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
land
coupler
Troposphere
ice
Troposphere
Clouds
PBL
Convection
Radiation
Mixing
Management of complexity
But, complex and costly
Dynamics / Physics
Advection
ocean
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
Representative Equations
Discrete/Parameterize
Theory/Constraints
Primary Products (i.e. A)
Derived Products (F(A))
Emissions, SST, …
e
e
DA/Dt = P –LA – n/HA+q/H
(An+Dt – An)/Dt = …
(ed, ep)
∂ug/∂z = -(∂T/∂y)R/(Hf0) Scale Analysis
T, u, v, F, H2O, O3 …
Pot. Vorticity, v*, w*, …
(eb, ev)
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

(A,U)
Grid Point (i,j)
Choice of where to
Represent Information
Gridded Approach
Orthogonal?
Uniform area?
Adaptive?
Unstructured?
Choice of technique to
approximate operations in
representative equations
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)
Banff, May 2005
Grid Point (i+1,j)
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
∫
Banff, May 2005
Line Integral
around discrete
volume
“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
Banff, May 2005
S. Pawson, primary contact
Importance of your decisions
(Precipitation in full GCM)
Spectral Dynamics
Community Atmosphere
Model / “Eulerian”
Finite Volume Dynamics
Community Atmosphere
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
Emissions, SST, …
Data
e
Boundary Conditions
DA/Dt = P –LA – n/HA+q/H
e
(OPfOT + R)x = Ao – OAf
(An+Dt – An)/Dt = …
∂ug/∂z = -(∂T/∂y)R/(Hf0)
e
Discrete/Error Modeling
Scale Analysis
Constraints on Increments
Ai ≡ T, u, v, F, H2O, O3 … (eb, ev)
Pot. Vorticity, v*, w*, … Consistent
(eb, ev) reduced
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?
Model Forecast
Sat
Sat
Bal
Sat
Bal Sat
Sat
Sat
Sat Bal
Bal
Sat
Sat
Ship
Sat
Ship
Sat
Ship
Sat
Bal
Satellite
Balloon
Ship
O – The Observation Operator
Banff, May 2005
To Measured
Quantity
Space and Time
Interpolation
(OPfOT + R)x = Ao – OAf
Rad
Rad
Rad Geo
GeoRad
Rad
Rad
Rad Geo
Geo
Rad
TemRad
Rad
Tem
Rad
Tem
Rad
Geo
Radiance
Geopotential
Temperature
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
Data Stream 1
(Assimilation)
Data Stream 2
(Monitoring)
Quality Control
Observation minus Forecast
Statistical
Analysis
Error
Covariance
Model
Forecast
Forecast / Simulation
Banff, May 2005
Analysis
&
(Observation
Minus
Analysis)
What does an assimilation system look like?
(Goddard Ozone Data Assimilation System)
Ozone Data
TOMS/SBUV
POAM/MIPAS
Obs - Forecast
Ozone Data
Sciamachy
MLS
Analysis
Increments
Forecast &
Observation
Error Models
Statistical
Analysis
Q.C.
Short-term
Forecast
(15 minutes)
Long-term forecast
“Analysis”
Analysis
HALOE
Sondes
Winds
Temperature
Tracer
Model
BALANCE, BALANCE, BALANCE!
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
Input Fields “ONE WAY COUPLER”
Winds, Temperature, …
Convective Mass Flux, Water, Ice, …
Turbulent Kinetic Energy …
Diabatic Heating …
Atmospheric “Model” History Tape
Wet/Dry
Solver
React. Rate
J Rates
Emissions
PBL
Convection
Mixing
Advection
Transport / Chemistry
The Transport Application
Residual Circulation
(u*,v*)
Wave Transport
MIXING
Banff, May 2005
Planetary
Synoptic
(u,v)
PDFs of total ozone: observations & CTM
DAS-driven
•Means displaced
•Spread too wide
GCM-driven
•Means displaced
•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
Diabatic
Kinematic
Diabatic
Kinematic
Kinematic
(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
Schoeberl, Douglass, Zhu and Pawson (JGR, 2003)
Transport have we reached a wall?
TRANSPORT with winds from assimilation
Residual Circulation
(u*,v*)
D
D
MIXING
Wave Transport
Planetary
Synoptic
C(u,v)
– 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,
O3 – T,
O3 – T,
O3 – T,
Chemistry (P and L) – Seconds – hours – 
Transport (U) – Hours – Days – 
Diabatic forcing – Days – Months – 
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 # 2
Self-comparison
Non-Satellite # 1
Self-comparison
DATA
GOOD
BAD
SUSPECT
Comparison to
“Expected”
Self-comparison
Intercomparison
Satellite # 1
GOOD!
O
MODEL
Forecast
Memory of earlier
observations
“Expected Value”
MONITOR
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
• Model failure
• New extreme of variability
Banff, May 2005
What we’re really interested in!
Link to the adaptive observing
What to Observe?
What to Process?
Obs - Forecast
Data
Anomalies
Analysis
Increments
Features
Statistical
Analysis
Q.C.
Short-term
Forecast
(15 minutes)
Long-term forecast
Tracer
Model
Banff, May 2005
“Analysis”
Analysis
Winds
Temperature
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