Comparison of Least Squares Collocation and Gaussian Process Regression for Estimating Geoid Heights in the Khomas Region of Namibia from Trigonometric and DEM-Derived Heights

Abstract

This study evaluates geometric geoid modelling within the Khomas Region of Namibia using Least Squares Collocation (LSC) and Gaussian Process Regression (GPR) with three Digital Elevation Models (SRTM v3, ASTER GDEM v3, Copernicus DEM GLO-30) and three global gravimetric geoids (EGM96, EGM2008, XGM2019e). LSC integrates trigonometric and DEM-derived orthometric heights into a corrector surface, and GPR is trained on the LSC output to produce a datum-consistent geoid. Validation against 12 Fundamental Town Survey Marks shows that LSC achieves the closest agreement with Global Navigation Satellite System (GNSS)-derived ellipsoidal heights (Root Mean Square Error (RMSE) = 0.26 m; Mean Absolute Error (MAE) = 0.14 m). Among the gravimetric models, EGM96 ranks closest to LSC (RMSE = 0.27 m). SRTM and Copernicus DEM yield nearly identical values with LSC, while ASTER produces a lower mean due to a positive elevation bias. LSC-GPR consistency is confirmed across all DEM scenarios (RMSE 1.05-1.12 m; r > 0.98). A persistent ≈ 1.3 m offset between the geometric and gravimetric geoids indicates a long-wavelength datum discrepancy, while the wider geometric spread reflects terrain signals unresolved by the global gravimetric geoids. The study provides a regional benchmark for future alignment with the International Height Reference System/Frame (IHRS/IHRF).

Keywords:
Geoid Modelling; Least Squares Collocation; Gaussian Process Regression; Statistical Estimation Theory; Digital Elevation Models; Multi-DEM Analysis

1. Introduction

The geoid, defined as the equipotential surface of the Earth’s gravity field that best corresponds to mean sea level (MSL), serves as the fundamental reference for orthometric heights and is indispensable for surveying, engineering design, hydrology, and precise GNSS-based positioning (Albayrak et al., 2024; Odumosu et al., 2018; Moritz, 1980). Because the geoid cannot be observed directly, its determination requires the integration of heterogeneous information sources, such as terrestrial gravity observations, GNSS/levelling data, and global geopotential models (Heiskanen and Moritz, 1967; Hosseini-Asl et al., 2021). In many regions of Africa, including Khomas in Namibia, the lack of dense gravity measurements and the limited availability of rigorously adjusted levelling networks impose severe constraints on producing high-accuracy geoid models (Merry, 1983; Arabelos and Tziavos, 1991; Corner and Durrheim, 2018).

In central Namibia, the available vertical control is sparse and heterogeneous. Trigonometric height measurements offer localized, relatively high-quality reference values, though their historical metadata (particularly regarding gravity corrections, orthometric adjustments, and levelling compensations), are incomplete or unknown (Owolabi, 2025). Conversely, elevations obtained from publicly available Digital Elevation Models (DEMs) like the Shuttle Radar Topography Mission (SRTM; Farr et al., 2007), the Advanced Spaceborne Thermal Emission and Reflection Radiometer Global Digital Elevation Model (ASTER GDEM; Tachikawa et al., 2011), and the Copernicus Digital Elevation Model (CopDEM; European Space Agency, 2024) offer spatially continuous data across the whole area. Orthometric heights referenced to a global geopotential model (EGM96 for SRTM and ASTER, EGM2008 for the Copernicus DEM) are how these products are distributed, and factors such as terrain effects, sensor-specific biases, and the resolution of the underlying reference geoid limit their accuracy (Farr et al., 2007). An initial look at the current datasets shows that there are considerable differences between trigonometric heights and those derived from the DEM, with the trigonometric data spanning 2,229 m, the SRTM and CopDEM-equivalent surface 2,220 m, and ASTER 2,230 m (Table 2). The differences noted here make it necessary to use a careful integration method to unify these conflicting datasets into a consistent height reference surface (Scalco et al., 2018; Cakir, 2025).

Before pursuing regional refinement, it is important to examine whether modern gravimetric geoids already meet the vertical accuracy needs of the Khomas Region. Several global geoid models exist, but their inability to represent local gravity-field variations and terrain-related effects limits their suitability for precision applications. We therefore compare three gravimetric global models: EGM96 (Lemoine et al., 1998), EGM2008 (Pavlis et al., 2012), and XGM2019e (Zingerle et al., 2020), with local GNSS-derived geoid heights at 12 Fundamental Town Survey Marks (FTSMs) located across Windhoek in Khomas Region. These 12 were the points physically found for GNSS observations at the time of this study; hence, only 12 stations were used for validation. Nonetheless, the 12 stations used are well distributed across Windhoek (Figure 2c). The validation is limited to accessible stations within the Windhoek area during the self-funded GNSS observation campaign. A region-wide validation across the full spatial extent of Khomas (including the more remote Khomas Hochland and the southern periphery), would require dedicated field-survey funding and is identified as a priority for future work.

Direct comparison of the global geopotential models with local vertical control reveals large discrepancies at validation points, with RMSE values that are non-tolerable for modern vertical-datum applications. This motivates the development of a regionally fitted and locally validated geoid solution that aligns with the local vertical datum (Al Shouny et al., 2022). It is acknowledged that newer-generation models (the GOCO and XGM families and combined satellite-only solutions) exist; XGM2019e is selected here as a current high-degree combined model that is freely distributed via the International Centre for Global Earth Models (ICGEM).

It is important to emphasise from the outset that the geoid sought in this study is a geometric geoid, not a purely gravimetric one. The aim is to generate a geoid surface that is consistent with the existing local vertical reference frame, i.e., MSL-based heights, rather than to compute an independent gravity-based model of the geoid. Accordingly, the specific objectives of this study are to (i) investigate the optimal integration strategy for trigonometric and DEM-distributed heights within the LSC and GPR-based machine-learning (ML) framework; (ii) evaluate the sensitivity of the resulting geometric geoid to the choice of DEM (SRTM, ASTER GDEM, and Copernicus DEM); and (iii) evaluate the accuracy and reliability of the resulting geometric geoid model for the Khomas Region using GNSS-derived geoid heights at the 12 FTSM stations.

A major contribution of this work is the detailed statistical analysis of the misfit between three Global Geopotential Models (GGMs), EGM96, EGM2008, and XGM2019e, and Namibia’s local vertical control network. This comparison, expressed through mean offsets, standard deviations, RMSE, and Mean Absolute Error (MAE), reveals that the global gravimetric models fail to capture key local gravity-field characteristics. It also simultaneously establishes a benchmark against which the improvements from the regional refinement can be evaluated. Furthermore, by extending the analysis to three independent DEM sources, this study provides the first systematic multi-DEM evaluation of geometric geoid modelling in central Namibia, supporting the generalisability of the proposed LSC-GPR framework under varying DEM scenarios.

2. Conceptual Framework

The methodological framework adopted in this study consists of a multi-stage workflow designed to derive a geometric geoid model for the Khomas Region by optimally integrating trigonometric heights with DEM-distributed orthometric elevations. The general approach involves (i) data acquisition and pre-processing, (ii) formulation of geoid undulation differences relative to the long-wavelength reference geoid, (iii) estimation of the local geoid correction using LSC, with GPR subsequently trained on the LSC output to produce a smoothed geometric geoid, and (iv) validation using GNSS-derived ellipsoidal heights at independent FTSM control points. The methodology is summarised in the workflow diagram (Figure 1).

Figure 1:
Workflow for geometric geoid modelling - LSC and GPR with three DEM scenarios.

2.1 Least Squares Collocation (LSC) for Optimal Combination of Trigonometric and DEM Heights

Least Squares Collocation (LSC) provides a rigorous stochastic framework for combining heterogeneous geodetic observations with different accuracies, spatial resolutions, and error behaviours. In this study, LSC is used to estimate the local geoid correction ΔN, which, when added to the long-wavelength reference geoid N _ref ^, yields a geometric geoid $N\mathrm{}=\mathrm{}Nref\mathrm{}+\mathrm{}\Delta N$ that is consistent with the local Namibian vertical datum (Klees and Prutkin, 2010; Odumosu et al., 2021).

2.1.1 Geometric Model Formulation

Throughout this paper, the following symbols are used consistently. H denotes orthometric height; h denotes ellipsoidal height; N denotes the geometric geoid undulation that we seek to estimate; N _ref denotes the geoid undulation derived from the chosen reference Global Geopotential Model (EGM96 for Scenarios i and ii; EGM2008 for Scenario iii); $\mathrm{}\Delta N\mathrm{}=\mathrm{}N\mathrm{}-\mathrm{}{N}_{\mathrm{r}\mathrm{e}\mathrm{f}}$ denotes the corrector surface estimated by LSC; and N _LSC and N _GPR denote the geometric geoid surfaces produced by the LSC and GPR estimators, respectively. All N _ref values were synthesized in the tide-free permanent-tide system on the WGS84 ellipsoid using the official ICGEM service (https://icgem.gfz-potsdam.de), including the zero-degree term (ZDT). This standardisation ensures that all geoid surfaces compared in this paper share a single reference framework, such that the ≈ 1.3 m mean offset reported in Section 4.4 cannot be attributed to inconsistencies in the tide system, ellipsoid, or zero-degree term. Hence, it is most plausibly interpreted as a long-wavelength datum separation between the Namibian MSL-based vertical reference and the global tide-free geopotential reference (see Section 3.1). For the geometric geoid determination, the classical height relationship applies:

$h\mathrm{}-\mathrm{}H\mathrm{}=\mathrm{}N$ (1)

where h is the ellipsoidal height, H is the orthometric height, and N is the geoid undulation. Trigonometric heights, being tied to the local MSL framework, can be related to ellipsoidal heights as:

$h\mathrm{}-\mathrm{}{H}_{Trig}\mathrm{}-\mathrm{}N\mathrm{}=\mathrm{}{\mathrm{\epsilon }}_T$ (2a)

Similarly, DEM-distributed orthometric heights can be related to ellipsoidal heights by re-referencing through the appropriate reference geoid (EGM96 for SRTM and ASTER, EGM2008 for the Copernicus DEM):

$h-H_{DEM}-N_{ref}={\epsilon }_{DEM}$ (2b)

where:

H_Trig - orthometric heights from trigonometric observations;

ε_T - systematic distortions due to observational and computational errors in the trigonometric heights;

H_DEM - orthometric heights distributed by the DEM provider (SRTM, ASTER, or CopDEM);

N_ref - geoid undulation from the reference GGM used by the DEM (EGM96 or EGM2008 as applicable);

ε_DEM - errors due to the DEM elevation product.

Subtracting (2a) from (2b) yields:

$\left(H_{DEM}-H_{Trig}\right)+\left(N-N_{ref}\right)+\left({\epsilon }_{DEM}-{\epsilon }_T\right)=0$ (3)

Hence:

$\left(H_{DEM}-H_{Trig}\right)+\left(N-N_{ref}\right)+\left({\epsilon }_{DEM}-{\epsilon }_T\right)=0$ (4)

$\Delta N=N-N_{ref}=(H_{DEM}-H_{Trig})-({\epsilon }_{DEM}-{\epsilon }_T)$ (5)

LSC is then used to estimate the corrector surface (ΔN), and, by extension, N can be correctly recovered.

2.1.2 Stochastic Model and Covariance Modelling

Given the conventional ordinary least-squares assumption that the observed signals are (Guimarães et al., 2014):

$L=Ax+v$ (6)

where L is the observation vector (H _DEM − H _Trig ), A is the design matrix, x are the unknown parameters (systematic components), and v is the noise. LSC generalises ordinary least squares by incorporating a stochastic structure:

$x=C_{xs}·C_{ss}^{-1}·L$ (7)

where C _xs is the cross-covariance between unknowns and observations and C _ss is the signal covariance. A key requirement for LSC is an appropriate covariance function (Jarmolowski, 2016). The covariance function reflects the behavior of the anomalous fields (here, the differences between the trigonometric and DEM-distributed heights). In this study, we adopt the modified Gauss-Markov model:

$C\left(r\right)=\left[C_0\right(1+{\left(\frac{r}{a}\right)}^2)+b]·exp(-\frac{r-a}{b})$ (8)

where r is the horizontal distance between sample stations in metres, a is the correlation length, b is a curvature parameter for long-range behaviour, and C₀ is the variance. Hyper-parameter tuning is performed using Bayesian/Levenberg-Marquardt optimisation to ensure robust estimation under sparse data conditions.

2.1.3 Estimation of Geoid Corrections by LSC

The corrected geoid at a prediction location P is then given by:

$\Delta N_p=C_{xs}\bullet C_{ss}^{-1}\bullet L$ (9a)

$N_p=N_{ref,p}+\Delta N_p$ (9b)

This approach adapts naturally to the topography and enables a high-fidelity representation of local gravity-field variations that are not resolved in the global geoid models.

2.2 Gaussian Process Regression (GPR) for Geoid Undulation Estimation

To enhance the stability and coherence of the geometric geoid estimation, a secondary geoid surface was produced using GPR trained on the LSC-derived geoid heights. The goal was to obtain a smooth, non-parametric representation of the geometric geoid that retains datum consistency while reducing local noise (Kaloop et al., 2022). GPR is a non-parametric Bayesian Machine Learning (ML) technique well suited for spatial prediction problems. It offers flexibility as a non-parametric estimator, natural uncertainty quantification, and strong performance in modelling smooth and continuous geoid signals. However, GPR tends to over-smooth localized terrain variations, so its accuracy declines in rugged areas without kernel adaptation. Also, its computational demand increases substantially with dataset size (Konakoglu and Akar, 2021).

2.2.1 Conceptual Foundation

A Gaussian Process is defined as:

$\Delta N\left(x\right)~GP\left(m\left(x\right),k\left(x,x^{\text{'}}\right)\right)$ (10)

where m(x) is the mean function (often assumed zero for simplicity) and k(x, x’) is the covariance function (kernel) defining the spatial similarity between points x and x’. For the GPR, the input features (x) are the station coordinates (Easting, Northing) and the output (y) is the geoid correction ΔN.

2.2.2 Kernel Specification, Training, and Prediction

A Matérn (ν = 2.5) kernel was used in this study:

$k\left(x,x^{\text{'}}\right)={\sigma }_f^2\left[1+\sqrt{5}·\frac{‖x-x^{\text{'}}‖}{l}+\frac{{5‖x-x^{\text{'}}‖}^2}{3l^2}\right]·\exp \left[-\sqrt{5}·\frac{‖x-x^{\text{'}}‖}{l}\right]$ (11)

where ${}_f^2$ controls the vertical variability and l controls the spatial smoothness. The GPR model was trained using the station coordinates as input (Equation 12) and the LSC-derived geoid undulation as the target (Equation 13):

$X=(E_i,N_i)$ (12)

$y=\Delta N_{LSC}(E_i,N_i)$ (13)

$\Delta N_{GPR,i}=GPR(E_i,N_i)$ (14)

$N_{GPR}=N_{ref,i}+\Delta N_{GPR}$ (15)

where ΔN_LSC(E, N) is the LSC-derived geoid undulation at the Easting (E) and Northing (N).

2.3 Study Area

The Khomas Region of Namibia (Figure 2), encompassing approximately 36 964 km², features a diverse topography characterised by steep mountainous highlands rising over 2 000 m above sea level (Atlas of Namibia, 2025). In the central and southern areas of the region, the land is elevated and the geology is complex, leading to localized gravitational anomalies. In contrast to these mountain ranges, the Namib Desert along the western coast and the Kalahari Desert to the east feature relatively flat landscapes that affect the gravitational field in a different manner (Mendelsohn and Huntley, 2023).

The combination of this intricate landscape and diverse land uses, including urban development in Windhoek, agriculture, and conservation areas, makes geoid modelling particularly challenging. Due to the topographical variability in the region, the accuracy of geoid models is influenced, since standard global models might not account for local gravitational anomalies effectively (Abd-Elmotaal et al., 2020). To enhance geospatial data accuracy and aid sustainable regional development, it is essential to create a high-resolution, locally optimal geoid model that reflects the unique topography and land use of the Khomas Region.

Figure 2:
Study area: (a) continental context; (b) topography of Namibia with used benchmarks overlaid; (c) zoomed-in view of the Khomas Region showing the FTSMs (all in Windhoek).

3. Data and Methods

The primary datasets for this study comprise the three-dimensional coordinates of Namibia’s Trigonometric Beacons (TBs), Fundamental Town Survey Marks (FTSMs) and Town Survey Marks (TSMs) located within the Khomas Region (Figure 2b and c). A total of 1 004 TBs, 21 FTSMs, and 296 TSMs were obtained from the Office of the Surveyor-General, Department of Surveys and Mapping (DSM), Ministry of Lands and Resettlement (Government Gazette No. 4044, 2008). The coordinates were supplied in the Lo projection spanning across the country (Lo 22/13 - 22/21), with orthometric heights forming the third dimension.

Trigonometric heights in Namibia are traditionally referenced to the local MSL framework. The precision of the trigonometric heights depends on the original levelling and triangulation standards, and the dataset typically exhibits centimetre- to decimetre-level uncertainties (Owolabi, 2025). Detailed statistics and error estimates of all data used are provided in Table 1.

Table 1:
Data used in this study.

Data	Source	No. of points / resolution	Accuracy	Scenario
Trigonometric Beacons (TB)	Ministry of Lands (DSM)	1 004	Not specified	All
FTSM and TSM	Ministry of Lands (DSM)	317	Not specified	All
SRTM v3 (SRTMGL1)	USGS EarthExplorer	1 arc-second (30 m)	≥ 10 m	Scenario (i)
ASTER GDEM v3	NASA / METI	1 arc-second (30 m)	≈ 8-15 m	Scenario (ii)
Copernicus DEM GLO-30	ESA / DLR TanDEM-X	1 arc-second (30 m)	< 4 m (LE90)	Scenario (iii)
EGM96	icgem.gfz-potsdam.de	Degree/order 360 (~55 km)	≈ ± 50 cm	Ref (i, ii)
EGM2008	icgem.gfz-potsdam.de	Degree/order 2 190 (~9 km)	≈ ± 10 cm	Ref (iii)
XGM2019e	icgem.gfz-potsdam.de	Degree/order 5 399 (~4 km)	≈ ± 10 cm	Validation

Three Global Geopotential Models, EGM96, EGM2008, and XGM2019e, were employed for comparative assessment and external validation. Supporting data, including fully normalised spherical-harmonic coefficients and the associated Legendre functions, were obtained from each model’s official ICGEM distribution (Table 1).

3.1 DEM Height Derivation for the Three Scenarios

The three publicly distributed DEMs used in this study (SRTMGL1 v3, ASTER GDEM v3, and Copernicus DEM GLO-30) are all delivered as orthometric heights referenced to a global geopotential model: EGM96 for SRTM and ASTER, and EGM2008 for the Copernicus DEM (Farr et al., 2007; Tachikawa et al., 2011; European Space Agency, 2024). In our workflow, these distributed orthometric heights are used directly as H _DEM . Where ellipsoidal heights are required for the LSC formulation, they are recovered by inverse geoid re-referencing as $h\mathrm{}=\mathrm{}HDEM\mathrm{}+\mathrm{}{N}_{\mathrm{r}\mathrm{e}\mathrm{f}}$, where N _ref is the geoid undulation of the appropriate reference GGM. To evaluate the sensitivity of the LSC-GPR framework to the choice of elevation input, three scenarios were formulated.

Scenario (i) Trigonometric heights combined with SRTM heights using EGM96 as the reference geoid. The SRTM elevations represent radar-derived heights from the SIR-C/X-SAR sensors and are characterized by a global vertical absolute accuracy of ≤ 10 m at the 90 % confidence level. Regional studies report RMSE values between 3 and 5 m in parts of Africa with similar terrain (USGS, 2019). The LSC corrector ΔN is therefore computed relative to EGM96.

Scenario (ii) Trigonometric heights combined with ASTER GDEM v3 heights using EGM96 as the reference geoid. ASTER GDEM v3 uses optical stereo photogrammetry (VNIR band) rather than C-band radar. Published comparisons in semi-arid Africa report a systematic positive bias of 1.5-3 m over highlands due to vegetation/canopy penetration differences, with approximately 2.5 m of random noise (Cakir, 2025; Tachikawa et al., 2011).

Scenario (iii) Trigonometric heights combined with Copernicus DEM GLO-30 heights using EGM2008 as the reference geoid. The Copernicus DEM is derived from TanDEM-X interferometric SAR data and is natively distributed in EGM2008 (European Space Agency, 2024; Marešová et al., 2021). Because the corresponding native CopDEM tiles were not acquired for the full benchmark point set, CopDEM-equivalent orthometric heights were obtained by re-referencing the SRTM-derived ellipsoidal surface to EGM2008 in two steps: (1) $h\mathrm{}=\mathrm{}{H}_{\mathrm{S}\mathrm{R}\mathrm{T}\mathrm{M}}+\mathrm{}{N}_{\mathrm{E}\mathrm{G}\mathrm{M}96}EGM96$; then (2) $H_{\mathrm{C}\mathrm{o}\mathrm{p}\mathrm{D}\mathrm{E}\mathrm{M}}=\mathrm{}h\mathrm{}-\mathrm{}{N}_{\mathrm{E}\mathrm{G}\mathrm{M}2008}$.

By using the same SRTM-derived ellipsoidal surface in both Scenario (i) and Scenario (iii) and changing only the reference geoid, this construction isolates the effect of the reference-geoid choice on the resulting geometric geoid. Hence, any difference between the two LSC solutions is therefore attributable solely to the smooth, long-wavelength difference between the EGM96 and EGM2008 geoid models over the Khomas Region.

All N_ref values were synthesised using the official ICGEM service (https://icgem.gfz-potsdam.de) in the tide-free permanent-tide system on the WGS84 ellipsoid and including the zero-degree term (ZDT). The ZDT accounts for the difference between the GM and the reference radius of the ellipsoid and the corresponding parameters of the global model and is essential for absolute geoid-height comparison. All ellipsoidal heights from GNSS observations are likewise on WGS84 in the tide-free system. The Namibian orthometric heights are referenced to the local MSL framework. Therefore, the ≈ 1.3 m mean offset between the geometric and gravimetric geoids reported in Section 4.4 represents the long-wavelength offset between the local MSL-based datum and the global tide-free geopotential reference, including the ZDT.

3.2 Data Treatment (Pre-processing)

Pre-processing operations were performed on the collected data to ensure dataset consistency. This ensures that the various datasets refer to exactly the same spatial location. The basic operations were coordinate-projection transformations: (a) conversion from “Lo projected coordinates” to UTM zone 33S; (b) conversion between UTM zone 33S and latitude/longitude. After pre-processing, all datasets were converted to UTM-zone-33S projected coordinates for the LSC operation. (c) Care was also taken to ensure that the offset between the TB, FTSM, and TSM marks and the actual ground surface was appropriately accounted for. Preliminary statistics of the input height datasets are presented in Table 2.

Table 2:
Preliminary statistics of the input height datasets.

Statistic	Mean (m)	Std Dev (m)	Min (m)	Max (m)	Range (m)
HTrig	1 493.01	453.26	252.80	2 481.63	2 228.83
HSRTM	1 487.61	453.86	247.26	2 466.86	2 219.60
HSRTM − H	−5.39	7.79	−37.43	10.19	47.62
HASTER	1 489.11	454.54	243.18	2 473.62	2 230.44
HASTER − H	−3.90	8.21	−41.26	16.42	57.68
HCopDEM	1 487.66	453.86	246.81	2 467.41	2 220.60
HCopDEM − H	−5.34	7.80	−37.93	10.69	48.62

3.3 Description of Methodology

This study applied a multi-level approach to geoid determination in the Khomas Region by comparing LSC with GPR. For each of the three DEM scenarios, geoid heights were first estimated by integrating trigonometric levelling heights with the orthometric heights distributed by each DEM provider (SRTM v3 and ASTER GDEM v3 (referenced to EGM96) and CopDEM-equivalent heights (referenced to EGM2008), using Equations (4)-(5). LSC was then implemented following Moritz’s (1980) stochastic gravity-field framework, using empirically derived local covariance functions to account for spatial correlation and observation noise. LSC and GPR geoid surfaces were estimated on a 2 km grid covering the entire Khomas administrative boundary (9 232 grid nodes across approximately 36 964 km²), using 87 trigonometric benchmark stations within Khomas plus stations within a 50 km buffer for training.

The second approach employed Gaussian Process Regression (GPR) to estimate geoid heights from the LSC geoid, following the formulation in Equations (12)-(15). GPR was trained using 80 % of the data to interpolate geoid values on the same regular grid, while the remaining 20 % was reserved for validation and testing (Rasmussen and Williams, 2006). A trend-residual decomposition was employed to ensure stable prediction in areas with sparse station coverage. To do this, a second-order bivariate polynomial trend surface was fitted by ordinary least squares to the LSC-derived geoid heights at the 87 trigonometric stations. This surface is then subtracted to isolate the local residual signal. GPR with a Matérn (ν = 2.5) kernel was then fitted to these residuals, and the polynomial trend was restored after prediction to yield the final GPR geoid surface.

To evaluate the accuracy of the LSC- and GPR-derived geoid models, GNSS-levelling was carried out at the 12 FTSMs in the Khomas Region. These points constitute the part of the FTSM control network for the Region and were intentionally used for independent leave-out validation. Model performance was quantified using RMSE, MAE and bias to enable an objective assessment of how well each method represents the geoid in the Khomas Region relative to the established global models.

4. Results and Discussion

This section presents the geometric geoid surfaces (and the associated geoid grids) generated using LSC and GPR across three DEM scenarios (SRTM, ASTER GDEM, and the Copernicus DEM), together with three global gravimetric geoid models (EGM96, EGM2008, and XGM2019e). All models were evaluated over the Khomas administrative region. Statistical summaries, spatial patterns, and difference fields are used to interpret the behaviour of the geoid models and their implications for the local vertical datum. In what follows, the geometric geoid surfaces produced by LSC and GPR are evaluated and discussed in relation to the global geopotential models (Figure 3).

4.1 Regional Characteristics of the Geoid Models

Descriptive statistics of the geoid heights at the 87 Khomas stations (Table 3) reveal clear distinctions between the geometric models across the three DEM scenarios and among the gravimetric models. The LSC geoids display mean undulations of approximately 33.36 m (SRTM), 32.14 m (ASTER), and 33.26 m (CopDEM); the corresponding GPR geoids exhibit nearly identical means (33.38 m, 32.17 m, and 33.29 m respectively). This close agreement across all scenarios confirms that both estimators capture the same spatial variability regardless of the DEM input (Phinyo et al., 2021).

Table 3:
Descriptive statistics of the geoid models at the 87 Khomas trigonometric stations.

Model	Mean (m)	Std (m)	Min (m)	Max (m)	Range (m)	Scenario
LSC (SRTM)	33.362	5.516	25.496	53.301	27.804	(i)
GPR (SRTM)	33.380	4.912	26.553	50.317	23.764	(i)
LSC (ASTER)	32.136	5.816	22.386	56.335	33.949	(ii)
GPR (ASTER)	32.173	5.144	23.771	52.729	28.958	(ii)
LSC (CopDEM)	33.264	5.527	25.806	53.348	27.542	(iii)
GPR (CopDEM)	33.289	4.879	26.937	50.156	23.219	(iii)
EGM96	32.094	1.149	29.370	33.660	4.290	-
EGM2008	32.043	1.170	29.309	33.887	4.578	-
XGM2019e	32.009	1.105	29.360	34.073	4.712	-

By contrast, the gravimetric geoids cluster tightly around mean values of about 32.0-32.1 m with markedly lower variability (σ ≈ 1.1-1.2 m). The coherence among EGM96, EGM2008, and XGM2019e reflects their common long-wavelength gravity-field structure and globally referenced equipotential surfaces. The systematically higher means of the geometric geoids relative to the gravimetric models suggest a long-wavelength datum offset between the local MSL-derived height system and the global geopotential reference.

However, Table 3 also shows that the geometric-gravimetric differences are not spatially uniform. The minima of the geometric geoids are several metres lower than those of the gravimetric models, and their maxima are tens of metres higher. The asymmetry of the min/max differences therefore indicates that the discrepancy between the two model categories has two superposed components: (i) a long-wavelength systematic component, of the order of ≈ 1.3 m on the regional mean, which is consistent with a vertical-datum offset between the Namibian MSL-based local datum and the global tide-free geopotential reference (further discussed in Section 4.7); and (ii) a high-frequency, terrain-correlated component that produces the much wider spread of the geometric geoids (σ ≈ 5-6 m) relative to the gravimetric geoids (σ ≈ 1.1-1.2 m). The latter component reflects (a) the propagation of DEM-trigonometric height residuals into the LSC corrector surface, (b) short-wavelength gravity-field signals that the long-wavelength GGMs (especially EGM96, truncated at degree/order 360) cannot resolve, and (c) localised terrain effects in the Khomas Hochland that are absent from the gravimetric solutions. Section 4.4 quantifies these components, and Section 4.7 discusses their implications for the modernisation of the local vertical datum.

4.2 Internal Consistency: LSC vs GPR Across DEM Scenarios

The pairwise comparison statistics (Table 4) confirm a high internal consistency between the LSC and GPR geoids across all three DEM scenarios. The LSC-GPR RMSE values range from 1.05 m (SRTM and ASTER) to 1.12 m (CopDEM), with correlation coefficients exceeding r = 0.98 in all cases. This shows that the GPR trend-residual approach preserves the datum-consistent LSC signal while suppressing local noise.

Table 4:
Pairwise comparison statistics at the 87 Khomas stations.

Comparison	RMSE (m)	Mean (m)	Std (m)	MAE (m)	r
SRTM: LSC vs GPR	1.053	−0.019	1.053	0.714	0.986
ASTER: LSC vs GPR	1.047	−0.037	1.046	0.747	0.989
CopDEM: LSC vs GPR	1.119	−0.025	1.119	0.759	0.985
SRTM: LSC vs EGM96	6.354	1.267	6.227	4.668	−0.554
ASTER: LSC vs EGM96	6.488	0.042	6.488	5.038	−0.520
CopDEM: LSC vs EGM2008	6.358	1.221	6.240	4.682	−0.543
EGM96 vs XGM2019e	0.266	0.085	0.252	0.212	0.976
EGM2008 vs XGM2019e	0.178	0.034	0.174	0.150	0.990

Similarly, the gravimetric model category exhibits strong internal consistency. The EGM2008-XGM2019e comparison yields an RMSE of only 0.178 m (r = 0.990), while the EGM96-XGM2019e comparison shows slightly larger differences (RMSE = 0.266 m) but remains well correlated (r = 0.976). In contrast, comparisons between the LSC-derived geometric model and gravimetric models (EGM96 or EGM2008) yield RMSE values of approximately 6.3-6.5 m with weak, negative correlations, reflecting the combined effect of the long-wavelength datum offset and the terrain-correlated residuals discussed in Section 4.1. This is further confirmed in Figure 4, where the GPR-LSC difference fields are spatially smooth, with values of order ≈ 1 m, while the LSC-GGM difference fields range from approximately −5 to +12 m across the supported region of the Khomas validation network.

Figure 3:
Geoid undulation maps across the Khomas Region.

Figure 5 compares all models against XGM2019e. The geometric geoid models (LSC, GPR) exhibit large, spatially structured differences across the Khomas Region, with values ranging from approximately −8 m in the south-western interior to +20 m in the north-eastern highlands. The corresponding RMSE values at the 87 trigonometric stations range from 5.7 to 6.4 m (Table 4). EGM96 and EGM2008, by contrast, agree closely with XGM2019e with RMSE = 0.27 m and 0.18 m respectively, at the 87 stations. This reflects their shared long-wavelength gravimetric foundation.

Figure 4:
Pairwise difference fields across the Khomas Region.

Figure 5:
Differences of all models with respect to the XGM2019e reference geoid.

4.3 Sensitivity to the Choice of DEM

The multi-DEM analysis shows that the choice of DEM affects the absolute level of the resulting geometric geoid but does not affect its internal structure or LSC-GPR coherence. The SRTM and CopDEM scenarios yield nearly identical LSC geoids (mean ≈ 33.3 m at the Khomas stations). This outcome is expected since CopDEM-equivalent orthometric heights were derived from SRTM via the geoid re-referencing approach in Section 3.1. The near-identical results therefore confirm that the LSC-GPR framework is largely insensitive to the choice of reference geoid (EGM96 vs EGM2008) in the DEM height-conversion step. It should be noted, however, that the Copernicus DEM is an independent product derived from the TanDEM-X mission (2010-2015) using X-band SAR interferometry and referenced to EGM2008 (European Space Agency, 2024; Marešová et al., 2021); for production applications, direct extraction of CopDEM heights at benchmark stations is recommended, as this may reveal additional differences attributable to the independent sensor and temporal baseline.

The ASTER scenario shows a reduced mean geoid undulation (≈ 32.1 m, compared with ≈ 33.3 m for SRTM and CopDEM). This approximately 1.2 m shift is attributable to the systematic positive bias of ASTER GDEM relative to SRTM in semi-arid, highland terrain (Cakir, 2025; Tachikawa et al., 2011). Since ASTER GDEM uses optical stereo photogrammetry (VNIR band) while SRTM uses C-band radar, the canopy/vegetation penetration differences introduce a systematic elevation-dependent bias of 1.5-3 m over the highlands (Okolie and Smit, 2022). This positive ASTER bias partially compensates for the DEM-trigonometric height discrepancy, producing a geometric geoid that is closer in absolute value to the gravimetric models.

Despite the shift in absolute level, the ASTER scenario shows the same high internal consistency between LSC and GPR (RMSE = 1.047 m; r = 0.989) as the other scenarios, confirming that the LSC-GPR framework is DEM-agnostic in terms of its interpolation performance. The wider spread observed in the ASTER geoid (range = 33.95 m for LSC) compared with SRTM (27.80 m) and CopDEM (27.54 m) is consistent with the higher random error component reported for ASTER GDEM in semi-arid terrain (≈ 2.5 m, Tachikawa et al., 2011). This random component, in addition to the systematic positive bias discussed above, propagates from the DEM-trigonometric height residuals into the LSC corrector surface. Looking at the distributions in Figure 6, the geometric models (LSC, GPR) exhibit a wider spread than the gravimetric models. This reflects both the short-wavelength terrain-related signal absorbed by the geometric corrector and the propagation of DEM and trigonometric input-data residuals (Section 4.1), unlike the gravimetric models, which contain only the long-wavelength structure of their underlying spherical-harmonic expansions. Figure 7 confirms the high internal consistency of the geometric framework: the LSC-GPR correlations exceed r = 0.98 (R² > 0.96) across all three DEM scenarios, while the GGMs are likewise tightly inter-correlated with one another (e.g. EGM2008-XGM2019e R² = 0.98).

Figure 6:
Distribution of geoid undulations at the 87 Khomas stations. (a) Kernel density estimates; (b) violin/box plots.

Figure 7:
Pairwise scatter plots with linear regression across the Khomas stations. Dashed grey lines indicate the 1:1 relation; red lines are the linear regression fits.

4.4 Geoid Corrections (ΔN) Relative to the Reference Geoid

To explicitly quantify the datum relationship between the geometric solutions and the global geoid, geoid corrections were computed using Equations (9) and (15). The mean offset is consistent across the SRTM and CopDEM scenarios, where the geometric geoids (LSC and GPR) exceed the gravimetric geoids by ≈ 1.3 m (SRTM) and ≈ 1.2 m (CopDEM) across the study area. The much smaller mean offset in the ASTER scenario (≈ 0.04 m) occurs because the systematic positive DEM bias of ASTER absorbs most of the long-wavelength datum component.

As discussed in Section 4.1, the discrepancy between the geometric and gravimetric geoids comprises two components. The first is a long-wavelength datum offset between the local MSL-derived datum and the global geopotential (Merry et al, 2005; UN-GGIM, 2023), of approximately 1.3 m on the regional mean for the SRTM and CopDEM scenarios. In the ASTER scenario, this offset is reduced to ≈ 0.04 m because the systematic positive DEM bias of ASTER absorbs most of the datum component (Section 4.3). The second is a much larger regional-scale spatial signal that is not present in the gravimetric models. This signal accounts for the wider spread of the geometric solutions (σ ≈ 5-6 m for geometric versus σ ≈ 1.1-1.2 m for gravimetric models). The geometric geoids exceed the gravimetric ones in the elevated north-eastern Khomas Hochland and fall below them in the lower central-western interior (Figures 3 and 4). This pattern shows that the geometric framework absorbs terrain-correlated gravity-field signals and DEM-trigonometric height residuals that the long-wavelength GGMs cannot resolve (Wonnacott and Merry, 2011). The close similarity between ΔN_LSC and ΔN_GPR across all scenarios confirms that the smoothing applied in the GPR step does not distort the underlying datum relationship.

4.5 Profile and Residual Analysis

Smoothed Easting transects through the centre of Khomas (Figure 8) provide a one-dimensional view of geoid behaviour across the region. In all three DEM scenarios, the LSC and GPR profiles track each other closely, while the gravimetric models (EGM96, and XGM2019e) appear as smoother, systematically offset curves. This visualization confirms the long-wavelength datum separation between the two model categories and illustrates that the geometric models capture higher-frequency terrain-correlated undulations that the long-wavelength gravimetric models cannot resolve.

Figure 8:
Smoothed Easting transects through the middle of Khomas for the three DEM scenarios.

Figure 9 presents the pairwise difference distributions, overlaid with kernel-density estimates and fitted normal curves for all three DEM scenarios. The GPR-LSC internal differences (left column) exhibit near-zero means (μ = −0.01 to 0.04 m) and a narrow spread (σ ≈ 1.0-1.1 m), confirming high internal consistency regardless of DEM choice. The geometric-gravimetric comparisons (middle and right columns) show broader, positively skewed distributions with means of 1.2-1.3 m and σ ≈ 5-6 m, further reflecting the long-wavelength datum offset combined with the terrain-correlated component discussed in Section 4.4. Shapiro-Wilk statistics (W = 0.80-0.90; p < 0.001) reject normality in most comparisons, indicating that the residual structure is governed by spatially correlated, terrain-dependent signals rather than random noise (Wonnacott and Merry, 2011; Merry et al, 2005).

Figure 9:
Difference distributions with kernel density estimates and fitted normal curves.

4.6 Validation of Ellipsoidal Heights at FTSM Control Points

Validation was performed by converting the FTSM orthometric heights to ellipsoidal heights using each model’s geoid heights. Residuals were then computed as:

$\Delta h=h_{GNSS}-(H_{FTSM}+N_{model})$ (16)

Table 5:
Validation of models against GNSS-determined ellipsoidal heights at the 12 FTSM Control Stations.

Model	Mean Res. (m)	RMSE (m)	MAE (m)	Min (m)	Max (m)
Geometric - Scenario (i): SRTM
LSC_SRTM	−0.005	0.26	0.14	−0.39	0.38
GPR_SRTM	0.17	0.71	0.52	−1.05	1.44
Geometric - Scenario (ii): ASTER
LSC_ASTER	−0.02	0.28	0.16	−0.42	0.41
GPR_ASTER	0.14	0.68	0.49	−0.98	1.38
Model	Mean Res. (m)	RMSE (m)	MAE (m)	Min (m)	Max (m)
Geometric - Scenario (iii): CopDEM
LSC_CopDEM	−0.008	0.26	0.14	−0.40	0.38
GPR_CopDEM	0.16	0.70	0.51	−1.03	1.42
Gravimetric
EGM96	0.003	0.27	0.15	−0.40	0.38
EGM2008	−0.03	0.46	0.29	−0.83	0.89
XGM2019e	−0.02	0.46	0.29	−0.90	0.88

As shown in Table 5 and Figure 10, the LSC geometric geoids achieve the best agreement with GNSS-determined ellipsoidal heights across all three DEM scenarios, with RMSE values of 0.26-0.28 m and near-zero mean residuals across the 12-FTSM validation set. The SRTM and CopDEM scenarios yield virtually identical validation statistics (RMSE = 0.26 m), consistent with their near-identical geoid fields (Section 4.3); the ASTER scenario shows marginally degraded performance (RMSE = 0.28 m) attributable to the additional ≈ 2.5 m random noise propagated from the ASTER elevation product (Tachikawa et al., 2011).

Notably, EGM96 attains comparable accuracy to the LSC solutions (RMSE = 0.27 m; MAE = 0.15 m). This confirms that this gravimetric model remains well suited to the local vertical datum in Khomas. The higher-degree gravimetric models EGM2008 and XGM2019e perform less favourably at the FTSM stations (RMSE = 0.46 m) than the lower-degree EGM96. This is consistent with the fact that, in regions of sparse terrestrial gravity coverage such as central Namibia, the high-degree coefficients of EGM2008 and XGM2019e are themselves affected by commission errors that are not present in EGM96 (whose maximum degree of 360 truncates well below the affected band). It must also be acknowledged that the scarcity of gravimetric information affects EGM96 as well (through both unresolved short-wavelength signals (omission errors) and limitations in the underlying terrestrial data). Nevertheless, its lower maximum degree reduces the commission-error component in the validation residuals at the FTSM stations (Merry et al, 2005).

Figure 10:
Overlay plot of residuals across FTSM stations.

The GPR geoids, while preserving datum consistency with the LSC solutions (Section 4.2), exhibit the largest residuals (RMSE = 0.68-0.71 m), with station-specific deviations exceeding 1 m at the most rugged FTSM stations (Figure 10). This confirms that the Matérn (ν = 2.5) kernel smoothing, although effective for spatial prediction over the 2 km grid, reduces local fidelity at individual control points. This is a well-documented trade-off between interpolation smoothness and pointwise accuracy in GPR-based geoid modeling (Rasmussen and Williams, 2006).

4.7 Implications for the Vertical Datum and Geoid Modelling

The results reveal a systematic vertical-datum offset between the Namibian MSL-based system and the global geoid models at the long-wavelength scale. The LSC and GPR geometric methods successfully estimate and absorb this offset, producing geoid surfaces that are consistent both internally and with the GNSS/FTSM validation. The close agreement between the LSC and GPR surfaces across all three DEM scenarios demonstrates that the GPR-based geometric geoid model is robust and scientifically sound when trained on a datum-consistent signal.

The multi-DEM analysis confirms that the LSC-GPR framework is methodologically DEM-agnostic. The near-identical SRTM and CopDEM results follow from the derivation in Section 3.1, where CopDEM-equivalent heights were obtained by re-referencing SRTM via the difference between the EGM96 and EGM2008 reference geoids (Pavlis et al., 2012). The systematic ASTER shift reflects well-documented sensor biases: ASTER’s optical stereo retrieves the top of canopy while SRTM’s C-band radar partially penetrates it, yielding positive ASTER offsets in vegetated and highland terrain (Hirt et al., 2010; Tachikawa et al., 2011; Moudrý et al., 2019). These results show that the framework can accommodate different DEMs provided the appropriate reference geoid is specified, and that the choice of DEM primarily affects the absolute level of the geometric geoid rather than its internal structure or LSC-GPR coherence.

The estimated long-wavelength datum offset of ≈ 1.3 m between the local MSL-based vertical reference and the global tide-free geopotential reference is highly relevant for the modernisation of Namibia’s national vertical reference frame and its future alignment with the IHRS/IHRF (UN-GGIM, 2023; Mphuthi and Odera, 2025). Although GGMs are inherently limited by their spatial resolution and cannot fully represent local terrain effects, the application of rigorous terrain corrections within a future gravimetric solution is recommended once dense terrestrial gravity data become available for the Khomas Region. The use of high-resolution DEMs (Copernicus GLO-30 and, where available, sub-30 m national LiDAR products), will be essential for adequately representing the short-wavelength terrain effects that long-wavelength GGMs cannot resolve.

The geometric geoid models developed in this study demonstrate decimetre-level agreement with GNSS-derived ellipsoidal heights across Khomas. This level of performance shows that the method can act as a reliable surrogate in regions where dense GNSS-levelling infrastructure is unavailable.

5. Conclusion

This study has presented an integrated evaluation of geometric and gravimetric geoid models across the Khomas Region of Namibia, combining LSC, GPR, and three Global Geopotential Models (EGM96, EGM2008, and XGM2019e). The analysis was extended by using three DEM sources (SRTM v3, ASTER GDEM v3, and the Copernicus DEM GLO-30), thereby providing the first systematic multi-DEM evaluation of the LSC-GPR framework in central Namibia. This strengthens the generalisability of the approach in rugged regions with sparse GNSS-levelling data. By reconstructing the GPR geoid using a training signal derived directly from LSC, the study introduces a controlled machine-learning formulation that isolates methodological behaviour from data-induced distortions.

The resulting GPR-LSC geoid demonstrates near-perfect spatial coherence with the LSC solution across all three DEM scenarios (RMSE: 1.05-1.12 m; r > 0.98), preserving the regional geoid structure while smoothing the high-frequency artefacts associated with topography and DEM-related errors. This confirms that non-parametric ML algorithms, when anchored to reliable datum-consistent targets, are capable of reproducing complex regional geoid fields even in topographically challenging environments.

The comparison with GNSS-derived ellipsoidal heights at the FTSM control points further highlights the strengths and limitations of the modelling techniques. The LSC geoid remains the most accurate and internally consistent model, achieving residuals at the 2-3 dm level and therefore providing the most reliable geometric realisation of the local vertical datum. EGM96 outperforms the GPR-derived geoid in the validation, producing residuals comparable to LSC.

Consequently, the study establishes the following. First, the existence of a ≈ 1.3 m mean offset between all geometric models and the gravimetric geoids confirms a systematic long-wavelength discrepancy between the Namibian MSL-based height system and the global geopotential reference. Invariably, the wider spread of the geometric geoids reflects the short-wavelength, terrain-correlated component that the long-wavelength GGMs cannot resolve. This decomposition into long- and short-wavelength components is essential when interpreting the offset and is highly relevant to Namibia’s future alignment with the IHRS/IHRF. Second, the multi-DEM analysis demonstrates that the LSC-GPR framework is robust and DEM-agnostic. This is seen with the SRTM and CopDEM producing nearly identical results, while ASTER’s systematic positive bias shifts the geometric geoid closer to the gravimetric level without affecting internal LSC-GPR coherence. Third, the demonstrated performance of the LSC-GPR framework shows that geometric geoid modelling remains a viable, scientific approach in data-sparse and topographically complex regions. This is because it provides a credible alternative for medium-precision engineering, hydrological modelling, and spatial planning where centimetre-level accuracy is not essential.

Several priorities for future work emerge from this study. (i) Expansion of the validation network beyond Windhoek to broaden the geographic coverage of model assessment in Namibia. (ii) Direct extraction of native CopDEM GLO-30 orthometric heights at the benchmark stations for production-grade applications to allow detection of differences attributable to the X-band sensor and the CopDEM-specific processing chain. (iii) The application of rigorous terrain corrections within a future gravimetric solution should be pursued once dense terrestrial gravity data become available for the Khomas Region, supported by the use of high-resolution DEMs (Copernicus GLO-30 and sub-30 m national LiDAR products where available).

ACKNOWLEDGEMENT

The authors gratefully acknowledge the Office of the Surveyor-General, Department of Surveys and Mapping (DSM), Ministry of Lands and Resettlement, Namibia, for providing the trigonometric and town survey benchmark data used in this study. The GNSS field observations were conducted through a self-funded campaign, and the authors thank the surveying personnel who assisted during data collection. The Global Geopotential Models and Digital Elevation Models were obtained from the International Centre for Global Earth Models (ICGEM), USGS EarthExplorer, NASA/METI, and the European Space Agency.

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DATA AVAILABILITY

The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request. The Global Geopotential Models (EGM96, EGM2008, XGM2019e) and the SRTM, ASTER GDEM, and Copernicus DEM elevation products are freely available from their respective public repositories cited in the References section.

Edited by

Silvana Philippi Camboim and Ana Cristina Oliveira Cancoro de Matos.

Publication Dates

• Publication in this collection
  19 Oct 2026
• Date of issue
  2026

History

• Received
  08 Dec 2025
• Accepted
  29 June 2026