Keywords

1 Introduction

The availability of a high-accuracy and resolution gravimetric geoid model has gained increased importance as it is crucial for height and depth determination in a variety of applications such as construction, surveying, and geosciences while it is also essential for geophysical studies as it provides important information about the Earth’s gravity field. In the Kingdom of Saudi Arabia (KSA), extensive related research has been carried out during the last 15 years in an effort to compute a Kingdom-wide geoid modeling using mainly gravity data from the General Authority for Geospatial information (GEOSA) and Arabian American Oil Company (ARAMCO). KSA-Geoid2009 was a geoid model based on EGM08 (Pavlis et al. 2012) and fitted to GNSS/Levelling geoid heights of 5,405 (5,028 from ARAMCO and 377 from GDMS) BMs. The collocation was the computation technique. The accuracy (STD of residuals to BMs) is higher than 10 cm around the GPS/Levelling BMs and increases to 1.3 m, may be 2.0 m within areas having sparse distribution of BMs. KSA-Geoid2015 computed by CC Tscherning and R Forsberg, is a gravimetric geoid based on land, ship-borne, satellite altimetry gravity data and EGM08 and DIR-R5 (Bruinsma et al. 2013) GGMs (up to degree/order 720). The SRTM30 PLUS (30′ × 30′) digital terrain mode has been used to compute terrain effects. The final KSA2015 geoid was obtained after fitting the gravimetric geoid to 4,157 GNSS/Levelling BMs. KSA-Geoid 2015 refers to the (old) SV71, tied to the MSL in Jeddah 1969, datum. KSA-Geoid2017, computed by R Forsberg and M Ayan, was based on the same principles of the KSA-Geoid2015 encompassing two new major data sets in the south-west Red Sea coastal region along with some advancements in terms of the RTM computations and estimation of geoid-quasi geoid differences. For KSA-Geoid2017 the RCR was used, EIGEN6C4 (Foerste et al. 2014) as a reference field, new DTU15 satellite altimetry data and more than a half million gravity data points from both new (GEOSA) and older (ARAMCO) data sources. The geoid was fitted to the new Jeddah2014 VRF system through a set of 280 GPS/levelling points along the new GEOSA first order levelling network.

Over the last years KSA has invested significant resources and manpower towards collecting various types of gravity data, as well as GNSS/Levelling observations aiming at the development of a new high-accuracy gravimetric geoid. However, when estimating a gravimetric geoid, it is important to ensure that the input data are homogeneous in terms of the horizontal, vertical, and gravity reference systems. As it is customary in all related work, when a geoid model is to be determined, it relies on all available gravity data, i.e., data from both historical and modern campaigns. This work is divided into two sections. In the first one, the creation of an accurate, consistent and homogeneous gravity database for both land and marine areas over KSA, by selecting and merging all the available gravity data sets is described, while the second one refers to the determination of a gravimetric only geoid model for KSA using the aforementioned new gravity database and the remove-compute-restore (RCR) technique evaluating Stokes’ integral in the frequency domain with a 2D spherical Fast Fourier Transform (FFT) and the Wang-Gore modification.

2 Homogenization of Land and Marine Gravity Data

Initially, a pre-processing of the land and marine gravity data was carried out, following appropriate methodological and theoretical tools to achieve both the quality check and homogenization of the data regarding the geodetic reference system (GRS), the vertical reference datum and the tide conventions. Apart from the system homogenization, residuals of each dataset to GGMs have been evaluated to detect blunders, while a least-squares collocation (LSC) based blunder detection and removal procedure (Vergos et al. 2005) was carried out. The overall aim of this pre-processing analysis is the construction of a homogeneous and consistent gravity database, where the geodetic system (GRS) of all data will be GRS80 while the gravity reference system (GrGS) will be IGSN71.

In the frame of the geoid computation, all existing land and marine gravity data over KSA were collected along with the necessary information (format, reference system, defining standards, etc.) for each dataset. The land gravity data came from two major datasets, i.e., one from ARAMCO and another from the General Directorate for Military Survey (GDMS), with each one consisting of several independent gravity campaigns. First, the gravity data were cross-checked for the geodetic reference system (GRS), given that they can refer to GRS30, GRS67 or GRS80. As already mentioned, the entire geoid processing will be carried out in GRS80 and thus all the appropriate transformations were done. The second homogenization process referred to the transformation from the Potsdam to the IGSN71 gravity reference system. Since old gravity observations refer to the Potsdam system, rather than IGSN71, for common points in both systems relative transformation techniques were developed and applied. Finally, as some gravity observations where in the form of simple Bouguer anomalies, Bouguer anomaly (BA) corrections were computed and restored to form free-air gravity anomalies. The simple Bouguer plate correction as −0.0419 ρH was used with ρ being the average density and H the orthometric height of the station. As in most campaigns the used density value was not known, several tests with ρ 1 = 2.2 g·cm −3 , ρ 2 = 2.4 g·cm −3 and ρ 3 = 2.67 g·cm −3 have been performed. The so-formed free-air gravity anomalies were then tested against free-air gravity anomalies from XGM2019e and other local data to conclude on the density value that provided the closest, in terms of the std. of the differences, results. The same pre-processing strategy was followed for the marine gravity data that were divided into two datasets depending on the wider areas that they are located, i.e., the Arabian Gulf and the Red Sea. After the evaluation and validation of the data for the GRS, the vertical datum and the tide conventions used, a final homogeneous gravity database referring to GRS80 and IGSN71 was created. At the end, all gravity data (old and new) referred to the KSA Gravity Reference Frame (KSA-GrRF), defined by absolute gravity values at absolute gravity stations of the KSA Gravity Base Network (KSA-GBN) observed over the entire KSA territory. An additional transformation from IGSN71 to the KSA-GrRF has been conducted for all old gravity data and all gravity values utilized for gravimetric geoid computations are in one unified KSA-GrRF. Table 1 tabulates the data holdings for the land and marine gravity datasets available before and after the data homogenization, clean-up and removal of double entries.

Table 1 Land and marine gravity datasets available and data holdings
Full size table
Fig. 1
figure 1

Distribution of complete land (magenta), airborne (grey), altimetry (blue from SIO and red from DTU), shipborne marine (black) and fill-in (brown) gravity data for the KSA-Geoid21 project

Full size image

The method of spectral evaluation (Gruber et al. 2012; Vergos et al. 2014) of GGMs using GPS/Levelling data is a standard tool during the last decade to achieve a fast evaluation of the spectral contribution of GGMs w.r.t. in-situ data. In the frame of the KSA-Geoid2021, the latest GOCE-only, GRACE/GOCE and combined GGMs have been evaluated using EGM2008 as ground truth in order to evaluate which GOCE-based GGM and to what d/o provides the overall best improvement relative to EGM2008. Among the GGMs evaluated, XGM2019e (Zingerle et al. 2019) provided the overall best results and was used as the reference field for the determination of the KSA-Geoid21 gravimetric geoid. The evaluation of the land gravity data consisted of two steps. First, LSC was employed in a blunder detection and removal step, during which the land gravity data were splitted in two halves, using the one as ground truth and the other as observations. Then the test was repeated in the opposite way (see Tscherning 1991; Vergos et al. 2005). During that test, 70 points in the ARAMCO database have been identified as blunders and 18 points in the GDMS one. Finally, for the areas where the newly acquired airborne gravity data (SGL 2021) overlapped with historic gravity campaigns, a similar LSC-based scenario for blunder detection and removal has been followed, using the airborne gravity data as ground truth to evaluate the land observations. In both LSC-based tests, the comparisons are performed with residual free-air gravity anomalies, using XGM2019e (Zingerle et al. 2019) as the reference field and modelling the topographic effects with a residual terrain model (RTM) reduction based on high-order (up to d/o 90,000) effects (Rexer et al. 2016). After all these pre-processing steps, the common and homogeneous database contained a total number of 2,010,766 land, airborne and shipborne points to be used for the determination of the gravimetric geoid model for the Kingdom as depicted in Fig. 1. At sea, the available marine gravity data have been complemented by altimetry-derived gravity anomalies from DTU2018 (Andersen and Knudsen 2019), up to 20 km from the coastline, and SIO29.1 (Sandwell et al. 2014) for the rest of the marine areas. In the neighbouring countries were no gravity data were available, EGM2008 to its full d/o has been used as fill-in. Table 2 tabulates the statistics for the original, reduced and residual gravity data in the final database.

Table 2 Statistics of the original, reduced and residual gravity data in the Aramco database (565,752 point values), GDMS database (5,492 point values) in the shipborne marine database (245,813 point values) and in the altimetry database (771,500 point values). Unit: [mGal]
Full size table

3 Geoid Determination with the Remove-Compute-Restore Procedure

The practical determination of the gravimetric geoid was performed using the remove-compute-restore approach (RCR) in the frequency domain employing the FFT evaluation of Stokes’ kernel function and a Wong-Gore modification (Wong and Gore 1969; Sideris 2013). The latter was mandatory especially for the geoid modelling over the Kingdom, given the large extent of the study area in both latitude and longitude. The modification by Wong-Gore accounts for long-wavelength errors in the residual gravity anomalies, after the reduction of the original gravity data to a GGM and the removal of the topographic effects. Then we can determine residual geoid heights (Ν res) and restore the contribution of the GGM (N GGM) and the topography (N topo), to derive the final gravimetric geoid with RCR as:

$$ {N}_{grav}={N}_{res}+{N}_{GGM}+{N}_{topo}. \vspace*{.5pt} $$
(1)

The FFT evaluation was carried out with GravSoft’s spfour program, during which the number of reference parallels can be selected, as well as the modification of the Stokes kernel. For the number of the of reference parallels used four options were tested (1, 3, 6 and 9). For the Wong-Gore modification, which is performed for a specific d/o and then linearly tapered to another higher d/o, all pairs formed from d/o 60 to d/o 300 have been tested. Since FFT needs gridded residual gravity anomalies, the grid was generated based on the irregular residual gravity anomalies over the Kingdom and prediction on a grid with LSC. To evaluate the different gravimetric geoid models resulting from the combination of number of parallels and modification degrees, evaluation with a set of available, high-accuracy, GNSS/Leveling dataset by GEOSA was performed. The

Table 3 Statistics of the final gravimetric geoid, quasi-geoid, and their validation [m]
Full size table

best results were achieved with a Wong-Gore modification between d/o 80 and 100 and a multiband solution with 3 bands. Figure 3 depicts the final geoid height differences between the final gravimetric geoid and the GEOSA GNSS/Leveling geoid heights. In the same processing line, the quasi-geoid over the Kingdom has been determined and from that the geoid was once again estimated using the analytical evaluation of the quasi-geoid to geoid separation by Flury and Rummel (2009). Table 3 tabulates the final gravimetric geoid (see Fig. 2), the quasi-geoid, the difference between the gravimetric geoid and that determined from the quasi-geoid model. The latter has a std. of 1 mm only, showing the consistency of the processing steps followed. KSA-Geoid21GRA shows an absolute difference to the 3,522 GEOSA GNSS/Leveling at the 13.6 cm level (see Fig. 3), while the relative difference is at the 6 ppm for distances up to 10 km and 1–5 ppm over distances ranging from 10 to 2,000 km. These results are achieved before any deterministic and/or stochastic fit. Compared to the previous KSAGeoid2017 gravimetric geoid model, the refined KSAGeoid21 model shows an improvement by ~7 cm in terms of the std. to the GNSS/Levelling data, despite the fact that this is not directly comparable as KSAGeoid2017 was validated against a much smaller number of 287 BMs. The deterministic and stochastic treatment of the residuals to the GNSS/Leveling BMs are discussed in the development of the KSA-Geoid21 Hybrid model (KSA-GEOID21GEOSA).

Fig. 2
figure 2

The final gravimetric geoid model KSA-Geoid21GRA for KSA

Full size image
Fig. 3
figure 3

Geoid height differences between the final gravimetric geoid and the GEOSA GNSS/Leveling geoid heights

Full size image

4 Conclusions

The creation of an accurate, consistent, and homogeneous gravity database for both land and marine areas over the Kingdom of Saudi Arabia (KSA) has been outlined, followed by the determination of a gravimetric-only geoid model for KSA using the new gravity database. To construct the final gravity database, many pre-processing steps have been conducted to quality control the data and homogenize them in terms of the geodetic reference system, the vertical reference datum, and the tide conventions. The geoid prediction was carried out with an RCR approach and was based on an FFT evaluation of Stokes’ integral with a Wang-Gore modification. The validation of the developed gravimetric geoid over 3,522 GNSS/Leveling benchmarks resulted in external absolute accuracies at the 13.6 cm level and relative accuracies at the 1–5 ppm over distances ranging from 10 to 2,000 km.