**A Model of Diffuse Broadband
Solar Irradiance for a Cloudless Sky**

**Warwick****
Grace of Grace Research Network**

**11 April 2006**

*Published in Australian
Meteorological Magazine June 2006*

**Abstract**

An analytic model is
presented for the broadband diffuse irradiance received at the earth’s surface
through a cloudless homogeneous atmosphere containing light absorbing agents and isotropic scattering
agents. The model is extended to include a contribution from light re-scattered
after ground reflection. The model is
validated, with some qualification, against (a) a

**Motivation**

Sophisticated and
comprehensive spectral and broadband models of radiation requiring numerical
solution (radiation transfer equation models) are well established in the study
of direct and diffuse light (Mishchenko et al 2002, 2003; Bird and Riordan,
1984). At the other extreme are well-supported and simple empirical broadband models
for diffuse irradiance in terms of the direct irradiance (Peterson and Dirmhirn 1981 and Campbell and
Norman 1998, hereafter PD and CN respectively). The highly idealised broadband model
developed in this paper provides an analytic expression for diffuse irradiance
which is comparable in simplicity to the empirical models. This model also
allows an *a priori* indication of the
empirical constants in the empirical models. Compared to the more sophisticated
numerical models, an analytic model of this nature provides an easy way to
evaluate the sensitivity of diffuse irradiance to environmental factors of
zenith angle, albedo, atmospheric transmittance, and the relative proportions
of scattering and absorption agents in the atmosphere. As such it is suitable
for illustrative or pedagogic purposes.

**Introduction**

At

Diffuse irradiance in
clear (cloud free) skies is typically 10% of the global irradiance. Air molecules
scatter light while aerosols may act to either absorb and scatter light. Scattering
by air molecules (Rayleigh scattering) is predominantly forward and backward of
a photon’s pre-scattering direction; scattering by aerosols (Mie scattering) is
predominantly transverse. Scattering and absorption are wavelength dependent
and height dependent. Table 1 shows typical amounts of scattering and
absorption due to the various atmospheric constituents. Water vapour is one of
the largest causes in variability of absorption
and scattering. Background information
on light transmission through the atmosphere is available at the web sites of
the Australian Bureau of Meteorology and other relevant international agencies.
Comprehensive texts include Kondratyev
1969, Iqbal 1983, Goody and Yung 1989, Bird and Riordan 1984, and Schwerdtfeger 1995.

Constituent |
Percent absorbed |
Percent scattered |

Ozone |
2 |
0 |

Water vapour |
8 |
4 |

Dry air |
2 |
7 |

Upper dust |
2 |
3 |

Lower dust |
0 |
0 |

Total |
~
14 |
~ 14 |

**Table
1** *Typical
fractions of solar radiation absorbed or scattered in passing through cloudless
atmosphere due to various constituents in the atmosphere (from Solar Radiation
Monitoring Laboratory, **University** of **Oregon**, 2002)*

In determining the
radiation, knowledge of the variation of the solar constant and the solar zenith angle (the angle of the sun
to the vertical) are required. Useful formulas of sufficient accuracy are
available from many sources (eg, Smithsonian Tables (List 1984), de Wit 2000)).
Half-hourly ground based observations of
irradiance are available from the Bureau of Meteorology with the convenience
that these are already tabulated against local solar time (ie, the equation of
time, the longitude distance to the reference meridian for the local time zone,
and any effect of daylight saving are already accounted for). Figure 1 compares the daily variation of the extra-terrestrial,
global, direct and diffuse irradiance for a cloud-free day at

Figure 1 *Observed half-hourly exposures of global, direct and diffuse irradiance
and calculated extra-terrestrial (ET) irradiance in MJ/m ^{2 }at *

**.**

**Figure
2** *Typical cloud-free diffuse irradiance.
Half-hourly exposures in MJ/m ^{2 }observed
at *

**Figure 3** *Schematic of main features of the model. Incoming solar radiation, Q,
at the top of the atmosphere may undergo absorption, scattering within the
atmosphere or reflection from the ground. Once scattered or reflected, light
may undergo subsequent scattering.*

**The Model**

The model as depicted
at Figure 3 assumes independence of wavelength and that all scattering is
isotropic. Also assumed is an homogeneous atmosphere of finite depth, *H*, containing absorbing agents and
scattering agents such that the extinction of the solar beam conforms to Beer’s
Law (Eq 1). In the first instance, it is assumed that there is no ground
reflection, ie, surface albedo is zero. The intensity (*q*, the
radiation energy per unit area on a surface normal to the beam) attenuates
exponentially with distance traveled *l*
such that

*dq = - kq dl (1)*

where the constant *k* represents the concentration of
intercepting agents: *k _{a}*
represents absorbers and

*dq = + k sec**n** q dy . (1a)*

Integrating Equation* *1*a *and noting that* q = Q *at* y = H, *then

* q = Q e ^{- k H }*

*dq = Q
k sec**n** e ^{-
k H }*

*Direct
(beam) irradiance (normal to sun) **at
the surface **= Q * *e ^{- k H}*

*Direct
irradiance **on horizontal
surface** ** **= Q cos **n** * *e ^{- kH}*

The product *kH* is known as the optical depth of the
atmosphere, and the term *e ^{- kHsec}*

The* *amount intercepted in the direct beam on
a horizontal unit area (that is, integrating Equation 1*c* through a vertical column of height *H* ) is therefore *Q cos**n** *(1 – *e ^{- kH }*

*S _{o}* = ½

Assume that this is distributed
uniformly within the column (reasonable for clear sky) so that the downward
scattered irradiance contribution *dS _{o}*
from a unit horizontal area element in the column of thickness

*dS _{o}* = ½

This amount may be envisaged
as being emitted isotropically from a thin horizontal uniform slab. The flux
density reaching the surface from such a source is known to attenuate as *w* where

* _{}* .

and where *2* is the zenith angle for any scattered beam in
question. Now* *it is apparent that * w* is a function __not__ of* **2* ,* *but
of* *the product* k _{a }*and

_{} (5)

where* β **.** *1.66. And thus Equation 3 is modified by the
attenuation factor, *w*, to become

_{} _{} . *(6)*

The derivation of the expression
for *w* at Equation 4 is well
established (eg Iqbal 1983, Schwerdtfeger 1995 and Coakley 2003) and is not provided
here. However, intuitively one might expect that *w* would involve exponential attenuation in *k _{a} y. *Substituting
Equation 5 into Equation 6 and integrating throughout the vertical column gives

_{} _{} *(7)*

and thus _{} . *(8)*

Anticipating that _{}* *(typical values are ~0.2), then Equation 8 may be approximated as

*S _{o} = *0.5

Equation 9 is therefore
an expression for zero-albedo diffuse irradiance. With increasing *k _{a}H*, Equation 9 will tend to
under-estimate. Replacing

*S _{o}
=
*0.5

so that the diffuse
irradiance is an analytic function of
zenith angle, *n*, of optical depth, * k _{ }H*, and the
scattering ratio,

Compared to photons
scattered once only, those photons undergoing second and subsequent scattering
have on average an increased path length. So the absorptive attenuation
expected from Equation 5 will be an under-estimate of absorption.

Explicitly, the
sources of error in the model as expressed by Equation 9 or 9*a* are as follows. The effect of multiple
scattering means that the model over-estimates diffuse irradiance when *D** *< 1. With increasing turbidity most of the
initial absorption and scattering occurs in the upper part of the atmosphere
and so the assumption of uniformity will break down - leading to an over-estimate by the model.
Thirdly, the simplifying approximation introduced at Equation 9 causes the
model to under-estimate.

For convenience, allow
*S _{o} *to denote the

With argument similar
to that above, the interception (*Int*)
of ground-reflected radiation is given by

*Int* = (*1 – e ^{-kH
}*

As before, the
downward scattered contribution *dS** _{1}* at the surface from a thin horizontal uniform
scattering slab is:

*dS** _{1}* = ½

*dS** _{1}* = ½

and *w *is the atmospheric depletion of the
beam that is scattered from the slab (* _{}*). The final expression for

*S** _{1}*
= 0.5

Since diffuse irradiance *S ≈
S _{0
}+ S*

then_{ }_{} . *(14)*

The neglect of the
reflected *S _{o}, S*

_{}. *(14a)*

Equations14 and 14*a* clearly show that *S* increases with *D* or *A,*
as intuitively expected. The effect of changes in *n* or *kH*
are less obvious but are easily plotted for inspection.* * An alternative perspective
is that the analytic model as a function of *n* requires a knowledge of the three parameters *T _{z}*

A comparison of the variants
of the model with symmetrical reflection (Equation 14) and with isotropic
surface reflection (represented by Equation 14*a*) is provided at Figure 4 with parameters *T _{z}* = 0.8,

**Figure 4** *Comparison of Fresnel and Lambert variants of the analytic model with
parameters T _{z} = 0.8, *

**Validation**

Validation was
performed in three ways: firstly, against output from a Monte Carlo model, having
the same essential physics but not incorporating the several approximations
used above, over a range of values of *T _{z}*,

*Monte Carlo** model*

A Monte Carlo model is
easily constructed (less than a page of coding) and conceptually simple in that
for the same model conditions, the atmosphere is comprised of 100 layers of
thickness, *h*, where *h = *0.01 *H*. Incoming solar radiation is
represented by the *Q cos**n** *term above and in the form of *N*
(~ 10 ^{5} ) photons which are ‘fired’ into the top layer from above.
Within any particular layer, each photon has a probability either of passing
through uneventfully, or of being intercepted. The probabilities for either of
these events are easily related to *kh sec**n*. If a
photon is intercepted it must then either be absorbed or scattered, with
probabilities related to *D* . If it is scattered, then under the isotropic
scattering condition, it scatters with equal probability at any solid angle. If
it is neither scattered nor absorbed, then the photon passes into an adjacent
layer where it is subject to the same process as before. A photon reaching the ground is reflected symmetrically
with probability equal to the albedo, otherwise it is absorbed. Each photon is
tracked until either it is absorbed or it exits from the top of the atmosphere.
Counts of photons corresponding to the irradiance components of direct, diffuse,
absorbed, and that returned to space are easily made.

The *T _{z} , *

For a range of *T _{z} *,

_{} *(15)*

and _{} *(16)*

Comparing the analytic
model for zero albedo with this correction factor against the

Figure 6a shows similar
comparative plots using the analytic model (without any correction factor) with
albedo of 0.25, 0.5 and 0.75 with *T _{z}*
limited to 0.6, 0.4 and 0.2. The comparisons are still quite good although
the analytic model tends to strongly under-estimate with higher values of

_{} *(17)*

_{} *(18)*

As shown by the
relevant comparison at Figure 6b, the corrected analytic model provides quite
good agreement. As expected *F _{A}* reduces to 1 when

**Figure
5a** *Comparison of analytic model (lines) with **Monte Carlo** model (diamonds) for a
range of T _{Z} , *

**Figure
5b**
Comparison *of corrected
analytic model with *

**Figure
6a** *Similar to Figure 5a but with albedo A =
0.25, 0.5 and 0.75 in rows from the top to bottom and T _{Z}
= 0.4, 0.6 and 0.8 in columns from left to right. With lower transmittance, relatively more
diffuse irradiance is received. With higher albedo, the irradiance increases
and the increase is greater with larger *