the framework of this
model through a combination of propagation, absorption and antenna pattern
factors.

We have demonstrated that the seasonal-diurnal patterns observed in radio
noise (with a maximum in the region of the solar terminator) is associated
with variations in the total absorption of the signal over the propagation
trajectory to the border of the dead zone. We have also shown that an observed
decrease in the noise intensity in the winter noon area in the EKB and SAS
radars (not detected with the BKS radar) can be explained by an additional
attenuation of the signal by the antenna beam pattern at high elevation
angles.

The agreement with observations is particularly good near the terminator where
the noise level is greatest. This has allowed to us compare the absorption
computed from different D region models and compare the resulting noise levels
with observations. We found that the absorption levels, when integrated along
the ray paths, were too high when using the standard IRI-2016 D-layer model.
However, the agreement with observations based on the use of the FIRI
[McKinnell Friedrich (2006)] and Danilov [Danilov . (1995)] D-layer models was
much better.

The above notwithstanding, in spite of a good qualitative agreement of the
seasonal-diurnal model variations with observations, quantitative simulation
results demonstrate that the model predicts far less noise than observed away
from the terminator during sunlit conditions. There are only two possible
explanations: either the model produces too much absorption, or there is
another source of noise that does not involve ground-based anthropogenic
sources. In the former case, one would have to face the fact that the model
produces realistic amounts of absorption at other times, particularly near the
terminators. In the latter case there are two possibilities, both of which
must be present but were not considered for our model based on ground sources:
cosmic background noise, which comes from above and would favor larger
elevation angles, and instrumental noise, i.e., saturated noise counts. The
ability to measure absorption variations from radar noise measurements depends
on the surrounding anthropogenic noise level - the higher the noise level, the
stronger absorption events the radar can study. For radars with low
anthropogenic noise level, such as KER, it may be difficult to study strong
vertical absorption events from the noise data.

The main message to carry from the present study is that 1) the mechanism
proposed in [Berngardt . (2018), Berngardt . (2019)] does very well near the
terminator where the noise level is at its strongest Also, 2) there are
additional noise sources to worry about, be they instrumental or of cosmic
origin. Finally 3) the present model should provide a useful tool for the
assessment of absorption on a day-to-day basis. This means that we could use
the present numerical model or its successors to extract important information
from the dynamics of noise measured by pole-oriented coherent scatter radars
in the 8 to 20 MHz range.

It should be kept in mind that the results presented here can only be
considered as preliminary, as they use reference ionospheric model IRI-2016 to
calculate the propagation trajectory. For more correct calculations of the
propagation path one really ought to take into account actual ionospheric
refraction, estimated either from radar data (using, for example, ground-
scatter signals in combination with measured elevation angles) or from
ionospheric electron density measurements by ionosondes. This approach will be
considered in future studies.

###### Acknowledgements.

The data of EKB radar were obtained using the equipment of Center for Common
Use ’Angara’ http://ckp-rf.ru/ckp/3056/ and available at
http://sdrus.iszf.irk.ru /ekb /page_example /simple. The work of EKB radar was
financially supported by the Ministry of Science and Higher Education of
Russian Federation. Work of OB and AM was supported by joint RFBR-CNRS grant #
21-55-15012 NCNI_a. JPSM receives support from the Canadian NSERC. The authors
acknowledge the use of SuperDARN data. SuperDARN is a collection of radars
funded by national scientific funding agencies of Australia, Canada, China,
France, Italy, Japan, Norway, South Africa, United Kingdom and the United
States of America. We thank all participants in the worldwide SuperDARN
collaboration for the distribution of SuperDARN data via
http://vt.superdarn.org/ tiki-index.php?page=Data+Access. The Saskatoon
SuperDARN radar operations are specifically funded by Canadian Space Agency
contracts and by a Major Science Initiative from Canada Foundation for
Innovation. The SuperDARN Kerguelen radar is operated by IRAP (CNRS, Toulouse
University and CNES) and is funded by IPEV and CNRS through INSU and PNST
programs. The SuperDARN Blackstone radar is operated by Virginia Tech with
funding from the National Science Foundation under award AGS-1935110.

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