*5.1. Convergence*

The automatic registration algorithm converges for both the synthetic and the real (Southern Ghana) case. However, some differences can be noticed when comparing the optimization results shown in Tables 1 and 3.

The minimization method (L-BFGS-B) needs more iterations for the synthetic than for the real case for all steps *i* except for *i* = 4. Iterations on the coefficient *β* were needed for step *i* = 2 and *i* = 5 for the synthetic case, while the real case needed it for its finer step *i* = 4. The real case is noisier than the synthetic one, but the displacement between the field *u* and *v* is more straightforward. Indeed, for the synthetic case, the mapping *T* has to describe both a rotation and a shear. For the real case, the mapping *T* only has to represent a translation. On the coarser grids, the translation does not violate the constraints on the barrier. On the other hand, a rotation is more likely to violate the constraints and so to require iterations on *β* on coarser grids too. This shows that the number of iterations of both the minimization and barrier method depends on the input fields *u* and *v*, especially on the mapping complexity. It also depends on the coefficients *C*1, *C*<sup>2</sup> and *C*3, but the influence of the chosen coefficient values is limited (results not shown here).

For the synthetic case, the decrease in the cost function is more important for the first steps. This can be explained by better first guesses for the finer grids. On the contrary, for the real case, the decrease is more important in the last steps. While the events were identical in the synthetic case, they have different shapes in the real one. On the coarser grid, these differences are masked by the strong smoothing. They become more visible on the finer grid on which there is less smoothing. The sharper features being more sensitive to small position errors results in a higher cost function. The finer morphing grids allow the mapping to take the shape difference into account, on top of the position error. The reduction of the cost function is thus becoming more important.

The intermediate mappings *Ti* give information about the impact of the steps *i* = 1, ..., *I*. They were evaluated by looking at the MAE of the warped fields *u*warp (Tables 2 and 4). In the real case, most of the MAE decrease is reached after the first iteration (divided by around 2). The decrease after the subsequent iterations is more marginal. The mapping on the coarser morphing grid *D*<sup>1</sup> is already able to capture reasonably well the displacement, that is, the translation toward the South-West. Thus, the finer morphing grids induce less improvement. In the synthetic case, the MAE is improved greatly after the first iteration too (divided by 7). However, the subsequent iterations continue to decrease the MAE (divided by 3 after step 2 and by 2 after step 3). The first morphing grids are too coarse to describe

accurately the complex displacement, which combine a rotation and a shear. Hence, increasing the resolution of the morphing grids improves the mapping *Ti* and thus the MAE. The more complex the displacement is, the finer the morphing grid needs to be and so the higher *I* has to be (i.e., the more steps *i* we need).

The computational time increases exponentially with the number of steps *i*. All computations shown here were done on a personal computer. For the synthetic case, the first four steps (i.e., *i* = 1, 2, 3 and 4) were completed in approximately 2 min, while the fifth iteration (*i* = 5) needed between 4 to 10 min (depending on the computer computational capacity). The real case requires fewer iterations than the synthetic one and so a shorter computational time (∼1 min for *I* = 4 and ∼4 min for *I* = 5).
