uncertainties evaluation for aerosol optical properties aldo amodeo cnr-imaa

Geneva, 20-23 September 2010EARLINET-ASOS Symposium

Second GALION Workshop

Uncertainties evaluationfor aerosol optical properties

Aldo AmodeoCNR-IMAA

Uncertainties evaluationfor aerosol optical properties

Aldo AmodeoCNR-IMAA



OUTLINE

General concepts

Source of errors in lidar measurements

The problem of the calculation of the statistical error

The problem of the calculation of the sistematic error



Basic concepts

Lidar measurements, as for all the measurements, need the estimation of the associated error, because every measurement without error could have no meaning.

The determination of the error is not a simple task, especially when several operations are applied in the data analysis: smoothing, averaging, background subtraction, gluing, analysis algorithm.

Two kinds of errors can be distinguished: statistical error and systematic error.



a) Statistical error mainly due to the to signal detection [background of sky and dark current of detector]

(Theopold and Bösenberg, 1988); directly related to this kind of error, there is the error introduced by operational procedures

such as signal averaging during varying atmospheric extinction and scattering conditions (Ansmann et al., 1992; Bösenberg, 1998);

b) Systematic error

due to uncertainties related to instruments, fixed parameters in the retrieval,… the systematic error associated with the estimate of temperature and pressure profiles

(Ansmann et al., 1992); the systematic error associated with the estimate of the ozone profiles in the UV (Ansmann

et al., 1992); the systematic error associated with the wavelength dependence parameter k (Ansmann et

al., 1992; Whiteman, 2000); the systematic error associated with the multiple scattering (Ansmann et al., 1992;

Wandinger, 1998; Whiteman, 2000); extinction uncertainties (up to 50% for heights below Zovl) are caused by the overlap

function (Wandinger and Ansmann, 2002).

SOURCES OF ERRORS



Error calculation

Parameter•Raw lidar signal•Extinction coefficient•Backscatter coefficient

Acquisition technique•Photoncounting•Analog



Gaussian distribution (suitable for analog mode)

If the variable can assume in principle continuous values, if a measurement is affected by many source of random errors, and systematic errors are negligible, the measured values will be distributed according a bell curve, centred on the true value of x.If measurements are affected by not negligible systematic effects, the distribution of the measurements will not centred around the true value.Variables affected only by statistical errors are described by Gaussian (or normal) distribution:

22/2

,2

1)(

Xx

X exG

X: true value of x, centre of the distribution, mean value after many measurements.

: distribution width standard deviation after many measurements

N

iix

NxX

1

1

2

11

1

N

iix xx

N

X



Poisson distribution (suitable for photoncounting mode)

The Poisson distribution describes experiments in which are counted events that happen randomly, but with a defined mean rate. The variable is discrete.If we count during a time interval T, the probability to observe events is given by the Poisson function:

: expected mean number of events within the time T:

Standard deviation of the observed number :

!

)(

ePTtimethewithincountsP

= 1 = 4 = 10

The horizontal axis is the index .The function is defined only at integer values of .

The connecting lines are only guides for the eye and do not indicate continuity.

)( trialsmanyafter

When is large, the Poisson distribution P() is well approximeted by Gauss distribution with the same mean and standard deviation: andX



This kind of information is contained in the measured standard deviation (z) of the lidar signal.

Possible techniques of evaluation

AnalyticalNumerical (Montecarlo)Calculation of the standard deviation among the single solutions

Statistical error



Error calculation



•Square root of the counts.•The error on the subtracted background raw signal should include the propagation of the error on the background.

where n is the number of bins used to calculate the background.

nk

kiiBNS N

nNwhereNN

12

B



532 nm, photon counting, laser rep. Rate 50Hz

1

10

100

1000

10000

100000

1000000

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Height (m)

Sum

med

Cou

nts

Sum 10min

Sum 20min

Sum 30min

Sum 40min

Sum 50min

Sum 60min




0

2

4

6

8

10

12

14

16

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Height (m)

Per

cen

tag

e d

evia

tio

n

PercErr_10min

PercErr_20min

PercErr_30min

PercErr_40min

PercErr_50min

PercErr_60min



Signal data binning 4 points532 nm, photon counting, laser rep. Rate 50Hz

1

10

100

1000

10000

100000

1000000

10000000

0 50 100 150 200 250 300 350 400 450 500

Height (m)

Co

un

ts

Sum Aver 30min

532 Sum 30min_4pBinned

Each point is obtained by:•summing the counts contained in a certain number of bins•associating as height the mean of the height range relative to the binned points.



Signal data binning 8 points


1

10

100

1000

10000

100000

1000000

10000000

0 50 100 150 200 250 300 350 400 450 500

Height (m)

Co

un

ts

Sum Aver 30min




Signal data binning


1

10

100

1000

10000

100000

1000000

10000000

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Height (m)

Co

un

ts

Sum Aver 30min





Signal data binning


0.01

0.1

1

10

100

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Height (m)

Rel

ativ

e st

and

ard

dev

iati

on

(%

)

532 PercErr 30min

532 PercErr 30min_4pBinned




Error calculation



•Standard deviation calculated on the averaging time interval.•The error on the subtracted background raw signal should include the propagation of the error on the background (sky and electronic)

22BSN



1064 nm, analog, laser rep. Rate 50Hz

0.1

1

10

100

1000

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Height (m)

Ave

rag

ed s

ign

al i

nte

nsi

ty (

mV

)

Aver 10min

Aver 20min

Aver 30min

Aver 40min

Aver 50min

Aver 60min




0.001

0.01

0.1

1

10

100

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Height (m)

Sta

nd

ard

dev

iati

on

(m

V)

DevSt 10min

DevSt 20min

DevSt 30min

DevSt 40min

DevSt 50min

DevSt 60min




0

5

10

15

20

25

30

35

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Height (m)

Rel

ativ

e st

and

ard

dev

iati

on

(%

)

PercErr_10min

PercErr_20min

PercErr_30min

PercErr_40min

PercErr_50min

PercErr_60min



1064 nm, analog acquisition, laser rep. Rate 50Hz

0.1

1

10

100

1000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Height (m)

Sig

nal

in

ten

sity

(m

V) 1064 Aver 30min

1064 Aver 30min_4pBinned

1064 Aver 30min_8pBinned




0.1

1

10

100

1000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Height (m)

Rel

ativ

e er

ror

(%)

1064 PercErr 30min






1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Height (m)

Err

or

on

th

e si

gn

al (

mV

)

1064 DevSt 30min_8pBinned

1064 DevSt 30min_8pBinned -Back

Error comparison with the background subtraction

22BackgroundSignalS

Where Background is the standard deviation of the calculated background.




0.1

1

10

100

1000

10000

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Height (m)

Rel

ativ

e er

ror

on

th

e si

gn

al (

mV

)


1064 PercErr 30min_8pBinned-Back

22BackgroundSignalS



Error calculation





ERROR CALCULATIONIN THE

AEROSOL EXTINCTION COEFFICIENT RETRIEVING

The aerosol extinction coefficient, aer can be determined from the N2 (or O2)

Raman backscattering signals through the application of the expression (Ansmann et al., 1990; Ansmann et al., 1992):

k

R

Rmolmol

aer

zzzPz

zNdzd

z

0

02

0

1

,,ln

,

P(z)power received from distance zat the Raman wavelength

transmitted laser wavelength

N(z) atmospheric number density

mol extinction coefficient due to absorption and Rayleigh scattering by atmospheric

gases, and where particle scattering is assumed to be proportional to -1.



HOW TO CALCULATE THE ERROR?Analytical or Numerical techniques?

PROBLEMS:difficulty in error propagationwhen handling procedures, suchas signal smoothing, are applied.

ANALYTICAL

k

R

0

0aer

1

zP

zP

dz

d

z,

NUMERICAL (Montecarlo techniques)

ADVANTAGES:no difficulty related to error

propagation calculation, whatever signal handling procedure is used.

THE PRINCIPLE

This procedure is based on a random extraction of new lidar signals, each bin of which is considered as a sample element of a given probability distribution with the experimentally observed mean value and standard deviation. The extracted lidar signals are then processed to retrieve a set of solutions from which the standard deviation as a function of the height is estimated.

It is important to know the signal standard deviation for each height and the type of distribution function. In the case of photon-counting, this is a Poisson distribution.



1..b b+1..k

Extracted signals

Solutions fromextracted signals

Solution with errors equal to the deviations from

the solutions obtained from the extracted signals.

Measuredsignal Solution

Numericaltechnique



Random extractorsSeveral procedures exist. Generally they start from the random generation of numbers according to the uniform distribution and transform the extracted numbers in numbers following the desired distribution.

Examples of simple algorithms for some extractors:

Gaussian distributionIf u1 and u2 are uniform on (0,1), then

are independent and Gaussian distributed with mean 0 and =1

Poisson distributionIterate until a successful is made:•begin with k=1 and set A=1 to start•generate u uniform in (0,1)•replace A with uA•if now A<exp(-) where is the Poisson parameter, accept nk=k-1 and stop;•otherwise increment k by 1, generate a new u and repeat, always starting with the value of A left from the previous try.

212211 ln22cosln22sin uuzanduuz



COMPARISON BETWEEN EXTINCTION ERRORCALCULATED BY ANALITYC AN MONTECARLO TECHNIQUES

Extinction calculated bySLIDING LINEAR FIT

BzAzPz

zNln

2

so that it is simple to calculate the derivative in the formula.

The ANALYTICAL error is:

k

R

0

0aer

1

Bz,



EXTINCTION ERROR DEPENDENCE ON THE USED ALGORITHM(Calculated by Montecarlo technique)

0

500

1000

1500

2000

2500

0 10 20 30 40 50

CASE 2: 90 m Fixed ResolutionSEVERAL TECHNIQUES

Statistical Error [%]

Hei

ght

[m]

7 pts Sliding Linear fit 5 pts Sliding Average 4 pts Data Binning

11 pts 2ndord. Sav.-Gol.0

500

1000

1500

2000

2500

0 10 20 30 40

CASE 2SLIDING LINEAR FIT

Statistical relative error [%]

Hei

ght

[m]

3 pts - Res:45 m 5 pts - Res:60 m 7 pts - Res:90 m 9 pts - Res:120 m 11 pts - Res:135 m



Error calculation

Parameter•Raw lidar signal•Extinction coefficient•Backscatter coefficient (Raman/elastic, elastic only)


Technique: AnalyticalNumerical (Montecarlo)Calculation of the standard deviation among the single solutions



ERROR CALCULATIONIN THE

AEROSOL BACKSCATTER COEFFICIENT RETRIEVING

The aerosol backscatter coefficient, aer can be determined from the ratio between the two lidar signals at laser and Raman wavelengths L and R (Ansmann et al.,

1992):

zTzT

zTzT

zP

zPzCzz

LaerLmol

RaerRmol

R

LRmolLmolLaer ,,

,,

,

,,,, *

The constant C* includes instrumental and geometrical system properties and is retrieved by normalizing lidar signal at a reference height z0 that is aerosol free:

00

00

0

0

000

*

,,

,,

,

,

,

1,,

zTzT

zTzT

zP

zP

zzzC

RaerRmol

LaerLmol

L

R

RmolLmolLaer

dzT

z

0,exp,



CONTRIBUTIONS TO THE BACKSCATTER STATISTICAL ERROR

22

,

,zz

z

zELASTICRAMAN

Laer

Laer

22

2

,

,

,

,

zP

zP

zT

zTz

R

R

Raer

RaerRAMAN

22

2

,

,

,

,

zP

zP

zT

zTz

L

L

Laer

LaerELASTIC

z

aeraer

aer dzT

zT0

),(,

,



Error calculation

Parameter•Raw lidar signal•Extinction coefficient•Backscatter coefficient (Raman/elastic, elastic only)




1064 nm analogical - 30 min integration time

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

-1000 1000 3000 5000 7000 9000 11000 13000

Height (m)

Inte

nsi

ty



1064 nm analogical - 30 min integration time

0.00E+00

1.00E+02

2.00E+02

3.00E+02

4.00E+02

5.00E+02

6.00E+02

7.00E+02

8.00E+02

9.00E+02

0 2000 4000 6000 8000 10000 12000

Height(m)

Per

cen

tag

e er

ror

on

th

e si

gn

al



Backscatter @ 106 nm - Iterative method

-5.00E-07

0.00E+00

5.00E-07

1.00E-06

1.50E-06

2.00E-06

2.50E-06

3.00E-06

0 2000 4000 6000 8000 10000 12000

Height (m)

Aer

oso

lack

scat

ter

(m-1

sr-1

)



Percentage error on the aerosolbackscatter @1064 nm

1.00E-01

1.00E+00

1.00E+01

1.00E+02

1.00E+03

1.00E+04

1.00E+05

1.00E+06

0 1000 2000 3000 4000 5000 6000

Height (nm)

Per

cen

tag

e er

ror

(%)



Signal temporal averaging

Binning

Background subtraction

Processing

Error propagation:

analytical

numerical

2

11 )(

1

1)(

n

ii xzx

nz

b

zz

')'(

22 )'()'( BS zz

Background determination

21

)'(1

1)'(

Bnj

ji

BiB

B xzxn

z

)'(zSFinal

Example of possible a procedure of analysis and error propagation for analog signals

Numerical technique could be used in the more useful step of the analysis

n

zz

)()( 1

1

)()'(1

2i

bk

kii zz

S

S

B

BB

n

zz

)'()'(

22 )'()'( BS zz



Summation of the signals

Binning

Background subtraction

Processing

Error propagation:

analytical

numerical

)()(1

1 zNzn

ii

)()'(1

i

bk

kii zNz

s

s

22 )'()'( BzzS

Background determination )'(

1 1

i

nk

kii

BB zN

n

BB

B

)'(zSFinal

Example of possible a procedure of analysis and error propagation for photoncounting signals

Numerical technique could be used in the more useful step of the analysis



If you use other techniques (smoothing, merging or other) or apply procedures in different order take care to apply the right error propagation procedure.

For example, in the case of merging, take into account the product for the normalization of the two signals and also the possible difference in typology between the two signals: analog and digital.



Some systematic errors

● Influence of the air-density profile● Influence of the Angstrom-exponent parameter● Influence of the lidar ratio assumption for the Klett retrievals atdifferent wavelengths● Influence of the calibration on backscatter retrievals at differentwavelengths● Errors due to a depolarization dependent receiver transmission

fromEARLINET ASOS training course for the retrieval of optical aerosol properties (Ina Mattis)Thessaloniki, 25-26 February 2008



Influence of the air density profile

● temperature gradient → small effect● absolute temperature → larger effectβ par ~ SigRatio − β molβ mol ~ number density of air molecules ~ T



Influence of the air density profile on Raman extinction profiles

● temperature gradient → large effect● absolute temperature → large effect



Influence of the absolute temperature on Raman extinction profiles

effect of the absolute temperature increases with height (optical depth)



Influence of the air density profile on Raman lidar-ratio profiles



Influence of the Angstrom exponent on Raman extinction retrievals



Influence of the lidar-ratio assumptionon Klett backscatter retrievals



Influence of the lidar-ratio assumptionon Klett backscatter retrievals

largest effect at smaller wavelength



Influence of the calibration on backscatterretrievals at different wavelengths



Influence of the calibration on backscatterretrievals at different wavelengths

largest effect at larger wavelength



Errors due to adepolarization dependent receiver transmission



Acknowledgements

EARLINET-ASOS projectfunded by the European Commission (EC)

under grant RICA-025991

uncertainties evaluation for aerosol optical properties aldo amodeo cnr-imaa

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