** CFCH**

**3.2 Sea Surface Height and Ocean Bottom Pressure**

**3.2.1 Radar Altimetry**

A radar altimeter essentially measures the round trip time of a radar pulse emitted from the
satellite which is then reflected by the sea surface. Additionally, tracking techniques such
as DORIS and GPS , are used in determining a precise orbit of the satellite. The principle
product are sea surface heights (h_{ssh}), which represent the height of the sea surface relative
to a chosen ellipsoid.h_{ssh}is derived from the measured range, the orbit and from a suite of
instrumental and geophysical corrections (See Fig3.2and Table3.2)

h_{ssh} =h_{orbit}−hrange+

### ∑

i

h_{corr,i}. (3.3)

In this work, along track Jason-1 and Jason-2 data from the Open Altimeter Database are used (Schwatke et al.,2010). The along track data have been gathered in mean along-track bins, with a length of about 5.8 km. Since the locations of these bins do not change over time, an advantage exists when setting up normal equations. The data have been corrected with the usual instrument errors and the EOT11a tidal model (Savcenko and Bosch,2012). Furthermore, radial orbit errors have been estimated and corrected for by applying a multi-mission crossover analysis (Bosch and Savcenko, 2007). The dynamic atmosphere correction comes from the MOG2D model (Carrère and Lyard,2003). A mean sea surface height model (Rio and Hernandez,2004) is also subtracted and provided at the locations of the bins.

In this thesis, the time variable effects of sea level and gravity are jointly investigated.

Therefore, the sea level anomalies (sea surface height relative to the mean sea surface,h_{mssh})
are decomposed as follows:

h_{ssh}−h_{mssh}= hIB+h_{steric}+ [S^{˜}+U]_{elas}+ [S^{˜}+U]_{GI A}+h_{dyn}. (3.4)

Figure 3.2: Radar altimetry measure-ment principle. The return trip of the radar pulse from the mi-crowave altimeter, is a measure of the difference between the satellite and the sea surface. Additionally, to aid in the precise orbit determi-nation, the satellites measures the signal emitted from the crust-fixed DORIS beacons (Doppler Orbitog-raphy and Radio-positioning Inte-grated by Satellite).

**Radar altimetry measurement principle**

**Radar,**
**DORIS.**

It should be stressed that the sea level anomaly, instead of the commonly used dynamic topography (sea surface height relative to a mean geoid) is decomposed here. The terms above will be elaborated upon in the following paragraphs.

Table 3.2: Altimetry corrections and as-sociated order of magnitude. Values taken from (Rosmorduc et al., 2009;

Fu and Cazenave,2001)

Name Magnitude

Instrument corrections

Ultra-stable oscillator drift 1cm Satellite center of gravity sat. dep.

(Radial) orbit correction few cm Geophysical corrections

Ocean tides 1-20 m

Solid Earth tides 50 cm

Ocean Pole tide 2 cm

Tidal loading 30 cm

Ionosphere 50 cm

Wet troposphere 50 cm

Dry troposphere 2.3 m

Surface corrections

Dynamic Atmosphere correction 15 cm

Sea state bias 50 cm

**Sea level response to atmospheric pressure**

In Eq. 3.3, a high frequency dynamic atmosphere correction is included, to account for
the quick ocean response to atmospheric forcing. On the time scales longer then 10 days,
which are the representative scales of this work, the correction can be assumed to be a
steady-state response which is commonly denoted as the Inverted Barometer (IB) response
to atmospheric pressure, h_{IB}. Correcting the sea level anomalies with the IB response
es-sentially adds the local atmospheric surface pressure as a column of sea water to the ocean
surface:

h_{IB} = ^{P}^{atm}−P_{re f}

gρ_{w} . (3.5)

In themodifiedIB response, the reference pressure,P_{re f}, is taken to be the oceanic mean of
the atmospheric pressure

P_{re f} = ^{1}
Aoce

Z

OceP_{atm}dω. (3.6)

The integral arises from the assumption that the ocean can be treated as incompressible w.r.t. atmospheric pressure variations, implying that only variations relative to the oceanic mean of the pressure field are relevant. The corrected sea level anomalies are less noisy, but now contain an atmospheric component. In contrast, GRACE data generally has its atmospheric component removed, such that care must be taken when combining altimetry with GRACE. This will be further discussed below.

**Mass versus Steric Induced Variations**

The volumetric sea level variations,h_{ster}, are induced by changes in temperature and
salin-ity (see Sec. 2.5). These volume changes are visible in the sea level anomalies, but are not
(directly) related to the mass changes. Since these changes do not induce ocean bottom
pressure variations, they cannot be detected by GRACE. As such, the altimetric sea level
anomalies are highly complementary to GRACE data.

Besides these volumetric effects, there are mass related changes in sea level (see Sec.2.4), which originate from a changing continental surface load. Both the change in the loading distribution as well as the addition or removal of water from the ocean play a significant role here.

The orbit of a satellite altimeter is provided relative to a time-fixed reference frame (for
example the TOPEX ellipsoid), such that deformations of the ocean floor will also be sensed
by the altimeter. The observed sea surface change is commonly denoted as geocentric sea
level change. This is essentially the variation of the geoid, with additional uniform layer
changes in order to impose the conservation of mass. The term[S^{˜}+U]_{elas}, thus represents
the geocentric sea surface change due to elastic surface loading effects. In this thesis, these
geocentric sea level variations are computed using the sea level equation as described in
Sec. 2.4.1to ensure a gravitationally consistent sea level response, conversing mass on a
global scale.

A similar term,[S^{˜}+U]_{GI A}, corresponds to the GIA induced geocentric sea level change.

On the timescales considered, this will be treated as a secular variation. Two effects play a role here. Firstly, the slow adjustment of the solid Earth, induce a non-uniform trend in the geoid. Secondly, as the mantle material is flowing back to the former glaciation areas (Laurentide and Fennoscandia), the ocean floor is sinking on average. The latter effect accounts for the slight offset between the GIA induced geoid surface and the geocentric sea level (i.e. the two equipotential surfaces do not share the same potential value).

**Dynamic Sea Level Variations**

Finally, there are dynamic variations of the sea level,h_{dyn}, which do not cancel out on the
time scales considered. This term contains the (net) effects of the wind and pressure driven
sea level variations and its associated (time-varying) currents. The dynamic sea surface
height as stated here can therefore not be simply thought of as the dynamic topography,
since the steric height, which is not included inh_{dyn}, would also contribute to the dynamic
topography. In fact, the sum ofhIB+h_{dyn} introduce variations in ocean bottom pressure.

In this thesis, h_{IB}+h_{dyn} is not the signal of interest, but it is reduced by subtracting the
(monthly) ocean bottom pressure variations from the GRACE de-aliasing model (GAC).

However, before correcting, the GAC product is modified by subtracting the oceanic mean
of the GAC product itself. This step prevents the introduction of artificial mean sea level
changes in the altimeter residuals. After correcting, the remainder of h_{dyn}+h_{IB} is simply
treated as measurement residual. After the correction, the altimetry data still contain the
signal of interest, namely the steric variations and the time-varying passive sea level
re-sponse to terrestrial surface loading.

In summary, the along-track corrected altimeter measurements, flow as observations in the fingerprint inversion as

∆h(* _{θ,}λ*) =h

_{ssh}(

*)−h*

_{θ,}λ_{mssh}(

*)−*

_{θ,}λT_{GAC}(* _{θ,}λ*)−

^{1}Aoce

Z

Oce

T_{GAC}(*θ*^{0},*λ*^{0})dω^{0}

. (3.7)