2.2.3. Correction for the Refraction Bias

Geolocated signal photons located below the water surface are not corrected for the refraction effect that redirects the light, and thus the LiDAR beams. It induces a positioning bias for photons in the water layer that would impact the bathymetry estimate.

A relevant coordinate system is important to compute simple correction formulas. In this paper, the results presented were obtained using the coordinate system defined by [21] and the correction formulas were recomputed from this point. Corrections were applied in a satellite-centered coordinate system, where Z is the vertical direction (opposite to the direction of local gravity), and Y is orthogonal to Z (in the horizontal plane) and oriented along the azimuth of the pointing vector [21]. The resulting geometry is presented in Figure 3.

#### 2.2.4. Validation of ICESat-2 Seabed Ellipsoidal Heights

The Litto3D® dataset was used to validate the seabed photon ellipsoidal heights corrected from the refraction bias. While the ICESat-2 photons' geographic coordinates are projected onto a local tangent plane (ENU) during the refraction bias correction, the vertical references are different. Soundings measured by the SHOM bathymetric LiDAR are provided with orthometric heights. The latter were converted into ellipsoidal heights (relative to the IAG GRS 80 ellipsoid) using Circe 5.2.1 (IGN free software). Data visualization and extraction were conducted in GlobalMapper software, and the points were processed using Spyder 4.15 python interpreter (an open-source MIT environment) to qualify the accuracy. The point density from Litto3D® is considerably higher compared to the ICESat-2 dataset, it was interpolated to allow a comparison with the ICESat-2 dataset.

**Figure 3.** (**a**) Schematic diagram for the change of coordinate systems and the computation of the refraction bias in the horizontal plane; (**b**) in the vertical plane (figure adapted from [21]); and (**c**) correction formulas.
