Miscellaneous Thoughts on Ocean Data Assimilation

Transcription

1 Miscellaneous Thoughts on Ocean Data Assimilation Mike Dowd Dept of Mathematics & Statistics (and Dept of Oceanography) Dalhousie University, Halifax, Canada STATMOS Summer School in Data Assimila5on, Boulder, Colorado, May

2 DA for Oceanography (Physical and Biological) What are the problems? What are the key features? What are some Data Assimilation (DA) issues and approaches used or being considered (and that we have not really covered here)?

3 The State Space Framework for DA x t = d(x t 1,θ) + e t x t ~ [x t,θ x ] or t 1 y t = h(x t,φ) + v t y t x t ~ [y t,φ x t ] dynamics observa)ons t = 1,,T General Features: - A spa)o- temporal problem. High dimensional state space. Actual degrees of freedom? - State vector indexes space as defined on model grid, but could switch space index with )me index and run sequen)ally (later ). - Dynamics relies on numerical ocean models (d operator). Markovian - > sequen)al - Observa)ons indirectly related to the ocean state (h operator), are ojen themselves models (with complex error distribu)ons and sampling protocols) Goal is to es5mate the ocean state x t and the model parameters θ.

4 Ocean Physics/Circulation: Governing Equations Solved numerically via community ocean model code (finite difference, finite element, finite volume) Applied to global, basin-scale, regional, and coastal domains Grid size are O(1-10km), time steps are O(minute)

5 Global Ocean Circulation Model: SST snapshot

6 Coastal Circulation Model

7 Physical Oceanographic DA Features: Nonlinear. Mul)ple interac)ng scales, complex geometry, and boundary condi)ons. Goals Online state es)ma)on for opera)onal oceanogrpahy, i.e. the filtering/predic)on problem. BUT also many scien)fically mo)vated retrospec)ve studies, i.e. reconstruc)on of the past ocean state. i.e. the smoothing problem. Sampling or observing array design of emerging interest (OSSEs)

8 Biological Oceanography: Biogeochemical Models Systems of ODEs: (complex dynamics for biology) dx dt = f (X,θ) + e(t) O- D model Systems of PDEs: (biology within ocean circulation) (complex dynamics and high dimensionality) 1- D or 3- D models X i t + u! X i (K X i ) = f i (X 1,..., X m,θ) + e(t), i = 1,...,m

9 Biogeochemical Dynamics dp dt = g { N;k N }γ P λ P P g P;k P dz dt = εg P;k P dn dt dd dt { } IZ λ Z Z + n Z { } IZ + n P = φd + βg { P;k P } IZ g { N;k N }γ P + νλ Z Z + n N = φd + λ P P + (1 ε β)g{ P;k P } IZ + (1 ν)λ Z Z + n D

10 Snapshot of Ocean Chlorophyll (Phytoplankton)

11 Biological Oceanographic DA Features: Built on physical circula)on models as interac)ng non- conserva)ve tracers. Uncertainty in the biological model structure, and appropriate level of complexity. Primary Goals Parameter es)ma)on, state es)ma)on, as well as system iden)fica)on to aid in scien)fic understanding. Some interest in opera)onal systems è Emerging area, and hence a variety of approaches being considered with respect to DA and dynamical models used. TRADITIONALLY, DEVELOPMENT OF OCEAN DA HAS BEEN DRIVEN MORE BY THE DYNAMICAL MODELS, AND LESS BY THE OBSERVATIONS

12 Measurements of the Physical Ocean

13 Measurements of Ocean Biology 5me series (moored instruments - plankton, nutrients) spa5al data (satellite chorophyll) 5me- space series (glider robots - plankton) OBSERVATION REVOLUTION IN OCEAN SCIENCES, STILL A PARTIALLY OBSERVED SYSTEM AND UNDERSAMPLED COMPLEX SPATIO- TEMPORAL SAMPLING PROTOCOLS

14 Variational DA Approaches Strong Constraint (assumes no model error ): minimize J(θ) = T 2 y t h(x t ) 1 Σv w.r.t θ t=0 subject to x t = d(x t 1,θ) Sta5s5cally, nonlinear (generalized least- squares) regression, with observa)on errors assumed Gaussian. NOTE: For DA, parameters are ojen ini)al or boundary condi)ons. Covariance matrices contain dynamical balances (e.g. geostrophy) The bread and bu[er of prac)cal data assimila)on (3DVAR, 4DVAR). For opera)onal DA, embedded with sequen)al DA cycle. Key aspect is Compu)ng minimiza)on θ J using adjoint equa)ons to faciliate

15 Given the cost function: 4DVAR and Adjoints Consider the prototypical DA problem of es)ma)ng the ini)al condi)ons x 0 J(x 0 ) = 1 2 (x 0 x 0 p ) T Σ 0 1 (x 0 x 0 p ) T (y t Hx t ) T Σ 1 v ( y t Hx t ) t=1 subject to x t = Dx t 1 We define the Lagrange function as: L = J + T t=1 λ T t 1 (x t Dx t 1 ) L λ t = 0 x t = Dx t 1 + e t L = 0 λ t 1 = D T λ t + H T Σ 1 v (y t Hx t ) x t L = 0 x 0 = x p 0 + Σ 0 Dλ 0 x 0 Forward Model Adjoint Model (backward in )me) Parameter es)ma)on

16 A harder varia)onal DA problem.. incorpora)ng dynamical model error Weak Constraint: Sta5s5cally: min J(θ) = T t=1 2 y t h(x t ) 1 Σv wrt x t and θ + x t d(x t 1,θ) Σe 1 2 If linear and Gaussian, random- beta regression (Duncan and Horne 1972, JASA), and the same as the Kalman smoother. Nonlinear case is a much more complex problem can use representers (i.e. solve in observa)on space), or op)miza)on based on calculus of varia)ons (i.e. inverse problems).

17 PROS Remarks on Variational DA Varia)onal DA works for large scale GFD systems. Flexible. Allows for joint es)ma)on for parameters and state Dynamical balances readily incorporated Can use approximate gradients - > incremental approach CONS Loss/Cost func)ons are user defined (usually least squares based, or otherwise assume MV normality) Li[le emphasis on error es)mates Uniqueness of solu)on? ISSUES: ill- posedness, iden)fiability, need for regulariza)on (smoothness priors) Deriva)on of tangent linear and adjoint equa)ons is model- specific, and maintenance of adjoint equa)ons for complex code is non- trivial.

18 Sequential Methods for DA Single stage transition of system from time t-1 to time t prediction measurement x t = d(x t 1,θ) +ν t y t nowcast [x t 1,θ y 1:t 1 ] forecast nowcast [x t,θ y 1:t 1 ] [x t,θ y 1:t ] time = t-1 time = t time - Recursive estimation of system state through time - Forecast and Measurement steps

19 Nudging ˆx t t = ˆx t t 1 + K(y t Hˆx t t 1 ) Many ad- hoc methods used for nudging (= adjus)ng forecast toward measurements). Major dynamical Issues! Now used mainly to correct for model drij from climatology. For linear Gaussian state space model, Kalman filter provides op)mal K (and state covariance propaga)on) - Nonlinear extensions available, e.g. extended Kalman filter. - NonGaussian extensions available, e.g. Gaussian sum filter. Problems with nonlinear dynamics (e.g. extended KF has instabili)es in covariance propaga)on, Evensen s PhD) Ensemble Kalman filter has supplanted these methods.

20 The Ensemble Kalman Filter!x (i) (i) t t = x t t 1 + K(y (i) (i) t Hx t t 1 ), i = 1,,n Works for large systems. Easy to implement Gaussian Approxima)on uses KF upda)ng equa)on NEEDS (spa)al) localiza)on due to spurious long range correla)ons (e.g. the LETKF, or parametric approxima)on/distance based filtering of covariance matrices) Some problems with highly nonlinear dynamics, nonlinear observa)on operators, plus dynamical balances, mass conserva)on

21 Ensemble Mean PON A Comparison Ensemble Std Dev PON depth Stochastic Simulation Stochastic Simulation depth Ens Kalman Filter Ens Kalman Filter depth Particle Filter Particle Filter time Both PF and enkf underperformed but for different reasons

22 Ensembles approaches rely on stochashc dynamic ensemble predichon Key Issue: small ensemble must cover the part of large state space

23 How to do Stochastic Simulation? 1. Dynamical noise: disturbance terms, or random forcing, added to each of the state variables (act as sources/sinks) 2. Stochastic parameters: Model parameters can follow a probability distribution or a stochastic process. 3. Initial and Boundary Conditions: Random initial states and/ or boundary forcing may be incorporated (NWP ensemble forecasting). Like treating these as parameters. 4. Multimodel Ensembles: incorporates structural model errors via using multiple models each with multiple configurations (used in climate modelling) Treatment of Model error is a neglected area?

24 Example: Stochastic Parameters Realization of O-D ODE based PZND model max growth parameter follows AR(1) Frequent transitioning across bifurcation point aperiodic/episodic dynamical dependencies maintained Represent equatorial Pacific

25 We want to generate large large ensembles, GFD models are computahonally expensive What about cheap evaluators for the dynamics, i.e. model surrogates?

26 Idea: The Transition Density and Copulas We want: x t ~ [x t x t 1,θ] - predic)ve/transi)on density We have: x t = d(x t 1,θ) + e t - a numerical model to generate samples Rather than a sample- based representa5on of model errors, what about a parametric representa5on? Must be: accurate, flexible, easy to sample from, high dimensions Idea: create mul)variate distribu)on for [x t x t 1,θ] using copulas - control the marginal distribu)ons, and dependence separately - construc)on and sampling from copulas is straighnorward - can be derived from ensembles and their predic)on errors using method of moments.

27 5 20 De Det Chl Residual Residual Residual 1.0 P(x1,, xn ) = C(P(x1 ),, P(xn )) Oxy Residual 0.15 Based on 1D PZND model of BATS site DIN 5 Depth= 250m Density: Oxy 6.5. SPECIFICATION OF THE STATE-SPACE MODEL Chl Phy Det DIN Depth= 150m Phy Density: MARGINALS Residual e 6.15: kernel density estimates for residuals at -50, -150 and -250 metres deep. read and magnitude of the residuals is quite different at different layers and for t state variables, requiring all 1, 750 marginal distributions to be parametrised ely. Also, nearby residuals are highly correlated. Finally, the residuals are not Samples from predic)ve density, or model errors For example the residuals imply a negative model bias for deep DET and shallow symmetrical about zero implying that the h(xt xt 1 ) is biased for some states and ates and a positive model bias for deep PHY. It is possible to centre h t (xt xt 1 ) at 1 ) b to account for the bias b, but this was not done in this application because CORRELATION mperfection of the data available for estimating b. Figure 6.16: Contour plot for the empirical residual correlation.

28 Idea: Emulator Based DA IDEA: Replace complex dynamical model with an emulator to facilitate computationally efficient data assimilation. Estimate seasonal cycle of two key biological parameters (C:chl ratio, Zooplankton grazing). APPROACH Approximate 3-D biological ocean model with a polynomial chaos based emulator Estimate time-varying parameters by minimizing a discrepancy (metric: adaptive grey block distance) between SeaWiFS satellite chlorophyll images and its model predicted field.

29 Most of our sequenhal eshmahon procedures come from Hme series. AdaptaHons for spaho- temporal problems?

30 Idea: A Location Particle Smoother for Spatio-Temporal Systems IDEA: run par)cle filter through )me, but apply par)cle smoother in the spa>al domain at every )me step. - At each )me, the )me- domain predic)ve distribu)on is used as proposal distribu)on, followed by sequen)al importance resampling with loca)on filter/smoother sweeping through space. Successful for mul)variate state es)ma)on for 1- D biological model (Bermuda Atlan)c Time Series site) Surface Depth Oxygen OXY LPS Deep year one year two

31 Performance of LocaHon ParHcle Smoother Prediction PDF Filter PDF: LPS Simula)on study: Lorenz96 system 10 true states generated 10 replicates per state Cauchy observa)on error (plus different error types, mis- specifica)ons wrt variance and tail density) Frequency σ 2 =.08 σ 2 =.015 Frequency proportion locations covered proportion locations covered Filter PDF: Particle Filter Filter PDF: EnKF Performance Metrics: Percent coverage of true state by 95% credible intervals Average ensemble variance (σ 2 ) Frequency σ 2 è collapse σ 2 =.025 Frequency proportion locations covered proportion locations covered PROS: outperforms EnKF for non- Gaussian likelihood CONS: less computa)onally efficient, no be[er for Gaussian errors

32 What about the (stahc) parameter eshmahon?

33 Approaches for Parameter Estimation 1. State Augmentation: append parameters to the state x t = x t θ t Can use standard sequential MC methods with iterative filtering 2. Likelihood Methods: Use sample based likelihoods T [y 1:T θ] = L(θ y 1:T ) = [y t x t,θ][x t y 1:t 1,θ]dx t t=1 3. Bayesian Hierarchical: particle MCMC, SMC^2

34 Sample Based Likelihood Surface L(θ y 1:T ) 1 n T t=1 ( n [y t x (i),θ] ) i=1 t t 1

35 Concluding Remarks Ocean DA has used a variety of approaches. Successful methods require consideration of problem specific features. Plug and play is unlikely to work (though modularity of methods are important). Physical DA and Biological DA have different goals (i.e. it is not all just filtering). Challenge is to incorporate full suite of observations, large-scale nonlinear models, characterize uncertainty, allow for sampling design, model building/ selection, and multi-model inference. Emerging interest in (hierarchical) Bayesian framework for estimation in dynamical systems. Useful prior information exists, and counters data paucity. But computational issues remain due to expensive GFD models (can cheap, approximate evaluators help?)