*2.1. Definitions*

Let *<sup>u</sup>* and *<sup>v</sup>* be two signals (or images) defined on a domain *<sup>D</sup>* <sup>⊂</sup> IR<sup>2</sup> and *<sup>T</sup>* : (*x*, *<sup>y</sup>*) <sup>∈</sup> IR<sup>2</sup> → *Tx*(*x*, *y*), *Ty*(*x*, *y*) <sup>∈</sup> IR<sup>2</sup> be a mapping function. The goal of **image registration** is to determine a spatial mapping *T* such that, ∀(*x*, *y*) ∈ *D*,

$$\begin{aligned} \boldsymbol{u}(\mathbf{x}, \mathbf{y}) &\quad \approx \quad \boldsymbol{u} \circ (\boldsymbol{I} + \boldsymbol{T}) \, (\mathbf{x}, \mathbf{y}) \\ &\approx \quad \boldsymbol{u} \left[ (\boldsymbol{I} + \boldsymbol{T}) \, (\mathbf{x}, \mathbf{y}) \right] \\ &\approx \quad \boldsymbol{u} \left[ \mathbf{x} + T\_{\mathbf{x}} (\mathbf{x}, \mathbf{y}), \mathbf{y} + T\_{\mathbf{y}} (\mathbf{x}, \mathbf{y}) \right] \end{aligned} \tag{1}$$

where *I* is the identity function.

There can be several mappings *T* that meet the requirement *v* ≈ *u* ◦ (*I* + *T*). Especially in areas without rainfall, the mapping *T* is not unique. We define three criteria to characterize one optimal mapping:

$$T \quad \approx \quad 0 \tag{2}$$

$$
\nabla T \quad \approx \quad 0 \tag{3}
$$

$$
\nabla \cdot T \quad \approx \quad 0 \tag{4}
$$

That is, the optimal mapping has to be as small, smooth and divergent-free (i.e., it is not shrinking or expanding the field) as possible.

Several approaches have been used to define the optimality of the mapping. For the FCA method applied to precipitation, Reference [19] use smoothness and barrier conditions. Contrary to our condition on the magnitude (Equation (2)), their barrier does not impact small scale displacements. Using the FCA for data assimilation, References [20,22] added two more constraints, one on the magnitude and one on the divergent. Reference [25] did not use any magnitude or barrier approach and only had constraints on the gradient and the divergence. Our constraints on the magnitude and on the smoothness are the same as those used in Reference [32]. Constraints on the divergence were used in several similar field distortion methods [20,22,25]. Thus, we also added a third constraint on the divergence in order to observe its impact. A short sensitivity study on the impact of these three coefficient is presented in Appendix B.

**Image warping** is the distortion of an image based on a spatial transformation of the domain. Warping can be used to transform an image into another one by using the spatial mapping *T* obtained from the registration method. The mapping *T* is gradually applied to the original image *u* as follows:

$$\mu\_{\text{warp}(\lambda)} = \quad \mu \circ (I + \lambda T) \quad \quad 0 \le \lambda \le 1 \tag{5}$$

Warping works well when the residual *v* − *u* ◦ (*I* + *T*) is small, which is not the case when the images *u* and *v* have different intensities for example. It is a spatial transformation. It only acts on the coordinates, it does not modify the intensity of the image *u*. On the other hand, **Cross-dissolving** only acts on the intensity. It fades two images *u* and *v* into each other:

$$
\mu\_{\text{diss}(\lambda)} = \quad \mu + \lambda(v - \mu) \quad 0 \le \lambda \le 1 \tag{6}
$$

**Image morphing** combines warping and cross-dissolving to account for both the spatial distortion and the difference in intensity:

$$
\mu\_{\text{morphisms}(\lambda)} = \quad (\mu + \lambda r) \circ (I + \lambda T) \qquad 0 \le \lambda \le 1 \tag{7}
$$

where *r* is the residual:

$$\|\sigma\|\| = \|v \circ (I + \lambda T)^{-1} - u\|\quad 0 \le \lambda \le 1\tag{8}$$

With this formula of *uλ*, we obtain *u*morph(0) = *u* and *u*morph(1) = *v*.
