Embed Size (px)
Citation preview
Method
1) In general, comparing the influence on the velocity error from the drifting error with from the shifting error when the mid-depth velocity is derived from the ARGO trajectory.2) Based on the Kalman Filter method, presents an new method to extrapolate the positions when ARGO float diving or resurfacing in a measurement cycle in order to reduce the estimation error about the
derived mid-depth velocity. 3) Quantitatively considering the uncertainty of the estimation method itself, especially to the extrapolation scheme for the float positions diving and resurfacing.4) To find the process or outline with which the credible estimation of the mid-depth velocity from ARGO trajectory can be acquired as many as possible.
Objectives
i. ARGO DATA: from November 2001 through October 2004, the ARGO floats data in delayed form have been downloaded from the two websites of the global data centers (ftp: usgodae1.usgodae.org/pub/outgoing/argo and ftp.ifremer.fr/ifremer/argo), which includes trajectory, meta, profile data files.
ii. ECCO/MIT DATA: current states estimate from the MIT/ECCO as a part of Estimating the Climate and Circulation of the Ocean (available: http://www.ecco-group.org/), which can be meaningfully constrained by global observations and then is potent to estimate the time-evolving, three-dimensional physical state of the full oceanic circulation.
Data used
IntroductionThe Array for Real-time Geostrophic Oceanography (ARGO) project not only creates the actuality to observe the temperature
and salinity profiles in the mid- and upper-ocean at real time, but also provides a unique opportunity to estimate the absolute velocity at mid-depth of global oceans. However the estimation can only be made based on float surface trajectories. Moreover, the positions information when the float diving and resurfacing is not available, which creates difficulty to estimate mid-depth current and leads to great estimation error. It is important that the analysis and estimation about the errors concerned should be comprehensive and as accurate as possible.
Unique opportunity to estimate the absolute current, especially at mid-depth!Fig.1. Distribution of active Argo floats on March 7, 2006 ( downloaded from
http://www.argo.ucsd.edu/index.html )
2 ) Error Analysis.hypotheses about the current in ocean:1) the velocity drifting on surface is m times as the velocity at mid-depth. 2) the average velocity integrated from the parking depth to surface is n times as the velocity at mid-depth.In generally, the parking time of ARGO float can be ~9 days, and the addition of the decent time and the ascent time can be 4 hours.
Davis et al. (1992) have presented a rigorous method to the ALACE float trajectory. By means of statistical approach, Schmid et al. (2001) further discuss that of Davis for ARGO float. Then based on the least squares principle, Park J.J et al. (2004) simplify the extrapolation process according to the concept of Davis et al. (1992). Nevertheless, these methods to extrapolate the positions still exist many limitations in practical application, such as the uncertainty about the method itself, and the instability when extrapolating and so on. So we present a newly analysis scheme for the extrapolation which is base upon the Kalman Filter (Kalman, 1960).
Denoted by (X, Y ) is a spatial location on ocean’s surface. The forecasted (by some methods or models), observed and estimated positions in a surface trajectory of an ARGO float are written as (Xtf
, Ytf), (X
to
, Yto), (X
te
, Yte ) at time t
respectively. Then Kalman filter formulations can be expressed as follow:
1 ) Extrapolation scheme.
P r e d i c t i o n T i m e ( h )2 4 6
Freq
uenc
y (%
)
0
2 0
4 0
6 0| X f - X o |
| Y f - Y o |T i m e F r e q u e n c y
T i m e b e f o r e P r e d i c t i o n ( h )2 4 6
Dev
iatio
n (k
m)
0
1
2
3| X f - X o || Y f - Y o |T i m e F r e q u e n c y
T h e N u m b e r o f L o c a t e d P o i n t s b e f o r e P r e d i c t i o n
4 6 8 1 0 1 2 1 4 1 6 1 8 2 0
Fore
cast
Err
or (
km)
0 . 0
. 5
1 . 0
1 . 5
Freq
uenc
y (%
)
0
2 0
4 0
6 0
E x f
E y fP o i n t s F r e q u e n c y
( a . ) ( b . )
( c . )
Fig.3. Mean absolute error of u (blue line) or v (red line) changes along (a) the time before prediction, (b) the prediction time. Where the shaded area presents the frequency for the time respectively. The rms of prediction error varies along the number of fixed points before forecasting in (c), where the shaded area denotes the frequency of the points number.
E x t r a p o la t in g T im e b e f o r e D e s c e n t ( h )
0 5 1 0 1 5 2 0
. 2
. 4
. 6
. 8
1 0
2 0
3 0
4 0Z o n a l M e r id i o n a lF r e q u e n c y o f t h e t im e
E x t r a p o la t in g T im e a f t e r A s c e n t ( h )0 2 4 6 8 1 0 1 2
Mea
n R
ms
of E
rror (
cm s
-1)
. 2
. 4
. 6
. 8
Freq
uenc
y of
Tim
e (%
)
1 0
2 0
3 0
4 0
Z o n a l M e r i d io n a lF r e q u e n c y o f t h e t im e
Fig.4. Mean rms error of the estimated mid-depth velocity only results from the extrapolation error in zonal (solid histogram) and meridional(diagonal histogram) as functions of: (a) for diving, where the solid line with light blue means the frequency of that time; (b) for resurfacing, where the solid line with light blue means the frequency of that time.
−
+
=
f
t
ft
ot
ot
tft
ft
et
et
YX
YX
YX
YX
K
1)( −+= RPPK ft
ftt
ftt
et PKIP )( −= 11 ++ += t
Tet
ft QMMPP
=
+
+e
t
et
ft
ft
YX
YX
M1
1
So the positions when float diving and resurfacing on surface would be extrapolated by this procedure as well as their errors including the uncertainty of method-self.
∆−−∆+∆+=
∆−+∆+∆+=
−+
−+
)]}cos(1[)sin({
)]}cos(1[)sin({
11
11
tfutfvf
tvYY
tfvtfuf
tuXX
nnnnn
nnnnn
βα
βα
−+−+−=−−++−=
−+
−+
nenn
enn
enn
enn
fn
nenn
enn
enn
enn
fn
yxbxbyayayxybybxaxaxεεεεεεεεεεεε
11
11
)1()1(
++
=
++
=
−−
−−
−−
−−
22
22
22
22
)()()()()()()()(
on
fn
on
on
fn
fne
n
on
fn
on
on
fn
fne
n
EyEyYEyYEyY
ExExXExXExX
><+><>><<
>=<
><+><>><<
>=<
−−−−
−−−−−−
−−−−
−−−−−−
on
on
fn
fn
on
on
fn
fne
nen
on
on
fn
fn
on
on
fn
fne
nen
yyyyyyyyyy
xxxxxxxxxx
1111
111111
1111
111111
,,,,,
,,,,,
εεεεεεεεεε
εεεεεεεεεε
Y=1.0
Freq
uenc
y (%
)
0
3
6
9
12||X f-Xo||M ean : 1.2 km
ExfM ean : 1.6 km
Error (km )1 2 3 4 5
Freq
uenc
y ( %
)
0
3
6
9
12||Yf-Yo||M ean : 1.0 km
EyfM ean : 1.7 km
Y=0.5
||X f-Xo||M ean : 1.2 km
ExfM ean : 1.4 km
Error (km )1 2 3 4 5
||Yf-Yo||M ean : 1.0 km
EyfM ean : 1.4 km
Y=0.1
Freq
uenc
y (%
)
0
3
6
9
12||Xf-Xo||M ean : 1.1 km
ExfM ean : 1.1 km
Error (km )1 2 3 4 5
Freq
uenc
y ( %
)
0
3
6
9
12||Y f-Yo||M ean : 1.0 km
EyfM ean : 1.1 km
Y=1.0
Freq
uenc
y (%
)
0
3
6
9
12||X f-Xo||M ean : 1.2 km
ExfM ean : 1.6 km
Error (km )1 2 3 4 5
Freq
uenc
y ( %
)
0
3
6
9
12||Yf-Yo||M ean : 1.0 km
EyfM ean : 1.7 km
Y=0.5
||X f-Xo||M ean : 1.2 km
ExfM ean : 1.4 km
Error (km )1 2 3 4 5
| | X
f
Fig.2. Comparison the average of the absolute error with the mean rms of prediction error for different γ value (left is 0.1, middle is 0.5, right is 1.0). Lower is for v component, upper is for u component.
−
−=
><><
=
−
−
en
en
en
en
n
n
n
n
vvuu
yx
yQxQ
1
12
2
/1/1
)()(
γγ
εε
Assume the prediction variance having:
Presumes:• The float trajectory on surface is a sum of linear drift, inertial motion, and noise, and the former two terms are dominant. So our prediction model for the series positions on surface is stemmed from their combination, then the noise is equivalent to the uncertainty of model-self.• The prediction error, observation error and model error are regarded as uncorrelated, and their distributions are normal.
Linear drift:
Inertial motion:
∆=∆+∆∆−
=−
∆=∆+∆∆−
=−
−−−−
−−−
−−−−
−−−
tvttt
YYYY
tuttt
XXXX
nnnn
nnnn
nnnn
nnnn
1212
212
1212
212
)(
)(
∆−−∆=−
∆−+∆=−
−−−
−−−
)]cos(1[)sin(
)]cos(1[)sin(
112
112
tff
utff
vYY
tff
vtff
uXX
nnnn
nnnn
=+
=−
0
0
fudtdv
fvdtdu
)(,,2,
,2,2,,)1(2
,)1(2,)1(2,)1(,
222
1212
1112
1
21111111112
12211
12111211222
1
xQyybyybyyb
yxbayxbaxxayxab
yxabxxaaxxaxx
nen
enn
en
enn
en
enn
en
ennn
en
ennn
en
enn
en
ennn
en
ennn
en
ennn
en
enn
fn
fn
+><+><−><+
><−><+><+><−−
><−+><−+><−>=<
−−−−−−−−−
−−−−−−−−−−−−−−−
−−−−−−−−−−−
εεεεεε
εεεεεεεε
εεεεεεεε
)(,,2,
,2,2,,)1(2
,)1(2,)1(2,)1(,
222
1212
1112
1
21111111112
12211
12111211222
1
yQxxbxxbxxb
xybaxybayyaxyab
xyabyyaayyayy
nen
enn
en
enn
en
enn
en
ennn
en
ennn
en
enn
en
ennn
en
ennn
en
ennn
en
enn
fn
fn
+><+><−><+
><+><−><+><−+
><−−><−+><−>=<
−−−−−−−−−
−−−−−−−−−−−−−−−
−−−−−−−−−−−
εεεεεε
εεεεεεεε
εεεεεεεε
JipingJiping XieXie1, 21, 2 and and JiangJiang ZhuZhu11
11Institute of Atmospheric Physics, Chinese Academy of Sciences, BInstitute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, 100029; eijing, 100029; 22Institute of Postgraduate Research, Institute of Postgraduate Research, Chinese Academy of Sciences, Beijing; Chinese Academy of Sciences, Beijing; (( EE--mail: mail: [email protected] [email protected] ))
Estimation of The MidEstimation of The Mid--Depth Currents from Depth Currents from ARGO Floats in PacificARGO Floats in Pacific
i. By virtue of the hypotheses concerned about m and n and the climatological current estimate of the MIT/ECCO monthly, comparison for the different affections resulting from the drifting error and shifting error for the estimation of mid-depth velocity by ARGO float in general, suggests that the drifting error would be a primary source of estimation error about the velocity at most of Pacific.
ii. Based on the Kalman Filter, the extrapolation error about the positions when float diving and resurfacing is about to 1.1 km on average, then the corresponding error of mid-depth velocity is equivalent to 0.2 cm s-1
over ~9 days parking time so that the accuracy is competitive to the other results (Park J.J. et al., 2004; Park J.J. et al., 2005). Moreover, the approximately evaluation of relative error could be proposed, whichfurther ensures their reliability and will be helpful to further be applied in other study fields.
iii. During this extrapolation scheme from Kalman Filter, the uncertainty of method-self can be explicitly quantified, and the linear drift and the inertial motion are combined by a clear and easy way, which all lead to the extrapolation method being simple and flexible so that it may be utilized in global oceans.
iv. Through the accuracy with relative error estimation less than 25%, we can further analyze the currents in 1000 and 2000 dB. The mean flow fields in the two depths and the time variability along the equator, 30°N and 38°N all can be explored by these velocities. So our procedure of estimation mid-depth velocity from ARGO float is valid in which: firstly to Q.C. the estimated velocity by directly method, then to correct thedrifting error by the Kalman Filter method, thirdly to evaluate its relative error by virtue of the climatlogical monthly m, n values, finally to acquire the mid-depth velocity with the relative error less than 25%.
v. In the end, there are some problems have been explored. (1) More rigorous and vigorous Q.C is need, especially to the type of APEX floats. (2) The diving and resurfacing time may be missing in the trajectory file, nearly to 42.8% and 37.5% in 1000 and 2000 dB respectively, which restricts the correction for the drifting error. (3) The shifting time in a cycle which can be inferred to evaluation about relative error, but considerable floats did not provide this information in their metafile. Consequently, in order to derive the mid-depth velocities from ARGO float with more accuracy and more great, the complete metadata and trajectory data will advance the application about the velocity estimation rapidly.
Summary
s 3 s 4 s 5
t 3 t 4 t 5
e 3 e 5
T r u e f i r s t p o i n tT r u e l a s t p o i n t
E s t i m a t e d f i r s t p o i n t
E s t i m a t e d l a s t p o i n t
s 3 s 4 s 5
t 3 t 4 t 5
e 3 e 5
T r u e f i r s t p o i n tT r u e l a s t p o i n t
E s t i m a t e d f i r s t p o i n t
E s t i m a t e d l a s t p o i n t
Fig.7. Schematics about the extrapolated positions by the method of Kalman Filter and with their estimation errors. Note: the red points mean the positions by extrapolating, other points indicate the corresponding true positions on
surface and on parking depth respectively.
( ) 543
53
543
53 )1)((ttt
nttVttt
eeV
VV
TT
TE
++−+
++++
=−
543
25
23
225
23 )))1(())1((/)(
tttntntVee
R adEa ++
−×+−×++=
22
22 )()(
tt
tetee vu
vvuuR
+
−+−= )()( 2
22
22
22
2
vRvu
vuR
vuu
R aee
ea
ee
ee ×
++×
+≈
2543
25
232
)()()()(
tttee
e +++
>=< εThen the rms of extrapolation error for the true mid-depth velocity could be described by the above formula : , which can be regarded as the mainly error source about the mid-depth velocity excluding the shifting error. In fact, by virtue of the monthly climatological current field and the above two hypotheses, we can make the approximately evaluation about its relative error, as following. Consequently, by virtue of the monthly climatological current field and the above two hypotheses, we can make the approximately evaluation about its relative error, as following.
Reference:Davis, R. E., D. C. Webb, L. A. Regier, and J. Dufour, 1992: The Autonomous Lagrangian Circulation Explorer (ALACE). J. Atmos. Oceanic Technol., 9, 264-285.Davis, R.E., 1998: Preliminary results from directly measuring mid-depth circulation in the tropical and South Pacific. J. Geophys. Res., 103, 24 619-24 639.Kalman, R.E., 1960: A new approach to linear filtering and prediction problems. Transactions of the AMSE-Journal of Basic Engineering. 83D,95-108. Park, J.J, K., Kim, and W.R., Crawford, 2004: Inertial currents estimated from surface trajectories of ARGO floats, Geophys. Res. Lett., 31, L13307, doi:10.1029/2004GL020191.Park, J.J., H. Kim,Brian A. King,Stephen C. Riser,2005:An Advanced Method to Estimate Deep Currents from Profiling Floats, Journal of Atmospheric and Oceanic Technology: Vol. 22, No. 8, pp. 1294–1304. depth circulation in the tropical Atlantic. J. Mar. Res., 59, 281-312. Reid, J. L., 1997: On the geostrophic circulation of the Pacific Ocean: Flow Schmid, C., R. L. Molinari, and S. L. Garzoli, 2001: New observations of the intermediate patterns, tracers, and transports. Progress in Oceanography, 39, 263-352.
Fig.10 Mean velocity vectors (cm s-1) based on weighted average between 2º×2º bin and 3º×3º bin where the weight in the former is two times as the latter’s during from November 2001 to October 2004 in Pacific, with the color scales to reveal speed (unit: cm s-1). (a) In 1000 dB. (b) In 2000 dB.
Estimation of the currentsAfter Q.C., there are 23723 velocity sectors derived by the direct estimation from the trajectory in Pacific during
this time, and there are 3701 velocity sectors should be eliminated mainly due to their measurement cycle for profiles may be out of order. The valid velocities by direct estimation through Q.C. in 1000 and 2000 dB respectively are 14568 and 9155. Further, in order to correct the drifting error for the mid-depth velocities, it is necessary to acquire the times (T
M) at the moment when the float diving and resurfacing. But there are many T
Mnot found in trajectory
file, especially in 1000 dB where there are 6242 fixed pairs missing that information, referred to 42.8%. As to in 2000 dB depth, there are 3430 fixed pairs missing that, namely up to 37.5%.
Fig.9. Distribute the diving positions of pair points to estimate the mid-depth velocity with colors coded by the approximate relative error of current. (a) In 1000 dB. (b) In 2000 dB.
T o ta l
V e loc ity (cm s-1 )
0 5 10 15 2 0 25
Freq
uenc
y (%
)
3
6
9
1 2
1 5
1 8
Rel
ativ
e E
rror
(%)
30
60
90
12 0
15 0
18 0
Fre que ncy o f V e loc ityM ean R e la tive E rro r
1000 dB
0 5 10 15 2 0 25
Freq
uenc
y (%
)
3
6
9
1 2
1 5
1 8
Rel
ativ
e Er
ror (
%)
30
60
90
12 0
15 0
18 0
F requ ency o f V e lo c ityM ea n R e la tive E rro r
2000 dB
Freq
uenc
y (%
)
3
6
9
1 2
1 5
1 8
Rel
ativ
e E
rror
(%)
30
60
90
12 0
15 0
18 0
Fre que ncy o f V e loc ityM ean R e la tive E rro r
Fig.8 To the corrected mid-depth current in Pacific the approximate evaluation about the mean relative error showed by the red line as function of the velocity magnitude with the frequency in blue histogram. Note: the upper is about the current in 1000 dB, the middle is in 2000 dB, and the lower is about all the currents including the two depths.
From the Fig.10., the whole mean flow fields in the two mid-depths in this area are explored. Obviously, there are two significant features in that. Firstly, the strong eastward currents are located at near the west equator in both 1000 dB and 2000 dB. But the former is partial to west and stronger over 16 cm s-1. Secondly, most of all the mean currents outside the tropical equatorial region in the two depths become weak except the western boundary current near the coast of Japan. Meanwhile, the westward current near 40ºN starts from 170ºW and passes through the dateline, and then turn southwest to 160ºE, which is clarified in Fig.10b due to its strength is distinguished with the surrounding velocities. These circulation characteristics are similar to the geostrophic circulations in the Fig.5e and Fig.5g of Reid (1997).
Fig.12 Zonal velocities from the ARGO float with relative error less than 25% from 35ºN to 41ºN (a) in 1000 dB and (b) in 2000 dB. Note: the positive means eastward, otherwise westward. And the velocity with red is over 10 cm s-1, with green is less 5 cm s-1, then the other is with blue.
Fig.13. Meridional velocities from the ARGO float with relative error less than 25% from 35ºN to 41ºN (a) in 1000 dB and (b) in 2000 dB. Note: the positive means northward, otherwise southward. And the velocity with red is over 10 cm s-1, with green
is less 5 cm s-1, then the other is with blue.
Zonal and meridional currents analysis along 38ºN, between 35ºN and 41ºN.
Fig.11. Velocity vectors with relative error less than 25% distribute in the Northwest Pacific in the two mid-depths (left is in 1000 dB, right is in 2000 dB). According to these vectors, their initial trajectories in trajectory file are showed in (b) and (d) respectively. Note: the velocity vector is coded by its trajectory’s color.
Fig.6. Relative error evaluation of v component at 1000 dB distributes in January (upper) and July (lower) for different estimate schemes. The left [(a), (b)] include the float’s drifting error on surface, the right [(c), (d)] exclude the error. Note: black solid line represents 20% isoline.
Fig.5. Relative error evaluation of u component at 1000 dB distributes in January (upper) and July (lower) for different estimate schemes. The left [(a), (b)] include the float’s drifting error on surface, the right [(c), (d)] exclude the error. Black solid line represents 30% isoline.
