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A potential-geometry approach to computation of Sea Surface Topography using satellite altimetry observations

A. A. Ardalan
Department of Surveying and Geomatics Eng.,
Center of Excellence in Geomatics Eng. And Disaster Management,
University of Tehran
[email protected]

G. Joodaki
Department of Surveying and Geomatics Eng.,
Center of Excellence in Geomatics Eng. And Disaster Management,
University of Tehran

V. Nafisi
Department of Surveying and Geomatics Eng.,
Center of Excellence in Geomatics Eng. And Disaster Management,
University of Tehran

M. Jafari
Department of Surveying and Geomatics Eng.,
Center of Excellence in Geomatics Eng. And Disaster Management,
University of Tehran

 

Abstract
Nowadays satellite altimetry observations have proven their value in the global oceanographic studies, and have become standard tool for global ocean tide modeling. One of the outputs of ocean tide models is the zero frequency tide constituent, which is also known in marine geodesy as Mean Sea Level (MSL). Mean Sea Level is not an equipotential surface but it is as main role in the definition of the geoid, reference equipotential surface of the earth, and even its realization at the sea areas. On the other hand, thanks to modern achievements in gravity field observation on the surface of the earth, in the air, and in the space near the earth, such as CHAMP, GRACE and GOCE, we have access to high-resolution and high-accuracy geopotential model. Those geopotential models if be introduced to an ellipsoidal Bruns formula, can produce reliable geoid solution at the sea area. In this paper having available the new MSL model computed by the geodesy team of the University of Tehran, for the Persian Gulf and Oman Sea, and having available the ellipsoidal harmonic coefficients of SEGEN (Ardalan, Grafarend 2000) a marine geoid has been computed point-wise at the 720 points along-track of TOPEX/POSEIDON altimetry satellite in the Persian Gulf and Oman Sea. Having also MSL value at those points, Sea Surface Topography (SST) at the 720 points is computed. This newly computed SST, which can be considered as the result of a geometry-potential space (geometry coming from satellite altimetry and potential from geopotential model) approach to SST computations has been tested against on the SST solutions available for the same region.

Introduction
Using tide gauge observations at coastal areas or mounted under water, tidal information can be obtained. But considering the size of oceans and water areas of the Earth, such type of point-wise observations could not ever provide a uniform coverage of the sea areas. On the other hand altimetry satellites are capable of providing sea level variations at a global scale and therefore they can be used as a substitute of tide gauge observations at the offshore and open seas.

Modern altimetry satellites are equipped with range-measuring instruments, which can measure the distance between satellites to the sea surface with up to few centimeters accuracy. For this reason the altimeter transmits a short pulse of microwave radiation with known power toward the sea surface. This pulse interacts with the rough sea surface and part of the incident radiation reflects back to the altimeter. Besides altimetry satellites are equipped with variety of positioning systems that could provide the geocentric position of the altimetry satellites to a very high degree of precision. In the Table 1 a summery of some satellite altimeter is presented.

Satellite Measurement precision (cm) Orbit accuracy (cm)
GEOS-3 25 500
Seasat 5 100
Geosat 4 30-50
ERS-1 3 8-15
TOPEX/POSEIDON 2 2-3
ERS-2 3 7-8

Table 1: Summary of past and present satellite altimeter measurement precisions and orbit accuracies

One of the most important products of the satellite altimetry observations is Sea Surface Topography (SST). This surface is the difference between Mean Sea Level (MSL), which is also known as static sea surface, and the geoid. In fact if the sea surface were affected only by gravitational forces should have be formed as an equipotential surface, called geoid. In reality due to various numbers of non-gravitational forces, as well as the tidal force, we have a dynamic sea surface. Figure 1 shows a schematic summary of the various contributions to the sea surface topography relative to a reference ellipsoid

Fig 1: summary of the various contributions to the sea surface height h relative to a reference ellipsoid. The components of h include the geoid undulations hδ , tidal variations hT , atmospheric pressure hα and the dynamic sea surface height hd associated with surface current.

The tidal part of the dynamic sea surface can be modeled and having removed these time varying effect one arrives at the MSL, which is static but still not an equipotential surface of the Earth due to non-gravitational affects such as winds, currents, salinity and temperature variations, for example. Therefore tide gauge observations could only provide us with tidal constituents and MSL; however connection of individual coastal tide gauge observations requires that the zero point of the tide gauges be connected to each other. This might be accomplished via precise leveling along the coast for the tide gauges which are close to each other. However distant tide gauges and especially those which are separated by water, like tide gauges at the different continents could not be connected that way. An alternative approach to the connection of the zero points of the tide gauge stations is via a reference surface, to which the MSL's could be compared. This reference surface could be considered as the one nowadays used as the height datum, i.e. the geoid. Therefore if we would like to connect the zero points of the world-wide tide gauges to each other via geoid, we would need to now SST at individual tide gauge stations.

In this paper we have computed Geoid undulation in the test area, Persian Gulf and Oman Sea and having computed marine geoid, SST at the test area has been computed by subtracting Geoid undulation from MSL.

The potential-geometry approach
The potential-geometry approach for computation of SST from satellite altimetry observations can be summarized as a five steps algorithm:

  1. From the satellite altimetry observations using one of the standard tidal analysis techniques temporal sea level variations observed by satellite altimetry can be decomposed into tidal constituents and a constant (time invariant) part. The time invariant part is called Mean Sea Level (MSL).
  2. Using one of the geopotential models, gravity potential values at the ellipsoidal points can be computed. For this we must use one of the current best geopotential models.
  3. Knowing the geoid's potential value, as one of the fundamental geodesy parameters, the difference between geoid's potential and the potential value at the reference ellipsoid can be computed. This difference corresponds to the separation between geoid and reference ellipsoid and as such could be considered as the marine Geoid in potential space.
  4. Using 'Ellipsoidal Bruns Formula' marine Geoid values in potential space can be transferred into geometry space.
  5. Having computed marine Geoid and MSL the SST can be computed by subtracting marine Geoid from MSL in point-wise manner.

Numerical Results
In this section we present our numerical results derived by application of the algorithm explained in the previous section, which is applied to our test area "Persian Gulf and Oman Sea".

Figure 2 shows gravity potential of the 720 computational points on the reference ellipsoid, computed based on the ellipsoidal geopotential model SEGEN computed by Ardalan and Grafarend (1999).

Fig 2: Gravity potential of ellipsoid at 720 computational points, computed based on the ellipsoidal geopotential model SEGEN.

Now we can compute the difference between the geoid's potential value and the computed gravity potential values at the reference ellipsoid, point-wise. In this way we arrived at point-wise estimation of marine geoid in terms of potential units, or as we call it, "Geoid in potential space". Having computed Geoid in potential space, Bruns type formulas, or transformations, can be used to convert the computed potential difference into height differences i.e. Geoid undulations. Figure 3 shows the Geoid values at our test region once transformed from the potential space into the geometry space as was mentioned.

Having computed the gravity potential at the surface of reference ellipsoid, now we can compute the gravity-geoid, i.e. the geoid based on the observables of the type modulus of gravity intensity. To keep the ellipsoidal approximation we use the following non-linear ellipsoidal Bruns formula in jacobi ellipsoidal coordinates ¦,I,f,h(Ardalan,1999):

where is the difference between Geoid's potential and actual potential on the surface of the reference equipotential surface and e2 = a2 – b2.
In our computations we have used SEGEN 1999 ellipsoidal geopotential model for computation.

Fig 3: The marine Geoid computed at our test area.

Finally, using available Geoid and the MSL in geometry space, we can estimate a point-wise SST by subtraction of marine Geoid from MSL. Figure 4 shows the SST computed at our test area.

Fig 4: SST values at 720 computational points in geometry space in meters.

Table 2 shows the minimum, maximum, mean and standard deviation of computed SST and Geoid undulation in the Persian Gulf and Oman Sea.

Table 2: statistics of computed SST and Geoid undulation

  Min(m) Max(m) Mean(m) Std(m)
SST -0.71733111000 2.0847420400000 0.42293953983204 0.45458725100690
Geoid -53.929867040 -17.07566889000 -34.5583051729069 8.84708093903374

Concluding Remarks
In this paper we presented an algorithm for computation of the SST, using satellite altimetry observations and geopotential model. Our approach is characterized by computation of potential on the ellipsoid, where in gravity potential space the geoid can be presented by its potential value alone. This transformation enabled us to derive Geoid undulation in potential space as the difference between ellipsoidal potential value and geoid's potential value. In the next step, using ellipsoidal Bruns formula the computed Geoid in potential space is transformed into geometry space in order to have Geoid undulation as the geometrical deviation between geoid and ellipsoid. Having marine Geoid and MSL in geometry space the SST has been computed as the difference between these two heights. Future works in this computation procedure will be verification and validation of the computed geoid and SST values from other approaches.

References

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