Underwater Barrier Sensing Under a Network False-Alarm Budget

We study joint sensor placement and detector operating-point allocation for guarding a region against an adversarial crossing. Each sensor may adjust its detection threshold, while all sensors draw on a shared network-level false-alarm budget. For a Gaussian signal-excess model of a square-law detector, we give a necessary and sufficient condition for the negative-log miss hazard to be concave in a sensor's false-alarm probability. At fixed deployment, this yields a concave max-min allocation whose saddle point equalizes marginal barrier gain per unit alarm against a worst-case occupation measure. A perspective reformulation supplies linear cuts for the mixed-integer joint problem, and a separate certificate lower-bounds the continuum barrier strength independently of the graph used for path search. On a deep-water case study based on the canonical Munk profile, reallocating a fixed alarm budget raises the graph design objective by 14.9% at unchanged sensor count and alarm load, and the certificate establishes the barrier requirement on the interpolated continuum field, which the graph value alone cannot.