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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

land

ice

Troposphere

coupler

Troposphere

ocean

Dynamics / Physics

Advection

Mixing

Radiation

Convection

PBL

Clouds

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

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

Simulation Environment(General Circulation Model, “Forecast”)(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

Choice of technique to approximate operations in representative equations

Rood (1987, Rev. Geophys.)

Gridded Approach

Orthogonal?

Uniform area?

Adaptive?

Unstructured?

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:

- Divergence theorem: for the advection-transport process
- Green’s theorem: for computing the pressure gradient forces
- 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

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

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

Data

Boundary Conditions

Emissions, SST, …

e

(OPfOT + R)x = Ao – OAf

DA/Dt = P –LA – n/HA+q/H

e

Discrete/Error Modeling

(An+Dt – An)/Dt = …

e

Constraints on Increments

∂ug/∂z = -(∂T/∂y)R/(Hf0)

Scale Analysis

(eb, ev) reduced

Ai≡ T, u, v, F, H2O, O3 …

(eb, ev)

Inconsistent

Pot. Vorticity, v*, w*, …

Consistent

Assimilation EnvironmentO 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

Sat

Rad

Sat

Rad

Bal

Geo

Sat

Rad

Sat

Rad

Bal

Geo

Sat

Rad

Space and Time

Interpolation

To Measured

Quantity

Sat

Rad

Bal

Geo

Sat

Rad

Bal

Geo

Sat

Rad

Sat

Rad

Ship

Tem

Sat

Rad

Ship

Tem

Sat

Rad

Ship

Tem

Sat

Rad

Bal

Geo

Satellite

Balloon

Ship

Radiance

Geopotential

Temperature

Model Forecast

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

(Assimilation)

Statistical

Analysis

Analysis

&

(Observation

Minus

Analysis)

Error

Covariance

Quality Control

Data Stream 2

(Monitoring)

Model

Forecast

Forecast / Simulation

Banff, May 2005

TOMS/SBUV

POAM/MIPAS

Ozone Data

Sciamachy

MLS

Forecast &

Observation

Error Models

Statistical

Analysis

Statistical

Analysis

Analysis

Q.C.

Winds

Temperature

Tracer

Model

Short-term

Forecast

(15 minutes)

Long-term forecast

What does an assimilation system look like?(Goddard Ozone Data Assimilation System)Obs - Forecast

Analysis

Increments

HALOE

Sondes

“Analysis”

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

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

Advection

Mixing

Convection

PBL

Emissions

J Rates

React. Rate

Solver

Wet/Dry

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

Synoptic

(u*,v*)

(u,v)

MIXING

Banff, May 2005

PDFs of total ozone: observations & CTM

GCM-driven

DAS-driven

- Means displaced
- Spread too wide

- Means displaced
- Half-width ok

Too much tropical-extratropical mixing in DAS

Douglass, Schoeberl, Rood and Pawson (JGR, 2003)

UKMO

DAO

DAO

GCM

Kinematic

Diabatic

Kinematic

Diabatic

Kinematic

Kinematic: considerable vertical and horizontal dispersion

Diabatic: vertical dispersion reduced (smooth heating rates)

Three-dimensional trajectory calculationsUKMO

UKMO

DAO

DAO

GCM

(50 days)

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

D

Residual Circulation

- 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?

Wave Transport

Planetary

Synoptic

D

C

(u*,v*)

MIXING

(u,v)

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

MIPAS data

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

Analysis

Q.C.

Quality Control: Interface to the ObservationsSatellite # 1

Self-comparison

DATA

GOOD

BAD

SUSPECT

Comparison to

“Expected”

Satellite # 2

Self-comparison

GOOD!

Intercomparison

Non-Satellite # 1

Self-comparison

O

“Expected Value”

MODEL

Forecast

Memory of earlier

observations

MONITOR

ASSIMILATE

What we’re really interested in!

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

Analysis

Statistical

Analysis

Analysis

Q.C.

Winds

Temperature

Tracer

Model

Short-term

Forecast

(15 minutes)

Long-term forecast

Link to the adaptive observingObs - Forecast

What to Observe?

What to Process?

Data

Anomalies

Analysis

Increments

Features

“Analysis”

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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