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488 Analytical Techniques for Atmospheric Measurement
and using equation A.23, it follows that
I
internal
= I
0
1 −R
i=1
R
i
= I
0
(A.28)
This means that the initial loss of intensity by reflection at the exterior wall is exactly
compensated by multiple reflections on the interior wall surface. Equation A.28 can be
derived separately for the vertical and parallel polarisation components of light and is
valid for all possible angles of incidence 0 ≤
0
≤ 90
. Thus the result is likewise valid
for the solar actinic flux transmitted into the photolysis reactor.
A.3 Spectral spectrometer bandwidth
When monochromatic radiation, for example a narrow atomic emission line, passes
through an imaging spectrometer, the transmitted spectrum will exhibit a broadened line
profile when the radiation reaches the detector. This broadening is generally caused by
the optical imaging of the entrance slit into the plane of detection and to some extent by
diffraction of the spectrometer (e.g. Demtröder, 2003). The resulting spectral response of
the spectrometer can be represented by an instrumental line-shape function ILS and
is characterised by a full-width at half-maximum (FWHM) called the bandwidth of
the instrument (Figure 9.45).
Any spectrum of light F
(monochromatic or broadband) passing the spectrometer
reaches the plane of detection as a broadened spectrum, which is the spectral convolution
of F
and ILS, denoted as F
⊗ILS:
F
⊗ILS =
0
F
ILS −
d
(A.29)
F
λ
(λ)
F
λ
(λ )
⊗ ILS(λ )
λ
i
λ
i
ILS(λ)
Spectrometer
Monochromatic
radiation
Δ
λ
δλ
Detector
element
Plane of
detection
Figure 9.45 Schematic illustration of the instrumental line-broadening of monochromatic radiation
transmitted through a spectrometer. F
is the line spectrum of the incident radiation. F
⊗ILS
is the broadened line spectrum when the transmitted radiation reaches the plane detection. is the
full-width at half-maximum (FWHM) of the instrumental line shape function (ILS) and is the spectral
width of the radiation that is actually selected and detected by a detector element.
Measurement of Photolysis Frequencies in the Atmosphere 489
When this spectrum falls onto a detector element covering a narrow interval centred
at a specific wavelength
i
, the mean detected radiation flux F
i
is given by:
F
i
=
1
i
+
1
2
i
−
1
2
F
⊗ILSd (A.30)
In case of a monochromator, F
i
is selected by an exit slit in front of a photoelectric
detector. The exit slit width determines , which is usually chosen to be equal to .
In case of a spectrograph, a many-element detector array is exposed to the dispersed
spectrum F
⊗ILS. Here corresponds to the width of a single detector element
(pixel). In general, the pixel width is smaller than the spectrograph bandwidth and several
adjacent pixel cover the instrumental line profile. It should be noted, however, that the
spectral resolution of the spectrograph is determined by , not by the pixel resolution.
The term instrumental line-shape function must be distinguished from the term instru-
mental slit function, which is the spectral convolution of the ILS with the response
of the exit slit (in case of a monochromator) or a pixel (in case of a spectrograph).
The slit function can be recorded as a smooth signal, when the spectrometer receives
monochromatic radiation that is tuned continuously over wavelength. If the monochro-
matic radiation has a fixed wavelength, monochromators can alternatively scan their
wavelength setting to obtain the slit function. The FWHM value of the slit function is
equal to if ≤ (for further details, see e.g. Demtröder, 2003).
A.4 Correction of the angular response of an actinic-flux
receiver 2 sr
A simple model of the atmospheric radiance L
near ground can be used to describe
the dependence of Z
H
of an upward looking radiometer on atmospheric parameters
(Hofzumahaus et al., 1999). It is assumed that the diffuse sky radiance is isotropic,
which is often a reasonable approximation for UV radiation in the lower troposphere
(cf. Figure 9.4a). The same assumption is made for the upwelling albedo contribution
(Lambertian reflector assumption). The radiance L
can then be expressed as follows
(Madronich, 1987b):
L
= L
0
+L ↓+L ↑ (A.31)
L
0
= F
0
0
0
(A.32)
L ↓=
F ↓
2
(A.33)
L ↑=
F ↑
2
(A.34)
where the symbols have the meanings as defined in Appendix A.1, and L
0
and F
0
are
the spectral radiance and actinic flux of the direct radiation from the direction
0
0
490 Analytical Techniques for Atmospheric Measurement
of the sun ( represents a Dirac delta function). L ↓ and F ↓ describe the downwelling
diffuse radiation, while L ↑ and F ↑ represent the upwelling parts.
If one inserts equations A.31–A.34 into equation 9.33 and makes use of the parameters
F
and
F
(defined in Appendix A.1), the following relationships are obtained:
Z
H
=
F
Z
p
+
1 −
F
2
upper
hemisphere
Z
p
d +
F
2
lower
hemisphere
Z
p
d (A.35)
F
2 sr =F
0
+F ↓ (A.36)
where =
0
is the solar zenith angle,
F
is the fraction of the direct actinic flux,
F
is
the ratio of the upward to downward actinic flux, and F
2 sr is the total downward
actinic flux that the radiometer aims to measure.
Obviously Z
H
has three contributions: the first term is related to the direct solar
radiation, the second one to the diffuse sky radiation, and the third one to the ground
albedo. The parameters
F
and
F
can be obtained from measurements or must be
estimated by a model. Appendix A.1 explains how
F
can be estimated in a simple way
from ground-albedo values A
s
.
A.5 Equivalent plane-receiver concept
In this part, the validity of the equivalent plane-receiver (EPR) concept is explained
following the description by Hofzumahaus et al. (1999).
First consider the situation where uniform and well-collimated radiation from a point
source at a very large distance is incident onto the surface S of the quartz diffuser
(Figure 9.46). In this case, E
is constant over the beam cross section and along the optical
axis z over the length of the receiver dome. According to equation 9.38, the measured
signal I
is then related to the incident irradiance E
by the detection sensitivity D
0
that has been defined by equation 9.31. In order to understand the physical meaning of D
ds
ds ′
S
S ′
E
λ
= constant
α
Quartz diffuser
Virtual plane
Z
Figure 9.46 Optical receiver irradiated by a collimated beam collinear with the rotational symmetry
axis of the optical receiver. S
and ds
are projections of the outer diffuser surfaces, S and ds, onto a
plane perpendicular to the optical axis (from Hofzumahaus et al., 1999, used with permission of Optical
Society of America).
Measurement of Photolysis Frequencies in the Atmosphere 491
we next examine the spectral photon flux
that is incident onto the detector surface S.
Each surface element ds (taken to be a ring element for the rotationally symmetric
receiver) receives a specific photon flux d
depending on the angle of incidence :
d
= E
ds
(A.37)
where ds
is the projection of ds onto a plane perpendicular to the optical axis:
ds
= ds cos (A.38)
The contribution dI
of the flux d
to the total measurement signal I
can be described
by:
dI
= rd
(A.39)
Here, r represents the local detection sensitivity of ds at the geometric position r on
the receiver surface. The term r depends on the efficiency of the transfer of incident
radiation from point r through the diffuser into the quartz fibre bundle leading to the
spectrometer.
Integration of equation A.39 over the whole surface S and substitution of d
(equation A.37) yield:
I
=
S
rE
ds
(A.40)
Since E
is assumed to be constant, we find by comparison with equation 9.38:
D
0
=
S
rds
(A.41)
We can interpret this integral to be the detection sensitivity of a virtual plane receiver
with the active surface S
, where S
is the projection of S of the real receiver onto a
plane perpendicular to the optical axis z (see Figure 9.46). Note that the position of the
virtual receiver plane along the z axis plays no role as long as the radiation is perfectly
collimated.
In the next step we consider the situation in which the receiver optic is exposed to
the radiation of a point source at a relatively close distance (Figure 9.21) such that the
irradiance E
varies along the optical axis from the tip to the end of the receiver optic
according to the inverse-square law. With this assumption the irradiance E
at any point r
on the surface S can be described as
E
= E
z
0
z
0
z
0
+r
2
(A.42)
where z
0
is the working distance from the lamp to the tip of the dome and z
0
+r is the
distance to the surface element dsr (Figure 9.47). We can now apply the approximation
1
1 +x
2
≈ 1 −2x if x1 (A.43)
492 Analytical Techniques for Atmospheric Measurement
z
z
0
r
ds ( r )
ξ
Figure 9.47 Receiver irradiated by a lamp at the working distance z
0
(cf. Figure 9.21). The surface
element ds
r is displaced relative to the reference point of the receiver by the distance
r along the
optical axis (from Hofzumahaus et al., 1999, used with permission of Optical Society of America).
which has an error of ≤ 1% for x≤006. Upon substitution of x =r/z
0
we find
E
= E
z
0
1 −
2r
z
0
(A.44)
Inserting this relationship in Equation A.40 yields for the measurement signal:
I
= E
z
0
S
rds
−
2
z
0
S
rrds
(A.45)
This is the signal that the spectroradiometer measures if the receiver optic is irradiated
by a lamp positioned at the distance z
0
.
In the next step we define an equivalent distance z
0
+z for the virtual plane receiver
(discussed above) such that the plane receiver with the sensitivity D
0
yields exactly
the same measurement signal I
(equation A.45) as the real detector at the working
distance z
0
:
I
= D
0
E
z
0
+z (A.46)
= D
0
E
z
0
z
0
z
0
+z
2
(A.47)
Applying approximation A.43 for small z/z
0
yield:
I
= D
0
E
z
0
1 −
2z
z
0
(A.48)
Inserting equation A.41 and comparison with equation A.45 yields:
z =
S
rrds
S
rds
(A.49)
Thus z is the mean displacement of the active surface elements ds along the optical axis
weighted by the local sensitivity r and the effective area ds
of each surface element.
Note that z is independent of z
0
and only depends on properties of the receiver optic.
Measurement of Photolysis Frequencies in the Atmosphere 493
This demonstrates that the real receiver optic exposed to an irradiance lamp at distance
z
0
can indeed be thought to be replaced by an equivalent plane receiver, which has the
active surface S
, the same detection sensitivity D
0
as the real detector, and follows the
inverse-square law if z
0
+z is used as the effective distance from the lamp. Note that
the EPR concept is based on an approximation that is valid only if the longitudinal
dimension of the receiver is small compared with the working distance z
0
.
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