ELEvoHI: Ensemble CME Arrival Forecast

• ELEvoHI is an advanced ensemble-based model that forecasts CME arrival times and speeds by coupling elliptical CME front geometry with a drag-based propagation method.
• The framework integrates heliospheric imaging data through geometric inversion and kinematic fitting to deliver real-time, operational space weather predictions.
• The ensemble approach quantifies uncertainties by sampling parameter variations and observational errors, enhancing forecast reliability for CME impacts.

The ELEvoHI (ELlipse Evolution model based on Heliospheric Imager data) is an advanced, ensemble-capable framework for predicting coronal mass ejection (CME) arrival times and speeds at heliospheric targets. By coupling wide-angle heliospheric imaging (HI) with a self-similar elliptical CME front geometry and a drag-based model (DBM) for the propagation, ELEvoHI delivers operationally viable forecasts with well-quantified uncertainties. The model is tailored for real-time space-weather monitoring using data from single or multiple vantage points and incorporates ensemble techniques to robustly assess effects of parametric and observational ambiguities.

1. Physical and Mathematical Foundations

ELEvoHI integrates a geometric representation of the CME front as an expanding ellipse with an analytic treatment of its interplanetary dynamics dominated by aerodynamic drag. The key steps are:

• Elliptical Front Geometry: The model assumes the CME front in the ecliptic plane is an ellipse, parameterized by the semi-major axis $a$ and semi-minor axis $b$, with an inverse aspect ratio $f = b/a$, an angular half-width $\lambda$, and a propagation longitude $\phi$ (Rollett et al., 2016, Amerstorfer et al., 2020, Hinterreiter et al., 2021). The ellipse expands self-similarly throughout the forecast period.
• Elongation-to-Distance Conversion (ELCon): HI observations provide time–elongation tracks $\epsilon(t)$. ELCon inverts these into apex heliocentric distances $r(t)$, by solving the Sun–observer–CME line-of-sight geometry for the point where the line of sight tangentially meets the ellipse.
• Drag-Based Propagation (DBM): CME acceleration is governed by

$\frac{dv}{dt} = -\gamma\,(v-w)|v-w|$

where $v$ is the apex speed, $w$ is the (locally or globally estimated) solar wind speed, and $\gamma$ is the drag coefficient, which formally depends on CME mass, cross-section, and ambient density (Amerstorfer et al., 2017, Amerstorfer et al., 2020). The analytic solution provides the time–distance evolution beyond $r \sim 20–30\,R_\odot$, where Lorentz forces are negligible.

• Ellipse Evolution Propagation (ELEvo): The best-fit geometric and dynamic parameters are used to propagate the elliptical front out to the target (e.g., 1 AU), extracting arrival time and speed for the apex and flanks (Rollett et al., 2016, Amerstorfer et al., 10 Dec 2025).

2. Algorithmic Architecture and Data Workflow

The forecasting pipeline follows a sequence grounded in the physics above:

1. Data Acquisition & Preprocessing: HI images (science or beacon quality) are converted into ecliptic J-maps, from which leading-edge elongations $\epsilon(t)$ are manually or semi-automatically tracked (Bauer et al., 2021). A dedicated Python tool automates these steps from NASA STEREO/HI archives, outputting ready-to-fit elongation series.
2. Geometric Inversion (ELCon): For each ensemble member (specified by $\phi, \lambda, f$), $\epsilon(t)$ is converted into $r(t)$. Errors due to track ambiguity or analyst choices are quantified by repeat tracking (typical $\sigma_r \leq 0.55\,R_\odot$ for HI1 FOV) (Amerstorfer et al., 10 Dec 2025).
3. Kinematic Fitting (DBM Fit): For each trial of the ambient wind speed $w$ (sampled from model medians and plausible $\pm 100\,\mathrm{km\,s^{-1}}$ ranges), the analytic DBM is fit to $r(t)$ to estimate optimal $\gamma, v_0, r_0$.
4. Forward Propagation (ELEvo): The elliptical front is numerically advanced using the best-fit parameters, yielding an arrival time ($t_{\mathrm{arr}}$) and speed ($v_{\mathrm{arr}}$) at the target longitude (Amerstorfer et al., 2020, Amerstorfer et al., 2017).
5. Ensemble Assembly: The full set of parameter permutations forms an ensemble, supporting uncertainty quantification and probability-of-hit diagnostics (Amerstorfer et al., 10 Dec 2025).
6. Data Assimilation and Constraint: When upstream in situ data are available (e.g., from Solar Orbiter), ensemble filtering can reject members incompatible with measured CME arrival time, drastically reducing forecast MAE (Laker et al., 2023).

3. Ensemble Modeling, Input Parameterization, and Uncertainty Quantification

ELEvoHI's ensemble mode captures a range of plausible CME geometries and environmental parameters:

• Parameter Space Sampling:
  • $\phi: \pm 10^\circ$ about best-fit (in $2^\circ$ steps)
  • $\lambda:$ ecliptic half-width, typical $\pm 10^\circ$ in $5^\circ$ steps
  • $f:$ inverse aspect ratio in $0.05–0.1$ increments over 0.7–1.0 (Hinterreiter et al., 2021, Bauer et al., 2021)
• Ambient Wind Uncertainty: $w$ is sampled in 9–17 values about the sector median (from WSA-HUX output) (Amerstorfer et al., 2020, Hinterreiter et al., 2021) or provided from in situ at L1.
• Ensemble Size: Standard runs use 220–350 members per event, forming a high-resolution probabilistic forecast (Amerstorfer et al., 2017, Hinterreiter et al., 2021).
• Uncertainty Reporting: Ensemble median and interquartile/standard deviation define the forecast and its spread. Hit/miss probabilities are derived as the fraction of ensemble members predicting impact at the target (Amerstorfer et al., 10 Dec 2025).
• Effects of Solar Wind Structure: Forecast uncertainty and systematic bias correlate with the local standard deviation in background wind within the CME's angular sector. Structured solar wind can increase arrival-time error and modeling bias (Barnard et al., 2021).

4. Model Evolution: Rigid vs. Deformable Fronts and Mass Estimation

ELEvoHI 2.0 introduces a deformable CME front that reacts in real time to spatial and angular solar wind variations:

• Rigid Ellipse Regime: Up to a transition distance ($r_{\mathrm{trans}}$, typically $65\,R_\odot$), the CME front is a self-similar ellipse propagated as detailed above.
• Deformable Front (Beyond $r_{\mathrm{trans}}$): The ellipse is discretized into N segments ($\theta_j$), each evolving its own $r_j(t)$ and $v_j(t)$ under localized drag, with $\gamma_j = c_D\,A_j\,n_j\,/\,M$. Solar wind speed $w_j$ and density $n_j$ are interpolated from a global heliospheric model (e.g., HUX, HUXt, EUHFORIA) (Hinterreiter et al., 2021). This enables spatially resolved predictions and more faithful arrival-time forecasts for flanks as well as apex.
• CME Mass Estimation: At the transition, the mass is recovered directly from the observationally determined $\gamma$, local density, and the cross-sectional area:

$$M = c_D\,A(r_{\mathrm{trans}})\,n(r_{\mathrm{trans}},w) / \gamma$$

This approach anchors mass estimates to remote HI and ambient model data (Hinterreiter et al., 2021).

5. Empirical Performance Metrics and Operational Considerations

Performance is benchmarked across several CME event sets using STEREO/HI (science and beacon) and supporting ensemble simulations:

• Arrival-Time Error (MAE/STD):
  • With science data: $8.2\pm5.5$ h (15-event set) (Amerstorfer et al., 2020); $8.81\pm3.18$ h (10-event set) (Bauer et al., 2021)
  • With beacon data: $11.36\pm8.69$ h (Bauer et al., 2021)
  • Ensemble filtering using upstream Solar Orbiter in situ data sharply improves MAE (from 10.4 h to 2.5 h) (Laker et al., 2023)
  • Observations from L5 yield minimal arrival-time error (average CME: $8.2\pm1.2$ h MAE) (Barnard et al., 2021)
  • Impact of HI data assimilation: Progressive inclusion of HI elongations beyond 35° reduces MAE to ~9 h (Amerstorfer et al., 10 Dec 2025)
• Arrival Speed Error (MAE/STD):
  • Science data: $59\pm31\,\mathrm{km\,s^{-1}}$; beacon data: $106\pm61\,\mathrm{km\,s^{-1}}$ (Bauer et al., 2021)
• Uncertainty Drivers: Increasing elongation coverage (i.e., tracking the CME further in HI) consistently decreases MAE, while rapid updates as HI data arrives support operational warning needs. Solar wind model accuracy directly limits ultimate skill (Amerstorfer et al., 10 Dec 2025, Rollett et al., 2016).
• Human Factors: Analyst-to-analyst variability in front tracking can introduce $\sim0.5\,R_\odot$ error in HI1 and propagate into arrival time uncertainty. Ensemble methods mitigate but do not eliminate this variability (Amerstorfer et al., 10 Dec 2025).
• Comparison with Other Models: ELEvoHI delivers superior arrival speed accuracy and comparable or improved timing compared to point-source, harmonic mean, or constant-speed geometric methods (Rollett et al., 2016).

6. Model Limitations, Current Debates, and Prospects

• Assumptions: The underlying drag model assumes constant background solar wind over the CME path, neglects internal CME magnetic structure, and presumes self-similar geometric expansion at least until $r_{\mathrm{trans}}$ (Hinterreiter et al., 2021).
• Ambiguity in Initial Conditions: Choice of propagation longitude/direction ($\phi$) and width ($\lambda$) has a direct impact on the forecast, and HI-only determinations (FPF) and coronagraph-based (GCS) approaches each have trade-offs. For weak/faint CMEs, HI tracking can outperform GCS-based direction estimates due to data availability (Amerstorfer et al., 10 Dec 2025).
• Sensitivity to Solar Wind Model Structure: Arrival time errors scale with the spatial variability of background wind; ensemble or real-time solar wind characterization should be part of best practices (Barnard et al., 2021).
• Ongoing Improvements: Future developments include deeper data assimilation (e.g., Kalman filtering), automated HI front tracking, more realistic solar wind input via physics-based, 3D MHD or data-driven ensembles, and support for next-generation missions (Vigil, PUNCH) with high-cadence, high-fidelity HI data (Amerstorfer et al., 10 Dec 2025, Bauer et al., 2021).
• Deformable Front and Mass Estimation: Adoption of the deformable front paradigm and HI-derived mass estimation are recent advances, improving timing and speed accuracy, especially for CMEs interacting with structured solar wind (Hinterreiter et al., 2021).

7. Summary Table: Core Components and Typical Parameter Choices

Component	Range / Options	Comment
Propagation angle	GCS: $\phi_0\pm10^\circ$; FPF: varied	Direction from coronagraph (GCS) or HI (FPF)
Half-width ($\lambda$)	GCS: $\lambda_0\pm10^\circ$; FPF: 35–55°	Choice affects arrival forecast for flank hits
Inverse aspect ($f$)	0.7–1.0 in 0.05–0.1 steps	Ellipse flattening; $f=1$ circular
Solar wind speed ($w$)	Median in sector, ±100 km/s range	Sampled based on WSA/HUX or in situ data
Drag parameter ($\gamma$)	Fitted per event, typ. $10^{-8}$–$10^{-7}$ km$^{-1}$	Encodes CME–solar wind coupling
Ensemble size	220–350	Permutations over above ranges; increased for HI data assimilation
Deformation	Only rigid ellipse (v1.0); deformable (2.0)	Local drag, wind with HI2/EUHFORIA for 2.0

ELEvoHI’s modular, physics-driven, and ensemble-based architecture supports robust, real-time CME arrival prediction with clear quantification and propagation of geometric, observational, and environmental uncertainties. Its flexibility enables integration with forthcoming observations from new heliospheric vantage points and supports ongoing advances in operational space weather forecasting (Amerstorfer et al., 10 Dec 2025, Rollett et al., 2016, Hinterreiter et al., 2021, Bauer et al., 2021, Amerstorfer et al., 2020, Barnard et al., 2021, Amerstorfer et al., 2017, Hinterreiter et al., 2021, Laker et al., 2023).