Keywords

1 Introduction

The new high-accuracy and resolution gravimetric geoid model for KSA, KSA-GEOID21GRAV (Vergos et al. 2023), is a gravimetric geoid model evaluated employing the Remove Compute Restore (RCR) approach (Barzaghi et al. 2018), XGM2019e (Zingerle et al. 2019) as a reference field and residual terrain model topographic corrections from a global model (Rexer et al. 2018). It is based on more than 808,000 land and marine gravity data from two mainly sources (Arab American Petroleum Company – ARAMCO and General Authority for Surveying & Geospatial Information – GEOSA former GASGI), satellite altimetry data from the DTU18 model (Andersen and Knudsen 2019), and a new dataset of airborne gravity data covering almost the 68% of the KSA territory. All these data were pre-processed, evaluated and validated in terms of consistency with the IGSN71 as gravity reference system, KSA-GRF17 as geodetic reference frame, KSA-VRF14 as vertical reference frame and refer to the tide free system. A homogeneous database containing a total number of 2,010,766 land, airborne and shipborne gravity data has been used for the determination of the gravimetric geoid employing a classical FFT-based solution to evaluate Stokes’ kernel function and a Wong-Gore modification (Wong and Gore 1969; Sideris 2013). Its overall agreement with GNSS/Levelling data from GEOSA reached a standard deviation of 13.6 cm level and relative accuracies at the 1–5 ppm over distances ranging from 10 to 2,000 km. In the frame of the determination of the final Hybrid geoid model for KSA, a hybrid deterministic and stochastic approach is followed employing 3,522 GAGSI GNSS/Leveling benchmarks (BMs), so as to provide a geoid model (KSA-GEOID21GASGI) appropriate for surveying and engineering applications as best fit to the BMs.

2 KSA-Geoid21 Hybrid Geoid Modeling

Following the initial validation of the final gravimetric geoid model, the next stage was to use a deterministic parametric model to reduce and remove biases and trends in the gravimetric model, relative to the GNSS/Levelling geoid heights at selected BMs. The determination of the hybrid geoid is in essence a geometric fit of the gravimetric geoid to the available GNSS/Levelling data, hence a geoid solution that best fits the latter and provides small residuals. It should be mentioned that KSA uses a new geopotential-based Vertical Reference Frame which is called Jeddah2014 and is tied to epoch 2014.75. In this work, the determination of the hybrid geoid model for the KSA is based on the high-quality GPS/Levelling data from GEOSA (former GASGI) (3,522 BMs). For the deterministic part of the fit, simple north-south and east-west bias and tilt models have been tested, as well as the classical 4- and 5-parameters transformation models (Tziavos et al. 2012; Vergos et al. 2014). Nevertheless, despite the fact that their estimated parameters practically have no physical meaning, the selected parametric models refer to second and third order polynomial ones, as the goal was to minimize the residuals to the GNSS/Leveling data as much as possible and let the stochastic part of the transformation model treat the remaining, unbiased, residuals. The observation equation of the differences between the gravimetric and GNSS/Levelling geoid height in this parametric LSC is given as (Moritz 1980):

$$ {\ell}_i=\left({h}_i^{GPS}-{H}_i\right)-{N}_i^{grav}={N}_i^{GPS}-{N}_i^{grav}, $$
(1)

where ℓ i denotes the observation, \( {h}_i^{GPS} \) the ellipsoidal height, H i the orthometric height, \( {N}_i^{grav} \) the gravimetric geoid height and \( {N}_i^{GPS} \) the so-called GNSS geoid height. In matrix notation it becomes

$$ \boldsymbol{b}=\boldsymbol{Ax}+\boldsymbol{s}+\boldsymbol{v}, $$
(2)

where, A is the design matric, x is the matrix of the unknowns, s denotes stochastic signal, v denotes the errors of the observations b. With Eq. (2) we can easily treat first the deterministic part to first absorb any systematic differences between the various types of heights and then estimate the stochastic residual signal with least-squares collocation. The unknown deterministic parameters of the transformation model are determined as:

$$ \hat{\boldsymbol{x}}={\left({\boldsymbol{A}}^{\boldsymbol{T}}\boldsymbol{PA}\right)}^{-\textbf{1}}{\boldsymbol{A}}^{\boldsymbol{T}}\boldsymbol{Pb}, $$
(3)

where \( \hat{\boldsymbol{x}} \) denotes the adjusted unknowns and P is the weight matrix. In the next step, after the removal of the deterministic part, an appropriate covariance function is estimated and employing LSC the stochastic signal is estimated and the hybrid geoid heights are computed from the gravimetric geoid heights as a combination of stochastic and deterministic modeling:

$$ {N}_{Hybrid}={N}^{grav}+{a}_i^T\hat{x}+\hat{s}. $$
(4)

The deterministic part \( {\boldsymbol{a}}_{\boldsymbol{i}}^{\boldsymbol{T}}\boldsymbol{x} \) depends on the chosen parametric model and in the case of the second order polynomial model becomes (Kotsakis and Katsambalos 2010)

$$ \begin{array}{ll}{\boldsymbol{a}}_i^T\boldsymbol{x}&={x}_0+{x}_1{\left({\varphi}_i-{\varphi}_0\right)}^0{\left({\lambda}_i-{\lambda}_0\right)}^1{{cos}}^1{\varphi}_i\\&\quad +{x}_2{\left({\varphi}_i-{\varphi}_0\right)}^1{\left({\lambda}_i-{\lambda}_0\right)}^0{{cos}}^0{\varphi}_i\\&\quad +{x}_3{\left({\varphi}_i-{\varphi}_0\right)}^1{\left({\lambda}_i-{\lambda}_0\right)}^1{{cos}}^1{\varphi}_i\\&\quad +{x}_4{\left({\varphi}_i-{\varphi}_0\right)}^2\\&\quad +{x}_5{\left({\lambda}_i-{\lambda}_0\right)}^2 \end{array}$$
(5)

2.1 KSA-Geoid21GASGI Hybrid Geoid Determination

Table 1 Statistics of geoid height differences between the gravimetric geoid model and GNSS/levelling BMs. Units [m]
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Table 2 Relative differences with baseline length for the geoid model after the parametric fit. Units: [ppm]
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Fig. 1
figure 1

Empirical and analytical covariance functions of the adjusted residuals between KSA-Geoid21 gravimetric geoid and GEOSA GNSS/levelling geoid heights

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As already mentioned, the hybrid geoid is based on the KSA-Geoid21GRAV and a set of 3,522 GNSS/Levelling BMs over KSA. Before the practical determination of the transformation model a 3σ test has been performed to remove possible blunders in the BM database. During the 3σ test, 23 points have been removed so that after the 3σ test the std. of the differences between the gravimetric geoid and the GNSS/Levelling BMs reduced to 13.3 cm and the mean to −10.0 cm (see Table 1). In the practical evaluation of the various parametric models tested, their fit has been evaluated in terms of the std. after the fit, the system condition number and coefficient of determination. For the simple NS-tilt and WE-tilt models the std. is at 10.8 cm and 13.0 cm, respectively, for the 4- and 5-parameter Helmert transformation models it reaches the 10.5 cm and 10.4 cm, and for the second and third order polynomial the 9.7 cm and 8.6 cm. As the goal was to model with the deterministic part the residuals and provide a smooth signal for the prediction with LSC, it was decided to use the second order polynomial model to treat the deterministic part. It provided both a reasonable reduction of the std. (from 13.3 to 9.7 cm), an adjusted coefficient of determination at the 0.467 level and a condition number of the system of normal equations at the 109.196. Note that the third order polynomial model may provide a smaller std. but the condition number was at the 1.4 × 106 which shows that the model results in over-parametrization, hence it was deemed as not appropriate.

To validate the adjusted residuals after the second order polynomial fit, the absolute and relative differences between the gravimetric geoid heights and GPS/Levelling geoid heights have been computed. 98.9% of the differences are lower than the \( 2 cm\sqrt{dist(km)} \) error, 92.3% are lower than the \( 1 cm\sqrt{dist(km)} \) error and 71.4% are lower than the \( 0.5 cm\sqrt{dist(km)} \) error. These statistics show the significant improvement in the GRAV-Geoid21 with most of the baseline differences (92%) being below the 1 cm error, showing that there are only a few exceptions with errors larger than 1 cm Kingdom-wide. Table 2 summarizes the relative differences as a function of baseline length for the adjusted gravimetric geoid, where relative accuracies smaller than 1.9 ppm are found for distances larger than 10–20 km and for shorter baselines the relative accuracy is at the 7.5 ppm level.

The next step for the determination of the hybrid KSA-Geoid21GASGI model was the stochastic treatment of the adjusted, with the deterministic second order polynomial model, residuals of the gravimetric geoid model. An empirical covariance function of the stochastic signal to be modelled (see Fig. 1) was estimated and to that a Gauss-Markov analytical covariance function models have been fitted, so that the auto- and cross-covariance matrices needed for the prediction of the stochastic signal of the hybrid geoid model using LSC will be carried out. Following Eq. (4) the hybrid geoid is determined combining the estimated deterministic and stochastic modeled signals. Figure 2 presents the hybrid KSA-Geoid21GASGI model, which is to be used in accordance with the KSA VRF and GRF, while Fig. 3 depicts the hybrid geoid standard error. It provides a standard error of 0.199 cm (see Table 3) while its fit to the GASGI GPS/Levelling BMs has a zero mean a std. of 0.02 m.

Fig. 2
figure 2

The KSA-Geoid21GASGI hybrid geoid model

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Fig. 3
figure 3

The associated KSA-Geoid21GASGI hybrid geoid model standard error

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To evaluate the possible improvement of the new KSA-Geoid21GASGI hybrid geoid model over the previous model KSA-GEOID17, an extended set of 17,528 GNSS levelling dataset comprising of observations over BMs from GEOSA (former GASGI), ARAMCO and MOMRA (Ministry of Municipalities and Rural Affairs) has been used. Over these BMs we have evaluated the level of Improvement/Deterioration of the new geoid model compared to KSA-GEOID17 based on the absolute values of the residuals to the GNSS/Levelling geoid heights. Figure 4 summarizes the results of this analysis where it can be seen that for 75.2% of the BMs there is an improvement, with a mean value at the 1.1 cm level, while for the rest 24.8% there is a deterioration of the difference, with a mean at the 0.7 cm level. The main improvement is found over the south-eastern part of the Kingdom where the ARAMCO BMs are situated, reaching 83.8% of the BMs. For the MOMRA BMs improvement is found for 74.8% of the BMs, while for the GEOSA BMs there is a mean improvement of 0.4 cm for 43% of the BMs and a mean deterioration of 0.6 mm for 57% of the BMs. The reason that for the GEOSA (former GASGI) BMs the improvement is not a significant as for the other two datasets is that these BMs have been used in the development of the KSA-GEOID17 model, which is a hybrid one as well, so it is expected to fit well.

Table 3 Statistics of the final KSA-Geoid21GASGI hybrid geoid models, its errors and differences to GASGI GPS/levelling and GRAV-Geoid21
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Fig. 4
figure 4

Improvement/deterioration of the new KSA-Geoid21 GASGI hybrid geoid model compared to KSA-GEOID17 over the extended set of GNSS/levelling BMs

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A final evaluation test for the new hybrid geoid model was performed by acquiring new real time kinematic (RTK) data, both in network RTK and single-base modes depending on the network coverage, has been conducted. A total number of 149 BMs have been surveyed with the new hybrid geoid models providing residuals with a mean value of −2.3 cm and std. of 7.4 cm and the KSA-GEOID17 having a mean of −2.0 cm and a std. of 8.4 cm. The largest residuals are found, as expected, outside the coverage of the KSA positioning service, where network corrections in the form of a virtual reference station are not available and single-base RTK solutions are provided. Given that these results are achieved in RTK mode, hence the errors in ellipsoidal height determination are higher, the uniform quality of the hybrid KSA-Geoid21GASGI is confirmed.

3 Conclusions

In this work the estimation of the hybrid geoid model KSA-Geoid21GASGI is described. Based on the high-accuracy and resolution gravimetric geoid model for KSA, KSA-GEOID21, with external absolute accuracies at the 13.6 cm level, the hybrid KSA-Geoid21 model was estimated. This was based on a deterministic second-order polynomial parametric model to reduce and remove biases and trends in the gravimetric model relative to the GNSS/Levelling geoid heights followed by the estimation of the residual stochastic part with LSC. The hybrid geoid model reaches a standard error of 2.0 cm and relative accuracies of 1.9 ppm for distances larger than 10–20 km. Compared to the previous hybrid geoid model, KSA-GEOID17, it provides a mean improvement of 1.1 cm for 75.2% of the BMs and a mean deterioration of 0.7 cm for 24.8% of them. Finally, even in RTK mode, the hybrid geoid model gives a std. of 7.4 cm which is 1 cm better than that of the previous model.