# Sensitivity Analysis: Differential Calculus of Models

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## Abstract

Models in remote sensing—and in science and engineering, in general—are, essentially, functions of discrete model input parameters, and/or functionals of continuous model input parameters. In this sense, the sensitivities of model output parameters are, essentially, derivatives—partial derivatives with respect to discrete input parameters, and variational derivatives with respect to continuous input parameters. Specificity of models, as compared to functions and functionals in general, is due to the fact that they have two mandatory components. The first component describes the object or process of study, as is. It is a system of a differential equation, or equations, with initial and/or boundary conditions, which is referred to as a *forward problem*. The second component describes the procedure of deriving the output parameters of the model, which simulate the observed quantities, *observables*, from the solution of the forward problem. In contrast to the first component, it is just an analytic expression, which is referred to as an *observables procedure*. In rare practical cases, the forward problem has an analytic solution, and the output parameters are analytic functions or analytic functionals of the input parameters. Correspondingly, analytic evaluation of sensitivities is possible. In most practical cases, only numerical forward solutions are available, and specific approaches of sensitivity analysis considered in this monograph become indispensable.