Remote sensing has become an essential component of the geosciences (the study of Earth and its system components). Remote sensing measurements are almost always energies measured in selected parts of the electro-magnetic spectrum. That is, the geophysical variable of interest is only observed indirectly; a forward model relates the energies to the variable(s) of interest and other elements of the state. The first derivative of that forward model with respect to the state is known as the Jacobian. In this chapter, we review the importance of the Jacobian to inferring the state, and we use it to diagnose which state elements may be difficult to estimate. We develop the Statistical Significance Filter and flag those state elements that consistently fail to get through the filter.
Cressie, N. (2018). A statistical analysis of the Jacobian in retrievals of satellite data. In B. S. Daya Sagar, F. Agterberg & Q. Cheng (Eds.), Handbook of Mathematical Geosciences: Fifty Years of IAMG (pp. 117-130). Switzerland: Springer Open.