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, May 2005
Plan of Presentations
• Models and Modeling
• Data Assimilation
– What is it?
– Why?
• 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
•   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

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.

• 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
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
(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
(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
• 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
– 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

To Measured

Quantity
Interpolation
Sat
Bal      Sat Bal                    Geo     Rad Geo
Sat
Bal                               Geo

Model Forecast                            Balloon                           Geopotential
Ship                              Temperature

O – The Observation Operator

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

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
•   Forecast initialization
•   “Background,” a priori profile, for retrievals
•   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

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

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
• 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
stratosphere
• Model + Data capture synoptic
information
Monitoring Data System

EP TOMS

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
– 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
GOOD!
SUSPECT

Non-Satellite # 1   Self-comparison
O
“Expected Value”
MODEL
Forecast
Memory of earlier                                             MONITOR
observations                                                      ASSIMILATE
• Good Data
– Normal range of expectation
– Spatial or temporal consistency
–   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
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

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