Indirect mapping ocean tides by means of land and island-based tidal gravity measurements is presented. The inverse scheme of linear programming is used for indirect mapping of ocean tides. Open ocean tides were measured by the numerical integration of Laplace's tidal equations

Kuo, J. T.

English

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t .
The Inverse Problem: Ocean Tides Derived from Earth Tide
Observations
John T. Kuo
Lamont-Doherty Geological Observatory, Columbia University
Palisades, New York 10964
21506
Summary. Evidence has been accumulated to
the point that open ocean tides can indeed be
mapped by solving the inverse problem using land-
based and island-based tidal gravity observations,
supplemented by shore and island ocean tide-gauge
observations and a few deep ocean-bottom observa-
tions. Past efforts have been towards a better
understanding of open ocean tides both through
numerical integration of Laplace's tidal equations
and through direct observations of tides in the
open oceans. The co-amplitude and co-phase tidal
charts principally of the tidal constituent M2
calculated by numerical integration of Laplace's
tidal equations generally fail to agree among
themselves, and with the tidal observations on
mid-ocean islands. The problem of open ocean
tides remains open.
During the last few years, we have indirectly
mapped both the M2 and O^ ocean tides in the
northeastern Pacific Ocean, based on the inverse
scheme of "Linear Programming." A comparison
between the inversion results of the H^ an^ °i
ocean tides and the three ocean-bottom observations
made by Scripps, which were not included in the
inversion scheme, gives an excellent agreement
and assures that the proposed technique of indirect
mapping of open ocean tides by means of tidal
gravity observations can contribute significantly
to attack the classic problem of open ocean tides.
Ever since we proposed the possibility of
indirectly mapping ocean tides by means of land-
and island-based tidal gravity measurements (Kuo
et. al. , 1970), skepticism has been raised
principally by physical oceanographers, concerning
its actual feasibility, typically such as the
recent one by Zetler (1978):
"It has been demonstrated that land-based
earth tides observed on a gravimeter near
an ocean are modified by the ocean tidal
loading. It is more controversial, how-
ever, whether it is possible to map the
open ocean tides by solving the inverse
problem using land-based gravity measure-
ments, shore constraints, and a few
ocean - bottom stations. Certainly it
can be done with an infinite of stations
of perfect precision; the number and
precision of land and ocean measurements
necessary to achieve required accuracy
have not yet been determined."
Evidence has been accumulated to the point that
open ocean tides can indeed be mapped by solving th<
inverse problem using land- and island-based tidal
gravity measurements, coupled with shore and island
ocean tidal measurements and a few ocean-bottom
measurements. There is definitely no need to
have an infinite of stations of perfect precision.
The degree of precision of tidal gravity measure-
ments, from our experience, must be 1% or better,
Proc. of the 9th GEOP Conference, An Iiiifrnaliuntil Symposium on the Applications of
GeutlcsyioGeaJynamics. Ocwher2-5.1978. Depl. of Geodetic Science Rept. No. 280. The
Ohio State Univ., Columbus, Ohio 43210.
which is perfectly achievable.
Much effort has been directed toward a better
understanding of open ocean tides, both through
numerical integration of Laplace's tidal equations,
(Pekeris and Accad, 1969; Zahel, 1970; Hendershott
and Munk, 1970; and others) and through direct
measurements of tides in the deep oceans
(Nowroozi et al., 1969; Filloux, 1971, Munk et al.,
1970; and others). The co-amplitude and co-phase
tidal charts principally of the tidal constituent
M2 calculated by numerical integration of Laplace's
tidal equations are apparently quite sensitive to
the boundaries of the ocean basins and the law of
friction, and generally fail to give close agreement
with the tidal observations on mid-ocean islands.
Moreover, the agreement among various tidal charts
is still poor as shown in Figures 1 and 2,
comparing the tidal charts for example, the «2
and QI in the northeastern Pacific Ocean.
During the last few years, we have indirectly
mapped both the «2 and O^ ocean tides in the north-
eastern Pacific Ocean, based on the inversion
scheme of "Linear Programming." A comparison
between the inversion results of the M2 and 0-^
ocean tides and of the three ocean-bottom observa-
tions made by Munk et al. (1970) and Irish et al.
(1971), which were not included in the inversion
scheme, gives an excellent agreement.
The basic data used in the inversion are
principally from the tidal gravity observations
made on North America and on the islands in the
northeastern Pacific Ocean and from the tide gauge
observations on the coasts and islands. A total
of 17 tidal gravity stations and 62 coastal and
island tide gauge stations was used for the north-
eastern Pacific Ocean. The distributions of the
tidal gravity stations and the tide gauge stations
of the contiguous continents and islands of the
northeastern Pacific used in the inversion are
shown in Figure 3.
The M2 and 0^ worldwide tide maps of Tiron
et.al. (1967), designated as Tiron M2 Map and
Tiron QI Map, respectively, as shown in Figures 4
and 5, were adopted as the starting models and
were digitized at 2° by 2° spacings. As a matter
of fact, Kuo and Jachens (1977) have shown that a
starting model is not of crucial importance in the
inversion. For oceanic regions within 2.5° of the
tidal gravity observational stations, these maps
were modified to conform with coastal observations.
The predicted tidal gravity effects arising solely
from ocean tides were calculated by numerical
convolution of Tiron M2 and O^ Maps with linear
combination of a mass loading Green's function for
an oceanic crust model (Farrell, 1972) and, in
addition, of a Newtonian attraction Green's
function for a density coating layer on the
surface of a sphere, which is not accounted for in
the formulation of the loading Green's function by
Farrell (1972) ..
A fourth order two-dimensional polynomial was
selected as the highest order polynomial surface
319
https://ntrs.nasa.gov/search.jsp?R=19790013335 2020-03-21T23:58:31+00:00Z
Figure 1. The M2 ocean tidal constituent as derived
by empirical (Dietrich, 1944) and Laplace
tidal equations by various authors.
DIETRICH
(1944)
140 120 100
B06DANOV
(1968)
140 W I2O W IOO |6O 140 120
Figure 2. The Oi ocean tidal constituent as derived
by empirical (Dietrich, 1977) and Laplace
tidal equations by various authors.
Thecorrection applied to Tiron M2 and 0^ Maps,
limited distribution of the available tidal
gravity data does not warrant a surface of order
higher than fourth.
The inversion scheme is based on "Linear
Programming." The reason of choosing this Linear
Programming Inversion is very simple. Linear
programming is concerned with the optimum operation
of interdependent variables, that is the minimiza-
tion of a linear objective function, whose
variables satisfy a system of linear inequalities
(Danzig, 1977). Unlike the other numerical solu-
tions of minimization problems by iterative proce-
dures, the solution to a linear programming problem,
if exists, is unique and gives a true minimum for
the entire system, i.e. a global minimum. Thus,
linear programming procedure can be ideally used
to seek the optimal inverse solution subject to
constraints imposed by the tidal gravity and
ocean tide gauge observations.
Figures 6 and 7 give the new M2 and O^ tide maps
for the northeastern Pacific Ocean, bounded on the
west by 160°W on the south by 15°N, and on the
north and east by North America, resulting from the
application of the linear programming inversion.
The most prominent feature of the new M2 map is the
existence of a single amphidromic system in the
region, centered at approximately 26°N latitude,
137°W longitude and rotates counterclockwise. The
amplitude contours display a low amplitude trough
of less than 20 cm. The amplitude increases
smoothly north of the trough attaining magnitudes
320
i r
e
J I I •I J_
60° N
40° N
20°N
20° S
180°
Figure 3.
160°W 140°W 120°W 100°W
Distribution of tidal gravity, tide gauge
and ocean-bottom tide stations in the north-
eastern Pacific Ocean.
greater than 120 cm near the head of the Gulf of
Alaska. The prominent feature of the new O^ map
is the modification of both the coamplitude and
cophase lines from that of the Tiron 0^ Map. The
new QI map shows a delay of the cophase lines,
compared with the cophase lines of the Tiron DI
Map. Except that the coamplitude line of 25 cm
remains nearly the same as Tiron O^ Map, all the
other coamplitude lines of 10, 15, and 20 cm are
curved somewhat to conform with the west coast of
North America.
The goodness of the inversion results must be
critically tested to insure the validity of the
inversion procedure. It can be best accomplished
by comparing the inversion results with observations
The observations from the three ocean-bottom sites,
Kathy, Filloux, and Josie II, shown as solid
circles in Figure 8 were not included as the basic
data in the inversion procedure, and are far re-
moved from possible local perturbation introduced
by islands or submarine topographic features.
The comparison between the values obtained by the
various tidal charts, the inversion procedure and
these corresponding observed values is given in
Tables I and II.
The agreement between the inversion results and
the observations at all three sites is better than
6 cm in amplitude and better than 5° in phase for
M2- Although there are no available observed data
of QI for the station Josie II, the agreement for
Q! at the stations, Kathy and Filloux is better
than 1 cm in amplitude and 1° in phase. The
inversion results appear to be quite insensitive
to uncertainties in knowledge of worldwide ocean
tidal distributions and to possible biases in tidal
gravimeter calibration. The agreement at Kathy and
Filloux for both M2 and Oj_ is embarrassingly good,
probably somewhat fortuitous, since the starting
321
-60°N
TIRON ETAL(1967)
M2
-40°N
160°W 140°W 120°W 100°W
Figure 4. Tiron et al., (1967), Original M2 Tidal
Map.
TABLE I
COMPARISONS OF THE INVERSED RESULTS OF M2 AND QI
WITH THE THREE OCEAN-BOTTOM TIDAL OBS. (MUNK ET.AL., 1970) AND AVAILABLE
TIDAL CHARTS AT THE STATIONS, KATHY, FILLOUX AND
JOSIE II. -
STATION
Kathy
124°25.8'W M2
27°45.0'N
Filloux
129°01.1'W M2
24°46.9'N
Josie II
.144°59.7'W M2
34°00.3'N
OBSERVED (0)
A(cm)/G(deg)
28.6/128.0
17.5/199.0
18.8/107.0
15.6/201.3
26.6/267.0
Not determined
D
-/1 15
-/190
-/92
-/190
•-/75
-/210
AVAILABLE
T
31/180
15/219
20/186
12/220 <
27/261
15/245
COTIDAL
B
41/172
21/208
35/172
22/210
15/232
25/235
CHARTS A/G
PSA Z . H* .
75/305 35/305 60/160 f
52/300 32/320 55/157
25/262 35/270 30/197
INVERSION
A/G
28.3/132.9
18.0/199.0
17.8/104.5
15.5/200.0
20.0/270.0
17.0/224.0
*D = Dietrich, 1977; T = Tiron et.al., 1967; B = Bogdanov et al.,. 1968
P&A = Pekeris & Accad, 1969; Z = Zahel, 1970; H = Ilendershott, 1973.
t
O = Observed (Hurik'et al., 1970)
322
TIRON ET ALU967)
0]
KATHY
FILLOUX o
• •
60°N
40°N
20°N
160°W 140°W 120°W 100°W
Figure 5. Tiron et al., (1967), Original Oi Tidal
Map.
TABLE II
DIFFERENCES
STATION
Kathy M2
Filloux M2
°1
Josie II M2
°1
D-0 T.-O B-O PSA - 0 ZTO H-O INVrO
-/-13" +2.4/+S2 +12.4/+44 +46.4/+177 +6.4/+177 +31.4/+32 -0.3/+4.9
-/-9 -2.4/+20 +3.S/+9 " - - - +0.5/0
-/-IS +1.2/+79 +16.2/+65 +33 . 2/+193 +13.2/+213 +36.2/+50 -1.0/-2.5
-/-11.3 -3.6/+18.7 6.4/+S.7 - - - -0.1/-1.3
-/-192 0.4/-6 -11.6/-35 -1.6/-5 +8 . 4/+3 +3.4/-70 -6.6/+3.0
Not Observed - - -
model could not be read reliably to one degree and were used as the bases for interpolating between
one cm. widely spaced direct ocean tidal observations rath-
The crucial difference between the present er than the traditional empirical procedures
procedure and those of past workers is that land- based on the numerical integration Laplace tidal
based and island-based tidal gravity observations equations. For comparison, Figure 9 shows a map
323
160 140
Figure 6. M2 obtained'from inversion for the north—
eastern Pacific Ocean.
- 20
100 W
derived by Munk et al. (1970) based on ocean-
bottom tidal measurements and empirical inference.
The inversion results of the M2 and O^ tides in
the northeastern Pacific Ocean assure that earth
tidal gravity can contribute significantly to
attack one of the remaining classic geodynamic
problems, - the open ocean tides. The worldwide
open ocean tides can be mapped by means of solving
the inverse problem of ocean tides for a series of
tidal gravity observations, complemented by
coastal, shelf, and island tidal gauge observations
and a limited number of ocean-bottom tidal
observations.
Acknowledgements. The author gratefully
acknowledges the support of the Office of Naval
Research under the contract TO-0605, and of l-he
National Science Foundation under the grant
NSF-EAR-76-24383'.
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-60°N
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KATHY-
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160°W
Figure 8.
140 120
40
20
100
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325
MUNK ET AL
1970
160 140 120 100 W
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. .'•
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326


The inverse problem: Ocean tides derived from earth tide observations

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