| The paper is made up of two parts. Firstly, Hadamard's three spheres theorem of Sub-Solution for Sub-Laplacian operation△Gu≥0is obtained,and is proved on the basis of fundamental solution and Maximum principle of sub-Laplacian operator. Secondly,application of a class of Partial Differential Equation is studied. Based on the acoustic approximation of the elastic wave equation, the calculating skills of some terms of the objective function in frequencydomaininversion are discussed.The first part consists of two chapters,in which we discuss the Hadamard's three-spheres-theorem of Sub-Laplacian operators on Carnot group.In the first chapter, we briefly introduce the development of the Maximum principle of Differential Equation and Hadamard's three spheres theorem, the meaning of selecting this question, and some concepts about Carnot group.In the second chapter, based on the fundamental solution of Sub-Laplacian operation and the Sub-mean property, Maximum principle and Hadamard's three spheres theorem on Carnot group are proved, and we show that maximal value function M(r)=max|x-1og|=ru(ξ) is a convex function with respect to |x-1og|2-Q.The second part consists of two chapters also, in which we discuss an optimalizationfor waveform inversion of the frequency-domain.In the third chapter, we briefly introduce the development of and background of the inverse problem of Partial Differential Equation.In the fourth chapter, based on the elasticity theory , we establish the wave equation for the pressure wavefield in the nonhomogeneous medium. Then the relation of covariance matrix CD in the data space with the model covariance matrixCM of the objetive function is found. Finally we show the deduction that the gradient direction of the data misfit (?) can be computed by forward modeling Green function to avoid the formidable calculation of the frechet F, and we show the iteratively algorithm.
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