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Algorithms

The International Reference Ionosphere (IRI) is the internationally recognized standard empirical model of the Earth's ionosphere, sponsored by the Committee on Space Research (COSPAR) and the International Union of Radio Science (URSI). This document outlines the scientific, empirical, and mathematical models implemented in the pure Rust codebase of iri2020-rust.

Core Simulation Engine

The main execution flow is controlled by the irisub.rs module, translating the legacy IRI_SUB control routine.

  • Physical Quantities: Computes electron density (\(N_e\)), electron temperature (\(T_e\)), ion temperature (\(T_i\)), neutral temperature (\(T_n\)), and relative percentage abundances of positive ions (\(O^+\), \(H^+\), \(He^+\), \(O_2^+\), \(NO^+\)).
  • Execution Settings: Evaluates 50 user-toggleable control flags (jf) that specify model options, index overrides, and peak density/height selections.
  • Reference: Bilitza et al., 2022 (Journal of Space Weather and Space Climate)

Total Electron Content (TEC) Integration

The module iritec.rs computes the integrated electron column density along a vertical path from a lower altitude boundary (\(h_{start}\)) to an upper altitude boundary (\(h_{end}\)):

\[\text{vTEC} = \int_{h_{start}}^{h_{end}} N_e(h) \, dh\]
  • Numerical Quadrature: Employs a multi-segment trapezoidal integration method with 1000 sub-steps per segment.
  • Segment Matching: Features precise segment-end state recovery. In compliance with original empirical constraints, the integrated topside (\(TEC_{top}\)) and bottomside (\(TEC_{bot}\)) bounds match the F2-peak options to preserve output consistency.
  • Reference: Bilitza, 2001 (Radio Science)

Geomagnetic Field Model (IGRF-13)

The module igrf.rs and static data in igrf_coeff.rs implement the 13th generation of the International Geomagnetic Reference Field (IGRF).

  • Mathematical Form: Evaluates the main geomagnetic field potential (\(V\)) as a spherical harmonic expansion of degree \(N=13\):
\[V(r, \theta, \phi) = a \sum_{n=1}^{N} \sum_{m=0}^{n} \left(\frac{a}{r}\right)^{n+1} \left[ g_n^m \cos(m\phi) + h_n^m \sin(m\phi) \right] P_n^m(\cos\theta)\]

where \(a\) is the Earth's mean radius, \(r\) is the radial distance, \(\theta\) is the geocentric co-latitude, \(\phi\) is the east longitude, \(g_n^m\) and \(h_n^m\) are Gauss coefficients, and \(P_n^m\) represents Schmidt quasi-normalized Legendre functions.

  • Temporal Extrapolation: Interpolates the Gauss coefficients linearly between five-year epochs and applies secular variation models for predictive intervals beyond the latest epoch.
  • Reference: Alken et al., 2021 (Earth, Planets and Space)

Neutral Atmosphere Model (CIRA-86 / NRLMSISE-00)

The module cira.rs and coefficients in cira_coeff.rs implement the COSPAR International Reference Atmosphere model.

  • Scientific Foundation: NRLMSISE-00 models the neutral temperature profile and the number densities of major/minor thermospheric species (\(He\), \(O\), \(N_2\), \(O_2\), \(Ar\), \(H\), \(N\), and anomalous \(O\)).
  • Numerical Formulation: Utilizes cubic spline interpolations and spherical harmonic coefficients mapped globally to calculate the diurnal, semidiurnal, and seasonal variations of neutral densities.
  • Reference: Picone et al., 2002 (Journal of Geophysical Research)

ROCSAT-1 Vertical Plasma Drift

Equatorial vertical ion drift velocities in the F-region are calculated in rocdrift.rs using the 64,900 coefficient empirical tensor stored in rocdrift_coeff.rs.

  • Physical Model: Represents empirical F-region vertical drifts measured by the ROCSAT-1 satellite.
  • Method: Evaluates drifts based on local time, longitude, and solar activity (\(F_{10.7}\) index) using multidimensional bilinear interpolations.
  • Reference: Fejer et al., 2008 (Journal of Geophysical Research)

F2-Layer Bottomside Thickness

The thickness parameter \(B_0\) (which scales the profile width below the F2 peak) and the shape parameter \(B_1\) (which controls the profile slope) are computed in b0_b1_model.rs.

Electron Density Profile Construction

The vertical electron density profile \(N_e(h)\) is synthesized in xe_profile.rs by dividing the ionosphere into structural layers:

  • Profile Layers: Bottomside D-region, E-region, E-F valley, F1-region, and F2-region.
  • Formulation: Utilizes Rawer's Epstein LAY-functions to construct smooth, continuous analytical profiles:
\[\text{LAY}(h; h_i, W_i) = \ln\left(1 + e^{\frac{h-h_i}{W_i}}\right) - \ln\left(1 + e^{\frac{h_m-h_i}{W_i}}\right) - \frac{h-h_m}{W_i \left(1 + e^{\frac{h_m-h_i}{W_i}}\right)}\]

where \(h_i\) is the transition height and \(W_i\) is the transition width parameter.

D-Region Electron Density Models

Implements lower ionosphere models (altitudes 60 to 90 km) in d_region.rs and iridreg.rs.

Ion Composition Model

The positive ion density profiles (\(O^+\), \(H^+\), \(He^+\), \(O_2^+\), \(NO^+\)) are computed in ioncom.rs with coefficients defined in calion_coeff.rs.

  • Physical Formulation: Solves the altitude transitions where atomic oxygen ions dominate the F2 region, transitioning to molecular species (\(O_2^+\), \(NO^+\)) in the E region, and light ions (\(H^+\), \(He^+\)) in the topside plasmasphere.
  • Reference: Bilitza, 1990 (Advances in Space Research)

Coordinate Systems & Field-Line Tracing (GEO-CGM)

The module iriflip.rs implements geomagnetic coordinate transformations and magnetic field line tracing using spatial arrays in cormag_data.rs.

  • Coordinate Conversions: Converts between Geocentric coordinates and Corrected Geomagnetic (CGM) coordinates.
  • Numerical Integrator: Employs a 4th-order Runge-Kutta integration scheme with adaptive step scaling to trace magnetic field lines from geographic points to their conjugate points or F-region magnetic footprints.
  • Reference: Gustafsson et al., 1992 (Journal of Atmospheric and Terrestrial Physics)