Differentiability of the Functional in the Problem of Reconstructing the Density of a Gravitational Anomaly

This article addresses the problem of reconstructing the density distribution of a gravitational anomaly from measurements of the vertical component of the gravitational field gradient. The study is motivated by the growing demand for reliable mathematical methods in gravity exploration and inverse modeling. The direct problem is formulated using the Poisson equation with homogeneous boundary conditions on an extended computational domain, while the inverse problem is posed as the minimization of a smooth quadratic misfit functional. A sensitivity analysis is performed to derive the functional gradient, and the Gateaux derivative is rigorously established through the solution of an adjoint boundary value problem. The analysis reveals a fundamental feature of the adjoint formulation: the right-hand side of the adjoint equation contains the derivative of the Dirac delta function. This singularity makes the direct application of classical gradient-based optimization methods numerically impractical, despite the formal differentiability of the objective functional. The obtained results provide a theoretical justification for abandoning conventional gradient optimization techniques in this class of inverse gravimetric problems and motivate the use of derivative-free optimization methods. The proposed framework establishes a mathematical foundation for the development of robust numerical algorithms for gravitational density reconstruction and outlines promising directions for future research.