Events characterized by large values in georeferenced processes can take various forms, such as occurring in isolated patches or stretching contiguously over large areas, and can further vary with the spatial location and the extremeness of the events. We use excursion sets above threshold exceedances in data observed over a two-dimensional grid of rectangular pixels to propose a general family of area coefficients that assess spatial-extent properties relevant for risk assessment, and study candidate coefficients from this family. These coefficients are defined locally and interpreted as a spatial distance from a reference site where the threshold is exceeded. We develop statistical inference and discuss robustness to boundary effects and resolution of the pixel grid. To statistically extrapolate coefficients towards very high threshold levels, we formulate a semiparametric model and estimate a parameter characterizing how coefficients scale with the quantile level of the threshold. Such coefficients can be useful for exploratory analysis but also as covariates in generative models for spatial random fields with nonstationary dependence, for example using the INLA-SPDE approach. The new coefficients are illustrated through applications to gridded daily temperature in continental France (reanalysis and unconditional climate model simulations). Joint work with Ryan Cotsakis (UNIL) and Elena di Bernardino (Université Côte d'Azur).