A rotation total variation regularization for full waveform inversion

Proof.

(i) Since C 1 ⁢ ( Ω ) is dense in W 1 , 1 ⁢ ( Ω ) , it suffices to show equation (2.14) for m ∈ C 1 ⁢ ( Ω ) . In this case, for any 𝝍 ^ ∈ Ψ ^ , Green’s theorem (integration by parts) gives

Due to 𝝍 ^ ⁢ ( 𝒙 ) = R θ ⁢ Λ ⁢ 𝝍 ⁢ ( 𝒙 ) ∈ B 𝜶 , θ ⁢ ( 0 ) and | 𝝍 ^ | ≤ 1 , from the Fréchet transform (2.5), we have

To show the reverse inequality, we take

and observe that

and 𝝍 ¯ ∈ C ⁢ ( Ω , R 2 ) with | 𝝍 ¯ ⁢ ( 𝒙 ) | < 1 for all 𝒙 ∈ Ω . By multiplying 𝝍 ¯ by a suitable characteristic function compactly supported in Ω, one can construct 𝝍 ^ ∈ Ψ ^ ∩ C 0 ∞ ⁢ ( Ω ) , ensuring that the left-hand side of equation (B.1) approaches

arbitrarily closely.

For (ii) and (iii): For any 𝝍 ^ ∈ Ψ ^ and m ∈ L 1 ⁢ ( Ω )

Taking the supremum over 𝝍 ^ ∈ Ψ ^ , we obtain

where | Ω | is the measure of Ω, since Ω is a bounded convex region in R 2 . ∎

Proof.

(i) Let { m k } k = 0 ∞ ⊂ L 1 ( Ω ), and m k ⇀ m ¯ . For 𝝍 ^ ∈ Ψ ^ , div ⁡ 𝝍 ^ ∈ C ⁢ ( Ω ) , we have

(B.6)

Taking the supremum over 𝝍 ^ ∈ Ψ ^ , we obtain

(ii) For any m 1 , m 2 ∈ L 1 ⁢ ( Ω ) , and t ∈ [ 0 , 1 ] , we have

(B.8)

Taking the supremum in the above inequality over 𝝍 ^ ∈ Ψ ^ gives the convexity of 𝒥 RTV , β 2 0 ⁢ ( m ) . ∎