Question

An overdetermined linear inverse problem is expressed as \( Gm = d \), where \( G \) is the data kernel, \( m \) is the vector of model parameters and \( d \) is the vector of observed data. If damping is applied to the inverse problem and the resultant generalized inverse is represented by \( G^{-g} \), the \textbf{model resolution matrix can be expressed as:}

• A. \( G^{T} G^{-g} \)
• B. \( G^{-g} G^{T} \)
• C. \( G^{-g} G \)
• D. \( G G^{-g} \)

Hint:

In inverse theory, the model resolution matrix is always expressed as \( R_m = G^{-g} G \), while the data resolution matrix is \( R_d = G G^{-g} \). Remember: model resolution involves \( G^{-g} G \).

Answer 1

The Correct Option is C

Step 1: General inverse solution
For the linear inverse problem: \[ Gm = d \] the solution is obtained using a generalized inverse \( G^{-g} \): \[ m = G^{-g} d \] Step 2: Define estimated model parameters
The estimated model parameters are written as: \[ m_{\text{est}} = G^{-g} d \] Since the observed data is \( d = G m \): \[ m_{\text{est}} = G^{-g} G m \] Step 3: Model resolution matrix
The model resolution matrix \( R_m \) is defined as: \[ m_{\text{est}} = R_m \, m \] Comparing with the above, we have: \[ R_m = G^{-g} G \] Step 4: Interpretation
- The resolution matrix measures how well the true model \( m \) can be recovered from the estimated solution.
- An ideal resolution matrix is the identity matrix \( I \), but in practice, due to noise and damping, it deviates from \( I \). \[ \boxed{R_m = G^{-g} G} \]