This work presents recent methodological developments in geophysical assimilation research. We revisit the meaning of the term solution of a mathematical model representing a geophysical system, and we examine its operational formulations. We argue that an assimilation solution based on epistemic cognition (which assumes that the model describes incomplete knowledge about nature and focuses on conceptual mechanisms of scientific thinking) could lead to more realistic representations of the geophysical situation than a conventional ontologic assimilation solution (which assumes that the model describes nature as is and focuses on form manipulations). Conceptually, the two approaches are fundamentally different. Unlike the reasoning structure of conventional assimilation modeling that is based mainly on ad hoc technical schemes, the epistemic cognition approach is based on teleologic criteria and stochastic adaptation principles. In this way some key ideas are introduced that could open new areas of geophysical assimilation to detailed understanding in an integrated manner. A knowledge synthesis framework can provide the rational means for assimilating a variety of knowledge bases (general and site specific) that are relevant to the geophysical system of interest. Epistemic cognition-based assimilation techniques can produce a realistic representation of the geophysical system, provide a rigorous assessment of the uncertainty sources, and generate informative predictions across space-time. The mathematics of epistemic assimilation involves a powerful and versatile spatiotemporal random field theory that imposes no restriction on the shape of the probability distributions or the form of the predictors (non-Gaussian distributions, multiple-point statistics, and nonlinear models are automatically incorporated) and accounts rigorously for the uncertainty features of the geophysical system. In the epistemic cognition context the assimilation concept may be used to investigate critical issues related to knowledge reliability, such as uncertainty due to model structure error (conceptual uncertainty).