Assimilation of high-frequency radar currents in theLigurian Sea

Alexander Barth(1), Jacopo Chiggiato(2),

Aida Alvera Azcarate(1), Baptiste Mourre(2),

Jean-Marie Beckers(1), Jochen Horstmann(2), Michel Rixen(2)

(1) GHER, University of Liege, Belgium, (2) NURC, Italy.

COSS-TT, Miami, 10-12 January 2012

ModelI ROMS nested in

MediterraneanOcean ForecastingSystem

I 1/60 degree resolu-tion and 32 verticallevels

I Currents: Western& Eastern CorsicanCurrent, NorthernCurrent, inertial os-cillation

I Two WERA HFradar systems (Pal-maria, San Rossore)by NATO UnderseaResearch Centre(NURC)

Observations

I Frequency of ν = 12.359 MHz and coupled to a wave length of λb = 12.13 m,

I Radial currents are used for the assimilation

I Azimuthal resolution of 6 degrees

I Currents are averaged over 1 h

Radial currents on 2010-07-06 21:30 relative to the Palmaria site: left panel showsWERA measurements and right panel shows ROMS results without assimilation.

Observations

Radial currents on 2010-07-06 01:30 relative to the San Rossore site: left panel showsWERA measurements and right panel shows ROMS results without assimilation.

Observation operator

I Radial currents are extracted by:

uHF =kb

1− exp(−kbh)

∫ 0

−hu(z) · er exp(kbz)dz (1)

• kb = 2πλb

• er is the unit vector pointing in the direction opposite to the location ofthe HF radar site

• positive values: current away from the system

• essentially represent an average over the upper meters.

I Smoothed in the azimuthal direction by a diffusion operator to filter scales smallerthan 6 degrees

Model errors covariance

I Estimated by ensemble simulation where uncertain aspect of the model are per-turbed

I Perturbed zonal and meridional wind forcing

I Perturbed boundary conditions (elevation, velocity, temperature and salinity)

I Perturbed momentum equation (ε)

du

dt+ Ω ∧ u = − 1

ρ0∇hp+

1

ρ0∇ · Fu +∇h ∧ ε ez (2)

• where ∇h = ex∂∂x

+ ey∂∂y

• does not create horizontal convergence or divergence (linked to barotropicwaves)

• can create mesoscale flow structures (absent or misplaced)

Ensemble spin-up

06/29 06/30 07/01 07/02 07/03 07/04 07/05 07/060

0.02

0.04

0.06

0.08

0.1

time

m/s

surface velocity spread

instantaneous velocity

daily averaged velocity

I Ensemble of IC is created by a 7 day ensemble integration starting from the sameIC but with perturbed forcing (ensemble spin-up)

I Spin-up should create mesoscale circulation features

Velocity spread

Surface velocity ensemble spread after 7 days (m/s)

8.5 9 9.5 10 10.5 11

42.6

42.8

43

43.2

43.4

43.6

43.8

44

44.2

44.4

0.030

0.050

0.070

0.100

0.150

0.200

0.300

I Velocity spread after 7 days

I Largest uncertainties near eddies

Spatial correlation

8.5 9 9.5 10 10.5 11

42.6

42.8

43

43.2

43.4

43.6

43.8

44

44.2

44.4

−0.4

−0.2

0

0.2

0.4

0.6

0.8

I Correlation of temperature at a specific point (magenta circle) and other surfacegrid points

I Resulting length-scale is about 50 km

Spatial correlation

8.5 9 9.5 10 10.5 11

42.6

42.8

43

43.2

43.4

43.6

43.8

44

44.2

44.4

−0.2

0

0.2

0.4

0.6

0.8

I Correlation of zonal velocity at a specific point (magenta circle) and other surfacegrid points

I Resulting length-scale is about 10 km

I Adequately observing surface velocity would require measurements with higherspatial resolution that the resolution of temperature measurements

Temporal correlation

−15 −10 −5 0 5 10 15−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

delta t (days)

co

rre

latio

nautocorrelation of radial velocity (9 E,43 N )

periodicity of 16 h (period of inertial oscillations is 17.6 h)

Data assimilation scheme

I Time dimension embedded in estimation vector x

I Different definitions of estimation vector are possible:

• x = (model trajectory), i.e. model state at all time instances

• x = (uncertain forcing fields), here IC, BC, wind and stochastic error termat all time instances

• x = (model trajectory, uncertain forcing fields)

I The optimal x is given by the Kalman analysis (using non-linear observationoperators as in Chen and Snyder (2007)):

xa = xb + A (B + R)−1 (yo − h(xb)) (3)

I where the matrices A and B are covariances estimated from the ensemble.

A = cov(xb, h(xb)) =⟨(x− 〈x〉) (h(x)− 〈h(x)〉)T

⟩(4)

B = cov(h(xb), h(xb)) =⟨(h(x)− 〈h(x)〉) (h(x)− 〈h(x)〉)T

⟩(5)

where 〈·〉 is the ensemble average.

Smoother scheme

I For a linear model and an infinite large ensemble, equation (4) minimizes,

J(x) = (x− xb)TPb−1(x− xb) + (yo − h(x))TR−1(yo − h(x)) (6)

or

J(x) = (x− xb)TPb−1(x− xb) +

∑n

(yon − (h(x)n))TRn

−1(yon − (h(x)n)) (7)

where n refers to the indexed quantifies at time n. This is the cost function fromwhich 4D-Var and Kalman Smoother can be derived.

I Approach is closely related to Ensemble Smoother (van Leeuwen, 2001), 4D-EnKF (Hunt et al., 2007) and AEnKF (Sakov et al., 2010) where model tra-jectories instead of model states are optimized and to the Green’s method withstochastic “search directions”

Observations

Analysis

IC, BC, wind, momentum error

radial surface velocity

model

IC, BC, wind, momentum error

radial surface velocity

model

IC, BC, wind, momentum error

radial surface velocity

model

Free model run

… … …

radial surface velocity

IC, BC, wind, momentum error

radial surface velocity

model

pert

urba

tions Ensemble run

x h(.) h(x)

Twin experiment

Scheme of a twin experiment:

I model is run with initial conditions (IC), boundary conditions (BC), forcing fields(e.g. here winds fields) that are assume to be the ”true” solution.

I pseudo-observations are extracted from this simulation.

I perturbation are applied to IC, BC and forcing fields.

I Based on those perturbed fields and the extracted pseudo-observation we deter-mine if the ”true” solution can be recovered.

Variable RMS(xf ,xt) RMS(xa,xt)

Temperature 0.080 0.067Salinity 0.0063 0.0057u-wind 0.61 0.40v-wind 0.60 0.54

I RMS for temperature, salinity and currents is a volume average.

I Assimilation window is here 48 hours.

Estimation of trajectory versus estimation of forcing fields

10−2

10−1

100

101

0.4

0.5

0.6

0.7

0.8

0.9

1

representation error

no

rma

lize

d e

rro

r

Optimizing trajectory

Optimizing forcing

I Both approaches equivalent for linear system (and additive noise)

I Unrealistic “ensemble extrapolation” when too small observation errors are used→ model trajectory and forcing fields are inconsistent

Error statistics for Palmaria Site

Without assimilation(positive values: currentaway from the magentadot)

With assimilation

Forecasts

0 10 20 30 40 50 60 70 80 90 1000.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

lead time (hours)

RM

S d

iffe

rence (

m/s

)

Free

Forecast with DA

I Impact of data assimilation on current forecast

I Comparison with surface currents from Palmaria

I HF radar assimilation improves the strength of the Northern Current and thisimprovement persists for some time.

Simulation with atmospheric model (WRF)

I blue arrows: WRF 10mwind vectors, red arrows:in situ wind measurementsfrom ICOADS (Inter-national ComprehensiveOcean-Atmosphere DataSet).wind_LS2.mp4

I 3 WRF domains at 30, 10,3.33 km resolution (two-waynesting). The limit of thosedomains are shown in black.

I 30-km grid model nested(one-way) into the GlobalForecast System

I 28 vertical layers

Model results with different wind forcings

I Total RMS dif-ferences (m/s):

Pal. Ros.COSMO 0.14 0.11

WRF 0.13 0.14

Figure 1: Radial surface current RMS difference

Conclusions

I Embedding the time dimension into the state vector leads to a smoother scheme(which is very simple to implement)

I Smoother schemes can be used to estimate the optimal model trajectory orforcing field

I Both approaches are not equivalent for non-linear systems or multiplicative noise

I The challenge is to make consistent analyzes

I derive “optimal” perturbation first → rerun the model with corrected forcing

I The source code of smoother schemes is available at http://modb.oce.ulg.

ac.be/alex or by email (a.barth@ulg.ac.be).

I Future: what can we learn about the model error from the correction terms?

References

Barth, A., A. Alvera-Azcarate, K.-W. Gurgel, J. Staneva, A. Port, J.-M. Beckers, andE. V. Stanev, 2010: Ensemble perturbation smoother for optimizing tidal boundaryconditions by assimilation of High-Frequency radar surface currents - application tothe German Bight. Ocean Science, 6, 161–178, doi:10.5194/os-6-161-2010.URL http://www.ocean-sci.net/6/161/2010/

Barth, A., A. Alvera-Azcarate, J.-M. Beckers, J. Staneva, E. V. Stanev, andJ. Schulz-Stellenfleth, 2011: Correcting surface winds by assimilating High-Frequency Radar surface currents in the German Bight. Ocean Dynamics, 61, 599–610, doi:10.1007/s10236-010-0369-0.URL http://hdl.handle.net/2268/83330

Burchard, H. and K. Bolding, 2002: GETM – a general estuarine transport model.Scientific documentation. Technical Report EUR 20253 EN, European Commission.

Chen, Y. and C. Snyder, 2007: Assimilating Vortex Position with an Ensemble KalmanFilter. Monthly Weather Review , 135, 1828–1845.

Hunt, B. R., E. J. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation forspatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230,112–126.

Kalnay, E., 2002: Atmospheric Modeling, Data Assimilation and Predictability . Cam-bridge University Press, 1 edition.

Sakov, P., G. Evensen, and L. Bertino, 2010: Asynchronous data assimilation with theEnKF. Tellus, 62A, 24–29.

Staneva, J., E. V. Stanev, J.-O. Wolff, T. H. Badewien, R. Reuter, B. Flemming,A. Bartholoma, and K. Bolding, 2009: Hydrodynamics and sediment dynamics inthe German Bight. A focus on observations and numerical modelling in the EastFrisian Wadden Sea. Continental Shelf Research, 29, 302–319.

van Leeuwen, P. J., 2001: An Ensemble Smoother with Error Estimates. MonthlyWeather Review , 129, 709–728.