Lectures on Modeling and Data Assimilation

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					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



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



                   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, …


                           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].



                                   Banff, May 2005
Conceptual/Heuristic Model
                                               •Observed characteristic
                                               behavior
                                               •Theoretical constructs
                                               •“Conservation”
                                                   •Spatial Average or
                                                   Scaling
                                                   •Temporal Average or
                                                   Scaling


                                               Yields
                                               Relationship between
                                               parameters if observations and
                                               theory are correct
Plumb, R. A. J. Meteor. Soc. Japan, 80, 2002

                           Banff, May 2005
Big models contain little models
                                                                 Atmosphere

          atmos                                          Thermosphere
                                                                 Mesosphere
                                                                 Stratosphere
                                                           Troposphere
land     coupler       ice


                                              Troposphere

          ocean                       Dynamics / Physics




                                                                   Convection
                                Advection



                                                     Radiation




                                                                                      Clouds
                                            Mixing




                                                                                PBL
 Management of complexity
 But, complex and costly
                             Where‟s chemistry and aerosols?
   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

                    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                        Sink of energy from below
      Mesosphere                         Relaxation to mean state
      Stratosphere                             Stratosphere
     Troposphere                         Geopotential @ 100 hPa


                               A mechanistic model to study stratosphere


                       Banff, May 2005
              Simulation Environment
            (General Circulation Model, “Forecast”)
   Boundary Conditions           Emissions, SST, …              e
 Representative Equations    DA/Dt = P –LA – n/HA+q/H           e
  Discrete/Parameterize         (An+Dt – An)/Dt = …        (ed, ep)
    Theory/Constraints        ∂ug/∂z = -(∂T/∂y)R/(Hf0)    Scale Analysis

 Primary Products (i.e. A)     T, u, v, F, H2O, O3 …       (eb, ev)
 Derived Products (F(A))      Pot. Vorticity, v*, w*, …   Consistent


(eb, ev) = (bias error, variability error)

Derived Products likely to be physically consistent, but to have
significant errors. i.e. The theory-based constraints are met.

                              Banff, May 2005
       Representative Equations
• ∂A/∂t = – UA + M + P – LA – n/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
   –   n is “deposition velocity”
   –   q is emission
   –   H is representative length scale for n and q


• All terms are potentially important – answer is a “balance”

                               Banff, May 2005
   Discretization of Resolved Transport

• ∂A/∂t = – UA

                                           Grid Point (i,j)
                            (A,U)




                     Choice of where to
                    Represent Information
Gridded Approach
Orthogonal?          Choice of technique to
Uniform area?      approximate operations in
Adaptive?           representative equations
Unstructured?      Rood (1987, Rev. Geophys.)
Discretization of Resolved Transport
        Grid Point (i,j+1)                  Grid Point (i+1,j+1)

                             (A,U)      (A,U)
                                         


                                         
                             (A,U)      (A,U)


        Grid Point (i,j)                    Grid Point (i+1,j)




                      Banff, May 2005
Discretization of Resolved Transport
        Grid Point (i,j+1)                 Grid Point (i+1,j+1)

                                 (U)
                                  

                                  
                             (U) (A) (U)
                                  
                                 (U)


        Grid Point (i,j)                   Grid Point (i+1,j)

                       Choice of where to
                       Represent Information
                       Impacts Physics
                       • Conservation
                       • Scale Analysis Limits
                       • Stability
 Discretization of Resolved Transport

• ∂A/∂t = – UA
                                         Line Integral
                                     ∫   around discrete
                                         volume




                   Banff, May 2005
  “Finite-difference” vs. “finite-volume”
• 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:
 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)
Lin and Rood (1996 (MWR), 1997 (QJRMS)), Lin (1997 (QJRMS), 2004 (MWR))
                              Banff, May 2005
The importance of your decisions
Importance of your decisions
  (Tape recorder in full Goddard GCM circa 2000)

                                          FINITE-VOLUME
                                           Slower ascent

                                           Faster mean
                                           vertical velocity


                                          FINITE-DIFFERENCE
                                            Faster ascent

                                            Slower mean
                                            vertical velocity

                                      S. Pawson, primary contact
                   Banff, May 2005
     Importance of your decisions
       (Precipitation in full GCM)




  Spectral Dynamics    Finite Volume Dynamics
Community Atmosphere   Community Atmosphere
  Model / “Eulerian”   Model / “Finite Volume”

                             Precipitation in California (from P. Duffy)

                          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

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




                   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)
                                   Banff, May 2005
              Assimilation Environment
           Model                                                Data
       Emissions, SST, …                    e            Boundary Conditions
 DA/Dt = P –LA – n/HA+q/H                   e          (OPfOT + R)x = Ao – OAf
      (An+Dt – An)/Dt = …                  e           Discrete/Error Modeling
    ∂ug/∂z = -(∂T/∂y)R/(Hf0)          Scale Analysis   Constraints on Increments
   Ai ≡ T, u, v, F, H2O, O3 … (eb, ev)                    (eb, ev) reduced
    Pot. Vorticity, v*, w*, … Consistent                    Inconsistent
O is the “observation” operator; Pf 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?
       What do these things mean?
                                  (OPfOT + R)x = Ao – OAf


                 Space and Time      Sat                                 Rad
                                      Sat                                 Rad




                                                           To Measured
                                       Sat     Bal                         Rad Geo
                                      Bal Sat                                Rad
                                                                          Geo Rad




                                                             Quantity
                 Interpolation
                                           Sat
                                            Sat                                Rad
                                     Bal      Sat Bal                    Geo     Rad Geo
                                                Sat                               Rad
                                                 Sat
                                              Ship Sat                             Rad
                                                                                Tem Rad
                                          Ship      Sat                      Tem      Rad
                                     Ship            Sat                 Tem           Rad
                                             Bal                               Geo

                                          Satellite                         Radiance
Model Forecast                            Balloon                           Geopotential
                                          Ship                              Temperature

                            O – The Observation Operator

                                        Banff, May 2005
   What do these things mean?
             (OPfOT + R)x = Ao – OAf




Radius of
Influence                    Correlation aligned with flow?
            Errors: Variance and Correlation
                     Banff, May 2005
Figure 5: Schematic of Data Assimilation System

                    Observation minus Forecast



   Data Stream 1                                    Statistical



                            Quality Control
                                                    Analysis            Analysis
   (Assimilation)                                                          &
                                                       Error          (Observation
                                                     Covariance
                                                                         Minus
   Data Stream 2
                                                      Model             Analysis)
    (Monitoring)                                     Forecast




                                              Forecast / Simulation


                                              Banff, May 2005
      What does an assimilation system look like?
                   (Goddard Ozone Data Assimilation System)

                    Ozone Data       Ozone Data           Obs - Forecast
                   TOMS/SBUV         Sciamachy
                   POAM/MIPAS           MLS
                                                           Analysis
     Forecast &
                                                           Increments
    Observation
    Error Models
                                                                           HALOE
                           Statistical
                           Analysis
                                                      “Analysis”
                                                       Analysis            Sondes


                             Q.C.             Winds
                                            Temperature

       Short-term           Tracer
        Forecast            Model
      (15 minutes)                                    BALANCE, BALANCE, BALANCE!

Long-term forecast                  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
                             Banff, May 2005
ECMWF, ERA-40
                Banff, May 2005
The transport application
                                   A ( space, time )

            Chemistry Transport Model                                          (CTM)

∂A/∂t = – UA + M + P – LA – n/HA+q/H

                      Transport / Chemistry




                                                                React. Rate
                      Convection




                                          Emissions
Advection




                                                                                       Wet/Dry
                                                      J Rates
             Mixing




                                                                              Solver
                                   PBL


Input Fields “ONE WAY COUPLER”
Winds, Temperature, …
Convective Mass Flux, Water, Ice, …
Turbulent Kinetic Energy …
Diabatic Heating …


            Atmospheric “Model” History Tape
     The Transport Application
Residual Circulation                       Wave Transport
                                           Planetary
            (u*,v*)                                    (u,v)
                                  MIXING   Synoptic




                       Banff, May 2005
PDFs of total ozone: observations & CTM




     DAS-driven                          GCM-driven
     •Means displaced                    •Means displaced
     •Spread too wide                    •Half-width ok

     Too much tropical-extratropical mixing in DAS
       Douglass, Schoeberl, Rood and Pawson (JGR, 2003)
Three-dimensional trajectory calculations
    UKMO
    UKMO       UKMO
               UKMO         DAO
                            DAO        DAO
                                       DAO         GCM
                                                   GCM        (50 days)
    Diabatic   Kinematic   Diabatic   Kinematic   Kinematic




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
    Schoeberl, Douglass, Zhu and Pawson (JGR, 2003)
   Transport have we reached a wall?
TRANSPORT with winds from assimilation
Residual Circulation               D          Wave Transport

               D                                          C(u,v)
                                              Planetary
     (u*,v*)                MIXING
                                              Synoptic


    – 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?
                     Banff, May 2005
  Major assimilation issue: Bias
Primary Products Errors, (eb, ev) = (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.
        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?



                     Banff, May 2005
MIPAS Ozone assimilation
           • 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
Monitoring Data System

                           EP TOMS
                           Going Bad




                          Adjustment
                          To change in
                          observing
                          system


        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
                      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?

                      Banff, May 2005
            Quality Control:      Statistical
                                  Analysis
     Interface to the Observations
                                                                                   Q.C.


  Satellite # 1     Self-comparison




                                       Intercomparison




                                                                   Comparison to
                                                         DATA




                                                                    “Expected”
  Satellite # 2     Self-comparison                      GOOD
                                                         BAD
                                                                                   GOOD!
                                                         SUSPECT

Non-Satellite # 1   Self-comparison
                                                                       O
                             “Expected Value”
    MODEL
    Forecast
Memory of earlier                                             MONITOR
  observations                                                      ASSIMILATE
      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                        What we’re really interested in!
       • Model failure
       • New extreme of variability

                             Banff, May 2005
      Link to the adaptive observing
What to Observe?                                           Obs - Forecast
                       Data
What to Process?
                     Anomalies
                                                            Analysis
Features                                                    Increments


                           Statistical
                           Analysis
                                                   “Analysis”
                                                    Analysis

                              Q.C.             Winds
                                             Temperature

            Short-term       Tracer
             Forecast        Model
           (15 minutes)
    Long-term forecast     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

                            Banff, May 2005