# An Assessment of the GOCE High-Level Processing Facility (HPF) Released Global Geopotential Models with Regional Test Results in Turkey

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Practical Need of Global Geopotential Model Validation Results

_{o}) of regional vertical datum and height system unification are not explicitly undertaken in this article, [37] is recommended for further reading on the detailed numerical evaluations of a number of GGMs on the height system unification between Turkey and Greece.

#### 2.2. Overview the Tested Global Geopotential Models

- DIR models contain GRACE observations in the lower to medium degrees of the expansions (in addition to GOCE gravity gradient data).
- As a priori gravity field information EIGEN5C (≤d/o 360) was introduced in DIR RL1, whereas the ITG-GRACE2010S (≤d/o 150) solution was the background model in RL2.
- In RL3, RL4 and RL5 models, GRACE was combined with GOCE and SLR data on the basis of the normal equations (including the SLR data (LAGEOS-1 and -2)) in order to improve the gravity field solution, since the very low-degree harmonics (in particular degrees 2 and 3) cannot be estimated with GRACE and GOCE data).
- In order to overcome the polar gap of GOCE’s observations caused by the orbit inclination of the satellite (inclined at 96.7° [18]), a spherical cap regularization (SCR) in accordance to [44] was applied using EIGEN-51C (≤d/o 240). In the later releases after RL1, the SCR was applied using GRACE and LAGEOS data. In the RL3, RL4, RL5 of DIR models, additionally, the predecessor release was used as a priori information (≤d/o 240, 260, 300, respectively, in the releases) and a Kaula regularization [44] was used beyond degrees 200 (for RL3, RL4) and 180 (for RL5).
- In the DIR RL1 solution, the gravity tensor elements, which are measured with GOCE on-board SGG, were accumulated with the relative weights of Txx 1.0, Tyy 0.5 and Tzz 1.0. They were equally weighted in combination for the releases after RL1. In RL4, RL5 and RL6 models, the off-diagonal tensor elements Txz were also included.
- In RL4 and RL5 models, the spectral bandwidth of the bandpass filter used to filter the SGG observations was extended by 1.7 MHz toward the lower frequency domain.
- Reference [16] notes that contrary to the TIM approach, the DIR approach employs the GOCE gravity gradient information as restricted in a certain spectral band, which is close to the gradiometer measurement bandwidth (5–100 MHz [45]). However, the filtering procedure in the TIM approach allows the use of information on the gradient observations over the entire spectrum.

- The TIM models were constrained using Kaula’s rule regularization applied to (near-) zonal coefficients, improving the signal-to-noise ratio (constraints for high degree coefficients).
- The off-diagonal tensor element Vxz was included in the models TIM RL3, RL4, RL5 and RL6. The stochastic models for the gravity gradient observations rely on optimum weighting based on variance component estimation.
- TIM RL6e includes additional terrestrial gravity field observations over GRACE’s polar gap areas (>83° S, N) [46].

- The SPW RL1 model includes the first GOCE quick-look model as the a priori information and the EGM2008 was used for error calibration of the estimated gravitational potential along the orbit that affects the low degrees of the solution. However, the later releases, RL2 and RL4, were not corrected using any a priori model, thus they are the GOCE-only models. EIGEN5C and EIGEN6C3stat models were respectively used in SPW RL2 and RL4 models’ computations for signal covariance modeling, in addition to FES2004 for ocean tide modeling.
- In all SPW models, the off-diagonal tensor elements Vxz were included.

#### 2.3. Synthesis of the Gravity Field Parameters Using GGMs Spherical Harmonic Coefficients

#### 2.4. Assessments of the Global Geopotential Models Using Spectral Enhancement Method (SEM)

_{2}from a higher resolution Earth geopotential model (e.g., ${\ell}_{2}=2190$ when using EGM2008). The very short-wavelength components of the gravity field, represented with spectral bands beyond degree ${\ell}_{2}$ (e.g., ${\ell}_{2}=2190$ in our case) until the maximum possible degree (e.g., 216,000), are completed as the residual terrain effect (RTE) of topographic data on the estimated geoid heights (see Equation (4)).

#### 2.5. Spectral Analysis of GGMs

_{2}—usually ${\ell}_{2}={\ell}_{max}$ as the maximum degree of the harmonic expansion equation) and shows the increase in overall power with increasing degree of $\ell $:

#### 2.6. GGMs’ Contribution in High-resolution Regional Geoid Modeling

## 3. Results and Discussion

#### 3.1. Internal Error Estimates of Tested GGMs

#### 3.2. Overview of the Regional Accuracies of the GGMs

#### 3.2.1. Assessment of GGMs’ Accuracies in Turkey

_{w}≥ 6 occurs every 1–2 years, particularly around the North Anatolian Fault Zone. All these activities continuously distort the geodetic networks by time and even have a minor effect on the change of the gravity field; however, the latter is not significant for the scope of this study. Hence, the first geodetic network (Dataset I) was exposed to the Izmit earthquake of 1999 (7.6 Mw) and the northwest part of this geodetic network, including the 81 benchmarks of Dataset II, have been remeasured and adjusted since then. Although the networks were positioned at the same geodetic datum, the physical establishments of the network stations were distorted by time and these distortions negatively influence their quality as control data in model evaluations. As a conclusion, the ground-truth data employed to validate the qualities and fitting performances of the GGMs should be of high quality, sufficiently dense, homogeneously distributed over the test area and as up-to-date as possible, especially in areas under the influence of dynamic Earth activities, such as Turkey.

#### 3.2.2. Testing the GGMs Contribution in Detailed Geoid Modeling

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Error amplitudes (accumulated) as a function of the spherical harmonic degrees “$\ell $” in terms of geoid heights: (

**a**) direct (DIR RLx) models; (

**b**) time-wise (TIM RLx) models; (

**c**) space-wise (SPW RLx) models; (

**d**) GOGRA04S, GOCO05S and EGM2008 models.

**Figure 2.**Distribution of 30 benchmarks (Dataset I) on Shuttle Radar Topography Mission (SRTM) topography.

**Figure 4.**Validation of global geopotential models (GGMs) regarding the standard deviations of the geoid undulation residuals (${N}_{GPS/lev.}-{N}_{GGM}^{enh.}$) in meters (for Dataset I): (

**a**) DIR RLx models; (

**b**) TIM RLx models; (

**c**) SPW RLx models; (

**d**) comparison among the last versions of the satellite-based and combined models.

**Figure 5.**Validation of GGMs regarding the standard deviations of the geoid undulation residuals (${N}_{GPS/lev.}-{N}_{GGM}^{enh.}$) in meters (Dataset II): (

**a**) DIR RLx models; (

**b**) TIM RLx models; (

**c**) SPW RLx models; (

**d**) comparison among the last versions of the satellite-based and combined models.

**Figure 6.**Standard deviations of the geoid height residuals in cm (note that the geoid heights (${N}_{GGM}^{enh.}$), derived from the enhanced GGMs over optimum (blue bar) and maximum (orange bar) degrees for (

**a**) the first dataset; (

**b**) the second dataset.

**Figure 7.**Comparison of RL5 and RL6 models: (

**a**) the standard deviations of the geoid undulation residuals (${N}_{GPS/lev.}-{N}_{GGM}^{enh.}$) in meters (Dataset II), (

**b**) the standard deviations of geoid undulations in cm, and please note that the geoid heights (${N}_{GGM}^{enh.}$ ), derived from the enhanced GGMs over the optimum (blue bar) and the maximum (orange bar) degrees of the models.

**Figure 8.**The experimental Turkey geoid, calculated using the remove–compute–restore (RCR) approach based on the reference geopotential model DIR-RL5.

Model | (Max. d/o) | Data | Difference w.r.t. Previous Version | Literature |
---|---|---|

DIR_RL1(240) | GOCE (2 m), CHAMP (6 y) | [18,39] |

The model is more accurate than GRACE models for d/o 130-150 and up, less accurate for the lower degrees. | ||

DIR_RL2(240) | GOCE (8 m) | [18,39] |

SST ≤ d/o 130 | ||

DIR_RL3(240) | GOCE (18m), GRACE (6.5y), SLR (6.5y) | [18,39] |

Combined GRACE as normal equations. and LAGEOS as normal equations., SGG components: equal relative weights 1.0. | ||

DIR_RL4(260) | GOCE (33 m), GRACE (9 y), SLR (>10 y) | [51] |

SGG: inclusion of Txz-component.8.3–125 MHz filter. GRACE: a blended combination of GRGS-RL02 ≤ d/o 54 and GFZ-RL05 from d/o 55 to 180. | ||

DIR_RL5(300) | GOCE (48 m), GRACE (>10 y), SLR (>10 y) | [51] |

The GRACE contribution up to d/o 130. Very low-degree harmonics of (esp. d/o 2 and 3) cannot be estimated accurately with GRACE and GOCE data, therefore LAGEOS-1 and -2 normal equations are used in the combination | ||

DIR_RL6(300) | GOCE (48 m), GRACE (>10 y), SLR (>10 y) | [52] |

Based on improved filtering of the reprocessed gradients of the full mission observations | ||

TIM_RL1(224) | GOCE (2 m) | [40] |

The model is independent of any other gravity field information. In the low degrees, it is not competitive with GRACE models. Kaula reg. is to improve SNR. | ||

TIM_RL2(250) | GOCE (8 m) | [18] |

No change in data processing strategy w.r.t. its predecessor. | ||

TIM_RL3(250) | GOCE (18 m) | [18] |

In addition to Vxx, Vyy, Vzz, the inclusion of Vxz component in the SGG observations. | ||

TIM_RL4(250) | GOCE (32 m) | [18] |

Applying the short-arc integral method to the kinematic orbits. | ||

TIM_RL5(280) | GOCE (48 m) | [53] |

SST ≤ d/o 150, SGG ≤ d/o 280, Kaula ≥ 201 | ||

TIM_RL6(300) | GOCE (48 m) | [54] |

SGG ≤ d/o 300, digital decorrelation filter to observation equations | ||

TIM_RL6e(300) | GOCE (48 m) | [46] |

Included terrestrial gravity field observations over GOCES’s polar gap areas | ||

SPW_RL1(210) | GOCE (2 m) | [18,41] |

Applying an iterative procedure based on the Wiener orbital filter to reduce time correlated noise of the gradiometer. Including Txz in the gravity gradients. | ||

SPW_RL2(240) | GOCE (8 m) | [55] |

No corrections to any a priori model. Using EIGEN5C for signal covariance modeling and FES2004 for ocean tide modeling. | ||

SPW_RL4(280) | GOCE (32 m) | [56] |

Using EIGEN6C3stat for signal covariance modeling. | ||

SPW_RL5(330) | GOCE (48 m) | [57] |

Using EIGEN6C4 and GOCO05S for signal covariance modeling. | ||

GOCO05S(280) | GOCE (48 m), GRACE (~10 y), CHAMP (6y), SLR (>10y) | [58] |

Combined satellite gravity field model, estimated with over 800,000,000 observations. | ||

GOGRA04S(230) | GOCE (48 m), GRACE (>10y) | [59] |

In the solution: the normal equations combine SST ≤ d/o 120, SGG ≤ d/o 230, GRACE ≤ d/o 180 | ||

EGM2008(2190) | GRACE, Gravity (terrestrial, airborne), Altimetry Data | [22] |

Dataset | I | II |
---|---|---|

Size of area | 6° × 19° | 3° × 4° |

Number of BMs | 30 | 81 |

BM density | 1 BM/200 km | 1 BM/45 km |

The distribution of BMs | Homogenous and quite sparse | Homogenous and sparse |

^{1} Coordinate datum | ITRF96 | ITRF96 |

2D coordinate accuracy | ±1.0 cm | ±1.0 cm |

H-accuracy | ±2.0 cm | ±2.0 cm |

^{1} Vertical datum | TUDKA99 | TUDKA99 |

H-accuracy | ±2.5 cm | ±2.0 cm |

Topography | 0–5000 m | 0–2500 m |

Reference | [76] | [76] |

^{1}The datasets include first order GPS network points of Turkey National Fundamental GPS Network with Helmert orthometric heights (third-order leveling network) in regional vertical datum. BM: benchmark.

GGMs and Detailed Geoid Model | Ref. GGM (d/o) | Statistics | |||||
---|---|---|---|---|---|---|---|

Min | Max | Mean | Std. | ||||

Optimum d/o of GGMs | DIR RL5 (155) | − | − | −192.8 | 61.8 | −72.8 | 61.0 |

expTG-Opt-1 | DIR RL5 (155) | Before fit | −174.1 | −94.4 | −127.9 | 16.5 | |

After fit | −37.2 | 31.6 | 0.0 | 9.8 | |||

TIM RL5 (155) | − | − | −192.8 | 61.2 | −72.7 | 61.1 | |

expTG-Opt-2 | TIM RL5 (155) | Before fit | −174.5 | −94.6 | −128.2 | 16.5 | |

After fit | −37.2 | 31.5 | 0.0 | 9.8 | |||

SPW RL5 (155) | − | − | −204.8 | 62.7 | −72.6 | 63.7 | |

expTG-Opt-3 | SPW RL5 (155) | Before fit | −164.9 | −83.8 | −119.6 | 16.3 | |

After fit | −34.3 | 29.7 | 0.0 | 11.4 | |||

GOCO05S (155) | − | − | −192.8 | 61.1 | −72.9 | 60.8 | |

expTG-Opt-4 | GOCO05S (155) | Before fit | −174.3 | −94.4 | −128.0 | 16.5 | |

After fit | −37.2 | 31.6 | 0.0 | 9.8 | |||

EGM2008 (155) | − | − | −89.9 | 171.0 | 23.1 | 63.1 | |

expTG-Opt-5 | EGM2008 (155) | Before fit | −170.3 | −89.7 | −124.2 | 16.5 | |

After fit | −37.2 | 31.6 | 0.0 | 9.8 | |||

Maximum d/o of GGMs | DIR RL5 (300) | − | − | −42.2 | 67.4 | 8.7 | 26.4 |

expTG-M-1 | DIR RL5 (300) | Before fit | −162.2 | −66.6 | −112.9 | 19.4 | |

After fit | −35.0 | 30.2 | 0.0 | 9.7 | |||

TIM RL5 (280) | − | − | −50.2 | 67.3 | 8.6 | 27.2 | |

expTG-M-2 | TIM RL5 (280) | Before fit | −164.8 | −69.8 | −115.2 | 19.6 | |

After fit | −34.3 | 30.0 | 0.0 | 9.6 | |||

SPW RL5 (330) | − | − | −57.5 | 88.8 | 9.1 | 31.3 | |

expTG-M-3 | SPW RL5 (330) | Before fit | −161.8 | −75.4 | −115.9 | 18.4 | |

After fit | −31.1 | 27.7 | 0.0 | 10.7 | |||

GOCO05S (280) | − | − | −48.2 | 62.7 | 8.0 | 26.2 | |

expTG-M-4 | GOCO05S (280) | Before fit | −163.6 | −68.2 | −113.9 | 19.7 | |

After fit | −34.4 | 30.9 | 0.0 | 9.6 | |||

EGM2008 (280) | − | − | −44.7 | 69.3 | 8.3 | 26.4 | |

expTG-M-5 | EGM2008 (280) | Before fit | −163.6 | −68.2 | −113.9 | 19.7 | |

After fit | −34.6 | 30.8 | 0.0 | 9.6 | |||

EGM2008 (300) | − | − | −43.1 | 66.6 | 8.7 | 25.7 | |

expTG-M-6 | EGM2008 (300) | Before fit | −163.6 | −68.2 | −113.9 | 19.7 | |

After fit | −33.0 | 33.6 | 0.0 | 11.7 | |||

EGM2008 (330) | − | − | −38.6 | 59.8 | 8.3. | 23.8 | |

expTG-M-7 | EGM2008 (330) | Before fit | −151.0 | −63.7 | −104.5 | 18.6 | |

After fit | −31.1 | 29.0 | 0.0 | 10.8 |

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## Share and Cite

**MDPI and ACS Style**

Erol, B.; Işık, M.S.; Erol, S.
An Assessment of the GOCE High-Level Processing Facility (HPF) Released Global Geopotential Models with Regional Test Results in Turkey. *Remote Sens.* **2020**, *12*, 586.
https://doi.org/10.3390/rs12030586

**AMA Style**

Erol B, Işık MS, Erol S.
An Assessment of the GOCE High-Level Processing Facility (HPF) Released Global Geopotential Models with Regional Test Results in Turkey. *Remote Sensing*. 2020; 12(3):586.
https://doi.org/10.3390/rs12030586

**Chicago/Turabian Style**

Erol, Bihter, Mustafa Serkan Işık, and Serdar Erol.
2020. "An Assessment of the GOCE High-Level Processing Facility (HPF) Released Global Geopotential Models with Regional Test Results in Turkey" *Remote Sensing* 12, no. 3: 586.
https://doi.org/10.3390/rs12030586