  We present a generalization of continuous position measurements that accounts
for a spatially inhomogeneous measurement strength. This describes many real
measurement scenarios, in which the rate at which information is extracted
about position has itself a spatial profile, and includes measurements that
detect if a particle has crossed from one region into another. We show that
such measurements can be described, in their averaged behavior, as
stochastically fluctuating potentials of vanishing time average. Reasonable
constraints restrict the form of the measurement to have degenerate outcomes,
which tend to drive the system to spatial superposition states. We present the
results of quantum-trajectory simulations for measurements with a step-function
profile (a "which-way" measurement) and a Gaussian profile. We find that the
particle can coherently reflect from the measurement region in both cases,
despite the stochastic nature of the measurement back-action. In addition, we
explore the connection to the quantum Zeno effect, where we find that the
reflection probability tends to unity as the measurement strength increases.
Finally, we discuss two physical realizations of a spatially varying position
measurement using atoms.
