We are concerned with the problem of optimally measuring an accessible signal under a total cost constraint, in order to estimate a signal which is not directly accessible. An important aspect of our formulation is the inclusion of a measurement device model where each device has a cost depending on the number of amplitude levels that the device can reliably distinguish. We also assume that there is a cost budget so that it is not possible to make a high amplitude resolution measurement at every point. We investigate the optimal allocation of cost budget to the measurement devices so as to minimize estimation error. This problem differs from standard estimation problems in that we are allowed to design the number and noise levels of the measurement devices subject to the cost constraint. Our main results are presented in the form of tradeoff curves between the estimation error and the cost budget. Although our primary motivation and numerical examples come from wave propagation problems, our formulation is also valid for other measurement problems with similar budget limitations where the observed variables are related to the unknown variables through a linear relation. We discuss the effects of signal-to-noise ratio, distance of propagation, and the degree of coherence (correlation) of the waves on these tradeoffs and the optimum cost allocation. Our conclusions not only yield practical strategies for designing optimal measurement systems under cost constraints, but also provide insights into measurement aspects of certain inverse problems.