**Appendix B. Sensitivity Study for the Regulation Coefficients** *C*1**,** *C*<sup>2</sup> **and** *C*<sup>3</sup>

The automatic registration is based on the minimization of the cost function *J*, which is composed of an observation term and a background term. The background term *Jb* consists of the three criteria defining the 'optimal' mapping *T*. The three criteria concern the magnitude (Equation (2)), the smoothness (Equation (3)) and the divergent (Equation (4)) of *T*. These three criteria can be tuned with the regulation coefficients *C*1, *C*<sup>2</sup> and *C*<sup>3</sup> respectively. In this section, we look at the influence of these three coefficient on the automatic registration.

For this sensitivity study, we use the synthetic case described in Section 3.1. In the result section (Section 4), we empirically set the coefficients to *C*<sup>1</sup> = 0.1 ans *C*<sup>2</sup> = *C*<sup>3</sup> = 1. Here, we consider four more cases. We first look at the impact of the individual coefficients, setting the two other to zeros. Then we examine the influence of their intensity by multiplying them by five. The only difference between the results shown here and the ones in Section 3.1 are the regulation coefficients.

The impact of the coefficients on the morphed and warped signals can be seen in Table A1. It shows the mean absolute error (MAE) of both signals for the four cases described above and for the original case from Section 3.1. The MAEs are staying similar. The three cases with only one coefficient have a slightly higher MAE for the morphed signal than the original case, but slightly lower for the warped signal. Multiplying the three coefficients by five have the inverse effect. The MAE of the morphed signal is lower but that of the warped signal is slightly higher. The coefficients seem to have a limited impact on the warped and morphed signal.


**Table A1.** MAE of the warped (*u*warp) and morphed *u*morp signals compared to the target field (*v*) for the different sensitivity runs (for *I* = 4).

We now look at the impact of these coefficients on the mapping *T*. Figures A1–A4 show the mapping *T* and the resulting grid deformation for the four new cases. They can be compare to Figure 7 for the original case. The main differences occur in area without precipitation. Without the smoothing constraints, larger distortion occurs near the boundary near the lower-right corner (visible in Figures A1 and A3 but not in Figure A2). Similarly, the grid appears very stretched on the right upper side. The case with only the magnitude constraint (*C*<sup>1</sup> = 0.1) exhibit the smooth transformation (Figure A1). The divergent-free condition favours rotation-like patterns, in order to conserve the grid cell's volume. The weight of the coefficient also have an impact on the mapping *T*. By comparing Figures 7a and A4a, one can see that the distortion is similar over the rainy area, while the spatial displacement is smaller over the non-rainy area. The distortion is smoother and very similar to the regular one in the upper-left corner, i.e away from the rainy peak.

To summarize, without the three weak constraints the distorted grid shows unnatural distortion. However, the regulation coefficients mainly affect the mapping *T* in areas without rainfall. This is reflected in the small influence it has over the morphed or warped field (Table A1).

**Figure A1.** Sensitivity study: *C*<sup>1</sup> = 0.1 and *C*<sup>2</sup> = *C*<sup>3</sup> = 0. (**a**) Mapping *T* obtained from the automatic registration and (**b**) its effect on the pixel grid. (**a**) Mapping *T*. (**b**) Pixel grid *Dn* before (in red) and after (in blue) distortion by the mapping *T*.

*Remote Sens.* **2019**, *11*, 2557

**Figure A2.** Sensitivity study: *C*<sup>2</sup> = 1 and *C*<sup>1</sup> = *C*<sup>3</sup> = 0. (**a**) Mapping *T* obtained from the automatic registration and (**b**) its effect on the pixel grid. (**a**) Mapping *T*. ((**b**) Pixel grid *Dn* before (in red) and after (in blue) distortion by the mapping *T*.).

**Figure A3.** Sensitivity study: *C*<sup>3</sup> = 1 and *C*<sup>1</sup> = *C*<sup>2</sup> = 0. (**a**) Mapping *T* obtained from the automatic registration and (**b**) its effect on the pixel grid. (**a**) Mapping *T*. (**b**) Pixel grid *Dn* before (in red) and after (in blue) distortion by the mapping *T*.
