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Underwater optical communication (UWOC) is a potential solution for broadband connectivity in oceans and seas for underwater applications providing high data rate transmission with low latency and high reliability. Recent measurement campaigns suggest generalized Gamma distribution as a viable model for oceanic turbulence. In this paper, we analyze the performance of a UWOC system by modeling the vertical underwater link as a multi-layer cascaded channel, each distributed according to independent but not identically distributed (i.ni.d.) generalized Gamma random variables and considering the zero bore-sight model for pointing errors. We derive analytical expressions for probability density function (PDF) and cumulative distribution function (CDF) for the signal-to-noise ratios (SNR) of the combined channel and develop performance metrics of the considered UWOC system using outage probability, average bit error rate (BER), and ergodic capacity. We also derive the asymptotic expressions for outage probability and average BER to determine the diversity order of the proposed system for a better insight into the system performance. We use Monte-Carlo simulation results to validate our exact and asymptotic expressions and demonstrate the performance of the considered underwater UWOC system using measurement-based parametric data available for turbulent oceanic channels.