Quantifying discrepancies in the three-dimensional seasonal variations between IGS station positions and load models

Abstract

Seasonal deformation related to mass redistribution on the Earth’s surface can be recorded by continuous global navigation satellite system (GNSS) and simulated by surface loading models. It has been reported that obvious discrepancies exist in the seasonal deformation between GNSS estimates and modeled loading displacements, especially in the horizontal components. The three-dimensional seasonal deformation of 900 GNSS stations derived from the International GNSS Service (IGS) second reprocessing are compared with those obtained from geophysical loading models. The reduction ratio of the weighted mean amplitude of GNSS seasonal signals induced by loading deformation correction is adopted to evaluate the consistency of seasonal deformation between them. Results demonstrate that about 43% of GNSS-derived vertical annual deformation can be explained by the loading models, while in the horizontal components, it is less than 20%. To explore the remaining GNSS seasonal variations unexplained by loading models, the potential contributions from Inter-AC disagreement, GNSS draconitic errors, regional/local-scale loading and loading model errors are investigated also using the reduction ratio metric. Comparison of GNSS annual signals between each IGS analysis center (AC) and the IGS combined solutions indicate that more than 25% (horizontal) and 10% (vertical) of the annual discrepancies between GNSS and loading models can be attributed to Inter-AC disagreement caused by different data processing software implementations and/or choices of the analysis strategies. Removing the draconitic errors shows an improvement of about ~ 3% in the annual vertical reduction ratio for the stations with more than fifteen years observations. Moreover, significant horizontal discrepancies between GNSS and loading models are found for the stations located in Continental Europe, which may be dominated by the regional/local-scale loading. The loading model errors can explain at least 6% of the remaining GNSS annual variations in the East and Up components. It has been verified that the contribution of thermoelastic deformation to the GNSS seasonal variations is about 9% and 7% for the horizontal and vertical directions, respectively. Apart from these contributors, there are still ~ 50% (horizontal) and ~ 30% (vertical) of the GNSS annual variations that need to be explained.

Appendices

Appendices

1.1 Appendix A: Proof of Inter-AC disagreement

The GNSS residual time series of IG2 and each AC2 solution can be assumed as

$$ {\text{IGS}} = D + X, $$

(9)

$$ {\text{AC}}_{i} = D + X_{i} , $$

(10)

where D represents the common signals between IG2 and all AC2 solutions, which includes error-free GNSS displacements and common systematic error between them. X and X_i indicate errors induced by different data processing strategies and software models for IG2 and each AC2 solution, respectively. Then, the 1-ratiow(AC) in Sect. 4.1 can be written as

$$ 1 - {\text{ratiow}}\left( {{\text{AC}}} \right) = \frac{{wtma\left( {X - X_{i} } \right)}}{{wtma\left( {D + X} \right)}} \times 100\% . $$

(11)

If X_A and X_Ai indicate the annual variation in X and X_i, then the annual variation difference between X and X_i should be

$$ X_{A} - X_{Ai} = A\cos (2\pi t + \varphi ) - A_{i} \cos (2\pi t + \varphi_{i} ) = (A\cos \varphi - A_{i} \cos \varphi_{i} )\cos 2\pi t - (A\sin \varphi - A_{i} \sin \varphi_{i} )\sin 2\pi t $$

(12)

where A and φ indicate the annual amplitude and phase of X, A_i and φ_i indicate the annual amplitude and phase of X_i. The weighted mean amplitude (wtma) of annual variations of the difference between IG2 and each AC2 solution can be calculated as

$$ wtma(X - X_{i} ) = wtma(X_{A} - X_{Ai} ) = \sqrt {(A\cos \varphi - A_{i} \cos \varphi_{i} )^{2} + (A\sin \varphi - A_{i} \sin \varphi_{i} )^{2} } = \sqrt {A^{2} + A_{i}^{2} - 2AA_{i} \cos (\varphi - \varphi_{i} )} . $$

(13)

As $- 1 \le \cos (\varphi - \varphi_{i} ) \le 1$, the range of the wtma(X-X_i) is

$$ \left| {A - A_{i} } \right| \le wtma(X - X_{i} ) \le \left| {A + A_{i} } \right|. $$

(14)

The IG2 solution can be considered as an average of all AC2 solutions. Thus, the seasonal error of IG2 can be roughly considered as

$$ A \approx \frac{{\sum\nolimits_{i = 1}^{n} {A_{i} } }}{n}, $$

(15)

where n represents the number of ACs. According to Eqs. (14) and (15), we can get

$$ \sum\limits_{i = 1}^{n} {wtma\left( {X - X_{i} } \right)} \le \sum\limits_{i = 1}^{n} {(A + A_{i} )} = n \cdot A + \sum\limits_{i = 1}^{n} {A_{i} } \approx 2 \cdot \sum\limits_{i = 1}^{n} {A_{i} } . $$

(16)

The Inter-AC disagreement (here represented by Ad) should be

$$ Ad = \frac{{\sum\nolimits_{i = 1}^{n} {wtma(X_{i} )} }}{n \cdot wtma(D + X)} = \frac{{\sum\nolimits_{i = 1}^{n} {A_{i} } }}{n \cdot wtma(D + X)} \ge \frac{{\sum\nolimits_{i = 1}^{n} {wtma(X - X_{i} )} }}{2n \cdot wtma(D + X)} = \frac{{\sum\nolimits_{i = 1}^{n} {\left[ {1 - {\text{ratiow}}(AC_{i} )} \right]} }}{2n}. $$

(17)

As a result, half of the average values of 1-ratiow(AC) for all ACs are the minimum value of the Inter-AC disagreement of annual variations.

See Figs. 10, 11, 12 and 13; Tables 6 and 7.

Fig. 10

Annual variations in surface displacements derived from IG2 for European stations (217 stations) in the East, North and Up components. The color of the dots represents the annual amplitudes. The direction of the arrows represents the annual phases (anti-clockwise from East)

Fig. 11

The distribution of 900 GNSS stations included in this study. The red dots represent the 134 common stations between IG2 and all AC2 solutions

Fig. 12

Annual variations in surface displacements derived from EM2 (left panels) and JP2 (right panels) solutions in the East (top panels) and North (bottom panels) components. The color of the dots represents the annual amplitudes, and the direction of the black arrows represents the annual phases which are anti-clockwise from East

Fig. 13

Reduction ratios of GNSS annual variations induced by GRACE correction for global stations (884 stations) in the East (top), North (middle) and Up (bottom) components. The color of the dots represents the magnitude of the reduction ratio. The black and white dots represent reduction ratios exceeding the maximum or minimum of the scale

Table 6 Weighted means of annual amplitudes of IG2 and IG2-AC2, as well as 1-ratiow(AC) of the annual variations for the common stations between each AC2 and IG2 solutions in global and European regions, respectively

Table 7 Average and RMS of annual amplitude (mm) of IG2 and each AC2 solutions as well as AOH for stations located in European region (217 stations) and other regions (683 stations)