4.2.1. Convergence

We first look at the convergence of the automatic registration procedure when applied to this real (noisy) case. Table 3 shows the number of iterations (nit) needed by the L-BFGS-B method to converge for each *β*, as well as the cost function before and after the minimization. Domain *D*<sup>4</sup> (for *β* = 1) necessitates the highest number of iterations and *D*<sup>1</sup> the lowest (1463 and 10, respectively). Iterations

on the coefficient *β* are preformed only on domain *D*4. Contrary to the synthetic case, the reduction of the cost function is important on the finer morphing grid. It is divided by two on *D*<sup>2</sup> and *D*<sup>3</sup> and by five on *D*4. In this case, the background term *Jb* is higher than the mapping error term *Jm* on all domains *Di*. The cost function after optimization *Jp*(*T*<sup>∗</sup> *<sup>i</sup>* ) increases from domain *D*<sup>1</sup> to *D*4, mostly due to the background term *Jb*. The important increase of the mapping error *Jm*(*T*<sup>∗</sup> *<sup>i</sup>* ) on domain *D*<sup>4</sup> is mostly due to the smoothing that reveals sharper features.

**Table 3.** Optimization results after each step *i* = 0, ..., 4 for the southern Ghana case. The number of iterations needed for the barrier approach (*β* iterations) and for the L-BFGS-B method are given separately. The total number of iterations correspond to the sum of the L-BFGS-B iterations for each *β* iterations. The cost function *Jp* is evaluated before and after optimization (i.e., for the first guess *<sup>T</sup>***fg** *i* and the 'optimal' grid *T*∗ *<sup>i</sup>* ). The latter has also been separated into three terms, the mapping error (*Jm*), the background error (*Jb*) and the penalization term (not shown here because of its value close to zero).


#### 4.2.2. Validation

The mapping *T* obtained from the automatic registration and its effect on the pixel grid *Dn* is shown in Figure 9. The field is shifted toward the South-West. The deformations are more important in the center and in the South of the domain, while being very small or null near the boundary. The area near the boundary corresponds to the padding area which is filled with zeros. The regulation terms are thus dominating the cost function in this area, especially the first one that ensure the mapping to be as small as possible. In contrast, the first term of the cost function is dominant in the center of the domain where the rainfall event is located. The second and third regulation terms ensure that the transition between these two areas is smooth.

This mapping *T* is then applied to the field *u*, the satellite estimate from IMERG, to correct the location of the rain event. Figure 10 shows the warped field *u*warp and the TAHMO measurements. One can see that the location of the rainfall event is corresponding more to the gauge data than before the warping (Figure 4). We define the center of the event by the grid cell with the maximum precipitation. Using the kriged gauge field *v* as the truth, we compute the position error of the event's center before and after the warping. It decreases from 55,365 km to 22,096 km. This remaining error of 22,096 km is due to an error in the longitude. Indeed, the maximum rainfall is at the correct latitude, but is two grid cells to the East of the actual peak. This error in longitude can be explained by the internal structure of the event. By comparing the pre-processed fields in Figure 6, one can see that the peak is located in the eastern part of the event for IMERG (*u*), but in the West for the kriged gauge (*v*). This difference could be due to the kriging and gauge network density (no gauge measurement is available near the peak of the warped field).

**Figure 9.** (**a**) Mapping *T* obtained from the automatic registration (with *C*<sup>1</sup> = 0.1 and *C*<sup>2</sup> = *C*<sup>3</sup> = 1) and (**b**) its effect on the pixel grid, for the southern Ghana case. (**a**) Mapping *T*. (**b**) Pixel grid *Dn* before (in red) and after (in blue) distortion by the mapping *T*.

**Figure 10.** Warped signal *u*warp (background) and TAHMO measurements (circles).

To investigate further the automatic registration, we compute the MAE and RMSE between the warped field and the target one, using the mappings *Ti* obtained at each step *i* (Table 4). Applying the mapping *T*1, defined on the coarsest morphing grid, already greatly reduces the MAE and the RMSE compared to the original errors. Increasing the resolution of the morphing grid, further decreases the MAE and RMSE, except for the RMSE for *i* = 2. While the cost function is divided by five on domain *D*<sup>4</sup> (see Table 3), the reduction of the error from *D*<sup>3</sup> to *D*<sup>4</sup> is much smaller.

The target field *v* has been obtained by kriging the gauge data. Hence, some interpolation errors were introduced, especially in the areas far from the gauges. Comparison to the whole field *v* is not representative, since it contains some large uncertainties. In the second part of Table 4, the warped field *u*warp is estimated at the station locations (shown in Figure 11 for *T*4) and is directly compared to the gauge measurements. As previously, a large part of error reduction is occurring on domain *D*1. In total (i.e., for *i* = *I* = 4), the RMSE has been divided by almost two, and the MAE by 1.5.


**Table 4.** MAE and RMSE of the warped (*u*warp) signal compared to the kriged TAHMO field (*v*) and to the gauge measurements, obtained at different steps *i*, for the southern Ghana case.

Figure 11 shows the warped field at the station locations. The stations at which the gauge data and the warped field disagree are numbered. At the other stations, both TAHMO and the warped field estimate zero precipitation. The precipitation amounts at the 18 numbered stations according to TAHMO, IMERG-Late (*u*), IMERG-Final and the warped field (*u*warp) are shown in Figure 12. They are ordered in decreasing order of precipitation with respect to TAHMO measurements. For stations 1 and 2 (with highest rainfall intensity), the warped signal is much closer to the measurements than IMERG-Late and IMERG-Final. However, *u*warp underestimates the intensity and estimates more rainfall at station 2 than 1 (the opposite of TAHMO). This underestimation can be explained by two factors. First, the maximum of IMERG-Late *u* on our domain (=45 mm/h) is lower than the one recorded by TAHMO. Second, we used linear interpolation for the warping. The rainfall at station 3 is underestimated by the three satellite estimates. At the stations 4 and 5, both IMERG-Late and IMERG-Final underestimate while *u*warp overestimates. IMERG misses the precipitation at station 6, while *u*warp overestimates by almost a factor four. The remainder of the stations has very low or no precipitation according to TAHMO. The three satellite estimates are similarly low (less than 1 mm/h) at these stations, with a few exceptions: IMERG-Late and IMERG-Final at station 14 and *u*warp at stations 8 and 12. The overestimation by *u*warp at station 8 corresponds to a second lower peak present in the original field *u* but not in the target field *v*.

**Figure 11.** Warped signal *u*warp at the stations location. The stations that differ from the TAHMO measurements are numbered.

**Figure 12.** Rainfall (in mm/h) at 18 TAHMO stations according to TAHMO, IMERG-Late, IMERG-Final and the warped signal (*u*warp).

#### **5. Discussion**
