EOF Analysis of the TOPEX/POSEIDON Altimetric Data in the Equatorial Atlantic

Carlos Augusto de Sampaio França & Afranio Rubens de Mesquita
Instituto Oceanográfico da USP


Empirical Orthogonal Functions (EOF) analysis applied to the Sea Level Height Anomalies (SSHA) derived from three years of TOPEX/POSEIDON data (cycles 011 to 120) shows two dominant modes in the Equatorial Atlantic (10ºS to 10ºN, 60ºW to 20ºE). The first mode explains about 40 % of the SSHA variance and consists of two regions of opposing phases separated by a nodal line between Northeast tip of Brazil and the Westernmost part of Guinea Gulf. The second mode explains around 30% of the SSHA variance and consists of two near zonally oriented regions with a nodal line crossing the area around 5 to 6ºS. Additionally, the two modes show enhanced variance in the vicinity of the North Brazil Current retroflection (3 to 6ºN; 50 to 60ºW) and the second mode enhanced variance in the Eastern side of the region symmetrically around the Equatorial line. Both modes have time history dominated by the annual harmonic with a secondary contribution of the semi-annual signal. The results agree with observations of other researchers in the area and with numerical simulations obtained through the use of a 3D free-surface model under development.

Empirical Orthogonal Functions

Being U a n-dimensional vector of random variables with covariance matrix C , Pi the ith eigenvector and corresponding eigenvalue l i of C. Being P the n x n matrix of eigenvectorsand L the n x n diagonal matrix of eigenvalues:

C P= L    (1)

The orthogonal transformation

Y = P U    (2)

generates a new set of uncorreleted random variables called the Principal Components of U (Rao, 1965).

If Nt is a n-dimensional vector of variables observed at t times and ^C its sample covariance matrix then Pi a eigenvector of ^C with its corresponding l i ith Empirical Orthogonal Function of Nt^ and

Yit = Pi Nt    (3)

its Time History


EOF analysis were applied to the TOPEX/POSEIDON altimetric data and to Numerical Simulations of the sea level obtained by a Free-surface 3-d Numerical Model under
development at IOUSP, in the Equatorial Atlantic (10ºN-10ºS; 60ºW-20ºE).

The EOF ranked by its eigenvalues show that the first three correspond to more than 80 % of the altimetric observations variability and 94 % of the simulations (Table 1).
TOPEX/POSEIDON % Numerical Solution %

Table 1. Variance explained by the first three EOFs of the sea level in the Equatorial Atlantic.

The Figures show the structure (P1= EOF-1) of the first EOF of the sea level observed by the altimeter and simulated by modeling.

EOF-1 for TOPEX/POSEIDON (Fig. 1) is a long wavelength dipole between Northwest and Southeast of the area with a nodal line between the Northeastern tip of the Brazilian coast and the Westernmost region of the Guinea Gulf. The mode is amplified in the Intertropical Convergence Zone (ITCZ) region. EOF-1 for numerical Solution (Fig. 2) shows the same pattern but enhanced amplitude of the mode along the axis of the North Brazil Current.

The Time History ( Y1t ) of the EOF-1 (Fig. 3) for both satellite and simulated sea level are dominated by the annual signal. Its amplitude is positive and maximum during February-March and negative minimum. September-October so that the sea level is high in the Southeast Equatorial Atlantic and low in the ITCZ region during February-March and reversed during September-October.

Fig. 1.

Fig. 2.

Fig. 3.

Second EOF is also a long wavelength structure. For TOPEX/POSEIDON (Fig. 4) it shows a nodal line around 5 to 6ºS, positive values to the South, and negative to the North with a
trough at the Equator more pronounced towards the Guinea Gulf. For the Numerical Solution the same pattern holds (Fig. 5). The differences between them are located along 2 to 4ºN
and the Northwest corner of the area where they reverse phases but both show enhanced variability.

The Time History ( y2t ) of the EOF-2 (Fig. 6) for both simulation and altimetry are also dominated by the annual period with a secondary contribution of the semi-annual. They show a
positive maximum around June-July and another extreme between December-January in the negative side. The Time History for this mode obtained from TOPEX/POSEIDON data
shows considerably more variability in higher frequencies than the one obtained through the analysis of the simulation with seasonal forcing.


The analysis of the TOPEX/POSEIDON altimetric data has shown characteristics of the sea surface topography variability in the Equatorial Atlantic, dominated by the seasonal
cycle. Duchene & Frankignoul (1991) have analyzed the dynamic topography obtained by using historical T-S data and Arnault & Cheney (1994) have analyzed Geosat altimetric data
and model simulation in the Equatorial Atlantic through EOF analysis. Even using different kind of data and sampling their results and the ones presented here are strikingly similar.
The only region where the present analyses do not agree in all details is near the North Brazilian coast and the North Brazil Current retroflection. This is obviously due to the
meso-scale variability known to occur in the region (Bruce & Kerling, 1984).

Fig. 4.

Fig. 5.

Fig. 6.

The agreement between the analysis of observations and simulations shows the ability of the numerical model under development at the IOUSP in reproducing the seasonal cycle in the Equatorial Atlantic. Also, it shows the potentiality of the use of EOF analysis as an objective test for tuning the model.


The sea level variability in the Equatorial Atlantic observed by the TOPEX/POSEIDON altimeter and simulated by the Numerical Model is dominated by the two first EOF modes which have its time history dominated by the annual cycle.

Being able to capture global characteristics of the sea level in the Equatorial area, the EOFs have shown to be a good tool for tunning the present numerical model.


Arnault, S. & Cheney, R. E. 1994. Tropical Atlantic sea level variability from Geosat (1985-1989). J. geophys. Res. , 99(C9) : 18 ,207-18 ,223.

Bruce, J. G. & Kerling, J. L. 1984. Near equatorial eddies in the North Atlantic. Geophys. Res. Letts, 11:779-782.

Duchene, C. & Frankignoul, C. 1991. Seasonal variations of surface dynamic topography in the tropical Atlantic: Observational uncertainties and model testing. J. mar. Res.,

Rao, C. R. 1965. Linear Statistical Inference and its Applications. New York. John Wiley & Sons, 625p.