Harvard - MURI

Allan R. Robinson, Pierre F.J. Lermusiaux,

Patrick J. Haley and Wayne G. Leslie

Division of Engineering andApplied Sciences

Department of Earth andPlanetary Sciences

Table of Contents1. Top three tasks to carry out/problems to address2. What we need most?3. Recent results relevant to MURI effort

• Quantitative Autonomous Adaptive Sampling • Multi-Scale Energy and Vorticity Analysis• Multi-Model Adaptive Combination

http://www.deas.harvard.edu/~robinsonhttp://www.deas.harvard.edu/~pierrel

Top Three Tasks to Carry Out/Problems to Address

1. Determine details of three metrics for adaptive sampling (coverage, dynamics, uncertainties) and develop schemes and exercise software for their integrated use

2. Carry out cooperative real-time data-driven predictions with adaptive sampling

3. Advance scientific understanding of 3D upwelling/relaxation dynamics and carry out budget analyses as possible

What Do We Need Most?

• Effective collaboration

• Integrated software

• Good quality data with error estimates

Determine details of three metrics for adaptive sampling and develop schemes and exercise software for their integrated use

1. The three metrics:

i. Coverage (maintain synoptic accuracy)

ii. Dynamics (maximize sampling of predicted dynamical events)

iii. Uncertainty (minimize predicted uncertainties)

2. Integrate these adaptive sampling metrics and schemes with platform control and LCS metrics and schemes

• Multiple platforms of different types used together in overall conceptual framework

3. Adaptive sampling schemes and software in pre-exercise simulations

• Continue development of ESSE and MsEVA nonlinear adaptive sampling

• Implement simple glider/AUV models within HOPS for i) measurement model and ii) data predictions

• Continue development error models for HOPS and for glider/AUV/ship/aircraft data (with experimentalists)

Carry-out real-time data-driven predictions with adaptive sampling

1. Work in real-time with a committed general team of experimentalists and carry out adaptive sampling• Link and/or integrate HOPS with control theory and LCS software

2. Carry out real-time HOPS/ESSE (sub)-mesoscale field and uncertainty predictions with integrated 3-metrics adaptive sampling• 1-way and/or 2-way nested HOPS simulations (333m into 1km into 3km)

• Sub-mesoscale effects including tidal effects

3. Efficient measures and assessment of predictive skill• Real-time forecast skill and hindcast skill of fields and uncertainties

• Theory and software to measure skill of upwelling center/plume estimate: e.g. shape/size of plume, scales of jet and eddies at plume edges, thickness of boundary layers, surface/bottom fluxes

4. Real-time physical-acoustical DA with MIT and real-time biological-physical DA as possible with collaborators

Advance scientific understanding of 3D upwelling/relaxation dynamics and carry-out budget analyses on several scales

1. Develop and implement software for momentum, heat and mass budgets

• On several scales and term-balances: e.g. point-by-point, time-dependent plume-averaged, Ms-EVA, etc.

• Compare data-based budgets to data-model-based budgets

2. Science-focused studies of sensitivity of upwelling/relaxation processes

• e.g. effects of atmospheric conditions and resolution, idealized geometries, tides/internal tides or boundary layer formulations on plume formation and relaxation

3. Improve model parameterizations based on model-data misfits (local and budgets)

4. Estimate predictability limits for upwelling/relaxation processes

What Do We Need Most?

• Effective collaboration, rapid and efficient communication and real integrated system and system software

• Effective integration of software

– LCS with HOPS

– Glider/AUV models with HOPS

• Good forcing functions and good initial conditions

• Real-time inter-calibration data stations to avoid false circulation features

• Occasional and simultaneous sampling by pairs of platforms, efficiently scheduled by real-time control groups

• Documented feedback from experimentalists

• Both in real-time and after experiment

Quantitative Adaptive Sampling via ESSE

1. Select sets of candidate sampling paths/regions and variables that satisfy operational constraints

2. Forecast reduction of errors for each set based on a tree structure of small ensembles and data assimilation

3. Optimization of sampling plan: select sequence of paths/regions and sensor variables which maximize the predicted nonlinear error reduction in the spatial domain of interest, either at tf (trace of ``information matrix’’ at final time) or over [t0 , tf ]

- Outputs:

- Maps of predicted error reduction for each sampling paths/regions

- Information (summary) maps: assigns to the location of each sampling region/path the average error reduction over domain of interest

- Ideal sequence of paths/regions and variables to sample

Which sampling on Aug 26 optimally reduces uncertainties on Aug 27?

4 candidate tracks, overlaid on surface T fct for Aug 26

ESSE fcts after DA of each track

Aug 24 Aug 26 Aug 27

2-day ESSE fct

ESSE for Track 4

ESSE for Track 3

ESSE for Track 2

ESSE for Track 1DA 1

DA 2

DA 3

DA 4

IC(nowcast) Forecast DA

Which sampling on Aug 26 optimally reduces uncertainties on Aug 27?

1. Define relative error reduction as: (27 - 27 ) / 27…..for 27 > noise

0………………for 27 noise

2. Create relative error reduction maps for each sampling tracks, e.g.:

track i

3. Compute average over domain of interest for each variable, e.g. for full domain: Best to worst error reduction: Track 1 (18%), Pt Lobos (17%), …., Track 3 (6%)

4. Create “Aug 26 information map”: indicates where to sample on Aug 26 for optimal error reduction on Aug 27

Large-scale Available Potential Energy (APE) Large-scale Kinetic Energy (KE)

• Both APE and KE decrease during the relaxation period• Transfer from large-scale window to mesoscale window occurs to account for

decrease in large-scale energies (as confirmed by transfer and mesoscale terms)

August 18 August 19 August 20

August 21 August 22 August 23 August 23August 22August 21

August 20August 19August 18

Windows: Large-scale (>= 8days; > 30km), mesoscale (0.5-8 days), and sub-mesoscale (< 0.5 days)

Multi-Scale Energy and Vorticity Analysis

Dr. X. San Liang

• Multiscale window decomposition in space and time (wavelet-based) of energy/vorticity eqns.• For example, consider Energetics During Relaxation Period:

Approaches to Multi-Model Adaptive ForecastingCombine ROMS/HOPS re-analysis temperatures

to fit the M2-buoy temperature at 10 m

By combining the models x1 and x2 we attempt to:1. eliminate and learn systematic errors2. reduce random errors

• Approach utilized here: neural networks• A neural network is a non-linear operator which can be

adapted (trained) to approximate a target arbitrary non-linear function measuring model-data misfits:

Sigmoidal Transfer Function

ii) Single Sigmoidal layer:

Oleg Logoutov

i) Linear least-squares:

d

Two fits tested

Top: Green – HOPS/ROMS reanalysis combined via neural network trained on the first subset of data (before Aug 17).

Bottom: Green – HOPS/ROMS reanalysis combined via adaptive neural network also trained on the first subset of data (before Aug 17), but over moving-window of 3 days decorrelation

Neural Network Least Squares Fit

Linear Least Squares Fit

IndividualModels

Equal Weights

• Observed (black) temp at the M2mooring• Modeled temp at the M2mooring:

ROMS re-analysis, HOPS re-analysis

Extra Vugrafs

ESSE Surface Temperature Error Standard Deviation Forecasts

Aug 12 Aug 13

Aug 27Aug 24

Aug 14

Aug 28

End of Relaxation Second Upwelling period

First Upwelling periodStart of Upwelling

ESSE: Uncertainty Predictions and Data Assimilation

1. Dynamics: dx =M(x)dt+ d ~ N(0, Q)2. Measurement: y = H(x) + ~ N(0, R)

3. Non-lin. Err. Cov. evolution:

4. Error reduction by DA:

QTxxxMxMTxMxMxxdtdP )ˆ)(ˆ()(())ˆ()()(ˆ(/

)()()( PKHIP where K is the reduced Kalman Gain

• ESSE retains and nonlinearly evolves uncertainties that matter, combining,

i. Proper Orthogonal Decompositions (PODs) or Karhunen-Loeve (KL) expansions

ii. Time-varying basis functions, and,

iii. Multi-scale initialisation and Stochastic ensemble predictions

to obtain a dynamic low-dimensional representation of the error space.• Linked to Polynomial chaos, but

with time-varying error KL basis:

P(0)=P0

Adaptive sampling schemes via ESSE

Adaptive Sampling Metric or Cost function:

e.g. Find Hi and Ri such that

dtt

ttPtrMinortPtrMin

f

RiHif

RiHi 0

,,))(())((

Dynamics: dx =M(x)dt+ d ~ N(0, Q)Measurement: y = H(x) + ~ N(0, R)

Non-lin. Err. Cov.:

QTxxxMxMTxMxMxxdtdP )ˆ)(ˆ()(())ˆ()()(ˆ(/

Adaptive Sampling: Use forecasts and their uncertainties to predict the most useful observation system in space (locations/paths) and time (frequencies)

Modeling of tidal effects in HOPS

• Obtain first estimate of principal tidal constituents via a shallow water model1. Global TPXO5 fields (Egbert, Bennett et al.)

2. Nested regional OTIS inversion using tidal-gauges and TPX05 at open-boundary

• Used to estimate hierarchy of tidal parameterizations :i. Vertical tidal Reynolds stresses (diff., visc.) KT = ||uT||2 and K=max(KS, KT)

ii. Modification of bottom stress =CD ||uS+ uT || uS

iii. Horiz. momentum tidal Reyn. stresses (Reyn. stresses averaged over own T)

iv. Horiz. tidal advection of tracers ½ free surface

v. Forcing for free-surface HOPS full free surface

T section across Monterey-BayTemp. at 10 m

No-tides

Two 6-day model runs

Tidal effects• Vert. Reyn.

Stress• Horiz.

Momentum Stress

CHL Aug 20

CHL Aug 21

CHL Aug 22

Post-Cruise Surface CHL forecast (Hindcast)

• Starts from zeroth-order dynamically balanced IC on Aug 4

• Then, 13 days of physical DA

• Forecast of 3-5 days afterwards

CHL Aug 20,

20:00 GMT