M o d e li n
g S
ys
t
Te
-F o llo w i
r rai n
ng
O
R es
e a rc h C o mm u ni
ty
4D Variational Data Assimilation
Observation Operators
Hernan G. Arango
ce
m
em
g
Ocean
Re
i
al
on
an
M o d e lin g S y s
te
ROMS 4DVAR ALGORITHMS
•
•
Strong Constraint
 Conventional (S4DVAR): outer loop, NLM, ADM
 Incremental (IS4DVAR): inner and outer loops, NLM, TLM, ADM (Courtier et al.,
1994)
 IS4DVAR_OLD: inefficient old conjugate gradient algorithm
(is4dvar_ocean_old.h, descent.F)
 IS4DVAR: new conjugate gradient algorithm, CONGRAD, Fisher 1997
(is4dvar_ocean.h, cgradient.h)
 IS4DVAR, LANCZOS: conjugate gradient and Lanczos algorithm, Fisher
1997 (is4dvar_ocean_lanczos.h, cgradient_lanczos.h)
Weak Constraint
 Indirect Representer Method (W4DVAR): inner and outer loops, NLM, TLM,
RPM, ADM (Egbert et al., 1994; Bennett et al, 1997)
 Physical Space Statistical Analysis (W4DPSAS): inner and outer loops, NLM,
TLM, ADM (Courtier, 1997)
Strong Constraint, Incremental 4DVAR
Let’s introduce a new minimization variable v, such that:
xk = B-1/2(xk + xk-1 – xb)
xk = B1/2vk + xk-1 – xb
yielding
J(vk) = ½(vk)Tvk + ½(Hxk – dk-1)TO-1(Hxk – dk-1)
The gradient of J in minimization-space, denoted v J, is given by:
v J = vk + BT/2HTO-1(Hxk – dk-1) = vk + BT/2x Jo => vk + W -1/2LT/2GS
The background-error covariance matrix can be factored as:
B = SCS => S(GL1/2W -1/2)(W -1/2LT/2G)S
where S is the background-error standard deviations, C is the
background-error correlations which can be factorized as C = C1/2CT/2,
G is the normalization matrix which ensures that the diagonal
elements of C are equal to unity, L is a 3D self-adjoint filtering
operator, and W is the grid cell area or volume.
Basic IS4DVAR Procedure
(1)
Choose an x(0) = xb(0)
(2)
Integrate NLROMS t  [0, ] and save x(t)
(NLM at OBS)
Inner Loop
Outer Loop
(a) Choose a x(0)
(b) Integrate TLROMS t  [0, ] and compute J
(TLM at OBS)
(c) Integrate ADROMS t  [0, ] to yield
(ADM forcing at OBS)
(d) Compute
(e) Use a descent algorithm to determine a “down gradient”
correction to x(0) that will yield a smaller value of J
(f) Back to (b) until converged
(3)
Compute new x(0) = x(0) + x(0) and back to (2) until converged
Incremental, Strong Constraint 4DVar
(IS4DVAR)
• Given a first guess (a forward trajectory)…
• And given the available data…
Incremental, Strong Constraint 4DVar
(IS4DVAR)
• Given a first guess (a forward trajectory)…
• And given the available data…
• IS4DVAR computes the changes (or increments) to
the initial conditions so that the forward model fits
the observations.
4DVAR Observations NetCDF
File
•
•
•
•
•
•
•
•
Utility/obs_initial.F
Utility/obs_read.F
Utility/obs_write.F
Utility/obs_scale.F
Utility/obs_depth.F
Utility/extract_obs.F
Adjoint/ad_extract_obs.F
Adjoint/ad_misfit.F
Metadata
Dimensions:
survey
weight
datum
Number of different time
Number of interpolation weight
Observations counter, unlimited dimension
Variables:
Nobs(survey)
survey_time(survey)
obs_type(datum)
obs_time(datum)
obs_lon(datum)
obs_lat(datum)
obs_depth(datum)
obs_Xgrid(datum)
obs_Ygrid(datum)
obs_Zgrid(datum)
obs_error(datum)
obs_value(datum)
Number of observations per time survey
Survey time (days)
State variable ID associated with observation
Time of observation (days)
Longitude of observation (degrees_east)
Latitude of observation (degrees_north)
Depth of observation (meters or level)
X-grid observation location (nondimensional)
Y-grid observation location (nondimensional)
Z-grid observation location (nondimensional)
Observation error, assigned weight
Observation value
dimensions:
Observations NetCDF
survey = 1 ;
tate_variable = 7 ;
datum = UNLIMITED ; // (79416 currently)
Variable
ζ
u
v
u
v
temp
salt
variables:
char spherical ;
spherical:long_name = "grid type logical switch" ;
int Nobs(survey) ;
Nobs:long_name = "number of observations with the same survey time" ;
double survey_time(survey) ;
survey_time:long_name = "survey time" ;
survey_time:units = "days since 2000-01-01 00:00:00" ;
survey_time:calendar = "365.25 days in every year" ;
double obs_variance(state_variable) ;
obs_variance:long_name = "global (time and space) observation variance" ;
obs_variance:units = "squared state variable units" ;
int obs_type(datum) ;
obs_type:long_name = "model state variable associated with observation" ;
obs_type:units = "nondimensional" ;
double obs_time(datum) ;
obs_time:long_name = "time of observation" ;
obs_time:units = "days since 2000-01-01 00:00:00" ;
obs_time:calendar = "365.25 days in every year" ;
double obs_depth(datum) ;
obs_depth:long_name = "depth of observation" ;
obs_depth:units = "meter" ;
double obs_Xgrid(datum) ;
obs_Xgrid:long_name = "x-grid observation location" ;
obs_Xgrid:units = "nondimensional" ;
double obs_Ygrid(datum) ;
obs_Ygrid:long_name = "y-grid observation location" ;
obs_Ygrid:units = "nondimensional" ;
double obs_Zgrid(datum) ;
obs_Zgrid:long_name = "z-grid observation location" ;
obs_Zgrid:units = "nondimensional" ;
double obs_error(datum) ;
obs_error:long_name = "observation error, assigned weight, inverse variance" ;
obs_error:units = "inverse squared state variable units" ;
double obs_value(datum) ;
obs_value:long_name = "observation value" ;
(i1,j1,k1)
obs_value:units = "state variable units" ;
8
5
7
6
4
1
ID
1
2
3
4
5
6
7
2
3
(i2,j2,k2)
Processing
• Use hindices, try_range and inside routines to
transform (lon,lat) to (,)
• Find how many survey times occur within the data
set (survey dimension)
• Count observations available per survey (Nobs) and
assign their times (survey_time)
• Sort the observation in ascending time order and
observation time for efficiency
• Save a copy of the observation file
• Several matlab scripts to process observations
ROMS GRID
• Horizontal curvilinear orthogonal coordinates
on an Arakawa C-grid
• Terrain-following coordinates on a staggered
vertical grid
Curvilinear Transformation
Staggered C-Grid, RHO-points
Staggered C-Grid, U-points
Staggered C-Grid, V-points
Vertical Terrain-following Coordinates
Vieste
(Italy)
Dubrovnik
(Croatia)
Depth
(m)
Longitude
Parabolic Splines Reconstruction
PARALLEL TILE PARTITIONS
8x8
Ny
} Nx
East-West MPI Communications
AsendE
TILE L
ArecvE
ArecvW
AsendW
TILE R
receive
Jend
Jend
Nonlinear
With Respect
To Tile R
i-2 i-1
i
i+1
i-2 i-1
i
i+1
Jstr
Jstr
send
Istr
V iL =
V iL+1 =
TILE L
Iend
R
Vi
V iR+1
AsendE
Istr
V iR-2 =
V iR-1 =
ArecvE
ArecvW
Iend
L
V i-2
V iL-1
TILE R
AsendW
ad_send
Jend
Jend
With Respect
To Tile R
Adjoint
i-2 i-1
i
i+1
i-2 i-1
i
i+1
Jstr
Jstr
ad_receive
ad_V iL-2 =
Istr
Iend
L
R
ad_V i-2 + ad_V i-2 ; ad_ViR-2 = 0
ad_V iL-1 = ad_V iL-1 + ad_V iR-1 ; ad_ViR-1 = 0
Istr
ad_V iR
=
ad_V iR
+
Iend
L
i
L
; ad_V i
=0
ad_V iR+1 = ad_V iR+1 + ad_V iL+1 ; ad_V iL+1 = 0
North-South MPI Communications
V jB+2 = V jT+2
ad_V jT+2 = ad_V jT+2 + ad_V jB+2 ; ad_V jB+2 = 0
V jB+1 = V jT+1
ad_V jT+1 = ad_V jT+1 + ad_V jB+1 ; ad_V jB+1 = 0
TILE T
TILE T
With Respect
to Tile B
Jend
Jend
j+2
j+2
AsendS
Nonlinear
j
j-1
send
j-1
receive
TILE B
ArecvS
Istr
Iend
ad_receive
Iend
ad_send
TILE B
j+2
j+2
ArecvN
j+1
Jend
j
ArecvS
Istr
AsendS
j+1
Jstr
j
Jend
AsendN
j-1
ArecvN
j+1
j
AsendN
j-1
Jstr
Jstr
Istr
V jT
Istr
Iend
B
= Vj
V jT-1 = V jB-1
ad_V jB
= ad_V jB
Iend
+
T
j
T
; ad_V j
=0
ad_V jB-1 = ad_V jB-1 + ad_V jT-1 ; ad_V jT-1 = 0
Adjoint
j+1
Jstr
Observations
LEO-15
THE STATE UNIVERSITY OF NEW JERSEY
RUTGERS
ul
li c
a
Ri
ve
NJSOS
Kilometers
LEO-15 Research Stations
M
LEO
Beach
Haven
r
0
1
2
Long Beach
Great
Field
Field
Station
Station
Bay
Longterm Ecosystem Observatory
Little Egg
Inlet
C
ab
le
Depth
Land and
Wetlands
1 meter
3 meters
6 meters
10 meters
14 meters
16 meters
18 meters
22 meters
Node A
Node B
Brigantine
Inlet
LEO-15
Research
Area
New Jersey Shelf
Observing
System
Brigantine
3km x 3km
1996-Present
30km x 30km
1998-2001
Satellites,
Aircraft, Surface
RADAR, Glider AUVs
300km x 300km
Beginning 2001
Assumptions
• All scalar observations are assumed to be at
RHO-points.
• All vector observations are assumed to be rotated
to curvilinear grid, if applicable. Vector
observations are always measured at the same
location.
• All observation horizontal locations are assumed
to be in fractional curvilinear grid coordinates.
• Vertical locations can be in fractional levels (1:N)
or actual depths (negative values).
• Removal of tidal signal?
• Filtering of non-resolved processes?
Observation Operators
• Currently, all observations must be in terms of
model state variables (same units):
– 2D configuration: zeta, ubar, vbar
– 3D configuration: zeta, u, v, T, S, …
• There is not a time interpolation of model
solution at observation times:
time - 0.5*dt < ObsTime < time + 0.5*dt
• There is not observations quality control
(screening) inside ROMS, except ObsScale.
• No observation constraints are implemented
(Satellite SST measurements)
Observation Interpolation
• Only spatial linear interpolation is coded.
• If land/sea masking, the interpolation
coefficients are weighted by the mask.
• If shallower than z_r(:,:,N), observations are
assigned to the surface level.
• If deeper than z_r(:,:,1), observations are assigned
to bottom level.
Recommedations
• Create a NetCDF file for each observation type.
• Use a processing program to meld NetCDF
observation files (nc_4dvar_meld.m).
• Keep a master copy of each observation file, in
case that you are running your application at
different resolutions.
• Decimation of observations. Finite volume
representation?
BACKGOUND
ERROR COVARIANCE
Model/Background Error Covariance, B
• Use a generalized diffusion squared-root operator
(symmetric) as in Weaver et al. (2003):
1/2
B = S C S = S (G L
-1/2
W
-1/2
) (W
T/2
L
G)
• The normalization matrix, G, ensure that the diagonal
elements of the correlation matrix, C, are equal to unity.
They are computed using the exact (expensive) or
randomization (cheaper) methods.
• The spatial convolution of the self-adjoint filtering operator,
1/2
L , is split in horizontal and vertical components and
discretized both explicitly and implicitly.
• The model/background standard deviation matrix, S, is
computed from long (monthly, seasonal) simulations.
-1/2
• The grid cell area or volume matrix, W
time invariant.
, is assumed to be
Model/Background Error Correlation (C)
Horizontal
Vertical (implicit)
Vdecay = 100 m
Hdecay = 100 km
Model/Background Error Correlation Normalization
Coefficients (G)
SSH
EAC
Temperature
EAC
Bottom Level
job_is4dvar.sh
set Di r =/ home/ ar ango/ ocean/ t oms/ r eposi t or y/ Pr oj ect s/ doubl e_gyr e
# Copy nonl i near model i ni t i al condi t i ons f i l e.
cp - p ${ Di r } / Dat a/ gyr e3d_bck. nc gyr e3d_i ni . nc
# 10 year mean
# Set backgr ound- er r or covar i ance st andar d devi at i ons f i l e.
set STDname=. . / Dat a/ gyr e3d_st d. nc
# Set backgr ound- er r or covar i ance nor mal i zat i on f act or f i l e
set NRMname=. . / Dat a/ gyr e3d_nr m_100k100i . nc
# Set obser vat i ons f i l e.
#set OBSname=gyr e3d_zSST_obs. nc
set OBSname=gyr e3d_zuvTS_obs. nc
# Get a cl ean copy of t he obser vat i on f i l e.
# i mpor t ant si nce t hi s f i l e i s modi f i ed.
Thi s i s r eal l y
cp - p ${ Di r } / OBS/ $OBSname .
# Modi f y S4DVAR t empl at e i nput scr i pt and speci f y above f i l es.
set I S4DVAR=i s4dvar . i n
i f ( - e $I S4DVAR) t hen
/ bi n/ r m $I S4DVAR
endi f
cp s4dvar . i n $I S4DVAR
subst i
subst i
subst i
subst i
subst i
subst i
t ut e
t ut e
t ut e
t ut e
t ut e
t ut e
$I S4DVAR
$I S4DVAR
$I S4DVAR
$I S4DVAR
$I S4DVAR
$I S4DVAR
ocean_st d. nc
ocean_nr m. nc
ocean_obs. nc
ocean_hss. nc
ocean_l cz. nc
ocean_mod. nc
$STDname
$NRMname
$OBSname
gyr e3d_hss. nc
gyr e3d_l cz. nc
gyr e3d_mod. nc
#subst i t ut e $I S4DVAR ocean_hss. nc Runs/ gyr e3d_hss_01. nc
#subst i t ut e $I S4DVAR ocean_l cz. nc Runs/ gyr e3d_l cz_01. nc
build.sh
# anal yt i cal sour ce f i l es, i f any. See t he t empl at es i n User / Funct i onal s.
set env MY_HEADER_DI R
${ MY_PROJECT_DI R}
#set env MY_ANALYTI CAL_DI R
# Put t he bi nar y t o execut e i n t he f ol l owi ng di r ect or y.
set env BI NDI R
${ MY_PROJECT_DI R}
# Put t he f 90 f i l es i n a pr oj ect speci f i c Bui l d di r ect or y t o avoi d conf l i ct
# wi t h ot her pr oj ect s.
set env SCRATCH_DI R
${ MY_PROJECT_DI R} / Bui l d
# Go t o t he user s sour ce di r ect or y t o compi l e. The opt i ons set above wi l l
# pi ck up t he appl i cat i on- speci f i c code f r om t he appr opr i at e pl ace.
cd ${ MY_ROMS_SRC}
# St op i f act i vat i ng bot h MPI and OpenMP at t he same t i me.
i f ( ${ ?USE_MPI } & ${ ?USE_OpenMP} ) t hen
echo " You cannot act i vat e USE_MPI and USE_OpenMP at t he same t i me! "
exi t 1
endi f
# Remove bui l d di r ect or y.
i f ( $cl ean == 1 ) t hen
make cl ean
endi f
# Compi l e ( t he bi nar y wi l l go t o BI NDI R set above) .
i f ( $par al l el == 1 ) t hen
make $NCPUS
el se
make
endi f
s4dvar.in
Hdecay( i sFsur ) ==Hdecay( i sVst r ) == 100. 0d+3
V- moment um st r ess
Hdecay( i sUvel ) == 100. 0d+3
Hdecay( i sVvel ) == 100. 0d+3
Hdecay( i sTvar ) == 100. 0d+3 100. 0d+3
Hdecay( i sTsur ) == 100. 0d+3 100. 0d+3
! sur f ace
!
!
!
!
3D U- moment um
3D V- moment um
1: NT t r acer s
1: NT sur f ace t r acer s f l ux
! Ver t i cal , i sot r opi c decor r el at i on scal es ( m) .
Vdecay( i sUvel ) == 100. 0d0
Vdecay( i sVvel ) == 100. 0d0
Vdecay( i sTvar ) == 100. 0d0
! 3D U- moment um
! 3D V- moment um
! 1: NT t r acer s
100. 0d0
! I n t he weak- const r ai nt al gor i t hms, t he f or ci ng er r or covar i ance mat r i x
! i s est i mat ed by scal i ng t he backgr ound er r or covar i ance mat r i x. These val ues
! pr ovi de t he scal i ng f act or s f or each st at e var i abl e.
Cf scal
Cf scal
Cf scal
Cf scal
Cf scal
Cf scal
Cf scal
Cf scal
Cf scal
!
!
!
!
!
e( i
e( i
e( i
e( i
e( i
e( i
e( i
e( i
e( i
sFsur )
sUbar )
sVbar )
sUst r )
sVst r )
sUvel )
sVvel )
sTvar )
sTsur )
=
=
=
=
=
=
=
=
=
Pol ynomi al coef f i
nor mal i zat i on of
i s t he pol ynomi al
coef f i ci ent . The
expect ed. Coef f i
npN2 =
Pcoef _N2 =
0. 3d0
0. 3d0
0. 3d0
0. 3d0
0. 3d0
0. 3d0
0. 3d0
0. 3d0
0. 3d0
!
!
!
!
!
!
!
!
!
0. 3d0
0. 3d0
ci ent s
t r acer
or der
val ues
eci ent
f or Br unt - Vai sal a f r equency squar ed ( N2) used f or
mi sf i t cost f unct i on. The f i r st val ue i n each l i ne
wher eas t he second val ue i s t he pol ynomi al expansi on
can be ent er ed i n any or der , [ 0: npN2] val ues ar e
ar e der i ved f r om a t ypi cal Nor t h At l ant i c pr of i l e.
9
2D var i abl e at RHO- poi nt s
2D var i abl e at U- poi nt s
2D var i abl e at V- poi nt s
sur f ace U- moment um st r ess
sur f ace V- moment um st r ess
3D var i abl e at U- poi nt s
3D var i abl e at V- poi nt s
NT t r acer s
NT sur f ace t r acer s f l ux
! degr ee of pol ynomi al
ORDER
0
1
2
3
COEFFI CI ENT
0. 00014377915637d+00
8. 22721559660467d- 07
2. 01838193105922d- 09
2. 50856113444129d- 12