This paper presents results of a study on an inversion transformation ℐA for backscattering of ultrasonic waves from an obstacle embedded in a solid. For a rigid sphere the resulting function A(t) is derived which is related to the cross-sectional area intercepted at any given time by a transverse plane moving across the obstacle in the same direction as the incident wave. The maximum value of A(t) is the total backscattering cross section. From this study emerges a proposed inversion algorithm for cavities. The use of the inversion algorithm is demonstrated for “exact” theoretical data obtained from Opsal as well as experimental data collected on an ellipsoidal cavity and a double cavity. Associated with the inversion of the experimental data is an iterative technique which optimizes the construction of A(t) and therefore the estimates for the size and shape of the scatterer.