# Ionospheric and geomagnetic response to the total solar eclipse on 21 August
2017

Amalia Meza<EMAIL_ADDRESS>Guillermo Bosch  María Paula Natali
Bernardo Eylenstein  Laboratorio de Meteorología espacial, Atmósfera
terrestre, Geodesia, Geodinámica, diseño de Instrumental y Astrometría
(MAGGIA), Facultad de Ciencias Astronómicas y Geofísicas (FCAG), Universidad
Nacional de La Plata (UNLP), Paseo del Bosque s/n, B1900FWA, La Plata,
Argentina  Consejo Nacional de Investigaciones Científicas y Técnicas
(CONICET), Godoy Cruz 2290, C1425FQB, Buenos Aires, Argentina  Instituto de
Astrofísica de La Plata (UNLP - CONICET), La Plata, Argentina  Observatorio
Geofísico Trelew, Trelew, Chubut, Argentina.

###### Abstract

Solar eclipses provide an excellent opportunity to study the effects of a
sudden localized change in photoionization flux in the Earth’s ionosphere and
its consequent repercussion in the Geomagnetic field. We have focused on a
subset of the data available from the North American 2017 eclipse in order to
study VTEC measurements from GNSS data and geomagnetic field estimations from
INTERMAGNET observatories near the eclipse path. Our simultaneous analysis of
both datasets allowed us to quantify the ionosphere and magnetic field
reaction to the eclipse event with which allowed us to compare how differently
these take place in time. We found that studying the behaviour of VTEC
differences with respect to reference values provides better insight of the
actual eclipse effect and were able to characterize the dependence of
parameters such as time delay of maximum depletion and recovery phase. We were
also able to test models that link the ionospheric variations in a
quantitative manner. Total electron content depletion measured from GNSS were
fed into an approximation of Ashour-Chapman model at the locations of
geomagnetic observatories and its predictions match the behaviour of magnetic
field components in time and magnitude strikingly accurately.

###### keywords:

total solar eclipse, VTEC from GNSS, geomagnetic field variation , LaTeX,
Elsevier , template

###### MSC:

[2010] 00-01, 99-00

††journal: Journal of LaTeX Templates

##  1 Introduction

The geomagnetic field exhibits variations in timescales that range from
fractions of seconds to millions of years. Some of them are internally
generated (in the Earth’s core, or in the Earth’s crust) and others are
external, created in the ionosphere-magnetosphere system. Among these the
regular daily variation of geomagnetic field during quiet time periods is a
common feature of geomagnetic field measurements; the current system
associated with the geomagnetic daily variation is typically termed the solar
quiet (Sq) current system. The Sq current system appears due to the
ionospheric wind dynamo.

The ionospheric dynamo is produced by movement of charged particles of the
ionosphere across Earth’s magnetic field. Tidal effects of the Sun and the
Moon and solar heating are both responsible for this effect. It is therefore
controlled by two parameters: the distribution of winds and the distribution
of electrical conductivity in the ionosphere. The orbital parameters of Earth,
Moon, and Sun; the solar cycle; solar flares; and solar eclipses, are some of
the external sources that influence the ionospheric dynamo. In particular any
process that alters ionospheric conductivity affects the electric current.

On the illuminated side of Earth the dominant source of ionization is solar UV
radiation. Solar flares and eclipses are very important and unique events to
analyze the dynamo current answer to short timescale perturbations. Solar
flares emit far UV together with soft and hard X-rays that penetrate deeper in
the atmosphere, temporarily ionizing the E region and D region. A solar
eclipse produces the opposite effect on ionospheric conductivity, as
ionization decreases when the Moon’s shadow crosses the Earth’s atmosphere.
Recombination of ionospheric electrons and ions in the absence of light
quickly reduces the conductivity.

Solar eclipses can be accurately predicted, therefore its effects on the
ionospheric plasma and geomagnetic field can be planned to be studied using
complementary techniques, e.g GNSS, ionospheric radiosonde, magnetometers,
etc. A solar eclipse produces an abrupt variation of the atmospheric
conditions. During this event the photoionization is strongly reduced, the
temperature of the atmosphere also falls and a cold spot with a well-defined
edge can then be defined. After that, the photoionizing flux recovers its
previous magnitude, and the atmosphere is heated again to the diurnal level
[Knížová and Mošna, 2011].

Consequently, solar eclipses produce a series of phenomena that can be
studied, such as: temporal and spatial analysis of the sudden electron density
(or total electron content) decay [Chernogor, 2013, Kumar et al., 2013,
Lyashenko and Chernogor, 2013], the abrupt geomagnetic variation [Kim and
Chang, 2018] and the generation of gravity and acoustic waves [Jakowski et
al., 2008, Chen et al., 2011]. The changes in the electron density
distribution and the total electron content (TEC) have been studied using
different techniques such as the Faraday rotation measurements [Tyagi et al.,
1980, Das Gupta et al., 1981], ionosonde networks [Le et al., 2009, Kurkin et
al., 2001], incoherent scatter radars [MacPherson et al., 2000, Cherniak and
Lysenko, 2013] and GNSS systems [Afraimovich et al., 1998, Ding et al., 2010,
Cherniak and Zakharenkova, 2018].

Eclipse effects is considered an external source of the geomagnetic field
variation, as detected in the middle of twentieth century [e.g. Kato et al.,
1956]. Later, other studies highlighted the relationship between the
geomagnetic component variability and the electric current obstruction effect.
One of the first studies was presented by Takeda and Araki [1984], in which
they show signatures of additional currents and fields generated by the
obstruction of Sq current system due to the ionospheric conductivity
depression during the eclipse. A decrease or increase on the magnetic field
component’s have been reported by many authors [e.g. Nevanlinna and Hakkinen,
1991, Brenes et al., 1993, Malin et al., 2000]. In spite of all of them taking
place during different local time, and at different geographical position, the
eclipse effect on the geomagnetic field is very evident.

A few investigations carried out simultaneous measurements of ionospheric and
magnetospheric parameters to study the effect of the solar eclipse on them
[e.g. Walker et al., 1991] that employ measurements of ionograms,
magnetograms, and microbarographs to study the solar eclipse of March 18th,
1988 in East Asia. Momani et al. [2010] used GPS, Incoherent Scatter Radar and
Earth’s magnetic field observations the Earth’s magnetic field to study the
total solar eclipse on August 1st 2008 over Northern Hemisphere.

In particular the effect of the total eclipse of the Sun on 21 August 2017 on
the terrestrial ionosphere was studied exhaustively by several authors [e.g.
Cherniak and Zakharenkova, 2018, Dang et al., 2018, Goncharenko et al., 2018,
Bullett and Mabie, 2018, Reinisch et al., 2018, Cnossen et al., 2019, Wang et
al., 2019]. The latter concludes that throughout the course of the moon’s
shadow over a particular observation site, the disturbance winds at the site
change direction and consequently their effects on the electron densities of
the F2 region also vary. These winds push the plasma down during the eclipse
and transport it up to the upper ionosphere after the eclipse. The combination
of chemical processes, wind transport, and ambipolar diffusion cause the time
lag and the asymmetric characteristic (rapid decline in Ne and slow recovery
from eclipse effects) of the topside ionosphere. Consequently, geographic
position, local time, geomagnetic conditions constrain the rate of electron
content depletion and its corresponding recovery times.

However, almost all studies mentioned above have studied either ionospheric or
geomagnetic effects independently. Hvoz̆dara and Prigancová [2002] proposed a
mathematical model based on the classical Ashour-Chapman model to explain the
geomagnetic field components’ variation. They quantify these in terms of the
position of both the quasi-circular spot of the ionospheric conductivity
decrease and the location of the geomagnetic observatory. In this work we
propose to simultaneously study both ionospheric and geomagnetic response to
the 2017 North America Solar Eclipse, combining information provided by GNSS
measurements and quantifying its relation to observed geomagnetic
perturbations.

##  2 Data and Methodology

The VTEC data computed using GNSS measurements and the three geomagnetic field
component using magnetometers observations are used in this analysis. A brief
description of these data and their variability are presented in this section.
Figure 1 shows the geographical distribution of the GNSS stations and the
location of the geomagnetic observatories. Victoria, Newport, and Boulder
geomagnetic observatories lie close to the totality path at a distance shorter
than 500 km, so we will restrict our analysis of the North America 2017
eclipse to the area relevant to these observatories indicated with a red
rectangle in Figure 1.

Figure 1: GNSS stations (blue circles) and geomagnetic observatories (red
circles). The blue and light-blue lines correspond to the total and the limits
of eclipse totality path, respectively. The red rectangle indicates the area
where VTEC maps were calculated

###  2.1 The VTEC GNSS

To analyze the response of the ionosphere to the sudden decrease of radiation
from the sun during the solar eclipse, the vertical total electron content
(VTEC) is computed. The VTEC values are obtained from the observations
recorded by more than 400 ground-based GNSS receivers located in Western
United States. All stations belong to the NOAA Continuously Operating
Reference Stations (CORS) Network (NCN), managed by NOAA/National Geodetic
Survey (ftp://geodesy.noaa.gov/cors/rinex/).

These observations were pre-processed with the Bernese GNSS Software version
5.2 [Dach R., 2015], using models recommended by the International Earth
Rotation and Reference Systems Service (IERS) [Petit and Luzum, 2010]. Ocean
tidal loading corrections were applied, following Letellier [2004], together
with atmospheric tidal loading displacements provided by van Dam et al.
[2010], and absolute phase-centre corrections for satellites and receivers, as
issued by the IGS.

The Bernese GNSS Software was modified to obtain the phase-code delay
ionospheric observable ($\tilde{L}_{I,\mathrm{arc}}$, Equation 2) along with
the geographic latitude and the sun-fixed longitude of the ionospheric pierce
point, zenith distance ($z^{\prime}$), azimuth angle, and time for each
satellite over each GNSS station displayed in Figure 1.

The ionosphere is approximated by a single shell of infinitesimal thickness
with equivalent STEC, located at 450 km above the Earth surface. The
intersection point of the line receiver-satellite with the ionospheric layer
is named ionospheric pierce point. An obliquity factor ($1/\cos z^{\prime}$ )
is used to map VTEC into STEC (integrated electron density along the signal
path); being $z^{\prime}$ the zenithal distance of the slant path at the
ionospheric piercing point.

$\mathrm{STEC}=\frac{1}{\cos z^{\prime}}\mathrm{VTEC}$ (1)

The code-delay ionospheric observable is modeled using an arc-dependent bias,
$\tilde{c}_{\mathrm{arc}}$, which accounts for receiver and satellite inter-
frequency bias and the ambiguity term. Following Ciraolo et al. [2007] and
Meza et al. [2009] the observation equation can be written as:

$\tilde{L}_{I,\mathrm{arc}}=\mathrm{STEC}+\tilde{c}_{\mathrm{arc}}+\varepsilon_{L}$
(2)

where $\tilde{L}_{I,\mathrm{arc}}$ is in TECU (10${}^{16}\,el$ m-2). Daily
solutions are computed to estimate the arc-dependent bias which is therefore
removed from $\tilde{L}_{I,\mathrm{arc}}$ to obtain STEC and VTEC through
equations 2 and 1 respectively.

The derived VTEC determinations were analyzed with an ad-hoc Python code that
iterates selecting data relevant to individual one degree longitude and
latitude bins. Within each bin the code performs the following tasks:

  1. 1.

Calculates the time evolution of the VTEC values averaging along 1.5 minute
intervals all VTEC determinations for the corresponding coordinate pair.
(VTECEcl, black points in Figure 2 upper panel).

  2. 2.

Uses astronomical ephemeris (PyEphem, https://rhodesmill.org/pyephem/) to
derive timing and percentage of eclipse obscuration at a reference altitude of
120 km [Yamazaki and Maute, 2017] (green dotted vertical lines and green
points respectively in Figure 2 upper panel). This reference height was chosen
to match the altitude of the ionosphere E-layer, where the electric field and
Sq currents are mostly affected by changes in electron density during the
eclipse. The timing of maximum occultation ($t_{0}$) will be used as reference
for estimating time delays of other ionospheric effects.

  3. 3.

Fits a polynomial function to the VTEC variations (red dashed curve) during
the eclipse to determine the timing of the maximum drop measured on the VTEC
curve due to the occultation ($t_{1}$, red dotted vertical line in Figure 2)
and its corresponding delay with respect to $t_{0}$ (${\Delta
t}_{1}=t_{1}-t_{0}$).

  4. 4.

Calculates a masked average of VTEC values from the immediately previous and
following days (VTECRef, blue points in Figure 2 upper panel). The masked
average consists of masking time intervals without data and consequently using
the only available data point when the other one is missing or performing the
average if both are present. In this way we can obtained a well sampled VTEC
reference to derive the change in VTEC ($\Delta$VTEC, black points in Figure 2
bottom panel) by comparing the observed behaviour against VTECEcl.

  5. 5.

Fits a Skewed Gaussian distribution (SkG, Ashour and Abdel-hameed [2010] and
references therein) to the $\Delta$VTEC values during the eclipse event (blue
dashed line in Figure 2 bottom panel) in order to derive both the timing
($t_{2}$, blue dotted vertical lines) and the maximum value of the VTEC
difference, defined as $\Delta$VTECmax = max( VTECRef \- VTECEcl). The use of
the SkG allows to account for the varying behaviour of $\Delta$VTEC curve and
derive additional parameters such as skewness, which will address the relation
between depletion and recovery times.

  6. 6.

Obtains spatially resolved information of $\Delta$VTECmax and ${\Delta
t}_{2}=t_{2}-t_{0}$, i.e. the time delay between the maximum Solar obscuration
and $\Delta$VTECmax.

Figure 2: Top panel: Black dots show VTEC values together with polynomial fit
(red dashed curve) to the VTEC variations during eclipse and the red dotted
line indicates the time of the maximum VTEC drop ($t_{1}$). Blue dots indicate
the masked average VTEC values from previous and following days as described
in text. Green curve shows the solar obscuration during the eclipse
(referenced at right axis), with green dashed vertical lines indicating the
first contact, maximum occultation and last contact. Bottom panel: To ease
comparison with top panel, the y-axis has been inverted as $\Delta$VTEC is
defined positive. Black dots show $\Delta$VTEC values calculated from the
difference between reference and eclipse days. The best fit using an
exponentially modified Gaussian distribution is plotted as a blue dashed line.
The blue dashed vertical line indicates the time of the maximum $\Delta$VTEC
derived from the fit ($t_{2}$).

###  2.2 The geomagnetic field

As stated before, we focused on geomagnetic values corresponding to data
collected at the three ground based observatories closest to the totality
path. The Boulder (BOU) and Newport (NEW) Observatories are operated by the
United States Geological Survey (USGS), and the Victoria (VIC) Observatory is
in turn supported by the Geological Survey of Canada (GSC). All three
participate in the International Real-time Magnetic Observatory Network
(INTERMAGNET). Consequently the geomagnetic values were obtained from the
INTERMAGNET data site. Considering that the eclipse signature in the
geomagnetic field, as seen by Malin et al. [2000] and Hvoz̆dara and Prigancová
[2002], is noticed as a smooth variation of its components within a 1-hour
interval, we chose to work with the available data at a sampling rate of 1
minute. These values are usually obtained by applying a digital Gaussian
filter to a higher sample rate data set centered in the minute, thus
eliminating short-term disturbances such as errors due to the instrument. The
retrieved data were already originally available in the three geomagnetic
field components of interest ($X,Y,Z$). In order to reject superposed
disturbances of much shorter period, the minute data will be further smoothed
by a 15-min window moving average.

The geomagnetic field variability is defined as the difference between the
values obtained during the eclipse event and the reference values. The latter
are obtained calculating the mean value of the five nearest geomagnetic quiet
days [Momani et al., 2010].

To examine patterns of the geomagnetic field variations induced by solar
eclipse, we analyzed data within a two hours time interval centered in the
maximum occultation . In order to eliminate the intrinsic regular daily
variabilities of the eclipse day and the reference day, during the two hour
window selected, the linear trend was removed using a first order polynomial
fit [Malin et al., 2000]. Finally the geomagnetic field variations, produced
by the solar eclipse, $\Delta X$, $\Delta Y$ and $\Delta Z$ are computed and
shown in the left column of Figure 7.

###  2.3 Relationship between VTEC and geomagnetic field

The classical Ashour-Chapman model, with Hvoz̆dara and Prigancová [2002]
modifications, is considered to analyse the geomagnetic components variability
and its relationship with the VTEC variation in the region of eclipse
obscuration.

Low-conductivity ionospheric spot is used as the Ashour-Chapman model of a
thin current sheet model with the arbitrarily directed undisturbed electric
field $\bf{E_{0}}$. In the present work the angle between x-axis (to
geographic North) and $\bf{E_{0}}$, $\epsilon$, is different from zero and the
direction of the equivalent Sq current system is assumed similar to
$\bf{E_{0}}$ (in this first approximation the Hall conductivity is not taken
into account for determination). Dedicated Ionospheric Field Inversion
(DIFI-3) model, time-varying spherical harmonic representation of the quiet-
time Sq and equatorial electrojet field, is used to $\epsilon$ determination
(https://geomag.colorado.edu/difi-calculator).

Our analysis is based on the mathematical explanation described in Hvoz̆dara
and Prigancová [2002], Appendix A. In their paper, the authors model the
geomagnetic effect due to changes in the local ionospheric conductivity linked
to the TEC decrement originated by the eclipse. They do this by means of a
cilindrical coordinate system ($r$, $\phi$, $z$) with origin on the eclipse-
induced conductivity spot and its z axis normal to the Earth’s surface. In
this system, the magnetic potential field can be written as:

$\mathrm{{\Omega}=-I\,a\,\sin\,\phi\,W(r,z)}$ (3)

where:

$\begin{split}I&=\frac{1-\kappa}{1+\kappa}\,I_{0}[A/m]\\\
W(r,z)&=\int_{0}^{\infty}s^{-1}J_{1}(sa)J_{1}(sr)e^{-sz}ds\end{split}$ (4)

$J_{1}$ is the Bessel function of the first kind and index 1.

Being $\mathbf{H}$ the disturbing magnetic field which is related with the
corresponding potential $\mathbf{\Omega}$ [Ashour and Chapman, 1965]. Then

$\mathbf{H}=-\mathrm{grad}\,{\Omega}$ (5)

The geomagnetic disturbance is defined by
$\mathrm{\mathbf{b}=\mu_{0}\,\mathbf{H}}$ where $\mu_{0}=400\,\pi$ and
$\mathbf{b}$ is in nT unit. Its cartesian components are:

$\begin{split}b_{x}&=\mu_{0}(H_{r}\,\cos\alpha-H_{\varphi}\,\sin\alpha)\\\
b_{y}&=\mu_{0}(H_{r}\,\sin\alpha+H_{\varphi}\,\cos\alpha)\\\
b_{z}&=\mu_{0}H_{z}\end{split}$ (6)

Table 1 shows the values defined to calculate the geomagnetic disturbance for
the different geomagnetic stations. The angle $\epsilon$, the distance from
the observatories and the center of the eclipse-induced conductivity spot,
$\delta$, and the degree of the TEC decrease caused by the solar eclipse,
$\kappa$.

Table 1: Parameters used in the theoretical model of geomagnetic eclipse-disturbance: $\epsilon$ is the angle between x-axis (to geographic North) and $\bf{E_{0}}$; $\delta$ is the distance from the observatories and the center of the eclipse-induced conductivity spot, and $\kappa$ is the degree of the VTEC decrement caused by the solar eclipse | $\epsilon$ | $\delta$ | $\kappa$
---|---|---|---
VIC | 130 | 390 | 0.63
NEW | 116 | 450 | 0.61
BOU | 97 | -206 | 0.57

The magnetic field variations $\mathbf{b}$ $=(b_{x},b_{y},b_{z})$ are adjusted
for the electromagnetic induction effect, in order to compare with the
geomagnetic field components variability from the geomagnetic observatories
Hvoz̆dara and Prigancová [2002]. Consequently, the $\Delta X$ , $\Delta Y$ and
$\Delta Z$, predicted by the model are expected to be 1.5 $b_{x}$, 1.5 $b_{y}$
and 0.3 $b_{z}$ respectively.

##  3 Results and Discussions

###  3.1 VTEC variation

Figure 3 shows the $\Delta$VTECmax during the eclipse, displayed as a
percentage value from the reference VTEC value. No evident trend can be seen
longitudinally, besides a local peak about 110°W. Figure 4 shows the
$\Delta$VTECmax as function of the geographical longitude, confirming that no
dependence on occultation percentage is present either. Liu et al. [2019]
studied the effects of an annular eclipse centered in Taiwan and found a
strong correlation between maximum obscuration and $\Delta$VTECmax in
percentage. Their figure 6i shows that $\Delta$VTECmax is directly
proportional to maximum obscuration (ranging between 40% and 90%), although
there is evidence of saturation at high occultation. Bottom/right panel in
Figure 4 confirms this, as no visible trend with $\Delta$VTECmax is present
when focusing on occultations larger than 75% .

Figure 3: Geographical distribution of relative $\Delta$VTECmax (in percentage
units). Lower and upper limits of eclipse totality path, together with
locations of geomagnetic observatories are also plotted as reference. Figure
4: Relative $\Delta$VTECmax plotted against maximum eclipse occultation (left
panel) and geographical longitude (right panel).

There is a noticeable difference among $\Delta$VTECmax north and south of the
totality path. We can interpret this considering that the integral of the
electron content in the ionosphere results from the balance between transport,
ionization and loss processes; the eclipse reduces the electron temperature,
decreases the pressure and consequently induces a downward drift of plasma
from the topside ionosphere [Ding et al., 2010, Cherniak and Zakharenkova,
2018]. An explanation of the latitudinal distribution of $\Delta$VTECmax is
that the eclipse switches off the ionization source, therefore the
recombination is more effective specially at lower latitudes where the neutral
mass density is higher [Cherniak and Zakharenkova, 2018].

Figure 5: Analysis of time delay distribution between maximum eclipse
occultation and ionospheric response. Both panels show trends of ${\Delta
t}_{1}$ in blue outline and ${\Delta t}_{2}$ in filled red. Left panel
highlights the overall difference between time delays and right panel displays
distinct dependence of time delays with geographical longitude.

Another interesting result worth discussing is the analysis of the time delay
between solar occultation and the ionospheric response measured from VTEC. The
presence of a time delay between occultation and ionospheric response has
already been presented in previous researches [e.g. Jakowski et al., 2008,
Boitman et al., 1999, Liu et al., 2019, Cherniak and Zakharenkova, 2018,
Momani et al., 2010]; although showing shorter delays. As outlined in Section
2.1 we measure time delays from the $\Delta$VTEC curve as this is more
strongly linked to the eclipse effect itself.

The variations in the local VTEC minimum, cited in references above are
actually due to a combination of the eclipse occultation and the rapid
increase in VTEC expected at mid-morning. The apparent increase in ${\Delta
t}_{1}$ seen in Figure 5 is not due to the eclipse itself but to the fact that
the daily VTEC increase in the ionosphere is slowing down as it reaches its
peak at about 20hrs UT. For comparison purposes we also show in Figure 5 the
time delay distribution derived from the VTEC curve which resemble values
cited in the literature. Furthermore, we were able to reproduce to longitude
bin average of the time delay as calculated from the VTEC curve in the right
panel of Figure 5. Blue outline bins show almost identical behaviour as in
Cherniak and Zakharenkova [2018], a small (2 to 3 minutes) offset is present,
mostly due to the fact that the cited authors refer to eclipse ephemeris at
ground level while we do it 120 km above. Filled red bins show, besides larger
values as already mentioned a slow but constant decreasing pattern as the
eclipse advances eastward.

We also analyzed the presence of trends among the asymmetry in the
$\Delta$VTEC curve measured via the $\gamma$ parameter of the skewed Gaussian
profile. A $\gamma$ value of 0 indicates a symmetric Gaussian profile and
positive values show recovery times larger than depletion ones. Figure 6
displays a visible trend with $\gamma$ values diminishing asymmetry eastwards
even within the scatter. This indicates that the recovery time is noticeable
larger than depletion time at the west coast but this difference as the shadow
moves eastward. Several effects can contribute to this behaviour, as the moon
shadow changes its shape and reduces its apparent speed when its cone axis
passes closest to Earth’s center.

Figure 6: Skewness variation plotted against geographic longitude where the
strong reduction of asymmetry in the $\Delta$VTEC profile can be seen. We have
added a color scale for eclipse occultation making it evident there is no
dependence on this parameter.

###  3.2 Geomagnetic field variation

The geomagnetic variability during total solar eclipse is analysed over the
observatories affected by an occultation larger than 80 %: VIC has the 89,1 %
at 17:20 UT (10:20 LT), NEW has the 84,6 % at 17:26 UT (10:26LT), and BOU has
the 92,4% at 17:46 UT (11:46 LT), its geographical locations are shown in
Figure 1.

In this section we will discuss the comparison of the observed geomagnetic
variability and the one proposed by the theoretical model described in Section
2.3 and displayed in the right column of Figure 7. These modeled geomagnetic
disturbances along the Cartesian components are defined along one hour
interval centered at the time of maximum occultation ($t_{0}$), which is
highlighted by the green box .

Top row in Figure 7 displays BOU observatory geomagnetic variability: the
$\Delta X$ component shows positive values with its maximum arising near
$t_{0}$, the $\Delta Y$ component has positive and then negative values before
and after $t_{0}$ respectively, and the $\Delta Z$ component has the lower
amplitude, showing positive values with its maximum near $t_{0}$. Its
corresponding right panel shows the very good agreement for this eclipse
configuration. Middle and bottom rows (VIC and NEW observatories) show similar
behaviours among themselves, which is quite expected due to close geographical
coordinates and relative location to the eclipse path, but quite different as
the one shown at BOU. The $\Delta X$ component is positive and its maximum
value takes place before $t_{0}$, the $\Delta Y$ component is negative and its
minimum value happens close to $t_{0}$, and the $\Delta Z$ Component records
lower magnitude variations, where its maximum difference is negative and
occurs after $t_{0}$. This distinct observed behaviour is still well described
qualitatively by the model in their corresponding right panels, but the
agreement is not as good as for BOU observatory.

Malin et al. [2000] and Momani et al. [2010] had also found perceptible
differences in geomagnetic response at different observatories locations.
Regarding the model predictions, dissimilar performance among observatories
can be attributed to a combination of observing conditions and model
limitations. VIC and NEW stations lie farther from eclipse path and smaller
geomagnetic disturbances are therefore expected which might be subject to
contamination by other sources of variations. Model assumptions (such as
cylindrical symmetry and ionospheric isotropy) can also contribute to these
discrepancies.

As a reference, we have also added the $\Delta$VTEC values in the left column
of Figure 7 which allows a straightforward comparison between the VTEC and
geomagnetic variations.

From the visual inspection of Figure 7 it can be readily seen that eclipse
effects impact on the VTEC and the geomagnetic field with very different
timescales. We interpret this as due to the fact that the variations of
electric current which flow mainly in the E layer are the ones responsible for
the slight changes in the geomagnetic field observed from ground level
observatories. This implies that the time variation of eclipse effects on the
magnetic field depends mostly on processes taking place in the E layer. On the
other hand, even though the VTEC determinations are integral determinations
over the line of sight, the F2 layer provides the major contribution to it.
Therefore, an explanation of the observed delay of the VTEC variability on
Figure 7 is that in the F2 layer the recombination process time is longer than
E layer, because the eclipse effect is controlled by the transport of plasma
[Bienstock et al., 1970].

Figure 7: Geomagnetic variability detected from BOU, VIC, and NEW
observatories. The left column shows the measurements of $\Delta X$, $\Delta
Y$, and $\Delta Z$, in blue, red, and yellow color respectively. The black
curve shows the $\Delta$VTECmax measurement. The right column shows the
geomagnetic variability based on the Ashour-Chapman thin current sheet model.

##  4 Concluding remarks

This work examines the ionospheric and geomagnetic response to the 2017 North
America Solar Eclipse. The VTEC obtained from GNSS observations were used to
compute the VTEC variability during the eclipse; the $\Delta$VTEC, which is
the difference between VTEC during the eclipse and reference days, and its
maximum ($\Delta$VTECmax) and also the maximum drop of VTEC curve due to the
occultation. The values of $\Delta$VTECmax range from 20% to 45% along the
eclipse path within an area of 70% obscuration, and the maximum values stretch
equatorward from the totality. The $\Delta$VTECmax lags 22-32 min after the
maximal eclipse time (${\Delta t}_{1}$). The values of VTEC drop had a 14-22
min time delay after the maximum occultation (${\Delta t}_{2}$).

The ${\Delta t}_{2}$ values and the recovery times concerning the depletion
times ( $\gamma$ parameter) are strongly related to local time. They have
larger values at westwards longitudes when the eclipse is at mid-morning. When
the eclipse is carried out close to midday, the depletion and recovery times,
and ${\Delta t}_{1}$ and ${\Delta t}_{2}$, become more similar.

For the first time, a total solar eclipse was simultaneously studied from both
ionospheric and geomagnetic points of view. The degree of the TEC decrease
caused by the solar eclipse was used in a mathematical model based on the
Ashour-Chapman model to predict the geomagnetic disturbance. Quantitatively
the model and the Cartesian geomagnetic components variabilities of the three
geomagnetic observatories, with larger occultation than 84%, were comparable
and consistent.

##  5 Acknowledgments

We wish to acknowledge the thorough review and useful comments provided by the
anonymous referees, which have helped to improve the final version of this
article. The results presented in this paper rely on data collected at
magnetic observatories. We thank the national institutes that support them and
INTERMAGNET for promoting high standards of magnetic observatory practice
(www.intermagnet.org).

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#  Site-specific Deep Learning Path Loss Models based on the Method of Moments

Conor Brennan  School of Electronic Engineering
Dublin City University
Dublin, Ireland
<EMAIL_ADDRESS>
https://orcid.org/0000-0002-0405-3869  Kevin McGuinness  School of Electronic
Engineering
Dublin City University
Dublin, Ireland
<EMAIL_ADDRESS>
https://orcid.org/0000-0003-1336-6477

###### Abstract

This paper describes deep learning models based on convolutional neural
networks applied to the problem of predicting EM wave propagation over rural
terrain. A surface integral equation formulation, solved with the method of
moments and accelerated using the Fast Far Field approximation, is used to
generate synthetic training data which comprises path loss computed over
randomly generated 1D terrain profiles. These are used to train two networks,
one based on fractal profiles and one based on profiles generated using a
Gaussian process. The models show excellent agreement when applied to test
profiles generated using the same statistical process used to create the
training data and very good accuracy when applied to real life problems.

###### Index Terms:

Propagation, rural, method of moments, surface integral equation, FAFFA,
machine learning, convolutional neural network.

##  I Introduction

The modelling of EM wave propagation over terrain is a central problem of
wireless network design. The physical scale of the problem has meant that
radio planners have historically relied on empirical curve-fitting approaches
[1] or knife edge models[2]. More accurate formulations, based on for example,
surface integral equations (IE) exist [3], but are slow if un-accelerated. A
variety of acceleration techniques exist, most notably the Fast Far Field
Approximation (FAFFA) [4]. These computational efficiencies can be optimised
using the Tabulated Interaction Method (TIM) [5, 6] which offers rapid,
accurate simulation but is currently somewhat restricted in its formulation to
relatively smooth surfaces. The purpose of this paper is to develop a model
which, like the TIM, is both a) similar in accuracy to the FAFFA and b) an
order of magnitude faster to implement. Crucially we seek to develop a model
which is capable of being extended to more general problems in the future, an
extension which is difficult for the TIM. Machine learning (ML) potentially
offers a framework to achieve this goal. It has been widely applied to
propagation problems in recent years, including propagation in indoor [7, 8]
and urban [9, 10, 11, 12] scenarios. Rural deployments have also been
considered such as in [13, 14, 15] some based on relatively simple propagation
models and others on the parabolic equation method [16]. In this work we seek
to develop an accurate site-specific model, that is one that can take in
specific terrain height information as input and generate predictions for path
loss along that particular profile. A key issue facing all ML techniques is
access to representative, accurate training data in sufficient quantity to
produce reliable models. This is particularly an issue in propagation
modelling, where measured data is expensive in terms of hardware and man-
hours. A popular alternative is to use synthetic data based on simulation, as
was done in several of the works cited above. This is the approach taken in
this paper too, whereby propagation over tens of thousands of realistic
profiles is efficiently analysed using the FAFFA. These data constitute a
training set which can then be used to develop an accurate, yet
computationally efficient, ML model. Synthetic data is justified in this
instance on the reasonable basis that integral equation models have
demonstrated very good agreement with measured data [3]. In any event, several
comparisons to measured data are presented in section V so that the reader can
gauge the performance. The paper is organised as follows. Section II briefly
describes the surface electric field integral equation and its efficient
solution via the method of moments and FAFFA algorithm. The process used to
develop the data sets is presented in section III while the development of the
ML models is described in IV. Results are presented in section V and we close
with conclusions and an outline of potential future work in section VI.

##  II EFIE Formulation

In order to generate training data the path loss over a set of artificially
synthesised terrain profiles is computed. The Electric Field Integral Equation
(EFIE), solved with the method of moments, is used to compute the path loss
over each profile.

Figure 1: Geometry for the EFIE

Figure (1) depicts a 2D problem where $TM^{z}$ incident fields emanate from a
line source and impinge on a 1D surface. A time variation of $e^{\jmath\omega
t}$ is assumed and suppressed. For ease of implementation the terrain is
assumed to be perfectly reflecting in this paper, a reasonable assumption at
grazing incidence, but this is not a fundamental restriction and will be
relaxed in future work. Under these assumptions the total field, $E_{z}^{t}$
at a general point $\bf r$ above the terrain surface can be written as

$E_{z}^{t}\left(\vec{r}\right)=E_{z}^{s}\left(\vec{r}\right)+E_{z}^{i}\left(\vec{r}\right),$
(1)

where $E_{z}^{i}$ is the known incident field, i.e the field from the source
that would exist in the absence of the scatterer while $E_{z}^{s}$ is the
unknown scattered field caused by the presence of the terrain. The scattered
field can be expressed in terms of an integral involving the surface electric
current, $J_{z}$, as

$E_{z}^{s}\left(\vec{r}\right)=-\frac{k_{0}\eta_{0}}{4}\int_{C}J_{z}\left({\vec{r}}^{\prime}\right)H_{0}^{(2)}\left(k_{0}\left|{\vec{r}}-{\vec{r}}^{\prime}\right|\right)dl^{\prime},$
(2)

where the integral takes place over the boundary of the scatterer (in this
case the terrain surface) and $H^{(2)}_{0}$ is a zero-order Hankel function of
the second kind. Applying the boundary condition of zero tangential fields for
a point ${\vec{r}}$ on the scatterer surface yields the following equation for
$J_{z}$

$E_{z}^{i}\left(\vec{r}\right)=\frac{k_{0}\eta_{0}}{4}\int_{C}J_{z}\left({\vec{r}}^{\prime}\right)H_{0}^{\left(2\right)}\left(k_{0}\left|{\vec{r}}-{\vec{r}}^{\prime}\right|\right)~{}dl^{\prime}.$
(3)

The method of moments is used to discretise the EFIE. The surface current is
expanded using $N$ pulse basis functions, $f_{n}$, as

${J_{z}\left(\vec{r}\right)}\simeq\sum_{n=1}^{N}j_{n}f_{n}\left({\vec{r}}\right),$
(4)

and point matching is applied at the basis domain centres to obtain a $N\times
N$ dense linear system

${\bf Zj}={\bf v}.$ (5)

Equation (5) can be solved in a variety of ways but it is computationally very
expensive to do so when one considers that $N$ can be of the order of hundreds
of thousands for a typical profile at radio frequencies. Assuming forward
scattering (approximating $\bf Z$ as a lower-triangular matrix) allows (5) to
be solved using a straightforward, but slow, process of back substitution
given by

$Z_{mm}j_{m}=V_{m}-\sum_{n<m}Z_{mn}j_{n}\mbox{ for }m=1\dots N.$ (6)

Forward scattering is a reasonable assumption for the gently undulating
terrain profiles considered in this paper, but nonetheless is an approximation
that will be relaxed in future work.

###  II-A FAFFA Acceleration

Solution via (6) is effective but very slow and not suitable for the
generation of training data which requires path-loss analysis for thousands of
profiles. To expedite the process the Fast Far Field Approximation (FAFFA) was
implemented. The FAFFA proceeds by assembling local collections of
neighbouring pulse basis functions into $M$ groups. With such a decomposition
(6) can be equivalently written as

$\displaystyle Z_{mm}j_{m}$ $\displaystyle=$ $\displaystyle
V_{m}-\sum_{l^{\prime}<l}\sum_{n\in G_{l^{\prime}}}Z_{mn}j_{n}-\sum_{n\in
G_{l},n<m}Z_{mn}j_{n}$ $\displaystyle\mbox{ for }l=1\dots M,m\in l.{}$

The essence of the FAFFA is to replace the independent interactions between
individual basis functions in separate groups with approximate interactions
written in terms of a small number of computations that are extensively re-
used. The approximation can be derived with reference to figure (2).

Figure 2: Geometry for the FAFFA

Pulse basis functions $m$ and $n$ are situated in two groups, $G_{l}$ and
$G_{l^{\prime}}$ with group centres $l$ and $l^{\prime}$ respectively. A
simple consideration of the geometry shows that

$\displaystyle\left|{\vec{r}}_{mn}\right|\simeq\left|{\vec{r}}_{ln}\right|-{\vec{r}}_{ml}\cdot{\hat{r}}_{ln}$
(8)
$\displaystyle\simeq\left|{\vec{r}}_{ln}\right|-{\vec{r}}_{ml}\cdot{\hat{r}}_{ll^{\prime}},$
(9)

which for $\left|{\vec{r}}_{mn}\right|$ large allows the associated matrix
element to be approximated as

$Z_{mn}\simeq Z_{ln}e^{\jmath
k_{0}{\vec{r}}_{ml}\cdot{\hat{r}}_{ll^{\prime}}}.$ (10)

Inserting this into (II-A) yields

$\displaystyle Z_{mm}j_{m}$ $\displaystyle=$ $\displaystyle
V_{m}-\sum_{l^{\prime}<l}e^{\jmath
k_{0}{\vec{r}}_{ml}\cdot{\hat{r}}_{ll^{\prime}}}\sum_{n\in
G_{l^{\prime}}}Z_{ln}j_{n}{}$ (11) $\displaystyle-\sum_{n\in
G_{l},n<m}Z_{mn}j_{n}.$

for $l=1\dots M,m\in l$. The key computational advantage of the FAFFA is that
for each pair of groups $G_{l}$ and $G_{l^{\prime}}$ once the summation
$\sum_{n\in l^{\prime}}Z_{ln}j_{n}$, representing the fields scattered from
group $G_{l^{\prime}}$ to the centre of group $G_{l}$, has been computed it
can then be stored and repeatedly used to efficiently approximate the fields
scattered from group $G_{l^{\prime}}$ to every point $m$ in group $G_{l}$. In
practice the groups are linear segments connecting sampled terrain heights.
Consequently the final sum on the right hand side of (11), representing the
interactions within a group, takes the form of a discrete convolution which
can be efficiently computed using a Fast Fourier Transform, yielding an
additional saving. Once the surface current has been computed using (11) total
fields, and thus path-loss, at selected points above the terrain profile can
be computed using (1) and (2). Finally, in order to use this 2D simulation to
make real-world predictions, the total electric field at each point is
multiplied by $\frac{1}{\sqrt{R}}$, where $R$ is the distance from the
transmitter. This ensures that the power density decays as $\frac{1}{R^{2}}$
in free space, and serves as a heuristic conversion from a 2D field to a 3D
field so as to best compare to measured data.

##  III Machine Learning Data Set

Synthetic data was created and used to develop two distinct ML models. In both
cases this involved randomly creating 8000 profiles and solving for the
electric field at sampled locations above each profile. In order to focus on
an examination of the ability of ML to model the physical scattering effects
of the terrain we kept some parameters constant in each realisation. These
parameters were frequency (assued to be 970$MHz$), transmitter height and
location (10.4$m$ over the leftmost terrain point), and receiver height
(2.4$m$ over the terrain at sampled points $50m$ apart). Each profile
comprised 256 sampled $(x,y)$ values where $x$ is range (in increments of
50$m$) and $y$ is height (chosen randomly in one of two ways, outlined below).
The FAFFA was applied to efficiently solve for the fields at the sampled
receiver locations. Each of the 8000 elements of the data set thus comprised
256 $(x,y)$ points denoting the field point locations and a vector of 256
corresponding path loss values in dB. We developed two distinct ML models, one
trained using synthetic profiles which were realisations of a Gaussian random
process, while the second was trained using realisations created using a
fractal-generating algorithm.

The first model, referred to as $ML_{GP}$, was based on a data set of size
8000 created using profiles which were realisations of a Gaussian random
process. The root mean square height was set to 20$m$ while the correlation
length was $800m$. The second model, referred to as $ML_{F}$ was based on a
data set of size 8000 created using profiles which were random fractals
created using the Diamond-Square algorithm [17] with variance set to 30 and
fractal parameter $H=1.2$. Some typical realisations (along with model
validation results) for both profile types are shown in Fig. (4).

##  IV ML model development

The goal of the machine learning model is to predict a $D$-dimensional path
loss profile associated with a given $D$ dimensional terrain profile. In this
case $D$ refers to the number of $50m$ linear segments used to describe the
profile. Here, we propose to use a deep neural network,
$f_{\theta}:\mathbb{R}^{D}\to\mathbb{R}^{D}$, with parameters $\theta$ and
train it using stochastic gradient methods on the synthetic training data.
There are various model architectures that could be used for this, including
convolutional neural networks (CNN), recurrent neural networks (RNN), and
Transformer-based architectures. In this work we opted for a CNN based model,
as unlike RNNs, these models can produce the entire path loss profile in a
single pass and are often simpler to train than Transformer-based approaches.
The network architecture is shown in Fig. 3 and is based on U-Net [18], which
is widely used in medical image and general semantic segmentation. The
architecture is modified to handle 1D signals by replacing all 2D
convolutional and batch normalization layers with their 1D counterparts. Three
dropout layers ($p=\frac{1}{2}$) are also added to around the central
bottleneck (after the 3rd and 4th downsampling layers and after the first
upsampling layer) to provide regularization and reduce overfitting. Upsampling
is done by linear interpolation without transposed convolutions. The output
layer of the network produces a single scalar value for each input and no
activation function is used on this layer.

Figure 3: Network Architecture

We reduce the number of parameters in the network by approximately 50% by
halving the number of channels on the internal layers when compared to the
original U-Net model. We also widen the kernel size on the initial two
convolution layers from 3 to 11 to provide more spatial context.

###  IV-A Adam Optimiser, calibration and data augmentation

The network was trained using a mean squared error loss function for 25 epochs
over the training data using the Adam optimiser [19] with an initial learning
rate of $10^{-2}$ and a weight decay $\lambda=10^{-5}$. The batch size was set
to 128. The model is evaluated on the validation set after each epoch and the
model with the lowest validation loss is retained for testing. The learning
rate is stepped down by a factor of 10 when the validation loss has not
reduced for 10 epochs. The initialization and batch normalization layers in
neural networks are typically calibrated so that the output layer produces
values that are approximately distributed according to a standard normal
distribution before training. If left unchanged, this will cause the loss
value to be very large in the early epochs, as the average value of the target
is approximately -134 dB. To compensate, we set the bias value of the output
layer to -134dB, which makes the network have an error close to that of a
regressor that always produces the mean value (around 630 dB2). This
accelerates training since the network does not need to spend a long time
during training moving the network predictions towards the range and scale of
the targets. Since the path loss is independent of the absolute height of the
terrain profile and source/receiver points, (in that it only depends on the
relative distance between the totality of points), the network should produce
a result that is invariant to changes in the absolute height of the input
profile. To encourage this, we perform random data augmentation at training
time to modify the absolute height of the input profile (keeping the relative
vertical displacement of the source and receivers fixed at 10.4$m$ and $2.4m$
respectively). Specifically, we add a random scalar
$\epsilon\sim\mathcal{N}(0,30)$ to each input before passing it through the
network. Other possible approaches to making the network produce outputs that
are invariant to profile height shifts are to either normalize the profile to
always start at height 0, or to use the profile derivatives as inputs instead
of the profile heights. Empirically, however, we found the data augmentation
approach more effective, likely because this approach also provides a degree
of additional regularization.

###  IV-B Uncertainty prediction

In addition to a point prediction of the path loss, in some applications it is
useful to also provide an estimation of the uncertainty of the predictions. To
facilitate this, we also trained a variation of the networks that also
estimates the variance by modifying the network to produce two outputs:

$\mu_{\theta}(\mathbf{x})=f_{\theta}(\mathbf{x})_{1},\quad\log\sigma^{2}_{\theta}(\mathbf{x})=f_{\theta}(\mathbf{x})_{2},$
(12)

where now $f_{\theta}:\mathbb{R}^{D}=\mathbb{R}^{D\times 2}$ and the model
predicts the log of the variance to ensure that the predicted variance is
always positive. Assuming that the distribution of the target conditioned on
the terrain profile
$p(\mathbf{y}\mid\mathbf{x})\sim\mathcal{N}(\mu_{\theta}(\mathbf{x}),\sigma_{\theta}(\mathbf{x})I)$
gives a negative log likelihood loss of:

$-\log
p(\mathbf{y}\mid\mathbf{x})=\frac{1}{2}\sum_{k=1}^{D}\left(\frac{(y_{k}-\mu_{\theta}(\mathbf{x})_{k})^{2}}{\sigma^{2}_{\theta}(\mathbf{x})_{k}}+\log\sigma^{2}_{\theta}(\mathbf{x})_{k}\right).$
(13)

This variant of the model is again trained to minimize the expected value of
the above loss using stochastic estimates on batches of size 128. It takes 3
times longer to train than the version that only produces point estimates (75
epochs).

###  IV-C Computation times and Validation

Both model variants are relatively fast to train, requiring less than 5
minutes (NVIDIA GeForce RTX 3090). Inference takes approximately 2ms per
profile on both GPU and CPU hardware (AMD Ryzen 9 5950x). Inference can also
be done in batches, leading to approximately 10-20$\times$ better throughput
in our experiments (batch size 128). In each case the 8000 profiles were
randomly split into 7500 for model development and 500 for validation.

Figure 4: Validation examples for $ML_{GP}$(left) and $ML_{F}$(right). In each
case profile is on top (red) while FAFFA (black) and $ML_{F},ML_{GP}$ (red)
pathloss predictions are below.

Fig. (4) shows some randomly selected validation tests. Two examples with
Gaussian profiles are shown on the left while the right shows two examples of
fractal profiles. The results show the ability of the ML model to accurately
solve problems involving profiles of the same type.

##  V Results

The validation results of section IV-C confirm that the ML model can rapidly
reproduce the predictions of a full-wave EM solver with good accuracy when
applied to profiles of a similar type (i.e. Gaussian or fractal). To be of use
in radio planning the models must generalise and be applicable to more general
real-world profiles. To do this we ran the trained models for several profiles
for which measured data was available. The results are shown in Figs (5). The
terrain profile is shown on top while the bottom plot shows the measured path
gain and predictions using the FAFFA, the two ML models and a knife edge model
based on Deygout’s method. Good agreement for the two ML models is noted with
$M_{F}$, the model based on fractal data, outperforming $M_{GP}$ the model
based on Gaussian profiles. This is perhaps un-surprising given the fractal-
like appearance of real terrain profiles. Table I provides statistics about
the performance of the models. $\overline{\epsilon}_{F}$ and $\sigma_{F}$ are
average error and standard deviation of ML predictions relative to the FAFFA
predictions, while $\overline{\epsilon}_{M}$ and $\sigma_{M}$ are statistics
(for all models) based on a comparison to the measured data. In the context of
the ML models the most important data are arguably $\overline{\epsilon}_{F}$
and $\sigma_{F}$. This is because the ML models are trained on the FAFFA model
and not the measurements. The FAFFA prediction thus serves as the upper-bound
on their performance and their ability to match the measured data is thus
limited to the performance of the FAFFA model. Any enhanced agreement with
measurements over that of FAFFA, while superficially welcome, would not
necessarily be reproduced for other profiles and could not be considered
indicative of an enhanced predictive capability. Nonetheless it should be
noted that, when compared to the measurements, both ML models have accuracy
which is similar to the FAFFA, while having a greatly reduced computational
burden which is similar to the knife edge model.

TABLE I: Model Error Statistics Model | Hjorringvej | Jerslev
---|---|---
| $\overline{\epsilon}_{F}$ | $\sigma_{F}$ | $\overline{\epsilon}_{M}$ | $\sigma_{M}$ | $\overline{\epsilon}_{F}$ | $\sigma_{F}$ | $\overline{\epsilon}_{M}$ | $\sigma_{M}$
MLF | 1.6 | 6.49 | 7.41 | 6.15 | 0.51 | 3.19 | 3 | 5.68
MLGP | 0.34 | 7.66 | 6.15 | 6.68 | 5.14 | 6.4 | 7.63 | 7.11
FAFFA | - | - | 5.81 | 7.42 | - | - | 2.49 | 6.39
Deygout | - | - | 14.11 | 9.41 | - | - | 9.79 | 8.03

Figure 5: Profile and Path loss for Hjorringvej (top 2) and Jerslev (bottom
2).

Finally Fig. (6) shows an example of the $ML_{F}$ prediction for a real
profile, where the uncertainty interval is shown in grey (extending to two
standard deviations around the $ML_{F}$ prediction in red). The computation of
the model variance in the model is discussed in section (IV-B).

Figure 6: $ML_{F}$ prediction (red) with uncertainty regions (grey) for
Hjorringvej profile. FAFFA prediction in blue.

##  VI Conclusions and future work

This paper has presented two ML models for predicting path loss over rural
terrain. The ML models are based on deep learning convolutional networks
trained with synthetic data. The training data was obtained by generating path
loss over thousands of randomly generated profiles, which were generated using
either a Gaussian process or a fractal-generating algorithm. The training path
loss data was obtained using the FAFFA, an accelerated method of moments
solver. In both cases the agreement between the ML models and the validation
data was excellent. When applied to real life profiles the model trained on
fractal profiles was more accurate with both ML models outperforming a knife
edge model. Future work will concentrate on identifying a more representative
data set and developing a more general model (which will be valid for multiple
frequencies, arbitrary transmitter and receiver heights etc).

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