Denote the inversion domain as the object domain D and the data domain S as the domain where the sources and the receivers are located, combination of domain D and domain S are called total domain T (see Figure 1). Total field ( ) j p r satisfies the Helmholtz equation[2,3]: 2 2( ) ( ) ( , ), s j j k r p r Q r r ⎡ ⎤ Δ + =− ⎣ ⎦ r T∈ (10) Where Q is the source term. We split the total field into its incident and scattered parts: inc sct j j j p p p = + (11) The incident field satisfies the equation: ( ) 2 2( ) ( ) , , inc S b j j k r p r Q r r ⎡ ⎤ ∇ + =− ⎣ ⎦ r T∈ (12) Subtracting equation (12) from equation (10), then the scattered field satisfied the equation: 2 2 2 ( ) ( ) ( ) ( ), sct b j b j k r p r k r w r ⎡ ⎤ ∇ + =− ⎣ ⎦ r T∈ (13) Where ( ) w r j are the contrast sources, defined as: ( ) ( ) ( ) j j w r r p r χ = (14) In which the contrast function ( )rχ is given by: 2 2 2 2 ( ) ( ) ( ) ( ) 1 ( ) ( ) b b b c r c r k r r k r c r χ − − − ⎡ ⎤ − = − = ⎢ ⎥ ⎣ ⎦ r D∈ (15) Here, ( ) ( ) 2 c r k r ρ = − is the velocity of the scattering object, and ( ) ( ) 2 c r k r b b ρ = − is the velocity of the background medium. Equation (13) can be written using operator notation: 2 ( ( )) ( ) ( ), sct b j b j H p r k r w r r T =− ∈ (16) The solution of equation (16) can be written as: 1 2 ( ) ( ) ( ) ( ) , sct j b b j b j p r H krwr Lwr rT − ⎡ ⎤ ⎡ ⎤ = − = ∈ ⎣ ⎦ ⎣ ⎦ (17) Introducing an operator s M that interpolates the field values defined at the finite-difference grids to the appropriate receiver position, the data equation for the contrast function χ can be written as: ( ) { [ ( )]} sct s j b j p r M L w r′ = r S r T ∈ ′∈ , (18)
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