Underwater localization is challenging as its efficacy is affected by propagation delays, motion-induced doppler shift, phase and amplitude fluctuations, multipath interference etc that are inherent in underwater acoustic channels. In this paper, we consider a recently proposed Underwater Positioning Scheme, which offers unique localization only in a finite region. We quantify the conditions for unique localization and propose a variant that offers unique localization with high probability regardless of the reference and unknown node deployment. We demonstrate the trade-offs between both schemes in terms of localizability space, localization latency and energy consumption.