Evaluation and homogenization of a marine gravity database from shipborne and satellite altimetry-derived gravity data over the coastal region of Nigeria

2.1 Altimetry derived-gravity data

In this study, we utilized marine satellite altimetry-derived gravity datasets, namely DTU21GRA [9] and SSv29.1 data [21], provided as grids with a 1-arc-minute resolution. EGM2008 [13] was employed as a reference field model to compute the DTU21GRA and SSv29.1. Among the DTU series models, DTU21GRA stands out for its focus on the near-coastal areas. Simultaneously, SSv29.1 distinguishes itself from other Sand well and Smith (SS) series models by incorporating two additional years of Sentinel-3A/B datasets in its computation. However, the critical difference between DTU21GRA and SSv29.1 lies in their choice of estimation algorithm. DTU21GRA incorporates residual sea surface heights (SSHs) into its estimation method [9] whereas SSv29.1 utilizes the residual slopes of the SSH obtained through mathematical differentiation of neighboring altimeter data [21].

The SSv29.1 gravity model can be downloaded from ftp://topex.ucsd.edu/pub/archive/grav/. The DTU21GRA gravity model was obtained from https://ftp.space.dtu.dk/pub/. The DTU21GRA, SSv29.1 gravity model and their differences are illustrated in Figure 1. The differences between DTU21GRA and SSv29.1 gravity values show a minimum, maximum, mean offset, and SD of −46.85, 79.93, 0.02 and 4.12 mGal, respectively. Larger differences were observed around coastal areas and islands, demonstrating the poor accuracy of satellite altimetry near coastal areas.

Figure 1:

Free-air gravity anomalies from the DTU21GRA model (a) SSv29.1 model (b) and the differences between DTU21GRA and SSv29.1 gravity values (c) in the coastal region of Nigeria.

2.2 Global geopotential models

The combined GGMs were generated by integrating satellite gravity measurements with terrestrial, airborne, shipborne gravity, and satellite altimetry observations. This integration results in a detailed gravity field with a refined spatial resolution. Five recently combined GGMs were used, as summarized in Table 1. These datasets were accessed using the International Center for Global Earth Models (ICGEM) Web Service [22]. The ICGEM calculation service was used to calculate free-air anomalies (FA) from the GGMs. The Geodetic Reference System 1980 (GRS80) ellipsoid was employed, and the zero-tide convention was consistently applied [22].

The combined GGMs used were SGG-UGM-2 [11], GOCE and EGM2008 Combination model (GECO) [23] and European Improved Gravity models (EIGEN-6C4) [24]. The surface gravity data utilized in the computation of the above-mentioned GGMs were the same as those utilized in EGM2008 [13] for the terrestrial regions. Marine gravity data obtained through satellite altimetry were used to compute SGG-UGM-2 and GECO, whereas DTU10GRA [25] satellite altimetry gravity data contributed to the computation of EIGEN-6C4 over the coastal region of Nigeria [25]. XGM2019e_2159 [12] used terrestrial gravity anomaly data obtained from the United States (US) National Geospatial-Intelligence Agency (NGA) over Nigeria’s Land and DTU13GRA [13] satellite altimetry data and other datasets over the coastal region of Nigeria. Finally, in the computation of EGM2008 [13], the authors employed terrestrial gravity anomalies over land and altimetry-derived gravity anomalies from the Danish National Space Center (DNSC07) [26] along the coastal region of Nigeria. Table 1 lists the characteristics of the five combined GGMs used in this study.

In Table 1, S represents satellite-based data (e.g., Gravity Field and Steady-State Ocean Circulation Explorer (GOCE)) mission [27] Laser Geodynamics Satellite (LAGEOS), and Gravity Recovery and Climate Experiment (GRACE) data [28], G represents ground data (e.g., terrestrial, shipborne, and airborne measurements), A represents altimetry, and T represents topographic-based data [22].

2.3 Shipborne gravity data

This study utilized shipborne gravity data from the BGI. The dataset comprised five marine surveys conducted on different ships [6], as shown in Table 2.

These surveys collectively covered a large part of the study area in the coastal region of Nigeria, with longitudes ranging from 2.5° E to 10° E, and latitudes ranging from 2.5° N to 6.5° N. A total of 2,462 data points were collected, with shipborne gravity anomalies ranging from −84.40 mGal to 131.81 mGal and an average value of 15.17 mGal. Figure 2 illustrates the spatial distribution of the shipborne gravity data, and Figure 3 illustrates the different survey legs using different colors. The distribution of the shipborne gravity dataset in the study area was not homogeneous for geoid modelling research, with a significant data gap. To ensure the reliability of our analysis, we considered the possibility of residual errors in the shipborne gravity data used in this study arising from various sources, such as instrumental drifts, navigational errors, discrepancies in ties to harbor base stations, and inconsistent use of reference systems [29], 35]. In addition, the accuracy of shipborne data is highly dependent on the quality of ship navigation. Hence, it is essential to thoroughly evaluate the consistency of these data and identify any outliers before integrating them with selected altimetry-derived gravity data.

Figure 2:

Distribution of the available shipborne gravity data within the coastal region of Nigeria.

Figure 3:

Geographical distribution of five surveys using different colours within the study area.

The cruises mentioned above (Table 2) were linked to the local network with absolute gravity points on land [36]. During the cruises, gravity data were collected using a GRAF-Askania GSS2-12, vibrating string accelerometer (VSA) sea gravimeter, and a continuous recording sea gravimeter, which were equipped with navigational instruments such as satellite/sextant, satellite navigation and gyroscope, and the US Transit satellite positioning system [29], 30], 32], 34].

It is important to highlight that there are other shipborne gravimetry campaigns performed in the coastal region of Nigeria by the United States Geological Survey (USGS) between July 22 and August 20, 1987, utilizing Lacoste Air-Sea gravimeter and Global Positioning System (GPS) technology in compliance with a request from the Defense Mapping Agency, encompassing the African coast, with the aim of augmenting gravity coverage in areas where it has been insufficient or inadequate. However, this dataset is not publicly accessible [37].

2.4 High-resolution terrain data

In this study, the General Bathymetric Chart of the Oceans (GEBCO) with a spatial resolution of 15 arcsec [38] was used to compute the topographic potential effect over the coastal region of Nigeria, which represents a very short-wavelength signal from the terrain data (Figure 4). A residual terrain model (RTM) [39] was computed using GEBCO data. The RTM computations within the ocean domain were executed using the TC toolbox in the GRAVSOFT package [40]. Given that bathymetry correction pertains to the disparities between ocean crust/sediment density and seawater density, we adopted a seawater density value of 1.028 g/cm³ and an ocean crust reference density of 2.9 g/cm³, resulting in a density difference of 1.872 g/cm³ [41]. We employed inner and outer integration radii of 10 km and 200 km, respectively. The RTM reduction utilizes two distinct bathymetry surfaces: a detailed surface and a reference surface over the data region. The effect of the detailed bathymetry surface was estimated from GEBCO 15 arc-second data [38]. A reference surface representing the mean elevation of the area was obtained by applying a suitable low-pass filter to the detailed surface. In our study, a smoothed grid of 5 min (a resolution corresponding to the gravity model SGG-UGM-2) was estimated from the detailed grid GEBCO 15 arc-seconds data, which we named GEBCO 5′ in this study [42]. The RTM results, utilizing the 1.872 g/cm³ density for the coastal region of Nigeria exhibit mean and SD of 0.46 mGal and 0.24 mGal respectively.

Figure 4:

High-resolution GEBCO 15 arc seconds model over the coastal region of Nigeria.

2.5 DTU21 MSS and CNES-CLS22 MDT

This study used the DTU21MSS model [8] because it used five years of Sentinel-3A data and Sentinel-3B data for two years. Furthermore, the SAMOSA+ physical re-tracker, an advanced waveform re-tracker, was utilized for pre-processing CryoSat-2 data to compute DTU21MSS. The selection of the SAMOSA+ re-tracker over others, such as the MWaP re-tracker or empirical retrackers, is justified by its generally lower root-mean-square error (RMSE) [43]. For validation purposes, the CNES-CLS22 MDT [44] was chosen because it uses the latest Gravity Observation Combination (GOCO06s) geoid model [45], which is based on the comprehensive reprocessing of the entire GOCE mission [27] and 14 years of GRACE data [28]. This model also incorporated the new CNES-CLS 2022 MSS [46]. Overall, the precision analysis of the CNES-CLS22 MSS showed a 40 % improvement compared with the 2015 model [46]. Additionally, in computing the CNES-CLS22 MDT, drifter and high-frequency radar data were processed to retain only the geostrophic components [44]. The CNES-CLS22 MDT and DTU21 MSS datasets are shown in Figure 5.

Figure 5:

DTU21MSS (left) and CNES-CLS22 MDT (right) over the coastal region of Nigeria.

3.1 Accuracy assessment of shipborne and altimetry-derived gravity data

In this section, we address the fact that the survey was conducted by different institutions (see Table 2) over various years, using different instruments. Consequently, linear drift and systematic bias must be corrected before validation. Additionally, these shipborne data contain several gross errors and outliers owing to varying measurement conditions. Therefore, it is necessary to eliminate these limitations based on preliminary evaluations using altimetry-derived gravity data (DTU21GRA and SSv29.1).

This assessment involves the adoption of a method similar to that outlined by Wessel and Watts [29], which addresses the inherent linear drifts within a shipborne gravity dataset by utilizing crossover (XO) points that commonly arise because of temporal variations in gravimeters, thereby introducing systematic errors in gravity measurements.

The conventional approach to mitigating this distortion involves connecting the gravimeter to the local network upon the conclusion of the cruise, followed by checking the differences between the meter readings and base station values. Any offset indicates potential linear drift.

To compute the drift error of a gravimeter utilizing XO points, the drift error was characterized as linear over time, and the drift factor was determined via a least-squares adjustment based on multiple XO points. A design matrix ‘A’ for the least-squares adjustment was formulated, where each row represents an XO point along with the measurement days and a constant term. The XO points between the track segments were calculated by computing the differences between consecutive gravity measurement points, and a linear interpolation was applied at the crossover points.

As previously discussed, it is necessary to remove the systematic bias in shipborne gravity measurements from the BGI before validation. This method involves editing these data by comparing them with altimetry-derived gravity data. We calculated the discrepancies between the altimetry-derived gravity data and shipborne gravity measurements using linear interpolation.

To mitigate this systematic error (mean offset), linear corrective models were applied to the residual [47] as expressed in Eq. (1). After eliminating linear drift and systematic bias, gross errors within each survey leg were identified and eliminated using rejection criteria. Specifically, data points showing residuals exceeding twice the average SD were identified as outliers and subsequently removed from the dataset.

The transformation models involve parameters related to the random noise term (vi), latitude (φi), and longitude (λi). The transformation parameters were ai = 0, 1, …, 2.

Finally, to ensure the reliability and accuracy of the remaining shipborne gravity data, we adopted a cross-validation (XV) approach, following the method described by Zaki et al. [48]. The XV method provides an unbiased statistical estimate of the error. This process involved excluding one observation at a time and utilizing each excluded point for validation. The remaining data were used for the interpolation. We used the 2-sigma approach as a rejection criterion, in which points exhibiting differences between the interpolated and observed values exceeding twice the SD of the residuals were removed as outliers. In this section, 436 data points were removed during the aforementioned process, representing 17.7 % of the entire dataset.

This study used the Kriging interpolation method for the XV technique because it is a widely employed interpolation method for detecting biased values [49].

3.2 Comparison of refined shipborne with predicted gravity models

The refined shipborne gravity measurements were compared with various estimated gravity models (DTU21GRA, SSv29.1, SGG-UGM-2, XGM2019e, GECO, EIGEN-6C4, and EGM2008) after eliminating the linear drift, systematic bias, and outliers. The discrepancies between the shipborne free-air anomaly (FA) and FA derived from various predicted gravity models were compared by interpolating the FA values from the various predicted gravity models into the positions of the shipborne FA measurements using linear interpolation. To identify the most suitable altimetry gravity model for integration, the model with the lowest SD and Root Mean Square (RMS) fit was selected.

3.3 Integrating the refined shipborne with DTU21GRA gravity data

This study focused on establishing a coherent gravity field covering the coastal region of Nigeria by integrating shipborne gravity data with selected altimetry-derived gravity data, primarily for geoid modelling purposes.

Following the refinement process of the shipborne marine gravity data, which entailed the removal of data points (as elaborated in Sections 3.2 and 4.2), we merged this refined shipborne gravity data with the selected altimetry-derived gravity data (DTU21GRA). The DTU21GRA 1′ × 1′ marine dataset was converted into point data, where each record at the grid node was treated as an individual data point, resulting in 3,371 records. The integration of refined shipborne data with the DTU21GRA [9] marine gravity anomaly grid was accomplished using the least squares collocation (LSC) technique, as reported by Kamto et al. [50], Zaki et al. [48], and El-Fiky [51]. At any arbitrary point P, the predicted value of the gravity anomaly, (S _p ), is given by (2).

where C _ll is the covariance matrix of the gravity measurements and C _pl denotes the cross-covariance vector between the estimated signal and measurements L.

First, we computed the residuals between the refined shipborne gravity data and DTU21GRA model. Gridding of the resultant residuals was then performed using the LSC method and the second-order Gauss–Markov covariance model used in LSC Eq. (3). The residuals were gridded onto a 1 arc-minute resolution, with the same resolution as the DTU21GRA grid. Consequently, a second-order Gauss–Markov covariance model with a 30-km correlation length and a white noise of 1 mGal was applied during the LSC after testing over a range of 10–50 km and 1–5 mGal [52].

where C₀ denotes the empirical covariance determined from the input data, α is the correlation length parameter, and S is the distance between the points under consideration. Finally, the grid of the residuals was added to the pre-gridded DTU21GRA [9] altimetry gravity anomaly values to obtain an enhanced altimetry dataset.

Validation was performed independently by examining the combined datasets at randomly selected scattered shipborne gravity stations, accounting for approximately 5 % of the data, which were not included in the LSC process. A total of 100 points were selected based on their geographical distributions in the northern, eastern, western, and southern regions of the study area. None of the 100 points selected for testing were considered outliers. This was because the test points were selected from the refined shipborne data and were not affected by outliers or gross errors.

3.4 The effect on geoid determinations over the coastal region of Nigeria

In this section, two gravimetric geoid models for the study area are generated: one uses refined shipborne data and the other uses the combined gravity dataset. The MDT was computed using the DTU21MSS model [8]. This was validated using CNES-CLS22 MDT [44]. The computed gravimetric geoid model adopted a methodology similar to that outlined by El-Fiky [51], and Moka et al. [7]. The approach for gravimetric geoid modelling is based on the Remove-Compute-Restore (RCR) technique, exemplified by Sanso and Sidera [53]; Hofmann-Wellnhof and Moritz [54], coupled with the RTM reduction method [39]. The RCR procedure was employed to compute the gravimetric geoid model for the coastal regions of Nigeria. Within this procedure, the components of the short and very short wavelengths of the functional of the disturbing potential were derived from the GGM and high-resolution GEBCO 15 arc seconds dataset [38], respectively. These effects were removed from shipborne gravity data and combined gravity data during the initial stage, resulting in residual gravity anomalies. Δg _res (see Eq. (4))

where Δg _FA represents the free-air shipborne gravity anomaly (both refined shipborne and combined gravity datasets), the long-to-short wavelength component of the gravity anomaly computed from GGMs is Δg _GGM , and Δg _RTM is the very short wavelength contribution of the gravity anomaly induced by local topography.

Before the marine geoid computation, the GGM was required as a reference model in the RCR procedures. Hence, selecting ideal GGMs is crucial for producing precise gravimetric geoid models. In this study, the SGG-UGM-2 [11] GGM at a spherical harmonic (SH) degree truncated up to degree and order (d/o) 360 was selected for the geoid computation to recover the long-to-short wavelength component of the gravity signal when modelling the gravimetric marine geoid, as it approximates the gravity field well over the coastal region of Nigeria (see Table 6 in Section 4.2).

The determination of the very short-wavelength component of the gravity signal is achieved through the RTM reduction method when using the shipborne free-air gravity anomaly for geoid determination. The gravimetric marine geoid model's computation involves integrating reference geoid heights from the GGM, residual geoid heights, and the very short-wavelength component of the geoid heights. This computational task was performed using the GRAVSOFT package [40].

4.1 Accuracy assessment of shipborne and altimetry-derived gravity data

In this section, the results based on the method described in Section 3.1 were presented. Our analysis revealed that after the residual linear drifts inherent within the shipborne gravity dataset using crossover adjustment for each leg of the surveys were addressed, survey 1 demonstrated a noticeable drift of −0.185 mGal/day. Likewise, Surveys 2, 3, 4, and 5 exhibited drift behaviors, with drift rates recorded as −0.0082, −0.020, −0.028, and 0.070 mGal/day, respectively (see Figure 7).

Figure 7:

Bar chart of estimated drift rate in mGal/day for each survey leg.

The statistical results of the differences between each survey leg and the altimetry-derived gravity data from the DTU21GRA [9] and SSv29.1 [21] models before and after addressing the residual linear drift and data editing are listed in Tables 3 and 4, respectively. The use of XO adjustment to address linear drift and data editing resulted in the subsequent elimination of linear drift, systematic errors (mean offset), and gross errors in each leg of the survey.

The mean offsets identified in Surveys 1, 3, and 5, initially measured at 14.16, 18.29, and 13.56 mGal for both altimetry gravity models (Table 3), have been eliminated (Table 4).

The large offsets observed in surveys 1, 3, and 5 (Table 3) may be attributed to incorrect ties to the land gravity datum or different land gravity datum [29].

This removal of the mean offset led to a reduction in the RMS values across all surveys (Table 4). Notably, the RMS values for Surveys 1, 3, and 5, which were originally around 14.59 mGal, 18.48 mGal, and 17.14 mGal for both models, have dropped to approximately 2.78 mGal, 2.32 mGal, and 4.67 mGal, respectively. These findings demonstrate that by addressing linear drift using XO adjustment and data editing, even a fifty-year old shipborne dataset can be corrected to attain an accuracy within a few mGal, rendering it suitable for combination with modern satellite data. Through this process, 358 shipborne gravity measurement points were eliminated from five survey legs. The remaining shipborne gravity dataset after removing the outliers, consisting of 2,104 points, demonstrates a mean and SD of −2.33, and 27.81 mGal, respectively. Consequently, all marine ship surveys are deemed acceptable and can be used for the next stage of refinement of shipborne gravity data using the XV techniques. Seventy-eight (78) shipborne gravity points with residual values greater than twice the SD were identified as gross errors, and outliers were removed after the XV method was applied. Following the XV procedure, 2026 refined shipborne gravity remained. Table 5 summarizes the statistical analyses conducted on the shipborne gravity data.

4.2 Comparison of refined shipborne with predicted gravity models

The evaluation results for the refined shipborne and gravity models are listed in Table 6. Among the gravity models, the DTU21GRA [9] gravity model demonstrated better agreement in fitting the shipborne gravity data, with the least SD and RMS differences of 3.14 and 3.17 mGal, respectively. On the other hand, the EIGEN-6C4 [24] model produced the largest SD and RMS values of 4.50 and 4.51 mGal, respectively. The findings of this study highlight the consistency of altimetry-derived gravity data with existing refined shipborne gravity data in the coastal region of Nigeria, which is attributed to the use of new satellite data and advanced data-processing techniques. We found that DTU21GRA [9] outperformed other models in the same region when compared with shipborne data.

Consequently, DTU21GRA [9] was chosen as the preferred altimetry-derived gravity model for further integration with refined shipborne gravity data for two reasons: the low SD value of 3.14, and the algorithm used to obtain gravity data from altimetric observations, that is, sea surface height (SSH). DTU21GRA [9] is the only gravity model that integrates Sentinel-3A/B SAR altimeter measurements in the coastal region of Nigeria. Abdallah et al. [55] obtained similar findings for the Red Sea when shipborne gravity data were compared with satellite altimetry-derived gravity data (DTU21GRA and SS v29.1). Their analysis revealed that the DTU21GRA model outperformed the SS v29.1 model, with SD and RMS values of 7.37 mGal and 8.73 mGal, respectively. In this study, when we compared our results with those of Abdallah et al. [55], our results showed an improvement of approximately 57.4 % in terms of SD and RMS, which is attributed to the processing strategy applied to the shipborne gravity data in our research.

4.3 Integrating the refined shipborne with DTU21GRA gravity data

In this section, the results based on the method described in Section 3.3 are presented. The refined shipborne data were integrated into the DTU21GRA [9] marine gravity anomaly grid by using the LSC technique. This analysis utilized 1926 records obtained by excluding 78 outliers identified through XV procedures and 100 points reserved for validation from the original dataset of 2,104 refined shipborne gravity (Table 5 and Figure 9). The statistical results of these data before and after the integration process reflect an improvement gained by employing LSC, as shown in Table 7. Where the SD dropped from 30.83 mGal to 26.84 mGal.

The free-air gravity anomalies from DTU21GRA [9] and the combined dataset with their differences are illustrated in Figure 8. The results of their differences exhibit minimum, maximum, mean, and SD values of −10.46, 10.64, 0.35 mGal, and 1.11 mGal, respectively.

Figure 8:

Free-air gravity anomalies from the DTU21GRA model (a) and combined gravity data (b) differences between the two models (c) along the coastal region of Nigeria: units [mGal].

The comparison between the selected 100 shipborne observations (Figure 9) and the DTU21GRA [9] before and after the LSC procedures demonstrated an improvement in the fit, as shown in Table 8. The SD values drop from 2.91 to 2.24 mGal. On the other hand, Table 9 shows the comparison between the complete refined shipborne gravity data and DTU21GRA [9] before and after the integration process.

Figure 9:

The distribution of 100 testing shipborne station units; [mGal].

Both the mean offset and the SD values decreased from 0.43 to −0.02 mGal and 3.14 to 2.69 mGal, respectively, which reveal an improvement in the final combined gravity data. This combination established a unified and consistent gravity field over the coastal region of Nigeria, ensuring the absence of data voids (Figure 8b).

4.4 The effect on geoid determinations over the coastal region of Nigeria

In this section, the results based on the method described in Section 3.4 are presented.

The marine geoid model computed using refined shipborne gravity data utilized the DTU21 MSS [8] to calculate the refined MDT. Similarly, the marine geoid model computed using the combined gravity datasets employed DTU21 MSS to calculate the combined MDT. Table 10 presents a comparison between the refined MDT and combined MDT against the CNES-CLS22 MDT [44]. Figure 10 illustrates the marine geoid computed using the combined dataset.

Figure 10:

Gravimetric geoid model computed with combined gravity dataset. Contour interval: 1 m.

Both the mean offset and SD values decreased from 0.080 to 0.076 mGal and from 0.023 to 0.016 m, respectively, indicating an improvement in SD when evaluated against the CLS22 MDT [44].

These outcomes affirm the efficacy of incorporating recently published satellite altimeter-derived gravity data (DTU21GRA) into the regional geoid modelling process, which is particularly beneficial in island regions, as pointed out by Wu et al. [56]. Additionally, including Sentinel-3A/3 B Synthetic Aperture Radar (SAR) altimetry data in the computation of DTU21GRA [9] further augments regional accuracy, thereby contributing to the observed improvements.