4d variational data assimilation observation operators 4d variational data assimilation observation...
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ean M od
earch C o m
r a i n - F o l l o w
M o d e l i n g
4D Variational Data Assimilation4D Variational Data AssimilationObservation OperatorsObservation Operators
4D Variational Data Assimilation4D Variational Data AssimilationObservation OperatorsObservation Operators
Hernan G. ArangoHernan G. Arango
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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)
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Strong Constraint, Incremental 4DVAR
Let’s introduce a new minimization variable v, such that:
J(vk) = ½(vk)Tvk + ½(Hxk – dk-1)TO-1(Hxk – dk-1)
v J = vk + BT/2HTO-1(Hxk – dk-1) = vk + BT/2x Jo => vk + W -1/2LT/2GS
B = SCS => S(GL1/2W -1/2)(W
-1/2LT/2G)S
xk = B1/2vk + xk-1 – xb
xk = B-1/2(xk + xk-1 – xb)
yielding
The gradient of J in minimization-space, denoted v J, is given by:
The background-error covariance matrix can be factored as:
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.
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Basic IS4DVAR Procedure
(1) Choose an x(0) = xb(0)
(2) Integrate NLROMS t [0, ] and save x(t) (NLM at OBS)
(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
Out
er
Loop
Inne
r Lo
op
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• Given a first guess (a forward trajectory)…• And given the available data…
Incremental, Strong Constraint 4DVar(IS4DVAR)
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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.
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4DVAR Observations NetCDF 4DVAR Observations NetCDF FileFile
• 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
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Metadata
Dimensions:
survey Number of different timeweight Number of interpolation weightdatum Observations counter, unlimited dimension
Variables:
Nobs(survey) Number of observations per time survey survey_time(survey) Survey time (days) obs_type(datum) State variable ID associated with observation obs_time(datum) Time of observation (days) obs_lon(datum) Longitude of observation (degrees_east) obs_lat(datum) Latitude of observation (degrees_north) obs_depth(datum) Depth of observation (meters or level) obs_Xgrid(datum) X-grid observation location (nondimensional) obs_Ygrid(datum) Y-grid observation location (nondimensional) obs_Zgrid(datum) Z-grid observation location (nondimensional) obs_error(datum) Observation error, assigned weight obs_value(datum) Observation value
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Observations NetCDFdimensions:
survey = 1 ; tate_variable = 7 ; datum = UNLIMITED ; // (79416 currently)
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" ; obs_value:units = "state variable units" ; 2
1
3
5
4
8
6
7
(i1,j1,k1)
(i2,j2,k2)
Variable IDζ 1u 2v 3u 4v 5
temp 6salt 7
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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
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ROMS GRIDROMS GRID
• Horizontal curvilinear orthogonal coordinates on an Arakawa C-grid
• Terrain-following coordinates on a staggered vertical grid
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Curvilinear Transformation
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Staggered C-Grid, RHO-points
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Staggered C-Grid, U-points
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Staggered C-Grid, V-points
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Vertical Terrain-following Coordinates
Dubrovnik(Croatia)
Vieste(Italy)
Longitude
Depth(m)
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Parabolic Splines Reconstruction
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}
} Nx
Ny
PARALLEL TILE PARTITIONS
8 x 8
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East-West MPI Communications
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Observations
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L o n g B e a c h
B e a c h H a v e n
K ilo m ete rs
0 1 2
L itt le E g g In le t
C a b l e
N o d e AN ode B
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Mullica R iver
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L E O -15 R esearch S ta tion s
B rigan tin eIn let
L an d andW etla nds1 m ete r 3 m ete rs6 m e ters10 m ete rs14 m ete rs16 m e ters18 m eters22 m ete rs
D ep th
LEO-15 LEO NJSOS
Longterm Ecosystem Observatory
30km x 30km 1998-2001
New Jersey ShelfObserving System
Satellites, Aircraft, SurfaceRADAR, Glider AUVs 300km x 300km
Beginning 2001
RUTGERSTHE STATE UNIVERSITY OF NEW JERSEY
3km x 3km1996-Present
Station Field
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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?
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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)
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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.
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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?
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BACKGOUNDERROR COVARIANCE
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Model/Background Error Covariance, B
• Use a generalized diffusion squared-root operator (symmetric) as in Weaver et al. (2003):
B = S C S = S (G L1/2
W-1/2
) (W-1/2
LT/2
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, L1/2
, 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.
• The grid cell area or volume matrix, W-1/2
, is assumed to be time invariant.
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Model/Background Error Correlation (C)
Horizontal
Hdecay = 100 km
Vdecay = 100 m
Vertical (implicit)
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Model/Background Error Correlation NormalizationCoefficients (G)
SSH Temperature
Bottom Level
EAC EAC
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job_is4dvar.sh
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build.sh
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s4dvar.in