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Theoretical Analyses and Numerical Tests of Variational Data Assimilation wit
h Regularization Methods
Huang SixunHuang Sixun P.O.Box 003, Nanjing 211101,P.R.ChinaP.O.Box 003, Nanjing 211101,P.R.China Email: Email: [email protected]
Canada-China Workshop on Industrial Mathematics
HongKong Baptist University, 2005
It is well known that numerical predictiIt is well known that numerical prediction of atmospheric and oceanic motions is reon of atmospheric and oceanic motions is reduced to solving a set of nonlinear partial diduced to solving a set of nonlinear partial differential equations with initial and boundafferential equations with initial and boundary conditions, which is often called direct prry conditions, which is often called direct problems. In the recent years, a variety of metoblems. In the recent years, a variety of methods have been proposed to boost accuracy hods have been proposed to boost accuracy of numerical weather prediction, such as vaof numerical weather prediction, such as variational data assimilation(VAR), etc. riational data assimilation(VAR), etc. VAR is using all the available information (e. VAR is using all the available information (e.g., observational data from satellites, radars,g., observational data from satellites, radars, and GPS, etc.) to determine as accurately as and GPS, etc.) to determine as accurately as possible the state of the atmospheric or oceapossible the state of the atmospheric or oceanic flow.nic flow.
ContentsContents
Part A Theoretical aspects
A.1 What’s the variational data assimilation?
A.2 Idea of adjoint method of VAR
A.3 3D-VAR
A.4 4-D VAR
B.1 variational assimilation for one-dimensional ocB.1 variational assimilation for one-dimensional oc
ean temperature modelean temperature model B.2 ENSO cycle and parameters inversionB.2 ENSO cycle and parameters inversion B.3 Assimilation of tropical cyclone(TC) tracksB.3 Assimilation of tropical cyclone(TC) tracks B.4 Inversion of radarB.4 Inversion of radar B.5 Inversion of satellite remote sensing data and its nB.5 Inversion of satellite remote sensing data and its n
umerical calculationumerical calculation B.6 Generalized variational data assimilation with noB.6 Generalized variational data assimilation with no
n- differential termn- differential term B.7 Variational adjustment of 3-D wind field B.7 Variational adjustment of 3-D wind field B.8 The model of GPS dropsonde wind-finding systeB.8 The model of GPS dropsonde wind-finding syste
mm
Part B Applications
A.1 What’s the variational data assimilation?
Talagrand 1995 Assimilation: using all the available in
formation, determine as accurately as possible the state of the atmospheric or oceanic flow
Variational Data Assimilation: study assimilation through variational analytical method(adjoint method)
Data assimilation undergoes the Data assimilation undergoes the following stagesfollowing stages
Stage 1 Objective AnalysesStage 1 Objective Analyses Interpolating observational data Interpolating observational data at irregular observational points to at irregular observational points to regular grid points by statistical regular grid points by statistical methods, which would be taken as methods, which would be taken as initial fieldsinitial fields
Stage 2 InitializationStage 2 Initialization Filtering high frequency Filtering high frequency
components in initial fields so as to components in initial fields so as to reduce prediction errorsreduce prediction errors
Stage 3 3D –VARStage 3 3D –VAR
Adjusting initial field Adjusting initial field xx00 so that so that xx00 is compatiblis compatible with observations e with observations yy and background and background xxb b , i.e. t, i.e. to make the following cost function minimumo make the following cost function minimum
HH----observation operator( nonlinear operator)----observation operator( nonlinear operator) yy---observational field---observational field xxbb- --- background field- --- background field BB---covariance matrix of background---covariance matrix of background OO---covariance matrix of observation---covariance matrix of observation
T 1 T 10 0 0
1 1( ( ) ) ( ( ) ) min
2 2b bJ[x ] (x x ) B (x x ) H x y O H x y
Stage 4 4D-Stage 4 4D-VARVAR
Case 1Case 1 State equationsState equations
FF is the classical PDO is the classical PDO Observation Observation XobsXobs [0,T] [0,T] Cost functionalCost functional
C---linear operatorC---linear operator It means that gives the “true value of tIt means that gives the “true value of t
he field at the point (in space and /or in timhe field at the point (in space and /or in time) of observatione) of observation
This is optimal control of PDEsThis is optimal control of PDEs
0
,
t T
0t 0
XF(t, X)
tX X X
，
2
2
0 ((0, ) )
1[ ] min!
2obs
L TJ X C X X
C X
Case 2Case 2
ModelModel
ww((tt) is assumed to have 0 mean and covarianc) is assumed to have 0 mean and covariance matrix error e matrix error QQ((tt))
information information background fields background fields xxbb
covariance matrix of covariance matrix of background errorbackground error
00t
x F(x(t)) w(t)
x x
T0 0 bE( ) )b(x x )(x x B
observational data observational data y y
ee((tt) is assumed to have 0 mean and cova) is assumed to have 0 mean and covariance matrix riance matrix OO((tt). ). ee((tt) is white proces) is white process, and also assumed to be uncorrelated s, and also assumed to be uncorrelated with the model error with the model error ww((tt).).
cost functional cost functional
( ( )) ( )y H x t e t
TT 1 T 10 0 0 0
T 1
0
1 1[ ] ( ( )) (( ( ))d
2 21
( ) ( )d min!2
b b
T
J x (x x ) B (x x ) y H x O y H x t
w t O w t t
A.2 Idea of adjoint method of A.2 Idea of adjoint method of VARVAR
As an example, we consider the inversion of IBVC for the following problem
----- observational data
the cost functional is
0t 0
( , ), 0
(1)
given
xF t x t T
tx x
x
obsx
2
T 2obs0 L ( )0
1J[x ] x x dt min!
2
Idea: solving an Idea: solving an optimization problem by optimization problem by
descent algorithmdescent algorithm
kxxkk
kJx
00)(x 0
10
k0x
0x
J0x
iteration
Approximate solutions
convergence
This can done in the following three steps:
step1.Derivation of the following tangent linear model(TLM)
Let u be disturbed to u U
. The corresponding
solutions of Eqns (2) are X and X~
.
Setting 0
ˆ limX X
X
, yielding TLM for X
0
ˆˆ ˆ ˆ( , ) , , homogeneous ,t
XF t X X X u BVC
t
from which X is solved as utRX ˆ)0,(
, where ( ,0)R t is the
resolvent operator of TLM.
step2 Determining of the directional derivative of J
along the direction u
The G - Calculus is
0 0
* *
0 0
ˆˆ ˆ ˆ[ ; ] ( , ) ( , ) ( , ( ,0) )
ˆ ˆ( ( ,0)( ), ) ( ( ,0)( ) , ).
T Tobs obs
u
T Tobs obs
J u u J u X X X dt X X R t u dt
R t X X u dt R t X X dt u
Then,
*
0
( ,0)( )T
obsuJ R t X X dt
which *( ,0)R t is the adjoint operator of R .
step3 Introducing the adjoint system
The adjoint system of TLM is
*( , ) , 0, adjoint .obst T
PF t X P X X P BVC
t
Using the relationship between ),(* tR and the resolvent
operator ),( tS of the adjoint eqnations, we have
),(* tR = ),( tS .
Then, we get
( 0 )u J P .
Observations
obsX
0( , ), , t
XF t X X u BVC
t
*( , )
0, adjoint
obs
t T
PF t X P X X
tP BVC
)(tP (0)u J P
1 ( ) .k
k ku kU
U U J
][][ 1 kk UJUJ
[ ]kJ U
ok
First assimilation on
[0, ]T , giving optimal
value
*
0tX U
*0
( , )
, t
XF t X
t
X U BVC
Prediction ( )
[0,2 ]
X t
t T
Second assimilation on
]2,[ TT giving an optimal
value t TX *1U
Using obsX
on ]2,[ TT
Prediction ( )
[ ,3 ]
X t
t T T Third assimilation on
[2 ,3 ]T T giving an optimal
value 2t TX *2U
continue
Descent algorithm
Compare J
Some key difficulties of adjoint method of VAR
(1) Ill-posedness
During iteration, the cost functional oscillates,
and decreases slowly so as to lead too low accurac
y. The reason: ill-posedness
(2) Error of BVC
The boundary 1 2
1 is closed boundary 1
1u g
, 1g is given by
measure
2 is open boundary，where there exists inflow
and outflow, but BVC 2
2u g
is obtained from
nesting grid model, which is artificial. Therefore, the
boundary error will cause prediction error.
2
1 1
2
(3) Local observations In some cases, especially in the oceans, observations are not incom
plete, e.g., observations are obtained from ships, sounding balloons, which will lead to calculation unstable, and therefore is worth studying further.
(4) Variational data assimilation with non-differentiable te
rm (on-off problem)
The adjoint method holds only with differentiable term; for system
s containing non-differentiable physical processes( called as “on-off”
) , a new method must be developed.
If H is linear operator , we obtain the optimal estimate
And the error estimate matrix is
[ ]J u[ ]J u
T 1 T 11 1( ( ) ) ( ( ) ) min!
2 2b bJ[x ] (x x ) B (x x ) H x y O H x y
A3 3D -VARA3 3D -VAR
1 1 1ˆ [ ] ( )T Tb bx x B H O H H O y Hx
21 1
2( ) ( )T TJ
P B BH O HBH HBx
Some key difficulties in Some key difficulties in 3D-VAR 3D-VAR
H H is an on observational operatoris an on observational operator
Prob.1: How to find Prob.1: How to find H H ??
Prob.2: Prob.2: H H is not a surjection. How to deal with it ?is not a surjection. How to deal with it ?
BB is non-positive is non-positive
OO is non-positive is non-positive
The hypothesis of unbiased errors is a difficult one in practice, because there often The hypothesis of unbiased errors is a difficult one in practice, because there often
as significent biases in the background fields(caused by biases in the forecast as significent biases in the background fields(caused by biases in the forecast
model) and in the observations ( or in the observational operators)model) and in the observations ( or in the observational operators)
The hypothesis of uncorrelated errorsThe hypothesis of uncorrelated errors
HH is a nonlinear operator, which leads to is a nonlinear operator, which leads to J J = min! is not unique, i.e. ill-posedness= min! is not unique, i.e. ill-posedness
A4 4D-VARA4 4D-VAR ModelModel
Model
00
)())((
xx
twtxFx
tt
)())()(( tQtwtwE T )(tw is while prosess
Information 1. )())(( tetxHy )())()(( tOteteE T )(te is while prosess
2. background bx BxxxxE Tbb )))((( 00
dtxHytOtxHyxxBxxxJT
t
Tb
Tb
0
))()(()))(((2
1)()(
2
1][ 1
01
00
min))(()()()(2
1
00
1 dtwxFxdttwtOtwT
t
TT
t
T
)(
)()(
00tBxx
tQxFx
btt
Adjoint model is
0
))()((])(
[][ 1
Tt
TT xHytOx
xH
x
F
If we suppose ,then the direct equIf we suppose ,then the direct equations and adjoint equations are not coations and adjoint equations are not coupled, except at the initial time tupled, except at the initial time t00
0Q t
B.1 variational assimilation for one-dimensional ocB.1 variational assimilation for one-dimensional oc
ean temperature modelean temperature model B.2 ENSO cycle and parameters inversionB.2 ENSO cycle and parameters inversion B.3 Assimilation of tropical cyclone(TC) tracksB.3 Assimilation of tropical cyclone(TC) tracks B.4 Inversion of RadarB.4 Inversion of Radar B.5 Inversion of satellite remote sensing data and its nB.5 Inversion of satellite remote sensing data and its n
umerical calculationumerical calculation B.6 Generalized Variational Data Assimilation for NoB.6 Generalized Variational Data Assimilation for No
n- Differential Systemn- Differential System B.7 Variational Adjustment of 3-D Wind Field B.7 Variational Adjustment of 3-D Wind Field B.8 The model of GPS Dropsonde wind-finding systeB.8 The model of GPS Dropsonde wind-finding syste
mm
Part B Applications
B.1 variational assimilation for one-dimensional ocean temperature model
The one-dimensional heat-diffusion model for describing the vertical distribution of
sea temperature over time is,
Here is sea temperature, is the vertical eddy diffusion coefficient,
is the sea water density, is the sea water specific heat capacity, is the light
diffusion coefficient, is the depth of ocean upper layer, is the transmission
component of solar radiation at sea surface, is the net heat flux at sea surface.
It is known that there exists the unique solution of the model if the initial boundary
condition and the model parameters are known and smooth.
0
0
0
00
0 0
( ) exp( ), ( , ) (0, ) (0, )
( ),
( ) [ ] , 0
p
t
z z Hp p
IT TK z z t H
t z z C
T U z
IT Q t TK K
z C C z
( , )T T t z ( , )K K t z
0pC
H 0I
( )Q t
0( , )K I
Assume , are known constants, the initial boundary conditions , and model parameters , are not known exactly, e.g., they have unknown errors and need to be determined by data assimilation. Now a set of observations of sea temperature is given on the whole domain. A convenient cost functional formulation is thus defined as
Where is a stable functional and is a
regularization parameter. The problem is: Find the optimal
initial boundary conditions and model parameters
, such that J is minimal.
0 and pc( ), ( )U z Q t 0( , ), ( )K t z I t
obsTJ
2 2 20 0 0 0 0
1 1[ , , , ] ( ) d d ( , )( ) d d
2 2
H H
obs
TJ U K Q I T T z t K t z z t
z
2
0 0
1( , )( ) d d
2
H TK t z z t
z
( ( ), ( ))U z Q t
0( , ), ( )K t z I t
pC pC
5 10 15 20 25 30Iteration Number
0.2
0.3
0.4
0.5
0.6
J
0
001.0
Iteration number
Decreasing of the cost functional J with iteration number
5 10 15 20 25 30Iteration Times
0.054
0.056
0.058
0.06
rm
No@K-KtD
0
001.0
iteration number
KE
The norm of eddy diffusion coefficient error True
KE K K
B.2 ENSO cycle and parameters B.2 ENSO cycle and parameters inversioninversion
ENSO: ENSO:
The acronym of theThe acronym of the EEl l NNino -ino -SSouthern outhern OOscillationscillation phenomenon which is the most prominent international oscillation phenomenon which is the most prominent international oscillation of the tropical climate system.of the tropical climate system.
The phase of the Southern Oscillation on El The phase of the Southern Oscillation on El NinoNino High temperature over eastern Pacific; High High temperature over eastern Pacific; High
surface pressure over the western and low surface pressure over the western and low surface pressure over the south-eastern tropical surface pressure over the south-eastern tropical Pacific coincide with heavy rainfall, unusually Pacific coincide with heavy rainfall, unusually warm surface waters, and relaxed trade winds in warm surface waters, and relaxed trade winds in the central and eastern tropical pacificthe central and eastern tropical pacific
The phase of the Southern Oscillation The phase of the Southern Oscillation on La Ninaon La Nina Surface pressure is high over the eastern Surface pressure is high over the eastern
but low over the western tropical Pacific, but low over the western tropical Pacific, while trades are intense and the sea while trades are intense and the sea surface temperature and rainfall are low in surface temperature and rainfall are low in the central and eastern tropical Pacificthe central and eastern tropical Pacific
La Nina
: Sea Surface Temperature Anomaly (SSTA): Sea Surface Temperature Anomaly (SSTA)
: thermocline depth anomaly : thermocline depth anomaly
: a monotone function of the air-sea coupling coefficient: a monotone function of the air-sea coupling coefficient
: external forcing: external forcing
: constants . : constants .
0
0
1 2 3 1
2
0
0
( )
(2 )
t t
t t
T a T a h a T T h f
h b h T f
T T
h h
T
hb
( 1, 2)if i
, ia
A nonlinear dynamical system for A nonlinear dynamical system for ENSO:ENSO:
obsTObtain the time series of T and h (denoted by and from the observational data set TAO (Tropical Atmosphere and Oceans)
obsh
Observation:
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000-1
0
1
2
SSTA-
--H20A
-1 -0.5 0 0.5 1 1.5 2-0.5
0
0.5
1
SSTA
H20A
(a)
(b)
1991 1992 1993 1994 1995 1996 1997 1998 1999 2000-1
0
1
2
-1 -0.5 0 0.5 1 1.5 2-0.5
0
0.5
1
SSTA
H20A
(a)
(b)
The time series of The time series of T T (solid (solid line) and line) and h h (dotted line); (dotted line);
The phase orbit of The phase orbit of T T and and h h (Running clockwise as the (Running clockwise as the
time goes on)time goes on)
Now, we seek optimal parameter and external Now, we seek optimal parameter and external forcing , such that the solution satisfies forcing , such that the solution satisfies
:: the terminal control termthe terminal control term
: the control parameter.: the control parameter.
b 1 2( ), ( )f t f t
( , )T h
0
2 2
0 0, 1 2
2 2
1, , , ,
2
min2
e
e
t obs obs
t
obs obst
J b T h f f T T W h h dt
T T W h h
2 2
2 e
obs obstT T W h h
-1.5 -1 -0.5 0 0.5 1-1
-0.5
0
0.5
SSTA
H20A
(c)
Blue : the observed valueBlue : the observed valuered : the value predicted by the original modelred : the value predicted by the original modelblack: the value predicted by the improved model whenblack: the value predicted by the improved model whengreen: the value predicted by the improved model whengreen: the value predicted by the improved model when
0 0.4
B. 3 Assimilation of tropical cyclone(TC) tracks
A TC is regarded as a point vortex, whose motion satisfies
Here , , are the velocity and coordinates of TC center respectively, and is the force exerted on TC, but don’t include the Coriolis force . Suppose that over the interval, the observational TC track is .
0
0
0 0 0 00 0 0 0
,
( ) ( )
- ( ) ( )
, , ,
x
y
t t t t
dx dyu v
dt dtdu
f y v F tdtdv
f y u F tdtx x y y u u v v
0 t T ( , )u v ( , y)x
( , )x yF F
( ), ( )obs obsX X t Y Y t
Now, the goal is: to determine the optimal initial velocity
and forces , such that the corresponding solution
makes the functional
minimal. are referred to as the regularization parameters,
is the restraint parameter at the terminal.
0 0, u v
( ), ( )x yF t F t
( ), ( ), ( ), ( )x t y t u t v t
2 2 2 210 0 0 0
2 2 2 22
0
1[ , , ( ), ( )] [( ( ) ( )) ( ( ) ( )) ] [ ]
2 2
[( ) ( ) ] [( ( ) ( )) ( ( ) ( )) ]2
T T
x y obs obs
T
obs obs
J u v F t F t x t X t y t Y t dt u v dt
du dvdt x T X T y T Y T
dt dt
1 2, 0
0
Table1. Main Characteristics of 4 TCs
TCs Beginning Time
Ending Time
Beginning Position
Ending Position
Tracks
9804 25 Aug. 14:00
7 Sept. 14:00
24.1N 132.9E
48.1N 166.0E
Zigzag Track
9806 16 Sept. 20:00
20 Sept. 12:00
21.2N 132.8E
29.5N 120.9E
Northeast- Westward
9807 18 Sept. 14:00
23 Sept. 2：00
16.1N 118.8E
41.8N 143.5E
Northeastward
0004 6 Jul. 8:00
11 Jul. 8:00
19.6N 119.9E
38.3N 123.7E
Northeast- Northward
Inversion of TC 9804 Inversion of TC 9804
tracktrack
128 133 138 143 148 153 158 163 16820
25
30
35
40
45
50
longitute
latitu
de
true pathassimilation path
,X YF F
,X YF F
0 20 40 60 80 100 120-2
-1.5
-1
-0.5
0
0.5
1
1.5x 10
-3
time
Fx F
yFxFy
Retrieved forces for TC 9804 ,X YF F
B.4 Inversion of RadarB.4 Inversion of Radar
The Definition of RadarThe Definition of Radar
Radar is an acronym for “Radar is an acronym for “RaRadio dio DDetecting etecting AAnd nd R Ranginganging”.”.
Radar systems are widely used in air-traffic Radar systems are widely used in air-traffic control, aircraft navigation, marine navigation control, aircraft navigation, marine navigation and weather forecasting.and weather forecasting.
The Definition of Doppler The Definition of Doppler RadarRadar
Doppler radarDoppler radar ：： the radar the radar can detect both reflectivity can detect both reflectivity intensity and radial velocity intensity and radial velocity of the moving objects with tof the moving objects with the “Doppler effect”.he “Doppler effect”.
The right graphic show ：
The forming process of reflectivity
Weather radars send out radio waves from an antenna
Objects scatter or reflect some of the scatter or reflect some of the
radio waves back to the antennaradio waves back to the antenna
More waves sent back, higher reflectivity the object have; less
waves, lower reflectivity.
The forming process of radial velocity
Frequency change in returning radio waves are also are also measured.measured.
Waves from an object Waves from an object moving toward the moving toward the antenna change to a antenna change to a higher frequencyhigher frequency ; ; moving away change moving away change to a lower frequency.to a lower frequency.
The computer useThe computer uses the frequency chs the frequency changes to show direanges to show directions and speeds ctions and speeds of the winds.of the winds.
The 2-D horizontal wind is governed by the following conservation of reflectivity factor of Radar and of mass in the polar coordinates
where are time , redial distance and azimuth respectively, is the reflectivity factor of Radar, are redial and azimuthal velocity respectively.
is eddy diffusion coefficient. is given by diagnosis. The inversion domain is
2 2
2 2 2
0
1 1( )
( )( , , ) ............. continuous equation
( , , ) ( , )
( , , ) ( , ), ( , , ) ( , ), 1, 2i i
r
r
t
r r i i
v v kt r r r r r r
vrvD t r
r r rt r r
t r t t r t r i
, ,t r ( , , )t r
,rv v
k ( , , , )w
D t r zz
1 2 1 2(0, ) ( , ) ( , ) T r r
Suppose that the observational data are known, the aim is to determine 2-D wind and . This is a ill-posed problem. We introduce the following functional
where 、 、 、 and are weight coefficients.
1 2 3 4
2 211
222
2 233
( , , ), ( , , ), min!
1( ) ( ) observational control
2 2( )
( ) weak restraint2
( ) regul2
r
obs obsr r
r
r
J v t r v t r k
d v v d
vrvD d
r r r
v v d
J J J J
J
J
J
2 2544
arization term
( ) ( ) background restraint2 2
v v d S S dJ
2 431
,obs obsrv
,rv v k
true
true vortex wind field
125.00k / 4t T retrieved
retrieved vortex wind field
124.47k / 4t T
The error between the retrieved vortex wind field
and true wind field
/ 4t T
B.5 Inversion of satellite remote sensing data and its numerical calculation
•With the use of techniques in nonlinear problems, the IDP
(improved discrepancy principle) method has been proposed to
the optimal smooth factor (parameter ) in the inversion
process of atmosphere profiles from satellite observation. This
method has also been used to inverse atmospheric parameters
from the observation of new generation geostationary
operational environmental satellite(GOES-8). Results show
that this method is more accurate than that in use.
If the atmosphere scatter effect is ignored, then the infrared radiance of the earth atmosphere system that goes to satellite sensor is
R ---- ---- the spectral radiance of a channel(given) B ---- Plank function ---- the total atmosphere transmittance abov
e the pressure level ---- surface emissivity ---- reflected radiation of the sun ---- surface value of physical quantities
0 0(0, ) (1 ) ( , )
s sP P
s s sR B Bd p Bd p p R
R
s
X=(T,q,Ts,,…) ,Ts ---surface temperature
q--- water vapormixed ratios
Tikhonov
regularization discrepancy
principle
0 0(0, ) (1 ) ( , )
s sP P
s s sR B Bd p Bd p p R
Nonlinear equation
F(X) = Y
Initial guess
X0
Linearized
0 0
0 0
( )
( )
F X X Y Y
Y F X
1 1( , )X
linearize the eqn.at X1
1 1
1 1
( )
( )
F X X Y Y
Y F X
2 2( , )X
0 1 2 3 4 5
100
200
300
400
500
600
700
800
900
1000
Pre
ss
ure
/ H
Pa
RMSE / K
First guessempirical retrievalIDP retrieval
0 5 10 15 20 25 30 35
100
200
300
400
500
600
700
800
900
1000
Pre
ss
ure
/ H
Pa
RMSE / K
First guessempirical retrievalIDP retrieval
B.6 Generalized Variational Data AssiB.6 Generalized Variational Data Assimilation with Non-Differential Termmilation with Non-Differential Term
The simple ordinary differential equation with non-differential term:The simple ordinary differential equation with non-differential term: （（ Zou X.Zou X. ，， 19931993 ）： ）：
Here Here is Heaviside function. is Heaviside function.
Problem:Problem:
Supposing the equation has a unique solution and the observation is Supposing the equation has a unique solution and the observation is known, our goal is to find the initial value and critical value that known, our goal is to find the initial value and critical value that can make functional can make functional
0x
0 0
c
t
dxF x H x x G x
dtx x
H
cxobsx
2
0 0, min!
T obscJ x x x x dt
Step1. Introduce a weak form:Step1. Introduce a weak form:
Step2. The weak form is disturbed as the following :Step2. The weak form is disturbed as the following :
Here is the time at that time Here is the time at that time
0 0
( ) ( ) ( ) ( )c
T T Tdxp t dt F x p t dt G x p t dt
dt
0 0
( ) ( ) ( ) ( )c
T T Tdxp t dt F x p t dt G x p t dt
dt
0 0 0ˆ ˆ ˆ
ˆ ˆc
T TT
T
c c c
F xdpx p t x dt x p t dt
dt xG x
x p t dt G x px
c c cx x
Step3. Introduce the adjoint system:Step3. Introduce the adjoint system:
Step4. Obtain the gradients of the functional:Step4. Obtain the gradients of the functional:
0
c obsc c
c
t T
dp t G xF GH x x p t H x x x x
dt x x F x
p
00
c
cx c
c
cx c
c
G xJ p p
F x
G xJ p
F x
0q cq J0q cq JInitial valueInitial value
Times Times of the of the iteratiiterati
onon
Results of the Results of the inversioninversion
TestTest11
0.050.05 0.250.25 1.0211.02166
1919 0.250.25002002
0.46650.466599
9.4902e-9.4902e-009009
TestTest22
0.050.05 0.550.55 0.6430.6438585 66 0.250.25 0.45760.4576 8.6545e-8.6545e-
011011
TestTest33
0.420.42 0.550.55 0.4050.4050606 1212 0.250.25
0010010.45730.4573
881.6551e-1.6551e-
009009
TestTest44
0.420.42 0.250.25 0.1490.1490808 22
0.370.37381381 0.250.25 0.075750.07575
66
Experiments:Experiments:
The track of the cost functional descending in the The track of the cost functional descending in the process of iteration process of iteration
0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
qc
q0
Experiment 1Experiment 2Experiment 3Experiment 4
B.7 Variational Adjustment of 3-D Wind B.7 Variational Adjustment of 3-D Wind FieldField
The vertical velocity of an air parcel is a very important quantity in atmos
pheric sciences. However, its magnitude is so small that it can not be measure
d accurately by meteorological apparatus, but rather inferred from the fields m
easured directly, such as the horizontal velocity, temperature , pressure, and so
on.
Three commonly used methods for inferring the vertical velocity are the k
inematical method, the adiabatic method, and the variational analysis met
hod (VAM) suggested by Sasaki(1969,1970). However, It turns out that Sasa
ki’s VAM can not adjust 3-D wind field well for observational wind containin
g high frequency components, even if filtering is applied. Here we combine V
AM with the regularization method and filtering to deal with this problem (G
VAM).
Suppose that is an observational horizontal wind field. Our aim is to seek an analytic field satisfying the equation of continuity
and make the functional
minimal. Here and
satisfies the boundary conditions
( , )u v
( , , )u v w
0u v w
x y p
* 2 2 2[ , , ] ( ) ( ) 2 ( ) d d d ( ) d d du v w v u
J u v w u u v v x y p x y px y p x y
3,( , , ) ( )b Tx y p p p R w
,b T
b Tp p p pw w w w
The discrepancy between the observed horizontal
wind velocity and the true value in the plane
2/)( bT ppp , where the vertical coordinate is
velocity (same to all figures below). (a)
229.0184 ~
2
L
Tuu ; (b) 229.0184 ~
2
L
Tvv
0
1
2
3
0
1
2
3-3
-2
-1
0
1
2
3
0
1
2
3
0
1
2
3-3
-2
-1
0
1
2
3
(b)(a)
0
1
2
3
0
1
2
3-3
-2
-1
0
1
2
3
0
1
2
3
0
1
2
3-3
-2
-1
0
1
2
3
The discrepancy between the analytic horizontal
wind velocity by VAM and the true field.
(a) 220.0319
2
L
Tuu ; (b) 220.0725
2
L
Tvv
(a) (b)
0
1
2
3
0
1
2
3-0.5
0
0.5
0
1
2
3
0
1
2
3-3
-2
-1
0
1
2
3
The discrepancy between the analytic horizontal
wind velocity by GVAM and the true field under
the first kind of boundary conditions. (a)
2
2 26.8556e-10T
Lu u ; (b) 2
2 29.5751e-10T
Lv v .
(a) (b)
0
1
2
3
0
1
2
3-3
-2
-1
0
1
2
3
0
1
2
3
0
1
2
3
-3
-2
-1
0
1
2
3
The discrepancy between the analytic horizontal
wind velocity by GVAM and the true field under
the second kind of boundary conditions.
(a) 220.0156
2
L
Tuu ; (b) 220.0173
2
L
Tvv .
(a)
(b)
B.8 The model of B.8 The model of GPS Dropsonde wind-finding GPS Dropsonde wind-finding
systemsystem
Introduction to Vaisala Dropsonde RD93
2
2
2
2
2
2
2 2 22 3
2 2 22 3
2 21 2 3
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) (
fx
fy
fz
dv dyd x dx dx dzd fx fx fy fzdt dt dt dt dtdt
dvd y dy dydx dzd fy fx fy fzdt dt dt dt dtdt
dv dyd z dz dx dzd fz fx fydt dt dt dt dtdt
c v v v v
c v v v v
c v v v v
2)fz
'
( )
1fm m g
m m
2
mm m
2
3 8( )fd
m m
1
12 1 1 1
1
1
1
12 1 1 1
1
1
11 2 1 2 1
1
( )( )
( )( )
( )( )
r
r
dxx
dtdx
u t x u Vdtdu
udtdy
ydtdy
ydtdy
v t y v Vdtdv
vdtdz
zdtdz
t z Vdtddt
0 0| , | ,ft t t fx x x x
0 0| , |ft t t fy y y y
0 0| , | 0ft t tz z z
0 0 0 0| , | , | 0ft t t tu u v v
2 2 2 21 1 1 1 2 1 2 1 1
0
1[ , , , , ] [( ) ( ) ( ) ( )
2
ft
obsobs obs obsJ u v w x x y y z z x x
2 22 1 1 3 1 1( ) ( ) ]obs obsy y z z dt
2 2 2 21 1 1 1 2 1 2 1 1
0
1[ , , , , ] [( ) ( ) ( ) ( )
2
ft
obsobs obs obsJ u v w x x y y z z x x
2 2 2 2 22 1 1 3 1 1 4 1 1 5 1( ) ( ) ( ) ]obs obsy y z z x y z dt
Now we can get the following adjoint equations and initial boundary conditions
1 obsdPx x
dt
2 1 121 2 1 1 1 2 1 1 1[ ( ) ] [ ( )( ) ]r r r
dPP P V x u V Q y v x u V
dt
12 2 1 1 2 1 1 4 1[ ( )( ) ] ( )obs
rR z w x u V x x x
2 1 132 1 1 1 2 1 1 1[ ( ) ] [ ( ) ( )]r r r
dPP v x u V Q y v V x u
dt
12 2 1 1[ ( )( ) ]rR z w x u V
1 obsdQy y
dt
1 2 122 1 1 1 1 2 1 1 1[ ( )( ) ] [ ( ) ]r r r
dQP x u y v V Q Q y v V v
dt
12 2 1 1 2 1 1 5 1[ ( )( ) ] ( )obs
rR z w y v V y y y
And it’s boundary conditions are：：
2 0| 0tP 2 | 0ft tP 3 | 0
ft tP 2 0| 0tQ 2 | 0ft tQ
3 | 0ft tQ 2 0| 0tR 2 | 0
ft tR 3 | 0ft tR
the gradients of J ：
1 3 2 2u J P P 1 2 2 3v J Q Q
1 2 2 3w J R R
1 2 2 1[ ( ) [ ( ) ]r rJ P x u V Q y v V
2 2 1[ ( ) ]rJ R z V
For our numerical computations, the following
conditions are assume: 31.2 /f kg m , 5 21.49 10 /m s , and
the dropsonde parameters are
1d m ,380 /o kg m , 29.8 /g m s , the initial high of
dropsonde are 2000m.
（ a ） x-axis position
Fig. the position of dropsonde
（ b ） y-axis position
Fig the position of dropsonde
（ c ） z-axis position
Fig. the position of dropsonde
Fig. the comparison between two kinds of cost functional decrease
(a) Compared without stabilized fuction ， 2
1 2.71041Td d L
c c
dcFig. the comparison between the true value, initial value and retrieval value of in the x-axis
(b) Compared with stabilized fuction ， 2
1 0.538083Td d L
c c
dcFig. the comparison between the true value, initial value and retrieval value of in the x-axis
(a) Compared without stabilized function ， 2
1 1.288726Td d L
c c
dcFig. the comparison between the true value, initial value and retrieval value of in the z-axis
(b) Compared with stabilized fuction ， 2
1 0.376459Td d L
c c
Fig. the comparison between the true value,initial value and retrieval value of in the z-axis dc
(a) Compared without stabilized fuction ，
Fig. the comparison between the true value,initial value and retrieval value of wind (x direction)
2
1 17.93860T
Lu u
(a) Compared with stabilized fuction ，
2
1 4.880377T
Lu u
Fig. the comparison between the true value,initial value and retrieval value of wind (x direction)
(a) Compared without stabilized fuction ， 2
1 32.979271T
Lv v
Fig. the comparison between the true value,initial value and retrieval value of wind(y direction)
2
1 6.921217T
Lv v
Fig. the comparison between the true value,initial value and retrieval value of wind(y direction)
(b) Compared without stabilized fuction ，
2
1 2.897249T
Lw w
(a) Compared without stabilized fuction ，
Fig. the comparison between the true value,initial value and retrieval value of updraft flow
2
1 1.839537T
Lw w
Fig. the comparison between the true value,initial value and retrieval value of updraft flow
(b) Compared without stabilized fuction ，
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