Where is calibration (So)for

aerosol optical depth measurements?

OrPractical sun spectral radiometer

calibration methods.Or

Learning from mistakes!

History Indicates

‘The three most important issues in sun photometry are:

• calibration;

• calibration; and

• calibration.’

Michalsky (2000)

Experience Suggests

• Mauna Loa’s don’t exist except at Mauna Loa!

• Subjective estimates of determining what is a valid ‘Langley’ are dubious.

• Sites with low incidence of extended clear sun periods have a low chance of success with atmospheric methods.

• Quantitative filters are better than qualitative filters.

Why is the calibration of sun spectral radiometers so difficult?

• There is no WRR and absolute spectral radiometers in field use!

• We are mainly interested in a derived quantity(aerosol optical depth) that requires more precision than irradiance measurement.

• Lack of stable detector-based reference.• The stability of the commonly used filter radiometers

is poor.• We need to have reference of spectral irradiance

AND the irradiance at the top of the atmosphere.

Spectral Irradiance Signal Equation

S(t) = (r0 /r)2 S0 exp(-mi (t) i (t)) + F(t,S, )

S(t) = signal measured at time t at wavelength S0 = signal at the top of the atmosphere at 1 AU

F(t,S, ) = circumsolar contribution to the signalmi (t) i (t) = extinction components (molecular, aerosol, ozone,

…)(r0 /r)

2 =earth sun distance correction for S0

Short cuts to avoid my mistakes: Atmospheric methods

• For properly designed instruments we can reduce the uncertainty in the measured sun signal.

• If is the small F() can be ignored.

• In molecular, aerosol, ozone-only regions of the spectrum and < 10 nm the Bouguer-Beer-Lambert ‘law’ may apply; hence

ln S(t) = 2 ln(r0 /r) + ln S0 -mm (t) m (t)

-ma (t) a (t) -mo (t) o (t)

• Then can use Least Squares Regression to estimate ln S0.

Least squares regression (LSQ)

For a set of n pairs of (x,y) observations where we assume the relationship is

y(t) = A x(t) + B

The LSQ allows us to estimate A and B by

<A> = [(x(t) – X)(y(t)- Y)]/[ (x(t)-X)2]

<B> = Y - <A> X

where X and Y are the mean values.

Other LSQ parameters

2 = [(y(t) - <A>x(t) - <B>)2]/[n – 2] = variance of the estimate in y(t)

= standard error of the estimate in y(t)

Sx = [(x(t) – X)2]/[n-1]

Sy = [(y(t) – Y) 2]/[n-1]

r = <A> Sx/ Sy = correlation coefficient, with

r2 = coefficient of determination

Atmospheric methods

The atmospheric methods assume that a simple linear expressiony(t) = A x(t) + B

of known variables x and y for a period of times t is a valid model of how the atmosphere is interacting with the radiation extinction.

Most importantly use of LSQ for determining the calibration So

assumes that the variation of A and/or B is randomly and normally distributed about the model.

Given that the atmosphere is rarely static and rather dynamic it is important to select x and y terms that ensure that A and B have little potential variation and uncertainty.

Classical ‘Langley’ mm = i mi

F() = 0.0Using

x(t) = mm(t)

y(t) = ln S(t)

and derive<A> = <>

<B> = <ln S0>+ 2 ln(r0 /r)

The classical model assumes that variation in is very small over t hence it is essential to restrict the period of observations to very clear sun periods where the mm/ t is high but there is little uncertainty in the signal;usually between mm = (2.0, 6.0). Also need to know mm(t).

Classical Langley PFR N01 27 Sept 2000 PMOD

y = -0.6221x + 1.5235

R2 = 0.9992

y = -0.0839x + 1.2634

R2 = 0.9934

y = -0.2617x + 1.4074

R2 = 0.9978

y = -0.4364x + 1.3878

R2 = 0.9987

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Molecular Air Mass

Ln S

W 368

W 412

W 500

W 862

Linear (W 368)

Linear (W 862)

Linear (W 500)

Linear (W 412)

Air mass differences (Robinson, 1966)

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6 7 8 9 10

Aerosol Airmass

Aero

sol -

Oth

er

Aerosol - Molecular

Aerosol - Ozone

Modified ‘Langley’

x(t) = ma(t)

y(t) = ln S(t) + mo(t) o(t) + mm(t) m (t)

and derive <A> = <a> and

<B> = <ln S0> + 2 ln(r0 /r)

The modified model assumes that variation in a is very small over the measurement period: restrict the period of observations to very clear sun periods where the mm/ t is high but there is

little uncertainty in the signal; usually between ma = (2.0, 6.0).Also need to know mo(t), o(t), mm(t), and m (t).

Mofified Langley PFR N01 27 Sept 2000 Davos

y = -0.0688x + 1.2595

R2 = 0.9901

y = -0.1305x + 1.3995

R2 = 0.9908

y = -0.1895x + 1.5127

R2 = 0.9907

y = -0.1659x + 1.3784

R2 = 0.9907

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Aerosol Air Mass

Ln S

+m*m

o+m

*oz

W 368

W 412

W 500

W 862

Linear (W 862)

Linear (W 500)

Linear (W 368)

Linear (W 412)

368 nm Classic - Modified Alice Springs June 1998-Jan 1999

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

Standard Error of Modified fit

Ln S

o(cl

assi

c) -

Ln S

o(m

odifi

ed)

868 Alice Springs Classic - Modified

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

Standard Eroor of Modified Fit

Ln S

o(cl

assi

c) -

Ln S

o(m

odifi

ed)

Other information on the Classical and Modified ‘Langley’

These methods explicitly give more weight to periods of high airmass because they are minimizing

(ln S(t) - 2 ln(r0 /r) +< ln S0 > - m (t) <>)2

and not the variance in optical depth which is the parameter one expects to be varying randomly and having a normal distribution through the period of observation.

Unweighted ‘Langley’

x(t) =1/ma(t)

y(t) = (ln S(t) + mo(t) o(t) + mm(t) m (t))/ma

and derive <A> = <ln S0> + 2 ln(r0 /r) and

<B> = <a>

In this case the minimization is of a, however it does

assume that variation in a is very small over the

measurement period hence it is essential to restrict the period of observations to very clear sun periods where the mm/ t is high but there is little uncertainty in the signal;

usually between ma = (2.0, 6.0). Also need to know mo(t),

o(t), mm(t), and m (t).

Un-weighted Langley PFR N01 27 Sept 2000

y = 1.2642x - 0.0705

R2 = 0.9997

y = 1.3915x - 0.1706

R2 = 0.9987

y = 1.5283x - 0.195

R2 = 0.9986

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6

1/(Aerosol Air Mass)

(Ln

S +

m*M

ol +

m*O

z)/M

a

W 368

W 412

W 500

W 862

Linear (W 862)

Linear (W 412)

Linear (W 368)

Varying aerosol conditions

All the previous methods assume that there are no trends in the optical depth components over the range of airmass, particularly a. The next methods remove the restriction on a

‘constant’ optical depth, and utilise the interaction between aerosols and radiative propagation.

We know that most natural aerosol size distributions will produce a smoothly variation of aerosol optical depth with wavelength. Furthermore, if the relative size distribution of the aerosols remains the same the aerosol optical depth is

directly proportional to the total number of particles. e.g. a () = (/1000)-

Aerosol optical depth

a () = c Qex(x) (dn/dln r) dln r

where

x = 2r/,

Qex is the extinction efficiency

(that is dependent on the complex refractive index of the aerosol)

and

dn/dln r is the aerosol number density

Aerosol optical depth a function of

Under most circumstances a single aerosol set of aerosol properties influences the extinction across wavelength.

It is also known that the most optically active aerosols

(r = 0.21.0 m) are those with the longest lifetime.

It is clear that there is good correlation between a(1) and

a(2) and any change in the natural aerosol distribution

will impact on most wavelengths (0.3 – 3 µm).

Relational methods

These methods require the calibration of one wavelength channel to be known, and then in various conditions the signals from that channels can be related to other channels.

Two useful methods are:

(1)     ratio-Langley and

(2)     general method

Ratio-Langley method x(t) =mm(t)

y(t) = (ln S1(t)/ Sr(t)

and derive<A> = <1 - r>

<B> = <ln S10/ Sr0>

It is easy to show that the variability in (1 - r) is much less

than the variability in either 1 or r in periods of moderately

varying aerosol optical depth and the uncertainty deceases as 1 r. This method is particularly useful when comparing

similar nominal wavelengths with the resultant <1 - r>

indicating minor differences between 1 and r .

(This method is not new – it is the primary method used in Dobson Spectrometers for deriving calibration and ozone parameters).

General method This method uses the ‘aerosol optical depth’ derived from a well-calibrated channel (or in some cases the aureole derived aerosol extinction).

For a well calibrated channel we can derive the aerosol component of extinction at r as

-ma (t) ar (t) = ln Sr(t) +mm (t) mr (t) +mo (t) or (t) -2 ln(r0 /r) - ln S0r

x(t) = - ma(t) ar (t) ,

y(t) = ln S(t) + mo(t) o(t) + mm(t) m (t)

and derive

<A> = <a/ar>and <B> = <ln S0> + 2 ln(r0 /r)

The general model assumes that variation in the relative aerosol optical depths (a/ar)is very small over the measurement period, even if there are

moderate variations in the absolute magnitude. Like the other methods the period of observations is restricted to clear sun periods where the mm/ t

is high but there is little uncertainty in the signal; usually between ma =

(2.0, 6.0). Also need to know mo(t), o(t), mm(t), and m (t) for both the

reference and calibrated channels.

General PFR N01 27 Sept 2000 Davos

y = x + 1.2185

R2 = 1

y = 1.905x + 1.3225

R2 = 0.9986

y = 2.748x + 1.3993

R2 = 0.9978 y = 2.4065x + 1.2791

R2 = 0.9981

0.6

0.7

0.8

0.9

1

1.1

1.2

-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0

- (Ma * Tau) at 862 nm

Ln S

+ m

*Mol

+ m

* Oz

W 368

W 412

W 500

W 862

Linear (W 862)

Linear (W 500)

Linear (W 368)

Linear (W 412)

Summary PFR N01 27 Sept 2000 Davos

Method 368 nm 412 nm 500 nm 862 nm

Classic 0.124 0.106 0.085 0.054

Modified 0.113 0.0970 0.087 0.050

Un-Weighted 0.128 0.109 0.095 0.055

Ratio-Langley 0.0167 (0.018) -0.022 -0.025

General 0.0016 (-0.002) 0.0011 0.0105

Checks and Balances

It is important to realise that no single calibration derived from any of these calibration methods at typical sites

will produce a useful value!

Only by multiple calibrations typically over a month or two with at least 20

good points will a valid calibration be produced.

868 nm Alice Springs April 1999 - July 2000

1.7

1.72

1.74

1.76

1.78

1.8

0 100 200 300 400 500 600

Day since 31 Dec 1998

Ln S

o (8

68 n

m)

General

Modified

Single wavelength methods

Method x(t) y(t) <A>Gradient

<B>Intercept

Classic Langley

mm(t) ln S(t) ln S0+

2 ln(r0 /r)

Modified Langley

ma(t) ln S(t)

+ mo(t) o(t)

+ mm(t) m (t)

aln S0 +

2 ln(r0 /r)

Weighted Langley

1/ma (ln S(t)

+ mo(t) o(t)

+ mm(t) m (t))

/ma

ln S0 a

Relational methods Method x(t) y(t) <A>

Gradient<B>

Intercept

Ratio-Langley

mm(t) ln (S(t)/Sr(t)) - arln (S0/S0r )

General -ma(t) ar(t) ln S(t)

+ mo(t) o(t)

+ mm(t) m (t)

a/ar ln S0 +

2 ln(r0 /r)

Quotes to Remember

‘The accuracy of results is inversely proportional to the promptness of

publication’

Robinson (1970)

‘Science is the search for ignorance’, Feynman (1963)

So what men knew about seasons was true and sometimes not; it was questionable but not

knowable, discernible by its origins but obscure in its presence and in its moment of

ending. It was like so many things men accepted without wonder. Yet in that

uncertainty lay the pivot point of existence.

 Fortress of Eagles

C J Cherryh