11institutetext: IMCCE, Observatoire de Paris, CNRS, Université PSL, Sorbonne
Université, Université de Lille, 77 av. Denfert-Rochereau, 75014 Paris, France
11email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext: Planetary Environment and
Asteroid Resource Laboratory, Origin Space Co. Ltd., 5 Shihua Rd, 518048
Shenzhen, China
22email<EMAIL_ADDRESS>
# Accuracy of meteor positioning from space- and ground-based observations

Hongru Chen 1122 Nicolas Rambaux 11 Jérémie Vaubaillon 11

###### Abstract

Aims. The knowledge of the orbits and origins derived from meteors is
important for the study of meteoroids and of the early solar system. With an
increase in nano-satellite projects dedicated to Earth observations or
directly to meteor observations (e.g., the Meteorix CubeSat), we investigate
the stereoscopic measurement of meteor positions using a pair of cameras, one
deployed in space and one on the ground, and aim to understand the accuracy
and the main driving factors. This study will reveal the requirements for
system setups and the geometry favorable for meteor triangulation.

Methods.  This Letter presents the principle of the stereoscopic measurement
from space and the ground, and an error analysis. Specifically, the impacts of
the resolutions of the cameras, the attitude and orbit determination accuracy
of the satellite, and the geometry formed by the moving target and observers
are investigated.

Results. To reach a desirable positioning accuracy of 1 km it is necessary to
equip the satellite with high-accuracy sensors (e.g., star tracker and GPS
receiver) to perform fine attitude and orbit determination. The best accuracy
can occur when the the target is at an elevation of $30^{\circ}$ with respect
to the ground station.

###### Key Words.:

meteorites, meteors, meteoroids – space vehicles – techniques: image
processing – methods: analytical

##  1 Introduction

Meteoroids are remnants of the early solar system. When a meteoroid enters
Earth’s atmosphere, it generates a luminous trace known as a meteor.
Determining meteor trajectories allows us to determine the origin of the
meteoroid (e.g., parent body and age, Drummond, 1981; Olsson-Steel, 1986;
Jenniskens, 2006; Šegon et al., 2017; Guennoun et al., 2019; Ryabova et al.,
2019). In addition, knowledge of meteoroid flux density can constrain
environment models, such as the IMEM2, which helps us to plan safe space
activities in the entire solar system (Divine, 1993; Lemcke, 1997; McBride,
1997; Soja et al., 2019). Knowledge of the flux density near Earth is
especially important for planetary defense. Many ground-based tracking
networks (e.g., FRIPON, CAMS, DFN, EDMOND, and AMOS) have been developed with
the purpose of determining the flux density (Colas et al., 2014, 2018;
Rudawska et al., 2015; Tóth et al., 2015; Day et al., 2016; Jenniskens et al.,
2016). Moreover, space-borne observation of meteors has recently been
investigated. The space-proof camera SPOSH was developed for meteor detection
(Oberst et al., 2011). Two projects have been launched with the purpose of
observing meteors from space, namely the SCUBE CubeSat and the METEOR
experiment on the International Space Station (ISS) (Ishimaru et al., 2014;
Arai et al., 2019). The CubeSat project Meteorix is planned for detecting
meteors and re-entering space debris (Rambaux et al., 2018, 2019).

As a meteor is a transient event, the reconstruction of its orbit generally
requires observations from distributed stations. General methods for meteor
triangulation have been well documented (e.g., Wray, 1967). Scenarios and
algorithms that employ multiple images taken from space only or from the
ground only have been intensively studied (Öpik & Singer, 1959; Bronshtehn,
1981; Ceplecha, 1987; Borovicka, 1990; Gural, 2012; Hajduková et al., 2017;
Egal et al., 2017; Vida et al., 2020; Sansom et al., 2019; Chen et al., 2020).
However, a combination of space- and ground-based observations has not yet
been explored. With the establishment of country- to continental-scale meteor
networks and the emergence of space-borne meteor observations, the chance to
capture meteors from both the ground and space is increasing with time.
Moreover, any Earth observation satellite can participate in meteor
observations in conjunction with ground stations; the number of Earth
observation satellite is increasing thanks to the establishment of new
satellite constellations (e.g., the Starlink project). In particular, the
future Meteorix can work with any station of ground networks (e.g., FRIPON,
CAMS, and DFN) to enable and enhance meteor orbit determination. The long
baseline between the satellite and the ground station seems favorable in terms
of stereoscopic measurements. Nevertheless, the positioning accuracy is
subject to the geometry formed by the moving target and observers; the setup
of the systems, such as the attitude and orbit determination accuracy of the
satellite; and image resolution.

This Letter presents an error analysis against the above-mentioned factors,
and shows the requirements for system setups and optimal geometry. After
briefly recalling the space- and ground-based meteor observations (section 2),
we present the principle of stereoscopic measurement and methods of deriving
position errors for the scenario of interest (section 3), the results showing
the influences of different error sources as well as the necessary system
setup (section 4), and a discussion on the impact of the observation geometry
(section 5).

##  2 Basic observation settings

Concerning the space-borne observation, we refer to the design of the Meteorix
project. The CubeSat is equipped with a camera and a detection chain that
features near real-time detection at low power consumption (Lacassagne et al.,
2016). The camera is placed at the end of the 3U CubeSat and pointed toward
the nadir. The science orbit is set to a sun-synchronous orbit, which is the
most common orbit that maximizes launch opportunities. It is assumed that the
orbit is nearly circular with an altitude around 500 km. The period is 94.7
min. The local time of the ascending node is set to 10:30. Consequently, the
night time lasts around 32 min per orbit. It has been estimated that this
setup allows a detection of around 100 meteors in one year (Rambaux et al.,
2019). The selected image sensor has 2048$\times$2048 pixels, and the camera
has a field of view of 40∘. For the ground-based observation, we adopt the
specification of a current FRIPON station: field of view of 180∘ (like many
other all-sky networks, such as CAMS and AMOS), resolution of roughly
1000$\times$1000 pixels, and axis of the camera pointed toward the zenith.

The usual observable altitude of meteors lies between 80 and 120 km (Ceplecha
et al., 1998). The speed of a meteor with respect to the ground is in the
range from 11 to 72 km/s. Considering a frame rate of 10 fps, to capture the
motion of the slowest meteors the position accuracy should be better than 1.1
km.

##  3 Space and ground stereoscopic measurement

###  3.1 Principle

With only one camera the direction from the camera to the target can be
measured, while the distance cannot be. With simultaneous observations by
stations distributed in space (e.g., Meteroix CubeSat) and on the ground
(e.g., FRIPON), the distance between the target and the two stations as well
as the three-dimensional coordinates can be determined, as depicted in Fig. 1.
Without loss of generality, the coordinate system is chosen such that the
origin, $O$, coincides with the location of the ground station, the $x$-axis
points to the CubeSat, the $y$-axis is perpendicular to the $x$-axis and
aligned with the direction to the meteor, and the $z$-axis completes the
right-handed coordinate system. Let
$\left[x_{1},y_{1},z_{1}\right]^{\text{T}}$ denote the coordinate of the
CubeSat. The true coordinate will be $\left[x_{1},0,0\right]^{\text{T}}$. Let
$\left[x_{m},y_{m},z_{m}\right]^{\text{T}}$ denote the coordinate of the
targeted meteor, and $\bf{i}$, $\bf{j}$, and $\bf{k}$ denote the unit vector
along the $x$-, $y$-, and $z$-axis, respectively. The position vector of the
meteor is

${\bf{r}}={\bf i}x_{m}+{\bf j}y_{m}+{\bf k}z_{m}.$ (1)

Let $\varphi_{1}$ and $\varphi_{2}$ represent angles between the directions
from the observers to the meteor and the baseline between the two observers,
as depicted in Fig. 1. The direction from an observer to the target can be
derived from the two-dimensional coordinates on the image. The coordinates of
the meteor can be computed from

$\begin{gathered}x_{m}=\frac{{{x_{1}}}}{{C+1}},\hfill\\\
y_{m}=\frac{{{x_{1}}\tan{\varphi_{2}}}}{{C+1}},\hfill\\\ z_{m}=0,\hfill\\\
\end{gathered}$ (2)

where

$C=\frac{\tan{\varphi_{2}}}{\tan{\varphi_{1}}}.$ (3)

Figure 1: Stereoscopic measurement by a CubeSat and a ground station, and the
defined coordinate system.

###  3.2 Measurement errors

The attitude and orbit determination errors of the CubeSat and image
resolutions will translate into errors of coordinates of the meteor.
Differentiating Eq.(1) yields

$\delta{\bf{r}}={\bf i}\delta x_{m}+x_{m}\delta{\bf i}+{\bf j}\delta
y_{m}+y_{m}\delta{\bf j}+{\bf k}\delta z_{m}+z_{m}\delta{\bf k}.$ (4)

Considering only in-plane (i.e., the $x$-$y$ plane) errors, orbit
determination error $\delta y_{1}$ along ${\bf j}$ results in a change of the
reference frame, which is expressed as

$\begin{gathered}\delta{\bf i}=\frac{\bf j}{B}\delta y_{1},\hfill\\\
\delta{\bf j}=-\frac{\bf i}{B}\delta y_{1},\hfill\\\ \end{gathered}$ (5)

where $B=x_{1}$, representing the length of the baseline between the two
cameras. For out-of-plane errors the last term of Eq.(4) is zero as $z_{m}$ is
zero. Let $\delta{\bf{r}}$ also be expressed as

$\delta{\bf{r}}={\bf i}\delta r_{i}+{\bf j}\delta r_{j}+{\bf k}\delta r_{k},$
(6)

where

$\begin{gathered}\delta r_{i}=\delta x_{m}-\frac{y_{m}}{B}\delta
y_{1},\hfill\\\ \delta r_{j}=\delta y_{m}+\frac{x_{m}}{B}\delta
y_{1},\hfill\\\ \delta r_{k}=\delta z_{m}.\hfill\end{gathered}$ (7)

Differentiating $x_{m}$ and $y_{m}$ in Eq. (2) with respect to the variables
yields

$\begin{gathered}\delta
x_{m}=\frac{1}{{C+1}}\delta{x_{1}}+\frac{{B\tan{\varphi_{2}}}}{{{{(C+1)}^{2}}{{\sin}^{2}}{\varphi_{1}}}}\delta{\varphi_{1}}-\frac{{B{{\sec}^{2}}{\varphi_{2}}}}{{{{(C+1)}^{2}}\tan{\varphi_{1}}}}\delta{\varphi_{2}},\hfill\\\
\delta
y_{m}=\frac{{\tan{\varphi_{2}}}}{{C+1}}\delta{x_{1}}+\frac{{B{{\tan}^{2}}{\varphi_{2}}}}{{{{(C+1)}^{2}}{{\sin}^{2}}{\varphi_{1}}}}\delta{\varphi_{1}}+\frac{{B{{\sec}^{2}}{\varphi_{2}}}}{{{{(C+1)}^{2}}}}\delta{\varphi_{2}}.\hfill\\\
\end{gathered}$ (8)

The direction error $\delta\varphi_{1}$ is a result of in-plane image
resolution, $\delta\varphi^{\parallel}_{res1}$; the in-plane attitude
determination error, $\delta\varphi^{\parallel}_{att}$; and the position error
of the CubeSat along the $y$-axis, which is expressed as

$\delta\varphi_{1}=\delta\varphi^{\parallel}_{res1}+\delta\varphi^{\parallel}_{att}+\frac{1}{B}\delta
y_{1}.$ (9)

Similarly, $\delta\varphi_{2}$ is extended to

$\delta\varphi_{2}=\delta\varphi^{\parallel}_{res2}-\frac{1}{B}\delta y_{1},$
(10)

where $\delta\varphi^{\parallel}_{res2}$ represents the in-plane direction
error due to the image resolution of the ground station.

It is tricky to determine the offset $\delta z_{m}$ along ${\bf k}$ when the
measured directions from the cameras to the target do not intersect in three-
dimensional space due to errors. Let $\delta z_{I}$ and $\delta z_{II}$ denote
the out-of-plane errors attributed to the CubeSat and the ground station,
respectively. The $z$ coordinate error $\delta z_{m}$ can be approximated by

$\delta z_{m}=\frac{\delta z_{I}+\delta z_{II}}{2}.$ (11)

Here $\delta z_{I}$ is a result of the out-of-plane image resolution,
$\delta\varphi^{\perp}_{res1}$; attitude determination error,
$\delta\varphi^{\perp}_{att}$; and orbit determination error, $\delta z_{1}$,
which is expressed as

$\delta
z_{I}=\frac{{BC}}{{(C+1)\cos{\varphi_{1}}}}(\delta{\varphi^{\perp}_{res1}}+\delta{\varphi^{\perp}_{att}})+{\delta
z_{1}},$ (12)

and $\delta z_{II}$, due to the out-of-plane image resolution of the ground
station, $\delta\varphi^{\perp}_{res2}$, is expressed as

$\delta
z_{II}=\frac{B}{{(C+1)\cos{\varphi_{2}}}}\delta{\varphi^{\perp}_{res2}}.$ (13)

It can be reasonably assumed that in-plane and out-of-plane direction errors
vary independently. Uncertainties of errors have the relationship,

$\begin{gathered}\sigma\varphi^{\parallel}_{res1}=\sigma\varphi^{\perp}_{res1}=\left(\frac{{{\theta_{1}}}}{{px_{1}}}\right),\hfill\\\
\sigma\varphi^{\parallel}_{res2}=\sigma\varphi^{\perp}_{res2}=\left(\frac{{{\theta_{2}}}}{{px_{2}}}\right),\hfill\\\
\sigma{\varphi^{\parallel}_{att}}=\sigma{\varphi^{\perp}_{att}},\hfill\\\
\end{gathered}$ (14)

where $\sigma$ indicates the 1$\sigma$ uncertainty of the variable following
it, $\theta_{1}$ and $\theta_{2}$ respectively denote the fields of views, and
$px_{1}$ and $px_{2}$ respectively denote the pixels (in a line) of the
CubeSat and the ground station. It is also assumed that distribution of orbit
determination error in space is homogeneous:

$\sigma x_{1}=\sigma y_{1}=\sigma z_{1}.$ (15)

The position uncertainty $\sigma r$ is defined as the root sum of variances of
the three position components, which is expressed as

$\sigma r=\sqrt{{\sigma^{2}}{r_{i}}+{\sigma^{2}}{r_{j}}+{\sigma^{2}}{r_{k}}},$
(16)

where $\sigma^{2}$ indicates the variance of the variable following it.
According to Eq.(7-13) and given that the error sources are uncorrelated,
${\sigma^{2}}{r_{i}}$, ${\sigma^{2}}{r_{j}}$, and ${\sigma^{2}}{r_{k}}$ can be
computed from

$\displaystyle{\sigma^{2}}{r_{i}}={}$
$\displaystyle{\left({\frac{1}{{C+1}}}\right)^{2}}{\sigma^{2}}{x_{1}}+{\left({\frac{{B\tan{\varphi_{2}}}}{{{{(C+1)}^{2}}{{\sin}^{2}}{\varphi_{1}}}}}\right)^{2}}({\sigma^{2}}\varphi_{res1}^{||}+{\sigma^{2}}\varphi_{att}^{||})$
(17)
$\displaystyle+{\left({\frac{{B{{\sec}^{2}}{\varphi_{2}}}}{{{{(C+1)}^{2}}\tan{\varphi_{1}}}}}\right)^{2}}{\sigma^{2}}\varphi_{res2}^{||}$
$\displaystyle+{\left({\frac{{\tan{\varphi_{2}}}}{{{{(C+1)}^{2}}{{\sin}^{2}}{\varphi_{1}}}}+\frac{{{{\sec}^{2}}{\varphi_{2}}}}{{{{(C+1)}^{2}}\tan{\varphi_{1}}}}-\frac{{\tan{\varphi_{2}}}}{{(C+1)}}}\right)^{2}}{\sigma^{2}}{y_{1}},$
$\displaystyle{\sigma^{2}}{r_{j}}={}$
$\displaystyle{\left({\frac{{\tan{\varphi_{2}}}}{{C+1}}}\right)^{2}}{\sigma^{2}}{x_{1}}+{\left({\frac{{B{{\tan}^{2}}{\varphi_{2}}}}{{{{(C+1)}^{2}}{{\sin}^{2}}{\varphi_{1}}}}}\right)^{2}}({\sigma^{2}}\varphi_{res1}^{||}+{\sigma^{2}}\varphi_{att}^{||})$
(18)
$\displaystyle+{\left({\frac{{B{{\sec}^{2}}{\varphi_{2}}}}{{{{(C+1)}^{2}}}}}\right)^{2}}{\sigma^{2}}\varphi_{res2}^{||}$
$\displaystyle+{\left({\frac{{{{\tan}^{2}}{\varphi_{2}}}}{{{{(C+1)}^{2}}{{\sin}^{2}}{\varphi_{1}}}}-\frac{{{{\sec}^{2}}{\varphi_{2}}}}{{{{(C+1)}^{2}}}}+\frac{1}{{(C+1)}}}\right)^{2}}{\sigma^{2}}{y_{1}},$
$\displaystyle{\sigma^{2}}{r_{k}}={}$
$\displaystyle{\left({\frac{{BC}}{{2(C+1)\cos{\varphi_{1}}}}}\right)^{2}}({\sigma^{2}}\varphi_{res1}^{\bot}+{\sigma^{2}}\varphi_{att}^{\bot})$
(19)
$\displaystyle+{\left({\frac{B}{{2(C+1)\cos{\varphi_{2}}}}}\right)^{2}}{\sigma^{2}}\varphi_{res2}^{\bot}+\frac{1}{4}{\sigma^{2}}{z_{1}}.$

###  3.3 Reduction of geometries

Figure 2 depicts the geometry of space- and ground-based observations.
Altitudes of the CubeSat, $h_{sat}$, and meteors, $h_{met}$, vary within small
ranges (around 500 km and between 80 and 120 km, respectively). In the
following analyses, $h_{sat}$ is set to 500 km and $h_{met}$ 100 km to
represent a general situation. The observation geometry can then be defined by
the directions from the ground station to the CubeSat and the meteor. For
convenience, the direction of the CubeSat with respect to the zenith of the
ground station, $\alpha$, and the direction of the observed meteor from the
CubeSat with respect to the camera axis, $\beta$, are chosen to generalize the
observing geometry of interest. As $\alpha$ and $\beta$ are limited by the
fields of view of the two observers, possible observation geometries can thus
be represented by an $\alpha$-$\beta$ map. Taking into account the round
surface of the Earth (where the radius of the Earth $R_{\oplus}=6371$ km is
used), $B$, $\varphi_{1}$, and $\varphi_{2}$ entering in Eq. (17-19) are
computed as follows. First,

$B=-{R_{\oplus}}\cos\alpha+\sqrt{{{({R_{\oplus}}+{h_{sat}})}^{2}}-{R_{\oplus}}^{2}{{\sin}^{2}}\alpha}.$
(20)

The triangle formed by the observers and target does not necessarily lie on
the vertical plane. The angle $\varphi_{1}$ is actually within an interval
expressed as

${\varphi_{1}}\in\left[{\arcsin\left({\frac{{{R_{\oplus}}\sin(\pi-\alpha)}}{{{R_{\oplus}}+{h_{sat}}}}}\right)-\beta,\arcsin\left({\frac{{{R_{\oplus}}\sin(\pi-\alpha)}}{{{R_{\oplus}}+{h_{sat}}}}}\right)+\beta}\right].$

As $\beta$ is to be varied in the range of the field of view (e.g., -20∘ to
20∘ in this work), which can cover situations at two ends of the intervals,
the problem can be reasonably simplified by fixing $\varphi_{1}$ for a given
$\beta$ to

${\varphi_{1}}=\arcsin\left({\frac{{{R_{\oplus}}\sin(\pi-\alpha)}}{{{R_{\oplus}}+{h_{sat}}}}}\right)-\beta.$
(21)

Then

${\varphi_{2}}=\arcsin\left(\frac{{{L_{1}}}}{{{L_{2}}}}\sin{\varphi_{1}}\right),$
(22)

where the distances from the meteor to the CubeSat, $L_{1}$, and the ground
station, $L_{2}$, are computed from

$\begin{gathered}{L_{1}}=({R_{\oplus}}+{h_{sat}})\cos\beta-\sqrt{{{({R_{\oplus}}+{h_{met}})}^{2}}-{{({R_{\oplus}}+{h_{sat}})}^{2}}{{\sin}^{2}}\beta},\hfill\\\
{L_{2}}=\sqrt{{L_{1}}^{2}+{B^{2}}-2{L_{1}}B\cos{\varphi_{1}}}.\hfill\end{gathered}$
(23)

Figure 2: Simplified geometry for meteor observation.

##  4 Results

In order to provide a guideline for system setups, we look at the influences
of the orbit determination error (e.g., $\sigma x_{1}$), attitude
determination error $\sigma{\varphi_{att}}$, image error of the CubeSat
$\theta_{1}/px_{1}$, and image error of the ground station
$\theta_{2}/px_{2}$. The baseline setup assumes $\sigma x_{1}=10$ km,
corresponding to coarse orbit determination based on TLE data only (Smith,
2019), $\sigma{\varphi_{att}}=4^{\circ}$, corresponding to a coarse attitude
determination based on magnetometer and sun sensors only (KU Leuven, 2018),
$\theta_{1}=40^{\circ}$, $px_{1}=2048$, $\theta_{2}=180^{\circ}$, and
$px_{2}=1000$, corresponding to the baseline design of Meteorix and FRIPON.
Then $\alpha$ ranges from 0 to 90∘, and $\beta$ from -20∘ to 20∘.

The position accuracy $\sigma r$ computed from Eq. (16-19) is presented in
Fig. 3 as a function of $90^{\circ}-\alpha$ (i.e., elevation of the CubeSat)
and $\beta$. The four panels correspond to the cases considering orbit
determination error only (upper left), attitude determination error only
(upper right), image error of the CubeSat only (lower left), and image error
of the ground station only (lower right). The lower left panel shows that the
CubeSat image resolution is good enough to not cause $\sigma r>$ 0.5 km,
except in the singular-geometry region (see next section for discussion). The
upper right panel shows that coarse attitude estimation results in a $\sigma
r$ around 40 km for most observation geometries. Therefore, the CubeSat should
perform fine attitude estimation where a star tracker is necessary. In this
case, $\sigma{\varphi_{att}}=0.04^{\circ}$ is expected (KU Leuven, 2018), and
the contribution of $\sigma{\varphi_{att}}$ to $\sigma r$ decreases to 0.4 km.
Even with this improvement the resulting $\sigma r$ can still go beyond 15 km,
due to the orbit determination error (see the upper left panel). It is
recommended to perform fine orbit determination as well. A GPS receiver can
enable a fine orbit determination at an accuracy of 0.1 km (Smith, 2019),
which will reduce the error contribution to 0.15 km for most geometries. The
resulting $\sigma r$ can go up to 2 km for some situations, given the effect
of the image resolution of the ground station. To widely achieve a desired
$\sigma r$ of 1.1 km, it is recommended to increase the image resolution of
the ground station, for instance to 2048$\times$2048 pixels. With an upgraded
resolution, the minimum $\sigma r$ will go down to 1 km for most geometries.
The position uncertainty $\sigma r$ following the recommended setups is shown
in Fig. 4. The pattern is similar to the lower left panel of Fig. 3, which
means the contribution of $\sigma\varphi_{2}$ due to the image resolution of
ground station is dominant.

Figure 3: Positioning uncertainty (color levels) contributed by baseline orbit
and attitude determination errors (upper panels) of the CubeSat, and baseline
image resolution of the CubeSat and the ground station (lower panels). Figure
4: Positioning uncertainty (color levels) for the recommended system setup,
and elevation (gray lines) of the meteor with respect to the ground station.

##  5 Discussion

Figure 4 also displays the elevation of the meteor with respect to the ground
station, which is derived from $\alpha$ and $\beta$. The shaded area where the
meteor is below the horizon is not a valid area for observation, and thus is
excluded from our discussion. In the figure, an elevation $>90^{\circ}$ (which
is not a common usage) indicates that the meteor and the CubeSat are on
different sides of the zenith of the ground station.

Figure 4 shows that $\sigma r$ strongly depends on the elevation of the
meteor. Following the recommended setup, $\sigma r<2$ km is widely achieved
except for the elevation interval $\left[65^{\circ},90^{\circ}\right]$
(labeled case 2 in the figure). The corresponding geometry is referred to as
the singular geometry. When the meteor is at an elevation of around
$30^{\circ}$ (labeled case 1) and also captured by the CubeSat camera, the
geometry is optimal with $\sigma r$ ranging from 0.5 to 0.8 km, which depends
on the observing directions from the CubeSat. Another optimal region appears
in the elevation interval $\left[110^{\circ},140^{\circ}\right]$ (labeled case
3), which corresponds to the commonly used elevation
$\left[40^{\circ},70^{\circ}\right]$ with the CubeSat observing from the other
side of the zenith.

Figure 5 schematically visualizes the singular and optimal geometries revealed
in Fig. 4. With only one camera, the error (green band) resulting from the
angular size of a pixel is infinite along the observing direction (green
dotted line), which should be constrained by the observation from another
direction. If two directions are aligned, the error is still infinite. Because
of the limited field of view of a CubeSat, it will be roughly on top of the
target when capturing the target. When the meteor is near the zenith of the
ground station (case 2 in Fig. 5) and the CubeSat observes it from the top,
the error along the observing direction is not effectively constrained.
Therefore, the resulting radius of the error zone (i.e., the intersecting
green zone) is large. This explains the singularity associated with the meteor
elevation interval $\left[65^{\circ},90^{\circ}\right]$. When the elevation of
the meteor is low ($30^{\circ}$ in case 1 and $40^{\circ}$ in case 3), the
included angle between the two observing directions is close to 90∘, leading
to effective triangulation and small errors. Even though the error across the
observing direction is proportional to the square of the distance, and thus
large for low-elevation observations, the resulting range of errors, which is
mainly contributed by the error along the observing direction, is still
minimized for low elevations.

Figure 5: Intersecting error zones depending on the geometry.

##  6 Conclusions

To achieve a desirable positioning accuracy of 1 km, the CubeSat should bring
high-accuracy sensors (e.g., star tracker and GPS receiver) to perform fine
attitude (i.e., accuracy $\sim 0.05^{\circ}$) and orbit (i.e., accuracy $\sim$
100 m) determination. In addition, the image resolution of FRIPON is
recommended to upgrade to 2048$\times$2048 pixels. The performance of meteor
triangulation is best (i.e., with smallest position error) when the meteor
elevation is around $30^{\circ}$. With the recommended system setup, as long
as a meteor is captured by both the CubeSat and the ground station, and at an
elevation $<65^{\circ}$ with respect to the ground station, the position and
the orbit of the meteor can be determined.

###### Acknowledgements.

We acknowledge the ESEP (Exploration Spatiale des Environnements Planétaires)
postdoctoral fellowship, DIM ACAV+ Région Île-de-France, Janus CNES, and IDEX
Sorbonne Universités for funding this research.

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11,700 km Transmission at 4.8 bit/4D-sym via Four-dimensional Geometrically-
shaped Polarization-Ring-Switching Modulation

Sjoerd van der Heide(1), Bin Chen(1)(3), Menno van den Hout(1), Gabriele
Liga(1),

Ton Koonen(1), Hartmut Hafermann(2), Alex Alvarado(1), and Chigo Okonkwo(1)

(1)Department of Electrical Engineering, Eindhoven University of Technology,
The Netherlands.

(2)Mathematical and Algorithmic Sciences Lab, Paris Research Center, Huawei
Technologies France SASU, France.

(3)School of Computer Science and Information Engineering, Hefei University of
Technology, China<EMAIL_ADDRESS>
###### Abstract

Using a novel geometrically-shaped four-dimensional modulation format, we
transmitted $11\times 200$ Gbit/s DWDM at 4.8 bit/4D-sym over 7,925 km and
11,700 km using EDFA-only and hybrid amplification, respectively. A reach
increase of 16% is achieved over PM-8QAM.

Keywords: Advanced Modulation, Coding and Multiplexing. Transmission
experiments for long haul, core and metro applications including data-center
interconnect

##  1 Introduction

The non-linear Shannon limit for single-mode transmission systems remains
despite the strong growth in traffic. In recent years, signal shaping via
probabilistic shaping (PS) or geometric shaping (GS) have been demonstrated to
close the gap to the non-linear Shannon limit by employing non-uniform
probability or non-equidistant constellation points whilst maximising
achievable information rate (AIR), respectively. Both techniques have been
extensively demonstrated with PS shown to require complex coding techniques,
whilst GS only requires straightforward modifications of the mapper and
demapper.

Multidimensional constant modulus modulation formats have been shown to
minimize the non-linear interference noise by minimizing the signal power
variations [1, 2]. Recently, the  four-dimensional 64-ary polarization-ring-
switching (4D-64PRS) format was introduced in [3], where the 4D coordinates
and labeling were jointly optimized. Numerical results in [3] show that
4D-64PRS outperforms other modulation formats with spectral efficiency (SE) of
6 bit/4D-sym, including polarization-multiplexed (PM)-8QAM, 4D-64SP-12QAM [4]
and 4D-2A8PSK [2]. Results were presented in both linear and non-linear
channel for a bit-interleaved coded modulation (BICM) system [3]. 4D-64PRS was
reported to yield over 0.8 dB better sensitivity than PM-8QAM by maximizing
generalized mutual information (GMI) [3].

In this work, we experimentally show a 16% reach increase over PM-8QAM, which
is considered a viable candidate for beyond 200G per optical carrier ultra
long-haul transmission [5, 6, 7]. Using a novel four-dimensional 4D-64PRS
modulation format, both EDFA-only and hybrid EDFA plus Raman amplified long-
haul transmission scenarios are evaluated. 4D-64PRS is shown to outperform
both PM-8QAM and 6b4D-2A8PSK in an 11-channel dense wavelength division
multiplexing (DWDM), net 200 Gbit/s/channel standard single-mode fiber (SSMF)
recirculating loop transmission experiment. It is shown that pre-FEC reach
gains are preserved for post-FEC.

##  2 Experimental transmission setup

Fig. 1: Experimental setup. Inset (a) shows a Stokes-space representation of
the 4D-64PRS modulation format, indicating 64 constant-modulus points in 16
distinct SOPs. Inset (b) shows the received spectrum after 106 circulations
(8,000 km) of EDFA-only amplification. Note that the CUT is depicted in the
center position but is tested in all 11 positions in the experiment.

(a)

(b)

(c)

Fig. 2: Experimental results using EDFA-only amplification. (a) Average GMI
per channel versus total launch power after 7,550 km. (b) Per-channel
performance versus BERs (top) and GMIs (bottom) measured for all 11 channels
individually after 7,925 km showing BERs below the FEC threshold $4\cdot
10^{-2}$[2] (top) and GMIs above 5.1 bit/4D[8] (bottom) for all 11 channels
for 4D-64PRS. (c) BER versus transmission distance for the center channel at
launch power of 9.5 dBm.

Fig. 1 depicts the experimental transmission setup. The transmitted signal is
modulated using eithr PM-8QAM, 6b4D-2A8PSK [2], or 4D-64PRS [3]. Fig. 1 inset
(a) shows the Stokes representation of the 4D-64PRS format, which exhibits 16
distinct SOPs and consists of 64 4D-symbols. Each distinct SOP represents four
4D-symbols on the Poincaré sphere indicating constant modulus in 4D. Sequences
containing 216 symbols are shaped using a root-raised-cosine (RRC) filter with
1% roll-off at 41.79 GBd, generated offline and uploaded to a 100-GSa/s
digital-to-analog converter (DAC). The positive ends of the differential DAC
outputs are connected to the optical-multi-format transmitter (OMFT) which
consists of an external cavity laser (ECL), a dual-polarization IQ-modulator
(DP-IQM), an automatic bias controller (ABC) and RF-amplifiers. The CUT, which
can be defined at any of the 11 tested C-band channels, is modulated by the
OMFT and subsequently amplified. The loading channels are provided by the
negative outputs of the DAC and modulated on the tones provided by 10 ECLs
using a DP-IQM. These loading channels are amplified, split into even and odd,
decorrelated by 10,200 (50 m) and 40,800 symbols (200 m), and multiplexed
together with the CUT on a 50-GHz grid using an optical tunable filter (OTF).
Bandwidth limitations due to transmitter electronics are compensated for
partly through digital filters and partly using the OTF as proposed in [9].

The 11-channel 50-GHz-spaced DWDM signal is amplified and through an acousto-
optical modulator (AOM) enters the recirculating loop which consists of a
loop-synchronous polarization scrambler (LSPS), a 75-km span of SSMF, an
erbium-doped fiber amplifier (EDFA), an AOM and an OTF used for gain
flattening. Fig. 1 inset (b) shows the optical spectrum after 106
circulations, which corresponds to 8,000 km of transmission using only EDFA-
amplification. Optionally, a hybrid amplification scheme can be used by adding
a 750 mW 1480 nm Raman pump in a backward configuration. Part of the output of
the recirculating loop is measured by a power meter, part is analyzed using an
optical spectrum analyzer (OSA) and the rest is amplified, filtered by a
wavelength selective switch (WSS) and digitized by a coherent receiver
consisting of a local oscillator (LO), a 90-degree hybrid, four balanced
photo-diodes and an 80-GSa/s analog-to-digital converter (ADC). Offline
digital signal processing (DSP) includes front-end correction, frequency-
offset compensation, chromatic dispersion compensation, multiple-input
multiple-output (MIMO) equalization with in-loop blind phase search (BPS),
error counting and GMI evaluation.

##  3 Results

(a)

(b)

(c)

Fig. 3: Experimental results using a hybrid of EDFA and Raman amplification.
(a) Average GMI per channel versus total launch power after 12,530 km. (b)
Per-channel performance versus BERs (top) and GMIs (bottom) measured for all
11 channels individually after 11,700 km showing BERs below the FEC threshold
$4\cdot 10^{-2}$[2] (top) and GMIs above 5.1 bit/4D[8]. (c) BER versus
transmission distance for the center channel at launch power of 6.5 dBm.

Transmission scenarios using EDFA-only amplification (Fig. 2) and a hybrid of
EDFA and Raman (Fig. 3) are evaluated. Fig. 2a shows that a total launch power
of 9.5 dBm maximizes the average GMI per channel resulting in a 0.21 bit/4D
increase for 4D-64PRS with respect to PM-8-QAM. At this optimal launch power,
we demonstrate transmission below the pre-FEC-threshold of $4\cdot 10^{-2}$
and above the GMI threshold of 5.1 bit/4D, thus enabling error-free 7,925 km
transmission of net 200 Gbit/s per channel after 25.5% overhead and for all 11
channels. The GMI threshold 5.1 bit/4D (0.85 NGMI) is based on a spatially-
coupled type LDPC code[8] and the corresponding BER threshold of $4\cdot
10^{-2}$ is derived in [2].

A reach increase of 16% (+1,150 km), for 4D-64PRS compared to PM-8QAM is shown
for a pre-FEC bit error rate (BER) of $4\cdot 10^{-2}$ in Fig. 2c. Moreover,
the 16% reach increase is preserved for the post-FEC gain using an off-the-
shelf DVB-S2 low-density parity-check code (LDPC) with 25% overhead and code
length $n=64800$. In addition to the gains shown versus PM-8QAM, 4D-64PRS is
also shown to outperform 6b4D-2A8PSK. BER and GMI calculations were done using
over 42 million bits for all plotted data points. Note that 63 million bits
were used for LDPC decoding.

For the hybrid amplification scenario, the relative gains are similar to the
EDFA-only case as shown in Fig. 3. A GMI-increase of 0.21 bit/4D at 7 dBm
launch power is observed in Fig. 3a. Further measurements were carried out at
6.5 dBm total launch power, near the optimal launch power. Fig. 3b shows all
channels were able to transmit over 11,700 km successfully. Fig. 3c shows a
pre-FEC reach increase of 17% and a post-FEC reach increase of 16%, which is
consistent with the EDFA-only amplification scenario.

##  4 Conclusions

The novel geometrically-shaped 4D-64PRS modulation format is experimentally
compared to other notable 6 bit/4D modulation formats such as PM-8QAM and
6b4D-2A8PSK, with 4D-64PRS outperforming both of them. Experimental results
show a reach increase of 16% at 7,925 km for SSMF and EDFA-only amplification
and 11,700 km of using hybrid of both EDFA and Raman amplification. LDPC
decoding is performed on the experimental data to demonstrate a post-FEC 16%
reach increase at a net SE of 4.8 bit/4D.

_
Partial funding from the Dutch NWO Gravitation Program on Research Center for
Integrated Nanophotonics (Grant Number 024.002.033). Research supported by
Huawei France through the NLCAP project. The work of A. Alvarado is supported
by the Netherlands Organisation for Scientific Research (NWO) via the VIDI
Grant ICONIC (project number 15685). Fraunhofer HHI is acknowledged for
providing their Optical-Multi-Format Transmitter._

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