Data mining is routinely used to organize ensembles of short temporal observations so as to re-construct useful, low-dimensional realizations of the underlying dynamical systems. By analogy, we use data mining to organize ensembles of a different type of short observations to reconstruct useful realizations of bifurcation diagrams. Here the observations arise not through temporal variation, but rather through the variation of input parameters to the system: typical examples include short one-parameter steady state continuation runs, recording components of the steady state along the continuation path segment. We demonstrate how partial and disorganized "bifurcation observations" can be integrated in coherent bifurcation surfaces whose dimensionality and topol./parametrization can be systematically recovered in a data-driven fashion. The approach can be justified through the Whitney and Takens embedding theorems, allowing reconstruction of manifolds/attractors through observations. We discuss extensions to different types of bifurcation observations (not just one-parameter continuation), and show how certain choices of observables create analogies between the observation of bifurcation surfaces and the observation of vector fields. Finally, we demonstrate how this observation-based reconstruction naturally leads to the construction of transport maps between input parameter space and output/state variable spaces.