# Robust mixture regression with Exponential Power distribution

CHEN XIAO Hong Kong Baptist University

(July 2020)

###### Abstract

Assuming an exponential power distribution is one way to deal with outliers in
regression and clustering, which can increase the robustness of the analysis.
Gaussian distribution is a special case of an exponential distribution. And an
exponential power distribution can be viewed as a scale mixture of normal
distributions. Thus, model selection methods developed for the Gaussian
mixture model can be easily extended for the exponential power mixture model.
Moreover, Gaussian mixture models tend to select more components than
exponential power mixture models in real-world cases, which means exponential
power mixture models are easier to interpret. In this paper, We develop
analyses for mixture regression models when the errors are assumed to follow
an exponential power distribution. It will be robust to outliers, and model
selection for it is easy to implement.

Keywords: Robust mixture regression, model selection, exponential power
distribution

##  1 Introduction

Finite mixtures of regression models have been broadly used to deal with
various heterogeneous data. In most cases, we assume the errors follow normal
distributions, and parameters are estimated by the maximum likelihood
estimator (MLE). However, this model will be very sensitive to outliers.

There are many methods to handle outliers in mixture of linear regression
models. Trimmed likelihood estimator (TLE)(Neykov et al.,2007) uses the top
$N\alpha$ data with the largest log-likelihood. The choice of $\alpha$ is very
important for the TLE. It should not be too small or too large. But it’s not
easy to choose $\alpha$ Yao(2014) proposed a robust mixture of linear
regression models by assuming the errors follow the t-distribution. Song(2016)
dealt with the same problem using Laplace distribution.

Assuming an exponential power distribution is one way to deal with outliers in
regression and clustering. (Salazar E et.al., 2012; Liang F et.al. 2007;
Ferreira, M.A.,2014). Gaussian distribution and Laplace dsitribution are
special cases of exponential distribution. leptokurtic distributions have
heavier tails than Gaussian distributions, which provide protection against
outliers. And an exponential power distribution is a scale mixture of normal
distributions. This makes it easier to understand.

Outliers are not the only problem. Choose the model selection is also a
challenge for mixture model. Gaussian mixture models tend to select more
components than exponential power mixture models in real world case, which
means the interpretability is greatly reduced.[7] One possible explanation is
that exponential power distribution is a scale mixture of Gaussian
distributions.

In this paper, robust mixture regression with Exponential power distribution
and model selection for it are introduced.

##  2 Exponential Power(EP) Distribution

The density function of the Exponential Power(EP) distribution $(p>0)$ where
the mean is 0 is

$f_{p}(e;0,\eta)=\frac{p\eta^{\frac{1}{p}}}{2\Gamma\left(\frac{1}{p}\right)}\exp\left\\{-\eta|e|^{p}\right\\},$
(1)

where $\Gamma(\cdot)$ is the Gamma function. When $0<p<2$, the distributions
are heavy tailed, which means it will provide protection against outliers.
When p=1, Equation(1) defines Laplace distribution. When p=2, Equation(1)
defines Gaussian distribution.

##  3 Robust regression with mixture of exponential power distribution

We assume that $(x_{i},y_{i})$ follows one of the following linear regression
model with probability $\pi_{k}$,

$y_{i}=x^{\prime}_{i}\beta_{k}+e_{ik}\qquad k=1,\ldots,K$ (2)

where $\sum_{k}^{K}\pi_{k}=1$. $e_{ik}$ follows Exponential Power (EP)
distributions $f_{p_{k}}\left(e_{ik};0,\eta_{k}\right)$

Assume $\bm{\Theta}=\left\\{\bm{\pi},\bm{\eta},\bm{\beta}\right\\}$

The complete likelihood function is

$\mathbb{P}(\mathbf{y};\mathbf{x},\bm{\Theta})=\prod_{i}^{N}\prod_{k=1}^{K}\left[\pi_{k}f_{p_{k}}\left(y_{i}-x_{i}^{\prime}\beta_{k};0,\eta_{k}\right)\right]^{z_{ik}}$

where$z_{ik}\in\\{0,1\\}$ is an indicator variable. It implies that $y_{i}$
comes form $k$th linear regression model. And $\sum_{k=1}^{K}z_{ik}=1$.

And the complete log-likelihood function is

$\log\mathbb{P}(\mathbf{y};\mathbf{x},\bm{\Theta})=\sum_{i}^{N}\sum_{k=1}^{K}z_{ik}\left[\log\pi_{k}+\log
f_{p_{k}}\left(y_{i}-x_{i}^{\prime}\beta_{k};0,\eta_{k}\right)\right]$

In practice, the number of components has to be estimated. So we use penalty
function to select K automatically.(Huang, T et.al., 2017)

$l(\Theta)=\log\mathbb{P}(\mathbf{y};\mathbf{x},\Theta)-P(\pi;\lambda)$

where$P(\pi;\lambda)=n\lambda\sum_{k=1}^{K}D_{k}\log\frac{\epsilon+\pi_{k}}{\epsilon}$

$\epsilon$ is a very small positive number, $\lambda$ is a positive tuning
parameter , and $D_{k}$ is the number$\text{ of free parameters in the
}k^{th}\text{ component. }$.

##  4 Algorithm

We propose a Generalized Expectation–Maximization algorithm to solve the
proposed model.

In the $\mathrm{E}$ step, conditional expectation of $z_{ijk}$ given $e_{ij}$
is computed by the Bayes’ rule:

$\begin{array}[]{l}\qquad\gamma_{ik}^{(t+1)}=\frac{\pi_{k}^{(t)}f_{p_{k}}\left(y_{i}-x_{i}^{\prime}\beta_{k}\mid
0,\eta_{k}^{(t)}\right)}{\sum_{l=1}^{K}\pi_{l}^{(t)}f_{p_{l}}\left(y_{i}-x_{i}^{\prime}\beta_{k}\mid
0,\eta_{l}^{(t)}\right)}\par\end{array}$ (3)

$\begin{aligned}
Q\left(\bm{\Theta},\bm{\Theta}^{(t)}\right)=&\sum_{i,k}\gamma_{ik}^{(t+1)}\left[\log
f_{p_{k}}\left(y_{i}-x_{i}^{\prime}\beta_{k}\mid
0,\eta_{k}^{(t)}\right)+\log\pi_{k}\right]\\\
&-n\lambda\sum_{k=1}^{K}D_{k}\log\frac{\epsilon+\pi_{k}}{\epsilon}\end{aligned}$

Then in the M step, $\bm{\Theta}$ are updated by maximizing
$Q\left(\bm{\Theta},\bm{\Theta}^{(t)}\right)$.

$\pi_{k}^{(t+1)}=\max\left\\{0,\frac{1}{1-\lambda
KD_{k}}\left[\sum_{i}\gamma_{ik}^{(t+1)}-\lambda D_{k}\right]\right\\}$ (4)

$\eta_{k}^{(t+1)}=\frac{N_{k}}{p_{k}\sum_{i,}\gamma_{ik}^{(t+1)}\left|y_{i}-x_{i}^{\prime}\beta_{k}\right|^{p_{k}}}$
(5)

where $N_{k}=\sum_{i,}\gamma_{ik}^{(t+1)}$.

On ignorning the terms not involving ${\beta}_{k}$, we have

$Q(\beta_{k})=-\sum_{i=1}^{N}\sum_{k=1}^{K}\gamma_{ik}^{(t+1)}\eta_{k}^{(t+1)}\left|y_{i}-x_{i}^{\prime}\beta_{k}\right|^{p_{k}}$
(6)

Then we use MM algorithm to estimate $\beta_{k}$

$\begin{split}\\{(y_{i}-x^{\prime}_{i}\beta_{k})^{\prime}(y_{i}-x^{\prime}_{i}\beta_{k})\\}^{\frac{p_{k}}{2}}\leq\\{(y_{i}-x^{\prime}_{i}\beta_{k}^{(t)})^{\prime}(y_{i}-x^{\prime}_{i}\beta_{k}^{(t)})\\}^{\frac{p_{k}}{2}}\\\
+\frac{p_{k}}{2}\\{(y_{i}-x^{\prime}_{i}\beta_{k}^{(t)})^{\prime}(y_{i}-x^{\prime}_{i}\beta_{k}^{(t)})\\}^{\frac{p_{k}}{2}-1}\left((y_{i}-x^{\prime}_{i}\beta_{k})^{\prime}(y_{i}-x^{\prime}_{i}\beta_{k})-(y_{i}-x^{\prime}_{i}\beta_{k}^{(t)})^{\prime}(y_{i}-x^{\prime}_{i}\beta_{k}^{(t)})\right)\end{split}$

Thus

$\hat{\beta_{k}}=\arg{min}_{\beta_{k}}\sum_{i=1}^{N}\sum_{k=1}^{K}\gamma_{ik}^{(t+1)}\eta_{k}^{(t+1)}W_{ik}^{(t)}(y_{i}-x^{\prime}_{i}\beta_{k})^{\prime}(y_{i}-x^{\prime}_{i}\beta_{k}),$
(7)

where
$W_{ik}^{(t)}=\frac{p_{k}}{2}\\{(y_{i}-x^{\prime}_{i}\beta_{k}^{(t)})^{\prime}(y_{i}-x^{\prime}_{i}\beta_{k}^{(t)})\\}^{\frac{p_{k}}{2}-1}$.

Equation (7) leads to the update

$\beta_{k}^{(t+1)}=\left(\sum_{i=1}^{N}\sum_{k=1}^{K}\gamma_{ik}^{(t+1)}\eta_{k}^{(t+1)}W_{ik}^{(t)}\bm{x}_{i}\bm{x}^{\prime}_{i}\right)^{-1}\left(\sum_{i=1}^{N}\sum_{k=1}^{K}\gamma_{ik}^{(t+1)}\eta_{k}^{(t+1)}W_{ik}^{(t)}\bm{x}_{i}y_{i}\right).$
(8)

We can also use Newton-Raphson method to updata $\beta$.

##  5 Simulation

In this section, we demonstrate the effectiveness of the proposed method by
simulation study.

Since TLE performed worse than the robust mixture regression model based on
the t-distribution [4] and exponential power distribution can cover Laplace
distribution, we only compare

1.the robust mixture regression model based on the t-distribution (MixT),

2.the robust mixture regression model based on the exponetional power
distribution (MixEP).

To assess different methods, we record the mean squared errors (MSE) and the
bias of the parameter estimates. In terms of the mixture switching issues, we
use label permutation that minimizes the difference between predicted
parameters and the true parameter values.

Example $1.$ Suppose the independently and identically distributed samples
$\left\\{\left(x_{1i},x_{2i},y_{i}\right),i=1,\ldots,n\right\\}$ are sampled
from the model

$Y=\left\\{\begin{array}[]{ll}0+X_{1}+\epsilon_{1},&\text{ if }Z=1\\\
0-X_{1}+\epsilon_{2},&\text{ if }Z=2\end{array}\right.$

where $Z$ is a component indicator with $P(Z=1)=0.5,X_{1}\sim N(0,1),X_{2}\sim
N(0,1),$ and both $\epsilon_{1}$ and $\epsilon_{2}$ follow the same
distribution as $\epsilon.$ We consider the following four cases for the error
density function of $\epsilon$ :(Weixin Yao et.al., 2014)

Case I: $\epsilon\sim t_{1}$ , which means $t$ -distribution with degrees of
freedom 1 ,

Case II: $\epsilon\sim
0.95\mathrm{N}(0,1)+0.05\mathrm{N}\left(0,5^{2}\right)-$,which is a
contaminated normal mixture,

Case $\mathrm{III}:\epsilon\sim N(0,1)$ with $5\%$ of high leverage outliers
being $X_{1}=2+\epsilon,$ and $Y=10+\epsilon$

Case IV: $Y=\left\\{\begin{array}[]{l}0+X_{1}+\gamma_{1}+\epsilon_{1}\\\
0-X_{1}+\gamma_{2}+\epsilon_{2}\end{array}\right.$ ,$\begin{aligned}
\epsilon\sim
N(0,1),P\left(\gamma_{1}\in(4,6)\right)=P\left(\gamma_{2}\in(4,6)\right)=0.1,\end{aligned}$
standard normal distribution with 10% outlier contamination.

CaseI: The number of samples is N=400.

Figure 1: CaseI

Figure 1 shows the simulation data in CaseI. We replicate this experiment 50
times, and the comparison of different methods are shown in Figure 2 and Table
1.

(a) MixEP

(b) MixT

Figure 2:  Boxplots for the biass of the estimators of beta MSE
---
Method | $\beta_{00}=0$ | $\beta_{01}=1$ | $\beta_{10}=0$ | $\beta_{11}=-1$
MixT | 0.0169 | 0.01936 | 0.01491 | 0.0203
MixEP | 0.3973 | 0.2175 | 0.2594 | 0.4337
bias
Method | $\beta_{00}=0$ | $\beta_{01}=1$ | $\beta_{10}=0$ | $\beta_{11}=-1$
MixT | -0.0169 | 0.01393 | 0.03333 | 0.001208
MixEP | 0.13677 | -0.2754 | -0.007769 | 0.2658
Table 1: CaseI:The mean squared errors (MSE)and the bias of the parameter
estimates for each estimation method.

CaseII: The number of samples is N=400.

Figure 3: CaseII

Figure 3 shows the simulation data in CaseII. We replicate this experiment 50
times, and the comparison of different methods are shown in Figure 4 and Table
2.

(a) MixEP

(b) MixT

Figure 4:  Boxplots for the biass of the estimators of beta MSE
---
Method | $\beta_{00}=0$ | $\beta_{01}=1$ | $\beta_{10}=0$ | $\beta_{11}=-1$
MixT | 0.0111 | 0.0105 | 0.006161 | 0.01433
MixEP | 0.0189 | 0.02219 | 0.01943 | 0.02024
bias
Method | $\beta_{00}=0$ | $\beta_{01}=1$ | $\beta_{10}=0$ | $\beta_{11}=-1$
MixT | 0.02289 | -0.009907 | -0.01731 | 0.02171
MixEP | -0.0116 | -0.0408 | 0.02623 | 0.03788
Table 2: CaseII:The mean squared errors (MSE)and the bias of the parameter
estimates for each estimation method.

CaseIII: The number of samples is N=400.

Figure 5: CaseIII

Figure 5 shows the simulation data in CaseIII. We replicate this experiment 50
times, and the comparison of different methods are shown in Figure 6 and Table
3.

(a) MixEP

(b) MixT

Figure 6:  Boxplots for the biass of the estimators of beta MSE
---
Method | $\beta_{00}=0$ | $\beta_{01}=1$ | $\beta_{10}=0$ | $\beta_{11}=-1$
MixT | 0.01188 | 0.01052 | 0.01857 | 0.02060
MixEP | 0.03807 | 0.03744 | 0.02973 | 0.02881
bias
Method | $\beta_{00}=0$ | $\beta_{01}=1$ | $\beta_{10}=0$ | $\beta_{11}=-1$
MixT | -0.01858 | -0.00432 | -0.006022 | -0.008520
MixEP | 0.06676 | -0.08744 | -0.01664 | 0.03178
Table 3: CaseIII:The mean squared errors (MSE)and the bias of the parameter
estimates for each estimation method.

CaseIV: The number of samples is N=400.

Figure 7: CaseIV

Figure 7 shows the simulation data in CaseIV. We replicate this experiment 50
times, and the comparison of different methods are shown in Figure 8 and Table
4.

(a) MixEP

(b) MixT

Figure 8:  Boxplots for the biass of the estimators of beta MSE
---
Method | $\beta_{00}=0$ | $\beta_{01}=1$ | $\beta_{10}=0$ | $\beta_{11}=-1$
MixT | 3.6119 | 0.1376 | 2.4279 | 0.1662
MixEP | 0.0411 | 0.02742 | 0.06486 | 0.04627
bias
Method | $\beta_{00}=0$ | $\beta_{01}=1$ | $\beta_{10}=0$ | $\beta_{11}=-1$
MixT | 0.8222 | -0.1153 | 0.6159 | 0.1979
MixEP | 0.08276 | -0.02965 | 0.1790 | 0.05284
Table 4: CaseIV:The mean squared errors (MSE)and the bias of the parameter
estimates for each estimation method.

It’s hard to choose the dimension of freedom for Mixture of t-distribution. In
the experiments shown above, the freedom leading to best results are used.
However, the freedom selected by BIC could lead to bad results. In addition,
selecting the number of components is also a problem for Mixture regression
with t-distribution.

##  6 Conclusion

The adaptively trimmed version of MIXT worked well when there are high
leverage outliers, but it can deal with drift. We can’t select a proper
dimension of freedom for MIXT using BIC when faced with drift.

Adaptively removing abnormal points can improve the estimation accuracy in
MixEP. It’s a natural idea is to build a model to cover the drift. This will
be the future work.

## References

  * [1] Huang, T., Peng, H., Zhang, K. (2017). MODEL SELECTION FORGAUSSIAN MIXTURE MODELS. Statistica Sinica, 27(1), 147-169. Re-trieved November 4, 2020, from
  * [2] Lange, K. (2016). MM optimization algorithms. Society for Industrial and Applied Mathematics. Weixing Song, Weixin Yao, Yanru Xing,
  * [3] Robust mixture regression model fitting by Laplace distribution, Computational Statistics and Data Analysis, Volume 71, 2014, Pages 128-137, ISSN 0167-9473, https://doi.org/10.1016/j.csda.2013.06.022.
  * [4] Weixin Yao, Yan Wei, Chun Yu, Robust mixture regression using the t-distribution, Computational Statistics and Data Analysis, Volume 71, 2014, Pages 116-127, ISSN 0167-9473, https://doi.org/10.1016/j.csda.2013.07.019.
  * [5] N. Neykov, P. Filzmoser, R. Dimova, P. Neytchev, Robust fitting of mixtures using the trimmed likelihood estimator, Computational Statistics and Data Analysis, Volume 52, Issue 1, 2007, Pages 299-308, ISSN 0167-9473, https://doi.org/10.1016/j.csda.2006.12.024.
  * [6] Wennan Chang and Xinyu Zhou and Yong Zang and Chi Zhang and Sha Cao,A New Algorithm using Component-wise Adaptive Trimming For Robust Mixture Regression, 2020. https://arxiv.org/abs/2005.11599
  * [7] X. Cao, Q. Zhao, D. Meng, Y. Chen and Z. Xu, Robust Low-Rank Matrix Factorization Under General Mixture Noise Distributions, IEEE Transactions on Image Processing, vol. 25, no. 10, pp. 4677-4690, Oct. 2016, doi: 10.1109/TIP.2016.2593343.
  * [8] Liang F, Liu C, Wang N: A robust sequential Bayesian method for identification of differentially expressed genes. Statistica Sinica 17: 571–597. 2007
  * [9] Salazar E, Ferreira MAR, Migon HS: Objective Bayesian analysis for exponential power regression models. Sankhya - Series B 74: 107–125. 2012 Ferreira, M.A., Salazar, E. Bayesian reference analysis for exponential power regression models. J Stat Distrib App 1, 12 (2014).


11institutetext: Astronomy & Astrophysics Section, Dublin Institute for
Advanced Studies, Dublin, D02 XF86, Ireland. 22institutetext: School of
Physics, Trinity College Dublin, College Green, Dublin 2, Ireland.
22email<EMAIL_ADDRESS>33institutetext: National Centre for Radio
Astrophysics, Tata Institute of Fundamental Research, Pune 411007,
Maharashtra, India.

# Imaging-spectroscopy of a band-split type II solar radio burst with the
Murchison Widefield Array

Shilpi Bhunia 11 Eoin P. Carley 2211 Divya Oberoi 33 Peter T. Gallagher 11

(Received …)

Type II solar radio bursts are caused by magnetohydrodynamics (MHD) shocks
driven by solar eruptive events such as Coronal Mass Ejections (CMEs). Often
both fundamental and harmonic bands of type II bursts are split into sub-
bands, generally believed to be coming from upstream and downstream regions of
the shock; however this explanation remains unconfirmed. Here we present
combined results from imaging analysis of type II radio burst band-splitting
and other fine structures, observed by the Murchison Widefield Array (MWA) and
extreme ultraviolet observations from Solar Dynamics Observatory
(SDO)/Atmospheric Imaging Assembly (AIA) on 2014-Sep-28. The MWA provides
imaging-spectroscopy in the range of 80-300 MHz with a time resolution of 0.5
s and frequency resolution of 40 kHz. Our analysis shows that the burst was
caused by a piston-driven shock with a driver speed of $\sim$112 km s-1 and
shock speed of $\sim$580 km s-1. We provide rare evidence that band-splitting
is caused by emission from multiple parts of the shock (as opposed to the
upstream/downstream hypothesis). We also examine the small-scale motion of
type II fine structure radio sources in MWA images. We suggest that this
small-scale motion may arise due to radio propagation effects from coronal
turbulence, and not because of the physical motion of the shock location. We
present a novel technique that uses imaging spectroscopy to directly determine
the effective length scale of turbulent density perturbations, which is found
to be 1 - 2 Mm. The study of the systematic and small-scale motion of fine
structures may therefore provide a measure of turbulence in different regions
of the shock and corona.

###### Key Words.:

techniques : imaging spectroscopy - Sun: activity- Sun : shock waves- Sun :
turbulence

##  1 Introduction

Type II solar radio bursts are produced by magnetohydrodynamics (MHD) shocks
driven by solar eruptive events such as coronal mass ejections (CMEs), flares,
and jets (Nelson & Melrose, 1985; Nindos et al., 2011; Gopalswamy et al.,
2018; Zucca et al., 2018; Maguire et al., 2021; Jebaraj et al., 2021;
Alissandrakis et al., 2021). The shock waves accelerate electrons, producing
Langmuir waves and subsequently plasma emissions at the fundamental ($f_{p}$)
and second harmonic ($2f_{p}$) of the plasma frequency, given by $f_{p}\approx
8980\sqrt{n_{e}}$, where $n_{e}$ is electron number density (cm-3). Often the
dynamic spectra of type IIs exhibit various kinds of fine structures (FS). For
example, the fundamental and harmonic bands of type II bursts drift to lower
frequencies over time, and can sometimes split into sub-bands, known as ‘band-
splitting’. Type IIs can exhibit FS on a variety of temporal and spectral
scales, and altogether such structure can give insight to shock kinematics,
geometry, propagation, and turbulence in the solar atmosphere. Such properties
are highly important for shock particle acceleration physics.

In a study based on 112 metric type II bursts Vrsnak, B. et al. (2001)
reported only 20% of type II bursts show the band-splitting phenomena. It is
not known in literature why such a small fraction of the type II bursts show
such structure. Despite this small fraction, band-splitting shows common
features across different events. For example, the relative frequency split
between the sub-bands $\triangle f/f$ is found to be within the range of
0.1-0.5 and this value remains constant over the time duration of the event
(Vrsnak, B. et al., 2001).

It is still unclear what causes the band-splitting of the type II bursts, but
there are two widely accepted theories. The first popular mechanism was
suggested by Smerd et al. (1974, 1975) where the high frequency sub-band (HFS)
and low frequency sub-band (LFS) of type II are caused by coherent plasma
emissions simultaneously coming from the downstream (behind) and upstream
(ahead) regions of the  same shock front. This is commonly used to derive
shock compression ratio, Mach numbers (e.g. Vrsnak, B. et al., 2001; Zucca et
al., 2018; Maguire et al., 2020) and the coronal magnetic field strength (e.g.
Kumari et al., 2017, 2019). There have been some imaging studies that have
found evidence supporting this interpretation (e.g. Zimovets et al., 2012;
Zucca et al., 2018; Chrysaphi et al., 2018). In these studies, band-splitting
was most often associated with a CME-driven shock, and the sub-band sources
were closely spaced in imaging (Chrysaphi et al. (2018) reported the apparent
spatial separation between two sub-band sources at 32 and 40 MHz for
fundamental emission to be $\sim$0.2$\pm$0.05 R$\odot$).

There have also been results that are not in the favour of the
upstream/downstream scenario. Du et al. (2014) found that the spectral
features first appeared in the HFS seconds earlier than in the LFS which
according to the upstream/downstream scenario should have happened the other
way around, that is the spectral structures should first appear in LFS as the
emission should first come from low density upstream (shock ahead) region.
According to review work by Cairns (2011), in the downstream region the
electron beam distributions are unstable to the production of enhanced
Langmuir waves. Also, recent observations (Zucca et al., 2018; Magdalenić et
al., 2020; Morosan et al., 2020; Kouloumvakos et al., 2021) of type II dynamic
spectra have shown the burst having multiple sub-bands which can not be
explained by conventional upstream/downstream scenario. Hence, an alternative
mechanism can be used to explain the band-splitting phenomenon, where
spatially separated parts of the shock front produce radio emissions at LFS
and HFS (McLean, 1967). Until now there have been evidences supporting the
upstream/downstream scenario whereas there has been very little evidence for
this theory (Zimovets & Sadykov, 2015).

While type IIs band-splitting is the most common fine structure, recent
observations with modern instruments with an excellent time, frequency, and
angular resolution have revealed many different FS in shock radio bursts
(Magdalenić et al., 2020). These FS are said to be a signature of turbulence
that shocks encounter while propagating through the inhomogeneous turbulent
corona (Chen et al., 2018; Carley et al., 2021). The propagation effects
modify the intrinsic properties (shape, size, positions, brightness
temperature) of the observed radio source (Kontar et al., 2017, 2019; Sharma &
Oberoi, 2020; Zhang et al., 2021; Ryan et al., 2021; Murphy et al., 2021).
There have been several studies (Fokker, 1965; Steinberg et al., 1971; Riddle,
1972; Zhang et al., 2021) which model the extent of the scattering effects on
the radio waves in the corona and some which compare the simulations with the
observations of solar radio bursts (e.g. Kontar et al., 2019). There is
currently an ongoing debate on the length scales of turbulence that cause
radio wave scattering. If the scattering happens where the spectrum is purely
Kolmogorov-like, then the characteristic scale length of the density
fluctuations is considered to be a combination of the inner and outer scale of
the turbulence spectrum (Thejappa et al., 2007). Alternatively, it is also
thought that the radio scattering happens near the inner scale. In the
literature, there is no consensus on which length-scale to use (Bastian,
1994). The lack of consensus is partly due to the lack of direct diagnostics
of this effective length scale.

In this paper, we analyze a variety of fine-scale structures and fine-scale
motions of type II radio bursts using the Murchison Widefield Array (MWA;
Tingay et al., 2013). Firstly, we provide rare imaging of band-splitting that
does not agree with the upstream/downstream scenario, supporting the McLean
(1967) model. Secondly, we developed a new technique that uses MWA imaging
spectroscopy to examine fine-scale source motions of the type II burst and
examine this in the context of coronal turbulence. Section 2 provides an
observational overview of the type II burst, Section 3 describes the methods
of the data analysis. The results and discussions are presented in Sections 4
and 5, respectively, followed by conclusions in Section 6.

##  2 Observations

Figure 1: a) a) GOES-15 light curves indicating a M5.1 class flare. The two
vertical dashed lines show the time range of the Learmonth dynamic spectra. b)
The dynamic spectra of the type II radio burst observed by the Learmonth
spectrograph. It is clear that the fundamental band show band-spitting where
the high and low frequency branches are indicated by FU and FL, respectively.
The corresponding band-splitting in harmonic band is marked by HU and HL,
respectively. (c,d,e) The three images show the position of MWA radio contours
overlaid on the AIA 193 Å channel base image at 02:49:30 UT. The purple,
orange, and blue contours are the MWA 132.86, 120.06, and 108.54 MHz contours
(50%, 60%, 70%, 80%, 90% of the peak intensity), respectively. They show the
position of the radio sources of different frequencies at the same time,
indicated by purple, orange, and blue points on the Learmonth spectra.

On 28/09/2014, a Geostationary Operational Environmental Satellite-15
(GOES-15)111https://space.oscar.wmo.int/satellites/view/goes_15 M5.1 class
solar flare started at 02:39:00 UT from the active region (AR) NOAA 12173 (Fig
1a). This flare was also associated with a slow CME which was first seen by
the Large Angle and Spectrometric Coronagraph C2 (LASCO; Brueckner et al.,
1995) at 03:24:05 UT. According to the LASCO C2
CDAW222https://cdaw.gsfc.nasa.gov/CME_list/nrl_mpg/2014_09/140928_c2.mpg
catalogue, the linear speed of the CME was 215 km s-1 on the plane-of-sky.
This eruption event was observed in extreme ultraviolet (EUV) wavelengths by
the Atmospheric Imaging Assembly (AIA; Lemen et al., 2012) onboard the Solar
Dynamics Observatory (SDO; Pesnell et al., 2012).

At 02:45 UT, the peak time of the X-ray flare, Learmonth
Spectrograph333Learmonth Observatory; https://www.sws.bom.gov.au/Solar/3/1
observed a type II burst, the radio signature of a shock during this eruptive
event (Fig 1b). The dynamic spectra shows both fundamental (fp) and harmonic
($\sim$2fp) of the burst. Type II radio bursts often show band-splitting (e.g.
Vrsnak, B. et al., 2001), where each of the fundamental and harmonic emissions
are split in to two sub-band which show the same spectral drifts. We refer to
these sub-bands as the high frequency sub-band (HFS) and low frequency sub-
band (LFS). Visually it can be difficult to characterize the HFS and LFS in
the type II harmonic band. To make the band-splitting clearer the Learmonth
dynamic spectra had been smoothed using a small kernel (2x2 pixels) in Fig 1b.
The fundamental band shows the clear band-splitting where the HFS and LFS are
indicated by $F_{U}$ and $F_{L}$. We see the corresponding band-splitting in
the harmonic band, indicated by $H_{U}$ and $H_{L}$ . The drift rates computed
for both branches in the harmonic band are found to be very similar. The drift
rates of the HFS and LFS are -0.160 and -0.158 MHz/s, respectively. Hence we
regard the two observed sub-bands to correspond to the band-splitting, often
seen in type II radio bursts.

This type II burst was observed using the MWA, a low-frequency radio
interferometer capable of providing a time and spectral resolution of 0.5 s,
and 40 kHz respectively, across its 80-300 MHz observing band. The MWA
observed the harmonic band of this radio burst in 6 different spectral pickets
of 2.56 MHz bandwidth each. These observations spanned the frequency range
from 79.10 to 133.36 MHz with a spectral resolution of 40 kHz. These spectral
bands of MWA observations are marked with black rectangular boxes on the
Learmonth dynamic spectra in Fig 1b.

##  3 Methods

The high fidelity solar radio images from the MWA data were obtained using the
Automated Imaging Routine for Compact Arrays for the Radio Sun (AIRCARS;
Mondal et al., 2019). AIRCARS is a self-calibration based radio imaging
pipeline optimised for spectroscopic snapshot imaging for centrally condensed
arrays like the MWA. Its efficacy has been demonstrated in multiple recent
works requiring high imaging quality and applications ranging from the first
ever detection of mSFU level metrewave impulsive emissions from the quiet Sun
(Mondal et al., 2020a) to detection of spatially resolved gyrosynchrotron
emission from CMEs (Mondal et al., 2020b), multiple investigations of quasi-
periodic-pulsations in active solar radio emissions (Mohan et al., 2019a, b;
Mohan, 2021b, a; Mondal & Oberoi, 2021) and for developing precise absolute
solar flux density calibration techniques (Kansabanik et al., 2022). Imaging
and calibration was done using AIRCARS at a time resolution of 0.5 s and
spectral resolution of 40 kHz for 8 spectral channels in each of the 6 MWA
spectral sub-bands. We have applied independent gains obtained from self-
calibration to each spectral channel.

Flux calibration was done using the fortuitous presence of Virgo-A in the very
large field of view of the MWA. The flux density of Virgo-A at our frequencies
of observation was obtained using a linear spectral fit to the data available
from the NASA/IPAC Extragalatic Database444NED; https://ned.ipac.caltech.edu
and a model for the MWA beam available from Sokolowski et al. (2017).

Figure 1c-e shows some examples MWA radio images of the type II emission at
132.86 (purple), 120.06 (orange) and 108.54 (blue) MHz. The radio contours are
overlaid on the AIA 193 Å base image of the eruption event at 02:49:30 UT. The
frequencies and time of these radio sources are also marked on the Learmonth
spectra by purple, orange, and blue circles. The radio contours show that the
type II harmonic emissions have a single compact source which can be well
described by 2D elliptical Gaussian. Hence, a Gaussian model was fit to each
of the AIRCARS images for all 6 spectral bands by doing chi-square
minimization using the Levenberg–Marquardt algorithm (LEVENBERG, 1944;
Marquardt, 1963) and the parameters of the best fit Gaussian models used to
extract the information about the location, size, shape, and intensity of the
burst emission. The fitting shows the emission source has semi-major axis in
the range 6′–10′and semi-minor axis lying between 5′–7′. The uncertainty in
the position of the fit was calculated using CASA imfit software which uses
the standard method described in Condon (1997). According to this method, the
uncertainty in position depends on full width at half maximum (FWHM) of major
and minor axes, peak intensity, the RMS noise, and the pixel width of the
direction coordinate. The uncertainties in position are found to be very
small, less than 1.2″, due to the high signal to noise ratio of the data.

The positional information from the Gaussian fits is used in the following
sections to understand the kinematics of the split band sources and the fine
scale motion of type II.

Figure 2: a) The ratio image of the eruptive event with SDO/AIA in 171, 193,
211 Å channels at 02:49:11 UT. The blue circle and green triangle points on
the image are the position of radio sources of different frequencies at the
same time from HFS and LFS, respectively (shown in panel b). The lines connect
the HFS and LFS points at the same time. It is clear that both band sources
are moving in different directions from each other and they are quite
separated in the position. The top (b) and bottom (c) figure in the right
panel show the Learmonth and MWA dynamic spectra of the type II harmonic band.
The rectangular panels on the Learmonth spectra highlight the regions of the
type II MWA has observed.

##  4 Results

###  4.1 Kinematics of the split-band radio sources

Figure 2b-c shows the harmonic band of the type II burst in the Learmonth and
MWA spectra. Points of different frequencies at the same time have been chosen
from HFS (blue circles) and LFS (green triangle) of the harmonic band in the
dynamic spectra. Figure 2a shows radio source positions (from the 2D Gaussian
fits) of these frequencies and times overplotted on the AIA image, a three
colour running ratio image of the eruption produced by AIA 171, 193, 211 Å
channels at 02:49:11 UT. Each AIA image was first normalised by its exposure
time and the time difference in images for the ratio was $\sim$3 mins. The
observations show that the LFS and HFS are spatially separated by up to around
485″($\sim$372 Mm), and get progressively closer at 157″($\sim$120 Mm) on the
plane-of-sky (POS). The uncertainty in the position of the sources was
considered to be the average position fluctuations of the radio sources
$\sim$30″(see Fig 7). The separation of split-band sources occurring
simultaneously is shown by the points connected by purple lines in Fig 2a. The
split-band sources also have noticeably different kinematics, with LFS (green
triangles) propagating towards south west, the most prominent direction of the
eruption, and HFS (blue circles) propagating in a lateral, east-west
direction. Towards the end of the type II, the HFS is positioned at a greater
radial distance (in the plane-of-sky). The separation in space of the LFS and
HFS by $\sim$372 Mm is clearly indicative of the two radio sources being
generated in different regions of the corona. This observation clearly
contradicts the upstream/downstream scenario e.g. the HFS and LFS sources
should be closely positioned in space and move in the same direction. In
Section 5, we further discuss these source kinematics with regard to the
various hypotheses for the band-splitting phenomena and explain that the
source positions and separations that we observe do not support the upstream-
downstream hypothesis for this event.

###  4.2 Large-scale kinematics of the shock

As the next step, in Fig 3, the speeds of the radio sources and the eruption
event in EUV observations were calculated to understand the relationship
between the eruptive active region and the type II burst.

Since the propagation of LFS sources was closely following the direction of
the eruption event at around 60 degrees (green line on the AIA ratio image in
the Fig 2), these sources have been chosen to calculate the speed of the radio
sources.

The speed of the eruption in EUV was calculated in several directions
indicated by lines on Fig 2a at 20, 40, 60, 80, and 120 degrees from the black
dotted line (in the plane of the sky). The brightest point on the EUV front
for these directions is chosen for each time using point-and-click method. The
distance of these points was calculated from the location of the active region
origin and linear fits are performed over these distances using linear least-
squares regression, shown in Fig 3. It was found that the speed of the EUV
front is not the same in every direction and varies from 20 - 112 km s-1. The
uncertainty in the distance of these points is calculated from point-and-click
uncertainty over several attempts. The eruption was most prominent towards 60
degrees and, where its speed was 112$\pm$3 km s-1.

In Fig 3 the green triangles indicate the position of the radio sources over
time. For the position uncertainties for each radio source, the average
fluctuations of the sources has been taken into account which is $\sim$0.5
arcmin or 23 Mm (Fig 7). To determine the speed, the distance of the radio
sources has been calculated from the active region position and the linear
regression has been performed to get the speed of the radio sources and the
error in the speed. The type II radio source speed (shock speed) is found to
be 580$\pm$50 km s-1, which is much faster than the maximum driver speed of
120 km s-1 observed in EUV555Note the plane-of-sky CME linear speed was 215 km
s-1, reported by SOHO/LASCO CDAW catalogue. This speed deprojected by the
active region angle from the sky-plane is $\sim$ 430 km s-1..

Given the much larger shock speed than driver speed, we expect that the shock
causing this type II burst is the piston driven mechanism (Vršnak & Cliver,
2008; Nindos et al., 2011; Magdalenić et al., 2020). This is discussed further
in Section 5.1.

Figure 3: The kinematics of the radio sources with respect to eruption. Given
that the speed of the shock is much higher than the maximum speed of the
eruption event, the shock originating this burst is expected to be a piston
driven shock where the shock speed can exceed the driver speed.

###  4.3 Fine-scale motion of radio sources

Figure 4: a) The running ratio image of the eruption in AIA 193, 211, and 171
Å channels at 02:49:11 UT. The position of the type II sources of 133.36 MHz
(red) and 133.22 MHz (purple) have been plotted over the image for a duration
of 5 min, represented by both red and purple color bars. The dynamic spectra
in panels (b) and (c) on the right show the peak amplitude of the Gaussian
fitted source. The arrows represent the instantaneous change in direction
($\theta_{i+1}-\theta_{i}$) of the source motion for different frequencies
(131.58, 131.74, 131.94, 132.08, 132.86, 133.02, 133.22 and 133.36 MHz) in the
channel. The blue arrows represent when the change in angle lies between 0 to
$\pi$ radian. Similarly, the red arrows represent when the change in direction
lies between 0 to -$\pi$ radian. Panel (c) shows a zoomed-in view of the
region marked in panel (b). In panel (c), one can see at each time the arrows
point in almost the same direction across nearby frequency channels,
exhibiting the correlated fine scale motion.

Figure 4(a) shows the harmonic type II source motion at 133.22 (purple) and
133.36 (red) MHz with respect to the active region for the duration of the
burst, starting at 02:46:12 UT and ending at 02:49:50 UT. The background image
is the three colour ratio image of the eruption in AIA 193, 211, and 171 Å
channels at 02:49:11 UT. Clearly, the fine scale motion of the radio sources
shows a complex chaotic character. We describe this motion by the parameter
$\Delta\theta$, the instantaneous change in source direction at time t+1 given
by $\Delta\theta_{t+1}=\theta_{t+1}-\theta_{t}$. This direction change is
expressed in radians and shown for each frequency channel (131.58, 131.74,
131.94, 132.08, 132.86, 133.02, 133.22, and 133.36 MHz) by the arrows on the
dynamic spectra in Fig 4b-c. The background dynamic spectra (grey color scale)
in both figures represent the peak amplitude of the Gaussian fitted source at
the corresponding frequency. Figure 4 (c) shows a zoomed-in view of the region
marked by the black rectangle on the Fig 4(b). While the arrows indicate
random instantaneous position changes at each time step, we observed that the
position changes behave similarly for adjacent frequencies. This implies that
the fine-scale motion is correlated across nearby frequencies. To investigate
this systematic correlated behavior, we produce a cross correlation of
instantaneous direction and amplitude changes for sources in adjacent
frequency channels. For example, $\@vec{f}(t)$ represents the direction and
magnitude of the instantaneous change in position over time for one frequency,
while $\@vec{g}(t)$ represents the same for another frequency. We use a dot-
product correlation to look for correlated position change (magnitude and
direction) between nearby frequencies, given by

$\@vec{f}*\@vec{g}(\tau)=\int\@vec{f}(t)\cdot\@vec{g}(t+\tau)dt=\int
f(t)g(t+\tau)\cos(\phi(t))dt.$ (1)

where $\phi(t)$ is the change of direction between $\@vec{f}(t)$ and
$\@vec{g}(t)$. This equation will show a large response when the position
shift direction and magnitude of the two time series for two different
frequencies are the same. Figure 5 shows the dot-product correlation between
133.36 MHz and seven other frequencies (133.22, 133.02, 132.86, 132.08,
131.94, 131.74 and 131.58 MHz). The dashed line represents $3\sigma$ above the
background noise for each of the correlation plot. We consider there is a
significant correlation only when the correlation peaks are above $3\sigma$.
We can see a significant correlation at zero time lag between 133.36 and
133.22 MHz. The correlation steadily drops with increasing frequency
difference from 133.36 MHz and drops below significance beyond the nearest
three frequencies.

In Fig 6, we show the correlation strength as a function of frequency
difference ($\Delta f$) for four coarse channels. The frequency difference
($\Delta f$) is the difference between the reference frequency (fref; chosen
to be the highest frequency in the channel) and each frequency in the same
channel used for the correlation. Each series of points (indicated by
different color) represents the correlation between fref and f${}_{ref}+\Delta
f$ frequency for each different coarse channel. The color scale of the circles
for each coarse channel represents the $\Delta f$ for eight frequencies in
each channel. Clearly, the correlation strength decreases for larger $\Delta
f$ from fref for each coarse channel.

The phase calibration solutions are independent across frequency channels, so
the fine-scale motion among different frequency sources would naively be
assumed to be uncorrelated. However a correlation is found to exist in fine
scale motion among nearby frequency channels and this correlation must have a
physical cause in the corona. Section 5.2 discusses this physical cause and
its potential relation to radio source scattering in the corona due to
turbulence.

Figure 5: Dot product correlation (blue) of the magnitude and $\Delta\theta$
between 133.36 MHz and seven other frequencies (133.22, 133.02, 132.86,
132.08, 131.94, 131.74 and 131.58 MHz). Dot product correlation represents a
cross correlation of instantaneous direction and amplitude changes for sources
in adjacent frequency channels. The dashed line represents the $3\sigma$ above
the background noise for each of the correlation plot. We regard the
correlation to be significant only if it exceeds $3\sigma$. Clearly, there
exists a significant correlation between the top three frequency pairs at zero
time lag, which drops below $3\sigma$ level with increasing separation between
frequencies

Figure 7 shows histograms of the instantaneous position shift magnitude
($\Delta s_{t+1}=s_{t+1}-s_{t}$) of sources for a single frequency from all
six coarse channels. The $x$-axis of all the plots is in log-space and a
Gaussian has been fitted over the histogram for all channels to calculate the
mean (black solid line) and standard deviation (black dotted line) of $\Delta
s$. The distributions show a clear log-normal distribution in each coarse
channel, and provide a clue to the potential physical cause of the stochastic
motion observed in images as discussed in Sec. 5.2.

Figure 6: Normalized correlation strength vs. frequency difference for all 4
coarse channels. Here the frequency difference ($\Delta$f) is the difference
between the reference frequency (fref) and each of the other frequencies for
the dot product cross correlation in MHz. For each channel, fref is chosen to
be the highest frequency and each circle points represent the correlation
strength for each frequency in that channel. The correlation strength
decreases with increasing $\Delta$f. Figure 7: Histogram plot showing the
probability density of instantaneous magnitude change in position ($\Delta
s_{t+1}=s_{t+1}-s_{t}$) of sources for a single frequency from all six
channels. The blue curved line in each plot represents the Gaussian fitting
done on the histogram. The mean and 1 sigma of the Gaussian are indicated as
solid and dotted black line, respectively.

##  5 Discussion

###  5.1 What causes band-splitting in this type II event?

In this work, we analyse a split-band metric type II burst that spatially and
temporally relates to the 28th September 2014 eruption event, a M5.5 solar
flare followed by a slow CME. The EUV observations of this event show the
propagation of the expanding loops of the active region (Fig 2, left panel).
It is found that the disturbance caused by the expansion is not propagating
uniformly in all directions and the maximum speed progressed towards the
south-west at $\sim$112 km s-1. However, it should be noted that there is no
direct evidence of the shock wave in the AIA observations. The direct evidence
of this shock front, a split band type II burst, was found in the dynamic
spectra from the Learmonth observatory and the MWA.

The LFS sources are found to propagate towards the same direction as the
eruption and the speed of the radio sources is found to be around 580 km s-1.
This is considerably faster than the driver speed of $\sim$112 km s-1 derived
from the EUV observations. It is therefore likely that the shock associated
with the eruption was a piston driven shock (Vršnak & Cliver, 2008). In such a
scenario, a subsonic driver can cause a shock. The plasma is pushed outwards
by a 3D piston impulsively, and the plasma ahead of the piston (expanding
magnetic loops) cannot pass behind the driver. As a result, the shock ‘races
away’ from the driver, and shock speed is greater than the driver speed. This
is in contrast to the bow shock scenario, in which plasma can pass behind the
driver, and the shock and driver speed are the same.

High fidelity interferometric imaging of MWA provides a golden opportunity to
image the source location of the split bands of type II at the same time.
There have only been a few studies that achieved this (Zimovets et al., 2012;
Zucca et al., 2014; Zimovets & Sadykov, 2015; Chrysaphi et al., 2018). This
allows us to understand the cause of band-splitting and also to compare the
two widely accepted band-splitting theories. The most popular interpretation
due to Smerd et al. (1974, 1975) is that the high and low frequency split
bands are coming from downstream (behind) and upstream (ahead) regions of the
shock front at the same time, respectively. The density compression across the
shock (see Section 2.1.2) leads to a separation of plasma frequency of
upstream-downstream sources ($f_{p}\sim 8980\sqrt{n_{e}}$). There has been
some evidence supporting this interpretation (Chrysaphi et al., 2018; Zimovets
et al., 2012; Zucca et al., 2014). There have also been results that are not
in the favor of this upstream/downstream scenario. Du et al. (2014) found that
the spectral features first appeared in the HFS seconds earlier than in the
LFS which, according to the upstream/downstream scenario, should happen the
other way around. According to the upstream/downstream scenario, there should
also be a strong correlation between the shock velocity and the frequency
ratio between the sub-bands, but Du et al. (2015) show only a very weak or no
correlation between these two parameters. The review work by Cairns (2011)
also argues that in the downstream region the absence of energetic electrons
with respect to the ambient plasma does not lead to the growth of Langmuir
waves in the region, not supporting the upstream/downstream scenario. An
alternative mechanism was proposed by McLean (1967) where the band-splitting
originates due to multiple parts of the shock front encountering coronal
environments of different physical properties, such as variations in electron
density or magnetic field. There have been very few observational studies that
support this view (Zimovets & Sadykov, 2015).

In our analysis, it is observed that the apparent location of the split band
sources are initially rather far apart ($\sim$372 Mm in POS) then get
gradually closer with time ($\sim$120 Mm in POS). They also appear to move in
different directions with time. For the reasons outlined below, the large
observed spatial separation and their motion cannot be attributed to
propagation effects alone. This emission is at the harmonic (f = 2fp), where
the impact of the propagation effects is much less dramatic as compared to the
fundamental. For instance, Zhang et al. (2021) show that depending on the
anisotropy and density fluctuations, a harmonic source at 35 MHz can suffer
absolute maximum position offset of $\sim$ 0.1 $R_{\odot}$ ($\sim$69 Mm). The
observed frequency range of the type II burst studied here higher (79-134
MHz), implying that the propagation effects should be smaller still. Even when
the propagation effects lead to large changes in the apparent locations and
sizes of the observed sources (e.g. Sharma & Oberoi, 2020), these changes are
expected to be similar at nearby frequencies. The difference in observing
frequencies for the HFS and LFS is not large. It lies in the range of 10 – 20
MHz leading to a $\Delta f/f\sim 15\%$. Despite their small spectral
separation, the observed differences in locations and motions of the HFS and
LFS sources are large. This implies that these differences can not arise due
to propagation effects alone and must come from these sources being
intrinsically well separated.

To understand the differences in these regions in terms of the geometry of the
magnetic field lines, the extrapolation of the field lines is done with the
Potential Field Surface extrapolation (PFSS) model (Stansby, 2019) using the
Global Oscillation Network
Group666https://gong2.nso.edu/archive/patch.pl?menutype=s (GONG) observations
at 02:14 UT. The PFSS field lines are overlaid on the AIA image of the
eruption event at 02:49:30 UT in 193 Å channel in the Fig 8. Both sub-band
sources of the type II are on different magnetic field loops, with the LFS
closer to the active region core than the HFS. As discussed earlier there can
be a significant difference in the true and apparent position of sources. It
is however clear from Fig 8 that the separations are so large that these
sources lie in different coronal environments.

This implies that the radio emissions must come from different parts of the
shock-front causing two sub-bands in the type II dynamic spectrum (Fig 2b).
The analysis by Zucca et al. (2018); Kouloumvakos et al. (2021) also suggests
a similar conclusion.

Figure 8: Magnetic field lines extrapolated with PFSS model overlaid on the
region associated with the eruptive event on the AIA 193 Å image at 02:49:30
UT. In our region of interest the field lines mostly are closed magnetic field
loops (white). Clearly, LFS and HFS sources are associated with different
loops.

This is rare evidence against the commonly used upstream-downstream
interpretation of split-band sources used widely to estimate the shock Mach
number (Vrsnak, B. et al., 2001) and the strength of the magnetic field
(Kumari et al., 2017, 2019). Our work cautions against using such a diagnostic
for split-band sources and reiterates the need for full imaging-spectroscopy
in interpreting type II fine structure.

###  5.2 Fine-scale motions: what is the case of the perturbations?

Figure 4 shows the complex motion of the type II sources over time. Our
analysis showed this motion, although disordered and random, exhibits a
systematic correlation across nearby frequencies. Figure 5 shows that the
source motion is well correlated across a bandwidth of 0.5 MHz for a specific
spectral channel and this is true for all other channels. As discussed later
in this section, the correlations in motion cannot arise due to calibration
effects, as independent calibration solutions are applied to each frequency
separately. This then poses the question of what causes disordered,
instantaneous position shifts to be correlated across radio sources at nearby
frequencies?

Recent studies have shown that low frequency radio sources can appear to be
displaced from their true location due to propagation effects arising from the
density turbulence in the corona (e.g. Kontar et al., 2019; Zhang et al.,
2021). Furthermore, previous studies have shown that stochastic fluctuations
in turbulent velocity fields can show a log-normal distribution (e.g. Arneodo
et al., 1998; Mouri et al., 2009). For each frequency channel we observe here,
the distribution of instantaneous magnitude change in position ($\Delta
s_{t+1}$) exhibits such a log-normal distribution (Fig 7). It is possible that
the instantaneous and stochastic source displacements we observe here may
arise due to turbulent density fluctuations. For example, radiation at nearby
frequencies launched from the same neighbourhood is expected to encounter
essentially the same turbulent structures in the corona and consequently
follow similar ray paths. Hence, we interpret this observed correlation over
small spectral spans as arising due to propagation effects like scattering and
refraction. An additional possibility is that these apparent motions are
arising due to small scale local variations of the shock surface itself.
Simulations have shown that a turbulent corona can cause a shock surface to
become rippled and inhomogenous, with shock properties such as shock strength
and acceleration efficiency being inhomogenous (e.g. Andreopoulos et al.,
2000; Zank et al., 2002; Lowe & Burgess, 2003; Neugebauer & Giacalone, 2005;
Burgess, 2006; Giacalone & Neugebauer, 2008; Lu et al., 2009; Vandas &
Karlický, 2011; Guo & Giacalone, 2010). If this is indeed the case, the root
cause for the observed rapid and small scale changes in the position of the
emission is still the turbulence in the background corona. This implies the
observed spectral separation over which there is correlated motion carries
information of the length scale of the coronal turbulence affecting the shock
surface.

We relate the spectral separation of correlated fine channels to a spatial
separation using a coronal electron density model. We have chosen the Saito
density model (Saito et al., 1970). A correlation among the fine channels
separated by 0.5 MHz represents a correlation among radio sources separated in
space by 1 - 2 Mm. We note that though the absolute coronal heights might
differ by larger amounts between different coronal electron density models,
the differences between the coronal heights corresponding to two nearby
frequencies are much smaller. For example, we followed the same procedure with
the Newkirk density model (Newkirk, 1961) and the resulting spatial separation
is the same. So these numbers hold independent of the electron density model
used. We interpret this as the correlation length scale of turbulence
fluctuations. For example, there is a perturbation in the corona of 1-2 Mm
which causes an instantaneous position shift of radio sources with similar
frequencies. If sources are separated by more than this length scale, the
correlation disappears, as we show in Fig 5.

There has been much debate in recent literature on the length scales of the
coronal turbulence that causes radio wave scattering. For example, Bastian
(1994) assume to this scale length to be the inner-scale of coronal
turbulence, on the order of 1-10 km in our case. However, Thejappa et al.
(2007) theoretically determined this effective length scale to be
$h=l^{1/3}_{i}l^{2/3}_{o}$ where $l_{i}$ and $l_{o}$ are the inner and outer
scale respectively. They use empirical expressions for these scales, with
$l_{i}=(r)^{\pm 0.1}$ where $r$ is the heliocentric distance in unit of solar
radii and $l_{o}=8.82\times 10^{-2}\bigg{[}\frac{R}{R_{o}}\bigg{]}^{0.82}$ AU.
Considering the heliocentric distance of 133.36 MHz radio source to be 1.51
R$\odot$ (according to 1 fold Saito density model), in our case these
expressions lead to an effective scale length of $h\sim 0.7$ Mm, which is in
the vicinity of our observationally determined scale length of 1-2 Mm. This
provides evidence that the random fine-scale motion of radio sources in our
observed type II radio burst is due to density turbulence, and that the scale
length of Thejappa et al. (2007) applies in our case. Recent observations
works (Chen et al., 2018; Carley et al., 2021) have found the density
turbulence spectrum to be Kolmogorov like and Carley et al. (2021) found the
spatial scales of the turbulence spectrum to range from $0.2$ Mm to $62$ Mm
placing our determined effective length scale in the inertial range of the
turbulence.

####  5.2.1 Excluding other systematic effects for the fine-scale motion

While analysing the fine scale motion of the type II sources we have to be
sure if these source motions are due to a coronal phenomenon (turbulence in
our hypothesis) and not due to ionospheric effects or imaging analysis
artifacts. Below we discuss the reasons why we can exclude these effects with
confidence.

  1. 1.

Ionospheric effects: While traversing the Earth’s ionosphere, low frequency
radio sources can suffer apparent angular position offsets due to ionospheric
propagation effects. It is conceivable that the observed correlated motion
across nearby frequencies can arise due to such effects. However the timescale
of the observed fluctuations, about 1 s, is much faster than those associated
with ionospheric variations (¡50 s)(Jordan et al., 2017). Also, the position
offsets due to ionospheric refraction are inversely proportional to $f^{2}$
(e.g. Loi et al., 2015), where $f$ is the observing frequency. This implies
that low frequency radio waves will be affected more than high frequencies and
also provides a quantitative scaling which should hold for ionospheric
propagation effects. To check for this relationship, in Fig 7 we show
histogram plots of the instantaneous position shift magnitude ($\Delta
s_{t+1}=s_{t+1}-s_{t}$) of sources for a single frequency from all six coarse
channels. A Gaussian has been fitted to the histograms. The mean of the
Gaussian model is shown by the black solid line and its $\sigma$ is also
mentioned. The $1/f^{2}$ trend expected due to ionospheric propagation is not
seen in either of these quantities, implying that they are not likely to arise
due to ionospheric propagation effects.

  2. 2.

Imaging artifacts: During the process of imaging using self-calibration, it is
possible for an arbitrary phase to be introduced which can then lead to
position offsets in the image plane (Pearson & Readhead, 1984). For the
analysis presented here, the self-calibration for each of the different
frequencies was done only for a single time slice and the solutions obtained
applied to the entire observing duration. In addition, self-calibration for
each of the frequencies was done independently and the self-calibration based
position shifts were corrected by moving the centroid of the self-calibrated
image to that of the image obtained prior to self-calibration. More
importantly, a single arbitrary shift in the image plane. arising due to self-
calibration, can anyway not give rise to the time varying shifts seen in these
data.

  3. 3.

An independent check: Out of an abundance of caution, we use the fortuitous
presence of Virgo-A in the large MWA field-of-view to investigate the possible
contamination from both these effects. If the observed shifts correlated
across nearby frequencies are a systematic effect arising out of either of the
imaging process or ionospheric propagation, then a similar effect should also
be seen for Virgo-A. An independent Gaussian model was fit to Virgo-A for each
time and frequency slice. The location of the peak of the Gaussian model was
subjected to analysis identical to that done for the location of the type II
source. The position of Virgo-A was found to be very stable across time and
frequency and no small scale motions correlated across neighbouring
frequencies could be found. This provides an independent confirmation that the
observed instantaneous position shifts in the type II emission are intrinsic
and do not arise due to any ionospheric propagation effects or imaging
artifacts.

##  6 Conclusions

The first part of this work focused on the understanding of how shocks can
cause split bands in the harmonic band of a particular type II event using
imaging spectroscopy of MWA and EUV observations of AIA/SDO. Imaging the radio
sources from these sub-bands revealed that the sources are separated in space
by up to $\sim$372 Mm in the plane-of-sky. The radio emissions are likely
generated in different coronal loops ahead of the shock front, generating
plasma emission at different frequencies during the burst. This provides rare
imaging evidence against the popular interpretation that the band-splitting is
caused by emission upstream and downstream of the shock front and calls into
question the widespread use of this interpretation to calculate important
properties such as density jump across the shock, Mach number, and magnetic
field. Our study demonstrates the importance of imaging for verifying which of
these competing theories are behind the band-splitting in type IIs.

In the second part of the work we have used imaging-spectroscopy to study the
effect of small scale perturbations to fine scale motion of type II sources.
At low radio frequencies, these perturbations are believed to arise due to
coronal density inhomogeneities which the radiation encounters while
propagating out through the corona. We observed that the fine-scale motion of
the sources exhibited a systematic correlation across nearby frequencies. To
investigate this we developed a new technique of cross correlation of
instantaneous change in direction and magnitude of the radio sources of nearby
frequency channels (dot-product correlation). We interpret such correlations
as arising due to small scale density perturbations from coronal turbulence.
These perturbations lead to similar displacement of sources at nearby
frequencies, spaced up to 0.5 MHz apart and corresponds to a spatial
separation of $\sim 1-2$ Mm at coronal heights of interest. If this is
interpreted as the effective length scale of turbulence, our observational
work agrees with the theoretical findings of (Thejappa et al., 2007).

Hence, our works provide a unique way to directly characterize the turbulence
in different regions of shocks and the corona. One can also potentially obtain
the intensity of these fine structures for a large range of frequencies and
get the power density spectra, giving insight into the spectral
characteristics of coronal turbulence (Chen et al., 2018; Carley et al.,
2021). Such studies can provide the range of length scales of the coronal
density perturbations and provide an opportunity to check if the determined
length scale lies in the inertial range. This will be an interesting avenue to
explore in future studies.

###### Acknowledgements.

This scientific work makes use of the Murchison Radio-astronomy Observatory
(MRO), operated by the Commonwealth Scientific and Industrial Research
Organisation (CSIRO). We acknowledge the Wajarri Yamatji people as the
traditional owners of the Observatory site. Support for the operation of the
MWA is provided by the Australian Government’s National Collaborative Research
Infrastructure Strategy (NCRIS), under a contract to Curtin University
administered by Astronomy Australia Limited. We acknowledge the Pawsey
Supercomputing Centre, which is supported by the Western Australian and
Australian Governments. SB is supported by the Hamilton PhD Scholarship at
DIAS. EPC is supported by the Schrödinger Fellowship at DIAS. DO acknowledges
support of the Department of Atomic Energy, Government of India, under the
project no. 12-R&D-TFR-5.02-0700. We would like to thank Dr. Peijin Zhang for
his valuable discussions. This research has also made use of NASA’s
Astrophysics Data System (ADS). This research has made use of the NASA/IPAC
Extragalactic Database (NED), which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with the
National Aeronautics and Space Administration. The authors would like to
acknowledge AIA, GOES and Learmonth team for open access to their data.We
thank the referee for the helpful comments to improve the paper.

## References

  * Alissandrakis et al. (2021) Alissandrakis, C. E., Nindos, A., Patsourakos, S., & Hillaris, A. 2021, A&A, 654, A112
  * Andreopoulos et al. (2000) Andreopoulos, Y., Agui, J. H., & Briassulis, G. 2000, Annual Review of Fluid Mechanics, 32, 309
  * Arneodo et al. (1998) Arneodo, A., Manneville, S., & Muzy, J. 1998, European Physical Journal B, 1, 129
  * Bastian (1994) Bastian, T. S. 1994, ApJ, 426, 774
  * Brueckner et al. (1995) Brueckner, G. E., Howard, R. A., Koomen, M. J., et al. 1995, Sol. Phys., 162, 357
  * Burgess (2006) Burgess, D. 2006, The Astrophysical Journal, 653, 316
  * Cairns (2011) Cairns, I. H. 2011, in The Sun, the Solar Wind, and the Heliosphere, ed. M. P. Miralles & J. Sánchez Almeida, Vol. 4, 267
  * Carley et al. (2021) Carley, E. P., Cecconi, B., Reid, H. A., et al. 2021, The Astrophysical Journal, 921, 3
  * Chen et al. (2018) Chen, X., Kontar, E. P., Yu, S., et al. 2018, The Astrophysical Journal, 856, 73
  * Chrysaphi et al. (2018) Chrysaphi, N., Kontar, E. P., Holman, G. D., & Temmer, M. 2018, The Astrophysical Journal, 868, 79
  * Condon (1997) Condon, J. J. 1997, Publications of the Astronomical Society of the Pacific, 109, 166
  * Du et al. (2014) Du, G., Chen, Y., Lv, M., et al. 2014, The Astrophysical Journal, 793, L39
  * Du et al. (2015) Du, G., Kong, X., Chen, Y., et al. 2015, The Astrophysical Journal, 812, 52
  * Fokker (1965) Fokker, A. D. 1965, Bull. Astron. Inst. Netherlands, 18, 111
  * Giacalone & Neugebauer (2008) Giacalone, J. & Neugebauer, M. 2008, The Astrophysical Journal, 673, 629
  * Gopalswamy et al. (2018) Gopalswamy, N., Yashiro, S., Mäkelä, P., et al. 2018, ApJ, 863, L39
  * Guo & Giacalone (2010) Guo, F. & Giacalone, J. 2010, The Astrophysical Journal, 715, 406
  * Jebaraj et al. (2021) Jebaraj, I., Kouloumvakos, A., Magdalenic, J., et al. 2021, in AGU Fall Meeting Abstracts, Vol. 2021, SM35D–2002
  * Jordan et al. (2017) Jordan, C. H., Murray, S., Trott, C. M., et al. 2017, Monthly Notices of the Royal Astronomical Society, 471, 3974
  * Kansabanik et al. (2022) Kansabanik, D., Mondal, S., Oberoi, D., Biswas, A., & Bhunia, S. 2022, ApJ, 927, 17
  * Kontar et al. (2019) Kontar, E. P., Chen, X., Chrysaphi, N., et al. 2019, The Astrophysical Journal, 884, 122
  * Kontar et al. (2017) Kontar, E. P., Yu, S., Kuznetsov, A. A., et al. 2017, Nature Communications, 8, 1515
  * Kouloumvakos et al. (2021) Kouloumvakos, A., Rouillard, A., Warmuth, A., et al. 2021, ApJ, 913, 99
  * Kumari et al. (2017) Kumari, A., Ramesh, R., Kathiravan, C., & Wang, T. J. 2017, Sol. Phys., 292, 161
  * Kumari et al. (2019) Kumari, A., Ramesh, R., Kathiravan, C., Wang, T. J., & Gopalswamy, N. 2019, ApJ, 881, 24
  * Lemen et al. (2012) Lemen, J. R., Title, A. M., Akin, D. J., et al. 2012, Sol. Phys., 275, 17
  * LEVENBERG (1944) LEVENBERG, K. 1944, Quarterly of Applied Mathematics, 2, 164
  * Loi et al. (2015) Loi, S. T., Trott, C. M., Murphy, T., et al. 2015, Radio Science, 50, 574
  * Lowe & Burgess (2003) Lowe, R. E. & Burgess, D. 2003, Annales Geophysicae, 21, 671
  * Lu et al. (2009) Lu, Q., Hu, Q., & Zank, G. P. 2009, The Astrophysical Journal, 706, 687
  * Magdalenić et al. (2020) Magdalenić, J., Marqué, C., Fallows, R. A., et al. 2020, The Astrophysical Journal, 897, L15
  * Maguire et al. (2020) Maguire, C. A., Carley, E. P., McCauley, J., & Gallagher, P. T. 2020, A&A, 633, A56
  * Maguire et al. (2021) Maguire, C. A., Carley, E. P., Zucca, P., Vilmer, N., & Gallagher, P. T. 2021, The Astrophysical Journal, 909, 2
  * Marquardt (1963) Marquardt, D. W. 1963, Journal of the Society for Industrial and Applied Mathematics, 11, 431
  * McLean (1967) McLean, D. J. 1967, PASA, 1, 47
  * Mohan (2021a) Mohan, A. 2021a, A&A, 655, A77
  * Mohan (2021b) Mohan, A. 2021b, ApJ, 909, L1
  * Mohan et al. (2019a) Mohan, A., McCauley, P. I., Oberoi, D., & Mastrano, A. 2019a, ApJ, 883, 45
  * Mohan et al. (2019b) Mohan, A., Mondal, S., Oberoi, D., & Lonsdale, C. J. 2019b, ApJ, 875, 98
  * Mondal et al. (2019) Mondal, S., Mohan, A., Oberoi, D., et al. 2019, The Astrophysical Journal, 875, 97
  * Mondal & Oberoi (2021) Mondal, S. & Oberoi, D. 2021, ApJ, 920, 11
  * Mondal et al. (2020a) Mondal, S., Oberoi, D., & Mohan, A. 2020a, ApJ, 895, L39
  * Mondal et al. (2020b) Mondal, S., Oberoi, D., & Vourlidas, A. 2020b, ApJ, 893, 28
  * Morosan et al. (2020) Morosan, D. E., Palmerio, E., Räsänen, J. E., et al. 2020, A&A, 642, A151
  * Mouri et al. (2009) Mouri, H., Hori, A., & Takaoka, M. 2009, Physics of Fluids, 21, 065107
  * Murphy et al. (2021) Murphy, P. C., Carley, E. P., Ryan, A. M., Zucca, P., & Gallagher, P. T. 2021, A&A, 645, A11
  * Nelson & Melrose (1985) Nelson, G. J. & Melrose, D. B. 1985, Type II bursts., ed. D. J. McLean & N. R. Labrum, 333–359
  * Neugebauer & Giacalone (2005) Neugebauer, M. & Giacalone, J. 2005, Journal of Geophysical Research: Space Physics, 110
  * Newkirk (1961) Newkirk, Gordon, J. 1961, ApJ, 133, 983
  * Nindos et al. (2011) Nindos, A., Alissandrakis, C. E., Hillaris, A., & Preka-Papadema, P. 2011, A&A, 531, A31
  * Pearson & Readhead (1984) Pearson, T. J. & Readhead, A. C. S. 1984, ARA&A, 22, 97
  * Pesnell et al. (2012) Pesnell, W. D., Thompson, B. J., & Chamberlin, P. C. 2012, Sol. Phys., 275, 3
  * Riddle (1972) Riddle, A. C. 1972, PASA, 2, 98
  * Ryan et al. (2021) Ryan, A. M., Gallagher, P. T., Carley, E. P., et al. 2021, A&A, 648, A43
  * Saito et al. (1970) Saito, K., Makita, M., Nishi, K., & Hata, S. 1970, Annals of the Tokyo Astronomical Observatory, 12, 51
  * Sharma & Oberoi (2020) Sharma, R. & Oberoi, D. 2020, The Astrophysical Journal, 903, 126
  * Smerd et al. (1974) Smerd, S. F., Sheridan, K. V., & Stewart, R. T. 1974, in Coronal Disturbances, ed. G. A. Newkirk, Vol. 57, 389
  * Smerd et al. (1975) Smerd, S. F., Sheridan, K. V., & Stewart, R. T. 1975, Astrophys. Lett., 16, 23
  * Sokolowski et al. (2017) Sokolowski, M., Colegate, T., Sutinjo, A. T., et al. 2017, PASA, 34, e062
  * Stansby (2019) Stansby, D. 2019, dstansby/pfsspy: pfsspy 0.1.2
  * Steinberg et al. (1971) Steinberg, J. L., Aubier-Giraud, M., Leblanc, Y., & Boischot, A. 1971, A&A, 10, 362
  * Thejappa et al. (2007) Thejappa, G., MacDowall, R. J., & Kaiser, M. L. 2007, ApJ, 671, 894
  * Tingay et al. (2013) Tingay, S. J., Goeke, R., Bowman, J. D., et al. 2013, PASA, 30, e007
  * Vandas & Karlický (2011) Vandas, M. & Karlický, M. 2011, A&A, 531, A55
  * Vrsnak, B. et al. (2001) Vrsnak, B., Aurass, H., Magdaleni´c, J., & Gopalswamy, N. 2001, A&A, 377, 321
  * Vršnak & Cliver (2008) Vršnak, B. & Cliver, E. W. 2008, Sol. Phys., 253, 215
  * Zank et al. (2002) Zank, G. P., Zhou, Y., Matthaeus, W. H., & Rice, W. K. M. 2002, Physics of Fluids, 14, 3766
  * Zhang et al. (2021) Zhang, P., Wang, C., & Kontar, E. P. 2021, ApJ, 909, 195
  * Zimovets et al. (2012) Zimovets, I., Vilmer, N., Chian, A. C. L., Sharykin, I., & Struminsky, A. 2012, A&A, 547, A6
  * Zimovets & Sadykov (2015) Zimovets, I. V. & Sadykov, V. M. 2015, Advances in Space Research, 56, 2811
  * Zucca et al. (2018) Zucca, P., Morosan, D. E., Rouillard, A. P., et al. 2018, A&A, 615, A89
  * Zucca et al. (2014) Zucca, P., Pick, M., Démoulin, P., et al. 2014, The Astrophysical Journal, 795, 68