Mat foundations are often used as a means of protecting buildings and other structures from excessive distortion due to differential settlements in the underlying ground. Once soil bearing capacity concerns have been addressed, the analysis of these foundations becomes a soil-structure interaction problem where the bearing pressure from the mat induces settlement in the underlying ground while localized settlement distorts the mat and redistributes the bearing pressure. An accurate representation of this soil-structure interaction is necessary to facilitate computations of the shear and flexural stresses in the mat and to develop an appropriate structural design. However, modeling and characterizing this system has long been a source of confusion and contention among both geotechnical and structural engineers. The soil response is typically characterized using the modulus of subgrade reaction, ks (also known as the coefficient of subgrade reaction) which describes a certain mechanical soil-structure interaction model known as a Winkler foundation. However, ks is arguably one of the most misunderstood and misapplied parameters in geotechnical practice, and proper assessment of this parameter is more complex and nuanced than might be expected. Further complexities are introduced when locally subsiding ground is present. This is because the Winkler model assumes settlement occurs in the soil only in response to an applied bearing pressure, whereas local subsidence introduces additional settlement (with associated shear and flexural stresses in the mat) which is independent of that caused by the applied structural loads. Methods of modeling and characterizing the subsurface conditions for the purpose of developing design values of ks to be used in mat foundation analysis and design are proposed, then these methods are extended using gap elements to accommodate sites with locally subsiding ground. These methods are compatible with standard geotechnical assessment techniques as well as standard structural analysis and design software packages.