# Physics-based model of solar wind stream interaction regions: Interfacing
between Multi-VP and 1D MHD for operational forecasting at L1

R. Kieokaew Institut de Recherche en Astrophysique et Planétologie, UPS, CNES,
CNRS, Toulouse, France<EMAIL_ADDRESS>R. F. Pinto 11footnotemark: 1
Departement d’Astrophysique/AIM, CEA/IRFU, CNRS/INSU, Univ. Paris-Saclay &
Univ. de Paris, Gif-sur-Yvette, France E. Samara CmPA, KU Leuven, Leuven,
Belgium C. Tao National Institute of Information and Communications Technology
(NICT), Koganei, Tokyo, Japan M. Indurain11footnotemark: 1 B. Lavraud
11footnotemark: 1 Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux,
CNRS, Pessac, France A. Brunet ONERA/DPHY, Université de Toulouse, F-31055
Toulouse, France V. Génot 11footnotemark: 1 A. Rouillard 11footnotemark: 1 N.
André 11footnotemark: 1 S. Bourdarie 77footnotemark: 7 C. Katsavrias
Department of Physics, National and Kapodistrian University of Athens, 15784
Athens, Greece F. Darrouzet Royal Belgian Institute for Space Aeronomy (IASB-
BIRA), Brussels, Belgium B. Grison Institute of Atmospheric Physics CAS, Dept
of Space Physics, Prague, Czech Republic I. Daglis 88footnotemark: 8

###### Abstract

Our current capability of space weather prediction in the Earth’s radiation
belts is limited to only an hour in advance using the real-time solar wind
monitoring at the Lagrangian L1 point. To mitigate the impacts of space
weather on telecommunication satellites, several frameworks were proposed to
advance the lead time of the prediction. We develop a prototype pipeline
called “Helio1D” to forecast ambient solar wind conditions (speed, density,
temperature, tangential magnetic field) at L1 with a lead time of 4 days. This
pipeline predicts Corotating Interaction Regions (CIRs) and high-speed streams
that can increase high-energy fluxes in the radiation belts. The Helio1D
pipeline connects the Multi-VP model, which provides real-time solar wind
emergence at $0.14$ AU, and a 1D MHD model of solar wind propagation. We
benchmark the Helio1D pipeline for solar wind speed against observations for
the intervals in 2004 - 2013 and 2017 - 2018. We developed a framework based
on the Fast Dynamic Time Warping technique that allows us to continuously
compare time-series outputs containing CIRs to observations to measure the
pipeline’s performance. In particular, we use this framework to calibrate and
improve the pipeline’s performance for operational forecasting. To provide
timing and magnitude uncertainties, we model several solar wind conditions in
parallel, for a total of 21 profiles corresponding to the various virtual
targets including the Earth. This pipeline can be used to feed real-time,
daily solar wind forecasting that aims to predict the dynamics of the inner
magnetosphere and the radiation belts.

_Keywords_ Space weather $\cdot$ solar wind $\cdot$ corotating interaction
region

##  1 Introduction

The Sun and the solar wind are the main drivers for space weather at Earth and
in the heliosphere. Nowadays, forecasting solar wind conditions becomes a
crucial task as the human society is increasingly dependent on space
technologies that are under the influence of space weather. The major drivers
that can have profound effects are the Coronal Mass Ejections (CMEs), caused
by solar eruptions, and Corotating Interaction Regions (CIRs), formed when a
fast solar wind or high-speed stream (HSS) from the coronal holes takes over a
slower solar wind. Although CMEs cause more severe events, their effects often
disappear in a day or two while CIR/HSS events have significant effects that
last several days after their arrivals (e.g. Burns et al., 2012). During solar
minima when CMEs are absent, CIRs cause low- to intermediate-strength
geomagnetic storms and occasionally lead to strong geomagnetic storms (e.g.
Richardson et al., 2001; Alves et al., 2006; Tsurutani et al., 2006; Zhang et
al., 2008; Kilpua et al., 2017; Chi et al., 2018). These CIR-driven storms
also have impacts down to the ionosphere and thermosphere, leading to
equatorial ionization anomaly (e.g. Oyedokun et al., 2022) and total electron
content variability (e.g. Chakraborty et al., 2020). Furthermore, CIRs impact
the dynamics of CMEs (e.g. Liu et al., 2019), broaden solar energetic particle
spectrum (e.g. Zhao et al., 2016; Wijsen et al., 2019), and modulate galactic
cosmic rays (e.g. Iucci et al., 1979; Rouillard and Lockwood, 2007). Cranmer
et al. (2017); Kilpua et al. (2017); Vršnak et al. (2017); Richardson (2018)
and Yermolaev et al. (2018) comprehensively reviewed our current physical
understanding and the geoeffectiveness of CIRs.

Communication satellites in the geostationary orbit and medium and low Earth
orbits are in the vicinity of the Earth’s radiation belts - the regions
encircling the near-Earth space and containing significant fluxes of high-
energy electrons and ions. CIRs can drive radiation belt electron variability
(e.g. Lam et al., 2009; Hudson et al., 2021), leading to electron flux
enhancements (e.g. Blake et al., 1997; Li et al., 1997); electron acceleration
especially in the presence of strong southward Interplanetary Magnetic Field
(IMF Bz) component (Blake et al., 1997), in which in certain cases reaching
relativistic level (Paulikas and Blake, 1979; Pinto et al., 2018; Hudson et
al., 2021; Pandya et al., 2019). The high speed streams that follow CIRs have
been demonstrated to be more effective (compared to CMEs) in producing multi-
MeV electron enhancements up to more than 7 MeV (Horne et al., 2018;
Katsavrias et al., 2019). This is due to fact that they produce intervals of
combined southward IMF and solar wind velocity over 500 km/s, which lead to an
enhanced magnetic reconnection rate at the dayside magnetopause (Borovsky and
Denton, 2006; Miyoshi et al., 2013).

Since the beginning of the space technology era, there have been a number of
reports of spacecraft anomalies and even failures, for example, at the
geostationary orbit due to the elevated fluxes of several MeV electrons that
persisted for several days following CIRs (see a review by Lanzerotti, 2007).
To better mitigate impacts on those satellites, we need to develop a system
for continuously predicting the solar wind and radiation belt conditions. With
the real-time monitoring of the solar wind at the Lagrangian L1 point, e.g. by
Advanced Composition Explorer (ACE), the continuous, real-time prediction of
the radiation belt conditions via modeling is limited to one hour in advance
or less as the solar wind takes about 40 - 60 minutes from the ACE spacecraft
to arrive at the magnetosphere. Using the Kp that is a 3-hour index indicative
of global auroral activities, the SPACECAST (Horne et al., 2013) model, for
instance, was used to provide a forecast up to 3 hours in advance based on the
real-time ACE data. To further improve our current capability, we need
continuous solar wind prediction with a significant lead time.

Numerous efforts have been made in modeling real-time solar wind conditions at
Earth. The majority of models consists of two main parts: (1) a young solar
wind emergence modeling and (2) a solar wind modeling and/or propagation from
near Sun to Earth. The former part is initiated by a reconstruction of the
coronal magnetic fields based on synoptic or syncronic maps from ground
observations; this is achieved by employing a simplified analytical model with
a magnetostatic potential field source surface (PFSS) extrapolation. The PFSS
extrapolation combined with the Wang-Sheeley-Arge (WSA) model is a widely-used
predictor of the solar wind in the corona/low heliosphere (Arge and Pizzo,
2000; Arge et al., 2004). In combination with the global 3-D
magnetohydrodynamics (MHD) “ENLIL” — the physics-based solar wind model
(Odstrčil et al., 1996; Odstrcil and Pizzo, 1999; Odstrcil et al., 2004) — the
WSA-ENLIL has been used to provide daily forecasts of solar-wind streams. The
WSA-ENLIL tool has also been extended to include CMEs using an ad-hoc
hydrodynamic cone model or the spheromak model (Taktakishvili et al., 2009,
2011, 2018). The Space Weather Prediction Centre of the National Oceanic and
Atmospheric Administration (SWPC/NOAA) has put the WSA-ENLIL into operation to
provide daily solar wind predictions with a lead time of 1 - 4 days (Pizzo et
al., 2011). Alternative to the WSA model is the MHD Around a Sphere (MAS) that
is a global, time-dependent model based on the resistive MHD equations (Linker
et al., 1999; Riley et al., 2012). The WSA-ENLIL and the MAS-ENLIL tools have
been extensively benchmarked by Owens et al. (2005); Lee et al. (2009);
MacNeice (2009a, b); Jian et al. (2011a, b, 2015). For the solar-wind
propagation part, several less-numerically intensive options are also
available, though they are not fully put into operation at the time of this
writing. A few models include Heliospheric Upwind eXtrapolation (HUX) (Riley
and Lionello, 2011), ballistic propagation (Dósa et al., 2018), and Tunable
HUX (Reiss et al., 2020). MacNeice et al. (2018) and Reiss et al. (2019, 2020)
compiled several combinations of the existing models and established
frameworks for cross-validation.

Recently, more efforts have been made to combine and benchmark European-based
models to develop operational solar-wind forecasting services. The Multi-VP is
a coronal model developed by Pinto and Rouillard (2017) that computes multiple
solutions of 1D solar wind flux tubes in sub-domains of interest. This
approach allows a much faster calculation compared to global models with
better accuracy at regions of interest, e.g. at the sub-Earth point, with a
lead time of 1 – 3 days (Rouillard et al., 2020). The Multi-VP model has been
coupled with EUHFORIA (European Heliospheric Forecasting Information Asset;
Pomoell and Poedts, 2018) and validated for HSS modeling in comparison with
WSA-EUHFORIA (Samara et al., 2021). In addition, the overall performance of
solar wind modeling with EUHFORIA has been assessed by Hinterreiter et al.
(2019) and Samara et al. (2022). Most recently, the Heliocast — a global MHD
model comprising both coronal and heliospheric parts — has been developed by
Réville et al. (under revision in JSWSC at the time of this writing) to
provide a daily solar wind forecast with a lead time up to 5 days for the
French organization for applied research in space weather111OFRAME, see
http://www.meteo-espace.fr/.. Moreover, an empirical solar wind forecast
(ESWF; Vršnak et al., 2007; Milošić et al., 2022) model for CIR/HSS prediction
with a lead time of 4 days has been developed using an empirical relation
found between the areas of coronal holes as observed in Extreme Ultraviolet
data and the solar wind speed at 1 AU (Robbins et al., 2006; Vršnak et al.,
2007). This model has been put into operation by the European Space Agency.

In this work, we consider the coupling of the Multi-VP model and a 1D MHD
model (Tao et al., 2005) for the first time to develop an automated pipeline
named “Helio1D” for solar wind CIR forecasting. The 1D MHD model was
originally developed for modeling solar wind conditions at Jupiter using the
in situ observations at Earth as inner boundary; it was later extended to
provide operational service for solar wind conditions at other solar-system
planets (André et al., 2018), see HelioPropa222http://heliopropa.irap.omp.eu/.
Here, we model multiple solar wind solutions at several targets around the
sub-Earth point in parallel. In essence, we develop a pipeline for daily solar
wind forecasting at Earth 4 days ahead with a systematic characterization of
timing and amplitude uncertainties. Using 13 years of data covering parts the
solar cycles 23 and 24, from 2004 to 2013 and from 2017 to 2018, we benchmark
the pipeline results as well as calibrate it to develop a prototype for
operational daily solar wind forecasting. The pipeline is integrated into the
operational forecasting service prototype on radiation belt environmental
indicators for the safety of space assets as a part of the European Union
Horizon 2020 SafeSpace333https://www.safespace-h2020.eu/ project.

The evaluation or validation of modeling results is crucial to understand the
efficiency of the forecast. Several works considered standard metrics such as
root mean square, mean square error, and Pearson correlation coefficient to
compare modeling results to the observations. With such metrics, a skill score
is also computed to assess whether the model performs better than a baseline
model. For CIR/HSS and CME modeling, some works also considered event-based
verification that allow the characterization of a true positive (predicted and
observed), a false positive (predicted but not observed), a false negative
(not predicted but observed) to construct contingency tables (Woodcock, 1976)
and compute the probability of detection (e.g. Reiss et al., 2016). Most
recently, an alternative approach using the Dynamic Time Warping (DTW)
algorithm, a technique commonly employed in time-series analysis, was applied
to characterize the performance of solar wind modeling (Samara et al., 2022).
We consider these metrics and approach for verification of the pipeline’s
results.

The paper is organized as follows. Firstly, we introduce the Multi-VP and 1D
MHD models and then describe the interfacing between them for the Helio1D
pipeline. Secondly, we describe metrics and techniques for quantifying the
performance of Helio1D. Thirdly, we provide results from the pipeline with the
assessments of the results using the classic metrics and the DTW algorithm.
Fourthly, we discuss the pipeline calibration with a newly developed approach
with the DTW algorithm. Finally, we discuss operational forecasting and
provide conclusions and prospects.

##  2 Automated CIR forecasting pipeline

We develop a pipeline to connect the solar wind forecasting from near-Sun to
Earth. Here, we briefly introduce the models for the solar wind formation
(Multi-VP) and solar wind propagation (1D MHD). We then describe the
interfacing of the two models for the automated Helio1D pipeline.

###  2.1 Multi-VP

MULTI-VP (Pinto and Rouillard, 2017) is a coronal model that covers the
heliocentric distances over which all solar wind streams are formed and
accelerated: between 1 and about 30 solar radii ($R_{\odot}$). The main
advantages are that it is data-driven, while taking into account the
thermodynamics of the wind flows across the highly stratified low solar
atmosphere, and hence it produces physically correct and realistic estimations
of the state of the solar wind across its domain. MULTI-VP determines a full
set of physical quantities consisting of solar-wind speed ($\mathbf{V}$),
density ($n$), temperature ($T$), magnetic field ($\mathbf{B}$), and secondary
quantities without requiring empirical scaling laws. In addition, MULTI-VP
actually computes a set of individual wind streams from surface to high corona
that can then be put together to build the full three-dimensional solar wind
solutions from multiple 1D solar-wind flux tubes. This brings an enormous
advantage with respect to more traditional 3D MHD models in terms of required
computing time, and also by not being subject to the strong diffusion of the
gradients in the transverse directions. In practice, it also lets us compute
solar wind solutions restricted to certain regions of interest rather than in
the full solar atmosphere.

For the 1D MHD model, MULTI-VP is required to produce time-series of the solar
wind properties at the sub-Earth point at 0.14 AU (at 30 $R_{\odot}$).
Internally, this output is defined via a data object that lists which magnetic
field lines (and hence individual wind streams) are sampled, as well as their
coordinates and geometry, and properties of the input magnetogram. Each time a
forecast is performed, we select a number of positions around the sub-Earth
point (i.e. at 0.14 AU in Sun-Earth line) stretching up to 15 degrees in
azimuth and 15 degrees in latitude. The spread in latitude is aimed to cover
positional errors propagated from the magnetic field reconstruction. The
spread in longitude is aimed to cover the azimuthal range that corresponds to
an elemental time-series three days of solar rotation with respect to Earth,
which translates to one day behind and two days ahead of the modeling at sub-
Earth point. The next forecast proceeds in the same way, producing another
elemental time-series that partially overlaps the previous in time. This
procedure is repeated to continuously build a long-term time-series that can
be used as an input to the 1D MHD model.

####  2.1.1 Multi-VP data

In order to characterize the pipeline performance with regard to various
phases of the solar cycle, we need sufficiently long datasets. The longest
historical magnetogram series available is that from the Wilcox Solar
Observatory (WSO). Here, we obtain mainly two long datasets from Multi-VP
processed using the WSO magnetograms. The first set extends from Carrington
Rotation (CR) 2024 to CR 2139, corresponding to data from December 2004 to
August 2013. This 8 year and 7 month-long interval covers the declining phase
(2004 – 2008) and the solar minimum (2009 – 2010) of solar cycle 23 as well as
the ascending phase (2011 – 2013) of solar cycle 24. The second set extends
from CR 2192 to CR 2210 and corresponds to data from June 2017 to November
2018. These two datasets are fed into the 1D MHD model.

###  2.2 1D MHD model

From $0.14$ AU (30 $R_{\odot}$) onwards, the solar wind is propagated through
the heliosphere using a 1D MHD model (Tao et al., 2005) that takes the solar
wind time-series as time-varying boundary conditions, the solar wind
parameters — mainly the radial plasma velocity and tangential velocity — vary
as a function of time (see more below). The 1D MHD model propagates the solar
wind using an ideal MHD fluid approximation to the target position while
taking into account the interaction between fast and slow streams in the
radial direction. The code solves the ideal MHD equations under the influence
of solar gravity in a one-dimensional spherically symmetric coordinate system.
The 1D MHD equations are solved using the Coordinated Astronomical Numerical
Software (CANS)444developed by T. Yokoyama at the University of Tokyo and
collaborators, see documentation at
http://www.astro.phys.s.chiba-u.ac.jp/netlab/pub/index.html., an early version
of CANS+ code (Matsumoto et al., 2019). The 1D MHD code was developed by Tao
et al. (2005) to propagate the solar wind observations at Earth to upstream of
Jupiter, and was further extended by the French plasma physics data
center555Centre de Données de Physique des Plasmas, France. See
http://heliopropa.irap.omp.eu/. to cover other solar-system planets or target
spacecraft positions (André et al., 2018). The code is robust and widely used
for solar wind propagation and CIR modeling in the heliosphere (e.g. Palmerio
et al., 2021; Nilsson et al., 2022).

The MVP output at 30 $R_{\odot}$ were used as boundary conditions for the
initiation of the 1D-MHD code, which then propagated these results up to the
L1 point at 1 AU. The coordinate system in the 1D MHD code is equivalent to
the Radial-Tangential-Normal (RTN) system, where the $X$-axis ($R$) is
pointing radially outward from the Sun in the equatorial plane to the Earth,
the $Z$-axis ($N$) is the solar rotation axis, and the $Y$-axis ($T$)
completes the orthonormal system. The outer boundary is set to 1.4 AU where
the derivatives of all physical parameters diminish. The grid spacing and the
time step are chosen to be $(1.4-0.14)/400$ AU and $150$ s, respectively. To
adopt the 1D simulation, we keep the tangential magnetic field $B_{y}$
component to physical values while the $B_{x}$ is fixed to 0.001 nT to meet
the solenoidal criterion, and the $B_{z}$ is set to zero (see Tao et al.
(2005) for discussion). We retrieve Multi-VP data at 30 minute cadence and
resample them to 1 hour cadence; these hourly-averaged data are then linearly
interpolated to meet the CFL (Courant-Friedrichs-Lewy) condition for every
time step.

###  2.3 Helio1D - Interfacing Multi-VP and 1D MHD

To automate the interfacing between the two models, several steps are taken.
Multi-VP provides solar wind time series with a full set of physical
parameters including plasma number density, velocity, temperature, magnetic
field in the RTN coordinates. We first format the time series to fit the input
format of the 1D MHD model. Here, we take the input at a 1-hour time
resolution and output the propagated time series at L1 with the same 1-hr
resolution. To successfully run the 1D MHD code, two criteria are required to
be met: (1) the data length must cover at least a solar rotation, i.e., 27.25
days, in order to produce a consistent CIR formation, and (2) the parameter
values must be physical (e.g. non-negative plasma number density). In certain
conditions, the Multi-VP model may provide unphysical solar wind values due to
the poor quality of the magnetograms. We apply a set of criteria to remove
unphysical values (see Appendix) and fill gaps using linear interpolation
between two available data points. These aspects are further described in
Section 6.

We set up the Helio1D pipeline with two modes for running (a) long-term
historical data consisting of several Carrington rotations and (b) short-term
data of 1-month long for daily, operational forecasting as shown in Fig 1. The
mode-(a) allows us to quantify the performance of the pipeline as well as
identify calibrations that may improve the pipeline performance for
operational forecasting. We first benchmark the pipeline using long-term
Multi-VP data in Section 4.

Figure 1: Diagram of the Helio1D pipeline.

##  3 Measuring the performance of Helio1D

In this section, we first introduce the metrics and methodology for
characterizing the pipeline performance. We then give results and discuss
calibration for the pipeline.

###  3.1 Standard metrics

We choose three standard metrics - root mean square error (RMSE), mean square
error (MAE), and Pearson correlation coefficient ($r$) - to measure the
performance of the pipeline. RMSE is a classic point-by-point metric that
compares the modeled value and observed value for a given point. It is defined
as

$RMSE=\sqrt{\frac{1}{T}\Sigma_{t=1}^{T}\left[V_{m}(t)-V_{o}(t)\right]^{2}}$
(1)

where $V_{m}(t)$ and $V_{o}(t)$ are the modeled and observed values,
respectively, at a given time $t$ while $T$ is the number of time elements.
Mean absolute error (MAE) is an alternative metric that measures a similar
property. It is defined as

$MAE=\frac{1}{T}\Sigma_{t=1}^{T}|V_{m}(t)-V_{o}(t)|$ (2)

where the symbols have the same meanings as defined for the RMSE. The units of
the RMSE and MAE are the same as the unit of parameter values $V_{m}(t)$ and
$V_{o}(t)$. Smaller values of RMSE and MAE indicate better model prediction
when compared to observations. Zero values of both metrics indicate a perfect
model prediction.

We also introduce the Pearson correlation coefficient (Pcc) to roughly
quantify the similarities between the two time series. In principal, Pearson
correlation measures the linear relationship between two datasets. Using the
observed and modeled time series ($V_{o}(t)$ and $V_{m}(t)$, respectively),
the formula for $Pcc$ is

$Pcc=\frac{\Sigma_{t=1}^{T}\left(V_{m}(t)-\langle
V_{m}(t)\rangle\right)\left(V_{o}(t)-\langle
V_{o}(t)\rangle\right)}{\sqrt{\Sigma_{t=1}^{T}\left(V_{m}(t)-\langle
V_{m}(t)\rangle\right)^{2}}\sqrt{\Sigma_{t=1}^{T}\left(V_{o}(t)-\langle
V_{o}(t)\rangle\right)^{2}}}.$ (3)

The $Pcc$-value ranges from $-1.0$ to $1.0$, with a value of $1.0$ being a
perfect positive linear correlation between the two time series and $-1.0$
being a perfect negative linear correlation between the two.

Additionally, to evaluate the model performance in a more concrete way using
above metrics, we introduce the skill of a forecast as

$Skill=1-\frac{MSE_{pred}}{MSE_{ref}}$ (4)

where $MSE_{pred}$ is the mean squared error of the prediction compared to the
observation and $MSE_{ref}$ is the mean squared error of the prediction
compared to a reference baseline model. Here, we will use the average of the
observation as a reference baseline model (i.e., the climatological mean in
climate studies). The observations of solar wind data at L1 are taken from the
High Resolution OMNIweb database (King and Papitashvili, 2005). The average of
the OMNI data of the corresponding interval are taken as a reference baseline
model. Skill score equal to unity means a perfect forecast while a zero skill
means that the model performs the same as the baseline model. On the other
hand, a negative skill means that the model performs worse than the baseline
model. The aforementioned metrics will be used throughout this report for the
quantification of the performance of the model.

###  3.2 Dynamic Time Warping Technique

Although the classic metrics — RMSE, MAE, skill score, and $Pcc$ — provide
quantification of the performance of the model, they do not provide in-depth
information such as time lags, i.e., the difference in arrival times between
the modeled and observed stream interfaces, that are crucial for model
evaluation. For this reason, we also consider an alternative technique called
Dynamic Time Warping (DTW). The DTW technique compares two sequences by
finding an optimal alignment by which one sequence may be stretched or
compressed (hence, “warped”) in the temporal domain to match the other under
certain constraints (e.g., Müller, 2007). The technique was first introduced
by Bellman and Kalaba (1959) in adaptive control processes and later found
several applications notably in speech and pattern recognition (e.g., Sakoe
and Chiba, 1978; Myers et al., 1980). More recently, DTW has been applied in
space weather in particular for comparing modeled geomagnetic indices and
solar wind data to observations (e.g., Laperre et al., 2020; Owens and
Nichols, 2021; Bunting and Morgan, 2022; Nilsson et al., 2022; Samara et al.,
2022). We first introduce its formulation and then describe the methodology
for our application.

The DTW technique finds the warping path between two sequences, e.g.,
$X=\\{x_{1},x_{2},...,x_{|X|}\\}$ and $Y=\\{y_{1},y_{2},...,y_{|Y|}\\}$,
through the DTW cost matrix by choosing the path that minimizes the total
cumulative cost compared to all other paths. The DTW cost matrix is defined
as:

$D(i,j)=\delta(i,j)+min\\{D(i-1,j-1),D(i-1,j),D(i,j-1)\\}$ (5)

where $i,j$ are the indices of $X$ and $Y$, respectively, and
$\delta(i,j)=|x_{i}-y_{j}|$ is the Euclidean distance between the element
$x_{i}$ of series $X$ and the element $y_{j}$ of series $Y$. Here, the optimal
warping path (i.e., alignment) can be found via back-tracing in the DTW cost
matrix; this process is carried out by choosing the previous points with the
minimum cumulative distance (e.g., Berndt and Clifford, 1994; Keogh and
Pazzani, 2001; Górecki and Łuczak, 2013). A main disadvantage of DTW is that
an element $x_{i}$ may be mapped to several elements of $Y$ — this problem is
called “pathological mapping” or “singularities” (e.g., Sakoe and Chiba,
1978). To alleviate this issue, certain constraints have been added such as
the so-called windowing, slope weighting, and step patterns (see Keogh and
Pazzani, 2001, and references therein). Various variations of DTW technique
have also been developed to optimize the alignments between the sequence
elements/points (e.g. Keogh and Pazzani, 2001; Chu et al., 2002; Keogh and
Ratanamahatana, 2005; Salvador and Chan, 2007; Efrat et al., 2007; Furtună,
2008; Zhu et al., 2012; Yadav and Alam, 2018, and references therein).

Choosing an appropriate DTW technique is crucial as different algorithms can
lead to different results. Here, we choose the FastDTW algorithm developed by
Salvador and Chan (2007) due to its multi-level approach. First, the time
series are sampled down to a very coarse resolution. A warp path is found in
this coarse resolution and projected onto a higher resolution. The warp path
is then refined and projected again to a higher resolution. The process of
projecting and refining is repeated until a warp path is found for the
original resolution time series. Most importantly, since the technique
initially finds a warp path in the coarse resolution, it is particularly
appropriate to capture large-scale features in the time series, notably “CIRs”
with large speed gradients.

To compute a metric similar to the skill score using the (classic) DTW
technique, Samara et al. (2022) defined the Sequence Similarity Factor (SSF)
to quantify the similarity between the modeled and observed time series as:

$\displaystyle SSF=\frac{DTW_{score}(O,M)}{DTW_{score}({O,\langle
O\rangle})},$ (6)

where $DTW_{score}(O,M)$ is the DTW cost calculated between the observed ($O$)
and modeled ($M$) time series, and the $DTW_{score}({O,\langle O\rangle})$ is
the DTW cost between the observed series and the baseline. Here, the baseline
is given as the average of the observed time series ($\langle O\rangle$). The
SSF is zero when the model forecast is perfect; it equals to unity when the
forecast performs exactly the same as the baseline; and it is higher than
unity when the forecast is worse than the baseline.

The DTW cost (equation 5) calculated between two sequences is the sum of the
length (i.e., the Euclidean distance) between all the optimal alignment pairs.
Therefore, the existence of singularities, where a data point $x_{i}$ has
several possible optimal alignments to $y_{j}$, increases the DTW cost. The
baseline is an average of the observed time series — it is a constant that is
not varied with time for a considered interval. The DTW mapping between a
model with time variation and a constant sequence (i.e., a straight line)
would have several singularities. As a result, we cannot directly compare the
DTW cost between two sequences with time variation to the DTW cost between a
time-varying sequence and the (constant) baseline. For this reason, we propose
two new metrics as follows. First, a normalized cost or normalized distance
can be defined as the ratio of the DTW cost to the number of all alignment
pairs. This quantity has the unit of the quantity under consideration (similar
to RMSE and MAE). Its value corresponds to the average distance of all the
mapped alignments that encodes both temporal and spatial differences between
the two sequences; we call this quantity “DTW normalized distance”. The
normalized SSF is then defined as:

${SSF}_{normalized}=\frac{{DTW}_{score}(O,M)/{DTW}_{length}(O,M)}{{DTW}_{score}({O,\langle
O\rangle})/{DTW}_{length}({O,\langle O\rangle})},$ (7)

where ${DTW}_{length}$ represents the number of all the alignments between two
sequences. When the ${SSF}_{normalized}$ is smaller than unity, the forecast
model performs better than the baseline. When it is less than unity, the
forecast model performs worse than the baseline. We consider these two newly
defined metrics besides the classic metrics in Section 4.1.

##  4 Results

###  4.1 Performance of Helio1D using the long-term data

Table 1: List of the Helio1D long-series intervals and the point-by-point metrics calculated using the Helio1D series in comparison with OMNI data for the same intervals. Interval |  RMSE (km/s) |  MAE (km/s) |  FastDTW distance (km/s) | Pcc | Skill score |  $SSF_{normalized}$
---|---|---|---|---|---|---
Jan – June 2005 | 175 | 145 | 89 | 0.05 | -0.79 | 0.77
July – Dec 2005 | 179 | 145 | 105 | -0.07 | -1.22 | 1.03
Jan – June 2006 | 124 | 98 | 62 | 0.17 | -1.00 | 0.84
July – Dec 2006 | 178 | 151 | 81 | -0.25 | -1.36 | 0.80
Jan – June 2007 | 155 | 125 | 69 | 0.03 | -1.11 | 0.77
July – Dec 2007 | 112 | 94 | 71 | 0.43 | -0.36 | 0.94
Jan – June 2008 | 133 | 108 | 72 | -0.02 | -2.24 | 1.16
July – Dec 2008 | 124 | 105 | 60 | -0.03 | -2.75 | 1.22
Jan – June 2009 | 91 | 70 | 55 | 0.10 | -0.78 | 1.08
July – Dec 2009 | 75 | 59 | 42 | 0.12 | -1.15 | 1.01
Jan – June 2010 | 103 | 80 | 67 | 0.32 | -0.27 | 0.87
July – Dec 2010 | 142 | 122 | 108 | -0.07 | -1.17 | 1.30
Jan – June 2011 | 159 | 128 | 97 | 0.07 | -0.38 | 0.83
July – Dec 2011 | 156 | 129 | 90 | 0.07 | -0.32 | 0.79
Jan – June 2012 | 167 | 134 | 101 | 0.08 | -0.30 | 0.80
July – Dec 2012 | 140 | 110 | 79 | 0.16 | -0.19 | 0.84
Jan – June 2013 | 184 | 150 | 95 | 0.22 | -0.10 | 0.66
July – Dec 2017 | 145 | 117 | 79 | 0.03 | -1.02 | 0.95
Jan – June 2018 | 123 | 97 | 58 | 0.01 | -0.82 | 0.79
July – Nov 2018 | 102 | 80 | 59 | 0.17 | -0.74 | 0.90

Table 1 shows the RMSE, MAE, FastDTW normalized distance, Pcc, Skill score,
and normalized SSF ($SSF_{normalized}$) from the comparison between the
Helio1D and the in situ observations at L1 for each 6-month interval. The OMNI
data were obtained at a 5-min resolution, and they are resampled to a 1-hour
resolution to match the cadence of the Helio1D data. We find that the
intervals in 2009 and the first half of 2010 have relatively low RMSE, MAE,
and FastDTW distance, with the interval of July – Dec 2009 having the lowest
values for all three metrics. Overall, the FastDTW distance has a lower value
than RMSE and MAE for each interval. The Pcc values are mostly close to zero,
and reach negative values for certain intervals, indicating that there is no
linear correlation between Helio1D and OMNI data (i.e., the two data are
qualitatively dissimilar). The interval of July – Dec 2007 has the best Pcc
value of 0.43, showing somewhat linear correlation between the Helio1D model
and the OMNI data; this interval is shown in Figure 3. The skill score is
negative for all intervals, indicating that the Helio1D performs worse than
the baseline model despite the relatively low errors or even positive Pcc, for
some intervals. In contrast, the normalized SSF of most intervals has a value
of less than unity (as highlighted in bold), indicating that the Helio1D
indeed performs better than the baseline model. We discuss these different
measures in Section 7. To investigate the Helio1D performance in various
phases of the solar cycle, we plot these metrics against the average number of
the sunspots in Figure 2.

Figure 2: The average RMSE (dot), MAE (triangle), and FastDTW normalized
distance (square) versus the average sunspot numbers (cross) for eace 6-month
interval. The values of the three metrics are shown on the left Y-axis (black)
and the average sunspot numbers are shown on the right Y-axis (red). The data
for 2005 - 2013 (solar cycle 23) and 2017 - 2018 (solar cycle 24) are
separated by a vertical grey dashed line. The individual metric values for
each solar cycle are connected by dotted lines.

Figure 2 shows the performance of the Helio1D characterized by the RMSE, MAE,
and FastDTW distance for the intervals shown in Table 1 (left y-axis, black)
compared to the average sunspot numbers of the corresponding intervals (right
y-axis, red). The sunspot number data were obtained as the monthly mean total
sunspot number from the Royal Observatory of Belgium666Source: WDC-SILSO,
Royal Observatory of Belgium, Brussels (https://www.sidc.be/silso/datafiles).
The data in 2005 – 2013 correspond to the solar cycle 23 and the data in 2017
– 2018 correspond to the solar cycle 24. Here, we have more complete data for
solar cycle 23 than the solar cycle 24. Based on the average sunspot numbers,
we infer that our data for the solar cycle 23 comprise the declining phase
from 2005 to 2007, the solar minimum from 2008 to mid-2009, the ascending
phase from mid-2009 to 2011, and the solar maximum from 2012 to 2013. The rest
of our data from mid-2017 to late 2018 correspond to the declining phase of
the solar cycle 24.

We find that all the metric values (shown in black) vary with the average
sunspot numbers (shown in red). During the declining phase of the solar cycle
23, all the metric values proportionally decrease with the average sunspot
numbers. During the ascending phase of the solar cycle 23, all the metric
values roughly increase with the average sunspot numbers. The most striking
features are during the solar minimum and the solar maximum. The metric values
reach lowest values during the solar minimum and/or the late declining phase
and early ascending phase. Here we find the minimum RMSE, MAE, and FastDTW
distance six months after the lowest average sunspot number in Jan – June
2009. The highest errors are found during the solar maximum, which are not
surprising as the solar magnetic fields often undergo changes and they are
thus less predictable. During the declining phase of the solar cycle 24, we
find that the errors proportionally decrease following the average sunspot
numbers. These results demonstrate that the performance of the Helio1D varies
with the solar cycle — it performs best during the solar minimum and it
performs worst during the solar maximum. This finding is consistent with the
performance of solar wind modeling by other existing models (e.g. Hinterreiter
et al., 2019). We further discuss these results in Section 7.

Figure 3: Comparison of CIR modeling from the 1D MHD code (magenta) to the
observations (black) from the OMNI database during May 2007 and April 2008 —
(a) radial bulk flow velocity ($V_{x}$); (b) tangential magnetic field
component ($B_{y}$); (c) ion number density ($n$); and (d) ion temperature
($T$).

To understand how the Helio1D pipeline qualitatively performs, we next
visualize an example of the model output against the observations at L1.
Figure 3 shows the Helio1D output in magenta and the OMNI data in black. In
Fig 3a, the bulk flow velocity displays a recurring pattern of the stream
interaction regions (i.e., CIRs) characterized by the transition from slow to
fast wind, clearly noticeable between August and November 2007. The stream
interfaces also collocate with the polarity change in $B_{y}$, shown in Fig
3b, and the enhancement of density and temperature in Fig 3c and Fig 3d,
respectively. Comparing to the observations, we find that the modeled solar
wind speed agrees qualitatively well and that the model correctly produces
consistent CIR formation. Nevertheless, the number density and temperature in
Fig 3c and Fig 3d at the stream interfaces are generally higher than the
observed values. These higher peaks are produced as a consequence of over-
compression at the stream interfaces due to the ideal MHD plasma assumption in
the 1D MHD code, which limits dissipation. Despite some over-compression and
temporal uncertainties, we conclude that the interfacing between Multi-VP and
1D MHD works reasonably well. We further perform qualitative assessments of
the pipeline performance and discuss calibration in Section 5.

From the long-term results (not shown), we find that the Helio1D pipeline
correctly produces the expected large-scale modulation of CIRs especially in
the bulk flow velocity, demonstrating that the Helio1D pipeline correctly
produces long-term solar wind fluctuations. On a shorter timescale, there are
some mismatches between the amplitudes of the fluctuations of the Helio1D
results and the observations. We quantify these differences in detail in
Section 4.2.

###  4.2 Performance of Helio1D with the FastDTW technique

Figure 4: Example of the FastDTW mapping and performance metrics. (a) The
FastDTW mapping (cyan) between the predicted CIR structures with Helio1D
(purple) and the OMNI (black) data. (b) Helio1D, OMNI, and the baseline model
(red) of the same interval. Relevant numbers indicative of the model
performance are noted.

Since the classic metrics only indicate crude information of the model
performance when comparing to real data, we consider applying FastDTW
algorithm. Fig 4 shows an example of the FastDTW mapping between Helio1D and
OMNI time series for the stream interfaces during 27 December 2005 and 14
February 2006. The OMNI data (black) show two clear stream interfaces and two
corresponding high-speed streams. It can be seen that the Helio1D correctly
produces the second stream interface along with the adjacent high-speed stream
period compared to the observation despite some time delay and differences in
the finer-scale structures.

Fig 4a shows the FastDTW alignments (cyan) that map large-scale features
consisting in the stream interfaces and high-speed stream. The normalized
FastDTW distance is about $53$ km/s for this interval. As there can be several
alignments for a data point (e.g., at local peaks), the number of alignment
pairs is equal or larger than the length of the data. We indicate the number
of the alignment pairs (called “path length”) to the length of the data in Fig
4a. Fig 4b shows the Helio1D, OMNI, and baseline model (red) of the same
interval. We note the RMSE of $119.4$ km/s and MAE of $79.4$ km/s, measured
between the Helio1D and OMNI data. The skill score, obtained from equation 4,
is $-1.23$ indicating that the Helio1D performs worse than the baseline model.
Nevertheless, the normalized SSF (equation 7) of $0.8$, obtained from the
comparison of the FastDTW mapping between OMNI and Helio1D to the FastDTW
mapping between OMNI and baseline model, indicates that the Helio1D model
performs better than the baseline model.

This example shows that the Helio1D model prediction is still useful despite
the negative skill score. We propose that the FastDTW algorithm can be used
for mapping large-scale features, in particular, of the stream interfaces and
high-speed streams. Here, the alignments can be used for extracting detailed
information (i.e., the normalized FastDTW distance and normalized SSF)
indicative of the model performance. We further exploit the alignments for
model calibration in Section 5.

As illustrated in Fig 4a, the FastDTW alignments optimally match large-scale
features in the time series. For a given FastDTW map path, we may define the
time difference as $\Delta t=t_{OMNI}-t_{1D}$ and the velocity difference as
$\Delta V=V_{OMNI}-V_{1D}$, where the subscript OMNI means the observations
and the subscript 1D means the Helio1D results. Here, the statistical
information on $\Delta t$ and $\Delta V$ from the FastDTW mapping are useful
for investigation of the model performance in terms of timing and magnitude
difference. For each mapping path, $\Delta t>0$, i.e., $t_{OMNI}>t_{1D}$,
means that the Helio1D model result is ahead of the observation; while $\Delta
t<0$ means that the Helio1D model result is behind (e.g., lagged or delayed)
with respect to the observation. Similarly, $\Delta V>0$, i.e.,
$V_{OMNI}>V_{1D}$, means that the Helio1D model result underestimates the
observed speed; while $\Delta V<0$, i.e.,$V_{1D}>V_{OMNI}$, means that the
Helio1D model result overestimates the observed speed. We can measure the
model performance by constructing histograms of distribution of $\Delta t$ and
$\Delta V$ as follows.

Fig 5 shows histograms of the $\Delta t$ and $\Delta V$ of all the data
obtained from FastDTW alignments on the velocity time series from 2005 to
2013. Using the Gaussian distribution fit on the histogram of $\Delta t$ in
Fig 5a, the median of the $\Delta t$ is about $2.7$ hours; the full width at
half maximum (FWHM), i.e., $~{}2.355\sigma$, where $\sigma$ is the standard
deviation, is about $31.8$ hours. In Fig 5b, we apply the Cauchy-Lorentzian to
the $\Delta V$ distribution to extract their statistical properties as it best
fits with the shape of the distribution. We find that the $\Delta V$ has the
median of $0.53$ km/s and the FWHM of $34.1$ km/s. This means that there is
almost no systematic bias for the modeling of the velocity amplitude by the
Helio1D, with the majority of the modeling having $\Delta V$ within $\pm 34.1$
km/s. In brief, considering several years of statistics covering parts of the
solar cycle 23, the Helio1D model results show minimal time and velocity
differences compared to the observations.

Figure 5: Histograms of (a) time delay and (b) velocity difference obtained
from FastDTW alignments between the Helio1D and OMNI data from 2005 to 2018.
The Gaussian (a) and the Cauchy-Lorentzian (b) distribution fits (red solid
lines) are applied to obtain median (red vertical dashed line) and FWHM of the
time delay and the velocity difference, respectively.

Using the FastDTW alignment information, we also explore the model performance
for different types of the solar wind. In Fig 5b, for instance, we find that
the $\Delta V$ distribution has slight skew or an asymmetry towards the
negative values. This indicates that the Helio1D model likely overestimates
the solar wind speed. To understand the model performance in detail, we
consider histograms of $\Delta t$ and $\Delta V$ for the individual 6-month
intervals in Table 1. Particularly, we categorize the solar wind schemes into
three types: (1) slow wind with $V<400$ km/s, (2) moderate wind with $400\leq
V\leq 500$ km/s, and (3) fast wind with $V>500$ km/s. This categorization is
based on the observed solar wind. Fig 6 shows an example of distributions of
$\Delta t$ (a) and $\Delta V$ (b) with individual distribution fit functions
to the slow (yellow), moderate (green), and fast (blue) solar winds, in
addition to all solar wind (black). Table 1 in Appendix summarizes the
statistical information of all the 6-month intervals.

Figure 6: Histograms of (a) time delay and (b) velocity difference obtained
from FastDTW alignments between the Helio1D and OMNI data from Jan to June
2006. Blue, red, and green bars represent slow, intermediate, and fast winds,
respectively. (a) The Gaussian distribution fits are applied to obtain median
($\mu$) and FWHM of the time delay. (b) The Cauchy-Lorentzian distribution
fits are applied to obtain $\mu$ and FWHM of the velocity difference. The
median and FWHM for all solar wind speeds are also given. See appendix for
$\mu$ and FWHM all 6-month intervals.

We now discuss our fitting results to the histograms of $\Delta t$ and $\Delta
V$ obtained from the FastDTW algorithm in Fig 6 for the data in Jan - June,
2006. For the slow and moderate winds, the median time differences are within
a few hours, with $\Delta t_{slow}=-1.5$ hours and $\Delta t_{moderate}=3.8$
hours. For the fast wind, in contrast, we obtain a rather large time delay
with $\Delta t_{fast}=17.5$ hours. Overall, the FWHM of time delay is about
$30$ hours for all solar wind speed ranges. For the velocity difference, we
find that the slow and moderate winds have small medians of $\Delta V$, with
$\Delta V_{slow}=6.2$ km/s and $\Delta V_{moderate}=3.5$ km/s. For the fast
wind, however, the median of $\Delta V$ is bigger, with $\Delta V_{fast}=74$
km/s. This example shows that the Helio1D indeed performs differently for
various solar wind speed ranges. For this interval, for instance, the Helio1D
model underestimates the speed of the fast wind. To investigate whether this
particular behavior persists, we next consider the median $\Delta t$ and
$\Delta V$ of the individual 6-month intervals throughout the solar cycle 23.

Fig 7 shows bar plots of the median $\Delta t$ in the left panels and $\Delta
V$ in the right panels, separated for all wind types (a, b), fast wind (c, d),
and slow wind (e, f). The bar plots are shown against the average sunspot
numbers of the corresponding intervals to indicate the phases of the solar
cycle. We note that the moderate wind does not show a particular trend and
thus excluded from this plot. Overall, we find that the Helio1D has a positive
$\Delta t$ (Fig 7a), indicating that the Helio1D timing is ahead of the
observations. The largest positive $\Delta t$ is found during the ascending
phase and the solar maximum (Jan - June 2012 in Fig 7a). This positive $\Delta
t$ is larger for the fast wind (40 hours) compared to the slow wind (25 hours)
as seen in Figs 7c and 7e. For the velocity difference for all wind types, we
find that the Helio1D provides a minimal velocity difference within 10 km/s
except during the ascending phase and solar maximum as seen in Fig 7b. When
considering the different wind types, we find that the median $\Delta V$ of
the fast wind (Fig 7d) is markedly different from that of the slow wind (Fig
7f). In particular, during the solar minimum and early ascending phase, there
is a large positive $\Delta V$, indicating that the Helio1D underestimates the
speed of the fast wind ($V_{1D}<V_{OMNI}$) by 50 - 150 km/s. In contrast,
during the same intervals, there is mostly negative $\Delta V$ for slow wind.
In other words, the Helio1D overestimates the speed of the slow wind
($V_{1D}>V_{OMNI}$). This overestimation can go up to 60 km/s as seen during
the ascending phase.

In brief, we find that the timing of the Helio1D results (i.e., stream
interfaces) is often ahead of the observations (consistent with Fig. 3). This
positive $\Delta t$ is even larger for the fast wind. In terms of the
velocity, we find that the Helio1D model underestimates the fast wind speed
while overestimates the slow wind speed. This effect is clear during the solar
minimum and the early ascending phase.

Figure 7: Bar plots of the median time difference ($\Delta t$; left panels)
and velocity difference ($\Delta V$; right panels) from the FastDTW technique
applied to the 6-month intervals in 2005 - 2013, over-plotted by the average
sunspot numbers (red cross, right Y-axis) indicative of the solar cycle. The
bar plots are shown for (a, b) all wind types, (c, d) slow wind, and (e, f)
fast wind.

##  5 Pipeline calibration

Figure 8: Calibration plots obtained from the FDTW alignments between Helio1D
(horizontal axis) and OMNI (vertical axis) time series for (a) solar wind
speed, (b) number density, and (c) temperature.

As mentioned in Section 4.1, Helio1D usually overestimates the number density
and temperature of plasma at stream interaction regions as a consequence of
over-compression at the stream interfaces. To alleviate this issue, we
consider a post-calibration of the data to lower the extreme peaks in the
Helio1D outputs. As shown in Section 4.2, FastDTW provides the alignments
between the modeled and observed solar wind speeds. Since the FastDTW maps
similar structures within the two time series, we create a calibration plot
for the mapped values between the Helio1D and OMNI data. Fig 8 shows a kernel
density estimation (KDE) plot of the mapped solar wind speeds from the Helio1D
(horizontal axis) and the OMNI data (vertical axis), which shows their
smoothed 2D density probability histogram. The brighter color in the plot
corresponds to a higher probability density. Here, the velocity data in Fig 8a
has a high probability density that mostly follows the line of slope of 1.
This demonstrates that there is overall no bias from the Helio1D modeling
(similar to Fig 6b), and that the magnitude of the solar wind speed from
Helio1D mostly matches the OMNI data.

The FastDTW alignments information allows us to map not only velocity but also
plasma density and temperature values between the modeled and observed time
series. In Figs 8b and 8c, we compare the plasma number densities and
temperatures between the Helio1D and OMNI using the mapped path obtained from
the FDTW application on the solar wind speed. It is clearly visible that the
number density and the temperature from Helio1D are higher than observed
values. To lessen these overestimations, we consider using linear regression.
Here, we apply a linear fit onto the data shown as the red solid line in Fig
8b. Similarly, we apply a linear fit function to the plasma temperature in Fig
8c. The linear fit functions of both number density and temperature are found
to have a slope of 0.653. Using these linear functions, we find that the
magnitudes of the peaks of the number density and temperature better match
those of the observations (not shown). We thus conclude that our calibration
on the plasma number density and temperature with the linear function
significantly improves the Helio1D modeling. We apply these calibrations for
the number density and temperature of the Helio1D pipeline for operational
forecasting described in Section 6.

##  6 Operational forecasting

We now shift our focus to the development of a stable “prototype” service for
solar wind forecasting with Helio1D. To continuously predict solar wind
conditions, several technical aspects as mentioned in Section 2.3 are
addressed. First and foremost, characterization of model uncertainties is
critical to assess the modeling results in order to plan for mitigation of
plausible impacts of extreme events. We provide model uncertainties using an
ensemble modeling in Section 6.1. Secondly, the 1D MHD model requires that the
length of input time series covers at least one solar rotation in order to
produce consistent CIR formation. The Multi-VP model has a setup that provides
for the daily solar wind emergence comprising 72 hours of solar wind
conditions covering from the present day ($D$) to the next two days ($D+2$).
Therefore, the solar wind emergence time series from Multi-VP must be
concatenated to produce a continuous time series with a minimum length of
27.25 days. Lastly, there can be data gaps and/or unphysical values arising
from either the inputs for MultiVP (i.e., magnetogram), due to the lack of
data or the presence of invalid data points, or numerical errors. The
implementation of automatic procedures to tackle these issues are detailed in
Section 6.2.

###  6.1 Helio1D ensemble modeling with 21 virtual targets

We perform an ensemble forecasting of the MULTI-VP and 1-D MHD models by
considering a range of heliospheric latitudinal and longitudinal
uncertainties. For example, a solar wind structure that is ahead or behind in
time compared to the observation can be accounted for by considering the time-
series at some nearby heliospheric longitudes. Also, the magnitude of the
solar wind speed profiles is different at different heliospheric latitudes as
a function of the warping of the Heliospheric Current Sheet and nearby
velocity gradients. Using daily magnetograms from WSO, we set up the Multi-VP
model to automatically generate daily one-dimensional solar wind profiles that
cover from $D$ to $D+2$ at the sub-Earth point and the surrounding virtual
targets. The surrounding virtual targets are set to spread up to $15^{o}$ from
the sub-Earth point in latitude and longitude; they are set at $5^{o}$ apart
from each other in latitude and longitude; these chosen points constitute the
star-grid pattern with the total of 21 points including sub-Earth point (see
Appendix). The spatial cuts through these 21 points on the 2D map of solar
wind emergence from Multi-VP along the ecliptic are then translated to
temporal solar wind profiles. These 21 time-series inputs are automatically
fed into the 1D MHD to propagate them in parallel from 0.14 AU to 1 AU. The 1D
MHD model is rather computationally inexpensive; the model running with 21
virtual targets remains fast compared to 3D MHD modeling in general.

Figure 9: Helio1D ensemble time series outputs using the 21 virtual spacecraft
compared to the OMNI data for March to April, 2018. The ensemble Helio1D
outputs are shaded in green and the output at sub-Earth point is plotted in
blue. The observation data are represented by a black line.

Fig 9 shows results from the Helio1D pipeline with the 21 virtual points
between March and May, 2018. The Helio1D data at the sub-Earth point are shown
in blue, and the observation data are shown in black. The green shade
highlights the spread of values between the minimum and maximum among the 21
virtual targets at each hour. Panels (a) - (d) show the solar wind speed,
tangential magnetic field, plasma number density, and plasma temperature,
respectively. We note that the post-calibration with the linear functions for
number density and temperature (found in Section 5) has been applied. Here,
the ensemble spread in green shade provides an error bar. In this example, we
find that the stream interfaces modeled for the sub-Earth point appear to lag
behind the observations for about two days. Nevertheless, the timing
uncertainties from the values at the virtual points indeed cover the timing of
the arrivals of observed stream interfaces. Overall, we find that most of the
observed data points fall within the error spread for both magnitude and
timing, although there are some underestimations of the speed of the high-
speed stream (consistent with our findings in Section 4.2). We conclude that
the Helio1D modeling using the 21 virtual targets improves the modeling
compared to using only one virtual point targeting at Earth.

###  6.2 Solar wind data concatenation & Data gaps

The main technical problems deal with the automated interfacing between Multi-
VP and 1D MHD model. Unlike the model benchmarking and case-by-case analysis,
we rely on automatic modeling for both the Multi-VP and 1D MHD models. Here,
the Helio1D pipeline is scheduled to run in-house every day. First, the Multi-
VP is set up to produce a daily forecast with a data length of 3 days,
covering from day $D$ to $D+2$. We first concatenate these Multi-VP data for
each virtual target by averaging the data from the day $D-30$ to the present
day $D$ to make the time series input to be sufficiently long. After the
concatenation process, this 1-month solar wind profile at 0.14 AU is
subsequently propagated by the 1D MHD model to provide the solar wind profile
at 1 AU. Since the solar wind takes time to propagate to the Earth, we gain an
extra lead time of 2 – 7 days depending on the solar wind speed. To provide a
total lead time of 4-days from Helio1D, we limit the extra time gained from
the 1D MHD propagation modeling to 2 days. This setup has been done to
automatically interface the Multi-VP and 1D MHD to provide daily solar wind
nowcast and forecast from day $D$ to $D+4$ (see branch (b) in Fig 1). In an
absence of the magnetogram data, the magnetogram of the previous solar
rotation is programmed to be fetched. This procedure is implemented based on
the assumption that the coronal structures remain unchanged compared to the
previous solar rotation (i.e., when the consistent, recurring CIRs are
expected).

Furthermore, one of the models, or both, may terminate earlier than expected
and give incomplete outputs. This is because, in reality, there can be data
gaps arising from the lack of daily data (e.g., magnetogram data), or
unphysical values in the data. The latter can come from numerical errors from
one of the codes or poor quality of the raw inputs. To prevent the Helio1D
pipeline from early termination, we perform an automatic search for data gaps
and unphysical values (after performing the fetching of previous magnetogram
when the present-day magnetogram is unavailable as mentioned above). The data
gaps are then filled using linear interpolation between two available data
points. The unphysical values identified using the criteria (see Appendix) are
subsequently removed. The resulting data gaps are then filled using linear
interpolation. To remove unphysical features that may arise from these
procedures, we perform a running average on the data using a window of $\pm 3$
hours centered around the data point. Finally, when either the input or daily
output is shorter than expected, we fill the data with latest available values
to complete the length of expected output. For practical purpose, we provide
daily solar wind time series extending from day $D-3$ to $D+4$ for a given day
$D$.

Example plots of the daily data from the Helio1D are publicly available; they
can be accessible from http://swift.irap.omp.eu/ and, in the near future,
through the ESA Space Safety Program via
www.esa.int/Space_Safety/Space_weather.

##  7 Discussion

We developed the Helio1D pipeline for solar wind modeling by interfacing
Multi-VP for the coronal part and 1D MHD for the interplanetary part. The
interfacing has been done for the first time as an effort to connect two
developed models for operational forecasting purpose. Using the long-term
Multi-VP data produced with the WSO magnetograms in 2005 - 2013 and 2017 -
2018, we evaluated the pipeline performance in various phases of the solar
cycles 23 and 24. In addition to classic metrics such as RMSE, MAE, and skill
score, we devised a new metric based on the FastDTW algorithm. Using the long-
term data and the metrics, we discuss our main findings as follows.

  * 1.

The performance of the Helio1D pipeline is dependent on the phases of the
solar cycle. Using the RMSE, MAE, and FastDTW distance, measured for each
6-month interval, we find that their values positively correlate with the
average sunspot numbers of the previous 6-month interval. The RMSE, for
example, reaches a minimum value of 80 km/s during the solar minimum, and
reaches a maximum value of 180 km/s during the solar maximum. Overall, the
Helio1D pipeline provides minimal magnitude errors during the late declining
phase and the solar minimum.

  * 2.

The Helio1D pipeline produces consistent CIR formation. For each CIR, the
solar wind speed profile at the stream interface qualitatively agrees with the
observations (Fig 3a). Thus, the model sufficiently includes the large-scale
physics of the CIRs. However, due to the limited dimensionality and ideal MHD
assumption, the compression at stream interfaces (resulting in extreme peaks
of number density and temperature) is rather strong compared to the
observations. This effect was found to be common to 1D MHD models used for
solar wind propagation (e.g., Zieger and Hansen, 2008). To alleviate this
effect, we perform a post-calibration to lower the peaks at stream interfaces
using a linear function. This linear function was found using the FastDTW
alignments of the solar wind profiles, which particularly mapped stream
interfaces.

  * 3.

Using the FastDTW alignments, we also evaluated the timing and magnitude
differences between the Helio1D and the observation data. For the data in the
solar cycle 23, we find minimal $\Delta t$ of 2.7 hours and $\Delta V$ of 0.5
km/s. However, when separating the data into shorter intervals of 6-months and
categorizing the solar wind types based on its speed, we find different
results summarized as follows.

    * 3.1

For all wind types, we find that the Helio1D timing is often ahead of that of
the observation for about 10 hours. This timing error is highest six months
after the average sunspot number reached the maximum in late 2011. Meanwhile,
the $\Delta V$ for all wind types remains low throughout the solar cycle
except for the ascending phase of the solar cycle in 2010.

    * 3.2

The fast wind has higher $\Delta t$ and $\Delta V$ than those of the slow
wind. The Helio1D timing of fast wind is often 10 - 20 hours earlier than that
of the observation, and this timing difference goes up to 40 hours six months
after the highest sunspot number. For $\Delta V$, the Helio1D pipeline mostly
underestimates the speed of the fast wind for about 50 - 100 km/s during the
declining phase and solar minimum; this difference goes up to $\sim 150$ km/s
during the late ascending phase.

    * 3.3

The timing of the slow wind agrees with that of the observations within $\pm
10$ hours, except during the solar maximum. For $\Delta V$, the Helio1D
pipeline often overestimates the speed of the slow wind for about 20 - 40
km/s. The $\Delta V$ is largest with a value of $-60$ km/s in late 2010.

Comparing to existing models that have been benchmarked using large data sets,
the performance of the Helio1D pipeline is in agreement with other equivalent
models. Owens and Riley (2017) utilized solar wind emergence at 0.14 AU from
MAS based on Carrington maps (e.g., WSO) of the photospheric magnetic field.
Particularly, they used a large ensemble (N = 576) of solar wind time series
from MAS, where the ensemble members are produced from sampling solar wind
solutions within a range of latitudes about the sub-Earth point. The solar
wind flows at 0.14 AU are then propagated using the HUX model (Riley and
Lionello, 2011) that is based on the fluid momentum equation (e.g., Pizzo,
1978) and takes into account a residual solar wind acceleration (Schwenn,
1990). Using a long interval of data from 1996 to 2016, they find that the
median RMSE of the upwind ensemble propagation is 107 km/s. Reiss et al.
(2020) performed a forecasting of ambient solar wind using the WSA model and
the HUX tool as well as the Tunable HUX (Reiss et al., 2020) to map solar wind
flows from near Sun to Earth for the period 2006 - 2015. Their RMSE is ranging
from 90 to 122 km/s while their MAE is ranging from 72 to 93 km/s. Their skill
scores are negative for WSA/HUX (greater than -0.6) and positive (less than
0.06) for WSA/THUX. Compared to their studies, our Helio1D pipeline
performance characterized by the average RMSE and MAE are in agreement with
their models despite somewhat larger values by about a few 10 km/s.
Nevertheless, RMSE and MAE are rather crude metrics; they do not tell all
qualities of the prediction and the measurement, e.g., as discussed by
MacNeice et al. (2018) and several others.

Regarding to the variation of the Helio1D performance with phases of the solar
cycle, we find that our findings are generally consistent with other 1D MHD
models. Zieger and Hansen (2008) performed an extensive validation of solar
wind propagation using a 1D MHD model. They find that the variation of the
coronal structure on the timescale of a solar rotation, characterized by the
recurrence index of solar wind speed, plays an important role in the
prediction efficiency. This explains our Helio1D results such that, in an
absence of variation, i.e., during the late declining phase, the model
generally predicts CIR formation consistent with the observations. In
contrast, in a presence of strong variations, i.e., when there are CME
emergences during the ascending phase and the solar maximum, the model
generally predicts poor results compared to the observations.

In terms of timing, we find that there is a bias on the solar wind stream
arrival time such that the Helio1D solar wind streams usually arrive 10 hours
earlier than the observations. This bias is stronger for the fast wind
compared to the slow wind. This effect may come from the assumption of an
ideal MHD plasma in the 1D MHD model that ignores finer-scale physics and/or
interactions that could slow down fast wind or accelerate slow wind in the
interplanetary space. Moreover, the limited dimensionality of the simulation
does not take into account the propagation in other directions; this may
result in the timing mismatch. In terms of magnitude, the Helio1D
underestimates the speed of the fast wind while overestimates the speed of the
slow wind. The bias on the fast and slow wind magnitudes may come from the
lack of the physics on solar wind acceleration. In HUX model, an empirical,
ad-hoc solar wind acceleration is incorporated into the model (Schwenn, 1990).
Since the 1D MHD code was originally developed for solar wind propagation in
the outer heliosphere (Tao et al., 2005), this physics was deemed less
important and it was not added into the model. We note that this aspect must
be addressed in a future improvement of the 1D MHD model for the inner
heliosphere; this aspect out of scope of this work which focuses on
interfacing the matured models.

Our work devised a new exploitation of the FastDTW technique for a detailed
assessment of the Helio1D performance. In particular, the FastDTW alignments
encoded optimal time delays and magnitude differences between the modeled
stream interfaces and observed structures. We highlight that the FastDTW can
be used to provide more qualitative measures than RMSE, MAE, and skill score,
for example. We also demonstrated that the FastDTW (normalized) distance
correlates with the RMSE and MAE. Nevertheless, the FastDTW technique is not
without caveats. The main constraint of DTW techniques in general is the
pathological mapping or the singularities. Therefore, the obtained $\Delta t$
and $\Delta V$ from the DTW applications are not unique but instead depending
on the constraints that have been added. In our case, we restrict that the
alignments along the time axis must be within 96 hours (corresponding to the
time that a solar wind with an intermediate speed should take to travel from
0.14 to 1 AU). Furthermore, we chose the FastDTW technique in particular
because it searches to first map the structures at coarse scales, then refine
to smaller scales. We emphasize that this algorithmic feature is desirable for
mapping large-scale solar wind structures such as CIRs or CMEs.

Finally, we highlight the work that needs to be taken to develop an automated
solar wind forecasting pipeline of Helio1D. In addition to the pipeline
calibration to alleviate the pipeline caveats, there are other aspects that
must be addressed. This comprises (1) providing model uncertainties and (2)
dealing with data gaps and bad data. The Helio1D modeling of an ensemble of 21
solar wind solutions was shown to be a reasonable method to provide worst case
scenario especially for the timing of stream interface arrivals. We note that
the ensemble members can be further exploited to obtain an optimized forecast;
we leave this aspect for future work. Most importantly, to develop a reliable
service, we implemented strategies to tackle with data gaps and bad data
points so that the pipeline can automatically run and provide continuous
nowcast and forecast. This aspect should be further tested in order to
evaluate their impacts on accuracy, performance, and stability of the
operational forecasting pipeline. Such a task is of particular importance; it
requires dedicated tests on this prototype pipeline in various situations
before employing it for applications in real world.

##  8 Summary

We developed a prototype pipeline “Helio1D” that automatically interfaces the
coronal model “Multi-VP” and the solar wind propagation model “1D MHD” to
provide nowcast and forecast of ambient solar wind containing CIRs at L1. The
operational prototype pipeline provides daily solar wind modeling with a lead
time of 4 days. In this work, we benchmarked and extensively tested the
Helio1D pipeline using 10 years of data spanning from late 2004 - 2013 and
2017 - 2018. We evaluated the performance of the pipeline using classic
metrics including RMSE, MAE, and skill score. In particular, we devised a new
exploitation of the FastDTW technique to map large-scale solar wind structures
of CIRs. We demonstrated that this technique can be used to characterize the
detailed performance of the solar wind modeling with Helio1D. For instance, we
characterized the statistical information on timing and magnitude differences
for velocity profils of stream interfaces. Using this new approach, we
assessed the pipeline performance for various phases of the solar cycle 23 and
investigated the modeling bias for fast and slow solar winds. We find that the
Helio1D pipeline performs best during the declining phase and solar minimum.
However, the Helio1D pipeline often underestimates the speed of the fast wind
while overestimates the speed of the slow wind, especially between the late
declining phase and early ascending phase of the solar cycle 23. Furthermore,
the solar wind structures modeled by Helio1D often arrives earlier than the
observations especially for the fast wind. These caveats plausibly arise from
the simplistic assumptions in the 1D MHD model, comprising the limited
dimensionality, the lack of dissipation, and the solar wind acceleration
physics, for example. Nevertheless, the Helio1D pipeline models consistent CIR
formation while remains computationally light, which is desirable for
operational forecasting.

To transition from case-by-case benchmarking to operational solar wind
forecasting, we implemented (1) the pipeline calibration to alleviate the
over-compression at stream interface due to the ideal MHD assumption, (2) the
ensemble modeling of 21 solar wind solutions at virtual targets including the
sub-Earth point to provide timing and magnitude uncertainties, and (3) the
procedures to remove unphysical data points and automatically fill data gaps.
The latter two procedures were discussed to be crucial in providing
continuous, reliable service while providing forecasting uncertainties.
Alternatively, the Helio1D pipeline can be adapted to use other more reliable
magnetogram sources and with higher resolution, e.g., from Air Force Data
Assimilative Photospheric Flux Transport (ADAPT; Hickmann et al., 2015; Arge
et al., 2013; Worden and Harvey, 2000). We emphasized that the implementation
of procedures for making the pipeline fully operational, in addition to the
pipeline benchmarking, are critical and they must be further tested in several
situations before employing the pipeline for real applications.

## Acknowledgements

The SafeSpace project has received funding from the European Union’s Horizon
2020 research and innovation programme under grant agreement No 870437. The 1D
MHD model of solar wind propagation initially developed by Tao et al. (2005)
and used for Helio1D is duplicated from the Heliopropa service provided by
CDPP (http://heliopropa.irap.omp.eu) and developed during the Planetary Space
Weather Services (PSWS) Virtual Activity of the Europlanet H2020 Research
Infrastructure funded by the European Union’s Horizon 2020 research and
innovation programme under grant agreement No 654208 and extended during the
Sun Planet Interactions Digital Environment on Request (SPIDER) Virtual
Activity of the Europlanet H2024 Research Infrastucture funded by the European
Union’s Horizon 2020 research and innovation programme under grant agreement
No 871149. Work at IRAP is supported by the French National Centre for
Scientific Research (CNRS), Centre national d’études spatiales (CNES), and the
University of Toulouse III (UPS).

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