# The tight bound for the strong chromatic indices of claw-free subcubic
graphs ††thanks: Supported by NSFC 11771080.

Yuquan Lin and Wensong Lin111Corresponding author. E-mail address:
<EMAIL_ADDRESS>
School of Mathematics, Southeast University, Nanjing 210096, P.R. China

###### Abstract

Let $G$ be a graph and $k$ a positive integer. A strong $k$-edge-coloring of
$G$ is a mapping $\phi:E(G)\to\\{1,2,\dots,k\\}$ such that for any two edges
$e$ and $e^{\prime}$ that are either adjacent to each other or adjacent to a
common edge, $\phi(e)\neq\phi(e^{\prime})$. The strong chromatic index of $G$,
denoted as $\chi^{\prime}_{s}(G)$, is the minimum integer $k$ such that $G$
has a strong $k$-edge-coloring. Lv, Li and Zhang [Graphs and Combinatorics 38
(3) (2022) 63] proved that if $G$ is a claw-free subcubic graph other than the
triangular prism then $\chi_{s}^{\prime}(G)\leq 8$. In addition, they asked if
the upper bound $8$ can be improved to $7$. In this paper, we answer this
question in the affirmative. Our proof implies a linear-time algorithm for
finding strong $7$-edge-colorings of such graphs. We also construct infinitely
many claw-free subcubic graphs with their strong chromatic indices attaining
the bound $7$.

Keywords: strong edge coloring; strong chromatic index; claw-free; subcubic
graph.

##  1 Introduction

Let $G=(V(G),E(G))$ be a finite undirected simple graph. For $v\in V(G)$, let
$N(v)=\\{u\in V(G):uv\in E(G)\\}$ denote the open neighborhood of $v$ and
$d(v)=|N(v)|$ be the degree of $v$. Let $\Delta(G)=\max\limits_{v\in
V(G)}d(v)$ denote the maximum degree of $G$. For convenience, we use the
abbreviation $[1,n]$ for $\\{1,2,\dots,n\\}$, where $n$ is any positive
integer.

Let $e$ and $e^{\prime}$ be two edges of $G$. If $e$ and $e^{\prime}$ are
adjacent to each other, we say that the distance between $e$ and $e^{\prime}$
is $1$, and if they are not adjacent but both of them are adjacent to a common
edge, we say they are at distance $2$. Given a positive integer $k$, a strong
$k$-edge-coloring of $G$ is a mapping $\phi:E(G)\to[1,k]$ such that for any
two edges $e$ and $e^{\prime}$ that are at distance $1$ or $2$,
$\phi(e)\neq\phi(e^{\prime})$. The strong chromatic index of $G$, denoted by
$\chi^{\prime}_{s}(G)$, is the minimum integer $k$ such that $G$ has a strong
$k$-edge-coloring.

The concept of strong edge coloring, first introduced by Fouquet and Jolivet
[10], can be used to model the conflict-free channel assignment in radio
networks [21, 20]. In 1985, Erdős and Nešetřil [7, 8] proposed the following
conjecture about the upper bound of $\chi^{\prime}_{s}(G)$ in term of
$\Delta(G)$, which if true, is the best possible.

###### Conjecture 1.1

(Erdős and Nešetřil [7, 8]) If $G$ is a graph with maximum degree $\Delta(G)$,
then

$\chi^{\prime}_{s}(G)\leq\begin{cases}\begin{array}[]{cl}\dfrac{5}{4}\Delta(G)^{2},&\text{if}\
\Delta(G)\ \text{is even,}\\\
\dfrac{5}{4}\Delta(G)^{2}-\dfrac{1}{2}\Delta(G)+\dfrac{1}{4},&\text{if}\
\Delta(G)\ \text{is odd.}\end{array}\end{cases}$

The conjecture is clearly true for $\Delta(G)\leq 2$. The case $\Delta(G)=3$
was verified by Andersen [1] in 1992, and independently by Horák, Qing, and
Trotter [13] in 1993. Furthermore, if $G$ is a subcubic planar graph then
$\chi^{\prime}_{s}(G)\leq 9$ [16]. For $\Delta(G)=4$, an upper bound of $23$
was proved by Horák [12] in 1990. It was improved to $22$ by Cranston [5] in
2006 and more recently Huang, Santana and Yu [14] obtained the upper bound
$21$. Recall that the conjectured bound is $20$ for $\Delta(G)=4$. Although
the general case when $\Delta(G)=4$ has not been solved yet, the bound $20$
holds for some special cases. In 1990, Faudree, Schelp, Gyárfás and Tuza [9]
proved that $\chi^{\prime}_{s}(G)\leq 4\Delta(G)+4$ for any planar graph $G$,
which implies that Conjecture 1.1 holds for planar graphs with maximum degree
$4$. And this bound for planar graphs with maximum degree $4$ was further
improved to $19$ by Wang, Shiu, Wang and Chen [22] in 2018.

In 2015, Bensmail, Bonamy and Hocquard [2] studied the upper bound on
$\chi^{\prime}_{s}(G)$ in terms of maximum average degree for the class of
graphs with maximum degree $4$, where they gave some sufficient conditions
under which Conjecture 1.1 is true. In 2018, Lv, Li and Yu [17] strengthened
Bensmail, Bonamy and Hocquard’s results. And they proved that for any graph
$G$ with $\Delta(G)=4$, if there are two vertices of degree $3$ whose distance
is at most $4$, then $\chi^{\prime}_{s}(G)\leq 20$.

For graphs with maximum degree $5$, Zang [24] gave the upper bound 37 (recall
that the conjectured bound is 29), which is the only progress as we know. And
for larger $\Delta(G)$, the problem is widely open. In 1997, Molloy and Reed
[19] used probabilistic techniques to prove that $\chi^{\prime}_{s}(G)\leq
1.998\Delta(G)^{2}$ for sufficiently large $\Delta(G)$. And an improvement to
$1.93\Delta(G)^{2}$ was provided by Bruhn and Joos [4] in 2015. Recently,
Bonamy, Perrett and Postle [3] further strengthened this bound to
$1.835\Delta(G)^{2}$. The current best known upper bound is
$1.772\Delta(G)^{2}$ which was showed by Hurley, de Joannis de Verclos and
Kang [15] in 2021.

A graph is called claw-free if it has no induced subgraph isomorphic to the
complete bipartite graph $K_{1,3}$. In 2020, Debski, Junosza-Szaniawski and
Śleszyńska-Nowak [6] presented the following upper bound for the strong
chromatic indices of claw-free graphs.

###### Theorem 1.2

(Debski, Junosza-Szaniawski and Śleszyńska-Nowak [6]) For any claw-free graph
$G$ with maximum degree $\Delta(G)$,
$\chi^{\prime}_{s}(G)\leq\frac{9}{8}\Delta(G)^{2}+\Delta(G)$.

A graph with maximum degree less than or equal to $3$ is called a subcubic
graph. In 2022, Lv, Li and Zhang [18] proved that, for any claw-free subcubic
graph $G$ other than the triangular prism, $\chi^{\prime}_{s}(G)\leq 8$.
Please see Figure 3 for the triangular prism (also called the $3$-prism).
Notice that the $3$-prism is a claw-free cubic graph with its strong chromatic
index being equal to $9$. In the same paper, the authors left the problem
whether this bound can be improved to $7$. This paper solves this problem and
the main result is the following theorem.

###### Theorem 1.3

Let $G$ be a claw-free subcubic graph. If each component of $G$ is not
isomorphic to the triangular prism, then $\chi^{\prime}_{s}(G)\leq 7$.

Remark 1. Recall that when $\Delta(G)=3$, the upper bound in Conjecture 1.1 is
$10$, the upper bound in Theorem 1.2 is $\frac{105}{8}$, and the upper bound
proved by Lv, Li and Zhang [18] is $8$, while our bound in Theorem 1.3 is $7$.

Remark 2. A graph $G_{0}$ on five vertices with strong chromatic index $7$ was
presented in [18] (see Figure 3). This shows the sharpness of the upper bound
$7$. In fact, there are other claw-free subcubic graphs with their strong
chromatic indices being equal to $7$. It is not difficult to verify that the
graph $H_{0}$ shown in Figure 3 has the strong chromatic index $7$. Therefore,
by Theorem 1.3, any claw-free subcubic graph containing $H_{0}$ as a subgraph
has the strong chromatic index $7$. This implies that there are infinitely
many claw-free subcubic graphs with their strong chromatic indices attaining
the upper bound $7$.

Figure 1: The $3$-prism

Figure 2: The graph $G_{0}$

Figure 3: The graph $H_{0}$

Remark 3. We would like to point out that our proof of Theorem 1.3 implies a
linear-time algorithm that can produces a strong $7$-edge-coloring of any
claw-free subcubic graph other than the $3$-prism.

The remainder of this paper is dedicated to the proof of Theorem 1.3 which is
organized as follows. In Section 2, after stating some definitions and
notations, we explain how to order the edges of a connected claw-free subcubic
graph and apply the greedy algorithm to obtain a partial strong $7$-edge-
coloring of the graph. Section 3 consists of a series of lemmas, each of which
shows that a particular partial strong $7$-edge-coloring of a claw-free
subcubic graph can be extended to a strong $7$-edge-coloring of the whole
graph. Finally, Section 4 summarizes our works and suggests some future
research directions.

##  2 Preliminaries and notations

Let $G$ be a connected claw-free subcubic graph not isomorphic to the
$3$-prism. To prove Theorem 1.3, we shall first find a subgraph of $G$ and
construct a strong $7$-edge-coloring of this subgraph, and then try to extend
the coloring to the whole graph $G$. The subgraph we find should have the
property that any two of its edges at distance greater than $2$ must also be
at distance greater than $2$ in $G$. We next introduce some notations and
preliminary facts that we will use in our proofs.

We use $\alpha$, $\beta$, $\gamma$ to denote colors and $\phi$, $\psi$,
$\sigma$ to denote edge colorings. Given two distinct edges $e$ and
$e^{\prime}$ of $G$, we say that $e$ sees $e^{\prime}$ in $G$ if they are
distance $1$ or $2$ apart. An edge coloring of a graph $G$ is good, if it is a
strong edge coloring of $G$ using at most $7$ colors. A good partial coloring
of a graph $G$ is a good coloring $\phi$ of some subgraph $H$ of $G$ such that
$\phi(e)\neq\phi(e^{\prime})$ if $e$ and $e^{\prime}$ see each other in $G$.

Let $\phi$ be a good partial coloring of $G$. We say that $e$ sees a color
$\alpha$ in $\phi$, if $e$ sees an edge $e^{\prime}$ for which
$\phi(e)=\alpha$. For $e\in E(G)$, let $F_{\phi}(e)$ denote the set of colors
that $e$ sees in $\phi$. In addition, let $\bar{E}_{\phi}$ denote the set of
edges in $G$ not already assigned colors by $\phi$ and $\bar{G}_{\phi}$ denote
the subgraph of $G$ induced by $\bar{E}_{\phi}$. For $e\in\bar{E}_{\phi}$, let
$A_{\phi}(e)$ denote the set of colors that $e$ does not see in $\phi$. It is
clear that $A_{\phi}(e)=[1,7]\setminus F_{\phi}(e)$ for any
$e\in\bar{E}_{\phi}$.

In order to construct a good partial coloring of a claw-free subcubic graph
$G$, we first apply the similar method in [23] to order the edges of $G$, and
then use the greedy algorithm to color them in the order, left only a few
particular edges uncolored.

Suppose $S$ is a nonempty subset of $V(G)$. For a vertex $v\in V(G)$, the
distance from $v$ to $S$, denoted by $d_{S}(v)$, is equal to
$\min\limits_{w\in S}\\{d(v,w)\\}$, where $d(v,w)$ is the distance between $v$
and $w$ in $G$. Let $I$ be the maximum distance from a vertex of $G$ to $S$.
For $i=0,1,\dots,I$, let $D_{i}=\\{v\in V(G):\ d(v,S)=i\\}$. A mapping $d_{S}$
from $E(G)$ to nonnegative real numbers is defined as: for any edge $e$ with
two end vertices $u$ and $v$, $d_{S}(e)=\frac{1}{2}(d_{S}(u)+d_{S}(v))$.
Suppose $R=(e_{k_{1}},e_{k_{2}},\dots,e_{k_{m}})$ is an ordering of the edges
of $G$. For any two integers $i$ and $j$ in $[1,m]$, if $i<j$ implies
$d_{S}(e_{k_{i}})\geq d_{S}(e_{k_{j}})$, then we say that the edge ordering
$R$ of $G$ is compatible with the mapping $d_{S}$. It is clear that any edge
joins two vertices that are either in the same $D_{i}$, or one in $D_{i}$ and
the other in $D_{i+1}$ for some $i$, and that each vertex in $D_{i}$ with
$i\geq 1$ is adjacent to at least one vertex in $D_{i-1}$. Let $e=xy$ be an
edge with $d_{S}(x)\leq d_{S}(y)$. If $d_{S}(e)\geq 1$, then $e$ is adjacent
to an edge $e^{\prime}=xz$ with $d_{S}(z)=d_{S}(x)-1$.

We are now ready to produce a good partial coloring of a claw-free subcubic
graph.

###### Lemma 2.1

Let $G$ be a connected claw-free subcubic graph other than the $3$-prism and
$S$ a nonempty subset of $V(G)$. The greedy algorithm, coloring the edges of
$G$ in an order $R$ that is compatible with the mapping $d_{S}$, will produce
a good partial coloring of $G$ with only edges in $\\{e\in E(G):d_{S}(e)<1\\}$
being left uncolored.

Proof. Let $e=xy$ be an edge with $d_{S}(e)\geq 1$. Suppose $d_{S}(x)\leq
d_{S}(y)$. Let $e^{\prime}=xz$ be a neighbor of $e$ with
$d_{S}(z)=d_{S}(x)-1$. It is clear that all edges incident to the vertex $z$
are behind the edge $e$ in the order $R$. And so, when $e$ is going to be
colored by the greedy algorithm, all edges incident to $z$ are not colored.

If $d(x)=2$ or $d(x)=3$ and the third neighbor $w$ of $x$ satisfies
$d_{S}(w)=d_{S}(x)-1$, then $e$ sees at most $5$ colors at that moment,
implying that $e$ can be colored properly. Thus we assume that $d(x)=3$ and
the third neighbor $w$ of $x$ satisfies $d_{S}(w)\geq d_{S}(x)$.

If $d_{S}(y)=d_{S}(x)\geq 1$, then $y$ has a neighbor $y^{\prime}$ with
$d_{S}(y^{\prime})=d_{S}(z)$ (It is possible that $z=y^{\prime}$). As $G$ is a
claw-free subcubic graph, it is easy to check that $e$ sees at most $6$ colors
(please refer to Figure 4 (a) and (b)). Thus $e$ can be colored properly.

Figure 4: The illustrations of Lemma 2.1

If $d_{S}(y)=d_{S}(x)+1\geq 2$, then $d_{S}(w)=d_{S}(x)$ or
$d_{S}(w)=d_{S}(x)+1$. Again, because $G$ is a claw-free subcubic graph, $e$
sees at most $6$ colors (please refer to Figure 4 (c) and (d)) and so $e$ can
be colored properly. Therefore, the lemma holds.

In the proof of Theorem 1.3, we will often choose some subset $S$ of $V(G)$.
With this subset $S$, by Lemma 2.1, we can obtain a good partial coloring
$\phi$ of $G$ with only edges $e$ with $d_{S}(e)<1$ being uncolored. And then
our task is to extend $\phi$ to a good coloring of the whole graph $G$.

There are two main techniques in our proofs, one is using Hall’s theorem [11],
the other is the modified caterpillar tree method which is slightly different
from the caterpillar tree method used by Horák, Qing and Trotter [13] in
proving the upper bound $10$ for the strong chromatic indices of cubic graphs.

By Hall’s theorem [11], $\\{A_{\phi}(e):e\in\bar{E}_{\phi}\\}$ has a system of
distinct representatives (abbreviated SDR) if and only if $|\cup_{e\in
M}A_{\phi}(e)|\geq|M|$, for every $M\subseteq\bar{E}_{\phi}$. Whenever
$\\{A_{\phi}(e):e\in\bar{E}_{\phi}\\}$ has a SDR, $\phi$ can be easily
extended to a good coloring of $G$. In this situation, we will say that we can
obtain a good coloring of $G$ by SDR.

##  3 The proof of Theorem 1.3

It is sufficient to prove Theorem 1.3 for connected graphs. Let $G$ be a
connected claw-free subcubic graph not isomorphic to the $3$-prism. If $G$ is
isomorphic to the complete graph $K_{4}$ or the graph $K_{4}^{\Delta}$ (i.e.,
the graph obtained from $K_{4}$ by replacing each vertex with a $3$-cycle, as
shown in Figure 5), then it is easy to check that $\chi^{\prime}_{s}(G)\leq
7$. Please refer to Figure 5 for a good coloring of $K_{4}^{\Delta}$. Thus in
this section, we always assume that $G$ is a connected claw-free subcubic
graph that is not isomorphic to any graph in $\\{\mbox{the
$3$-prism},K_{4},K_{4}^{\Delta}\\}$.

Figure 5: The two graphs $K_{4},K_{4}^{\Delta}$ and a good coloring of
$K_{4}^{\Delta}$

###### Lemma 3.1

If $G$ has a vertex of degree $1$, then $\chi^{\prime}_{s}(G)\leq 7$.

Proof. Let $v_{0}$ be a vertex of degree $1$ in $G$ and $e_{0}$ be the edge
incident with $v_{0}$. Set $S=\\{v_{0}\\}$. Then, by Lemma 2.1, $G-v_{0}$ has
a good coloring. Since $e_{0}$ sees at most $5$ edges in $G$, we can extend
this good coloring of $G-v_{0}$ to $G$.

###### Lemma 3.2

If $G$ has a vertex of degree $2$, then $\chi^{\prime}_{s}(G)\leq 7$.

Proof. Let $v_{0}$ be a vertex of degree $2$ in $G$ with two neighbors $v_{1}$
and $v_{2}$. If $v_{1}v_{2}\in E(G)$ or $d(v_{1})=2$, then put
$S=\\{v_{0}\\}$. By Lemma 2.1, $G-v_{0}$ has a good coloring $\phi$ with
$|A_{\phi}(v_{0}v_{1})|\geq 2$ and $|A_{\phi}(v_{0}v_{2})|\geq 1$. We can
obtain a good coloring of $G$ by SDR easily. So by symmetry, we may assume
that $v_{1}v_{2}\notin E(G)$ and $d(v_{1})=d(v_{2})=3$. Let
$N(v_{1})=\\{v_{0},u_{1},u_{1}^{\prime}\\}$ and
$N(v_{2})=\\{v_{0},u_{2},u_{2}^{\prime}\\}$. Since $G$ is a claw-free subcubic
graph, we must have $u_{1}u_{1}^{\prime}\in E(G)$ and $u_{2}u_{2}^{\prime}\in
E(G)$. This implies that
$\\{u_{1},u_{1}^{\prime}\\}=\\{u_{2},u_{2}^{\prime}\\}$ or
$\\{u_{1},u_{1}^{\prime}\\}\cap\\{u_{2},u_{2}^{\prime}\\}=\emptyset$.

If $\\{u_{1},u_{1}^{\prime}\\}=\\{u_{2},u_{2}^{\prime}\\}$ then $G$ is
isomorphic to $H_{1}$ (please see Figure 6 for $H_{1}$), and so
$\chi^{\prime}_{s}(G)=7$. Thus we assume that
$\\{u_{1},u_{1}^{\prime}\\}\cap\\{u_{2},u_{2}^{\prime}\\}=\emptyset$. If
$d(u_{1})=2$, then the two neighbors of $u_{1}$ are $v_{1}$ and
$u_{1}^{\prime}$. By setting $S=\\{u_{1}\\}$, similar to the argument in the
previous paragraph, we can get a good coloring of $G$. Thus we assume
$d(u_{1})=3$. Symmetrically, we may also assume that
$d(u_{1}^{\prime})=d(u_{2})=d(u_{2}^{\prime})=3$. Let
$w_{1},w_{1}^{\prime},w_{2},w_{2}^{\prime}$ denote the third neighbors of
$u_{1},u_{1}^{\prime},u_{2},u_{2}^{\prime}$, respectively. Denote by
$E(\\{u_{1},u_{1}^{\prime}\\},\\{u_{2},u_{2}^{\prime}\\})$ the set of edges
with one vertex in $\\{u_{1},u_{1}^{\prime}\\}$ and the other in
$\\{u_{2},u_{2}^{\prime}\\}$. If
$|E(\\{u_{1},u_{1}^{\prime}\\},\\{u_{2},u_{2}^{\prime}\\})|=2$, then $G$ is
isomorphic to $H_{2}$ (see Figure 6) and we clearly have
$\chi^{\prime}_{s}(G)\leq 7$. We next deal with the remaining two cases
$|E(\\{u_{1},u_{1}^{\prime}\\},\\{u_{2},u_{2}^{\prime}\\})|=1$ and
$|E(\\{u_{1},u_{1}^{\prime}\\},\\{u_{2},u_{2}^{\prime}\\})|=0$.

Figure 6: The three graphs $H_{1},H_{2}$ and $H_{3}$

Case 1. $|E(\\{u_{1},u_{1}^{\prime}\\},\\{u_{2},u_{2}^{\prime}\\})|=1$.

Without loss of generality, assume that $u_{1}u_{2}\in E(G)$. If
$u_{1}^{\prime}$ and $u_{2}^{\prime}$ are adjacent to a common neighbor, then
$G$ must be isomorphic to $H_{3}$ (see Figure 6), implying that
$\chi^{\prime}_{s}(G)\leq 7$. Thus, we assume
$u_{1}^{\prime}u_{2}^{\prime}\notin E(G)$ and $w_{1}^{\prime}\neq
w_{2}^{\prime}$.

Set $S=\\{v_{0},v_{1},v_{2},u_{1},u_{2}\\}$. By Lemma 2.1, we get a good
partial coloring $\phi$ of $G$ with nine edges uncolored (please see Figure 7
for the names of the uncolored edges of $G$). Observe that
$|A_{\phi}(e_{3})|\geq 5$, $|A_{\phi}(e_{i})|=6$ for $i=1,2,4,5$ and
$|A_{\phi}(f_{j})|\geq 4$ for $j=1,2,3,4$.

Figure 7: Case 1 in the proof of Lemma 3.2

Let $\alpha$ be a color in $A_{\phi}(f_{1})\cap A_{\phi}(f_{3})$. By coloring
$f_{1}$ and $f_{3}$ with the same color $\alpha$, we extend $\phi$ to a new
good partial coloring $\psi$ of $G$, in which $|A_{\psi}(e_{3})|\geq 4$,
$|A_{\psi}(e_{i})|\geq 5$ for $i=1,2,4,5$ and $|A_{\psi}(f_{j})|\geq 3$ for
$j=2,4$. Observe that $A_{\psi}(e_{5})\cap A_{\psi}(f_{2})\neq\emptyset$, we
can further extend $\psi$ to another good partial coloring $\sigma$ of $G$ by
coloring $e_{5}$ and $f_{2}$ with the same color $\beta\in A_{\psi}(e_{5})\cap
A_{\psi}(f_{2})$. Then we have $|A_{\sigma}(e_{3})|\geq 3$,
$|A_{\sigma}(e_{i})|\geq 4$ for $i=1,2,4$ and $|A_{\sigma}(f_{4})|\geq 2$.

Now, if $A_{\sigma}(e_{2})\cap A_{\sigma}(f_{4})\neq\emptyset$, we can color
$e_{2}$ and $f_{4}$ with the same color $\gamma\in A_{\sigma}(e_{2})\cap
A_{\sigma}(f_{4})$ and then color $e_{3},e_{4},e_{1}$ by SDR. Otherwise, we
greedily color $f_{4},e_{3},e_{4},e_{1},e_{2}$ in this order.

Case 2. $|E(\\{u_{1},u_{1}^{\prime}\\},\\{u_{2},u_{2}^{\prime}\\})|=0$.

In this case, $w_{1},w_{1}^{\prime},w_{2},w_{2}^{\prime}$ are not necessarily
distinct. However, this does not affect the following arguments. Please see
Figure 8 for the names of vertices and edges. Set $S=\\{v_{0},v_{1},v_{2}\\}$.
By Lemma 2.1, there is a good partial coloring $\phi$ of $G$ with six
uncolored edges $e_{1},e_{2},f_{1},f_{2},f_{3},f_{4}$. It is clear that
$|A_{\phi}(e_{1})|=|A_{\phi}(e_{2})|=4$ and $|A_{\phi}(f_{j})|\geq 2$ for
$j=1,2,3,4$.

Figure 8: Case 2 in the proof of Lemma 3.2

We now make two easy but useful observations. One is that $A_{\phi}(f_{1})\cup
A_{\phi}(f_{2})\subseteq A_{\phi}(e_{1})=[1,7]\setminus F_{\phi}(e_{1})$ and
$A_{\phi}(f_{3})\cup A_{\phi}(f_{4})\subseteq A_{\phi}(e_{2})=[1,7]\setminus
F_{\phi}(e_{2})$, where
$F_{\phi}(e_{1})=\\{\phi(g_{1}),\phi(h_{1}),\phi(h_{2})\\}$ and
$F_{\phi}(e_{2})=\\{\phi(g_{2}),\phi(h_{3}),\phi(h_{4})\\}$. The other is
that, for any $i\in\\{1,2\\}$ and any $j\in\\{3,4\\}$, $f_{i}$ and $f_{j}$ do
not see each other.

As $|A_{\phi}(e_{1})|=|A_{\phi}(e_{2})|=4$, $A_{\phi}(e_{1})\cap
A_{\phi}(e_{2})\neq\emptyset$. According to the value of $|A_{\phi}(e_{1})\cap
A_{\phi}(e_{2})|$, we divide the proof into the following four subcases.

Subcase 2.1. $|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=1$. ($|A_{\phi}(e_{1})\cup
A_{\phi}(e_{2})|=7$.)

In this case, let’s explain why $\phi$ can be extend to a good coloring of $G$
by greedily coloring $f_{1},f_{2},f_{3},f_{4},e_{1}$ and $e_{2}$ in this
order. First, $f_{1},f_{2},f_{3},f_{4}$ can be colored properly. Secondly,
since $A_{\phi}(f_{3})\cup A_{\phi}(f_{4})\subseteq A_{\phi}(e_{2})$ and
$|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=1$, there is at least one color
available for $e_{1}$ when $e_{1}$ is to be colored. Finally, since
$A_{\phi}(f_{1})\cup A_{\phi}(f_{2})\subseteq A_{\phi}(e_{1})$ and
$|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=1$, after $e_{1}$ is colored, at least
one color is available for $e_{2}$.

Subcase 2.2. $|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=2$. ($|A_{\phi}(e_{1})\cup
A_{\phi}(e_{2})|=6$.)

Let $A_{\phi}(e_{1})\cap A_{\phi}(e_{2})=\\{\alpha_{1},\alpha_{2}\\}$. If
$\\{\alpha_{1},\alpha_{2}\\}\cap(A_{\phi}(f_{1})\cup
A_{\phi}(f_{2}))\cap(A_{\phi}(f_{3})\cup A_{\phi}(f_{4}))\neq\emptyset$,
without loss of generality, assume $\alpha_{1}\in A_{\phi}(f_{1})\cap
A_{\phi}(f_{3})$. Now, a good coloring of $G$ can be obtained by first
coloring $f_{1}$ and $f_{3}$ with $\alpha_{1}$ and then greedily coloring
$f_{2},f_{4},e_{1},e_{2}$ in this order. Thus, we assume that
$\\{\alpha_{1},\alpha_{2}\\}\cap(A_{\phi}(f_{1})\cup
A_{\phi}(f_{2}))\cap(A_{\phi}(f_{3})\cup A_{\phi}(f_{4}))=\emptyset$.

If $\\{\alpha_{1},\alpha_{2}\\}\cap(A_{\phi}(f_{1})\cup
A_{\phi}(f_{2}))=\\{\alpha_{1},\alpha_{2}\\}\cap(A_{\phi}(f_{3})\cup
A_{\phi}(f_{4}))=\emptyset$, then it is easy to see that $\phi$ can be
extended to a good coloring of $G$. Therefore, by symmetry, we may assume that
$\alpha_{1}\in A_{\phi}(f_{1})$ and $\alpha_{1}\notin A_{\phi}(f_{3})\cup
A_{\phi}(f_{4})$.

Suppose that $A_{\phi}(f_{1})\cup
A_{\phi}(f_{2})\neq\\{\alpha_{1},\alpha_{2}\\}$. If $\alpha_{1}\in
A_{\phi}(f_{2})$, then there exists a color $\beta\in(A_{\phi}(f_{1})\cup
A_{\phi}(f_{2}))\setminus\\{\alpha_{1},\alpha_{2}\\}$. We can color $f_{1}$
and $f_{2}$ to get a new good partial coloring $\psi$ of $G$ so that
$\\{\psi(f_{1}),\psi(f_{2})\\}=\\{\alpha_{1},\beta\\}$. And then we extend
$\psi$ to a good coloring of $G$ by greedily coloring
$f_{3},f_{4},e_{2},e_{1}$ in this order. If $\alpha_{1}\notin
A_{\phi}(f_{2})$, then there exists a color $\beta\in
A_{\phi}(f_{2})\setminus\\{\alpha_{1},\alpha_{2}\\}$. A good coloring of $G$
can be obtained by coloring $f_{1}$ with $\alpha_{1}$, $f_{2}$ with $\beta$
and then coloring the remaining edges in the order $f_{3},f_{4},e_{2},e_{1}$.

Now we assume $A_{\phi}(f_{1})\cup
A_{\phi}(f_{2})=\\{\alpha_{1},\alpha_{2}\\}$. Recall that $A_{\phi}(f_{3})\cup
A_{\phi}(f_{4})\subseteq A_{\phi}(e_{2})$, we must have
$A_{\phi}(f_{3})=A_{\phi}(f_{4})=A_{\phi}(e_{2})\setminus\\{\alpha_{1},\alpha_{2}\\}$.
Notice that $g_{1}$ sees a color $\alpha$ in $\phi$ if and only if $f_{1}$ or
$f_{2}$ also sees $\alpha$ in $\phi$, we can recolor $g_{1}$ with $\alpha_{1}$
to obtain a new good partial coloring of $G$, which we refer to it as $\psi$.
Observe that $A_{\psi}(f_{1})=A_{\psi}(f_{2})=\\{\phi(g_{1}),\alpha_{2}\\}$,
$A_{\psi}(e_{1})=(A_{\phi}(e_{1})\setminus\\{\alpha_{1}\\})\cup\\{\phi(g_{1})\\}$,
$A_{\psi}(f_{3})=A_{\psi}(f_{4})=A_{\phi}(f_{3})$ and
$A_{\psi}(e_{2})=A_{\phi}(e_{2})$. Now, we can get a good coloring of $G$ by
coloring $f_{1}$ with $\phi(g_{1})$, $f_{2}$ with $\alpha_{2}$, $f_{3}$ and
$f_{4}$ with the two colors in
$A_{\phi}(e_{2})\setminus\\{\alpha_{1},\alpha_{2}\\}$, $e_{2}$ with
$\alpha_{1}$ and $e_{1}$ with a color in
$A_{\phi}(e_{1})\setminus\\{\alpha_{1},\alpha_{2}\\}$.

Subcase 2.3. $|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=3$. ($|A_{\phi}(e_{1})\cup
A_{\phi}(e_{2})|=5$.)

Let $A_{\phi}(e_{1})\cap
A_{\phi}(e_{2})=\\{\alpha_{1},\alpha_{2},\alpha_{3}\\}$,
$A_{\phi}(e_{1})=\\{\alpha_{1},\alpha_{2},\alpha_{3},\beta\\}$ and
$A_{\phi}(e_{2})=\\{\alpha_{1},\alpha_{2},\alpha_{3},\gamma\\}$. Recall that
$A_{\phi}(f_{1})\cup A_{\phi}(f_{2})\subseteq A_{\phi}(e_{1})$ and
$A_{\phi}(f_{3})\cup A_{\phi}(f_{4})\subseteq A_{\phi}(e_{2})$, as
$|A_{\phi}(f_{j})|\geq 2$, we must have
$A_{\phi}(f_{j})\cap\\{\alpha_{1},\alpha_{2},\alpha_{3}\\}\neq\emptyset$ for
each $j=1,2,3,4$. We have two subcases to deal with.

Subcase 2.3.1. $(A_{\phi}(f_{1})\cup A_{\phi}(f_{2}))\cap(A_{\phi}(f_{3})\cup
A_{\phi}(f_{4}))\neq\emptyset$.

W.l.o.g., let $\alpha_{1}\in A_{\phi}(f_{1})\cap A_{\phi}(f_{3})$. By
assigning the color $\alpha_{1}$ to $f_{1}$ and $f_{3}$ and coloring $f_{2}$
(resp. $f_{4}$) with a color in $A_{\phi}(f_{2})\setminus\\{\alpha_{1}\\}$
(resp. $A_{\phi}(f_{4})\setminus\\{\alpha_{1}\\}$), we extend $\phi$ to a new
good partial coloring $\psi$. Observe that $|A_{\psi}(e_{1})|\geq 1$,
$|A_{\psi}(e_{2})|\geq 1$ and $|A_{\psi}(e_{1})\cup
A_{\psi}(e_{2})|=|(A_{\phi}(e_{1})\cup
A_{\phi}(e_{2}))\setminus\\{\alpha_{1},\psi(f_{2}),\psi(f_{4})\\}|\geq 2$.
Therefore, we can further extend $\psi$ to a good coloring of $G$ by SDR.

Subcase 2.3.2. $(A_{\phi}(f_{1})\cup A_{\phi}(f_{2}))\cap(A_{\phi}(f_{3})\cup
A_{\phi}(f_{4}))=\emptyset$.

In this subcase, it is clear that $|A_{\phi}(f_{1})\cup A_{\phi}(f_{2})|=2$ or
$|A_{\phi}(f_{3})\cup A_{\phi}(f_{4})|=2$. W.l.o.g., we assume that
$|A_{\phi}(f_{1})\cup A_{\phi}(f_{2})|=2$ (i.e.,
$A_{\phi}(f_{1})=A_{\phi}(f_{2})$), $\alpha_{1}\in A_{\phi}(f_{1})\cup
A_{\phi}(f_{2})$ and $\alpha_{3}\in A_{\phi}(f_{3})\cup A_{\phi}(f_{4})$.

If $A_{\phi}(f_{1})=A_{\phi}(f_{2})=\\{\alpha_{1},\alpha_{2}\\}$, then
$A_{\phi}(f_{3})=A_{\phi}(f_{4})=\\{\alpha_{3},\gamma\\}$. Recall that $g_{1}$
sees a color $\alpha$ in $\phi$ if and only if $f_{1}$ or $f_{2}$ also sees
$\alpha$ in $\phi$. Thus, we can always modify $\phi$ by recoloring $g_{1}$
with $\alpha_{1}$ to obtain a new good partial coloring $\psi$ of $G$, in
which $A_{\psi}(f_{3})=A_{\psi}(f_{4})=\\{\alpha_{3},\gamma\\}$ and
$A_{\psi}(e_{2})=\\{\alpha_{1},\alpha_{2},\alpha_{3},\gamma\\}$. Now, if
$\phi(g_{1})=\gamma$, we have
$A_{\psi}(f_{1})=A_{\psi}(f_{2})=\\{\alpha_{2},\gamma\\}$ and
$A_{\psi}(e_{1})=\\{\alpha_{2},\alpha_{3},\beta,\gamma\\}$, and we are back to
Subcase 2.3.1. While if $\phi(g_{1})\neq\gamma$, we have
$A_{\psi}(f_{1})=A_{\psi}(f_{2})=\\{\alpha_{2},\phi(g_{1})\\}$ and
$A_{\psi}(e_{1})=\\{\alpha_{2},\alpha_{3},\beta,\phi(g_{1})\\}$, implying
$|A_{\psi}(e_{1})\cap A_{\psi}(e_{2})|=2$. And we are back to Subcase 2.2.

If $A_{\phi}(f_{1})=A_{\phi}(f_{2})=\\{\alpha_{1},\beta\\}$, by symmetry, we
may assume that $A_{\phi}(f_{3})\cup
A_{\phi}(f_{4})\neq\\{\alpha_{2},\alpha_{3}\\}$ (i.e.,
$\\{\alpha_{3},\gamma\\}\subseteq A_{\phi}(f_{3})\cup A_{\phi}(f_{4})$). Also,
we can recolor $g_{1}$ with $\alpha_{1}$ to get a new good partial coloring
$\psi$ of $G$, in which $\\{\alpha_{3},\gamma\\}\subseteq A_{\psi}(f_{3})\cup
A_{\psi}(f_{4})$ and
$A_{\psi}(e_{2})=\\{\alpha_{1},\alpha_{2},\alpha_{3},\gamma\\}$. Now, if
$\phi(g_{1})=\gamma$, we have
$A_{\psi}(f_{1})=A_{\psi}(f_{2})=\\{\beta,\gamma\\}$ and
$A_{\psi}(e_{1})=\\{\alpha_{2},\alpha_{3},\gamma,\beta\\}$, and we are back to
Subcase 2.3.1. And if $\phi(g_{1})\neq\gamma$, then we have
$A_{\psi}(f_{1})=A_{\psi}(f_{2})=\\{\beta,\phi(g_{1})\\}$ and
$A_{\psi}(e_{1})=\\{\alpha_{2},\alpha_{3},\beta,\phi(g_{1})\\}$. As
$|A_{\psi}(e_{1})\cap A_{\psi}(e_{2})|=2$, we are back to Subcase 2.2.

Subcase 2.4. $|A_{\phi}(e_{1})\cap A_{\phi}(e_{2})|=4$. ($|A_{\phi}(e_{1})\cup
A_{\phi}(e_{2})|=4$.)

Let
$A_{\phi}(e_{1})=A_{\phi}(e_{2})=\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}$.
Erasing the colors of $g_{1}$ and $g_{2}$ in $\phi$ yields another good
partial coloring $\psi$ of $G$.

If $|A_{\psi}(g_{1})|=|A_{\psi}(g_{2})|=1$, then it is straightforward to
check that $A_{\phi}(f_{1})\cap A_{\phi}(f_{2})=\emptyset$,
$A_{\phi}(f_{1})\cup A_{\phi}(f_{2})=A_{\phi}(e_{1})$, $A_{\phi}(f_{3})\cap
A_{\phi}(f_{4})=\emptyset$ and $A_{\phi}(f_{3})\cup
A_{\phi}(f_{4})=A_{\phi}(e_{2})$. Now based on $\phi$, we can always color
$f_{1},f_{2},f_{3},f_{4}$ to obtain a new good partial coloring $\sigma$ with
$\\{\sigma(f_{1}),\sigma(f_{2})\\}=\\{\sigma(f_{3}),\sigma(f_{4})\\}$.
Consequently, $e_{1}$ and $e_{2}$ can be colored properly.

Now, w.l.o.g., we assume that $|A_{\psi}(g_{1})|\geq 2$. In this situation, it
is not difficult to verify that
$A_{\psi}(g_{1})\cap\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}\neq\emptyset$.
W.l.o.g., let $\alpha_{1}\in A_{\psi}(g_{1})$. By recoloring the edge $g_{1}$
with the color $\alpha_{1}$ in $\phi$, we obtain a new good partial coloring
$\sigma$ of $G$, in which
$A_{\sigma}(e_{1})=\\{\phi(g_{1}),\alpha_{2},\alpha_{3},\alpha_{4}\\}$ and
$A_{\sigma}(e_{2})=\\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\\}$.
Obviously, $|A_{\sigma}(e_{1})\cap A_{\sigma}(e_{2})|=3$ and so we are back to
Subcase 2.3.

As we have exhausted all cases, the lemma follows.

###### Lemma 3.3

Let $G$ be a connected claw-free cubic graph. If $G$ has a cut vertex, then
$\chi^{\prime}_{s}(G)\leq 7$.

Proof. Let $v_{0}$ be a cut vertex of $G$ and
$N(v_{0})=\\{u_{0},v_{1},v_{2}\\}$. As $G$ is a claw-free cubic graph, there
is exactly one edge in $G$ among the vertices in $N(v_{0})$. Without loss of
generality, let $v_{1}v_{2}\in E(G)$. Let $G_{1}$ be the component of
$G-v_{0}$ containing $u_{0}$ and $G_{2}$ the component of $G-v_{0}$ containing
$v_{1}v_{2}$. Please see Figure 9 for the names of vertices and edges in $G$.

Figure 9: A graph $G$ with a cut vertex $v_{0}$

By Lemma 3.2, $G_{1}$ (resp. $G_{2}$) has a good coloring, say $\phi_{1}$
(resp. $\phi_{2}$). We may assume that
$\\{\phi_{1}(f_{1}),\phi_{1}(f_{2}),\phi_{1}(f_{3})\\}=\\{\phi_{2}(h_{1}),\phi_{2}(h_{2}),\phi_{2}(h_{3})\\}$
as otherwise we can permute the colors among the edges of $G_{1}$. Combining
$\phi_{1}$ and $\phi_{2}$ yields a good partial coloring of $G$, calling it
$\phi$. It is clear that $|A_{\phi}(e_{0})|=2$, $|A_{\phi}(e_{1})|\geq 2$ and
$|A_{\phi}(e_{2})|\geq 2$.

If $|A_{\phi}(e_{0})\cup A_{\phi}(e_{1})\cup A_{\phi}(e_{2})|\geq 3$, then we
can obtain a good coloring of $G$ by SDR. Thus we assume that
$A_{\phi}(e_{0})=A_{\phi}(e_{1})=A_{\phi}(e_{2})=\\{\alpha_{1},\alpha_{2}\\}$.
Let $\beta=\phi(f_{4})$. Permute the two colors $\alpha_{1}$ and $\beta$ in
$E(G_{1})$. After the permutation, we have
$A_{\phi}(e_{0})=\\{\alpha_{2},\beta\\}$ and
$A_{\phi}(e_{1})=A_{\phi}(e_{2})=\\{\alpha_{1},\alpha_{2}\\}$. Since
$|A_{\phi}(e_{0})\cup A_{\phi}(e_{1})\cup A_{\phi}(e_{2})|\geq 3$, we complete
the proof by SDR.

###### Lemma 3.4

Let $G$ be a $2$-connected claw-free cubic graph. If $G$ contains a $4$-cycle
with a chord, then $\chi^{\prime}_{s}(G)\leq 7$.

Proof. Let $C=v_{1}v_{2}v_{3}v_{4}v_{1}$ be a $4$-cycle in $G$ with
$v_{1}v_{3}\in E(G)$. Because $G$ is not isomorphic to $K_{4}$,
$v_{2}v_{4}\notin E(G)$. Let $u_{2}$ and $u_{4}$ denote the third neighbor of
$v_{2}$ and $v_{4}$, respectively. Since $G$ is a $2$-connected claw-free
cubic graph, it is not difficult to see that $u_{2}\neq u_{4}$ and
$u_{2}u_{4}\notin E(G)$. If $N(u_{2})\cap N(u_{4})\neq\emptyset$, then $G$ is
isomorphic to $H_{4}$ (as shown in Figure 10) and so $\chi^{\prime}_{s}(G)\leq
7$. Thus we assume that $N(u_{2})\cap N(u_{4})=\emptyset$. Please see Figure
10 for the the names of vertices and edges in $G$.

Figure 10: The graph $H_{4}$ and a graph $G$ with a chorded $4$-cycle

Set $S=\\{v_{1},v_{2},v_{3},v_{4}\\}$. By Lemma 2.1, we get a good partial
coloring $\phi$ of $G$, in which $|A_{\phi}(e_{i})|=5$ for $i\in[1,4]$,
$|A_{\phi}(e_{5})|=7$ and $|A_{\phi}(f_{1})|=|A_{\phi}(f_{2})|=2$. If
$|A_{\phi}(e_{1})\cup A_{\phi}(e_{3})|\geq 6$, then a good coloring of $G$ can
be easily obtained by SDR. Thus we assume that $|A_{\phi}(e_{1})\cup
A_{\phi}(e_{3})|=5$. Observe that $A_{\phi}(e_{1})=A_{\phi}(e_{2})$ and
$A_{\phi}(e_{3})=A_{\phi}(e_{4})$, we may assume that
$A_{\phi}(e_{1})=A_{\phi}(e_{2})=A_{\phi}(e_{3})=A_{\phi}(e_{4})=[1,5]$. And
so $\\{\phi(g_{1}),\phi(g_{2})\\}=\\{\phi(g_{3}),\phi(g_{4})\\}=\\{6,7\\}$.

Erasing the colors of $g_{1}$ and $g_{2}$ in $\phi$ yields a good partial
coloring of $G$, calling it $\psi$. Now, if there exists some color $\alpha\in
A_{\psi}(g_{1})\setminus\\{6,7\\}$, we color $g_{1}$ with $\alpha$ and $g_{2}$
with $\phi(g_{2})$ to get a new good partial coloring $\sigma$ of $G$, in
which $A_{\sigma}(e_{1})=([1,5]\setminus\\{\alpha\\})\cup\\{\phi(g_{1})\\}$
and $A_{\phi}(e_{3})=[1,5]$. Clearly, $|A_{\sigma}(e_{1})\cup
A_{\sigma}(e_{3})|\geq 6$. So we can extend $\sigma$ to a good coloring of $G$
by SDR. Thus we assume $A_{\psi}(g_{1})=\\{6,7\\}$. Symmetrically, we may also
assume $A_{\psi}(g_{2})=\\{6,7\\}$. Notice that $h_{1}$ sees a color $\alpha$
in $\psi$ if and only if $g_{1}$ or $g_{2}$ also sees $\alpha$ in $\psi$, we
can recolor $h_{1}$ with $\phi(g_{1})$ in $\psi$. By further coloring $g_{1}$
with $\phi(h_{1})$ and $g_{2}$ with $\phi(g_{2})$, we get a new good partial
coloring $\sigma$ of $G$ with
$A_{\sigma}(e_{1})=([1,5]\setminus\\{\phi(h_{1})\\})\cup\\{\phi(g_{1})\\}$ and
$A_{\phi}(e_{3})=[1,5]$. Again we have $|A_{\sigma}(e_{1})\cup
A_{\sigma}(e_{3})|\geq 6$ and so complete the proof by SDR.

###### Lemma 3.5

Let $G$ be a $2$-connected claw-free cubic graph. If $G$ contains an induced
$4$-cycle, then $\chi^{\prime}_{s}(G)\leq 7$.

Proof. Let $C=v_{1}v_{2}v_{3}v_{4}v_{1}$ be an induced $4$-cycle in $G$. As
$G$ is a claw-free cubic graph not isomorphic to the $3$-prism,
$u_{1}u_{2}\notin E(G)$. Let $N(u_{1})=\\{v_{1},v_{2},w_{1}\\}$ and
$N(u_{2})=\\{v_{3},v_{4},w_{2}\\}$ (Please refer to Figure 11). Since $G$ is
$2$-connected, $w_{1}w_{2}\notin E(G)$.

Figure 11: A graph $G$ with an induced $4$-cycle $C=v_{1}v_{2}v_{3}v_{4}v_{1}$

Setting $S=\\{v_{1},v_{2},v_{3},v_{4}\\}$, by Lemma 2.1, we get a good partial
coloring $\phi$ of $G$, in which $|A_{\phi}(e_{i})|=6$ for $i=1,3$,
$|A_{\phi}(e_{i})|\geq 5$ for $i=2,4$ and $|A_{\phi}(f_{j})|=4$ for
$j=1,2,3,4$. Observe that there exists some color $\alpha\in
A_{\phi}(f_{1})\cap A_{\phi}(f_{3})$, by coloring $f_{1}$ and $f_{3}$ with
$\alpha$, we extend $\phi$ to another good partial coloring $\psi$ of $G$, in
which $|A_{\psi}(f_{2})|=|A_{\psi}(f_{4})|=3$,
$|A_{\psi}(e_{1})|=|A_{\psi}(e_{3})|=5$ and
$|A_{\psi}(e_{2})|=|A_{\psi}(e_{4})|\geq 4$.

Now, if there exists some $\beta\in A_{\psi}(f_{2})\cap A_{\psi}(f_{4})$, then
we can color $f_{2}$ and $f_{4}$ with $\beta$ and greedily color
$e_{2},e_{4},e_{1},e_{3}$ in this order to get a good coloring of $G$.
Otherwise, we have $|A_{\psi}(f_{2})\cup A_{\psi}(f_{4})|=6$, and so the
remainning six egdes can be colored properly by SDR. Thus the lemma holds.

At present time, the only case we need to deal with is that $G$ is a
$2$-connected claw-free cubic graph without $4$-cycles. It is clear that each
vertex is exactly on one $3$-cycle. Let $\Delta_{1}$ and $\Delta_{2}$ be two
$3$-cycles in $G$. Then $\Delta_{1}$ and $\Delta_{2}$ are vertex-disjoint and
there is at most one edge from $V(\Delta_{1})$ to $V(\Delta_{2})$. Moreover,
it is not difficult to see that $G$ has no induced odd cycles of length at
least $5$ and each induced even cycle has length at least $6$.

###### Lemma 3.6

If $G$ is a $2$-connected claw-free cubic graph without $4$-cycles, then
$\chi^{\prime}_{s}(G)\leq 7$.

Proof. Choose a minimum induced even cycle $C$ in $G$ and label the vertices
of $C$ as $v_{1},v_{2},\ldots,v_{2p}$ and the edges $e_{i}=v_{i-1}v_{i}$ for
$i=2,3,\ldots,2p$ and $e_{1}=v_{2p}v_{1}$. Clearly, $2p\geq 6$. As $G$ is a
claw-free cubic graph and has no $4$-cycles, we may assume that $v_{2j-1}$ and
$v_{2j}$ have the common neighbor $u_{2j}$ for each $j\in[1,p]$ and
$u_{2},u_{4},\ldots,u_{2p}$ are distinct. Since $C$ is a minimum induced even
cycle in $G$, $u_{2i}u_{2j}\notin E(G)$ for any $i,j\in[1,p]$. Let $w_{2j}$
denote the third neighbor of $u_{2j}$ for each $j\in[1,p]$. Also,
$w_{2},w_{4},\ldots,w_{2p}$ are distinct because $G$ is claw-free and cubic.
As for the names of vertices and edges in $G$, please refer to Figure 12.

Figure 12: A graph $G$ with an induced $2p$-cycle $(p\geq 3)$

Notice that if $2p\geq 8$, then $w_{2j}w_{2j+2}\notin E(G)$ for any
$j\in[1,p-1]$ and $w_{2}w_{2p}\notin E(G)$. And if $2p=6$, then there is at
most one edge in the subgraph of $G$ induced by $\\{w_{2},w_{4},w_{6}\\}$
since $G$ is not isomorphic to $K_{4}^{\Delta}$. So in either cases, we may
assume that $w_{2}w_{4}\notin E(G)$. This implies that the four edges
$h_{1},h_{2},h_{3},h_{4}$ are distinct.

Let $G^{\prime}$ be the graph obtained from $G$ by deleting
$v_{1},v_{2},\ldots,v_{2p},u_{2},u_{4},u_{6}$ and adding a new edge
$w_{2}w_{4}$. Observe that $G^{\prime}$ is a claw-free subcubic graph and each
component of $G^{\prime}$ has a vertex of degree less than $3$, by Lemmas 3.1
and 3.2, $G^{\prime}$ has a good coloring $\phi$. Ignoring $w_{2}w_{4}$ in
$\phi$ yields a good partial coloring of $G$. Let $\alpha=\phi(w_{2}w_{4})$.
And let $c_{i}=\phi(h_{i})$ for each $i\in[1,2p]$. Then
$\alpha,c_{1},c_{2},c_{3},c_{4}$ are distinct. As $|A_{\phi}(g_{6})|\geq 2$,
there exists a color $\beta\in A_{\phi}(g_{6})\setminus\\{\alpha\\}$. Without
loss of generality, assume that $c_{3}\neq\beta$.

Now based on $\phi$, we color $g_{2},g_{4}$ and $e_{6}$ with the same color
$\alpha$, color $g_{6}$ with $\beta$ and $e_{5}$ with $c_{3}$. Observe that
$f_{5}$ sees at most $5$ colors at this moment, we can color $f_{5}$ with some
color $\gamma\neq c_{4}$. This yields a new coloring $\psi$, which is indeed a
good partial coloring of $G$. Please refer to Figure 13 and Figure 14 for this
coloring.

Figure 13: $\psi$, $\bar{G}_{\psi}$ and $|A_{\psi}(e)|$ for each
$e\in\bar{E}_{\psi}$ when $2p\geq 8$

Figure 14: $\psi$, $\bar{G}_{\psi}$ and $|A_{\psi}(e)|$ for each
$e\in\bar{E}_{\psi}$ when $2p=6$

In the following, we will construct a good coloring $\sigma$ of
$\bar{G}_{\psi}$ so that $\sigma(e)\in A_{\psi}(e)$ for every
$e\in\bar{E}_{\psi}$. Then, this coloring $\sigma$ combining with the coloring
$\psi$ forms a good coloring of $G$, and we are done.

If $2p\geq 8$, then it is straightforward to check that $|A_{\psi}(e_{1})|\geq
5$, $|A_{\psi}(e_{2})|=6$, $|A_{\psi}(e_{3})|=5$, $|A_{\psi}(e_{4})|=4$,
$|A_{\psi}(e_{7})|\geq 2$, $|A_{\psi}(e_{i})|\geq 5$ for each
$i\in\\{8,9,\dots,2p\\}$, $|A_{\psi}(f_{4})|=3$, $|A_{\psi}(f_{6})|\geq 1$,
$|A_{\psi}(f_{7})|\geq 3$, and $|A_{\psi}(f_{i})|=4$ for each
$i\in\\{1,2,3,8,9,\dots,2p\\}$. (Please refer to Figure 13). And if $2p=6$, it
is clear that $|A_{\psi}(e_{1})|=3$, $|A_{\psi}(e_{2})|=6$,
$|A_{\psi}(e_{3})|=5$, $|A_{\psi}(e_{4})|=4$, $|A_{\psi}(f_{4})|=3$,
$|A_{\psi}(f_{6})|\geq 1$, and $|A_{\psi}(f_{i})|=4$ for each
$i\in\\{1,2,3\\}$. (Please refer to Figure 14).

Before we go further to the next step, we make two observations which are
simple but helpful in the following arguments (please refer to Figure 15 for
the illustrations of Observation A and Observation B).

Observation A:
$A_{\psi}(f_{1})=A_{\psi}(f_{2})=[1,7]\setminus\\{\alpha,c_{1},c_{2}\\}\subseteq
A_{\psi}(e_{2})=[1,7]\setminus\\{\alpha\\}$.

Observation B: $A_{\psi}(e_{4})=A_{\psi}(f_{4})\cup\\{c_{4}\\}$,
$A_{\psi}(f_{3})=A_{\psi}(f_{4})\cup\\{\gamma\\}$, and
$A_{\psi}(e_{3})=A_{\psi}(f_{4})\cup\\{c_{4},\gamma\\}$.

Figure 15: Observetion A and Observetion B

Now, if $2p\geq 8$, we color
$f_{6},e_{7},f_{7},e_{8},f_{8}\ldots,e_{2p},f_{2p},e_{1},f_{1},e_{2},f_{2},e_{3},f_{3},f_{4}$
in this order greedily to get a good partial coloring $\sigma$ of
$\bar{G}_{\psi}$, where only $e_{4}$ is uncolored. And if $2p=6$, we color
$f_{6},e_{1},f_{1},e_{2},f_{2},e_{3},f_{3},f_{4}$ in this order greedily to
get a good partial coloring $\sigma$ of $\bar{G}_{\psi}$, with only $e_{4}$
being uncolored. For simplicity, let $a_{i}=\sigma(e_{i})$ and
$b_{j}=\sigma(f_{j})$ for edges $e_{i},f_{j}\in\bar{E}_{\psi}$ that are
colored in $\sigma$ (please refer to Figure 16).

Figure 16: the good partial coloring $\sigma$ of $\bar{G}_{\psi}$

If
$A_{\psi}(e_{4})\setminus\\{a_{2},b_{2},a_{3},b_{3},b_{4}\\}\not=\emptyset$,
then the edge $e_{4}$ can be colored properly and we are done. We therefore
assume that $A_{\psi}(e_{4})\subseteq\\{a_{2},b_{2},a_{3},b_{3},b_{4}\\}$.

If $\gamma\not\in\\{a_{2},b_{2},a_{3}\\}$, then we may assume that
$b_{3}=\gamma$ as otherwise we can recolor the edge $f_{3}$ with the color
$\gamma$ to make $b_{3}=\gamma$. By Observation B, $\gamma\not\in
A_{\psi}(f_{4})$ and $\gamma\not\in A_{\psi}(e_{4})$. It follows that there
exists a color $b_{4}^{*}$ in
$A_{\psi}(f_{4})\setminus\\{a_{3},b_{3},b_{4}\\}$ and
$A_{\psi}(e_{4})=\\{a_{2},b_{2},a_{3},b_{4}\\}$. Now, we can recolor $f_{4}$
with $b_{4}^{*}$ and then color $e_{4}$ with $b_{4}$. This results in a good
coloring of $\bar{G}_{\psi}$. Therefore we assume that
$\gamma\in\\{a_{2},b_{2},a_{3}\\}$.

It follows from the assumption $\gamma\in\\{a_{2},b_{2},a_{3}\\}$ that
$b_{3}\not=\gamma$. Since $b_{3}\in A_{\psi}(f_{3})$, by Observation B,
$b_{3}\in A_{\psi}(f_{4})$. Therefore we must have $\\{b_{3},b_{4}\\}\subset
A_{\psi}(f_{4})$. The remainder of the proof is divided into two cases
according to whether $a_{3}$ is in $A_{\psi}(f_{4})$ or not.

Case 1. $a_{3}\not\in A_{\psi}(f_{4})$.

Since $a_{3}\in A_{\psi}(e_{3})$ and, by Observation B,
$A_{\psi}(e_{3})=A_{\psi}(f_{4})\cup\\{c_{4},\gamma\\}$, it is clear that
$a_{3}\in\\{c_{4},\gamma\\}$. Let
$A_{\psi}(f_{4})=\\{b_{3},b_{4},\alpha^{*}\\}$. Then
$A_{\psi}(e_{4})=\\{c_{4},b_{3},b_{4},\alpha^{*}\\}$. If $c_{4}$ (or
$\alpha^{*}$) is not contained in $\\{a_{2},b_{2},a_{3}\\}$, then we can color
the edge $e_{4}$ with $c_{4}$ (or $\alpha^{*}$) and obtain a good coloring of
$\bar{G}_{\psi}$. Thus we assume that
$\\{c_{4},\alpha^{*}\\}\subset\\{a_{2},b_{2},a_{3}\\}$. This together with
$\gamma\in\\{a_{2},b_{2},a_{3}\\}$ implies that
$\\{a_{2},b_{2},a_{3}\\}=\\{c_{4},\alpha^{*},\gamma\\}$. As
$a_{3}\in\\{c_{4},\gamma\\}$, $\alpha^{*}$ must be in $\\{a_{2},b_{2}\\}$. Now
we can recolor the edge $f_{4}$ with $\alpha^{*}$ and then color $e_{4}$ with
$b_{4}$. This completes the coloring of $\bar{G}_{\psi}$.

Case 2. $a_{3}\in A_{\psi}(f_{4})$. (So
$A_{\psi}(f_{4})=\\{a_{3},b_{3},b_{4}\\}$.)

In this case, by Observation B,
$A_{\psi}(e_{4})=A_{\psi}(f_{4})\cup\\{c_{4}\\}=\\{c_{4},a_{3},b_{3},b_{4}\\}$
and $a_{3}\not=\gamma$. Since
$A_{\psi}(e_{4})\subseteq\\{a_{2},b_{2},a_{3},b_{3},b_{4}\\}$, $c_{4}$ must be
in $\\{a_{2},b_{2}\\}$. As $\gamma\in\\{a_{2},b_{2},a_{3}\\}$ and
$a_{3}\not=\gamma$, $\gamma\in\\{a_{2},b_{2}\\}$. Therefore,
$\\{a_{2},b_{2}\\}=\\{c_{4},\gamma\\}$. Now we erase the colors of $f_{3}$ and
$f_{4}$ in $\sigma$ to get a new good partial coloring of $\bar{G}_{\psi}$,
where the only uncolored edges are $f_{3}$, $f_{4}$ and $e_{4}$. This coloring
is still denoted by $\sigma$. We next extend $\sigma$ to a good coloring of
$\bar{G}_{\psi}$.

If $a_{2}\in A_{\psi}(f_{2})$, then, by Observation A, we also have $a_{2}\in
A_{\psi}(f_{1})$. Notice that
$b_{1},a_{2}\notin\\{a_{2p},b_{2p},a_{1},b_{2},a_{3}\\}$, we exchange the
colors of $f_{1}$ and $e_{2}$ in $\sigma$. If $a_{2}=c_{4}$ and $b_{2}=\gamma$
(so $b_{1}\neq c_{4}$), we color $e_{4}$ with $c_{4}$. It follows that there
is at least one color available for $f_{3}$ and there are at least two colors
available for $f_{4}$. Hence we can color $f_{3}$ and $f_{4}$ properly. If
$a_{2}=\gamma$ and $b_{2}=c_{4}$ (so $b_{1}\neq\gamma$), we color $f_{3}$ with
$\gamma$. Then $e_{4}$ has at least one available color and $f_{4}$ has at
least two available colors. Therefore, coloring $e_{4}$ and then $f_{4}$ gives
a good coloring of $\bar{G}_{\psi}$.

If $a_{2}\notin A_{\psi}(f_{2})$, then there exists some color $b_{2}^{*}$ in
$A_{\psi}(f_{2})\setminus\\{a_{1},b_{1},b_{2}\\}$. We recolor $f_{2}$ with
$b_{2}^{*}$. Now, if $a_{2}=c_{4}$ and $b_{2}=\gamma$, we color $f_{3}$ with
$\gamma$. Recall that $\gamma\not\in A_{\psi}(e_{4})$, there is at least one
color available for $e_{4}$ and there are at least two colors available for
$f_{4}$. Thus $e_{4}$ and $f_{4}$ can be colored properly. If $a_{2}=\gamma$
and $b_{2}=c_{4}$, we recolor $e_{3}$ with $c_{4}$. Then both $e_{4}$ and
$f_{3}$ have at least two available colors and $f_{4}$ has three available
colors. It follows that we can obtain a good coloring of $\bar{G}_{\psi}$ by
SDR.

Combining all lemmas in this section, Theorem 1.3 holds.

##  4 Final Remarks

Let $G$ be any connected claw-free subcubic graph not isomorphic to the
triangular prism. This paper proves that $\chi^{\prime}_{s}(G)\leq 7$. And
this upper bound $7$ is sharp. In addition, our proof implies a linear-time
algorithm for finding a strong $7$-edge-coloring for such a graph.

If $G$ is a connected claw-free cubic graph not isomorphic to the triangular
prism, then it is easy to see that $\chi^{\prime}_{s}(G)\geq 6$. Therefore,
for such graph $G$, $\chi^{\prime}_{s}(G)\in\\{6,7\\}$. Let $H$ be a connected
cubic graph. Denote by $H^{\Delta}$ the graph obtained from $H$ by replacing
each vertex with a $3$-cycle. It is clear that $H^{\Delta}$ is a connected
claw-free cubic graph.

We end this paper by asking the following three questions.

Question 1: Let $k\geq 3$ be an integer and $H$ the $k$-prism. Is it true that
$\chi^{\prime}_{s}(H^{\Delta})=6$?

Question 2: Is it possible to characterize all connected claw-free cubic
graphs $G$ with $\chi^{\prime}_{s}(G)=6$?

Question 3: Is the strong list-chromatic index of any claw-free subcubic graph
other than the triangular prism at most 7?

Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

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# Reduction of rain-induced errors for wind speed estimation on SAR
observations using convolutional neural networks

Aurélien Colin1,2, Pierre Tandeo1,3, Charles Peureux2, Romain Husson2, Ronan
Fablet1,3 1 IMT Atlantique, Lab-STICC, UMR CNRS 6285, F-29238, France.2
Collecte Localisation Satellites, Brest, France.3 Odyssey, Inria/IMT, France

###### Abstract

Synthetic Aperture Radar is known to be able to provide high-resolution
estimates of surface wind speed. These estimates usually rely on a Geophysical
Model Function (GMF) that has difficulties accounting for non-wind processes
such as rain events. Convolutional neural network, on the other hand, have the
capacity to use contextual information and have demonstrated their ability to
delimit rainfall areas. By carefully building a large dataset of SAR
observations from the Copernicus Sentinel-1 mission, collocated with both GMF
and atmospheric model wind speeds as well as rainfall estimates, we were able
to train a wind speed estimator with reduced errors under rain. Collocations
with in-situ wind speed measurements from buoys show a root mean square error
that is reduced by 27% (resp. 45%) under rainfall estimated at more than 1
mm/h (resp. 3 mm/h). These results demonstrate the capacity of deep learning
models to correct rain-related errors in SAR products.

###### Index Terms:

Synthetic Aperture Radar, Deep Learning, Oceanography, Wind.

##  I Introduction

Synthetic Aperture Radar (SAR) is a powerful tool for studying the ocean
surface. C-Band SAR are sensitive to variations in sea surface roughness, and
have been used to detect various meteorological and ocean processes, referred
to as metocean, such as atmospheric or ocean fronts [1], icebergs [2], oil
surfactants from pollution [3] or generated by plankton [4], and some species
of seaweed [5]. They are particularly useful for studying waves [6] and
extreme events like cyclones [7, 8]. There has been particular attention given
to estimating wind speed using these sensors.

As the number of satellite missions with C-SAR sensors increases and archives
of these data accumulate, it is becoming easier to build large SAR datasets.
This paper focuses on the Sentinel-1 mission from the Copernicus program,
which consists of two satellites, Sentinel-1A (launched in 2014) and
Sentinel-1B (launched in 2016, which has been out of operation since December
2021). Sentinel-1C is planned to be launched in 2023. Ground Range Detected
Higher Resolution Interferometric Wide-swath (GRDH IW) observations have a
range of 250 km, an azimuth of about 200 km, and a resolution of 10 m/px.
These observations are mainly routinely acquired over coastal areas.
Systematic processes are used to produce geophysical products from these
observations, including wind speed estimates. Several Geophysical Model
Functions (GMFs) have been developed for this purpose, including CMOD3 [9],
CMOD4 [10], CMOD5 [11], CMOD5.N [12], CMOD6 [13], CMOD7 [14] and C_SARMOD2
[15]. These GMFs use the vertical-vertical polarization, which is Sentinel-1’s
default co-polarization in coastal areas. However, the co-polarization channel
saturates at high wind speed [16]. Therefore, the cross-polarization has also
been used to estimate the wind speed [7]. In particular, H14E [17] was found
to provide accurate wind speed measurements even in extreme events such as
category 5 hurricanes [18]. The cross-polarization has the advantage of lower
incidence and wind direction dependency [19]. GMFs have also been developed
for horizontal-horizontal polarization [20]. Bayesian nonparametric wind speed
estimation has also been proposed, offering the advantage of not relying on
wind direction priors that are sensitive to meteorological processes with
rapid temporal and spatial evolution [21].

However, these GMFs are sensitive to contamination from non-wind processes. In
particular, rainfall can either increase or decrease sea surface roughness
[22], making it difficult to correct for its effects.

Deep learning models, particularly Convolutional Neural Networks (CNNs), have
demonstrated their ability to detect rain signatures in SAR observations [23].
These models are known to be able to tackle denoising [24] and inpainting [25]
tasks because they use contextual information to estimate the original signal.
This paper is dedicated to estimating wind speed in rainy areas using a model
that does not require an explicit rainfall prior and only uses the parameters
available to GMFs.

In the first section, we present the SAR data used to train the model and the
ancillary information available. The second section describes the methodology
used to build the dataset, with special attention given to ensuring a balanced
representation of rainfall observations. The final section presents the
results on the training set and confirms them with in-situ measurements from
buoys, demonstrating the model’s ability to correct for rain overestimates.

##  II Dataset

The SAR measurements used in this chapter come from 19,978 IW observations
acquired globally between March 03rd, 2018 and the February 23rd, 2022,
inclusive. Each of these observations covers approximately 44 000 km² and has
a resolution of 100 m/px, downscaled from the GRDH products available at 10
m/px. The radiometric information used as input is the Sea Surface Roughness
(SSR) defined in [26] as the normalized radar cross section $\sigma_{0}$
divided by the $\sigma_{0}$ of a wind of 10 m/s and a direction of 45°
relative to the antenna look angle. The $\sigma_{0}$ of this neutral wind is
given by the GMF CMOD5.N.

Obtaining global information on rain that can be used in conjunction with SAR
observations can be difficult. A previous study conducted using a global
Sentinel-1 dataset found only 2,304 partial collocations with the satellite-
based radar GPM/DPR [27] out of 182,153 IW. ”Partial collocations” refers to
instances where at least 20x20 km of a swath is observed by the spaceborne
weather radar 20 minutes before or after the SAR observation. Coastal ground-
based radars like NEXRAD [28] could provide rainfall estimates, but they are
affected by topography and may not capture all wind regimes. Therefore, SAR-
based rain estimation is preferred to maximize the number of available
observations and simplify the collocation process. We used a recent SAR
rainfall estimator [23] that emulates NEXRAD’s reflectivity and proposes three
rainfall thresholds that roughly correspond to 1 mm/h, 3 mm/h, and 10 mm/h.

Ancillary information, such as incidence angle and satellite heading, is
retrieved from Sentinel-1 Level-2 products. It also includes collocations with
atmospheric models from the European Centre for Medium-Range Weather
Forecasts, which provide modelled wind speed and direction, as well as the
surface wind speed computed by the GMF. The atmospheric models have a spatial
resolution of 0.25x0.25 degrees and a temporal resolution of 3 hours [29],
while the GMF is computed at a spatial resolution of 1 km/px and corresponds
to the observation itself. The GMF used in Level-2 product is IFR2 [30] until
July 2019 and CMOD5.N [12] afterwards. The ancillary information contained in
the Level-2 products is available at a spatial resolution of 1 km/px. However,
it is interpolated to 100 m/px to match the grid of the radiometric channels.

##  III Methodology

This section presents the methodology for building the rain-invariant wind
speed estimator. We first describe the deep learning architecture of the
model, then we discuss the creation of the dataset, which is biased to have a
large number of rain examples. The final section describes the evaluation
procedure.

###  III-A Deep Learning Model

The architecture used in this chapter is the UNet architecture [31] depicted
in Fig. 1. UNet is an autoencoder architecture with the advantage of being
fully convolutional, meaning it has translation equivariance properties
(translations of the input result in translations of the output). In addition,
skip connections between the encoder and the decoder facilitate training,
especially by reducing the vanishing gradient issue [32]). Introduced in 2015,
UNet has been used in various domains and has demonstrated its importance for
segmentation of SAR observations [33, 23, 34].

The output of the model always contains a single convolution kernel, activated
by the Rectified Linear Unit (ReLU) function to ensure that the prediction is
in the interval [0, +$\infty$]. All convolution kernels in the hidden layers
are also activated by ReLU functions. The model is set to take input of
256x256 pixels during training, but since the weights only describe
convolution kernels, it is possible to use the model for inference on images
of any shape as long as the input resolution remains at 100 m/px. Variants of
the model are trained with different numbers of input channels. The
architecture is modified by changing the size of the first convolution kernel,
which is defined as a kernel of size (3, 3, c, 32), where c is the number of
input channels.

Figure Fig. 1: Architecture of the UNet model used for estimating the wind
speed.

###  III-B Dataset balancing procedure

In this section, we describe the process for building a balanced dataset. Our
goals are to (I) ensure that the wind distribution of each dataset is close to
the real-world distribution, (II) prevent information leak between the
training, validation, and test subsets, (III) ensure that the groundtruth wind
speed, obtained from an atmospheric model, accurately represents the real-
world wind speed, and (IV) include enough rain samples to allow the model to
learn from them.

##### Rain and rainless patches selection

As discussed earlier, rainfall estimation is provided by a deep learning model
at a resolution of 100 m/px, on the same grid as the SAR observation.
Therefore, it is possible to separate the observations into two areas,
$\mathcal{A}^{+}$ and $\mathcal{A}^{-}$, based on the 3 mm/h threshold from
the rainfall estimation.

$\displaystyle\mathcal{A}^{+}=\\{x:Rainfall(x)>=3mm/h\\}$ (Eq. 1)
$\displaystyle\mathcal{A}^{-}=\\{x:Rainfall(x)<3mm/h\\}$ (Eq. 2)

However, most SAR observations do not contain rain signatures. Collocations
with GPM’s dual polarization radar, a satellite-based weather radar, indicated
that the probability of rain rates higher than 3 mm/h was 0.5%. Thus, by
dividing the SAR observations into tiles of 256 by 256 pixels, we call ”rain
patches” the tiles with more than 5% of their surface predicted to have rain
rates higher than 3 mm/h, and ”rainless patches” those without rain
signatures. We denote $n_{+}$ as the number of rain patches and $n_{-}$ as the
number of rainless patches. To ensure that the model will learn regardless of
the rain-situation, we set $n_{+}=n_{-}$.

Fig. 2 indicates collocations between the reanalysis from ERA5 and the
satellite-based weather radar GPM/DPR. It shows the wind speed distribution
for rainfall higher than 3 mm/h (a), 30 mm/h (b), and the overall distribution
(orange curve). Stronger rainfall is associated with a lower probability of
strong wind speeds. However, the impact is mostly marginal at moderate rain
rates.

(a)

(b)

Figure Fig. 2: Distribution of ERA5 wind speed collocated with surface rain
rate from GPM/DPR for rainfall higher than 3 mm/h (a) and 30 mm/h (b). They
amount for 0.55% and 0.02% of the collocations, respectively. The orange curve
in the figure shows the wind distribution regardless of the rainfall.

##### Restriction to a priori Accurate Model Wind Speeds

Atmospheric models have been known to have coarse resolution (0.25x0.25
degrees spatially, 3 hours temporally) and to be unable to accurately depict
fine-scale wind fields. However, they are computed globally and independently
of the SAR observations. On the other hand, SAR-based wind fields from the
GMFs are known to be accurate on rainless patches, but are diversely affected
by rainfall. At low and moderate wind speeds, impinging droplets lead to an
overestimation of the wind speed. Under high wind speed conditions, dominant
attenuation results in an underestimation of the wind speed. Because of the
difficulty of collocating a sufficient quantity of high wind speed and
rainfall events, this study mainly aims to correct the wind speed
overestimation. We calculate $\Delta_{\mathcal{A^{-}}}$ as the discrepancy
between the GMF and the atmospheric model on rain-free pixels.

$\Delta_{\mathcal{A^{-}}}=MSE_{|\mathcal{A^{-}}}(Atm,GMF)$ (Eq. 3)

In our experiments, the threshold was set at $\Delta_{\mathcal{A}^{-}}<1$ m/s.
All patches containing a higher discrepancy between the two wind speed sources
were discarded. By ensuring agreement between the atmospheric model and the
SAR-based GMF on rainless patches, we assume that the atmospheric model is
close to the real wind speed and use it as a target for optimizing the deep
learning model.

##### Balancing to the real-world wind distribution

It should be noted that this condition ensures accurate modeled wind speeds
and rain distribution, especially because the rainfall estimator is known to
overestimate rainfall at high wind speeds.

We denote:

  * •

$P^{+}$ as the wind speed distribution on $n_{+}$.

  * •

$P^{-}$ as the wind speed distribution on $n_{-}$.

  * •

$P$ as the wind speed distribution on $n_{-}\cup n_{+}$.

Balancing the dataset to the real-world wind distribution translates to the
following condition:

$\forall x,P(x)=\frac{n_{+}P^{+}(x)+n_{-}P^{-}(x)}{n_{+}+n_{-}}$ (Eq. 4)

As we choose to keep all rain patches and to set $n_{+}=n_{-}$, Eq. 4 leads
to:

$\forall x,P^{-}(x)=\frac{1}{2}(P^{+}(x)-P(x))$ (Eq. 5)

For some wind speeds $x$, $P^{+}(x)$ is higher than twice $P(x)$. In these
cases, we relax the condition from Eq. 5 in order to avoid removing rain
patches. Fig. 3 depicts the wind speed distribution for rain and rainless
patches. The mean squared error between $P$ and $\frac{1}{2}(P^{+}+P^{-})$
reach 8.8%.

Figure Fig. 3: Wind speed distributions for rain (blue) and rainless (red)
patches. Orange curve correspond to the real world wind distribution.

The dataset can be further balanced to ensure that, for each wind speed, the
number of rain and rainless patches is equal. However, this leads to remove of
84% of the data. Appendix 1 compares the performance of this second dataset.
As did not provide improvements, this dataset is left out of the main
document.

####  III-B1 Training, Validation and Test Set Division

After extracting the patches following the distributions $P^{+}$ and $P^{-}$,
they are split into training, validation, and test sets. Each subset preserves
the same distributions. Furthermore, to avoid information leakage, if a patch
from one IW is in a subset, every patch from the same IW belongs to the same
subset. The stochastic brute forcing method described in Algorithm 1 draws
random IWs and computes the distribution of the validation and test subsets,
compares them to the overall distributions, and returns the solution that
minimizes the difference. In this algorithm, $\bar{P}_{e}$ indicates the wind
speed distribution multiplied by the number of patches in $e$ and divided by
the total number of patches. It ensures that the validation and test subsets
each contain approximately 10% of all the patches.

input : The list $L$ of all swath $i$ and their patch distribution $P_{i}$.

output : $L_{val}$ and $L_{test}$, the list of the swaths contained in the
validation and test sets.

1
$\mathcal{L}$ = MAE;

2 $e_{min}=+\infty$;

3 $n_{1}$ = len(L)/10;

4 $n_{2}$ = len(L)/3;

5 for  _$i\leftarrow 0$ to $1000000$ by $1$_ do

6 $n$ = random.integer(min=$n_{1}$, max=$n_{2}$);

7 $candidate$ = random.choice($L$, size=$n$, replace=False);

8 $c_{val}$ = $candidate[:n/2]$;

9 $c_{test}$ = $candidate[n/2:]$;

10
$e_{val}=\mathcal{L}(0.1P^{+},\bar{P}^{-}_{c_{val}})+\mathcal{L}(0.1P^{-},\bar{P}^{-}_{c_{val}})$;

11
$e_{test}=\mathcal{L}(0.1P^{+},\bar{P}^{+}_{c_{test}})+\mathcal{L}(P^{-},0.\bar{P}P^{-}_{c_{test}})$;

12

13 $e=\frac{2\cdot e_{val}\cdot\;e_{e_{t}est}}{e_{val}+e_{test}}$;

14 if  _$e <e_{min}$_ then

15 $L_{val}=c_{val}$;

16 $L_{test}=c_{test}$;

17 $e_{min}=e$;

18

19 end if

20

21 end for

Algorithm 1 Stochastic brute forcing

The initial 19,978 IW observations account for 105,164 rain patches and
2,094,370 rainless patches. At the end of the process, the distribution in the
subsets is as follow:

  * •

168,349 patches from 14,169 IWs in the training set;

  * •

20,944 patches from 1,763 IWs in the test set;

  * •

21,010 patches from 1,763 IWs in the validation set.

Before training the model, we compute the mean and standard deviation of each
channel on the training set and use them to normalize the inputs during
training, validation, and inference. The output, however, is not normalized.
We train the model for 100,000 weight updates (i.e. steps of the stochastic
gradient descent) to minimize the Mean Square Error (MSE) between the
prediction and the wind speed from the atmospheric model, with a batch size of
16 and a learning rate of $10^{-5}$ using the Adam optimizer.

###  III-C Evaluation procedure

To evaluate the impact of each input channel, we train various variants of the
model:

  * •

1 uses only the VV channel.

  * •

2 uses the VV channel, the incidence angle and the a priori wind direction.

  * •

3 uses both the VV and the VH channel, the incidence angle and the a priori
wind direction.

  * •

4 uses both the VV and the VH channel, the incidence angle, the a priori wind
direction and the wind speed prior.

  * •

5 uses only the wind speed prior.

All channels are interpolated to 100 m/px and concatenated on the same grid.

Incidence angles and wind directions from European Centre for Medium-Range
Weather Forecasts (ECMWF) atmospheric model are obtained from the Level-2
products and notably used for the computation of the GMF. Therefore,
architecture 2 contains the same inputs as the GMF though interpolated at a
finer resolution.

Each architecture is trained five times to reduce the impact of random
initialization on the evaluation results. The results is presented as the mean
and standard deviation over these five independent trainings.

We compare the results using the Root Mean Square Error (RMSE) and the Pearson
correlation coefficient (PCC). The PCC is formulated in Eq. 6.

$PCC_{Y,\hat{Y}}=\frac{\mathds{E}[(Y-\mu_{Y})(\hat{Y}-\mu_{\hat{Y}})]}{\sigma_{Y}\sigma_{\hat{Y}}}$
(Eq. 6)

The results are computed against both the groundtruths from the atmospheric
model, which provides a large test set, and against collocations with buoys,
which have good temporal resolution and are in-situ measurements.

##  IV Results

###  IV-A Benchmarking experiments

The performance of the models compared to ECMWF are calculated on the test
subset for each input variant and the baseline GMF. The results of this
analysis can be found in Table I. It appears that the most important input is
the GMF itself, as both 4 and 5 have better results than the other variants.
1, 2 and 3 are unable to achieve better results than the GMF, except under
strong rainfall, even though 2 and 3 have access to all the channels used by
the GMF.

|  | RMSE | PCC
---|---|---|---
|  |  |  |  |
MODEL AND CHANNELS |  RAIN RATE |  Balanced Dataset |  Neutral Dataset |  Balanced Dataset |  Neutral Dataset
|  $[0,1[mm/h$ |  $\Diamond$ 1.38 [0.016] |  $\Diamond$2.40 [0.019] |  $\Diamond$ 89.9% [0.19%] |  $\Diamond$ 74.2% [0.42%]
|  $[1,3[mm/h$ |  $\Diamond$ 1.64 [0.046] |  $\Diamond$2.78 [0.038] |  $\Diamond$ 92.7% [0.39%] |  $\Diamond$ 79.5% [0.58%]
|  $[3,10[mm/h$ |  $\Diamond$ 1.59 [0.038] |  $\Diamond$3.18 [0.070] |  $\Diamond$92.7% [0.38%] |  $\Diamond$ 78.9% [0.62%]
1 $[$VV$]$ |  $\geq 10mm/h$ |  $\Diamond$2.12 [0.052] |  $\bigstar$3.27 [0.135] |  $\Diamond$81.7% [1.03%] |  $\Diamond$ 74.3% [0.90%]
|  $[0,1[mm/h$ |  $\Diamond$ 0.87 [0.009] |  $\Diamond$2.26 [0.019] |  $\Diamond$ 96.2% [0.06%] |  $\Diamond$ 77.7% [0.24%]
|  $[1,3[mm/h$ |  $\Diamond$ 1.04 [0.086] |  $\Diamond$2.87 [0.061] |  $\Diamond$ 97.2% [0.47%] |  $\Diamond$ 77.6% [0.24%]
|  $[3,10[mm/h$ |  $\Diamond$1.19 [0.053] |  $\Diamond$3.39 [0.193] |  $\Diamond$ 96.2% [0.37%] |  $\Diamond$ 75.0% [3.04%]
2 $[$VV, INC, WDIR$]$ |  $\geq 10mm/h$ |  $\Diamond$2.27 [0.151] |  $\Diamond$ 3.86 [0.700] |  $\Diamond$81.7% [1.89%] |  $\Diamond$ 63.2% [9.19%]
|  $[0,1[mm/h$ |  $\Diamond$ 0.83 [0.002] |  $\Diamond$ 2.18 [0.006] |  $\Diamond$ 96.5% [0.02%] |  $\Diamond$ 79.3% [0.07%]
|  $[1,3[mm/h$ |  $\Diamond$ 0.93 [0.020] |  $\Diamond$2.73 [0.022] |  $\Diamond$ 97.7% [0.07%] |  $\Diamond$80.1% [0.29%]
|  $[3,10[mm/h$ |  $\Diamond$1.09 [0.022] |  $\Diamond$3.25 [0.052] |  $\Diamond$96.7% [0.07%] |  $\Diamond$ 78.0% [0.44%]
3 $[$VV, VH, INC, WDIR$]$ |  $\geq 10mm/h$ |  $\Diamond$2.13 [0.050] |  $\Diamond$3.68 [0.310] |  $\Diamond$83.9% [0.66%] |  $\Diamond$ 70.8% [2.95%]
|  $[0,1[mm/h$ |  $\bigstar$0.64 [0.007] |  $\bigstar$1.90 [0.028] |  $\bigstar$97.9% [0.04%] |  $\bigstar$85.0% [0.33%]
|  $[1,3[mm/h$ |  $\bigstar$0.63 [0.015] |  $\bigstar$2.29 [0.075] |  $\bigstar$98.9% [0.03%] |  $\bigstar$87.1% [0.56%]
|  $[3,10[mm/h$ |  $\bigstar$0.78 [0.040] |  $\bigstar$2.55 [0.132] |  $\bigstar$98.4% [0.08%] |  $\bigstar$87.1% [1.12%]
4 $[$VV, VH, INC, WDIR, GMF$]$ |  $\geq 10mm/h$ |  $\bigstar$1.63 [0.162] |  $\Diamond$3.37 [0.113] |  $\bigstar$90.9% [1.17%] |  $\Diamond$ 73.4% [2.31%]
|  $[0,1[mm/h$ |  $\Diamond$0.67 [0.003] |  $\Diamond$ 3.16 [2.650] |  $\Diamond$97.7% [0.02%] |  $\Diamond$ 80.7% [17.55%]
|  $[1,3[mm/h$ |  $\Diamond$0.68 [0.005] |  $\Diamond$ 4.17 [3.582] |  $\Diamond$98.8% [0.01%] |  $\Diamond$82.5% [12.67%]
|  $[3,10[mm/h$ |  $\Diamond$0.88 [0.021] |  $\Diamond$ 4.53 [3.739] |  $\Diamond$98.0% [0.03%] |  $\Diamond$83.4% [8.19%]
5 $[$GMF$]$ |  $\geq 10mm/h$ |  $\Diamond$1.94 [0.087] |  $\Diamond$ 4.79 [2.917] |  $\Diamond$87.9% [0.66%] |  $\Diamond$ 74.0% [8.19%]
|  $[0,1[mm/h$ |  $\Diamond$ 0.77 |  $\Diamond$ 2.41 |  $\Diamond$ 97.0% |  $\Diamond$ 81.2%
|  $[1,3[mm/h$ |  $\Diamond$ 0.84 |  $\Diamond$ 3.16 |  $\Diamond$ 98.1% |  $\Diamond$ 80.0%
|  $[3,10[mm/h$ |  $\Diamond$ 1.25 |  $\Diamond$ 3.42 |  $\Diamond$ 96.5% |  $\Diamond$ 81.9%
GMF |  $\geq 10mm/h$ |  $\Diamond$ 4.65 |  $\Diamond$ 3.70 |  $\Diamond$ 52.5% |  $\bigstar$ 75.4%
TABLE Table I: Comparison of the five variants of the model and the two datasets. RMSE and PCC are computed on the respective test set and for five training with random initialization. Results are given as mean and standard deviation in brackets. The best result for each metric is indicated by $\bigstar$. Results better than the GMF are in bold. (a) | (b)
---|---
|
|
(c) | (d)
|
|
(e) | (f)
|
Figure Fig. 4: Observation from the November 11th 2018 at 04:56:47. Sea
Surface Roughness (SSR) of the VV channel in dB (a), geophysical model
function estimated wind speed in m/s (b), deep learning estimated wind speed
in m/s (d), prediction of the rainfall estimator (c) and transects of the
green (e) and red (f) lines, with rainfall predicted higher than 10 mm/h
indicated as grey areas.

Examples of rain-induced wind speed overestimation are presented in Fig. 4.
Fig. 4.a shows the SSR on a logarithmic scale, with bright patterns appearing
in the observation and segmented as rainfall in Fig. 4.c. These patterns
correlate with high wind speeds estimated by the GMF in Fig. 4.b. The deep
learning method in Fig. 4.d ignores this spike in the SSR, but it also seems
to underestimate the wind speed in the bottom-left quarter of the observation.
Both of these behaviours are especially visible in the transect shown in Fig.
4.f.

###  IV-B Application to SAR observation with groundtruthed in-situ data

However, ECMWF wind speeds are model data and not in-situ, which can be
obtained using anemometers on buoys. Using the dataset created in [35], 4732
collocation points between Sentinel-1 and National Data Buoy Center (NDBC)
buoys are identified. The rain prediction model estimates that 4643 of these
points are rainless, 75 record rainfall of more than 1 mm/h, and 14 record
rainfall of more than 3 mm/h. On a side note, the height at which in-situ
measurements were taken varies, with most being between 3.8 m and 4.1 m above
sea level. As mentioned in [35], the SAR inversion and deep learning
prediction are both normalized to the altitude of the corresponding in-situ
measurement. Denoting $w$ the wind speed at 10 m estimated from the SAR, the
wind speed at elevation $h$ is given by the exponential law [36]:

$\color[rgb]{0,0,0}w_{h}=w\cdot\left(\frac{h}{10}\right)^{0.11}$ (Eq. 7)

Table II indicates that the performances of the deep learning are higher than
the GMF for both the Root Mean Square Error and the Pearson Correlation
Coefficient for all rain ranges. The RMSE decreases by 0.04, 0.37 and 1.33 m/s
for rainless, light rain, and moderate rain-situations, respectively. The bias
decreases by 0.26 and 1.01 m/s under light and moderate rain, but increases by
0.02 m/s for rainless situations. Table II also demonstrates the importance of
the dataset building scheme as a dataset composed of collocations without the
aformentionned sample selection, referred to as the ”neutral dataset,”
consistently has lower performances than the balanced dataset.

|  |  Balanced Dataset |  Neutral Dataset | GMF
---|---|---|---|---
Bias | $<1mm/h$ |  $\Diamond$ 0.73 [0.04] |  $\Diamond$ 1.32 [0.04] |  $\bigstar$ 0.71
$[1,3]mm/h$ |  $\bigstar$1.38 [0.04] |  $\Diamond$1.47 [0.04] |  $\Diamond$ 1.64
$>3mm/h$ |  $\bigstar$0.92 [0.07] |  $\Diamond$ 1.96 [0.29] |  $\Diamond$ 2.38
RMSE | $<1mm/h$ |  $\bigstar$1.44 [0.03] |  $\Diamond$ 1.76 [0.12] |  $\Diamond$ 1.48
$[1,3]mm/h$ |  $\bigstar$1.81 [0.04] |  $\Diamond$1.95 [0.18] |  $\Diamond$ 2.18
$>3mm/h$ |  $\bigstar$1.60 [0.10] |  $\Diamond$2.42 [0.21] |  $\Diamond$ 2.93
PCC | $<1mm/h$ |  $\bigstar$93.6% [0.16%] |  $\Diamond$ 92.9% [0.20%] |  $\Diamond$ 93.4%
$[1,3]mm/h$ |  $\bigstar$96.3% [0.22%] |  $\Diamond$ 93.4% [0.19%] |  $\Diamond$ 95.3%
$>3mm/h$ |  $\bigstar$95.9% [0.35%] |  $\Diamond$93.4% [2.04%] |  $\Diamond$ 91.3%

TABLE Table II: Bias, Root Mean Square Error, and Pearson Correlation
Coefficient of model 4, the GMF, for each rainfall level. The best result for
each metric and rainfall level is indicated by $\bigstar$. Results better than
the baseline are in bold. Results are given as mean and standard deviation in
brackets.

Fig. 5 presents a comparison of the performances against NDBC buoys for
various wind speeds. Unfortunately, collocations are scarce at very low and
high wind speeds, which are the intervals where both the GMF and deep learning
models have higher errors.

(a) | (b)
---|---
|
Figure Fig. 5: Evaluation of the Root Mean Square Error across the wind speed
(a) and number of collocation with NDBC buoys in rainless areas (b).

In the following, we observe two cases where rainfall was detected on the buoy
position at the time of observation.

####  IV-B1 2017-01-08 01:58:19 at NDBC 46054

The observation from 2017-01-08 01:58:19 covers the north of the Californian
Channel Islands (Fig. 6.a). Several meteorological buoys are dispersed over
the channel, including NDBC 46054 and NDBC 46053, which are indicated as red
dots. The wind speed over the area is mostly around 6 m/s, but a squall line
appears at the position of NDBC 46054 and spans over a dozen kilometers. A
surge in the air column reflectivity is recorded by weather radar from the
NEXRAD network. The ground station is located at 119 km. The GMF indicates
very high wind speeds, higher than 20 m/s (Fig. 6.d). The deep learning model
attenuates these values to between 6 m/s and 8 m/s (Fig. 6.e).

---

Figure Fig. 6: Sea surface roughness from the January 08th 2017 at 01:58:19 in
VV channel (a), zoom on an area of 35x35 km centred on the buoy NDBC 45054
(b), NEXRAD base reflectivity acquired at 01:57:58 (c), wind speed given by
the GMF (d) and by the deep learning model (e).

For NDBC 46054 (Fig. 7.a), only one measurement of wind speed and direction
per hour is available. It recorded a wind speed of 6.3 m/s eight minutes
before the SAR observation. The GMF and the deep learning model estimated wind
speeds of 15.1 m/s and 5.9 m/s, respectively. While the temporal resolution of
NDBC 46054 is one measurement per hour, NDBC 46053 records data every ten
minutes. Furthermore, the gust front appears to be moving toward the right
part of the observation. This can be seen in the time series in Fig. 6 as a
large variation in wind direction between 02:40:00 and 03:00:00. The variation
in wind speed seems to precede the variation in direction, first increasing
then decreasing to a lower wind regime. On NDBC 46053 (Fig. 7.ba), the GMF and
the deep learning model agree on a wind speed of 4.5 m/s, which is slightly
lower than the in-situ data of 5.4 m/s. Since the distance between NDBC 46054
and NDBC 46053 is approximately 60 km, the progression of the gust front can
be estimated to be around 90 km/h. With a width of around 5 or 6 km, the whole
system would pass the buoys in three minutes. This means that even NDBC 46053
may not have been able to accurately estimate the wind speed due to its low
temporal resolution. However, it is worth noting that even the gust speed at
NDBC 46054, defined as the maximum wind speed over a given number of seconds,
does not record a speed higher than 9 m/s.

(a)
---
(b)
Figure Fig. 7: Time series of the NDBC buoy wind measurements around January
08th 2017 01:58:19 for NDBC 46054 (a) and NDBC 46053 (b), and the estimation
from the GMF, the deep learning model and the atmospheric model.

####  IV-B2 SAR-20191006T232853 NDBC-41009

The observation from 2019-10-06 23:28:53 was recorded on the east coast of
Florida. While most of the swath covers the marshes around Orlando and Cap
Canaveral rather than the ocean, convective precipitation can be observed in
the right part of the image (Fig. 8.a). The cells are moving downward (north-
north-east), as indicated by the stronger gradient of the convective front.
Since the wind from the convection is opposing the underlying wind regime, an
area of lower wind speed appears as an area of lower backscatter. Rainfall was
detected by a weather radar from the NEXRAD network located at 64 km (Fig.
8.c). The GMF is impacted by these rain signatures and estimates a very high
local wind speed (Fig. 8.d). The deep learning model is less affected by the
rain signatures, but also appears to blur the low wind speed area (Fig. 8.e).

---
Figure Fig. 8: Sea surface roughness from October 06th 2019 at 23:28:53 in VV
channel in dB (a), zoom on an area of 35x35 km around the buoy NDBC 41009 (b),
NEXRAD observation acquired at 23:27:33 (c), wind speed given by the GMF (d)
and by the deep learning model (e).

The time serie from NDBC 41009 in Fig. 9 indicates that the lower
backscattering was indeed caused by a drop in wind speed rather than a change
in direction, as the latter does not significantly change during the passage
of the convective cell (possibly because the underlying wind regime is
strong). It does record a sudden drop in wind speed to 7.5 m/s one minute
after the SAR observation, while the GMF and the deep learning model estimated
wind speeds of 13.7 m/s and 8.9 m/s, respectively.

Figure Fig. 9: Time series of the NDBC buoy wind measurements around October
06th 2019 at 23:28:53 for NDBC 41009, and the estimation from the GMF, the
deep learning model and the atmospherical model.

##  V Conclusion

Previous studies have shown that high-resolution rain signatures can be
automatically extracted from SAR observations. Using this SAR rainfall
segmenter, we built a wind estimation dataset where 50% of the patches contain
rainfall examples. Samples were chosen so that a SAR-based and a SAR-
independent wind speed model agree on non-rain pixels, ensuring that their
estimates are close to the true wind speed. A UNet architecture was trained on
this dataset to estimate wind speeds based on the SAR-independent atmospheric
model. We tested several input combinations and found that the most important
parameter was the wind speed prior from the geophysical model function, which
the deep learning model had difficulty emulating.

Collocations with buoy in-situ measurements show that the model outperforms
the current Geophysical Model Function (GMF) on rain areas, reducing the
RootMean Square Error (RMSE) by 27% (resp. 45%) for rain rates higher than 1
mm/h (resp. 3 mm/h). On rainless areas, performances are similar with a small
reduction of the RMSE by 2.7%. However, since the buoys have a time resolution
of ten minutes, some quick sub-mesoscale processes, such as gust fronts, are
difficult to register. The limited spatial range of the buoys also makes it
challenging to observe rare phenomena. Future work should address these
concerns.

A secondary dataset was created differing with the main dataset by Eq. 4.
Here, the balancing policy is defined as:

$\forall x,P(x)=P^{+}(x)=P^{-}(x)$ (Eq. 8)

The distributions $P^{+}$ and $P^{-}$ are presented in Fig. 10. The balancing
is performed for every wind speed, which lead to remove 84% of the rain
patches -especially at 5 and 10 m/s- since the number of rain patches at 8 m/s
is limited.

Figure Fig. 10: Wind speed distribution for rain (blue) and rainless (red)
patches.

Models were optimized under the same training process as in subsection III-B1,
in particular with the same number of weight updates. Comparison with the
first balancing scheme display lower performances despite the more accurate
balancing, as indicated Table III.

|  |  Dataset I |  Dataset II | GMF
---|---|---|---|---
Bias | $<1mm/h$ |  $\Diamond$ 0.73 [0.04] |  $\Diamond$ 0.73 [0.08] |  $\bigstar$ 0.71
$[1,3]mm/h$ |  $\bigstar$1.38 [0.04] |  $\Diamond$1.40 [0.10] |  $\Diamond$2.18
$>3mm/h$ |  $\Diamond$0.92 [0.07] |  $\bigstar$0.83 [0.14] |  $\Diamond$ 2.38
RMSE | $<1mm/h$ |  $\bigstar$1.44 [0.03] |  $\Diamond$1.46 [0.06] |  $\Diamond$ 1.48
$[1,3]mm/h$ |  $\bigstar$1.81 [0.04] |  $\Diamond$1.85 [0.10] |  $\Diamond$ 2.18
$>3mm/h$ |  $\bigstar$1.60 [0.10] |  $\Diamond$1.61 [0.10] |  $\Diamond$ 2.93
PCC | $<1mm/h$ |  $\bigstar$93.6% [0.16%] |  $\Diamond$ 93.3% [0.15%] |  $\Diamond$ 93.4%
$[1,3]mm/h$ |  $\bigstar$96.3% [0.22%] |  $\Diamond$96.1% [0.23%] |  $\Diamond$ 95.3%
$>3mm/h$ |  $\bigstar$95.9% [0.35%] |  $\Diamond$95.0% [0.46%] |  $\Diamond$ 91.3%

TABLE Table III: Bias, Root Mean Square Error of model 4, the GMF, for each
rainfall level. The best result for each rainfall level is indicated by
$\bigstar$. Results better than the baseline are in bold. Results of deep
learning models are given as mean and standard deviation in brackets.

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|  Aurélien COLIN After graduating from the IMT Atlantique as an engineer, he
began a Ph.D and is currently studying the segmentation of various
meteorological and ocean phenomena on Synthetic Aperture Radar using deep
learning models.
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|  Pierre TANDEO was born in France in 1983. He received the M.S. degree in
applied statistics from Agrocampus Ouest, Rennes, France, and the Ph.D. degree
from the Oceanography from Space Laboratory at IFREMER, Brest, France, in
2010. Then, he spent two years as a Postdoctoral Researcher with the
Atmospheric Science Research Group, University of Corrientes, Argentina, and
three years at Télécom Bretagne, Brest, France. Since 2015, he is an associate
professor at IMT Atlantique, Brest, France, and a researcher at Lab-STICC,
CNRS, France. Since 2019, he is an associate researcher at the Data
Assimilation Research Team, RIKEN Center for Computational Science, Kobe,
Japan. His main research interests are focused on IA, data assimilation, and
inverse problems for geophysics.
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|  Charles PEUREUX is a research engineer at CLS since 2020 where he is mainly
involved with SAR oceanography. He previously worked at IFREMER where he
defended his thesis on the observation and modelling of the directional
properties o short gravity waves. Thereafter, he worked on the SKIM satellite
mission, especially for the production of high resolution sea state numeral
simulations.
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|  Romain HUSSON is a scientist and project engineer at CLS, in the
Environmental Monitoring and Climate Business Unit. After working as intern at
NASA JPL and as YGT at ESRIN on SAR wind-wave-current activities, he completed
in 2012 his PhD in Physical Oceanography at CLS Brest developing methods to
estimate synthetic swell field from SAR wave mode observations. Since then, he
has been involved in several applicative and R&D projects such as wind/wave
products for marine renewable energy and contributed to several ESA and
European projects such as Cal/Val activities for ENVISAT and S-1 mission
performance center as Level-2 expert.
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|  Ronan FABLET is a Professor at IMT Atlantique and a research scientist at
Lab-STICC in the field of Data Science and Computational Imaging. He is quite
involved in interdisciplinary research at the interface between data science
and ocean science, especially space oceanography and marine ecology. His
current research interests include deep learning for dynamical systems and
applications to the understanding, analysis, simulation and reconstruction of
ocean dynamics, especially using satellite ocean remote sensing data.
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