SUM-SINES EQUATION FOR ESTIMATING THE PERCENT OF SHADOW LENGTH OF TARGETS IN THE SATELLITE IMAGES

With the development of high resolution sensors in remote sensing satellites, the shadow phenomena has appeared clearly in satellite image. The shadow is a separated feature in the satellite image, some time it may be considered as a problem due to the loss of ray information at the shaded region, other may be considered as a criteria to the height of the body which has a shadow. The length of shadow depends on the height of the body, location on the earth surface, and the sun location in sky at the imaging time. The sun location is varying every hour during the day and every day during year. These varying is calculated by complex astronomic equations. In this article, we simplified these calculations to just on equation depend on one parameter, and examine this equation by field measurements. The suggested equation is sum-sine equation with an enough accuracy to be used in civil, ecological, gardens designing near the high buildings, or architectural purposes. The equation can be used to estimate the building height from the shadow length in the satellite image, as will as it may used to estimate shadow length from the known body height on the earth.


INTRODUCTION: LIGHT AND SHADOW
Light is the electromagnetic spectrum rays with the visible range coming naturally from the sun. The electromagnetic rays propagate in straight lines from the light source.
The opaque objects obscure light to reach area in the opposite side if the light source, so the shadow area still dark relatively and called the shadow. Therefore, the shadow is the darkness that caused by an object when it blocks the light from getting a corresponding surface (Satellite Imaging Corporation). It can be considered that the shadow occurs at the image when there are objects which obscure the light coming from the source partially or completely (Prasath and Haddad, 2006).

PREVIOUS WORKS
The phenomena of the shadow was appeared clearly with the increasing of resolution satellite sensors (IKONOS, QuickBird, WorldView, and other) after 2000, so many studies were written about shadow to remedy or use it as a function to know the properties of shaded body. Cooper (1969)  where, dn is the day number, dn is equals1 at the first of January and 32 at the first of February. Lin and Nevatia (1998) adopted complex algorithms to estimate the three-dimensional model from the two-dimensional image by triangular equations [R. 2].
Using SPOT satellite images Shettigara and Sumerling (1998) Won Seok, et al. (2007) suggested geometric method for estimate the building (height) from one satellite image separately, he depended some astronomic parameters in the remote sensing imaging system as solar angles, image scale, direction of the shadow, angle of satellite imaging.

114
Shurooq Mahdi Ali and Emad Ali Al-Helaly Karantzalos and Paragios (2008) were adopted a metadata or ground information to find the hidden dimension (height) of the object in a simple way not mathematically. Arévalo et al. (2008) determined the shadow in images of high resolution satellite Quick-Bird of resolution (0.6 meters), but said it is a way suitable for images satellites IKONOS and WorldView. In all the angle of the Sun was taken fixed, and Arévalo et al. (2008) supposed that the calculating of sun angle at over particular day along the year is very complex.

THE SHADOW GEOMETRY
The shadows are two dimensional forms that result from three dimensional opaque bodies, and depend on the geometrical properties of these bodies, and at the angle of incident rays of the sun.

Angle of the sun and its synchronization
Sun is the main source of all electromagnetic waves directly or indirectly, the sun appears moving in the sky along the day, and the shadow is changing according to the sun direction and orientation.
The angle of the sun is varying all the year and day hours because of the difference in the angle of the fall of radiation on the ground.
In addition, it is seemed moving in complex orbit on the sky from the observer on the Earth.
The incident angle of the sun light on the Earth's surface varies by distance from the equator because of the spherical shape of the planet as it is well known. The sun is at 21/22 of March perpendicular to the equator so that there are no shadows to the vertical structures there, but it will be inclined in areas northern or southern equator due the spherical shape of the planet (Al-Najim and Al-Naimy, 1981). Here we will discuss the relation between the sun angle and the scene on the earth, which affect the shadow length and direction.

The sun
The sun's diameter is about 1,390,000 kilometers and the average distance to the Earth's surface is about 149.5 million kilometers (Spencer, 1971). It generate various wavelengths of electromagnetic radiation from long waves of radios and short wavelength of cosmic rays .

Electromagnetism which coming from sun consists of approximately 46 % visible radiation and
The Earth is rounding at a biosphere orbit around the sun, though, the amount of rays intensity falling on the ground varies throughout the year. The earliest distance to the Earth is 147 million km at the beginning of January, and further distance is 152 million km at the beginning of July, Fig. 2 (Appleton, 1945).

Fig. 1. Earth's orbit around the sun (Muneer, 2004).
Accurately the distance (Iqbal, 1983) between the sun and earth (ro)= 1.496 ×10 8 km, or in more accurate 149597890±500 km (Sun et al., 2013) and (r) is the sun-earth distance for every day in the year.
The day angle (Г : rad) is calculated from the equation below: Where dn is the day number, as it is mentioned.
The Earth's axis of rotation across the year is varying to be about ±23.5 degrees (Iqbal, 1983) during the year, which affects the sun incident angle and it's difference on the surface of the earth, thus affects the direction of the shadow and its extension.
The angle, position and elevation of the sun can be organized by astronomical equations (Appleton, 1945) in the following manner.

Sun location
The angle of solar deviation (δ): it is the angle between the equatorial plane (the plane of Earth revolution about the sun) and the line between the center of the Earth and the center of the sun. It is also termed declination angle, Fig. 3.

Shurooq Mahdi Ali and Emad Ali Al-Helaly
This equation is came from a Fourier formula, was developed by Spencer, (1971), and it is the most accurate equation. It is derived from the apparent sun path from the ground.
The geographic latitude angle (ϕ): this is the earth point coordinate, it is positive to the north of the Earth, and zero at the equator line.

Sun
The change the in orbit inclination. The change the in angle of sun deviation (δ) The seeming sun path due to the different deviation angle (δ) The shadow of all objects is lengthened words toward the poles of the planet, even if they are in the same latitude, Fig. 4.   (Iqbal, 1983). and roughly the angle of the head's head at sunrise or sunset is 90 degrees. It is also named (zenith angle), and can be calculated from the equation below (Iqbal, 1983): The elevation angle of the relationship is calculated by (Arévalo et al., 2008): Now, we can calculate every sun coordinate, or angles depending, and shadow on the number of days for everybody when its orientation and dimensions are known.

THE SELECTED BUILDING
The selected building is a (Qasr Al-Safer hotel and restaurant) in Najaf, City. The orientation of it is (32 o 00' 02.49" N) and (44 o 21' 32.88" E). or (32.000691 N 44.359133 E).
The image in Fig. 7 is a band composed QuickBird imagery by WGS 84 map projections. And the seen above is imaged at 09:43 a.m. according to the key information file which is attached with the imagery CD.
The building is 25.30 m height over the adjacent street level, on the crossing of two streets (Al-Rawan and Al-Ameer) as shown in Fig. 8, and there are no high buildings around it except at one side. These properties make easy to measure the length of the shadow from two sides. By the electronic and magnetic composes, we found the direction of building walls to measure the direction of shadow according to it, by this we need no repeat the field measurements of shadow according to the eastern direction, Fig. 9.

THE SUN ORIENTATIONS DURING THE YEAR
The sun orientation in the sky is calculated from astronomy equations during the years as presented.

THE FIELDS MEASUREMENTS
A scale tape and simple instruments achieved the field measurements. The results are illustrated in the table below: North East

GRAPHICAL REPRESENTATION
We need represent the results graphically by Fig. 11 and Fig. 12. Fig. 11 represents the shadow length of the chosen building, and Fig. 12 is more general, because it represent the percent of shadow with the real height of any building.

Fig. 11. The change of shadow length during a year.
We depend the numbers that resulted from the astronomic equations for more accuracy possible.
The two curves are similar to the sine distribution but there is a skew to the left, this need some modification and similar equation take into account this skew.

THE SUGGESTED SUM-SINE EQUATION
The proper equation form to describe the percent shadow length curve in Fig. 12  By testing some other day numbers it is cleared that we can depend this accurate sum-sine equation to find the percent of shadow of any point in our Najaf city. But we can also find the Sum-Sine equation for any other city for getting the shadow percent of any known height building when it is needed to know the shadow extension on the street, architectural aims, or any other civil purposes.
The sum-sine equation provide practical calculation for any engineer or designer, and can be achieved by a simple hand calculator or a programmed Excel sheet instead of the astronomy equations.
This equation can be used in to ways, either for calculating the height of building from the shadow length that measured from the satellite image at the certain imaging time, or for calculating the shadow length from the known height of any building in our region.

CONCLUSION
It is recommend to use the suggested equation to calculate the percentage of shadow for any building in any time of the year from the satellite image. Also, it is possible to estimate the shadow percent for any formal building at any day in the year at the imaging time. However,