SIR-HUXt - a particle lter data assimilation scheme for assimilating CME time-elongation proles.

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SIR-HUXt - a particle ﬁlter data assimilation

scheme for assimilating CME time-elongation

proﬁles.

Luke Barnard, Mathew Owens, Chris Scott, Matthew Lang, Mike Lockwood

Department of Meteorology, University of Reading, UK

October 6, 2022

Abstract

We present the development of SIR-HUXt, the integration of a sequential im-

portance resampling (SIR) data assimilation scheme with the HUXt solar wind

model. SIR-HUXt is designed to assimilate the time-elongation proﬁles of CME

fronts in the low heliosphere, such as those typically extracted from heliospheric

imager data returned by the STEREO, Parker Solar Probe, and Solar Orbiter

missions.

We use Observing System Simulation Experiments (OSSEs) to explore the

performance of SIR-HUXt for a simple synthetic CME scenario of a fully Earth

directed CME ﬂowing through a uniform ambient solar wind, where the CME is

initialised with the average observed CME speed and width. These experiments

are performed for a range of observer locations, from 20◦to 90◦behind Earth,

spanning the L5 point where ESA’s future Vigil space weather monitor will

return heliospheric imager data for operational space weather forecasting.

We show that, for this CME scenario, SIR-HUXt performs well at constrain-

ing the CME speed, and has some success at constraining the CME longitude.

The CME width is largely unconstrained by the SIR-HUXt assimilations, and

further experiments are required to determine if this is related to the speciﬁc

CME scenario, or is a more general feature of assimilating time-elongation pro-

ﬁles. An analysis of rank-histograms suggests the SIR-HUXt ensembles are well

calibrated, with no clear indications of bias or under/over dispersion. Improved

constraints on the initial CME speed lead directly to improvements in the CME

transit time to Earth and arrival speed. For an observer in the L5 region, SIR-

HUXt returned a 69% reduction in the CME transit time uncertainty, and a

63% reduction in the arrival speed uncertainty. This suggests SIR-HUXt has

potential to improve the real-world representivity of HUXt simulations, and

therefore has potential to reduce the uncertainty of CME arrival time hindcasts

and forecasts.

arXiv:2210.02122v1 [astro-ph.SR] 5 Oct 2022

1 Introduction

Coronal Mass Ejections (CMEs) are large eruptions of magnetised plasma from

the Sun’s atmosphere. CMEs are the primary cause of severe and extreme space

weather at Earth, and so understanding the heliospheric evolution of CMEs, and

forecasting their arrival at Earth receives a signiﬁcant amount of research eﬀort.

Currently, numerical simulations of the real-world heliospheric evolution of coro-

nal mass ejections (CMEs) have signiﬁcant uncertainty. This is partly evidenced

by the large average errors in CME arrival time of ±10 h[Riley et al., 2018].

These uncertainties are caused by a range of factors, but appear to be domi-

nated by uncertainties in the initial and boundary conditions of the solar wind

models, with uncertainty in the ambient solar wind structure and CME parame-

ters introducing similar levels of uncertainty [Riley and Ben-Nun, 2021]. These

uncertainties therefore limit our ability to draw scientiﬁc conclusions from the

simulations of real-world CMEs, and limit the skill of space weather forecasts

of CMEs.

This motivates the pursuit of methods with which to reduce the uncertainty

on numerical simulations of real-world CMEs. Data Assimilation (DA) methods

have excellent potential for improving the representivity of solar wind numerical

models. The objective of DA is to combine information from simulations and

observations to provide an optimal estimate of the state of a dynamical system.

Heliospheric DA is still a relatively new research topic, but progress is beginning

to be made.

[Lang et al., 2017] explored how the Local Ensemble Transform Kalman ﬁl-

ter could be used to assimilate in-situ observations of solar wind plasma prop-

erties into the ENLIL magnetohydrodynamic (MHD) solar wind model, which

demonstrated clear improvements in the representivity of the ENLIL simula-

tions. The Buger Radial Variational Data Assimilation (BRaVDA) scheme was

developed in [Lang and Owens, 2019], in which a variational DA scheme was

coupled to the hydrodynamic (HD) HUX solar wind model [Riley and Lionello, 2011],

for the assimilation of observations of the solar wind speed. Experiments with

synthetic observations and solar wind speed observations from the STEREO

spacecraft showed that BRaVDA reduced the errors in the solar wind speed pre-

dictions at Earth. This work was extended by [Lang et al., 2021] to the HUXt

model, a HD solar wind model with explicit time-dependence [Owens et al., 2020,

Barnard and Owens, 2022], in which it was shown that over the period 2007-

2014, BRaVDA returned a 31% reduction in the RMSE of hindcasts of the solar

wind speed at Earth.

These works have so far focused on the assimilation of in-situ observations

of solar wind plasma properties, but progress has also been made on the assim-

ilation of remote sensing observations, such as those provided by heliospheric

imagers (HIs) [Eyles et al., 2008, Howard et al., 2008] and interplanetary scin-

tillation (IPS) [Fallows et al., 2022]. For example, [Barnard et al., 2020] showed

that an ensemble of solar-wind-CME simulations with the HUXt model could be

weighted by the time-elongation proﬁles of CMEs derived from the STEREO

Heliospheric Imager (HI) data. This weighting prioritised ensemble members

that more closely matched the observed time-elongation proﬁle, and led to up

to 20% improvements in hindcasts of the CMEs arrival time at Earth. Simi-

larly, [Iwai et al., 2021] demonstrated how assimilating Interplanetary Scintilla-

tion (IPS) observations of 12 halo CMEs into the SUSANOO-CME MHD model

led to improvements in the predicated Earth arrival times of these CMEs.

Although [Barnard et al., 2020] demonstrated that HI data contains useful

information on CMEs that can be used to constrain the HUXt solar wind sim-

ulations, they did not use formal DA methods. In this work, we present the

development of SIR-HUXt, which couples a sequential importance resampling

(SIR) particle ﬁlter DA scheme with the HUXt solar wind model. SIR-HUXt

is constructed to assimilate time-elongation proﬁles of a CMEs ﬂank, such

as those typically extracted from the STEREO-HI data [Davies et al., 2009,

Barnard et al., 2015, Barnard et al., 2017]. This is an important milestone to-

wards the development of DA schemes that can directly assimilate the HI inten-

sity data into solar wind numerical models. We present a ﬁrst test of SIR-HUXt

by using Observing System Simulation Experiments (OSSEs) to investigate the

performance of SIR-HUXt for a simple synthetic CME scenario and for a range

of observer locations relative to Earth.

This article proceeds with section 2 describing the models and methods we

use, including the HUXt numerical model, the background to the SIR algorithm,

and on OSSEs. Section 3 presents the results of the OSSEs, and our conclusions

are presented in section 4.

2 Methods and Data

2.1 HUXt

HUXt is an open source numerical model of the solar wind, developed in Python

[Owens et al., 2020, Barnard and Owens, 2022]. It is a 1D radial model that

uses a reduced-physics approach to produce solar wind simulations that emulate

the solar wind ﬂows produced by 3-D MHD models, but at a small fraction of

the computational cost.

The motivation for developing HUXt is that the models simplicity and com-

putational expense permits the development of certain experiments and tech-

niques that would typically be too expensive with 3-D MHD models. For ex-

ample, the particle ﬁlter data assimilation experiments in this study require

≈1065-day simulations of the inner heliosphere, which is currently an imprac-

tical demand of 3-D MHD solar wind models with widely available computing

resources.

In this work, HUXt is run with its default conﬁguration. The radial grid

spans 30 Rto 240 R, with a grid step of 1.5R. The time-step is 8.7 minutes.

There are 128 evenly space longitudinal bins, although to save on computation,

and as we are only examining Earth-directed CMEs, the simulation domain only

spans the longitude range of ±70◦.

CMEs are included in HUXt via the Cone CME parameterisation, in which

CMEs are represented as a time-dependent velocity perturbation to the model

inner boundary. Six parameters are required to specify the initiation of a Cone

CME; the initiation time; the speed; the angular width; the source longitude

and latitude; and the radial thickness of the perturbation. Further details

of the Cone CME parameterisation in HUXt are given in [Owens et al., 2020,

Barnard et al., 2021, Barnard and Owens, 2022]. CMEs are tracked through

HUXt simulations by inserting test particles into the ﬂow on the CME surface

at the model inner boundary. These test particles then passively advect with

the ﬂow and are followed at all time steps out to the model’s outer boundary.

Pseudo-observers are used with HUXt to compute the time-elongation pro-

ﬁle of the Cone CME ﬂank, to emulate the time-elongation data products typ-

ically derived from Heliospheric Imager observations e.g. [Davies et al., 2009,

Barnard et al., 2015, Barnard et al., 2017, Pant et al., 2016]. This is achieved

by computing the elongation of each particle on the CME boundary and ﬁnd-

ing the particle with maximum elongation in an observer’s ﬁeld of view. A

better solution would be to forward model the observations from Heliospehric

Imager instruments by performing Thomson scattering simulations with HUXt

output. However, the HUXt equations are derived from incompressible hydro-

dynamics, and so only the ﬂow speed is solved for, not the ﬂow mass density.

This prohibits a fully self-consistent forward modelling of Heliospheric Imager

data from HUXt simulations. Consequently, tracking the maximum elonga-

tion of the CME tracer particles is a necessary approximation. However, both

[Barnard et al., 2020] and [Chi et al., 2021] showed that this approach returned

time-elongation proﬁles that compared favourably to those extracted directly

from STEREO-HI images, which gives us conﬁdence this approximation is rea-

sonable.

2.2 Sequential Importance Resampling (SIR)

The objective of data assimilation is to provide an optimal estimate of the state

of a system by combining the information from both a model and observations

of the system, taking proper account of the uncertainties on each.

This can be expressed mathematically via Bayes’ theorem, which states that,

p(ψ|θ) = p(θ|ψ)p(ψ)

p(θ).(1)

The factors in this equation are typically separated into into several collo-

quially named terms. The ”prior”, p(ψ), is the probability density of the model

being in a speciﬁc state, in the absence of any other external information. The

”likelihood”, p(θ|ψ), which is the probability density of obtaining a set of ob-

servations θ, given a model state ψ. The ”evidence”, p(θ), is the probability

density of obtaining a set of observations although in most practical examples

the evidence becomes a normalising constant that can be ignored. Finally, the

”posterior”, p(ψ|θ), is the conditional distribution of model states given a set of

observations.

Computation of the posterior, or approximations to it, is the focus of data

assimilation. The posterior provides the optimal estimate of the state of the sys-

tem, representing the distribution of model states that are most consistent with

the observations. In practical geophysical examples, it is not possible to fully

characterise the posterior distribution, and diﬀerent data assimilation method-

ologies are used to infer certain properties of the posterior e.g. its mean, mode,

or variance [Le Dimet and Talagrand, 1986, Burgers et al., 1998]. A particle

ﬁlter is set of a data assimilation methodologies that aims to approximate the

full posterior distribution via an ensemble of ”particles” [Van Leeuwen, 2009,

Chorin and Tu, 2009, Ades and van Leeuwen, 2013, Browne and van Leeuwen, 2015,

Fearnhead and K¨unsch, 2017, Potthast et al., 2019].

Sequential Importance Resampling (SIR) is a method of particle ﬁltering

that can be used for sequential data assimilation [Van Leeuwen, 2009, Fearnhead and K¨unsch, 2017].

In SIR, the posterior is approximated by the analysis of an ensemble of simu-

lations, or ”particles”. The prior is approximated by generating an ensemble

of simulations that reﬂects the uncertainty in the models initial and boundary

conditions. The model evolves the ensemble forward in time, until a set of

observations are available. At the observation time, an analysis is performed

which weights each simulation in accordance with its agreement with the ob-

servations. Then, this weighted ensemble is used to generate a new ensemble

of simulations which are closer to the observations. The model then resumes

advancing the simulations forward in time, until the next set of observations

are available. The data assimilation proceeds in this way, performing sequential

analysis steps when observations are available. The posterior distribution, at

some speciﬁc time, is approximated by the distribution of the ensemble after an

analysis step.

In this work we develop SIR-HUXt, a coupling of an SIR scheme with the

HUXt solar wind model, with the objective of assimilating the time-elongation

proﬁles of CME fronts, such as those that can be derived from white light

heliospheric imaging. SIR-HUXt essentially functions as a form of parameter

estimation, returning estimates of the posterior of the Cone CME parameters

that are most consistent with the observed the time-elongation proﬁle of a CME.

The following subsections describe the speciﬁcs of the SIR algorithm used in

SIR-HUXt.

2.2.1 Initial Ensemble Generation

The initial ensemble is generated by perturbing a subset of the Cone CME

parameters only, following a similar method to [Barnard et al., 2020]. Specif-

ically, perturbations are applied to the Cone CME speed, angular width, and

longitude. We focus on these three parameters only as they are probably the

most important parameters for determining if and when a CME impacts Earth

[Pizzo et al., 2015, Riley and Ben-Nun, 2021], whilst considering all the Cone

CME parameters would be too computationally expensive for this proof-of-

concept study.

The random perturbations for each parameter are drawn from a uniform dis-

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SIR-HUXt-aparticlelterdataassimilationschemeforassimilatingCMEtime-elongationproles.LukeBarnard,MathewOwens,ChrisScott,MatthewLang,MikeLockwoodDepartmentofMeteorology,UniversityofReading,UKOctober6,2022AbstractWepresentthedevelopmentofSIR-HUXt,theintegrationofasequentialim-portanceresampling(SIR)d...

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