atmospheric data assimilation.pptx

7/28/2019 Atmospheric DATA ASSIMILATION.pptx

1/70

ATMOSPHERICDATA ASSIMILATION

By Roger Daley*

Journ al of the Meteoro log ical Soc iety o f Japan, Vol. 75, No. 1B, pp.319-329, 1997

(Manu scr ipt received 23 May 1995, in revis ed from 15 February 1996)

*Naval Research Laboratory, 7 Grace Hopper Avenue, Monterey CA 93943-5502, USA


2/70

DEFINITION

Data assimilation is an analysis technique

in which the observed information is

accumulated into the model state by taking

advantage of consistency constraints with laws of

time evolution and physical properties.

-F. Bouttier and P. Courtier


3/70

OBJECTIVE (1)

to produce a regular, physically consistent

4 dimensional representation of the state of the

atmosphere from a heterogeneous array of in situand remote instruments which sample imperfectly

and irregularly in space and time.

-Roger Daley


4/70

OBJECTIVE (2)

to provide a dynamically consistent

motion picture of the atmosphere and

oceans, in three space dimensions, with

known error bars.

-M. Ghil and P. Malanotte-Rizzoli


5/70

OBJECTIVE (3)

Extracts the signal from noisy observations

(filtering)

Interpolates is space and time (interpolation)

Reconstructs state variables that are not

sampled by the observation network

(completeness)

-R.Daley


6/70

KEEP IN MIND (1) What is the purpose of the DA weather

prediction, physical understanding, signal

detection, environmental monitoring, etc?

What are the physical characteristics of thephenomenon of interest?

What are its temporal and spatial

characteristics and what relations exist

between state variables? What are the characteristics of other

physical phenomena which might obscure

the desired signal?


7/70

KEEP IN MIND (2)

What are the characteristics of the observing

system?

Is the observing system largely under the

control of the scientist (as in field

experiment) or is it given?

Is it possible to influence the design of the

observing system, can DA techniques be

used in the observation system design?


8/70

KEEP IN MIND (3)

All models and observations are approximate

The resulting analyses will be approximate

The observations must be combined in some

optimal fashion

It is better to have enough observations tooverdetermine the problem

The model is used to provide the preliminary

estimate

The final estimate should fit the observations withintheir (presumed) observation error.


9/70

MAXIMUM LIKELIHOOD

ESTIMATION (1)

Zero dimensional/scalar case and a definevariable x

Observation x o and a forecast x f (produced bya model)

Observation error: o= x ox Model error: f= x fx These errors are assumed to be random,

unbiased, normally distributed.


10/70

MAXIMUM LIKELIHOOD

ESTIMATION (2)

A variable w/c is normally distributed with mean 0

and variance 2 has a probability distribution

() = 2 0.5 _1exp(2/22)

The joint probability distribution of errors is

(0, ) = 20.5 _1exp(2

/22 2

/22)

error variance


11/70


12/70

MINIMUM VARIANCE

ESTIMATION

Unbiased linear estimate ofxx e = cox o + cfxf(co, cf non-negative)

co + cf = 1 Unbiased linear estimate error:e= x ex Expected error variance ofx: = (co)22+ (cf)22 Best LinearUnibiasedEstimate

x e = (2x o + 2x f )(2+ 2)-1


13/70

MAXIMUM LIKELIHOOD ESTIMATE

VS.

MINIMUM VARIANCE ESTIMATE The maximum likelihood estimate

x a = x f + 2(2 + 2)-1(x o - x f)-finds the mode Best LinearUnibiasedEstimate

x e = (2x o + 2x f )(2+ 2)-1-finds the mean

When the error probabilities are normally-distributed as in() = 2 0

.5 _1exp(2/22) , the mean and themode and the minimum variance and maximum likelihoodestimates are the same.


14/70

THE L2 NORM (1)

In most meteorological practice, L2 norms are usedbecause they lead to linear analysis equations.

L2 norm estimation yields the mean, L1 estimationgives the median and Lestimation determines themid-range.


15/70

THE L2 NORM (2)

Example: estimation based on 5 observations.

Assume that each observation has the same

observation error variance 2

The 5 observation values:

-22.5, 1.1, 1.2, 1.3 and 650

It seems likely that there must have been severe

measurement problems in the first and lastobservations.


16/70

THE L2 NORM (3)

The mean value of the observations is 126.3 (L2)

The median value is 1.2 (L1)

The mid-range value is 313.95 (L)

In this example, minimization with respect to the L1norm gives the most credible estimate.

The L1 norm is much superior to the L2 norm when itcomes to detecting and removing gross errors.

In atmospheric data assimilation, there are situationswhere errors are not normally distributed.


17/70


18/70


19/70

All observing systems have their limitations, problems,

and failures, resulting in the reported measurementsbeing sometimes incorrect.

Such data must be identified and rejected by the data

assimilation system in order to avoid corruption of theanalysis.

Due to the amount of data handled this is done byautomatic routines, both in the form of preprocessing andduring the data assimilation stage.

Xiang-Yu Huang and Henrik Vedel


20/70


21/70

THREE-DIMENSIONAL

SPATIAL ANALYSIS


22/70


23/70

THE VECTOR CASE (2)

Assume that the observation and forecast error are

unbiased, normally-distributed and not mutually

correlated. That is,

= = 0

T is the matrix transpose.


24/70


25/70


26/70


27/70


28/70


29/70

GENERAL OBSERVATION NETWORKS(3) THE MODIFIED COST FUNCTION

Orig: J = 0.5[xo-xa]TR-1[xo-xa] + 0.5[xf-xa]T[Pb]-1[xf-xa]

Mod: J = 0.5{[yo-H(xa)]TR-1[yo-H(xa)] + 0.5[xf-xa]T[Pb]-1[xf-xa]

Reason: the observed and forecast variables are not

necessarily the same. Observed variable asyo and forecast variable as xf.


30/70

GENERAL OBSERVATION NETWORKS(4) THE MODIFIED MAXIMUM LIKELIHOOD ESTIMATE

Orig: xa = xf+ Pb[Pb + R]-1[xo-xf]

Mod: xa = xf+ PbHT[HPbHT+ R]-1[yoH(xf)]

Note: H(x) is frequently a non-linear operator, but it

can be linearized by defining the tangent linear operatorH = ()


31/70

GENERAL OBSERVATION NETWORKS(5) THE MODIFIED ANALYSIS ERROR COVARIANCE

Orig: [Pa]-1 = R-1 + [Pb]-1

Mod: [Pa]-1 = HTR-1H + [Pb]-1


32/70


33/70

THE OPTIMAL INTERPOLATION (OI)

TECHNIQUES (2)

OI is often simplified so that it does not produce a whole

domain analysis, but rather a number of local analyses at

each gridpoint or small grid volume.

OI methodology is sufficiently powerful to perform

credible multivariate analysis that is, where the

observations of one variable (temperature, say) are used

in the analysis of another variable (wind, say).

This is done by incorporating linear diagnostic relations

(such as geostrophic and hydrostatic balance) between

two variables in the forecast error covariance Pb


34/70


35/70


36/70


37/70

THE FORECAST/ANALYSIS CYCLE (1)

We modify the notation to introduce forecast x

and

analysis x vectors at time

We define a model, which marches forward in time from

time to time x = M (x )


38/70

THE FORECAST/ANALYSIS CYCLE


39/70

THE FORECAST/ANALYSIS CYCLE(3) THE CYCLE


40/70

DYNAMICALLY-GENERATED ANALYSIS WEIGHTS(1) THE KALMAN FILTER

While the forecast x

is responsive to all the complexitiesof atmospheric flow simulated by the model, the forecast

error covariancesbspecified in the OI and 3DVAr

algorithms are completely insensitive to the flow.

Modern assimilation techniques attempt to generate the

analysis weights dynamically, explicitly using the model

M.


41/70


One way of doing this, is through an explicit evolution

equation for the forecast error covariance.

Defining the tangential linear model =

()

and

assuming that model M is imperfect, the forecast error at timetn is related to the analysis error covariance at time tn-1 by

b =

T

Where Qn= is the model error covariance and

nis the model error


42/70


The fundamental equation of the Kalman filter algorithm:

b =

T

(an equation for the propagation of second moment error statistics)

The non-linear form of Kalman filter is referred to asextended Kalman filter (EKF)

The EKF is a sequential algorithm that makes use of the

past and present observations.

Statistical error moments higher than the second are

generated.


43/70


44/70


45/70

THANK

YOU


46/70


47/70


48/70

THE INVERSE PROBLEM (2)

The objective of an inverse problem is to find the bestmodel m such that (at least approximately)

d = G(m)

where G is an operator describing the explicitrelationship between the observed data, d , and the modelparameters.

In various contexts, the operator G is calledforwardoperator, observation operator, orobservation function.

In the most general context, G represents the governingequations that relate the model parameters to theobserved data (i.e. the governing physics).


49/70

SIMPLE KALMAN FILTER EXAMPLE (1)SOURCE: http://credentiality2.blogspot.com/2010/08/simple-kalman-filter-example.html

Kalman filters are a way to take a bunch of noisy

measurements of something, and perhaps also some

predictions of how that something is changing, and maybeeven some forces we're applying to that something, and to

efficiently compute an accurate estimate of that

something's true value.
http://credentiality2.blogspot.com/2010/08/simple-kalman-filter-example.htmlhttp://credentiality2.blogspot.com/2010/08/simple-kalman-filter-example.htmlhttp://credentiality2.blogspot.com/2010/08/simple-kalman-filter-example.htmlhttp://credentiality2.blogspot.com/2010/08/simple-kalman-filter-example.htmlhttp://credentiality2.blogspot.com/2010/08/simple-kalman-filter-example.htmlhttp://credentiality2.blogspot.com/2010/08/simple-kalman-filter-example.htmlhttp://credentiality2.blogspot.com/2010/08/simple-kalman-filter-example.htmlhttp://credentiality2.blogspot.com/2010/08/simple-kalman-filter-example.html


50/70

SIMPLE KALMAN FILTER EXAMPLE (2)

Let's say we want to measure the temperature in a room. We

think it's about 72 degrees, 2 degrees. And we have a

thermometer that gives uniformly random results within a

range of 5 degrees of the true temperature.

We take a measurement with the thermometer and it reads

75. So what's our best estimate of the true temperature?

Kalmanfilters use a weighted average to pick a point

somewhere between our 72 degree guess and the 75 degreemeasurement.


51/70


Here's how we choose the optimal weight, given the accuracy ofour guess and the accuracy of the thermometer:

weight =temperature variance

temperature variance + thermometer variance

0.29 =

+

If the weight is large (approaching 1.0), we mostly trust ourthermometer. If the weight is small, we mostly trust our guess

and ignore the thermometer. 29% weight means we'll trust our guess more than the

thermometer, which makes sense, because we think our guessis good to 2 degrees, whereas the thermometer was onlygood to 5


52/70


53/70


how confident are we in our estimate of 72.87 degrees?

estimate variance =temperature variancethermometer variance


1.43 = +

So we think our estimate is correct to 1.43 degrees

we have a guess that the temperature in the room is

72.87 degrees, 1.43 degrees.


54/70


Now from the guess that the temperature in the room is

72.87 degrees, 1.43 degrees.

And we still have a thermometer that tells the temperature 5 degrees.

That's basically the situation where we started, so we can run the wholealgorithm again:

First we compute the weight, using our new, more accurate guess

confidence:

weight = temperature variancetemperature variance + thermometer variance

0.22 =.43

.43+


55/70


56/70


And the new confidence level:

estimate variance =temperature variancethermometer variance


1.11 = .43.43+

So after the second measurement, we estimate that the

actual temperature is

72.46 degrees, 1.11 degrees


57/70


58/70

HIGH FREQUENCY INTERFERENCE (1)

The atmosphere has a number of timescales.

In general, only limited frequency band will be important

for a given atmospheric DA application.

For synoptic and planetary scale forecast/analysespurposes, it is timescales of approximately one hour to

one week which are of interest

For mesoscale or convective scale modeling, the

relevant timescales are minutes to hours For climate/environmental monitoring purposes,

timescales from weeks to decades are of interest.


59/70


60/70


In atmosphere, high frequency modes generally haverelatively low amplitude.

However, when models are integrated from analyses

produced from observations by some analysisalgorithm, high frequency oscillations of an amplitudemuch larger than observed in nature may be excited.

These oscillations may completely obscure thephenomena of interest or may cause perfectly goodobservations to be rejected.


61/70


62/70


Illustration of the effect of initialization.


63/70


64/70


65/70


66/70


67/70

THE PRIMITIVE EQUATIONS

V(dot)+f k + = ()

+=

+

=

The LHS of the equation s have been linearized about a state of

rest w ith a domain-averaged temperature f ield.


68/70

INTIALIZATION

Primitive equations are written as:

z (dot) + Wzz = Rz

y (dot) + Wyy = Ry

Notes: Amplitudes of high frequency modes denoted by vectorz and the

corresponding frequencies by the diagonal matrix Wz Amplitudes of slower (Rossby) modes denoted by vectory and the

corresponding frequencies by the diagonal matrix Wy

Rz and Ry are projections of Rv and R onto the fast and slowmodes, respectively and each is a non-linear function of both zand y


69/70


70/70

SUPPRESSION OF HIGH FREQUENCIES IN

KF AND 4DVAR

KF

-high frequencies can be suppressed using normal

mode theory of Q

4DVAR

-most successful is using the Machenhauer condition

as a constraint in the minimization of J = Jp + Jn (from n=0to N)

atmospheric data assimilation.pptx

Documents