** WP 3.3.4 Sea-level changes **

Gabriel Strykowski and Jens Nykjaer Larsen, Geodynamics Division, KMS

**1. Introduction**

The objective of WP 3.3.4 is to study possibility of separation of steric/non-steric sea level changes by a gravity satellite mission like GOCE. There are some conceptual problems with this task. Especially, regarding the additional information that is being used. In general, the gravitational attraction from a source is a product (linear apprioximation) of the effect of source geometry (sea surface topography) and of the mass density contrast to the surroundings.

*Mass density:* The only difference between the two types of the sea level changes (steric/non-steric) with respect to the gravitational signal at the satellite
height is a small difference in the mass density associated with the change of the water volume of the ocean above the so called seasonal thermocline. The
depths to the base of the seasonal thermocline are some 50 m - 300 m (Tomczak and Godfrey, 1994; Bigg, 1996) with few spots ranging to 900 m.
According to (Stammer, 1997), the maximal change of the sea surface height associated with the steric effect is of the order of some 0.10 m. Thus, the
range of the volume change is from some 0.10 m / 300 m = 0.0003 to some 0.10 m / 50 m = 0.002. This effect is counterbalanced by the corresponding
change of the mass density of the water column (the mass of the water column is constant). For a mass density of 1030 kg/m^{3} it means a change of some
0.31 kg/m^{3} to 2.06 kg/m^{3}. Other factors like pressure, temperature and salinity have the same or even bigger influence on the change of mass density then
the steric effect (Gill, 1982, Appendix A). Thus, without a detailed knowledge of all these factors it is not possible to isolate the steric effect even if it was
possible to estimate accurately the mass density of the water from the gravity signal at the satellite height.

*Source geometry: *Even if the above separation of different factors affecting the mass density of the water could be modelled perfectly, i.e. the salinity, the
pressure and the temperature were perfectly known, another problem remains. A separation in the gravity signal the effect of the source geometry from the
effect of the mass density contrast remains. One has to know one to estimate the other. How well do we need to know source geometry to estimate mass
density? In the above example, a detection of a 0.2% change of mass density contrast requires (to a linear approximation) a knowledge of undulations of
the topography at 0.2%-level, i.e. for the above signal magnitude of 0.10 m we need to know the sea surface topography at a level of 0.0002 m (which is
not possible at the present).

In conclusion: A separation of the steric and non-steric effects based only on measured gravity data, even enhanced by detailed sea surface topography information (e.g. altimetry) is not possible.

*Including other information*: A more fruitful way of separating steric- and non-steric sea surface topography (perhaps the only way in practise) is to
include the knowledge about the inducing mechanism, e.g. the seasonal heat from the sun. This is exactly the basis of the separation performed by
(Stammer, 1997), where the meteorological data from ECMWF (heat flux, wind stress field and the climatological seasonal surface temperature, salinity
boundary) as well as the state- of- the- art WOCE Parallel Ocean Climate Model were used. The independent measurements of the sea surface topogaphy
were provided by the TOPEX/POSEIDON altimetric mission. The role of these data was to constrain the geographical extension (for different time
periods) of the ocean topography anomalies associated with the steric- and the non-steric effect.

*The task of WP 3.3.4*: From the above discussion the task of WP 3.3.4 should be reformulated. It is not possible to separate the effect of steric and
non-steric sea level change without a detailed analysis like that of (Stammer, 1997). However, one can use gravity mission like GOCE to provide
information about the total anomalous sea surface topography. Thus, referring to the above, the role of gravity data of GOCE mission will be similar to the
information provided by the satellite altimetry data of the TOPEX/POSEIDON mission. The detailed formulation of WP 3.3.4 is described in Sec. 2.

Acknowledgement: Ph.D.-students: O. Leeuwenburgh and J.L. Høyer and prof. C.C.Tscherning contributed with ideas and discussion.

**2. The setup of the investigation**

The investigation conducted here will make use of two stuctures chosen from Fig.1a in (Stammer, 1997) showing the fall.93 sea surface height anomaly with respect to 2 years average. One structure is located in the Pacific Ocean and the other one in the Atlantic Ocean. The question asked will be whether the gravimetric signal generated by this anomalous sea surface topography will be visible in the gravity data of the GOCE mission at the satellite height.

The structure in the Pacific Ocean is approximately located 0^{o}N to 9^{o}N and 170^{o}E to 270^{o}E. It will be approximated by a set of rectangular prism of height
2 cm covering the above area. The other structure in the Atlantic Ocean is approximately located 9^{o}N to18^{o}N and 300^{o}E to 331^{o}E. It can be approximated
by a rectangular prism of height -2 cm.

Similarily to what was done in WP 3.3.5, 20 realizations of noise in the gravity field caused by error in the spherical harmonic coefficients (SHC) was
generated for each of the two areas and for the two types of error statistics GOCE and EGM96. Moreover, the gravity signal at the height of 300 km of the
gravity response from the anomalous SSH structures (see above) was computed. A mass density contrast (transition from the sea water to the free air) of
1034 kg/m^{3} was used.The objective of this simulation study is to investigate whether the signal from the structure will be seen above the noise level.

**3. The results**

The results of the investigation will be shown on the figures below.

Reference field and gravimetric response from the anomalous SSH structures (mass density contrast: 1034 kg/m^{3} ) at the height of 300 km:

Fig. 1a The reference anomalous gravity field (EGM96) at the height of 300 km over the test area in the Atlantic Ocean. (unit: mgal)

Fig. 1b The reference anomalous gravity field (EGM96) at the height of 300 km over the test area in the Pacific Ocean. (unit: mgal)

Fig. 1c The direct gravimetric respose of the anomalous SSH structure in the Atlantic Ocean at the height of 300 km. Structure height: 2 cm (unit: nanogal)

Fig. 1d The direct gravimetric response of the anomalous SSH structure in the Pacific Ocean at the height of 300 km. Structure height: -2 cm (unit: nanogal)

Two realizations of noise in the spherical harmonic coefficients of GOCE and EGM96 (see WP 3.3.5) at the height of 300 km:

Fig. 2a The realization of noise #1, height 300 km, SHC error statistics of GOCE, the test area in the Atlantic Ocean. (unit: microgal)

Fig. 2b The realization of noise #1, height 300 km, SHC error statistics of EGM96, the test area in the Atlantic Ocean. (unit: microgal)

Fig. 2c The realization of noise #1, height 300 km, SHC error statistics of GOCE, the test area in the Pacific Ocean. (unit: microgal)

Fig. 2d The realization of noise #1, height 300 km, SHC error statistics of EGM96, the test area in the Pacific Ocean. (unit: microgal)

Fig. 3a The realization of noise #2, height 300 km, SHC error statistics of GOCE, the test area in the Atlantic Ocean. (unit: microgal)

Fig. 3b The realization of noise #2, height 300 km, SHC error statistics of EGM96, the test area in the Atlantic Ocean. (unit: microgal)

Fig. 3c The realization of noise #2, height 300 km, SHC error statistics of GOCE, the test area in the Pacific Ocean. (unit: microgal)

Fig. 3d The realization of noise #2, height 300 km, SHC error statistics of EGM96, the test area in the Pacific Ocean. (unit: microgal)

Two realizations of noise in the spherical harmonic coefficients of GOCE and EGM96 at the height of 300 km and the anomalous SSH structure responses:

Fig. 4a The realization of noise #1 + anomalous SSH structure response, height 300 km, SHC error statistics of GOCE, the test area in the Atlantic Ocean. (unit: microgal)

Fig. 4b The realization of noise #1 + anomalous SSH structure response, height 300 km, SHC error statistics of EGM96, the test area in the Atlantic Ocean. (unit: microgal)

Fig. 4c The realization of noise #1 + anomalous SSH structure response, height 300 km, SHC error statistics of GOCE, the test area in the Pacific Ocean. (unit: microgal)

Fig. 4d The realization of noise #1 + anomalous SSH structure response, height 300 km, SHC error statistics of EGM96, the test area in the Pacific Ocean. (unit: microgal)

Fig. 5a The realization of noise #2 + anomalous SSH structure response, height 300 km, SHC error statistics of GOCE, the test area in the Atlantic Ocean. (unit: microgal)

Fig. 5b The realization of noise #2 + anomalous SSH structure response, height 300 km, SHC error statistics of EGM96, the test area in the Atlantic Ocean. (unit: microgal)

Fig. 5c The realization of noise #2 + anomalous SSH structure response, height 300 km, SHC error statistics of GOCE, the test area in the Pacific Ocean. (unit: microgal)

Fig. 5d The realization of noise #2 + anomalous SSH structure response, height 300 km, SHC error statistics of EGM96, the test area in the Pacific Ocean. (unit: microgal)

**4. Discussion and conclusions**

In the Introduction it was argued that the use of gravity data alone for the separation of steric- and non-steric ocean topography is not possible. Gravimetric signal at some height consist of a combined effect of source geometry (the sea surface topography) and the mass density contrast. These effects cannot be separated aquately from the gravity signal alone. Another problem is that even if it was possible to perform a perfect separation of the geometry and the mass density contrast, this would not be diagnostic for detecting a steric effect. There are other factors in the ocean that can affect the mass density variation of the sea water at the same order of magnitude and more then the steric effect. Thus, it is not possible to diagnose a steric effect without a detailed knowledge of the other factors.

Consequently, the task of WP 3.3.4 was reformulated. The gravity mission like GOCE is viewed as a way of providing information about the anomalous sea surface heights, similarily to e.g. TOPEX/POSEIDON satellite altimetry mission. In practise, there is a problem of separating the gravity effect of the anomalous SSH from the background. Hower, this in principle could be handled by repeating the measurements in time over some area. The changes of the gravitational attraction with time could be parametrized as the effect of changing SSH.

However, the example above show that the gravitational attraction of the anomalous SSH cannot be seen above the noise level, even for clearly improved gravity data of GOCE mission (as compared to EGM96), see Figs. 4a-d and Figs. 5a-d. For the Atlantic Ocean example, the magnitude of the gravity signal at the height of 300 km generated by the anomalous SSH structure (horizontal extension; NSxEW: 1008 km x 3472 km; thickness: 2 cm) is only some 0.3 microgal, see Fig. 1c. The standard deviation of the noise is (ensamble average for 20 realizations of noise) 141 microgal for EGM96 and some 4 microgal for GOCE, see Figs. 2a-b and Figs. 3a-b. Roughly speaking, for the EGM96 gravity field the structure would be visible above the noise level if it was 9.4 m thick and for the GOCE improved gravity field model if it was 0.27 m thick.

For the Pacific Ocean example, the magnitude of the gravity signal at the height of 300 km generated by the anomalous SSH structure (horizontal extension; NS x EW: 1008 km x 11200 km) is only some 0.6 microgals, see Fig. 1d. The standard deviation of the noise (ensamble average for 20 realizations of noise) is some 142 microgal for EGM96 and some 3 microgal for GOCE, see Figs. 2 c-d and Figs. 3c-d. For the EGM96 gravity field the structure would be visible above the noise level if it was 4.73 m thick and for the GOCE improved gravity field model if it was 0.10 m thick.

Thus, although the improvement in the gravity models by GOCE mission is quite remarkable it is not (as yet) possible to detect the signals at which one could study steric/non-steric sea level changes.

**Bibliography**

Bigg, G.R.: The oceans and climate. Cambridge University Press, 1996.

Gill, A.E.: Atmosphere-ocean dynamics. International Geophysics Series, vol. 30, Academic Press, 1982.

Gill, A.E. and P.P. Niiler: The theory of the seasonal variability in the ocean. Deap Sea Res., 20, 141-177, 1973.

Stammer, D.: Steric and wind-induced changes in TOPEX/POSEIDON large-scale sea surface topography observations. JGR, 102, C9, 20,987-21,009, 1997.

Tomczak, M. and J.S. Godfrey: Regional Oceanography: an Introduction. Pergamon, 1994.

Last update 1999.06.22 by cct.

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