# A Quasigeoid-Derived Transformation Model Accounting for Land Subsidence in the Mekong Delta towards Height System Unification in Vietnam

^{1}

^{2}

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^{*}

## Abstract

**:**

_{0}for the Vietnam Local Vertical Datum. The gravity potential of the Vietnam Local Vertical Datum is estimated equal to ${\mathrm{W}}_{0}^{\mathrm{LVD}}$ = 62,636,846.81 ± 0.70 m

^{2}s

^{−2}with the global equipotential surface realized by the conventional value W

_{0}= 62,636,853.4 m

^{2}s

^{−2}.

## 1. Introduction

_{0}= 636,853.4 m

^{2}s

^{2}, whereas the heights determined from levelling refer to the national Mean Sea Level (MSL), called the Vietnam Local Vertical Datum (VLVD). The MSL over 1950–2005 for a single tide gauge in the north of Vietnam, called Hon Dau (20°40’, 106°49’), was assigned to zero height on the VLVD. With its large mean bias, the gravimetric quasigeoid model does not allow the accurate transformation of GNSS ellipsoidal heights to physical heights in the VLVD. Most often, the gravimetric quasigeoid or geoid model is forced to fit onto the local vertical datum. Such a hybrid quasigeoid/geoid model is used to convert GNSS ellipsoidal heights to physical heights. Most countries make continuous efforts to determine and improve their hybrid geoid or quasigeoid model successfully. For comparison, Table 1 shows the accuracy of several local hybrid geoid or quasigeoid models, and notably of neighboring countries in Asia. The resulting standard deviations obtained for the most recent hybrid models range from a few cm up to 15 cm, depending on the quality of the available gravity and GNSS/levelling data for the hybrid geoid/quasigeoid determination.

_{0}, is considered equal to 62,636,853.4 m

^{2}s

^{−2}[27]. From this value, Height System Unification (HSU) can be realized to connect height systems together by determining potential differences referring to the conventional value. Knowing the gravity potential of LVD, the gravity potential difference of every LVD and the LVD offset values between all the LVDs can be determined. Therefore, determining the gravity potential value of LVD plays an important role in the HSU.

_{0}) for the VLVD.

## 2. Materials and Methods

#### 2.1. Offset Model Determination Methodology

^{GNSS/levelling}) and from the model GEOID_LSC (ζ):

_{0}represents the contribution of the zero-degree harmonic term to the GGM with respect to a specific reference ellipsoid [29].

_{0}and U

_{0}are the geocentric gravitational constant of the reference ellipsoid and the normal gravity potential, respectively. The WGS-84 ellipsoid is used as the reference ellipsoid for computation of GEOID_LSC, GM

_{0}= 398,600.4418 × 10

^{9}m

^{3}s

^{−2}and U

_{0}= 62,636,851.7146 m

^{2}s

^{−2}(report of the National Imagery and Mapping Agency (NIMA) [30]), while the Earth’s geocentric gravitational constant GM and the gravity potential Wo are set to GM = 398,600.4418109 × 10

^{9}m

^{3}s

^{−2}and W

_{0}= 62,636,853.4 m

^{2}s

^{−2}. The mean Earth radius R is taken equal to 6371 km, and the normal gravity γ at the surface of the ellipsoid is computed by using Equation (4-60) of Hofmann-Wellenhof and Moritz (2006) [29].

- linear in φ and λ model:$${\epsilon}^{\prime}={a}_{0}+{a}_{1}\phi +{a}_{2}\mathsf{\lambda}+\epsilon $$
- second-order polynomial model:$${\epsilon}^{\prime}={a}_{0}+{a}_{1}\phi +{a}_{2}\mathsf{\lambda}+{a}_{3}{\phi}^{2}+{a}_{4}\phi \mathsf{\lambda}+{a}_{5}{\mathsf{\lambda}}^{2}+\epsilon $$
- third-order polynomial model:$${\epsilon}^{\prime}={a}_{0}+{a}_{1}\phi +{a}_{2}\mathsf{\lambda}+{a}_{3}{\phi}^{2}+{a}_{4}\phi \mathsf{\lambda}+{a}_{5}{\mathsf{\lambda}}^{2}+{a}_{6}{\phi}^{3}+{a}_{7}{\phi}^{2}\mathsf{\lambda}+{a}_{8}\phi {\mathsf{\lambda}}^{2}+{a}_{9}{\mathsf{\lambda}}^{3}+\epsilon $$
- four parameter Helmert model:$${\epsilon}^{\prime}={a}_{1}cos\phi cos\mathsf{\lambda}+{a}_{2}cos\phi sin\mathsf{\lambda}+{a}_{3}sin\phi +{a}_{4}+\epsilon $$

#### 2.2. Data for the Determination of the Offset Model

#### 2.2.1. Gravimetric Quasigeoid Model (GEOID_LSC)

#### 2.2.2. GNSS/Levelling Data

## 3. Land Subsidence in Vietnam

#### 3.1. GNSS and InSAR Data

#### 3.2. Land Subsidence and Correcting GNSS/Levelling Data in the Mekong Delta

^{GNSS/levelling}), we performed the correction according to the following formula:

- using all 802 GNSS/levelling points and calculating the homogeneous distortion parameter (case 1);
- using all 802 GNSS/levelling points and calculating two distortion parameters for two regions: southern (<17° in latitude) and northern part (>17° in latitude) (case 2).

## 4. Results and Discussion

#### 4.1. Offset Model Estimation and Validation

#### 4.2. Estimation of the Geopotential Value W_{0} for the VLVD

_{0}for the VLDV using the differences in height anomalies derived from them. The geopotential number (C) is the potential difference between an equipotential surface (W

_{i}) and a reference equipotential surface. National datum from traditional levelling realizes by selecting as their zero point O a coastal tide gauge and setting it a geopotential value ${W}_{0}^{LVD}$, while a geoid/quasigeoid model realizes the origin of a global datum (W

_{0}). The geopotential number for point i can be written as [44,45]

_{0}must be omitted in this computation. The height anomalies from good 779 GNSS/levelling points after correcting the subsidence for the Mekong Delta was used for estimating ${\mathrm{W}}_{0}^{\mathrm{LVD}}$. Calculation of two tilt parameters for two regions, southern (<17° in latitude) and northern part (>17° in latitude), was also used in this case. The results are shown in Table 7 where the improvement in the STD between the null and third-order model with two parameters was significant. Therefore, the results calculated from the third-order model with two parameters were used to estimate ${W}_{0}^{LVD}$ employing Equation (14). A gravity potential ${W}_{0}^{LVD}$ = 62,636,846.81 ± 0.70 m

^{2}s

^{−2}for the LVD of Vietnam has been determined as an offset to the equipotential surface realized by the conventional value W

_{0}= 62,636,853.4 m

^{2}s

^{−2}.

## 5. Conclusions

_{0}for the existing LVD in Vietnam. The gravity potential of the VLVD is estimated equal to ${W}_{0}^{LVD}$= 62,636,846.81 ± 0.70 m

^{2}s

^{−2}with the global equipotential surface realized by the conventional value W

_{0}= 62,636,853.4 m

^{2}s

^{−2}. With this gravity potential value, we can thus connect the height system of Vietnam with the neighboring countries.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Global Navigation Satellite System (GNSS)/levelling data: yellow dots are first- and second-order levelling benchmarks, whereas purple dots denote third-order ones.

**Figure 3.**(

**a**) Differences between GNSS/levelling data and GEOID_LSC; (

**b**) Linear regression on the differences of GNSS/levelling data and GEOID_LSC in northern (>21° in latitude) and near coast (points within 50 km from the coastline) and (

**c**) Differences between GNSS/levelling data and DIR/EGM (d/o 719) plus RTM effects.

**Figure 4.**Magnitude of relative differences between GEOID_LSC with 803 GNSS/levelling points over 21,423 baselines (blue), fourth-order tolerance (orange) and third-order tolerance (purple).

**Figure 6.**(

**a**) Differences of GNSS/levelling data and GEOID_LSC in the southern region; (

**b**) Linear regression on the differences of GEOID_LSC and GNSS/levelling data in the southern region; (

**c**) InSAR results derived for the period 2006–2010 from ALOS-1 provided by Dr Laura E Erban [20] and (

**d**) InSAR results derived for the period 2015–2018 from Sentinel-1, project EMSN057 [41].

**Figure 7.**(

**a**) Distortions, (

**b**) residuals and (

**c**) differences between the 779 GNSS/levelling points and gravimetric quasigeoid model adding offset model.

No | Country, Region | Name | Year | STD (cm) | Geoid Type | Reference |
---|---|---|---|---|---|---|

1 | Australia | AUSGeoid98 | 2005 | 11.8 | Hybrid quasigeoid | [4] |

AUSGeoid09 | 2009 | 3.0 | Hybrid quasigeoid | [5] | ||

AUSGeoid2020 | 2018 | 2.7 | Hybrid quasigeoid | [6] | ||

2 | Japan | GSIGEO2000 | 2002 | 4.0 | Hybrid geoid | [7] |

GSIGEO2011 | 2014 | 1.8 | Hybrid geoid | [8] | ||

3 | South Korea | KNGeoid13 | 2013 | 5.4 | Hybrid geoid | [9] |

KNGeoid14 | 2014 | 5.2 | Hybrid geoid | [9] | ||

4 | Thailand | THAI12H | 2012 | 15.8 | Hybrid geoid | [10] |

5 | Philippines | PGM2016 | 2016 | 2.2 | Hybrid geoid | [11] |

6 | Peninsular (Malaysia) | VMGEOID04 | 2018 | 5.0 | Hybrid geoid | [12] |

Sabah and Sarawak (Malaysia) | EMGEOID05 | 2018 | 10.0 | Hybrid geoid | [12] | |

7 | Hong Kong | HKGEOID-2000 | 2004 | 1.7 | Hybrid geoid | [13] |

8 | Shenzhen | SZGEOID-2001 | 2004 | 1.4 | Hybrid geoid | [13] |

**Table 2.**Descriptive statistics of the absolute (residuals) and relative differences between the 812 GNSS/levelling stations and GEOID_LSC. Unit: (m).

Absolute Differences | ||||||

Mean | STD | Max | Min | Outlier Points | ||

ζ^{GNSS/levelling} – ζ^{LSC} (812 points) | 0.680 | 0.097 | 0.987 | 0.310 | 9 | |

ζ^{GNSS/levelling} – ζ^{LSC} (excluding outliers) (803 points) | 0.682 | 0.092 | 0.937 | 0.396 | 0 | |

ζ^{GNSS/levelling} – ζ^{DIR/EGM+RTM} (803 points) | 0.682 | 0.168 | 1.138 | 0.119 | ||

ζ^{GNSS/levelling} – ζ^{LSC} (North-east part) (190 points) | 0.705 | 0.077 | 0.879 | 0.459 | ||

ζ^{GNSS/levelling} – ζ^{LSC} (Southern, <11°) (120 points) | 0.634 | 0.092 | 0.937 | 0.402 | ||

Relative Differences | ||||||

Mean | STD | Max | Min | Outlier third-order | Outlier fourth-order | |

∆ζ^{GNSS/levelling} – ∆ζ^{LSC} (803 points) (21,423 baselines) | 0.087 | 0.071 | 0.518 | 0 | 8153 (38.06%) | 2052 (9.58%) |

**Table 3.**Relative differences between 803 GNSS/levelling points and GEOID_LSC, per baseline length (every 10 km) (NoB: Number of Baselines). Unit: (m).

10 km | 20 km | 30 km | 40 km | 50 km | 60 km | 70 km | 80 km | 90 km | 100 km | All | |

NoB | 96 | 760 | 1356 | 1776 | 2215 | 2505 | 2845 | 3068 | 3334 | 3468 | 21,423 |

mean | 0.055 | 0.065 | 0.077 | 0.082 | 0.084 | 0.089 | 0.089 | 0.091 | 0.090 | 0.093 | 0.087 |

STD | 0.045 | 0.055 | 0.063 | 0.066 | 0.070 | 0.072 | 0.073 | 0.072 | 0.073 | 0.074 | 0.071 |

Max | 0.219 | 0.302 | 0.407 | 0.394 | 0.451 | 0.451 | 0.518 | 0.449 | 0.467 | 0.428 | 0.518 |

min | 0.001 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Northern Part | |||||||||||

NoB | 34 | 231 | 376 | 530 | 647 | 751 | 854 | 868 | 1004 | 998 | 6294 |

mean | 0.044 | 0.075 | 0.080 | 0.084 | 0.083 | 0.089 | 0.092 | 0.094 | 0.093 | 0.093 | 0.089 |

STD | 0.030 | 0.061 | 0.066 | 0.066 | 0.069 | 0.071 | 0.071 | 0.069 | 0.072 | 0.072 | 0.070 |

Max | 0.120 | 0.295 | 0.322 | 0.316 | 0.390 | 0.400 | 0.468 | 0.396 | 0.392 | 0.404 | 0.468 |

min | 0.001 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Points Near the Coast (within 50 km from the coastline) | |||||||||||

NoB | 18 | 202 | 312 | 357 | 380 | 369 | 347 | 361 | 336 | 321 | 3003 |

mean | 0.042 | 0.055 | 0.067 | 0.073 | 0.072 | 0.079 | 0.078 | 0.074 | 0.074 | 0.75 | 0.073 |

STD | 0.037 | 0.049 | 0.055 | 0.066 | 0.067 | 0.067 | 0.065 | 0.060 | 0.069 | 0.069 | 0.064 |

Max | 0.138 | 0.290 | 0.388 | 0.333 | 0.418 | 0.363 | 0.384 | 0.408 | 0.467 | 0.352 | 0.467 |

min | 0.007 | 0 | 0 | 0 | 0 | 0.001 | 0 | 0 | 0 | 0 | 0 |

**Table 4.**Descriptive statistics of the differences between the GNSS/levelling data corrected in Mekong Delta and GEOID_LSC. Unit: (m).

Mean | STD | Max | Min | Outlier Points | |
---|---|---|---|---|---|

ζ^{GNSS/levelling} – ζ^{LSC} (Southern, <11°) (120 points) | 0.634 | 0.092 | 0.937 | 0.402 | |

ζ^{GNSS/levelling} – ζ^{LSC} (>11°) (683 points) | 0.690 | 0.089 | 0.933 | 0.396 | |

ζ^{GNSS/levelling_cor} – ζ^{LSC} (Southern, <11°) (120 points) | 0.664 | 0.085 | 0.937 | 0.459 | |

ζ^{GNSS/levelling_cor} – ζ^{LSC} (803 points) | 0.686 | 0.089 | 0.937 | 0.396 | 1 |

ζ^{GNSS/levelling_cor} – ζ^{LSC} (excluding outliers) (802 points) | 0.687 | 0.088 | 0.937 | 0.423 |

Mean | STD | Max | Min | Outlier Points | |
---|---|---|---|---|---|

Linear (802 points) | 0 | 0.087 | 0.285 | −0.296 | |

Second-order (802 points) | 0 | 0.085 | 0.265 | −0.326 | |

Third-order (802 points) | 0 | 0.082 | 0.262 | −0296 | |

Helmert model (802 points) | 0 | 0.086 | 0.285 | −0.316 | |

case 2 (Third-order) (802 points) | 0 | 0.078 | 0.252 | −0.288 | 6 |

case 2 (Third-order) outlier (796 points) | 0 | 0.075 | 0.225 | −0.223 |

**Table 6.**Descriptive statistics of the differences between the GNSS/levelling data and gravimetric quasigeoid adding offset model with baseline length < 100 km. Unit: (m).

Absolute Differences | ||||||

Mean | STD | Max | Min | Outlier Points | ||

ζ^{GNSS/levelling} – ζ^{LSC}-ε (796 points)(cross-validation) | 0 | 0.065 | 0.250 | −0.291 | 17 | |

ζ^{GNSS/levelling} – ζ^{LSC}-ε (779 points)(excluding outliers) (cross-validation) | 0 | 0.059 | 0.170 | −0.170 | ||

ζ^{GNSS/levelling} – ζ^{LSC}-ε (779 points) | 0 | 0.034 | 0.099 | −0.109 | ||

ζ^{GNSS/levelling} – ζ^{LSC}-ε (Hanoi) (32 points) | −0.004 | 0.047 | 0.109 | −0.092 | ||

ζ^{GNSS/levelling} – ζ^{LSC}-ε (HCMC) (29 points) | −0.001 | 0.055 | 0.104 | −0.139 | ||

Relative Differences | ||||||

Mean | STD | Max | Min | Outlier 3rd Order | Outlier 4th Order | |

∆ζ^{GNSS/levelling} – ∆ζ^{LSC}-∆ε (779 points) (20,243 baselines) | 0.026 | 0.020 | 0.109 | 0 | 377 (1.86%) | 14 (0.07%) |

∆ζ^{GNSS/levelling} – ∆ζ^{LSC}-∆ε (Hanoi) (469 baselines) | 0.024 | 0.018 | 0.074 | 0.003 | 8 | 0 |

∆ζ^{GNSS/levelling} – ∆ζ^{LSC}-∆ε (HCMC) (384 baselines) | 0.025 | 0.019 | 0.085 | 0.085 | 13 | 0 |

**Table 7.**Descriptive statistics of the differences between the GNSS/levelling data removed tilt effects and GEOID_LSC. Reference geopotential values ${W}_{0}^{LVD}$ for the VLVD with the global reference level realized by the conventional value W

_{0}= 62,636,853.4 m

^{2}s

^{−2}. Unit: (m).

Mean | STD | Max | Min | |
---|---|---|---|---|

Null model | 0.688 | 0.083 | 0.912 | 0.435 |

Third-order | 0.693 | 0.075 | 0.897 | 0.458 |

Third-order (two parameters) | 0.689 | 0.071 | 0.902 | 0.478 |

$\delta {W}^{LVD}$ | 6.60 ± 0.70 (m^{2}s^{−2}) | ${W}_{0}^{LVD}$ | 62,636,846.81 ± 0.70 (m^{2}s^{−2}) |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vu, D.T.; Bruinsma, S.; Bonvalot, S.; Remy, D.; Vergos, G.S. A Quasigeoid-Derived Transformation Model Accounting for Land Subsidence in the Mekong Delta towards Height System Unification in Vietnam. *Remote Sens.* **2020**, *12*, 817.
https://doi.org/10.3390/rs12050817

**AMA Style**

Vu DT, Bruinsma S, Bonvalot S, Remy D, Vergos GS. A Quasigeoid-Derived Transformation Model Accounting for Land Subsidence in the Mekong Delta towards Height System Unification in Vietnam. *Remote Sensing*. 2020; 12(5):817.
https://doi.org/10.3390/rs12050817

**Chicago/Turabian Style**

Vu, Dinh Toan, Sean Bruinsma, Sylvain Bonvalot, Dominique Remy, and Georgios S. Vergos. 2020. "A Quasigeoid-Derived Transformation Model Accounting for Land Subsidence in the Mekong Delta towards Height System Unification in Vietnam" *Remote Sensing* 12, no. 5: 817.
https://doi.org/10.3390/rs12050817