## 2 Methodology for C/V of Radar Altimeters
### Calculation of the altimeter Bias  
The calculation of the altimeter bias consisted of comparing the ellipsoidal height of the lake surface H, gained by combining the parameters distributed in the altimetry data (in green on Figure 3) with a combination of parameters collected during the cruises (in red on Figure 3), according to the experimental design. Water heights from ship cruise and from satellite were all calculated above the ellipsoid of reference.  
The water height above the ellipsoid was measured, instantaneously, both by the GPS and the altimeter in the middle of the track. Then we averaged the data 10 km before and after this point to determine the absolute bias, which was the averaged difference between both set of measurements. Since the geographical sampling of the measurements was different. In average, over the same distance we had two to three times more GPS points of measurements than in the altimetry ones. For each of the altimeter measurement, we selected the closest GPS measurement with a threshold distance of 300 m. Then, for each couple of measurements (altimetry, GPS) we calculated the height difference. In the vicinity of the lake's center, the slope of a geoid is very low, therefore, the induced error in the worst case remained within a few millimeters (see below for the so-called cross-track error). The average of all differences was then calculated, with the associated standard deviation, and it gave the value of the absolute bias and its precision. We chose to perform the calculation near the center of the lake mainly because it avoids the shoreline's effects on the altimeter waveform. It also allowed for diminishing the effect of a geoid slope on the distance between the GPS and the altimetry measurements (within 300 m threshold), and the cross-track shifting of the orbit described below.  
The height of the surface of Lake Issykkul varied significantly from the shore to the center of the Lake, and in the East to West direction. In particular, near the coast, the surface slope was steep (it might reach 10-15 cm/km [9]). Such slopes in height were not properly accounted for by any current geoid model and thus could not be cancelled by converting the ellipsoidal heights into orthometric heights. It was then mandatory that the ship collected water heights, exactly, at the nadir of the satellite passes. Yet, due to the cross-track orbital shift of 1 km, approximately, from cycle to cycle, which is inherent to any altimetry mission, it was impossible to predict, in advance, the exact position of the satellite along its theoretical track, at an accuracy greater than 1 km. Consequently, the shipborne measurements could be slightly shifted, with respect to the actual overpass. Yet, the slopes were minimum at the center of the lake, which corresponded to a minimum in the local geoid undulation. We, thus, limited this slope effect in the calculation of the absolute bias, by using only the measurements within \(\pm\)10 km, on both sides of this local minimum. We expected that the error due to this slope effect did not exceed 1 or 2 cm. By chance, during the last 3 campaigns analyzed here, the surveys were within 200-300 m away from the satellite tracks. The resulting potential error in the area where the biases were calculated, consequently, did not exceed a few millimeters.