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arxiv: 2606.31433 · v1 · pith:TS4OYSBTnew · submitted 2026-06-30 · ⚛️ physics.space-ph

A new model for long-term forecasting of Galactic cosmic rays

David Pelosi , Fernando Bar\~ao , Bruna Bertucci , Emanuele Fiandrini , Miguel Orcinha , Nicola Tomassetti This is my paper

Pith reviewed 2026-07-01 02:47 UTC · model grok-4.3

classification ⚛️ physics.space-ph
keywords galactic cosmic rayssolar modulationParker transport equationlong-term forecastingheliospherespace radiationdiffusion-advection
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0 comments X

The pith

A one-dimensional Parker transport model with proxy-derived parameters reconstructs galactic cosmic-ray fluxes across solar cycles and enables decadal forecasts.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a forecasting framework that solves the one-dimensional spherically symmetric Parker transport equation for charged particles in the heliosphere, incorporating diffusion, solar-wind advection, and adiabatic energy losses. Parameters are made charge-sign and rigidity dependent and are tied to solar activity through Hilbert-Huang transform filtering and cross-correlation with delayed solar proxies. The resulting reconstructions match multi-species measurements from PAMELA, AMS-02, and ACE across varied energy ranges and solar-activity phases. When the same parameterisation is driven by forecasted solar proxies, the model produces decadal-scale predictions of galactic cosmic-ray intensities.

Core claim

The charge-sign- and rigidity-dependent parametric description of the diffusion-advection processes in the one-dimensional Parker transport equation yields good overall agreement with the data, as shown by the reconstruction uncertainty. The robustness of this approach is validated across a broad set of multichannel datasets covering different particle species, energy ranges, and phases of solar activity, supporting its applicability to space radiation monitoring and forecasting. Furthermore, when coupled with solar-proxy forecasting models, it enables decadal-scale predictions of galactic cosmic-ray fluxes.

What carries the argument

The one-dimensional spherically symmetric Parker transport equation with a charge-sign- and rigidity-dependent parametric description of diffusion-advection processes, whose effective parameters are obtained via Hilbert-Huang filtering and cross-correlation with solar proxies.

If this is right

  • Reconstruction uncertainties remain low across multiple particle species and rigidity ranges when the parametric description is applied to historical data.
  • The framework can be used for space radiation monitoring because it reproduces observed fluxes during different phases of solar activity.
  • Coupling the transport model to independent solar-proxy forecasting models produces quantitative predictions of galactic cosmic-ray intensities on decadal time scales.
  • Such predictions support long-term planning and radiation-risk assessment for future space missions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the proxy-parameter correlations shift in future cycles, the model would require periodic recalibration using new flight data.
  • Extending the one-dimensional description to include latitudinal or azimuthal dependence could reduce residuals at high rigidities where drift effects matter.
  • Real-time assimilation of current solar-proxy values could turn the same machinery into a short-term nowcasting tool in addition to its long-term forecasting role.
  • Comparison of the predicted spectra with measurements from probes at different heliocentric distances would test whether the assumed spherical symmetry remains adequate.

Load-bearing premise

The cross-correlation relationships between solar proxies and the effective model parameters, derived from historical data, will continue to hold for future solar cycles outside the training interval.

What would settle it

Direct comparison of the model's decadal predictions against measured cosmic-ray fluxes from PAMELA, AMS-02 or ACE during the next solar cycle that lies entirely outside the current training interval.

Figures

Figures reproduced from arXiv: 2606.31433 by Bruna Bertucci, David Pelosi, Emanuele Fiandrini, Fernando Bar\~ao, Miguel Orcinha, Nicola Tomassetti.

Figure 1
Figure 1. Figure 1: Individual effects of propagation parameters on the proton flux near Earth, obtained by varying each parameter independently within the range determined by the fitting procedure, while keeping all others fixed. The gray dashed line indicates the proton LIS used for calibration. See the main text for a detailed discussion of how each parameter affects the modulated flux. a complexity penalty, thus favoring … view at source ↗
Figure 2
Figure 2. Figure 2: Bayesian Information Criterion (BIC) values obtained from fits to AMS-02 proton data and ACE/CRIS carbon data, for the different combinations of model parameters listed in Sect. 2.1. The parameterization K2 (Eq. 3), when coupled with ε, consistently yields the lowest BIC values, indicating a significant statistical preference. Its stable behavior over the entire time range, including the polarity-reversal … view at source ↗
Figure 3
Figure 3. Figure 3: Modulated energy spectra for protons measured by PAMELA (light blue) and AMS-02 (green), and for carbon measured by ACE (orange), shown for six selected BRs. The magenta lines indicate the best-fit fluxes, while the gray dashed lines represent the local interstellar spectra (LIS) for protons and carbon adopted in this work Boschini et al. (2020) as inputs to the model. The model shows good agreement with t… view at source ↗
Figure 4
Figure 4. Figure 4: Best-fit results for the model parameters k0, δ, and ε obtained from the fitting procedure described in Sec. 2.3, using proton measurements from PAMELA and AMS–02 combined with monthly carbon data from ACE. The dark blue points show the same parameters after smoothing with the HHT filtering algorithm described in Sec. 2.5. The bottom panel displays the temporal evolution of the monthly and smoothed sunspot… view at source ↗
Figure 5
Figure 5. Figure 5: Hilbert spectra of the model parameters, showing the frequency content of both the original time series ⃗q (left) and their smoothed reconstructions ⃗qs (right), obtained using the HHT-based algorithm described in Sec. 2.5. For both cases, the long-term trend has been removed. The hatched region indicates the data gap in the PAMELA dataset. The smoothing procedure effectively suppresses high-frequency fluc… view at source ↗
Figure 6
Figure 6. Figure 6: Top row: Cross-correlation functions ⃗f for each model parameter, obtained by minimizing the loss function defined in Eq. 15. The magenta points correspond to the transformed parameters ⃗q ∗ defined in Eq. 13, plotted against the proxy A/S(t − τ ). The fitted values of Nknots and b for each parameter are indicated in the respective panels. Bottom rows: The same functions ⃗f are plotted against S(t − τ ) fo… view at source ↗
Figure 7
Figure 7. Figure 7: Relative flux reconstruction uncertainty of the PgLis model for protons, computed as a function of the smoothed solar proxy S and kinetic energy Ek for both polarity phases, obtained using the bootstrap procedure described in Sec. 2.7. flux at 1 AU becomes straightforward once the solar in￾put proxy S and the corresponding magnetic polarity A are specified. At any given epoch t, the time-lag relation allow… view at source ↗
Figure 8
Figure 8. Figure 8: Top: PgLis predictions spanning several solar cycles at ∼ 440 MeV for protons, compared with long-term measurements from SOHO/EPHIN and BESS as validation datasets, and with PAMELA Adriani et al. (2013); Martucci et al. (2018) and AMS–02 Aguilar et al. (2021) data used for calibration. Middle: Same long-term prediction for multichannel proton fluxes at ∼ 1.3 GeV. Bottom: PgLis predictions compared with hel… view at source ↗
Figure 9
Figure 9. Figure 9: PgLis predictions compared with monthly ACE fluxes for carbon, oxygen, iron, and magnesium, spanning several solar cycles. The orange data points indicate the portion of the carbon dataset used for calibration, while the blue points represent the validation datasets. The shaded bands represent the 68% C.L. model uncertainty. The average relative reconstruction errors for each validation dataset are reporte… view at source ↗
Figure 10
Figure 10. Figure 10: PgLis predictions compared with monthly AMS-02 Aguilar et al. (2025) fluxes for carbon, oxygen, nitrogen, and lithium, shown at selected energy bins, from May 2011 to November 2022. These measurements serve as validation datasets. The model, coupled with the SSN forecasts from Asikainen & Mantere (2023), provides a prediction for the full 25th solar cycle. The shaded bands represent the 68% C.L. model unc… view at source ↗
Figure 11
Figure 11. Figure 11: Integrand QZ (Ek)· DZ /ΦZ (Ek)· JZ (Ek) of the effective dose rate in Eq. (18) for protons (blue), helium (red), and iron (green), computed from PgLis fluxes at a reference solar-minimum epoch. The integrand peaks in the range ∼ 100 MeV/n to ∼ 1 GeV/n, identifying the energy interval most responsible for the radiation dose. The strength of the present approach lies in its cali￾bration procedure, which com… view at source ↗
Figure 12
Figure 12. Figure 12: Monthly averages of the whole-body effective dose rate (in Sv yr−1 ), computed using ICRP fluence-to-dose coef￾ficients and quality factors, induced by GCRs (Z = 1–28) modeled with PgLis for an astronaut in ISS orbit during an EVA. The calculation assumes sex-averaged reference body proportions, accounts for the geomagnetic cutoff using Størmer’s dipolar approximation, and considers three shielding config… view at source ↗
read the original abstract

The modulation of galactic cosmic rays, driven by the evolution of the heliospheric magnetic field, strongly influences the intensity of cosmic rays reaching near-Earth space. Characterizing this process is crucial both for advancing our understanding of cosmic-ray transport and for assessing radiation exposure and related hazards in space environments. Here we present a newly developed forecasting framework built on a numerical description of charged particle transport in the heliosphere and its dependence on solar activity, designed for the long-term forecasting of galactic cosmic-ray fluxes. It solves a one-dimensional, spherically symmetric form of the Parker transport equation, including diffusion, solar-wind advection, and adiabatic energy losses. The model has been validated using multi-species flux measurements from space-based experiments: PAMELA, AMS-02, and ACE. Its strategy is based on Hilbert-Huang transform filtering and cross-correlation between delayed solar proxies and effective model parameters. Our charge-sign- and rigidity-dependent parametric description of the diffusion-advection processes yields good overall agreement with the data, as shown by the reconstruction uncertainty. The robustness of this approach is validated across a broad set of multichannel datasets covering different particle species, energy ranges, and phases of solar activity, supporting its applicability to space radiation monitoring and forecasting. Furthermore, when coupled with solar-proxy forecasting models, it enables decadal-scale predictions of galactic cosmic-ray fluxes, thereby supporting long-term planning and radiation-risk assessment for future space missions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript presents a one-dimensional spherically symmetric numerical model based on the Parker transport equation (diffusion, solar-wind advection, adiabatic losses) for long-term forecasting of galactic cosmic-ray (GCR) fluxes. Effective charge-sign- and rigidity-dependent diffusion and advection parameters are obtained via Hilbert-Huang transform filtering and cross-correlation with solar proxies on historical PAMELA, AMS-02, and ACE multi-species flux data; the resulting parametric description is reported to yield good reconstruction agreement, and coupling to solar-proxy forecasts is claimed to enable decadal-scale GCR predictions for space radiation applications.

Significance. If the derived proxy-parameter relationships prove stationary, the framework could supply a practical, data-constrained tool for decadal GCR forecasting that complements existing modulation models. The reported multi-channel validation across particle species, energies, and solar phases constitutes a concrete strength for monitoring applications.

major comments (2)
  1. [Abstract] Abstract: The central forecasting claim—that coupling the fitted model to solar-proxy forecasts 'enables decadal-scale predictions'—rests on the untested assumption that the Hilbert-Huang-derived lag relationships and rigidity/charge-sign scalings between proxies and effective parameters remain valid outside the training solar cycles. No hold-out validation across independent cycles or forward-test of transferability is described, directly undermining the decadal utility asserted in the abstract and conclusion.
  2. [Abstract] Abstract and validation description: Effective diffusion-advection parameters are obtained by cross-correlation and fitting to the same historical flux datasets (PAMELA/AMS-02/ACE) subsequently used to demonstrate 'good overall agreement' and 'reconstruction uncertainty.' This circularity means the reported validation quantifies consistency with the fitted relationships rather than independent predictive skill, which is load-bearing for any claim of applicability beyond the training interval.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central forecasting claim—that coupling the fitted model to solar-proxy forecasts 'enables decadal-scale predictions'—rests on the untested assumption that the Hilbert-Huang-derived lag relationships and rigidity/charge-sign scalings between proxies and effective parameters remain valid outside the training solar cycles. No hold-out validation across independent cycles or forward-test of transferability is described, directly undermining the decadal utility asserted in the abstract and conclusion.

    Authors: We acknowledge that no explicit hold-out validation on solar cycles fully independent of the fitting data is presented. The cross-correlations and scalings were derived from the available multi-cycle datasets (PAMELA covering cycle 24, AMS-02 spanning cycle 24 into 25, and ACE over multiple cycles), with consistency checked across species and phases. This provides support for the parametric form but does not constitute a forward test of stationarity. We agree the forecasting language should be qualified. In revision we will adjust the abstract and conclusions to state that the framework, when coupled to solar-proxy forecasts, offers a basis for decadal predictions whose reliability depends on the stationarity of the derived relationships, which remains to be verified. revision: yes

  2. Referee: [Abstract] Abstract and validation description: Effective diffusion-advection parameters are obtained by cross-correlation and fitting to the same historical flux datasets (PAMELA/AMS-02/ACE) subsequently used to demonstrate 'good overall agreement' and 'reconstruction uncertainty.' This circularity means the reported validation quantifies consistency with the fitted relationships rather than independent predictive skill, which is load-bearing for any claim of applicability beyond the training interval.

    Authors: The reported agreement is indeed a reconstruction using parameters fitted via proxy cross-correlation on the same historical fluxes. This is the standard procedure for determining effective transport coefficients in modulation models. The value of the validation lies in the demonstrated consistency of the single charge-sign- and rigidity-dependent parametric description across multiple species, energies, and solar phases. We will revise the manuscript to describe the comparison explicitly as reconstruction and to clarify that claims of applicability beyond the training interval rest on the assumption that the proxy-parameter relationships remain valid, without independent predictive testing in the present work. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a numerical solution of the 1D Parker transport equation whose effective diffusion-advection parameters are calibrated via Hilbert-Huang filtering and cross-correlation against historical PAMELA/AMS-02/ACE flux data. Validation consists of showing agreement between the calibrated model and the same multichannel datasets; forecasting is described as coupling the calibrated model to separate solar-proxy forecasts. No quoted equation or derivation step reduces the claimed GCR fluxes or decadal predictions to the input data by construction. The transport physics supplies independent content, the empirical calibration is standard, and no self-citation chain or uniqueness theorem is invoked to force the result. The stationarity assumption for future cycles is an extrapolation risk, not a circular reduction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim depends on the 1D spherical symmetry approximation, the existence of stable cross-correlation mappings between solar proxies and transport parameters, and the assumption that these mappings generalize beyond the fitted interval. No new particles or forces are introduced.

free parameters (2)
  • charge-sign and rigidity dependent diffusion coefficients
    Determined via cross-correlation with solar proxies and adjusted to match observed fluxes
  • advection and adiabatic loss scaling factors
    Effective parameters tuned within the parametric description to reproduce multi-species data
axioms (1)
  • domain assumption One-dimensional spherically symmetric form of the Parker transport equation governs GCR modulation
    Invoked as the numerical foundation for particle transport in the heliosphere

pith-pipeline@v0.9.1-grok · 5795 in / 1389 out tokens · 54878 ms · 2026-07-01T02:47:49.382006+00:00 · methodology

discussion (0)

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