Various embodiments include apparatus and methods to operate a drilling operation relative to formation boundaries. The apparatus and methods can include operating one or more transmitters in a borehole in a formation having a thickness between two boundaries, selecting thickness models based on applying responses from operating the one or more transmitters such that the thickness of the formation is between the two thickness models, and generating a value of a distance to a nearest boundary based on linearization of the thickness models with respect to a long distance investigation parameter and a short distance investigation parameter. Additional apparatus, systems, and methods are disclosed.