1. Banff, May 2005 Lectures on Modeling and Data Assimilation Richard B. Rood NASA/Goddard Space Flight System Visiting Scientist, Lawrence Livermore National Laboratory May 7 - 13, 2005 Banff, Alberta, CANADA

2. Banff, May 2005 Plan of Presentations Models and Modeling Data Assimilation –What is it? –Why? –Things to Think About Coupled Modeling

3. Banff, May 2005 Model and Modeling Model –A work or construction used in testing or perfecting a final product. –A schematic description of a system, theory, or phenomenon that accounts for its known or inferred properties and may be used for further studies of its characteristics. Types: Conceptual, Statistical, Physical, Mechanistic, …

4. Banff, May 2005 Types of Models (see also, Chapter 17, Peixoto and Oort, 1992) Conceptual or heuristic models which outline in the simplest terms the processes that describe the interrelation between different observed phenomena. These models are often intuitively or theoretically based. An example would be the tropical pipe model of Plumb [1996], which describes the transport of long-lived tracers in the stratosphere. Statistical models which describe the behavior of the observations based on the observations themselves. That is the observations are described in terms of the mean, the variance, and the correlations of an existing set of observations. Johnson et al. [2000] discuss the use of statistical models in the prediction of tropical sea surface temperatures. Physical models which describe the behavior of the observations based on first principle tenets of physics (chemistry, biology, etc.). In general, these principles are expressed as mathematical equations, and these equations are solved using discrete numerical methods. Good introductions to modeling include Trenberth [1992], Jacobson [1998], Randall [2000].

5. Banff, May 2005 Conceptual/Heuristic Model Plumb, R. A. J. Meteor. Soc. Japan, 80, 2002 Observed characteristic behavior Theoretical constructs “Conservation” Spatial Average or Scaling Temporal Average or Scaling Yields Relationship between parameters if observations and theory are correct

6. Big models contain little models landice ocean atmos coupler Atmosphere Thermosphere Mesosphere Stratosphere Troposphere Dynamics / Physics ConvectionAdvectionMixingPBLRadiationClouds Management of complexity But, complex and costly Where’s chemistry and aerosols?

7. Banff, May 2005 What are models used for? Diagnostic: The model is used to test the processes that are thought to describe the observations. –Are processes adequately described? Prognostic: The model is used to make a prediction. –Deterministic –Probabilistic

8. Banff, May 2005 What’s a mechanistic model? Mechanistic models have one or more parameters prescribed, for instance by observations, and then the system evolves relative to the prescribed parameters. Thermosphere Mesosphere Stratosphere Troposphere Sink of energy from below Relaxation to mean state Stratosphere Geopotential @ 100 hPa A mechanistic model to study stratosphere

9. Banff, May 2005 Simulation Environment (General Circulation Model, “Forecast”) Boundary Conditions Representative Equations Discrete/Parameterize Theory/Constraints Primary Products (i.e. A) Derived Products ( F (A)) DA/Dt = P –LA –  q/H (A n+  t – A n )/  t = … ∂u g /∂z = -(∂T/∂y)R/(Hf 0 ) T, u, v, , H 2 O, O 3 … Pot. Vorticity, v*, w*, … Emissions, SST, …   (  d,  p ) Scale Analysis (  b,  v ) Consistent (  b,  v ) = (bias error, variability error) Derived Products likely to be physically consistent, but to have significant errors. i.e. The theory-based constraints are met.

10. Banff, May 2005 Representative Equations ∂A/∂t = –  UA + M + P – LA – /HA+q/H –A is some constituent –U is velocity  “resolved” transport, “advection” –M is “Mixing”  “unresolved” transport, parameterization –P is production –L is loss – is “deposition velocity” –q is emission –H is representative length scale for and q All terms are potentially important – answer is a “balance”

11. Discretization of Resolved Transport ∂A/∂t = –  UA Gridded Approach Orthogonal? Uniform area? Adaptive? Unstructured? Choice of where to Represent Information Choice of technique to approximate operations in representative equations Rood (1987, Rev. Geophys.)  ( A,U ) Grid Point (i,j)

12. Banff, May 2005 Discretization of Resolved Transport  ( A,U )  ( A,U )  ( A,U )  Grid Point (i,j)Grid Point (i+1,j) Grid Point (i+1,j+1)Grid Point (i,j+1)

13. Discretization of Resolved Transport (U)(U) (U)(U) (U)(U) Grid Point (i,j)Grid Point (i+1,j) Grid Point (i+1,j+1)Grid Point (i,j+1) (A)(A) Choice of where to Represent Information Impacts Physics Conservation Scale Analysis Limits Stability (U)(U)

14. Banff, May 2005 Discretization of Resolved Transport ∂A/∂t = –  UA ∫ Line Integral around discrete volume

15. Banff, May 2005 1.Divergence theorem: for the advection-transport process 2.Green’s theorem: for computing the pressure gradient forces 3.Stokes theorem: for computing the finite-volume mean vorticity using “circulation” around the volume (cell) Finite-difference methods “discretize” the partial differential equations via Taylor series expansion – pay little or no attention to the underlying physics Finite-volume methods can be used to “describe” directly the “physical conservation laws” for the control volumes or, equivalently, to solve the integral form of the equations using the following 3 integral theorems: “Finite-difference” vs. “finite-volume” Lin and Rood (1996 (MWR), 1997 (QJRMS)), Lin (1997 (QJRMS), 2004 (MWR))

16. The importance of your decisions

17. Banff, May 2005 Importance of your decisions (Tape recorder in full Goddard GCM circa 2000) Slower ascent Faster mean vertical velocity Faster ascent Slower mean vertical velocity S. Pawson, primary contact FINITE-DIFFERENCE FINITE-VOLUME

18. Banff, May 2005 Importance of your decisions (Precipitation in full GCM) Precipitation in California (from P. Duffy) Spectral Dynamics Community Atmosphere Model / “Eulerian” Finite Volume Dynamics Community Atmosphere Model / “Finite Volume”

19. Banff, May 2005 Some conclusions about modeling Physical approach versus a mathematical approach –Pay attention to the underlying physics – seek physical consistency –How does my comprehensive model relate to the heuristic models? Quantitative analysis of models and observations is much more difficult than ‘building a new model.’ This is where progress will be made. –Avoid coffee table / landscape comparisons

20. Banff, May 2005 The Dark Path of Data Assimilation Basics of Assimilation Assimilation in tracer transport Ozone assimilation

21. Banff, May 2005 Data Assimilation Assimilation –To incorporate or absorb; for instance, into the mind or the prevailing culture (or, perhaps, a model) Model-Data Assimilation –Assimilation is the objective melding of observed information with model-predicted information. Attributes: Rigorous Theory, Difficult to do well, Easy to do poorly, Controversial (“Best” estimate)

22. Assimilation Environment Boundary Conditions (OP f O  + R)x = A o – OA f Discrete/Error Modeling Constraints on Increments (  b,  v ) reduced Inconsistent DA/Dt = P –LA – /HA+q/H (A n+  t – A n )/  t = … ∂u g /∂z = -(∂T/∂y)R/(Hf 0 ) A i ≡ T, u, v, , H 2 O, O 3 … Pot. Vorticity, v*, w*, … Emissions, SST, …    Scale Analysis (  b,  v ) Consistent DataModel O is the “observation” operator; P f is forecast model error covariance R is the observation error covariance; x is the innovation Generally assimilate resolved, predicted variables. Future, assimilate or constrain parameterizations. (T, u, v, H2O, O3) Data appear as a forcing to the representative model equation Does the average of this added forcing equal zero?

23. Banff, May 2005 What do these things mean? (OP f O T + R)x = A o – OA f Sat Bal Sat Bal Ship O – The Observation Operator Space and Time Interpolation Rad Geo Rad Geo Tem Satellite Balloon Ship Radiance Geopotential Temperature To Measured Quantity Model Forecast

24. Banff, May 2005 What do these things mean? (OP f O T + R)x = A o – OA f Errors: Variance and Correlation Radius of Influence Correlation aligned with flow?

25. Banff, May 2005 Figure 5: Schematic of Data Assimilation System Data Stream 1 (Assimilation) Data Stream 2 (Monitoring) Quality Control Model Forecast Statistical Analysis Forecast / Simulation Analysis & (Observation Minus Analysis) Observation minus Forecast Error Covariance

26. Banff, May 2005 Statistical Analysis Q.C. What does an assimilation system look like? (Goddard Ozone Data Assimilation System) Forecast & Observation Error Models “Analysis” Statistical Analysis Tracer Model Short-term Forecast (15 minutes) Winds Temperature Analysis Ozone Data Sciamachy MLS Ozone Data TOMS/SBUV POAM/MIPAS Obs - Forecast Analysis Increments HALOE Sondes BALANCE, BALANCE, BALANCE! Long-term forecast

27. Banff, May 2005 Why do we do assimilation? Global synoptic maps (Primary (Constrained) Product) Unobserved parameters (Primary - Derived Product) –Ageostrophic wind, constituents, vertical information, Derived products –Vertical wind / Divergence, residual circulation, Diabatic and Radiative information, tropospheric ozone, … Forecast initialization Radiative correction for retrievals “Background,” a priori profile, for retrievals Alternative to traditional retrieval Instrument/Data System monitoring Instrument calibration Observation quality control Model evaluation / validation

28. Banff, May 2005 ECMWF, ERA-40

29. The transport application Chemistry Transport Model (CTM) Transport / Chemistry PBLAdvectionMixingJ RatesConvectionReact. RateSolverEmissions Input Fields “ONE WAY COUPLER” Winds, Temperature, … Convective Mass Flux, Water, Ice, … Turbulent Kinetic Energy … Diabatic Heating … Wet/Dry Atmospheric “Model” History Tape ∂A/∂t = –  UA + M + P – LA – /HA+q/H A ( space, time )

30. Banff, May 2005 The Transport Application Residual CirculationWave Transport Planetary Synoptic MIXING (u,v) (u*,v*)

31. PDFs of total ozone: observations & CTM Douglass, Schoeberl, Rood and Pawson (JGR, 2003) GCM-driven DAS-driven Means displaced Half-width ok Means displaced Spread too wide Too much tropical-extratropical mixing in DAS

32. UKMO DAO GCM DiabaticKinematicDiabaticKinematic Schoeberl, Douglass, Zhu and Pawson (JGR, 2003) (50 days) Kinematic: considerable vertical and horizontal dispersion Diabatic: vertical dispersion reduced (smooth heating rates) GCM shows very little dispersion, regardless of method used Assimilated fields are excessively dispersive Three-dimensional trajectory calculations UKMO DAO GCM

33. Banff, May 2005 Transport have we reached a wall? –Derived quantities are not physically consistent Dynamic – Radiative equilibrium is not present –Bias acts as forcing and generates spurious circulations –Data insertion generates “noise” that grows and propagates  relation to bias –Temperature constraint too weak to define winds? Wallace and Holton (1968) –Thickness measurements too thick? Residual Circulation Wave Transport Planetary Synoptic MIXING    (u,v) (u*,v*) TRANSPORT with winds from assimilation

34. Major assimilation issue: Bias Primary Products Errors, (  b,  v ) = (bias error, variability error), errors usually reduced. Derived Products and unobserved parameters likely to be physically Inconsistent, errors likely to increase relative to simulation. Why? Consider Ozone and Temperature: How are they related? O3 – T, Chemistry (P and L) – Seconds – hours –  O3 – T, Transport (U) – Hours – Days –  O3 – T, Diabatic forcing – Days – Months –  O3 – T, Other constituents – Seconds – hours – days –  If adjust O3 and T by observations to be “correct” and if that “correction” is biased, then there has to be a compenradion somewhere in the Representative Equation. Usually it appears as a bias in unobserved parameters and leads to “inconsistent” results. Budgets do NOT balance.

35. Banff, May 2005 Ozone Assimilation Why? ( Rood, NATO ASI Review Paper, 2003) –Monitoring instrument behavior –Improving radiative calculation Models Retrievals –Tropospheric ozone? –… What? –Impact of new data, what does it mean?

36. Comparison of an individual ozone sonde profile with three assimilations that use SBUV total column and stratospheric profiles from: –SBUV –SBUV and MIPAS –MIPAS MIPAS assimilation captures vertical gradients in the lower stratosphere Model + Data capture synoptic variability and spreads MIPAS information MIPAS Ozone assimilation MIPAS data

37. Banff, May 2005 Monitoring Data System EP TOMS Going Bad Adjustment To change in observing system

38. Banff, May 2005 Summary (1) Good representation of primary products, T, wind, ozone Model-data bias, “noise” added at data insertion, data insertion as a source of gravity waves provides difficult challenges Derived products are often “non-physical,” and examples of improving primary products degrades derived products –Pushing model errors into the derived products –Need to incorporate Theory/Constraints into assimilation more effectively

39. Banff, May 2005 Summary (2) Can we really do “climate” with assimilated data sets? –Don’t do trends If I worked in data assimilation what would I propose? –Primary products: New data, Bias correction –Derived products/Use in “Climate” studies: Fundamental physics of model, model improvement –Error covariance modeling? Data assimilation technique?

40. Statistical Analysis Q.C. Quality Control: Interface to the Observations Satellite # 1 Satellite # 2 Non-Satellite # 1 MODEL Forecast Self-comparison Intercomparison “Expected Value” O Comparison to “Expected” DATA GOOD BAD SUSPECT GOOD! MONITOR ASSIMILATE Memory of earlier observations

41. Banff, May 2005 When Good Data Go Bad? Good Data –Normal range of expectation –Spatial or temporal consistency Bad data –Instrument malfunction … –Cloud in field of view … –Operator, data transmission error –The unknown unknown New phenomenon Model failure New extreme of variability What we’re really interested in!

42. Banff, May 2005 Statistical Analysis Q.C. Link to the adaptive observing “Analysis” Statistical Analysis Tracer Model Short-term Forecast (15 minutes) Winds Temperature Analysis Obs - Forecast Analysis Increments Long-term forecast Features Data Anomalies What to Observe? What to Process?

43. Banff, May 2005 Model - Data Assimilation Objective, Automated Examination and Use of Observing System. –All types of observations … need to write an observation operator. Requires Robust “Sampling” Observing System as a Foundation –There is no controversy of “sampling” versus “targeted” observations. They are each an important part of scientific investigation. Powerful Technique for Certain Applications. Provides Information that might be used in Adaptive Observing (or Data Processing). –From Quality Control Subsystem –From Forecast Subsystem