Introducing an operator D M that selects fields only on the object domain, we obtain the object equation for the contrast source jw : { } inc D j j b j p w M L w χ χ ⎡ ⎤ = − ⎣ ⎦ (19) We handle the inverse problem as a minimization of a cost function, being a linear combination of errors in the data equation and the object equation, the method alternatively constructs sequence of contrast sources by a conjugate gradient iterative method such that the cost function is minimized. And the contrast function is then determined to minimize the error in the object equation. The cost function is a superposition of the errors in the data equations and errors in the object equations: ( ) { } { } 2 2 2 2 ( , ) ( ) , S inc D j b j j j b j j j S D S D j j j inc j j j j S D f M L w p w M L w F w F w F w f p χ χ χ χ χ ⎡ ⎤ ⎡ ⎤ − − + ⎣ ⎦ ⎣ ⎦ = + = + ∑ ∑ ∑ ∑ (20) In which j f is the scattered measured data. The 2L -norms on domains S and domain D are defined as follow: 2 ( ) ( ) j j j S S v v r v r dr =∫ v r v r dr v D j j j D ∫ = ( ) ( ) 2 (21) Where the overbar denote the complex conjugate of a function. We minimize the cost function (20) by conjugate gradient method, When the updated value of χis obtained, we can produce the real velocity distribution of the medium. If the value of the cost function is not smaller than the prescribed error criterion, the update step will be repeated until convergence is achieved. The flow of the inversion and migration can be seen in Figure 2. Figure 2. Flowchart of the full wavefield inversion and migration method
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