2.2 Conjugate Gradient Algorithm
A unified framework by optimum method base on least-squares norm can be applied to most inversion problems.The full waveform inversion procedure in the intercept time（τ）-slowness（p）domain is used to minimize the misfit between the data and the calculated wave field，using a local iterative search algorithm（conjugate gradient）method similar to the waveform inversion in the frequency-slowness domain as Kormendi and Dietrich［17］.The forward calculation problem in the intercept time-slowness domain can be expressed simply by
d=f（m），
where d is the data vector，the calculated wavefield inτ-p domain.m is the model vector，which consists of the elastic parameters of each layer when we describe the subsurface media by stratified layers with homogeneous thickness.If S-wave velocities and densities are kept fixed in the inversion procedure，the model vector only consists of the P-wave velocities of each layer；m 0denotes a priori model，the initial elastic parameters；and dobsdenotes the observed wavefiled in theτ-p domain.Then the cost function of least-squares norm can be defined by
where
where CD is the data covariance matrix and CM the model covariance matrix.The superscript T denotes the transpose of a matrix.
Nonlinear iterative inversion finds a model to minimize the cost function with the guide of the gradient by such optimum methods as the steepest descent method，the quasi-Newton method and the conjugate gradient method.The conjugate gradient method is shown in the following steps［17］：
（1）Compute the synthetic data dnfrom model m nat iteration n
dn=f（mn），n=0，1，2，…；
（2）Compute the residuals between the synthetic data and the calculated data
Δdn=dn-dobs
and the discrepancies between the current model and the priori model
Δmn=mn-m0；(https://www.daowen.com)
（3）Compute the cost function
and check whether it satisfies the stopping condition；
（4）Compute the direction of the steepest ascent/gradient
where
is the matrix of the Frechet derivatives of the plane-wave response respect to the model parameters，which can be obtained analytically.Numerical solution of the Frechet derivatives is used in our program for simpleness to write the codes.The second term of above formula can be removed when the influence of the priori model in the calculation of gradient is not considered.This is adopted in our calculation.
（5）Compute the conjugate direction
（6）Compute the optimum stepμnto minimize the cost function；
（7）Compute the new model
