## 2 Materials and Methods
### Methods  
#### 2.3.1 Evaluation of Global DEMs for Costa Rica  
Nine DEMs were evaluated using: (1) the LVIS mean elevation of the lowest detected mode within the waveform (ZG) and (2) the levelling dataset. First, these two methods were compared (results are presented in Supplementary S1). For each LVIS point, the footprint diameter is about 20 m. We set several thresholds, 20 m, as well as 5, 10 and 30 m, to search for point pairs between the levelling and LVIS datasets. For such comparisons, both datasets should refer to a common reference surface. Orthometric heights of the levelling stations were transformed to ellipsoidal heights by adding EGM2008 geoid undulations, computed using the Geopot\(\vec{U}\) software [35]. For the first DEM evaluation approach, it is necessary to convert LVIS data from 1998, 2005, and 2019 provided in terms of ellipsoidal heights above the WGS84 ellipsoid to physical heights above EGM96. Only for the TanDEM-X DEM is no conversion required. The transformation was performed by subtraction of EGM96 geoid undulations computed with the National Geospatial-Intelligence Agency (NGA) interpolation software _intrt.f_ from the binary file of the 15\({}^{\prime}\) mesh size worldwide EGM96 geoid heights _WW15 MGH.DAC_[36]. For the second method, further consideration is needed. DEMs are generally provided in terms of physical heights above the EGM96 geoid, as an approximation for mean sea level. In general, most national height systems refer to an adjusted mean sea level determined by several years of tide gauge measurements. In this way, the physical heights of levelling points correspond to a reference surface that is usually different from any geoid model. Furthermore, the EGM96 geoid model is a low-resolution Earth Gravity Model (EGM), and substitution of the reference surface for a higher-resolution geoid model, such as EGM2008, or a local geoid model is desirable for all DEM users. In the case of EGMs, \"resolution\" does not correspond to the mesh grid size, but to the wavelength of the features of the gravity field that the models can represent, which is about 9 km for EGM2008 and about 56 km for EGM96. This is very different from the mesh size of the gridded EGMs provided by NGA (e.g., 1\({}^{\prime}\) or 15\({}^{\prime}\)). Furthermore, EGM models are coefficients of spherical harmonics that are used for a chosen ellipsoid and gravitational parameter \(GM\) to calculate various parameters of the gravity field. For example, if one uses the WGS84 ellipsoid parameters or GRS80 ellipsoid parameters in a spherical harmonic synthesis, the difference in geoid undulations would be 0.93 m. The article \"There is no such thing as \"The\" EGM96 geoid: Subtle points on the use of a global geopotential model\" by [PERSON] 1998 [35] gives an insight to the scientific community into this issue. In mountainous areas of Costa Rica, the difference between the EGM96 and EGM2008 geoid models reaches a significant value of 2.27 m, as can be seen in Figure 6. For the evaluation of DEMs with the second validation dataset, we computed the difference between EGM2008 and EGM96 at the 2233 levelling stations locations so that their height would refer to EGM96 as well. Figure 6: Comparison of gridded geoid undulations (N) for the study area: (**a**) EGM2008. (**b**) EMG96. (**c**) Difference between EGM2008 and EGM96: the mean/min/max difference is \(0.2/-0.9/2.3\) m; the Standard Deviation (STD) is 0.45 m.  
As seven out of nine evaluated DEMs are DSMs and include VB, the differences, especially for the LVIS dataset, do not follow a normal distribution. The shapes of the histograms of the differences appear to be a sum of two distributions, a bimodal distribution, with one peak corresponding to a vegetated case and the other to a bare-earth case. The statistical value commonly used to describe uncertainty, the Standard Deviation (STD), should be used only for normal probability distributions. When applied to other types of distributions, it provides misleading information.