ECE 583Lecture 10

Errors in Langley method; temporal effects, absorption

10-1 Langley method

Consider all of the assumptions that are inherent in theLangley method

PBiggest assumption is that the optical depth is constant with timePOther assumptions are

• Atmospheric composition is horizontally homogeneous• Airmass is known

PAssumes Beer’s law and voltage relationship is valid• Small wavelength interval (Beer’s law is monochromatic)• Responsitivity is relatively constant over the bandpass• Optical depth is relatively constant over the bandpass

P Instrument is not contributing to errors• Stable with time• Response is linear (or can be corrected)• Field of view effects are minimal• Digitization of the output

PAlso assumes that the attenuation is linear with airmass

10-2

VV

VV

m m0

0

VV

VV

m m0

0

VV

VV

m

m

0

0

Errors in Langley method

Analyze the sensitivity of the results with respect to each ofthe parameters in the voltage relation

PDifferentiation by parts gives

PSolving for the percent error in intercept

PSolving for the optical depth error gives

10-3

V V e m0

ii

i

V Vm

ln ln0

Regression “errors”

One issue that should be clear is that a standard linearregression will be biased

PTypical method is to find the “best” fit straight line through themeasurements via linear regression

PErrors depend on the airmass leading to more weight being given tovariations in optical depth at large airmass

ln(V)=-m given only errors are due to random atmosphericfluctuations

PDevelop a technique that does not suffer from this issue• Define

• Then

• Where i is the ith measurement

10-4

1

1N ii

N

i i

( )ii

N

1

2

dd V i

i

N

ln ( )0 1

2 0

Regression errors

The least squares fit minimizes the squared deviations fromthe averages

PAverage optical depth is

PFluctuation in optical depth is

PWant the V 0 such that this intercept for a straight line minimizes

PThus want,

10-5

dd V

dd V

dd Vi

i

N

ii

Ni

ln( ) ( )

ln ln0 1

2

1 0 02 0

ii

i

ii

i

Ni

ii

N

ii

NV Vm

ddV m N N

V Vm

ddV N m

ln ln ln ln0

0 1

0

1 0 1

1 1 1 1 1

ln

ln ln

VN

Vm

Vm m

Nm m

i

ii

Ni

ii

N

ii

N

ii

N

ii

N0

21 1 1

21 1

2

1

1 1

ln ln1 1

1 11

21

21 1

21 1

2

mVm m

Vm

Nm m

ii

Ni

ii

N

ii

Ni

ii

N

ii

N

ii

N

Unweighted fit

Taking the derivative with respect to the intercept voltage

P

PUsing the relationships for i and the average optical depth it can beshown that

PThen the best estimate of the intercept and optical depth are

10-6

ln

( )( )

V

ii

N

ii

N

ii

N

ii

N

ii

N

N

Nm m

m

Nm m

02

2

21 1

22

21

2

21 1

21 1

1

1 1

(ln ) ( )V0 113

Uncertainties are really what is of interest

Evaluate the uncertainties by using the standard deviation ofthe fit - (lnV 0 ) and ( )

PThen the variance on the intercept and optical depth estimates are

• Where is the standard deviation of the optical depth based on i

PThe unweighted case provides an improved estimate of the standarddeviation of the intercept and optical depth estimates

PFor the case of a large number of measurements

10-7 Renormalized example10-8

VV

VV

m m0

0

VV

m

Airmass limitations

Assume that the dominant errors are due to errors in theairmass

PErrors in airmass can be due to• Choice of airmass model• Errors in refraction computation (uncertainty in profiles)• Location error• Timing errors

PAssume that the errors are caused by timing uncertaintyPRecall

PThen in this case

• Want a limit of 0.1% in measurement error• Means that airmass error should be less than 0.002 for a clear sky case

10-9

VV

m

m dmdt

t

Airmass limitations

Modeling efforts show that it is straightforward to meet the0.1% error for m<5.0

PErrors in airmass can be reduced by • Precisely recording time to better than 1 second• Spherical geometry must be taken into account• Refraction must be included• Vertical distribution of atmospheric attenuators must be reasonably

modeledPWhere does the 1 second requirement come from?

• Recall

• Write the error in airmass as that due to an error in time

10-10

dmdt

smax

.0 0015 1

t smax.

..0 001

0 00150 67

Airmass timing

Analysis of airmass as a function of time for standard casesusing solar ephemeris shows

Pdm/dt is a maximum at high solar zenith angles (surnrise, sunset)PFurther, it can be shown that the error is largest in spring and fall

PThe 0.1% error/change in measurments due to timing error for a largeoptical depth of 1.0 means

PConclusion is that a 1 s accuracy is needed• Requirement is looser for smaller optical depths• Requirement is less for longer wavelengths (lower optical depths)

10-11

V V e V V VV V DD

DDVV

diff direct diff

direct

diff

direct

0

1, ,

,

,

,

( )

DDVV

m Pdiff

directaerosol aerosol aerosol( ) cos( )0

Diffuse light redux

The finite field of view of the sensor means that there is adiffuse light component in the measurement

PRecall the diffuse-to-direct ratio introduced previously

• Varies with FOV of the sensor• Also varies with wavelength, solar zenith (airmass), and optical depth

• P is the phase function• 0 is the single scatter albedo

10-12

DDVV Km Pd

a a1

2( )

Diffuse-light effects

Diffuse light causes two primary effects - added light andtemporal variations

PBoth effects can modeled reasonably well• Recover optical depth from the Langley method• Remove molecular optical depth and determine aerosol optical depths

(we’ll discuss this more later)• Compute the diffuse contribution from assumed aerosol model• Correct for diffuse effects to iterate on optical depth and intercept

PTemporal variations can be measured• Measure direct signal with one channel• Measure diffuse light with another channel• Develop a factor K that accounts for differences in center wavelength,

responsivities, and aerosol effects

10-13 Error in from Langley method

Plot here shows theoptical depth errordue to diffuse light

PModel-base resultsPVariety of Junge

exponents (recall thatlarge Junge values implysmaller particles)

PResults as a function ofFOV

10-14

Diffuse-light and Langley plots

Generate a model-basedLangley data set

PWavelength of 400 nm withmoderate aerosol amount

P4-degree full-field FOVPRetrieved intercept is 0.6% too lowPRetrieved optical depth to low by

0.013 in optical depth (4.3% inaerosol optical depth)

PThis assumed no temporalvariability in the aerosol content

10-15 Langley methodThe langley method requires that the optical depth not vary as

a function of timePGraph below shows the results from a day for which the Langley method

does not work wellPThis gives an incorrect value for the intercept and for optical depthPTucson from October 30, 2001

0 1 2 3 4 5 6 7Airmass

3

4

5

6

10-16

Not so good Langley plot

One of the biggest error source in the Langley method istemporal variations in atmospheric composition

Langley Plot

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0Airmass

-3.0

-2.0

-1.0

0.0

1.0 June 23, 2001Ivanpah Playa

400 nmLangley fit

870 nmLangley fit

10-17 Variation in optical depth

Can still determine an average optical depth as well asinstantaneous values

Optical Depth vs. Time

6.0 7.0 8.0 9.0 10.0 11.0 12.0Mountain Standard Time

0.0

0.2

0.4

0.6

0.8

1.0

1.2 Ivanpah PlayaJune 23, 2001

400 nm870 nm

10-18

Variation in Langley plot

Typically the varibility seen in a Langley plot is caused byvariations in atmospheric conditions

P Instrumental effects can play a role• Noise in the measurement• Temperature/thermal effects in the detectors

PAtmospheric variability can be caused by• Clouds• Wind conditions• Convection• Pollutants

PBest measurements take place• On mountaintops where lower aerosol loading means lower aerosol

variability• Mornings which are cooler and more stable with less convective mixing• Low wind conditions• Away from pollution sources

10-19 Residual optical depth

Examine the difference between the instantaneous opticaldepth and regression average

PGood day will showsmall randomvariations in theresidual

PAirmass effect cancause larger errors atsmaller airmass

PConvection alsomore prevalent atsmall airmass for amorning collection

PLarger number ofpoints at smallairmass also affectsthe regression

10-20

Residual plots

Residual optical depth plot shown here is still from a not sobad day

PAirmass range onthe regressionwas limited tobetween 2.0 and6.5

PShould haveconfidence thatthis is valid beforeblindly attemptingto restrict airmassrange

PMay date fromTucson shownhere correspondsto a hot daysusceptible toconvective effects

10-21 Residual plots

Plot here is one for which the Langley assumptions are notvery valid

PCould stillprovide areasonableestimate ofintercept but notlikely

PThis was a“clear” day in thesummer

10-22

( )( )

tm t

VV

1 0

0

Cm

VV

1 0

0

( ) ( ) ( ) ( )t t m t m tVV1 2

1 2

0

0

1 1

Residual plots

The only foolproof means of detecting temporal variations is toknow the interecpts ahead of time

PHowever, errors in the voltage intercept will cause an error in the retrievedoptical depth

PThe correct instantaneous optical depth is related to the computed opticaldepth by

P If the error in intercept is small the changes in optical depth will still beapparent

P Inverse airmass difference typically <0.1 for about 30 minutes difference• Assume 2% uncertainty in intercept• Changes greater than 0.002 in optical depth are discernible

10-23 Temporal variationsin optical depth

PDifficult to see the slowlyvarying atmospheric trends• High frequency atmospheric

changes• Sensor noise

PTemporal averaging allows theslowly varying effects to bedetermined

PSlowly-varying atmosphericchanges are more problematicfor the Langley retrievals

10-24

Temporal variationin optical depth

PSame data set as previousviewgraph except for shorterwavelength

PPlotted versus airmass andtime

PNote that there are similarfeatures between this bandand the longer wavelength

10-25

V V ei imi i

0

Q V Vii

N

0 02

1

*

i ii

i

i

ii

N

where N mVV1

1 1 0

1ln

*

Intercept correction method

Improved intercept retrieval can be obtained with informationon the change in optical depth over the measurements

PDefine an instantaneous intercept as

PGoal is to retrieve an optimal intercept V*0 that minimizes a performancefunction Q

PThis requires the instantaneous optical depth as

• V*0 can be found if the residual optical depth can be modeled ordetermined

• This information can be obtained from the diffuse component

10-26

d Vdm

cons tln( ) tan

ln( ) ln( )( )

V Vm t

0 0

Residual plot failure

There is an interesting result for which an optical depth thatvaries with time will still provide a good Langley plot

PConsider the case where the log of measurement is linear in airmassPThen

P “It can be shown” that the optical depth that satisfies this condition is

PWhere the primed quantities are the difference between the actual and theerroneous intercepts and optical depths

10-27 Intercept determination

The question then becomes what is the best way to determinethe intercepts for a given sensor

PThe intercepts are needed to allow determination of instantaneous opticaldepths

PFirst step is to find dates that are good for Langley analysis• No apparent variations in the Langley plot• Scatter is small at high airmass• No curvature to data at high airmass• All wavelengths without absorption have small standard deviations• Low optical depths at long wavelengths (<0.05 for Tucson)• Residuals at long wavelengths are small (<0.002)• Residuals show random scatter• Intercepts show reasonable agreement with past results

PCompute average of selected dates

10-28

Determination of InterceptTypically, several days of data are used to calibrate the solar

radiometerPGraph below shows results from several datesPDifferences can be caused be instrumental effects and atmospheric

variability

Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7 Band 8 Band 10350

400

450

500

550

600

650

700

Oct. 28 Oct. 29 Oct. 30Nov. 1 Nov. 5

10-29 Determination of interceptGraphs below summarize the variability on the dates shown in

the previous graph

November 1 November 5

October 28 October 29 October 30

10-30

Interceptdetermination

POne mechanism forevaluating a good Langleyday is comparisons betweendays

PCompare retrieved interceptwith standard deviation ofthe intercept (these areweighted results)

PCompare the retrievedintercept with the opticaldepths on the date

10-31 Interceptdetermination

PSelected dates fromprevious data set

PUpper graph spans a 16-month period

P Is the 7% decrease real oran artifact of the Langleyresults?

PLower results are for a 6-month period

10-32

Intercept vs. TimeThe whole reason for all of this work is to ensure that we can

track changes in instrument response over time

Band 1 Band 2 Band 3 Band 4 Band 5 Band 6 Band 7 Band 8 Band 10350

400

450

500

550

600

650

700

April 02 August 02 Nov. 02

10-33 Intercept versus time10-34

Interceptversus time

PScale the intercepts byband for easiercomparison

PNote that many of thedays produce similarchanges in interceptsrelative to other dates

PBand 8 results areanomalous indicating aninstrumental change

10-35 How well can we do?10-36

How well can we do?10-37 Comparisons between sensors10-38