Homework 4

Gradient regularization

An alternative technique is a solution of the regularized least-squares optimization problem

$$\min\left( \vert\mathbf{F} \mathbf{m} - \mathbf{d}\vert^2 + \epsilon^2 \vert\mathbf{R} \mathbf{m}\vert^2\right)\;,$$ (5)

where $\mathbf{d}$ is irregular data, $\mathbf{m}$ is model estimated on a regular grid, $\mathbf{F}$ is forward interpolation from the regular grid to irregular locations, $\epsilon$ is a scaling parameter, and $\mathbf{R}$ is the regularization operator related to the inverse of the assumed model covariance. In our experiment, $\mathbf{R}$ is the finite-difference gradient filter.

inter0
Figure 7. Rainfall data interpolated using regularization with the gradient filter.

Figure 7 shows the interpolation result after 10 and 100 iterations. 100 iterations are not enough to converge to an acceptable solution, which is evident from the correlation analysis in Figure 8.

inter0-100-pred Figure 8. Correlation between interpolated and true data values for regularization with 100 iterations.

Homework 4