Publisher Summary This chapter gathers various techniques and results concerning transmutation methods in the areas of linear stochastic estimation and inverse problems in geophysics. The remarkable similarity in patterns and structure between formulas and methods in these areas is displayed. Some new results on geophysical inverse problems are included and, when there is an underlying stochastic process, the role of minimization in characterizing transmutation kernels is shown to be equivalent in stochastic geometry to least squares filtering estimation. Given that many natural processes are governed by second order linear differential equations (at least to a first approximation) it is certainly no surprise to see that certain mathematical patterns and structures recur frequently in physics and applied mathematics. This chapter emphasizes patterns and structure common to these areas and to unify in a certain way the treatment of such typical problems.