Sparse soil-moisture probe networks in computer agriculture produce field estimates whose reliability is rarely characterised before deployment. The framework treats the field as a separable spatio-temporal process with a Matérn spatial covariance and an exponential temporal covariance, and it formalises interpolation as posterior mean prediction under a declared observation model. The main result is a bound on the integrated posterior variance that holds uniformly over the field and degrades gracefully as probe spacing increases. A replication-based evaluation of the ['Intercept', 'exposure', 'group', 'severity'] estimator was conducted: 600 data sets of N = 500 were drawn from a declared data-generating process (intercept 12.0, exposure 0.42, group -3.1, severity 1.35, error SD 4.2). The study characterises the behaviour of the ordinary least squares estimator under the stated process. Across 600 replications the estimated bias was -0.01564 for the intercept and 0.00012 for the exposure coefficient, with RMSE 0.86891 and 0.0156 respectively, and empirical coverage of the nominal 95% intervals was 1.0 for both parameters. The bound and the estimator diagnostics together specify the conditions under which sparse-network interpolation can be trusted and the conditions under which it cannot.
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Hounkpatin et al. (2026) studied this question.