Abstract:The research on seismic inverse problem stimulates the vigour of the classical elastic theory. It is the most advanced subject involving geophysics,geology,applied mathematics etc. Especially the seismic exploration,a branch of geophysics,can actually be considered as a inverse problem research;it analyses recorded wave field informations and then reconstructs formation and lithology structures so as to discover oil and gas.Inverse problem is more difficult than forward one because of ill-posed property. Theoretically and practically,the elastic wave inversion in this article is a leading edge subject. The recorded wave field is abstracted as element d of data space,which is nonlinear with element m of model space. The distributions of d and m can be described by using Gaussian probability density,and the least square solution represents the estimated maximum likelihood solution of inversion. To cope with inhomogeneous weak scattering,the relation between scattering field and model parame ters can be derived from elastic dynamics theory. Finally,the gradient vectors of associated problems are expressed as model parameters(Lame coefficient,wave velocity and density)in time-space and frequency domains,and the estimated solution expressions are also given here.