Automated spectral recomposition with application in stratigraphic interpretation

Numerical Method

Assuming nonlinear parameters of $\mathbf{m}$ , linear parameters of $\mathbf{a}$ can be obtained by solving the linear least-squares problem,

$$\mathbf{a}=\pmb{\psi}(\mathbf{m})^\dagger\mathbf{d}\;,$$ (5)

where $\pmb{\psi}(\mathbf{m})$ is the matrix composed of $\psi_i(m_i,f_j)$ and $\pmb{\psi}(\mathbf{m})^\dagger$ is the Moore-Penrose generalized inverse of the $\pmb{\psi}(\mathbf{m})$ matrix. Replacing this $\mathbf{a}$ in the original function, the minimization problem takes the form

$$\mathop{\mbox{min}}_{\mathbf{m}}\left\Vert(\pmb{I}-\pmb{\psi}(\mathbf{m})\pmb{\psi}(\mathbf{m})^\dagger)\mathbf{d}\right\Vert _2^2\;,$$ (6)

where the linear parameters have been eliminated (Golub and Pereyra, 1973). We use the Gauss-Newton method (Björck, 1996) to linearize the problem as follows:

$$d(f_i) \approx \sum_i{R_j(m_i,f_j)}+\sum_i\frac{\partial R_j}{\partial m_i}\Delta{m_i}$$

$$\approx \sum_i{a_i\;\pmb{\psi}(m_i,f_j)}+\sum_i[a_i'\;\pmb{\psi}(m_i,f_j)+a_i\;\pmb{\psi}'(m_i,f_j)]\Delta{m_i}.$$

Starting with initial values of ${m_i}$ , we are able to solve for ${a_i}$ and ${a_i'}$ using equation 5. Then we solve for the model increment $\Delta{ {m_i}}$ . After a number of iterations, summation of $\Delta {{m_i}}$ converges to the estimated value. The Gauss-Newton method is efficient. In most cases, approximately 20 iterations provide an acceptable convergence. Fitting more component frequencies helps minimize the residual. Geological factors can help the user decide how many components to include in the model. In addition, providing good initial values helps the algorithm avoid being trapped in a local minimum.

Automated spectral recomposition with application in stratigraphic interpretation