A fundamental requirement in realistic ocean simulations and dynamical studies is the optimal estimation of gridded fields from the spatially irregular and multivariate data sets that are collected by varied platforms. In this work, we derive and utilize new schemes for the mapping and dynamical inference of ocean fields in complex multiply-connected domains and study the computational properties of these schemes. Specifically, we extend a Bayesian-based multiscale Objective Analysis (OA) approach to complex coastal regions and archipelagos. Such OAs commonly require an estimate of the distances between data and model points, without going across complex landforms. New OA schemes that estimate the length of shortest sea paths using the Level Set Method (LSM) and Fast Marching Method (FMM) are thus derived, implemented and utilized in idealized and realistic ocean cases. An FMM-based methodology for the estimation of total velocity under geostrophic balance in complex domains is also presented. Comparisons with other OA approaches are provided, including those using stochastically forced partial differential equations (SPDEs). We find that the FMM-based OA scheme is the most efficient and accurate. The FMM-based field maps do not require postprocessing (smoothing). Mathematical and computational properties of our new OA schemes are studied in detail, using fundamental theorems and illustrations. We find that higher-order FMM’s schemes improve accuracy and that a multi-order scheme is efficient. We also provide solutions that ensure the use of positive-definite covariances, even in complex multiply-connected domains.