** Glacier clustersGlacier clusters**

**5.2 Inversion for Fingerprint Magnitudes**

**5.2.5 Steric Sea Level Changes**

Similar to the hydrological parameters, the estimated steric principal components from the inversion, can be compared to the ones as computed from the steric height changes derived fromIshii and Kimoto(2009) (see Section4.2.1). The time variation of the first 18 modes, explaining 81% of the variance, is plotted in Fig.5.21.

From Fig 5.21, it is clear that the estimated principal components follow the original ones closely. In particular, the inter-annual variations are well captured by the inversion.

Peculiar is that the higher modes from the Ishii data are more susceptible to oscillations

compared to their estimated counterparts. One can suspect that these oscillations point to spatial-temporal sampling problems in the original ARGO data. Clearly, the inversion results do not seem to be affected by this, and the altimeter data yields smoothly varying estimates.

From the inversion, steric sea level can be reconstructed by multiplying the estimated PCs with their corresponding EOFs. Here, both the EOFs fromIshii and Kimoto(2009) as well as the ’bootstrapped’ patterns are used. As discussed in Sec.4.2.1, these bootstrapped patterns are necessary to extract all the relevant signals from the altimetry, and prevent an underestimation of the steric signal.

For each grid point of the reconstructed steric height, an annual harmonic and trend is fitted. The corresponding trend and the RMS of the residual are plotted at the top of Figure 5.22. The reconstructed trend is very similar to the sea level trend from Jason-1 only (see Fig. 5.22). The typical structure in the North Pacific points to the influence of the Pacific Decadal Oscillation, which has been entering a cooling phase since 2003. The strong neg-ative trend in the equatorial Pacific also points to the strong La Niña, which occurred in 2010. Residual effects of the El Niño Southern Oscillation (ENSO) can still be spotted in the spatial RMS. Both oscillations have spatially similar patterns and are therefore difficult to separate on these time scales.

To study the steric changes in the deeper part of the ocean, the steric component of the
upper 700m of the ocean from the Ishii data is subtracted from the . The remaining sea
level variations should provide some hints on the structures which are associated with
deep ocean changes. Similar as above, the fitted trend, and residual*σ*is plotted in the
mid-dle section of Fig. 5.22. Although smaller in magnitude, the resulting trend has a much
more uniform structure. In particular, the large negative trend in the equatorial Pacific has
now become positive. Furthermore, a strong reduction of the RMS can be observed in the
equatorial Pacific, which hints to the removal of a large part of the ENSO.

Finally, the total reconstructed sea level has been subtracted from the gridded altimetry data. The associated trend and residual RMS is shown at the bottom of Fig. 5.22. The remaining trend shows little or no large scale signals, which suggests that most of the al-timetry trend signal has been absorbed in the unknowns. The standard deviation of the residual, on the other hand still shows regions with large variability originating from the altimetry data. However, these variations can be mostly attributed to dynamic effects such as meso- scale eddies and planetary rossby waves. Such features are in fact expected to end up in the altimetry residuals.

Regarding the retrieval of steric variations, the fingerprint inversion pursued in this the-sis come with a set of advantages compared to the more straightforward approach to derive steric changes from the difference of altimetry and GRACE (e.g.Lombard et al.,2007).

Firstly, in the inversion approach, the mass induced component and steric induced com-ponent are estimated simultaneously from GRACE and Altimetry. Although, the altimetry appear to have little influence on the mass-related parameters from a formal perspective (see Sec.4.2.3), it does change the estimated hydrology in terms of the associated mean sea

**−0.2**
**0.0**
**0.2**

**PC [−]**

**Steric PC_01 (28.1% / 28%) ** **Steric PC_02 (14.8% / 43%) **

**−0.2**
**0.0**
**0.2**

**PC [−]**

**Steric PC_03 (9.2% / 52%) ** **Steric PC_04 (3.8% / 56%) **

**−0.2**
**0.0**
**0.2**

**PC [−]**

**Steric PC_05 (3.4% / 59%) ** **Steric PC_06 (3.1% / 62%) **

**−0.2**
**0.0**
**0.2**

**PC [−]**

**Steric PC_07 (2.5% / 65%) ** **Steric PC_08 (2.3% / 67%) **

**−0.2**
**0.0**
**0.2**

**PC [−]**

**Steric PC_09 (1.7% / 69%) ** **Steric PC_10 (1.6% / 70%) **

**−0.2**
**0.0**
**0.2**

**PC [−]**

**Steric PC_11 (1.6% / 72%) ** **Steric PC_12 (1.5% / 73%) **

**−0.2**
**0.0**
**0.2**

**PC [−]**

**Steric PC_13 (1.4% / 75%) ** **Steric PC_14 (1.3% / 76%) **

**−0.2**
**0.0**
**0.2**

**PC [−]**

**Steric PC_15 (1.2% / 77%) ** **Steric PC_16 (1.2% / 79%) **

**−0.2**
**0.0**
**0.2**

**PC [−]**

**2004** **2006** **2008** **2010** **2012**
**Steric PC_17 (1.1% / 80%) **

**2004** **2006** **2008** **2010** **2012**
**Steric PC_18 (1.0% / 81%)**

**: Inversion ** **: PCA Ishii et al. 2009**

Figure 5.21: Time variation of the first 18 principal components of the steric sea level changes. The scales, as estimated by the inversion, are plotted along the principal com-ponents which were computed from the PCA of the dataset fromIshii and Kimoto(2009).

The percentages denote the variance explained (per mode/cumulative).

❘ ✁✂✄☎✆ ✝✞ ☎✆ ✝✟ ✁✠ ✟ ✡✠✆ ☛☞☞ ✌✍ ✝✎

❘ ✁✂✄☎✆ ✝✞ ☎✆ ✝✟ ✁✠ ✟ ✡✠✆ ☛☞☞ ✌✍ ✝✎

❊☎✆✞☞ ✟✄✏☎✑☎ ✠ ✟ ✟ ✒ ✓✔✔☞ ✕☛☞☞ ✌✍ ✝✎

❊☎✆✞☞ ✟✄✏☎✑☎ ✠ ✟ ✟ ✒ ✓✔✔☞ ✕☛☞☞ ✌✍ ✝✎

❆✖✆ ✟☞ ✆ ✝✍☞ ✟ ✄✏☎❊☎✆✞☛☞☞ ✌✍✝✎

❆✖✆ ✟☞ ✆ ✝✍☞ ✟ ✄✏☎❊☎✆✞☛☞☞ ✌✍✝✎

✲✗✘ ✘ ✗✘

s☛☞☞ ✎ s☛☞☞ ✎

s☛☞☞ ✎ s☛☞☞ ✎

s☛☞☞ ✎ s☛☞☞ ✎

✘ ✗ ✘ ✷ ✘ ✸ ✘ ✹✘ ✺ ✘ ✻ ✘ ✼✘ ✽✘

Figure 5.22: Reconstructed steric sea level trend and the standard deviation of the residual (after removing the mean, trend and annual harmonic curve per grid point). The middle plots shows the trend and standard deviation of the reconstructed steric sea level but now with the gridded steric height from the upper 700m ofIshii and Kimoto(2009) re-moved. The bottom plots display the trend and the standard deviation of the altimetric residual after removing the total fit (mass-induced, GIA and steric sea level and the effect of the nuisance parameters).

level change (not shown here).

Another advantage stems from the step-wise parameterization of the steric variations in the inversion, where the initial 100 EOFs from the Ishii data are augmented with an additional 60 bootstrapping EOFs derived from the altimetry residuals. This step-wise ap-proach allows a decomposition of the steric component into a shallow and a residual deep ocean component. Such decompositions provide valuable insights in the ongoing discus-sion of the deep ocean warming which is expected from the radiation imbalance of the Earth system. The results here are consistent with a signaificant warming in the deeper ocean.

Finally, in the straightforward approach (Altimetry minus GRACE), a large and
uncer-tain GIA correction has been applied to GRACE (~1.7^{mm}/yr in terms of ocean-averaged
equivalent water height), whereas the altimetry needs to be corrected only by ~0.3^{mm}/yr

(Lombard et al., 2007). The GIA correction in the inversion approach is consistently co-estimated from the data. Furthermore, since the mass induced sea level in the inversion is parameterized by equipotential surfaces, the effect of the GIA error remains small com-pared to the situation where equivalent water heights are computed directly from GRACE.