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Bathymetry, i.e., depth, imaging in a river is of crucial importance for shipping operations and flood management. With advancements in sensor technology and plentiful computational resources, various types of indirect measurements can be used to estimate high-resolution river bed topography. In this work, we image river bed topography using depth-averaged quasi-steady velocity observations related to the topography through the 2-D shallow water equations. The principal component geostatistical approach (PCGA), a fast and scalable variational inverse modeling method powered by low-rank representation of covariance matrix structure, is presented and applied to two riverine bathymetry identification problems. To compare the efficiency and effectiveness of the proposed method, an ensemble-based approach is also applied to the test problems. It is demonstrated that PCGA is superior to the ensemble-based approach in terms of computational effort and accuracy because of the successive linearization of the forward model and the optimal low-rank representation of the prior covariance matrix. To investigate how different low-rank covariance matrix representation by the two approaches can affect the solution accuracy, we analyze the direct survey data of the river bottom topography in the test problem and show that PCGA utilizes more efficient and parsimonious choice of the solution basis than the ensemble-based approach. Geostatistical analysis performed on the direct survey data also confirms the validity of the chosen covariance model and its structural parameters.