** Glacier clustersGlacier clusters**

**4.2.3 Combination and Weighting of Altimetry and GRACE**

The solution system for the common parameters, ˆ**x**c, can be constructed by reducing all
time varying parameters as described in App.C.2.

**N**_{cc}−**N**^{T}_{rc}**N**^{−}_{rr}^{1}**N**_{rc}
ˆ

**x**_{c} =**b**_{c}−**N**_{rc}^{T}**N**^{−}_{rr}^{1}**b**_{r}. (4.42)
The block diagonal structure of **N**_{rr} allows this system to be written as the sum of the
individual monthly systems, each separately reduced for its time varying coefficients:

### ∑

^{}

^{N}^{(}

^{cc}

^{i}

^{)}

^{−}

^{N}^{(}

^{rc}

^{i}

^{)}

^{T}

^{N}^{(}

^{rr}

^{i}

^{)−}

^{1}

^{N}^{(}

^{rc}

^{i}

^{)}

^{}

^{x}^{ˆ}

^{c}

^{=}

### ∑

^{}

^{b}^{(}

^{c}

^{i}

^{)}

^{−}

^{N}^{rc}

^{(}

^{i}

^{)}

^{T}

^{N}^{(}

^{rr}

^{i}

^{)−}

^{1}

^{b}^{(}

^{r}

^{i}

^{)}

^{}

^{.}

^{(4.43)}

Once the common parameters are estimated, they can be fixed to these values in each
of the monthly systems (according to App. C.3andC.4). This procedure guarantees that
the solution vector ˆ**x**ris the same as would have been obtained from solving the assembled
system from Eq. 4.41.

Although, numerically efficient, the drawback of this procedure is that the monthly error-covariances will be too optimistic. During the fixing step, the common parameters are as-sumed to be known, and the corresponding part of the normal matrix is ignored. To obtain the correct error-covariance matrix for month k, a post-processing step can be performed to obtain the full error-covariance as follows:

**C**^{(}^{k}^{)}=
"

**N**^{(}_{rr}^{k}^{)} **N**^{(}_{rc}^{k}^{)}
**N**^{(}_{rc}^{k}^{)}^{T} **N**^{(}_{cc}^{k}^{)}

# +

"

0 0

0 ∑i6∈k

**N**^{(}_{cc}^{i}^{)}−**N**^{(}_{rc}^{i}^{)}^{T}**N**_{rr}^{(}^{i}^{)−}^{1}**N**^{(}_{rc}^{i}^{)}

#!−1

. (4.44) In this expression, the block diagonal structure seen in Eq. 4.41has been exploited. The first term on the right hand side is simply the normal matrix of the month k. The second term is the combined and reduced matrix using data from all months except for monthk, whose contribution is already included in the first term.

**0.8**
**0.9**
**1.0**
**1.1**
**1.2**
**1.3**
**1.4**

σ**0**

**2003** **2004** **2005** **2006** **2007** **2008** **2009** **2010** **2011**
**Year**

**Estimated variance components**

**: GRACE** **: J1** **: J2**

Figure 4.18: Estimated VCE component (square root), in the fingerprint inversion for the contributions of GRACE, and the Jason-1 and Jason-2 missions. The discontinuity in the Jason-1 curve at the start of 2009 indicates the migration to the extended mission.

To compute the relative weighting of the GRACE and altimetry normal equation sys-tems, its variance components were estimated (App. C.6). It should be remarked that the variance components themselves were estimated without applying constraints and while keeping the common parameters fixed to their reference values. The GIA parameters were fixed to 1, which implies that the a priori GIA model was removed from the systems. Fig.

4.18shows the estimated variance components. The variance component associated with
the altimeters remains relatively constant at the level of(1.2)^{2} times the formal variances,
where Jason-2 appears to be more accurate compared to Jason-1. The variance factor for
GRACE fluctuates more strongly but exhibits the same time behavior as when estimated
for the surface loading inversion as indicated in Fig.4.2.

**10**^{−1}**10**^{0}**10**^{1}**10**^{2}**10**^{3}**10**^{4}

σ** [Gton]**

**Land Glaciers** **Greenland** **Antarctica** **Hydr.** **Ster.**

**Alaska** **Alps** **Brooksrange**

**Caucasus Himalaya**

**Iceland** **Kamchatka** **Kerg_Heard_Is** **New_Zealand** **Norway** **Nova_Zembla** **Oct_revo_Is_Rus** **Patagonia** **Qn_Elizabeth_Is** **Svalbard** **Tien_Shan** **Green_0001 Green_2001** **Green_0002** **Green_2002** **Green_0003** **Green_2003 Green_0004** **Green_2004** **Green_0005** **Green_2005** **Green_0006** **Green_2006 Green_0007** **Green_2007** **Green_0008** **Green_2008** **02_EAIS** **03_EAIS** **04_EAIS** **05_EAIS** **06_EAIS** **07_EAIS** **08_EAIS 09_EAIS** **10_EAIS** **11_EAIS** **12_EAIS** **13_EAIS 14_EAIS** **15_EAIS** **16_EAIS** **17_EAIS** **01_WAIS** **18_WAIS 19_WAIS** **20_WAIS** **21_WAIS** **22_WAIS** **23_WAIS 24_PENIN 25_PENIN 26_PENIN 27_PENIN**

**PCA01_WGHM** **PCA02_WGHM PCA03_WGHM** **PCA04_WGHM** **PCA05_WGHM** **PCA06_WGHM** **PCA07_WGHM PCA08_WGHM** **PCA09_WGHM** **PCA01_ISHII** **PCA02_ISHII** **PCA03_ISHII** **PCA04_ISHII PCA05_ISHII** **PCA06_ISHII** **PCA07_ISHII** **PCA08_ISHII PCA09_ISHII**

**0.001**
**0.01**
**0.1**
**1**
**10**

σ** [mm]**

**: J1+J2 reduced** **: J1+J2** **: GRACE+J1+J2** **: GRACE**

Figure 4.19: Monthly formal errors of the fingerprint inversion (September 2008), for several combinations of Jason-1, Jason-2, and GRACE. The solution of the ’reduced’

altimetry-only inversion contain the total effect of Greenland, Antarctica and the land glaciers as single parameters, and the first three hydrological modes. All errors are ex-pressed in terms of their ocean mass change and eustatic sea level contribution.

The formal errors of the time varying coefficients (diagonal of**C) are plotted in Fig.**4.19,
for the month September 2008. The errors are obtained from an unconstrained inversion of
the combination, a GRACE-only solution and a altimetry-only inversion. In addition the
errors of a simplified altimetry-only inversion are shown. In the latter setup, the drainage
basins in Greenland and Antarctica are merged to represent uniform changes.
Further-more, the combined effect of all the land glaciers, and the parameterization of the GIA are
represented by single parameters.

Although, the formal errors are still expected to be too small compared to the true errors, they are still a good relative measure of the obtainable accuracy. For example, from the combination, it can be seen that the land glaciers can be estimated relatively well, com-pared to the other mass parameters. Small narrow basins in Greenland (number 2005, 2006) and on the Antarctic peninsula (number 25, 26), are associated with much larger er-rors compared to their larger counterparts.

It is also clear that the errors of the mass related parameters are almost entire

deter-mined by the GRACE contribution. Consequently, the altimetry, shows virtually no skill in determining these parameters. Errors for the altimetry-only inversion are so large that no usable results can be expected from such an inversion. Even when the parameterization is strongly simplified, the altimetry-only errors are still several hundreds of Gton. The rea-son for this unstable behavior is that the altimeter can only indirectly sense the continental mass changes. The associated sea level changes are much weaker compared to the direct effect of the mass on the potential field, and generally decrease as one moves away from the mass source.

**GRC+J1+J2 Error correlation**
**GRC+J1+J2 Error correlation**

**Alaska****Alps**
**Brooksrange****Kamchatka****Caucasus****Himalaya****Iceland**
**Kerg_Heard_Is****New_Zealand****Nova_Zembla****Norway**
**Oct_revo_Is_Rus****Qn_Elizabeth_Is****GIA_COMPLEM****PCA01_WGHM****PCA02_WGHM****PCA03_WGHM****PCA04_WGHM****PCA05_WGHM****PCA06_WGHM****PCA07_WGHM****PCA08_WGHM****PCA09_WGHM****PCA01_ISHII****PCA02_ISHII****PCA03_ISHII****PCA04_ISHII****PCA05_ISHII****PCA06_ISHII****PCA07_ISHII****PCA08_ISHII****PCA09_ISHII****GIA_FENNO****GIA_GREEN****Green_0001****Green_2001****Green_0002****Green_2002****Green_0003****Green_2003****Green_0004****Green_2004****Green_0005****Green_2005****Green_0006****Green_2006****Green_0007****Green_2007****Green_0008****Green_2008****Tien_Shan****GIA_LAUR****Patagonia****24_PENIN****25_PENIN****26_PENIN****27_PENIN****GIA_ANT****Svalbard****01_WAIS****18_WAIS****19_WAIS****20_WAIS****21_WAIS****22_WAIS****23_WAIS****02_EAIS****03_EAIS****04_EAIS****05_EAIS****06_EAIS****07_EAIS****08_EAIS****09_EAIS****10_EAIS****11_EAIS****12_EAIS****13_EAIS****14_EAIS****15_EAIS****16_EAIS****17_EAIS****HTX_J1****HTY_J1****HTX_J2****HTY_J2****HTZ_J1****HTZ_J2**

**Alaska****Alps** **Brooksrange**
**Caucasus Himalaya**

**Iceland** **Kamchatka** **Kerg_Heard_Is** **New_Zealand** **Norway****Nova_Zembla** **Oct_revo_Is_Rus****Patagonia** **Qn_Elizabeth_Is** **Svalbard** **Tien_Shan** **Green_0001 Green_2001 Green_0002 Green_2002 Green_0003 Green_2003 Green_0004 Green_2004 Green_0005 Green_2005 Green_0006 Green_2006 Green_0007 Green_2007 Green_0008 Green_2008**

**02_EAIS 03_EAIS 04_EAIS 05_EAIS 06_EAIS 07_EAIS 08_EAIS 09_EAIS 10_EAIS 11_EAIS 12_EAIS 13_EAIS 14_EAIS 15_EAIS 16_EAIS 17_EAIS 01_WAIS 18_WAIS 19_WAIS 20_WAIS 21_WAIS 22_WAIS 23_WAIS** **24_PENIN 25_PENIN 26_PENIN 27_PENIN**

**PCA01_WGHM PCA02_WGHM PCA03_WGHM PCA04_WGHM PCA05_WGHM PCA06_WGHM PCA07_WGHM PCA08_WGHM PCA09_WGHM** **PCA01_ISHII PCA02_ISHII PCA03_ISHII PCA04_ISHII PCA05_ISHII PCA06_ISHII PCA07_ISHII PCA08_ISHII PCA09_ISHII**

**GIA_ANT** **GIA_COMPLEM**
**GIA_FENNO GIA_GREEN** **GIA_LAUR**

**HTX_J1 HTY_J1 HTZ_J1 HTX_J2 HTY_J2 HTZ_J2**

**Land Glaciers**

**Greenland**

**Antarctica**
**Hydr.**

**Ster.**

**GIA**
**CN−CF**

**−100**

**−80**

**−60**

**−40**

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

**%** **100**

Figure 4.20: Formal error correlation of the estimated parameter scales, at full resolution, for September 2008. The GIA parameters (trends) are constrained by data over the entire time interval.

In contrast, the errors associated with the steric patterns, are very well constrained by the altimetry data. This is also the reason why the estimation of 160 steric parameters can

be performed without problems.

From the above, it would be tempting to conclude that in order to estimate steric sea level change it would suffice to compute the mass component from GRACE only and remove it in a post-processing step from the altimetry. Principally, there is nothing wrong with this procedure. However when one compares GRACE-only solutions with the combination so-lutions (not shown here), one sees that the addition of altimetry data do cause significant changes of the mass related parameters as well. The leverage, which the altimetry has on the mass parameters, can obviously not be easily spotted from the diagonal of the error-covariance, which stays approximately constant regardless of the addition of altimetry.

The correlation matrix of**C**can also provide useful insight in the stability of the
inver-sion problem. Wherever correlations occur, one can expect separability issues between the
parameters. The correlations reflect (1) the similarity of the used fingerprint patterns, and
(2) an unfavorable sampling by GRACE and/or altimetry. Fig. 4.20shows a section of the
monthly correlation matrix for September 2008.

From the correlation matrix we see that the strongest correlations (|*ρ*| > 0.7) occur for
drainage basins which are small and in each other vicinity. The land glaciers appear to be
well separable but some smaller correlations exist with Greenland for neighboring glacier
clusters such as Iceland, Svalbard and Queen Elizabeth Is. Furthermore, although the
glacier cluster areas were excluded from the hydrology patterns, one can still see some
correlations with the hydrological parameters.

Problems also occur for the CF offsets of the altimetry, when estimated for each month
and per satellite separately. In particular the Z component is expected to cause problems
due to its correlation with the steric parameters. This can be expected as the Z-component
is ill-constrained by the altimetry as the sampling is limited to±66^{◦} latitude and outside
the maximum sea ice mask. For this reason, the altimeter offsets will be estimated as mean
parameter over the entire time interval as will be explained below.

At first sight, the correlations of the GIA parameters (estimated as trends) appear not so large. However,these correlation are especially problematic, as they can strongly influence the secular behavior of affected basins. For example, the Antarctic component of the GIA, is correlated with the mass pattern of the Antarctic basins (1, 2, 3, 17, 18, 19). Furthermore, the complementary GIA patterns is strongly correlated to the glaciers in Iceland, and cor-relations also occur with the Greenland basins. These corcor-relations are very problematic for the inversion and require additional constraints to be applied to the GIA. The choice of these constraints will be discussed in the following section.