Question

Consider a geophysical inverse problem of the form \( \mathbf{d} = \mathbf{G} \mathbf{m} \), where \( \mathbf{G} \) is the forward operator, \( \mathbf{m} \) is the model vector and \( \mathbf{d} \) is the observed data vector. The Earth model parameters can be estimated using \( \mathbf{m}_{{est}} = \mathbf{H} \mathbf{d} \), where \( \mathbf{H} \) is the pseudoinverse of \( \mathbf{G} \). Given \( \mathbf{G} = \mathbf{U} \Sigma \mathbf{V}^T \) as the singular value decomposition of \( \mathbf{G} \), and assuming \( \mathbf{G} \) is full rank, which of the following options is/are correct?

• A. \( \mathbf{H} = \mathbf{U}^T \Sigma^{-1} \mathbf{V} \)
• B. \( \mathbf{H} = \mathbf{V} \Sigma^{-1} \mathbf{U}^T \)
• C. \( \mathbf{H} = \mathbf{U}^T \Sigma^{-1} \mathbf{V}^T \)
• D. \( \mathbf{H} = \mathbf{U} \mathbf{U}^T \mathbf{V} \Sigma^{-1} \mathbf{U}^T \)

Hint:

The Moore-Penrose pseudoinverse of a matrix \( \mathbf{G} \) is given by \( \mathbf{H} = \mathbf{V} \Sigma^{-1} \mathbf{U}^T \) for the singular value decomposition \( \mathbf{G} = \mathbf{U} \Sigma \mathbf{V}^T \).

Answer 1

The Correct Option is B, D

Step 1: Using the pseudoinverse formula.
For a full rank matrix \( \mathbf{G} \), its Moore-Penrose pseudoinverse \( \mathbf{H} \) is given by: \[ \mathbf{H} = \mathbf{V} \Sigma^{-1} \mathbf{U}^T. \] Step 2: Conclusion.
Thus, the correct formula for \( \mathbf{H} \) is option (B) and (D).