Abstract:
Estimating field-scale indigenous nitrogen supply
from soil organic carbon (SOC) requires propagating sparse
point observations across a spatially connected landscape. This
paper formalises that propagation as a diffusion process on a
graph. We represent a paddy-field monitoring network as a
weighted graph G = (V, E), define the combinatorial graph
Laplacian L = D − A, and show that the heat equation on
G has closed-form solution x(t) = exp(−tL)x0, converging to
the graph mean at a rate governed by the algebraic connectivity
(the Fiedler value). We then pose field-scale SOC/indigenous-
nitrogen-supply reconstruction from partial observations as a
Tikhonov-regularised graph-diffusion problem, derive its closed-
form solution and its relationship to Gaussian-field/ harmonic-
function semi-supervised learning. A worked numerical example
on a twelve-node illustrative paddy-catchment network seeded
within the SOC range documented for Sri Lankan paddy top
soils demonstrates the full pipeline: Laplacian spectral decom-
position, heat-kernel diffusion, and Tikhonov reconstruction at
four withheld nodes, achieving a mean absolute error of 0.189
percentage points of SOC. We compare this to empirical Bayesian
Kriging cross-validation error reported for a real national soil
database and discuss the conditions under which graph diffusion
is complementary to, rather than a replacement for, geostatistical
interpolation.
