SPECTRAL ANALYSIS OF SEA SURFACE TOPOGRAPHY
OBSERVED BY TOPEX/POSEIDON ALTIMETRY USING
2D
DFT
Waldemar Popiński
Department of Standards, Central Statistical Office
w.popinski@stat.gov.pl
Wiesław Kosek
Space Research Centre, Polish Academy of Sciences
kosek@cbk.waw.pl
Abstract. A method of spectral analysis and filtering of sea level anomalies (SLA) obtained from
TOPEX/POSEIDON altimetric measurements is presented. Such sea level anomalies at a particular
time moment create sea surface topography which can be analysed using the two dimensional
discrete Fourier transform (2D DFT).
The original data called 'cycles' are defined on a rectangular grid in latitude-longitude domain and
are transformed using the two dimensional DFT to the corresponding wave-number domain with
coordinates representing number of waves per degree in the latitude and longitude directions,
respectively. Filtering sea surface topography is achieved by multiplying the transformed data by a
bivariate filter transfer function and next by inverting the DFT.
These analyses enable detection of the most energetic waves in sea surface topography and
estimating how much the sea level anomalies at different time moments are correlated along the
meridians and parallels. It has been found that the correlation and coherence of sea surface
topography along the parallels is higher than along the meridians for the whole wave-number range
and independently of time.
Two dimensional continuous Fourier Transform
The two-dimensional continuous Fourier Transform of an integrable bivariate function is defined as
F (u, v)   f ( x, y ) exp( 2i (ux  vy)) dxdy,
where u , v are the numbers of wave crests per distance unit in x and y directions, respectively,
and its inverse transform is given by the formula:
f ( x, y )   F (u , v) exp( 2i (ux  vy)) dudv.
Two dimensional discrete Fourier Transform
For a given complex-valued function f (k1, k2 ), k1  0,1,...N1  1, k2  0,1,..., N2  1, defined over a
two-dimensional grid, equidistant in both coordinates, we can define its two-dimensional Discrete
Fourier Transform (2D DFT)
F (n1 , n2 ) 
N 2 1 N 1 1
  f (k , k ) exp( 2i
n 2  0 n1  0
1
2
k1n1
kn
 2i 2 2 )
N1
N2
and the inverse transform is defined as
f (k1 , k2 ) 
1
N1 N 2
N 2 1 N 1 1
  F (n , n ) exp( 2i
n 2  0 n1  0
1
2
k1n1
kn
 2i 2 2 )
N1
N2
We can use the 2D DFT to filter the discrete function f (k1 , k 2 ) according to the formula:
f ' (k1 , k2 )  FFT 1[ F (n1 , n2 ) A(u, v)] ,
where
(1  u 2 u 2 )(1  v 2 v 2 ) for | u | u, | v | v
A(u, v)  
0
otherwise

is the paraboloidal filter transfer function defined for | u | 1 2 , | v | 1 2 and u , v are filter
window half-length parameters, FFT is the Fast Fourier Transform operator.
Filtration of sea surface topography by 2D DFT
In our investigations we used the 2D DFT to analyse SLA data. The SLA data were computed by
the Center for Space Research - The University of Texas at Austin from the T/P satellite altimeter
measurements corrected for instrumental and geophysical corrections. These data contain gridded
1o  1o sea level heights ranging from  65 o to 65 o in latitude and from 0 o to 359 o in longitude.
These data are stored in gridded files according to the T/P repeat cycle number (T/P repeat period is
equal to 9.92 days) starting from September 23rd 1992.
To use the FFT the dimensions N1 , N 2 must be integer powers of 2. In order to satisfy this
condition and to avoid wrap-around effects of filtering we applied zero-padding all around the
border of two-dimensional array storing the analyzed data. The sea level anomaly data are defined
on the equiangular  ,  grid with dimensions N   130 , N   360 , respectively, so we put them in
the middle part of the two-dimensional array with dimensions N1  256  2 8 , N 2  512  2 9 . The
remaining part of the array which includes also the land area is zero-padded.
The SLA data corresponding to the cycles with the numbers of 110, 120,..., 230 from 1995 Sep. 8th
to 1998 Dec. 11th, were filtered by the 2D DFT high pass filter (HPF) with 3 different cutoff wavenumbers of 0.05, 0.1 and 0.2 deg 1 . These cutoff wave numbers, the same in  and  directions,
correspond to the angular wavelengths of 20 o , 10 o and 5 o , respectively. Such filtration enables
presentation of sea surface topography details in different spectral bands covering rapid variations
( 5 o ) or longer period variations ( 20 o ). The maps of the high pass filtered sea surface topography
are shown together with the maps of SLA data in Figures 1-13. When the cutoff wave-number of a
HPF diminishes, then the results of sea surface topography filtration are more similar to the original
SLA data. Increasing the cutoff wave-number reveals more rough details in sea surface
topography. The results of sea surface topography filtration show that the roughness of the sea
surface along the meridians is much higher than along the parallels.
All the maps except for the maps corresponding to cycle numbers of 120, 130 and 160 (1995
December, 1996 March and 1997 January) (Figs. 2, 3, 6) show very long ridge on the Pacific
Ocean north to the equator at the latitude around  5 o and rows at both sides of this ridge (Fig. 1, 4,
7-12). Usually, these ridges and rows become more regular when the cutoff wave-number of the
filter increases. The height difference between the top of the ridge and bottom of the row can reach
the value of the order of 20-30 cm.
Analysis of sea level anomalies in September 1995 shows very regular ridge along the equatorial
Pacific Ocean for all filter cutoff periods (Fig. 1) which disappears between the end of 1995 and the
beginning of 1996 (Fig. 2, 3). During this time interval an increase of the sea surface level of the
order of 30-40 cm in the Red Sea and East Chinese Sea is visible (Fig. 1, 2).
Cycle 130 (1996, March) corresponds to the moment of the small La Nina event of 1996 (Fig. 2, 3).
During this time interval there is a 30 cm increase in the sea surface level in the western and
decrease of -20 cm in the eastern Pacific Ocean. After this time interval the ridges and rows on the
both sides of the equator along the equatorial Pacific Ocean begin to form (Fig. 4, 5) but they
diminish in the beginning of 1997 (Fig. 6). In this time interval there is a 30-40 cm increase of the
sea surface level of the Black Sea (Fig. 6).
In April 1997 before the El Nino 97/98 event there are two ridges visible in the Pacific Ocean at
 3o N and  10 o N with one row between them (Fig. 7). During this time interval there is a 30-40
cm increase in the sea surface level of the East Chinese Sea, Red Sea and at the south coast of
Borneo. On the map corresponding to the cutoff wave-number of 0.2 deg 1 one can see that a ridge
along the equatorial Pacific Ocean begins to form (Fig. 7) which is visible very well during the El
Nino 97/98 event (Figs. 7-11).
During the El Nino 97/88 event there is decrease of the sea surface level in the western and increase
of the sea surface level in the eastern Pacific Ocean (Fig. 8, 9, 10). The decrease of the sea surface
level in the western Pacific Ocean and eastern Indian Ocean is accompanied by increase of the sea
surface level in the western Indian Ocean (Fig 9, 10) as well as increase of the order of 20-30 cm in
the Black, Caspian, North and Baltic Sea (Fig. 10). There is a -20-10 cm deep row on the Pacific
along the equator area, accompanied by two ridges at both sides that are visible better when the
filter cutoff wave-number increases (Fig. 10).
After the El Nino event the equatorial row in the Pacific Ocean is accompanied by the ridge at
 3o N (Fig. 11). The height difference between the top of the ridge and bottom of the row is of the
order of 30 cm and it decreases with the increase of the filter cutoff wave-number as one could
expect. In this time interval there is a 20 cm increase of the sea surface level in the eastern Indian
Ocean (Fig. 11). In September 1998 there is decrease of about 20 cm in the sea surface level in the
central Pacific Ocean near the equator, accompanied by a 10-20 cm increase of the sea surface
level at the east coast of Asia and North America (Fig. 11). The increase of the filter cutoff wavenumber reveals one row on the equator and one ridge to the north at the central eastern Pacific
Ocean (Fig. 12).
Conclusions
Results of high pass filtration of the sea level anomalies show that the sea surface waves along the
parallels have greater periods than those along the meridians all over the oceans. The maps of sea
level anomalies filtered by the HPF with the cutoff wave-number of 0.05 deg 1 are very similar to
the original sea level anomalies data as one could expect.
High pass filtration of sea surface topography reveals regular ridges and rows in the equatorial
Pacific Ocean. They become more regular when the high pass filter cutoff wave-number increases.
The height distance between the top of the ridges and bottom of the rows can reach the value of 2030 cm and it decreases with the filter cutoff wave-number.
As can be seen on the maps corresponding to the cutoff wave-number of 0.2 deg 1 before the El
Nino 97/98 event the equatorial ridge begins to form, which persists during the whole El Nino
event.
Acknowledgements. This paper has been supported by the grant No 51/T12/97/13 of the Polish
Committee of Scientific Research directed by Prof. B. Kołaczek.
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