The fractal properties of sea surface topography derived from TOPEX/POSEIDON (1992-1996)

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The fractal properties of sea surface topography derived

from TOPEX/POSEIDON (1992±1996)

Tian-Yuan Shih*, Jin-Tsong Hwang, Tzong-Jer Tsai

Department of Civil Engineering, National Chiao-Tung University, 1001 Ta-Hsueh Road, Hsin-Chu, Taiwan Received 12 March 1998; received in revised form 2 December 1998; accepted 11 December 1998

Abstract

Fractal dimension with various algorithms has increasing applications in characterizing both linear and areal features. In this study, the variabilities of sea surface topography derived from TOPEX/POSEIDON are studied. The variogram method is applied to compute fractal dimensions for each scene. Cumulatively, 29 scenes taken from 1992, 111 scenes taken from 1993, 110 scenes from 1994, 98 scenes taken from 1995 and 38 scenes taken from 1996 are analyzed. The spatial resolution of each scene is two degrees along both longitudinal and latitudinal directions. The annual averages of fractal dimensions are 2.528, 2.527, 2.523, 2.523 and 2.524 for 1992, 1993, 1994, 1995 and 1996, respectively. # 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Fractal dimension; Sea-surface topography

1. Introduction

Fractal dimension is considered an elegant technique

for describing rugged systems (Kaye, 1989).

Represented as a real number, fractal dimension can function as an index re¯ecting the cursiveness of lines, roughness of surfaces and other `irregularities' in Euclidean space. In an extensive review, Cox and Wang (1993) categorized fractal dimension applications in the ®eld of earth sciences into several general cat-egories. These categories include (a) testing whether or not a feature is fractal, (b) characterizing surface geo-metry to determine internal properties and (c) using fractal geometry to study formation and degradation processes and (d) using fractal slopes to determine multiple processes and the scales over which they are dominant. In this study, the variability of sea surface

topography derived from TOPEX/POSEIDON is ana-lyzed with fractal dimension. The objectives of this study are to characterize the global pattern of the stu-died sea surface topography (SST) to determine whether it is fractal in nature, whether this pattern is isotropic and how this global pattern changes with time.

2. Methods

Seven schemes are available to compute the fractal dimensions for surfaces: (1) the divider method, (2) the box method, (3) the triangular method, (4) the slit-island method, (5) the power spectral method, (6) the variogram method and (7) the size distribution method. The ®rst four methods apply directly to a simple geometric pattern and the latter three methods to a functional representation (Cox and Wang, 1993). For practical implementation, a number of algorithms are reported. For instance, Jaggi et al. (1993)

* Corresponding author. Fax: +886-35-716-257. E-mail address: tyshih@cc.nctu.edu.tw (T.-Y. Shih)

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plemented the line-divider method, the variogram method and the triangular method. Sarkar and Chaudhuri (1992) proposed a computationally simple algorithm and compared it with four other implemen-tations. Ouchi and Matsushita (1992) devised area-scal-ing, a scheme similar to the triangular method.

Fractal dimensions computed with dierent methods tend to vary systematically. This has been well docu-mented in Cox and Wang (1993) and also reported in Lam (1990) and Tate (1998). Actually, dierent im-plementations based on the same fractal dimension estimation method may result dierently in a systema-tic manner owing to dierences in handling problems, such as the remainder problem, curve-®tting, orien-tation of the measurement plane, size and direction of the sample, etc. Estimates of fractal dimension for real-world topography are not yet reproducible, between either investigators or methods (Evans and McClean, 1995).

After reviewing the seven methods listed in Cox and Wang (1993), the variogram method is selected in this study. Semivariance is the primary tool of modern geostatistics, a ®eld of analysis developed from regio-nalized variable theory for the modeling of continuous, non-deterministic surfaces exhibiting spatial depen-dence (Burrough, 1986; Isaaks and Srivastava, 1989). This technique is based on the idea that the statistical variation of data is a function of distance. The vario-gram relates distances between sample points to the variance of the dierences in the data. The semivar-iances are used to ®t an approved mathematical func-tion. The parameters of a ®tted model may include a

range (a), a nugget (C0) and a sill (C+C0). The range

indicates a spatial scale of the pattern, the nugget reveals information on variability between adjacent pixels, the sill gives information on the total variability of the area considered and the type of variogram model or the shape of the variogram reveals infor-mation on the spatial behavior of the data (Burrough, 1986; Jong and Burrough, 1995). Variograms are used as a description of texture for images (Lark, 1996) and applied to analyze resolution related issues (Atkinson, 1997). In the variogram method, the fractal dimension is estimated based on a geostatistical analysis, i.e. the variogram of semivariance function. For a pro®le

along an array z(xi), the semivariance v(h ) can be

esti-mated as

vh _2n1 Xn

i1

zxi ÿ zxi h2 1

where n is the number of pairs of discrete points separ-ated by a distance h. The fractal dimension D is obtained by measuring the slope of the log±log plot of estimated semi-variance against the sampling interval.

D 3 ÿslope₂ : 2

Whereas topographic surfaces do not exhibit pure fractal behavior, they often exhibit self-similarity across a limited range of scales. The calculations pre-sented here are limited to variation within a range of 20 units (408), within which fractal behavior was expected. Although the choice of the sampling interval and the slope determination of the log±log plot remains a dicult task, variogram method can be easily adopted for measures of anisotrophic data sets. In this study, data is analyzed with two sets of orthog-onal directions: the grid directions (north±south, east± west) and the diagonal directions (north-east to south-west, south-east to north-west). Fractal dimensions are computed for all four directions and all data pairs are used in the case of no direction dierentiation.

The computational procedure of the variogram method without direction dierentiation is listed as fol-lows:

1. Read the data to be analyzed, in this study, the SST.

2. Specify the maximum distance between two points,

Pairs_distance. In this study, the

Pairs_distance is speci®ed as 90, because the longest distance between two points on the globe is 1808 and the pixel resolution is 28. The minimum distance used is one.

3. Initialize the log±log plot array,

x[Pairs_distance][3]. In x[.][0], the

value of the logarithm of distance,log d, is stored.

The height dierence variances of the corresponding distance and the total occurrence of that distance

will be computed and stored in x[.][1] and

x[.][2]respectively in the next step.

k=0;

for (i=0; i < Pairs_distance; i++) for (j=i; j < Pairs_distance; j++){ x[k][0]=sqrt(i i+j j); if (x[k][0] > 0.0) x[k][0]=log(x[k][0]); x[k][1]=0.0; x[k][2]=0.0; k++;}

4. Accumulate the height dierences between two pix-els, then compute the variances. There are 180 col-umns and 71 pixels in each column for each SST

data set used in this study. That is,n_lat=71and

n_long=180.

for (i=0; i < n_lat; i++){ for(j=0; j < n_long; j++){ k=0;

for(ii=0; ii < Pairs_distance; ii++)

T.-Y. Shih et al. / Computers & Geosciences 25 (1999) 1051±1058 1052

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for(jj=ii; jj < Pairs_distance; jj++) { if(((i+ii) < 71)&&((j+jj) < 180)) { difference=SST[i+ii][j+jj]ÿSST[-i][j]; x[k][1]=x[k][1]+differen-ce differenx[k][1]=x[k][1]+differen-ce; x[k][2]=x[k][2]+1; } if(((i+jj) < 71)&&((j+ii) < 180)) { difference=SST[i+jj][j+ii]ÿ SST[i][j]; x[k][1]=x[k][1]+differen-ce differenx[k][1]=x[k][1]+differen-ce; x[k][2]=x[k][2]+1; } k++; }}} k=0;

for (i=0; i < Pairs_distance; i++) for(j=i; j < Pairs_distance; j++){ x[k][1]=x[k][1]/x[k][2];

if(x[k][1] > 0.0) x[k][1]=log(x[k][1]); k++;}

5. Conduct a least squares regression between log d

and the variances. The slope of this regression is then used to calculate the fractal dimension,

D=3ÿslope/2.

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Fig. 2. C ontour plot of scene 1993-01-01 .P lotted with GMT (We ssel and Smith, 1991).

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3. Data

TOPEX/POSEIDON is a joint project conducted by the United States' National Aeronautics and Space Administration (NASA) and the French Space Agency, Centre National d'Etudes Spatiales (CNES), or studying global circulation from space (AVISO, 1992). The primary sensor for the TOPEX/ POSEIDON mission is a dual frequency Ku/C band NASA radar altimeter (NRA). The measurements taken simultaneously at two frequencies, 13.6 GHz (Ku band) and 5.3 GHz (C band), are combined to obtain altimeter height of the satellite above the sea surface, including the wind speed, wave height and

ionospheric corrections. The data used in this study are the dynamic sea surface heights derived from NRA measurements. Besides wind speed and water vapor corrections, the measured sea surface heights are further applied with the ocean tide model correction and geoid height. Restated, the tide and geoid com-ponents are removed from the surface height. Further discussions on data processing and quality of derived sea surface height can be found in Hwang (1996).

The duration period of the TOPEX/POSEIDON sat-ellite is around ten days. However, the frequencies of the frames used in this study are one frame for each 3.33 day period. Each frame contains ten days of data, is spatially smoothed with average ®lters and is densed to 2 degrees by 2 degrees resolution. Data con-densation is performed using the weighted average method. Data within a 300 km radius of each grid

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node are averaged using an exponential weighting function. Some screening processes are also conducted during the gridding procedure, such as the deletion of measurements exceeding the mean surface height by more than three meters (Christensen, E.J., 1995, pers. comm.).

This data of sea surface heights is obtained from JPL with ftp via network. Each frame is stored as a separate digital ®le. The grid ®les are arranged in rows of latitude, with latitude and the 180 nodes for that latitude written in a single record. There are ninety-one such records for latitude ranging from +908 to

Fig. 4. Fractal dimension of ®ltered scenes.

Table 1

Statistical indices of fractal dimension for original SST data (29 scenes in 1992, 111 scenes in 1993, 110 scenes in 1994, 98 scenes in 1995, 38 scenes in 1996)

Direction conditions 1992 1993 1994 1995 1996

mean S.D. mean S.D. mean S.D. mean S.D. mean S.D. EW 2.612 0.005 2.610 0.004 2.605 0.005 2.605 0.004 2.602 0.007 NS 2.520 0.004 2.518 0.005 2.516 0.006 2.514 0.006 2.516 0.005 NW±SE 2.548 0.005 2.549 0.005 2.546 0.006 2.543 0.006 2.546 0.004 NE±SW 2.519 0.005 2.519 0.005 2.514 0.006 2.513 0.005 2.515 0.004 No direction 2.528 0.005 2.527 0.004 2.523 0.005 2.523 0.005 2.524 0.005

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ÿ908. The ninety-one data records are followed by ninety-one records containing the ¯ag values, which describes the node as land, sea, or ice (Norman, R., 1995, pers. comm.). These grid ®les are then processed to extract the sea surface heights by over-writing all land and ice pixels with a default value. In this study, the default value is zero. The latitude of the selected study area ranges from +708 to ÿ708 because the region poleward of 708 is covered primarily by ice and land. Because of the nature of the data sets provided, in this research, the unit used for planimetric coordi-nates is 28 and height is measured in mm.

4. Results and discussion

For each frame, ®ve fractal dimensions (one for

each direction condition) are computed via the vario-gram method. A typical log±log plot is shown in Fig. 1. Due to the nature of the earth, the meridian of longitude 1808 East is the same as the longitude 1808 West meridian. That is, the longest distance between two meridians is 1808. If this fact is not considered, the distance-variance plot would be symmetric. On a log±log plot, the right-hand side of the curve exhibits a sharp drop. At the end, the variance equals zero. That is, the distance of 3608 is essentially 08. In this study, the symmetric case is avoided by taking the roundness of the earth into account.

A least squares procedure is applied to obtain the slope. Only data points within a lag of 20 units (408) are introduced in the curve ®tting. The correlation

index r2 _{obtained from all curve ®tting cases ranges}

from 0.975 to 0.999.

Table 2

Statistical indices of fractal dimension for ®ltered SST data (29 scenes in 1992, 111 scenes in 1993, 110 scenes in 1994, 98 scenes in 1995, 38 scenes in 1996)

Direction conditions 1992 1993 1994 1995 1996

mean S.D. mean S.D. mean S.D. mean S.D. mean S.D. EW 2.429 0.007 2.426 0.005 2.422 0.006 2.422 0.005 2.423 0.005 NS 2.410 0.005 2.411 0.007 2.410 0.008 2.408 0.007 2.411 0.003 NW±SE 2.456 0.007 2.459 0.007 2.457 0.007 2.454 0.007 2.457 0.003 NE±SW 2.426 0.005 2.428 0.006 2.424 0.008 2.423 0.006 2.426 0.003 No direction 2.418 0.006 2.420 0.005 2.417 0.007 2.416 0.006 2.418 0.003 Table 3

Fractal dimensions with dierent vertical scaling (test data: 01/01/1993)

Rescaled cases Range of data EW NS NW±SE NE±SW No direction max. (mm) min. (mm) Original 1642.909 ÿ2459.195 2.608 2.524 2.555 2.526 2.544 Case 1 255 0 2.608 2.524 2.555 2.526 2.544 Case 2 200 0 2.608 2.524 2.555 2.526 2.544 Case 3 100 0 2.608 2.524 2.555 2.526 2.544 Table 4

The intercepts with dierent vertical scaling (test data: 01/01/1993)

Rescaled cases Range of data EW NS NW±SE NE±SW No direction max. (mm) min. (mm)

original 1642.909 ÿ2459.195 4.442 4.939 5.010 4.999 4.789 case 1 255 0 1.664 2.161 2.232 2.221 2.011 case 2 200 0 1.421 1.918 1.990 1.978 1.768 case 3 100 0 0.728 1.225 1.296 1.285 1.075

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As shown in Fig. 2, there are some high frequency signals in the original 2 28 data frames. Therefore, a 3 3 median ®lter is applied to each frame to further smoothen the sea surface heights. Fractal dimensions are computed for the ®ltered data sets as well. As expected, the fractal dimensions for the ®ltered sets are relatively lower than those of the original data-sets. In Figs. 3 and 4, the fractal dimensions are plotted against the days in a year. The characteristics of seasonal changes are very similar for all ®ve direc-tion condidirec-tions. The trends in the ®ltered and original cases are also similar. The annual average and the as-sociated standard deviation of fractal dimensions are listed in Tables 1 and 2. A comparison of the fractal dimension time sequence for the ®ltered SST with the one for the original dataset reveals a similar ¯uctuation pattern. The fractal dimensions for the ®ltered data are less than those for the original, which can be accounted for by the reduced roughness caused by the removal of noise.

In this study, the unit used for planimetric coordi-nates is 28 and height is measured in mm. A previous study indicated that fractal dimensions are insensitive to the vertical or horizontal exaggerations and the intercept value in the log±log plot re¯ects the degree of vertical exaggeration. (Ouchi and Matsushita, 1992). An experiment was performed with the scene 01/01/ 1993 to verify this phenomenon. The fractal dimen-sions and the intercepts in the log±log plot are listed in Tables 3 and 4. The results are precisely as expected.

5. Concluding remarks

Results obtained in this study demonstrate that the sea surface heights derived from TOPEX/POSEIDON present a fractal nature. The average fractal dimension for the omni-direction case is 2.525 for the original datasets and 2.418 for the ®ltered datasets. Slight dierences occur between the fractal dimensions com-puted for dierent direction conditions. The time series for both ®ltered and original datasets present similar ¯uctuations, implying that there are structural patterns in the time domain. However, this could be due to the scale of the analyzed grid data. The distance measure adopted in this research is the Eucledian distance with latitude and longitude. Fractal dimensions with dier-ent distance measures, such as the length of the great circle, are also computed. The characteristics of the fractal dimension time series remain, but the lower cor-relation index value (typically in the range of 0.5 to 0.7) indicates that other non-fractal processes, most likely the numerical eect in the computation of vario-gram, have larger in¯uences on the geodesics.

Acknowledgements

The authors wish to thank Dr. E.J. Christensen and Dr. R. Norman of JPL for kindly providing the SST data, as well as the anonymous reviewers for their comments and constructive suggestions that improved the paper.

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