We now turn our attention to the properties of the earth itself that are measured by the various synthetic aperture radar programs. In focusing on the properties of the earth surface detected by imaging radars, we start by analyzing a more traditional radar situation in which we detect at point of target. We will then generalize that to distributed habits such as crops, forests, another ground cover types of interest in remote sensing. Consider the situation shown in the diagram in which we start with the concept of an isotropic radiator on the left hand side, which is a transmitter which radiates equally in all directions, a simple light bulb can be thought of as an approximation to an isotropic radiator. In the next couple of slides we will work through the details of this diagram. We look first at the transmission of radiation from the isotropic source to the target. The power transmitted. P sub script t spreads out over the surface area of a sphere at a distance are not the power density over that spherical surface is given by the first equation here, which is just the power transmitted divided by the surface area of the sphere. When a real antenna is used rather than an isotropic radiator, the power density towards the target is increased by the gain of the antenna. The gain is just a number that tells us how much better the antenna is in the preferred direction compared with isotropic behavior. It is dimensionless. Now we have to describe the properties of the target in such a way that we can use it in our calculations. We describe it by an artificial area Sigma, called its radar cross section, which is defined as a cross sectional area at right angles to the incoming beam such that the area intercepts power from the incoming power density and re radiates it isotropically. The power available for isotropic re radiation is given by the first equation on this slide. The power density then produced back at the platform as a result of isotropic radiation from the target is as given by the second equation. In order to describe the actual power received back at the right app platform after having been scattered from the target, we introduce a property of the receiving antenna called its aperture. Which is also an area at right angles to the incoming beam of power density and which extracts power from that theme as seen in the last equation n this slide, the aperture has dimensions of area. The aperture of an antenna is a property we've used when describing its ability to extract power from an incoming wave front of power density. That is, when receiving. An antenna as a receiver can also be described in terms of its receiver gain, G subscript are, which is related to the aperture of the antenna as per the top expression on this slide, if we use receive again rather than aperture, the equation for receive power from the last slide becomes as shown in the center of the slide here. This is a famous equation. It's called the radar range equation. Note that there is suit powered varies inversely with the fourth power of range. It shows explicitly as should be expected that the power level received at the radar is a linear function of the target property. The radar cross section. Note that the equation has been derived on the basis of a point target rather than the distributed landscape which we will consider shortly. By point targets do arise in radar remote sensing. Any isolated strong reflector smaller than the size of a pixel can behave that way. Examples include houses, isolated trees, and targets intentionally located in the scene to help calibrate the radar. We will see examples of those later. We didn't have to see how we can modify the radar equation to account for the imaging of distributed regions, such as natural grasslands, forests, crops, oceans and so on. To do that, we represent the landscape by a collection of infinitesimal elements, as shown in the figure on this slide, each has an effective area of DS and a radar cross section of the Sigma. On the average, the region. Is it really that cross section per unit area of the signature? Yes, this is called the scattering coefficient. It is given the symbol sigma superscript o and it has units of meters squared per meter squared. That means in principle, it is really dimensionless. But it is important in practice to maintain the seemingly strange units of middle square can be described as we will see later. We're now treat each of the infinitesimal resolution elements as though, it were a single isolated point scatterer and use the right our equation which were previously as in the equations shown here within integrate all those contributions over the resolution cell to get the received power from the whole element as saying. The received power is a linear function of the scattering coefficient as it was for the radar cross section before. This is important, because it means that the measured reserved power is in proportion to the scattering coefficient of a surface. The scattering coefficient is usually expressed in the log rhythmic units of decibels and its value is related to a reference level. In this case, the scattering coefficient of one meter squared per meter squared. Base 10 logarithm so used and it is common to refer to the scattering coefficient as sigma nought. Radar cross sections for point targets are also expressed in decidbels reference to one meter squared as seen in the second equation. The disability notation is very useful in practice not just because the scattering coefficient and radar cross section can vary over several orders of magnitude, but also because values can be computed easily as we see on the next slide. This slide shows a range of scattering coefficients expressed in normal form. And indeed, the form along with some sample calculations which shall have the day bay form can be used to compute knew values easily. Because we have radars with different polarization configurations, we add subscripts to the scattering coefficient to indicate which transmit receive configuration was used to measure it. For quad or fully polarized radars, the four scattering coefficients are usually brought together into a sigma nought matrix. In more advanced treatments of radar imaging, we like to refer to the transmit receive situation in terms of electric fields rather than powers and power densities. At derivations here, we're based on the concept of power transmission and reception. But as we saw when discussing polarization, the actual signal is carried as a combination of propagating electric and magnetic fields. Just as we expressed the power view in terms of scattering coefficients, we can do the same with fields and in particular with the electric fields involved rather than scattering coefficients within describe the scattering events in terms of the elements of the scattering matrix indicated here. Incidentally, this matrix equation tells us why the second subscript on the matrix element here and in all treatments is the transmitted or incident polarization and the first subscript is a scattered or received polarization. If they were the other way around, the matrix vector multiplication would not work. Summarizing this lecture. The transmitted right our signal is treated by assuming isotropic radiation multiplied by the gain of the transmitting antenna. Gain is just a number, which tells us how much better the antenna is compared with isotropic transmission in the direction of interest. The target is described by an area called its radar cross section, which is defined by assuming that the power scattered from the target is isotropically reradiated. Distributed targets are described by the radar cross section per unit area, which we call the scattering coefficient both radar cross section and the scattering coefficient are polarization dependent. Both radar cross section and scattering coefficient are usually expressed in decibels. Finally, we can also describe the scattering behavior of a target in terms of electric fields rather than powers. In which case, we define and use the scattering matrix of the target. The first two questions Yeah, are important in understanding why the scattering coefficient of the landscape image by remote sensing radars, is a function of look, or incident angle. The last question shows you that radar image in space, essentially on two inverse square laws applied at the same time.