3D-Var Revisit3D-Var Revisitededandand

Quality Control of Surface Temperature DataQuality Control of Surface Temperature Data

Xiaolei ZouXiaolei Zou

Department of MeteorologyDepartment of Meteorology

Florida State UniversityFlorida State University

zou@met.fsu.eduzou@met.fsu.edu

June 11, 2009June 11, 2009

OutlineOutline

• 3D-Var Formulation3D-Var Formulation• Statistical FormulationStatistical Formulation• AnalysisAnalysis• Practical ApplicationsPractical Applications

Part I:

• MotivationMotivation• EOF analysisEOF analysis

• QC for QC for TTss

Part II:

Part I Part I

3D-Var Revisited3D-Var Revisited

FactsFacts

1) All background fields, observation operators and observations have errors.

2) There is no truth. Errors in background, observation operator and observations can only be estimated approximately.

Produce the best analysis by combining all available information.

The GoalThe Goal

QuestionsQuestions

1) What is the measure of the best analysis?

2) How to combine all available information?

Variational FormulationVariational Formulation

J(x0 ) =12(x0 −xb)T B−1(x0 −xb) +

12(H(x0 )−yobs)T (O+ F)−1(H(x0 )−yobs)

A scalar cost function is defined:

x0 ← analysis of the atmospheric state

yobs ← observations

H ← observation operator

xb ← background

B ← background error covarnace matrix O ← observation error covarnace matrix F ← forward model error covarnace matrix

where

Statistical FormulationStatistical Formulation

€

pobs(y | yobs)

pb (x0 | xb )

€

H(x0)€

yobs

€

xb

Available information

pH (y | H (x ot ))

PDF

Write the PDFs for all three sources of information as:

€

pobs

€

pb€

pH

€

σ(x0,y) = pobs pb pH

Joint PDF:

PDF of the a posteriori state of information

The Bayes TheoremThe Bayes Theorem

The marginal PDF of the a posteriori state of information:

is the PDF of the a posteriori state of information in model space.

€

σ(x0) = σ (x0,y)dy∫ = pb (x0 | xb ) pobs(y | yobs)∫ pH (y | H(x0))dy

Application of Bayes TheoremApplication of Bayes Theoremto Data Assimilationto Data Assimilation

σ (x0 ),Data assimilation derives some features of the PDF, which is the a posteriori state of information in model space.

• The maximum likelihood estimate

~ analysis

• The covariance matrix of this estimate

~ analysis error covariance A

σ (x0a ) = max

x0

σ (x0 )

x0a

A = σ(x0∫ ) x0 −x0a( )

Tx0 −x0

a( )dx0

Assuming All Errors Are Gaussian,Assuming All Errors Are Gaussian,

The PDF for yobs:

pobs (y yobs ) =C1 exp −12

y−yobs( )TO−1 y−yobs( )

⎛⎝⎜

⎞⎠⎟

pb (x0 xb ) =C2 exp −12

x0 −xb( )TB−1 x0 −xb( )

⎛⎝⎜

⎞⎠⎟

pH (y H (x0 )) =C3 exp −12

y−H(x0 )( )TF−1 y−H(x0 )( )

⎛⎝⎜

⎞⎠⎟

The PDF xb:

The PDF for H(x0):

Bayes Estimate Under Bayes Estimate Under Gaussian AssumptionsGaussian Assumptions

pobs (y yobs ) =C1 exp −12

y−yobs( )TO−1 y−yobs( )

⎛⎝⎜

⎞⎠⎟

pb (x0 xb ) =C2 exp −12

x0 −xb( )TB−1 x0 −xb( )

⎛⎝⎜

⎞⎠⎟

pH (y H (x0 )) =C3 exp −12

y−H(x0 )( )TF−1 y−H(x0 )( )

⎛⎝⎜

⎞⎠⎟

σ (x0 ) = pb (x0 | xb ) pobs (y | yobs )∫ pH (y | H (x0 ))dy

σ (x0 ) = C exp −1

2x0 − xb( )

TB−1 x0 − xb( ) + H (x0 ) − yobs( )

TO + F( )

−1H (x0 ) − yobs( )( )

⎛⎝⎜

⎞⎠⎟

Maximum Likelihood EstimateMaximum Likelihood Estimate

σ (x0 ) = C exp −1

2x0 − xb( )

TB−1 x0 − xb( ) + H (x0 ) − yobs( )

TO + F( )

−1H (x0 ) − yobs( )( )

⎛⎝⎜

⎞⎠⎟

= C exp −1

2J(x0 )

⎛⎝⎜

⎞⎠⎟

Maximizing

€

σ(x0)

€

J(x0)Minimizing

€

⇔

The PDF of the a posteriori state of information in model space:

Statistical Estimate Variational Calculus

Gaussian and Non-Gaussian signalsGaussian and Non-Gaussian signals

The signals are sampled at 10000 points. PDFs are constructed at an interval of (ymax −ymin) /100.

y =rand(x)

y =4sinπx1000

⎛⎝⎜

⎞⎠⎟

Gaussian and Non-Gaussian signalsGaussian and Non-Gaussian signals

y =4sinπx1000

⎛⎝⎜

⎞⎠⎟+ rand(x)

y =0.4sinπx1000

⎛⎝⎜

⎞⎠⎟+ rand(x)

3D-Var & 3D-Var Analysis

The 3D-Var data assimilation solves a general inverse problem using the maximum likelihood estimate under the assumptions that all errors are Gaussian.

The 3D-Var analysis is the maximum likelihood estimate if all errors are Gaussian.

Zero Gradient: A necessary ConditionZero Gradient: A necessary Condition

J(x0 ) =12(x0 −xb)T B−1(x0 −xb) +

12(H(x0 )−yobs)T (O+ F)−1(H(x0 )−yobs)

∇J(x0 ) = B−1(x0 − xb ) + HT (O + F)−1(H (x0 ) − yobs )

B−1(x0 −xb) +HT (O+ F)−1(H(x0 )−yobs) =0

J(x0 + Δx0 )−J (x0 ) = ∇Jx0

( )TΔx0

∇J(x0 ) = 0

a linear operatora nonlinear operator

H {

Analytical Expression of Solution Analytical Expression of Solution with a Linear Modelwith a Linear Model

B−1(x0* −xb) +HT (O+ F)−1(H(x0

* )−yobs) =0

H is linear: H (x0 ) =Hx0

x0* −xb +BHT (O+ F)−1(Hx0

* −yobs) =0

x0* =xb + HTR−1H +B−1( )

−1HT O+ F( )−1 yobs −Hxb( )

Analytical Expression of Solution Analytical Expression of Solution with an Approximate Linear Modelwith an Approximate Linear Model

B−1(x0* −xb) +HT (O+ F)−1(H(x0

* )−yobs) =0

B−1 x0* −xb( ) +HT (O+ F)−1 H x0

* −x( )b− H(xb)−yobs( )( ) =0

H (x0* ) ≈H(xb) +H x0

* −xb( )

x0* =xb + HT O+ F( )−1 H +B−1

( )−1

HT O+ F( )−1 yobs −H xb( )( )

=xb +BHT HBHT +O+ F( )−1

yobs −H xb( )( )

Analysis ErrorAnalysis Error

When linear approximation is valid,When linear approximation is valid, the a posteriori PDF is approximately Gaussian, with the analysis as its mean and the following covariance matrix:

A = HTR−1H +B−1( )−1

=B−BHT HBHT +R( )−1

HB

3D-Var Analysis

A−1 =HT O+ F( )−1 H +B−1

A-1 is referred to as an information content matrix. When the analysis error is small, the value of ||A-1|| is large, the information content is large.

A = HTR−1H +B−1( )−1

A−1 ≥ B−1 , A−1 ≥ O+ F( )−1

The information content of the 3D-Var analysis is greater than the information content in either the background field or the observations that were assimilated.

3D-Var Practice

• Develop System Decision on variables and resolutions Estimate of background error covariance• Assimilate Data Decision on observations to be assimilated Understanding of the observations Estimate of observation errors Comparison between observations and background Development of the observation operator Estimate of model errors• Obtain Solution Minimization (preconditioning, scaling) Advanced computing (parallelization, data intensive computing platforms)

What does 3D-Var data assimilation involve?

J(x0 ) =12(x0 −xb)T B−1(x0 −xb) +

12(H(x0 )−yobs)T (O+ F)−1(H(x0 )−yobs)

F

xb yobsx0

H

B O

Choice of analysis variable

What data to assimilation?

Which model to use?

What background to start with?

How to estimate elements in B?

Where to find their values?

How to quantify it?

Model Space Observed Space

x0

3D-Var analysis

+

What need to be done before and after conducting 3D-Var

experiments?

3D-Var3D-VarInput Data Output Analysis

Quality Control Diagnosis of Analysis

What need to be done before and after conducting 3D-Var experiments?

• Quality Control Knowing the data Knowing the major difference between data and background field Remove errorneous data Eliminate data that render errors non-Gaussian • Diagnosis of 3D-Var analyses Check the convergence Examine the analysis increments Estimate analysis errors Assess forecast impact Provide physical and dynamical explanations to the numerical results one obtains

When Working with Real-Data,When Working with Real-Data,

The key things areThe key things are

• Knowing the dataKnowing the data before inputting them before inputting them

into a 3D-Var system by a careful QC!into a 3D-Var system by a careful QC!• Kowing the systemKowing the system after a 3D-Var after a 3D-Var

experiment by a careful analysis of experiment by a careful analysis of

the 3D-Var results!the 3D-Var results!

Examing 3D-Var ResultsExaming 3D-Var Results

Analysis - obsAnalysis - obsone-week average one-week average

resultsresults

q p

u v

Differences between Differences between model and obs.model and obs.beforebefore and and afterafter a a 3D-Var experiment3D-Var experiment

pb - pobs and pa-pobs

29

σ ob2

σ a2

σ o2

σ b2

σ oa2

Inferred from

calculated

σ b2 = σ ob

2 − σ o2 σ a

2 = σ oa2 − σ o

2and

Part IIPart II

Quality Control of Surface Quality Control of Surface Temperature DataTemperature Data

31

Motivations

• Surface data are abundant • Very little surface data are assimilated in operational systems• Surface data are important to thunderstorm prediction

Challenges

• Existing data assimilation systems have short or no memory of surface data• Diurnal cycle dominants the variability of surface variability and is not described with sufficient accuracy in large-scale analysis which is used as background in mesoscale forecast• Background errors are non-Gaussian

32

A Total of 3197 Surface Stations

The number of missing data at each station in January 2008 is indicated by color bar.

33

Improving Surface Data Assimilation

Key steps:

1) Inclusion of more surface data

2) Improved QC

3) Vertical interpolation based on the atmospheric

structures within the boundary layer

Surface layer

Mixed layer

3) Incorporation of dynamic constraint

34

EOF Modes for Ts Constructed from Station Observations

First Second Third

Fourth Fifth Sixth

35

EOF Modes for Ts Constructed from Station Observations (cont.)

Seventh Eighth

Ninth Tenth

36

Explained Variances

Surface Data (blue)NCEP analysis (red)

37

10

14

18

-10

0

10

-10

-5

0

5

-4

0

4

-4

0

4

-4

0

4

Fi rst

Second

Thi rd

Fourth

Fi f th

Si xth

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29Ti me (uni t: day)

Principal Components (PCs)

38

Principal Components (PCs)

-3

0

3

-3

0

3

-2

0

2

-2

0

2

Seventh

Ei ghth

Ni nth

Tenth

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29Ti me (uni t: day)

39

Dominant Oscillations in January 2008P

erio

d (

un

it:

day

)

Obs.

NCEP

EOF mode EOF mode EOF mode

Per

iod

(u

nit

: h

our)

Per

iod

(u

nit

: h

our)

Longer-periodoscillation

Diurnal oscillation Shorter-periodoscillation

40

Diurnal Oscillation

-10

0

10

-4

0

4

-4

0

4

-6

0

6

-6

0

6

-6

0

6

-6

0

6

Tenth

Ni nth

Si xth

Fi f th

Fourth

Thi rd

1 2 3 4 5 6 7Ti me(uni t: day)

Second

41

Longer-Period Oscillations

-12

0

12

-12

0

12

-12

0

12

-12

0

12

-12

0

12

Si xth

Seventh

Ei ghth

Ni nth

Tenth

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Ti me (uni t: day)

42

Diurnal Oscillationand

Longer-Period Oscillations

Phase difference

Amplitude difference

43

PC Differences between Surface Data and NCEP Analysis

Second Third

Fourth Fifth

Sixth

Time (unit: day)

Time (unit: day)

Blue line: First WeekRed line: Last Week

44

Frequency Distributions of Diurnal Cycle Modes

First Week

Third

FourthFourth

Fifth

Fifth

SixthFre

qu

ency Second

Tobs-TNCEP (unit: K)F

req

uen

cy

Fre

qu

ency

Fre

qu

ency

Fre

qu

ency

Last Week First Week Last Week

Fourth

Sixth

Tobs-TNCEP (unit: K)

January 2008

Second

Fourth

Sixth

Third

Fifth

45

Frequency Distributions (modes 2-6)

First Week Last Week

Entire Month

Fre

qu

ency

Fre

qu

ency

Tobs-TNCEP (unit: K)

Tobs-TNCEP (unit: K)Tobs-TNCEP (unit: K)

Fre

qu

ency

Sum of Modes 2-6

46

Statistical Measures

Mean Variance

Kurtosis Skewness

47

QC Procedure

Step 1:

1) Historical extremum checkΔT > The average of NCEP analysis of each station pluses (minuses) 15-times its variance

• Temporal consistency check ΔT > 50℃ in 24-hours interval1) Bi-weight check

Z-score > 3• Spatial consistency checkT > The average of linear fit to highly correlated

stations pluses (minuses) 4-times its variance

48

QC Procedure (cont.)

Step 2:

The Z-score of the difference between station observation and background field must less than 4

Step 3:

The Z-score of the difference between station observation and background field excluding the contribution from diurnal cycle must less than 2

49

( a )( a )

Step 2

( b )( b )

Step 3

Background

Obs

.

Obs

.

Background

50

Frequency Distribution before and after QC

First Week Last Week

Entire Month

Tobs-TNCEP (unit: K)

Tobs-TNCEP (unit: K)Tobs-TNCEP (unit: K)

Fre

qu

ency

Fre

qu

ency

Fre

qu

ency

51

Frequency Distribution with and without Contribution from Modes 2-6

First Week Last Week

Entire Month

Tobs-TNCEP (unit: K)Tobs-TNCEP (unit: K)

Tobs-TNCEP (unit: K)

Fre

qu

ency

Fre

qu

ency

Fre

qu

ency

52

Correlations and RMS Differences of the PCs before and after QC

53

Data Number Removed at Each Station

Step One Step Two

Step Three All Three QC Steps

54

Percentage of Data Removed by QC

Step 1 Steps 1-2 Steps1- 3

55

Percentage of Data Removed by QC

Time (day)

56

Variation of the Statistical Measureswith QC Steps

Mea

n (

un

it:

K)

Std

. (u

nit

: K

)

Sk

ewn

ess

Ku

rtos

is

Step 1 Step 2 Step 3 Step 1 Step 2 Step 3Ori. No DC Ori. No DC

57

Time Evolution of Standard Deviationbefore and after QC

Std

. (K

)

Time (unit: day)

58

Time-Zone Dependence of Diurnal Oscillation

Time (unit: hour)

Tem

per

atu

reT

emp

erat

ure

Weekly mean Ts at seven surface stations selected within different time zones: Zone 1: 55.03E, 36.42N Zone 2: 65.68E, 40.55N Zone 3: 82.78E, 41.23N Zone 4: 98.9E, 40.0N Zone 5: 110.05, 41.03 Zone 6: 128.15E, 40.89N Zone 7: 141.17E, 39.7N

Surface Obs.

NCEP Analysis

59

Average Time at Which Ts Reached the Maximum in the First Week of January 2008

Tim

e (U

TC

) T

ime

(UT

C)

60

Global Diurnal Cycle

NCEP ECMWF (ERA-Interim)

Surface ObservationsTime (UTC)

Time (UTC)

Tem

pera

ture

(K

)

Tem

pera

ture

(K

)

Time (UTC)

January 1-7, 2008

SummarySummary

• Diurnal cycle dominates the temporal variability of

surface data

• Large-scale analysis contains a significant phase error

(~10-85 degrees) of the diurnal cycle

• A three-step QC procedure is developed to identify

outliers in surface-station temperature data which have

a non-Gaussian frequency distribution

More details can be found inMore details can be found in

Qin, Z.-K., X. Zou, G. Li and X.-L. Ma, 2009: Quality control of surface temperature data with non-Gaussian background errors. Quart. J. Roy. Meteor. Soc., Submitted.

Zou, X. and Qin, Z.-K., 2009: Diurnal cycle in global analysis. J. Geo. Letter., to be submitted.