01
IMPLEMENTEDPropagate geometry
OMM/TLE → SGP4 → Earth-fixed state. Solve visibility, range, range rate, azimuth, elevation, and antenna off-axis angles.
R = ∥rₛ − rᵍ∥ · Ṙ = û · (vₛ − vᵍ)COMPUTATION PIPELINE
Every number has a path.
Trace orbit state to received I/Q, then follow calibrated RF integrals into a time-varying thermodynamic estimate and an explicit serve-or-withhold decision.4forward stages5inverse methods
FORWARD PASS
State → observable
02
IMPLEMENTEDTraverse atmosphere
Intersect each path with neutral-atmosphere pressure surfaces, cloud slabs, rain depth, and the three-dimensional reconstruction grid. Ionospheric propagation is outside the implemented operator.
Aₘᵥ = path length of ray m in voxel v03
HYBRIDForm complex channel
Accumulate spreading, antenna and polarization gain, gas/cloud/rain loss, excess phase path, and geometric plus atmospheric residual Doppler.
H = |H| exp[−j 2πf(R + ΔL)/c]04
IMPLEMENTEDEmulate receiver
Mix to complex baseband, wipe geometric Doppler, sample the ADC, estimate SNR and Eb/N₀, then evaluate illustrative QPSK lock and BER.
z[n] = h*x · exp(j2πΔfn/fₛ) + w[n]OBSERVATIONSA · φ · fD · τamplitude, phase, Doppler, delay
JOINT JACOBIANJ = ∂h / ∂[x, η]atmosphere and signed nuisance state
ATMOSPHEREx̂(x,y,z,t)retrieval, uncertainty, abstention
INVERSE PASS
Observable → atmospheric state
MethodUpdate familyBehaviorBest role
ARTSequential additiveFast nonnegative ray correctionsInteractive baseline
SIRTSimultaneous additiveBalances row and voxel sensitivitySmooth baseline
MARTSequential multiplicativeNaturally preserves positivityPositive fields
TV-ARTART + total variationRetains fronts while suppressing noiseBrowser twin
Joint 4DNonlinear weak constraintT / RH / liquid + clock / orbit / gainOffline research
JOINT LEO RF WEATHER RETRIEVAL
One RF event estimates weather and calibration states together.
Same-event delay and link-qualified attenuation drive a P.453 / P.676 / P.840 inverse for temperature, relative humidity, cloud liquid, dry pressure, clocks, orbit error, receiver gain, and beam gain. No weather history enters the retrieval.Temperature
0.475deg C RMSE | mean +/- 0.020 over four final-v2 trials96.1% lower than no-RF 12.097 | gate < 1
Relative humidity
2.939percentage points RMSE | mean +/- 0.664 over four final-v2 trials90.6% lower than no-RF 31.209 | gate < 4
Precipitable water
0.030mm RMSE | mean +/- 0.010 over four final-v2 trials99.8% lower than no-RF 17.842 | gate < 1
Dry pressure
1.146hPa RMSE | estimated jointlyStandardized RF residual MSE 0.910 | no measured pressure supplied
METHOD COMPARISON
Calibration and time coupling both matter.
MethodTRHPWVClass
No RFStructural atmosphere only
12.09731.20917.842baseline4D, no calibrationAtmosphere absorbs RF nuisance error
3.06616.4910.406baselineJoint 3DIndependent epochs with nuisance calibration
0.5323.2560.031baselineJoint 4DDynamic atmosphere and nuisance trajectory
0.4842.9950.029activeOracleTrue nuisance removed; undeployable bound
0.4242.5690.027boundWithout joint calibration, nuisance error is absorbed as weather. Joint 4D improves all three retrieval targets over independent-epoch joint 3D and approaches the oracle-calibrated bound.
STATE + GEOMETRY
Altitude diversity comes from limb and direct paths.
DIRECT PATHS768ground-to-LEO links
LIMB PATHS768tangent-localized links
CANDIDATE BANDS1412-190 GHz
LOSS GATE40 dBunusable samples omitted
4 x 4 x 4 cells | 6 epochs at 1 hourDry pressure is estimated jointly and coupled vertically by a nonlinear hypsometric constraint; no measured pressure is supplied to the inverse.
Candidate bands are sensing-design frequencies, not a claim that today's commercial LEO radios expose all 14 channels.
FROZEN SEVEN-EVENT CENSUS
Every public-cube event meets every weather gate and the RF closure gate.
Pooled event RMSE | mean +/- sample standard deviation
Temperature0.425+/- 0.046 deg C
Relative humidity2.445+/- 0.543 points
PWV0.041+/- 0.026 mm
Worst event: 0.496 deg C, 3.583 RH points, 0.083 mm PWV.
Protocol and numerical audit
Weather gates7/7no trial deletion
RF closure7/7standardized residual MSE 0.899 +/- 0.010
Solver convergence6/7iteration criterion
Beryl-Houston reached the 20-iteration cap while still meeting all weather gates and the measurement-closure gate; it is retained and counted as a solver failure.
A predeclared inclusion rule ran the unchanged final-v2 configuration on every distinct compatible public cube. No case-specific tuning or trial deletion was allowed. The seven-event census tests breadth under a frozen configuration, but only NYC and Phoenix were untouched after the final-v2 design freeze.
PAIRED GEOMETRY ABLATION
Direct paths alone fail T and RH; curved limb paths recover altitude structure.
Direct only | 768 direct + 0 limb
Temperature1.515deg C RMSE
Relative humidity9.826percentage points RMSE
PWV0.117mm RMSE | 9,990 observations
Fails temperature and RH gates.
Direct + curved limb | 768 direct + 768 limb
Temperature0.437deg C RMSE
Relative humidity2.294percentage points RMSE
PWV0.038mm RMSE | 17,227 observations
Passes all requested gates.
Limb augmentation reduces T / RH / PWV RMSE by 71.1% / 76.7% / 67.6%. Same public truth cube, RF and nuisance seed, direct-observation noise, 14 bands, 40 dB gate, priors, and solver; no retuning.
PHOENIX ROBUSTNESS MATRIX
Group-robust weighting keeps corrupted RF from becoming weather.
ConditionEstimatorDecisionTRHPWV
Nominal Phoenixjoint 4DSERVE0.4952.5360.035
5% burst corruptionGaussianWITHHOLD1.5967.0570.329
5% burst corruptiongroup HuberSERVE0.5752.9610.059
30% link lossjoint 4DSERVE0.5472.9180.049
2x nuisance amplitudejoint 4DSERVE0.4392.2980.036
Combined stressgroup HuberSERVE0.6833.5620.080
TRUTH SEPARATION
Public weather drives the RF experiment, not the retrieval.
RF OSSE driven by public NOAA HRRR analysis cubes. HRRR supplies hidden atmospheric truth only for RF synthesis and scoring; it is not an inverse-model input. This is not field validation or a commercial-LEO accuracy claim.
Truth roleRF synthesis + scoring only
Inverse inputssame-event RF + trajectory + identities
Compute pathstatic evidence JSON in the public twin
LIMITS OF THE EVIDENCE
What this result does not establish.
- The receiver observations are simulated from public HRRR analyses, not collected from operational LEO links.- The 14 frequencies are candidate sensing channels; link-budget gating removes any modeled attenuation above 40 dB.- HRRR is model-analysis truth rather than independent radiosonde or radiometer truth.- Rain, ice, ray bending, scintillation, terrain, and hardware-specific channel effects remain outside this final-v2 experiment.
Mean offline solve time is 34.3 s; the website serves only these precomputed numbers and adds no runtime inversion cost.
Open machine-readable evidenceTHE IDENTIFIABILITY RULE