Low-Earth-orbit navigation (scenario kind family)

7 scenario kinds of the engine v0.38.0. Advisory: analysis output from a research and engineering tool, not type-approved navigation equipment and not a safety-of-life service. Every label here is the engine's or the verification matrix's; a kind is MODELLED unless the matrix says otherwise (how a label is earned: the README's Evidence section).

Topic pages: LEO-PNT, LEO-SIGNAL, LEO-PASS, LEO-NAVMSG. Run one: kshana scenarios/leo-pnt-end-to-end.toml. All families: KINDS.md.

leo-signal#

LEO-PNT signal designs: parameterised low Earth orbit (LEO) positioning, navigation and timing (PNT) signals for any system, band and structure. A SIGNAL is a band (centre frequency, transmit bandwidth, ITU allocation RNSS/RDSS/MSS/EESS/non-RNSS/other) and components (acquisition, data, pilot; any subset), each with a modulation (BPSK(n) with fractional n, BOC(m,n), MBOC(6,1,p) or a flat OFDM-like band), a power share, optional FDMA sub-carrier offsets, a code length and a data rate. Signals come from compiled-in public preset files (xona-pulsar, iridium-stl, starlink-soo, centispace, generic-c-band, generic-bands; each with its source URL and every unpublished value labelled REPRESENTATIVE) or are written inline. PER SIGNAL: the band-limited power spectral density and the fraction of each component's power the transmit band passes (BPSK closed form beside the numeric value), the RMS (Gabor) bandwidth, the delay-lock-loop code-tracking thermal-noise jitter against C/N0 and early-late spacing from the band-limited coherent (or non-coherent) early-late formula of Betz & Kolodziejski 2009 with its unlimited-band textbook form and its vanishing-spacing Gabor bound, the ranging accuracy in metres, the acquisition search space (Doppler bins from the orbit's maximum Doppler plus oscillator and user terms, times code bins), square-law detection probability and the mean serial and code-parallel acquisition time, the spectral separation coefficient of the signal into GPS L1 C/A, Galileo E1, GPS L5, Galileo E5a, E5b and E5 AltBOC and of each into the signal with the C/N0 degradation, and the CW, wideband and matched J/S the tracked component tolerates, from the spectrum kind's own SSC machinery. A BAND TRADE: first-order ionospheric delay (40.3 TEC/f^2), free-space loss, ranging accuracy at equal C/N0 and at equal EIRP, and jammer tolerance. Optional SHAPE CHECKS hold a design's spectrum against a described measurement (MODELLED consistency). VALIDATED: the BPSK power-in-band (90.3 % in the main lobe) and Gabor-bandwidth closed forms, the band-limited jitter reducing to the textbook coherent early-late form and to its Gabor bound, the offset BPSK spectral separation coefficient against its Parseval closed form, the AltBOC spectrum's unit area, and the maximum LEO Doppler against the published Xona Pulsar (32-34 kHz) and Iridium (36 kHz) figures. MODELLED: every design, power split, C/N0 and jammer level; EFQPSK, CSK and OFDM spectra are approximated.

Required fields: none. Optional: presets, signals, receiver, acquisition, compatibility, trade, shape_checks, psd_points.

Ledger: MODELLED 2 · VALIDATED 2. Full page, with the inputs, results, evidence and limitations: leo-signal.

kshana scenarios/leo-band-trade.toml

leo-pass#

LEO pass and link budget, band by band, for LEO-PNT (positioning, navigation and timing from low Earth orbit), system-agnostic: a user (ground, maritime, air or indoor, static or moving) and one or more LEO satellites from a DESIGNED PASS (a circular orbit placed so the user sees one pass of a chosen maximum elevation at a chosen time, sun-synchronous if asked), explicit ELEMENTS, a TLE (two-line element set) through the engine's SGP4, or a WALKER constellation built by the constellation-design code (the highest passes reported). Bands come from a named preset (generic multi-band UHF/L/S/C, generic C band, Xona Pulsar X1/X5, Iridium STL, Starlink as a Doppler-only signal of opportunity, CentiSpace, and an optional Celeste IOD preset marked WORKSHOP) or are written into the scenario; each preset states whether its numbers are PUBLIC (with URL), REPRESENTATIVE or WORKSHOP. Per satellite, band and epoch: elevation, azimuth, range, range rate and range acceleration; free-space path loss; EIRP and the satellite pattern gain (isoflux, Gaussian beam or flat); user antenna gain against elevation (patch, hemispherical or isotropic); gaseous attenuation (ITU-R P.676-10 Annex 2), rain attenuation (ITU-R P.838-3 coefficients and the P.618-14 procedure, from a rain rate and rain height), tropospheric scintillation (ITU-R P.618-14 section 2.4.1), building entry loss for an indoor user (ITU-R P.2109-2), polarisation mismatch, system noise temperature and C/N0 (carrier-to-noise density); Doppler and Doppler rate in closed form from the range rate and range acceleration, each checked against central differences; the first-order ionospheric delay 40.3 STEC/f^2 from a Klobuchar or vertical-TEC slant TEC scaled to the part of the ionosphere below the satellite. Band pairs get the ionosphere-free coefficients, noise amplification and code noise at the pass peak. MEO GNSS satellites (Galileo or GPS presets) are evaluated with the same antenna and noise model from their interface-document received powers, so the LEO bell-shaped pass and the flat GNSS carriers share one plot. An optional [iot] section gives time to first fix, energy per fix and battery life against duty cycle for a stated receiver power budget (MODELLED). Each band pair also gives the slant TEC its geometry-free combination recovers at the peak and that estimate's code-noise sigma (ionosphere sounding). An optional [spoofer] section counterfeits the signals of a claimed position that jumps or is pushed from the true one after an onset and runs two monitors with stated false-alarm and missed-detection probabilities: Doppler and pass-geometry consistency (measured range rates against those predicted from the orbits and an independent prior position and velocity, a generalised least-squares chi-square test over a window, on the GNSS, LEO and all channels) and cross-band consistency (the epoch-to-epoch step of each LEO band pair's model-corrected geometry-free combination), reporting per-epoch statistics and when each monitor detects (MODELLED). VALIDATED components: free-space loss, the P.838-3 coefficients against its Table 5, the P.618-14 rain and scintillation procedures and the P.2109 building entry loss against the ITU-R Study Group 3 validation examples, the IS-GPS-200 L1/L2 group-delay ratio, and the static-user maximum Doppler of a circular orbit against arXiv 2509.19551 Table 1. MODELLED: EIRPs, patterns, the Chapman fraction of TEC below the satellite and the whole pass.

Required fields: none. Optional: epoch, duration_s, step_s, user, atmosphere, ionosphere, satellite, leo_constellation, gnss, iono_free_pairs, iot, spoofer, name, description, skyline.

Ledger: MODELLED 7 · VALIDATED 5. Full page, with the inputs, results, evidence and limitations: leo-pass.

kshana scenarios/leo-pass-iridium.toml

leo-navmsg#

LEO navigation message: the broadcast ephemeris and clock of a low Earth orbit (LEO) positioning, navigation and timing satellite, for any orbit, carrier and model, with optional named presets (Xona Pulsar, Iridium, Starlink, CentiSpace, a representative C-band system, the ATOMIC zero-clock model, Celeste IOD). CONTENT: auxiliary data (SVID, issue of data, band, signal health), synchronisation (week, time of week, a second-order clock polynomial af0-af2 about toc), ephemeris in one of four models -- the Galileo OS SIS ICD 16-parameter Keplerian set; that set plus along-track, cross-track and radial correction polynomials; the Liu et al. 2025 22-parameter model; an ECEF polynomial with no clock terms (ATOMIC zero-clock) -- and other services (a Klobuchar broadcast set, NeQuick-G ai0-ai2 with the effective ionisation level, and system-time-to-UTC parameters with the ICD leap-second logic). FITTER: a truth orbit integrated with zonal J2-J6 or EGM2008 gravity to the chosen degree and drag, and a seeded free or steered clock; Levenberg-Marquardt fit of the Keplerian set on non-singular elements, then linear least-squares correction polynomials; clock fitted net of the relativistic term the user adds. ANALYSES: fit-interval-trade (signal-in-space range error, SISRE, with global-average weights for the orbit altitude, versus fit interval and update period, orbit-only and with clock), model-comparison (four models, parameter count, encoded bits, and Kshana's fit at the Liu et al. altitudes beside the published figures), midpass-update (the highest pass over a user, each message switch's position, clock and user-range jump, worst-geometry jump, PASS/FAIL against a continuity threshold), encode-decode (Kshana's own documented binary frame with CRC-24Q, a per-field half-step quantisation budget, a corrupted-frame check, a RINEX-4-style record export and import labelled a Kshana extension, and a CSV table). VALIDATED: the SISRE weights against the published MEO/GEO table (Montenbruck et al. 2018); the Galileo user algorithm against RTKLIB on real Galileo broadcast ephemerides; CRC-24Q against its catalogue check value and the RTCM 1005 example frame. MODELLED: every fit and SISRE figure is representation error against a Kshana-integrated truth (no orbit determination or prediction error); the Liu et al. comparison uses different orbits; the encoding and text formats are Kshana's own.

Required fields: none. Optional: name, description, seed, analysis, preset, orbit, clock, message, trade, comparison, midpass.

Ledger: MODELLED 5 · VALIDATED 6. Full page, with the inputs, results, evidence and limitations: leo-navmsg.

kshana scenarios/leo-navmsg-celeste-iod.toml

leo-pvt#

Fused medium-Earth-orbit (MEO) and low-Earth-orbit (LEO) positioning, navigation and timing (PNT) over ANY constellations: each [[system]] is a GNSS preset of the constellation-design kind, a LEO preset (xona-pulsar, iridium-stl, starlink-sop, centispace, generic-c-band, the atomic-zero-clock ephemeris model, or another preset file, each in its own file with its sources marked PUBLIC, WORKSHOP or DERIVED), Walker shells or explicit element sets, with a carrier, a chip rate, a carrier-to-noise density (C/N0) envelope from the elevation mask to the zenith, a signal-in-space range error (SISRE), a Doppler sigma and a clock model (its own receiver clock, so the inter-system bias is estimated, or a known broadcast offset). Four modes. DOPPLER: batch Gauss-Newton positioning from range rate with analytic partials, clock-drift and optional velocity states and an optional height constraint, single- or multi-satellite, the error against window length, the Doppler, Doppler-rate and jerk envelope of each LEO system, and the single-pass along-track and cross-track accuracy against the cross-track offset (the cross-track error grows without bound as the pass goes overhead; a single pass also has a mirror solution across the ground track). A Doppler-only signal of opportunity (starlink-sop) runs here. JOINT: per-epoch weighted least-squares pseudorange fixes from GNSS alone, LEO alone and both, per-measurement sigmas from a delay-lock-loop thermal-noise model at C/N0 plus the SISRE (or a fixed sigma, so a signal model can feed them), the estimated inter-system biases, and PDOP/HDOP/VDOP against the number of LEO satellites added. POLAR: satellites in view and DOP against latitude for GNSS, LEO and both. TIMING: time transfer to Coordinated Universal Time (UTC) from LEO satellites at a known position with a two-state clock filter whose process noise comes from the engine's oscillator classes, against C/N0 and the oscillator, and the IS-GPS-200 system-time-to-UTC expression; with [timing] trace = true every row also reports its time error and predicted sigma at every epoch. VALIDATED: the largest Doppler of an overhead-to-horizon pass against the published Iridium (up to 36 kHz) and Xona Pulsar X1 (32 to 34 kHz) figures. MODELLED: everything else; the DOP is checked against a hand-derived case, the fixes against their covariance, and the Doppler model against the range-acceleration identity.

Required fields: none. Optional: mode, seed, duration_s, step_s, user, dll, doppler, joint, polar, timing, system, skyline.

Ledger: MODELLED 3 · VALIDATED 2. Full page, with the inputs, results, evidence and limitations: leo-pvt.

kshana scenarios/leo-pvt-fjord-skyline.toml

leo-ppp#

Precise point positioning (PPP) convergence with GNSS only and with LEO augmentation: a float extended Kalman filter on ionosphere-free code and carrier phase with the static station coordinates, a white receiver clock, one inter-system bias per extra system, a random-walk zenith wet delay mapped by 1/sin(elevation) and a float ambiguity per satellite arc; the precise orbit-and-clock error is white and common to the code and phase of a satellite, so each satellite is one correlated two-row update in a Joseph-stabilised form. Measurements are simulated from the same model with seeded noise over several sites and seeds, and each case (GNSS only first, then each [[case]] of LEO systems, any constellation or preset) reports the median convergence time (the first epoch after which the horizontal and vertical errors stay below the thresholds, 10 cm by default), the fraction converged, the median error curves and the filter consistency (the run-averaged position NEES against the two-sided 95% chi-square band). Optionally beside the published Li et al. (2019, J. Geod. 93:749, doi 10.1007/s00190-018-1195-2) mid-latitude figures: 9.6 min for multi-GNSS shortened to 7.0, 3.2, 2.1 and 1.3 min with 60, 96, 192 and 288 LEO satellites. MODELLED: a consistency comparison of the trend, not a reproduction of that study's constellations, noise or stations; no ambiguity resolution, cycle slips, phase wind-up, tides or time-correlated product errors.

Required fields: none. Optional: seed, duration_s, step_s, runs, criterion_horizontal_m, criterion_vertical_m, compare_li_2019, noise, site, gnss, case.

Ledger: MODELLED. Full page, with the inputs, results, evidence and limitations: leo-ppp.

kshana scenarios/leo-ppp-convergence.toml

ntn-positioning#

Positioning from a 5G non-terrestrial network (NTN) downlink in the mobile-satellite service (MSS) S band (3GPP band n256, downlink 2170 to 2200 MHz; default carrier 2172.5 MHz): the Cramér-Rao bound (CRB) on time of arrival, c / (2 pi beta sqrt(2 (C/N0) T)) with beta the root-mean-square (Gabor) bandwidth of the flat OFDM spectrum (B/sqrt(12)), and on the Doppler of a complex tone, sqrt(3 / (2 pi^2 (C/N0) T^3)), for each signal (default a 5 MHz New Radio channel with 4.5 MHz occupied and a 200 kHz narrowband channel with 180 kHz occupied); then, over any LEO constellation, downlink time-of-arrival fixes with an unknown receiver clock (seeded, with the bound plus a network synchronisation error as the sigma), their PDOP and formal 3D sigma, and a single-satellite Doppler fix over the longest pass with the height held. MODELLED: the bandwidth-to-bound step is the textbook closed form, checked against numerical integration of the flat and band-limited BPSK spectra (internal consistency, no published worked figure pinned), and the positioning accuracy is a bound on a multipath-free channel with a stated synchronisation error, not an achieved result.

Required fields: none. Optional: seed, carrier_hz, signal, integration_s, doppler_integration_s, cn0_dbhz, sync_error_m, user, duration_s, step_s, system.

Ledger: VALIDATED. Full page, with the inputs, results, evidence and limitations: ntn-positioning.

kshana scenarios/ntn-5g-positioning.toml

leo-pnt-chain#

One LEO-PNT (positioning, navigation and timing from low Earth orbit) system end to end, for any constellation and signal: the leo-signal design named by signal_design (a public preset or an inline [signal] design) sets a [pass] band's centre, bandwidth, chip rate and EIRP split, the leo-pass link budget gives the tracked C/N0 and band-limited code jitter over the pass, a C/N0 line in sin(elevation) and the [navmsg] message's broadcast SISRE (fitted at the pass satellite's altitude) feed every LEO system of the [fusion] leo-pvt joint fix, and optionally the [ppp] leo-ppp cases; every hand-off is listed with its value and unit. MODELLED.

Required fields: signal_design, pass, fusion. Optional: name, description, seed, signal, navmsg, od_sisre_m, ppp.

Ledger: MODELLED. Full page, with the inputs, results, evidence and limitations: leo-pnt-chain.

kshana scenarios/leo-pnt-end-to-end.toml