# the geoid - School of GeoSciences

```High resolution
sea surface topography
for the northern Atlantic
using the
Iterative Combination Method
School of GeoSciences
University of Edinburgh
Geoid has been
(mean) sea
z = 0.66  0.2 m
geoid
h = 6371440.27 
(zero height)
N = 6371439.61 
surface
0.2 m
0.02 m
Why is the Geoid
so poorly
determined?
Raw
marine gravity data
have
•
incomplete
coverage
and are
• inaccurate
Raw
marine gravity data
have
•
incomplete
coverage
and are
• inaccurate
The problem of gravity data coverage
To compute a geoid you
need complete coverage of detailed gravity
over local region
plus lower resolution global coverage
The geoid
is a weighted surface integral
of gravity
N ( xo , yo ) 

g ( x , y ) F ( x  xo , y  yo ) dx dy
Marine gravity data gaps
must be filled
interpolated
gravity
measured
gravity
gint(x,y)
~2m
free interpolation
is very noisy
Patching
with
altimetric gravity anomalies
MDT from geoid EDINP
&
KMS04
Manual patching with altimetric anomalies
is better but still noisy
There is a fundamental inconsistency with using
altimetric gravity anomalies for patching
Need a more rigorous way to combine altimetry,
an MDT model
and along-track gravity
By analogy with a real gravity anomaly and the geoid,
define an altimetric gravity anomaly as a vertical
derivative of the sea surface height
g

g alt


‘real’ gravity anomaly

N
g
z
g
N
g
z
h
g
z

 g
z
 g
gravity effect of sea surface topography
KMS altimetricgravity anomalies
 250 mGal
& Per Knudsen,1998
Gravity effect of composite model of dynamic sea
surface topography
 2 mGal
Although the bias
of altimetric gravity anomalies
is small,
for geodetic oceanography the
2 mGal is the signal
and the
250 mGal
is
noise
Using an
altimetric gravity anomaly as a proxy for
real gravity
makes
MDT zero
(or alternatively represents it by an a priori long-wavelength model)
g alt

‘real’ gravity anomaly
h
 g
z
g  g
gravity effect of sea surface topography
The Iterative Combination Method
•
generates a model of sea surface topography
and interpolates gravity from ship tracks to
a grid in a way that is rigorous and mutually
consistent
Constraint on gravity interpolation
The area integral of gravity
interpolated
into between-track gaps
geoid that fits the
altimetric sea surface minus the
MDT model
must generate a
N  h  
geoid derived
from
infilled gravity
sea surface
height
derived from
altimetry
MDT model
Constraint on the MDT model
The
model for mean dynamic sea surface
topography must generate a
gravity effect that fits altimetric gravity minus
real gravity
along ship tracks
g  g alt  g
observed
along-track gravity
gravity
derived from altimetry
± fixed
fixed
gravity effect
of MDT model
iteratively
modified
Constraint on the MDT model
The
model for mean dynamic sea surface
topography must generate a
gravity effect that fits altimetric gravity minus
real gravity
g 
along ship tracks
g alt  g
observed
along-track gravity
gravity
derived from altimetry
± fixed
fixed
gravity effect
of MDT model
iteratively
modified
The Iterative Combination Method
•
generates a model of sea surface topography
and interpolates gravity from ship tracks to
a grid in a way that is rigorous and mutually
consistent
FFT
(combined
geoid) n
(combined
gravity) n
datum
weighted
combination
(combined
gravity) n+1
start
initial
gravity
initial
pseudogeoid
FFT
weighted
combination
next
iteration
datum
(gravity-based
geoid)n+1
Pseudo - geoid
Altimetric
Mean Sea
Surface
GOCINA
Composite Mean
–
Dynamic
Topography
–
(extended)

Residual Pseudo - geoid
GRACE
geoid
GGM01c
(n  90)
Pseudo - geoid
Altimetric
Mean Sea
Surface
GOCINA
Composite Mean
Dynamic
Topography
GRACE
geoid
GGM01c
(n  90)
(extended)
removed &
restored just to
make quantities
small
Pseudo - geoid
Altimetric
Mean Sea
Surface
GOCINA
Composite Mean
Dynamic
Topography
(extended)
Initial MDT model
gets iteratively
modified
GRACE
geoid
GGM01c
(n  90)
Pseudo - geoid
Altimetric
Mean Sea
Surface
GOCINA
Composite Mean
Dynamic
Topography
(extended)
fixed
GRACE
geoid
GGM01c
(n  90)
altimetric mean sea surface
extended composite MDT
residual pseudo-geoid
Pseudo - geoid weight
Weight of pseudogeoid determined
from expected
uncertainty in
composite MDT
model
FFT
(combined
geoid) n
(combined
gravity) n
datum
weighted
combination
(combined
gravity) n+1
start
initial
gravity
initial
pseudogeoid
FFT
weighted
combination
next
iteration
datum
(gravity-based
geoid)n+1
Gravity
Composite of ship, airborne and land
free-air anomalies,
altimetric free-air anomalies,
–

Residual Gravity
GRACE freeair
anomalies
GGM01c
(n  90)
Residual gravity
Gravity weight
wg

z
x
2
 y2  z2
± zero weight
where no data

3
2
Convergence of iterative weighting
algorithm
Parameter change per iteration
log(standard deviation)
2
MDT (cm)
1
gravity (mGal)
0
-1
-2
0
2
4
6
8
10
Iterative
combination
Method MDT
purely
terrestrial
data - no
GRACE
input
Impact of GRACE data
Long wavelength components
of the finally interpolated free-air gravity
must match GRACE free-air anomalies
Long wavelength components
of altimetric sea surface minus MDT model
must match GRACE geoid
Residual between
surface gravity and
GRACE GGM01s freeair anomalies,
smoothed with a
450km low-pass filter
Make surface gravity
consistent with
residual
Residual between
composite MDT and
MDT consistent with
GRACE GGM01s
smoothed with a
450km low-pass filter
Make composite MDT
consistent with
residual
ICM MDT
on a 2 km grid
smoothed to 450 km
With this smoothing,
the MDT model is
constrained by
GRACE and satellite
altimetry
Shorter
wavelengths, not
controlled by
GRACE, may be due
to geoid errors or
MDT errors
However, their rms
residual is only
2.3 cm
Some residuals
correspond to an
intensification on
Other residuals have
the characteristic
‘bulls eye’ pattern of a
gravity field error
ICM MDT
on a 2 km grid
smoothed to 450 km
This corresponds to
all residuals being
gravity errors
Unsmoothed ICM
MDT on a 2 km grid
This corresponds to
all residuals being
corrections to the
MDT model
ICM MDT on a 2 km
grid
Smoothed
MDT
smoothed to
ll >>7575kmkm
Some gravity field
errors suppressed
by smoothing
Geostrophic current velocities deduced from ICM MDT models
GRACE-corrected GOCINA composite MDT
ICM MDT with 75 km smoothing
Mean surface flow from Lagrangian drifters (Jakobsen et al,
2003)
Drifters assimilated into circulation model (Nøst & Isachsen,
2003)
ICM model
with 75 km
smoothing
Conclusions
We have determined precise and
oceanographically realistic MDT and surface
ocean current flows for the northern Atlantic,
with no hydrographic input.
The ICM method has given rms geoid/MDT
errors < ~3 cm for the whole ocean.
End of slide show
Pre-iteration inputs
Residual pseudo-geoid
Residual gravity
Weighting
functions
Geodetic oceanography
Compute the shape of the geoid (zero height) from
measurements of gravity
Deduce the shape of the sea surface by
measuring the two-way reflection time of microwaves
from a satellite-borne altimeter
The difference is sea
surface topography
mean
static
geoid
(zero height)
dynamic
sea surface
topography
mean
sea surface
Satellite altimetry
captures the
time variation
Small difference between very
large quantities
sea surface
 = 0.66 m
geoid
(zero height)
N = 6371439.61 m
h = 6371440.27 m
Oceanography & global gravity models
Fred Whipple (1962) Started NASA program
Izsak (1966) Smithsonian Standard Earth
– satellite orbital perturbations
Rapp (1977, 1987, 1991) Ohio State University
– satellites plus 5, 1, ½ averages of surface gravity
1996 Earth Gravity Model (NASA , DIMA, OSU consortium)
discarded all marine gravity as too unreliable!
Oceanography & global gravity models
Over the oceans
EGM96 used ‘pseudo-gravity’ anomalies
Pseudo-gravity anomalies
revolutionised
marine geology and geography
David Sandwell’s images of ocean tectonics
Bill Haxby’s bathymetric charts
Loosing the plot ...
h  N
‘real’
gravity anomaly
 2U
 2U

x 2
N ( xo , yo ) 
?

 
 U
gravity effect
U
of
g 
z sea surface topography
 2U
 2U


 0
2
2
y
z
gN
g ( x , y ) F ( x  xo , y  yo ) dx dy
g pseuodogravity  g  g
Not having a good geoid limits
the accuracy and resolution
1996 Earth Gravity Model
The Edinburgh North Atlantic
Gravity Model
•
reprocessing all marine gravity data
•
•
dealing rigorously with pseudo-gravity anomalies
Pseudogravity
captures
profile shape
successfully
with some
smoothing of
the shortest
wavelength
features
Offset
Mainly due to
survey
datum error
but
also to
gravity effect
of sea
surface
topography
Summary statistics
3500 km by 3500 km square
(Easting & Northing of Lambert’s conic projection, centred on Iceland)
1.693M point values
* 967 ‘surveys’
* 13169 line segments
* 8793 intersections
* 778 connected ‘surveys’
* 465862 points on connected network
Use of
VERIFY
Polynomial
captures
profile
successfully
KMS99 has
same datum
but smoothes
structure
Gravity meters only measure gravity differences
Gravity meters may also suffer from instrumental drift
Use a least-squares minimisation
of the mismatch at track cross-over points
by adding a datum shift for each survey
and, for long and well crossed surveys,
a correction for instrumental drift
Cross-over error statistics (mGal)
standard deviation
weighted standard deviation
7.17
4.52
2.88
0.28
Still need
to patch
the gaps
Can image the variability without knowing
an absolute baseline (the geoid)
FFT
(combined
geoid) n
(combined
gravity) n
datum
weighted
combination
(combined
gravity) n+1
start
initial
gravity
initial
pseudogeoid
FFT
weighted
combination
next
iteration
datum
(gravity-based
geoid)n+1
Convergence of iterative weighting
algorithm
Parameter change per iteration
log(standard deviation)
2
MDT (cm)
1
gravity (mGal)
0
-1
-2
0
2
4
6
8
10
Raw output MDT
on 2 km grid
Smoothed MDT
l > 75 km
Can trace
individual
current streams
Now have errors
comparable with
those of
altimetry
 = 2.3 cm
Now have errors
comparable with
those of
altimetry
 = 2.3 cm
Grace errors
for l > 450 km
How good is the GRACE model?
How good is the GRACE model?
-28
CHAMP EIGEN3p
GRACE+CHAMP GFZ
GRACE +TEG Texas
GRACE - Texas
EGM96
ln(power)
-29
-30
-31
0
20
40
60
80
100
120
degree n
Presentation at Frascati
140
160
180
200
How good is the GRACE model?
-26
-27
egm96
ggm01s
ggm02s
-28
-29
-30
0
40
80
120
Update: GGM02s is grossly wrong for n > ~120
160
How good is the GRACE model?
egm96
-29
ggm01s
ggm02s
-29.5
-30
0
20
40
60
80
100
120
140
GGM02s is detectably different from GGM01s
for n > ~ 50
160
GGM02s – GGM01s Geoid difference
Constructed with
low pass cosinebell taper from
n=80
to
n = 100
No smoothing
25 cm
Geoid difference
Constructed with
low pass cosinebell taper from
n=80
to
n = 100
400 km
smoothing
9 cm
Geoid difference
Constructed with
low pass cosinebell taper from
n=80
to
n = 100
800 km
smoothing
5 cm
Geoid difference
Constructed with
low pass cosinebell taper from
n=80
to
n = 100
1600 km
smoothing
2 cm
mean sea surface topography
= mean sea surface height – geoid
  h  N
This week’s satellite image
of the
North Atlantic Circulation:
Our new geodetic technique
to monitor climate change
GOCINA:Gravity & Ocean Circulation in the North Atlantic
Per Knudsen, Ole Anderssen - Danish Space Agency
Fabrice Hernandez, Marie-Louise Rio - CNRS-CLS
Johnny Johannesson - Nansen Environment Centre Bergen,
Dag Solhein - Norwegian Mapping Authority,
Keith Haines - Earth System Science Centre Reading
Introduction
•
Why is the North Atlantic so important for climate change?
•
What do we know about the present day ocean circulation?
•
What is geodetic oceanography?
Deep water
formation in
the North
Atlantic drives
the conveyor
belt
Stopping deep water formation in the North Atlantic
switches off the conveyor belt

. . . switches off the conveyor belt . . .
Quickly!

Need to be able to monitor the Global Ocean Circulation
What do we know about today’s ocean circulation?
Matthew Fontaine Maury, 1859
SEA DRIFT AND WHALES: on which the
movements of the sea as indicated by the
THERMOMETER are shown
World Ocean
Circulation
Experiment
WOCE
Intense, one-off
hydrography
Characterise surface ocean currents by
dynamic sea surface topography
- represents pressure variations
analogous to a weather map
Wind direction
- parallel to isobars
Wind speed
- related to
isobar spacing
Dynamic sea surface
topography - MDT
geoid
(zero height)

Pressure
variations
sea surface
For classical oceanography,
dynamic sea surface topography
is computed from vertical hydrographic profiles of
temperature and salinity
This is equivalent
to not being able to measure atmospheric pressure
but having to base weather forecasting on
Levitus MDT – null depth at 700 m
Hydrographic images
lack time resolution and
completeness of spatial coverage
Must use hydrographic data
to control the evolution of
Global Ocean Circulation Models
OCCAM Global Ocean Circulation Model
assimilating hydrographic data
(¼ resolution)
CLS (GOCINA)
OCCAM
CLS (RIO-03)
Nansen
Global
ocean circulation
models
do not agree
in detail
~20 cm mismatch
Best current estimate
is an average of
the highest resolution
GOCMs that
assimilate
hydrographic
observations
The GOCINA
composite model
Resolution ~ 200 km
Contour interval 2 cm
```

21 cards

21 cards

35 cards

30 cards

33 cards