This work covers and combines three themes concerning the Weibull distribution in the heavy-tailed region, as is relevant to sea clutter: (i) Characterizing the Weibull distribution as a compound clutter model and deriving a computationally tractable form for its implied texture distribution. (ii) Computing the distribution of a positively weighted sum of independent identically distributed (iid) Weibull random variables -- facilitated by the compound formulation, with emphasis on the tail. (iii) Applying the results for the case of a discounted sum of Weibull random variables to modelling the detection threshold of radar systems employing scan-to-scan feedback integration in the presence of Weibull sea clutter plus thermal noise.