The inverse problem with an application to the earthquake source inversion has been investigated. The kinematical model is used to describe the earthquake source and the Body Force Equivalent (B.F.E.) is considered as a mathematical model in the inversion. It is shown that the integral part representation of the source radiation, due only to displacement discontinuity, is independent of the force behavior at the fault. Also, the presence of subsequent force discontinuity will not affect the integral representation. Under the assumption of force continuity, it is shown that the dislocation vector is the eigenvalue solution of a certain linear operator.; To obtain a solution a least-squares technique has been employed. The least squares (LSQ) approach has been investigated in some detail from both the theoretical and the numerical point of view. Tikhonov regularization has been employed to provide a stable and meaningful solution. The regularization error and the rate of convergence are also discussed.; The theoretical results have been implemented into a practical application through Singular Value Decomposition (S.V.D) of the LSQ system matrix. For the purpose of source inversion the LSQ model is improved by introducing shifting optimization. Based on the spatial properties of dislocation, expansion of the dislocation into a series of an appropriately chosen set of functions is also presented.; As an application, the source inversion for the Imperial Valley earthquake of 1979 in California is presented. Detailed discussions and comparison are included between different LSQ models. Also, the effects of different regularization parameters on the dislocation have been considered.