D-LEGOS / NUMERICAL EXPERIMENT / 6 NOVEMBER 2026

Reconstructing a field from sparse observations

A coupled scalar model is evaluated against withheld states to examine whether sparse observations improve reconstruction, preserve the total quantity and strengthen subsequent prediction.

This is a controlled, dimensionless scalar-diffusion experiment. It does not use Caribbean meteorological observations.

Explore results ↓Research note ↗Source code ↗

RECONSTRUCTION AND OPEN-LOOP FORECAST

Numerical comparison

Eight synthetic observations constrain the first 40 steps. The remaining 40 steps test prediction without further observations.

State to compare Observation gain 0.65 Spatial influence (cells) 1.7 Update interval

Model step 0 / 80

Initial state — no observations yet

Reference state Generated only for the synthetic test

Conserved observation update Model under evaluation

Absolute error Same fixed colour scale in every frame

Outlined circles = eight observation cells. Small squares = eight separate held-out evaluation cells. Colour indicates a dimensionless scalar, not measured temperature or rainfall.

Held-out analysis RMSE · steps 1–40—

Held-out forecast RMSE · steps 41–80—

Maximum total-budget drift—

Where error accumulates

Held-out-cell RMSE at each step. Shading marks the period with observations; the right side is a forecast without new data.

Model onlyConserved updateUnconstrained updateWrong observationsPersistence

Compare every candidate

Lower is better for RMSE. A near-zero scalar-total drift is a conservation check, not proof of forecast accuracy.

Candidate	Analysis RMSE	Forecast RMSE	Max budget drift

MODEL AND OBSERVATIONS

Numerical method

A reference scalar field evolves on a closed 8 × 8 grid with paired neighbour exchanges. A candidate model uses a different diffusion coefficient and initial spatial pattern. Eight cells supply observations during the analysis interval; another eight are reserved for evaluation.

A distance-weighted correction updates the candidate field. Subtracting the spatial mean of that correction preserves the scalar total. The unconstrained variant omits this adjustment. The misassigned-observation control tests the effect of incorrect sensor locations.

Mathematical formulation, results and sources

INTERPRETATION

Reconstruction is not prediction

Error is measured on held-out cells, first while observations are supplied (steps 1–40), then without further updates (steps 41–80). This separates agreement with available observations from subsequent predictive skill.

The conserved quantity is a dimensionless scalar. These results are a numerical component study, not validation of atmospheric mass, moisture or energy budgets. The corresponding Caribbean reconstruction study is described in the research note.