INTERACTIVE VERTICAL RESOLUTION LAB
Excess delay is a weighted sum of layer refractivity, and the weights are just geometric path lengths. In the flat-Earth limit that path length is Δh / sin ε for every layer alike — the elevation only changes a common scale factor. Stack a hundred elevations and you get a hundred copies of one number, the zenith total delay. The operator is rank 1.
A spherical shell makes the true chord non-separable, so the weight vectors do rotate apart — by about 0.27° between 60° and 20° elevation, and roughly 12.6° by the time you reach 2°. Two measurements a fraction of a degree apart cannot separate two unknowns once noise is present, which is why the information sits exactly where multipath and mapping-function error are worst.
For a receiver at elevation ε ≥ 0 the tangent radius is p = r₀·cos ε ≤ r₀, so the tangent point lies behind the receiver, underground. Lift the receiver above the layers and it moves into the atmosphere, where it sweeps downward and each arc becomes near-independent. Separate the receivers horizontally instead and their rays cross. Both are geometry changes; neither is a satellite-count change.
OBSERVING GEOMETRY
THE CONTROLS THAT MATTER
Every ray climbs monotonically away from the receiver, on the left, so each one integrates the whole column above it at once. Shallower rays are longer everywhere, not preferentially low down.
Both axes are fixed by the scale buttons alone, never by the configuration, so altitude and pointing can be compared without the frame moving underneath them. The width is the span of the widest ray the frame can hold — the one grazing at 0.3 km. An up-looking ray is genuinely short next to an occultation ray, and at this fixed scale it looks it.
Every ray here is a straight line in space — this model has no refraction. The bend is the horizontal axis. Plotting altitude against ground-track distance flattens a curved Earth, so a straight chord falls away from the horizontal by x²/2R: about 0.8 km over 100 km of track, but 12.6 km over 400 km. An up-looking ray crosses only tens of kilometres, so the sag stays invisible against its climb; an occultation ray crosses hundreds, so the sag becomes the whole shape.
INFORMATION CONTENT
Bars are singular values on a log scale. Only the bars clearing the floor add usable information; everything below is drowned by the 1.00 mm delay noise.
A layer with a kernel diagonal near zero is being supplied by the prior, not by the measurement. Up-looking geometry leaves almost the whole column in that state.
L = 10⁻⁶ Σⱼ Nⱼ Δsⱼ(ε)Δsⱼ = √((R+hⱼ₊₁)² − p²) − √((R+hⱼ)² − p²), p = (R+h_rx)·cos εA = Ãᵀ(ÃÃᵀ + I)⁻¹Ã, DOFS = tr(A)d_Fresnel = 2√(λD) = 478 m at 1.575 GHzIn the flat-Earth limit Δsⱼ = Δhⱼ/sin ε, identical in shape for every layer, so the operator collapses to rank 1 and all elevations measure one number. Curvature breaks that only near the horizon; a tangent point breaks it completely. DOFS is an information measure, not an accuracy claim: it counts what the geometry constrains, and says nothing about bias, calibration, or field validation.