Active-learning prioritization of rain-gauge densification under budget constraints: a computer-science evaluation of flood-forecast skill gain per instrument dollar

Rain-gauge networks in data-sparse basins are typically densified by spatial coverage criteria rather than by the marginal value of each additional instrument to flood-forecast skill. the present analysis formalises gauge densification as a sequential decision problem in which a learner selects the next gauge location from a candidate set, observes a proxy label, and updates a surrogate of forecast-relevant rainfall error, subject to a fixed budget. We derive a cost-normalised active-learning criterion, the marginal skill gain per instrument dollar, and show that under a submodular surrogate the greedy policy admits a constant-factor approximation guarantee relative to the optimal budget-feasible subset. The framework is evaluated on a real reanalysis record for Kampala, Uganda: NASA POWER monthly MERRA-2 air temperature at 2 m, 312 monthly observations from 2000 to 2023, with mean 22.443 C, standard deviation 1.043 C, and a linear trend of -0.1564 C per decade. The trend is small relative to interannual variability, which we use to motivate the variance-reduction objective that the active learner optimises. We report the design of the prioritisation procedure, the approximation bound, and the conditions under which cost-normalised greedy selection diverges from coverage-maximising selection. The contribution is a computer-science formalisation of instrument placement as budgeted active learning, with an explicit skill-per-dollar objective and a provable guarantee, rather than a new empirical forecast study.