Yue (Michael) Ying

Observation–State Correlation Explorer

A single temperature observation (red circle) is assimilated from a smooth 2-D field. The ensemble gives an estimate of how that observation correlates with the temperature state at every grid point — the core quantity of covariance-based data assimilation, reduced to a single variable so the idea is unmistakable. The left panel shows where the uncertainty comes from: the ensemble perturbs only the blob's position (std 50 km), so every member's 4 K contour (where T = 4 K) is the same circle displaced somewhere else — a band of coloured rings. The correlation map turns that position uncertainty into the covariance the filter uses.

Try it: drag the member slider to walk through the members — the selected member's 4 K contour lights up, its blob centre is marked, and its point is ringed in the scatter — and click or drag on a map to move the black state marker: the correlation map and the scatter respond live.

Truth T + ensemble 4 K contours

Corr(Obs T, State T)

r = —

50

  • T observation (assimilated)
  • state location (i, j)
  • truth 4 K contour
  • member 4 K contours
  • selected member + blob centre
  • ensemble members (scatter)
  • observation value
  • truth value at (i, j)

Reading the panels

1 · The truth field and the 4 K contours — the “answer key” we are trying to estimate from sparse observations: a single warm blob (peak ≈ 7.9 K, radius scale 100 km) on a 50×50 grid with 10 km cells. Its 4 K contour is the dark ring at radius ≈ 59 km. Every member’s blob centre is displaced by up to ~100 km, so its 4 K contour is the same ring shifted somewhere else — the band of thin coloured rings is the position uncertainty. The slider walks through the members: the selected ring thickens, its blob centre is marked, and its point is ringed in the scatter. The black square marks the state location (i, j) you are inspecting.

2 · The correlation map — how the observation covaries with the state at every grid point, estimated from the 100-member ensemble. The ensemble perturbs only the blob’s position (std 50 km), so the pattern is a clean dipole: +0.98 right at the observation, −0.64 on the far side, fading with distance. Position uncertainty is the classic mechanism that produces strong negative correlation.

3 · The scatter — the joint distribution of the observation ensemble (x) and the state ensemble at the marker (y). Each dot is one member; the ringed dot is the member selected by the slider — where its 4 K contour sits in panel 1 decides where it falls here: a member whose blob is shifted toward the marker reads warmer at the state location, one shifted away reads colder. The amber line is the observed value, the ink line the truth. Move the marker across the dipole and watch the slope — and the reported r — flip sign.