A Reduced One-Dimensional Source–Transport–Observation Analysis for Soil-Gas Interpretation at the Soil–Atmosphere Interface

Soil-gas observations can retain a source-related response while remaining ambiguous with respect to source amplitude, source depth, transport state, water state, and measurement support. This study develops a reduced one-dimensional source–transport–observation analysis for gas-continuous unsaturated soils. Integral-normalized source kernels, a finite-volume solver, a raw scaled-coordinate Jacobian, common-threshold nuisance projection, and an auxiliary noise-scaled check are combined in a reproducible workflow. At a display tolerance of 0.05, the primary structural convention normalizes parameter columns over the complete 11-row observation universe before extracting observation subsets. Under this convention, a carbon dioxide (CO2) surface-flux observation retains one projected amplitude direction; within-set normalization reduces that restricted result to zero. The CO2 profile, ideal-state-constraint set, and full diagnostic set retain ranks of 1/2, 2/3, and 3/4, respectively, under both conventions. The noise-scaled calculation places one projected source response at or above the illustrative one-standard-deviation threshold within a numerical tolerance of 1 × 10−10. A published-data worked example shows why water and carbonate context are required before a gas-phase CO2 deficit is interpreted as a source decrease. The result is a pre-field screening method, not a site-calibrated inversion.