# Shipborne GNSS-Determined Sea Surface Heights Using Geoid Model and Realistic Dynamic Topography
## 5 Derivation and Validation of Shipborne GNSS-Based Sea Surface Heights
### Validation and Least-Squares Adjustment of the Results  
Ideally, the filtered and corrected residuals should be near-zero. However, such idealistic results cannot be expected, due to errors in the total station, tape, GNSS, and TG measurements; estimated corrections' inaccuracies; and deficiencies in the used VLM, HDM, and geoid models. For instance, an examination of the residuals' statistical properties (Figure 14a) shows biases between the results of different campaigns (notice mean values). A likely explanation is errors in tape measurements conducted in the harbor, for reducing surveys' GNSS results to the sea surface (cf. Equation (8)). Since such measurements are subjective (the surveyor had to assess the optimal measure from the moving sea surface) and the vessel swayed slightly during the measurements, the tape-determined values likely induce the most significant errors. Figure 13: Empirically estimated (**a**) squat and (**b**) static draft corrections. The residuals' standard deviation estimates vary between 1.8 cm and 5.5 cm (Figure 14a), demonstrating, in general, the satisfactory performance of the results, especially considering the variety of different datasets combined. The highest standard deviation estimate describes the C4 campaign, which is expected, as only two GNSS antennas (instead of four) could be used for the \(iSSH_{CoM}^{Final}\) solution (cf. Section 5.1). Thus, the vessel's high-frequency attitude changes likely contaminated the solution. On the other hand, although the second half of campaign C1 was conducted in windy and wavy conditions (due to a rising storm; recall the discussion in Section 4.1), the resulting standard deviation estimate was only 2.8 cm, suggesting that the approach used could successfully eliminate the vessel's high-frequency attitude changes and marine conditions' influence on GNSS measurements. Another principal measure is the consistency of the residuals, which were, hence, compared at the campaign-internal intersections, by defining two criteria: (_i_) distance between points less than 250 m, and (_ii_) time between intersections (also consecutive points) more than 30 min. The two criteria had to be true at the same time. According to the first criterion, nearby parallel routes could also yield an intersection (geoidal heights and DT remain relatively unchanged over short distances), to avoid a lack of detected intersections when no actual route crossings existed. Moreover, if a vessel stopped for more than 30 min (e.g., to collect water samples) and did not drift significantly (more than 250 m), the stopping was counted as an intersection (the low-pass filtering influence disappears with 30 min: 53 measurements \(\times\) 30 s = 26.5 min). The resulting statistics (Figure 14b) indicate a good consistency, with generally near-zero mean differences, and standard deviation estimates varying between 1.8 cm and 3.4 cm. Notably, the NKG2015 geoid model errors do not affect these comparisons, since the model is static (i.e., not a time-dependent variable). The defined criteria were further employed to compare residuals of different campaigns at their intersections, following the principles described above. Notice in Figure 15 that the absolute values of mean differences (sign difference is due to the subtraction order) and standard deviation estimates differ slightly, for example, if campaign C1 is compared to C2 and vice-versa. Since the algorithm searches intersections by moving along the validated campaign, the two tests may yield different points for comparisons, due to the second intersection search criterion (i.e., the algorithm moves along different routes, detecting different points based on the definitions). Generally, these assessments show a good agreement between the residuals of different campaigns, in terms of standard  
Figure 14: Statistical properties of (**a**) filtered and corrected residuals and (**b**) their differences at campaign–internal intersections. The black lines are mean values, and colored bars denote standard deviation estimates.