# Online Evasive Strategy for Aerial Survey using Sierpinski curve

Ashay Wakode  Arpita Sinha  Indian Institute Of Technology Bombay, Mumbai,
Maharashtra 400076 India (e-mail: ashaywakode@gmail.com).  Indian Institute Of
Technology Bombay, Mumbai, Maharashtra 400076 India (e-mail:
<EMAIL_ADDRESS>
###### Abstract

This paper deals with the aerial survey of a closed region using the Space-
Filling curve, particularly Sierpinski curve. The specified region is
triangulated, and the Sierpinski curve is used to explore each smaller
triangular region. The entire region may have one or more obstacles. An
algorithm is presented which suggests evasive manoeuvre (detour) if an
obstacle is detected. The algorithm is online; that is, it does not require
prior knowledge of the location of obstacles and can be applied while the
robotic system is traversing the designated path. The fractal nature of the
Sierpinski curve and simple geometric observations were used to formulate and
validate the algorithm. The non-uniform coverage and multiple obstacle
problems are also dealt with towards the end.

###### keywords:

Autonomous Vehicles, Coverage Path Planning, Online Obstacle evasion, Robotic
Exploration, Space-Filling curve, Trajectory and Path Planning

##  1 Introduction

Space-Filling curves are a special type of curves that pass through every
point of closed interval onto which they are mapped. Space-Filling curves are
made out of smaller similar curves, which is referred to as the fractal nature
of space filling curves (Sagan (1994), Bader (2012)). Due to these interesting
properties, space-filling curves are widely used in applications where the
agent needs to explore/survey a region and gather data. Other applications of
Space-Filling curves can be found in computer graphics (Dafner et al. (2000),
Memon et al. (2000)), Computer Aided-Design (Bader (2012)), parallel computing
(Zumbusch (2000)) and many more.

With advent of Unmanned Aerial Vehicles (UAVs), the robotics
exploration/survey task has been simplified. Survey using UAVs is superior due
to the following advantages :

  1. 1.

The search agent no longer need to worry about the region’s topography.

  2. 2.

The surface of the region being explored remains undisturbed.

  3. 3.

Cheaper and less bulky UAVs can be deployed in huge numbers, reducing cost and
increasing efficiency in terms of both energy and time.

A survey of a region using aerial vehicles is theoretically better but
requires robust algorithms/strategies for implementation in the real world.
This paper deals with one such strategy for aerial survey using the Sierpinski
curve.

The robotics exploration problem is a sub-problem of the Coverage Path
Planning problem (CPPP). In CPPP, a robotic agent needs to traverse through a
certain region while doing tasks like mowing, painting, or cleaning, to name a
few. There are many solutions available for CPPP which can be used for robotic
exploration problem and hence for aerial survey: Boustrophedon decomposition,
Grid-based methods, Graph-based methods, and many more (Galceran and Carreras
(2013)).

Space-Filling curves provide a novel solution to robotic exploration problem
because of their useful properties. An exhaustive search can be carried out
since it passes through every point of the region. The fractal nature allows
us to design algorithms/strategies easily since a space-filling curve is a
combination of other similar space-filling curves. Space-Filling curves can
span both 2D and 3D space, and algorithms built for a 2D region can be easily
extended to 3D space.

Spires and Goldsmith (1998) suggest the use of a swarm of robots for
exploration, each robot following a space-filling curve. This approach is
efficient in terms of energy, robust to failures, and assures coverage in
finite time, But the presence of obstacles is not considered. Tiwari et al.
(2007) and Ban et al. (2013) propose a solution to the exploration problem
using a space-filling curve. However, apriori location of obstacles is
required making them unsuitable for online implementation. Nair et al. (2017),
Joshi et al. (2019) give out online algorithms for robotic exploration using
Hilbert’s space-filling curve. However, the obstacles must occupy only single
and double neighboring grid locations. But, this approach cannot be used for
the Sierpinski curve. So, No online exploration strategy exists for an agent
following a Sierpinski curve.

We have formulated a strategy for aerial survey when the search agent uses the
Sierpinski curve as a guiding path while avoiding obstacles; when the location
of obstacles is unknown. The suggested strategy is online; that is, the
whereabouts of the obstacles are unknown when the search is initiated, and the
strategy comes into play once it detects the obstacles. The details of
strategy implementation in multiple obstacle situations and non-uniform
coverage scenarios are established. Finally, the strategy is put together as a
pseudo-code for real-life applications.

Section 2 introduces various preliminaries on the Sierpinski curve and other
tools required in the following sections. Section 3 formulates the problem and
presents the strategy. Various lemmas for validating the result are produced
in this section. Section 4 touches upon multiple obstacle scenarios. Section 5
gives out the pseudo-code for implementation of the strategy, and the non-
uniform coverage problem is dealt with in brief. Lastly, Section 6 concludes
the paper.

##  2 Preliminaries

###  2.1 Sierpinski Curve

Sierpinski curve is a mapping from $\mathbf{I}=[0,1]$ to $\mathbf{T}$, where

$\bf{T}=\\{(x,y):y\geq 0,x-y\geq 0,x+y-2\leq 0\\}$ (1)

In this paper, we will use approximations of the Sierpinski curve. The
approximate Sierpinski curve can be constructed by dividing T into smaller
triangles and connecting them in a certain fashion, see figure 1. I is also
divided into smaller sub-intervals, which are then mapped to the smaller sub-
regions, see figure 2. Here, the refinement of tessellation used for
constructing the Sierpinski curve is quantified by iteration number (denoted
as $n$). In figure 1, $n=1$ Sierpinski curve has right triangle divided
$2^{2}=4$ times, $n=2$ Sierpinski curve has right triangle divided $2^{3}=8$
times, and so on. As $n\rightarrow\infty$, approximate Sierpinski curve tends
to Sierpinski curve.

A robotic agent with point size sensing radius will require to follow the
Sierpinski curve with iteration number going to $\infty$. However, a search
agent with a finite non-zero sensing radius going from one center of triangle
to another following the approximate Sierpinski curve can explore T entirely,
given it can explore the entire triangular cell it is present in.

Figure 1: First 6 iterations of the Sierpinski curve shown using yellow piece-
wise straight line Figure 2: Mapping between I and T

###  2.2 Grammar-based construction of Sierpinski curve

Figure 3: Patterns $S$, $Z$, $R$, and $P$

The fractal nature of the Sierpinski curve can be used for its construction.
Consider patterns $S$, $Z$, $R$, and $P$, as shown in figure 3. Here, $S$ is
the $n=1$ iteration of Sierpinski curve. Now, using the following production
rules, further odd-numbered iterations of the Sierpinski curve can be
constructed:

$S\Rightarrow S\nearrow R\rightarrow P\searrow S$ (2) $R\Rightarrow R\nwarrow
Z\uparrow S\nearrow R$ (3) $Z\Rightarrow Z\swarrow P\leftarrow R\nwarrow Z$
(4) $P\Rightarrow P\searrow S\downarrow Z\swarrow P$ (5)

Higher iterations can be constructed in the following way:

$\displaystyle n=1:S$ $\displaystyle n=3:S\nearrow R\rightarrow P\searrow S$
$\displaystyle n=5:(S\nearrow R\rightarrow P\searrow S)\nearrow(R\nwarrow
Z\uparrow S\nearrow R)\rightarrow$ $\displaystyle(P\searrow S\downarrow
Z\swarrow P)\searrow(S\nearrow R\rightarrow P\searrow S)$

Similar patterns and production rules exist for even-numbered iteration
Sierpinski curves (Bader (2012)).

There is another way to look at the fractal nature of the Sierpinski curve. It
is easy to observe that the iteration 2 Sierpinski curve can be obtained by
putting two iteration 1 Sierpinski curves along the shorter side symmetrically
and joining the ends. To generalize, iteration $n$ Sierpinski curve can be
obtained by combining two iteration $n-1$ Sierpinski curves. It is not hard to
prove the previous statement. The patterns and the production rules introduced
previously can be easily used to prove the statement.

###  2.3 Extension to Generalised Sierpinski curve

The Sierpinski curve generated in the previous subsection spans an isosceles
right triangle (T). However, we are interested in the Sierpinski curve
spanning a random triangle, referred to as the Generalised Sierpinski curve
(abbreviated as GSc). Patterns and production rules similar to the ones
introduced in the last subsection exists for GSc.

Figure 4: Projective Transformation

Working with GSc using patterns and production rules can get complicated.
Instead we introduce a simple method for generating GSc. Here, the Sierpinski
curve for T is generated and converted to GSc spanning a known triangle
(denoted as U) using projective transformation (see Kanatani et al. (2016)),
as shown in figure 4. Projective transformation is a linear map given by
equation 6.

$\begin{bmatrix}x\prime\\\ y\prime\\\
1\end{bmatrix}=\begin{bmatrix}\alpha&\beta&\gamma\\\ \epsilon&\zeta&\eta\\\
0&0&1\end{bmatrix}\begin{bmatrix}$x$\\\ $y$\\\ 1\end{bmatrix}$ (6)

Here, $(x,y)\in\textbf{T}$ mapped to $(x\prime,y\prime)\in\textbf{U}$.
$\alpha$, $\beta$, $\gamma$, $\epsilon$, $\zeta$ and $\eta$ are constants for
given U, and can be calculated by mapping vertices of U, which are already
known.

##  3 Main Result

###  3.1 Problem Formulation

Figure 5: Sensing mechanism of the aerial vehicle is of the form of a cone,
sensing radius on ground can be changed by varying height,
$H_{1}>H_{2}\implies R_{1}>R_{2}$

The search agent, which in this case is an aerial vehicle, travels through U
using the path marked by GSc. The sensing radius of the search agent and the
iteration of GSc are to be decided by the user. The sensing radius and the
iteration of GSc are chosen such that the search agent is able to search the
entire triangular cell it is present in (See figure 5). The iteration of GSc
depends on how rigorously the user wants the search agent to survey a certain
part of the region (Further details can be found in Section 5).

In this paper, we assume the obstacles to block single triangular cells in the
GSc, and that the search agent detects those obstacles before bumping into
them. We further assume that the search agent is equipped with a sensing
mechanism to detect obstacles.

While traversing the GSc, the search agent at $i^{th}$ triangular cell detects
obstacle at $i+1^{th}$ triangular cell. So, the search agent has to go around
the obstacle cell to reach $i+2^{th}$ triangular cell and thus successfully
evade the obstacle. The search agent may have multiple choices of evasive
maneuvers, and the best will be the least lengthy.

Figure 6: Possible geometries of 3 consecutive triangular cells in T

So, the problem boils down to planning evasive maneuvers between the $i^{th}$
and $i+2^{th}$ triangular cells, with the $i+1^{th}$ cell blocked by an
obstacle. The possible cases of 3 consecutive triangular cell geometries in
the Sierpinski curve are listed in figure 6.

###### Lemma 1

Case 1-5 are the only possible 3 consecutive triangular cell geometries in the
Sierpinski curve.

The fractal nature of Sierpinski curve can be used here. It is easy to observe
that the case 2-5 are the only cases possible in pattern $S$, $R$, $Z$ and
$P$. Finally, the geometries generated after joining $S$, $R$, $Z$ and $P$
also contains case 1-5 only. Therefore, cases 1-5 are the only possible 3
consecutive triangular geometries in the Sierpinski curve.$\blacksquare$ Since
projective transformation is a linear map, the previous statement also holds
for GSc, with the perpendicular bisectors in cases 1-5 turning into medians
(Line joining a vertex to the mid-point of the opposite side).

Here, case 1 is tagged $f$. Cases 2 and 3 are similar; since they are water
images, they are tagged as $h$. Similarly, Cases 4 and 5 are tagged as $k$.
Finally, some nomenclature:

  1. 1.

$W_{i}$ : Centre of the $i^{th}$ triangular cell.

  2. 2.

$C$ : Common vertex shared by all 3 triangular cells in each case 1-5.

  3. 3.

$B_{i,j}$ : Vertex shared by $i^{th}$ and $j^{th}$ triangular cells only.

  4. 4.

$D_{i,j}$ : Vertex of $i^{th}$ triangular cell not shared with $j^{th}$
triangular cell.

Finally, algorithm 1 can be used to tag 3 consecutive triangular cells
beginning with $i^{th}$ cell. Algorithm for tag generation

1:Input : $i$, $C$, $D_{i,i+1}$, $D_{i+1,i}$, $B_{i,i+1}$

2:Output : Tag

3:if $A$($D_{i,i+1}$, C, $D_{i+1,i}$) $=0$ then

4: Tag is $f$

5:else if $A$($D_{i,i+1}$, $B_{i,i+1}$ , $D_{i+1,i}$) $=0$ then

6: Tag is $h$

7:else

8: Tag is $k$

9:end if

###  3.2 Proposed Strategy and Validity of Proposed Strategy

Table 1: Evasive Strategies Sr. No. | Evasive Maneuver | Change of Path | $L$
---|---|---|---
1 |  |  $w_{i}$ $\rightarrow$ C $\rightarrow$ $w_{i+2}$ | $L_{1}=\frac{\mid C-w_{i}\mid+\mid w_{i+2}-C\mid}{\mid w_{i+2}-w_{i}\mid}$
2 |  |  $w_{i}$ $\rightarrow$ C $\rightarrow$ $w_{i+2}$ | $L_{2}=\frac{\mid C-w_{i}\mid+\mid w_{i+2}-C\mid}{\mid w_{i+2}-w_{i}\mid}$
3 |  |  $w_{i}$ $\rightarrow$ $B_{i,i+1}$ $\rightarrow$ $B_{i+1,i+2}$ $\rightarrow$ $w_{i+2}$ | $L_{3}=\frac{\mid B_{i,i+1}-w_{i}\mid+\mid B_{i+2,i+1}-B_{i,+1}\mid+\mid w_{i+2}-B_{i+2,i+1}\mid}{\mid w_{i+2}-w_{i}\mid}$
4 |  |  $w_{i}$ $\rightarrow$ C $\rightarrow$ $w_{i+2}$ | $L_{4}=\frac{\mid C-w_{i}\mid+\mid w_{i+2}-C\mid}{\mid w_{i+2}-w_{i}\mid}$
5 |  |  $w_{i}$ $\rightarrow$ $B_{i,i+1}$ $\rightarrow$ $B_{i+1,i+2}$ $\rightarrow$ $w_{i+2}$ | $L_{5}=\frac{\mid B_{i,i+1}-w_{i}\mid+\mid B_{i+2,i+1}-B_{i,i+1}\mid+\mid w_{i+2}-B_{n+2,n+1}\mid}{\mid w_{i+2}-w_{i}\mid}$

Evasive maneuvers are proposed in Table 1. $L$ quantifies the length of
maneuvers. Case 2 has been used for visualizing maneuvers for tag $h$. Case 4
has been used for visualizing maneuvers for tag $k$. The algorithm for online
implementation is presented in Section 5.

###### Lemma 2

Only the first two and the last two triangular cells in GSc have all three
vertices on the boundary of U. Also, only $2^{nd}$ and $3^{rd}$ triangular
cells (both from start and end of GSc) share two vertices, both of which lie
on the boundary.

The lemma is proved for the Sierpinski curve spanning T. Since the
transformation from T to U is linear, the statement would hold for GSc.

Figure 7: Combination of two $n=2k+1$ iteration Sierpinski curve to get
$n=2k+2$ iteration Sierpinski curve; Vertices of interest are highlighted in
blue; Triangular cells of interest are numbered; Easy to see lemma 2 holds for
both

Using mathematical induction. Lemma holds for $n=1,2$. Using fractal nature of
the Sierpinski curve, iteration $n=2k+2$ (even) Sierpinski curve can be
constructed using two $n=2k+1$ iteration Sierpinski curves, and interestingly
lemma holds in both cases (See figure 7). Similar argument holds for $n=2k+1$
(odd) as well. Therefore, the lemma holds.$\blacksquare$

###### Lemma 3

Using the evasive strategies enlisted in Table 1, a search agent can evade any
obstacle lying entirely inside a single triangular cell, except when the
obstacle is present inside the one of first 3 or last 3 triangular cells of
GSc.

Table 1 lists evasive techniques for nodes with tags $f$, $h$ and $k$. The
tags are given to geometries of cases 1 to 5 (refer figure 6). Cases 1-5 are
the only possible geometries of three consecutive triangular cells in GSc, as
proved in lemma 1. So, it remains to be shown that strategies enlisted in
table 1 are always possible for each tag. Hence, the following cases :

Figure 8: Tag $f$ geometry

  1. 1.

Tag $f$ (See figure 8) : This will be impossible if $C$ lies on the boundary.
Assuming $C$ lies on the boundary, there would be two line segments
originating from $C$ with an interior angle $>180^{\circ}$, which is
infeasible. Hence $C$ will never lie on the boundary. Therefore, the evasive
strategy for this case will always be possible.

Figure 9: Tag $h$ geometry; Obstacle represented by black triangular cell

  2. 2.

Tag $h$ (See figure 9) : Let the cell with the obstacle be denoted as $\Delta
DEF$. The possible boundary lines which can stop the search agent from
following the evasive maneuvers are labeled as $m$, $n$, and $o$.

Firstly, assume only lines $o$ and $m$ are present. Then the cell preceding
$\Delta DEF$ will have all of its vertices on the boundary, and hence it will
be either $1^{st}$ or $2^{nd}$ cell from start or end (from lemma 2), which
implies $\Delta DEF$ will be $2^{nd}$ or $3^{rd}$ cell; this contradicts the
assumption that the cell with an obstacle is not one of the first or last
three cells in GSc.

Secondly, assume only lines $o$ and $n$ are present. In this situation, the
robotic system can follow the evasive strategy 2.

Thirdly, assume only lines $m$ and $n$ are present. Here, $\Delta DEF$ and the
succeeding triangle share two vertices that lie on the boundary. This implies
$\Delta DEF$ is either the $2^{nd}$ or $3^{rd}$ cell from start or end (from
lemma 2). Therefore, a contradiction.

Lastly, assuming the presence of all lines $o$, $m$, and $n$ will imply that
$\Delta DEF$ is the $2^{nd}$ triangle from start or end (from lemma 2). Again,
a contradiction.

Figure 10: Tag $k$ geometry; Obstacle represented by black triangular cell

  3. 3.

Tag $k$ (See figure 10) : Proceeding on the same lines, Let the cell with the
obstacle be denoted as $\Delta HIJ$. The possible boundary lines which can
stop the search agent from doing an evasive maneuver are labeled as $x$, $y$,
and $z$.

Firstly, assume only lines $x$ and $z$ are present. The cell succeeding
$\Delta HIJ$ will have all of its vertices on the boundary, and hence it will
be either $1^{st}$ or $2^{nd}$ cell from start or end (from lemma 2), hence
$\Delta HIJ$ will be $2^{nd}$ or $3^{rd}$ cell. A contradiction.

Secondly, assume only lines $z$ and $y$ are present. In this situation, the
robotic system can follow evasive strategy no. 4.

Thirdly, assume only lines $x$ and $y$ are present. Here, $\Delta HIJ$ and the
preceding cell share two vertices which lie on the boundary. Which implies
$\Delta HIJ$ is either $2^{nd}$ or $3^{rd}$ triangle from start or end (from
lemma 2). Therefore, a contradiction. Lastly, assuming the presence of all
lines $x$, $y$, and $z$ will imply that $\Delta DEF$ is $2^{nd}$ cell from
start or end (from lemma 2). Again, a contradiction.

Therefore, The lemma is proved.$\blacksquare$

##  4 Multiple Obstacle Scenario

The strategy presented can be extended to situations wherein multiple
obstacles are present. The alternate path for evading a single obstacle
depends only on the triangular cells preceding and after the triangle with an
obstacle. So, if the $j^{th}$ triangular cell has an obstacle, then the
strategy presented will not work if the $j+1^{th}$ triangle also has an
obstacle. Nevertheless, the strategy can be applied in a usual way if the
$j+2^{th}$ triangle has an obstacle. So, the strategy will work for any number
of disjoint obstacles. This is shown in figure 11.

Figure 11: Triangle explored using iteration 4 Sierpinski curve; Red
triangular cells: locations blocked by obstacles; Red points: locations at
which obstacle is detected and the evasive maneuver is initiated.

##  5 Implementation & Non-Uniform Coverage Scenario

###  5.1 Real-life Implementation

Sierpinski curve explores a triangle, but in real life, the region to be
explored may not be a triangle. It will be a closed curve. A closed region can
be approximated by polygon (referred to as $P$) using fast and efficient
algorithms; see Sklansky and Gonzalez (1980). $P$ can be triangulated to get
$T_{P}$, using existing methods with time complexity of $O(n\log n)$ for
Siedel’s method (Seidel (1991)), $O(n)$ for Toussaint’s method (Toussaint
(1984)), and many more. The choice of method for triangulation rests with the
user.

Figure 12: Black closed curve: Region to be explored; Brown Polygon:
Approximate Polygon with triangulation; Dark green vertices and dashed edges:
Graph to be explored

A dual graph $G(T_{P})$ of $T_{P}$ is generated. A dual graph of a
triangulated polygon is a graph with vertices for triangles, and the vertices
denoting triangles sharing sides are joined using edges (Berg et al. (1997)).
Lastly, exploring the entire region will need the search agent to visit
vertices of $G(T_{P})$ and use the Sierpinski curve to explore the triangles
denoted by vertices. Traversing between the vertices of $G(T_{P})$ will be a
Traveling Salesman Problem, whose solution can be found offline (see Figure
12).

The aerial survey of a scalene triangle using the Sierpinski curve requires
function $Y$, which goes to zero on the edges, and is non-zero everywhere. $Y$
can be formulated by multiplying line equations of edges of the triangle.
Function $Y$ ensures that the robotic system does not cross the boundary and
go out of the region to be explored. Assuming the search agent at the $i^{th}$
triangular cell, and obstacle is detected at $i+1^{th}$ triangular cell. Using
$Y$, algorithm 5.1 gives a path to be followed to $i+2^{th}$ triangular cell.

Algorithm for online implementation

1:Input : $n$, Search agent at $i^{th}$ cell, Obstacle detected at $i+1^{th}$
cell, $C$, $B_{i,i+1}$, $B_{i+1,i+2}$

2:Output : Detour to get to $i+2^{th}$ triangular cell

3:if $Tag(i)=f$ then

4: Follow strategy 1 from Table 1

5:end if

6:if $Tag(i)=h$ then

7: if $Y(C)\neq 0$ then

8: if $L_{2}$$<$$L_{3}$ then

9: Follow strategy 2 from Table 1

10: else if $Y(B_{i,i+1})\times Y(B_{i+1,i+2})\neq 0$ then

11: Follow strategy 3 from Table 1

12: end if

13: else

14: Follow strategy 3 from Table 1

15: end if

16:end if

17:if $Tag(i)=k$ then

18: if $Y(C)\neq 0$ then

19: if $L_{4}$$<$$L_{5}$ then

20: Follow strategy 4 from Table 1

21: else if $Y(B_{i,i+1})\times Y(B_{i+1,i+2})\neq 0$ then

22: Follow strategy 5 from Table 1

23: end if

24: else

25: Follow strategy 5 from Table 1

26: end if

27:end if

###  5.2 Non-Uniform Coverage

In robotic exploration scenarios, the search agent may have some prior
knowledge about the region. Search agent may be aware of certain parts of the
region that have a higher probability of discovering entities of interest like
minerals or resources. In such a situation, it is wise for the search agent to
explore the higher probability parts rigorously, giving those parts more time
and surveying them more closing. Such scenarios are referred to as Non-Uniform
Coverage Planning.

Sadat et al. (2014) discusses non-uniform coverage scenarios in aerial survey.
Sadat et al. (2015) go a step forward and use Hilbert’s space filling curve
for exploration. Finally, with Hilbert’s space-filling curve, Nair et al.
(2017) introduced strategies for obstacle evasion in a non-uniform coverage
context.

This paper investigates obstacle evasion for the Sierpinski curve in non-
uniform scenarios using strategy presented in Section 3. The parts of the
region which are required to be explored rigorously are traversed using a
higher iteration Sierpinski curve, Which allows the aerial vehicle to spend
more time on such parts. Nevertheless, the distance between surface of the
region and the aerial vehicle reduces as sensing radius is reduced, implying
lesser loss of information due to errors.

Figure 13: Shortcut Heuristic Figure 14: Non-Uniform Coverage of region A,
with 1 and 2 as parts; Search agent jumps to the purple point while going from
1 to 2, then goes to yellow point to start exploring 2, due to SH

Here, we use Shortcut Heuristic (denoted as SH) for traveling between parts
(see Sadat et al. (2014)). In SH, a search agent visits all the triangular
cells at the lowest level of a part, then visits the nearest triangular cell
of the next part and goes on visiting all the triangular cells at the lowest
level in that part; this is done recursively to cover the entire region (See
figure 13). Finally, figure 14 shows an example of non-uniform coverage using
the Sierpinski curve.

##  6 Conclusion

The Aerial Survey problem using the Sierpinski curve has been considered.
Strategies to evade obstacles online were presented. Simple geometric
properties were used to devise evasive strategies. The suggested strategies
introduced are proven to be exhaustive. The evasion plan is also shown to be
helpful in non-uniform coverage situations. The strategy can be used for two
or more disjoint obstacles covering triangular cells of the Sierpinski curve.
The presented strategy cannot be used for obstacles covering two or more
arbitrary triangular cells, but this can be a good starting point. Hence
further work demands consideration of arbitrary sized and shaped obstacles.

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11institutetext: Department of Physics, University of Helsinki, P.O. Box 64,
FI-00014, Helsinki, Finland
22institutetext: NASA Postdoctoral Program Fellow, NASA Goddard Space Flight
Center, Greenbelt, MD 20771, USA
22email<EMAIL_ADDRESS>
# Type II radio bursts and their association with coronal mass ejections in
solar cycles 23 and 24

A. Kumari 1122 D. E. Morosan 11 E. K. J. Kilpua 11 F. Daei 11

(Received ; accepted )

###### Abstract

Context. Metre wavelength type II solar radio bursts are believed to be the
signatures of shock-accelerated electrons in the corona. Studying these bursts
can give information about the initial kinematics, dynamics and energetics of
CMEs in the absence of white-light observations.

Aims. In this study, we investigate the occurrence of type II bursts in solar
cycles 23 and 24 and their association with coronal mass ejections (CMEs). We
also explore the possibility of occurrence of type II bursts in the absence of
a CME.

Methods. We performed statistical analysis of type II bursts that occurred
between 200 - 25 MHz in solar cycle 23 and 24 and found the temporal
association of these radio bursts with CMEs. We categorised the CMEs based on
their linear speed and angular width, and studied the distribution of type II
bursts with ‘fast’ ($speed\leavevmode\nobreak\ \geq 500km/s$), ‘slow’
($speed\leavevmode\nobreak\ <500km/s$), ‘wide’ ($width\leavevmode\nobreak\
\geq 60^{o}$) and ‘narrow’ ($width\leavevmode\nobreak\ <60^{o}$) CMEs. We
explored the type II bursts occurrence dependency with solar cycle phases.

Results. Our analysis shows that there were total 768 and 435 type II bursts
in solar cycle 23 and 24, respectively. Of these, 79 $\%$ were associated with
CMEs in solar cycle 23 and 95% were associated with CMEs in solar cycle 24.
However, only 4$\%$ and 3$\%$ of the total number of CMEs were accompanied by
type II bursts in solar cycle 23 and 24, respectively. Most of the type II
bursts in both cycles were related to ‘fast and wide’ CMEs (48 $\%$). We also
determined a typical drift rate and duration for type II bursts, which is 0.06
MHz/s and 9 min, respectively. Our results suggest that type II bursts
dominate at heights $\approx 1.7-2.3\pm 0.3\leavevmode\nobreak\ R_{\odot}$
with a clear majority having an onset height around 1.7 $\pm
0.3\leavevmode\nobreak\ R_{\odot}$ assuming the four-fold Newkirk model.

Conclusions. The results indicate that most of the type II bursts had a white-
light CME counterpart, however there were a few type II which did not have a
clear CME association. There were more CMEs in cycle 24 than cycle 24.
However, the number of type II radio bursts were less in cycle 24 compared to
cycle 23. The onset heights of type IIs and their association with wide CMEs
reported in this study indicate that the early CME lateral expansion may play
a key role in the generation of these radio bursts.

###### Key Words.:

Sun: coronal mass ejections (CMEs), Sun: – activity, Sun: – radio radiation,
Sun: – sunspots, Sun: – corona

##  1 Introduction

Coronal mass ejections (CMEs) can be powerful drivers of plasma shocks that
can accelerate electrons to high energies. Accelerated electron beams can in
turn generate radio bursts at metre and decimetre wavelengths through the
plasma emission mechanism. Radio shock signatures associated with CMEs are
categorised as type II radio bursts (Smerd, 1970; Dulk, 1970; Klassen et al.,
2002; Gopalswamy et al., 2005; Ramesh et al., 2023). Type II radio bursts are
characterised as lanes of emission drifting from high to low frequencies in
the solar dynamic spectra, with a duration of a few minutes to approximately
one hour. These lanes have a frequency ratio of 1 to 2 representing emission
at the fundamental and harmonic of the plasma frequencies and (Nelson &
Melrose, 1985). Radio imaging and spectroscopy combined with coronal modelling
have shown that type II bursts are associated to CME-driven shock waves (e.g.,
Smerd, 1970; Ramesh et al., 2010; Zucca et al., 2018; Kumari et al., 2019).
Type II bursts can also consist of numerous fine structures. For example,
herringbone bursts are narrow bursts in dynamic spectra superimposed on type
II bursts or occurring on their own, drifting towards higher and lower
frequencies (Roberts, 1959; Cairns & Robinson, 1987; Carley et al., 2015).
Herringbones are believed to be the radio signature of individual electron
beams accelerated by the CME shock (Zlobec et al., 1993; Morosan et al.,
2019). A wide variety of other fine structures composing the emission lanes of
type II bursts have recently been documented in Magdalenić et al. (2020).

Figure 1: Solar radio dynamic spectra of a high frequency type II burst on
November 04, 2015. The duration of the burst is $\sim 12$ minutes and it shows
a frequency drift of $\sim 0.2$ MHz/s from 400 MHz to 55 MHz. The ‘cyan’ color
lines show the fundamental and harmonic bands in the spectra.

Type II bursts are the most common radio bursts accompanying CMEs, hence these
bursts are often used to study CME kinematics, energetics and dynamics near
the Sun (Cunha-Silva et al., 2015; Kumari et al., 2017c; Ma & Chen, 2020). The
advancement of space-based coronagraphs and ground-based spectrographs in the
past two decades helped to understand the relationship between CMEs and meter
wavelength type II radio bursts. Metric type II bursts start typically at
$\approx 1.5\leavevmode\nobreak\ R_{\odot}$ (Lin et al., 2006). Claßen &
Aurass (2002) studied the association of metric and interplanetary (IP) type
II bursts with CMEs observed with LASCO for 63 events during 1997-2000, where
the authors related the type II origin to blast wave shocks or the leading
edge/flanks of the CMEs. Gopalswamy (2006) performed a similar study for
metric and decameter-hectometer (DH) type II bursts during 1996-2004, where
the author concluded that the metric type II bursts are mostly associated with
slow CMEs and DH type II are associated with CMEs with speed $>1100$ km/s.
Kahler et al. (2019) also studied metric and DH type II bursts and their
association with solar energetic particles (SEPs) and found that it is
uncommon for these events to be associated with ‘fast’ and ‘narrow’ CMEs.
Recently, Pohjolainen & Talebpour Sheshvan (2021) studied isolated DH type II
radio bursts, which were found to be associated with a shock near the CME
leading front.

However, at present there is no comprehensive long term statistical study of
metric type II bursts and their relationship with CMEs that covers over one
solar cycle. Recently, Morosan et al. (2021) studied moving radio bursts, that
include type II and moving type IV bursts, and their association with CMEs,
and found that moving radio sources are almost exclusively associated with CME
eruptions. Patel et al. (2021) studied the decameter-hectometer type II bursts
and concluded that CME initial speed are related to the duration of type II
bursts. With the end of solar cycle 24 and the availability of white-light CME
and type II radio bursts data for more than two decades, we take this
opportunity to study the association of type II bursts with CMEs for the last
two solar cycles, 23 and 24, and determine general properties of these CMEs
and the associated radio emission. This article is arranged as follows:
section 2 has the details of observational data and the data analysis methods.
We discuss the results obtained with our analysis in section 3. We present the
summary and conclude the paper in the last section 4.

Figure 2:  The histograms showing the distribution of occurrence of: (a) all
CMEs; (b) all type II bursts; (c) all type II bursts which were accompanied
with white-light CMEs; and d) all type II bursts without any association with
CME; from January, 1996 to December, 2019, which corresponds to solar cycle 23
and 24. The dashed red profile shows the Gaussian fit to the the histograms.
The two peaks of these Gaussian peaks correspond to the solar maxima in the
respective solar cycles.

##  2 Observations and Data Analysis

We used the CMEs listed in the Coordinated Data Analysis Workshop (CDAW;
Yashiro et al., 2004, 2008; Gopalswamy et al., 2009b)
database111https://cdaw.gsfc.nasa.gov/, which lists the CMEs manually detected
with the the Large Angle and Spectrometric Coronagraph (LASCO) onboard the
Solar and Heliospheric Observatory (SOHO; Brueckner et al., 1995). This list
contains the CMEs detected with LASCO coronagraphs since January, 1996. It
includes CME properties such as linear and second order speed, position angle,
acceleration, mass, and kinetic energy. We also used the Cor1 and Cor2
coronagraphs from the Sun-Earth Connection Coronal and Heliospheric
Investigation (SECCHI; Howard et al. (2008)) onboard the Solar Terrestrial
Relationship Observatory (STEREO). For the CMEs observed with the STEREO
spacecrafts, we used the list provided on the Solar Eruptive Event Detection
System
(SEEDS)222http://spaceweather.gmu.edu/seeds/secchi/detection_cor2/monthly/
website. The CMEs listed here are detected automatically using software
packages, which extracts the position angle, angular width, velocity, peak,
and average brightness (Olmedo et al., 2008). We also used CME list provided
by Vourlidas et al. (2017), where the authors used the dual-viewpoint CME
catalog from the SECCHI/COR telescopes333http://solar.jhuapl.edu/Data-
Products/COR-CME-Catalog.php. These catalogues contain STEREO detected CMEs
from 2009 - 2014.

For radio data, we used the event lists provided by the Space Weather
Prediction Center (SWPC)444https://www.swpc.noaa.gov/products/solar-and-
geophysical-event-reports. These event lists contain details of the solar
eruptions such as X-ray and optical flares, radio burst types and their
proprieties, the start and end time of the eruption, the class of the flares,
frequency of radio bursts etc. We also used the sunspot data to investigate
the correlation between the appearance of sunspots and the occurrence of type
II bursts. The sunspot data was taken from SILSO World Data Center
(1996-2019)555http://www.sidc.be/silso/datafiles, which has daily, monthly
mean and 13 month averaged sunspots data available.

In this work, we investigate the relation between type II bursts and CMEs in
solar cycle 23 and 24. To do this, we extracted the type II bursts information
from the SWPC event lists from January, 1996 - December, 2019. This exclusive
type II burst list contains all the type II bursts in solar cycle 23 and 24,
with details such as, start and end time, start and end frequency, reporting
spectrograph, and the associated active region (AR). An example of a type II
burst is shown in Figure 1a. This figure shows a high frequency, high drift
type II burst that occurred on November 04, 2015. The solar dynamic spectra
shown here is with the ground-based Compound Astronomical Low-cost Low-
frequency Instrument for Spectroscopy and Transportable Observatory
(CALLISTO)666http://www.e-callisto.org/ spectrometer at the Gauribidanur Radio
Observatory (GRO)777https://www.iiap.res.in/centers/radio. The fundamental and
harmonic bands are the type II bursts and they are marked in the figure.

To find the type II associations with CMEs, we used the criteria that the
start time of the associated CME should be within 2 hours of the start time of
the type II. This time difference between the start times of type IIs and CMEs
was chosen in order to compensate for disc events where the CME may be seen
later in white light images compared to limb events (e.g., Kumari et al.,
2021; Morosan et al., 2021; Kumari, 2022). Two new list was prepared based on
the aforementioned criteria, which contain the information of: i) the type II
bursts associated with CMEs; ii) the type II bursts without any association
with CMEs. These two lists were used to further carry out the current study.

Table 1: Yearwise distribution of CMEs, type II radio bursts and their association. Year | CMEs | % CMEs | Type IIs | % Type IIs | Type IIs | % Type IIs | Type IIs | % Type IIs
---|---|---|---|---|---|---|---|---
|  |  |  |  | with CMEs | with CMEs | without CME | without CME
Total | 30125 | 100.0% | 1203 | 100.0% | 1024 | 85.2% | 179 | 14.8%
1996 | 206 | 1.5% | 4 | 0.5% | 2 | 0.3% | 2 | 1.2%
1997 | 385 | 2.7% | 21 | 2.7% | 16 | 2.6% | 5 | 3.1%
1998 | 716 | 5.1% | 60 | 7.8% | 24 | 3.9% | 36 | 22.6%
1999 | 1016 | 7.2% | 100 | 13.0% | 75 | 12.3% | 25 | 15.7%
2000 | 1664 | 11.9% | 90 | 11.7% | 72 | 11.8% | 18 | 11.3%
2001 | 1499 | 10.7% | 160 | 20.8% | 136 | 22.3% | 24 | 15.1%
2002 | 1700 | 12.1% | 116 | 15.1% | 96 | 15.8% | 20 | 12.6%
2003 | 1130 | 8.1% | 71 | 9.2% | 62 | 10.2% | 9 | 5.7%
2004 | 1102 | 7.9% | 69 | 9.0% | 58 | 9.5% | 11 | 6.9%
2005 | 1248 | 8.9% | 49 | 6.4% | 45 | 7.4% | 4 | 2.5%
2006 | 1047 | 7.5% | 19 | 2.5% | 15 | 2.5% | 4 | 2.5%
2007 | 1442 | 10.3% | 7 | 0.9% | 6 | 1.0% | 1 | 0.6%
2008 | 863 | 6.1% | 2 | 0.3% | 2 | 0.3% | 0 | 0.0%
Total (SC23) | 14018 | 100.0% | 768 | 100.0% | 609 | 79.3% | 159 | 20.7%
2009 | 746 | 4.6% | 1 | 0.2% | 1 | 0.0% | 0 | 0.0%
2010 | 1117 | 6.9% | 13 | 3.0% | 13 | 3.1% | 0 | 0.0%
2011 | 1990 | 12.4% | 72 | 16.5% | 68 | 16.4% | 4 | 20.0%
2012 | 2177 | 13.5% | 71 | 16.3% | 69 | 16.6% | 2 | 10.0%
2013 | 2338 | 14.5% | 73 | 16.9% | 72 | 17.3% | 1 | 5.0%
2014 | 2478 | 15.4% | 110 | 25.3% | 102 | 24.6% | 8 | 40.0%
2015 | 2058 | 12.8% | 54 | 12.4% | 51 | 12.9% | 3 | 15.0%
2016 | 1393 | 8.7% | 15 | 3.4% | 15 | 3.6% | 0 | 0.0%
2017 | 786 | 4.9 | 18 | 4.1% | 6 | 4.1% | 1 | 5.0%
2018 | 476 | 2.9% | 2 | 0.5% | 2 | 0.5% | 0 | 0.0%
2019 | 543 | 3.4% | 6 | 1.4% | 1 | 1.2% | 1 | 5.0%
Total (SC24) | 16107 | 100.0% | 435 | 100.0% | 415 | 95.4% | 20 | 4.6%

##  3 Results

The yearly distribution of CMEs and type II radio bursts, and the percentage
of type II bursts with and without CMEs are shown in Table 1. There were 768
and 435 type II bursts reported during SC 23 and SC 24, respectively. There
were 14018 and 16107 CMEs reported during SC 23 and SC 24, respectively (based
on LASCO white-light observations). There were additional 5944 CMEs recorded
with the STEREO coronagrapghs (A and B combined) in SC 24.

The results in Table 1 are also illustrated in Figure 2 which shows the
histograms of occurrence of CMEs and type II bursts in cycles 23 and 24. We
fit a Gaussian to these distributions which showed that both CMEs and type II
events peaks during solar cycle maxima, i.e during the years 2000-2002 and
2013-2014 for cycle 23 and 24, respectively. We note that for cycle 23, there
were more CMEs during the declining phase than rising phase of cycle. Almost
$\approx 24\%$ of the CMEs in cycle 23 were recorded during the decline phase.
In contradiction, there were more CMEs during the rising phase of cycle 24
than the declining phase. However, for type II radio bursts, both cycles
showed similar trend, where there were more bursts during the beginning of the
cycle compared to the end of the cycle. This indicates that the type II bursts
occurrence follow the same pattern as sunspot cycle.

###  3.1 Type II Bursts and associated CMEs

In cycle 23, $\approx 79\%$ type II bursts were associated with a white-light
CME using our criteria, while in cycle 24 the percentage was considerable
higher, $\approx 95\%$s. However, the fraction of CMEs associated with type II
bursts were more similar and very low, only $\approx 4\%$ and $\approx 3\%$ in
SC23 and SC 24, respectively. The lower association of type II bursts to CMEs
in SC 23 could be explained by two possibilities: i) the unavailability of the
STEREO spacecraft that were available during SC 24 to provide different view
points making it possible to identify more CMEs especially those originating
from the far-side of the Sun; and ii) the loss of contact of LASCO during 1998
when no white-light CME identifications could be made. We note that the loss
of contact with STEREO-B during SC4, we could not use it for type II and CME
association. Figure 2 (bottom panel) shows the distribution of occurrence of
type II bursts with and without any CME association. The Gaussian fits show a
similar pattern where the peaks coincide with then peak of the solar cycle.
Table 1 also shows the type II-CME association for both the cycles.

Figure 3:  The histograms showing the variation of occurrence of type II
bursts with: (a) Fast; (b) Slow; (c) Wide; (d) Narrow; (e) Fast and Wide; (f)
Fast and narrow; (g) Slow and Wide; (h) Slow and Narrow CMEs for solar cycle
23 and 24.

###  3.2 Type II Bursts associated with type of CMEs

We categorised the CMEs on the basis of their linear speeds and angular width
as ‘Fast’, ‘Slow’, ‘Wide’ and ‘Narrow’ CMEs (Kumari et al., 2021; Morosan et
al., 2021). The CMEs which had linear speed $\geq 500$ km/s were classified as
‘Fast’ CMEs, and the rest were kept in the ’Slow’ CME category. Similarly,
CMEs with angular width $\geq 60^{o}$ were designated as ‘Wide’ CMEs and the
rest were ‘Narrow’ CMEs. Table 2 contains the list of type II bursts
associated with different type of CMEs.

We found that most of the type II bursts were associated with ’Fast and Wide’
CMEs, irrespective of the cycle ($\approx 48\%$). Figure 3 shows histograms of
the distribution of type II bursts associated with various CME types. The
Gaussian distributions to the dataset peak during solar maximum for ’Fast’,
’Slow’, ’Wide’, and ’Narrow’ CMEs for both solar cycles. We note that ’Fast’
and ’Slow’ CMEs were almost equally distributed in the past two cycles,
however most of the type II bursts were associated with ’Wide’ CMEs. The
bottom panel of Figure 3 also shows the type II bursts for a combination of
speed and width of CMEs. There were a very few type II bursts associated with
’Fast and Narrow’ CMEs. Figure 3 and Table 2 show that type II bursts
association with CMEs follows the same pattern in both solar cycles.

We note that we have considered the linear speed of CMEs as mentioned in the
CME catalogues used in the present study. A constant linear speed has
projection effects and it does not include the complete kinematics of CMEs in
the solar atmosphere. However, we used an average linear speed for CMEs since
we have more than thirty thousand CMEs included in our study. We acknowledge
that this could lead into a few misclassification for type of CMEs in the
present study. Several previous studies have reported that type II bursts are
often related to the flanks of CMEs (Holman & Pesses, 1983; Gopalswamy et al.,
2009a; Kumari et al., 2017c). The CME speeds at the leading edge and flanks
may be vary and individual case studies needs to be carried out to verify the
CME shock speed and type II association.

Figure 4:  The histograms showing the distribution of occurrence type II
bursts with: (a) all Halo CMEs; (b) all partial Halo CMEs; (c) all partial
Halo and Halo CMEs; and d) all other CMEs; from January, 1996 to December,
2019, which corresponds to solar cycle 23 and 24. The dashed red profile shows
the Gaussian fit to the the histograms. The two peaks of these Gaussian peaks
correspond to the solar maxima in the respective solar cycles.

###  3.3 Type II Bursts associated with Halo and Partial Halo CMEs

We studied the association of type II radio bursts with halo CMEs. partial
halo and halo CMEs are included into the ’Wide’ category. The CMEs with width
$W=360^{o}$ are classified as full halos and those with width $120^{o}\leq
W<360^{o}$ are classified as partial halos. Table 3 lists the type II bursts
associated with partial and full halo CMEs in cycle 23 and 24. We found that
there were more type II bursts associated with partial and full halo CMEs in
cycle 24 $26\%$) than cycle 23, irrespective of the fact that there were less
type IIs in the latter cycle. This The distribution halo CMEs were almost
equally distributed as partial and full halos for both the cycle. Figure 4
shows the distribution of type II bursts associated with partial halo, full
halo and other CMEs. The Gaussian fits to the distribution also follow similar
pattern as previous distribution i.e the peaks occur during the maxima of the
respective solar cycles.

Figure 5: The histograms showing the variation drift rates and duration of
type II bursts solar cycle 23 ((a) and (b), respectively) and 24 ((c) and (d),
respectively). Figure 6: The histograms showing the variation of the start
(left panel) and end (right panel) frequencies of type II bursts in cycle 23
and 24. We have used the fundamental start frequency of all the type II bursts
as mentioned in section 3.5. For the type II bursts which were a fundamental-
harmonic pair, we have divided the start frequencies by a factor of two.

###  3.4 Type II Bursts and Their Properties

We analysed the spectral (time-frequency) properties of the type II bursts for
solar cycle 23 and 24 using the type II bursts list generated in section 2. We
estimated the duration of the type II bursts using the start and end time
($t_{start}$ and $t_{end}$, respectively) as noted in the SWPC event lists. We
also estimated the drift rates of these bursts using the start and end
frequencies ($f_{start}$ and $f_{start}$, respectively) as recorded in the
solar dynamic spectra. Figure 5 shows the distribution of drift rate and
duration of the type II bursts in the last two cycles. More than $90\%$ of the
type II bursts had drift rates higher than 0.05 MHz/s in both cycles for
frequency range 200 - 25 MHz. The behaviour of these type II bursts are very
similar to moving type IV bursts as reported by Kumari et al. (2021).
Similarly, almost $80\%$ bursts had duration less than 20 minutes. We note
that type II bursts are slowing drifting features on the solar dynamic
spectra, and these can have different drift rates, in turn different speeds in
the solar atmosphere. In the present study, we have considered a constant
drift rate for type II bursts, which can underestimate or overestimate the
shock speed calculated from the radio burst.

Figure 7:  Left panel: The relation between CME linear speeds as mentioned in
catalogue and the type II speed estimated with four-fold Newkirk’s density
model and duration of the bursts. Right panel: Distribution of type II
formation heights using the aforementioned density model.

###  3.5 Heights of Type II bursts

We converted the start and end frequencies obtained for the type II busts from
the SWPC events into heliocentric distances. We used the four-fold Newkirk’s
density model for frequency to heliocentric distance conversion (Newkirk,
1961; Mann et al., 1995). We used this model and enhancement factor to account
for the enhanced densities above active regions. There is always an ambiguity
in using a particular density model as different density models and
enhancement factors can reflect a range of possible heights of the radio
sources. We began with an assumption that all the type II bursts were a
fundamental-harmonic pair, hence the start frequency of the burst was divided
by a factor of two for a more accurate height estimation. The end frequency of
the bursts was kept as it is since in the end frequency of type II bursts is
the end frequency of the fundamental lane. The speeds of the type II bursts
were estimated using the height estimate and the time duration (see section
3.4) of the bursts. Figure 6 shows the histogram of start and end frequencies
of all the type II bursts in studied in this manuscript. For the type II
bursts which were a fundamental-harmonic pair, we have divided the start
frequencies by a factor two.

We first compared the speed of the type II bursts to that of the associated
CME. Figure 7a shows the linear correlation between the shock speed from radio
bursts and CME speed from whitelight coronagraph data. There was almost no
correlation (cc =0.03) found between type II speed and CME linear speed. The
most likely cause for this lack of correlation is that the speed of the metric
type II bursts reflects the speed at lower coronal heights ($\leq
2.5\leavevmode\nobreak\ R_{\odot}$) compared to the CMEs linear speed
mentioned in the CDAW catalogue which is obtained at much higher heights
($\geq 2.5\leavevmode\nobreak\ R_{\odot}$). The type II speeds are also likely
to be indicative of the lateral or flank speeds of CMEs and not the radial
outward speed since a lot of metric type IIs have been reported to occur at
the CME flank (Chrysaphi et al., 2018; Morosan et al., 2019; Majumdar et al.,
2021). Another likely cause of low correlation could be due to the assumption
of constant plane-of-sky CME speed. Figure 7b shows the distribution of type
II heights over the years. We found that by using four-fold Newkirk’s density
model, for a large majority of type IIs, their starting height is at $\approx
1.7\pm 0.3\leavevmode\nobreak\ R_{\odot}$ as shown by the significant peak in
7b (for frequency range $200-25$ MHz). The error bar indicates the variation
in the type II height estimate while considering other density models, such as
the Baumbach (Baumbach, 1937, two-four fold), Saito (Saito et al., 1977, two-
six fold) models. This error bar is also similar to the standard deviation in
the type II height estimates using a four-fold Newkirk’s model. There were
only a few metric type II bursts which started at heights higher than $\approx
2.3R_{\odot}$ ($\leq 5\%$) which corresponds to frequencies below 50 MHz.
Similarly, there were also fewer type II bursts starting at low corona heights
below 1.7 R⊙. The majority of type IIs have an onset height between 1.7 and
2.3 R⊙ which indicates the height of plasma oscillations due to the electrons
accelerated at the magneto-hydrodynamics (MHD) shocks in the solar corona.

Figure 8:  Left panel: The number of CMEs (per year) and type II bursts (per
year) from 1996-2019. The number of type IV bursts are very less as compared
to the number of CMEs however, there is a very strong correlation ($94\%$)
between the occurrence of the the white-light feature and radio emission in
for cycle 24. There is a marginal correlation between these two quantities in
cycle 23. This may be because of lesser CMEs detected by LASCO in the previous
cycle. Right panel: The number of type II (per year) and sunspots (per year;
adapted from SILSO World Data Center (1996-2019)) from 1996-2019. These two
quantities show a strong correlation in cycle 23 and 24 ($\approx 94\%$ and
$\approx 98\%$, respectively).

###  3.6 Type II Bursts, CMEs and Sunspot Number Correlation

Finally, we studied the relation between CMEs, type II bursts and sunspot
number (SSN) during the past two solar cycles. Figure 8a shows the correlation
between number of CMEs and type II bursts recorded per year. The correlation
coefficient is low for cycle 23 ($\approx 0.61$), however, type II bursts are
highly correlated with CMEs for solar cycle 24 (the coefficient is $\approx
0.95$). The sunspot numbers and type II bursts are also highly correlated in
both cycles ($\approx 0.94$ and $\approx 0.98$, respectively). Figure 8 also
shows that these three phenomenon are quite well correlated for cycle 24.
Michalek et al. (2019) had also shown that sunspot number and CMEs were poorly
correlated for cycle 23 when compared to cycle 24. The correlation was
improved when they separated poor CME events from their list.

Table 2: Number and percentage of type II radio bursts associated with CMEs with different properties: ’Fast’, ’Slow’, ’Wide’, ’Narrow’, and a combination of these properties. Category | Total | Cycle 23 | Cycle 24
---|---|---|---
| No. | % | No. | % | No. | %
Total | 1024 | 100.0% | 609 | 59.5% | 415 | 40.5%
Fast | 554 | 54.1% | 347 | 57.0% | 207 | 49.9%
Slow | 470 | 45.9% | 262 | 43.0% | 208 | 50.1%
Wide | 809 | 79.0% | 482 | 79.1% | 327 | 78.8%
Narrow | 215 | 21.0% | 127 | 20.9% | 88 | 21.2%
Fast & Wide | 495 | 48.3% | 313 | 51.4% | 182 | 43.9%
Fast & Narrow | 59 | 5.8% | 34 | 5.6% | 25 | 6.0%
Slow & Wide | 314 | 30.7% | 169 | 27.7% | 145 | 34.9%
Slow & Narrow | 156 | 15.2% | 93 | 15.3% | 63 | 15.2%
Table 3: Number and percentage of type II radio bursts associated with partial halo and halo CMEs. Category | Total | Cycle 23 | Cycle 24
---|---|---|---
| No. | % | No. | % | No. | %
Total | 1024 | 100.0% | 609 | 59.5% | 415 | 40.5%
Partial Halo CMEs | 241 | 23.5% | 131 | 21.5% | 110 | 26.5%
Halo CMEs | 252 | 24.6% | 144 | 23.6% | 108 | 26.0%
Partial Halo & | 493 | 48.1% | 275 | 45.1% | 218 | 52.5%
Halo CMEs
Other CMEs | 531 | 51.9% | 334 | 54.9% | 197 | 47.5%
Table 4: Typical parameters values for type II bursts and CMEs in solar cycle 23 and 24.  Parameter | All Type II | Type II with CMEs | Type II without CMEs
---|---|---|---
| Cycle 23 | Cycle 24 | Total | Cycle 23 | Cycle 24 | Total | Cycle 23 | Cycle 24 | Total
Type II Duration (minutes) |  |  |  |  |
Mean | 10.4 | 10.7 | 10.5 | 10.7 | 10.8 | 10.7 | 7.5 | 8.4 | 8.2
Median | 9.0 | 9.0 | 9.0 | 9.0 | 9.0 | 9.0 | 6.0 | 6.0 | 6.0
Std. Dev. | 6.8 | 9.7 | 7.9 | 8.1 | 6.6 | 10.1 | 6.4 | 6.9 | 4.4
Type II Drift Rate (MHz/s) |  |  |  |  |
Mean | 0.08 | 0.09 | 0.08 | 0.10 | 0.11 | 0.11 | 0.08 | 0.08 | 0.08
Median | 0.05 | 0.06 | 0.06 | 0.05 | 0.06 | 0.06 | 0.05 | 0.06 | 0.06
Std. Dev | 0.03 | 0.02 | 0.03 | 0.03 | 0.02 | 0.03 | 0.03 | 0.02 | 0.03
Type II Speed (km/s) |  |  |  |  |
Mean | 700 | 840 | 751 | 645 | 848 | 729 | 632 | 621 | 626
Median | 538 | 671 | 592 | 490 | 671 | 563 | 475 | 456 | 461
Std. Dev | 568 | 716 | 630 | 773 | 945 | 815 | 773 | 848 | 811
Type II Height ($R_{\odot}$) |  |  |  |  |
Mean | 1.77 | 1.76 | 1.77 | 1.78 | 1.77 | 1.78 | 1.77 | 1.83 | 1.74
Median | 1.71 | 1.62 | 1.69 | 1.73 | 1.64 | 1.69 | 1.69 | 1.85 | 1.67
Std. Dev. | 0.27 | 0.29 | 0.28 | 0.26 | 0.29 | 0.28 | 0.27 | 0.27 | 0.28
CME Width (Degrees) |  |  |  |  |
Mean | – | – | – | 160 | 173 | 165 | – | – | –
Median | – | – | – | 110 | 125 | 115 | – | – | –
Std. Dev. | – | – | – | 126 | 122 | 123 | – | – | –
CME Speed (km/s) |  |  |  |  |
Mean | – | – | – | 768 | 670 | 728 | – | – | –
Median | – | – | – | 584 | 499 | 546 | – | – | –
Std. Dev. | – | – | – | 489 | 546 | 526 | – | – | –

##  4 Discussion

We present the first comprehensive long-term analysis of the occurrence of
metric type II radio bursts (low coronal radio bursts) during two solar cycles
and their association with CMEs of various properties. We found that $\approx
79\%$ and $\approx 95\%$ of the type II bursts were associated with CMEs in
solar cycle 23 and 24, respectively. However, only $\approx 4\%$ and $\approx
3\%$ of CMEs were associated with type II bursts in solar cycle 23 and 24,
respectively. The results indicate that most of the type II bursts had a
white-light CME counterpart, however, there were a few type IIs with no clear
CME association. Some studies suggest that such type IIs may be related to
flare-driven blast waves (Magdalenić et al., 2012; Kumar et al., 2016; Morosan
et al., 2023). There were more CMEs in cycle 24 than cycle 23. However, the
number of type II radio bursts were less in cycle 24 compared to cycle 23,
thus a greater number of CMEs does not indicate that a greater number of type
IIs would occur in a solar cycle. The type II occurrence also depends upon the
CME properties, such as the lateral expansion, speed, and the coronal
conditions such as the ambient magnetic fields (B),
Alfv$\acute{\text{e}}$n-Mach number (MA), Alfv$\acute{\text{e}}$n speed
($V_{a}$), etc. (Mann et al., 2003; Lin et al., 2006).

Almost $48\%$ of the type II bursts were associated with ’Fast and Wide’ CMEs.
This trend is similar to type IV bursts association with CMEs in cycle 24
(Kumari et al., 2021) and it was also found in the case of moving radio bursts
(mostly type IIs and type IVs) in the study of Morosan et al. (2021). There
were a very few type II bursts associated with ‘Slow and Narrow’ CMEs. Morosan
et al. (2021) suggested that less radio bursts associated with ’Slow and
Narrow’ CMEs are observed due to the fact that the generation of moving bursts
(mostly consisting of type II and type IV bursts) is related to the lateral
expansion during the early acceleration phase of a solar eruption.

Our analysis shows that almost half of the type II bursts in both solar cycles
were accompanied with partial and full halo CMEs, essentially they were
accompanied by very wide CMEs. Since it is well known that these CMEs can be
potentially responsible for geomagnetic storms (Zhang et al., 2003), these
type II bursts can be used to study the early kinematics of the geoeffective
halo CMEs. These radio bursts could also be useful in case of stealth CMEs
whose early signatures are not visible in coronagraph images, however, so far
there has not yet been an account of a type II bursts associated with a
stealth CME. The correlation between type IIs and sunspots is also high
indicating that type IIs are associated with CMEs resulting from eruptive
flares from sunspots and less likely to be associated with phenomena such as
filament eruptions that do not require the presence of sunspots (Feynman &
Hundhausen, 1994; Choudhary & Moore, 2003).

In comparison with other statistical studies of radio bursts, there were less
type II bursts (435) reported in cycle 24 than type IV bursts (447; Kumari et
al., 2021). This could be because for type IV emission, shock is not an
essential requirement unlike for type II events (Salas-Matamoros & Klein,
2020). In turn, a higher number of type II bursts were associated with CMEs
compared to type IV bursts. Both type IIs and type IVs are, however, mostly
associated with ‘Fast and Wide’ CMEs based on this current and previous
studies (Kahler et al., 2019; Kumari et al., 2021; Morosan et al., 2021).

Our results suggest that type II bursts dominate at a height of $\approx
1.7-2.3\pm 0.3\leavevmode\nobreak\ R_{\odot}$ (between $200-25$ MHz) with a
clear majority having an onset height around 1.7 $\pm 0.3\leavevmode\nobreak\
R_{\odot}$ assuming the four-fold Newkirk model. This height range indicates
that there may be a critical height for type II occurrence for the majority of
solar eruptions. Similar results were found in a theoretical study by Lin et
al. (2006) that suggests that type IIs typically occur at a height of 1.5 R⊙,
which is similar to the heights reported in the present work for the majority
of type II bursts. The authors modelled a coronal mass ejection (CME) with a
terminal speed of approximately 1000 km/s, using magnetic field strength and a
constant of magnetic reconnection rate to obtain this height. Lin et al.
(2006) suggested that type II bursts can not appear lower than a critical
height, which depends upon then coronal environment and it determines the
start frequency of the burst. The critical height is likely highly dependant
on the global Alfvén speed in the corona. For example, at a height of
$\sim$1.2 R⊙ near active regions, the Alfvén speed is usually greater than
1000 km/s (Parker, 1988; Aschwanden, 2005; Morosan et al., 2016; Régnier,
2015) and a CME or any other disturbance is too slow to drive a shock at those
heights. Below 1.2 R⊙, only five type IIs were found, confirming that CMEs may
generally be too slow to become super Alfvénic. Even the fastest reported
CMEs, require some time to expand in order to reach a speed of 1000 km/s in
the low corona (e.g., Mann et al., 2003; Kumari et al., 2017a; Morosan et al.,
2019; Pohjolainen & Talebpour Sheshvan, 2021). As the CMEs accelerate and
expand, they can then reach regions of low Alfvén speed (e.g., Warmuth et al.,
2004; Morosan et al., 2022) outside the active region associated with the CME
eruption. Since type IIs are believed to be the signature of CME-driven
shocks, it is likely that CME shocks also form at similar heights. The typical
type II burst speed was found to be $\approx 750km/s$ and type II bursts
appear to have a higher speed if they have a white-light CME association (see
Figure7a). Table 4 shows the typical parameters such as speed, duration, drift
rates for the type II bursts and speed and width for CMEs in SC23 and SC24.
The large standard deviation here means that the estimated speeds, drift
rates, duration, etc. are widely spread out. Since we have CMEs and type II
bursts for all the phases of the solar cycles, it is expected that the
deviation in mean parameters will be large. However, it is worth noting that
the median value is relatively close to the mean value (Michalek et al.,
2019).

When the CME propagates through the solar corona through the supercritical
shock region, the shock wave creates density irregularities in the plasma,
which can excite Langmuir waves (Schmidt & Cairns, 2012; Mann et al., 2022).
Our analysis shows that $\approx 1.7-2.3\pm 0.3\leavevmode\nobreak\ R_{\odot}$
(between $200-25$ MHz) represents this supercritical shock region capable of
generating electron beams that could excite Langmuir waves. The shock
formation heights that generate type II solar radio bursts can vary depending
on the properties of CME and the surrounding coronal plasma. To verify
independently, we used the Alfvén speed, shock height and density relation for
type II bursts (for split-band type II radio bursts Smerd et al., 1975; Vršnak
et al., 2004a). The type II height is directly proportional to start frequency
of the burst and magnetic field and inversely proportional to Alfvén speed.
For an ambient magnetic field of range $1.5-2.0$ G (typical B values estimated
for type II bursts starting around 200 MHz; see for example Dulk & McLean,
1978; Kumari et al., 2017b, and the references therein), and Alfvén-Mach
number in the range $1.59<MA<2.53$ (Mann et al., 2022), we get shock formation
height of $\approx 1.7\leavevmode\nobreak\ R_{\odot}$. The lateral expansion
during the early acceleration phase of a ’Fast and Wide’ CME (Morosan et al.,
2021; Majumdar et al., 2021) can lead to the formation of supercritical shock
in the height range $\approx 1.7\pm 0.3\leavevmode\nobreak\ R_{\odot}$, which
can lead to excitation of Langmuir waves needed for type II solar radio burst
emission. For type II bursts, the enhanced Langmuir wave level above the
thermal one can be achieved rapidly (Kouloumvakos et al., 2021; Mann et al.,
2022), compared to type III solar radio bursts which require some travel time
for the electrons to reach the region where the Langmuir waves are excited
(Reid, 2020). This instability distance depends upon the spectral index,
electron beam velocity, the size of the acceleration site, and the temporal
injection profile. For type II bursts, this happens close to the electron
acceleration sites (Schmidt & Cairns, 2012; Morosan et al., 2022). We note
that a few high frequency (heights $\leq 1.5\leavevmode\nobreak\ R_{\odot}$)
were observed in the past two solar cycles, but the occurrence of these type
II bursts were $<1\%$ of the total type II bursts. We note that individual
case studies of high frequency type II bursts are required to shred light on
this (Mondal et al., 2020; Kouloumvakos et al., 2021).

This study extensively uses the reports from SWPC which lists the manual
detection of solar radio bursts from the Radio Solar Telescope Network (RSTN)
for 24-hour solar observations. However, we acknowledge that RSTN
spectrographs have frequency limits and lower sensitivities compared to the
new-age non-solar dedicated radio telescopes, such as the LOw Frequency ARray
(LOFAR; van Haarlem et al., 2013) and the Murchison Widefield Array (MWA;
Tingay et al., 2013). As our study makes use of two solar cycle radio bursts
and CMEs data, we rely on the SWPC, CDAW and SEEDS catalogues for directly
extracting the onset time, duration, linear speed, angular width, start and
end frequencies, etc. parameters for the events. Since these catalogues are
combinations of manual (SWPC and CDAW) and automatic detection (SEEDS) of
events, we understand that there may be chances of misclassification of
events. We note that we have used linear speed for CMEs and constant drift
rate for type II bursts, which may also lead into error in classification of
CMEs and type II speeds. Currently, there are only two low frequency solar-
dedicated imaging instruments, the upgraded Nançay Radioheliograph (NRH;
Kerdraon & Delouis, 1997) and the Gauribidanur RAdioheliograPH (GRAPH; Ramesh
et al., 1998), which covers the solar observations between
$\leavevmode\nobreak\ 40-450$ MHz from $2.5-16$ UT at spot frequencies. This
still leaves a big temporal window where no solar radio imaging observations
are available at present. With the absence of solar-dedicated imaging radio
instruments, specially at low radio frequencies, one way to estimate the shock
generation locations in the ‘middle’ corona is by using existing density
models, such as Newkirk (1961) (see section 3.5 for details). This method has
been used for in several previous type II radio bursts studies in the absence
of imaging observations (Vršnak et al., 2004b; Vasanth et al., 2014; Ramesh et
al., 2022). This study also highlights the importance of having a low radio
frequency solar-dedicated imaging instruments for identifying the shock
generation locations in the solar corona.

This study should be extended in the future to include imaging observations of
type II radio bursts. With dedicated solar radio imaging instruments like the
upgraded NRH, the GRAPH, the expanded Owens Valley Solar Array (EOVSA; Gary et
al., 2018), the Mingantu Spectral Radio Heliograph (MUSER; Yan et al., 2021)
and the upcoming Daocheng Solar Radio Telescope (DSRT; Yan et al., 2022), and
other powerful non solar radio imaging instruments like the LOFAR, the MWA,
the Long Wavelength Array (LWA; Ellingson et al., 2009) and the upgraded Giant
Metrewave Radio Telescope (GMRT; Swarup, 1991; Gupta et al., 2017), multiple
datasets can be combined to determine the origin and source regions of these
radio bursts with better accuracy. With continuous spectroscopic imaging, the
source locations of these bursts relative to the CME and their speed relative
to the CME flanks can also be investigated in detail to further investigate
the formation heights and typical speeds of type II bursts that were also
identified in this study using simple empirical models.

###### Acknowledgements.

A.K. and D.E.M. acknowledge the University of Helsinki Three Year Grant. D.E.M
acknowledges the Academy of Finland project ‘RadioCME’ (grant number 333859).
E.K.J.K. and A.K. acknowledge the European Research Council (ERC) under the
European Union’s Horizon 2020 Research and Innovation Programme Project SolMAG
724391. E.K.J.K acknowledges the Academy of Finland Project SMASH 310445. All
authors acknowledge the Finnish Centre of Excellence in Research of
Sustainable Space (Academy of Finland grant number 312390). A.K’s research was
supported by an appointment to the NASA Postdoctoral Program at the the NASA
Goddard Space Flight Center (GSFC).

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