Geophvs. J . Inr. (1995) 122, 551-568
Gravity anomalies derived from Seasat, Geosat, ERS-I
and TOPEX/POSEIDON altimetry and ship gravity:
a case study over the Reykjanes Ridge
Accepted 19% February 20. Received I Y Y S February 20: in original form I Y Y 4 Novciiihcr 7
SUMMARY
We have employed least-squares collocation to derive a gravity lield o v e r the
Reykjanes Ridge using Seasat. Gcosat/ERM. ERS-I and TOPEXIPOSEIDON
altimeter data combined with ship gravity. To avoid 21 crossover adjuslincnt to
correct for orbital errors we used mean geoid gradients. which were obtained by
averaging gradients over 60, 10 and 36 rcpeat cycles for Geosat/ERM. ERS-1 and
TOPEX/POSEIDON, respectively. The average standard deviations for the
Geosat/ERM, ERS-I and TOPEX/POSEIDON mean _gradients are I . I ? . 3.25 and
2.01 p r a d respectively. The standard deviations of the ship gravity, which were
assigned based on weightings derived from an analysis of crossing differences. range
from 6.72 to 20.17 mgal. The necessary covariances for the least-squares collocation
computations were derived using the law of covariance propagation. Before merging
with the altimeter data, the ship gravity for each leg was adjusted by removing a
quadratic polynomial in time in order to match a satellite-only gravity field in a
least-squares sense. The result of the adjustment suggests that most of thc ship
gravity data collected in the 1960s and the 1970s have average offsets of about
14mgal. which is close to the offset of the Potsdam Datum. The rms difference
between the satellite-only gravity, derived using least-squares collocation. and the
adjusted ship gravity is 7.10mga1, smaller than the rms difference o f 12.47mgal
between the satellite-only gravity derived by a Fourier transform method and the
adjusted ship gravity. The rms difference between the combined gravity field.
derived from both altimetry and ship gravity, and the adjusted ship gravity is
2.65 mgal. suggesting that the former has successfully absorbed the high-frequency
component of the gravity signal provided by the latter. The combined gravity field
thus features a regionally uniform medium resolution from altimetry and a locally
high resolution from ship gravity. The averagc accuracy estimate given by
least-squares collocation for the combined gravity is 5.76 mgal. The combined
gravity field reveals detailed tectonic structures over the Reykjanes Ridge related to
the interaction of the Iceland hot spot and sea-floor spreading at the ridge.
Key words: altimetry, covariance function. geoid gradient. least-squares collocation.
ship gravity.
INTRODUCTION
Over the past two decades, altimeter measurements from
Geos-3, Seasat, and Geosat have made significant
contributions to geodesy. geophysics and oceanography.
* N~~
at: ~
~of civil ~ ~
University, Hsinchu 30050. Taiwan.
~
~ ~ ~ t ii o n Chiao
a ~l~ i-ung
~
t~
Ongoing missions such as ERS-I and TOPEX/POSEIDC)N
(T/P) continue to collect altimeter data, adding t o the
coverage of existing altimeter data in both space and rime.
Apart from the noise level for Geos-3 o f about 1 metre. all
other missions yielded range measurements with noises o f
the order of a decimetre or lower. The 2-D resolution o f
satellite-derived
~
~i
~ ~gravity
~
. is~ primarilyt controlled by the
spacing between adjacent satellite ground tracks. McAdoo
5s I
A Marl\\ ( l W 2 ) pointed out that the current 2-D resolution
i \ ahout IS hni for gravity dcrived from the Geosat Geodetic
Mission (CiM) Atimetry. and 66 km using the first six
nionthz of ERS-1 Interim Geophysical Data Records.
Including Seasat and Geosat/ERM data in this analysis
would probably increase the resolution only marginally.
Oncc data from more closely spaced ERS-1 ground tracks
become available. the resolution anywhere will be
comparable with that now obtainable with Geosat G M data,
which is currently only available for latitudes south of 30 “S.
None the less. if one is seeking to obtain the highest possible
resolution. all sources of gravity information should be
considered. An obvious additional source is ship gravity,
which is abundant in many regions of tectonic interest and
contains a wealth of short-wavelength information. However, in nearly all previous studies that derived gravity fields
bascd o n satellite altimetry. ship gravity was used only as an
independent source for checking the quality of the
computed gravity anomalies, rather than as an additional
data set.
Least-squares collocation (LSC) provides a natural way of
combining altimetry and ship gravity. Examples of the use of
least-squares collocation to calculate gravity anomalies are
the works by Rapp (1979, 1985), Hwang (1989), Knudsen
(1991), and Rapp & Basic (1992), who all used altimeter
data alone. Another commonly used method for altimetrygravity conversion is that based o n Fourier transforms (FT),
examples being Haxby et al. (1983), Freedman & Parsons
(1986), Sandwell and McAdoo (1988, 1990), Sandwell (1992)
and McAdoo & Marks (1992). T h e FT method, while being
very efficient computationally, is not easily adaptable to
include more than one type of potential field data. The LSC
method, on the other hand, needs more computer time
but has the capability to combine heterogeneous data
and t o give accuracy estimates for the computed gravity
anomalies.
Recent applications of the Fourier transform methods
(Sandwell 1992: McAdoo & Marks 1992) have made use of
geoid gradients or deflections of the vertical derived from
the sea-surface height measurements. In most of the
previous studies using LSC, the sea-surface heights, which
are approximately equivalent to geoidal heights, were used
directly. The errors in sea-surface heights caused by
non-spatially correlated orbit errors can be treated by a
simple crossover adjustment to remove the biases or biases
plus linear trends of the arcs in the work area (Knudsen
1987). By using deflections of the vertical, however, it is
possible to remove the bias of an arc without crossover
adjustment with only negligible errors left in the data
(Sandwell 1984). In this study, we will use geoid gradient
(i.e. deflection of the vertical with the sign reversed) as the
altimeter data type when employing the LSC method. The
LSC method is able to combine directly geoid gradients
from satellites with different inclinations, rather than
resorting to the iterative procedure that is needed in the FT
method (Sandwell 1992).
In what follows, we derive a gravity field for the area
surrounding the Reykjanes Ridge (Fig. 1) in order t o
illustrate the application of least-squares collocation to
calculate gravity anomalies using both geoid gradients and
ship gravity. The geoid gradient values are derived from
Seasat, Geosat/ERM, ERS-1 and T/P altimetry, and the
ship gravity is from a global compilation of the holdings of
the major geophysical data centres. The area chosen has the
advantage, because of its high latitude, that the satellite
tracks have about one-half of the cross-track spacing seen at
the equator. It also has good ship-track coverage. In order
to combine ship gravity with the altimeter data, it is first
necessary to correct the ship gravity for long-wavelength
errors. W e describe a procedure for doing this by cornparing the ship gravity with gravity anomalies derived from
satellite data alone.
THE METHOD O F LEAST-SQUARES
COLLOCATION
The LSC method, used in this study to calculate gravity
anomalies from satellite altimeter data and ship gravity data,
has been well documented elsewhere, for example
Tscherning & Rapp (1974) and Moritz (1980), and we
provide only a brief outline here. Following Rapp & Basic
(1992). we first remove a reference field prior to applying
least-squares collocation and then restore it afterwards; the
reference field used was the OSU91A geopotential model
(Rapp, Wang & Pavlis 1991). The expressions used are
In the above two equations, E and Ag are vectors containing
residual geoid gradients and residual gravity anomalies, i.e.
after removal of the reference field. The covariance matrices
C,.z., C,,,
and CAgwa,are for gradient-gradient, gravity
anomaly-gradient and gravity anomaly-gravity anomaly
respectively, and contain an error part implied by the errors
in the coefficients of the reference field, and a signal part
implied by a degree variance model; here we use that of
Tscherning & Rapp (Tscherning & Rapp 1974, eq. (25A);
Appendix A). The superscript ‘s’ denotes a covariance
matrix between the predicted value and the observables (in
E , A ~ ) and ‘0’ denotes a covariance between the
observables. The diagonal matrices D, and D, contain the
noise variances of geoid gradients and ship gravity
anomalies, respectively; &c
are the a posteriori estimates of
variances in the predicted gravity field. Agrcf is the reference
gravity anomaly computed from the OSU91A model. If ship
gravity data is lacking in a prediction cell, or if a gravity field
derived only from altimeter data is required, only the
matrices C&, C:* and D,, and the vector E will be present
in eqs (1) and (2).
A n important issue concerns the optimum maximum
harmonic degree for the reference field. In a theoretical
study, Wang (1993) suggested that, when using an FFT
method for geoid-gravity conversion, the optimum maximum degree should be the highest degree of the reference
field used, provided that, in the computation of Agrcl, the
geopotential coefficients are scaled by the quotients between
degree variance and the sum of degree variance and error
degree variance. The use of the quotients will effectively
Gravity anomalies from altimeter data
553
65"
60"
55"
50'
45"
345"
~
31 5"
320"
325'
330"
335"
340'
Figure 1. Bathymetry (solid lines, contour interval = 500 m) and sea-floor age (dotted lines, contour interval = 10 Ma) over the area studied.
downweight the high-degree coefficients, as these
coefficients will normally have larger errors than the lower
degrees. A similar situation can be found in the LSC process,
where we take into account the error degree variances of the
field in computing the covariance functions (see Appendix
A). Although the use of a high-degree reference field will
increase the errors of the reference values, the inclusion of
error degree variances will balance the signal and error
budgets in such a manner that the predicted gravity anomaly
has the property of minimum error variance. Also, the
maximum degree must be compatible in resolution with the
inversion cell size, which is a half-degree plus a data border
of the same width for this study. This discussion leads to the
choice of degree 360 as the maximum degree for the
OSU91A field.
The required covariance functions can be derived using
the law of covariance propagation (Moritz 1980, p. 86). A
detailed derivation of the covariance functions needed in
this study are given in Appendix A. Fig. 2 shows some of the
global, i.e. unscaled, covariance functions, with the reference
field of OSU91A to degree 360 removed. In the actual
calculations, the inversion was applied to a half-degree cell
with data selected from the cell and a half-degree data
border about the cell (see Hwang 1989; Rapp & Basic 1992).
The global residual covariance functions (see eq. A3) were
scaled to local residual covariance functions [to be used in
(1) and ( 2 ) ] using the weighted mean of two values, one
from the ratio between the variance of geoid gradients
(see
estimated in the cell and the value C,,(O) = C,,,(O)
Appendix A for notation), the other from the ratio between
the variance of the ship gravity anomalies and the value
CAKAR(O).
Fig. 3(a) shows the histogram of the scale factors
554
C. Hwang and B. Parsons
800
/
-200
O0
spherical
1
distance (degree)
2
3
I
(a)
60
1
1
20
,
1
- longitudinal component
- - transverse component
for the Reykjanes Ridge from 2400 cells when using
altimeter data alone, whereas Fig. 3(b) shows that for the
case where both altimeter and ship gravity were used. From
Fig. 3 we see that the scale factors are mainly less than
unity. sbggesting that the factor A in the Tscherning-Rapp
degree variance model (see eq. A2) and the error degree
variances of OSU91A are too large for the gravity field over
the Reykjanes Ridge (for a geoid gradient the former has
the dominant effect on the total covariance, based on a
comparison between signal and error covariances). A scale
factor will not change the correlation length of the
covariance function, but will scale the variance and the
curvature parameter [see Moritz (1980, p. 174) f o r the
definitions of the three quantities]. Moreover. a scale factor
of less than one will reduce the standard deviation of the
predicted gravity anomaly as compared to the case of using
the unscaled global covariance functions.
A n inversion-free algorithm was used to implement eqs
(1) and (2). Rewriting eqs (1) and (2) for the case of
multi-point prediction, the residual gravity anomalies,
contained in vector s, and their error covariance matrix,
contained in matrix C,. can be calculated formally as
s = C,,C-'I = (Gp1C,,)7'(Gp'I)= B"y.
(31
Z;, = C, - C,,,Cp'C, = C,, - BTB,
(4)
where C,, is the transpose of C,,, the matrix of covariances
between the predicted gravity values and the observed
quantities; G is the lower triangular matrix in the Cholesky
decomposition of C, namely, C=GGT, where C is the
matrix containing covariances relating each observable to
the others plus the error covariances: and I is a vector
containing the observations, i.e. geoid gradients or ship
gravity, with a reference field removed: C, is the covariance
matrix of gravity anomalies. The vector y = G-'I, along with
each column of B=G-IC,, is obtained by forward
substitutions. Furthermore, let B = (b,, b,, . . . ,b,,), where b,
is the ith column vector of 6. Then, we can obtain the error
variance of the ith component in s by calculating only the
norm of b,, namely,
a:, = C,, - b,'b,,
(5)
which saves substantial computer time.
ALTIMETER MEASUREMENTS
AND DATA AVERAGING
.....
-10
L ,
0
1
2
3
spherical distance (degree)
(4
Figure 2. Global covariance functions, with the reference field of
OSU9lA to degree 360 removed: (a) covariance between gravity
anomaly C,,,,,
(b) covariance between the longitudinal component
of geoid gradient and gravity anomaly C,*, (c) covariances of the
longitudinal and transverse components of geoid gradient C,, and
C
.,,
See Appendix A for notation.
In using satellite altimeter data, the systematic errors and
data noise must be taken into account and we shall
investigate these first in order to select a suitable altimeter
data type. It is known that the sea-surface height, h,
obtained by differencing the ellipsoidal height and the
altimeter range, consists of the geoidal height ( N ) , the
time-dependent and time-independent sea-surface topography, the radial orbit error, and errors due to improper
geophysical corrections (Wunsch & Gaposchkin 1980). The
radial orbit error has its dominant energy at the zero
frequency (the bias), and at a frequency of one cycle per
revolution has a wavelength of 40000 km (Sandwell 1984).
For an area without energetic ocean circulation, such as the
Reykjanes Ridge, the sea-surface topography, if obtained by
averaging data over a sufficiently long time, also has a
Gravity anomalies from altimeter data
60
555
1
1
40
20
0
0
1
scale factor
(a)
2 0
1
scale factor
(b)
2
Figure 3. Histograms of factors for scaling the global covariance functions: (a) scale factors for the altimeter data alone, and (b) scale factors
for altimeter and ship data together.
dominant feature at long wavelengths. Thus in an area of
size no larger than a few thousand km2, the sea-surface
height may be expressed as
h =N + a
+ bs,
(6)
where s is the along-track distance and a and b represent the
bias and linear trend representing the separation between N
and h caused by sea-surface topography and the dominant
orbit errors. By taking the along-track gradient of the
sea-surface height, we get the geoid gradient, plus an error
term b, namely
(7)
Now the theory of LSC requires that the data used should
be centred (Moritz 1980, p. 76). To achieve this we may in
practice treat each of the satellite passes over the work area
separately to remove the mean value of ah/&. By doing this
the bias b will be automatically eliminated. Thus geoid
gradients as derived from the de-meaned ah/& are in
principle free from the systematic error introduced by the
long-wavelength sea-surface topography and the once-perrevolution orbit error. This error-free condition is nearly
satisfied when using a small prediction cell (for example less
than 1" X 1" as in this study), where b is very likely to be a
constant.
Furthermore, it is known that the gravity anomaly, which
differs from gravity disturbance by a very small amount, and
geoid gradient are both distance derivatives of the Earth's
disturbing potential (Heiskanen & Moritz 1985). Thus, in
theory, predicting the gravity anomaly from the geoid
gradient is more stable than doing so using geoidal heights.
Another. advantage of using geoid gradients is that we do
not need to adjust the sea-surface height as in Knudsen
(1987). Sandwell (1992, p. 438) and Sandwell & Zhang
(1989) provide further discussions on the advantages of
using geoid gradient as the altimeter data type. Furthermore, by analogy with the treatment of crossover-adjusted
sea-surface heights, which are correlated due to the
application of orbit corrections, we have ignored the
correlation between two successive along-track gradients
and kept the matrix D, in eqs (1) and ( 2 ) diagonal.
The altimeter data used in this study are from four
satellite missions: Seasat, Geosat/ERM, ERS-1 and
TOPEX/POSEIDON, in the form of geoid gradients. In
order to enhance the signal-to-noise ratio and remove data
gaps we average the geoid gradients from a number of cycles
as in Sandwell & McAdoo (1990). The sea-surface heights
derived from the one-per-second range measurements were
used for the averaging. The Geosat/ERM gradients were
averaged over 60 17-day repeat cycles (Cheney et a!. 1987).
For ERS-1, the gradients were averaged over the first 10
35-day cycles, using the Interim Geophysical Data Records
produced by Cheney, Lillibridge & McAdoo (1991). The
T / P gradients were from AVISO (1992) and were averaged
over the first 36 10-day cycles. For the averaging process, we
first estimate geoid gradients for each pass (half a
revolution) by taking along-track differences between two
successive sea-surface height measurements that are less
than 2 s apart. As the nominal ground track for a satellite
pass, we choose points at integral seconds before and after
the equator crossing of the pass. The standard errors in the
averaged geoid gradients were also calculated. Seasat does
not have repeat tracks, so a uniform standard deviation of
10prad was assigned to the gradient data using the
approximate formula a: = ((T: + a ; ) / d 2 = 2a2/d2, where
(T = 5 cm is Seasat's
noise level, and d =6.73 km is the
approximate along-track spacing. Owing to the large
difference in accuracy between the Seasat and Geosat data,
we removed collinear Seasat tracks (within 7 k m of
Geosat/ERM tracks) to avoid possible spurious effects due
to the close spacing. Fig. 4 shows the ground tracks for each
satellite over the Reykjanes Ridge. The improvement of the
Gravity anomalies f r o m altimeter data
1100
900
0)
300
557
same time interval, but the former has a higher standard
deviation than the latter. The standard deviations of the
average gradients from the first 10 cycles of T/P (about 100
days) have a mean value of 4.37 p r a d , higher than that from
the 10 ERS-1 cycles (see Table l ) , despite the fact that T / P
has better instrumental accuracy than ERS-1. Factors such
as the time-varying component of the sea-surface will create
a data noise which cannot be removed o r reduced if the time
of data averaging is not sufficiently long. Also, Yale &
Sandwell (1994) found that sea-surface heights measured
from individual altimeter passes increase in wavelength
resolution for the satellites according to the order T/P,
Geosat, ERS-1, suggesting that the pre-stacked noise level
of ERS-1 is the largest and that of T/P is the smallest.
The consistency of the LSC method using geoid gradients
as the potential field data, with the covariance functions
described above. was tested with the following experiment.
W e computed the north-south and west-east components
of geoid gradients on a 2' X 2' grid using gradient data from
the four satellites. Next we interpolated the two components
to the actual subsatellite points and resolved the along-track
gradients using E,, = t pcos a,, + v,, sin a,,. where F,, is the
along-track gradient, a,, is the track azimuth, and tP,vP are
the north-south and west-east components of the gradient
at point P. Fig. 7 shows the histograms of the differences
between the predicted and the observed along-track
gradients for Seasat, Geosat/ERM, ERS-1 and T/P. The
statistics show that the rms differences in the four cases are
3.13, 1.30, 1.47 and 1.68prad with the means being 0.05,
-0.04, -0.05 and 0.13 p r a d . The discrepancies are all within
the noise levels of the individual satellite data sets, and are
about 5 to 15 per cent of the rms powers of the signals. The
large difference for Seasat is due t o the assigned 1 0 p r a d
noise, which gives less weight to the Seasat data in the LSC
computation of geoid gradients as compared to the others.
In general, data with larger errors will be more difficult to
'reproduce' in the LSC process than those with less noise.
Furthermore, despite the fact that ERS-1 has a larger noise
level than the T/P (see Table l ) , the difference between the
predicted and observed gradients in the former case is
smaller. This is probably due to the denser data coverage of
ERS-1, causing the predicted gradients to be dominated by
the ERS-1 observables.
-m-
-@
-
;
'
loo
-100
56
I
58
60
62
latitude (degree)
64
1100
900
THE AZIMUTH OF A N ALONG-TRACK
GEOID GRADIENT
.-
0
8 300
In calculating the covariance functions used in the LSC
calculations (Appendix A), the azimuth of an along-track
geoid gradient is needed. One way to calculate the azimuth
is numerical differentiation, as follows:
100
Ax
a = t a n - ' -,
AY
56
58
60
62
latitude (degree)
64
Figure 5. Stacked geoid gradients from pass a467 of ERS-1 (upper)
and from pass 170 of TOPEX/POSEIDON (lower), which all travel
across the Reykjanes Ridge. The ERS-1 gradients are plotted every
cycle beginning at Cycle 1, while the TOPEX/POSEIDON
gradients are plotted every third cycle beginning at Cycle 3. The
bottom line in each of the plots corresponds to the average
gradients (based on 10 cycles for ERS-1 and 36 cycles for T/P).
where, in a planar approximation, Ax = R cos ($)(A2 - A,),
Ay = R ( & - 4,), with 4, and A, being latitudes and
the mean latitude.
longitudes at the endpoints and
Because the distance between two successive subsatellite
points is small-it is 6 to 7 km for most of the satellites-this
numerical approximation may yield reasonable results.
However, it is more convenient computationally to use a
theoretical expression for the azimuth. The approximate
mean motion of the satellite relative to the ascending node
6
558
C. Hwang and B. Parsons
ERS-1
ERM
70
n
8 50
W
240
40
:ilL
S
2
I
,I
70
60
30
W
$
I
20
+
10
10
0
0
0 1 2 3 4 5 6 7 8
std. dev. (microrad)
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
std. dev. (microrad)
std. dev. (microrad)
70
6o
TIP
ERS-1
ERM
I
TIP
70 I
ruylit
I
1
?--
70
1
50
40
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
std. dev. (microrad)
std. dev. (microrad)
std. dev. (microrad)
(b)
Figure 6. Histograms of the standard deviations of the geoid gradients from Geosat/ERM, ERS-I and T/ P for (a) global coverage, and (b) the
Reykjanes Ridge area alone.
Table 1. Percentages of standard deviations for the geoid gradients
within I, 2, 3, 4 and 5 prad for (a) the global coverage and (b) the
Reykjanes ridge.
(a)
Satellite
<I
<2
<3
<4
Geosat
ERS-1
TIP
72.27
2.10
0.16
95.67
22.79
63.16
98.82
58.36
94.17
99.06
80.67
97.71
<5
99.45
90.56
98.82
Mean
0.97
3.10
2.03
(b)
Satellite
<I
<2
<3
<4
(5
Mean
Geosat
ERS-1
T/ P
42.64
1.13
0.06
97.66
17.02
62.62
99.27
51.07
96.41
99.55
76.68
98.20
99.74
89.51
98.95
1.12
3.25
2.01
is given by
li=&+M,
(9)
where ch and M are the average velocities of the perigee and
the mean anomaly, respectively, which can be calculated by
(Kaula, 1966):
w = 3nC2,a%
2 2 2 (1 - 5 cos2 i),
4(1-e ) a
Gravity anomalies from altimeter data
-10 -5 0 5 10
difference (microrad)
(a)
559
-10 -5 0
5 10
difference (microrad)
(b)
meridian
0 5 10
difference (microrad)
(c)
-10 -5
-10 -5 0
5 10
difference (microrad)
(d)
Figure 7. Histograms of differences between predicted and
observed along-track geoid gradients for (a) Seasat, (b)
Geosat/ERM, (c) ERS-I and (d) TOPEX/POSEIDON.
Figure 8. Geometry used to calculate the .azimuth a of an
along-track geoid gradient at subsatellite point P ; X , Y , Z are the
inertial coordinates and GM is the Greenwich Meridian.
the orbital plane. The secular precession rate
orbital plane can be determined from
f2
of the
with
n=&
where a and e are the semimajor axis and the eccentricity of
the satellite's orbital ellipse respectively, i the inclination, a,
the semimajor axis of the Earth's reference ellipsoid and C,,,
the second zonal harmonic coefficient (for example
C,,, = -1082.6255596 X 10-' from GEM-T3). In practice, we
may use a = R + H , where R =6371 km and H is the
satellite's mean altitude ( H = 800, 800, 782, 1336km for
Seasat, Geosat, ERS-1 and T / P respectively). As shown in
Fig. 8, the azimuth can be determined from the direction of
flight of the satellite projected onto the ground. The two
components of the projected velocity of the satellite at the
subsatellite point in the local coordinate system are
v, = R(U sin p - wb cos +), v y = RU cos 0. Thus
tan
v
U sin p - wk cos (CI
U cos p
=L=
VY
where
(14)
and (CI i s the geocentric latitude, which can be determined
from the geodetic latitude 4 using tan (CI = (1 - e2) tan 4,
with e being the eccentricity of the reference ellipsoid.
wk = w , - f2 is the Earth's rotational velocity relative to
The formula for a given here is similar to that in Sandwell
(1992), but derived with a different approach; we used this
formula to compute the azimuths of the along-track geoid
gradients needed in the expressions for the covariance
matrices.
SHIP GRAVITY
The ship gravity used consists of the free-air gravity
anomalies in a global data base at Oxford compiled from the
holdings of the major data centres. Fig. 9 shows the data
distribution over the Reykjanes Ridge. Wessel & Waits
(1988) have pointed out that ship gravity is affected by
several sources of long-wavelength errors, for example drift
of the gravimeter, uncertainty about the reference field
used, absence of base-station ties. In the following section
we describe an empirical method of correcting for these
long-wavelength errors by comparison with a gravity field
derived from altimetry alone. Here we are concerned with
the problem of assigning standard deviations to the ship
gravity for use in the least-squares collocation calculations.
We start with the results of the crossover adjustment of
Wessel & Watts (1988), and make an assumption that the
adjusted ship gravity anomalies contain only random noise
due to instruments and are free from the systematic errors
discussed by Wessel & Watts (1988). At a crossover point p ,
the discrepancy d g is the difference between the meas-
560
C. Hwang and B. Parsons
3
320"
325'
330'
335"
340"
345"
320"
325'
330"
335'
340"
345"
65'
60'
55"
50'
45'
31 5'
Figure 9. Ground rack coverage of ship gravity for the Reykjanes Ridge area. The thick lines correspond to the tracks for cruises c2112 (left
and v2302 (right).
urements g:, and g: from two cruises (or legs) j and k :
d:,"
= R:, -
s;.
(16)
In the absence of systematic errors, the expectation of the
crossover discrepancy is then zero, namely,
-
E ( d f )= d
= 0,
(17)
and its variance o n using error propagation is
V ( d f )= u;+ u;,
(18)
where we assume that all measurements on a cruise have the
same error variance and the measurements o n two different
cruises are uncorrelated. Moreover, the mean squared
crossover discrepancy for the adjusted ship gravity can be
expressed as
where n is the number of crossovers. Assuming that the
process leading to crossing errors is stationary, namely, all
the crossovers are treated as repeat realizations of some
event at a single point, then eq. (19) is immediately
transformed to the formula for the estimation of the
variance of the crossover errors V(d:f) ( n must be
sufficiently large). If we further assume that all cruises have
a uniform standard deviation, namely a; = af = a for any
j , k , then from eqs (18) and (1Y) we obtain
2v2 = s2.
(20)
Using S = 13.96 mgal, which is the rms crossover discrepancy of the adjusted ship gravity from Wessel & Watts
(1988), then CT = 9.87 mgal, which represents the overall
standard deviation of the ship gravity.
In Wessel & Watt's (1988) study, the quality of the ship
gravity from a particular cruise is represented by the cruise's
weight, which ranges from 1 to 10. The weights are derived
from a crossover analysis of the ship gravity using a DC shift
and linear drift model, and the higher the weight the better
the quality. W e will now use the weights to assign standard
deviations to different cruises. In the theory of parameter
estimation, weights are commonly taken to be inversely
proportional to error variance, i.e.
where v:, is a scale factor and w is the weight. In the
so-called Gauss-Markoff model, the method for estimating
a: may be found in, for example, Koch (1987, Section 3).
Gravity anomalies from altimeter data
In the present study, we assume the average standard
deviation u is the weighted mean given by
C.
w= I
n,
where nw is the number of cruises with weight w . In Wessel
& Watts’ study the number of cruises with weights from 1 to
10 are 30, 31, 17, 27, 707, 13, 13, 9, 11, 2, respectively. This
information, together with the assumption that the numbers
of crossovers on all cruises are the same, leads to the
estimate u o =20.17 mgal. With this value, we can then
assign a standard deviation to each cruise, resulting in the
weight-standard deviation relationship in Table 2. For
cruises not listed in Table A1 of Wessel & Watts, the
weights are assigned according to the years of data
collection as follows: before 1965, the weights are identically
1; from 1965 on, the weights are increased by 1 every two
years; after 1981, the weights are identically 10. Such an
estimate of weights is based upon the ship’s navigation
systems, which have improved over time and are the major
factor governing the accuracy of ship gravity. There are 68
ship cruises over the Reykjanes Ridge, 46 of which have
over 100 gravity measurements in the area.
A remaining question concerns the proper relative
weightings of altimeter gradients and ship gravity in the LSC
solution. Assuming that a 1 Frad error in geoid gradient
corresponds to a 0.98 mgal error in gravity, we find that the
standard deviations of the ship gravity are higher than those
of the averaged geoid gradients from altimetry given in
Table 1. However, to make a fair comparison, one would
have to use the noise level of geoid gradients from a single
satellite pass. Seasat’s 10 prad noise and Geosat/GM’s
6 prad noise (Sandwell 1992) show that non-repeat satellite
data and ship data may have roughly similar noise levels,
and thus it will be appropriate to use the currently estimated
standard deviations of the altimeter gradients and ship
gravity for weightings. A rigorous treatment of the problem
of properly weighting individual data sets might resort to a
method similar in principle to MINQUE described in Rao &
Kleffe (1988).
In order to investigate further the problem of estimating
the noise level of the ship gravity, we carried out the
following experiment. We resampled the gravity data at a
2 km interval (the average spacing) along cruise ~2115,
which has a weight of 5 and an almost straight ground track.
A spectral analysis of the data was then made to determine
the power spectra as functions of wavelength. It was found
that the spectra become ‘flat’ or white for wavelengths less
Table 2. Weights of ship cruises and associated standard deviations (in mgal) of
ship gravity measurements.
Weight Std. dev.
1
20.17
2
14.26
3
11.65
4
10.09
5
9.02
Weight Std. dev.
6
7
8
9
10
8.24
1.62
7.13
6.12
6.38
561
than 6 km, giving an rms power of 1.17 mgal for the noise.
This value may reflect the intrinsic measurement noise of
the instrument. However, the overall accuracy of ship
gravity belonging to a cruise is governed by many other
factors, for example, for some types of gravimeter, the
accuracy with which cross-coupling to horizontal accelerations can be determined, and we cannot obtain an accurate
estimate of the noise level without better knowledge of
these error sources. This should explain why the noise ‘floor’
for c2115 is inconsistent with the assigned standard deviation
of 9.02 mgal given in Table 2.
T H E A D J U S T M E N T OF S H I P G R A V I T Y
USING SATELLITE-ONLY GRAVITY
It is also necessary to account for the long-wavelength errors
in ship gravity, such as mechanical drift of the gravimeter,
off-levelling, incorrect ties to base stations and inconsistent
use of a reference field, as pointed out in Wessel & Watts
(1988). Fig. 10 shows examples of comparisons between
observed ship gravity and gravity anomalies interpolated
onto the same ship track from a gravity field on a 2’ X 2’
grid predicted using least-squares collocation from the
altimeter-derived gradients alone. The ship gravity is clearly
offset from the satellite-only gravity, and the offset varies
with time. If the ship gravity and altimetry are to be used
together to predict a combined gravity field, these
differences must be corrected for.
The geoid gradients derived from the sea-surface heights
for the four satellites refer to different ellipsoids: the Seasat
data were adjusted by a bias and trend model (Liang 1983)
and the height system will refer to the ‘master’ arcs used in
the adjustment; the Geosat/ERM data refer to the GRS80
ellipsoid (Cheney et af. 1987); the ERS-1 data also refer to
the GRS80 system (Cheney et af. 1991); the T / P data refer
to an ellipsoid with a semimajor axis of 6 378 136.3 m and a
flattening of 1/298.257 (AVISO 1992). The difference in the
semimajor axis of the reference ellipsoid between two
different satellite data sets has not been a problem since, in
an area the size of the Reykjanes Ridge, any differences are
basically constant and hence will be removed upon
differencing the sea-surface heights to obtain gradients. Thus
the reference geopotential field used in the LSC will govern
the definition of the ellipsoidal system for the satellitederived gravity anomalies. Hence the gravity anomalies
derived from the altimeter data in this study will refer to the
gravity system implied by the OSU91A model. Note that
0.87 mgal should be added to the ship gravity anomalies to
account for the atmospheric effect before comparison with
the satellite gravity (Rapp 1979).
The deviation between the satellite gravity and ship
gravity anomalies resulting from the use of different
ellipsoidal systems can be characterized as
Sg,
= a,,
+ a , sin’ d
based on the formula for normal gravity (Heiskanen &
Moritz 1985). As an example, consider a ship cruising along
a meridian with a constant speed V , and 4 = rb0 + A&,
where $o is the starting latitude. Using the relationships
sin2 4 = (1 - cos 24)/2 and A 4 = V t / R , with t being time,
and the Maclaurin series for the cosine function up to the
562
C. Hwang and B. Parsons
100
I
_ _ _ _ raw ship gravity
-satellite-only gravity
adjusted ship gravity
combined gravity
~
6
7
8
9
elapsed time (days)
(a)
- _ _ _ raw ship gravity
-satellite-only gravity
adjusted ship gravity
combined gravity
~
-50
0
1
2
elapsed time (days)
(b)
Figure 10. Ship gravity before and after adjustment compared with satellite-only and combined gravity along-track for cruises (a) v2302 and
(b) c2112.
second order, we can transform (23) into a quadratic
polynomial in time:
6g, = b,, + b , f
+ b2t2.
(24)
The error in ship gravity due to drift of the gravimeter
will have a similar form to (24). Other sources such as
incorrect tie-ins to base stations and off-levelling will
probably contribute to a DC shift (for the latter we assume a
constant ship speed and a constant heading). Thus we expect
that most of the net difference between satellite gravity and
ship gravity can be accounted for over moderate periods of
time by the following expression:
6g,
= d,,
+ d t + d,t2,
I
(25)
where do will be termed the bias. The following procedure
was then used to adjust the ship gravity. We first derived a
satellite-only gravity field on a 2 ' X 2 ' grid using the
altimeter-derived gradients alone. For each cruise a
quadratic in time was determined that best fitted the
difference between the satellite-only gravity and the ship
gravity, and this was subtracted from the ship gravity. Table
3 lists the estimated coefficients of the quadratic for each
cruise. Because d , and d, represent the total effect of the
different error sources, and are the result of a least-squares
fit, for some cruises these terms can be very large as
compared to the DC shifts and drift rates given in Table A1
of Wessel & Watts (1988).
For cruises with fewer than 100 data points falling within
Gravity anomalies from altimeter data
Table 3. Results of adjustment of ship gravity to satellite-only
gravity using quadratic polynomials.
dl
ch611
d084a
d084b
d091a
d093a
d093b
d131a
d131b
d131c
dut02
f2181
f2281
g9008
jchOl
kea37
kea38
kea39
kea40
kkt75
sh676
ss013
ss014
stOlb
v2302
v2303
v2305
v2702
v2703
v2706
v2707
v2801
v2804
v2805
v2909
v29 10
v2911
v3008
v3009
~3012
1966
1977
1977
1978
1978
1978
1982
1982
1982
1970
1981
1981
1990
1969
1970
1971
1971
1971
1975
1976
1965
1965
1979
1966
1966
1966
1969
1969
1969
1969
1970
1970
1970
1972
1972
1972
1973
1973
1973
197
172
189
66
150
169
229
253
255
223
269
291
278
218
289
149
174
206
248
306
223
247
276
227
239
289
179
200
268
294
187
266
285
205
224
259
149
178
282
-19.00
-3.13
-1.88
1.81
1.72
0.83
-15.55
2.93
5.60
-12.39
8.10
-5.40
2.32
-12.98
-14.13
-11.73
-11.73
-11.80
-13.11
-6.81
-7.39
-12.67
5.42
-13.96
-8.52
-2.58
-13.86
-10.70
-10.76
-16.80
-11.70
-25.42
-13.20
-10.77
-12.20
-10.18
-14.15
-11.53
-11.28
0.09
2.51
0.90
-5.18
0.46
3.48
-0.64
-0.26
-0.57
0.45
-0.57
0.27
0.02
-8.91
-0.97
-0.24
-3.01
2.60
0.80
0.47
0.00
0.11
0.05
-1.79
0.16
2.16
0.88
0.43
-0.69
2.11
0.43
-3.33
-0.16
-1.35
0.07
2.49
-0.78
-2.32
-0.19
0.04
-3.89
-0.15
-0.04
-0.18
0.10
d2
0.00
0.35
0.20
1.04
0.01
-0.34
-0.05
0.01
0.01
0.00
-0.03
0.01
0.02
-3.01
-0.25
-0.03
0.20
-0.15
0.12
0.03
0.22
0.00
0.00
4.92
0.01
-0.76
0.41
0.37
-1.02
0.08
-0.04
0.13
0.01
1.53
-0.09
0.33
0.00
1.34
0.09
-0.01
10.55
-0.05
0.02
-0.01
0.00
weight
3
5
5
5
10
2
1
7
7
8
8
8
10
10
10
7
9
9
10
5
5
5
5
5
5
7
5
5
8
5
3
5
5
5
5
5
5
5
5
5
5
5
5
6
5
563
40
n30
8
W
x
0
c 20
a,
3
-
m
2 10
0
-40
-30
-20 -10
0
10
20
30
40
difference (mgal)
(a)
40
20
301
-40
-30
-20
-10
0
10
20
30
40
difference (mgal)
(b)
Note. The year and day correspond to the date calculated from
the same initial time as used in the polynomial of eq. (25). The
coefficients d,,, d , and d, are measured in mgal, mgal day and
mgal day-*, respectively.
'
the area studied, the adjustment sometimes yielded
unrealistically large coefficients, and it was decided that
these cruises should be excluded from further use. It was
noted that recent cruises such as cd0.52 and g9008 show
offsets that are small compared to the average offset of
10.40 mgal from all cruises. The negative DC shifts shown in
Table 3.suggest that the gravity data collected from the
cruises in the 1960s and the 1970s are consistently larger
than the satellite gravity and, in particular, the D C shifts
seem to agree with the 14 mgal offset of the Potsdam Datum
-40
-30 -20
-10
0
10
20
30
40
difference (mgal)
(c)
Figure 11. Histograms of differences between the satellite-only
gravity and the ship gravity (a) before adjustment of the ship gravity
and (b) after adjustment. Also shown (c) is the histogram of the
differences between the combined gravity and the adjusted ship
gravity.
564
C. Hwang and B. Parsons
(Dehlinger 1978, p. 35). Fig. 11 shows histograms of the
differences between ship and satellite-only gravity before
and after the adjustment. Fig. 11 indicates that the
differences before the adjustment are biased towards
negative values and then are normalized by the adjustment.
The overall statistics give a mean and standard deviation
of the differences between the satellite-only gravity and the
ship gravity before the adjustment of -4.01 and 10.68 mgal,
which were reduced to 0.123 and 7.10mgal after the
adjustment. These figures should be compared with the rms
power of 34.15 mgal for the adjusted ship gravity. The
statistics of the comparisons between the adjusted ship
gravity from individual cruises and the satellite-only gravity
will depend on the sampling interval. If the ship gravity was
sampled at an interval not significantly smaller than the grid
of the satellite-only gravity and the short-wavelength gravity
variations along the ship-track are relatively weak, then the
difference is small. For instance, the comparison along cruise
kea38 shows an rms difference of 3.08 mgal. If, on the other
hand, the sampling interval of the ship gravity is much
smaller than the grid and the short-wavelength gravity signal
is strong, then the difference will be large. For instance, the
comparison along cruise g9008 shows an rms difference of
8.70 mgal.
We also made other satellite-only solutions using various
combinations of satellites. The comparisons between the
satellite-only gravity and the adjusted ship gravity are
summarized in Table 4. Also shown in Table 4 is a
comparison made for the global gravity field by Sandwell &
Smith (1992), which was derived from the Seasat, Geosat
and ERS-1 gradient data using a Fast Fourier Transform
(FET) method (third row in Table 4). The average D C shift
of the ship data relative to their gravity is 10.89 mgal. From
Table 4 it is clear that the solution without T / P data is close
to that derived from using all four satellites and has better
accuracy than that of Sandwell & Smith. This comparison
also shows that the LSC method yields better results than
the FFT method over the area studied. It is noted that the
comparison made by Neumann et al. (1993) over the central
South Atlantic shows that, when the dense Geosat/GM data
is included in the calculation, Sandwell & Smith’s satellite
gravity yields a standard deviation of 7 mgal with respect to
the ship gravity. Furthermore, the contribution of the dense
ERS-1 data is clearly demonstrated by the numbers in rows
2 and 5 of Table 4, where we see that the increase in the
discrepancy due to the lack of ERS-1 data is about 30 per
cent.
Table 4. Differences (in mgal) between
satellite-only gravity and adjusted ship gravity.
Satellites and
method
Mean
Std. dev.
1, 2, 3 , 4, Isc
1, 2, 3, fft
1, 2, 3 , Isc
1, 2, 4, Isc
0.12
-0.14
7.10
12.47
7.62
9.22
0.15
0.10
Satellite identification: 1 = Seasat, 2 = Geosat,
3 = ERS-I, 4 = TjP.
Isc: least-squares collocation; fft: fast Fourier
transform.
G R A V I T Y A N O M A L I E S FROM ALTIMETER
GRADIENTS A N D ADJUSTED SHIP
G R A V I T Y B Y LSC
Using the adjusted ship gravity and altimetric gradients
together we have produced a combined gravity field on a
2’ X 2‘ grid. A colour image of the gravity field is shown in
Fig. 12, which was generated using the GMT program
GRDIMAGE (Wessel & Smith 1991). The combined
gravity was compared with the adjusted ship gravity,
yielding a mean difference and standard deviation of 0.02
and 2.65 mgal, respectively. Along-track comparisons for
legs c2112 and v2302 are shown in Fig. 10. The combined
gravity is almost coincident with the adjusted ship gravity in
this figure. Fig. 11 shows the histogram of differences
between the combined gravity and the adjusted ship gravity.
These differences are narrowly centred at zero. This is not
unexpected as we have incorporated the adjusted ship
gravity in the solution. Thus the LSC has actually
‘reproduced’ the ship gravity, despite the large standard
deviation, 9.02 mgal, assigned to the ship gravity on the two
cruises, which is significantly larger than the average noise
level of the gradient data. A possible explanation is that,
once the systematic errors are removed, the ship gravity has
an intrinsic noise level much smaller than that given in Table
2, as suggested by the spectral analysis described in a
previous section.
The LSC method also gives accuracy estimates for the
predicted values (see eq. 2). The average and rms values of
the accuracy estimates for the satellite-only gravity are 5.92
and 7.16 mgal respectively, compared to the average noise
level of 2.35 prad (equivalent to 2.30 mgal in gravity) for the
altimeter gradients. Thus the satellite-only gravity has noise
greater than the data used. This is explained by the fact that
we are predicting gravity anomalies through covariance
functions using gradients that can be calculated at large
distances from the points of prediction. The combined
gravity yields accuracy estimates better than those from
satellite-only gravity, with average and rms values of 5.76
and 6.99mgal respectively. The average accuracy of the
combined gravity is still poorer than the average noise of the
gradient data, but better than the average noise of the ship
gravity. We conclude that, given a new set of ship gravity
measurements, the difference from the combined gravity
field would be of the order of about 2 to 3mgal where the
locations of the ship gravity measurements are close to
locations of existing data points-either altimetric gradients
or ship gravity.
D I S C U S S I O N A N D CONCLUSIONS
There are a number of features of considerable interest that
can be seen in the gravity field shown in Fig. 12. The gravity
anomalies around the Reykjanes Ridge and Iceland are
predominantly positive. This is due in part to the
long-wavelength component of the Earth’s gravity field,
which may have an origin deep in the mantle, but also to the
gravity anomalies associated with shallow bathymetry
around Iceland (Sclater, Lawver & Parsons 1975; Cochran &
Talwani 1978). This relationship between depth and gravity
anomalies, and the extensive melting on Iceland, indicates
that there is a hot, convective upwelling located beneath
315"
65
320'
325
a
330'
335"
340
65"
O
60"
60"
55
55"
50"
50"
315"
320"
325"
330"
335"
340"
W
D
c
e2 3 x 4
-35-28-21 -14 -7 0
7 14 21 28 35 42 49 56 63
mgal
Figure 12. Colour image of the combined gravity field. The image is shaded relief, with illumination and viewpoint of an observer in the
south-east.
Gravity anomalies f r o m altimeter data
Iceland, providing the dominant influence on the tectonics
of the area.
At mid-ocean ridges with slow spreading rates, like that
observed for the Reykjanes Ridge, the axial topography is
normally characterized by a valley; in contrast, at fast
spreading rates a small axial high is observed (Macdonald
1982). Studies of gravity anomalies at mid-ocean ridges
(Small & Sandwell 1989: Owens & Parsons 1994) show that
gravity reflects the axial valley o r high at the different
spreading rates, and hence can be used as a proxy for the
topography. Fig. 12 shows that, on the Reykjanes Ridge
close to Iceland, an axial high can be seen despite the slow
spreading rate. Moving farther south along the ridge, a
transition between the axial high and an axial valley can be
clearly seen at about 59 ON. A recent model used to explain
axial topography is the one developed by Chen & Morgan
1990): at slow-spreading ridges cooling close to the axis
penetrates into the mantle immediately beneath the crust,
producing a strong, high-viscosity layer there. It is the
deformation of this layer that results in the axial topography
(Lin & Parmentier 1989; Chen & Morgan 1990). At
sufficiently high spreading-rates, the cooling at the ridge axis
does not reach the base of the crust or the mantle beneath.
and the absence of a strong mantle layer, together with a
hotter, low-viscosity lower crust, removes the stresses that
cause the axial valley. The near-axis temperature structure
can be influenced by factors other than the spreading-rate,
for example changes in the mantle temperature at depth,
and changes in the crustal thickness (Phipps Morgan &
Chen 1993). It is expected that both of these latter
parameters will vary with distance from Iceland along the
ridge. Hence, the location of the transition between axial
high and axial valley o n the Reykjanes Ridge, and the way
the amplitudes of valley o r high vary with distance from
Iceland, provides constraints on the variations in mantle
temperature and crustal thickness.
The latitude of the transition between axial valley and
axial high also seems to mark a division between features
seen off-ridge. North of the transition, a series of gravity
highs and lows can be seen fanning out on either side of the
Reykjanes Ridge. The overall eifect is of a V or series of Vs
straddling the Reykjanes Ridge; these features in the gravity
field must correspond to the V-shaped ridges described by
Vogt (1971). These V-shaped ridges cut across isochrons and
can be explained in terms of regions of increased
melt-production periodically migrating down the ridge at
velocities of the order of 10-20cmyrp'. It is generally
accepted that the cause of the excess melting in Iceland is
the higher temperatures associated with a convective
upwelling beneath Iceland (e.g. White & McKenzie 1989).
Numerical studies of convection, used to interpret the
intermediate-wavelength depth and gravity anomalies
associated with hotspots and mid-plate swells, often show
instabilities in the convective flow, especially in the presence
of a near-surface low-viscosity layer (Robinson & Parsons
1988). These instabilities produce disturbances in the
temperature structure of the convection, which could
produce changes in the amount of crust generated if
entrained at the mid-ocean ridge. The amount of melting is
very sensitive to temperature (McKenzie & Bickle 1988),
and only a small temperature change would be required to
generate sufficient additional crust to explain the V-shaped
565
ridges. The velocities expected in the convective circulation
associated with a mantle plume are of the order of
10-30 cm yr-' (Watson & McKenzie 1991), similar to the
rates at which the source of the V-shaped ridges must
migrate down the Reykjanes ridge. If the origin of the
V-shaped ridges is related to time dependence in the
temperature structure of a convective upwelling centred on
Iceland, then the spacing of the ridges, and their along-strike
continuity seen in the gravity field, provides observations
constraining that time dependence.
South of the transition between axial high and axialvalley, the off-ridge gravity anomalies are more coherent
and of greater amplitude than those t o the north. A possible
explanation is that, if an axial valley has existed at these
latitudes for some time, the off-axis features reflect
segmentation of the ridge, like the segmentation observed
elsewhere in the North Atlantic (e.g. Sempere, Purdy &
Schouten 1990; Lin ef af. 1990). A complete discussion and
interpretation of the features revealed in the combined
gravity field for the Reykjanes Ridge will be presented
elsewhere, but the wealth of coherent short-wavelength
detail apparent in Fig. 12 underlines the value of using all
available gravity information.
T h e general signature of the gravity field around the
Reykjanes Ridge is largely due to the satellite altimetry, as
the colour image of the satellite-only field (not shown here)
and that of the combined field (Fig. 12) appear to be quite
similar overall. However, much of the detailed structure has
come from the ship data. For instance, the transition from
axial high to medium valley, occuring at about 59"N and
328", is better defined in the combined field than in the
satellite-only field. Also seen in the combined field are
several segments o n the ridge axis, which are less
pronounced in the satellite-only field. The combined gravity
thus features a regionally uniform medium resolution from
altimetry and locally high resolution from ship gravity.
In the study described in this paper we have discussed
techniques for combining altimeter measurements from
different satellites and ship gravity to calculate a gravity field
using least-squares collocation. Because least-squares
collocation can make use of any type of geopotential data, it
provides a natural way of combining geoid gradients from
satellites with different inclinations, something that requires
less straightforward, iterative methods when using Fourier
transform techniques. One might argue that, once dense
altimeter data with global coverage from the ERS-1 168-day
repeat mission becomes available, there will be less need to
make use of ship gravity. However, increasing numbers of
detailed marine gravity surveys exist, particularly, for
example, in relation to investigations of the mid-ocean ridge
system. The resolution of these detailed ship surveys
remains better than that which can b e obtained even with
dense altimeter coverage, at least until multiple cycles of the
dense altimeter ground tracks are available, allowing
averaging and consequent noise reduction. The method
described above, of referencing ship gravity to a
satellite-only gravity field, should provide a basis for
removing systematic differences between satellite-derived
gravity and ship gravity surveys, and for producing
combined gravity fields with the best-possible resolution.
Readers interested in the gravity field please send e-mail to
[email protected].
566
C. Hwang and B. Parsons
ACKNOWLEDGMENTS
We made extensive use of the GMT software of Wessel &
Smith (1991) in displaying altimeter and ship data.
Discussions with Tony Watts and Jenny Collier on the ship
gravity were very useful. This work was begun while the first
author was a postdoctoral fellow in the Department of Earth
Sciences at Oxford, supported by a grant from Amoco
Exploration (UK). The research has also been supported by
National Science Council of Republic of China, project no.
NSC83-0410-E-009-015.
REFERENCES
AVISO,
1992.
AVISO
User
Handbook:
Merged
TOPEXIPOSEIDON Products. AVI-NT-02-101 -CN, Edn 2.1,
The French Space Agency (CNES).
Chen, Y. & Morgan, W. J., 1990. Rift valley/no rift valley transition
at mid-ocean ridges, J . geophys. Res., 95, 17 571-17 581.
Cheney, R.E., Douglas, B.C., Agreen, R.W., Miller, L., Porter,
D.L. & Doyle, N., 1987. Geosat altimeter geophysical data
record user handbook. NOAA Technical Memorandum NOS
NGS-46, National Oceanographic and Atmospheric Administration, US Dept. of Commerce, Rockville, MD.
Cheney, R.E., Lillibridge, J. & McAdoo, D., 1991. Preliminary
analysis of ERS-I fast delivery altimeter data. Office of ocean
and earth sciences, NOAA National Ocean Service, Rockville,
MD.
Cochran, J.R. & Talwani, M., 1978. Gravity anomalies, regional
elevation, and the deep structure of the North Atlantic, J .
geophys. Res., 83, 4907-493 1.
Dehlinger, P., 1978. Marine Graviry. Elsevier, New York.
Freedman, A.P. & Parsons, B., 1986. Seasat-derived gravity over
the Musicians Seamounts, J . Geophys. Res., 91, 8325-8340.
Haxby, W.F., Karner, G.D., LaBrecque, J.L. & Weissel, J.K., 1983.
Digital images of combined oceanic and continental data sets
and their use in tectonic studies, EOS Trans. Am. geophys. Un.
64,995-1004.
Heiskanen, W. & Moritz, H., 1985. Physical Geodesy, reprint,
Institute of Physical Geodesy, Tech. University of Graz,
Austria.
Hwang, C., 1989. High precision gravity anomaly and sea surface
height estimation from Geos-3/Seasat altimeter data. Rept. 399,
Dept. of Geod. Sci. and Surv., Ohio State University,
Columbus, OH.
Kaula, W.M., 1966. Theory of satellite geodesy, Blaisdell Co.,
London.
Knudsen, P., 1987. Estimation and modeling of the local empirical
covariance function using gravity and satellite altimeter data,
Bull. GPodhsique, 61, 145-160.
Knudsen, P., 1991. Simultaneous estimation of the gravity field and
sea surface topography from satellite altimeter data by least
squares collocation, Geophys. J . Int., 104, 307-317.
Koch, K.R., 1987. Parameter estimation and hypothesis testing in
linear models, Springer-Verlag, New York.
Liang, C.K., 1983. The adjustment and combination of Geos-3 and
Seusat altimeter data. Rept. 346, Dept of Geod. Sci. and Surv.,
Ohio State University, Columbus, OH.
Lin, J. & Parmentier, E.M., 1989. Mechanisms of lithospheric
extension at mid-ocean ridges, Geophys. J . Int., 96, 1-22.
Lin, J., Purdy, G.M.. Schouten, H., Sempcre. J.-C. & Zervas, C.,
1900. Evidence from gravity data for focused magmatic
accretion along the Mid-Atlantic ridge, Nature, 344, 627-632.
McAdoo, D.C. & Marks, K., 1992. Gravity ovcr the southern ocean
from Geosat data, J . geophys. Res., 97, 3247-3260.
Macdonald, K.C., 1982. Mid-ocean-ridges: fine scale tectonic,
volcanic and hydrothermal processes within the plate boundary
zone, Ann. Rev. Earth planet. Sci., 10, 155-190.
McKenzie, D. & Bickle, M.J., 1988. The volume and composition of
melt generated by extension of the lithosphere, J. Petrol., 29,
625-679.
Pliipps Morgan, J. & Chen, Y., 1993. Dependence of ridge-axis
morphology on magma supply and spreading rate, Nature, 364,
706-708.
Moritz, H., 1972. Advanced least square, Rept 175, Dept of Geod.
Sci. and Surv., Ohio State University, Columbus, OH.
Moritz, H., 1980. Advanced Physical Geodesy, Abacus Press.
Neumann, G.A., Forsyth, D.W. & Sandwell, D.T., 1993.
Comparison of marine gravity from shipboard and high-density
satellite altimetry along the Mid-Atlantic Ridge, 30.5"-35.5"S,
Geophys. Res. Lett., 20, 163-1642,
Owens, R. & Parsons, B., 1994. Gravity fields over mid-ocean ridges
from GEOSAT GM data: variations as a function of speading
rate, Geophys. Res. Lett., 21, 2837-2840.
Rao, C.R. & Kleffe, J., 1088. Estimation of variance components
and applications, J. Wiley, New York.
Rapp, R.H., 1979. Geos-3 data processing for the recovery of geoid
undulations and gravity anomalies, J . geophys. Res., 84,
3784-3792.
Rapp, R.H., 1985. Detailed gravity anomalies and sea surface heights
derived from Geos-3/Seasat altimeter data, Rept. 365, Dept. of
Geod. Sci. and Surv., Ohio State University, Columbus, OH.
Rapp, R.H. & Basic, T., 1992. Oceanwide gravity anomalies from
Geos-3, Seasat and Geosat altimeter data, Geophys. Res. Lett.,
19, 1979-1982.
Rapp, R.H., Wang, Y.M. & Pavlis, N., 1991. The Ohio State 1991
geopotential and sea surface topography harmonic coefficient
models, Rept. 410, Dept of Geod. Sci. and Surv., Ohio State
University, Columbus, OH.
Robinson, E.M. & Parsons, B., 1988. Effect of a shallow
low-viscosity zone on the formation of mid-plate swells, J .
geophys. R e x , 93, 3144-3156.
Sandwell, D.T., 1984. A detailed view of the South Pacific geoid
from satellite altimetry, J . Geophys. Rex, 89, 1089-1 104.
Sandwell, D.T., 1992. Antarctic marine gravity field from high
density satellite altimetry, Geophys. J . Int., 109, 437-448.
Sandwell, D.T. & McAdoo, D.C., 1988. Marine gravity of the
southern ocean and Antarctic margin from Geosat, J. geophys.
Res., 93, 10389-10396.
Sandwell, D. & Zhang, D.T., 1989. Global mesoscale variability
from the Geosat exact repeat mission; Correlation with ocean
depth, J. geophys. Res., 94, 17 971-17 984.
Sandwell, D. & McAdoo, D., 1990. High-accuracy, high resolution
gravity profiles from 2 years of the Geosat Exact Repeat
Mission, J . geophys. R e x , 95, 3049-3060.
Sandwell, D.T. & Smith, W.H.F., 1992. Global marine gravity from
ERS-1, Geosat and Seasat reveals new tectonic fabric. EOS
Trans. A m . geophys. Un. 73, 133.
Sclater, J.G., Lawver, L.A. & Parsons, B., 1975. Comparison of
long-wavelength residual elevation and free-air gravity
anomalies in the North Atlantic and possible implications for
the thickness of the lithospheric plate, J . geophys. Rex, 87,
3949-3958.
Sempere, J.-C., Purdy, G.M. & Schouten, H., 1990. Segmentation of
the mid-Atlantic ridge between 24" and 34"40'N, Nature, 344,
427-43 1.
Small, C. & Sandwell, D.T., 1989. An abrupt change in ridge axis
gravity with spreading rate, J . geophys. Res., 94, 17 383-17 392.
Strange, T.M., 1991. The determination and geophysical applications of free air gravity anomalies using satellite altimetry and
ship gravity. PhD thesis, Oxford University, UK.
Tscherning, C., 1976. Covariance expressions for second and lower
order derivatives of the anomalous potential, Rept. 225, Dept of
Geod. Sci. and Surv., Ohio State University, Columbus, OH.
Gravity anomalies f r o m altimeter data
Tscherning, C.C. & Rapp, R.H., 1974. Closed Covariance
expressions for gravity anomalies, geoid undulations and
deflections of the vertical implied by anomaly degree variance
models. Rept. 208, Dept. of Geod. Sci. and Surv., Ohio State
University, Columbus, OH.
Vogt, P.R., 1971. Asthenosphere motion recorded by the ocean
floor south of Iceland, Earrh Planet. Sci. Left., 13, 153-160.
Wang, Y.M., 1993. On the optimal combination of potential
coefficient model with terrestrial gravity for FFT geoid
computation, Manuscripta Geodaetica, 18, 406-416.
Watson, S. & McKenzie, D., 1991. Melt generation by plumes: a
study of Hawaiian volcanism, J . Petrol., 32, 501-537.
Wessel, P. & Watts, A.B., 1988. On the accuracy of marine gravity
measurements, 1.geophys. Res., 93,393-413.
Wessel, P. & Smith, W.H., 1991. Free software helps map and
display data, EOS Trans. Am. geophys. Un., 72,441-446.
White, R.S. & McKenzie, D.P., 1989. Magmatism at rift zones: the
generation of volcanic continental margins and flood basalts, J .
geophys. Rex, 94,1685-7129.
Wunsch, C. & Gaposchkin, E.M., 1980. On using satellite altimetry
to determine the general circulation of the oceans with
applications to geoid improvement, Rev. Geophys. Space f h y s . ,
18,725-745.
Yale, M.M. & Sandwell, D.T., 1994. Along-track resolution of
individual and stacked ERS-I, TOPEX and Geosat/ERM
profiles, EOS, Trans. Am. geophys. Un., 75 (16) (1994
supplement).
567
is then
In the LSC calculations, covariance functions between
various quantities are needed. A brief derivation of these
functions is presented below.
where En is the anomaly error degree variance of the
reference field, and N M A X is the maximum harmonic
degree of the field. The evaluation of K((1,) or K((1,)and its
derivatives for the degree variance model given in eq. (A2)
may be done by an analytical method as in Tscherning &
Rapp (1974).
In the present study the required isotropic covariance
functions are covariances of the longitudinal and transverse
components of the geoid gradient, I , m: C,, C,,,,,; the
covariance between the longitudinal component of geoid
gradient and gravity anomaly, C,AK;and the covariance for
gravity anomalies, C,,,,. Their relationships with K ( +) and
its derivatives can be found in, for example, Moritz (1980,
pp. 108-109), or Tscherning & Rapp (1974, p. 25). Since
these covariances are functions of only the spherical
distance, they can be pre-calculated and tabulated with a
suitable step-size of (1, (e.g. 0.01'). We used a program
developed by Tscherning (1976) to calculate these
covariances. In the LSC computation the actual values of
the covariances at some (1, are then linearly interpolated
from the table. Note that in this study we use geoid
gradients as the data type, while in the classical treatment of
covariance functions by Tscherning & Rapp (1974) and
Moritz (1980, p. 108) deflection of the vertical was used. The
two quantities bear opposite signs.
A1 Isotropic covariance functions
A2 Covariances for geoid gradients
First of all we look for isotropic covariance functions,
namely, functions that are dependent on only the spherical
distance between two arbitrary points. All the covariance
functions that are needed can be derived from the
covariance function of the Earth's disturbing potential
K(I,!J), where (1, is the spherical distance, by the law of
covariance propagation (Moritz 1980, p. 108). The
covariance K((1,) on a sphere of radius R may be
represented by a series in Legendre polynomials (Moritz
1980, p. 96) as
As shown in Fig. A l , at P the longitudinal and transverse
components of geoid gradient are 1 and m, respectively. At
Q, the counterparts are I' and m'. The geoid gradient along
a given azimuth at P or Q can be determined by these two
components as
A P P E N D I X A: C O V A R I A N C E FUNCTIONS
+ m sin ( a c p- a p Q ) ,
cos (aev- m,.)+ m i sin ( m e , - a / , )
F~ = 1 cos
eQ = I '
=
( a c I, mp,)
-1' cos (aEv- a Q P ) m'sin ( a t Q
- aQp).
(A41
(A5)
north
where Tp, T, are the disturbing potentials at P and Q
respectively, c, is the anomaly degree variance of degree n,
and P, is the Legendre polynomial of degree n. In this study
we used Model 4 of Tscherning & Rapp (1974, eq. 68) for
c,, 1.e.
c,
=
A(n
-
( n - 2)(n
f
1)
+ B ) f+',
with A.= 425.28 mga12, B = 24, s = 0.999 617. When a
reference field is used, its error must be taken into account.
The covariance function for the residual disturbing potential
Figure Al. Sketch illustrating definitions of azimuths and the
directions of longitudinal and transverse geoid gradients used in
calculating covariance functions between points P and Q.
568
C. Hwang and B. Parsons
Therefore, the covariance between
E~
and
E,
c,, = COV ( E p , F Q ) = E { E p E y }
= -c//cos (at,!ap,) cos ( a c ,- ' y e p )
- Ln
sin ( a c / >
sin (a,p 'yap),
have employed the property that cov (Ag,, m Q P )= 0. On
using the planar approximation we have
is
CAgc = -cos
-
- UPQ)
-
(A6)
where we have employed the properties that cov (I, m ) = 0,
and that the azimuths are constants with respect to the
expectation operator E. If the spherical separation between
P and Q is small, the planar approximation that
app = aPQ- K can be used, leading to
A4 Special cases
When the spherical distance is zero, we have
1
C,,(O)
A3 Covariance for gravity anomaly and geoid gradient
The covariance between the longitudinal component of
geoid gradient from P to Q, and the gravity anomaly at Q, is
CIA, = cov (I,,
AgQ) = cov (lp,? Agp)
= cov ( I Q p ,
A g p ) = cov ( A g p , Ipp) = C,,,.
= EIAg,",,
= cos
(a,,
cos (at,- Q Q P ) + m,, sin (ac',- @ Q P ) l }
- ",P)C/,g
(A91
and the covariance between the geoid gradient at P and
gravity anomaly at Q is
c,,
= cos ( a e /? "ry)Cta,1
(A101
where l,,
mQp are the longitudinal and transverse
components of geoid gradients at Q (pointing to P ) , and we
= - lim
R2 +.o
K'(9) 1
sin 9 R2
___ - - lim
+-(I
K"(9)
~-
-
1
~
cos I) R2
K " ( 0 ) C/,(O).
This shows that the variance of geoid gradient is invariant
with respect to azimuth. At a crossover of two satellite
ground tracks, the spherical distance between geoid
gradients along the descending and ascending tracks is zero.
The two along-track gradients can be expressed as
(A8)
Therefore the covariance between the gravity anomaly at P
and the geoid gradient along a given azimuth at Q is
c,,,
('41 1)
(atc,- " P Q ) C / A g .
E, =
6 cos a , + 77 sin a d ,
E~ =
6 cos ad + 77 sin a d ,
(A.14)
where E , and E~ are the geoid gradients along the ascending
and descending tracks respectively, and 6 and 7) are the
north and east components of geoid gradient respectively.
The covariance between E , and ed is
Ctdtd= E{E,E,} = CJO) cos a , cos ad + C,,,,(O) sin a , sin a d
=
C,,(O) cos ( a , - ad).
Moreover, if 9 = 0, we have C,,,
(A.15)
=0
because C,,g
= 0.