Observability of Finite-Depth Double-Diffusive Exchange from Sparse Temperature-Salinity Measurements

Double-diffusive interfaces can support exchange histories that are spatially organized but only sparsely observed. A resolved three-dimensional calculation contains the route by which scalar gradients broaden, remain compact, connect with remote parts of a finite-depth layer, or organize horizontally, whereas field products usually provide profiles, repeated casts, autonomous-float records, or hydrographic sections. We ask which route-relevant features of finite-depth double-diffusive exchange remain observable after that measurement reduction. Four controlled finite-depth exchange histories are treated as known truth fields and sampled with vertical-profile, profile-bundle, vertical-coarsening, Argo-style, and section-like observing formats. The resulting observables are compared with selected Ice-Tethered Profiler, Argo, and CCHDO/GO-SHIP products to place the synthetic measurements in realistic observing contexts. Isolated profiles are weak route identifiers: at the final comparison time, strict case accuracy remains below 0.48, route-family accuracy is about 0.53--0.56 in leave-one-out profile classification, and 22 of 36 profiles are closer to a different-route profile than to a same-route profile under the current feature set. Profile bundles are substantially more informative. In the bundle-enumeration framework, the one-profile late-time route-family baseline is 0.694, while two-, three-, and four-profile bundles reach route-family accuracies of 0.917, 0.961, and 0.994. Coarse vertical sampling can preserve broad route separation while distorting local interface-width estimates, and section-like sampling adds horizontal-scale information only when station spacing resolves the relevant mode. The useful observing unit for hidden finite-depth double-diffusive exchange is therefore an ensemble or section, not an isolated cast.