** OBP GPS**

**4.1.2 Combination and Weighting of GRACE, GPS and OBP**

After preparing the data, the information from GRACE, GPS and OBP may be merged in a
combined normal equations system,**N**_{comb},**b**_{comb}(see App.C.6). In addition to the surface
loading coefficients, 7 Helmert parameters and the mass correction parameter are included
in the unknown vector **x**_{comb}. The individual normal equation systems are padded with
zeros to accommodate these parameters before combining them according to

**N**_{comb} = ^{1}

*σ*_{GRC}^{2} **N**_{GRC}+ ^{1}

*σ*_{GPS}^{2} **N**_{GPS}+ ^{1}

*σ*_{OBP}^{2} **N**_{OBP},
**b**_{comb} = ^{1}

*σ*_{GRC}^{2} **b**_{GRC}+ ^{1}

*σ*_{GPS}^{2} **b**_{GPS}+ ^{1}

*σ*_{OBP}^{2} **b**_{OBP},
[_{l}_{0}^{T}_{Pl}_{0}]_{comb} = ^{1}

*σ*_{GRC}^{2}

[_{l}_{0}^{T}_{Pl}_{0}]_{GRC}+ ^{1}
*σ*_{GPS}^{2}

[_{l}^{T}_{0}_{Pl}_{0}]_{GPS}+ ^{1}
*σ*_{OBP}^{2}

[_{l}^{T}_{0}_{Pl}_{0}]_{OBP}_{.} _{(4.7)}
The observation groups are scaled by the corresponding a priori*σ’s. For the joint *
in-version schemes used here, these are estimated by means of VCE (Variance Component
Estimation, see App. C.6). The resulting time variation of*σ*_{OBP},*σ*_{GRC}, and*σ*_{GPS}, is plotted
in Fig. 4.2. Clearly,*σ*_{GPS}remains more or less constant in time, indicating a posteriori error
of about 1.4 times the formal error. The variance component of the FESOM derived bottom

pressure fluctuates most strongly, and displays a seasonal signal. A possible explanation can be that seasonal signal may be slightly underestimated in the model.

The error model of the FESOM data contains only diagonal elements, which conse-quently means that correlations among the model nodes are unaccounted for. To mitigate this effect, an empirically derived factor of five is additionally used to downweight the model error-covariance, after the VCE weights are computed. This reduces the weight the model has in the inversion and also increases the correlations of the joint inversion with in situ bottom pressure recorders slightly (see Sec.5.1.5). It is expected that this downweight-ing can be avoided when appropriate error-covariance functions become available for the model error.

**0.5**
**1.0**
**1.5**
**2.0**
**2.5**

σ**0**

**1200** **1250** **1300** **1350** **1400** **1450** **1500** **1550** **1600**
**GPS week**

**Estimated variance components**

**: GRACE** **: GPS** **: FESOM**

Figure 4.2: Estimated variance components, *σ*0, of the joint inversion of GRACE (RL05),
GPS and FESOM data.

**0**
**10**
**20**
**30**

**degree n**

**GRACE contrib. [%]**

**0**
**10**
**20**
**30**

**degree n**

**GRACE contrib. [%]**

**GPS****week 1353**

**0 25 50 75100**

**OBP contr. [%]**

**OBP contr. [%]**

**GPS****week 1353**

**0 25 50 75100** **0**

**10**
**20**
**30**

**degree n**

**GPS contrib. [%]**

**0**
**10**
**20**
**30**

**degree n**

**GPS contrib. [%]**

**GPS****week 1353**

**0 5 10 15 20**

**0**
**10**
**20**
**30**

**degree n**

**−30 −20 −10** **0** **10** **20** **30**
**order m**

**0**
**10**
**20**
**30**

**degree n**

**−30 −20 −10** **0** **10** **20** **30**
**order m**

**GPS****week 1600**

**0 25 50 75100**

**−30 −20 −10** **0** **10** **20** **30**
**order m**

**−30 −20 −10** **0** **10** **20** **30**
**order m**

**GPS****week 1600**

**0 25 50 75100** **0**

**10**
**20**
**30**

**degree n**

**−30 −20 −10** **0** **10** **20** **30**
**order m**

**0**
**10**
**20**
**30**

**degree n**

**−30 −20 −10** **0** **10** **20** **30**
**order m**

**GPS****week 1600**

**0 5 10 15 20**

Figure 4.3: Diagonal data contributions at the level of surface loading coefficients for
GRACE, GPS and OBP in the joint inversion. *σ*_{OBP}^{2} is additionally downweighted by
a factor 5. Note the scale difference for GPS.

To get an impression on how much each data type contributes to the inversion, the di-agonal of the redundancy matrix (App. C.6) is plotted in Fig. 4.3. Since the weights also

change over time, two weeks have been considered with strongly differing FESOM weights (see Fig. 4.2). The formal contributions of the datasets to the lowest degree coefficients have been zoomed in for clarity.

From the triangle plots it becomes clear that the FESOM data contributes mostly to the inversion at the higher degrees. Where most of the geophysical signal is occuring, the lower and mid degrees, GRACE is the major contributor. At first sight, the GPS data appear to have only a minor contribution to the joint inversion. It plays however an important role for the degree 1 coefficients, as it is the only data source which provides coverage over the continents. Furthermore, all surface loading coefficients are global parameters. Therefore, by eye-balling Fig. 4.3it is hard, if not impossible, to see that some data types contribute more strongly in different regions (e.g. the GPS contribution will be larger in Europe be-cause of the denser GPS network there).

**0.001**
**0.01**
**0.1**
**1**
**10**
**100**
**1000**

**mm****2**

**Signal and error degree variances (2006)**

**0.001**
**0.01**
**0.1**
**1**
**10**
**100**
**1000**

**mm****2**

**0** **5** **10** **15** **20** **25** **30**

**degree**

**Signal and error degree variances (2006, non−seasonal)**

**: WGHM+GAC** **: GRC+GPS+OBP** **: GRACE**

Figure 4.4: Degree variances of the signal and error of the combination solution, a GRACE-only solution, and a background model containing hydrology and the GAC product.

The shaded regions span the variations of the weekly solutions over the year 2006. A solid and dashed boundary indicate the signal and noise respectively. The combination solution and GRACE have no background models restored.

A peculiar feature is seen in the sectorial bands (n=m) implying a relatively strong con-tribution of GRACE accompanied by a weak concon-tribution of OBP. At first sight, one might suspect that the area weighting of the FESOM data may have been too strong, and that the

equatorial band, which can be represented by a linear combination of sectorial harmonics, is now under-weighted with respect to the other nodes. However, a closer inspection of the normal matrices indicates that the cause lies in the covariance information of GRACE.

From the formal error of the weekly GRACE-only solutions, one can see that that sectorials are more accurate than their immediate neighbors (not shown here). A secondary line, es-sentially the sectorial shifted by three orders, is also visible in GPS week 1600. This feature is also related to the GRACE covariance information. The exact cause of these features re-mains unclear, but it is probably related to the weekly ground track patterns of GRACE.

Since OBP and GPS data are combined, the formal errors of the joint inversion are ex-pected to be smaller than those of a single inversion. Additionally, the GRACE errors, increasing for higher degrees, are likely to be dampened. This can can be seen from Fig.

4.4, where degree variances of both the signal and error are plotted. Since different weeks are not behaving in the same way, the spread of the weekly solutions in the entire year 2006 are plotted as regions. A secondary plot shows the non-seasonal signal, where a seasonal fit, performed at the coefficient level, was removed first. Clearly, the signal of the com-bination solution is smaller than that of the GRACE signal at the higher degrees. This is the consequence of the information from FESOM, which effectively constrains the noisier GRACE data at the higher degree part of the spectrum, as can be expected from Fig. 4.3.

This behavior is also reflected in the formal errors of the combination solution, which lie below the errors from GRACE. The removal of the seasonal signal, decreases the signal mostly in the lower degrees.

**GRC+OBP+GPS Error correlation**
**GRC+OBP+GPS Error correlation**

**M**_{0}**T**_{X}**T**_{Y}**T**_{Z}**S**_{C}**R**_{X}**R**_{Y}**R**_{Z}**C**_{10}**C**_{11}**S**_{11}**C**_{20}**C**_{21}**C**_{22}**S**_{21}**S**_{22}**C**_{30}**C**_{31}**C**_{32}**C**_{33}**S**_{31}**S**_{32}**S**_{33}

**M****0** **T****X** **T****Y** **T****Z** **S****C** **R****X** **R****Y** **R****Z** **C****10** **C****11** **S****11** **C****20** **C****21** **C****22** **S****21** **S****22** **C****30** **C****31** **C****32** **C****33** **S****31** **S****32** **S****33**

**Helmert Param.**

**Geocenter**

**OBP+GPS Error correlation**
**OBP+GPS Error correlation**

**M**_{0}**T**_{X}**T**_{Y}**T**_{Z}**S**_{C}**R**_{X}**R**_{Y}**R**_{Z}**C**_{10}**C**_{11}**S**_{11}**C**_{20}**C**_{21}**C**_{22}**S**_{21}**S**_{22}**C**_{30}**C**_{31}**C**_{32}**C**_{33}**S**_{31}**S**_{32}**S**_{33}

**M****0** **T****X** **T****Y** **T****Z** **S****C** **R****X** **R****Y** **R****Z** **C****10** **C****11** **S****11** **C****20** **C****21** **C****22** **S****21** **S****22** **C****30** **C****31** **C****32** **C****33** **S****31** **S****32** **S****33**

**Helmert Param.**

**Geocenter**

**−100**

**−80**

**−60**

**−40**

**−20**
**0**
**20**
**40**
**60**
**80**

**%** **100**

Figure 4.5: Correlations of the error-covariance matrix of a joint inversion from GRACE,
GPS and modeled OBP data versus the correlation obtained from a GPS and OBP
com-bination. The week considered is GPS week 1353 (centered at the 14^{th} Jan 2004).

Up to now, only the diagonal part of error-covariance matrix has been discussed. How-ever, the full error-covariance information from the inversion may also provide interesting insights. Fig. 4.5shows a subsection of the formal error-correlation matrix of two types

of the joint inversion. On the left hand side, all data types are used, whereas at the right hand side the GRACE data is removed. Clearly, the correlation between the surface load-ing coefficients is lower when GRACE data is used. This illustrates that the GRACE data allows a better separation of the coefficients, at least on the formal side. Since GRACE per-forms well at the low degrees coefficients this is something which can be expected. A more surprising feature hides in the decreased correlations of the degree 1 coefficients. Although GRACE cannot measure the degree 1 surface loading coefficients directly, its addition in the joint inversion does improve the separability of the degree 1 coefficients from themselves as well as from other parameters (e.g. model bias and other surface loading coefficients).

This feature is important but will be hardly visible in the formal (diagonal) errors of the solution.