A comparison of sequential data assimilation schemes for ocean prediction with HYCOM Twin Experiments A. Srinivasan, University of Miami, Miami, FL E. P. Chassignet, COAPS, Florida State University, Tallahassee, FL O. M. Smedstad, QinetiQ North America, Stennis Space Center, MS, USA T. M. Chin, University of Miami, Miami, FL F. Counillon, Nansen Center, Bergen, Norway J. M. Brankart and P. Brasseur LEGI, Grenoble, France W. C. Thacker, AOML, NOAA, Miami, FL J. A. Cummings, NRL Monterey, CA LOM-29 - June 2 nd, 29 p.1/31
Assimilation Schemes for HYCOM Multivariate Optimal Interpolation (MVOI) J. A. Cummings; Lorenc 1981, Daley 1991, Cummings 25 Ensemble Optimal Interpolation (EnOI) F. Counillon; Evensen 23, Oke 22 Ensemble Reduced Order Information Filter (EnROIF) T. M. Chin; Chin etal 1999,1 Fixed basis variant of the SEEK filter (SEEK) J. M. Brankart and P. Brasseur; Pham et al. 1998 Objective: To assess the performance of these schemes in identically configured twin experiments assimilating synthetic altimeter and surface temperature obs. Domain & Model Configuration: 1/12 Gulf of Mexico nested in the 1/12 North Atlantic LOM-29 - June 2 nd, 29 p.2/31
Data Update Equation Common linear formula for updating the model-forecast x f to obtain data-analysis x a : x a = x f + K(y Hx f ) (1) y is the data to be assimilated H is the observation operator K is an optimization parameter often called the gain matrix K = P f H T (HP f H T + R) 1 (2) P f is the forecast error covariance R is the observation error covariance Main difference among the four methods is in the numerical representation of P f LOM-29 - June 2 nd, 29 p.3/31
MVOI P f is separated into several variances and correlations C h = (1 + s h )exp( s h ) (3) C v = (1 + s v )exp( s v ) (4) C f = (1 + s f )exp( s f ) (5) C b = C f C h C v (6) where C h and C v are the horizontal and vertical correlations and s h and s v are scaled distances. analysis on z levels (42 zlevels between -25 m) state variables: U, V, T, S and intf. Pressure dynamical method (Cooper-Haines 1996) for vertical projection simultaneous 3D analysis obs errors uncorrelated LOM-29 - June 2 nd, 29 p.4/31
EnOI In EnOI, the forecast covariance matrix is essentially the sample covariance of an ensemble of model states P f EnOI = 1 M 1 M m=1 (x f m x f ) ( x f m x f ) T (7) where x f m is the m th sample of the forecast ensemble, x f is the ensemble mean, and M is the number of samples. The analysis update is computed as: x a = x f + αp f H T (αhp f H T + R) 1 (y Hx f ) (8) where α (, 1] is a parameter introduced to allow different weights for the ensemble and measurements. State Varaibles: UT, VT, UB, VB, PB, DP, T, S, SSH, SST Vertical Projection: static correlations; analysis in observation space LOM-29 - June 2 nd, 29 p.5/31
EnROIF EnROIF uses a Markov random field (MRF) to model the forecast error process as e(i, j) = ( i, j) N γ(i, j, i, j) e(i i, j j) + δ(i, j) (9) where N specifies a set of local grid locations, γ is the regression coefficient (small matrix), and δ(i, j) is a white noise with unit variance. Computation for the state analysis x a in ROIF is performed as L a (x a x f ) = H T R 1 (y Hx f ) (1) where the sparsely banded analysis information matrix L a = L + H T R 1 H is numerically inverted. State Varaibles: DP, SSH, SST Vertical Projection: static correlations; 2D vertically decoupled analysis LOM-29 - June 2 nd, 29 p.6/31
Fixed basis SEEK In SEEK filter, P f, is assumed to be of low rank, M and is represented by dominant modes of empirical orthogonal functions (EOFs) as P f SEEK = 1 M 1 M m=1 S f ms ft m (11) where S f m, m = 1,..., M, are the M most dominant EOF modes. In the SEEK analysis the Kalman Gain is rewritten using the Sherman-Morrrison-Woodberry matrix identity as K = S f [I + ( HS f ) R 1 ( HS f ) T ] 1 ( HS f ) R 1 (12) State Varaibles: UT, VT, UB, VB, PB, DP, T, S, SSH, SST Vertical Projection: static correlations; analysis in reduced space LOM-29 - June 2 nd, 29 p.7/31
Twin Experiments: Initial States 3 o N Truth State Aug 3 1999 SSH Assimilation Run State Aug 3 1999 SSH 3 o N Latitude Latitude 96 o W 92 o W 88 o W 84 o W 8 o W 96 o W 92 o W 88 o W 84 o W 8 o W temp. sect. 25.44 N temp. sect. 25.44 N depth [m] 5 1 depth [m] 5 1 15 15 95 9 85 8 Longitude 95 9 85 8 Longitude LOM-29 - June 2 nd, 29 p.8/31
Twin Experiments: Data 3 o N altimeter 3 Aug. 9 Sep. 1999 3 o N MCSST 3 Aug. 1999 Latitude Latitude 96 o W 92 o W 88 o W 84 o W 8 o W Longitude 96 o W 92 o W 88 o W 84 o W 8 o W 76 o W Longitude altimeter data Aug Dec. 1999 1 MCSST data Aug Dec. 1999 1 data count 5 data count 5 5 1 Days 5 1 Days LOM-29 - June 2 nd, 29 p.9/31
Twin Experiments: SSH.22.2.18 SSH (Forecast) RMS Error MVOI EnOI SEEK EnROIF.16 RMS error [m].14.12.1.8.6.4.2 2 4 6 8 1 12 14 Days LOM-29 - June 2 nd, 29 p.1/31
Twin Experiments: SSH Sea Surface Elevation (m) (Forecast Truth) Oct 18, 1999 (day 5) MVOI EnOI 3 o N 3 o N 96 o W 92 o W 88 o W 84 o W 8 o W 96 o W 92 o W 88 o W 84 o W 8 o W SEEK EnROIF 3 o N 3 o N 96 o W 92 o W 88 o W 84 o W 8 o W 96 o W 92 o W 88 o W 84 o W 8 o W.2.2.2.2 LOM-29 - June 2 nd, 29 p.11/31
Twin Experiments: SSH Sea Surface Elevation (m) (Forecast Truth) Dec 7, 1999 (day 1) MVOI EnOI 3 o N 3 o N 96 o W 92 o W 88 o W 84 o W 8 o W 96 o W 92 o W 88 o W 84 o W 8 o W SEEK EnROIF 3 o N 3 o N 96 o W 92 o W 88 o W 84 o W 8 o W 96 o W 92 o W 88 o W 84 o W 8 o W.2.2.2.2 LOM-29 - June 2 nd, 29 p.12/31
Twin Experiments: SST.8.7 SST (Forecast) RMS Error MVOI EnOI SEEK EnROIF.6 RMS error [degc].5.4.3.2.1 2 4 6 8 1 12 14 Days LOM-29 - June 2 nd, 29 p.13/31
Twin Experiments: SST Sea Surface Temperature (degc) (Forecast Truth) Oct 18, 1999 (day 5) MVOI EnOI 3 o N 3 o N 96 o W 92 o W 88 o W 84 o W 8 o W 96 o W 92 o W 88 o W 84 o W 8 o W SEEK EnROIF 3 o N 3 o N 96 o W 92 o W 88 o W 84 o W 8 o W 96 o W 92 o W 88 o W 84 o W 8 o W.2.2.2.2 LOM-29 - June 2 nd, 29 p.14/31
Twin Experiments: Unobserved Variables [ u,v, t,s] RMS error [m/s] U velocity (Forecast) RMS Error.35.3.25.2.15.1.5 5 1 15 RMS error [m/s] V velocity (Forecast) RMS Error.35.3.25.2.15.1.5 5 1 15 MVOI EnOI SEEK EnROIF Temperature (Forecast) RMS Error 2 Salinity (Forecast) RMS Error.35 RMS error [degc] 1.5 1.5 5 1 15 Days RMS error [psu].3.25.2.15.1.5 5 1 15 Days LOM-29 - June 2 nd, 29 p.15/31
Twin Experiments: Temperature(degC) section 25.4 N (Forecast Truth) Aug 31, 1999 (day 2) MVOI EnOI depth [m] 2 4 2 4 6 95 9 85 Longitude 6 95 9 85 Longitude SEEK EnROIF depth [m] 2 4 2 4 6 95 9 85 6 95 9 85 1 1 1 LOM-291 - June 2 nd, 29 p.16/31
Twin Experiments: Temperature(degC) section 25.4 N (Forecast Truth) Oct 18, 1999 (day 5) MVOI EnOI depth [m] 2 4 2 4 6 95 9 85 Longitude 6 95 9 85 Longitude SEEK EnROIF depth [m] 2 4 2 4 6 95 9 85 6 95 9 85 1 1 1 LOM-291 - June 2 nd, 29 p.17/31
Twin Experiments: depth of 2 deg isotherm (m) (Forecast Truth) Aug 31, 1999 (day 2) MVOI EnOI 3 o N 3 o N 96 o W 92 o W 88 o W 84 o W 8 o W 96 o W 92 o W 88 o W 84 o W 8 o W SEEK EnROIF 3 o N 3 o N 96 o W 92 o W 88 o W 84 o W 8 o W 96 o W 92 o W 88 o W 84 o W 8 o W 1 1 1 LOM-29 1 - June 2 nd, 29 p.18/31
Twin Experiments: depth of 2 deg isotherm (m) (Forecast Truth) Oct 18, 1999 (day 5) MVOI EnOI 3 o N 3 o N 96 o W 92 o W 88 o W 84 o W 8 o W 96 o W 92 o W 88 o W 84 o W 8 o W SEEK EnROIF 3 o N 3 o N 96 o W 92 o W 88 o W 84 o W 8 o W 96 o W 92 o W 88 o W 84 o W 8 o W 4 2 2 4 6 8 4 2 2 4 LOM-29 6 8 - June 2 nd, 29 p.19/31
Twin Experiments: 3D U&V u velocity day 2 u velocity day 5 depth [m] 2 4 6 8.2.4 2 MVOI 4 EnOI 6 SEEK EnROIF 8.1.2 v velocity day 2 v velocity day 5 depth [m] 2 4 6 2 4 6 8.2.4 rms error [m/s] 8.1.2 rms error [m/s] LOM-29 - June 2 nd, 29 p.2/31
Twin Experiments: 3D T&S Temperature day 2 Temperature day 5 depth [m] 2 4 6 8 2 4 RMS error [degc] Salinity day 2 2 MVOI 4 EnOI 6 SEEK EnROIF 8 1 2 RMS error [degc] Salinity day 2 depth [m] 2 4 6 8.2.4 RMS error [psu] depth [m] 2 4 6 8.1.2 RMS error [psu] LOM-29 - June 2 nd, 29 p.21/31
Twin Experiments: Layer Thickness Mean Layer Thickness 3 Layer 5 6 Layer 6 Truth MVOI 2 1 5 1 15 Layer 7 6 4 2 4 2 5 1 15 Layer 8 6 4 2 EnOI SEEK EnROIF depth [m] 5 1 15 Layer 9 25 2 5 1 15 Layer 1 3 25 15 5 1 15 Layer 11 36 34 32 3 5 1 15 Days 2 5 1 15 Layer 12 45 4 35 5 1 15 Days LOM-29 - June 2 nd, 29 p.22/31
Twin Experiments: Layer Thickness Mean Layer Thickness 55 Layer 13 75 Layer 14 Truth MVOI 5 45 5 1 15 Layer 15 11 1 9 7 65 6 5 1 15 Layer 16 11 1 EnOI SEEK EnROIF depth [m] 8 5 1 15 Layer 17 1 8 6 4 5 1 15 Layer 19 5 45 4 9 5 1 15 Layer 18 22 2 18 16 5 1 15 Layer 2 65 6 55 35 5 1 15 Days 5 5 1 15 Days LOM-29 - June 2 nd, 29 p.23/31
Twin Experiments: Net Transports.6.4 Net transports across 26 N MVOI EnOI SEEK EnROIF.2 Transport (SV).2.4.6.8 2 4 6 8 1 12 14 Days LOM-29 - June 2 nd, 29 p.24/31
Twin Experiments: State Variables Impact of State Variables.2.18.16 EnOI_142 SEEK_154 SEEK_155 SEEK_156 SEEK_157 MVOI_12 MVOI_116 RMS error [m].14.12.1.8.6.4.2 2 4 6 8 1 12 Days LOM-29 - June 2 nd, 29 p.25/31
Twin Experiments: Ensemble Size and Covariance Rank Impact of Ensemble Size.2.18 EnOI_146 EnOI_142 SEEK_155 SEEK_162 SEEK_163 SEEK_151.16 RMS error [m].14.12.1.8.6.4.2 2 4 6 8 1 12 Days LOM-29 - June 2 nd, 29 p.26/31
Twin Experiments: Localization Impact of Localization.2.18 EnOI_142 EnOI_144 EnOI_145 SEEK_153 SEEK_154.16 RMS error [m].14.12.1.8.6.4.2 2 4 6 8 1 12 Days LOM-29 - June 2 nd, 29 p.27/31
Twin Experiments: Data Window Impact of Data Window.2.18 MVOI_12 MVOI_121 EnOI_142 SEEK_155 SEEK_164.16 RMS error [m].14.12.1.8.6.4.2 2 4 6 8 1 12 Days LOM-29 - June 2 nd, 29 p.28/31
Twin Experiments: Assimilation Strength Impact of Assimilation Strength.2 EnOI_142 EnOI_143 EnOI_141.18.16 RMS error [m].14.12.1.8.6.4.2 2 4 6 8 1 12 Days LOM-29 - June 2 nd, 29 p.29/31
Computational Costs Scheme Scalability Memory Req. Typical Wall Clock/update MVOI O(m 3 /V ) 22 min EnOI O(m 3 ) nn 31 min EnROIF sn 8 min SEEK O(r 3 ) nn 14 min n ==> no of members in the ensemble N ==> size of the forecast model s ==> size of the MRF neighborhood r ==> covariance rank (== no of members in our experiments) v == no of analysis volumes in MVOI all experiments were done on 8 core/2.6 GHz Intel/12GB RAM) LOM-29 - June 2 nd, 29 p.3/31
Summary All schemes are able to reduce errors in both observed and unobserved variables. 3D multivariate (MVOI, EnOI, SEEK) are clearly better than decoupled 2D scheme used currently in EnROIF Best performance is obtained when all state variables are used in the estimation space Subsurface correction based on correlations(enoi and SEEK) are better balanced and seem more effective than the dynamical method(mvoi) The nearly identical performance of EnOI and SEEK is both expected and reassuring Several parameters (ensemble size, covariance rank, radius of localization etc.) have to be explored for a given problem to find a good solution that is also cost effective. LOM-29 - June 2 nd, 29 p.31/31