4D Variational Data Assimilation Observation Operators 4D Variational Data Assimilation Observation Operators Hernan G. Arango

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: J(  v k ) = ½(  v k ) T  v k + ½(H  x k – d k-1 ) T O -1 (H  x k – d k-1 )  v J =  v k + B T/2 H T O -1 (H  x k – d k-1 ) =  v k + B T/2  x J o =>  v k + W -1/2 L T/2 GS B = SCS => S(GL 1/2 W -1/2 )(W -1/2 L T/2 G)S  x k = B 1/2  v k + x k-1 – x b  x k = B -1/2 (  x k + x k-1 – x b ) 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 = C 1/2 C T/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) = x b (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 Outer Loop Inner Loop

Given a first guess (a forward trajectory)… And given the available data… Incremental, Strong Constraint 4DVar (IS4DVAR)

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: surveyNumber of different time weightNumber of interpolation weight datumObservations 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

Observations NetCDF dimensions: 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 :00:00" ; survey_time:calendar = " 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 :00:00" ; obs_time:calendar = " 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" ; (i1,j1,k1) (i2,j2,k2) VariableID ζ 1 u2 v3 u4 v5 temp6 salt7

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 Dubrovnik (Croatia) Vieste (Italy) Longitude Depth (m)

Parabolic Splines Reconstruction

} } Nx Ny PARALLEL TILE PARTITIONS 8 x 8

East-West MPI Communications With Respect To Tile R Nonlinear Adjoint With Respect To Tile R

North-South MPI Communications Nonlinear Adjoint With Respect to Tile B

Observations

LEO-15 LEO NJSOS Longterm Ecosystem Observatory 30km x 30km New Jersey Shelf Observing System Satellites, Aircraft, Surface RADAR, Glider AUVs 300km x 300km Beginning 2001 RUTGERS THE STATE UNIVERSITY OF NEW JERSEY 3km x 3km 1996-Present Station Field

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): B = S C S = S (G L 1/2 W -1/2 ) (W -1/2 L T/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, L 1/2, 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.

Model/Background Error Correlation (C) Horizontal Hdecay = 100 km Vdecay = 100 m Vertical (implicit)

Model/Background Error Correlation Normalization Coefficients (G) SSHTemperature Bottom Level EAC

job_is4dvar.sh

build.sh

s4dvar.in