# Decoding Orders for Securing Untrusted NOMA

Sapna Thapar, Deepak Mishra, and Ravikant Saini S. Thapar and R. Saini are
with the Department of Electrical Engineering, Indian Institute of Technology
Jammu, Jammu $\&$ Kashmir 181 221, India (e-mail<EMAIL_ADDRESS>ravikant.saini@iitjammu.ac.in). D. Mishra is with the School of Electrical
Engineering and Telecommunications, the University of New South Wales (UNSW)
Sydney, NSW 2052, Australia (e-mail: d.mishra@unsw.edu.au).This research work
has been supported by the TCS RSP Fellowship.

###### Abstract

This letter focuses on exploring a new decoding order to resolve the secrecy
issue among untrusted non-orthogonal multiple access users. In this context,
firstly, we identify the total number of possible decoding orders
analytically. Then, we propose a decoding order strategy ensuring positive
secrecy rate for all users, and obtain the number of feasible secure decoding
orders based on this proposed strategy numerically. Thereafter, we present a
suboptimal policy to avoid the computational complexity involved in finding
the best secure decoding order. Numerical results illustrate that the
suboptimal solution provides a performance gain of about $137\%$ over the
relevant benchmark.

Index Terms$-$ 5G communications, physical layer security, non-orthogonal
multiple access, untrusted users.

##  I Introduction

Non-orthogonal multiple access (NOMA) is confronted with critical secrecy
issues due to the broadcast nature of wireless transmission, and successive
interference cancellation (SIC) based decoding at the receivers [1]. Physical
layer security (PLS) has recently emerged as a revolutionizing approach for
security provisioning over wireless links in NOMA [2].

Utilizing the concept of PLS, existing works have mostly focused on securing
NOMA against external eavesdroppers [2]. Due to the inherent possibility of
decoding other users’ data while performing SIC, the design of secure NOMA
when users are assumed to be untrusted is a challenge. Untrusted users’
network is a practical and hostile situation where users have no mutual trust,
and each user focuses on securing its own data from all others, which leads to
complex resource allocations [3], [4]. In this context, assuming only far
users as untrusted, sum secrecy rate of trusted near users is investigated in
[5] for a NOMA system. In [6] also, the far user is assumed as untrusted and
secrecy outage probability of trusted near user is analyzed. However, near
user has to decode far user’s data to apply SIC, this results in a crucial
security risk for the far user in case of untrusted near user. In this regard,
secure decoding order for a two-user untrusted NOMA is proposed in [4] which
ensures positive secrecy rate for both the users.

Compared to the two-user case [4], the selection of secure decoding order for
multi-user untrusted scenario is a combinatorial and complicated problem.
Besides, [4] assumed an ideal setup with perfect SIC, which is not practical
due to various implementation problems such as decoding error and complexity
scaling [1], [7]. To this end, considering imperfect SIC at receivers, we
study secure decoding order selection problem for multi-user untrusted NOMA,
which to the best of our knowledge, has not been investigated yet in the
literature.

The key contributions are as follows: (1) A novel decoding order strategy for
multi-user untrusted NOMA system is proposed. The key idea is to obtain
positive secrecy rate at each user. (2) For a system having $N$ untrusted
users, we analytically obtain the total number of possible decoding orders.
Based on the proposed decoding order strategy, we present insight on the count
of possible secure decoding orders that ensure positive secrecy rate to all
users. (3) To reduce the computational complexity involved in exploring the
best secure decoding order, we explore such favourable secure decoding orders
that can provide higher secrecy rate for each user. We also offer a suboptimal
decoding order policy. (4) Numerical results are provided to validate the
proposed approach and quantify the achievable secrecy rate performance gain by
the suboptimal solution over the relevant benchmark.

##  II Secure NOMA with Untrusted Users

###  II-A System Model and NOMA Principle

We consider the power-domain downlink NOMA system with a base station (BS) and
$N$ untrusted users. The $n$-th user is denoted by $U_{n}$, where
$n\in\mathcal{N}=\\{1,2,...,N\\}$. Each node in the system is equipped with a
single antenna. The Rayleigh fading channel gain coefficient between BS and
$U_{n}$ is denoted by $h_{n}$. The channel power gain $|h_{n}|^{2}$ follows
exponential distribution with mean $\lambda_{n}=L_{c}d_{n}^{-e}$, where
$L_{c}$, $e$, and $d_{n}$ denote path loss constant, path loss exponent, and
distance from BS to $U_{n}$, respectively [6]. Without loss of generality,
channel power gains are sorted such as
$|h_{1}|^{2}>|h_{2}|^{2}>...>|h_{N}|^{2}$. The BS superimposes users’
information signals and broadcasts the generated mixture with total
transmission power $P_{t}$. $\alpha_{n}$ denotes power allocation (PA)
coefficient, i.e., the fraction of $P_{t}$ assigned to $U_{n}$ satisfying
$\sum_{p\in\mathcal{N}}\alpha_{p}=1$. At receiver side, $U_{n}$ extracts the
desired signal from the superimposed signal by cancelling the interfering
signals through the SIC process. Without loss of generality, we assume
received additive white Gaussian noise for all users with zero mean and
variance $\sigma^{2}$. We consider an imperfect SIC process where SIC could
not be performed ideally due to practical limitations. Therefore, residual
interference from imperfectly decoded signals exists while decoding later
users. The residual interference can be expressed by a factor $\zeta$,
$(0\leq\zeta\leq 1)$, where $\zeta=0$ represents perfect SIC while $\zeta=1$
refer to totally unsuccessful SIC [7].

###  II-B Proposed Decoding Order Representation

While performing SIC, each user decodes its data as well as other users’ data
in a certain sequence. Thus, the collection of such sequences allocated for
each user in the system can be defined by “decoding order” of the system.
According to conventional decoding order strategy [1], a stronger user first
decodes data of all weaker users in ascending order of users’ channel gain
strength and then decodes its own data, and a weaker user never decodes the
stronger user’s data. Conversely, in the case of untrusted users, each user
can decode data of itself as well as all other users. Note that SIC on the
receivers’ side is a physical layer capability which enables the receiver to
decode packets that arrive collectively. Therefore, each user can decode data
of any user at any stage [5]-[4], [8] which results in a large number of
decoding orders, which we will discuss in more detail in Section III-A.
Considering these possibilities into account, let us represent the decoding
order for an $N$-user system as an $N\times N$ matrix $\mathbf{D}_{o}$, where
$o$ is an index indicating the $o$-th decoding order. The $m$-th column of
matrix $\mathbf{D}_{o}$, where $m\in\mathcal{N}$, is represented by a column
vector $\mathbf{d}_{m}$ of size $N\times 1$, which demonstrates the SIC
sequence followed by $U_{m}$. More specifically, $k$-th element of
$\mathbf{d}_{m}$, i.e., $[\mathbf{d}_{m}]_{k}=n$ signifies that $U_{m}$
decodes data of $U_{n}$ at $k$-th stage, where $n,k\in\mathcal{N}$ and
$[\mathbf{d}_{m}]_{1}\neq[\mathbf{d}_{m}]_{2}\hskip 2.84544pt\neq...\hskip
2.84544pt\neq[\mathbf{d}_{m}]_{N}$.

###  II-C Achievable Secrecy Rates at Users in Untrusted NOMA

The signal-to-interference-plus-noise-ratio, $\gamma^{o}_{nm}$, when $U_{m}$
decodes data of $U_{n}$, where $m,n\in\mathcal{N}$, can be defined as

$\gamma^{o}_{nm}=\frac{\alpha_{n}|h_{m}|^{2}}{\big{(}\zeta\sum_{i\in\mathcal{U}_{om}^{bn}}\alpha_{i}+\sum_{j\in\mathcal{U}_{om}^{an}}\alpha_{j}\big{)}|h_{m}|^{2}+\frac{1}{\rho_{t}}}.$
(1)

$\mathcal{U}_{om}^{bn}$ denotes set of users’ indices, $i$, which are decoded
before decoding of $U_{n}$ by $U_{m}$, where
$i\in\mathcal{N}\setminus\\{n\\}$. Similarly, set $\mathcal{U}_{om}^{an}$
denotes those users’ indices, $j$, which will be decoded after decoding of
$U_{n}$ by $U_{m}$, where $j\in\mathcal{N}\setminus\\{n\\}$. Thus,
$\mathcal{U}_{om}^{bn}$ and $\mathcal{U}_{om}^{an}$ are disjoint sets such
that $\mathcal{U}_{om}^{bn}\cap\mathcal{U}_{om}^{an}=\varnothing$ and
$\mathcal{U}_{om}^{bn}\cup\mathcal{U}_{om}^{an}=\mathcal{N}\setminus\\{n\\}$.
$\rho_{t}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{P_{t}}{\sigma^{2}}$ is BS
transmit signal-to-noise ratio. The corresponding achievable data rate is
given by

$R^{o}_{nm}=\log_{2}(1+\gamma^{o}_{nm}).$ (2)

Utilizing the concept of PLS, we investigate the achievable secrecy rate
$R^{o}_{sn}$ at $U_{n}$, which is given by [2], [3]

$R^{o}_{sn}=\\{R^{o}_{nn}-\underset{m\in\mathcal{N}\setminus\\{n\\}}{\max}R^{o}_{nm}\\}^{+},$
(3)

where $\\{\triangle\\}^{+}=\max(0,\triangle)$. Note that the key idea of
achieving positive secrecy rate at $U_{n}$ is to ensure that the condition
$R^{o}_{nn}>R^{o}_{nm}$, simplified as $\gamma^{o}_{nn}>\gamma^{o}_{nm}$, must
be satisfied.

##  III Decoding Order Strategy for Secure NOMA

###  III-A Possible Decoding Orders for Untrusted NOMA

Below we first identify the total number of possible decoding orders for an
$N$-user untrusted NOMA system.

###### Proposition 1

The total number of possible decoding orders for an $N$-user untrusted NOMA
system is equal to $(N!)^{N}$.

###### Proof:

In an untrusted NOMA system, each user can decode its data as well as other
users’ data. Thus, for an $N$-user system, each user has a total of $N$ stages
to decode data of all users. Considering the possibility to decode itself and
other users’ data at any stage, the number of permutations at each user is
$N!$. As a result, we see that the count of total permutations for an $N$-user
system, i.e., the number of possible decoding orders is $\underbrace{N!\times
N!\hskip 2.84544pt...\hskip 2.84544ptN!}_{N\text{times}}=(N!)^{N}$. ∎

Let us define a set of total possible decoding orders for an $N$-user system
as $\mathcal{T}=\\{\mathbf{D}_{o}|1\leq o\leq(N!)^{N}\\}$.

###  III-B Conventional NOMA Strategy in Untrusted Scenario

Extending the conventional decoding order strategy [1] in untrusted scenario,
all users decode signals in ascending order of users’ channel gain strength.
Thus, $\gamma^{o}_{nm}$ can be written as

$\gamma^{o}_{nm}=\frac{\alpha_{n}|h_{m}|^{2}}{\big{(}\zeta\sum_{i=n+1}^{N}\alpha_{i}+\sum_{j=1}^{n-1}\alpha_{j}\big{)}|h_{m}|^{2}+\frac{1}{\rho_{t}}}.$
(4)

Below we provide a key result on the inability of conventional NOMA strategy
to secure an untrusted NOMA system.

###### Lemma 1

Utilizing the conventional decoding order strategy of NOMA in an untrusted
scenario, the data of a weaker user cannot be secured from its respective
stronger user.

###### Proof:

To analyze the secrecy performance for a weaker user against the stronger
user, we study secrecy rate $R^{o}_{sn}$ at $U_{n}$ against $U_{m}$ as defined
in (3), where $m<n$ and $m,n\in\mathcal{N}$. The required condition
$\gamma^{o}_{nn}>\gamma^{o}_{nm}$ given in Section II-C for positive
$R^{o}_{sn}$ at $U_{n}$ gives $|h_{n}|^{2}>|h_{m}|^{2}$ which is infeasible as
$|h_{m}|^{2}>|h_{n}|^{2}$ is considered. Thus, positive secrecy rate for a
weaker user against the stronger user cannot be obtained. ∎

Next, we provide a result on the secrecy rate performance for stronger users
against weaker users by Corollary 1.

###### Corollary 1

With conventional NOMA strategy, the stronger user’s data is always safe from
its respective weaker user.

###### Proof:

Using (3), we investigate $R^{o}_{sn}$ at $U_{n}$ against $U_{m}$, where $m>n$
and $m,n\in\mathcal{N}$. We find that $\gamma^{o}_{nn}>\gamma^{o}_{nm}$,
required condition for positive $R^{o}_{sn}$, gives a feasible condition
$|h_{n}|^{2}>|h_{m}|^{2}$, which ensures positive secrecy rate for $U_{n}$. ∎

###### Remark 1

It can be easily inferred from the aforementioned results that by extending
the conventional decoding order strategy in an untrusted scenario, only the
strongest user’s data is secured from all other users, and therefore, this
strategy is not suitable for a secure NOMA system with untrusted users.

###  III-C Proposed Decoding Order Strategy for Secure NOMA

Considering the motive to secure each user’s data from all others, below we
propose a new decoding order strategy.

###### Theorem 1

If $U_{n}$ decodes its data at $k$-th stage and $U_{m}$ decodes data of
$U_{n}$ at $k^{\prime}$-th stage, where $m,n,k,k^{\prime}\in\mathcal{N}$,
$m<n$, and $k^{\prime}<k$, then the weaker user’s data can be safeguarded
against the stronger user with a suitable constraint on PA .

###### Proof:

We find $\gamma^{o}_{nn}$ and $\gamma^{o}_{nm}$ by using (1) to analyze the
secrecy rate $R^{o}_{sn}$ defined in (3) for $U_{n}$ against $U_{m}$, where
$m<n$ and $m,n\in\mathcal{N}$. For positive $R^{o}_{sn}$ at $U_{n}$, the
required condition $\gamma^{o}_{nn}>\gamma^{o}_{nm}$ given in Section II-C
gives

$\frac{\zeta\sum_{i\in\mathcal{U}_{om}^{bn}}\alpha_{i}+\sum_{j\in\mathcal{U}_{om}^{an}}\alpha_{j}+\frac{1}{\rho_{t}}}{\zeta\sum_{i\in\mathcal{U}_{on}^{bn}}\alpha_{i}+\sum_{j\in\mathcal{U}_{on}^{an}}\alpha_{j}+\frac{1}{\rho_{t}}}>\frac{|h_{m}|^{2}}{|h_{n}|^{2}}.$
(5)

If $U_{n}$ decodes its data at $k$-th stage and $U_{m}$ decodes data of
$U_{n}$ at $k^{\prime}$-th stage, where $k^{\prime}<k$ and
$k,k^{\prime}\in\mathcal{N}$, then it means $U_{m}$ decodes data of $U_{n}$ at
least one stage before the stage of $U_{n}$ decoding itself. As a result,
$\mathcal{U}_{om}^{bn}$ has at least one less interfering term in comparison
to $\mathcal{U}_{on}^{bn}$, and $\mathcal{U}_{om}^{an}$ has at least one more
interfering term compared to $\mathcal{U}_{on}^{an}$. Thus, due to more
interfering terms in $\mathcal{U}_{om}^{an}$, we always find a PA condition on
solving (5). This result shows that we can obtain positive secrecy rate for
the weaker user against the stronger user with a suitable PA constraint [4]. ∎

Next, a key result on the secrecy rate performance for stronger users against
weaker users is given by Corollary 2.

###### Corollary 2

By using the proposed decoding order strategy stated in Theorem 1, the
stronger user’s data can also be secured from the respective weaker user.

###### Proof:

For positive $R^{o}_{sn}$ at $U_{n}$ against $U_{m}$ when $m>n$, the required
condition $\gamma^{o}_{nn}>\gamma^{o}_{nm}$ gives either a feasible condition
$|h_{n}|^{2}>|h_{m}|^{2}$ or a suitable PA constraint. This result ensures
that positive secrecy rate at $U_{n}$ can be obtained. ∎

Based on the proposed decoding order strategy given in Theorem 1, a set of
decoding orders can be obtained which ensure positive secrecy rate for each
user against all other users. This set is denoted by $\mathcal{S}$ and is
termed as set of secure decoding orders. Let us analytically define the set of
these secure decoding orders for an $N$-user untrusted NOMA system as
$\mathcal{S}=\\{\mathbf{D}_{o}|1\leq o\leq(N!)^{N},\text{and}\hskip
2.84526pt[\mathbf{d}_{n}]_{k}=n,[\mathbf{d}_{m}]_{k^{\prime}}=n,m<n,k^{\prime}<k,m,n,k,k^{\prime}\in\mathcal{N}\\}$.
Thus, $\mathcal{S}\subset\mathcal{T}$. Note that since the motive is to secure
data of each user from all others, we further focus on the secure decoding
orders obtained based on the proposed decoding order strategy in Theorem 1.

###### Remark 2

It is obtained numerically that the count of secure decoding orders based on
the proposed decoding order strategy for $2$, $3$, and $4$ users are $1$,
$12$, and $3036$, respectively.

###### Remark 3

We have studied the system with $N\leq 4$ due to two major reasons: first, the
number of total decoding orders is enormous with more users due to which
excessive computational complexity occurs in finding secure decoding orders
and second, the number of users should not be too large in NOMA because it is
an interference-limited system and implementation complexity increases at the
transmitter side and receiver side with an increase in the number of users
[9]. However, the investigation can be extended for more users.

##  IV Low-Complexity Suboptimal Design

After identifying set $\mathcal{S}$ of secure decoding orders, we focus on
finding an optimal secure decoding order that can maximize the minimum secrecy
rate among users. The corresponding optimization problem over set
$\mathcal{S}$ can be formulated as

$\displaystyle
OP:\underset{\mathbf{D}_{o}\in\mathcal{S}}{\max}\quad\underset{n\in\mathcal{N}}{\min}\quad
R^{o}_{sn},\quad$ $\displaystyle\text{s.t.}\quad C1:R^{o}_{sn}>0.$

$OP$ is a complex combinatorial problem because a feasible PA is to be
obtained for each decoding order that satisfies $C1$ and optimization is to be
performed over the set $\mathcal{S}$ of secure decoding orders which keep
increasing with $N$ as given in Remark 2. Therefore, to reduce this
complexity, we select such favourable secure decoding orders from set
$\mathcal{S}$ that can provide higher secrecy rate for each user by the
approach given below.

###  IV-A Favourable Secure Decoding Orders

###### Proposition 2

The favourable secure decoding orders providing relatively higher secure rate
are those in which each user decodes its own data after decoding the data of
all others.

###### Proof:

For a decoding order $\mathbf{D}_{o}$, the secrecy rate $R^{o}_{sn}$ for each
user $U_{n}$ defined in (3) can be increased by increasing $R^{o}_{nn}$, where
$n\in\mathcal{N}$. From (2), we note that $R^{o}_{nn}$ increases with an
increase in $\gamma^{o}_{nn}$, and a maximum value of $\gamma^{o}_{nn}$ is
obtained when interference from other users is minimum. If $U_{n}$ decodes its
data at the last stage, then $\mathcal{U}_{on}^{an}=\varnothing$ and
$\mathcal{U}_{on}^{bn}=\mathcal{N}\setminus\\{n\\}$. As a result, a maximum
value of $\gamma^{o}_{nn}$ is obtained since only the residual interference
due to imperfect SIC remains in the denominator of $\gamma^{o}_{nn}$. Thus,
decoding of own data at the last stage by each user can improve secrecy rate
performance. ∎

Let us represent a set of such favourable secure decoding orders in which each
user decodes its data at the end as $\mathcal{L}$. For all remaining secure
decoding orders of set $\mathcal{S}$ in which no user decodes its data at the
last stage, we define a set $\mathcal{O}$. Here
$\mathcal{L}\subset\mathcal{S}$, $\mathcal{O}\subset\mathcal{S}$,
$\mathcal{L}\cap\mathcal{O}=\varnothing$ and
$\mathcal{L}\cup\mathcal{O}=\mathcal{S}$. Since the achievable secrecy rate at
each user is less in the secure decoding orders of set $\mathcal{O}$ in
comparison to the secure decoding orders of set $\mathcal{L}$, we neglect the
set $\mathcal{O}$ to reduce the computational complexity in finding the
optimal solution. Next, an analytical insight on the count of favourable
secure decoding orders is provided.

###### Proposition 3

For an $N$-user system, the total number of favourable secure decoding orders
is $((N-1)!)^{N}$.

###### Proof:

In case of decoding own data at the last stage, each user has $(N-1)$ stages
to decode other users’ data. Thus, the number of permutations at each user
$(N-1)!$. As a result, the count of favourable secure decoding orders for
$N$-user system is $\underbrace{(N-1)!\times(N-1)!\hskip 2.84544pt...\hskip
2.84544pt(N-1)!}_{N\text{ times}}=((N-1)!)^{N}$. ∎

From Proposition 3, we observe that secure decoding orders in set
$\mathcal{L}$, are still in large number. Therefore, to avoid any
computational complexity, we now design a suboptimal policy.

###  IV-B Suboptimal Decoding Order Policy

Observing the dependence of secrecy rate performance on PA, we consider two PA
schemes such as (i) lesser PA to the weaker user in comparison to the stronger
user (LPWU), and (ii) lesser PA to the stronger user in comparison to the
weaker user (LPSU). Let us define PA coefficient $\alpha_{n}$ for $U_{n}$ as

$\alpha_{n}=\frac{1}{{(|h_{n}|^{2})^{\beta}}\big{(}\sum_{p\in\mathcal{N}}\frac{1}{(|h_{p}|^{2})^{\beta}}\big{)}},$
(6)

where $\beta$ is a real number $(-1\leq\beta\leq 1)$ with $\beta<0$ and
$\beta>0$, respectively, representing LPWU and LPSU schemes. Considering both
the PA schemes, our proposed suboptimal decoding order policy for an $N$-user
system follows two steps:

$\bullet$ Step 1: For the first $(N-1)$ stages, each user decodes data of
other users in the sequence of weakest to strongest user (W-S) for LPWU
scheme, and in the order of strongest to weakest user (S-W) for LPSU scheme
$\rightarrow$ In the case of LPWU scheme, the possibility of decoding weaker
users’ data by stronger users is higher due to their better channel gain and
more PA. In such a way, the chance of decoding the weakest user’s data by all
other users is maximum. For this case, if the weakest user’s data is decoded
at the first stage by stronger users, then the achievable data rate decreases
due to interference by other users. As a result, the secrecy rate for the
weakest user improves. Considering this concept, we find that the (W-S)
approach should be followed at all users. Similarly, LPSU scheme represents
the maximum PA to the weakest user. In this case, with a focus to increase the
secrecy rate for stronger users, (S-W) approach should be followed.

$\bullet$ Step 2: At $N$-th stage each user decodes its data in both PA
schemes $\rightarrow$ As explained in Proposition 2, decoding own data at last
stage by each user improves secrecy performance.

The detailed methodology is outlined in Algorithm 1.

Algorithm 1 Suboptimal decoding order policy.

1:$N$, $\beta$, $\mathcal{N}$

2:suboptimal secure decoding order

3:Define an $N\times N$ matrix $\mathbf{D}_{o}$ for $o$-th decoding order as
explained in Section II-B

4:for each $m$-th column vector $\textbf{d}_{m}$ of matrix $\mathbf{D}_{o}$ do

5: Obtain set $\mathcal{N}$ excluding the element $m$, such as
$\mathcal{N}=\mathcal{N}\setminus\\{\text{m}\\}$

6: for each $q$-th element $t$ of set $\mathcal{N}$ do

7: Set $[\textbf{d}_{m}]_{q}=t$

8: Set $[\textbf{d}_{m}]_{N}=m$ $\triangleright$ $m$-th user decodes its own
data at end

9: if $\beta<0$ then $\triangleright$ LPWU scheme - (W-S)

10: Sort the first $N-1$ elements of $\textbf{d}_{m}$ in descending order

11: else if $\beta>0$ then $\triangleright$ LPSU scheme - (S-W)

12: Sort the first $N-1$ elements of $\textbf{d}_{m}$ in ascending order

13: for each stage $k$ of matrix $\mathbf{D}_{o}$ do

14: Set $[\mathbf{D}_{o}]_{k,m}=[\textbf{d}_{m}]_{k}$

15:return $\mathbf{D}_{o}$ $\triangleright$ $\mathbf{D}_{o}$ as a suboptimal
secure decoding order

We denote the suboptimal decoding order for LPWU and LPSU schemes as
$\widetilde{\mathbf{D}}$ and $\widehat{\mathbf{D}}$, respectively. Note that
the proposed algorithm is based on the sorting of elements. So, the
computational complexity can be expressed as O(n log(n)).

##  V Numerical Results and Discussion

In this work, users are distributed over a square field with length $l=500$m
and BS is placed at its center. We have used $L_{c}=1$, $e=3$, and small scale
fading is assumed to have Rayleigh distribution with mean value $1$. Besides,
$\zeta=0.1$, $P_{t}=10$dBm and $\sigma^{2}=-90$dBm. Numerical results for each
decoding order have been obtained in terms of the minimum secrecy rate that
can be ensured to each user.

We first validate the performance of our proposed approach in Fig. 1. For
this, the optimal secrecy rates achieved among secure decoding orders in set
$\mathcal{L}$ and $\mathcal{O}$ are depicted for different PAs. The results
reveal that the secure decoding orders in which all users decode their data at
the last stage outperform that of other decoding orders. Fig. 1 validates that
$\widetilde{\mathbf{D}}$ and $\widehat{\mathbf{D}}$, respectively, are the
suboptimal decoding orders for LPWU and LPSU schemes, except for some values
of $\beta$. $\widetilde{\mathbf{D}}$ appears to work better than
$\widehat{\mathbf{D}}$ until a certain point $\beta=0.2$ is reached, because,
at the low value of $\beta$, the PA for weaker users is not enough to
compensate the interference caused. Fig. 1 also exhibits that the secrecy rate
achieved through our proposed suboptimal solution match the optimal
performance. Thus, the accuracy of suboptimal policy is about $90\%$.

Next, Fig. 2 presents the performance comparison of proposed suboptimal
decoding orders with the benchmark. The benchmark scheme considered here is
the average secrecy rate of all secure decoding orders in set $\mathcal{S}$.
We observe that an average performance gain of about $137\%$ is achieved by
suboptimal solution over the benchmark scheme. Fig. 2 also shows that the
secrecy rate performance degrades on increasing the number of users. This is
because with an increase in the number of users, the achievable data rate at
each user decreases due to interference by other users.

Figure 1: Validating the accuracy of proposed approach with a variation of
secrecy rate performance for different PAs. Figure 2: Comparison of the
proposed suboptimal decoding order policy with the benchmark average secrecy
rate of all secure decoding orders in set $\mathcal{S}$.

##  VI Concluding Remarks

We have proposed a decoding order strategy for solving secrecy issues in
multi-user untrusted NOMA system. To avoid the computational complexity in
finding the best decoding order out of the total possible secure decoding
orders based on the proposed strategy, we have recommended a suboptimal
policy. Significant improvement of $137\%$ in secrecy rate is achieved using
the suboptimal solution over the benchmark.

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# Propagation of Interplanetary Shocks in the Heliosphere111

Munkhjargal Lkhagvadorj Wigner Research Centre for Physics, Department of
Space Physics and Space Technology
Konkoly-Thege Miklós út 29-33.
H$-$1121 Budapest, Hungary  Eötvös Loránd University, Faculty of Science
Pázmány Péter sétány 1/A.
H$-$1117 Budapest, Hungary  Institute of Physics and Technology, Department of
Theoretical and High Energy Physics
Peace Avenue 54b, Bayanzurkh District
13330 Ulaanbaatar, Mongolia  Gábor Facskó Wigner Research Centre for Physics,
Department of Space Physics and Space Technology
Konkoly-Thege Miklós út 29-33.
H$-$1121 Budapest, Hungary  Milton Friedman University, Department of
Informatics
Kelta utca 2.
H-1039 Budapest, Hungary  Andrea Opitz Wigner Research Centre for Physics,
Department of Space Physics and Space Technology
Konkoly-Thege Miklós út 29-33.
H$-$1121 Budapest, Hungary  Péter Kovács Wigner Research Centre for Physics,
Department of Space Physics and Space Technology
Konkoly-Thege Miklós út 29-33.
H$-$1121 Budapest, Hungary  David G. Sibeck NASA Goddard Space Flight Center
8800 Greenbelt Rd, Greenbelt
MD 20771, USA Gábor Facskó<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Interplanetary shocks are one of the crucial dynamic phenomena in the
Heliosphere. They accelerate particles to high energies, generate plasma
waves, and can trigger geomagnetic storms in the terrestrial magnetosphere
disturbing significantly our technological infrastructures. In this study, two
IP shock events are selected to study the temporal variations of the shock
parameters using magnetometer and ion plasma measurements of the STEREO$-$A
and B, the Wind, Cluster fleet, and the ACE spacecraft. The shock normal
vectors are determined using the minimum variance analysis (MVA) and the
magnetic coplanarity methods. During the May 7, 2007 event, the shock
parameters and the shock normal direction are consistent. The shock surface
appears to be tilted almost the same degree as the Parker spiral, and the
driver could be a Co–Rotating Interacting Region (CIR). During the April 23,
2007 event, the shock parameters do not change significantly except for the
shock $\theta_{Bn}$ angle, however, the shape of the IP shock appears to be
twisted along the perpendicular direction to the Sun-Earth line as well. The
driver of this rippled shock is Stream Interaction Regions (SIRs)/CISs as
well.

Interplanetary Shocks (829) — Solar Wind (1534) — Planetary magnetospheres
(997) — Magnetic storms (2320)

††journal: ApJ

##  1 Introduction

The solar corona is hotter than the photosphere, the chromosphere, and the
transient layers beneath it. As a result, the high temperatures ionize atoms,
creating a plasma of free-moving electrons and ions, the so-called solar wind.
Historically, (Parker, 1958) predicted the existence of the solar wind and
coined terms describing it. He deduced it based on German astronomer Ludwig
Bierman’s observation of how the comet tail always points away from the Sun
(Biermann, 1957). The existence of the solar wind was confirmed by the Mariner
2 spacecraft (Snyder & Neugebauer, 1965). The solar wind is a collisionless
plasma, whose speed is both supersonic and super-Alfvenic. The latter means
that its speed exceeds the Alfvén speed, which is a representative speed of
magnetohydrodynamic waves in a plasma. Due to abrupt change of the solar wind
speed from supersonic to subsonic (e.g. by encountering an obstacle), a shock
wave is emerged. Interplanetary (IP) shock waves are common in the
heliosphere, which is a bubble-like region of space surrounding the Sun and
extending far beyond the orbits of the planets and is filled with the solar
wind. There are many varieties of shocks such as planetary bow shocks, shocks
that are risen due to the stream interaction regions (SIR), which are called
co-rotation interaction region (CIR) when extending beyond 1 AU (see for
example in Koban et al., 2023), and coronal mass ejection (CME) driven shocks.
IP shocks are one of the main and efficient accelerators of energetic
particles in the heliosphere (Tsurutani & Lin, 1985; Keith & Heikkila, 2021).
These accelerated particles can enter the geomagnetic field and are hazardous
to astronauts and satellites. IP shocks driven by CMEs can trigger large
geomagnetic storms that can damage oil and gas pipelines and interfere with
electrical power infrastructures (Gonzalez et al., 1994). GPS navigation and
high-frequency radio communications are also affected by ionosphere
irregularities brought on by geomagnetic storms (Cid et al., 2014) and can
cause internet disruptions around the world (Jyothi, 2021; Facsko et al.,
2024). Therefore, IP shocks are important in determining and understanding
space weather.

Some dayside magnetospheric transient events (Zhang et al., 2022, see
references therein) generated by the interaction of a discontinuity frozen in
the solar wind and the terrestrial bow shock. This fact raises the questions
of how stabile the discontinuities are, how long these phenomena travels with
the solar wind, and whether they evolve during their travel, or show some wave
dispersion (Facskó et al., 2005, István Ballai, personal communication).
Dayside magnetospheric transient events are generated by tangential (TD) and
rotational discontinuities (RD). However, there are catalogues for neither TDs
nor RDs. Therefore, it is easier to study IP shocks because many were observed
and listed previously (see Section 3).

Reshetnyk et al. (2010) studied the 2D curvature of two IP shock events.
Reshetnyk et al. (2010)’s paper showed that we do not know so much about the
shape of discontinuities especially, the IP shocks. Facsko et al. (2010)
extended the previuos study to 3D. More events and more discontinuities (IP
shocks) were needed. A special automated shock identification method was
developed (Facsko et al., 2025), however, finally, the appropriate events to
study the temporal development and 3D shape of IP shocks were found in an open
database developed by Nikbakhsh (2014); Kilpua et al. (2015). Finally, the
origin of the IP shock are also very interesting. Based on out current
knowledge the shock could be CME driven, the result of SIR/CIR development, or
caused by a flare. The determination of the 3D shape of IP shocks with a
backward projection might explain the origin of these very important events.

The shape of an IP shock front depends on the inhomogeneities of the solar
wind through the shock is propagating (Heinemann & Siscoe, 1974; Odstrčil &
Pizzo, 1999; Dryer et al., 2004). Russell et al. (1983a, b) studied IP shock
using two- and four-spacecraft measurements assuming that the IP shocks were
planar locally. Terasawa et al. (2005) observed $\mathrm{15-20}^{\circ}$ shock
surface ripple using Geotail (Nishida, 1994) and Advanced Composition Explorer
(ACE; Stone et al., 1998) data. Szabo (2005) reported discrepancies of IP
shock normal directions observing the same IP shock using ACE and Wind (Lin et
al., 1995; Wilson et al., 2021), or Wind and IMP–8 (Team, 1974) spacecraft.
Lepping et al. (2003) determined IP shock normals using Wind and IMP–8
observations at 1 AU (Astronomical Unit $\approx 1.5\times
10^{8}\,\mathrm{km}$). The average radius of the IP shock curvature was a few
million km. Neugebauer & Giacalone (2005) collected and analyzed 26 IP shocks
between the Earth’s bow shock and $L_{1}$ Lagrangian point. The average radius
of the IP shock curvatures was a bit larger than Lepping et al. (2003)’s
result, however, it also was a few million km. This result agreed with
previous simulation results (Neugebauer & Giacalone, 2005).

Lario et al. (2008) observed the same IP shock at 1 AU using ACE measurements
and at 5.4 AU using Ulyesses observations. However, Lario et al. (2008)
studied the solar energetic particle (SEP) events. The interplanetary solar
wind properties changed the properties of the SEP events in this study.

The Helios probe (Howard, 1975) observed IP shocks to evolve and change their
behaviour at various heliospheric distances in the inner heliosphere (Lai et
al., 2012) and between 1 and 5 AU by the Ulysses mission (Wenzel et al., 1992;
Burton et al., 1992; Zhao et al., 2018). Trotta et al. (2023) studied
shocklets using ACE, Wind, DSCOVR (Deep Space Climate Observatory), Time
History of Events and Macroscale Interactions during Substorms (THEMIS,
Angelopoulos, 2008) B and C measurements. Trotta et al. (2024) observed a CME
driven shock at 0.07 AU and 0.7 AU using Parker Solar Probe (PSP, Fox et al.,
2016), the Solar Orbiter (SolO, Marsch et al., 2005) observations. The PSP
crossed a thick magnetosheath at the flank according to the early evolution
phase of the CME. The SolO observed high energy (2 MeV) shock accelerated
protons and a much more disturbed shock environment than the one observed by
PSP. Therefore, the local small-scale environment of the shock was very
different at different locations of the heliosphere.

In this study, we focus on the macroscopic structure and parameters of the
observed IP shock events. The structure of this paper is as follows: we first
list and describe briefly the missions, instruments, and databases that we
used in this study in Section 2. We describe the applied analysis methods in
Section 4. Then we introduce the datasets that we analysed in Section 5. We
discuss and analyse the observations in Section 6. Finally, we summarize the
results of our study in Section 7.

##  2 Missions and instruments

In this paper, Solar Terrestrial Relations Observatory (STEREO; Kaiser et al.,
2008) A and B, Wind (Lin et al., 1995; Wilson et al., 2021), ACE, and the
Cluster (Credland et al., 1997; Escoubet et al., 2001) spacecraft magnetic
field, ion plasma, and spacecraft potential data are used.

###  2.1 The STEREO mission

NASA’s twin STEREO A and B spacecraft were launched on October 26, 2006, from
Kennedy Space Center. In heliospheric orbit at 1 AU, STEREO$-$A (Ahead) leads
while STEREO$-$B (Behind) trails Earth. The two spacecraft separate at
$44^{\circ}$ in heliographic longitude from each other annually. Both
spacecraft were equipped with two instruments and two instrument suites, with
a total of 13 instruments on each spacecraft.

The PLAsma and SupraThermal Ion Composition (PLASTIC) instrument measures
protons and alpha particle parameters, as well as the composition of heavy
ions in the solar wind plasma (Galvin et al., 2008). The In-situ Measurements
of Particles and CME Transients magnetic field experiment (IMPACT) suite of
instruments consists of seven instruments among which three are located on on
a 6-meter long boom while the other four are installed in the of the
spacecraft (Acuña et al., 2008). The IMPACT measures the parameters of
protons, heavy ions, and electrons, and the MAG magnetometer sensor in it
measures the in situ magnetic fields in a range of $\pm 512\,nT$ with 0.1 nT
accuracy (Kaiser & Adams, 2007).

We downloaded all data from NASA’s Coordinated Data Analysis Web (CDAWeb,
https://cdaweb.gsfc.nasa.gov/). We obtained the STEREO$-$A and B magnetic
field observation and the ion plasma data from the STEREO IMPACT with the time
resolution of 100 ms and the STEREO PLASTIC instruments, respectively. The
magnetic field and the plasma data of the STEREO$-$A and B are in the RTN
spacecraft coordinate system, where R is radially outward from the Sun, T is
along the planetary orbital, and N is northward direction.

###  2.2 The Wind mission

NASA’s Wind spacecraft was launched on November 1, 1994, (Lin et al., 1995;
Wilson et al., 2021). The Wind was initially sent to $L_{1}$ Lagrange point,
however, the reaching of $L_{1}$ was delayed in order to be able to study in
advance the terrestrial the magnetosphere and lunar environment, as well.
Following a sequence of orbital adjustments, the Wind spacecraft was
positioned in a Lissajous orbit close to the $L_{1}$ Lagrange point in early
2004 for studying the incoming solar wind (NASA WIND team, 2020).

The spacecraft is equipped with eight instruments, however, we used only the
Magnetic Field Investigation (MFI; Lepping et al., 1995) and the Three-
Dimensional Plasma and Energetic Particle Investigation (3DP Lin et al., 1995)
measurements in this study. The MFI consists of two magnetometers at the
12-meter boom measures magnetic field from 4 nT, to 65536 nT with a time
resolution of 22 or 11 vectors per second in the calibrated high-resolution
mode, while in the primary science mode the resolution can be three seconds,
one minute and one hour (NASA WIND team, 2022). The 3DP measures the solar
wind key parameters such as velocity, density, and temperature.

From CDAWeb, we got Wind magnetic and plasma data from the MFI and 3DP
instrumens with a time resolution of 3 s, respectively. The magnetic field and
the plasma data of the Wind are in the Geocentric Solar Ecliptic System (GSE)
coordinate system, where X-axis is pointing to the Sun from the Earth, Y-axis
is in the ecliptic plane showing opposite to the planetary motion, and Z-axis
is the northward direction.

###  2.3 The ACE mission

NASA’s ACE spacecraft was launched on August 25, 1997 (Stone et al., 1998).
The spacecraft is located at $L_{1}$ Lagrangian point similarly to the Wind
spacecraft. The spacecraft is equipped with nine primary scientific
instruments and one engineering instrument. We used in the study the
measurements of the Magnetometer (MAG; Smith et al., 1998), and the Solar Wind
Electron, Proton and Alpha Monitor (SWEPAM; McComas et al., 1998). The MAG
consists of twin triaxial flux-gate magnetometers such that magnetometer
sensors have between 3 and 6 vectors $s^{-1}$ resolutions for continuous
observation of the interplanetary magnetic field (Smith et al., 1998).

From CDAWeb, we downloaded the ACE magnetic and plasma data from the ACE MAG
with a time resolution of 1 s and from SWEPAM with a time resolution of 64 s,
respectively. The magnetic field and the plasma data are both in the RTN and
GSE coordinate systems.

###  2.4 The Cluster fleet

ESA’s Cluster constellations consist of four satellites, which were launched
on 16 July and 9 August 2000 (Credland et al., 1997; Escoubet et al., 2001).
The Cluster satellites orbit in a tetrahedral formation around Earth. The
perigee and apogee of the orbit are approximately four and 19.6 $R_{E}$,
respectively (Zhang et al., 2010). Each of the four satellites is equipped
with 11 identical instruments. We used the data of the Cluster Ion Composition
(CIS; Rème et al., 1997, 2001), the Fluxgate Magnetometer (FGM; Balogh et al.,
1997, 2001), and the Electric field and waves (EFW; Gustafsson et al., 1997,
2001) instruments in this study .

The FGM is composed of two tri-axial fluxgate magnetometers, which are
installed on one of the two 5-meter radial booms. It measures in the dynamic
range $\pm 65,536$ nT. At the highest dynamic level, the resolution is $\pm 8$
nT, and the time resolution is 100 vectors per second (Balogh et al., 1997,
2001; Organization, 2021). The CIS instrument measures three-dimensional ion
distribution, and it is composed of two distinct sensors: the Composition
Distribution Function (CODIF) sensor and the Hot Ion Analyzer (HIA) sensor
(Rème et al., 1997, 2001). The CIS experiment is not operational for Cluster-2
and the HIA sensor is switched off for Cluster-4 due to a problem with the
high voltage of the electrostatic analyzer (CIS team, 2021). Hence, for
Cluster-2 and Cluster-4, the EFW instrument measurement became more
indispensable for the analyses because we could obtain electron density in
another way, independently from ion and electron plasma measurements
(Gustafsson et al., 1997, 2001).

As the spacecraft travels through the plasma environment, it acquires an
electric charge as a result of contact with charged particles. This charging
process results in an electrical potential discrepancy between the spacecraft
and the plasma around it, a phenomenon referred to as spacecraft potential.
The EFW instrument measures the spacecraft potential and the electron density
of the plasma could be calculated using an empirical formula.

$N_{e}=200(V_{sc})^{-1.85},$ (1)

where $N_{e}$ is the calculated electron density and $V_{sc}$ is the Cluster
EFW spacecraft potential (Sandhu et al., 2016; Trotignon et al., 2010, 2011).

From CDAWeb, we acquired the magnetic data of Cluster satellites from the
Cluster FGM with a time resolution of 4 s for all the four Cluster satellites,
ion data from the Cluster CIS with a time resolution of 4 s for the Cluster
SC1 and SC3 satellites, and the spacecraft potential data from EFW with a time
resolution of 4 s for the Cluster SC2 and SC4 satellites where CIS
measurements were not available. All the Cluster data were in the GSE
coordinate system.

##  3 IP shock events and transformations

IP shock candidates are chosen from the shock lists in the Database of
Heliospheric Shock Waves maintained at the University of Helsinki
http://www.ipshocks.fi/. For the event selection, we chose the year 2007
because STEREO$-$A and STEREO$-$B were close to each other near the Sun-Earth
line. In the first event, May 07 of 2007, the selected spacecraft are
STEREO$-$A, STEREO$-$B, Wind, and the four Cluster satellites. For the second
event, April 23, 2007, the spacecraft are STEREO$-$A and B, ACE, and Wind.

After choosing the shock candidates and obtaining the data, we transformeddata
coordinates in such a way that all the coordinate systems are changed to the
Heliocentric Earth Ecliptic (HEE) coordinate system, where the X-axis is
toward the Earth from the Sun, Z-axis is the ecliptic northward direction,
while Y axis completes the right-handed Cartesian system. This system is fixed
with conserning the Sun-Earth line. To transform from the RTN and the GSE
coordinate systems to the HEE coordinate system, we used Transformation de
REpères en Physique Spatiale (TREPS; http://treps.irap.omp.eu/) online tool.
The TREPS tool, which is developed by the French Plasma Physics Data Centre
(CDPP), the national data centre in France for the solar system plasmas, is
based on SPICE (Spacecraft, Planet, Instrument, C-matrix, and Events)
information system kernels created by National Aeronautics and Space
Administration (NASA)/Navigation and Ancillary Information Facility (NAIF)
tool (Génot et al., 2018).

##  4 Analysis methods

Usually coplanarity method is applied to find the shock normal. The
coplanarity method is based on the magnetic coplanarity theorem, which states
that the magnetic field vectors on both sides of the shock and the shock
normal lie in the same plane. Similarly, the velocity on both sides or, in
other words, the velocity jump through the shock also lie in the same plane.
The method of magnetic coplanarity is straightforward to implement and, it
only magnetic field data is needed for applying the method (Paschmann & Daly,
1998; Schwartz, 1998).

We select two intervals upstream and downstream of the shock, calculate
average upstream and downstream magnetic field vectors using the values of the
interval, and determine the shock normal vector using the coplanarity method.
However, in this case, the magnetic field varies strongly, therefore the
calculated shock normal vector looks very sensitive for the selected intervals
if one use only the coplanarity method. Hence, the calculated shock normals
for each spacecraft pointed in different directions. The result cannot be
explained physically, therefore applying the coplanarity method only is
pointless in this case. Hence, we need the support of another method in this
case.

Therefore, we use Minimum Variance Analysis (MVA) (Sonnerup & Scheible, 1998;
Schwartz, 1998) and magnetic co-planarity (CP) methods to determine the IP
shock normals. The MVA technique is based on the assumption that variations in
the magnetic field would be observed when a single spacecraft passes through a
1D (in reality, it is a 2D or 3D transition layer) current layer or wavefront.
The divergence of the magnetic field constraint
$\left(\nabla\cdot\mathbf{B}=0\right)$, therefore the normal component of the
magnetic field must remain constant. If such a normal direction can be found,
then the variations in the magnetic field are zero or at the least have a
minimum variance (Sonnerup & Cahill, 1967; Sonnerup & Scheible, 1998; Wang et
al., 2024). MVA can only work if there are less fluctuations in the direction
normal to the shock than in the direction perpendicular to magnetic fields
upstream and downstream from the shock.

By comparing the two methods, the upstream and downstream time intervals of
the magnetic field measurements are set. We set up the following requirements
to accept the results:

  1. 1.

The angle between the shock normal vectors obtained by the minimum variance
analysis (MVA) and the magnetic coplanarity method must be less than
$15^{\circ}$.

  2. 2.

The ratio between the intermediate eigenvalue and the smallest eigenvalue
should be greater than 2.

The above methods have been used in previous studies for shocks (Bebesi et
al., 2010, 2019; Borodkova et al., 2019; Nikbakhsh, 2014; Shan et al., 2013;
Terasawa et al., 2005). Similar conditions have been used in other studies for
tangential discontinuities (Facskó et al., 2008, 2009, 2010).

We apply both methods for every single spacecraft. The shock normal vectors
calculated independently point to a similar direction. This result is
accaptable and could be explained physically. We suppose that the same shock
passes all spacecraft at different times and locations. The shock is supposed
to be a bit curved and perhaps ripped. Therefore, applying both methods leads
to a valuable and acceptable result.

##  5 Observations

With the visual inspection of the magnetic field and plasma data, we
identified jumps in magnetic field ($\mathbf{B}$), speed ($\mathbf{V}$),
density (N), and temperature (T). Using these quantities, we determined the
times when shock events occurred for each spacecraft data set. The common
origin of a shock event detected by different spacecraft can be concluded from
the similar time profiles of the magnetic field and plasma data recorded in
each spacecraft position near to the shock. From the positions of the
spacecraft and the times of shock arrivals to the different positions, the
shock geometry and the propagation parameters (velocity, direction) can be
deduced. The following case studies of two events are not orderly defined for
one another. First, we analyzed the shock event on May 7, 2007, and then the
event on April 23, 2007.

###  5.1 Event May 7, 2007

On May 7, 2007, the Wind spacecraft first detected the shock at 07:02:30
(UTC). After that, the STEREO$-$A spacecraft detected the shock at 08:11:30
(UTC), and then the STEREO$-$B spacecraft detected the shock at 09:42:00
(UTC). The four Cluster satellites detectd the shock as well. Cluster SC1
detected the shock at 08:27:55 (UTC), and Cluster SC3 detected the shock at
08:28:00 (UTC). The magnetic field and the plasma parameters are shown in
Figure 1 for the Wind (a), for the STEREO$-$A(b), for the STEREO$-$B (c), in
Figure 2 for the Cluster SC1 (a), and SC3 (b). There is no indication in the
shock database that Cluster SC2 and SC4 detected a shock in the analysed
period. However, since the four Cluster spacecraft are located close to each
other, they all had to encounter the given shock. Without plasma data on
Cluster SC2 and SC4 satellites, a shock cannot be confirmed based solely on
the magnetic field data. However, using the empirical formula described in
Section 2.4 the electron density could be estimated using EFW spacecraft
potential data. Therefore, comparing the magnetic field and the electron
plasma density profile, the shock is determined. The Cluster SC2 detected the
shock at 08:28:10 (UTC), according to the magnetic field and density plots
(Figure 2c). The Cluster SC4 detected the shock at the same time as SC2
(Figure 2d). Similarly, The electron density parameter is obtained by the
empirical formula (Eq. 1).

Here we list all magnetic field observations of the spacecraft for the event
and apply the MVA and the CP methods to them. The upstream and downstream time
intervals that most agree between the MVA and the CP for all the spacecraft
are shown in Table 1, and their corresponding magnetic field plots are shown
in Figure 3 for Wind (a), STEREO$-$A (b), STEREO$-$B (c), while in Figure 4
for Cluster spacecraft. Table 1 also shows the ratio between the smallest
eigenvalue $\lambda_{3}$ and the intermediate eigenvalue $\lambda_{2}$ as well
as the angle between the MVA normal and CP normals in the determined upstream
and downstream time intervals for each spacecraft.

The accepted upstream and downstream time intervals were properly selected
considering the angle difference between the results obtained by the two
methods is minimal and the eigenvalue criteria are sufficient for each
spacecraft data. Using the determined time intervals, the calculations of the
ratio between the upstream and downstream total magnetic fields, ion plasma
densities, and ion plasma temperatures as well as the bulk speed, and shock
$\theta_{Bn}$ angle are made. These parameters, the MVA normal, and the
magnetic CP normals are shown in Table 2. The shock parameters fulfil the
necessary shock criteria (Lumme et al., 2017). The results of the additional
parameter calculations are shown in Table 3.

Using the results, the 2D sketches of the IP shocks that were detected by the
spacecraft are shown in Figure 5. In these 2D sketches, the shock propagation
and normal vector orientation are shown in a temporal development manner.
Since four Cluster satellites are relatively close to one another, their
averaged position as well as normal vectors are shown in the general 2D
sketches (also in 3D sketches, see later). For explicitly showing normal
vector directions and positions of all Clusters satellites, it is suitable to
change their coordinates and normal vectors into a GSE coordinate system with
the positions in the Earth radii ($R_{E}$) unit, see Figure 6.

The 3D sketches of the shock normals are shown in Figure 7a, and the 3D
sketch, the STEREO$-$A, and B, and averaged Clusters positions are time-
shifted to the position of Wind to see the overall shape of this IP shock. In
order to assess the shape of the shock, we shifted the shock normals hence, as
they had been detected contemporary with the shock observation time in Wind.
For this, we used the average solar wind speeds and the time-differences
between the observations in Wind and in the three other missions. The shifted
shock normals are shown in Figure 7b. It turns out that the starting points of
all of the normal vector lay approximately in a plane surface, meaning that
the underlying shock exhibits a planar structure in the spatial scale of the
spacecraft distances. The plane fitted to the time-shifted normals is shown in
Figure 7c.

###  5.2 Event April 23, 2007

In this specific event, the STEREO$-$A spacecraft detected a shock first at
06:53:35 (UTC), then the ACE spacecraft detected the shock at 08:57:00 (UTC),
and the Wind spacecraft detected the shock at 09:12:00 (UTC). The magnetic
field and the plasma data are shown in Figure 8 for the STEREO$-$A (a), ACE
(b), and Wind (c).

STEREO$-$A and B have identical instruments, therefore, STEREO$-$B must have
detected the shock on that day even though there is no detected IP shock for
STEREO$-$B in the shock lists database mentioned in Section 3. So, by using
the average solar wind speed, 400 km/s, we concluded that the shock detection
time for STEREO$-$B should be around 13:00 to 15:00. Considering this, a
possible shock signature from STEREO$-$B is found around 13:21:30 even though
it is a faint signature. The magnetic field and the plasma plot are shown in
Figure 8d.

We use all magnetic field observations of the spacecraft about the event and
apply the MVA and the CP methods to them. In this event, the order between
upstream and downstream is swapped because it is a fast-reverse (FR) shock
event, which means the shock propagates toward the Sun against the flow
however is carried outward from the Sun by the bulk solar wind flow Kilpua et
al. (2015). The upstream and downstream time intervals that give the best
agreement between the results of the MVA and CP shock normal determination for
all spacecraft are shown in Table 4 Figure 9 shows the corresponding magnetic
field observations for STEREO$-$A (a), ACE (b), Wind (c) and STEREO$-$B (d)
missions. Table 4 also shows the ratio between the intermediate eigenvalue
$\lambda_{2}$ and the smallest eigenvalue $\lambda_{3}$ as well as the angle
between the MVA and CP normal vector determinations of the shock in the
upstream and downstream time intervals for each spacecraft.

Similarly to the Event May 7, the upstream and downstream time intervals are
defined correctly as well, considering the angle difference between the two
methods is minimal and the eigenvalue criteria are fulfilled for each
spacecraft data. Also, using the determined time intervals, in Table 5 we
summarize, for each mission, the ratio between the upstream and downstream
total magnetic fields, densities and temperatures, the differences between the
upstream and downstream solar wind bulk speeds. From the obtained
upstream/downstream ratios and bulk speed differences we conclude that the
criteria for detecting a sock event in the considered period is fulfilled
(Lumme et al., 2017) for each spacecraft. Even for STEREO$-$B, for which shock
was not indicated in the shock catalogue (see above), the numbers
unambiguously prove the existence of a shock event during the studied
interval. However, the almost quasi-parallel shock detected by the STEREO$-$B
appears to become quasi-parallel a few hours after the shock detection times
of STEREO$-$A, ACE, and the Wind based on the shock $\theta_{Bn}$ angle, see
Table 5, where the calculated results of additional parameters are shown.

The sketches of the IP shocks geometries that were detected by the spacecraft
are shown in Figure 10. In these 2D sketches, the shock propagation and normal
vector orientation are shown in a temporal development manner.

The 3D sketch of the IP shock geometries are shown in Figure 11. The 3D
sketch, the STEREO$-$A, and B positions are time-shifted to the position of
STEREO$-$A to see the overall shape of this IP shock. In order to assess the
shape of the shock, we shifted the shock normals hence, as they had been
detected contemporary with the shock observation time in STEREO$-$A. For this,
we used the average solar wind speeds and the time-differences between the
observations in STEREO$-$A and in the two other spacecraft. The shifted shock
normals are shown in the top right panel of Figure 11. The top view of the
time-shifted normals is shown in the bottom part of Figure 11.

##  6 Discussion

The IP shock parameters did not change significantly most separated couple of
probes, from STEREO$-$A to B in neither case (Table 2, 3, 5, 6). There is no
sign of wave dispersion on the scale of 40 million km, that is the distance of
the STEREO spacecraft.

###  6.1 Event May 7, 2007

The IP shock fitted plane is tilted $56.42^{\circ}$ from the Sun-Earth line
according to Figure 7c. The tilt appears to be almost the same as expected for
the Parker spiral impacting the Earth from the dawn side. Hence, it is the
reason why the Wind detected the shock first even though STEREO$-$A’s position
is relatively closer to the Sun’s direction. This period, 2007, was during the
solar minimum phase, and stream interaction region (SIR) or co-rotating stream
interactions (CIR) were dominant (Opitz et al., 2014), and this further proves
the result, see Figure 12. Furthermore, to see a correlation between fast-
forward shock and geomagnetic activity, we investigated the $K_{p}$-index
(Bartels, 1949; Rostoker, 1972; Matzka et al., 2021,
https://www.swpc.noaa.gov/noaa-scales-explanation) on May 07, 2007, as shown
in Figure 13. It appears that this forward shock-leading event disturbed the
magnetosphere, causing a G1-minor geomagnetic storm with the $K_{p}$\- index
peaking around 15:00 (UTC). The geomagnetic substorm happened about 6 hours
after the detection of the shock by the STEREO$-$B spacecraft.

###  6.2 Event April 23, 2007

In Figure 11 it seems that the shape of the shock is not uniform and is
twisted from the STEREO$-$A spacecraft to the other three spacecraft in the
Z-axis along the transverse direction (Y-axis) to the Sun-Earth line, see
Figure 11b. As seen from Figure 11c, the STEREO$-$A alone is on the dusk
(left) side of the Sun-Earth line while the Wind, the ACE, and the STEREO$-$B
are on the dawn (right). In this XY-plane view (Figure 11), the shock normal
vectors appear to be changing or slightly rotating their direction from the
STEREO$-$A to the Wind, the ACE, and the STEREO$-$B about the Z-axis, and the
shock is changing from the quasi-perpendicular to quasi-parallel based on the
shock $\theta_{Bn}$ angle. This IP shock is a fast reverse shock, meaning it
travels toward the Sun in a frame fixed to the solar wind even though the
shock propagates away from the Sun with the solar wind. The shock detection
time difference between the STEREO$-$A and the Wind/ACE is about two hours
while between STEREO$-$A and B is almost six hours, yet the shock orientation
is significantly changed from the STEREO$-$A to the other three spacecraft
indicating the change is spatial, not temporal. So, due to this nature, the IP
shock could be a local ripple. The ripples on the shock surface are known be
caused by ICME (interplanetary coronal mass ejections) shock drivers as they
do not propagate into homogeneous interplanetary medium (Acuña et al., 2008).
However, for this particular event the source is suggested to be a stream
interaction region (SIR) according to the catalogue given in https://stereo-
ssc.nascom.nasa.gov/pub/ins_data/impact/level3/STEREO_Level3_Shock.pdf.

We investigated the $K_{p}$-index of this event as seen in Figure 14. Similar
to the fast-forward shock event of May 07, 2007, a G1-minor geomagnetic storm
occurred on this day (see space weather scales of National Oceanic and
Atmospheric Administration (NOAA) https://www.swpc.noaa.gov/noaa-scales-
explanation), but the beginning of the geomagnetic storm happened three hours
before the shock detection time of the STEREO$-$A while the ending of the
storm was partially at the same time as the STEREO$-$A. As stated before,
SIR/CIRs form when the fast-moving solar wind catches the slow-moving solar
wind Richardson (2018). This sometimes forms a pair of shocks, one leading as
a fast-forward shock while a rarefied shock trails the solar wind as a fast-
reverse shock but oftentimes as just sole fast-forward or fast reverse shocks
(Jian et al., 2019). Nevertheless, since it always involves the fast-moving
solar wind, it makes total sense why the G1-minor geomagnetic storm with a
$K_{p}$-5 index happened before the detection of the shock because it seems
the fast-moving solar wind caused the minor storm before the fast-reverse
shock finally arrived at the spacecraft positions.

The normal orientation inconsistency can also be explained by the argument
that this shock is a fast-reverse shock because SIRs have characteristic tilts
such that forward waves direct towards the solar equatorial plane while the
reverse waves tend to move in the direction of the solar poles (Kilpua et al.,
2015). Therefore, this tilting nature may explain the XZ component tilted
orientation between the STEREO$-$A and the other three spacecraft – the Wind,
the ACE, and the STEREO$-$B.

##  7 Summary and conclusions

In this paper, two conjugated multi-spacecraft studies of interplanetary (IP)
shock propagation are presented. It aims to determine how IP shocks develop
and evolve in spatial and temporal propagation. The spatial separation of
observations is 40 million km. In this scale, we do not see any significant
development of the IP shocks. The shock parameters are similar within their
estimated error at each location. No sign of wave dispersion was observed.

The Event May 7, 2007 is a fast-forward (FF) shock event, appears planar and
tilted $56.42^{\circ}$ to the Sun-Earth line. The tilt of this planar shock
surface is almost identical to the usual propagation of the Parker spiral
impacting the Earth. Therefore, the origin of the shock is co-rotating stream
interactions (CIRs), which agrees with the detected CIR on that day. There is
no sign of temporal change in this scale. In the spatial range of 40 million
km, no change was observable. The G1-minor geomagnetic storm happened after
this shock was detected, indicating this shock event caused the geomagnetic
storm on that day.

The Event April 23, 2007 is a fast reverse (FR) shock event and, it appears
the shape of the shock is not uniform and twisted from the STEREO$-$A
spacecraft to the other spacecraft along the transverse direction to the Sun-
Earth line. The shock normal vectors also were observed to be changing along
the Y-axis. A G1-minor geomagnetic storm occurred on this day, similar to the
fast-forward shock event of May 7, 2007. Intriguingly, the onset of this storm
took place three hours before the shock detection time recorded by STEREO$-$A.
The termination of the storm partially coincided with the STEREO$-$A detection
time. The source of this shock is SIR/CIR. The G1-minor geomagnetic storm,
with a $K_{p}$-5 index, began before the shock detection. It seems that the
fast-moving solar wind instigated the minor storm before the arrival of the
fast-reverse shock at the spacecraft’s position. The orientation irregularity
can be potentially accounted for by the characteristic tilts of SIRs.

No multipoint observation was made before using so many and so different
probes. The 3D spatial curve and the origin of the IP shocks are determined.
These macroscopic features were visible only in heliospheric
magnetohydrodynamic simulations before. However, the known IP shock catalogues
of the STEREO, the Wind, the ACE and the Cluster missions provided older
observations. A natural continuation of this study is using PSP, the SolO, and
the BepiColombo (Milillo et al., 2010) measurements for joint observations.
The larger spatial separation of the IP shock observations might allow us to
observe the temporal development and dispersion of these shocks.

Acknowledgments This paper uses data from the Heliospheric Shock Database,
generated and maintained at the University of Helsinki
(https://www.ipshocks.fi). We would also like to acknowledge NASA CDAWeb, the
STEREO$-$A and B IMPACT, PLASTIC, the Wind MFI, 3DP, ACE MAG, SWEPAM, and the
Cluster FGM, CIS, EFW teams for providing data, and TREPS, designed and
developed by the French Plasma Physics Data Centre (CDPP), for coordinate
transformations for this study. This work was partially financed by the
National Research, Development, and Innovation Office (NKFIH) FK128548 grant.
ML was supported by the Stipendium Hungaricum Scholarship. The authors thank
Oleksiy Agapitov, Géza Erdős and Zoltán Németh for the useful discussions.

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Table 1: The parameters and main results of the minimum variance (MVA) and magnetic co-planarity (CP) analyses of magnetic field records of the considered missions recorded about the May 7, 2007 shock event. Here, $\Delta t_{up}$ denotes the defined upstream and $\Delta t_{down}$ denotes the defined downstream time intervals with MVA and CP analysis methods. $\mathbf{n}_{MVA}$ and $\mathbf{n}_{CP}$ are the normal vectors given by MVA and CP methods, respectively. $\lambda_{2}/\lambda_{3}$ indicates the ratio between the intermediate eigenvalue $\lambda_{2}$ and the smallest eigenvalue $\lambda_{3}$, and $\Delta\theta_{MVA-CP}$ is the angle between the MVA and CP normals on May 7, 2007.  Spacecraft | $\Delta t_{up}$ | $\Delta t_{down}$ | $\mathbf{n}_{MVA}$ | $\mathbf{n}_{CP}$ | $\lambda_{2}/\lambda_{3}$ | $\Delta\theta_{MVA-CP}$
---|---|---|---|---|---|---
| $\left[t_{up}^{start}-t_{up}^{end}\right]$ | $\left[t_{down}^{start}-t_{down}^{end}\right]$ |  |  |  |
| [hh:mm:ss-hh:mm:ss] | [hh:mm:ss-hh:mm:ss] |  |  |  | $\left[{}^{\circ}\right]$
Wind | 06:59:00 - 07:01:47 | 07:03:50 - 07:05:50 | [-0.76, -0.63, -0.14] | [-0.76, -0.63, -0.15] | 14.82 | 0.38
STEREO$-$A | 08:08:00 - 08:10:05 | 08:12:30 - 08:13:50 | [0.89, 0.44, 0.14] | [0.89, 0.44, 0.144] | 3.16 | 0.37
STEREO$-$B | 09:37:00 - 09:40:35 | 09:42:57 - 09:43:38 | [-0.85 -0.51 -0.09] | [-0.86 -0.50 -0.08] | 9.51 | 1.10
Cluster-1 | 08:26:10 - 08:27:00 | 08:28:00 - 08:29:00 | [0.85 0.53 -0.03] | [-0.85 -0.53 -0.03] | 167.90 | 0.21
Cluster-3 | 08:26:30 - 08:27:10 | 08:29:00 - 08:29:47 | [0.75 0.64 0.15] | [-0.76 -0.63 -0.12] | 22.43 | 1.92
Cluster-2 | 08:26:56 - 08:27:40 | 08:28:32 - 08:29:37 | [0.84 0.54 0.01] | [-0.84 -0.54 -0.01] | 91.56 | 0.10
Cluster-4 | 08:26:25 - 08:27:10 | 08:28:45 - 08:29:40 | [0.86 0.51 -0.08] | [0.84 0.53 0.11] | 238.63 | 2.49

Table 2: Resulting core parameters of studying the data of the Wind, STEREO$-$A, B, and Cluster spacecraft on May 7, 2007. $B_{d}/B_{u}$, $N_{d}/N_{u}$, and $T_{d}/T_{u}$ are the ratio of the upstream and downstream magnetic field, density, and temperature, respectively. $\Delta V$ is the velocity difference between the two sides of the shock. Furthermore, $\theta_{Bn}$ is the shock $\theta$ angle. Due to the CIS experiment is not operational for Cluster SC2, and the CIS HIA sub$-$instrument being switched off for Cluster SC4 spacecraft, the plasma parameters are not available for these satellites. The Cluster CIS HIA instruments SC1 and SC3 provide parallel ($T_{\parallel}$) and perpendicular ($T_{\perp}$) temperatures, respectively.  Spacecraft | $B_{d}/B_{u}$ | $N_{d}/N_{u}$ | $T_{d}/T_{u}$ | $\Delta V$ | $\theta_{Bn}$
---|---|---|---|---|---
|  |  |  | [km/s] | $\left[{}^{\circ}\right]$
Wind | $1.86\pm 0.10$ | $1.70\pm 0.06$ | $1.95\pm 0.12$ | $26.8\pm 1.7$ | 70.80
STEREO$-$A | $1.89\pm 0.02$ | $2.18\pm 0.90$ | $3.21\pm 2.4$ | $36.00\pm 6.00$ | 81.83
STEREO$-$B | $1.90\pm 0.07$ | $1.90\pm 0.6$ | $2.90\pm 6.00$ | $40.25\pm 11.00$ | 59.45
Cluster 1 | $1.53\pm 0.04$ | $1.63\pm 0.09$ | $1.10_{\parallel}\pm 1.30_{\parallel}$ | $31.20\pm 3.10$ | 86.62
|  |  | $1.40_{\perp}\pm 0.32_{\perp}$ |  |
Cluster 3 | $1.57\pm 0.05$ | $1.69\pm 0.08$ | $1.20_{\parallel}\pm 1.50_{\parallel}$ | $30.50\pm 2.80$ | 89.74
|  |  | $1.40_{\perp}\pm 0.40_{\perp}$ |  |
Cluster 2 | $1.53\pm 0.03$ | $1.42\pm 0.18$ | N/A | N/A | 88.19
Cluster 4 | $1.57\pm 0.05$ | $1.49\pm 0.19$ | N/A | N/A | 86.55

Table 3: Resulting additional parameters of studying the data of the three spacecraft on May 7, 2007. $V_{sh}$ is the shock speed, $C^{up}_{s}$ is the upstream sound speed, $V^{up}_{A}$ is the upstream Alfvén speed, and $C^{up}_{ms}$ is the upstream magnetosonic speed. Plasma $\beta_{up}$ based on upstrem parameters, Alfvén-Mach ($M_{A}$), and Magnetosonic-Mach ($M_{ms}$) numbers are also shown. Spacecraft |  $V_{sh}$ [km/s] |  $C^{up}_{s}$ [km/s] |  $V^{up}_{A}$ [km/s] |  $C^{up}_{ms}$ [km/s] | Plasma $\beta_{up}$ | $M_{A}$ | $M_{ms}$
---|---|---|---|---|---|---|---
Wind | 317.33 | $48.84\pm 2.73$ | $29.81\pm 10.91$ | $57.22\pm 6.14$ | $3.22\pm 2.38$ | $2.83\pm 1.20$ | $1.47\pm 0.35$
STEREO$-$A | 352.41 | $46.27\pm 3.30$ | $34.89\pm 15.00$ | $57.95\pm 9.41$ | $2.11\pm 1.80$ | $2.19\pm 1.12$ | $1.32\pm 0.42$
STEREO$-$B | 354.91 | $46.89\pm 5.85$ | $31.95\pm 7.33$ | $56.74\pm 6.35$ | $2.58\pm 1.34$ | $2.76\pm 1.07$ | $1.55\pm 0.52$
Cluster 1 | 342.02 | $43.88\pm 3.09$ | $25.59\pm 7.70$ | $51.31\pm 3.99$ | $3.26\pm 1.89$ | $3.84\pm 1.40$ | $1.99\pm 0.46$
Cluster 3 | 308.11 | $43.88\pm 2.63$ | $26.80\pm 7.64$ | $51.42\pm 3.98$ | $3.21\pm 1.83$ | $3.72\pm 1.35$ | $1.93\pm 0.46$

Table 4: The parameters and main results of the minimum variance (MVA) and co-planarity (CP) analyses of magnetic field records of the considered missions recorded about the April 23, 2007 shock event. Here, $\Delta t_{down}$ denotes the defined downstream and $\Delta t_{up}$ denotes the defined upstream time intervals with MVA and CP analysis methods. $\mathbf{n}_{MVA}$ and $\mathbf{n}_{CP}$ are the normal vectors given by MVA and CP methods, respectively. $\lambda_{3}/\lambda_{2}$ indicates the ratio between the intermediate eigenvalue $\lambda_{2}$ and the smallest eigenvalue $\lambda_{3}$, and $\Delta\theta_{MVA-CP}$ is the angle between the MVA and CP normals on April 23, 2007. Spacecraft | $\Delta t_{down}$ | $\Delta t_{up}$ | $\mathbf{n}_{MVA}$ | $\mathbf{n}_{CP}$ | $\lambda_{2}/\lambda_{3}$ | $\Delta\theta_{MVA-CP}$
---|---|---|---|---|---|---
| $\left[t_{down}^{start}-t_{down}^{end}\right]$ | $\left[t_{up}^{start}-t_{up}^{end}\right]$ |  |  |  |
| [hh:mm:ss-hh:mm:ss] | [hh:mm:ss-hh:mm:ss] |  |  |  | $\left[{}^{\circ}\right]$
STEREO$-$A | 06:46:00 - 06:52:23 | 06:57:03 - 07:02:34 | [-0.86, 0.02, -0.50] | [-0.86, 0.01, -0.51] | 3.29 | 1.07
ACE | 08:51:30 - 08:56:30 | 09:03:00 - 09:07:00 | [0.87, -0.19, -0.45] | [0.87, -0.23, -0.43] | 4.22 | 3.43
Wind | 09:04:00 - 09:10:00 | 09:12:20 - 09:13:15 | [-0.82, 0.43, 0.37] | [-0.81, 0.44, 0.39] | 3.38 | 1.46
STEREO$-$B | 13:15:45 - 13:20:00 | 13:24:00 - 13:26:00 | [-0.75, 0.59, 0.29] | [-0.73, 0.61, -0.30] | 8.18 | 1.67

Table 5: Resulting core parameters of studying the data of the STEREO$-$A, B, ACE, and Wind spacecraft on April 23, 2007. $B_{d}/B_{u}$, $N_{d}/N_{u}$, and $T_{d}/T_{u}$ are the ratio of the upstream and downstream magnetic field, density, and temperature, respectively. $\Delta V$ is the velocity difference on the sides of the shock. Here $\theta_{Bn}$ is the shock $\theta$ angle.  Spacecraft | $B_{d}/B_{u}$ | $N_{d}/N_{u}$ | $T_{d}/T_{u}$ | $\Delta V$ | $\theta_{Bn}$
---|---|---|---|---|---
|  |  |  | [km/s] | $\left[{}^{\circ}\right]$
STEREO$-$A | $1.47\pm 0.16$ | $2.00\pm 0.70$ | $1.48\pm 0.50$ | $46.00\pm 13$ | 88.40
ACE | $1.41\pm 0.08$ | $1.44\pm 0.16$ | $1.22\pm 0.09$ | $23.72\pm 5.57$ | 46.2
Wind | $1.35\pm 0.06$ | $1.51\pm 0.11$ | $1.06\pm 0.12$ | $19.00\pm 6.00$ | 61.85
STEREO$-$B | $1.73\pm 0.09$ | $1.98\pm 0.14$ | $1.34\pm 0.15$ | $21.20\pm 3.40$ | 29.39

Table 6: Resulting additional parameters of studying the data of the three spacecraft on April 23, 2007. $V_{sh}$ is the shock speed, $C^{up}_{s}$ is the upstream sound speed, $V^{up}_{A}$ is the upstream Alfvén-speed and $C^{up}_{ms}$ is the upstream magnetosonic speed. Plasma $\beta_{up}$, Alfvén$-$Mach number, and Magnetosonic Mach number are also shown. Spacecraft | $V_{sh}$ | $C^{up}_{s}$ | $V^{up}_{A}$ | $C^{up}_{ms}$ | Plasma $\beta_{up}$ | Alfvén-Mach | Magnetosonic-Mach
---|---|---|---|---|---|---|---
| [km/s] | [km/s] | [km/s] | [km/s] |  |  |
STEREO$-$A | 467.95 | $64.97\pm 7.74$ | $68.71\pm 29.20$ | $94.57\pm 21.85$ | $1.07\pm 0.94$ | $1.00\pm 0.67$ | $0.72\pm 0.41$
ACE | 412.36 | $58.28\pm 4.94$ | $52.82\pm 15.90$ | $78.66\pm 11.26$ | $1.46\pm 0.91$ | $1.49\pm 0.56$ | $1.00\pm 0.27$
Wind | 395.53 | $63.79\pm 5.15$ | $69.16\pm 21.71$ | $94.09\pm 16.34$ | $1.02\pm 0.66$ | $0.90\pm 0.42$ | $0.66\pm 0.26$
STEREO$-$B | 370.31 | $57.31\pm 5.38$ | $29.04\pm 12.28$ | $64.25\pm 07.34$ | $4.67\pm 4.04$ | $1.69\pm 1.08$ | $0.76\pm 0.37$

Figure 1: The panels show from top to bottom, the magnetic field magnitude as
well as its components, the total velocity, and its components, density, and
temperature. The dashed red line represents the exact shock time. The duration
of the plots is two hours. (b) The plot of the shock detected by the Wind
spacecraft on May 7, 2007, at 07:02:30 (UTC). FF stands for the fast forward
shock, which means the shock is travelling away from its driver. (b) The plot
of the shock detected by the STEREO$-$A spacecraft on May 7, 2007, at 08:11:30
(UTC). (c) The plot of the shock detected by the STEREO$-$B spacecraft on May
7, 2007, at 09:42:00 (UTC).

Figure 2: The plot of the shock detected by the Cluster SC1 (a), SC2 (b), SC3
(c), and SC4 (d) on May 7, 2007, at 08:27:55 (UTC). The panels show from top
to bottom, the magnetic field magnitude as well as its components, the total
velocity, and its components, density, and parallel and perpendicular
temperatures. The dashed red line represents the exact shock time. The
duration of the plot is two hours. Only the magnetic field magnitude and
density are shown for Cluster SC2 and SC4.

Figure 3: (a) The Wind magnetic field measurements on May 7, 2007. The
upstream $\Delta t_{up}$ is between (06:59:00 - 07:01:47), and the downstream
$\Delta t_{down}$ is between (07:03:50 - 07:05:50). (b) The STEREO$-$A
magnetic field measurements on May 7, 2007. The upstream $\Delta t_{up}$ is
between (08:08:00 - 08:10:05), and the downstream $\Delta t_{down}$ is between
(08:12:30 - 08:13:50). (c) The STEREO$-$B magnetic field measurements on May
7, 2007. The upstream $\Delta t_{up}$ is between (09:37:00 - 09:40:35), and
the downstream $\Delta t_{down}$ is between (09:42:57 - 09:43:38). FF stands
for the fast forward shock, which means the shock is travelling away from its
driver. The chosen intervals of the upstream and downstream magnetic fields
were used for minimum variance analysis and co-planarity methods. The blue and
red dashed lines each represent the upstream starting time and ending time and
the downstream starting time and ending time, respectively.

Figure 4: The Cluster SC1 (a), SC3 (b), SC2 (c), and SC4 (d), respectively. B
magnetic field measurements on May 7, 2007. The upstream $\Delta t_{up}$ is
between (08:26:10 - 08:27:00), and the downstream $\Delta t_{down}$ is between
(08:28:00 - 08:29:00). The symbols and details of the figures are the same as
3. The chosen intervals of the upstream and downstream magnetic fields were
used for minimum variance analysis and co-planarity methods.

Figure 5: 2D (a) XY, (b) XZ, and (c) YZ sketches of the propagation of the IP
shock through spacecraft – Wind (light blue), STEREO$-$A (green) and B (red),
and average position of four Cluster satellites (purple) on May 7, 2007. The
orange arrow represents the Sun-Earth line direction. The arrows on spacecraft
positions indicate the normal vector direction calculated from the co-
planarity method and the lines perpendicular to the normal vectors indicate
the shock surface orientations. The sizes of the lines are arbitrary. The
shock orientations for the four Cluster satellites are averaged.

Figure 6: 2D sketch in (a) XY, (b) XY, and (c) YZ plans of the propagation of
the IP shock through four Cluster satellites – SC1 (blue), SC2 (light blue),
SC3 (purple), and SC4 (red) on May 7, 2007. The orange arrow represents the
Sun-Earth line direction. The arrows on spacecraft positions indicate the
normal vector direction calculated from co-planarity and the lines
perpendicular to the normal vectors indicate the shock surface orientations.
The sizes of the lines are arbitrary.

Figure 7: A 3D sketch of the propagation of the IP shock through spacecraft–
Wind, STEREO$-$A, and STEREO$-$B, and average position of four Cluster
satellites on May 7, 2007. (a) Actual shock detected positions. (b) The
positions of STEREO$-$A, STEREO$-$B, and the average positions of four Cluster
spacecraft are shifted back in time to the shock detection time of the Wind
spacecraft. (c) The time-shifted positions are fitted with a plane. The arrows
indicate the normal vector direction and the planes perpendicular to the
normal vectors indicate the shock surface orientations. The sizes of the
planes are arbitrary.

Figure 8: (a) The plot of the shock detected by the STEREO$-$A spacecraft on
April 23, 2007, at 06:53:35 (UTC). (b) The plot of the shock detected by the
ACE spacecraft on April 23, 2007, at 08:57:00 (UTC). (c) The plot of the shock
detected by the Wind spacecraft on April 23, 2007, at 09:12:00 (UTC). (d) The
plot of the shock detected by the STEREO$-$A spacecraft on April 23, 2007,
around 13:21:30 (UTC). FR stands for the fast reverse shock, which means the
shock is moving toward its driver. The panels show from top to bottom, the
magnetic field magnitude as well as its components, the total velocity and its
components, density, and temperature. The dashed red line represents the exact
shock time. The duration of the plot is two hours.

Figure 9: (a) The STEREO$-$A magnetic field measurements on April 23, 2007.
The downstream $\Delta t_{down}$ is between (06:46:00 - 06:52:23), and the
upstream $\Delta t_{up}$ is between (06:57:03 - 07:02:34). (b) ACE magnetic
field measurements on April 23, 2007. The downstream $\Delta t_{down}$ is
between (08:51:30 - 08:56:30), and the upstream $\Delta t_{up}$ is between
(09:03:00 - 09:07:00). (c) The Wind magnetic field measurements on April 23,
2007. The downstream $\Delta t_{down}$ is between (09:04:00 - 09:10:00), and
the upstream $\Delta t_{up}$ is between (09:12:20 - 09:13:15). (d) STEREO$-$B
magnetic field measurements on April 23, 2007. The downstream $\Delta
t_{down}$ is between (13:15:45 - 13:20:00), and the upstream $\Delta t_{up}$
is between (13:24:00 - 13:26:00). FR stands for the fast reverse shock, which
means the shock is moving toward its driver. The chosen intervals of the
upstream and downstream magnetic fields were used for minimum variance
analysis and co-planarity methods. The red dashed lines each represent the
downstream starting time and ending time and the upstream starting time and
ending time respectively.

Figure 10: 2D (a) XY, (b) XZ, and (c) YZ sketches of the propagation of the IP
shock through spacecraft– STEREO$-$A, ACE, Wind, and STEREO$-$B. The orange
arrow represents the Sun-Earth line direction. The arrows on spacecraft
positions indicate the normal vector direction calculated from the co-
planarity method and the lines perpendicular to the normal vectors indicate
the shock surface orientations. The sizes of the lines are arbitrary.

Figure 11: (a) A 3D sketch of the propagation of the IP shock through four
spacecraft– STEREO$-$A, Wind, ACE, and STEREO$-$B. (b) The positions of Wind,
ACE, and STEREO$-$B are shifted back in time to the shock detection time of
STEREO$-$A. (c) The shifted positions are shown from the top view. The arrows
indicate the normal vector direction and the planes perpendicular to the
normal vectors indicate the shock surface orientations. The sizes of the
planes are arbitrary.

Figure 12: STEREO$-$A PLASTIC bulk velocity, the SWEA electron density, MAG
magnetic field magnitude, and the SEPT ion intensity from 110 keV to 2200 keV.
From February 1, 2007, to June 1, 2007, periods featuring well-established Co-
rotating Interaction Regions (CIRs) with noticeable boosts in ion energy are
highlighted with grey shading (Opitz et al., 2014, Figure 2).

Figure 13: $K_{p}$-index on May 07, 2007. The green bars indicate moderate
geomagnetic activity, while the red bar indicates a G1-minor geomagnetic
storm.

Figure 14: The $K_{p}$-index on April 23, 2007. The green bars indicate
moderate geomagnetic activity, while the yellow bar denotes intensifying
geomagnetic activity with the red bars symbolize a geomagnetic storm.