When the area of the interface between two phases is proportional to the mass of one of the phases (e.g. for a solid adsorbent, for an emulsion or for an aerosol), the specific surface area (a,s or preferably as) is defined as the surface area divided by the mass of the relevant phase.
Sources:
Green Book, 2nd ed., p. 63 (https://goldbook.iupac.org/files/pdf/green_book_2ed.pdf)
PAC, 1972, 31, 577. 'Manual of Symbols and Terminology for Physicochemical Quantities and Units, Appendix II: Definitions, Terminology and Symbols in Colloid and Surface Chemistry' on page 583 (https://doi.org/10.1351/pac197231040577)