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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.
References
Basic, T., and R. H. Rapp, Oceanwide prediction of gravity anomalies and
sea surface heights using Geos-3, Seasat, and Geosat altimeter data and
ETOPO5U bathymetric data, Rep. 416, Dep. of Geod. Sci. and Surv.,
The Ohio State Univ., Columbus, 1992.
Bettadpur, S. V., and R. J. Eanes, Geographical representation of radial
orbit perturbations due to ocean tides: Implications for satellite
altimetry, J. Geophys. Res., 99(C12), 24,883-24,894, 1994.
Brenner, A. C., C. J. Koblinsky, and B. D. Beckley, A preliminary
estimate of geoid-induced variations in repeat orbit satellite altimeter
observations, J. Geophys. Res., 95(C3), 3033-3040, 1990.
Callahan, P.S., GDR Users Handbook (draft 2), TOPEX/POSEIDON
Proj. Doc. 633-721, Jet Propul. Lab. Calif. Inst. of Technol., Pasadena,
October 1993.
Cartwright, D. E., Detection of tides from artificial satellites (review), in
Tidal Hydrodynamics, edited by B.B. Parker, pp. 547-567, John Wiley,
New York, 1991.
Cartwright, D. E., and R. D. Ray, Oceanic tides from Geosat altimetry, J.
Geophys. Res., 95(C3). 3069-3090, 1990.
Cartwright, D. E., and R. D. Ray, Energetics of global ocean tides from
Geosat altimetry, J. Geophys. Res., 96(C9), 16,897-16,912, 1991.
Cartwright, D. E., R. Spencer, J. M. Vassie, and P. L. Woodworth, The
tides of the Atlantic Ocean, 60N to 30S, Philos. Trans. R. Soc. London
A, 324, 513-563, 1988.
Colombo, O. L., Altimetry, orbits and tides, NASA Tech. Memo. 86180,
1984.
Lambeck, K., Geophysical Geode•y, Clarendon, Oxford, 1988.
Lanczos, C., An ite• ation method for the solution of the eigenvalue
problem of linear differential and integral operators, J. Res. Nat. Bur.
Stand. U.S., 45(4), 1950.
Mazzega, P., and S. Houry, An experiment to invert Seasat altimetry for
the Mediterranean and Black Sea mean surfaces, Geophys. J. Int., 96,
259-272, 1989.
McCarthy, D. D., (Ed.), IERS Standards (1992), IERS Tech. Note 13, Obs.
de Paris, Paris, July 1992.
Proudman, J., On the dynamical equations of the tides, I, II, III, Proc.
London Math. Soc., 18, 1-68, 1917.
Pugh, D. T., Tides, Surges and Mean Sea-Level, John Wiley, New York,
1987.
Rao, D. B., Free gravitational oscillations in rotating rectangular basins, J.
Fl•'.id Mech., 25, 523-555, 1966.
Rao, D. B., and D. J. Schwab, Two-dimensional normal modes in
arbitrary enclosed basins on a rotating earth: Application to Lakes
Ontario and Superior, Philos. Trans. R. Soc. London A, 281,63-96,
1976.
Ray, R. D., and B. V. Sanchez, Radial deformation of the Earth by
oceanic tidal loading, NASA Tech. Memo. 100743, 1989.
Ray, R. D., and B. V. Sanchez, Improved smoothing of an altimetric
ocean-tide model with global Proudman functions, Geophys. J. Int.,
118, 788-794, 1994.
Sanchez, B.V., Proudman functions and their application to tidal
estimation in the world ocean, in Tidal Hydrodynamics, edited by B.B.
Parker, pp. 27-39, John Wiley, New York, 1991.
Sanchez, B. V., and D. E. Cartwright, Tidal estimation in the Pacific with
application to SEASAT altimetry, Mar. Geod., 12(2), 81-115, 1988.
Sanchez, B. V., D. B. Rao, and P. G. Wolfson, Objective analysis for tides
in a closed basin, Mar. Geod., 9(1), 71-91, 1985.
Sanchez, B. V., R. D. Ray, and D. E. Cartwright, A Proudman-function
expansion of the M 2 tide in the Mediterranean Sea from satellite
altimetry and coastal gauges, Oceanol. Acta, •5(4), 325-337, 1992.
Schrama, E. J. O., and R. D. Ray, A preliminary tidal analysis of
TOPEX/POSEIDON altimetry, J. Geophys. Res., 99(C12), 24,799-
24,808, 1994.
Schureman, P., Manual of harmonic analysis and prediction of tides, Spec.
Publ. 98, Coast and Geod. Surv., U.S. Dep. of Comm., Washington,
D.C., 1958.
Schwab, D. J., and D. B. Rao, Barotropic oscillations of the
Mediterranean and Adriatic Seas, Tellus, Ser. A, 35, 417-427, 1983.
Schwiderski, E. W., On charting global ocean tides, Rev. Geophys., 18,
243-268, 1980.
Tapley, B. D., et al., Precision orbit determination for
TOPEX/POSEIDON, J. Geophys. Res., 99(C12), 24,383-24,404, 1994.
Uotila, U. A., Notes on adjustment computations, I, Dep. of Geod. Sci.
and Surv., The Ohio State Univ., Columbus, 1986.
Wang, Y. M., and R. H. Rapp, Geoid gradients for Geosat and
Topex/Poseidon repeat ground tracks, Rep. 408, Dep. of Geod. Sci.
and Surv., The Ohio State Univ., Columbus, 1991.
Wang, Y. M., and R. H. Rapp, Estimation of sea surface dynamic
topography, ocean tides, and secular changes from Topex altimeter
data, Rep. 430, Dep. of Geod. Sci. and Surv., The Ohio State Univ.,
Columbus, 1994.
Wiser, M., The global digital terrain model TUG87, internal report, Inst.
of Math. Geod., Tech. Univ. of Graz, Graz, Austria, 1987.
N. K. Pavlis, Hughes STX Corporation, 7701 Greenbelt Road, Suite
400, Greenbelt, MD 20770.
B. V. Sanchez, Space Geodesy Branch, NASA Goddard Space Flight
Center, Code 926, Greenbelt, MD 20771. (e-mail:
g5bvs @ gibbs.gsfc.nasa.gov)
(Received January 5, 1995; revised July 5, 1995;
accepted July 5, 1995.)
