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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 100, NO. C12, PAGES 25,229-25,248, DECEMBER 15, 1995

Estimation of main tidal constituents from TOPEX altimetry

using a Proudman function expansion

Braulio V. Sanchez

Space Geodesy Branch, NASA Goddard Space Flight Center, Greenbelt, Maryland

Nikolaos K. Pavlis

Hughes STX Corporation, Greenbelt, Maryland

Abstract. Tidal models for the main diurnal and semidiurnal constituents have been computed

from TOPEX altimeter data and a set of Proudman functions computed numerically in the space

defined by the ocean basins. The accurate modeling of the ocean tides is necessary in order to

interpret the height measurements of the ocean surface obtained from satellite altimeters. It is also

an interesting dynamical problem in its own right. The surface height field due to any tidal

constituent can be expanded in terms of the eigenfunctions of the velocity potential (Proudman

functions) with coefficients estimated in a least squares sense from a field of discrete data points

obtained from altimetry, tide gauges, bottom pressure sensors, etc. The Proudman functions

constitute a mass conserving orthogonal basis; their computation does not require any assumption

concerning friction or energy dissipation, only a numerical grid expressing the shape of coastline

and the bathymetry of the ocean basins. They have the space structure of standing waves and can

be identified as the zero-rotation gravitational normal modes. They have to be evaluated

numerically only once for each particular grid resolution. In this investigation the Proudman

functions were computed by means of finite differences in spherical coordinates over a 2øx2 ø grid

covering most of the world's oceans for a total of 8608 degrees of freedom. The data field used in

this study consists of approximately 15 months of TOPEX altimetry in the form of collinear

differences. Results for the major semidiurnal and diurnal constituents (M2, S2, N2, K2, K1, O1, P 1,

and Q1) have been obtained in terms of corrections to a priori values obtained by fitting

Schwiderski's (1980) models. The new models (Goddard Space Flight Center (GSFC94A)) are

tested at a set of "ground truth" data points. These tests indicate substantial improvement for most

of the constituents as compared with Schwiderski's solutions. Use of GSFC94A results in a 7.8-cm

reduction in the rms overlap difference of 15.5 cm. The GSFC94A model yielded a mean rms sea

surface variability of 7.9 cm, compared with the 9.4 cm obtained when using Schwiderski's model.

1. Introduction

Advances in computer technology and numerical techniques

allowed the development of global tidal models during the late

1960s and 1970s, culminating with the models of Schwiderski

[1980] in the late 1970s and early 1980s. The advent of satellite

altimetry has introduced a new era in the development of global

tidal models, beginning with the Geosat-based models produced

by Cartwright and Ray [1990, 1991]. The success of the

TOPEX/POSEIDON mission is providing the community of

tidalists with the opportunity to develop a new generation of

altimetry-based global tidal models. This paper reports on the

development of a model based on TOPEX altimeter data using

Proudman functions as a basis to represent the tidal height and

assimilate these data.

Proudman [ 1917] gave an account of a general method of

treating the tidal equations using the Lagrangian formulation. Rao

[ 1966] reformulated the method using the Eulerian approach. Rao

and Schwab [1976] and Schwab and Rao [1983] used the

technique to compute normal modes of closed basins. Sanchez et

Copyright 1995 by the American Geophysical Union.

Paper number 95JC02082.

0148-0227/95/95JC-02082505.00

al., [1985] computed a solution for the M2 tidal constituent in

Lake Superior using the Proudman functions as a basis for the

expansion. Cartwright et al. [1988] used Proudman functions

with data from a combination of coastal, island, and pelagic

measurements to compute the M2 and O1 tides in the Atlantic-

Indian ocean basins, as well as the M3 tide in the North Atlantic

Ocean. Sanchez and Cartwright [ 1988] produced solutions for the

M2 and O1 tidal constituents over the Pacific basin using Seasat

altimetry data, while Sanchez et al., [ 1992] obtained a solution for

the M2 tidal constituent in the Mediterranean basin using

Proudman functions and Geosat altimetry data.

Proudman's [ 1917] method represents the surface displacement

or tidal height as a linear combination of standing waves. The

basis functions for the expansion are the eigenfunctions

associated with the velocity potential, which are also the

gravitational normal modes of the basin at zero rotation. The

velocity potential eigenfunctions have to be computed only once

for a particular basin since their structure depends only on the

shape of the boundary (coastline) and the bathymetry .They

conserve mass and satisfy orthogonality over the space of the

basin.

The altimetry data provided by the TOPEX/POSEIDON

mission are of unprecedented quality and quantity from the

standpoint of ocean tidal analysis. As a consequence, high-quality

tidal solutions based on these data can be obtained by almost any

25,229

25,230 SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS

method. However, new approaches are of interest and,

furthermore, might offer certain advantages.

As reviewed by Cartwright [1991], the strategies for tidal

analysis from satellite altimetry data fall into the following three

general categories: (1) purely temporal analysis, (2) joint analysis

in time and longitude, and (3) general analysis in time and space.

Methods which fall into the last category are expected to take

full advantage of the space correlations of the tides, therefore

achieving higher accuracy. The Proudman function approach is a

member of this class. Specifically, previous applications have

shown the following.

1. Tidal aliasing problems can be alleviated, as shown by

Sanchez and Cartwright [1988], who obtained solutions for O1

and M2 based on 25 days of Seasat altimetry data. The

corresponding aliased periods were 14.8 days for Ol and 16.4

days for M 2.

2. It is possible to obtain fairly accurate solutions in areas

where the magnitude of the tide is comparable to the altimeter

noise and the data coverage is less than optimum. This was shown

by Sanchez et al. [ 1992], who obtained solutions for the M 2 tide

in the Mediterranean Sea from Geosat altimetry data. These

solutions show great improvement over previous solutions

obtained by Cartwright and Ray [1991] using the admittance

technique.

3. The Proudman-function solutions reduce significantly high-

frequency noise which is detrimental to accuracy. This was

shown by Ray and Sanchez [ 1994], who used Proudman functions

to smooth the Cartwright and Ray [ 1991] models. The resulting

(smoothed) solutions exhibit lower rms discrepancy at 80 "ground

truth" locations than the original models.

4. The method allows the assimilation of various data types,

such as satellite altimetry and tide gauges or bottom pressure

sensors. This was shown by Sanchez et al. [1992], who obtained

solutions for M 2 in the Mediterranean Sea, which combined

Geosat altimetry data and tide gauge data from 43 coastal

locations. The combined solutions show better error estimates

than the solutions using altimetry data alone.

5. The eigenvectors of the velocity potential constitute an

integral part of Proudman's [1917] methodology. The theory

provides a formulation for the calculation of gravitational normal

modes modified by rotation, as well as the rotational normal

modes modified by gravity. It can be modified to produce

synthetic forced solutions corresponding to the tidal constituents,

as done by Sanchez et al. [ 1985]. Comparisons of these synthetic

solutions with those obtained from satellite altimetry data could

be used to analyze dynamic parameters, such as friction.

2. Modeling Aspects

The objective of this investigation is to estimate ocean-wide

models for the major tidal constituents using overlap differences

of altimetric measurements of the ocean's surface. Three main

parts comprise this analysis strategy: (1) the basis functions used

to represent (geographically) the tidal height field, the Proudman

functions; (2) the mathematical model used to relate the

observables to the parameters of interest; and (3) the estimation

technique employed.

2.1. Proudman Functions

The mathematical development of the theory has been

presented before [Sanchez et al., 1985], and will not be repeated

here. It is sufficient to recall that partition of the transport field

into solenoidal and irrotational components and application of the

divergence operator yields the eigenvalue problem from which

the velocity potential eigenvectors are derived:

V.hV• r =-;•rOr (1)

The impermeability of the coast provides the associated boundary

conditions:

h--•- = 0 (2)

where h denotes the depth of the fluid and n stands for the

direction normal to the boundary. The subscript 7 has a range

determined by the number of degrees of freedom allowed by the

system.

For this study the numerical solution of the eigensystem (1)

has been obtained by means of finite differences expressed in

spherical coordinates. The grid used has a 2øx2 ø equiangular

resolution and extends from 69.25øN to 76.75øS latitude, as

40 •'• •o

20 ø

o

20 ø

40

60'

80 ø

160' 120' 80' 40' 0' 40' 80' 120' 160 ø

Figure 1. Spherical coordinate grid used in the development of the Proudman functions. Grid size is 2øx2 ø.

SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS 25,231

shown in Figure 1. The resulting eigensystem has 8608 degrees of

freedom. Notice that this grid excludes enclosed basins, such as

the Mediterranean, Hudson Bay, etc. Such basins require

individual analyses with grids of much finer resolution [Sanchez

et al., 1992], while this study is mainly focusing on tidal

estimation over open, deep ocean areas. The Arctic Ocean is also

excluded from this grid.

The Lanczos method [Lanczos, 1950] was used in the

formulation of the matrix image of the operator appearing in (1).

The Lanczos basis was created with normalization of every

auxiliary vector with respect to every previous one. This was

done for 8608 iterations which cover the total number of degrees

of freedom allowed by the system. The subroutines BISECT and

TINVIT from EISPACK were used in the solution. The hardware

consisted of CRAY supercomputers available at the Goddard

Space Flight Center (GSFC) and at Cray Research, Incorporated,

in Minneapolis, Minnesota. The solution was obtained for the

entire 8608 eigenvalues, but only the first 3000 eigenvectors were

created. This set is more than sufficient to provide a basis for the

representation of the tidal height field in a series expansion form,

given the speed with which such a series converges.

It can be shown that the velocity potential eigenvectors are also

the gravitational normal modes of the basin at zero rotation. As

such, the eigenvalues of the solution can be associated with a

frequency of oscillation according to the following relation:

v r = • (3)

where g is the acceleration due to gravity and v is the frequency

of oscillation. The periods associated with the frequencies of the

solution range from 105.6 hours corresponding to the second

eigenvalue to 16.9 mins for y = 8608. The first eigenvalue

corresponds to a constant eigenvector with infinite period. This

vector is not used in the estimation. These periodicities do not

enter into play when fitting a tidal height field with the

eigenvectors of the solution. However, the results show some

interesting coherence patterns between the zero-rotation periods

and the periods of the particular tidal constituents (see also

Sanchez, [1991]). Figure 2 shows these periods as a function of

wavenumber for the first 50 and for the first 900 eigenvalues (this

second range covers all the eigenvectors used in the tidal

solutions performed in this study).

The spatial structure of (I> r , for three selected wave numbers

7, is illustrated in Figures 3a through 3c. Wavenumbers 20 and 40

have been selected because they are among the most energetic in

the estimation of diurnal and semidiurnal tidal constituents from

TOPEX altimeter data. Wavenumber 500 has been included to

show the fine spatial structure (short spatial wavelengths) present

in eigenvectors of higher wavenumbers. The following

normalization has been applied:

__1 ii•y•l•dA:6• (4)

A basin

where A denotes the area of the basin and 6• is the Kronecker

delta. The dotted and solid contour lines in Figure 3 correspond to

negative and positive elevations, respectively. The zero-contour

lines (nodal lines) appear as bolder solid contours. The contour

interval is the same for all figures.

2.2. Mathematical Modeling

The following notation is introduced:

_h w instantaneous sea surface height above a reference ellipsoid.

h w mean (time-invariant) sea surface height.

r/g geocentric tide.

120

lOO

m 80

o

60

40

2O

120

100

m 80

o

v

60

40

2O

i i

o

o

o

o

o -

ø o

o

oo o

ooo,•

VUOoooooo -

øøøøøoooooooo• oooooooooooo•

I I I I

10 20 30 40 5o

Proudman function number

0 100 200 300 400 500 600 700 800 900

Proudman function number

Figure 2. Zero-rotation periods of the Proudman functions

for (a) wavenumbers 1-50 and (b) wavenumbers 1-900.

W t all temporal variations of h w other than tides.

h s satellite's ellipsoidal height.

p altimeter range observable.

It holds true that

h w -hs-P-h w -I-l•g -I-w t.

(5)

Equation (5) relates theoretical (true) quantities. Orbit

determination provides an estimate of h s (denoted (h s)c), while

the onboard altimeter provides a measurement of p (denoted

prn). In addition, an existing model of the geocentric tide enables

one to compute a value (r/g)C. These are contaminated by errors

so that

h s = (hs)C _/Xh• (6a)

p = pm _ Ap (6b)

•g = (•g)C -A•g (6C)

A computed value of h w (denoted (h w)c) is defined as

(hw) c = (hs)C - p rn

so that from (5), (6), and (7) one has

(hw) c -(77g) c -- h w q- Ah s -- Z•77g -- Z•p + W t .

(7)

(8)

25,232 SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNC'HONS

Figure 3. Spatial structure of selected Proudman functions for (a) y= 20, (b) y= 40, and (c) y = 500.

SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS 25,233

The quantity:

(hw ) r = (hw )C - (llg ) c (9)

will be referred to as residual sea surface height (rSSH). This is

computed from equation (9) and from the available (a-priori) tide

model. Accordingly, (8) becomes (with explicit notation for time-

dependent terms):

(hw)r(t) = •w +(Ahs-Arlg-AP+wt)(t). (10)

An overlap difference is defined as the difference between two

residual sea surface heights evaluated at the same location at two

different times:

(•w)r(tl,t2)=(hw)r(t2)-(hw)r(tl) (11)

so that from (1 O) one has

(•v )r(ti,t9.) : Ahs(tvt,)- Arlg(tvt9.)- Ap(ti,t9.) + wt(q,t ,).

(12)

Equation (12) provides the fundamental relationship between the

overlap difference and the terms pertaining to radial orbit error,

geocentric tide error, measurement error, and nontidal sea surface

variability. Tapley et al.. [1994] report 3 to 4 cm overall root

,•a, •qua• um•) radial orbit error for TOPEXdt'u•mt•or•.

Considering such high radial orbit accuracy, it was decided to

omit the orbit error term altogether from the modeling. Detailed

discussion on the radial orbit error characteristics for "frozen"

repeat orbits are given by Colombo [1984]. The tidal part of the

orbit error Ahs r introduces an additional complication to the

problem. Colombo [1984, section 5.7] pointed out that its short-

period component is most likely to arise from errors in the ocean

tide model used during the orbit determination. He showed that

this part, in the case of a "frozen" repeat orbit, is "lumped"

together with the geocentric tide term (see also Bettadpur and

Eanes [1994]). From overlap differences alone, one can only

estimate this lumped tide. An iterative process is required to

separate Ar/g from Ahs r and, furthermore, the "bottom-

referenced" tide Ar/o from the geocentric tide Ar/g [Colombo.,

1984, section 6.3]. The tidal solutions that are presented here

should be viewed as results from the first iteration in this process.

They are developed under the assumptions that (1) the satellite's

orbit is perfectly known radially, and (2) the solid-earth tide tSand

the ocean loading effect t59) are modeled perfectly (the data are

corrected for these effects using the corrections provided in the

geophysical data record).

In addition, the terms Ap and w t are combined into a random

variable e with geographically dependent statistical properties, as

tt(t) =-Ap(t) + wt(t). (13)

Accordingly. (12) becomes:

(•w)r(ti,to.) = -Arlo(tpto.)+•e(ti,to. ) . (14)

In this study the overlap differences are formed by subtracting

the residual sea surface height referring to a 'master' repeat cycle

(subscript m) from the corresponding value of every other cycle

(subscript j). Therefore, an overlap difference at location (•o,•)

takes the form (with explicit notation for the geographic

arguments)

(•hw)r (•0, •,; tm, ti )= (h w )r (•0, •,; ti ) - (h w)r (•0, •,; tm )

= Arlo(q,/•;t m)- Arlo(q,/•;t , )+ [e(q, •,;t,)- e(q,/•;tm)] (15)

Under the assumptions of linearity and time-invariance

[Colombo, 1984, section 5.3] the total ocean tide is the result of

superposition of the effects from each particular constituent •:,

i.e.,

Ar/o ((P, X;t)= Z At/o r ((p, X; t) (16)

and the frequency of each constituent can be calculated purely

from astronomical data. These assumptions are in close

agreement with reality over open, deep ocean areas. The

Proudman-function expansion of the tidal height correction for a

constituent •: is given by [ Sanchez et aL, 1992]

oo 1

Ar/or(tP ,Jr;t) = Z Z

i=0/•=0

where

tI)i(tp, g) i th Proudman function value at location (q,g);

w r (t) angular frequency (speed);

V* (t) astronomical phase;

fr (t) nodal modulation factor;

u r (t) nodal modulation angle.

and AR•5 are the Proudman-function coefficient corrections to be

estimated (/5 distinguishes inphase from quadrature terms). One

b = b(true)-b(a priori). 08)

In view of (16) and (17), (15) becomes

oo 1

(8'tw)r(9'&tm'tj)--ge(•O'&tm'tj) = Z Z Z ARk'd•i(q'X)

r i=0/•=0

where T m is the epoch at 0.00 hours • of the day in which the

instant t m hlls and t• = t m - T m (similarly for t 5.)(see also

Colombo [ 1984, p. 111 ].

Equation (19) is the line• mathematical model based on which

a least squares adjustment is performed to estimate the

Pmudm•-function coefficient corrections. In this study the

a-priori values of the Proudman-function coefficients were

obtained from least squares fits to Schwiderski [1980] gfidded

maps. The angular frequency of each constituent, as well as the

astronomical phase, were obtained based on the equations

defining the fundamental arguments of the International

Astronomical Union (IAU) 1980 nutation series [McCarthy,

1992]. The nodal modulation angles and hctors [Pugh, 1987,

section 4:2:2] were computed as prescribed by Schureman

[1958]. Here ur(t)is given as a linear combination of six

auxiliary angles (•,v,Q,v',R,v"), while fr(t) is a nonlinear

function (in general) of the angles (I,v,P). The necessary

information is summ•zed in Table 1 (for details, see Schureman

[•958]).

2.3. Estimation Technique

The mathematical model (19) is written in vector form as

(•hw)r({o,•;tm,t j)-•jE({O,•;tm,tj)= [,l•((p, •; t m )- A({o,•;tj )]. X

(20)

25,234 SANCHEZ AND PAVLIS: TOPEX TIDES USlNG PROUDMAN FUNCTIONS

Table 1. Tidal Constituent Information

Darwin's Doodson

Symbol Number • v (2 v' R

Factor f Alias Period, Number of Proudman

v" Formula d Functions

K 1 165.555 0 0 0 -1 0 0 (227) 173.2 500

O] 145.555 2 -1 0 0 0 0 (75) 45.7 500

M 2 255.555 2 -2 0 0 0 0 (78) 62.1 900

S 2 273.555 0 0 0 0 0 0 unity 58.7 900 (500)

P] 163.555 0 0 0 0 0 0 unity 88.9 500

Q• 135.655 2 -1 0 0 0 0 (75) 69.4 500

K 2 275.555 0 0 0 0 0 -2 (235) 86.6 900 (500)

N 2 245.655 2 -2 0 0 0 0 (78) 49.5 900 (800)

Variables are the six auxiliary angles used to compute ur(t). Formula numbers refer to Schureman, [1958]. Numbers in

parentheses in last column indicate use of a truncated set of coefficients.

where X is the vector of unknown Proudman-function coefficient

corrections and the elements of vector A are the terms

multiplying AR[p in (19). The ensemble of (20), for all available

overlap differences at a given location (q0, Z), takes the matrix

form

6F-bE =6A.X (21)

which leads to observation equations:

¾ = 6A. •- 6F (22)

where ¾ is the vector of residuals and •[ denotes the adjusted

values of the parameters [ Uotila, 1986]. Assuming

E{6E} = 0 (23a)

E•(9, Z) = E{6E.6E*} (23b)

and the weight matrix of the observations is defined by

x)= o0 ß x)] . (24)

where •02 is the a-priori variance of unit weight. Minimization of

the target function

<p = vT ß Pa=(q0, t). V (25)

subject to the condition (22), leads to the normal equations

N(q0, Z). •[- U(q0, Z) = 0 (26)

where

N(rp, Z) = 6AT. pa= (qo, Z ) .6A (27a)

U(r,o, A) = 5AT. Pa: (•o, A) ß •.

(27b)

Consider now that at a given ground track location (q0, Z), K

repeat cycles contribute valid rSSHs. One can form at most K-1

linearly independent overlap differences. One way to form these

differences is by subtracting one rSSH from all others. This is the

way selected here. The resulting overlap differences are error

correlated, even if the original rSSH are assumed to be

uncorrelated. In fact, if the errors of the rSSHs are assumed to be

uncorrelated and have equal variance ffø-(q0, Z), then the error

covariance matrix of the overlap differences is

-2

1 2

za(o,x ): 2 ,

1 1 1 . 2

(28)

that is, the overlap differences are 50% error correlated to each

other. In this case the weight matrix of the overlap differences can

be formed without inverting E•(q0, Z) since

x)= o0 ß x)]-'

1 K-1

=•- 1 K-1 .

0'2 (•0,/• ) -'•' K K

o .

I I I •1

K K K

(29)

Despite the simplicity of (29), the assumption of uncorrelated

errors for the rSSHs may not be a realistic one. The sea surface

variability component of the error w t is not necessarily behaving

as white noise, while errors in the instrument and media

corrections applied to the altimeter range are most likely to

introduce spatial and temporal correlations in Ap. However, the

assumption of temporally uncorrelated rSSHs can be avoided

without unduly increasing the computational effort. Following

Mazzega and Houry [ 1989], the altimeter noise Ap was assumed

to be white, i.e.

cOVAp(At) = D 2 ß •(At) (30)

where 6(At) is the Dirac delta, while the temporal correlation of

the sea surface variability was assumed to decay exponentially,

i.e.

COVw, (At)= V 2 .exp - •.2 • ' (31)

The following numerical values were used for the parameters

appearing in equations (30) an, d (31): D 2 = 25 cm 2, x = 25 days,

and V2= 0.10.[min(rr•,25)] •' cm 2'

Note that D refers to normal points, not 1-Hz data; crv was

estimated directly from the data for each ground track location

and is plotted in Plate 3 (see section 3.3). The empirical formula

defining V 2 was devised so as to avoid disproportionate

downweighting of the data in areas of high mesoscale variability.

It results in weight ratios of approximately 1 to 3.5 for areas of

high versus low variability. The variable x (corresponding to a

correlation time of 20.8 days) was kept constant over the entire

oceanic domain. It is fully recognized here that the modeling of

the stochastic properties of the errors can be improved, e.g., by

estimating directly from the data empirical covariance functions

for w t with varying correlation times depending on the location.

Such analysis was beyond the scope of this study.

SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS 25,235

On the basis of (30) and (31), a distinct error covariance matrix

was formed for the available overlap differences at each ground

track location. It should be mentioned here that latitude-

dependent weighting, such as the one described by Wang and

Rapp [1994], was not applied. Assuming that the data errors are

uncorrelated among different locations (i.e., permit correlation in

time but not in space), the error covariance matrix for the totality

of overlap differences over the entire ocean becomes block-

diagonal. Each block pertains to a particular ground track

location, and all operations to form the normal equations are

independent among different locations. This assumption of space

decorrelation can be justified only weakly by the use of normal

points (which are about 70 km apart along track). It is

necessitated by the size of the problem which prohibits the

formation of the full error covariance matrix (in this analysis such

a matrix would have dimension of about 1.8 million). According

to the above, the normal equations for the totality of overlap

differences were obtained as

N= ZN((o, g) (32a)

U= • U(•p, •t) (32b)

Certain areas of the oceanic domain are either void of data due

to the satellite's inclination or poor in coverage due to the

presence of ice. Such data gaps affect the estimation of

coefficients, especially of higher wavenumbers. To avoid

spurious estimates of these coefficients, an a priori constraint was

used. Per wavenumber, the corrections to be estimated were

assumed to have a priori value zero, with standard deviation equal

to 10% of the power implied by the coefficients obtained from the

fit of Schwiderski's[1980] model. A diagonal matrix Px was

defined with elements equal to the reciprocal values of the a

priori variances. The solution vector and its error covariance

matrix become [ Uotila, 1986]:

• = (N +Px)-•.U (33a)

Z i =c•).(N +PX) -1 (33b)

and the adjusted Proudman-function coefficients are

,• = ,•(a prod)+ •(. (34)

3. Altimeter Data Processing

The tidal solutions developed in this study are based on • 0

altimeter data collected by the dual-frequency TOPEX altimeter •.

during the first 47 repeat cycles of the TOPEX/POSEIDON •

mission. These data span the period from September 25, 1992, to •, -1

January 2, 1994. A substantial amount of preprocessing had to be •

applied to the original altimeter data in the form of geophysical •

data records (GDRs) to convert them into overlap differences .•,

suitable for input to the least squares estimator. These processing

steps are discussed here in some detail.

3.1. Formation of a 1-Hz Collinear Database

The first step in the processing consists of registering the 1-Hz

GDRs into a collinear database. The geographic locations that

define this database are the subsatellite locations corresponding to

a nominal 1 s -1 sampling rate along the ground track produced by

the orbit during repeat cycle 17. The orbital ephemeris estimated

at NASA Goddard Space Flight Center from satellite laser

ranging (SLR) and Doppler orbitography and radiopositioning

integrated by satellite (DORIS) data [Tapley et al., 1994] was

used to define the ground track of cycle 17. The altimeter data are

registered to these database locations in the form of SSHs as

follows (B. D. Beckley, private communication, 1993).

1. Altimeter 1-Hz GDRs are edited out if anyone of the

following flags is set [Callahan, 1993]: Alt_Badl (bit 3) (normal

ocean mode)' Alt_Badl (bit 6) (more than four points flagged in

Iono_Bad); Geo_Bad (bit 1) (land); or Geo_Bad (bit 7) (ice). In

addition, GDRs from cycles 1 through 12 have been edited out if

the off-nadir angle exceeded 0.45 ø . This was done to avoid use of

some data collected at the beginning of the mission when the

satellite's attitude control system was not properly calibrated

[Callahan, 1993, pp. 3-10].

2. GDRs passing the editing criteria are used next to

interpolate the SSH, along the reference track, at the database

locations. This is done only if both 1-Hz GDRs surrounding a

database location have passed the editing criteria. The

interpolated SSH is computed from a straight-line least squares fit

to the high-rate (10 Hz) SSHs. An iterative 3.5cr edit is employed

in this fit. SSH is not computed at a database location if more

than four (out of 10) high-rate SSH_Bad flags (Callahan, 1993,

pp. 3-16) are set. In addition to the SSH, all environmental,

geophysical and instrument corrections appearing on the GDR are

interpolated to the database location. Notice that the SSHs on the

database are not corrected for cross-track gradients of the sea

surface.

The 1-Hz altimeter database has been developed and is

maintained at NASA GSFC. It supports a variety of scientific

investigations which may have different requirements of data

accuracy/coverage. For this reason, it was decided to apply a

rather loose editing when forming the database (so that the

majority of observations can be retained) and let the user apply

additional editing which may be suitable to the particular

investigation.

3.2. Definition of a "Master" Normal Point Ground Track

To define the overlap difference as prescribed by (15), one

needs to define a master repeat cycle. This cycle has to be

' , , l, [ ,I [ i, ,i i i [ i ! ! i , , i

0 100 200 300 400 500

MJD - 48880.0

Figure 4. Average surface pressure difference from 1013.3

mbar over the ocean area sampled by TOPEX during cycles 1-

47 (extending from 66øN to 66øS). The dots represent these

averages computed on the basis of the TOPEX GDR dry

topospheric correction with a best fitting cubic spline shown

as a solid line. The linear trend is shown as a dashed line.

25,236 SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS

8O

6O

4O

2O

-2O

-4O

-6O

-8O

I

60 90 120 150 180 210 240

Longitude

1.0--

0.8-

O.6-

0.4-

0.2-

0.0_

I I I

27O 3OO 33O

-2 -1 0 1

(meters)

Figure 5. Location of 50,101 11-s normal points of TOPEX cycle 17 which define the "master" ground track.

360

lOO%

80%

60%

40%

20%

o%

I 5 9 13 17 21 25 29 33 37 41 45

Cycle

Figure 6. Percentage overlap of normal points with respect to TOPEX cycle 17.

SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN F[•CTIONS 25,237

lOO%

80%

60%

40%

20%

o%

3 8 13 18 23 28 33 38

K

Figure 7. Percentage of ground track locations where at least K overlap differences occur.

selected so as to provide the best overall geographic coverage

with valid observations. Any location void of data on the master

cycle will produce a gap, even if other cycles have valid data to

contribute. This selection process was based on considerations

pertaining to the completeness of coverage with valid data over

the open ocean and the ice concentrations over polar latitude

regions. A number of alternative editing scenarios were tested for

different cycles, and the resulting coverage and quality of data

were examined and compared. The adopted editing procedure

consists of the following acceptance criteria which were applied

to the data of the 1-Hz collinear database: (1) Sigma_SSH < 0.15

m, (2) significant wave height (SWH) < 10.0 m, (3) Sigma_SWH

< 1.0 m, (4) abs(ocean tide correction) < 2.0 m, (5)

Sigma_Naught < 25.0 dB, (6) -3.0 m < dry tropospheric

correction < -2.0 m, (7) -1.0 m < wet tropospheric correction <

0.0 m, (8) -0.5 m < ionospheric correction < 0.0 m, (9)

abs(solid-earth tide correction) < 1.0 m, (10) abs(SSH-

MSS*OSU92) < 3.0 m, and (11) ocean depth > 0 m (based on

TUG87 30'x 30' mean elevation database)

The Ohio State University mean sea surface (MSS) [ Basic and

Rapp, 1992] was corrected for a 0.40-m radius difference and a

0.24-m Y shift before performing criteria 10 above (see also

Callahan, [1993]). This corrected MSS will hereafter be denoted

MSS*(OSU92). The TUG87 30'x30' mean elevation database

compiled at the Technical University of Graz is described by

Wiser [1987]. The 1-Hz SSH data that pass the above criteria are

then corrected for: (1) instrument corrections, (2) solid-earth tide,

(3) ocean tide (Schwiderski [1980] model), (4) ocean loading, (5)

pole tide, (6) wet and dry tropospheric corrections, (7)

ionospheric correction, and (8) inverted barometer correction.

A residual 1-Hz SSH is then formed by subtracting the

MSS*(osu92) from the corrected TOPEX SSH. This accounts (to

the first order) for the cross-track gradient correction [Callahan,

1993, section 3.4.4], which is necessary because the spacecraft

does not overtly exactly its track [Brenner et al., 1990; Wang and

Rapp, 1991]. These residual SSHs are binned over intervals of

11 s to form a "normal point." This is computed from a straight-

line least squares fit with an iterative 3 (y edit. A normal point is

not computed if the rms of fit exceeds 0.5 m, or less than three

(out of five) 1-Hz rSSHs are accepted inside each 5-s subinterval

surrounding the center point, or the along-track slope of the

rSSHs exceeds (in absolute value) 10 arc sec.

The treatment of the inverted barometer effect requires some

discussion. For the formation of normal points the static inverted

barometer correction was applied to the 1-Hz SSHs, as described

by Callahan [ 1993, pp. 3-7]. This correction was calculated based

on a constant reference atmospheric pressure of 1013.3 mbar.

From the dry tropospheric correction provided with the data the

atmospheric pressure on the ocean's surface was computed for

every (accepted) 1-Hz data point. Normal point values of this

pressure were then computed in the same way as used for the

residual SSHs. The area-average value (over the part of the ocean

sampled by TOPEX) was then calculated for each repeat cycle.

The area-average surface pressure differences from 1013.3 mbar

are plotted as dots in Figure 4. As can be seen, the variation of the

surface pressure is characterized by a linear trend (which may

arise due to the limited duration of the data record), a strong

annual component and, superimposed shorter-period signatures.

A cubic spline interpolation was performed that yielded the solid

curve in Figure 4. This interpolator (after detrending) provided

values of the surface pressure difference (from 1013.3 mbar) at 2-

day intervals. On the basis of these values, an additional inverted

barometer correction term was applied to the normal point

residual SSHs so as to refer them to the variable (in time) surface

pressure reference instead of the constant 1013.3 mbar.

This editing and normal point formation procedure when

25,238 SANCHEZ AND PAVLIS: TOPEX TIDES USlNG PROUDMAN FUNCTIONS

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

Longitude

36O

,201

E 15

z

• lO

o

,,.. 5 •

o

• 0

10 20 30 40

(Count)

Plate 1. Number of normal point sea surface height (SSH) values available per ground track location

8o

60

40

20

-20

-40

-60

-80

,, , .. • '•'.•

0 30 60 90 120 150 180 210 240 270 300 330 360

Longitude

L 4

-40 -20 0 20 40

(cm)

Plate 2. Average value of the quantity (SSH TOPEX - MSS*OSU92) (see text) per ground track location. Units

are centimeters.

SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS 25,239

applied to the data of cycle 17 (526,160 1-Hz SSHs from the

collinear database) rejected 6615 1-Hz data points and produced a

set of 50,101 normal points. The location of these normal points

is shown in Figure 5. As can be seen, cycle 17 provides excellent

coverage over the whole oceanic domain sampled by TOPEX.

Cycle 17 spans the period from February 28 to March 10, 1993.

This period corresponds to minimal ice coverage over the

southern hemisphere, thus providing optimal data coverage over

the Antarctic circumpolar region. The ice coverage over the

northern hemisphere does introduce some gaps over the areas of

Newfoundland/Labrador Sea, the Bering Sea, and the Sea of

Okhotsk. These regions, however, are semienclosed basins, and

their absence from the solution is outweighed by the presence of

the Antarctic circumpolar region. Cycle 17 was thus selected to

define the master normal point ground track.

3.3. Formation of a Normal Point Collinear Database

The editing and normal point formation procedure described

above was applied to the TOPEX data from cycles 1 to 47. The

normal points formed were forced to be centered at the ground

track locations of Figure 5. In this manner, a collinear database of

normal points was constructed. The overlap differences input to

the adjustment were obtained from this database considering the

following two conditions. (1) For a master ground track location

to participate in the adjustment, at least two valid residual SSHs

should be available at that location (so that at least one overlap

difference could be defined). (2) Proudman functions had to be

defined at every ground track location. Their evaluation was

based on a bilinear interpolation algorithm.

Of the 50,101 locations of the master ground track, 28

locations were found to violate condition 1 and 857 violated

condition 2. Accordingly, the input file consisted of 49,216

ground track locations, with a total of 1,790,191 overlap

differences. The percentage of overlap of coverage with normal

points between cycle 17 and the other cycles is illustrated in the

bar diagram of Figure 6. The reduced overlap during cycles 1 to

16 is due to the antenna-sharing plan (TOPEX versus

POSEIDON altimeters) followed during the beginning of the

mission; this is also the reason for the gaps during cycles 20, 30,

and 41. The sinusoid signature after cycle 17 is due to the annual

growth and reduction of the ice coverage over the southern ocean.

On the basis of the available data, at a given ground track location

the maximum number of overlap differences is 43 (the minimum

is obviously 1). The bar diagram of Figure 7 gives the percentage

of ground track locations (over 49,216) where at least "K" overlap

differences are available. As can be seen, about 90% of all

locations have at least 25 (out of 43) valid overlap differences.

This information is illustrated geographically in Plate 1. Plate 2

displays the average value of the normal point residual SSH

(=SSH TOPEX - MSS*osu92) per ground track location. Over

the 49,216 ground track locations that participate in the

adjustment, these values have extrema of-198.2 to 121.5 cm,

with an ocean-wide average of-0.5 cm and an rms of 13.2 cm.

Plate 3 displays the rms variability (about the mean of Plate 2) per

ground track location. The mean rms (over 49,216 locations) is

9.4 cm. The extrema of the values in Plate 3 are 0.4 to 114.3 cm.

These statistics were obtained without performing any editing

pertaining to the number of residual SSHs available per ground

track location and by using the a priori Schwiderski [ 1980] model

to remove the ocean tide effect.

4. Results

The estimation of the coefficients of the Proudman functions

can be performed according to different solution designs

-8O

0 30 60 90 120 150 180 210 240 270 300 330 360

Longitude

• 14

E 12

z 10

• 8

• 6

'-' 4

o 2

• 0

5 10 15 20

Plate 3. Standard deviation (rms about the mean) of the quantity (SSH TOPEX - MSS*OSU92) per ground track

location. Units are centimeters.

25,240 SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS

,t

t

o

ß ,- o

o

o

o

o ,_

o

ß

o

SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS 25,241

involving the consideration of factors, such as the magnitudes of

the tidal constituents, the alias periods produced by the TOPEX

orbit, and the size of the matrices involved in the solution. On the

basis of a number of preliminary test solutions, it was decided to

model the semidiurnal constituents with 900 Proudman functions

and the diurnal with 500. This selection amounts to 11,200

unknown coefficients to be estimated simultaneously (ideally).

Because of the large size of such a system of normal equations, it

was decided to design the solution as outlined in Table 1: two

groups of tidal constituents were estimated separately; the first

group consisted of (M2, S2, K1, and Ol ) and the second of (N2,

K2, P1, and Q• ). This strategy yielded a total of 5600 unknowns

for each solution group. This was necessitated by the available

computer resources. Improved computing algorithms which take

Table 2. Percentage of the Total Variance Present in the Five

Most Energetic Vectors for Each Estimated Constituent

Wave number Schwiderski [ 1980] GSFC94A

40 15.2 14.2

2 12.1 12.1

35 5.4 6.6

52 5.0 5.7

44 4.7 4.6

S2

40 12.8 11.6

44 7.4 7.4

52 6.0 6.3

35 4.5 6.0

39 4.3 4.6

N2

2 25.8 26.6

40 11.7 11.3

39 5.0 5.7

35 4.3 5.3

52 3.5 4.0

2 16.4 13.5

40 10.7 9.7

44 6.0 5.6

52 4.7 5.2

39 3.8 4.3

20 11.5 12.6

13 12.3 11.5

15 10.1 9.8

18 6.0 7.2

24 6.0 4.8

O•

15 13.2 13.2

13 11.8 11.4

20 10.1 10.1

16 6.1 6.5

12 5.2 5.3

20 11.5 13.6

13 9.8 10.2

2 10.8 10.1

15 9.3 9.0

18 4.4 6.1

Q•

2 24.6 24.6

15 11.8 12.3

13 8.3 9.6

20 6.6 7.7

16 4.9 5.4

Ordering is according to decreasing percentages of the Goddard Space

Flight Center GSFC94A model.

Table 3. Cumulative rms Amplitude for Schwiderski's [1980]

Solution, the GSFC94A Model and the Estimated Corrections

to Schwiderski's model

Constituent Schwiderski GSFC94A Estimated

[ 1980] Correction

Proudman

function fit

M 2 41.79 42.72 6.73

S 2 15.76 16.13 3.20

N 2 9.81 9.86 1.54

K 2 4.76 5.01 0.91

K 1 14.09 15.09 1.90

O1 10.25 10.96 1.53

P• 4.75 4.99 0.73

Q• 2.46 2.59 0.40

Values are in centimeters.

advantage of massively parallel processor architecture are

currently being designed. These will enable the simultaneous

adjustment for all parameters in future solutions. Such an

adjustment is necessary in order to quantify the effect of the

current strategy on the results. A posteriori testing indicated that

the S2, K2, and N 2 solutions yield better ground truth

comparisons when using a truncated set of coefficients, as

indicated by the numbers in parenthesis in Table 1. These

truncated sets were estimated as part of the larger set, however.

The eight sets of estimated Proudman function coefficients

constitute the tidal model GSFC94A.

Plates 4a through 4d illustrate the rms power per wavenumber

for the estimated corrections to Schwiderski's [ 1980] model and

the GSFC94A model, as well as the errors associated with the

GSFC94A solution, for M2, S2, K•, and O1, respectively. The

plotted curves are the result of a 10-point moving average

smoothing applied to the original values, which otherwise appear

too noisy. Although the uncertainties associated with GSFC94A

are formal errors from the least squares adjustment, which have

not been calibrated, it is interesting to notice that for S2 the

signal-to-noise ratio approaches unity around wave number 550.

Comparisons with tide gauge data (independent information)

indicated that the S 2 model should be truncated to 500 Proudman

functions. This implies that the errors associated with GSFC94A

are rather representative estimates of the accuracy of the model

(but slightly optimistic).

The rms overlap difference (over 1,790,191 values

participating in the adjustment) was reduced from 15.5 cm before

to 13.4 cm after the adjustment. This represents a significant

reduction of 7.8 cm. The use of the GSFC94A model yielded a

mean rms sea surface variability of 7.9 cm, compared with the 9.4

cm obtained when using Schwiderski's [ 1980] model.

Some results of the estimation are shown in Tables 2 and 3.

Table 2 lists the percentage of the total variance present in the

five most energetic vectors of the GSFC94A solution, as well as

the same quantity for the corresponding vectors of the a priori

model [Schwiderski, 1980]. Table 3 gives the cumulative rms

amplitude for each constituent for the a priori model, the

GSFC94A model, and the estimated corrections, as based on the

total number of vectors used in the expansion (truncated sets for

S2, K 2, and N 2 as discussed above).

Examination of Tables 2 and 3 indicates that with two

exceptions (K• and P•), the ordering of the five most energetic

vectors remains the same for both the a priori and the new

25,242 SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS

75

60

45

30

15

-15

-30

-45

-60

-75

0 30 60 90 120 1 0 210 240 270 3 •0

Longi rude

Figure 8. Vector magnitude of the estimated corrections to Schwiderski's [1980] model for M 2. Contour

interval is 1.0 cm.

360

solution. The constituents of the GSFC94A model have always

slightly larger cumulative rms amplitude than those of

Schwiderski's [1980] model. As expected, the maximum

contribution to the change is due to M2, with 6.73 cm.

Some of the largest contributors to the variance of the solutions

are vectors number 2 and 40 for the semidiurnal constituents and

number 2, 13, 15, and 20 for the diurnal ones. The associated

periodicities for these zero-rotation gravity modes are as follows:

70.57 hours (2), 23.59 hours (13), 22.95 hours (15), 18.68 hours

(20), and 12.48 hours (40). The spatial structure of vectors 20 and

40 is shown in Figure 3. These vectors are selected by the data

entirely by spatial coherence, time resonance not being involved.

These results indicate that coherences in the semidiurnal range

are generally higher for the semidiurnal tides, and coherences in

the diurnal range are higher for the diurnal tides. Vector number 2

is the exception to this rule. It is a long-wavelength oscillation,

with nodal lines between the eastern tip of New Guinea and the

Aleutian peninsula in the Pacific and between Rio de Janeiro in

Brazil and Namibia in Africa in the Atlantic. This vector appears

prominently in the spectrum of three of the semidiurnals, both in

the model of Schwiderski [1980] and in the GSFC94A model.

Curiously, it is absent from the first 10 most energetic vectors of

S2. It is present in three of the diurnals but not in K 1. The

presence of this mode in the fits can be traced to the truncation of

the basin in the northern latitudes. Proudman functions computed

on global grids which include the Arctic Ocean have been used to

fit existing global hydrodynamic models, and the results do not

exhibit such an anomalous periodicity among the most energetic

vectors.

The vector magnitudes of the estimated corrections to

75

60

45

30

15

-15

-30

-45

-60

-75

30

60 90 1 0 150 180 210 240 270

Long i rude

Figure 9. Same as Figure 8, but for S 2. Contour interval is 0.5 cm.

300 3 0 360

SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS 25,243

6O

45

3O

15

-15

-3O

-45

-6O

I.O•J.

-75

0 0 90 1 0 210 240 270 3• 0

Long i t ud e

Figure 10. Same as Figure 8, but for K1. Contour interval is 0.5 cm.

Schwiderski's [ 1980] model are shown in Figures 8 through 11 for Weddell Sea areas and in the South Pacific between Antarctica

M2, S2, KI, and O1, respectively. The largest deep ocean and New Zealand, with valneq of q crn All fcmr constituents

corrections to M 2 occur in the Indian Ocean, the Arabian Sea, the

Bay of Bengal, and off the Amazon delta in the equatorial

Atlantic, with corrections in the 10-cm range. Large areas of the

central Pacific approximate those values, as well as an area south

of New Zealand. Blue water corrections to S2 are the largest in

the Indian Ocean, the Arabian Sea, the Bay of Bengal, and in the

South Atlantic off the South American coast, as well as in the

Weddell Sea. These areas show corrections in the 4 to 6-cm

range. Areas of the central Pacific have corrections in the 3 to

4-cm range. The largest open ocean corrections to K l are found in

the Weddell Sea (4-5 cm), in the North Pacific south of the

Aleutian Trench, in the Campbell Plateau area south of New

Zealand, and in the western Indian Ocean, with corrections of 3 -4

cm. Constituent O• shows its largest corrections in the Scotia Sea-

show large corrections in the Bering Sea, the Sea of Okhotsk, the

Patagonian shelf, and other continental shelf areas and

semienclosed seas. However, as can be seen from Figure 5, the

Sea of Okhotsk and most of the Bering Sea are void of data, and

thus the solution obtained for these areas is an extrapolation made

only possible through the use of Proudman functions. In these

areas, as well as over shallow water or continental shelf areas, the

model should be used with care.

The Weddell Sea is another area without much altimetry data,

where the Proudman functions have been used to extrapolate the

solution. The Weddell Sea corrections to Schwiderski [1980]

shown in Figures 8-11, exhibit reasonable magnitudes, and the

contours match well with the rest of the corrections. In addition,

the total tide maps displayed in Plates 5 and 6, for M 2 and K•

75

6O

45

3O

15

-15

-3O

-45

-6O

-75

3O

:

o 210

Long i t ud e

240 270

Figure 11. Same as Figure 8, but for O•. Contour interval is 0.5 cm.

25,244

SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS

show continuity and regularity between the Weddell Sea and

adjacent areas of the solution better constrained by the data. A

more rigorous conclusion pertaining to the accuracy of the model

over different geographic areas requires error maps based on error

covariance propagation.

Global maps of the amplitude and Greenwich phase lag for the

M 2 and K 1 constituents of the GSFC94A model (total bottom-

referenced tide) are shown in Plates 5 and 6. An important aspect

of the model is its smoothness, which can also be deduced from

Figures 8 through 11.

5. Comparisons With Tide Gauge Data

A globally distributed set of 104 tide gauge stations and

bottom pressure sensor locations has been recently compiled by

the international tidalist community as a ground truth set against

which to compare the results produced by the latest tidal models.

Although there are indications that some of the locations are of

questionable applicability and even the most reliable locations

have error bars associated with their values, they still provide as

accurate a set as presently possible in order to gain a general idea

of the quality of the results. Tables 4 and 5 present the results

obtained when evaluating Schwiderski's [1980] models and the

Proudman function models at the ground truth locations.

Table 4 shows the results obtained when all the available data

points are included. Not all the locations have data for all the

main constituents (P l, Q•, K2, and N 2 do not have ground truth

values at all the 104 locations). Comparison of Schwiderski's

[1980] results with the results from GSFC94A indicates an

improvement in all the main constituents, with the exception of

K2 for which the values are nearly equal. Notice the considerable

improvement in the main tide M2. Table 5 presents the results

obtained when some of the outliers are excluded, using a 2.5

sigma editing criterion for both the a priori and the new solution

(the number of accepted stations is also given for both

comparisons). Again, the GSFC94A solution shows definite

improvements for all the constituents, with the exception of K2.

Perhaps the proximity of the alias periods of K2 (86.6 days) and

P• (88.9 days), in conjunction with the small amplitude of K 2, is

causing problems in the estimation of it.

Plates 7 through 10 illustrate the ground truth comparison

results from the GSFC94A model for constituents M 2, S 2, K• and

Table 4. The rms Difference Between the Harmonic

Constants Estimated at 104 Ground Truth Tide Gauge

Stations and the Values Predicted by Schwiderski's

[1980] and the GSFC94A Models

Constituent Number of

Stations

Schwiderski

[1980]

Proudman

function fit

GSFC94A

M 2 104 4.27 2.38

S 2 104 1.72 1.25

N2 101 1.33 0.97

K2 100 0.65 0.66

K 1 104 1.67 1.51

O 1 104 1.25 1.09

P1 100 0.58 0.54

Q1 97 0.37 0.35

Values are in centimeters.

Table 5. The rms Difference Between the Harmonic

Constants Estimated at 104 Ground Truth Tide Gauge

Stations and the Values Predicted by Schwiderski's

[1980] and the GSFC94A Models

Constituent Number of Schwiderski Number of

Stations [ 1980] Stations

Proudman

function fit

!

GSFC94A

M 2 103 4.13 101 2.08

$2 104 1.72 104 1.25

N 2 99 1.18 97 0.74

K 2 99 0.61 98 0.62

K 1 101 1.35 101 1.10

O1 101 1.02 102 0.89

P• 96 0.49 97 0.49

Q1 94 0.31 94 0.32

The 2.5 sigma editing criterion was applied. Values are in centimeters.

O•, respectively. Most of the ground truth comparisons for M 2

fall under 2 cm, as shown by the histogram of Plate 7. The largest

discrepancies occur at the following three locations: Dzaoudzi in

the northern Mozambique channel; IAPSO.2.1.32, a shelf break

station at the end of the northwestern Bering Sea shelf; and

IAPSO. l.l.71 in the northeast Atlantic Ocean, off the Hebridean

shelf. Plate 8 shows that most of the S 2 differences are under 1.5

cm, with the largest deviations occurring at IAPSO.2.1.16,

located on the Bowie seamount in the northeast Pacific Ocean;

Port-Lou-103-92 in the Mascarene Islands region in the west

Indian Ocean; and Raoul Island in the southwest Pacific. The

comparisons for Ki are shown in Plate 9. Most of the

discrepancies are under 2 cm, the largest values occurring at

IAPSO. 2.1.32 and at Heard Island in the Kerguelen Plateau in the

south Indian Ocean. Finally, the results for O• (Plate 10) show

differences concentrated under 1 cm, the largest discrepancy

occurring at Heard Island.

Tide gauge comparisons for the Cartwright and Ray [1991]

models and GSFC94A have been computed by R. Ray (personal

communication, 1995) using a modified version of the previously

mentioned 104 ground truth locations. This set contains revised

harmonic constants for Dzaoudzi and Chichijima and excludes

the stations IAPSO.3.2.13 and Raoul Island for a total of 102 data

locations. The rms discrepancies (in centimeters) for the four

major constituents of the Cartwright and Ray [1991] models and

the corresponding values for GSFC94A are as follows: M2 (3.23,

2.18), S2 (2.22, 1.21), K• (1.89, 1.41), and O1 (1.22, 1.06). The

results of GSFC94A are better in every case, which is to be

expected since GSFC94A is based on TOPEX data, while the

Cartwright and Ray [1991] models are based on GEOSAT data.

S. Klosko (personal communication, 1995) has computed

spherical harmonic fits to the GSFC94A tidal height fields. The

global rate of tidal energy dissipation is given in terms of the

second-degree prograde terms C•m and •rn [Larnbeck, 1988].

The GSFC94A-implied ø values for the four major,, constituents are:

M2 (o3.28 cm, 128.77 ), S2 (1.07 cm, 130.43 ), K 1 (2.62 cm,

36.72 ), and O• (2.30 cm, 47.17 ø). These values are in good

agreement with the values reported by Schrama and Ray [ 1994,

Table 5], which were obtained based on a contemporary

TOPEX-derived tidal model. The main contributions to the

associated secular change in the mean motion of the Moon (units

SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS 25,245

30øN

o

30'S

6o's ]•

i1 i

t

60øE 120øE 180øE

120øW 60øW

0 10 20 30 40 50 60 70 80 90 100 110 120 130 400

Plate 5. Corange surface in centimeters and cotidal lines at intervals of 30 ø for the bottom-referenced M2

model implied by the GSFC94A solution.

30'N

o

30'S

60-s !

60øE

120øE 180øE 120øW 60øW

o

0 10 20 30 40 50 60 70 80 90 100 110 120 130 400

Plate 6. Same as Plate 5, but for K1.

25,246 SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS

•, 20

z

'• 10

o

,,_ 5

o

•, 0

0 1 2 3 4

Plate 7. The rms of the inphase and in-quadrature difference between GSFC94A model estimates and insitu

data at 104 tide gauge locations for M2. Units are centimeters.

:..; .

o 2••-..• ....• •- ..

-•o-

-•o•

-60

-80 _•

0 30 60 90 120 150 180 210 240 270 300 330 360

Lon9itude

•, 20

'• •5

• •0

o

•- 5

o

• O_

Plate 8. Same as Plate 7, but for S2.

SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FLTNCTIONS 25,247

•o .-.• • • "' .-.'/•-' ee -7 '•.-_•-•"-" ß .,--

•o- • •••L ß ' • '• •'•:•- .•

•-•o-( '• ' "'• -..;*'.* • •. *

_•o• • : ß •.

,. ...... ....... :

0 30 60 90 120 150 180 210 240 270 300 330 360

Longitude

E 15

z

• 10

o

,,_ 5-

o

0

Plate 9. Same as Plate 7, but for K I.

• •-•' •,----'• e- .c: •__._ =..= •-'•-.•,, ',.,/ - -.

•o- . ."- •"•.,•,.

•• ß e e •'-;•;:,• ,•'•- ß ß

.. /'-•,.,, .e- 00 ß

* .

-40 • ß

ß e ß ß

-60 ß

-8o , ,,---'"'•-"-•"T-'•, - ,,-..-- ,,-•---'"'-'-•'- -'•----- •

......... I I ' ! I ' ' I ...... i

0 30 60 90 120 150 180 210 240 270 300 330 360

Longitude

• 20

E •5

z

• •0

o

•_ 5-

o

• O.

2

Plate 10. Same as Plate 7, but for O].

25,248 SANCHEZ AND PAVLIS: TOPEX TIDES USING PROUDMAN FUNCTIONS

of seconds of arc per century squared) are M2 (-20.06),

N2 (-1.34), and O1 (-2.55).

6. Conclusions

Tidal models for the main diurnal and semidiurnal constituents

have been developed using TOPEX altimeter data (in the form of

overlap differences) and a functional representation based on the

eigenvectors of the velocity potential or Proudman functions.

Comparisons with independent data from a set of globally

distributed ground truth tide gauge stations and bottom pressure

sensors indicate substantial improvement over Schwiderski's

[1980] (a priori) models. Use of the new model (GSFC94A)

results in a 7.8-cm reduction in the rms overlap difference of 15.5

cm.

Possible future improvements should include the following: (1)

expansion of the data field to include 2 or more years of

TOPEX/POSEIDON altimeter data; (2) combination solutions

using altimetry and bottom pressure sensor data; (3) an iteration

scheme using the GSFC94A model as a-priori and correcting for

the radial orbit error arising from ocean tide model errors; (4)

application of a finer resolution grid in the shallower ocean

regions; (5) estimation of the long-period constituents when the

length of the data record allows it; (6) estimation of smaller-

amplitude constituents, perhaps by admittance techniques; (7)

computation of the associated tidal velocity fields; and (8)

computation of the associated ocean loading models. Finally,

propagation of the error covariance matrix associated with

GSFC94A (or future improved solutions) onto tidal height errors

is needed in order to study geographically the errors from these

solutions.

Acknowledgments. We thank the Space Geodesy Branch of the

Goddard Space Flight Center and the TOPEX/POSEIDON Project for

providing the necessary support. We thank Dennis Morrow of Cray

Research, Incorporated for providing the computational support necessary

to compute the Proudman functions. We thank the following individuals

from the Hughes STX Corporation: Brian Beckley for providing

supporting software to access the altimeter database that he maintains at

NASA Goddard Space Flight Center, Bill Cunningham and Mash

Nishihama for preparing a number of graphics, John Robbins for placing

the manuscript in camera-ready form, and Richard Ray for useful

discussions.

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(Received January 5, 1995; revised July 5, 1995;

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