1977) and in studies of planetary atmospheres (see e.g., Nixon et al. 2007). The fundamental problem is to determine the state vector, [**x**]{} of dimension $n$, often a vector of temperatures and mixing ratios at different altitudes (but could be other desirable variables), given some set of observations, [**y**]{} of dimension $m$, usually a vector of flux values at each wavelength. In the absence of any noise, they can be related through [**y=F(x)**]{}, where [**F(x)**]{} is a model that simulates the measurement at each wavelength given a representative atmosphere. In an idealized scenario, if the relationship between [**x**]{} and [**y**]{} is linear, we can linearize [**F(x)**]{} and write $${\bf y}={\bf F( x_a)}+{\bf K} ({\bf x}-{\bf x_a})$$ where [**K**]{} is the $m \times n$ Jacobian matrix whose elements are given by the Frechet derivative $$K_{ij}=\frac{\partial F_{i}({\bf x})}{\partial x_j}$$ with $F_i$ being the measurement in the $i^{th}$ channel, and $x_j$ the value of the $ j^{th}$ parameter. The vector [**x$_a$**]{} is the prior ([*a priori*]{}) state, which represents our best initial guess of the true state before the observations are made. The Jacobian describes the sensitivity of the measurement at each wavelength in a spectrum to a perturbation of a given parameter 