1. An overview of spectral methods for the optimal processing of satellite altimetry and other data
I.N. Tziavos1, M.G. Sideris2, G.S. Vergos1, V.N. Grigoriadis1, V.D. Andritsanos1
1Department of Geodesy and Surveying, Aristotle University of Thessaloniki, Univ. Box 440, , Greece
2Department of Geomatics Engineering, University of Calgary, AB, Canada
Objectives
Methodology
Present the achievements and breakthroughs during the last fifteen years of altimetric research at the Universities of Calgary and Thessaloniki in the determination of the gravity field and the geoid.
Main focus on the contribution of optimal combination methods, i.e., the FFT-based Multiple-Input Multiple-Output System Theory (MIMOST) and traditional Least Squares Collocation (LSC) employing single and multi-satellite altimetry as well as shipborne gravity and bathymetry/topography data.
Results for the areas of Canada, off-shore Newfoundland eastern Canada, and the Mediterranean Sea, eastern part of the Mediterranean basin.
Multiple Input Multiple Output System Theory (MIMOST)
Combination of multi-satellite altimetric (ERS1-GM and ERM, GEOSAT-GM and ERM, GFO, ENIVSAT) and shipborne gravity data for the determination of the marine gravity field and the geoid
Input errors: Simulated noise fields  white noise
Theoretical comparison between the two methods
 Theoretical frequency impulse response
 Estimated output spectrum
Error propagation is possible with MIMOST methods when the input noise power spectral density functions (PSDs) are known and the solution in the spectral domain is formally equivalent to LSC. Error propagation with heterogeneous data can be handled by MIMOST methods under the condition that signal and noise PSDs are available.
In the MIMOST solution the original transfer functions are modified by factors dependent on the noise-to-signal ratio and such the noise is filtered out of the input data and error estimates are derived for the predicted results.
The MIMOST solution is formally equivalent to stepwise LSC and can be computed in a stepwise manner. There is no matrix inversion in MIMOST and thus its efficiency exceeds by far that of LSC. With gridded data the efficiency of LSC can be greatly improved by employing the Fast Collocation procedure.
The algebraic simplicity of the MIMOST solution is lost when the data noise is not stationary. In practice, this is not a problem for LSC.
 Input observation spectrum
 Input error spectrum
 Prediction error spectrum
 Input observation PSD
 Input error PSD
 Optimal frequency impulse response
Least Squares Collocation
Traditional spherical least squares collocation (LSC) employing empirical and analytical covariance functions to estimate a combined solution from the available altimetric and gravimetric geoid models
 Prediction signal
Input and output signal cross-covariance matrix
Results from Eastern Canada
 Input signal
 Input signal cross-covariance matrix
 Observation noise
Statistics of the estimated marine geoid models in eastern Canada and comparison with 10-year stacked T/P data (m)
max
min
mean
rms
std
Nalt
1.66
45.05
27.24
28.64
±8.83
Ngr
1.78
45.00
27.20
28.60
Ncombined
1.77
44.90
28.62
±8.77
T/P – Nalt
-0.76
0.75
0.00
0.20
±0.20
T/P – Ngr
-1.12
0.95
0.39
±0.29
T/P – Ncombined
-0.93
0.84
0.24
±0.24
The accuracy of gravimetric geoid determination is ±10 cm worst than that of the altimetric models
The combination of altimetry SSHs with shipborne gravity data improves the gravimetric solution by 5cm and reduces the range of the differences with T/P by 60 cm
ERS1 and GEOSAT altimetric (left) gravimetric (center) and combined (right) geoid models
Results from Eastern Mediterranean
ERS1 and GEOSAT altimetric (left) and gravimetric (right) geoid models
MIMOST (left) and LSC (right) combined geoid models
Statistics of final 1 geoid models (m)
Statistics estimated geoid height at the GAVDOS TG
min
max
mean
rms σ
Ngrav
0.780
39.913
21.185
23.579
±10.352
Nalt
1.057
40.206
21.376
23.808
±10.484
Ncomb MIMOST
6.168
37.733
22.899
24.615
±9.127
Ncomb LSC
9.857
25.638
16.867
17.324
±3.951
GAVDOS
N (m)
σN (cm)
Ngrav
16.734
±1.41
Nalt
16.705
±0.91
Ncomb MIMOST
16.716
±0.65
Ncomb LSC
16.729
±0.40
Prediction error from the LSC combined geoid model
Ierapetra AC
Mid-MED Jet
Mid-Ionian Jet
Cretan AC
Jets from the Aegean merging the Cretan AC and “feeding” the Mid-Ionian Jet
Is this another feader of the Mid-MED Jet or another branch of it???
Statistics of the final 1 QSST model (m)
min
max
mean
rms σ
QSST
-0.510
0.657
0.014
0.238
±0.238
Statistics of estimated geostrophic velocities (m/s)
min
max
mean
rms σ us
-0.999
1.298
0.016
0.293
±0.292 vs
-1.211
1.292
-0.005
0.284
±0.284
total field
0.000
1.307
0.347
0.408
±0.214
QSST in the area under study from a combination of the altimetric and gravimetric geoid models (left) and direction and magnitude of the current velocity  circulation in the area (right)
Conclusions
The results from MIMOST and LSC are almost the same and differ at the sub-cm level which is considered as insignificant.
The combination of altimetry and shipborne gravity data improves the accuracy of the gravimetric only solution by about 5-10 cm.
The combined use of altimetry and marine/land gravity data improves the altimetric only solutions close to the coastline and over shallow water regions where the former present low data resolution and accuracy.
The multi-satellite solutions improve the resolution and accuracy of the single-satellite ones and lead to better estimates of the high-frequency part of the QSST.
The future of marine gravity field modeling lays in the synergy with other geosciences and combination of all available heterogeneous data sources.
LSC is an optimal estimator, but is problematic when large datasets are involved (significant amounts of memory and computer power are needed). On the other hand MIMOST is more efficient and provides exactly the same results with LSC.
Acknowledgement: This research was funded from (a) the Greek Secretariat for Research and Technology in the frame of the 3rd Community Support Program (Op. Sup. Pror ), Measure 4.3, Action (International Scientific and Technological Co-operation) and (b) the Ministry of Education under the O.P. Education ΙΙ program “Pythagoras ΙΙ – Support to Research Teams in the Universities”.
Session 3: Marine Geodesy, Gravity, Bathymetry – 15 Years of Progress in Radar Altimetry, March 13-18, Venice, Italy