In many practical situations, it is difficult, if not impossible, to perform direct measurements or calculations of the external forces acting on vibrating structures. Instead, vibrational responses can often be conveniently measured. This paper presents an inverse method for estimating a distributed random excitation from the measurement of the structural response at a number of discrete points. Part 1 is devoted to the presentation of the theoretical development. The force identification method is based on a modal model for the structure and a spatial orthonormal decomposition of the excitation field. The estimation of the Fourier coefficients of this orthonormal expansion is presented. As this problem turns out to be ill-posed, a regularization process is introduced. The minimization problem associated to this process is then formulated and its solution is developed. Part 2 is devoted to the application of this force identification method to a practical flow-induced vibration problem.