Surrogate modeling of the variability of metocean conditions in space and in time during hurricanes is a crucial task for risk analysis on offshore structures such as offshore wind turbines, which are deployed over a large area. This task is challenging because of the complex nature of the meteorology-metocean interaction in addition to the time-dependence and high-dimensionality of the output. In this paper, spatio-temporal characteristics of surrogate models, such as Deep Neural Networks, are analyzed based on an offshore multi-hazard database created by the authors. The focus of this paper is two-fold: first, the effectiveness of dimension reduction techniques for representing high-dimensional output distributed in space is investigated and, second, an overall approach to estimate spatio-temporal characteristics of hurricane hazards using Deep Neural Networks is presented. The popular dimension reduction technique, Principal Component Analysis, is shown to perform similarly compared to a simpler dimension reduction approach and to not perform as well as a surrogate model implemented without dimension reduction. Discussions are provided to explain why the performance of Principal Component Analysis is only mediocre in this implementation and why dimension reduction might not be necessary.