A geometric interpretation of analysis

By its simplicity and intuitive appeal, the geometric interpretation of analysis provides a complementary understanding of minimum variance estimation.  The geometric interpretation is made possible by using a Hilbert space representation of random variables.  In this presentation we will argue how actually a geometric approach can help to explore/discover new relationships, in identifying assumptions, and provide an alternative pathway of understanding the concept of analysis and estimation of error covariances.  
For example, relationships between analysis increments in cross-validation could be easily derived.  An interpretation of sequential observation processing also follows a simple interpretation.  Important considerations in establishing relationships for an arbitrary number of collocated data sets could also be established.  Then we examine how we can relax the assumption of an optimal analysis.  This will guide us in deriving a new diagnostic of observation statistics with correlated errors.