Compound flooding is a multidimensional consequence of the joint impact of multiple intercorrelated drivers, such as oceanographic, hydrologic, and meteorological. These individual drivers exhibit interdependence due to common forcing mechanisms. If they occur simultaneously or successively, the probability of their joint occurrence will be higher than expected if considered separately. The copula-based multivariate joint analysis can effectively measure hydrologic risk associated with compound events. Because of the involvement of multiple drivers, it is necessary to switch from bivariate (2D) to trivariate (3D) analyses. This study presents an original trivariate probabilistic framework by incorporating multivariate hierarchal models called asymmetric or fully nested Archimedean (or FNA) copula in the joint analysis of compound flood risk. The efficacy of the derived FNA copulas model, together with symmetric Archimedean and Elliptical class copulas, are tested by compounding the joint impact of rainfall, storm surge, and river discharge observations through a case study at the west coast of Canada. The obtained copula-based joint analysis is employed in multivariate analysis of flood risks in trivariate and bivariate primary joint and conditional joint return periods. The estimated joint return periods are further employed in estimating failure probability statistics for assessing the trivariate (and bivariate) hydrologic risk associated with compound events. The statistical tests found the fully nested Frank copula outperforms symmetric 3D copulas. Our work confirms that for practical compound flood risk analysis together with bivariate or univariate return periods, it is essential to account for the trivariate joint return periods to assess the expected compound flood risk and strength of influence of different variables if they occur simultaneously or successively. The bivariate (also univariate) events produce a lower failure probability than trivariate analysis for the OR-joint cases. Thus, ignoring the compounding impacts via trivariate joint analysis can significantly underestimate failure probability and joint return period.