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Global circulation fromSEASAT altimeter dataTheo Engelis aa Department of Geodetic Science andSurveying , The Ohio State University ,Columbus, OhioPublished online: 10 Jan 2009.

To cite this article: Theo Engelis (1985) Global circulation from SEASAT altimeterdata, Marine Geodesy, 9:1, 45-69, DOI: 10.1080/15210608509379515

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Global Circulation fromSEASAT Altimeter Data

Theo EngelisDepartment of Geodetic Science and SurveyingThe Ohio State UniversityColumbus, Ohio

Abstract A set of time-averaged sea surface heights at1° intervals, derived from the adjusted SEASAT altimeter data,and the GEML2 gravity field are used to estimate the long-wavelength stationary sea surface topography. In order to re-duce the leakage of energy in the estimated sea surface to-pography, the GEML2 field is augmented by the Rapp81 grav-ity field to generate geoidal undulations with wavelengthsconsistent with the ones of sea surface heights. These undu-lations are subtracted from the sea surface heights, and theresulting differences are subjected to filtering in order to re-cover sea surface topography with minimum wavelengths of6000 km and an estimated accuracy of 20-25 cm. These esti-mates agree well with oceanographic and other satellite-de-rived results.

The direction of current flow can be computed on a globalbasis using the spherical harmonic expansion of sea surfacetopography. This is done not only for the SEASAT/GEML2estimates, but also using the recent dynamic topography es-timates of Levitus. The results of the two solutions are verysimilar and agree well with the major circulation features ofthe oceans.

Marine Geodesy, Volume 9, Number 10149-0419/85/010045-00$02.00/0Copyright © 1985 Crane, Russak & Company, Inc.

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46 Theo Engelis

Introduction

In many early geoid-related determinations, it was assumed thatthe geoid is the equipotential surface that coincides with themean sea surface averaged over an extended period of time. Ofcourse, this consideration is incorrect at the submeter level sincethe mean sea surface deviates from the geoid, mainly becauseof the oceanic currents and the seasonal steric anomalies. Timevariations of the sea surface (i.e., eddies, etc.) create additionaldeviations that have a time-varying character. All these devia-tions are ideally subject to geostrophic and hydrostatic equilib-rium requirements and are formally called "sea surface topog-raphy" (SST). The permanent sea elevation is called "stationarySST," while its time variations are characterized by the term"time-varying SST."

The determination of the SST is of primary value for bothgeodesy and oceanography. In geodesy, an estimate of the sta-tionary SST can be used for the determination of the geoid froma mean sea surface realized from altimetry. In oceanography,both stationary and time-varying SSTs are required for the de-termination of the surface ocean circulation. Then, using thegeoid as a reference surface, ocean velocities at any depth canbe computed. A goal that has been established among geodesistsand oceanographers is the determination of SST to an accuracyof about 10 cm over any wavelength of interest.

The role of purely geodetic techniques in ocean circulationdetermination is primarily that of providing values of SST usingaltimeter data. Geodetic techniques are able to circumvent as-sumptions of a level of no motion and other problems existingin the oceanographic determinations and to provide consistentand synoptic data over the oceans on a global basis. The wholeprinciple of using an altimetric system for oceanographic studiesis fairly simple. An altimetric system uses a precision radar tomeasure the distance from the satellite to the ocean surface. Ifthe geocentric height of the satellite is also known, then it ispossible to determine the geocentric distance of the mean seasurface and its time variations. Given the geoid, the stationary

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Global Circulation from SEAS AT Altimeter Data 47

SST can be expressed, as follows:

is = i - N (1)

where % is the height of the mean sea surface (SSH) above areference ellipsoid, N is the geoidal undulation with respect tothe same ellipsoid, and is is the stationary SST.

In practice, this simple idea becomes fairly complicated, sincethe altimetric process is a function of many variables, each hav-ing sources of errors affecting the SST determination. A numberof these error sources can be modeled and removed from theraw altimeter measurements. These are the tropospheric andionospheric refraction effects, ocean tides, atmospheric pres-sure, and others. The most critical factors that can cause seriouslimitations in SST determination are basically the orbital errorsand the geoidal uncertainties. The major part of the orbital erroris of a long-wavelength nature (Wunsch and Gaposchkin, 1980).In contrast, the geoid errors are small at long wavelengths andincrease with decreasing wavelength. So adequate estimationprocedures are required for the reduction of the long-wavelengthorbital errors and for the usage of the geoid only up to wave-lengths with acceptable accuracies.

The SEASAT Data Base

SEASAT was launched into a nearly circular orbit at an altitudeof 800 km and an inclination of 108°. This inclination allowed forthe coverage of the global oceanic surface, excluding the Northand South Pole areas. In its three-month lifetime, SEASAT hasprovided approximately 4,000,000 observations 7 km apart fromeach other in an along-track direction. The coverage producedby SEASAT can be better understood in terms of coverage pro-duced by any consecutive 43 revolutions (three-day coverage).During the first two months, consecutive three-day orbits had adrift at about 1-2° at equator crossings while, during the lastmonth, the orbits had no drift. Groundtracks during this monthwere about 9° apart at equator crossings.

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48 Theo Engelis

The primary intent of the SEASAT altimeter data processingat Ohio State University was to reduce the long-wavelength or-bital errors and at the same time eliminate all time variations ofSST in order to determine a time-averaged sea surface. Detailson this processing are given in Rowlands (1981) and Rapp(1982a). The adjusted sea surface heights (SSHs) were then usedto generate point values on a 1° grid with an average accuracyof 30 cm (Rapp, 1982b). This computation was made by least-squares collocation, using as data only the five points closest tothe prediction point. The covariances used were taken to be thecovariances described in Tscherning and Rapp (1974). The ac-curacy estimate of 30 cm represents the uncertainties of theSSHs over the whole frequency spectrum. These uncertaintiesare due to a large number of errors that have not been modeledin the adjustment process, all of them being periodic with var-iable wavelengths. More specifically, these uncertainties rep-resent residual long-wavelength orbital errors, some mesoscaleSST variations in an averaged form, and all small-scale andmesoscale orbital errors. In addition, there is the presence ofsome altimeter noise. Although no extensive specific investi-gation on the errors by wavelength has been performed, it isobvious that the error spectrum contains most of its energy inthe higher frequencies and almost negligible energy in the longwavelengths, considering of course that the long-wavelength or-bital error modeling in the adjustment process was efficient.

These gridded SSHs refer to the GRS80 ellipsoidal model(Moritz, 1980) and contain the induced permanent tidal effect.In addition, they have been corrected, by subtracting 11 cm, toproperly handle the altimeter bias (Rapp, 1982b). This data setis the one used for the determination of the stationary SST.

The Oceanic Geoid

In order to obtain the deviations of the mean surface from thegeoid, the geoidal surface has first to be defined. Generally thereis an ambiguity in the definition of the geoid. A number of def-initions have been proposed, each identifying a different levelsurface as the geoid. A good discussion on the different defi-

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Global Circulation from SEAS AT Altimeter Data 49

nitions is given by Rizos (1980). The most appropriate definitionfor the determination of SST is the one called the "oceanic"definition of the geoid, according to which the "oceanic geoid"for a selected epoch of measurement is that level surface of theearth's gravity field for which the average nontidal SST is zero,as sampled globally in oceanic regions.

For the realization of the oceanic geoid, none of the geoidrepresentations derived under the assumption of its coincidencewith the mean sea surface can be used, since they implicitlycontain all the SST information. Also, the terrestrial gravityanomalies cannot be used, since their distribution is uneven andtheir accuracies variable. It is obvious that only gravity modelsderived exclusively from satellite dynamics are adequate to rep-resent the geoid for SST determinations. The major limitation inthe usage of these models is their errors, which tend to increasewith increasing degree. The most accurate satellite-derived grav-ity model is the GEML2 model (Lerch et al., 1982), which isgiven complete to degree and order 20 along with accuracy es-timates. The cumulative and by-degree accuracies of GEML2are given in Figure 1.

Using this model, the geoidal undulations can be computedfrom the well-known formula

N = No + — 2 (-) 2 KCnm - Cnmref) cos m\

+ Snm sin m\]Pnm(sin $) (2)

where

No = the zero degree undulation arising from the factthat the reference ellipsoid is an arbitrary one,

GM = the gravitational constant times the mass of the_ _ earth,Cnm, Snm = the fully normalized potential coefficients of the

_ gravity model,Cnmref = the fully normalized potential coefficients for

the Reference ellipsoid^ which are zero exceptfor C2Oref> C40ref, and C60ref;

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50 Theo Engelis

i 1 1 1 1 1 1 1 1 i2 11 6 8 10 \3 m 16 18 20

DEGREE

FIGURE 1. GEML2 cumulative and by-degree accuracies.

a = the equatorial radius of the reference ellipsoid,7 = the normal gravity,

$, X = the geocentric latitude and longitude, respec-tively, of the point of computation,

_ r = the geocentric radius to the point (4>, \ ) ,Pnm(sin <}>) = the fully normalized associated Legendre func-

tions.

In order to be consistent with the gridded SSHs that includethe induced permanent deformation, it is necessary for the po-tential coefficients (specifically C2,o) to include the correspond-ing effect. Since the GEML2 coefficient does not include this,a correction term must be added. Using the correction term givenby Moritz (1979), we have

C 2 , 0 ( C O R ) — ^ 2 , 0 G E M L 2 — 9 X 1 0 / V 5

The best reference ellipsoid that can be used for the undulation

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Global Circulation from SEAS A T Altimeter Data 51

computation must be geocentric and have a flattening of the ac-tual earth. In addition, we require a semi-major axis such thatthe average undulation No is defined to be zero in oceanic areas.The requirement of geocentricity is automatically satisfied bysuppressing the first-degree terms in the harmonic expansion,Equation (2). On the other hand, the flattening of the ellipsoidcan be computed using C2,O(COR) and the GM, to, and a of theGRS80 ellipsoid. Using formulas in Heiskanen and Moritz(1967), we obtain

/ " ' = 298.25657701

With this flattening, the GEML2 model, and the GRS80 pa-rameters, a set of geoid undulations on a 1° grid can be computedfrom Equation (2), with No = 0. It is obvious that all the GEML2geoid errors propagate directly into the SST estimates throughEquation (1). So the GEML2 model can be used only up to alimited degree, and, consequently, SST can be determined onlyup to that degree. Figure 1 and results by Engelis (1983) indicatethat the optimum degree of expansion is 6, providing an accuracyof 18 cm. The undulations obtained from Equation (2) have wave-lengths of the order of 6000 km and greater. On the contrary,the SSHs that are to be used in Equation (1), being determinedon a 1° grid, contain frequencies up to a maximum harmonicdegree 180.

In order to remove the high-frequency information of SSH(harmonic degrees 7 to 180) from Equation (1), a filtering of thesefrequencies has to be performed. This filtering can most con-veniently be done by transformation into the frequency domainand then removal of the undesired frequencies by truncating thespectrum at degree 6. The transformation to the frequency do-main of a function on a sphere is done by a spherical harmonicanalysis (Colombo, 1981). However, this procedure requires thatthe function be defined over the whole sphere. So the followingalternative definition of SST can be used

t,s = I - N in oceans

£, = 0 on land

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52 Theo Engelis

The assignment of zero is basically motivated by the fact thatall oceanic effects are absent on land, so a hypothetical meansea surface on land would coincide with the oceanic geoid. SSTis also defined to be zero in polar regions because no altimeterdata are available in these areas.

There is a potential problem in implementing Equation (3) andFiltering out the frequencies above harmonic degree 6. Indeed,if N are the geoid undulations from GEML2 and £ are the SSHsas determined from the adjusted SEASAT altimeter data, thenEquation (3) becomes

is = I - N = I + S£ - (N + hN + AAO in o c e a n s /tx

(4)

Is = 0 on land

where

8£ = the errors of I,8N = the errors of N arising from GEML2 up to degree 6,AiV = the undulation information above degree 6.

If, on the other hand, we try to implement Equation (3) by justusing t, and N, then we shall have

Is = I - N in oceans

t,s = 0 on land

which gives an inconsistent representation of the function on thesphere, having as a result the occurrence of leakage of energyamong the spherical harmonics that are to be determined fromthe spherical harmonic analysis (Tai and Wunsch, 1983).

In an attempt to reduce the leakage between the coefficients,Equation (4) has to be implemented in the best possible way.While there is no way to remove the errors §£ and 8iV, the re-sidual undulations, AN, from degree 7 to degree 180 can be de-termined by using the gravity model by Rapp (1981), which isgiven complete up to degree 180. However, this model intro-duces additional errors SN\ (the reported cumulative accuracy

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Global Circulation from SEAS AT Altimeter Data 53

of the model is on the order of 1.6 m). So Equation (4) becomes

t, = i-N + (ll-hN- 57V,)

is = 0

where TV is now computed from Equation (2), using a combinedmodel containing the GEML2 coefficients up to degree 6 andRapp's coefficients from degree 7 up to degree 180. This way,leakage between the coefficients is reduced during the sphericalharmonic analysis, since the only quantities that can give rise toleakage effects are the errors 8£, 8/V, and 8/V]. One should notethat Rapp's model implicitly contains all SST information, sinceit is based in part on altimetric data, so it cannot be used forSST recovery by itself. From degree 7 to degree 180, though, itcontains little SST information, since most of the SST energy isat long wavelengths. So TV, as determined from the combinedmodel, contain oceanic effects that do not give rise to any sig-nificant leakage. The use of GEML2 from degree 7 to degree 20was initially considered, but was rejected since its errors at thesedegrees are larger than the corresponding values of Rapp'smodel.

Best-Fit Ellipsoid to the Ocean Surface

Another form of leakage of energy is introduced in the sphericalharmonic coefficients of SST due to the fact that the estimatesI and N have been computed with respect to different referenceellipsoids. This effect can be completely avoided by implying thedefinition of the oceanic geoid. According to this definition

M(ls) = M((, - TV) = 0 (7)

where M is the averaging operator over the oceanic areas andis defined to be

_ SJOcosJ,Z COS (f>

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54 Theo Engelis

The sea surface heights available in this analysis refer to theGRS80 reference ellipsoid. In order to refer them to the samereference ellipsoid (a, f) as N, the following correction has tobe applied:

£ = £GRSSO ~ da + OGRSSO df sin2 § (9)

where

da = a - aCRSSo (10)

df = f - /GRSSO (11)

Substituting in Equation (8) and considering Equation (6), weobtain

M(£,) = M{1 ~ da + aGRs8o df sin2 <j>- N + Si - SN - 5iV,) = 0 (12)

which gives

da = Mil, + OGRSSO df sin2 4> — iV)+ M(8£ - 8iV - SNi) (13)

From Equation (13) it is obvious that an estimate of da is affectedby the average effect of the errors in sea surface heights andgeoid undulations. This average error is clearly nonzero, sincethe averaging is not global, but is expected to have a small mag-nitude. After making the computations for all 1° grid points forthe oceanic areas, an estimate da = —0.95 m was obtained. Sothe parameters (a, f) of the reference ellipsoid are

a = 6,378,136.05 m „„

/ " ' = 298.25657701

This ellipsoid provides the best reference for the oceanicgeoid. It has the property that the average undulation is zero.Also, it is the best-fit ellipsoid to the long-wavelength mean seasurface (SST of shorter wavelengths is implicitly contained in

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Global Circulation from SEAS AT Altimeter Data 55

the combined gravity model), since this ellipsoid has the flat-tening of the real earth and its semi-major axis is chosen so thatthe average nontidal long-wavelength SST is zero.

This best-fit ellipsoid to the ocean surface is generally non-geocentric. The nongeocentricity is due to oceanic effects andthe nonsymmetric location of land and oceans on the earth. Thecenter of the best-fit ellipsoid with respect to the center of massis defined by the origin shifts dx, dy, dz. These origin shifts canbe computed simultaneously with the correction to the GRS80ellipsoid, da, by minimizing in the least-squares sense the dif-ferences between £ and TV. The relationship between these dif-ferential quantities is (Heiskanen and Moritz, 1967)

I - N + aGRS80 df sin2 <j>= cos cf> cos X dx + cos cf) sin K dy + sin cf) dz + da (15)

where df is known from Equations (11) and (14).For the purpose of demonstrating the effect of neglecting the

higher frequency undulation information, as well as the effect ofthe model errors in the determination of da, dx, dy, and dz,several models have been considered for the realization of N inaddition to the one used in the previous section. The results forthe semi-major axis and the origin shifts of the reference ellip-soids are shown in Table 1 for all the implied models. It is obviousthat the differences between the second and third sets of resultsare due to the larger errors of GEML2 and the existing SSTinformation in Rapp's model (both, for degrees 7 to 20). Thedifferences between the third and fourth sets of results clearlyindicate the effect of neglecting the higher frequency (aboven = 20) undulation information.

Results from the Spherical Harmonic Analysis

Equation (12) implies that the actual quantities to be subject tofiltering are

Is = I - da + OGRSSO df sin2 <j> - iV in oceans , . ^(16)

is = 0 on land

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56 Theo Engelis

Table 1Semi-Major Axis and Origin Shifts of the Best-Fit Ellipsoids to the

Mean Sea Surface Using Different Models

Gravity Model a dx dy dz

Spherical Harmonic Analysis 6,378,136.05 -0.44 -0.12 0.40

GEML2: nmax = 6 6,378,136.01 -0.55 -0.11 0.73Rapp: n > 6

GEML2: nmax = 20 6,378,135.91 -0.59 -0.18 0.64Rapp: n > 20

GEML2: nmax = 20 6,378,135.69 -0.74 -0.23 0.47

GEM10B: «max = 36 6,378,135.84 -0.63 -0.12 0.58

Rapp:nmax = 36 6,378,135.92 -0.57 -0.14 0.66

where £5 contains

a) SST up to degree 6 (all higher frequencies are cancelledout because of the use of the combined model),

b) geoid undulation errors up to degree 6 arising fromGEML2,

c) geoid undulation errors from degrees 7 to 180 arising fromRapp's model,

d) all errors in SSH not removed during the orbital error re-moval process.

The global samples of Equation (16) have been harmonicallyanalyzed, and spherical harmonic coefficients up to degree 6have been retained. These coefficients provide SST estimatesaffected mainly by the GEML2 errors, so their accuracy is thecumulative accuracy of the GEML2 model up to degree 6 (18cm). This accuracy is of course a minimum bound, since all thelong-wavelength orbital errors that have, remained from the or-bital error removal process directly propagate to the SST esti-mates. Moreover, the leakage effects give rise to additional er-rors.

The SST harmonic coefficients, their power spectrum, and theGEML2 error power spectrum (computed from the providedmodel accuracies) are shown in Table 2. The existence of the

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Global Circulation from SEASAT Altimeter Data 57

first degree harmonic coefficients is simply another indication ofnongeocentricity of the mean sea surface. These origin shifts arealso given in Table 1.

Construction of Global Maps

The spherical harmonic coefficients of SST can reveal valuableinformation concerning the behavior of the mean sea surface.This information can be better obtained if interpretation of thecoefficients is done in conjunction with the observation of mapsconstructed from these coefficients and showing the behavior ofthe long-wavelength mean sea surface on a global basis.

There are basically two types of maps that can provide oceaniccirculation information in the best possible way. The first is acontour map showing the pattern of SST; the second shows thedirection of the flows of the oceanic currents.

For the construction of a contour map, the spherical harmoniccoefficients are used for a harmonic synthesis of SST point val-ues on a regular grid. Then a contour map is constructed on thebasis of this data set. It must be pointed out that, since the con-tour map reflects information of very long wavelengths, the con-tours are slightly distorted at the boundaries of the oceans andin closed oceanic regions. While interpreting a SST contour map,one should disregard any values on land, since they are mean-ingless.

An equally important type of map is the current direction map.Although it is of a qualitative nature, it provides a very instruc-tive picture of the pattern of the oceanic flows. The constructionof this kind of map is based on the evaluation of the gradientsof SST along latitude and longitude directions and then on thecomputation of flow velocity and its direction by means of thefollowing equations (Coleman, 1981;Rizos, 1981;Engelis, 1983):

i = - * - ^ (17)

y fR cos 4> dk y '

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Table 2SST Spherical Harmonic Coefficients (cnm, dnm) and Power Spectrum (CJJ); GEML2 Error Spectrum (el) and

Signal-to-Noise Ratio; Levitus-Derived Coefficients (c*m, d*m), Power Spectrum (cr*2), and Cross-Spectrum (Si)

Cnn

SEASAT/GEML2( + )a LEVITUS (1982)*

ol oi/el #Cnm

0.08.9

-14.8

-24.8-1.8

0.3

4.60.72.0

-4.2

i *

2.4

1.82.8

-1.6-2.4

0.0

303

632

5151

si01

001

01

2

0

123

0.022.9

-25.4

-23.02.7

13.9

1.43.77.1

-9.0

-7.1

0.7-11.1

3.2-2.511.9

1220

905

303 16

905

19

0.923

0.713

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4 0 3.0 1.11 -0.3 -0.4 3.5 0.82 -1.6 1.1 53 25 2 -3.6 0.7 38 0.453 -2.3 5.7 1.2 2.24 -1.5 0.6 0.0 -2.0

5 0 -10.5 -7.91 -1.7 -3.5 0.3 1.92 -0.6 9.3 282 144 2 -1.3 4.7 97 0.773 -4.0 2.3 -0.8 2.04 -1.3 3.2 -1.0 0.35 -4.3 -3.9 0.5 0.0

6 0 15.2 9.9<! 1 0.0 -3.5 -1.1 -1.5

2 3.9 -4.6 -3.7 -0.33 -2.2 1.2 567 121 4.7 0.4 -2.5 132 0.5124 13.7 -0.9 -0.6 0.25 3.9 5.5 3.0 -0.36 -42 52 02 (U

" Units are in cm for cnm, dnm and cm2 for o-2,, e2.b Units are in dyn-cm for c*nm, d*m and (dyn-cm)2 for cr*2.

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60 Theo Engelis

where / is the Coriolis force coefficient. Note that these equa-tions are valid under the assumption of a nonaccelerating surfacelayer and the absence of atmospheric pressure gradients and offrictional forces due to wind stress. Then the azimuth A andmagnitude v of the velocity of the flow are given by

A = tan"1 ( T ) (19)

v = (x2 + y2)m (20)

Equations (17) and (18) can be easily expressed with respectto SST spherical harmonic coefficients as follows:

S ^ i 4« < x , J • x \ % P n m (sin(j>) , o n

x= - ~ 2 2J (cnmcosm\ + dnmsmm\) (21)7«« = im=o "9

N n

y = ,R^ • 2 2 m ( ~ c«m sin m\ + dnm cos m\) Pnm (sin cf>)

(22)

where

dPnm (sin 9) — . — .I T — — = Pnim+\) (sin 9) - m tan 9 Pnm (sin 9) (23)

Note that there is a computational instability in determinationsof the current velocities at low latitudes due to the zero Coriolisforce coefficient at the equator. Using the SST harmonic coef-ficients up to degree 6 (Table 2), a contour map and a currentdirection map have been constructed and are shown in Figures2 and 3, respectively. These maps reveal the oceanic circulationcharacteristics for minimum wavelengths of about 6000 km.

In Figure 2 we can easily observe two distinct depressions inthe Southern Hemisphere. The first is in the Pacific Ocean, witha magnitude of 1.2 m, and the second is in the South IndianOcean, with a magnitude of 1.6 m. These depressions generatetwo well-defined gyres created by the cyclonic formations be-

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O\

90

LONGITUDE

30 60 90 120 150 180 210 240 270 300 330 0

-90 130

T60

I ' ^ I • ' f i i | i i | i • | i i | • i |

90 120 150 180 210 240 270 300 330 0

LONGITUDE

FIGURE 2. Sea surface topography for wavelengths up to degree 6. Contourinterval = 20 cm. [Based on SEASAT/GEML2( + ).]

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5\

90

j-60-

-90

30i

60 90. i ,

LONG120

. i .

ITUDE150i

180i •

210, i

240 270i

300 330i

-60

? f r r * --_...,,,• * * v 4 v •»

\ \ \ I 1 I 4

\ \ \ i v t t v

i i t t v \ \ -I U U V UM l I i n »

N ^ , - - . - ^ / / / ̂ " - , \ *w "̂ "̂ *»

0 30 60 90 120 150 180 210 240 270 300 330 0LONGITUDE

•90

-6o:

-90

FIGURE 3. Current directions for wavelengths up to degree 6. [Based onSEASAT/GEML2( + ).]

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Global Circulation from SEAS AT Altimeter Data 63

tween the subpolar and circumpolar currents. Two rises of 0.8and 1.2 m in the South Pacific create two more gyres in theSouthern Hemisphere, while in the Northern Hemisphere onemajor gyre occurs, created by the rise of 1.4 m in the CentralPacific. In Figure 3 one can recognize the flow directions of allthese gyres. Moreover, many of the major currents of the world,like the Kuroshio, North Pacific, California, Peru, East Aus-tralia, and others, seem to have patterns consistent with thoseshown in any classical oceanographic literature.

Comparison with Oceanographic Results

The most recent, elaborate, and comprehensive oceanographicinvestigation of the world's oceans on a global basis has beenprovided by Levitus (1982). Levitus has computed 33 sets ofdynamic topography estimates, corresponding to minimumwavelengths of 500 km and referring to 33 different equal-pres-sure surfaces whose pressure ranges from 6.25 to 5750 db. Allthese data sets have been provided on a tape by James Marsh(private communication, 1983). From all these data sets, the onewith respect to a 2250-dbar surface has been chosen for com-parison with the SEASAT-derived SST. The reason is that sucha surface seems most closely to approximate a level surface ona global basis. (Montgomery, 1969). This data set consists of33,856 1° x 1° dynamic topography estimates, well distributedin the oceanic areas of the world. This data set has been analyzedin a manner consistent with the SEASAT-GEML2-derived SSTestimates. As a first step, its mean value (2 dyn-m) was computedand subtracted for each individual 1° x 1° estimate. This op-eration corresponds to the adoption of the oceanic geoid in theSEASAT computations. Then zeroes were assigned on land andpolar regions, thus creating a global set of 64,800 values that washarmonically analyzed to provide SST harmonic coefficients.These coefficients up to degree 6 are shown in Table 2, togetherwith the SEASAT-derived coefficients. Of course, all coeffi-cients up to degree 180 are completely valid from this analysis,since no obvious error sources are present to impose limitationson the maximum degree of harmonic expansion. Comparing the

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64 Theo Engelis

two sets of coefficients in Table 2, we can see a good agreementbetween the zonal coefficients of degrees 2, 3, 4, and 5. How-ever, the first and sixth zonal terms show some discrepancies.Examining the tesseral coefficients, we can see that some ofthem agree very well but most are quite different. The SEASAT-derived tesseral coefficients have generally larger magnitudesand are responsible for the longitude-dependent features shownin Figure 2.

In a more elaborate comparison one can compute the cross-spectrum of the two coefficient sets, which indicates their cor-relation by degree as follows:

^ " (r r* A- d ti* \52 ^Ljnx = Q \ ^ nm*- nm ' "/im*-* nmj /*yA\

V(T2« CX*2«

where

cnm,dnm = the SEASAT/GEML2( + )-derived coefficients,c*nm, d*nm = the Levitus coefficients,

v2n, o-*2« = the degree variances of the two sets as com-

puted from Equation (13).

These correlations, which are shown in Table 2, decrease withincreasing degree.

Finally, a third way of comparing the two solutions is by con-structing a contour map and a current direction map based onthe Levitus coefficients up to degree 6. These maps are shownin Figures 4 and 5 and, if compared with Figures 2 and 3, showa much smoother behavior of the oceans with an almost completelatitude dependence, especially in south latitudes. However, theoverall pattern of the oceanic circulation is consistent betweenthe two sets of maps.

The differences occurring in all these comparisons can basi-cally be attributed to the following sources of errors and inef-ficiencies:

• the GEML2 errors which increase with increasing degree,• the possibility of a slightly differing behavior of the oceans

during the SEASAT mission, which was too short to trulyrecover the stationary SST,

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90

-90

30 60

LONGITUDE

90 120 150 180 210 240 270 300 330 0

30l

60 90 120LONGITUDE

I [ ^ ^ l '"""' r ' ' I ' ' I ' ' I ' ' I150 180 210 240 270 300 330 0

— 90

FIGURE 4. Dynamic topography with respect to a 2250-db surface for wave-lengths up to degree 6. Contour interval = 20 dyn-cm. (Based on Levitus data.)

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£

90

60 -

30LONGITUDE120 150

90

-9030 60 90 120 150

LONGITUDE180 210 240 270 300 330 0

FIGURE 5. Current directions with respect to a 2250-db surface for wave-lengths up to degree 6. (Based on Levitus data.)

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Global Circulation from SEAS AT Altimeter Data 67

• possible long-wavelength residual orbital errors that mighthave remained after the adjustment process,

• almost complete lack of oceanographic data in the SouthernHemisphere (Levitus, 1982, Figures 6-10),

• possible inadequacy of the oceanographic data to com-pletely recover features of increasing degree (eventual er-rors in oceanographic determinations should also be con-sidered, as well as some aliasing),

• noncoincidence of the 2250-dbar surface with an equipoten-tial surface.

The overall agreement in the comparison of the maps and ofthe harmonic coefficients themselves and, at the same time, theunderstanding of all the limiting factors mentioned above, in-dicate that the two determinations of SST are consistent.

Long-wavelength SST using SEASAT altimeter data has alsobeen computed by Tai and Wunsch (1984). Their resulting map,demonstrating SST up to degree 6, is in good agreement withthe map of Figure 2.

Conclusions

The synoptic determination of quasi-stationary SST and oceaniccirculation is possible by using satellite altimetry and geodetictechniques. For the time being though, there are several limi-tations for SST determination with the desired accuracy of 10•cm over any wavelength of interest.

The most serious limitation in the determination of stationarySST is the inadequacy of the existing gravity models. TheGEML2 model, which is considered to be the most accurate one,especially in the low degrees, allows for the determination ofSST only up to wavelengths of 6000 km, with an accuracy ofabout 20 cm.

The altimetric measurements, in turn, can cause additionalproblems. The appropriate modeling of the orbital errors is ofbasic importance. The error model, for the time being, usuallyconsists of correcting the satellite arcs for bias and tilt. This way,only the long-wavelength orbital errors are removed, whileshorter-wavelength errors are left unaffected. This fact is not

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68 Theo Engelis

critical for the moment, since the available gravity models canalso be used only in long wavelengths. A more efficient modelingis of course desirable.

An additional factor concerning the optimum determination ofSST is the satellite specifications. Indeed, an optimum combi-nation of the density and repeatability of the subsatellite tracksis critical for the determination of time-varying SST over desiredtime and space intervals. Moreover, the stationary part of SSTand seasonal variations are to be recovered by an extended life-time of the satellite. From this point of view, SEASAT did notcompletely respond to SST-determination requirements becauseof its premature failure. Indeed, maps such as that in Figure 2show only stationary SST over a three-month period. An inter-annual SST signature, although it is not expected to differ sig-nificantly from this quasi-stationary SST, can very well havefeatures of a broader character.

A detailed detection and analysis of the oceanic behavior, bothof stationary and time-varying nature and for periods of severalyears, will have to wait until the launch of TOPEX, which to-gether with the GRM mission, is expected to be able to providea synoptic and very accurate observation of the ocean's climateand activities.

References

Coleman, R. 1981. A geodetic basis for recovering ocean dynamic informationfrom satellite altimetry. Unisurv S-19. School of Surveying, The Univer-sity of New South Wales, Kensington, Australia.

Colombo, O. 1981. Numerical methods for harmonic analysis on the sphere.Department of Geodetic Science Report No. 310. Ohio State University,Columbus.

Engelis, T. 1983. Analysis of sea surface topography using SEASAT altimeterdata. Department of Geodetic Science and Surveying Report No. 343.Ohio State University, Columbus.

Heiskanen, W. A., and H. Moritz. 1967. Physical Geodesy. San Francisco:W. H. Freeman.

Lerch, F. H., S. M. Klosko, and G. B. Patel. 1982. A refined gravity modelfrom Lageos (GEML2). Geophys. Res. Lett. 9: 1263-1266.

Levitus, S. 1982. Climatological atlas of the world ocean. Professional Paper13. NOAA, Geophysical Fluid Dynamics Laboratory, Rockville, Mary-land.

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Global Circulation from SEASAT Altimeter Data 69

Montgomery, R. B. 1969. Comments on oceanic leveling. In Deep Sea Re-search, supplement to Volume 16, pp. 147-162. Oxford: Pergamon.

Moritz, H. 1979. Report of Special Study Group No. 5.39 of the InternationalAssociation of Geodesy fundamental geodetic constants. Draft submittedto XVII General Assembly of International Union of Geodesy and Geo-physics, Canberra.

Moritz, H. 1980. Geodetic reference system 1980. Bull. Geodes. 54.Rapp, R. H. 1981. The Earth's gravity field to degree and order 180 using

SEASAT altimeter data, terrestrial gravity data, and other data. Depart-ment of Geodetic Science and Surveying Report No. 322. Ohio State Uni-versity, Columbus.

Rapp, R. H. 1982a. A summary of the results from the OSU analysis ofSEASAT altimeter data. Department of Geodetic Science and SurveyingReport No. 335. Ohio State University, Columbus.

Rapp, R. H. 1982b. A global atlas of sea surface heights based on the adjustedSEASAT altimeter data. Department of Geodetic Science and SurveyingReport No. 333. Ohio State University, Columbus.

Rizos, C. 1980. The role of gravity field in sea surface topography studies.School of Surveying, Kensington, University of NSW, Sydney, Australia.

Rizos, C. 1981. On estimating the global ocean surface circulation from satellitealtimetry. Mar. Sci. 13: 865-870.

Rowlands, D. The adjustment of SEASAT altimeter data on a global basis forgeoid and sea surface height determination. Department of Geodetic Sci-ence and Surveying Report No. 325. Ohio State University, Columbus.

Tai, C. K., and C. Wunsch. 1984. An estimate of global absolute dynamictopography. J. Phys. Oceanogr., in press.

Tscherning, C , and R. H. Rapp. 1974. Closed covariance expressions for grav-ity anomalies, geoid undulations, and deflections of the vertical impliedby anomaly degree variance models. Department of Geodetic Science andSurveying Report No. 208. Ohio State University, Columbus.

Wunsch, C , and E. M. Gaposchkin. 1980. On using satellite altimetry to de-termine the general circulation of the oceans with application to geoidimprovement. Rev. Geophys. Space Phys., 18: 725-745.

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