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AN EFFICIENT ALGORITHM FOR COMPUTING THE CROSSOVERS IN SATELLITE
ALTIMETRY
IrJASA-CR-
182389)
Abi
EFFICIENT
ALGORITBtl
FOB
N88-15286
COBPUTING
THE
CROSSOVERS
IN
SATELLITE
ALTIHETEY
Final
Technical
Report
(Scripps
Institution
of
Oceanography)
15
p
CSCL
22A
Unclas
G3/43
01
186
22
Scripps Institution
of
Oceanography
A-030
La Jolla, California
92093
a
-2-
ABSTRACT
An
efficient algorithm has been devised to compute the crossovers in satellite altimetry. The significance of
the crossovers is twofold. First, they are needed
to
perform the crossover adjustment
to
remove the orbit
errorSecondly, they yield important insight
to
the oceanic variability. Nevertheless there is no published algorithm
to make this very time consuming task easier, which is the goal of
this
note. The success of the algorithm is predi-
cated on the ability
to
predict (by analytical means) the crossover coordinates
to
within
6
km
and
1
second of the
true values. Hence, only one
interpolation/ex@apolation
step
on
the data
is
needed
to
derive the crossover coordi-
nates in contrast to the many
interpolation/extrapolation
operations that are usually needed
to
arrive at the same
accuracy level
if
deprived of this information.
1.
INTRODUCTION
In satellite altimetry (in which the sea level relative to a reference ellipsoid
that
best
approximates the shape
of the
earth
is measured along the satellite ground track, e.g., Wunsch and Gaposchkin,
1980).
the tern "crossover"
refers to the intersection
of
two ground tracks. The coordinates of a crossover point
are
comprised of the location
(Le., latitude and longitude) and the two times when the satellite passes over the crossover point. The crossover
difference is the difference between the two
sea
level measurements. The crossovers
are
important in two aspects.
First, the crossover difference reveals the temporal variability of the
sea
level and
also
avoids the geoid uncer-
tainty (the geoid is
an
equipotential surface of the earth gravity field,
to
which a motionless Ocean would conform).
The
sea
level
varies
with
respect
to
the
reference ellipsoid
primarily
because
of
the
earth
gravity
field
and
secon-
darily due
to
the dynamical effects
of
the Ocean. Presently the uncertainty
of
the geoid
is
at least comparable
if
not
larger then the effects of
ocean
dynamics (e.g.,
Tai,
1983).
Analyses based on crossover differences have shown
and are continuing
to
reveal valuable insight on the
sea
level temporal variability (e.g., Cheney and Marsh,
1981;
Fu
and Chelton,
1985;
Cheney
et
al.,
1986.)
Secondly, the crossovers play a pivotal role in removing the satellite orbit error (i.e., the uncertainty of the
satellite's altitude)
by
the so-called crossover adjustment method (e.g., Tai and
Fu,
1986;
Tai,
1987).
The orbit
error, which usually causes the crossover difference
to
be
over
1
m
in size, cannot
be
taked lightly lest the crossover
difference should reveal not the sea level temporal variability but the variability produced by the orbit error. Furth-
,
.
"
.
'*e
'
.
'
-3-
ermore, the basin-scale geoid uncertainty
is
small enough for the basin-scale circulation
to
be
investigated (e.g., Tai
and Wunsch
1983,1984)
and the orbit error reduction is of paramont importance in
this
case.
For the exact repeat orbit, one can use along-track differences instead of crossover differences
to
reduce the
orbit error
if
one is only interested in the temporal variability. However, the crossover adjustment
has
to
be
per-
formed if one is interested in the absolute topography because ascending and descending tracks may have different
orbit error characteristics which are only evident in crossover differences.
There are many error
sources
in
satellite altimetry (e.g., Tapley
et
al.,
1982).
Thus, it
is
imperative that
any
estimate
be
deduced from many samples
so
as
to
minimize the effect of a particular error source or a particularly
bad error realization because the errors are more or less independent of each other; and it is not unusual
to
find
thousands or even millions of crossovers being treated in a single problem (e.g., Marsh
et
al.,
1982;
Rapp,
1983).
However, the simple prelude of finding the crossoven
can
consume more computing
resources
than solving the
problem itself if
an
inefficient algorithm
is
used Realizing the importance of a good algorithm, yet one would
search in vain
in
the literature for a published algorithm to compute the crossovers. Since the launch of Geosat
(March,
1985).
the matter
has
taken on added urgency. The initial
18
months of the Geosat Mission
is
classified, but
the crossover differences
in
this period are unclassified and would
be
available to the public
if
it were not such a
cumbersome job.
The purpose of this note is
to
make one
good
and tested algorithm available to the public. Hopefully a better
algorithm
will
be
published
as
the result of this note. In the following,
it
will be demonstrated that one can derive
the coordinates of a crossover point
to
within
1
second and
6
km
of the true
values
(1
km
if
an
ad hoc formula is
used) from the circular orbit approximation while compensating for the
earth's
oblateness. Thus only one
interpolation/extrapolation
step
is
needed to derive the coordinates of the crossover
from
the
data
as
oppose
to
three
or
four
interpolation/extrapolation
steps that are usually needed if deprived of this information.
2.
CIRCULAR ORBIT APPROXIMATION
The
orbit of the satellite
is
better described by
an
ellipse. However for the purpose of satellite altimetry, the
I
orbit is made
so
circular that the circular approximation would greatly simplify
the
problem while incur little error.
For example, the first eccentricity
(see
definitions in Section
3)
of the Seasat's orbit is merely
0.001,
while its coun-
2
I.
-4-
terpart
for the earth is 0.0818 (Le., one incurs much bigger error by assuming the
earth
is
spherical. The effect
of
the earth's oblateness will
be
discussed
in
Section
3).
If one assumes the orbit
is
circular and the earth is spherical, the location
of
the ground track
can
be
easily
derived
as
a function of time. This has been done
in
Tai and Fu (1986,
see
their eq.
(3).
Appendix
1,
and Figure 1).
We
can
adapt their formulation
to
the present case; and the relevant equations are
-
41
=
j
sin-'(sin
At
sini)
,
M
=
tan-'(tan Atcosi)
-
Wt
,
(1)
(2)
where the time,
t,
is
nondimensionalized
to
make
2x
correspond
to
the duration of one revolution;
6
is the latitude
(the overhead bar conveys the fact that it
is
the geocentric latitude,
see
Figure
1);
X
is the longitude and defined
to
be
in
the range of
[O,
2x);
i
is the inclination angle;
f2
is the
earth
rotation rate relative
to
the orbital plane. Quantities
with a subscript
o
are related to the equator crossing, and At
=
t-to
,
M
=
X
-
I,,
(Le.,
to
and
I,,
are
the equator cross-
ing time and longitude respectively). The adaptation
is
done
so
that equations (1) and
(2)
are valid for either an
ascending or a descending track with
j=1
if ascending and
j=-1
if
descending. Thus, the valid range
of
each vari-
able is: -i
S
6
S
i,
bd
5
d2,
and
S
(l+kR)
@,
where k=l if i >
@
(Le., retrograde), and k=
-1
if
i
<
1J2
(i.e.,
prograde).
2a.
Determine the equator crossing time and longitude
As
a first step, the algorithm requires that the data
be
sorted into ascending and descending tracks, and the
equator crossing time and longitude
be
determined for each hack. Because
data
gaps (e.g., over land) often prevent
the direct determination
of
these coordinates from
data,
there is
a
need
to
determine them analytically. From a point
(with cordinates
t,
6.
A)
along the track, one can determine from
(1)
that for the equator crossing time,
rlst
=
j
sin-' (sin
&ini
,
to
=t-At
,
and for the equator crossing longitude, from
(2)
and
(3),
A,,
=
A-tan"(tan
AZ
cosi
)
+
~t
.
(3)
(4)
Note
that
if
h,
should lie outside
[0,
211:). one can add or substract
211:
from
A,
to
make
it
fall
in
this range.
2b. Determine the coordinates
of
a
crossover
point
(1). Longitude
I
-5-
Let a subscript a
(
or d) convey the meaning of ascending (or descending). Then from the geometrical sym-
metry entailed
in
equations (1) and
(2).
it is
easy
to
see
that the longitude of the crossover point must lie right
in
the
middle between the two equator crossing longitudes of the two
tracks.
To
be
more specific,
fiF
if
boa
-
x&jl2(l+Q)x
,
Loa
+
hod
2
h=
where the sign in (6b) is such that
(KX
c
2x.
And if
(142)~
c
bo(l-h4dl
c
(l+Q)x, we have two possibilities:
(i) two crossover points
if
i
>
I&?,
i.e.,
(ii)
no crossover point if i
<
a.
(2)
Time
Knowing
h
(therefore
M),
one
can
solve for
At
using eq.
(2).
which can
be
transformed
to
a more convenient
form that avoids the evaluation of
tan-',
Le.,
f
(At)=tan(M+QAt)-runbtcosi
=O
.
(7)
Hence, it becomes a problem of finding the zero of the transcendental function
f.
One can
use
either the ascending
or the descending equator crossing longitude to form
M.
Care must
be
taken (Le., add or substract
2x)
to make
sure
that
k=l
if
i
>
I&?
and k=-1 if
ita.
S
(l+kn)a.
Also
note that
if
Ah
c
0
(or Ah
>
0).
At lies between
0
and kx/2 (or -ksc/2). Remember that
(3)
Latitude
The geocentric latitude can be derived immediately from
eq.
(1).
I
3.
EFFECTS
OF
THE EARTH'S OBLATENESS
As
mentioned
in
the previous section, the earth
is
far more elliptical
than
the orbit. The effects are twofold
(see
Fig.
1).
First,
the
subsatellite point (marked
as
g
in
Fig.
1,
where the line joining it with the satellite
is
perpen-
dicular to the local
earth
surface)
is
different from the point (marked
s
in Fig. 1) where +e line joining the satellite
with
the earth center intersects the
earth
surface. Secondly, the coordinate
data
are given in terms of the geographic
I
latitude (defined as the angle between the local vertical and the equatorial plane, Le., the latitude that can be deter-
2