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Optimal experimental design (OED) is the general formalism of sensor placement and decisions about the data collection strategy for engineered or natural experiments. This approach is prevalent in many critical fields such as battery design, numerical weather prediction, geosciences, and environmental and urban studies. State-of-the-art computational methods for experimental design, however, do not accommodate correlation structure in observational errors produced by many expensive-to-operate devices such as X-ray machines or radar and satellite retrievals. Discarding evident data correlations leads to biased results, poor data collection decisions, and waste of valuable resources. We present a general formulation of the OED formalism for model-constrained large-scale Bayesian linear inverse problems, where measurement errors are generally correlated. The proposed approach utilizes the Hadamard product of matrices to formulate the weighted likelihood and is valid for both finite- and infinite- dimensional Bayesian inverse problems. We also discuss widely used approaches for relaxation of the binary OED problem, in light of the proposed pointwise weighting approach, and present a clear interpretation of the relaxed design and its effect on the observational error covariance. Extensive numerical experiments are carried out for empirical verification of the proposed approach by using an advection-diffusion model, where the objective is to optimally place a small set of sensors, under a limited budget, to predict the concentration of a contaminant in a bounded domain.