# Centimeter Precision Geoid Model for Jeddah Region (Saudi Arabia)

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{FA}) using the GGM signal (Δg

_{GG}

_{M}). In this long-wavelength component, the low-frequency gravity signal due to the topographic masses is also taken into account. The higher frequency of the topographic effect is then modelled and removed from the data by computing the Residual Terrain Correction (Δg

_{RTC}) ([11,12]). At the end of the remove step, the so-called residual gravity anomalies ∆g

_{res}are obtained according to Equation (1)

_{res}) by applying, e.g., least squares collocation [13].

_{GGM}) coming from the GGM and the topographic high-frequency components (ζ

_{RTC}) of the height anomaly are then computed and added to the residual height anomaly (ζ), to obtain the gravimetric model of the height anomaly ζ in the restore phase.

_{B}is the Bouguer anomaly in gal and H is the orthometric height in km. This is an approximate formula, the validity of which has been discussed in [17]. However, given that the area under investigation is quite flat, formula (2) should be accurate enough for the purpose.

## 3. Data Sets

#### 3.1. GPS/Leveling Data

_{GPS/lev}. Moreover, this set of data can be used in the so-called hybrid geoid undulation estimate, where the gravity and the GPS/levelling data are used together to give a combined estimate of the geoid undulation.

#### 3.2. The Gravity Data Set

_{obs}have been reduced to free-air gravity anomalies, computing the normal gravity γ according to the Geodetic Reference System 1980 (GRS80) [18] using the following formula

_{FA}(P) = g

_{obs}(P) – γ(Q)

_{res}), which is about 7.7 km (0.07°), subsets of the gravity data were selected and the related statistics were computed, block by block, as the standard deviation σ. The measured gravity values that were out of the 2σ interval centered in the block mean have been marked as possible outliers (Δg

_{res}*). To confirm that these selected values were outliers, Student’s t-inference test at a significance level of 5% was performed. The hypothesis ${H}_{0}:\Delta {\widehat{g}}_{res}=\Delta {g}_{res}^{*}$ was tested, where $\Delta {\widehat{g}}_{res}$ represents the predicted value obtained using a least squares collocation interpolator based on the data included in a 0.07° cap centered in each investigated gravity point. If the test fails, the selected gravity value is confirmed as an outlier. In the collocation procedure, the following covariance function (Eq. 4) has been used:

_{0}is the variance of the data included in the current cap and α is the correlation length.

#### 3.3. The Global Geopotential Model

#### 3.4. Digital Terrain Model

^{−3}and 1030 kg m

^{−3}for the land and for the water, respectively. The first surface is the topographic surface, represented by the detailed DTM in Figure 2. The reference DTM is smoother than the first ones and should be correlated to the topography effect that is already taken into account by the used GGM (see Table 2). This surface is not known a priori and it depends on the maximum degree of the spherical harmonic expansion of the applied GGM. This smoother surface can be statistically identified by testing different smoothed versions of the available detailed DTM, computed by a moving average procedure based on different window sizes (in this work, from 5’ to 50’). The statistics of the residual gravity anomalies (Δg

_{res}) are then used in the choice of the best window size for the DTM averaging procedure. The smoothed reference DTM that gives the lowest values of the mean and standard deviation of the residual gravity anomalies is the proper reference DTM to be considered in the RTC computation. As the average DTM is correlated to the geopotential model used to compute the residuals, for each considered GGM, an analysis of the size of the averaging window has been performed. By using this approach, we optimized both the d/o of each considered GGM and the reference DTM to be used in the RCR procedure (Table 2).

## 4. Result and Discussion

#### 4.1. The Remove–Compute–Restore Results

_{res}show the different features of the gravity residuals in relation to the d/o of the used GGM. This can also be seen in the structure of the empirical auto-covariance functions that are described herein.

_{res}was performed using the least squares collocation (LSC) method ([13,14]). The LSC method is based on the estimate of the empirical auto-covariance function (ACF) of the residual gravity anomalies and its interpolation with a positive definite function, called the covariance function model ([30,31,32]).

_{res}(Table 4), estimated by LSC, also reflects the different behaviors based on the adopted GGM. The solutions based on the low d/o GGMs (up to 250) have height anomaly values that can reach 1 m with a mean value of about 30 cm or more. On the contrary, the height anomaly values ζ

_{res}estimated from the high-resolution GGMs (d/o 1800) have mean values of about 8 cm and maximum and minimum values smaller than 25 cm in absolute value. The solutions based on GOCO05c and XGM2016 have intermediate statistics with respect to the ones described above.

_{GGM}) coming from the different GGMs considered in this study and the topographic short-wavelength components ($\zeta $

_{RTC}) of the height anomaly were then computed and added to the residual height anomaly ($\zeta $), obtained by LSC.

#### 4.2. Validation of the Gravimetric Geoid Model

_{GPS}, that is N

_{GPS/lev}= h

_{GPS}–H. In Table 5, statistics of the differences between the gravimetric geoid undulations and N

_{GPS/lev}are reported. Independently of the GGM used, the differences present significant discrepancies. This is mainly due to the different reference systems in which the gravimetric and the GPS/levelling-derived geoid undulations are given. Thus, in order to enable a proper comparison, a datum transformation must be estimated. According to Section 2 [33], a three-parameter transformation (dx, dy, dz) using formula (5) can be used:

_{grav}= N

_{GPS/lev}+ ΔN(θ,λ) = N

_{GPS/lev}+ dx sinθ cosλ+ dy sinθ sinλ + dz cosθ

#### 4.3. The Hybrid Estimate of the Height Anomaly and Geoid Undulation

_{res}) are integrated into the LSC procedure with the residual GPS/levelling height anomalies ($\zeta $

_{GPS/lev_res}obtained by GPS/levelling geoid undulations and applying formula (2)), found by subtracting the long-wavelength component ($\zeta $

_{GGM}), using a global geopotential model and the topographic short-wavelength component ($\zeta $

_{RTC}). In the LSC estimator, the vector of the observations is now given by $y=(\mathsf{\Delta}{g}_{res},{\zeta}_{\mathrm{GPS}/\mathrm{lev}\_\mathrm{res}}$) and the covariance matrix C is a block matrix, composed of the gravity covariance matrix C

_{Δg}

_{Δg}, the covariance matrix C

_{ζζ}of the height anomalies and the cross-covariance matrix C

_{ζ}

_{Δg}. The covariance matrix of the gravity data is the same used in the pure gravimetric solution (Figure 1), whereas the covariance matrix C

_{ζζ}and the cross-covariance C

_{ζ}

_{Δg}are obtained via covariance propagation from C

_{Δg}

_{Δg}. On the main diagonal of the C

_{ζζ}matrix, the height anomaly variance ${\sigma}_{n}^{2}\left(\zeta \right)$ is added. This value is chosen by assuming that the standard deviation of the height anomalies obtained from GPS/levelling data is about 3 cm, which represents a realistic value, taking into account the precision in the GPS and the levelling observations given in Section 3.

_{GGM}) and the topographic high-frequency component ($\zeta $

_{RTC}) will be added in order to complete the restore step.

#### 4.4. Validation of the Hybrid Geoid Model

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Free-air gravity anomalies of the new measured database in the Jeddah Region of Saudi Arabia (red rectangle in the bottom left box). The black stars represent the suspected outliers; the red stars represent the confirmed outliers.

**Figure 2.**Assembled 3″ × 3″ grid of DTM for the Residual Terrain Correction computation, obtained by merging the SRTM3 3″ × 3″ and General Bathymetric Chart of the Oceans (GEBCO) 30″ × 30″ grids. The dashed black box represents the computation area.

**Figure 5.**The spatial distribution and histogram of the differences between gravimetric undulations and the GPS/levelling values after datum shift estimate (values in centimeters). This represents the solution based on the spacewise (GO_SPW)_R5 GGM.

**Figure 6.**Differences between the geoid undulations derived by the hybrid models, based on GO_SPW_R5 and direct (GO_DIR)_R5 GGMs. The white dots represent the whole gravity database used.

**Figure 7.**Differences between the geoid undulations derived by the hybrid models, based on GO_SPW_R5 and GO_XGM2016 GGM. The white dots represent the whole gravity database used.

**Table 1.**Scheme of the Remove–Compute–Restore (RCR) procedure for the estimate of the geoid undulation, using gravity anomalies Δg and the Molodensky theory. Free air (FA); derived by a Global Gravity Field Model (GGM); residual terrain correction (RTC); residual component (res). N is the geoid undulation; $\zeta $ is the height anomaly.

Remove step: | Δg_{FA} − ∆g_{GG}_{M} − ∆g_{RTC} = ∆g_{res} |

Compute step: | $\zeta $_{res} = Collocation(∆g_{res}) |

Restore step: | $\zeta $_{res} + $\zeta $_{GGM} + $\zeta $_{RTC} = $\zeta $ |

Geoid undulation | N = $\zeta $ + Δg_{B} H |

**Table 2.**Maximum degree/order (d/o) harmonic coefficients of the Global Gravity Models (GGMs) used in the computation and tested in relations to the spherical window size of the moving average of the detailed Digital Terrain Model (DTM). For each GGM, the fourth and fifth columns represent the chosen combination parameters (d/o and window size), giving the best statistics for the gravity residuals. In the last column, S means satellite, A altimetry and T gravity terrestrial data. The spherical models highlighted in grey are GGMs already tested in 2014 when the project was set up.

GGM | Maximum d/o Available | d/o Tested | d/o Used | Window Size for the DTM Averaging | Type of Data |
---|---|---|---|---|---|

EGM2008 [20]
| 2190 | 360, 720, 1080, 1440, 1800, 2190 | 1800 | 5’ | S–A–T |

EIGEN-6c2 [5] | 1949 | 360, 720, 1080, 1440, 1800, 2190 | 1800 | 5’ | S–A–T |

EIGEN-6c4 [6]
| 2190 | 360, 720, 1080, 1440, 1800, 2190 | 1800 | 5’ | S–A–T |

GOCO05c [7]
| 720 | 360, 720 | 360 | 35’ | S–A–T |

GO_SPW_R4 [5]
| 280 | 220, 230, 240, 250, 260, 270, 280 | 250 | 35’ | S |

GO_SPW_R5 [23]
| 330 | 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 320, 330 | 240 | 35’ | S |

GO_DIR_R4 [24]
| 260 | 220, 230, 240, 250, 260 | 240 | 35’ | S |

GO_DIR_R5 [24]
| 300 | 220, 230, 240, 250, 260, 270, 280, 290, 300 | 240 | 35’ | S |

GO_DIR_R6 [25] | 300 | 240 (437.172) | 240 | 35’ | S |

GO_TIM_R4 [2]
| 250 | 220, 230, 240, 250 | 230 | 35’ | S |

GO_TIM_R5 [24]
| 280 | 220, 230, 240, 250, 260, 270, 280 | 240 | 35’ | S |

GO_TIM_R6 [26]
| 300 | 220, 230, 240, 250, 260, 270, 280, 290, 300 | 240 | 35’ | S |

GO_TIM_R6e [27]
| 300 | 220, 230, 240, 250, 260, 270, 280, 290, 300 | 240 | 35’ | S |

XGM2016 [8]
| 719 | 360, 719 | 719 | 15’ | S–A–T |

**Table 3.**Statistics of gravity residuals of 4411 gravity points, at the end of the “remove” step: ∆g

_{FA}− ∆g

_{GGM}− ∆g

_{RTC}= ∆g

_{res}testing different GGMs. For reference, in the last row (all highlighted in grey) we report the statistic values of ∆g

_{FA}(mGal).

∆g_{res} | Mean | St. dev. | Min. | Max. | Root Mean Square |
---|---|---|---|---|---|

EGM2008 | 0.584 | 17.382 | −61.409 | 61.480 | 17.40 |

EIGEN-6c2 | 1.715 | 17.229 | −57.859 | 62.405 | 17.31 |

EIGEN-6c4 | 1.821 | 17.304 | −59.210 | 63.032 | 17.40 |

GOCO05c | 4.519 | 19.986 | −71.299 | 88.139 | 20.49 |

GO_SPW_R4 | 2.327 | 20.735 | −69.548 | 80.374 | 20.87 |

GO_SPW_R5 | 2.346 | 21.117 | −70.904 | 76.367 | 21.25 |

GO_DIR_R4 | 2.004 | 21.300 | −70.776 | 79.684 | 21.39 |

GO_DIR_R5 | 2.372 | 20.796 | −71.121 | 76.530 | 20.93 |

GO_DIR_R6 | 2.156 | 20.797 | −72.120 | -74.056 | 20.91 |

GO_TIM_R4 | 2.549 | 21.143 | −71.379 | 84.047 | 21.30 |

GO_TIM_R5 | 1.724 | 20.943 | −72.068 | 76.134 | 21.01 |

GO_TIM_R6 | 1.925 | 20.851 | −72.571 | 73.907 | 20.94 |

GO_TIM_R6e | 1.921 | 20.849 | −72.662 | 73.898 | 20.94 |

XGM2016 | 1.796 | 20.004 | −76.500 | 68.744 | 20.08 |

Δg_{FA} | −0.417 | 23.021 | −80.093 | 58.713 | 23.02 |

**Table 4.**Statistics of the residual height anomalies computed by least squares collocation ($\zeta $

_{res}) on 1′ × 1′ grid (7381 values) (cm).

N_{res} | Mean | St. dev. | Min. | Max. |
---|---|---|---|---|

EGM2008 | 9.0 | 8.3 | −14.0 | 26.0 |

EIGEN-6c2 | 8.0 | 7.4 | −13.0 | 24.0 |

EIGEN-6c4 | 8.0 | 7.9 | −13.0 | 25.0 |

GOCO05c | 6.0 | 11.4 | −34.0 | 27.0 |

GO_SPW_R4 | 31.1 | 29.0 | −31.0 | 90.0 |

GO_SPW_R5 | 36.8 | 28.5 | −19.0 | 96.0 |

GO_DIR_R4 | 32.5 | 34.6 | −37.0 | 104.0 |

GO_DIR_R5 | 36.8 | 29.5 | −21.0 | 98.0 |

GO_DIR_R6 | 38.5 | 28.0 | −18.0 | 98.0 |

GO_TIM_R4 | 29.0 | 32.5 | −44.0 | 91.0 |

GO_TIM_R5 | 33.2 | 29.2 | −21.0 | 96.0 |

GO_TIM_R6 | 37.5 | 28.3 | −20.0 | 98.0 |

GO_TIM_R6e | 37.6 | 28.3 | −20.0 | 99.0 |

XGM2016 | 28.0 | 18.9 | −47.0 | 47.0 |

**Table 5.**Statistics of the differences between the gravimetric geoid undulations (labelled with the name of the GGM used) and the Global Positioning System (GPS)/levelling values before the datum transformation (cm).

Nres | Number of Points | Average | St. dev. | Min. | Max. |
---|---|---|---|---|---|

EGM2008 | 435 | 92.1 | 10.0 | 63.2 | 117.7 |

EIGEN-6c2 | 435 | 82.7 | 10.8 | 54.1 | 107.2 |

EIGEN-6c4 | 435 | 83.3 | 10.0 | 54.8 | 107.2 |

GOCO05c | 435 | 71.5 | 16.9 | 23.8 | 104.7 |

SPW_R4 | 435 | 91.0 | 16.2 | 50.4 | 129.8 |

SPW_R5 | 435 | 92.2 | 16.5 | 50.7 | 131.1 |

DIR_R4 | 435 | 90.6 | 16.8 | 48.5 | 132.1 |

DIR_R5 | 435 | 91.8 | 16.8 | 49.1 | 132.1 |

DIR_R6 | 435 | 92.6 | 16.4 | 51.2 | 132.2 |

TIM_R4 | 435 | 91.0 | 17.7 | 45.4 | 132.4 |

TIM_R5 | 435 | 91.1 | 16.8 | 48.3 | 131.9 |

TIM_R6 | 435 | 94.6 | 16.4 | 53.4 | 134.2 |

TIM_R6e | 435 | 94.7 | 16.4 | 53.2 | 134.2 |

XGM2016 | 435 | 88.5 | 11.9 | 61.3 | 116.3 |

**Table 6.**Statistics of the differences between the gravimetric geoid undulations (labelled with the name of the GGM used) and the GPS/levelling values after the datum transformation (cm).

Nres | Number of Points | St. dev. | Min. | Max. |
---|---|---|---|---|

EGM2008 | 418 | 8.4 | −20.9 | 20.1 |

EIGEN-6c2 | 422 | 7.8 | −19.5 | 19.9 |

EIGEN-6c4 | 422 | 7.9 | −20.1 | 20.2 |

GOCO05c | 430 | 8.6 | −19.0 | 21.1 |

SPW_R4 | 426 | 7.8 | −18.8 | 19.4 |

SPW_R5 | 427 | 7.9 | −19.6 | 20.4 |

DIR_R4 | 426 | 7.9 | −19.4 | 20.2 |

DIR_R5 | 424 | 7.8 | −19.7 | 19.6 |

DIR_R6 | 427 | 7.9 | −19.9 | 20.4 |

TIM_R4 | 423 | 7.8 | −19.8 | 19.3 |

TIM_R5 | 427 | 8.0 | −20.0 | 20.4 |

TIM_R6 | 424 | 7.8 | −19.4 | 19.8 |

TIM_R6e | 426 | 7.9 | −19.8 | 20.1 |

XGM2016 | 426 | 7.5 | −18.3 | 19.0 |

**Table 7.**Statistics of the differences between hybrid geoid undulations and GPS/levelling values (cm). Set A includes GPS/levelling points used in the hybrid procedure; set B includes the remaining GPS/levelling points that are not taken into account in the hybrid solution.

N_{res} | Set of Data | Points | St. dev. | Min. | Max. |
---|---|---|---|---|---|

EGM2008 | A | 190 | 6.7 | −16.0 | 15.1 |

B | 87 | 7.9 | −20.1 | 17.2 | |

EIGEN-6c2 | A | 224 | 6.1 | −14.7 | 12.0 |

B | 103 | 8.4 | −20.9 | 20.0 | |

EIGEN-6c4 | A | 228 | 4.5 | −10.4 | 10.1 |

B | 106 | 4.8 | −11.8 | 9.9 | |

GOCO05c | A | 207 | 6.6 | −16.9 | 13.5 |

B | 98 | 8.2 | −18.9 | 16.7 | |

SPW_R4 | A | 222 | 3.0 | −7.0 | 6.3 |

B | 111 | 3.9 | −9.6 | 9.5 | |

SPW_R5 | A | 224 | 3.1 | −7.7 | 7.1 |

B | 109 | 3.7 | −8.3 | 8.3 | |

DIR_R4 | A | 224 | 3.1 | −7.6 | 7.4 |

B | 108 | 3.7 | −8.9 | 8.5 | |

DIR_R5 | A | 226 | 3.2 | −8.1 | 7.7 |

B | 110 | 3.8 | −9.1 | 8.0 | |

DIR_R6 | A | 225 | 3.1 | −7.8 | 6.9 |

B | 109 | 3.8 | −9.4 | 8.8 | |

TIM_R4 | A | 224 | 3.1 | −7.8 | 6.6 |

B | 109 | 3.8 | −8.3 | 8.6 | |

TIM_R5 | A | 224 | 3.2 | −8.1 | 7.2 |

B | 108 | 3.7 | −8.3 | 8.4 | |

TIM_R6 | A | 226 | 3.2 | −8.1 | 7.6 |

B | 111 | 3.9 | −9.3 | 8.8 | |

TIM_R6e | A | 228 | 3.3 | −8.4 | 7.2 |

B | 109 | 3.8 | −9.3 | 8.5 | |

XGM2016 | A | 233 | 3.2 | −7.0 | 6.5 |

B | 113 | 3.9 | −9.8 | 9.8 |

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**MDPI and ACS Style**

Borghi, A.; Barzaghi, R.; Al-Bayari, O.; Al Madani, S.
Centimeter Precision Geoid Model for Jeddah Region (Saudi Arabia). *Remote Sens.* **2020**, *12*, 2066.
https://doi.org/10.3390/rs12122066

**AMA Style**

Borghi A, Barzaghi R, Al-Bayari O, Al Madani S.
Centimeter Precision Geoid Model for Jeddah Region (Saudi Arabia). *Remote Sensing*. 2020; 12(12):2066.
https://doi.org/10.3390/rs12122066

**Chicago/Turabian Style**

Borghi, Alessandra, Riccardo Barzaghi, Omar Al-Bayari, and Suhail Al Madani.
2020. "Centimeter Precision Geoid Model for Jeddah Region (Saudi Arabia)" *Remote Sensing* 12, no. 12: 2066.
https://doi.org/10.3390/rs12122066