### 3.6.2 Description of the UNet-emulator architecture

We describe here with more details the Figure 3.2, which schematize the UNet architecture used for the emulator. On the figure, the different blocks are represented by colorful arrows. Each arrow is described below :

* Each blue arrow $\Rightarrow$ corresponds to a convolution block composed of a layer built with a set of convolutional 2 × 2 filters. The number of filters increases all along the contracting part (the number of filters is respectively 64, 128, 256, 512 and 1024 and is given on the top of each block in Figure 3.2). The outputs of this layer are then normalised with a batch normalisation layer (see e.g. Ioffe and Szegedy (2015)) to improve the statistical robustness of the layer. Finally the ReLu activation function completes this block.

$$ ReLu : R(z) = max(0, z) $$

* Each red arrow $\downarrow$ is a Maxpooling layer. It performs 2 × 2 pooling on each feature map, which simply divides by 2 the spatial dimension by taking the maximum of each 2 × 2 block. It is applied through all convolution block outputs in the encoding path. The size of the images is indicated on the side of each block on Figure 3.2.

* Each green arrow $\uparrow$ is a transpose convolution layer. It allows to perform up-sampling in the expansive part of the algorithm. It multiplies the spatial dimension by 2 by applying the same connection as the classical convolution but in the backward direction.

* The black arrow $\Rightarrow$ represents a fully connected dense network of 4 layers which is applied on the 1-dimensional inputs (Z).

* The grey arrow $\Rightarrow$ represents a simple concatenation layer which recalls the layers from the encoding path in the decoding one.

* Finally, the light blue arrow $\Rightarrow$ is the output layer, which is a simple convolutional layer with a single filter and a linear activation function.

### 3.6.3 Supplementary Material: Summary tables

We present here the summary tables of the different tests performed with the UNet-based emulators and for the two steps of the emulator evaluation.
<table><thead><tr><th rowspan="2"></th><th colspan="3">RMSE</th><th colspan="3">ACC</th><th colspan="3">Variance ratio</th><th colspan="3">Wasserstein distance</th></tr><tr><th>SQ 05</th><th>MEAN</th><th>SQ 95</th><th>SQ 05</th><th>MEAN</th><th>SQ 95</th><th>SQ 05</th><th>MEAN</th><th>SQ 95</th><th>SQ 05</th><th>MEAN</th><th>SQ 95</th></tr></thead><tbody><tr><td>MAX</td><td>0.70 (E6)</td><td>0.86 (E6)</td><td>1.26 (E30)</td><td>0.93 (E11)</td><td>0.97 (E11)</td><td>0.98 (E8)</td><td>0.95 (E9)</td><td>0.98 (E9)</td><td>1.00 (E8)</td><td>0.14 (E6)</td><td>0.24 (E6)</td><td>0.36 (E13)</td></tr><tr><td>MEAN</td><td>0.67</td><td>0.82</td><td>1.20</td><td>0.92</td><td>0.96</td><td>0.98</td><td>0.93</td><td>0.96</td><td>0.98</td><td>0.08</td><td>0.14</td><td>0.26</td></tr><tr><td>MIN</td><td>0.64 (E19)</td><td>0.8 (E11)</td><td>1.17 (E19)</td><td>0.91 (E13)</td><td>0.96 (E6)</td><td>0.98 (E6)</td><td>0.89 (E6)</td><td>0.92 (E6)</td><td>0.95 (E20)</td><td>0.04 (E9)</td><td>0.11 (E8)</td><td>0.18 (E28)</td></tr><tr><td>Main Emulator</td><td>0.67</td><td>0.83</td><td>1.22</td><td>0.92</td><td>0.96</td><td>0.98</td><td>0.93</td><td>0.95</td><td>0.98</td><td>0.08</td><td>0.13</td><td>0.19</td></tr><tr><td>UPRCM</td><td>0.43</td><td>2.10</td><td>8.1</td><td>0.79</td><td>0.95</td><td>0.99</td><td>0.68</td><td>0.92</td><td>1.16</td><td>0.14</td><td>1.82</td><td>7.88</td></tr><tr><td>Cheap</td><td>0.92</td><td>1.11</td><td>1.34</td><td>0.87</td><td>0.93</td><td>0.96</td><td>0.94</td><td>0.98</td><td>1.02</td><td>0.08</td><td>0.15</td><td>0.28</td></tr><tr><td>Emul-Hist</td><td>1.16</td><td>1.35</td><td>1.70</td><td>0.90</td><td>0.95</td><td>0.97</td><td>0.88</td><td>0.93</td><td>0.96</td><td>0.74</td><td>0.87</td><td>1.01</td></tr><tr><td>No.Smoothing</td><td>0.62</td><td>0.76</td><td>1.2</td><td>0.93</td><td>0.97</td><td>0.98</td><td>0.95</td><td>0.96</td><td>0.98</td><td>0.07</td><td>0.13</td><td>0.23</td></tr><tr><td>TAS_included</td><td>0.36</td><td>0.55</td><td>0.98</td><td>0.96</td><td>0.98</td><td>0.99</td><td>0.96</td><td>0.98</td><td>0.99</td><td>0.05</td><td>0.08</td><td>0.15</td></tr></tbody></table>

<table><thead><tr><th rowspan="2">RCM truth values</th><th colspan="3">CLIMATOLOGY</th><th colspan="3">PRESENT PERIOD 2006-2025</th><th colspan="3">99th Percentile</th></tr><tr><th></th><th></th><th></th><th>Number of days over 30°C</th><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Emul vs RCM truth</td><td>-2.166</td><td>9.879</td><td>16.518</td><td>0.000</td><td>3.356</td><td>23.500</td><td>12.138</td><td>23.817</td><td>28.580</td></tr><tr><td>MAX</td><td>SpaCor</td><td>SQ 05</td><td>Mean</td><td>SQ95</td><td>SpaCor</td><td>SQ 05</td><td>Mean</td><td>SQ95</td><td>SpaCor</td><td>SQ 05</td><td>Mean</td></tr><tr><td>MEAN</td><td>1.0 (E1)</td><td>-0.06 (E7)</td><td>0.06 (E16)</td><td>0.31 (E22)</td><td>0.96 (E11)</td><td>-5.27 (E22)</td><td>-0.57 (E22)</td><td>1.72 (E9)</td><td>1.00 (E3)</td><td>-0.61 (E22)</td><td>-0.19 (E22)</td><td>0.32 (E9)</td></tr><tr><td>MIN</td><td>1.00</td><td>-0.17</td><td>0</td><td>0.17</td><td>0.95</td><td>-8.35</td><td>-1.34</td><td>0.32</td><td>1.00</td><td>-0.87</td><td>-0.38</td><td>0.07</td></tr><tr><td>Main Emulator</td><td>1.0 (E1)</td><td>-0.30 (E1)</td><td>-0.07 (E12)</td><td>0.07 (E4)</td><td>0.93 (E6)</td><td>-12.22 (E6)</td><td>-1.92 (E6)</td><td>0.03 (E20)</td><td>1.00 (E9)</td><td>-1.11 (E2)</td><td>-0.6 (E6)</td><td>-0.19 (E20)</td></tr><tr><td>UPRCM</td><td>1.00</td><td>-0.08</td><td>0.03</td><td>0.17</td><td>0.95</td><td>-10.06</td><td>-1.68</td><td>0.06</td><td>1.00</td><td>-0.96</td><td>-0.45</td><td>0.06</td></tr><tr><td>Cheap</td><td>0.81</td><td>-2.88</td><td>0.53</td><td>8.02</td><td>0.41</td><td>-22.15</td><td>-2.80</td><td>0.08</td><td>0.77</td><td>-4.22</td><td>-0.32</td><td>7.4</td></tr><tr><td>Emul-Hist</td><td>1.00</td><td>-0.16</td><td>0.03</td><td>0.27</td><td>0.92</td><td>-7.77</td><td>-1.59</td><td>0.09</td><td>1.00</td><td>-0.84</td><td>-0.30</td><td>0.19</td></tr><tr><td>No.Smoothing</td><td>1.00</td><td>-0.39</td><td>-0.28</td><td>-0.09</td><td>0.83</td><td>-16.93</td><td>-2.79</td><td>0.02</td><td>1.00</td><td>-1.66</td><td>-0.98</td><td>-0.49</td></tr><tr><td>TAS_included</td><td>1.00</td><td>-0.14</td><td>-0.01</td><td>0.20</td><td>0.96</td><td>-8.59</td><td>-1.32</td><td>0.04</td><td>1.00</td><td>-0.77</td><td>-0.31</td><td>0.14</td></tr><tr><td></td><td>1.00</td><td>-0.12</td><td>-0.03</td><td>0.06</td><td>0.97</td><td>-6.24</td><td>-0.98</td><td>0.07</td><td>1.00</td><td>-0.65</td><td>-0.25</td><td>0.07</td></tr></tbody></table>

<table><thead><tr><th rowspan="2">RCM truth values</th><th colspan="3">CLIMATOLOGY</th><th colspan="3">FUTURE PERIOD 2080-2100</th><th colspan="3">99th Percentile</th></tr><tr><th></th><th></th><th></th><th>Number of days over 30°C</th><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Emul vs RCM truth</td><td>-0.19</td><td>11.53</td><td>17.89</td><td>0</td><td>16.80</td><td>112.29</td><td>14.04</td><td>25.7</td><td>30.55</td></tr><tr><td>MAX</td><td>SpaCor</td><td>SQ 05</td><td>Mean</td><td>SQ95</td><td>SpaCor</td><td>SQ 05</td><td>Mean</td><td>SQ95</td><td>SpaCor</td><td>SQ 05</td><td>Mean</td></tr><tr><td>MEAN</td><td>1.0 (E1)</td><td>-0.16 (E7)</td><td>0 (E16)</td><td>0.18 (E22)</td><td>0.98 (E3)</td><td>-30.67 (E11)</td><td>-5.75 (E11)</td><td>0.36 (E22)</td><td>1.00 (E26)</td><td>-0.89 (E22)</td><td>-0.35 (E22)</td><td>0.17 (E22)</td></tr><tr><td>MIN</td><td>1.00</td><td>-0.27</td><td>-0.08</td><td>0.07</td><td>0.97</td><td>-45.22</td><td>-8.06</td><td>0.04</td><td>1.00</td><td>-1.15</td><td>-0.57</td><td>-0.06</td></tr><tr><td>Main Emulator</td><td>1.00</td><td>-0.16</td><td>-0.04</td><td>0.09</td><td>0.95 (E13)</td><td>-61.91 (E6)</td><td>-10.63 (E6)</td><td>0 (E30)</td><td>1.00 (E1)</td><td>-1.44 (E2)</td><td>-0.83 (E6)</td><td>-0.24 (E6)</td></tr><tr><td>UPRCM</td><td>0.80</td><td>-2.87</td><td>0.5</td><td>7.79</td><td>0.98</td><td>-52.41</td><td>-9.23</td><td>0.01</td><td>1.00</td><td>-1.25</td><td>-0.63</td><td>-0.03</td></tr><tr><td>Cheap</td><td>1.00</td><td>-0.37</td><td>-0.14</td><td>0.09</td><td>0.96</td><td>-46.35</td><td>-9.08</td><td>0.01</td><td>0.78</td><td>-4.42</td><td>-0.34</td><td>7.29</td></tr><tr><td>Emul-Hist</td><td>1.00</td><td>-1.46</td><td>-1.27</td><td>-1.1</td><td>0.80</td><td>-100.01</td><td>-16.02</td><td>0</td><td>1.00</td><td>-1.45</td><td>-0.73</td><td>-0.11</td></tr><tr><td>No.Smoothing</td><td>1.00</td><td>-0.25</td><td>-0.13</td><td>0.06</td><td>0.98</td><td>-45.37</td><td>-8.17</td><td>0</td><td>1.00</td><td>-3.74</td><td>-2.48</td><td>-1.67</td></tr><tr><td>TAS_included</td><td>1.00</td><td>-0.14</td><td>-0.04</td><td>0.04</td><td>0.99</td><td>-26.54</td><td>-5.15</td><td>0.06</td><td>1.00</td><td>-1.04</td><td>-0.58</td><td>-0.15</td></tr></tbody></table>

<table><caption>**Table 3.3:** Summary of the results obtained in the 1st step of evaluation (UPRCM world). The scores are the one used to summarize the maps on figures 3.6,3.7, for the different versions of the emulator tested in the paper. The first line (RCM truth) is the RCM absolute values (except for the first section). On each sections, the three first lines give the mean and the range of the ensemble of emulators described in Section 3.4.4. For the ensemble minimums and maximums the name (E1 to E30) of the emulator corresponding to the value is given in brackets. “Main Emulator” is the one used for the results section (3.3). “Cheap” emulator is the one including only the variables at 700hpa described in Section 3.4.3. “Emul-Hist” is the emulator trained only on the historical period (section 3.4.2). “NO_smothing” and “TAS_included” correspond to the emulator trained with no prior smoothing of the data, and the one including TAS among the inputs (Section 3.4.3).</caption><thead><tr><th rowspan="2">RCM truth values</th><th colspan="3">CLIMATE CHANGE 2080-2100 vs 2006-2025</th><th colspan="3">99th Percentile</th></tr><tr><th></th><th></th><th></th><th>Number of days over 30°C</th><th></th><th></th><th></th></tr></thead><tbody><tr><td>Emul vs RCM truth</td><td>1.34</td><td>1.65</td><td>2.02</td><td>-0.08</td><td>13.45</td><td>90.14</td><td>1.01</td><td>1.88</td><td>2.79</td></tr><tr><td>MAX</td><td>SpaCor</td><td>SQ 05</td><td>Mean</td><td>SQ95</td><td>SpaCor</td><td>SQ 05</td><td>Mean</td><td>SQ95</td><td>SpaCor</td><td>SQ 05</td><td>Mean</td></tr><tr><td>MEAN</td><td>0.96 (E26)</td><td>-0.16 (E28)</td><td>-0.06 (E9)</td><td>0.05 (E1)</td><td>0.98 (E3)</td><td>-28.17 (E11)</td><td>-1.89 (E11)</td><td>0.31 (E22)</td><td>0.84 (E26)</td><td>-0.57 (E11)</td><td>-0.06 (E11)</td><td>0.48 (E11)</td></tr><tr><td>MIN</td><td>0.95</td><td>-0.19</td><td>-0.08</td><td>0.01</td><td>0.97</td><td>-39.67</td><td>-6.7</td><td>0.1</td><td>0.80</td><td>-0.73</td><td>-0.19</td><td>0.31</td></tr><tr><td>Main Emulator</td><td>0.95 (E1)</td><td>-0.21 (E22)</td><td>-0.11 (E6)</td><td>-0.02 (E6)</td><td>0.94 (E13)</td><td>-52.36 (E6)</td><td>-8.71 (E6)</td><td>0.01 (E29)</td><td>0.72 (E1)</td><td>-0.84 (E16)</td><td>-0.28 (E10)</td><td>0.22 (E16)</td></tr><tr><td>UPRCM</td><td>0.96</td><td>-0.18</td><td>-0.07</td><td>0.02</td><td>0.97</td><td>-44.98</td><td>-7.56</td><td>0.1</td><td>0.76</td><td>-0.77</td><td>-0.18</td><td>0.38</td></tr><tr><td>Cheap</td><td>0.82</td><td>-0.3</td><td>-0.03</td><td>0.15</td><td>0.31</td><td>-87.4</td><td>-12.56</td><td>1.54</td><td>0.62</td><td>-0.81</td><td>-0.03</td><td>0.64</td></tr><tr><td>Emul-Hist</td><td>0.94</td><td>-0.3</td><td>-0.17</td><td>0.07</td><td>0.96</td><td>-41.43</td><td>-7.47</td><td>0.11</td><td>0.73</td><td>-1.03</td><td>-0.43</td><td>0.21</td></tr><tr><td>No.Smoothing</td><td>0.87</td><td>-1.23</td><td>-0.99</td><td>-0.86</td><td>0.66</td><td>-84.12</td><td>-13.23</td><td>0.07</td><td>0.55</td><td>-2.34</td><td>-1.5</td><td>-0.92</td></tr><tr><td>TAS_included</td><td>0.96</td><td>-0.25</td><td>-0.11</td><td>-0.03</td><td>0.97</td><td>-38.92</td><td>-6.86</td><td>0.08</td><td>0.84</td><td>-0.81</td><td>-0.27</td><td>0.12</td></tr><tr><td></td><td>0.98</td><td>-0.09</td><td>-0.01</td><td>0.07</td><td>0.98</td><td>-23.52</td><td>-4.18</td><td>0.16</td><td>0.87</td><td>-0.51</td><td>-0.08</td><td>0.36</td></tr></tbody></table>
<table><thead><tr><th rowspan="2"></th><th colspan="3">RMSE</th><th colspan="3">ACC</th><th colspan="3">Variance ratio</th><th colspan="3">Wasserstein distance</th></tr><tr><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th></tr></thead><tbody><tr><td>MAX</td><td>1.44 (Em30)</td><td>2.27 (Em7)</td><td>2.85 (Em22)</td><td>0.70 (Em8)</td><td>0.76 (Em8)</td><td>0.81 (Em2)</td><td>0.96 (Em8)</td><td>1.01 (Em22)</td><td>1.09 (Em26)</td><td>0.29 (Em2)</td><td>0.82 (Em7)</td><td>1.23 (Em27)</td></tr><tr><td>MEAN</td><td>1.40</td><td>2.24</td><td>2.80</td><td>0.69</td><td>0.76</td><td>0.81</td><td>0.93</td><td>0.99</td><td>1.06</td><td>0.21</td><td>0.75</td><td>1.11</td></tr><tr><td>MIN</td><td>1.38 (Em14)</td><td>2.20 (Em12)</td><td>2.76 (Em4)</td><td>0.69 (Em7)</td><td>0.75 (Em10)</td><td>0.81 (Em1)</td><td>0.90 (Em6)</td><td>0.96 (Em6)</td><td>1.03 (Em20)</td><td>0.13 (Em12)</td><td>0.66 (Em12)</td><td>1.00 (Em30)</td></tr><tr><td>Main Emulator</td><td>1.46</td><td>2.25</td><td>2.83</td><td>0.69</td><td>0.76</td><td>0.81</td><td>0.95</td><td>1.00</td><td>1.07</td><td>0.29</td><td>0.78</td><td>1.12</td></tr><tr><td>GCM</td><td>1.76</td><td>3.34</td><td>10.17</td><td>0.61</td><td>0.72</td><td>0.78</td><td>0.73</td><td>0.97</td><td>1.28</td><td>0.24</td><td>2.17</td><td>9.73</td></tr><tr><td>Cheap</td><td>1.48</td><td>2.25</td><td>2.83</td><td>0.69</td><td>0.75</td><td>0.80</td><td>0.93</td><td>0.98</td><td>1.03</td><td>0.46</td><td>0.84</td><td>1.14</td></tr><tr><td>Emul-Hist</td><td>1.54</td><td>2.16</td><td>2.75</td><td>0.68</td><td>0.75</td><td>0.8</td><td>0.89</td><td>0.95</td><td>1.02</td><td>0.16</td><td>0.28</td><td>0.55</td></tr><tr><td>No Smoothing</td><td>1.50</td><td>2.29</td><td>2.96</td><td>0.59</td><td>0.73</td><td>0.80</td><td>0.83</td><td>0.94</td><td>1.07</td><td>0.18</td><td>0.53</td><td>0.92</td></tr><tr><td>TAS included</td><td>1.81</td><td>2.48</td><td>4.13</td><td>0.65</td><td>0.73</td><td>0.79</td><td>0.83</td><td>1.03</td><td>1.21</td><td>0.16</td><td>0.84</td><td>2.93</td></tr></tbody></table>

<table><thead><tr><th colspan="11">PRESENT PERIOD 2006-2025</th></tr><tr><th rowspan="2">RCM truth values</th><th colspan="3">CLIMATOLOGY</th><th colspan="3">ACC</th><th colspan="3">Variance ratio</th><th colspan="3">Wasserstein distance</th></tr><tr><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th></tr></thead><tbody><tr><td>Emul vs RCM truth</td><td>-2.17</td><td>9.90</td><td>16.52</td><td>0.00</td><td>3.36</td><td>23.50</td><td>12.14</td><td>23.82</td><td>28.58</td><td>0.29 (Em2)</td><td>0.82 (Em7)</td><td>1.23 (Em27)</td></tr><tr><td>MAX</td><td>1.00 (Em3)</td><td>0.41 (Em19)</td><td>1.00 (Em7)</td><td>1.41 (Em27)</td><td>0.96 (Em1)</td><td>-1.35 (Em22)</td><td>5.10 (Em22)</td><td>34.22 (Em22)</td><td>1.00 (Em7)</td><td>0.3 (Em22)</td><td>0.96 (Em22)</td><td>1.63 (Em5)</td></tr><tr><td>MEAN</td><td>1.00</td><td>0.27</td><td>0.93</td><td>1.31</td><td>0.95</td><td>-2.1</td><td>3.08</td><td>23.78</td><td>1.00</td><td>-0.01</td><td>0.73</td><td>1.34</td></tr><tr><td>MIN</td><td>1.00 (Em1)</td><td>0.13 (Em27)</td><td>0.83 (Em12)</td><td>1.20 (Em12)</td><td>0.94 (Em21)</td><td>-3.10 (Em21)</td><td>1.64 (Em6)</td><td>13.31 (Em20)</td><td>1.00 (Em27)</td><td>-0.35 (Em27)</td><td>0.50 (Em20)</td><td>1.05 (Em20)</td></tr><tr><td>Main Emulator</td><td>1.00</td><td>0.43</td><td>0.96</td><td>1.29</td><td>0.95</td><td>-1.61</td><td>3.23</td><td>24.06</td><td>1.00</td><td>0.26</td><td>0.83</td><td>1.43</td></tr><tr><td>GCM</td><td>0.70</td><td>-3.16</td><td>1.19</td><td>9.99</td><td>0.16</td><td>-22.53</td><td>-1.49</td><td>9.95</td><td>0.58</td><td>-4.63</td><td>0.77</td><td>9.38</td></tr><tr><td>Cheap</td><td>1.00</td><td>0.61</td><td>1.05</td><td>1.40</td><td>0.93</td><td>-4.75</td><td>0.33</td><td>17.63</td><td>1.00</td><td>0.032</td><td>0.61</td><td>1.28</td></tr><tr><td>Emul-Hist</td><td>1.00</td><td>0.10</td><td>0.66</td><td>0.98</td><td>0.86</td><td>-10.49</td><td>-2.05</td><td>0.12</td><td>1.00</td><td>-0.98</td><td>-0.17</td><td>0.46</td></tr><tr><td>No Smoothing</td><td>1.00</td><td>-0.05</td><td>0.68</td><td>1.20</td><td>0.73</td><td>-5.26</td><td>1.95</td><td>19.75</td><td>0.98</td><td>-0.97</td><td>0.52</td><td>2.39</td></tr><tr><td>TAS included</td><td>0.97</td><td>-1.38</td><td>0.55</td><td>3.12</td><td>0.61</td><td>-5.22</td><td>3.86</td><td>33.57</td><td>0.95</td><td>-1.77</td><td>0.60</td><td>2.88</td></tr></tbody></table>

<table><thead><tr><th colspan="11">FUTURE PERIOD 2080-2100</th></tr><tr><th rowspan="2">RCM truth values</th><th colspan="3">CLIMATOLOGY</th><th colspan="3">ACC</th><th colspan="3">Variance ratio</th><th colspan="3">Wasserstein distance</th></tr><tr><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th></tr></thead><tbody><tr><td>Emul vs RCM truth</td><td>-0.19</td><td>11.53</td><td>17.89</td><td>0.00</td><td>16.80</td><td>112.29</td><td>14.04</td><td>25.70</td><td>30.55</td><td>-0.08</td><td>13.45</td><td>90.14</td></tr><tr><td>MAX</td><td>1.00 (Em22)</td><td>0.17 (Em11)</td><td>0.73 (Em7)</td><td>1.18 (Em16)</td><td>0.97 (Em8)</td><td>-13.17 (Em22)</td><td>0.86 (Em22)</td><td>32.27 (Em22)</td><td>1.00 (Em4)</td><td>-0.33 (Em22)</td><td>0.41 (Em22)</td><td>1.25 (Em5)</td></tr><tr><td>MEAN</td><td>1.00</td><td>0.01</td><td>0.65</td><td>1.07</td><td>0.96</td><td>-26.27</td><td>-3.05</td><td>12.458</td><td>1.00</td><td>-0.70</td><td>0.14</td><td>0.99</td></tr><tr><td>MIN</td><td>1.00 (Em1)</td><td>-0.15 (Em27)</td><td>0.56 (Em12)</td><td>0.97 (Em30)</td><td>0.94 (Em2)</td><td>-30.84 (Em20)</td><td>-6.69 (Em20)</td><td>0.42 (Em20)</td><td>0.99 (Em27)</td><td>-1.00 (Em2)</td><td>-0.14 (Em20)</td><td>0.74 (Em20)</td></tr><tr><td>Main Emulator</td><td>1.00</td><td>0.16</td><td>0.69</td><td>1.11</td><td>0.96</td><td>-25.33</td><td>-2.62</td><td>12.25</td><td>1.00</td><td>-0.58</td><td>0.23</td><td>1.08</td></tr><tr><td>GCM</td><td>0.70</td><td>-3.28</td><td>1.00</td><td>9.59</td><td>0.07</td><td>-109.27</td><td>-12.11</td><td>21.34</td><td>0.56</td><td>-5.19</td><td>0.30</td><td>9.04</td></tr><tr><td>Cheap</td><td>1.00</td><td>0.3</td><td>0.70</td><td>1.03</td><td>0.93</td><td>-40.66</td><td>-8.31</td><td>0.18</td><td>1.00</td><td>-1.14</td><td>-0.29</td><td>0.69</td></tr><tr><td>Emul-Hist</td><td>1.00</td><td>-0.96</td><td>-0.50</td><td>-0.10</td><td>0.76</td><td>-99.23</td><td>-15.87</td><td>0.03</td><td>0.99</td><td>-3.26</td><td>-1.93</td><td>-0.77</td></tr><tr><td>No Smoothing</td><td>0.997</td><td>-0.64</td><td>0.22</td><td>0.77</td><td>0.54</td><td>-86.63</td><td>-9.42</td><td>23.84</td><td>0.98</td><td>-2.15</td><td>-0.35</td><td>1.59</td></tr><tr><td>TAS included</td><td>0.97</td><td>-1.43</td><td>0.39</td><td>2.86</td><td>0.39</td><td>-63.43</td><td>2.64</td><td>87.53</td><td>1.00</td><td>-2.13</td><td>0.17</td><td>2.33</td></tr></tbody></table>

<table><caption>**Table 3.4:** Summary of the results obtained in the 2nd step of evaluation (GCM world). The scores are the one used to summarize the maps on figures 3.6,3.7, for the different versions of the emulator tested in the paper. The first line (RCM truth) is the RCM absolute values (except for the first section). On each sections, the three first lines give the mean and the range of the ensemble of emulators described in Section 3.4.4. For the ensemble minimums and maximums the name (E1 to E30) of the emulator corresponding to the value is given in brackets. “Main Emulator” is the one used for the results section (3.3). “Cheap” emulator is the one including only the variables at 700hpa described in Section 3.4.3. “Emul-Hist” is the emulator trained only on the historical period (Section 3.4.2). “NO_sMOOTHING” and “TAS_included” correspond to the emulator trained with no prior smoothing of the data, and the one including TAS among the inputs (Section 3.4.3).</caption><thead><tr><th colspan="11">CLIMATE CHANGE 2080-2100 vs 2006-2025</th></tr><tr><th rowspan="2">RCM truth values</th><th colspan="3">Annual Mean Delta TAS</th><th colspan="3">ACC</th><th colspan="3">Variance ratio</th><th colspan="3">Wasserstein distance</th></tr><tr><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th><th>SQ 05</th><th>Mean</th><th>SQ95</th></tr></thead><tbody><tr><td>Emul vs RCM truth</td><td>1.33</td><td>1.65</td><td>2.02</td><td>-0.08</td><td>13.45</td><td>90.14</td><td>1.01</td><td>1.88</td><td>2.79</td><td>-0.08</td><td>13.45</td><td>90.14</td></tr><tr><td>MAX</td><td>0.91 (Em28)</td><td>-0.39 (Em28)</td><td>-0.26 (Em28)</td><td>-0.13 (Em11)</td><td>0.94 (Em8)</td><td>-28.15 (Em22)</td><td>-4.24 (Em22)</td><td>9.98 (Em22)</td><td>0.76 (Em12)</td><td>-1.06 (Em11)</td><td>-0.51 (Em11)</td><td>0.22 (Em4)</td></tr><tr><td>MEAN</td><td>0.89</td><td>-0.44</td><td>-0.28</td><td>-0.155</td><td>0.917</td><td>-37.72</td><td>-6.13</td><td>3.56</td><td>0.666</td><td>-1.19</td><td>-0.59</td><td>0.14</td></tr><tr><td>MIN</td><td>0.87 (Em22)</td><td>-0.47 (Em6)</td><td>-0.31 (Em6)</td><td>-0.18 (Em10)</td><td>0.88 (Em6)</td><td>-49.90 (Em6)</td><td>-8.33 (Em20)</td><td>0.27 (Em20)</td><td>0.47 (Em13)</td><td>-1.38 (Em13)</td><td>-0.66 (Em30)</td><td>0.02 (Em12)</td></tr><tr><td>Main Emulator</td><td>0.89</td><td>-0.41</td><td>-0.27</td><td>-0.14</td><td>0.91</td><td>-37.94</td><td>-5.92</td><td>3.58</td><td>0.63</td><td>-1.26</td><td>-0.59</td><td>0.15</td></tr><tr><td>GCM</td><td>0.56</td><td>-0.54</td><td>-0.19</td><td>0.07</td><td>0.02</td><td>-88.65</td><td>-10.62</td><td>13.54</td><td>0.04</td><td>-1.36</td><td>-0.47</td><td>0.48</td></tr><tr><td>Cheap</td><td>0.91</td><td>-0.49</td><td>-0.34</td><td>-0.20</td><td>0.91</td><td>-47.58</td><td>-8.64</td><td>0.18</td><td>0.96</td><td>-1.60</td><td>-0.91</td><td>-0.24</td></tr><tr><td>Emul-Hist</td><td>0.73</td><td>-1.44</td><td>-1.16</td><td>-0.99</td><td>-0.59</td><td>-94.60</td><td>-13.83</td><td>0.11</td><td>0.36</td><td>-2.59</td><td>-1.77</td><td>-0.96</td></tr><tr><td>No Smoothing</td><td>0.73</td><td>-0.76</td><td>-0.46</td><td>-0.27</td><td>0.37</td><td>-80.50</td><td>-11.37</td><td>9.01</td><td>0.60</td><td>-1.56</td><td>-0.87</td><td>-0.19</td></tr><tr><td>TAS included</td><td>0.68</td><td>-0.44</td><td>-0.17</td><td>0.10</td><td>0.56</td><td>-59.36</td><td>-1.22</td><td>56.57</td><td>0.29</td><td>-1.31</td><td>-0.43</td><td>0.43</td></tr></tbody></table>
# Chapter 4

## Study of the RCM-Emulator applicability to GCM ensembles

This chapter evaluates the robustness of the emulator introduced in Chapter 3 to various input sources. To simplify the reading, I tried to make it independent, implying that some parts recall and summarise the previous chapter. Moreover, I work with multiple RCM climate simulations and their driving GCM simulations. I have tried to simplify the naming and clearly identify the simulations I use at the beginning of each section. Nevertheless, It might be difficult to follow for a reader non-familiar with those climate models, and I apologize in advance for that.
## 4.1 Introduction

Evaluating and understanding the uncertainties associated with climate projections is crucial to deliver reliable information about climate change considering all possible futures. There are three main sources of uncertainty. The chaotic nature of the climate system induces a strong natural variability reproduced in climate models and which needs to be quantified. The future evolution of human activities is another source of uncertainty that the scientific community addresses with different scenarios of greenhouse gas emissions along the 21st century (Meinshausen et al, 2011). Finally, each climate model is a possible representation of the Earth System. They are built by different teams in different research centres worldwide with different approaches to solving the same problems. As there is no “best” model, the variety of models is essential, and the robustness of a result can come from the agreement of the majority of these models. Multiple studies are developing methods to make the best out of this large ensemble of GCM simulations (Knutti, 2010; Ribes et al, 2017). Thus, to provide reliable information about climate change, the scientific community cooperates to build large simulation ensembles with multiple Global Climate Models (GCMs) driven by different scenarios. Several members from the same GCM/Scenario pair are produced to consider the internal variability.

Regional Climate Models (RCM) are high-resolution limited area models nested in a GCM simulation. They are used to downscale GCM simulations, which are too coarse to reveal the consequences of climate change at local scales. RCMs are an additional source of uncertainty. Ideally, it is necessary to downscale with multiple RCMs each simulation from the GCM large ensembles to consider all possible futures and create reliable information about the local impacts of global climate warming. However, the high cost of RCMs only allows for downscaling some simulations, and the needed exploration of uncertainties is limited. RCMs ensembles are 4-dimensional matrices, and the driving scenarios, the choices of the GCM and the RCM, and the number of members are the four entries. CORDEX, is an international initiative which aims to coordinate the design and filling of such matrices over specific regional domains. Figure 4.1 illustrates the EURO-CORDEX domain and matrix, which results from about ten years of international cooperation.

<figure><img src="image_41.png" /><figcaption>**Figure 4.1:** EURO-CORDEX domain and matrix illustration.</figcaption></figure>

To tackle this issue, Chapter 3 introduces the RCM-Emulator, a hybrid downscaling approach combining both statistical and dynamical downscaling. The emulator
learns the downscaling function included in the RCM. It is the relationship between
low-resolution altitude variables - supposedly well represented by GCMs - and
high-resolution surface variables. To ensure perfect coherence between predictors
and predictands, the emulator is trained in a perfect model framework, where
both inputs and target come from the same RCM simulation. The emulator is
designed for a specific RCM and can emulate only this one. Chapter 3 presents the
evaluation of the emulator on a single RCM simulation driven by the same GCM
as the training simulation but in a different scenario. The present chapter aims to
extend this evaluation.

The main objective of the emulator is to downscale a maximum of GCM simula-
tions while respecting and preserving their diversity. It is then important to study
its transferability to different sources of inputs. Therefore, different levels of trans-
ferability can be considered.

1. Member-transferability: the simplest, it evaluates the emulator's ability to downscale another member from the same GCM/Scenario couple as the one used for training.

2. Scenario-transferability: It is the robustness to different forcing scenarios with the same GCM. In chapter 3, we started to validate this question, and the present chapter will further document it.

3. GCM-transferability: probably the most complex one, it is the ability of the emulator to provide a coherent downscaling of simulations from global models different than the one used to drive the training simulation.

This work is based on ALADIN63, the state-of-the-art regional climate model
developed at CNRM. In the EURO-CORDEX and Copernicus C3S projects, 10
ALADIN63 simulations were performed covering different driving GCMs and
scenarios. This small ALADIN63 matrix 4.1 allows us to explore levels 2 and 3 of
transferability as any of this simulation is a possible test set. Moreover, I think it
is legitimate to consider that the second level implies the first one, especially at the
beginning of the 21st century, where the scenarios are similar. It is worth recalling
here that the emulator is designed to learn the downscaling function of a given
RCM, which is why the evaluation focuses on ALADIN63 runs.

This transferability evaluation will be conducted in three steps. First, I evaluate
the scenario and GCM-transferability of the emulator for near surface temperature
introduced in the last chapter. Secondly, I will extend this analysis by training
other emulators of the same RCM (ALADIN63) using different training simulations
and check if it provides similar results. Furthermore, I will also investigate the
benefits of using simulations from multiple driving GCMs in the training set.
Following the methodology and the results from Chapter 3, I will address these
questions of transferability in a perfect model framework. This choice is deeper
commented in Section 4.2.3. Finally, I will use the emulator in application mode
and downscale the GCM simulations used to drive ALADIN63. I will try to assess
the quality of these emulated series by comparing them with both the GCM low
resolution and the RCM high-resolution temperature.
<table><caption>**Table 4.1:** The ALADIN63 matrix.</caption><thead><tr><th rowspan="2">DRIVING<br>SCENARIO</th><th colspan="4">DRIVING GCMS</th></tr><tr><th>CNRM-CM5<br>(CNRM)</th><th>MPI-ESM-LR<br>(MPI)</th><th>NorESM1-M<br>(NCC)</th><th>HadGEM-ES2<br>(HGM)</th></tr></thead><tbody><tr><td>Historical</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>RCP26</td><td>x</td><td></td><td></td><td></td></tr><tr><td>RCP45</td><td>x</td><td></td><td></td><td></td></tr><tr><td>RCP85</td><td>x</td><td>x</td><td>x</td><td>x</td></tr></tbody></table>

This chapter is divided into four sections. The first one presents the simulations used in this chapter and sets the methodology applied to evaluate the transferability of the emulator. After that, sections 4.3 and 4.4 present the results in perfect model evaluation and the application to GCM inputs. The last one discusses different choices made during the construction of the emulator.

## 4.2 Methodology

This section firstly introduces the available simulations to explore the transferability question. Then I recall the principle of the perfect model framework and justify why I use it for this chapter. Finally, I will expose the different metrics I will use.

### 4.2.1 The ALADIN63 matrix

In the EURO-CORDEX framework, ten scenario simulations were performed with ALADIN63 with different driving GCMs and scenarios. They are reported in Table 4.1. All GCMs used to drive ALADIN63 are part of the fifth phase of the Coupled Model Intercomparison Project (CMIP5). The first GCM is CNRM-CM5, which is developed at CNRM. Because they are developed in the same institute, ALADIN63 and CNRM-CM5 share very similar components that guarantee some consistency in the downscaling procedure<fnref n="1" />. Four simulations from CNRM-CM5 were downscaled by ALADIN63, one historical (1951-2005) and one for each RCP2.6, RCP4.5 and RCP8.5 scenarios (2006-2100) of greenhouse gas emissions. In chapter 1, the emulator is trained on the CNRM-HIST and RCP85, while the RCP4.5 was used to evaluate it. The other driving GCMs are MPI-ESM-LR, developed at the Max Plank Institute (Giorgetta et al, 2013), NorESM1-M, developed at the Norvegian Climate Center (Bentsen et al, 2013) and the HadGEM2-ES, the Met Office climate model (Collins et al, 2011). The historical and RCP8.5 runs were downscaled for these three models with ALADIN63. The CNRM, MPI, NCC and HGM acronyms will respectively refer to the CNRM-CM5, the MPI-ESM-LR, the NorESM1-M and the HadGEM2-ES GCMs.

In Section 4.4 I will use the same GCM simulations to test the emulator in application mode. As they all have different horizontal resolutions, I interpolated all of them on the CNRM grid at 1.4° resolution. All these simulations are available on

<footnote n="1">¹ALADIN63 relies on a more recent version of the code than CNRM-CM5, so the consistency is not “perfect” but the best we have</footnote>
<figure><img src="image_42.png" /><figcaption>**Figure 4.2:** Illustration of the inputs domain (blue box including the sides) and the target domain (green box).</figcaption></figure>

the mesocenter ESPRI facility, which is supported by CNRS, UPMC, Labex L-IPSL, CNES and Ecole Polytechnique. Concerning the HGM historical simulation, I could not find some input variables on the ESPRI nor the ESGF (ref) nodes, so I only work on the RCP85 simulation.

### 4.2.2 Recall on the main emulator and Experiment design

The ALADIN63-emulator from Chapter 3 is built to reproduce the daily near-surface temperature over a 64×64 grid points domain located between South of France and North-East of Spain and illustrated in Figure 4.2. The predictors are five variables at three different pressure levels<fnref n="2" /> plus some surface fields and the aerosols optical depth over a larger input domain of 16×16 GCM (CNRM-CM5) grid points. They are normalised daily according to their spatial means and standard deviations so that only the information about the atmosphere's spatial structure remains. The temporal information is given through a vector containing for each day the spatial means and standard deviations used for normalising each variable map, plus external forcings such as the greenhouse gas concentration and a seasonal vector. The emulator relies on a neural network architecture called UNet that has been adapted to the problem. More details can be found in the previous chapter.

The emulator is calibrated in the perfect model framework. It implies that, for the training phase, both predictors and predictands come from the same high-resolution simulation. The idea is to ensure a perfect correlation between the predictors and the target surface variable, such that the neural network focuses on the estimation of the downscaling function only. The predictors are then upscaled to a typical GCM resolution, and we apply a moving average window to kill any high-resolution

<footnote n="2">²i.e. different altitudes</footnote>
<figure><img src="image_43.png" /><figcaption>**Figure 4.3:** Training, evaluation and application of the emulator. Inspired and adapted from Wang et al (2021).</figcaption></figure>

information that could have remained. As explained in chapter 3, the same “filtering” will be applied to GCM data before giving them to the emulator as it allows to reach the effective resolution of the model and to work with more robust data. Figure 4.3 illustrates the training protocol. The perfect model framework also serves for the evaluation with a different simulation than the training set. Thus, the emulated series can be compared with the RCM high-resolution variable we target to emulate. Conversely, when the emulator is used to downscale a GCM simulation, it is impossible to evaluate the emulated series correctly. As the emulator does not reproduce the large scale biases between GCM and RCM, it is normal to have biases between the GCM-downscaled series and the true RCM. Therefore, separating these large scale biases’ local impact and the emulator’s possible error is difficult.

In Chapter 3 the ALADIN63-emulator is trained on CNRM-HIST and CNRM-RCP85 ALADIN63 simulations and evaluated on the CNRM-RCP45, in the perfect model framework. From now on, I will refer to it as the *main emulator*. The first part of this work (Section 4.3.1) analyses the main emulator on all other ALADIN63 simulations in perfect model framework. Following Figure 4.3, predictors are extracted from each of the ALADIN63 simulations (cf. Table 4.1), upscaled and filtered to obtained the *UPRCM* simulations. Those inputs are then given to the main emulator, which recreates high-resolution temperature series that are compared with the corresponding ALADIN63 simulation. The objective is then to assess how the emulator “digests” inputs from different sources. In section 4.3.2, I construct other ALADIN63-emulators using other training sets. The exercise aims to evaluate if it affects the results obtained in chapter 3 and section 4.3.1, and so how the emulator depends on the training set.
4.2.3 Justification of the perfect model framework

For the reasons above, I first address the transferability in the perfect model
framework. The RCM-emulator relies on a specific convolutional neural network
architecture. Another way to see the current chapter goals is to assess if the neural
network and the previous strategy will be robust to slightly different input spaces.
Indeed, the driving scenario modifies the atmosphere composition, which impacts
all variables differently. Furthermore, two GCMs rely on different assumptions
reflecting the choices of the teams building them and impacting the entire structure
of the system, the relationship between variables, the vertical dependence etc.. In
both cases, the input space from a new simulation to downscale will differ from the
training input space, and this chapter aims to study the robustness of the emulator
in such cases.

To validate the use of the perfect model framework it is necessary to assess
if the difference between two RCM simulations is representative from the one
between two GCM simulations. The driving RCP scenario is likely to impact
RCM and GCM similarly, so it seems reasonable to study the second level of
transferability within this framework. However, it is not intuitive that the in-
trinsic differences between two GCMs will reflect in the RCM simulations they drive.

The upper panel of Figure 4.4 shows the yearly mean times series (1950-2100, RCP8.5) for the temperature, the geopotential and the humidity at two pressure levels (500 and 850 hPa) over the input domain defined in chapter 1, for the 4 UPscaled RCM (UPRCM) simulations we have. They are an extract of the inputs given to the emulator. This figure helps to illustrate the kind of differences that can exist between the different ALADIN63 simulations. Firstly we can observe that the relationship between variables at a given pressure level differs between two simulations. For example, NCC-UPRCM is generally less humid and colder than MPI-UPRCM at 500hPa but has a similar altitude for the geopotential. Secondly, the vertical dynamic differs, as we can see by comparing the right and left panels. For instance, the MPI-UPRCM is more humid than any other simulation at 500hPa, but it is not true at 850hPa. Finally, we can also remark that the different simulations have different responses to the increase in greenhouse gas concentration imposed by the RCP85 scenario. The HGM-UPRCM (in green) notably simulates more significant changes than the others. Therefore, fundamental differences exist between these 4 UPRCM simulations in the internal relationship of the emulator inputs.

The lower panel shows the yearly mean times series of the same variables but for
the driving GCM simulations. The first comforting remark is that the differences
observed in the UPRCM simulations are in the same order of magnitude as the ones
we can see in the GCM simulations. For a given variable, the vertical structure
differs from one model to the other. When comparing right and left columns, it is
particularly visible for the MPI-GCM humidity or the CNRM-GCM temperature.
This result tends to validate the evaluation of transferability in the perfect model
framework. Moreover, we can observe how the RCM transforms the GCM large
scale. Regarding almost all six variables, the differences between the four simula-
tions are not the same as in the UPRCM world. The most remarkable example
is probably for the HGM simulations, where ALADIN seems to underestimate
the atmospheric warming of the original GCM simulation. Still, regarding the
temperature at 850hPa, CNRM-GCM, MPI-GCM, and NCC-GCM are much more
similar than their UPRCM versions.

Figure 4.4 confirms that the perfect model framework is a good opportunity to test whether the emulator built in Chapter 3 of this manuscript is robust to different input sources and all the intrinsic differences they carry.

4.2.4 Metrics, Benchmark and results presentation

As in the previous chapter, I use the interpolated low resolution near-surface tem-
perature from the corresponding input simulation as a naive reference. It helps us
to locate the emulator’s performance with respect to a representation of its driving
large scale. On the other hand, I will also evaluate the emulators on their training set
and use it as a reference for the best result possible as it is a level 0 of transferability.

As a matter of coherence, I use the same metrics as the ones Chapter 3 over slightly
different periods. I compute the maps for the mean and 99th and 1st quantiles over
three thirty-year periods: 1951-1980, 2006-2035, and 2071-2100 referred to as past,
present and future periods. I replaced in this chapter the number of days over 30°C
by the first quantile as it is essential to look at both sides of the distribution. These
maps are computed for the RCM truth, the Emulated series and the interpolated
UPRCM/GCM. In addition, I add the biases maps to highlight the differences
between the RCM and both the emulator and I-UPRCM maps. I also use the
four scores from the previous chapter, the RMSE, the variance ratio, the temporal
correlation (ACC) and the Wasserstein distance. These scores are computed over
the whole simulation, separating the historical (1950-2005) from the projection
(2006-2100) part.

These metrics are computed at the grid point scale, implying that I obtain a
map for each metric and each evaluated simulation. As in the previous chapter,
we use the spatial mean and superquantiles (5th and 95th) to summarise each
map. For the raw climatology and quantile maps, I also use the spatial correlation
and spatial RMSE to measure the ability of the emulator to reproduce the high
resolution spatial structure brought by the RCM. In this chapter, I will evaluate
many emulators on multiple metrics. Thanks to these summary metrics, I can
synthesise all these results summary plots.
<figure><img src="image_44.png" /></figure>

**Figure 4.4:** Yearly anomalies time series for 3 atmospherics variables (Specific Humidity, Geopotential altitude and Temperature, from up to down) at 2 pressure levels (500 and 850 hPa, from left to right). These series are shown for the 4 UPRCM simulations (upper panel) and their driving GCM simulations (lower panel). We show here the yearly mean over the input domain. The series are plotted as anomalies compared to CNRM-UPRCM climatology over the 1971-2000 period. In the upper panel the simulations are differentiated according to the driving simulation which is plotted in the lower panel.
## 4.3 Perfect Model evaluation

In this section, I address the reliability of the emulator under different sources of inputs and its capacity to reproduce the daily maps under different climates. We place ourselves here in the perfect model framework to evaluate the emulated series' accuracy with respect to their RCM true series. We divide this section into two parts. Firstly we will study the transferability of the emulator trained in Chap 1, and secondly, we will train 15 other emulators on multiple combinations of the training set.

### 4.3.1 Main emulator

The UNet-emulator built in chapter 1 was trained using the entire CNRM Historical and RCP85 simulations following the training procedure recalled in Section 4.2.2. It was then evaluated in the perfect model framework on the CNRM RCP45 simulation, showing an excellent ability to downscale a more intermediate RCP scenario than the one used for calibration. I extend this evaluation to all ALADIN63 simulations (see Sec 4.2.1). The results shown here are computed on the novel simulations but also on the CNRM-RCP45 to compare the new results with the ones from the previous chapter. I also downscaled the UPRCM-CNRM HIST and RCP85 used for calibration, which I consider a naive benchmark. Of course, the results on this simulation are expected to be good. As in Chapter 3 we use the interpolated UPRCM temperature as a benchmark. These two benchmarks will help us to analyse and compare the results on the different simulations. In the entire section the RCM simulations are named accordingly to their driving GCMs. The GCM simulations are not used in this section, so the names CNRM, MPI, NCC or HGM only refer to the ALADIN63 simulations.

#### Yearly mean series reconstruction

Figure 4.5 gives a first insight into the emulator’s ability to reproduce the different climates, as it plots the time series of the year means over the whole domain. The emulator produces, as the RCM, daily temperature maps, so these mean series are, in some sense, reconstructed large scale. In a matter of coherence, it is fundamental that the emulator respects and reproduces this kind of chronology and sensitivity aspects well, even if it is not designed for it.

The RCM and emulator series look perfectly similar, with the correct order between the different simulations. The gaps between the 3 CNRM simulations are well reproduced. The RCP85 warms much stronger after 2050, while the RCP26 is slightly cooling at the end of the century. Moreover, the MPI and NCC simulations appear warmer than the CNRM throughout the period, and the emulator respects this well. Finally, the HGM RCM simulation (in brown) is warming the most and exactly as in the emulator plot. These series tend to show that the emulator is robust to different input conditions and correctly follows its driving large scale. Now we can have a closer look into these emulated simulations.
<figure><img src="image_45.png" /><figcaption>**Figure 4.5:** Time series of the yearly mean near surface temperature over the target domain for the 6 entire (Historical + Scenario) simulations at our disposal. The upper panel shows the series computed on the RCM true simulations, while the lower panel shows the ones computed from the emulated simulations.</figcaption></figure>

## Grid points 30 years statistics

In a second step, we can look at temporal statistics at the grid point scale over 30 years periods. I compute the mean, 99th and 1st percentile maps over the past (1971-2000), present (2006-2035) and future (2071-2100) periods. As explained in Section 4.2.4 we summarise these maps with the spatial mean and spatial 5th and 95th super-quantiles, and I report these numbers in summary bar plots. The biases maps regarding the corresponding RCM truth maps are summarised similarly. Figures 4.6 and 4.7 show the summary plots for the present period and an illustration for one simulation. Figure 4.8 shows the summary plots for the past and future periods. The results are shown for both Scenario and GCM transferability, but we firstly comment the scenario-transferability (i.e. the left part of the plots).

The first conclusions on the yearly series are confirmed at the grid point scale. Concerning the Scenario-transferability, the emulator reproduces accurately the CNRM-RCP26 climatology and quantiles maps. It confirms the results obtained in the last chapter on CNRM-RCP45 and the ability of the emulator to reproduce more intermediate scenarios than the ones used for calibration. For both simulations, the emulator reproduces the spatial structures of the different maps very well, as the spatial correlations are always equal to one and the RMSE around 0.1°C on the present climatology (Figure 4.6). The climatology maps are identical, as the red
and green bars are indistinguishable and the biases very small. Figure 4.7 shows that the emulator reproduces less well the extreme percentile maps on these two simulations. The spatial RMSEs are higher (0.3/0.4°C), and the bias summary plots show an underestimation of the warmest extremes and inversely for the 1st percentile. The maps for the CNRM RCP26 1st percentile (Figure 4.7) help to understand these biases. The emulator captures the spatial structure but has a general warm bias. Remarkably, the worst points do not necessarily correspond to the coldest points, as the bias is more significant in the Rhone valley than on top of the Pyrenees. The analysis is the same for the RCP45 simulation with stronger biases because the first quantile map is colder in this present period than the RCP26.

<figure><img src="image_46.png" /><figcaption>**Figure 4.6:** Climatology on present climate. The upper panel shows the MPI simulation climatology maps for the RCM truth, the emulator and the interpolated UPRCM temperature. The emulator and i-UPRCM biases with respect to the RCM true map are also shown. These maps are summarised by their mean and super-quantiles. The spatial correlation and RMSE w.r.t the RCM truth are computed for the raw maps. The lower panel regroups these statistics for the 6 simulations. On the upper part for each simulation, the RCM (red), the emulator (green) and the interpolated UPRCM (blue), statistics are represented by an error bar. The lower (resp. upper) boundary of the bar shows the map's 5th (resp. 95th) super-quantile and the point the mean. The lower plot shows the same statistics for the bias maps. The middle table shows the spatial correlation and RMSE for the emulator maps.</figcaption></figure>
<figure><img src="image_47.png" /></figure>

<figure><img src="image_48.png" /></figure>

<figure><img src="image_49.png" /></figure>

**Figure 4.7:** Upper panel: Same as lower panel on Figure 4.6, for 1st and 99th quantiles. Center (resp. Lower) panel: 1st (resp. 99th) quantile maps for the CNRM-RCP26 (resp. NCC) simulation for the RCM truth, the Emulated series and the i-UPRCM, and the biases maps.

Figure 4.8 extend those remarks for future periods<fnref n="3" />. The spatial structure is always well reproduced with spatial correlation equal to 1. Notably, the green bars always match flawlessly with the red ones. In future periods, the effect of the different scenarios is well visible in the first percentile, for example, and the emulator respects those different evolutions. We can conclude that even if the emulator has difficulties reproducing the extremes, it generally fits well with the original RCM

<footnote n="3">³The historical is common to RCP85, RCP45 and RCP26 and has been used in the training set.</footnote>
series validating the scenario-transferability.

Moreover, one can observe that the results of reproducing the CNRM Hist and RCP85 are similar to those for the other CNRM RCPs. This brings confidence to the neural network training since it shows no over-fitting on the training set.

Regarding the transferability to other driving GCMs, the emulator reproduces the different maps correctly even if the results are less good than in the CNRM simulations. Figure 4.6 shows a cold bias (about - 0.2 °C) in the 3 emulated series which reflects in the RMSE. The spatial structure is well captured, as we can observe on the MPI maps given for illustration. The cold bias is similar over the entire domain with no spatial structure. Regarding Figure 4.7, the emulator has a clearer bias on the 99th percentile for the MPI and NCC simulations. The NCC maps (Figure 4.7) show a general bias which seems stronger on the reliefs. Conversely, the emulator is quite accurate on the 1st quantile.

A possible explanation is that the emulator has more difficulties in a warmer climate. The MPI and NCC are constantly warmer than CNRM (cf Fig 4.5) and the emulator provides slightly colder (about - 0.2 °C) simulations. The HGM simulation has a similar average temperature to the CNRM in the past period. Figure 4.8 shows that its emulated series has very good climatology and quantile maps. However, due to a stronger response to climate change, HGM gets warmer than CNRM at the end of the simulation. We can then observe that the HGM emulated climatology map is slightly colder than the RCM one. This cold bias might come from the highest temperature, which the emulator never saw in its training sample and has more difficulties reinventing. In Figure 4.8, the summary plots for the 99th percentile maps support this hypothesis, the emulator maps on MPI Hist+85, NCC Hist+85, and HGM 85 are too cold with RMSEs over 1°C. Notably, the emulator performs poorly on the representation of the MPI RCP85 99th percentile map with a significant cold bias. On the other hand, the emulator reproduces the colder extremes on these simulations very well. These results highlight the role of the emulator, which is not designed to extrapolate too far from its calibration set. Similarly, the emulator is less good at reproducing the coldest extremes in the HGM HIST and CNRM RCP26, the two coldest simulations.

Nevertheless, the emulator still provides very accurate maps. It recreates almost perfectly the temperature range over the maps as the red and green bars on all summary plots are always very close. Indeed the temperature difference between CNRM and MPI or NCC is about two degrees, while the emulator bias in only about 0.2 degrees. On the illustrating maps on Figures 4.6 or 4.7, the emulator maps are indistinguishable from the RCM ones. Moreover, the order between the simulations is well respected. In Figure 4.8, the HGM, MPI, and NCC simulations reach similar temperatures at the end of the century, while HGM HIST is colder than the others, and the emulator reproduces this aspect. To conclude on this part, the emulator seems robust to the GCM transferability as it produces realistic and coherent maps. Nevertheless, it seems complicated to extrapolate too far from its training set, notably to reproduce the same range of warmest extremes in a warmer climate. This is a problem that a better training set could probably overcome.
<figure><img src="image_50.png" /><figcaption>**Figure 4.8:** Summary plots for Climatology, 1st and 99th percentiles on past (1970-2000, "H" columns) and future (2071-2100, "85", "45", "26" columns) period.</figcaption></figure>
## Grid points climate change analysis

Figure 4.9 shows the summary plots for the climate change responses maps for the three statistics studied previously. The emulator reproduces well the mean climate change for all six simulations. The biases observed in the climatology maps for MPI and NCC simulations are constant over past and future periods, so it does not impact the delta. The warming in the HGM simulation is slightly underestimated (RMSE about 0.2°C), but the emulated simulation warms way more (≥ 1°C more) than any other, just as the RCM does. For all six simulations, the emulated maps have the right spatial structure of the warming, with the right spatial range.

It is interesting to notice in Figure 4.9 that, for example, the structure of the change for the first percentile is different between the HGM and the MPI simulations and the emulator correctly captures this difference. Indeed, one can observe that MPI warms more in the plains from South-West of France than HGM. On the other hand, HGM has a stronger warming just South of the Pyrenees. Furthermore, the spatial correlation between these two RCM maps is equal to 0.79 and accurately reproduced on the emulator maps. The results are less good concerning the 99th percentile with larger biases and less good spatial correlations (about 0.85 and even 0.76 for the MPI). Still, the emulator simulations have always the proper spatial range of warming, with, for instance, a more considerable spatial magnitude in the NCC map than in the HGM one. It is also noteworthy that the differences between the blue (i-UPRCM) and red (RCM) bars are not always the same, demonstrating that the downscaling transformation differs from one driving large scale to another.

## Grid point daily analysis

As a third step and to have a closer look at these emulated series, it is necessary to evaluate them daily. The four scores used in Chapter 1 and recalled in Section 4.2.4 are computed over the entire ten simulations (Historical + RCPs) on the daily time series in each grid point. The results are summarised in Figure 4.10, through summary plots where each map's mean and superquantiles are reported. These scores reflect the emulator's ability to reproduce the daily time series in all simulations as they compare the entire time series at each point.

Regarding the two CNRM simulations (RCP45 and 26), the emulator reproduces almost identically the daily time series all over the target domain. For instance, even the worst points have a Wasserstein distance less than 0.2 and a variance ratio higher than 90%. As expected from the climatology and quantile maps analyse, the scores on the MPI, HCM and NCC simulations are less “perfect”, which is probably somehow expected. The first remark is that the scores spatial means on these simulations are completely comparable with CNRM ones. The average RMSE is lower than 1°C, the variance ratio and correlation are almost equal to one and the mean Wasserstein is less than 0.2 with a lower bound similar to the CNRM simulations scores. However, some points are less well reproduced, as shown by the upper bound of the error bars on the Wasserstein distance plot and illustrated on the MPI scores maps. The MPI's Wasserstein map indicates that the emulator performs well everywhere but has more difficulties on the reliefs. It is not shown
here, but the analysis is the same for NCC and HGM Wasserstein distance.

<figure><img src="image_51.png" /><figcaption>**Figure 4.9:** Summary plots for climatology, 1st and 99th percentiles deltas between past (1970-2000) and future (2071-2100) period.</figcaption></figure>
Summary plots for daily scale scores

<figure><img src="image_52.png" /></figure>

<figure><img src="image_53.png" /><figcaption>**Figure 4.10:** Upper: Summary plots for the daily scores: RMSE, the deseasonalised temporal correlation (ACC), the ratio of variance (VAR-R) and the Wasserstein distance, for all 10 simulations, on their entire length. Lower: Illustration of the scores maps on the MPI RCP85 simulation.</figcaption></figure>

Another way to look at the transferability of the emulator is to analyse its ability to reproduce the transformation of the daily probability functions across models and time. Figure 4.11 plots the spatio-temporal probability density function of the daily temperature over all Pyrennees points higher than 1000m. It compares the true RCM (full line) and the emulated (dotted line) pdfs over a historical (1971-2000) and future (2071-2100) periods. It also plots the difference between the past and future pdfs to better visualise the transformation due to global GHG emissions. On each plot, two simulations are represented on each side of the x-axis. This figure shows that the emulator reproduces almost perfectly the different pdfs over both periods and all driving GCMs. It also matches exactly the climate change transformation. This figure also illustrates the greater Wasserstein distance (Fig. 4.10) for MPI simulation on the Pyrenees. We can see that the emulator has more difficulties in reproducing the right tail of the future MPI distribution. However, it
is observable here that even over these bad points, the pdf of the emulator looks
credible regarding the RCM one. Finally, it is remarkable that the driving GCM
has a non-negligible impact on the pdfs and that the emulated series reproduce this
aspect. The CNRM/MPI comparison illustrates well this aspect, with the appari-
tion of a dipole in the MPI future pdf, which does not exist in the CNRM simulation.

<figure><img src="image_54.png" /><figcaption>**Figure 4.11:** Comparison of the pdf transformation from the emulator and the RCM over the high Pyrenees points for all ALADIN63 simulations.</figcaption></figure>

Conclusion on the main emulator transferability

This section has been dedicated to evaluate the emulator on various simulations
according to multiple aspects. It was lengthy, but I believe it is necessary to
have better confidence in the tool introduced in Chapter 3. The strong take-home
message here is that the emulator is robust to other inputs as it provides simulations
close to original RCM ones. We have shown here that the emulator shows as
good performances on intermediate scenarios as on the training set for a given
driving emulator, confirming its truthfulness for the member and scenario level of
transferability.

The results are still very accurate concerning the GCM transferability, showing
the robustness of our methodology to eventual differences in the representation of
the atmospheric conditions. Notably, the emulator showed excellent accuracy in
reconstructing the complex spatial structure on the climatologic and quantile maps
across time and on the climate change maps. It shows, nevertheless, some limitations
in reproducing some extremes, especially in a globally warmer simulation. The
training set is probably an important factor here. We have seen that the simulations
drove by CNRM, used for training the main emulator, are generally colder than the
three other GCMs and it warms less. The emulator had to extrapolate on those and
showed its limits. The next section will be helpful to answer this question. Some
of these limitations might also come from memory effects, notably in the soil (snow
or humidity), not taken into account or reproduced by the emulator. For instance,
it has been shown in various studies that a deficit in soil moisture leads to stronger
summer hot extremes (Seneviratne et al, 2010; Hauser et al, 2017). As the emulator
downscales each day independently, there are no chances it takes this effect into
account. It could explain the less good performances in representing the highest
percentile.
<figure><img src="image_55.png" /><figcaption>**Figure 4.12:** Summary plots for the mean biases maps, in the cross validation experiment. The panels correspond to the evaluation simulation. Each color corresponds to the different emulators: it indicates the training simulations. The marker shape corresponds to the version of the emulator: the 'o' is the main version while the others are sensitivity test described in Sec. 4.5.1 and 4.5.2. The black 'plus' at the extreme right is the Large scale near surface temperature (I-UPRCM). The grey columns indicates if the evaluation simulation is part of the emulator training set.</figcaption></figure>

### 4.3.2 Cross-Validation on other training sets

In the previous section, we evaluated the emulator trained on CNRM HISTORICAL and RCP85 simulations on its capacity to reproduce all other ALADIN63 simulations. It is now interesting to see if these results are comparable when we use a different training set. In this section, I train 15 more ALADIN63-emulators using different combinations of the available simulations. First, on top of the main emulator, one emulator is built using each MPI, NCC and HGM 150 years of simulation (HIST+RCP85). Secondly, all possible combinations of two or three of these four simulations are used to calibrate one emulator. I also added emulators trained on CNRM Historical plus RCP45 and RCP26, which will allow us to assess the scenario transferability through warmer scenarios. All these emulators are then evaluated in the perfect model framework following the same protocol as in the previous section on all ALADIN63 simulations, including the calibration set. Indeed, we have seen in the previous section that the results obtained when emulating the training set are quite representative to what could be obtained on simulation driven by the same GCM and same or close scenario. In this section the use of CNRM, MPI, NCC and HGM acronyms refer to the corresponding
ALADIN63 simulation on the entire 150 years combining HISTORICAL and RCP85 scenario.

In this section, we will look at the biases of the emulators versus the RCM true
series for the mean climatology and the 99th and 1st quantiles, the variance ratio
and Wasserstein distance. The results are reported using summary plots in Figures
4.12, 4.13, 4.14. Each figure presents the results for a given metric. Each figure is
composed of 6 panels, where each panel corresponds to the evaluation simulation.
One colour represents one emulator, and the shaded rectangles identify the emulator
trained on the concerned evaluation simulation. For each emulator, on each panel,
I plotted consecutively the past (except for CNRM RCP45 and RCP26), present
and future periods. These figures contain a lot of results so we will discuss them
step by step.

One simulation based emulators

First, we can focus on the emulators trained on CNRM45 (Emul-45) and CNRM26
(Emul-26). They globally show similar results as the main emulator. Regarding
the climatology bias maps in Figure 4.12 the three CNRM emulators offer identical
performances to reproduce the three CNRM simulations in the past and present
climates. It confirms that, for perfect input conditions, an emulator train using
a given driving GCM and warming scenario can perfectly reproduce the RCM
temperature simulation driven by the same GCM and at the same or more
intermediate level of warming. It also confirms that our training methodology
prevents over-fitting, and the training set could reasonably be considered as an
evaluation set. It is observable, however, that the Emul-26 performs less well in the
future period of both RCP45 and 85 simulations. The same remark can be made
to a smaller extent for the Emul-45 on the future RCP85. It appears more evident
on the 99th quantile summary plot (Figure 4.13), where Emul-26 has considerable
local biases. This result is expected as the emulator is not supposed to extrapolate
and this test is similar to training only on the historical part (Sec 4.2 Chap1).
Regarding all metrics, Emul-45 performs less well than the two others and even on
its calibration set. It notably underestimates the local temporal variance leading
to biases on the quantile maps. The explanation have not properly been explored
here and a hidden bug is not to exclude. It might also be due to the randomness
of the training mentioned in Sec.X of Chapter 3, as the scores are consistent with
the range of emulators built in this same section. Considering these observations,
it is worth noting that these three emulators have similar performances on the
three other simulations. For example, they have similar errors on the Wasserstein
distance plots in Figure 4.14.

Secondly, we can look at the three other emulators trained on single driving GCM: Emul-MPI, Emul-NCC and Emul-HGM are trained using the upscaled ALADIN63 simulations driven by the MPI, NCC and HGM GCMs. They are represented in red, green and yellow on the summary plots. First, each performs well on its own training set, with the same defaults on the extremes as the main emulator. Each bias map shows very low local biases (visible from the size of the error bar), and quantiles are slightly under or over-estimated. The Emul-HGM has less good performance, with
a more extensive spatial range on all metrics. Looking now at their results on the other simulations, the first remarkable point is the similarity between the Emul-MPI and Emul-NCC. They have comparable scores on each other simulations and seem to produce similar maps on the other simulations as it is particularly remarkable on the 99th quantile summary plot (Fig 4.13). Emul-HGM shows poorer results on MPI and NCC simulations with much stronger local biases on the climatology and 99th quantiles maps and overestimating the temporal variance at some points (Fig 4.14). Finally, it is worth noting that these three emulators behave similarly on each of the three CNRM simulations. For instance, on the three simulations, they all have a slight warm bias on the climatology map and capture well the average 99th quantile but with significant local biases. These biases are coherent with the cold bias of the main emulator on these three simulations in the previous section. The conclusions on these three emulators are very close to the ones in Section 4.3. All these emulators are robust to other input sources, but they might produce some biases, mainly when the simulation to downscale represents a too different climate. Finally, they all reproduce adequately the climate change maps, which are not shown here but it is remarkable that eventual biases are consistent through time.

It seems reasonable to conclude here that for a given RCM, the emulator will present similar performances whatever the simulation used for training. Emul-MPI and Emul-NCC behave analogous than the Main emulator. They have excellent results regarding the entire distribution of data as long as the input space to downscale isn't too far from the training one. The more it deviates, the more complicated it is reproduce the series and notably the extremes. Emul-HGM has globally less good performances. It highlights that using a single simulation for training might be risky. It would be necessary to explore further why this specific emulator has different results. A possible explanation is that it is the GCM simulation that ALADIN63 transforms the most. In section 4.2.3, we showed for instance that HGM-GCM was warming way more than HGM-UPRCM across the century. This might have created some atmospheric conditions different from the three other ALADIN63 simulation.

## Multiple simulations based emulators

It is now natural to question what happens when two or more simulations driven by different GCMs are used for calibration. As we have four different driving GCMs, we used a maximum of three historical plus RCP85 simulations for training to keep one out of the training sample for the evaluation. Using different simulation from different driving GCM helps to enlarge the input space in the training phase. It is closer to the reality of a CORDEX matrix where the RCM are used to downscale multiple GCMs. In order to build the best emulator one can legitimately imagine that it is better to use all available simulations to train it. This experiment allows to test this aspect. On the summary plots, the shaded area shows the emulators including the evaluation simulation.
BIAS 99th Quantile

<figure><img src="image_56.png" /></figure>

BIAS 1st Quantile

<figure><img src="image_57.png" /></figure>

<figure><img src="image_58.png" /><figcaption>**Figure 4.13:** Same as Fig 4.12 for the 99th and 1st quantile biases maps.</figcaption></figure>
The first comment is that even when the driving GCM has been used for training, the results are similar or better when the emulator is trained on more simulations. It is evident on the Wasserstein and variance plots (Fig. 4.14) that on the grey areas, the range of the 5% most extremes points is getting smaller when going to the right of the colourful bars. The conclusions are more challenging on the actual evaluation set, i.e. the white areas on the plots. But when looking only at the CNRM evaluation simulations, the more we use simulations for training, the better the reproduction of the original RCM series. It is also the case for the HGM series, even if it is the most difficult to reproduce and the results are less good. Regarding the MPI simulation, the presence of the NCC simulation is of primary importance to improve the results, and the inverse is also true. However, it still comes that having more than one simulation for the training improves the outcomes. For example, the Emul-CNRM-HGM is better at reproducing MPI and NCC than Emul-HGM. On the other hand, when two simulations are similar, it brings nothing to combine them for training. Indeed Emul-MPI-NCC does not show significatively better results than Emul-NCC or Emul-MPI.

## Cross experiment conclusions

There are three important conclusions from this cross-validation experiment.

* **The scenario transferability works better from more extremes to intermediate scenarios than conversely.**

The main emulator is able to reproduce the CNRM RCP45 and 26 very well. The results are even comparable to the ones obtained on the calibration set. However, Emul-RCP45 and especially Emul-RCP26 showed difficulties to reproduce the end of the RCP85 simulation. It implies that training the emulator using a single GCM and the most extremes scenarios allows to downscale at least all simulations of the same GCM driven by similar or less warming scenarios.

* **All conclusion on the main emulator are still valid when using another simulation for training.**

It seems that, in most cases, the conclusion made regarding the Main in Chapter 3 and in section 4.3.1 are deployable to an other emulator trained on different simulations.

* **Having more than a single driving GCM for the simulations used for calibration helps for the GCM-transferability.**

We have seen that increasing the number of simulation for training the emulator enlarge the input space and generally improve the emulator robustness. At worst it did not improve the results when they were already good. On the other hand it helped to correct the bad poor results of the Emul-HGM.

On top of bringing more confidence in the emulator methodology, this section gives insight on how to design a matrix of RCM simulations to train the best emulator.
<figure><img src="image_59.png" /><figcaption>**Figure 4.14:** Same as Fig 4.12 for the Variance ratio and Wasserstein distance maps.</figcaption></figure>
## 4.4 Application on GCM data

In chapter 3, I tried to evaluate the main emulator when downscaling a GCM simulation. I downscaled the CNRM RCP45 simulation with the emulator and compared the result with the ALADIN63 simulation driven by it. The emulator was producing a coherent simulation, with a precise reproduction of the high resolution spatial structure and daily variability. However, it had a warm bias, which we found consistent with a large scale bias between the GCM and UPRCM inputs. Indeed the emulator came out to logically follow its driving large scale without reproducing the RCM large scale transformation. As there are daily but also statistical differences between the GCM and the RCM large scales, I concluded in Chapter 3 the RCM true series is not the correct reference to evaluate the GCM-emulated series. It is then truly complicated to assess the emulator in this exercise, whereas it sounds necessary to have complete trust in this downscaling approach as downscaling GCM simulation is the final aim of RCM-emulators. I propose looking at the downscaling of the GCM simulations used to drive the ALADIN63 simulations. I will try then to evaluate the emulated simulations by comparing them with the RCM truth and keeping an eye on the GCM near-surface temperature interpolated on the target domain. I remind the reader that I am missing the historical part of the HGM GCM simulation.

<figure><img src="image_60.png" /><figcaption>**Figure 4.15:** Yearly time series over the whole domain for the 6 simulations, for the RCM true series, the GCM-driving main emulator series and the GCM series.</figcaption></figure>

Figure 4.15 shows the yearly mean time series over the target domain for the RCM, the GCM-Emulated and the interpolated GCM. The RCM and emulator series are similar, but two significant differences arise. Firstly, the emulated HGM series is much warmer than the HGM RCM simulation all along the century, especially at
<figure><img src="image_61.png" /><figcaption>**Figure 4.16:** Summer (JJA) and Winter (DJF) mean time series for the RCM (red), the Emulated series from the GCM (green), the GCM (blue) and the UPRCM (yellow) over the target domain.</figcaption></figure>

the end since the emulator produces more intense warming. The emulated series follows the GCM and seems, in this aspect, more coherent than the RCM series. The HGM-GCM simulates a warmer climate at the end of the century than the three other GCMs over our target region, and the ALADIN63 simulations did not reproduce this.

Conversely, the emulated MPI simulation may be colder than it should be as it is closer to the emulated CNRM than the emulated NCC, while it is the opposite in both GCM and RCM series. It is particularly visible on the historical part. It might be explained by a cold bias of the emulated MPI in summer. Figure 4.16 shows the winter and summer time series means for the MPI simulation. It comes that the emulated MPI is much colder than the RCM series in summer. It can, at least partially, be explained by the large scale as the GCM is also colder than the RCM. Moreover, in summer, RCM and UPRCM are very close, with the UPRCM being sometimes slightly colder while the emulated series is much colder than the GCM. It is, of course, very complex to interpret. The low resolution near-surface temperature is not a predictor given to the emulator, and the UPRCM mean is naturally very similar to the RCM mean. Thus, many good reasons could explain the bigger differences between the GCM and emulator. For example, the GCM-MPI has a poorer resolution than the CNRM grid (where we interpolate all inputs), implying a poorer representation of the relief. Nevertheless, this cold behaviour of the emulator in summer for the MPI simulation could be suspect and necessitates more investigation to be understood. Here is probably the biggest challenge for the emulator evaluation as it is complicated to evaluate without a truth series as in perfect model evaluation.
<figure><img src="image_62.png" /><figcaption>**Figure 4.17:** Upper panel: Summary plots for the bias in mean climatology maps, past (1970-2000), present (2006-2035) and future (2071-2100) periods. Lower: Climatology maps for the HGM simulations with the RCM simulation, the GCM-emulated simulation CNRM emulator and the GCM simulation. It also shows the biases.</figcaption></figure>

I used all emulators built in Section 4.3.2 on these 4 GCM simulations to evaluate the training set's impact when downscaling GCM simulations. Figure 4.17 shows the bias summary plots in the climatology maps. The reference is the corresponding RCM temperature. Interestingly, all these emulators have similar biases, as shown by the spatial means and superquantiles. Surprisingly, training on the corresponding ALADIN63 simulation does not impact the performances. I show only the results for the climatological bias here, but the conclusion is precisely the same on all other metrics used in the previous sections. Globally we can see that the emulated series have larger mean and local biases with greater superquantiles than in the perfect model. The most substantial differences appear in the HGM
simulation, especially in the summer, as shown in Figure 4.17. The emulated
climatology is warmer over land and relief while there is a cold bias over the coasts.
This cold bias could come from missing sea surface temperature information. On
the other hand the warm bias seems consistent with the large scale. However, we
can see here again that the climatology map from the emulated series is coherent
with a good representation of the high-resolution spatial structure brought by the
RCM.

These similar results on all emulators show that the downscaling function learnt by
the emulators is globally similar on all training sets, and the emulator's production
always stays consistent with its driving large scale. It confirms the robustness of the
emulator, and if some issues need to be fixed, notably regarding some biases, the
general result here is that the downscaling of GCM simulations with the emulator
works and provides the adapted high resolution information of climate according to
the given large scale.

**4.5 Discussion**

**4.5.1 On the standardization method**

In Chapter 3 of this manuscript, I presented my choice for the standardization
method used to prepare the inputs for the neural network. I chose to separate the
temporal and spatial information into two sets of inputs. The first set regroups
all 2D variables (reported in Table 1chaptab1) normalized according to their daily
spatial mean and standard deviation. It results for every day in a collection of
maps, all centred in 0, highlighting the spatial anomalies or structure of the given
day. The second set contains somehow temporal information. It is a 1D array
including the collection of daily means and standard deviations used to normalize
the 2D variables, the yearly greenhouse gas concentration used to drive the RCM
simulation and a vector including a cosinus and a sinus representing the day of
the year. This set of 1D variables is also normalized according to its climatology
over 30 years. To summarize, for each day, the set of 1D variables indicates the
position of the day in comparison to others in terms of global climate (through
the GHG concentration), its place in the year and somehow an indication of the
climate variability mode over the input domain trough the set of spatial mean and
standard deviation. The 2D maps bring information on the daily spatial structure
completing the information on the climate mode of the day.

I made this choice because of the robustness to new inputs. Indeed, it is well
established that neural networks (ref) and especially convolutional neural networks
are very sensitive to input space changes. As shown in section 4.2.3, the relationship
between different variables and altitude levels can differ between 2 GCMs. Figure
4.4 shows these differences at the input domain scale. They are probably bigger
when we look at the grid point scale, especially because two GCMs have different
horizontal resolutions. My intuition here was that this normalization would help us
to prevent this effect.

A probably more intuitive way to normalize the inputs is to do it at the grid point
<table><caption>Table 4.2</caption><thead><tr><td>NAME</td><td>DESCRIPTION with respect to the main emulator</td></tr></thead><tbody><tr><td>CNRM_no850</td><td>No variables at 850 hectopascal</td></tr><tr><td>CNRM_nosfc</td><td>No surface variables</td></tr><tr><td>CNRM_nosfc_no850</td><td>No variables at 850 hectopascal & No surface variables</td></tr></tbody></table>

scale. I constructed an emulator called CNRM_std2 to test our standardization strategy and my hypothesis. This emulator is, in every aspect, trained in the same way as the main emulator as it takes the same set of inputs. It is trained on the same simulation (CNRM HIST+RCP85) and uses the same neural network. The only difference is the normalization procedure since each daily 2D map is directly normalized by the mean and standard deviation maps computed over a 30 years reference period of the corresponding variable. Following this, the set of 1D inputs contains only the external forcings (GHG, Solar and ozone constraints) and the cosine sinus vector for the day of the year. The results are reported in all plots from the cross-validation exercise (Fig. 4.12, 4.13, 4.14). The emulator CNRM_std2 results are very similar to our main emulator on the CNRM simulations, with sometimes slightly better results regarding the climatology plots, for example (Fig. 4.12). However, regarding the three other simulations driven by different GCMs, the main emulator performs better than the novel one. It is obvious for all metrics on the MPI simulation, which seems to be the most different from the CNRM one in Figure 4.4. To conclude here, the CNRM_std2 has, in this perfect model evaluation, stronger biases and a less good representation of the variances and pdfs on the simulation driven by different GCMs, which tends to confirm the hypothesis that the two steps normalization helps for the robustness of the emulator.

### 4.5.2 Some tests on inputs choices

The large set of simulations to evaluate the emulators is a good opportunity to make small tests on some input choices from chapter 1. For instance, I showed in Chapter 1 that including the surface temperature in the inputs set improves the emulator's performance in the perfect model evaluation but seemed to degrade the downscaling of GCM inputs. This is logical as the UPRCM near-surface temperature might still contain high-resolution information that the GCM near-surface temperature cannot include. However, I decided to keep the surface wind and sea pressure level, and this choice could be questionable. In the same spirit, I wanted to test whether removing the variable at 850hpa would change something, as it is still close to the surface, especially over the relief. To test these two points, I built three emulators described in table 4.2. As in Sec.4.5.1, the results are reported in the figures from the cross-validation and GCM sections (sections 4.3.2 and 4.4). It seems that excluding these variables slightly degrade the results, especially regarding the 99th quantiles (Fig. 4.13) or the variance ratio for the MPI, NCC and HGM simulations. Regarding the results on the GCM simulation, removing these variables could have improved the results even if it is difficult to evaluate. The impact is small, but it seems to reduce the spatial range of differences in the climatology maps around a global bias. It is not shown here, but the same comment can be made on the maps of variance ratio and Wasserstein distance, especially when both levels (surface and 850hPa)
are removed. This result should be kept in mind for future development and use of
the emulator, but it has not been deeper investigated in this work.

4.6 Conclusion

In this chapter, I tried to address the overriding question of the emulator's
transferability to other input sources. The aim of the emulator is to downscale
large ensembles of simulations. In the previous chapter, I introduced the emulator's
concept and methodology. I showed that it can reproduce a simulation close from
the training one (same driving GCM, intermediate scenario of GHG). However,
ensuring that the emulator works well for downscaling any GCM and any scenario
is essential. ALADIN63 is an ideal case study for the emulator since there are
three simulations driven by the same GCM and three different scenarios of GHG
concentration, and four simulations driven by four different GCMs following both
the historical and the RCP85 scenarios.

As in the previous chapter, I address this question in the perfect model framework where the inputs come from upscaling and smoothing the RCM simulations as described in Sec.4.2.3 in Chapter 3. This question of transferability remains a challenging exercise in the perfect model world, as the driving GCM strongly impacts the relationship between the input variables. Furthermore, these differences between the different UPRCM simulations are comparable to those observed in GCM simulations. Therefore, the perfect model framework is a good opportunity to evaluate the emulator's robustness under other inputs.

Firstly, I evaluate the emulator built in the last chapter on all available simulations. The results are globally excellent. The emulator produces the high resolution structure and variability brought by the RCM. Furthermore, it recreates the differences between all the simulations regarding temperature levels and warming strength at the entire domain, regional and grid point scales, even on the 1st and 99th quantile change maps. For instance, I show that each simulation simulates different density functions over the Pyrenees and transforms the regional climate differently across time. The emulated series replicate all these observed differences. This result gives us solid confidence in the tool, even if it probably could be improved notably in reproducing the most intense extremes. Indeed, I notably showed that the emulator underestimates the warmest extreme in a simulation globally warmer than the training set and it reflects in the mean climatology.

In a second step, I propose a cross-validation experiment to evaluate the impact of
the training set on the emulator’s performance. The ALADIN63 matrix is also an
excellent opportunity to build multiple emulators trained on different combinations
of the available simulations. This experiment strengthens the robustness of our
methodology as all emulators show akin results. They all correctly recreate each
simulation, with some biases or defaults but all in the same order of magnitude.
There is nevertheless a little damper as the emulators trained on the HGM
ALADIN63 simulations are significantly less good. The other conclusion from this
experiment is the importance of having the greatest spread in the training set,
including the most extreme scenarios and, when possible, various driving GCMs.
The aim is to cover the most extensive range of climates to minimize the risk of extrapolation in application mode. Moreover, it also prevents using a too specific simulation, as was the case here for the HGM simulation.

Finally, I tried to validate the emulator in application mode using GCM inputs. The emulator was able to downscale these GCMs simulations and provide consistent results. It was particularly interesting to see that the emulator recreates the strong sensitivity of the HGM simulation better than the RCM. However, the large scales biases between GCM and RCM make the evaluation very hard since there is no truth to confront the emulated series. In particular, the emulator seemed too cold in summer in the MPI simulation. Still, it is difficult to assess if it is coherent with the driving large scale or if there is an issue in the emulator construction. A potential limitation of the emulator which could lead to such biases is the absence of soil variables, such as soil moisture or snow accumulation, which can carry a strong memory effect not represented in the atmospheric variables. These variables are less well represented in the GCM, so it is impossible to use them directly as predictors. Therefore, I think this is a possible improvement method that should be investigated.

This chapter gives a lot of confidence in the emulator for the daily near-surface tem-
perature to apply it to actual GCM data and create big ensembles of high resolutions
temperature series. Of course, the emulator still needs improvements, notably re-
garding the extremes, but its current version provides already a robust basis for the
downscaling of new GCM simulations. Moreover, the results of this transferability
study could also be used as guidelines for future RCM simulations in order to give
the best chance to complete the matrix with RCM-Emulators. In the conclusions of
the manuscript (Chapter 6) I will give more details on this perspective.
# Chapter 5

## Precipitation downscaling with the RCM-Emulator : The role of the loss function

This chapter presents the use of the emulator for the downscaling of the daily amount of precipitation. It has been written as an article to be submitted. Therefore, some parts of the introduction and the methodology sections might be redundant with the previous chapters (notably Chapter 3).
## 5.1 Introduction

Precipitations are the principal source of usable fresh water on Earth. It is a key variable to study and predict because of its fundamental role in the Earth's system balance, in the ecosystem's life, and above all, human survival and activities rely directly on it. On the other hand, it is a potential source of catastrophic events. Heavy rainfalls are responsible for floods devastating cities and cultures. Severe droughts can cause colossal damage to ecosystems and agricultural activities and limit access to drinkable water. In the current context of global climate warming, the eventual changes in precipitations climatology and extremes are crucial to study for adaptation policies.

The study of rainfall is very complex. Precipitation is a non-continuous variable neither in time nor in space. It is necessary to look at precipitation series through different time and spatial scales to completely understand their nature. Their occurrence and intensity characterise them. For example, the monthly mean precipitation amount indicates seasonal cycle or large-scale variability but could underline very different situations. Moderate and regular rain or a dry spell followed by heavy precipitation will result in the same monthly mean but will impact the local ecosystems and hydrological cycle very differently. Precipitations can also be very local as they might strongly depend on the local topography, the distance to the sea or a lake, etc., and on small scale physical processes such as convective instability or cold pool (Caillaud et al, 2021). It is therefore necessary to study them at refined resolutions.

Global Climate Models (GCMs) represent the global Earth system. They are today the primary tool to study the impact of human greenhouses gases emissions on the climate. They divide the globe into 3d grids. Each cell represents a specific location with its characteristics (altitude, kind of surface, etc.). A set of physical equations is resolved at regular steps in each cell and between cells to represent the different processes driving atmospheric circulation. The horizontal size of the cell is the resolution of the model. Finer resolution implies more grid points and higher computational costs. GCMs have a resolution between 50 and 300km, according to the model. It is too coarse to accurately represent the precipitation in all its complexity, especially when looking at the most extreme events. For these reasons, multiple studies have shown the added value of Regional Climate Models (RCMs) for representing precipitations, especially extreme events (Torma et al, 2015; Giorgi et al, 2016; Fantini et al, 2018).

RCMs are similar in all aspects to GCMs, except that they represent only a limited area domain allowing a higher resolution (1 to 25 km). An RCM simulation receives at regular time interval information from a GCM simulation at the borders of its domain. The global circulation is then simulated by the GCM and passed to the RCM, which integrates it and produces a higher resolution simulation. The RCM is then used to downscale the GCM coarse resolution and we talk about dynamical downscaling (by contrast to the statistical downscaling introduced below). We say that the GCM drives or forces the RCM. The main limitation of RCMs is their high computational costs which do not allow for downscaling large ensembles of
GCM simulations, although it is essential to produce reliable information. Various studies (Déqué et al, 2012; Evin et al, 2021) have pointed out at least four sources of uncertainty associated with RCM simulations: the choice for the RCM and GCM with the defaults they carry, the uncertainty around the evolution of human activities and especially the emissions of GHG and finally, the internal variability of the climate. In order to deliver a reliable message about future climate, it is necessary to run numerous long simulations from different RCMs, driven by different GCMs and themselves forced by multiple economic scenarios. Multiple simulations with the same Scenario, GCM, RCM triplet are then necessary to differentiate the impacts of climate change and the internal variability.

In this study, I propose testing whether the RCM-emulator introduced in Chapter 3 is suitable for emulating daily precipitation in the RCM ALADIN63 (12km resolution). The concept of the RCM-emulator is to learn, with machine learning tools, the relationship between low-resolution large-scale variables describing the atmosphere circulation of a given day and a local high-resolution surface variable, such as precipitation daily amount. This downscaling function is learnt inside existing RCM simulations. The aim is to tackle the cost limitation of RCM by mimicking its downscaling function for a specific variable at a low computational cost and then by applying it to any GCM simulations. RCM-Emulators belong to the hybrid downscaling methods as they involve both statistical and dynamical downscaling. Statistical downscaling aims to estimate or reproduce the same large-scale/local-scale relationship with statistical methods. The fundamental difference is that statistical downscaling uses historical observation data to learn the downscaling function. By using historical and future RCM simulations in the training set, the RCM-emulator is more robust to the possible non-stationarity of this relationship. Moreover, it can also be applied over regions with no long series of observational data.

Various studies have proposed statistical downscaling methods to downscale GCM or RCM simulations. Maraun et al (2010) and more recently Gutiérrez et al (2019) present an overview of available statistical methods for downscaling precipitations. Some very recent studies (Baño-Medina et al, 2020, 2021; Vandal et al, 2019; Wang et al, 2021) have successfully used convolutional neural networks for this purpose. The RCM-emulator used in Chapter 3 and this study relies on a convolutional neural network architecture called UNet (Ronneberger et al, 2015). It has shown an excellent ability to emulate the temperature, notably in reproducing the complex spatial structure and daily variability brought by the RCM.

The second objective of this study is to look at the impact of the loss function on emulator prediction. If precipitations are complex to study, they are also challenging to model. Their intermittent nature implies a substantial variability in time and space and high non-linearity. Baño-Medina et al (2020) undertake this issue with a well-designed probabilistic neural network. It outputs the three parameters of a Bernoulli-Gamma distribution for each day and grid point. Wang et al (2021) use a GAN neural network architecture, which produces for each day a possible realisation. Here, I will compare two standard loss functions for regression problems with a third designed explicitly for the precipitation
downscaling task. It relies on an asymmetric regularisation parameter which penalises stronger the network when it under-evaluates higher precipitation intensities.

As in the rest of this manuscript, I train and evaluate the emulators using a perfect model framework. It implies that both predictors and predictands (precipitation here) come from the same RCM simulation. The target domain here is centred over the Alps and is four times bigger than in the preceding chapters. It will also help to address the emulator’s transferability to another domain and the needed computational time for a larger target domain. This chapter contains four main sections. In Section 5.2, I present the framework of the study: the data, the target domain and the associated predictors. I will also recall the neural network architecture and the concept of perfect model training and present the three losses functions used to calibrate three emulators and the evaluation tools I will use. Section 5.3 presents the detailed evaluation of the emulators in a perfect model framework, while Section 5.4 shows the results of the asymmetric emulator application on GCM simulations. Finally, in a discussion part (Section 5.5), I come back to the construction of the asymmetric loss function and the computational performances of the emulator.

## 5.2 Methodology

In this section, I set the framework used to build and evaluate the RCM emulator for precipitation. I firstly present briefly the simulations used in this work. Then I present the emulator base: the chosen target domain and predictors and the neural

Daily precipitation mean 1980-2000

<figure><img src="image_63.png" /></figure>

**Figure 5.1:** Illustration of the input (left) and target (right) domains through the climatology of the daily rainfall over the 1980-2000 period. The black line on the left panel shows the target domain while the input domain is the entire map. On the target domain: the red points are the 4 illustrating points on Figure 5.7 and 5.12. From North to South, there is Paris, a high point (2247 meters) in the Swiss Alps, Toulouse and Roma. The three blue boxes are the three regions used for the SAL evaluation in section 5.3.2: The north region, centred over Belgium, the Cevennes region (south-east France) and the Dinaric Alps.
<table><caption>**Table 5.1:** The ALADIN63 matrix.</caption><thead><tr><th rowspan="2">DRIVING<br>SCENARIO</th><th colspan="4">DRIVING GCMS</th></tr><tr><th>CNRM-CM5<br>(CNRM)</th><th>MPI-ESM-LR<br>(MPI)</th><th>NorESM1-M<br>(NCC)</th><th>HadGEM-ES2<br>(HGM)</th></tr></thead><tbody><tr><td>Historical</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>RCP26</td><td>x</td><td></td><td></td><td></td></tr><tr><td>RCP45</td><td>x</td><td></td><td></td><td></td></tr><tr><td>RCP85</td><td>x</td><td>x</td><td>x</td><td>x</td></tr></tbody></table>

network architecture. I will present the three loss functions used to train the three emulators for the inter-comparison. I will also shortly recall the perfect model framework approach used to train and evaluate the emulators. Finally, I will detail all the metrics used to evaluate the emulator under different aspects.

### 5.2.1 Datas : the ALADIN63’s matrix

The emulator proposed in this study relies on the regional climate model ALADIN63 (Nabat et al, 2020). A total of ten simulations have been published with this RCM over the whole Europe in the EURO-CORDEX framework (Vautard et al, 2020). They downscale four different GCMs and four different scenarios (cf Table 5.1). The CNRM-CM5 global climate model is developed in the same institute as ALADIN63, so they belong to the same family of models. CNRM-CM5 drove 4 ALADIN63 simulations, the historical (1951-2005) and three RCP scenarios (2.6, 4.5 and 8.5, on the period 2006-2100). MPI-ESM-LR, NorESM1-M and HadGEM2-ES are the three other GCMs used to drive ALADIN63 following the historical and RCP8.5 scenarios of greenhouses gases emissions. From now CNRM-CM5 will be referred to as CNRM, MPI-ESM-LR as MPI, NorESM1-M as NCC and HadGEM2-ES as HGM.

### 5.2.2 RCM-Emulator conception : Predictors, architecture and target.

In this chapter, I focus on the challenging emulation of daily precipitation from ALADIN63 at 11° horizontal resolution (about 12km). In order to also test the emulator’s transferability to another domain, I chose a more extensive domain centred over the Alps. The target domain is visible on the left side of Figure 5.1. It is four times bigger than the previous domain as it counts 128 × 128 grid points. It includes the entire Alps and goes from Sardaigna until the north of France and from the Pyrenees until Croatia. This domain is interesting because it has different areas with different precipitation regimes. For example, the Cevennes (South-East of France) region is known for its very extreme events in autumn, similarly to other coastal areas of the Mediterranean region. The reliefs receive more precipitation than plane regions, and they are regions where the higher resolution of RCMs brings a lot of added values, especially for the extremes. The flat regions of the north of the domain receive a lot of precipitation throughout the year but have less strong daily extremes than the southern regions. The emulator is trained to
<table><caption>Table 5.2: List of predictors</caption><thead><tr><th colspan="6">2D Variables</th></tr><tr><th>Field</th><th>Altitude Levels</th><th>Variables Notation</th><th>Units</th><th>Temporal Aggregation</th><th>Dimension</th></tr></thead><tbody><tr><td rowspan="3">Geopotential</td><td rowspan="3">850, 700, 500 hPa</td><td>zg500</td><td rowspan="3">m</td><td rowspan="3">Daily mean</td><td rowspan="3">[i, j]</td></tr><tr><td>zg700</td></tr><tr><td>zg850</td></tr><tr><td rowspan="3">Specific Humidity</td><td rowspan="3">850, 700, 500 hPa</td><td>hus500</td><td rowspan="3">K</td><td rowspan="3">Daily mean</td><td rowspan="3">[i, j]</td></tr><tr><td>hus700</td></tr><tr><td>hus850</td></tr><tr><td rowspan="3">Temperature</td><td rowspan="3">850, 700, 500 hPa</td><td>ta500</td><td rowspan="3">m/s</td><td rowspan="3">Daily mean</td><td rowspan="3">[i, j]</td></tr><tr><td>ta700</td></tr><tr><td>ta850</td></tr><tr><td rowspan="4">Eastward Wind</td><td rowspan="4">850, 700, 500 hPa</td><td>ua500</td><td rowspan="4">m/s</td><td rowspan="4">Daily mean</td><td rowspan="4">[i, j]</td></tr><tr><td>ua700</td></tr><tr><td>ua850</td></tr><tr><td>ua850</td></tr><tr><td rowspan="4">Northward Wind</td><td rowspan="4">850, 700, 500 hPa</td><td>va500</td><td rowspan="4">m/s</td><td rowspan="4">Daily mean</td><td rowspan="4">[i, j]</td></tr><tr><td>va700</td></tr><tr><td>va850</td></tr><tr><td>va850</td></tr><tr><td>Sea Level Pressure</td><td>Surface</td><td>psl</td><td>Pa</td><td>Daily mean</td><td>[i, j]</td></tr><tr><td>Total Aerosol Optical Depth forcing</td><td>Surface</td><td>TAOD</td><td></td><td>Monthly mean</td><td>[i, j]</td></tr><tr><th colspan="6">1-D Variables</th></tr><tr><td>Daily spatial means of 2D variables</td><td></td><td>x&#772;<br/>with x &isin; V2D</td><td></td><td>Daily</td><td>[#V2D]</td></tr><tr><td>Daily spatial standard deviation of 2D variables</td><td></td><td>x&#772;<br/>with x &isin; V2D</td><td></td><td>Daily</td><td>[#V2D]</td></tr><tr><td>Total Anthropogenic greenhouses gas forcings</td><td></td><td>ant_ghg</td><td></td><td>Yearly</td><td>[1]</td></tr><tr><td>Solar and Ozone forcings</td><td></td><td>sol, oz</td><td></td><td>Yearly</td><td>[2]</td></tr><tr><td>Seasonal Indicators<br/>Cos(2&pi;t/365) ; Sin(2&pi;t/365)</td><td></td><td>cos,sin</td><td></td><td>Daily</td><td>[2]</td></tr></tbody></table>
reproduce land and overseas precipitations, even if I will sometimes evaluate it only on land. Finally, the bigger domain will allow characterising the computational time needed for the emulator’s calibration.

The emulator I use in this chapter for the downscaling of precipitation is built exactly like the one for the temperature in the two last chapters. It can be seen as a classical machine learning problem

$$Y_t = F(\tilde{X}_t, \tilde{Z}_t)$$

where $(\tilde{X}_t, \tilde{Z}_t)$ are the low resolution predictors, $Y_t$ the high resolution target variable (here daily amount of precipitation) at day $t$ and $F$ the downscaling function that we aim to estimate with the neural network. The list of predictors is the same as in Chapter 3, with both sets of 1D and 2D inputs recalled in Table 5.2. The standardisation procedure is also the same. For each day, I normalise each 2D input spatially. The daily spatial mean and standard deviation are then given to the emulator through the set of 1D inputs, which also contains the external forcings (yearly greenhouse gases concentration, solar and ozone constraints) and the seasonal indicator (sinus, cosinus vector). More details can found in Chapter 3. The input domain is adapted to match the new target domain. It is a 22*16 grid point on the CNRM-CM5 grid (1.4°) centred over the target domain, (the whole map on Figure 5.1).

The neural network architecture is also the same as in the temperature emulator. It relies on the well-known UNet architecture (Ronneberger et al, 2015). The only differences with Chapter 3 are due to the size of the input and target domains. As shown on Figure 5.2, the first layer of the network reshapes the 2D inputs from [16, 22, 32] to [16, 16, 64] in order to obtain squared images before the encoding path. On the other side, the expanding path is extended to reach the target domain size.

### 5.2.3 Loss function for the neural network training

In this chapter, I propose a deeper look on the impact of the loss function on the emulator’s performance. The loss function is an essential part of the neural network training. In the training phase, the network sees examples of inputs and target variable pairs. For each day of the training set, it makes a prediction and compares it with the truth. The loss function evaluates the network prediction against the expected outcome. The network parameters are then updated according to the loss function results. This operation is repeated until the cost (i.e. the average loss results) stabilises. The best combination of parameters has the lowest cost over a validation set, different from the training set. This is then a minimisation problem to find the best estimate $\hat{F}$ such that :

$$\hat{F} = \arg \min_{\theta \in \Theta} L(\mathcal{D}, \theta) \qquad (5.1)$$

Where $\Theta$ is the ensemble of possible parameters, $\mathcal{D}$ the validation set and $L$ the loss function.
<figure><img src="image_64.png" /><figcaption>Figure 5.2: Illustration of the neural network architecture.</figcaption></figure>

Precipitations are particularly complicated to emulate with neural networks because of their distribution. Indeed, as illustrated in Figure 5.12, the distribution of precipitation looks like a highly left-skewed gamma distribution. There are many days with low precipitation and a few with very high precipitation. These different events contribute non equally to the mean, with a few days having more impact than the other ones. The emulator must reproduce well the entire distribution. The good reproduction of rare extreme events constitutes the substantial added value of RCM, so the emulator should reproduce them accurately. The loss function is therefore a way to rebalance the data and force the emulator to look more specifically into specific events.

In this Chapter, I compare three emulators, constructed with different loss functions:

* **Emul-MSE**: uses the classical mean squared error as loss function, as for the temperature emulator. It corresponds to the L2 norm.

$$L(y, \hat{y}) = \frac{1}{N \times T} \sum_{t=0}^{T} \sum_{i \in \mathcal{D}} (y_{i,t} - \hat{y}_{i,t})^2 \quad (5.2)$$

With $\mathcal{D}$ the ensemble of grid points, $N$ the number of grid points and $T$ the number of days.

* **Emul-MAE**: uses the mean absolute error. It corresponds to the L1 norm.

$$L(y, \hat{y}) = \frac{1}{N \times T} \sum_{t=0}^{T} \sum_{i \in \mathcal{D}} |y_{i,t} - \hat{y}_{i,t}| \quad (5.3)$$

* **Emul-ASYM**: uses a specific loss function that I designed for the precipitation problem. It is based on the MAE loss function plus an asymmetric term
<figure><img src="image_65.png" /><figcaption>Figure 5.3: Illustration of daily precipitation distribution, in the Cevennes box (cf Fig 5.1) all points and days are pooled.</figcaption></figure>

which penalizes stronger the emulator when it underestimates the true value while it was a raining day.

$$L(y, \hat{y}) = \frac{1}{N \times T} \sum_{t=0}^{T} \sum_{i \in \mathcal{D}} |y_{i,t} - \hat{y}_{i,t}| + \gamma_{i,t}^2 \times \max(0, y_{i,t} - \hat{y}_{i,t}) \quad (5.4)$$

With $\gamma_{i,t} = F_i(y_{i,t})$ and $F_i$ the cumulative distribution function of a random variable $Y_i$ following a gamma distribution

$$Y_i \sim \Gamma(\alpha_i, \beta_i) \sim \Gamma_i$$

where $\alpha_i$ and $\beta_i$ are fitted on the training set.

The MAE and MSE losses are the most commonly used loss functions for regression problems. The MAE loss sums the absolute distance between an observation and its prediction. It gives the same weight to each observation. Knowing that daily rainfalls are strongly left-skewed, with a vast number of observations with a small amount of precipitation, the EMUL-MAE should be able to suit these days well. However, the rare cases with large precipitations should be less well reproduced.

The MSE loss function gives more weight to the significant errors than the small ones. The MSE generally shows the best results in regression problems. It is equivalent to the maximum likelihood estimation in a Gaussian setting. It leads theoretically to the best estimate for normally distributed data knowing the inputs. In the case of precipitations, it is not likely to be the case because of their highly intermittent nature. So the MSE loss function might not be well adapted. Emul-MSE is the same emulator as the one of Chapter 3.

The choice of the asymmetric loss function is more widely discussed at the end of this chapter. The intuition comes from the results of both EMUL-MAE and
<figure><img src="image_66.png" /><figcaption>**Figure 5.4:** Illustration of the three loss functions according to the error ($y - \hat{y}$). For the ASYM loss, as it depends on the true prediction and the location, I illustrate it with $y = 20$ and 2 locations: Roma and the Alps point already mentioned (Fig 5.1).</figcaption></figure>

EMUL-MSE presented in section 5.3. The idea is to add a penalty when the emulator underestimates consequent precipitations. The more extreme this observation, the higher the penalty should be. The $\gamma_{i,t}$ parameter determines how extreme is a given observation and defines the weight accordingly. At each grid point, I estimate the parameters of a gamma distribution, which I fit on the rainy days (over 1mm) of the training set (using the scipy python package, Virtanen et al (2020)). The Gamma distribution has been widely used to described precipitation data (Katz, 1977; Vrac and Naveau, 2007) but other distribution could be considered. In order to make this parameter estimation more robust, I fit them yearly and then average these parameters over the years. It gives a map of the shape and scale parameters. The $\gamma_{i,t}$ parameter is then the evaluation of $y_{i,t}$ by the Cumulative Distribution Function (CDF) associated to the gamma distribution $\Gamma_i$ fitted at the point $i$.

### 5.2.4 Evaluation Metrics

In order to evaluate and compare the performances of the emulators I will evaluate them with respect to the daily precipitation series from the corresponding RCM simulation (cf Fig.5.5). I will use various metrics to compare the targeted ($Y$) and the predicted ($\hat{Y}$) series to have the most complete evaluation possible and understand the strengths and weaknesses of each emulators. The different metrics are detailed in the sections below.
<figure><img src="image_67.png" /><figcaption>Figure 5.5: Scheme of the training (left), perfect model evaluation (middle) and GCM world application (right) protocols.</figcaption></figure>

## Time series comparison

First of all I will evaluate in each grid point if the emulated time series matches the original RCM series through two metrics already used in this manuscript:

* **Temporal Anomalies Correlation.** This is the Pearson correlation coefficient after removing the seasonal cycle:

$$ACC(Y, \hat{Y}) = \rho(Y_a, \hat{Y}_a), \qquad (5.5)$$

with $\rho$ the Pearson correlation coefficient and $Y_a$ and $\hat{Y}_a$ are the anomaly series after removing a seasonal cycle computed on the whole series.

* **Ratio of Variance.** It indicates the performance of the emulator in reproducing the local daily variability. We provide this score as a percentage:

$$RoV(Y, \hat{Y}) = \frac{Var(\hat{Y})}{Var(Y)} * 100 \qquad (5.6)$$

Both metrics are computed at each grid point. According to what was done in the previous chapters of this manuscript, each map is summarised with its spatial mean and 5th and 95th super-quantiles. The super-quantile $\alpha$ is defined as the mean of all the values larger (resp. smaller) than the quantile of order $\alpha$, when $\alpha$ is larger (resp. smaller) than 0.5.

## Climatological scale metrics

It is necessary to evaluate the emulators at the climatological scale. I will look at three statistics over at least 20 years: the daily precipitation mean, the 99th quantile and the percentage of dry days. These three metrics are prints of the variable distribution from the mean and extreme sides. I will compare the emulators’ maps ($\hat{Y}$) with the RCM truth maps ($Y$) and compute the relative biases (eq. 5.7) from the emulated series maps against the RCM truth. The biases maps are then in percentage. When the biases are too strong, notably because of comparing very
small values, I use the simple bias ($\hat{Y} - Y$), expressed in mm/days. Again, the statistics are computed point-wise, and I summarise each map with the spatial mean and super-quantiles.

$$ \text{Relative bias} = \frac{(\hat{Y} - Y)}{Y} \times 100 \qquad (5.7) $$

These three statistics will be looked at in present climate but also in climate change context. Each statistic will be computed in a future period and the climate change statistic is the relative difference with respect to the past period. Then the simple bias is computed between RCM and emulator climate change statistics.

## PDF normalisation

Since the pdf for the rainfall are very heavy-tailed, it is difficult to compare them. I propose here to have a deeper look into the distributions thanks to the ASoP method introduced in Klingaman et al (2017) and used in multiple studies as Berthou et al (2020) or Vergara-Temprado et al (2020). It consists in computing the precipitation frequency following some well-chosen bins $b_n$ defined in Eq 5.8. The bins are such that they contain a similar number of events for bins over 1mm and as long as the number of events is sufficient.

$$ b_n = e \left( \ln(0.005) + \left[ n \frac{(\ln(120) - \ln(0.005))^2}{59} \right]^{\frac{1}{2}} \right) \quad \text{with } n \in [0, 100] \qquad (5.8) $$

Then we can look at each bin's contribution $C_n$ to the mean by multiplying each frequency by the corresponding bin's mean as described in Eq. 5.9. Both frequency and contribution are interesting in comparing the emulated series with the true RCM.

$$ C_n = f_n m_n \qquad (5.9) $$

where $f_n$ and $m_n$ are the frequency and the mean of bin $b_n$

I use the skill score proposed in Berthou et al (2020) to evaluate the difference between the emulators and the RCM truth contributions curves. The fractional contributions are the actual contributions divided by the total mean precipitation of the series. They give information on the shape of the distribution independently from the mean. The Fractional Contribution Skill Score (FCSS) sums the absolute difference in each bin between the fractional contributions of an emulator and the targeted true series. The area under the FC curve is equal to 1, so the FCSS is equal to 0 when the two distributions are identical and to 2 when there is no overlap between them. It measures the differences between the two distribution shapes independently from the series mean. This score is illustrated on Figure 5.12 and further commented in the results section 5.3.1.
$$ FCSS(Emul, RCM) = \sum_{n \in [0,100]} |FC_n^{Emul} - FC_n^{RCM}| \qquad (5.10) $$

$$ \text{where } FC_n = \frac{C_n}{\sum_n C_n} = \frac{C_n}{mean} $$

## SAL score

In order to further evaluate the performances of the Emulator, I use an object-oriented score introduced in Wernli et al (2008). The SAL score aims to evaluate the spatial structure of precipitation objects from a predicted map versus a reference. It compares two maps of precipitation at a given time step. It accounts for the objects' structure (S-component) and location (L-component) plus the total amplitude of precipitation (A-component). In perfect model evaluation, the Emulator should be able to reproduce the precipitation events accurately. This score indicates if the Emulator recreates objects like the RCM and if they have the same characteristics.

The first step is to identify the precipitation objects. To do so, I used the *pysteps* (Pulkkinen et al, 2019) python library, which integrates a SAL implementation. Each day, the objects are all the groups of 5 or more consecutive points with more precipitation than a minimum threshold equal to $R^* = \frac{1}{15}R^{95}$, $R^{95}$ being the 95th quantile on the map. Multiple objects can be detected every day. Then, the three components are computed aiming to differentiate objectively different precipitation objects. Figure 5.6, from Wernli et al (2008), illustrates the behavior of the SAL score between observed (O) and forcasted (F) objects on some idealized and basic cases.

The three SAL components are defined as follow:

* **Amplitude component**: It compares the normalized total amount of precipitation over the domain between the Emulator and the RCM truth.

$$ A = 2 \frac{D(R_{Emul}) - D(R_{RCM})}{D(R_{Emul}) + D(R_{RCM})} \qquad (5.11) $$
Where $D(R) = \frac{1}{N} \sum_{i \in D} R_i$

With $D$ the ensemble of all grid points (even with no rain), $N$ the number of grid points, and $R_i$ is the precipitation amount in the $i$th grid point. Then $D(R)$ is the average precipitation per grid point over the whole map. $A$ takes values between -2 and 2, and 0 is the perfect reproduction.

* **Location component**: It is composed of two parts $L = L_1+L_2$. $L_1$ measures the distance between the two maps' centre-of-mass of the total precipitation fields.

$$ L_1 = \frac{|x(R_{Emul}) - x(R_{RCM})|}{d} \qquad (5.12) $$

$x(R)$ is the centre-of-mass of the precipitation field, considering all precipitation points and not only the identified objects. $d$ is the maximum distance on
the map. The maximum value for $L_1$ is 1 since it is normalized by $d$. The second component aims to distinguish the individual objects. $L_2$ computes the normalized difference between the map average of the distances of the objects to the global centre-of-mass.

$$L_2 = 2 \frac{|r(R_{Emul}) - r(R_{RCM})|}{d} \qquad (5.13)$$
$$\text{Where } r(R) = \frac{\sum_{k \in \mathcal{O}} R_k |x - x_k|}{\sum_{k \in \mathcal{O}} R_k}$$

With $\mathcal{O}$ the ensemble of identified objects, $R_k$ the total amount of rain in object $k$, $x_k$ the centre-of-mass of the object $k$ and $x$ the total centre-of-mass of the map. The maximum value for $r$ is $d/2$, so $L_2$ ranges between 0 and 2. The case where $L_2$ is equal to 0 is if both maps contain at most a single object or if, on both maps, all objects have the same centre of mass. The L-component is then between 0 and 2.

* **Structure component :** On each object, we can compute a scaled volume :

$$V_k = \frac{R_k}{R_k^{max}} \quad \forall k \in \mathcal{O} \qquad (5.14)$$

which is the sum of all object's points divided by the object's maximum point. Then we compute $V$ the weighted mean of all scaled volume over the maps:

$$V(R) = \frac{\sum_{k \in \mathcal{O}} R_k V_k}{\sum_{k \in \mathcal{O}} R_k} \qquad (5.15)$$

And finally the S-component is the normalized difference in $V$ :

$$S = 2 \frac{V(R_{Emul}) - V(R_{RCM})}{V(R_{Emul}) + V(R_{RCM})} \qquad (5.16)$$

Wernli et al (2008) illustrates with several examples that this part of the score captures information on both size and shape of each object. It notably differentiates peaked objects, with intense precipitation but affecting few grid points, than an other one, larger but with comparable amplitude (i.e low intensity but affecting more points). It corresponds to examples (c) and (e) on Figure 5.6 where object $F$ would have smaller maximum. The structure component scales between -2 and +2.

The results are then represented through a diagram where each day is represented by a point with the S and A components on the x and y axis respectively, and the L component given by the color of the point. SAL diagram are visible in Figure 5.14 and commented in Section 5.3.2. Following the recommendation of Wernli et al (2009), I will apply this score on sub-domains of a maximum of 500km by side represented with blue squares on Figure 5.1.
<figure><img src="image_68.png" /><figcaption>**Figure 5.6:** Illustration of SAL scores with idealized examples, extracted from Wernli et al (2008).</figcaption></figure>

**5.3 Perfect model Evaluation**

This section is divided in two part. In a first evaluation step I evaluate and compare the three emulators in perfect model framework using the CNRM-ALADIN RCP45 simulation, from 2006 to 2100, as hidden truth. After a first impression on the emulators’ abilities through some examples, I will extend the analysis with climatological and daily scores. This section also helps to understand the impact of the loss function on the trained emulator. In a second step, I limit the evaluation to the Emul-ASYM and comment the SAL results which will help us to objectively determine if the emulator is able to create precipitation objects. In a second part I extend the analysis to all available ALADIN63 simulations (cf. Table 5.1) and study the emulator ability to reproduce their climate change projections.

**5.3.1 Comparison of the three emulators**

*First look into the emulators' prediction*

Before evaluating the emulators' performances with metrics, it seemed worthwhile to look into the raw series they produce. Figure 5.7 shows the time series at four grid points for the year 2022 in the evaluation simulation for the RCM truth and the three emulators. The four grid points show very different series. The Alps point series shows the strongest variability and intensities, with many days over 50 mm and almost no dry spell. The Paris series has minimal variability with numerous small precipitation days and low extremes compared with the other points. The
Toulouse and Roma series are the two most similar, with some dry periods and heavy precipitation days, especially in spring and autumn in Toulouse. Roma is pretty dry during spring and summer 2022 in this simulation and has a very strong rainfall event in fall.

<figure><img src="image_69.png" /><figcaption>**Figure 5.7:** Daily precipitation time series for four grid points. The RCM truth (in red) and the three emulators are plotted on each panel.</figcaption></figure>

The emulators series are very encouraging. They reproduce the four original series accurately, respecting each point's characteristics. They look like precipitation series as they appear to be able to produce days with no precipitation and days with heavy rainfall. All emulators capture the extreme autumn rainfall in Roma and the dry spell between May and June. The very high variability over the Alps point also appears well reproduced by the three emulators. On all four points, the three emulators seem to miss some extremes simulated by the RCM, as it occurs several times that the red line comes higher than the others. It also seems to be the case for the green curve (Emul-ASYM), but it is more difficult to see as it covers the other lines. However, it does not seem that Emul-MSE or Emul-MAE ever make stronger extremes than the RCM. At this point, it is impossible to decide if an emulator performs better than the others.

Figure 5.8 shows the precipitation field over the target for three days randomly
picked along the simulation. It shows the RCM truth, the three emulators and the
UPscaled precipitation field (UPRCM). The UPRCM helps to have an insight into
the input resolution and shows how the RCM and the emulators refine it, even if
precipitation is not part of the predictors. Several exciting points appear in this
figure. First of all, the emulators' prediction on each panel is very coherent with
the RCM. The precipitations are always well located with coherent intensity. It
seems, however, that the emulators are producing too smooth objects. On the RCM
maps, there are some very sharp and precise structures that the emulators fail to
reproduce with the same precision. For example, on the lower panel, there is a hole
with no rain over the southwest of France, which is missed by all emulators, even
if Emul-MAE and Emul-ASYM make less intense precipitation over this area. The
middle panel RCM map also shows very sharp structures that appear smoother
in the emulators' maps. Nevertheless, the extreme points are well located for the
three days.

<figure><img src="image_70.png" /><figcaption>**Figure 5.8:** 3 randomly chosen days illustrating the precipitation field of ALADIN63 at the Upscaled resolution (UPRCM), its native resolution (RCM truth). The three right-most plots show the precipitation field for each of the three emulators.</figcaption></figure>
In terms of intensities, the three emulators have mostly the correct spatial mean. Emul-ASYM reproduces better the spatial extremes as it has closer 95th superquانتiles than Emul-MSE and Emul-MAE, which are both under-estimating the spatial extremes on these three days. Emul-ASYM is overestimating the spatial SQ95 on the first panel, as it creates a more significant local extreme over the Alps than in the RCM map. It is, however, remarkable that this extreme is not inconsistent with the UPRCM map. Indeed it is interesting to notice the differences between the RCM and the UPRCM maps, which attest to the resolution's impact. The RCM is able to create sharp and well defined objects, with locally strong intensities. Regarding this aspect, the emulators seem to have an adequate capacity to refine the low-resolution maps and always recreate consistent high-resolution maps. Nevertheless, it seems that the objects created by the emulator are less *precise* than on the original RCM maps.

## Daily scale analysis

In a second step, and to extend the first observations from the previous section, we can look at some scores over the time series. Firstly, the upper panel on figure 5.9 shows the Pearson correlation coefficients calculated between the RCM and the emulators' series in each grid point. The three emulators appear to have similar performances regarding this aspect, with a reasonable deseasonalised correlation with the true series over the whole domain. The best correlations are over the reliefs with Pearson coefficients larger 0.9. The lowest correlation appears over the driest area (cf Fig. 5.11), like the south of the Pyrenees or the North-East corner of the domain, but the correlations are still around 0.75.

<figure><img src="image_71.png" /><figcaption>**Figure 5.9:** Temporal Anomalies Correlation (up) and Ratio of variance (bottom) computed on the entire evaluation simulation (2006-2100) for the three emulators.</figcaption></figure>
The lower panel on Figure 5.9 shows the variance ratio for the three emulators against the RCM truth. Emul-ASYM manages to reproduce in each point the RCM variance much better than the two others. Its variance ratio ranges from 80 to 120 percent, with a big part of the map being very light, showing about 100% variance reproduction. It slightly overestimates the relief's variance and slightly underestimates it over the regions with low rain average (cf Fig. 5.11). On the other hand, both Emul-MSE and Emul-MAE vastly underestimate the variance over the whole domain, even if Emul-MSE is slightly better.

It seems that the three emulators understand the large scale the same way, as they
can recreate the chronology of the original RCM series very accurately. They can
identify where and when the precipitations occur at the grid point scale, as shown by
the good correlation maps. However, the loss choice seems to substantially impact
the reproduction of the events' intensity as the emulators have different variance
ratio maps. Let us see if this is confirmed when we look at aggregated statistics.

Climatological scale analysis

In this section, we look at some aggregated statistics to evaluate if the series
produced by the emulator are statistically similar to the RCM one and how they
differ. Figure 5.11 shows three climatological metrics over 20 years in the present
period for the RCP4.5 simulation which is not in the training set. The upper panel
shows the average daily precipitation over 2006-2025, the middle one is the 99th
quantile, and the lower one shows the proportion of dry days. This figure illustrates
the impact of each loss function on the emulator.

The Emul-MSE mean is very similar to the RCM map. The spatial mean and superquantiles are the same. The bias map shows that it slightly underestimates the RCM values, but at maximum by 15% and over regions with low precipitations. However, it presents much poorer results on the other part of the distribution: it largely underestimates the 99th quantile (-15% on average) and the number of dry days (-10% on average). It is due to the nature of the mean squared error loss, mainly concentrating around the mean. The Emul-MAE is, meanwhile, very accurate for the representation of dry days, very slightly overestimating them. However, it fails to reproduce the mean and the 99th quantile maps, broadly underestimating them. The MAE loss gives the same weight to all errors. Since the number of dry days is the most represented (between 35 and 85% of the days are between 0 and 1 mm) they weigh much more in the emulator training, so it mainly focuses on them.

The Emul-ASYM aims to correct the EMUL-MAE by giving more weight to the
rainy days, proportionally to the amount of rain. It has similar performances to
Emul-MAE over the dry days' map, which is expected since both emulators have
the same loss function on this part of the distribution. However, the Emul-ASYM
mean and 99th quantile maps are also very accurate. It shows in both cases
less than 15% bias over the worst points and almost no bias on average over the
maps. Regarding both climatologic maps, it seems to slightly overestimate the
precipitation where it is raining the most and under-estimates at the driest points.
Nevertheless, these errors are shallow, and the Emul-ASYM is clearly the best
option if we aggregate the performances for the three metrics.
Daily precipitation mean 2006-2025

<figure><img src="image_72.png" /></figure>

Daily precipitation 99th quantile 2006-2025

<figure><img src="image_73.png" /></figure>

Percentage of dry days, 2006-2025

<figure><img src="image_74.png" /></figure>
**Figure 5.11:** (Upper) the mean map of daily precipitation accumulations over the 2006-2025 period, (middle) the 99th quantile map over the same period and (lower) the percentage of dry days. These three statistics are shown for the RCM and the three emulators. For each emulator and each metric, the relative bias maps are shown. The spatial mean and 95th and 5th superquantiles are given for each map.

On all maps in Figure 5.11, it is striking to see how well the emulators reproduce
the complex spatial structures. Emul-MAE and Emul-MSE have strong biases
that are uniform over the domain. All three statistics present locally different
patterns, and the emulators reproduce that. For instance, on the 99th quantile
maps, there is a strong pattern in the Cevennes, just south of the Massif Central
(France), which is much less intense in the daily mean map. It is the same for the
emulators' maps. The spatial structure over Italy is also very complex; there is a
thin line over the reliefs with more rainy days and higher extremes, which is also
perfectly reproduced by the emulators. Similar examples exist for the entire domain.

In order to extend this result, we can look at the entire distribution using the ASoP
method described in section 5.2.4. In Figure 5.12, the pdf analysis is detailed for the
four grid points previously used: Toulouse, a high point in the Swiss Alps, Paris and
Roma. The first column shows the events frequencies for each bin defined in section
5.2.4. Most days fall in bins under 0.1mm as the red curve comes from high on the
left part of the plots. The Emul-ASYM and the Emul-MAE reproduce this part well,
while the Emul-MSE underestimates the very low precipitations ($\le 0.1\,mm/day$)
and overestimates the ones between 0.01 and 10mm. It is less pronounced for
the Alps point, where the event distribution is more uniform across the bins with
more intensities 10mm and 100mm than the other three points. Emul-ASYM re-
produces the frequency of these stronger events better than the two other emulators.

The second column shows the actual contributions to the mean, which are the
frequencies multiplied by the bins' mean. The first remark is that Emul-ASYM
slightly overestimates the contribution of the precipitations around 10mm, which
probably led to the wet bias on the mean map of figure 5.11. Emul-MAE
produces insufficient rainfall over ~ 8mm as the right part of the distribution
is shifted to the left. The same remark applies to the Emul-MSE to a minor
extent, which has a better reproduction of the mean, confirming what we saw in
figure 5.11. Nevertheless, the Emul-ASYM matches better the right tail of the curve.

The last column illustrates the fractional contributions skill score by plotting
the difference between the emulators and the RCM distributions. The fractional
contributions are the actual contributions normalized by the mean of the series,
allowing us to compare only the shape of the distribution across the bins. It helps
to see that the Emul-MSE and Emul-MAE distribution are generally left-shifted,
with too many small precipitations and not enough big ones. The Emul-ASYM
curve generally looks better even if it tends to be right-shifted as it produces
too many heavy precipitations. The regularization term in the Emul-ASYM loss
appears to play its role pretty well as the distribution of precipitation is closer to
the real one but might sometimes be too strong.
<figure><img src="image_75.png" /><figcaption>**Figure 5.12:** Illustration of the probability density function analysis following the ASoP method (Klingaman et al, 2017) on four example grid points. Each line is a point and each column is a different step of the method. The first column shows the frequency of events in each bins, the second and the third the actual and the fractional contribution and the last column illustrates the skill score. The number in the last column pplots are the scores for each emulator at the corresponding point.</figcaption></figure>

The skill score measures the area between the emulators' fractional contribution and the RCM one, and we can see that the Emul-ASYM outperforms the others over these 4 points. It is interesting to notice that Emul-MSE and Emul-MAE perform better over the Alps point, where the precipitations are more uniformly distributed across the bins. Finally, Figure 5.13 shows that the Emul-ASYM skill score is better over the whole domain. It generalizes the distribution analysis and confirms that the specifically designed loss function is more adapted than the two others to reproduce the highly skewed distribution of precipitation.
<figure><img src="image_76.png" /><figcaption>Figure 5.13: Fractional Contribution Skill Score maps for the three emulators.</figcaption></figure>

## Conclusion on the comparison

Until here, we have analysed the role the loss function can play in the calibration of the emulator. Table 5.3 summarise the results obtained on the three emulator. They all demonstrated an excellent capacity to reproduce the daily precipitation time series with a good temporal correlation with the original RCM series. Nevertheless the loss function impacts strongly the intensity of the events. The MSE loss function penalizes strongly the large error which centers the prediction around the mean because of the chaotic nature of precipitations. Thus, if the mean daily precipitations is well represented the extremes are underestimated. On the other hand the MAE reproduces well the low precipitations but underestimate the intensity of larger events. Finally, the Emul-ASYM, thanks to a regularization term added in the loss function, managed to reproduce better the entire rainfall distribution at each grid point of the domain with notably a better reproduction of the extremes. Therefore, the loss function plays here as a cursor to set the event intensities, while the chronology of the series is captured from the predictors. From now on, we will consider only the EMUL-ASYM.

<table><caption>Table 5.3: Summary of the emulators’ comparison results</caption><thead><tr><th>Emulators</th><th>Temporal correlation</th><th>Average precipitation</th><th>Dry days</th><th>Heavy precipitation</th><th>Variance</th><th>PDF Shape</th></tr></thead><tbody><tr><td>MSE</td><td>++</td><td>++</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>MAE</td><td>++</td><td>-</td><td>++</td><td>-</td><td>-</td><td>-</td></tr><tr><td>ASYM</td><td>++</td><td>++</td><td>++</td><td>++</td><td>+</td><td>+</td></tr></tbody></table>

## 5.3.2 Deeper evaluation of Emul-ASYM

### Object oriented analysis

The illustration Figure 5.8 that the precipitation objects created by the emulator are smoother than in the RCM. The SAL method presented in section 5.2.4 is an objected-oriented evaluation approach which compares on two maps the object
similarities.

Following the recommendation of Wernli et al (2009), I limited the evaluation to three subdomains of about 500km by the side. The blue boxes represent them on figure 5.1. The first subdomain focuses on the Cevennes regions. This part of South France is well known for its extreme autumn precipitation events. These events are the object of multiple studies (Ribes et al, 2019; Caillaud et al, 2021) because of their strong socio-economic impacts. It is then important to assess whether the emulator is able or not to reproduce such events. The second domain is another hotspot for Mediterranean extreme precipitation events (Ivušić et al, 2021) located in Croatia, over the Dinaric Alps and the North of the Adriatic Sea. The last subdomain is centred around Belgium, including the South-East of England, the North-East of France and West of Germany. This region presents a different climatology with extreme events of smaller intensities occurring more in winter.

Figure 5.14 presents the SAL scores’ results. For each region, there are five SAL diagrams. The left most diagram represents the results for all raining days. Then going to the right we consider only days where the spatial 99th percentile of the RCM truth series is above an increasing threshold. The threshold and the number of considered days are indicated on each diagram. Thus, from left to right we consider only more and more extreme events. The first general comment is that over all these diagrams, the emulator reproduces accurately the extreme majority of the events. Indeed the red boxes regroup 90% of the days and they are always centred around 0 with most points are deep blue, showing good Location score. On the first column representing all raining days, the emulator underestimates the global amount of precipitation over the domain, with the red box being pulled down left. It can be explained by the fact that the emulator misses some small precipitation objects. The SAL metrics are one-sided: they evaluate how the predicted map matches the reference one. Knowing the chaotic nature of rainfall, it is perfectly fine that the emulator misses or add some small events. Besides, we fix the threshold according to the RCM true series. It is then logical that events, especially small ones, are missed or underestimated by the emulator. It is not shown here, but when we fix the threshold according to the emulated series, then the emulator overestimates the amplitude of some small RCM events and the red box is pushed up-right.

On the right of the figure, when we look at days with heavier precipitation, the amplitude gets centred around zero or slightly positive on the right-most column of the two Mediterranean regions. In addition, the emulator tends to produce larger objects with a positive S-component. However, the centre of the object is most of the time well located. It tends to generalize that the emulator produces smoother objects than the RCM, especially on significant intensities events.

There is a correlation between the amplitude and the structure metrics. It can attest that the emulator always creates objects consistent with the RCM. They are either smaller or bigger in terms of both shape and amplitude. On all maps, we can see some days with lousy location and structure scores but the correct amplitude. They are typical of days where the emulator produced too smooth objects and did not peak like the RCM. The emulator produces one large object with medium
intensity, while the RCM produces multiple peaked objects with high intensities. It
implies bad locations and structure scores but good amplitude.

<figure><img src="image_77.png" /><figcaption>**Figure 5.14:** SAL diagram for the three regions: Cevennes (up), North of the domain centred around Belgium (middle), and a region over Croatia and the North of the Adriatic sea. From left to right, the panel show the SAL results for days with maximum events intensities above an increasing threshold. Each point on the diagram represents a day with the A-Component on the y-axis, the S on the x-axis and the color give the Location score. The red box includes 90% of the points, and the black cross indicates the A and S median. The 5th, 50th and 95th quantiles are given in white on the colormap for the Location component.</figcaption></figure>

Generally speaking, the emulator manages to reproduce the precipitation objects
simulated by the RCM, even if they do not always have the perfect characteristics.
The emulator captures most of the extreme events with the most suitable charac-
teristics. The emulator seems nevertheless to produce smoother objects. A further
analysis, with an application to a hydrological impact study, should be conducted
to determine whether it is a fundamental limitation and how we could maybe adapt
the emulator.

ALADIN63 matrix extension

In order to give more robustness to the good performances of the Emul-ASYM, we
can extend the evaluation to all ALADIN63 simulations available for our target
domain. Figure 5.15 summarizes climatological maps as the ones shown on Figure
5.11. The three panels (from left to right) correspond to the three statistics we
looked at in Section 5.3.1: the mean amount of daily precipitation, the 99th
quantile and the percentage of dry days over the 2006-2025 period. On each panel,
the upper part shows the summary statistics for the raw maps of the RCM and the
emulator, and the lower part summarises the relative bias maps of the emulator
with respect to the RCM truth. On each panel, the columns correspond to a
simulation. Each bar shows the spatial mean of the map, the upper bound shows
the 95th super-quantile and the lower bound shows the 05th super-quantile. The
first column shows the results for the CNRM RCP85 simulation, which has been
used to train the emulator. The results on this simulation are given here as an
indicator and cannot be taken alone to evaluate the emulator’s performances.

<figure><img src="image_78.png" /><figcaption>**Figure 5.15:** Summary plots of the three climatological statistics regrouping the results on all ALADIN63 simulations. On each error bar, the lower (resp. upper) bound is the spatial 5th (resp. 95th) superquantile and the spatial mean is represented by the dot. The upper panels show the raw maps summary statistics for the RCM (in red) and the Emul-ASYM (in green), and the lower panels show them for the relative bias maps.</figcaption></figure>

The results are very encouraging as the performances of the emulator are very
similar across simulations, even if those are different in several aspects. For
instance, the 3 CNRM simulations have higher daily means than the three others
since the spatial mean and superquantiles are higher. It is less evident on the 99th
quantile maps, where only the NCC simulation produces “eye-visible” less intense
extremes. The emulator’s bars reproduce these aspects. As observed previously, the
emulator overestimates the average daily precipitation on the wettest points and
underestimates it over the driest points whatever the statistics, which stays valid for
all simulations. The biases on land points are similar to the ones observed for the
CNRM RCP45 simulation, showing that the emulator reproduces each simulation
with the same accuracy. In all these simulations, the emulator reproduces the three
parts of the distribution well over the whole domain.

<figure><img src="image_79.png" /><figcaption>**Figure 5.16:** The left panel shows the summary plots for the temporal variance, the upper panel summarizes the raw variance maps, and the lower one shows the ratio of variances. The right panel shows the Fractional Contribution Skill Score (FCSS) for the 2006-2100 period for the four simulations that have not been shown yet (CNRM 26, MPI, NCC, HGM).</figcaption></figure>

The analysis is the same regarding the variance maps summarized in Figure 5.16. The temporal variance differs according to the simulation. For example, the RCM simulation driven by NCC has a smaller variance than the CNRM simulations or the HGM. The emulator reproduces in each case the variance maps quite accurately. However, in every simulation, it strengthens the variance where it is the strongest. The variance ratio summary plot confirms that the analysis made for the CNRM RCP45 in section 5.3.1 extends to all other simulations. The emulator can reproduce the daily time series with globally acceptable variance at every grid point. In the worse cases, it misestimates the variance by about 20%. Figure 5.16 also shows the FC skill score maps for the four missing evaluation simulations (the RCP45 simulation in 5.13). Here again, we can observe that the emulators reproduce the shape of the precipitation distribution correctly at each grid point in all simulations. It is impressive to see how similar are these four maps. The emulator has similar
performances across all simulations at the grid point scale.

Climate change reproduction

In order to finalise the evaluation of the emulator in the perfect model framework, we can look at the climate change maps. To do so, we will look at the three statistics used in the previous sections: the mean daily precipitation, the 99th quantile and the percentage of dry days. In each simulation, I compute the relative changes in a future period (2070-2100) versus a past period (1950-1980). The changes in precipitation are likely to be different according to the seasons over western Europe so we will look at the seasonal climate change here. The different studies about changes in precipitation amount over the region project a decrease in summer precipitations, notably around the Mediterranean sea, and an increase of winter precipitation on the North. Besides, a possible increase in extreme precipitation, especially over northern Europe, is expected. The results for the four seasons and the three statistics on all simulations are summarised through summary plots in Figure 5.17 while the results for the MPI and HGM simulations are illustrated in Figure 5.18.

The first remark is that on all plots summarising the raw maps, the green bar
sticks very well to the red one, implying that the emulator correctly reproduces
the maps and the intensity of the local changes. It is particularly notable on
the summer plot, where the differences between the projections are the strongest.
The MPI and NCC simulations show a substantial decrease in the mean daily
precipitation over the entire map, associated with a global increase in the percentage
of dry days. On the other hand, the HGM simulation projects an increase in
average daily rainfall over some regions in summer. The emulator reproduces
each simulation specificity with mainly the right intensity. Figure 5.18 shows
summer and winter changes for the MPI and HGM simulations. It illustrates well
that the emulator correctly captures the big spatial pattern. Still, in summer,
we can observe that the emulator precisely places the regions where the HGM
simulation produces an increase in average rainfall. This increase matches an
increase of the 99th quantile in the same regions, and the emulator produces
the same relationship. Similar analysis can exist on the winter maps, conclud-
ing that the emulator reproduces the ALADIN63 simulation with excellent accuracy.

Nevertheless, the emulator's maps are more continuous than the RCM maps,
especially for the 99th quantile maps, which are a lot patchy. It results in significant
local biases between the emulator and the RCM maps. It partly explains the large
biases on the bias maps summary plots in Figure 5.17. Generally, the emulator
tends to overestimate some changes as we can see that the green bar is often longer
than the red one. The number given on top of the bias maps summary plots shows
the percentage of sign agreement between RCM and emulator over the grid points.
It shows that the emulator identifies well the changes as these numbers are very
high ( always above 75%, very often above 90%). Moreover, on the bias maps of
Figure 5.18, the hatching shows the points where RCM and Emulator disagree on
the signs. It is visible that they mostly correspond to points with minor changes.
Seasonal relative changes 2070-2100 vs 1950-80

<figure><img src="image_80.png" /><figcaption>**Figure 5.17:** Same as Fig 5.15 for the seasonal climate change (2070-2100 vs 1950-80) summary plots for the three statistics of interest: the daily precipitation mean, the 99th quantile and the percentage of dry days. The changes are the relative difference between the future period and the past one. The biases are simple bias between the Emulator and RCM relative change maps. On each bias summary plot the number indicates the % of points where RCM and Emulator agree on the sign.</figcaption></figure>
To conclude, the emulator can reproduce high-resolution climate change maps with the same strong spatial pattern and intensities. Another relevant remark, not shown here, is that Emul-MSE and Emul-MAE have the same ability as Emul-ASYM to reproduce the climate change maps. It means that each emulator keeps the same biases along the simulation, and the changes are mainly driven by the large scale, which the emulators understand well.

Conclusions on Emul-ASYM

Through sections 5.3.1 and 5.3.2 we have analysed the ability of the emulator trained
with the asymmetric loss function to reproduce the precipitation field simulated by
the RCM. The conclusion on the emulator performances are summarised here:

* The emulator is able to produce realistic precipitation time series well corre-
  lated to the RCM ones and with the right variability.

* The grid-point regularization term in the asymmetric loss function helps to re-spect and reproduce the entire complex distribution of precipitation every-where on the target domain.

* The emulator tend to underestimate the precipitation in generally dry regions and inversely in the wettest parts of the domain.

* The emulator creates coherent objects of precipitation, with generally the right characteristics even if they tend to be too smooth (i.e. less sharp and precise than the RCM objects).

* Those conclusion on a single evaluation series stand also for the other evalu-
  ation simulations that we have at our disposal, respecting both scenario and
  GCM transferability defined in Chapter 4.

* Finally the climate change maps obtained from the emulated series are almost identical to the RCM ones. It gives a lot of confidence to use the emulator in climate change context.

The emulator present therefore satisfactory results in perfect model evaluation and, even is there is space for improvements, we will study its behavior in GCM application.
Climate Change maps, 2070-2100 vs 1950-1980
Average Daily Precipitation

<figure><img src="image_81.png" /></figure>

**Figure 5.18:** Relative changes (in %) between 2070-2100 and 1950-1980 for the MPI and HGM driven simulations regarding (up) the mean map of daily precipitation accumulations, (middle) the 99th quantile map and (low) the percentage of dry days. These three statistics are shown for the RCM and the emulator, plus the simple bias map between the two. For each map, the spatial mean and 95th and 5th superquantiles are given. The hatching indicates the point where RCM and Emulator disagree on the sign.
## 5.4 GCM data application

In this section, I apply the emulator directly to GCM data as in its application mode. In order to have a reference to analyse those results, I will downscale with the emulator the GCMs used to drive ALADIN63: CNRM-CM5, MPI-ESM-LR, HadGEM2-ES and NorESM1 (cf. Table 5.1). I will focus only on the RCP85 simulations from 2006 to 2100. This section aims to evaluate the emulator’s behaviour when applied to GCM inputs. However, as shown in Chapters ?? and 4, the RCM simulations cannot be used as the exact reference to evaluate the emulator. The large-scale differences between an RCM simulation and its driving GCM imply low day-to-day correlation and long-term statistical differences. The difficult task here is to assess if the emulator produces a series consistent with the GCM large scale and the high resolution added value from the RCM.

### 5.4.1 Illustration of the daily GCM/RCM differences

Figure 5.19 shows five consecutive autumn days of the precipitation field in the CNRM RCP85 simulation. For each day, we see the RCM truth simulation and the emulated maps in perfect model and application mode. In perfect model mode, the emulator inputs come the UPscaled RCM predictors, and in the application mode, it takes GCM predictors. I also plot the UPRCM and GCM precipitation maps of the corresponding days. It is important to remember here that the low-resolution precipitation field is not a predictor. The UPRCM precipitation is simply the RCM map interpolated on the GCM grid, and I use it to compare with the GCM precipitation map.

It is striking that over these six days, the GCM emulated maps never look the same as the RCM and the UPRCM emulated maps. As shown in the previous section, the UPRCM’s emulator reproduces the RCM map very well. It recreates most objects in the right place and with the right intensity. We have seen that the objects are more spatially continuous than in the RCM maps. Looking at the maps produced by the emulator when it downscales GCM predictors, we can see that they are very different. The rainfall occurrence, location or intensity do not match the RCM map but is perfectly coherent with the low-resolution map from the GCM. For instance, on the first day, the RCM produces heavy precipitation on both sides of the Adriatic sea, while there is nothing on the GCM emulated nor in the GCM maps. Then a strong precipitation event arrives from south and stays blocked on the Alps at day 3. The GCM seems produces the same event but one day after the RCM and it stays south of the Alps. Remarkably, the GCM emulated maps are always perfectly coherent with the precipitation field produced by the GCM, which gives some confidence that the emulator creates realistic maps. Indeed, the emulator seems robust to GCM predictors, as it “understands” and follows the large-scale atmospheric conditions.

The sentiment after this little illustration, is that the emulator seems to do the job. It recreates high-resolution fields which look like the RCM ones, or at least the emulator in perfect model mode. Indeed, the downscaled objects from the GCM look similar to the UPRCM emulated maps. They seem smoother than what creates the
RCM, as we saw in section 5.3. Nevertheless, similarly to the perfect model mode, the emulator creates objects with realistic intensities or characteristics. These six days also illustrate the difficulty of evaluating the emulator in application mode without proper reference. Indeed these day-to-day mismatches make it impossible to differentiate an eventual emulator's bad behaviour and large-scale induced divergence.

<figure><img src="image_82.png" /><figcaption>**Figure 5.19:** Illustration of 6 consecutive days rainfall maps for the RCM, the emulator downscaling the UPRCM, the emulator downscaling the GCM, and both the UPRCM and GCM precipitation fields.</figcaption></figure>
5.4.2 Present climate analysis

Since studying time series reproduction does not make sense, we can look at
climatological statistics. As in the perfect model evaluation, I compute the annual
average daily rainfall, the 99th quantile and the percentage of dry days in the
present climate (2006-2025) in the four simulations. The results are shown as maps
in Figure 5.20 because there are too big local biases which cannot be analysed on
summary plots. For the same reason, we look at simple biases as relative bias can
get very high for small biases on small rain amount.

The first general comment on all these maps is that they all seem realistic, which
testify to the emulator’s robustness. Indeed, because of the well-known limits of
neural networks regarding the input space, it risked making completely unrealistic
series. The spatial extremes and mean on the raw maps are in the same order
as for the RCM maps. The second reassuring remark is that most biases differ
from one model to the other. The emulated CNRM is wetter in the Alps than its
corresponding RCM simulation, while it is the opposite for the HGM simulations.
It is furthermore notable that the three statistics agree on these two examples.
The emulated CNRM 99th quantile is higher over the northern Alps than in the
RCM simulation combined with fewer dry days, leading to a stronger average of
daily precipitation. The complete opposite analysis can be done for the HGM
simulations over the southern Alps. This coherence also gives some confidence in
the downscaling of the emulator.

On the other hand, some biases might be more questioning. For instance, all emu-
lated simulations undervalue the 99th quantile over the Cevennes in South France.
This specific region is well known for its extreme events, and it is an added value
from the RCM to reproduce them better. The emulator produces substantial ex-
treme in this region, but less intense than over the Alps, while there are comparable
on the RCM maps. Likewise, the emulated maps show a dry bias on all models
around the Adriatic sea and a wet bias on the North of the Pyrenees. Though, it
is difficult to assess without a dedicated study if this is indeed a limitation of the
emulator.

5.4.3 Climate change analysis

In order to complete the study of the emulator performances on downscaling GCM
simulations, it is necessary to look at its ability to create consistent high-resolution
climate change maps, as it is the primary purpose of the emulator. To do so,
I decided to look at the average precipitation changes in summer for the four
simulations used in this section. The aim here is really to illustrate what the
emulator produces. Figure 5.21 shows for each model the summer changes for the
RCM simulation, the GCM-emulated simulation, the bias between emulator and
RCM projections, and the bias map between UPRCM and raw GCM. The changes
are shown as the relative differences to the past period, while the biases are the
simple difference between the emulator (resp. GCM) and the RCM (resp. UPRCM)
maps.
Average daily precipitation, 2006-2025

<figure><img src="image_83.png" /></figure>

99th quantile, 2006-2025

<figure><img src="image_84.png" /></figure>

Percentage of dry days, 2006-2025

<figure><img src="image_85.png" /></figure>

**Figure 5.20:** Present (2006-2025) climate statistics of 4 simulations (CNRM RCP85, MPI, NCC and HGM) for (Upper) the mean map of daily precipitation accumulations, (middle) the 99th quantile map and (lower) the percentage of dry days. For each simulation, we see the RCM, the emulated one and the simple bias between them. The spatial mean and 95th and 5th superquantiles are given for each map.
The MPI simulations are the ones which agree the most. Both emulator and RCM are drying over the entire domain, with more intensity over the Mediterranean sea and the Atlantic Ocean coasts. The emulator is drying more over the Mediterranean than the RCM. Both Emulator and GCM bias maps are consistent. No strong bias over the land, and more pronounced on the south of the domain.

Summer average daily precipitation changes
2070-2100 vs 2006-2025

<figure><img src="image_86.png" /></figure>

**Figure 5.21:** Summer relative changes of average daily precipitation between future (2070-2100) and present (2006-2025) period for the 4 GCM simulations downscaled with the emulator: CNRM, MPI, NCC and HGM under RCP85 scenario. From left to right, the column shows: the RCM map, the emulated map (from GCM), the difference between emulator and RCM, and the difference between GCM and UPRCM. The spatial mean and 95th and 5th superquantiles are given for each map.
The HGM emulator is drying much more than the RCM, which completely matches the GCM, as both bias maps are brown, with some extreme differences from the RCM. Surprisingly the CNRM GCM simulation is wetter around the Mediterranean sea in the future. The emulator reproduces it well, as the bias maps are very close. The RCM produces an entirely different change, much more consistent with the other simulations.

Finally, the emulated NCC simulation is quite similar to the RCM one. It is drying over most of the domain, more intensively in the south, such as the RCM. The emulator changes are, however, less pronounced than in the RCM simulation, as shown by the bias map. It is getting wetter in the middle of the Mediterranean Sea, which looks weird, but there is low precipitation on the sea so that it might be simply noise. However, the emulator's bias map does not match the one from the GCM, which is surprising. The GCM occurs to be globally drier than the UPRCM, while we have the opposite from the emulator. Again, it is difficult to conclude the soundness of the emulator downscaling. Nevertheless, once again, the emulator produces realistic maps without evidence invalidating its application to GCM simulations.

# 5.5 Discussion

## 5.5.1 On the asymmetric loss

In this section, I wanted to reflect on the construction of the Emul-ASYM loss. The idea came from the behaviour of Emul-MAE. It predicted the low precipitation very well but largely underestimated the medium and large rainfall. It is mostly due to the unbalancing of the data. Figure 5.11 shows that over the domain, at least 35% of days are dry (less than 1mm), with some grid points reaching more than 80% of dry days. Emul-MAE reproduces better the most raining points as shown by the variance ratio (Fig. 5.9) and the FCSS (Fig. 5.13) maps. The aim was then to give more weight to the underrepresented raining days. I chose to handle this issue by penalising the neural network stronger when it underestimates a precipitation amount. It felt like the right answer to the Emul-MAE default. It is the asymmetric part done by the $max(0, y_{i,t} - \hat{y}_{i,t})$ parameter in equation 5.4. However, it needed to depend on the intensity of the rain. The more intense the event is, the rarer it is, so the stronger the emulator needs to be penalised if it underestimates it, especially because these events constitute an essential part of RCM's added value. It also depends on each grid point because of the strong spatial variability of extremes visible on the 99th quantile maps (Fig. 5.11). It is the role of the $\gamma_{i,t}$ parameter.

(Eq. 5.17) is the first assymetric loss function I tried with good results. It has fixed values of $\gamma$ according to a certain threshold based on the local quantile function. In each grid point $i$ we compute the $p^{th}$ quantile, and for every target value $y_{i,t}$ the associated $\gamma_{i,t}$ parameter can take two values if $y_{i,t}$ is higher or lower than the threshold. There are 3 parameters to define for this loss function and figure 5.22 shows different attempts. We can see that the value of $\gamma^{UP}$ strongly impacts the results. Too low is not enough to correct the Emul-MAE, while too
high leads to a considerable overestimation of the precipitations (expected by construction). The $\gamma^{LOW}$ influences mainly the results on the low precipitations days, where the MAE is already accurate. So 0 or a low value is preferable. The quantile choice also has a strong impact. It is not shown here, but by analysing the pdf curves of these tries, we could see an artificial bump after the given quantile.

$$L(y, \hat{y}) = \frac{1}{N \times T} \sum_{t=0}^{T} \sum_{i \in D} |y_{i,t} - \hat{y}_{i,t}| + \gamma_{i,t} \times \max(0, y_{i,t} - \hat{y}_{i,t}) \\ \text{With: } \gamma_{i,t} = \begin{cases} \gamma^{UP} & \text{if } y_{i,t} \ge Q_i(p) \\ \gamma^{LOW} & \text{if } y_{i,t} < Q_i(p) \end{cases} \quad p \in [0, 1] \quad (5.17) \\ \gamma^{UP}, \gamma^{LOW} \text{ and } p \text{ are fixed parameters to be defined and} \\ Q_i \text{ is the quantile function in point } i.$$

For this last reason, I searched something continuous which would give similar results. The Gamma distribution has been shown to fit precipitation data correctly. The idea was then to fit in each grid point a gamma distribution and use the associated cumulative function to set the $\gamma$ parameter according to the target value. Once again, I made multiple tests. The orange bar on Figure 5.22 shows the results when the gamma is fitted on all days. The star red bar shows the results for the loss function as used in this chapter, with the gamma fitted only on days over 1mm rainfall, and the brown bar shows the results without taking the square of the $\gamma$. Fitting the Gamma distribution on all days seems to lead to an undervaluation of the number of dry days, increasing the mean value. Taking the square of the $\gamma$ parameter seems to correct an intensification of the medium rainfall.

I also added on Figure 5.22, two others emulators based on the same loss function as the Emul-ASYM used in the rest of this chapter. The training of neural networks implies a part of randomness (see 3), which can lead to slightly different results. These two other versions help to put into perspective the results. Indeed, if Emul-ASYM overestimated the mean daily rainfall, a twin emulator shows different results and underestimates the 99th quantile map. It does not change the conclusion that Emul-ASYM produces accurate results in reproducing the entire local rainfall distribution. We can see that the three red bars perform better over the three statistics than the Emul-MAE and Emul-MSE and than all other tries. Nevertheless, it needs to be considered when analysing the best loss.

The discussion on the asymmetric loss function felt important, as I do not claim that the choice made here is the best. There could be a better and more robust way re-weight the training sample. We have seen that Emul-ASYM tends to overestimate the medium-range precipitation, probably because the associated $\gamma$ parameter is too strong. The problem was mainly over the relief where the Emul-MAE was doing a better job due to more balanced data. Maybe the regularisation parameter $\gamma$ should also consider the MAE performance. The point here is to present what I have reached as it constitutes a basis for more development.
Asymmetric loss versions: Comparison on present climate statistics

<figure><img src="image_87.png" /><figcaption>**Figure 5.22:** Summary plots of the three climatological statistics for different emulators trained with different versions of the asymmetric loss. The “V0” versions correspond to the equation 5.17 with the $\gamma$ parameter noted “b” here and $p$ the threshold quantile. See sec 5.5.1.</figcaption></figure>

5.5.2 On the computational efficiency

The target domain chosen for this chapter is four times larger than in the previous chapters. We saw that the RCM-emulator showed good abilities in this new domain, and even if it is not the same variable, it seemed to follow the large-scale predictors perfectly. This more extensive domain is an excellent opportunity to make a short evaluation of computational efficiency. In the previous chapters, the target domain is a square with 64 grid points side, and the input domain is a square of 16× 16 low-resolution grid points. In this chapter, the target domain size is 128×128, while the input domain includes 22 grid points in longitude and 16 in latitude. Both neural networks include about 25 million parameters to fit. Indeed the layers with the most parameters are at the bottom of the architecture.

Regarding available memory, it was impossible to train the precipitation emulator on the new domain with the same GPU as in Chapter 1. So I trained both emulators on a bigger GPU to compare the computational times. The GPU used here is an Nvidia Tesla V100 PCIe 16GB. Both emulators necessitate about 60 epochs to be trained. Actually, only 45 are necessary, as the neural network does not improve on the last 15 epochs. 1 epoch on the new domain takes about 130 seconds, which implies a total training time of 2 hours and 10min.

On the other hand, the emulator trained on the smaller domain needs 65 seconds
per epoch. The training of the emulator for the more extensive domain is two
times longer, while the domain is four times bigger. The computational resources
are then not linearly proportional to the target domain size. Moreover, the entire
ALADIN63 domain is of size 450× 450 grid points, less than four times bigger than
the target domain considered in this chapter. Even when we use 4 emulators to
cover the entire ALADIN63 domain, it will take less than 10 hours to train. It is not
comparable with the computational time of ALADIN63, which needs several weeks
to perform one simulation. Besides, the training of the emulator can probably be
more optimised as I am not an expert in GPU resource management. Finally it is
important to note here that once the emulator trained the producing time<fnref n="1" /> takes
only few seconds even on the bigger domain.

5.6 Conclusion

This chapter aims to propose a credible solution to the high computational costs of Regional Climate Models to build large ensembles of high-resolution precipitation projections. It relies on the RCM-emulator introduced in Chapter 3 for the case of temperature downscaling. RCM-emulators belong to the family of hybrid downscaling methods. They use RCM simulations to estimate the downscaling relationship between low-resolution and large-scale variables and a high-resolution surface variable. This chapter aims to see if it is suitable for the complex variable of precipitation, including the extreme parts of its distribution, and, if not, how it can be adapted.

The RCM-Emulator used here is exactly the same as in the rest of this manuscript. It takes the same set of predictors: various two-dimensional variables at different pressure levels and external forcing, such as greenhouse gas concentrations. The predictors are normalised exactly the same way. The neural network architecture is also the same, and the emulator is trained in the same perfect model framework where both predictors and targets come from the same RCM simulations. The predictors are then upscaled to low resolution so that they look like a GCM simulation output. The perfect model framework ensures a perfect match between input and output and grants the focus of the neural network on the downscaling function. The only difference with chapter 3 is the target domain which is more extensive and centred over the Alps here. It implied an adaptation of the input domain and a little extension of the neural network. Of course, the target variable is also different as we want to emulate the precipitation field from ALADIN63.

The same neural network is referred to as Emul-MSE in this chapter. In perfect model evaluation, it showed an excellent capacity to follow the large-scale atmospheric circulation and to recreate the precipitation times series with reasonable accuracy. However, the climatological analysis revealed a problem in the intensity. The emulator concentrates its prediction around the climatological mean, with too many medium-low precipitations and a clear underestimation of the number of dry

<footnote n="1">¹This does take into account the preparation of the data. The longest part is to actually get the data from online servers. Then the cropping, interpolation and smoothing can be done in few hours.</footnote>
days and the 99th quantile. It is due to the complexity of the precipitation variable, which is non-continuous and highly unbalanced. In this chapter, I proposed to adjust the loss function to tackle this issue.

I proposed an asymmetric loss function designed to penalise the emulator stronger in the training phase when it underestimates the true precipitation amount; the more extreme the rain, the stronger the penalty. It showed that the loss function is an excellent means of constraining neural networks to focus on specific aspects as it corrected the emulator prediction to fit better the precipitation intensity. This adapted emulator showed the same good properties as the original emulator in terms of the temporal match with the original RCM series in perfect model evaluation. However, it reproduces the temporal variance and the entire distribution better. At the climatological scale, it reproduces almost perfectly the maps of the daily average precipitation, the 99th quantile or the percentage of dry days, with notably the reproduction of the high-resolution spatial structure. It has some local biases but they never exceed 15%. Moreover, the precipitation objects recreated by the emulator are also coherent with the ones created by the RCM. They are well located and have the right intensity, but they tend to be more spatially continuous than on the RCM simulation. Nevertheless, this does not affect the statistical properties of the series, even at the grid point scale.

I extended the evaluation of the emulator to multiple RCM simulations. The perfect model evaluation grants the perfect match between the predictor and the expected output. So the emulator was expected to reproduce all differences between the RCM simulations. The abilities of the emulator extended perfectly to all simulations. It showed the same good skills and default for all simulations. The emulator tended to overestimate the precipitations where they are the strongest, which stayed the case, in the same proportions for all tested simulations. In the climate change context, the emulator has also shown an excellent ability to reproduce the complex spatial patterns of change for each simulation and the three statistics we watched. Furthermore, the loss function does not impact the climate change maps from the emulator. It does not impact how the neural network follows the large scale but corrects the intensity. If the emulator underestimates the extreme precipitation, it does it similarly throughout the simulation. The event's occurrence drives the change, and we saw that the loss function does not impact the temporal correlation.

The application of the emulator to GCM simulations is a real-world exercise. Nevertheless, it is very challenging to evaluate. Indeed, the emulator is constructed in a perfect model framework to ensure a perfect match between predictors and target variables. It focuses only on the downscaling function and can be applied to any GCM simulation. On the other hand, it implies that if there are any large-scale biases between an RCM simulation and its driving GCM, the emulator will reproduce it. Such biases have been demonstrated in chapters 3 and 4 and in various studies (Boé et al, 2020; Bartók et al, 2017; Taranu et al, 2022). There is then no high-resolution reference to evaluate the emulator when it downscales a GCM simulation. I applied the emulator on four RCP85 simulations from the four GCMs used to drive ALADIN63. The results are very consistent with the perfect model evaluation. The emulator seems to follow the GCM large
scale very accurately. It produces coherent and realistic climatological maps
with locally strong biases with the RCM. Looking at climate change maps, they
also might differ from the RCM ones. However, they stay coherent with the
GCM/RCM biases. In my opinion, the emulator fits the job for downscaling GCM
simulations. However, to have better trust in it, the emulated GCM simulations
should be looked at deeper and maybe with different approaches. In particular,
setting a GCM/RCM pair without large scale discrepancies may be the oppor-
tunity for a proper evaluation of the emulator when downscaling a GCM simulation.

In terms of computational efficiency, the more extensive domain implied to use a
GPU with more available memory to train the emulator. We could see that it takes
about two hours to train the emulator while the production takes less than a minute.
It is not comparable to the time necessary to produce a new ALADIN63 simulation.
Moreover, the emulator was easily transferable to a new variable and a new domain.
The loss function is a good way to constrain and force a neural network to specific
events and better fit the entire distribution of a variable as complex as precipitation.
I think the RCM-emulator is a promising way to create large ensembles of high-
resolution precipitation simulations by downscaling all existing GCM simulations.
# Chapter 6

# Conclusion & Perspectives

This final chapter aims to summarise the main conclusions of this work, to present the big questions raised and introduce some perspectives on improvements, applications or developments. This PhD targeted to propose a solution to enlarge the size of regional climate model (RCM) simulation ensembles. These models' increasing resolution and complexity imply high computational costs that strongly limit the number of simulations and, thus, the study of the associated uncertainties. The reliability of the regional climate change messages extracted from the actual RCMs ensembles is then insufficient. The correct exploration of all possible futures at local scales is necessary to impact studies and organise adaptation policies. To address this issue, I propose in this manuscript a hybrid downscaling approach as it relies on the two distinct families of downscaling: dynamical and statistical. While the former relies on RCMs, the latter aims to learn the statistical relationship between what I called all along this report large and local scales. The large scales correspond to some low-resolution fields in altitude or averaged surface fields over a vast domain that global climate models (GCMs) should represent correctly. On the other hand, the local scale are characterizing the whether that is actually experimented by humans. They range from one to hundred kilometres. They depend on the local environment and necessitate high-resolution climate models to be represented.

Therefore, I proposed here to build an RCM-Emulator that relies on the same idea as statistical downscaling. However, instead of targeting the *real* one, the emulator aims to learn the large scale/local scale relationship simulated by a given RCM. The advantage of this approach is that the relationship can be estimated in non-stationary climates and regions with no observations. Of course, the emulator does not aim to correct the relationship simulated by the RCM; it learns to reproduce it with its defaults. By constructing one emulator for all existing RCMs, it would be possible to downscale, for some variables, all existing GCMs simulations with all RCMs, resulting in large RCMs/GCMs matrices. Studying those ensembles would be an important step towards reliable local information on climate change.

In what follows, I propose to come back on the choices I made to build the RCM emulator, the reasons behind them, and the essential results and questions that this work has raised. This final chapter comports six sections. The first one recalls the critical choices in defining the RCM emulator framework. Sections 6.2 to 6.5 answer
the three other questions about the efficiency, the trustability and the applicability
of the emulator that I have set at the end of the introductory chapter. In those four
sections, I also present some perspectives of improvement or development directly
linked to the issues that stay open regarding those specific aspects. Finally, the last
section presents some more general perspectives.

6.1 RCM-Emulator concept

This work's first part consisted of defining the appropriate framework to build the
emulator. The aim was first to identify and isolate the downscaling function of the
RCM and then find the most appropriate tool and dataset to learn it. I decided
to describe the RCM dynamical downscaling as a combination of two functions.
The RCM receives large scale inputs from the GCM that it firstly transforms,
creating its own large scale, and in a second time it downscales this *new* large
scale to create high-resolution fields. There is no reason to assume that the large
scale transformation is the same for any GCM, and results from Chapter 4 tends
to confirm it. Therefore, we need to focus only on the downscaling function to
have a universal emulator. This choice implies some important assumptions on the
causes of the large scales transformation that I will discuss later. I proposed to
use a perfect model framework with both large scale and local scale coming from
the same RCM simulation. The idea is to create, from the RCM simulation, a
low-resolution large scale looking like a GCM output and learn the transfer function
to the high-resolution surface field. This strategy ensures a perfect coherence
between the large scale low-resolution predictors and the high-resolution local scale
predictands.

To estimate this relationship, I used a fully convolutional neural network architec-
ture called UNet that I adapted to fit our regression task. The efficiency of CNNs in
dealing with large images made them the perfect candidate to build the emulator.
Indeed both inputs and outputs of the emulator are gridded objects that can be
considered as images. Moreover, their ability to deal with large dimensional objects
allows us to make a minimal prior selection of needed inputs and leave the network
finding the right combinations for each output grid point. Various studies have
already used convolutional neural networks in statistical downscaling (Vandal et al,
2019; Baño-Medina et al, 2020) or hybrid frameworks (Serifi et al, 2021) with success.

I defined a common set of inputs that I kept for the whole study. It mainly includes
classical atmospheric variables (temperature, geopotential height, humidity and
wind) at three pressure levels plus some surface variables that should be accurate
in the GCM and the aerosols as they play an important role at the regional scale
(Nabat et al, 2020). I also included external forcings such as the concentration of
greenhouse gases and a vector representing the seasonal cycle. The flexibility of
the neural network architecture allows separating the predictors' spatial structure
information and the temporal part so that the neural network considers them
equally before reconstructing the high-resolution field. The choice of the input set
could still be discussed and maybe optimised. In particular, the predictors come
from the RCM simulation during the emulator calibration. It is crucial to ensure
that the predictors do not include some RCM high resolution signal. Otherwise, the downscaling function we try to learn is truncated and thus cannot be well reproduced in the application world. It is why the low-resolution target variable is not included in the predictors set (cf chapter 3). It could be that such local information remains in the inputs set I used and further analysis to detect it is necessary. We have also seen that a RCM-emulator trained with a minimal set of predictor can provide reasonable results. Regarding an optimisation of the input set, a sensitivity analysis to identify the influence of each predictor is a necessary study that I did not have time to explore in this work.

Thus once this is set, this work aimed to assess if the RCM emulator was a suitable option to fill up RCM matrices. To do so, we need to give some guarantees regarding three main aspects: the efficiency, trustability and applicability. In what follow, I detail those three aspects.

## 6.2 Efficiency

Efficiency is the first reason behind this work and the statistical RCM emulator. Indeed the objective is to bypass the extremely high cost of regional climate models. In this manuscript, we have seen that the neural network proposed could be trained on GPU within only a few hours, even on a large domain. The production of a new high-resolution series given a low-resolution simulation takes less than a minute. The longest part is finally to get the input fields from the simulation to downscale and interpolate them to the emulator input grid. This would also take some hours and could easily be automatised (at least the interpolation part). Regarding this aspect, the efficiency of the emulator is satisfactory.

Nevertheless, efficiency can be considered under other facets not raised in this work. An important point is the easiness of use and application. A very practical point, once trained, the emulator can be stored in a `.h5`<fnref n="1" /> file of less than 2GB. So it is easy to transport and can be used on GPU and CPU. A second point is the easiness of application to a slightly different problem. We have seen in the manuscript that the emulator was used on different domains and variables. The code available on a public GitHub page has not been changed. Indeed, I created a python function that designs the neural network architecture according to the size of inputs and target. The choice of the target variable does not impact the code unless specific adaptations, such as the loss function in the precipitation case, for example. Moreover, during the time of my PhD, two intern students in other research centres deployed our methodology to the MAR RCM over Antarctica and for different variables. Emma Amblard<fnref n="2" /> emulated the surface temperature and precipitation, while Marijn Van der Meer<fnref n="3" />, worked with a very different variable, the Surface Mass Balance. Both internships got excellent results showing the emulator methodology and concept's good transferability and ease of use.

<footnote n="1">¹Hierarchical Data Format 5</footnote>

<footnote n="2">²Internship at the IPSL, supervised by Cecile Agosta, Patrick Gallinari, Christoph Kittel and myself</footnote>

<footnote n="3">³Internship at the TU-DELFT, supervised by Stef Lhermite</footnote>
Nonetheless, there is still space for improvement. First, regarding the UNet
architecture I built, there is probably a way to improve computational efficiency.
A costing part of the architecture is the bottom of the “U”. I chose to go down
to a 1-dimensional encoding vector in order to put on the same level the temporal
information, which is 1D, and the spatial information, originally 3-D, before the
reconstruction. It implies many layers and parameters to fit to reach the target
dimension. One could modify this neural network architecture, making the encoded
array larger and improving the emulator’s computational cost. Besides, it would
allow getting the best out of the UNet architecture. Indeed, some studies have
developed a dilated UNet allowing multiple input sizes thanks to its fully convolu-
tional aspect. Until now, once trained, the emulator focuses on a specific domain
and can be applied only to this domain. One could imagine a RCM-emulator for
the entire RCM domain and able to consider only a subdomain. The orography
and longitude/latitude coordinates of the target domain would be added to the
input list, as well as the coordinate map of the input domain. Each training sample
would represent a piece of the domain. In application mode, one could imagine a
very interactive tool located on a server close to the data centres. The user could
choose its target domain of interest, the adapted input domain, and easily produce
high-resolution emulated simulations. A more complete tool, could allow the user
to choose the RCM to downscale, the training set, the input list and the target
variable to build a new emulator. Such a tool implies a lot of development (and
human resources), but I believe that it is realistic.

Finally, simpler or more efficient architectures could also be used. I believe that this
work, and the other pioneer works of CNN application to downscaling, open the
door to constructing different emulators. The diversity of large emulated ensembles
will benefit the production of reliable regional climate information.

6.3 Trustability

The second key point was to assess the trustability of the RCM emulator. The
objective is to have a tool reproducing the downscaling function from a given RCM
to apply it to other GCM simulations. It is then of primary importance to assess
if the emulator well estimates the function. Because it is trained in the perfect
model framework, the emulator *lives* in the RCM world, it is then easy to evaluate
if this large scale/local scale relationship is correctly learned. I firstly used an RCM
simulation similar to the training one (i.e. driven by the same GCM and the closest,
more intermediate scenario). I recreated some low resolution from it to evaluate the
emulator in the perfect model framework and to have a perfect reference to compare
the emulated series. We saw that the emulator perfectly captures the atmospheric
circulation in both precipitation and temperature cases. The emulator is always
perfectly coherent with its driving large scale (i.e. the simulation it downscales) and
notably shows excellent temporal correlations with the reference series. Moreover,
we saw that it also captures the high-resolution spatial structure brought by the
RCM. The reliefs, coastal areas or islands have specific climates that the emulator
well reproduced. In this work, I took time to deeply evaluate the emulator under
various aspects, similarly as it is done for RCMs. It was essential to assess if the
production of the emulator is good enough to be used in impact studies. The main
conclusion regarding this aspect is that the emulator produces realistic maps and
series that are almost indistinguishable from the RCM ones. We also have seen
that the statistical properties from the emulated series were almost identical from
the original RCM series, with notably a complete reproduction of the probability
density function. We also looked at indices of extremes, climatology mean and
quantiles, and the changes in future with good reproduction of the emulator. We
also validated the neural network algorithm by comparing to standard statistical
downscaling approaches used as emulator. The UNet-emulator showed globally
better performances even if those strategies also had reasonable results.

Nevertheless, there is still some room for improvement. We notably have seen
that the emulator encounters difficulties reproducing some temperature extremes
and the precipitation objects tend to be too smooth. For the former, we identified
that an adapted loss function could help to reproduce better the extremes of
precipitation. A similar strategy could easily be used to constrain the emulator
to focus more on temperature extremes. A possible explanation could also be the
lack of memory in the downscaling, as the days are considered separately, and the
chronology comes only from the predictors. For example, the presence of snow
impacts the near-surface temperature. However, the emulator cannot see it as it
results from the previous days, and the emulator does not consider it. Similarly,
a deficit in soil moisture can reinforce the high temperature extremes, and the
emulator can not consider this aspect either. Therefore, it might be a good idea
to include some temporal link between emulator predictions. This could be done
using Recurrent Neural Network and Long Short Term memory networks.

The too smooth representation of rainfall objects is explainable by the determinism
of the neural network used here. Indeed the probability that rainfall occurs in
a grid point or next to it is often the same. So the network, by minimising the
error, puts rain in both grid points. Nevertheless, we saw that the emulated
series' statistical distribution is similar to the original RCM one, with a good
reproduction of dry days and heavy rainfalls. A deeper analysis, including an
impact study, might be necessary to see if it is a real issue. If it is, I can see at
least one option to tackle this issue. Generative Adversarial Networks are neural
networks that produce one possible outcome instead of producing what minimises
the error. Precipitation nowcasting studies based on GAN showed their ability to
produce realistic rainfall images with precise and sharp objects. They are probably
a good option for our case. As a brief reminder, GANs are composed of two
fighting networks: the Generator generates images, and the Discriminator learns to
differentiate the generated images from the true ones. In our case, the Generator
would be the adapted UNet. The Discriminator can be considered simply as a loss
function, telling the Generator if the image is realistic. I started to develop this as-
pect, but the results were too preliminary (and not good enough) to be included here.