Modeling Solar Radiation at the Earth’s Surface

This section reviews the properties of solar radiation on Earth and summaries well-known models which are used to estimate the amount of radiation falling on a tilted plane.

Extraterrestrial solar radiation falling on a surface normal to the sun’s rays at the mean sun earth distance is given by solar constant (Isc). The current accepted value of Isc is 1367 W/m2.

When solar radiation enters the Earth’s atmosphere, a part of the incident energy is removed by scattering or absorption by air molecules, clouds and particulate mat­ter usually referred to as aerosols. The radiation that is not reflected or scattered and reaches the surface straight forwardly from the solar disk is called direct or beam radiation. The scattered radiation which reaches the ground is called diffuse radia­tion. Some of the radiation may reach a panel after reflection from the ground, and is called the ground reflected irradiation. In the Liu and Jordon approach the dif­fuse and ground reflected radiations are assumed to be isotropic. The total radiation consisting of these three components is called global or total radiation as shown in Fig. 2.1.

In many cases it is necessary to know the amount of energy incident on tilted surface, as shown in Fig. 2.1. However, measured total and diffuse radiation on horizontal surface are given in most available solar radiation databases. There are many models to estimate the average global radiation on tilted surfaces.

In this section we present the isotropic model developed by Liu and Jordan (Liu and Jordan 1963) which also estimates the average hourly radiation from the average daily radiation on a tilted surface.

The daily total radiation incident on a tilted surface HT can be written as

where HT, Ньт, Hd, T and H, T are daily total, beam, diffuse and ground reflected radiation, respectively, on the tilted surface.

In this model, (Liu and Jordan 1963) assumed that the intensity of diffuse ra­diation is uniform over the sky dome. Also, the reflected radiation is diffuse and assumed to be isotropic. Consequently, the daily total radiation on a tilted surface is given by

1 + cos б 1 — cos в

Ht = HbRb + Hd 2 + Hp 2 (2.2)

where Hb, Hd and H are daily beam, diffuse, total radiation, respectively, on a hori­zontal surface. в represents a tilt angle, p the ground albedo and Rb the ratio of the daily beam radiation incident on an inclined plane to that on horizontal plane. For the northern hemisphere and south facing surfaces Rb is given by

cos (ф — в) cos 5 sin®( + (d's sin (ф — в) sin 5 Rb (2.3)

cos ф cos 5 sin Щ + (Os sin Ф sin 5

where p, 5 and (os are the latitude, the declination and the sunset hour angle for the horizontal surface, respectively. (Ds is given by

cos = cos—1 (—tan ф tan S) (2.4)

(d's is the sunset hour angle for the tilted surface; it is given by

co's = min{cos— 1 (—tan^tanS),cos—1 (—tan(ф — в)tan5)} (2.5)

In the relation (2.3) (Ds and cd!. are given in radian.

The daily clearness index KT is defined as the ratio of the daily global radiation on a horizontal surface to the daily extraterrestrial radiation on a horizontal surface. Therefore,

(2.6)

where H0 is the daily extraterrestrial radiation on a horizontal surface. H0 is given by (Sayigh 1977; Kolhe et al. 2003)

where jd is the Julian day of the year.

Outside the atmosphere there is neither diffuse radiation nor ground albedo. H0 is then assumed to be composed only of the beam radiation. Similarly, for tilted surfaces, the daily extraterrestrial radiation above the location of interest HT0 is constituted only of direct component. Then, according to the relation

HbT = HbRb

HT0 can be computed as follows

Ht q = HoRb

2 Fractal Dimension Estimation

2.1 Preliminaries

Mathematically, any metric space has a characteristic number associated with it called dimension, the most frequently used is the so-called topological or Euclidean dimension. The usual geometrical figures have integer Euclidean dimensions. Thus, points, segments, surfaces and volumes have dimensions 0, 1, 2 and 3, respectively.

But what for the fractals objects, it is more complicated. For an example, the coastline is an extremely irregular line in such way that it would seem to have a surface, it is thus not really a line with a dimension 1, nor completely a surface with dimension 2 but, an object whose dimension is between 1 and 2. In the same way, we can meet fractals whose dimension ranges between 0 and 1 (Like the Cantor set which will be seen later) and between 2 and 3 (surface which tends to fill out a volume), etc. So, fractals have dimensions which are not integer but fractional numbers, called fractal dimension.

In the classical geometry, an important characteristic of objects whose dimensions are integer is that any curve generated by these elements contours has finite length. Indeed, if we have to measure a straight line of 1 m long with a rule of 20 cm, the number of times that one can apply the rule to the line is 5. If a rule of 10 cm is used, the number of application of the rule will be 10 times, for a rule of 5 cm, the number will be 20 times and so on. If we multiply the rule length used by the number of its utilization we will find the value 1 m for any rule used.

This result if it is true for the traditional geometry objects, it is not valid for the fractals objects. Indeed, let us use the same way to measure a fractal curve,

with a rule of 20 cm, the measured length will be underestimated but with a rule of 10 cm, the result will be more exact. More the rule used is short more the measure will be precise. Thus, the length of a fractal curve depends on the rule used for the measurement: the smaller it is, the more large length is found.

It is the conclusion reached by Mandelbrot when he tried to measure the length of the coast of Britain (Mandelbrot 1967). He found that the measured length depends on the scale of measurement: the smaller the increment of measurement, the longer the measured length becomes.

Thus, fractal shapes cannot be measured with a single characteristic length, be­cause of the repeated pattern we continuously discover at different scale levels.

This growth of the length follows a power law found empirically by Richardson and quoted by Benoit Mandelbrot in his 1967 paper (Richardson 1961)

L (n) x П а (2.10)

where L is the length of the coast, n is the length of the step used, the exponent а represents the fractal dimension of the coast.

Other main property of fractals is the self-similarity. This characteristic means that an object is composed of sub-units and sub-sub-units on multiple levels that re­semble the structure of the whole object. So fractal shapes do not change even when observed under different scale, this nature is also called scale-invariance. Mathe­matically, this property should hold on all scales. However, in the real world the self-similarity is only observed over some scales the objects are then statistically self-similar or self-affine.

2.2 Experimental Determination of the Fractal Dimension of Natural Objects

Fractal dimension being a measurement in the way in which the fractal occupies space, to determine it we have to draw up the relationship between this way of occupation of space and its variation of scale. If a linear object of size L is measured with a self-similar object of size l, then number of self similar objects within the original object N(l) is related to L/l as

<2лі)

where D is the fractal dimension. From where

For the self-similar fractals, L/l represents the magnification factor and l/L the reduction factor. Nevertheless, when one tries to determine fractal dimension of

natural objects, one is often confronted with the fact that the direct application of Eq. (2.12) is ineffective. In fact, the majority of the natural fractal objects existing in our real world are not self-similar but rather self-affine. The magnification factor and the reduction factor are thus difficult to obtain since there is not an exact self­similarity. Other methods are then necessary to estimate the fractal dimensions of these objects.

In practice, to measure a fractal dimension, several methods exist, some of which are general, whereas others are applicable only to special classes of fractals. This section, focuses on the more commonly used methods namely, Box-counting di­mension and Minkowski-Bouligand dimension which are based on the great works of Minkowski and Bouligand (Minkowski 1901; Bouligand 1928) and from which derive several other algorithms.

If one plots ln(N(є)) versus 1п(1/є), the slope of the straight line gives the esti­mate of the fractal dimension Db in the box-counting method.

Figure 2.2 gives an example illustrating this method. The object E (a curve) is covered by a grid of squares of side є1 = 1/20, and for this value of є total number of squares contained in the grid is 202 = 400 and the number of squares intersecting the curve E is 84 (Fig. 2.2a). In Fig. 2.2b, which is obtained using different values of є, the slope of the straight line fitted by a linear regression constitutes the fractal dimension of the curve E.

Minkowski-Bouligand dimension: This method is based on Minkowski’s idea of dilating the object which one wants to calculate the fractal dimension with disks of radius є and centered at all points of E. The union of these disks thus creates a

Minkowski cover.

Let Б(є) be the surface of the object dilated or covered and Dm the Minkowski - Bouligand dimension. Bouligand defined the dimension Dm as follows

or, rearranged

The fractal dimension can then be estimated by the slope of the log-log plot: ln(S(є) /є) = f (ln(1/є)) fitted by the least squares method. Figure 2.3a shows the Minkowski covering E(є) composed of the union of disks of radius є.

## Modeling Solar Radiation at the Earth’s Surface

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### Foreword

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