ArticlePDF Available

Estimation of main tidal constituents from TOPEX altimetry using a Proudman function expansion

Authors:

Abstract and Figures

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 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. In this investigation the Proudman functions were computed by means of finite differences in spherical coordinates over a 2°×2° 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. -from Authors
Content may be subject to copyright.
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].
... As a result of availability of versatile satellite altimetry information and also tide gauge information, tremendous efforts have been made towards global ocean tide modeling and therefore referring to only outstanding contributions may result in following lengthy list: Andersen 1995 [13], Andersen 1995 [14], Cartwright and Ray 1990 [10], Cartwright and Ray 1991 [15], [16], Desai and Wahr 1995 [17], Eanes 1994 [18], Eanes 2002 [19], Eanes and Bettadpur 1996 [20], Egbert et al. 1994 [21], Egbert 1997 [22], Egbert et al. 1999 [23], Egbert and Erofeeva 2002 [24], Egbert and Ray 2000 [25], Egbert and Ray 2001 [26], Egbert and Ray 2003 [27], Kagan and Kivman 1993 [28], Kantha 1995 [29], Knudsen 1994 [30], Krohn 1984 [31], Le Provost et al. 1994 [7], Le Provost et al. 1998 [32], Le Provost 2002 [33], Lefèvre et al. 2000 [8], Lefèvre et al. 2002 [34], Letellier 2004 [35], Letellier et al. 2004 [36], Ma et al. 1994 [37], Matsumoto et al. 1995 [38], Matsumoto et al. 2000 [39], Mazzega et al. 1994 [40], Ray et al. 1994 [41], Ray 1999 [42], Sanchez and Pavlis 1995 [43], Schrama and Ray 1994 [44], Schwiderski 1980 [6], Schwiderski 1980 [45], Schwiderski 1980 [46], Tierney et al. 2000 [47], and Wang and Rapp 1994 [48]. To be able to provide a systematic review over the mentioned contributions, we have arranged Table 1 and Table 2. Short technical specification of the abovementioned efforts towards global ocean tide modeling is provided in Table 1, while Table 2 is dedicated to a brief description over the computational procedure of the global ocean tide models. ...
... Table 3 provides a list of various approaches to ocean tide modeling along with a brief description, while Table 4 presents a list of various types of ocean tide models. [30] T/P no 4 −65 • ≤ φ ≤ +65 • 1 • × 1.5 • harmonic/spherical harmonics none FES94.1 [7] no yes 8 global 0.5 • × 0.5 • hydrodynamics/finite elements nonlinear Mazzega [40] T/P yes 8 −66 • ≤ φ ≤ +66 • 0.5 • × 0.5 • inversion none RSC94 [41] T/P no 60 −65 • ≤ φ ≤ +65 • 1 • × 1 • response/orthotides functions none SR95.0 [44] T/P no [29] T/P yes [38] T/P no 8 global 0.5 • × 0.5 • response linear GSFC94A [43] T/P no ...
Article
Full-text available
In this contribution an empirical approach to global ocean tide and Mean Sea Level (MSL) modeling based on satellite altimetry observations is presented with all details. Considering the fact that the satellite altimetry technique can provide sea level observations at the global scale, spherical harmonics defined for the whole range of spherical coordinates (0 ≤ λ ≤ 2π, and − π/2 ≤ ϕ≤ +π/2) could be among the possible choices for global ocean tide modeling. However, when applied for modeling of global ocean tide, spherical harmonics lose their orthogonality due to the following reasons: (1) Observation of sea surface is made over discrete points, and not as a continuous function, which is needed for having the orthogonality property of spherical harmonics in functional space. (2) The range of application of spherical harmonics for global ocean tide modeling is limited to the sea areas covered by satellite altimetry observations and not the whole globe, which is also required for the fluffiness of the orthogonality of spherical harmonics. In this contribution we show how a set of orthonormal base functions at the sea areas covered by the satellite altimetry observations can be derived from spherical harmonics in order to solve the lack of orthogonality. Using the derived orthonormal base functions, a global MSL model, and empirical global ocean tide models for six major semidiurnal and diurnal tidal constituents, namely, S2, M2, N2, K1, P1, and O1 as well as three long term tidal components, i.e., Mf, Mm, and Ssa, are developed based on six years of Jason-1 satellite altimetry sea level data as a numerical case study.
... It is a straightforward, practical, and feasible approach in computing accurate tide prediction, also offers less physical understanding and perspective on its dynamics (Savcenko and Bosch, 2012;Fok, 2012). Harmonic analysis is one of the approaches that can be applied for estimating empirical tides (Fok, 2012), whereas the others are Response method (Munk and Cartwright, 1966;Cartwright and Ray, 1990;Smith, 1999), and Proudman functions (Sanchez and Pavlis, 1995), with each having their advantages and disadvantages. Nevertheless, the empirical model is much relevant for the local analysis of tidal modelling from SALT alone (Fok, 2012) as the empirical model generally offers higher accuracy than that of the hydrodynamic model. ...
Article
Sustaining coastal ecosystems from the impact of rising sea levels is vital since they offer environmental services that are considered critical for humans’ well-being. The Coastal Vulnerability Index (CVI) is favoured in coastal hazard studies, as it provides a predictive approach to coastal changes through its simple numerical basis in classifying coastlines based on their potential to change. The recent widespread use of innovative technologies has contributed to accessible coastal components to investigate coastal hazards and analyse risk assessment. However, most of the studies related to CVI have rarely exploited these forms of technology despite being utilised in various geodetic applications. Hence, a comprehensive review is performed to evaluate the application of these technological advances, particularly on exploiting space geodetic technology to determine and classify coastal information from various analytical approaches. This paper systematised modern approaches utilizing space technology to form CVI to assess the significant processes affecting the vulnerability and evolution of coastal areas. It is envisaged that the exploitation of space technology will facilitate CVI development and promote effective coastal management strategies.
... Fok (2012) described each category as below: 1. Hydrodynamic models are derived by solving the Laplace tidal equations numerically, using bathymetry data as input of depths, and ocean tidal constants observed by tide gauges around the world as the boundary conditions 2. Assimilation models are derived either by assimilating altimeter data only, or in combination with tide-gauge data, into the hydrodynamic model. 3. Empirical models are derived from altimeter data only using (a) response analysis (Munk and Cartwright, 1966), (b) harmonic analysis, or (c) Proudman functions (Sanchez and Pavlis, 1995). The empirical models could be subdivided into (i) semi-empirical and (ii) purely empirical. ...
Conference Paper
Full-text available
Eight different tide model have been evaluated in this study, concerning several measurements, along Iranian coastline of the Persian Gulf and Gulf of Oman. Comparing results in frequency domain, it’s found that model discrepancies arise where shallow water stations are considered and maximum error belong to the shallowest part of the Gulf, where Pohl station located in. On the other hand, maximum error of tide models is limited to 10 cm in deeper part of the Gulf, indicating different tide models have close performance in deeper waters with minimum error arises from nonlinear nature of shallow waters. Considering time domain, it’s found that FES model including more shallow water constituents, result to better tidal level prediction. FES also present the best tidal currents in this area.
... Empirical-modelling approach, when compared to both hydrodynamic and the assimilated one, offers less physical insight but a simple and practical method for an accurate tidal analysis. Given adequately available altimetry data, the ocean tide model can be derived using: (1) harmonic analysis (Darwin 1883, Schureman 1941, (2) response analysis (Munk and Cartwright 1966) extended with orthotide formalism (Groves and Reynolds 1975) or (3) Proudman functions (Sanchez and Pavlis 1995). This approach could be subdivided into (1) semi-empirical and (2) purely empirical. ...
Article
Nowadays, data combination/fusion is a common practice to integrate multiple data sources into a spatially and/or temporally regular data set for facilitating analysis and interpretation. Given the advancement in active remote sensing for monitoring our changing environment (no matter ocean, land or ice), the data acquisition from various spaceborne platforms with different spatio-temporal resolutions is enormous; calling for a need of multisatellite data combination. Instead of equally or spatially weighted solution, this article illustrates a spatio-temporal combination approach that weights data both in space and in time simultaneously for ocean tide modelling using multisatellite altimetry data sets under different spatial and temporal resolutions. The tide models generated from this approach show substantial improvement near coastal regions when compared to other contemporary ocean tide models by using independent pelagic and coastal tide gauges serving as ground truth. The discrepancy of the best model is ~23%, three times larger than that of the deep ocean. Further improvement lies in more and better spatio-temporal observations in the near future. We anticipate this approach can be applied to other spatio-temporal data in a similar settings for spatio-temporal dependent unknown parameters, including geodetic, geophysical, oceanographic data and imagery under different spatial and temporal resolutions.
Article
Two representation methods for the Dynamic Ocean Topography (DOT) are compared. One uses surface spherical harmonics, the other Proudman functions, which form an ocean domain-specific orthonormal basis. The DOT implied by the temporally averaged output of the POCM_4B ocean circulation model, provided the data for the implementation and testing of the two methods. Using these data a spherical harmonic representation was developed, to degree 30, and a Proudman function decomposition employing 961 basis vectors, so that both representations involve an equal number of parameters. The input DOT field had an rms value of ±66.6 cm. The recovered rms DOT was ±66.1 cm for the spherical harmonic case, ±66.3 cm for the Proudman function case, while the rms difference between the two cases was ±4.2 cm. Although in an overall sense the two representations (with equal number of parameters) yield similar results, in the proximity of the ocean domain boundary the Proudman functions approximate the input DOT field better than the surface spherical harmonics.
Article
One of the tremendous achievements of the TOPEX/POSEIDON (T/P) satellite mission is the release of 12 new global ocean tide models during 1994. Virtually all of these models are based on the high-precision altimetry obtainable from the T/P satellite, and all of the models are superior to both the Cartwright and Ray (1990, 1991) model as well as to the Schwiderski (1980a, b) model when compared to tide gauges. In the present intercomparison, only global ocean tide models released during 1994 have been tested. -from Authors
Article
The author presents results from a high-resolution, data-assimilative, fully nonlinear barotropic tidal model of the global oceans that excludes the Arctic. The model assimilates, in waters deeper than 1000 m, altimetric tides derived from the analysis of 2 years of TOPEX altimetric data. It also assimilates tide gage data from coastal tide gages. The model domain includes that covered by the altimeter and extends to the Antarctic. The model results are compared with observational data from pelagic gages. These comparisons show that overall in the open ocean, in deep waters away from the margins of the primary basins, the model performance is comparable to other tidal models derived using TOPEX data, except for M2 and S2. -from Author
Article
This chapter discusses ocean tides and reviews the usual mathematical representations of ocean tides. It reviews the conclusions of the different evaluation tests applied: (1) comparison to pelagic and island gauge in situ data, (2) crossover residual analysis, (3) TOPEX/POSEIDON (T/P) and tide gauge time-series analysis, and (4) comparison to gravity-loading measurements. The chapter describes the characteristics of the ocean tides, focusing particularly on the major components in each species: the semidiurnal components, and the diurnal components. Highly accurate tidal prediction models are required for the oceanographic exploitation of altimetric data, but in turn, altimetry itself has brought a revolutionary instrument for observing tides at the global scale. The era of high-precision altimetry will certainly be associated in the future to the period where, after a few years of observation, the scientific community definitively mapped the characteristics of the ocean tides to their best possible accuracy.
Article
A global ocean model has been developed within the context of TOPEX/Poseidon (T/P) on the basis of a finite element hydrodynamic modeling approach combined with data assimilation based on the representer method. The solution produced at global scale represents a spectacular improvement over what was available before the era of T/P. Typically, the mean discrepancy between the main tidal components and a hundred in situ tide gauges is 1.7 cm for M2 and 1 cm or less for the other components. However, the accuracies of this solution and of that produced by different authors within the T/P tide working group are all worse near coastlines and over continental shelves. This is the case for our finite element solution (FES) FES94.1 solution over the Yellow Sea and the East China Sea (YS-ECS) where the discrepancy is 33 cm for the M2 tide and 9 cm for the K1 tide when compared to a set of 192 tide gauges distributed along the coastlines. This is due largely to the complex geometry of the basin and the limited knowledge of the bathymetry. The aim of this paper is to investigate how our FES model can be improved with a dedicated application focused on one of the most energetic coastal basins: the YS-ECS (~180 Gigawatts for M2, i.e. 8% of the global energy dissipated in the whole ocean). For this application, a finite grid was implemented with a resolution down to 5 km along the coasts and over the continental shelf break. Particular attention was paid to bathymetry by complementing the ETOP05 database with regional maps. Sensitivity tests to the tuning of the bottom friction and the nonlinear interactions between the diurnal and semidiurnal components allow us to investigate the impact of dissipation parameterization and to produce an optimal solution, without any data assimilation, for nine main components of the tidal spectrum (improvement by a factor of 2). M2 distance to the observations is now ~17.5 cm (global variance of 92 cm), and K1 is 4 cm (global variance of 21 cm). As nonlinear components are significant over these basins, the two main quarter-diurnal components are calculated. The improvements are explained by five points: (1) the choice of a mixed friction, (2) a refinement of the mesh, (3) a choice of reliable boundary conditions, (4) a refined topography and, (5) the use of a specific friction coefficient 1.5×10-3. Surprisingly, the energy budget for the M2 component leads to a dissipation similar to the value estimated by the global FES94.1. This is a major result of this study, which leads us to the conclusion that when forcing a regional model with sea level boundary conditions along the open limits, the simulated velocity field adjusts to the tuning of friction coefficients in order to dissipate the same amount of energy made available at its open boundaries.
Article
Full-text available
An empirical model has been derived for the semidiurnal principal lunar tide M2 in the Mediterranean Sea, excluding the Aegean Sea. The work is primarily an exercise in combining satellite altimeter data with tide-gauge data, through the use of specially constructed spatial basis functions. The resulting M2 model is shown to be superior to a previous altimeter-based model that had been derived without benefit of these functions. -from Authors
Book
Full-text available
Satellite altimeter data (Geos-3, Seasat and Geosat) and bathymetric data have been used to calculate an ocean wide set of free-air gravity anomalies and geoid undulations. An averaged, over one year (1986/1987), Geosat sea surface height track, based on improved, over GEM-T2, orbits was used as the master altimeter frame. The Geosat track was based on the averaging of 22 ERMs and taking into account geoid gradients (Wang and Rapp, 1992). In the computation the altimeter sea surface heights were reduced to geoid undulations using an Ohio State long wavelength spherical harmonic representation of sea surface topography (dynamic heights). The altimeter data in a region, usually a 8°x 8° area, was subjected to a bias alone crossover adjustment with a fitting to the geoid undulation implied by the OSU91 potential coefficient model. The average Geosat crossover discrepancies after this adjustment were ± 4 cm with poorer agreement with Seasat data (± 10 cm) and Geos-3 data(± 46 cm). The anomalies and undulations were predicted in a 0.5° cell using a two component remove/restore technique. The first part was the use of the OSU91A potential coefficient model to degree 360 as a reference while the second part involved the computation of the residual terrain model (RTM) effect on both anomalies and geoid undulations using the 5'x 5' ETOP05U data set and a Fourier computation program from Forsberg. The incorporation of the bathymetric data introduces a strength to the solution, especially in areas where the altimeter coverage is sparse. The initial predictions were subjected to several editing steps. Contour plots of the undulations were examined to identify obviously poor prediction. The predictions were also examined relative to magnitude and predicted standard deviations so that unacceptable predictions could be removed. A second edit was carried out by comparing predicted along track gradients along a Geosat track for comparison with observed values. Such comparison revealed significant discrepancies in some areas. The first edit predictions were then modified to correct most, but not all, of the problems. The final data set contained 2,312,964 geoid undulation and 2,325,669 gravity anomalies, on a 0.125° grid, with standard deviations for each point. A mean sea surface can be created by applying a sea surface topography correction to the predicted geoid undulation, and taking into account a proper tide reference system. The predicted point values were used to create 30' x 30', 1° x 1°, and 5° x 5° mean gravity anomalies and geoid undulations. The 1° x 1° anomalies were compared to terrestrial anomaly data with an agreement of ±9.2 mgals which is 14% less than with the previous (Hwang, 1989) prediction. A detailed description of a number of different data sets developed in this report are described in an appendix.
Chapter
Full-text available
Joseph Proudman (1917) gave an account of a general method of treating the dynamical equations of the tides. The equations were obtained and used in Lagrangian form. More recently, D. B. Rao (1966) reformulated the method using the Eulerian approach. In Proudman's method the eigenfunctions associated with the velocity potential (Proudman functions) are the basis functions for the expansion of the surface height field. The potential applications of the method to tidal estimation are investigated in the context of the Proudman functions as basis functions.
Article
This review article highlights the three-century development of our scientific understanding of ocean tides, culminating through myths, paradoxes, and controversies in a global tide model that now permits the prediction of the instantaneous total tide anywhere in the open oceans with an accuracy of better than 10 cm. Particular attention is drawn to the most recently developed computerized techniques comprehending hydrodynamical dissipation and secondary tide-generating forces, finite-differencing schemes, geometric boundary and bathymetry modeling, and hydrodynamical interpolation of properly selected empirical tide data. -from Author
Article
This review article highlights the three-century development of our scientific understanding of ocean tides, culminating through myths, paradoxes, and controversies in a global tide model that now permits the prediction of the instantaneous total tide anywhere in the open oceans with an accuracy of better than 10 cm. All major aspects of tidal research, including empirical, mathematical, and empirical-mathematical methods, are considered. Particular attention is drawn to the most recently developed computerized techniques comprehending hydrodynamical dissipation and secondary tide-generating forces, finite-differencing schemes, geometric boundary and bathymetry modeling, and hydrodynamical interpolation of properly selected empirical tide data. Numerous computer experiments are mentioned that were carried out by various researchers in order to evaluate the magnitudes of the featured effects. Further possible improvements are mentioned, especially in nearshore areas, in the Arctic Sea, and near Antarctica, where empirical tide and bathymetry data are either rough or marginal.