# Decorrelation of GRACE Time Variable Gravity Field Solutions Using Full Covariance Information

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## Abstract

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## 1. Introduction

## 2. Decorrelation Strategy and Setup of Experiments

## 3. Results from Closed-Loop Simulations

#### 3.1. The Closed-Loop Simulation Environment

#### 3.2. Determination of a Favorable Filter Design

## 4. Impact of Post-Processing Methods on the Phase of Seasonal Signals with the Closed-Loop Environment

## 5. Application to Real Data

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Normalized 2D representation of the filter Kernel ${W}_{\alpha}$ in the spatial domain, exemplarily shown for a geographical longitude of λ = 0 degrees, and latitude φ = 30 degrees north (center of figure). (

**b**) Transection through the normalized smoothing kernel. In blue the east-west section and in red the north-south section through the center of the kernel are shown.

**Figure 2.**(

**a**,

**b**) coefficients for estimates of the Kaula rule signal variance from DDK4 filtered ITSG-Grace2014 solutions (blue) and HIS model data (red).

**Figure 3.**Cumulative geoid errors in mm for different VADER filter radii (average) computed from the residuals of all monthly solutions (median is shown) with respect to the corresponding HIS signal using full error variance-covariance matrices (red), only SH order blocks (black), and only the diagonals (blue) compared with the Gaussian filtered solutions (dashed cyan).

**Figure 4.**Cumulative geoid error from residuals for different static and non-stationary states of the error covariance matrix and signal variance matrix fed into the design of the VADER filter. Colors indicate the respective result for specific filter strength for the individual months of a 143 month time series. Black x-marks indicate the median cumulative geoid error at the median average smoothing radius of the respective cluster of results. (

**a**) Stationary error covariance and cyclo-stationary signal variance. (

**b**) Non-stationary error covariance and cyclo-stationary signal variance. (

**c**) Stationary error covariance and stationary signal variance. In addition for comparison 143 individual monthly results for the Gaussian and empirical decorrelation approach.

**Figure 5.**(

**a**) SH degree RMS in mm geoid height of the residuals of the raw (gray) and the filtered solutions using the VADER (red) and the Gaussian filter (blue) in comparison with the mean HIS signal (black) for month 2009-08. Ocean tide errors are added in terms of ocean tide model difference signal (dashed lines). (

**b**) Cumulative geoid errors in mm for different filter radii (for the VADER filter the average radii are given) computed from the residuals of all monthly solutions (median is shown) with respect to the corresponding HIS signal filtered with VADER (solid lines) and Gaussian (dashed lines). Complete de-aliasing (blue) is compared with effects of ocean tide model differences (red) and of 20% of the non-tidal atmospheric and oceanic signals (black).

**Figure 6.**Example of real data (ITSG-Grace2014) filtering with different filter approaches for April 2004 (

**left**column) and September 2004 (

**right**column).

**Figure 7.**Estimated mass trends applying different DDK (black cross marks) and VADER (blue asterisk marks) filter strengths in Gigatons per year for the region of the Antarctic Peninsula. In purple color the IMBIE mass balance estimate based on the gravity method, in green color the average of all methods applied and compared in [2] are plotted.

**Table 1.**Mean Gaussian radii estimated from filter kernels at 0, 30, and 60 degrees latitude over 124 months (2003-02 until 2013-12) for the ITSG-Grace2014-VADER filter. Radii for the DDK results are computed as simple mean from numbers given in [17] for 0, 30, and 60 degrees latitude.

DDK | Mean Gaussian Radius (km) | VADER (α) | |
---|---|---|---|

7 | 232 | ||

6 | 267 | 254 | 0.1 |

5 | 285 | 287 | 0.5 |

4 | 333 | 335 | 1 |

3 | 360 | 436 | 5 |

2 | 475 | 492 | 10 |

**Table 2.**Performance metrics for different post-processing methods are shown. Evaluation is based on the cumulative geoid error from residuals of a 143 month closed-loop time series. All values are given in mm.

All Values mm Geoid | Median | Min | Max |
---|---|---|---|

S&W 250 km | 0.40 | 0.23 | 4.38 |

Gauss 350 km | 0.36 | 0.22 | 6.75 |

VADER M static N static (272 km) | 0.27 | 0.19 | 6.84 |

VADER M variable N static (274 km) | 0.27 | 0.18 | 7.14 |

VADER M static N variable (280 km) | 0.23 | 0.17 | 0.44 |

VADER M variable N variable (276 km) | 0.24 | 0.17 | 0.43 |

**Table 3.**RMS and mean value (in brackets) of phase shift in days derived from the 143 month closed-loop time series for four river basins.

River | VADER | VADER Med. | S&W | Gauss |
---|---|---|---|---|

Parana | 8.7 (−0.7) | 9.4 (−0.5) | 13.5 (−4.8) | 12.1 (−0.1) |

Mississippi | 1.4 (<0.1) | 1.4 (<0.1) | 3.3 (<0.1) | 1.8 (<0.1) |

Amazon | 1.9 (−0.7) | 1.8 (−0.6) | 3.1 (−1.9) | 7.4 (−1.5) |

Ganges | 8.0 (2.5) | 8.3 (2.5) | 29.8 (5.8) | 26.5 (5.9) |

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## Share and Cite

**MDPI and ACS Style**

Horvath, A.; Murböck, M.; Pail, R.; Horwath, M. Decorrelation of GRACE Time Variable Gravity Field Solutions Using Full Covariance Information. *Geosciences* **2018**, *8*, 323.
https://doi.org/10.3390/geosciences8090323

**AMA Style**

Horvath A, Murböck M, Pail R, Horwath M. Decorrelation of GRACE Time Variable Gravity Field Solutions Using Full Covariance Information. *Geosciences*. 2018; 8(9):323.
https://doi.org/10.3390/geosciences8090323

**Chicago/Turabian Style**

Horvath, Alexander, Michael Murböck, Roland Pail, and Martin Horwath. 2018. "Decorrelation of GRACE Time Variable Gravity Field Solutions Using Full Covariance Information" *Geosciences* 8, no. 9: 323.
https://doi.org/10.3390/geosciences8090323