** Glacier clustersGlacier clusters**

**4.2.2 Observation Equations of the Fingerprint Inversion**

**GRACE**

Changes in the Stokes coefficients from GRACE are linked to the mass related components of the fingerprints. The unknown time varying magnitude can be linearly linked to the Stokes coefficients of GRACE:

*δC*_{20}

...
*δC*_{n}_{max}_{n}_{max}

=_{A}

**x**_{ice}
**x**_{glac}
**x**_{hydro}

**x**_{gia}

+_{e.}_{(4.28)}

Since mass is conserved globally, and the GRACE data has its reference frame origin in the
center of mass of the Earth system, only coefficients with degrees larger than 1 are
consid-ered. This automatically transforms all of the fingerprints to the CM frame. The unknown
time varying magnitudes, are grouped in different mass contributions. **x**_{ice} represents the
factors associated with the ice sheets in Greenland and Antarctica,**x**_{glac} contain the
mag-nitudes of the remaining (clusters of) land glaciers, and **x**_{hydro} is linked to the terrestrial

6see also Eq.4.13for the spectral representation of a basin average

hydrological variations. The variations due to glacial isostatic adjustment effects are
con-tained in the vector**x**_{gia}.

The columns of the design matrix **A** are composed of the corresponding fingerprints
expressed in terms of normalized Stokes coefficients:

**A**(_{t}) =

*δC*_{20}^{Greenland1} *δC*_{20}^{Greenland2} · · · *δC*_{20}^{gia}^{N}(t−t_{0})

... ... ...

*δC*_{n}^{Greenland1}_{max}_{n}_{max} *δC*_{n}^{Greenland2}_{max}_{n}_{max} · · · *δC*_{n}^{gia}_{max}^{N}_{n}_{max}(t−t_{0})

. (4.29)

For convenience, the fingerprints of the ice sheets and the land glaciers are normalized
such that its mass contribution represents 1 Gton. In this way, the unknown factors**x**can
be directly interpreted as eustatic sea level contributions in the units Gtons. It should be
noted that no such normalization is applied in the case of the GIA contribution. Since the
present day GIA component of relative sea level rise is zero, a division by zero would be
the result. Furthermore, a normalization is also not recommended when using EOFs as
base functions (which is the case for the hydrological and steric parameters). Often, there
are EOFs which have a (almost) zero contribution to sea level. A normalization would then
induce large scale differences between the columns of**A.**

The GIA induced signal has a secular nature, the corresponding GIA potential rates are
therefore scaled by the time difference with respect to the reference timet0. For the release
05 GRACE data, the 1st of January 2005 is used, which is the reference time of the static
background model Eigen-6c. Currently, the GIA component is the only time dependent
factor in the design matrix. It is therefore smarter to store only the time independent part,
without (t−t_{0}), of the matrix and introduce this time dependency at the level of the
nor-mal equations as described in App. C.5.

Monthly normal equations of GRACE release 05 are available up to degree and order 180. In addition to the standard de-aliasing models, the time variable component of the Eigen-6C is removed a priori (in the standard GRACE solutions, this signal is restored). To make the signal consistent with the altimetry data, the time varying component is restored on the normal equation level in accordance with App. C.3. On top of that, I also apply an a priori correction which ensures that the background de-aliasing model has a zero mean over the ocean. This makes the signal content consistent with that of the altimeters, and it prevents the background models to introduce artificial changes in the ocean mean. The latter potentially cause (systematic) errors in the estimated parameters.

This large truncation degree of the normal equation systems does not allow a direct so-lution of all the Stokes coefficients. However using the transformation method of App.

C.5, the normal equation systems may be transformed in terms of fingerprint magnitudes, without the need for solving but with retaining the full information up to degree and or-der 150. Coefficients with degrees larger than 150 are eliminated from the normal equation systems.

**N**^{†}_{GRC} =**A**^{T}**NA,**

**b**^{†}_{GRC} =**A**^{T}**b,** (4.30)

[_{l}^{T}** _{Pl}**]

^{†}=

_{l}^{T}

_{Pl.}_{(4.31)}

The transformation operation drastically reduces the amount of unknowns, and thus the storage requirements, of the normal equation system from approximately 23000 to only about 125 (the number of mass related fingerprints).

**Altimetry**

As explained in Sec.3.2.1, an altimeter is sensitive to changes in the geoid height. In addi-tion, the range measurement is also affected by changes in the volume of the ocean basin and ocean-wide shifts due to the changing ocean mass. The altimetric observable for each surface loading fingerprint therefore consists of the (elastic) term ˜S+U. The GIA compo-nent has the same form.

Similar to GRACE, a linear observation equation can be constructed for the corrected altimetry data (see Eq.3.7):

∆hi

...

∆hM

=_{YB}

**x**_{ice}
**x**_{glac}
**x**_{hydro}

**x**_{gia}

+_{KC}^{}_{x}_{ster}^{}+_{P}

t_{x}
t_{y}
tz

CN−CF

+_{e.}_{(4.32)}

The columns of matrix **B** contain the geocentric sea level changes associated with the
unknown patterns:

**B**(t) =

S˜_{00}^{Greenland1} S˜^{Greenland2}_{00} · · · S^{˜}^{gia}_{00}^{N}(t−t_{0})
aδC_{10}^{Greenland1} aδC_{10}^{Greenland2} · · · aδC_{10}^{gia}^{N}(t−t_{0})

... ... ...

aδC_{n}^{Greenland1}_{max}_{n}_{max} aδC^{Greenland2}_{n}_{max}_{n}_{max} · · · aδC_{n}^{gia}_{max}^{N}_{n}_{max}(t−t_{0})

. (4.33)

here, the Stokes coefficients scaled by the Earth radiusarepresent the induced geoid changes.

To be as close to the altimeter reference system as possible, the geoid changes are provided
in the CF system, by choosing the appropriate degree 1 coefficients. The matrix**Y**maps the
patterns, expressed in spherical harmonics, to the locations of the altimeter measurements:

**Y**=

Y¯_{00}(*θ*_{1},*λ*_{1}) · · · Y^{¯}nmaxnmax(*θ*_{1},*λ*_{1})

... ... ...

Y¯_{00}(*θ*_{M},*λ*_{M}) · · · Y^{¯}_{n}_{max}_{n}_{max}(*θ*_{M},*λ*_{M})

. (4.34)

The amount of rows of**Y**can reach several hundred thousand. From a storage perspective
it is therefore advised not to compute the matrix**Y**explicitly.

The steric EOFs, provided on a geographic grid, are stacked in the matrix**C. These are**
then interpolated to the altimeter points by means of the (sparse) bi-linear interpolation

matrix**K.**

Finally, a set of correction parameters is introduced, which may absorb remaining differ-ences between the origin of the altimeter measurements (a CN frame) and the CF frame.

The estimated offset is expected to be small as the origin of the reference system of the al-timeters is expected to be close to the center of surface figure (CF). This issue is further elab-orated upon in Sec4.2.4. For each observation, the translation vector,

t_{x} t_{y} t_{z}T

CN−CF, is
projected onto the local radial direction by matrix**P**

**P**=

**e**^{T}_{1,h}

...
**e**^{T}_{M,h}

=

sin*θ*_{1}cos*λ*_{1} sin*θ*_{1}sin*λ*_{1} cos*θ*_{1}

... ... ...

sin*θ*_{M}cos*λ*_{M} sin*θ*_{M}sin*λ*_{M} cos*θ*_{M}

. (4.35)

When all the unknowns are stacked in one vector the resulting observation equation is described by:

∆hi

...

∆hM

=^{}**YB KC P**

**x**ice

**x**_{glac}
**x**_{hydro}

**x**_{gia}
**x**ster

**t**_{CN}−CF

+*e*=**D**

**x**ice

**x**_{glac}
**x**_{hydro}

**x**_{gia}
**x**ster

**t**_{CN}−CF

+*e.* (4.36)

For each altimeter, the normal equations can now be build by using the range error as a
diagonal error-covariance,**C**_{alt}=diag

*σ∆h*^{2}_{1}· · ·_{σ∆h}^{2}_{M}^{}. Similar to the GRACE processing,
the introduction of the secular time dependency is postponed until after the assembly of the
normal equations. It now also becomes clear what the advantage of the OpenADB altimeter
data is. In the so-called ’BINS’ format, the altimeter data is sampled in bins which do not
change position in time. This means that for each mission (here Jason 1, Jason 1 extended
mission, and Jason 2), a design matrix**D**can be precomputed (each one is approximately
700 Mb) and stored on disk. The normal equation systems are transformed according to

N_{alt}=_{D}^{T}_{C}^{−}_{alt}^{1}_{D,}_{(4.37)}

**b**_{alt}=**D**^{T}**C**^{−}_{alt}^{1}∆h, (4.38)

[_{l}_{0}^{T}_{Pl}_{0}]_{alt}=_{∆h}^{T}_{C}^{−}_{alt}^{1}_{∆h.} (4.39)
A screening of the data has been performed, before the assembly of the normal equation
systems to reduce the effect of outliers and seasonal sea-ice at the higher latitudes. For this
means, data with excessive large sea level anomalies and range errors (magnitudes larger
than 1 m), have been excluded. In addition, a sea-ice mask, derived from the maximum
sea-ice extent (Cavalieri et al.,1996) has been applied to the data. This step removes valid
measurements in the summer months, and reduces the maximum latitude reached. At the
same time however, a potential seasonal aliasing of high latitude signal is also avoided
(data at the higher latitudes is constrained only during the summer months).