 Hello and welcome to today's lecture. So today just to break the monotony before I begin the lecture, let us have a quick overview that is a summary of what we have been learning till now as part of three lectures of module 2. So let me quickly walk you through the topics covered so far. So till now as part of the second module, we have been discussing about what is synthetic aperture radar abbreviated as SAR, the basics and how images are being formed by SAR, how it operates. So a SAR image can be considered as a complex number and then we were trying to understand about key terminologies such as azimuth resolution, ground range resolution, slant range resolution, SWAT and we also discussed the concept of a chirped pulse. Remember, all these are terminologies which will eventually help you understand the bigger picture in the upcoming lectures. So what I will do is I shall be repeating these or referring to these terminologies wherever possible to let it sink in. So with this background, we are currently in the second module and the fourth lecture. Now whether our discussion is on imaging radars like SAR or non-imaging radars say like altimeters, the performance of a radar system needs to be defined that is how good does a radar give you a data. So that needs to be defined and it is usually done by something known as a signal to noise ratio SNR and a range resolution. Now here by signal to noise ratio, I am referring to the strength of the signal with respect to that of the noise. So remember the example in one of the earlier lectures when I asked you to imagine that you are in a crowded traffic junction and then you are trying to focus listen to your favorite song which is being played somewhere nearby. So the traffic junction as such is going to offer a lot of background noise whereas you are straining your ears in a particular direction to listen to your favorite song which is being played. Of course, we discussed about this in the context of chirped pulses but then it is useful here as well. So my point is always compare a radar antenna to your ears and the measured power of received echo is very important. And for that matter we should learn what it is dependent on, measured power of received echo. Very important and we should learn what it is dependent on. Now a question is displayed on the screen that is is it technically possible to measure the famed echo that is returned by target. So you will get the answer to this question slowly as we proceed through the lectures. So in this context, let me try to introduce something known as a radar equation to you, radar equation. So this is the fundamental relationship that governs the power of received echo. Remember whenever we talk about synthetic aperture radars or imaging radars, there is always a pulse, coherent pulse that is being sent from the instrument which hits the target and gets scattered. A part of it returns back, gets registered by the antenna and that is what you get as a complex number and SAR image is nothing but a collection of complex numbers. So till now our understanding is limited to this. So what we will do is we will try to understand what is the fundamental basic relationship that will govern the power of received echo. Because complex numbers are registered based on the power of the received echo, received from where, from the target and it is getting scattered by the target in all directions. It is not isotropic but then it is getting scattered in all directions and a few of this return echo managed to travel all the way through the atmosphere, get registered in the antenna and that is what gives you the data, SAR data, SAR image. So again at the relationship of radar equation let me reiterate it is very much applicable even if you are referring to imaging radars like SAR or non-imaging radars like altimeters. Remember we discussed about what is an imaging radar that gives you images and what is a non-imaging radar like altimeters. So for deriving a radar equation we need the help of a few key terminologies and unless we understand them we cannot fully appreciate a radar equation. So these terminologies are number one is radar cross section abbreviated as RCS radar cross section. Second is gain of an antenna and third is back scattering cross section. I am trying to introduce these terms now so that we can discuss in detail as part of this lecture. So let us start by scattering cross section. So here in front of you you can see a small diagram of the earth surface and assume we have a transmitter which is transmitting the signals that is causing the incident field that you see on the screen. The target is nothing but you assume it as a domestic aircraft which is flying in the troposphere. So this incident field is going to hit the target and of course the blue lines what you see are nothing but the scattered field the energy that is getting scattered in all directions and now we are trying to understand about scattering cross section. So here remember always when I say scattering I am always referring to redirection of incident electromagnetic energy by any object or any target redirection. Now assume we are talking about a discrete target not a continuous target discrete target. So as an electromagnetic wave hits this discrete target there is scattering happening and this scattered field is nothing but the difference in the total field when the target is present and the field that would exist if this target were absent. So let me try to repeat the scattered field what you see on your screen is nothing but the difference in total field when the target is present and the field that would exist if the target were absent. So here remember we are assuming that all the other conditions that is surrounding conditions remain the same. Here I have mentioned the example of a target as a domestic aircraft but different targets will tend to have different ways to scatter the electromagnetic radiation. So let us come to scattering cross section. Scattering cross section refers to how effectively a scatterer scatters the radiation. It is usually denoted by the Greek letter sigma scattering cross section. It helps you to understand the effectiveness of any scatterer effectiveness of any scatterer. So the incident energy as such is not going to be scattered equally in all directions as I mentioned it is not going to be isotropic. There will be some directions in which scattering will be more predominant and certain other directions where in scattering might be less. Now instead of mentioning scattering cross section it will be more appropriate for us to define something known as directional scattering. Now to define directional scattering we need the help of solid angle. So that brings us to the figure that you see on the screen solid angle. So how do we define a solid angle? In a three dimensional space that is in 3D the angle that is subtended by any part of the spherical surface at the center is known as solid angle. Let me try to repeat. Assume a three dimensional spherical space, spherical space. So the angle that is subtended by any part of the spherical space at the center is known as solid angle. The angle subtended by any part of the spherical surface at the center known as solid angle. Now solid angle of 4 pi is a full sphere. We are aware of that and typically we estimate solid angle by this relationship that is area by radius. In this case area by radius. So our aim is to define effectiveness of a scatterer and we mentioned that we need to define it in terms of a direction because there will be more scattering in some directions and less scattering in some other directions. So I am going to write sigma as sigma theta to denote directional scattering cross section, directional scattering cross section and it is defined as a function of angle of observation which is theta and it can be expressed by this relation that is numerator we have scattered power remember per unit solid angle in direction theta and denominator is intensity of incident plane wave that is the power that is scattered by total power that is incident. So you know if you watch closely the denominator is divided by 4 pi. This is to normalize as solid angle of 4 pi is a full sphere as we discussed solid angle of 4 pi is a full sphere hence to normalize we are dividing the denominator by 4 pi. Remember that a scattering cross section need not directly be equal to the physical cross sectional area of the target for example you know in the one of the earlier slides I showed you an aircraft as a target. So the physical cross sectional area of the target that need not be same as scattering cross section okay. Now that we have seen a directional scattering cross section which has units of meter square. Let us try to understand a slightly different variant of directional scattering cross section which we call as total scattering cross section sigma t total scattering cross section okay sigma t. So we define it as the ratio of total scattered power upon intensity of incident plane wave. So the total scattering cross section it tells you the amount of energy that is getting scattered okay just a different variant of sigma theta that is all. So we learnt about directional scattering cross section and total scattering cross section. So with this background let us try to understand what is radar cross section okay abbreviated as RCS radar cross section. So in active microwave remote sensing like radars the system is illuminating the target which means we have full control of how much electromagnetic energy is falling upon or is incident upon the target area. We are in full control of that because we are not depending upon the electromagnetic energy from the sun we are not depending upon the sunlight. So in radars we are fully in control of how much amount of energy is falling on or is incident on the target area. Now our aim is to quantify the amount of energy that is getting returned to the sensor okay and we can understand this in terms of the radar cross section that is RCS. So before I show you the relationship visually I want you to assume that a certain target area let us call it sigma based upon our understanding of scattering cross section. So assume this target area has intercepted the incident electromagnetic energy okay this target area has intercepted the incident electromagnetic energy and it has redirected the same isotropically for one instant you assume that scattering is happening equally in all directions. So target area sigma which has intercepted the incident electromagnetic energy and then it has redirected the same isotropically. Then the radar cross section is nothing but the intensity received by sensor by intensity incident on the target multiplied by 4 pi R square. Now you understand why I am multiplying it by 4 pi R square okay. Here let me give you a few cases okay as in assume that the target area is scattering very very little power back okay. So what will happen is that sigma will naturally approach 0. This can be as the target is either very very small or it is transparent to the incident radiation or it is absorbing all of the radiation. In such cases the power scattered will be small and the value of sigma shall closely approach 0 that is for case 1. What is case 1? When the target area is scattering very very little power back. Let me take you to case 2 for example assume the cross sectional area that is sigma is very very high okay sigma very very high. The target tends to scatter more energy in the direction of radar antenna. Now the cross section may be much larger than the target frontal area remember the cross section what we are discussing radar cross section it may be much larger than just the frontal area. So when we discuss about case 2 it is worthwhile to remember about corner reflectors remember corner reflectors which we briefly mentioned in parts of the earlier lecture under special scatterers which tend to scatter more energy in the direction of radar antenna. They are usually used in the calibration validation purposes and we also discussed about the corner reflectors that are available in hydrobath which is installed by ISRO okay. Corner reflectors. So moving on the values of sigma that is radar cross section we can compute them analytically okay they can be computed analytically for example we can estimate the value for a perfectly conducting sphere or a flat plate or a cylinder okay. Given here are few examples and I want you to focus on the last two which are nothing but the object shapes of corner reflectors dihedral corner reflector or trihedral corner reflector. So the RCS values are given in the case of dihedral reflectors it is proportional to the length of side and in the case of trihedral reflectors it is proportional to fourth power of edge length. So you know there are different objects wherein we can calculate the RCS values. Now let us translate this into real world scenario. So assume you work in say air traffic management and you are using radar remote sensing to estimate the scattering cross section of domestic aircrafts. Shown here are few examples of commonly accepted RCS values for different targets. We can have a small fighter plane which can have an RCS of 2 meter square. Our own human body has an RCS of 1 meter square. A bird which is having an RCS of 0.01 meter square. Similarly for different types of aircrafts whether they are being used for domestic purposes or for defense purposes each of these will be having different RCS values radar cross section values in units of meter square. So a couple of commonly accepted RCS values are shown here for different targets. Remember there are many more and these are just a small sample used for representation purposes. So one thing to note here is that for distributed targets. So till now we have been discussing about discrete targets. Now imagine a distributed target say the ocean or the bare ground. When we increase the measurement area that is ocean area is huge, bare ground area is huge. So when we increase this measurement area the power that is getting scattered also changes accordingly. Let me re-itrate for distributed targets like bare ground or oceans. Whenever we increase the measurement area the scattered power is also going to change accordingly and the radar cross section is also going to change. So we need some means to quantify all the natural scatterers. Some means some generic method by which we need to quantify all the natural scatterers. So earlier we were discussing about discrete targets and now I want you to fix your attention on distributed targets. We already know what is an instrument footprint is it not? Instrument footprint we are aware of that that is the area on the ground that is illuminated by a synthetic aperture radar system instrument footprint. So as this module is on SAR I am going to use the term SAR system area on the ground illuminated by a SAR system instrument footprint. Now in one single footprint there can be multiple targets, isn't it? Multiple targets. There can be buildings, there can be water body, there can be vegetation. So my point is we may have multiple targets within the same footprint and we need some measure which can represent that is which can normalize the values independent of the footprint size. So one way is to define the normalized backscatter coefficient. So the quantity that is commonly used in radar remote sensing including imaging radars we call it as sigma naught, sigma naught. It is also known by different names like normalized radar cross section NRCS and it is unitless. So what does it give us? Before I show you the expression I want you to understand what is sigma naught. What does it give us? The scattering cross section per unit area of surface. So whenever we divide the area actual geometric area on ground surface whenever we divide by the actual geometric area on the ground surface sigma naught becomes a target property. It is no longer dependent on the footprint size. So the expression for estimating sigma naught or normalized radar cross section NRCS is sigma by A. Remember it is strictly the property which we are interested in because it says something about the ground surface. Now imaging radars like SAR, synthetic aperture radar they do not measure sigma naught directly. No they do not measure sigma naught directly. Instead they measure something known as a radar brightness or beta naught. Now the actual geometrical area illuminated is something which we do not know. Let me repeat the actual geometrical area that is getting illuminated is something we do not know. Why? Why do I say that we do not know? Because of topographical variations. The actual area may be different from A. Predicted area is different. Actual area is different. And if we do not have information about the topography the radar brightness shall be used as a measure of backscatter. Radar brightness used as a measure of backscatter. Now say we have information about the topography through some means. Then we can get the value of sigma naught by using the expression beta naught by sine of theta i where theta i is nothing but the incident angle. And what is beta naught? It is nothing but the brightness, radar brightness beta naught. What is the difference between beta naught and sigma naught? The difference is in the area considered. Why? Because the actual geometrical area that is getting illuminated is something we do not know because of topographical variations. So, if we do not have information about topography whatever we get as radar brightness that is beta naught we can use it directly as a measure of backscatter. Now on the contrary if we have information about the topography to get the value of sigma naught that is normalized radar cross section we can use this simple expression. I will show you a diagram shortly wherein you are able to visually understand the difference between these sigma naught and beta naught. Now please remember that as the incident angle increases the scattered energy per unit surface area also changes. So, let me reiterate as the incident angle is increasing the scattered energy per unit surface area is also changing. So, the system as such is just comparing the outgoing radiation and the incoming radiation. The radar system as such is just comparing the outgoing radiation and incoming radiation and it cannot determine the amount of energy drop off due to differing incidence angles. Amount of energy drop off due to differing incidence angles cannot be determined by a radar system. So, for convenience we use another term known as gamma which is nothing but sigma naught by cos of theta i. We already know what is sigma naught. Now theta i is nothing but the incident angle. So, we started to understand about scattering cross section and we have covered sigma naught, beta naught and gamma. Of course, you will be understanding how to estimate this as part of the tutorials and we will be delving deeper about what each term means as the lecture progresses. This is just to give you a quick round of introduction on what these terms mean. Just to remind you why we are learning all this. It is to derive the radar equation for which we need to understand about radar cross section. We have already seen this and again back scattering cross section. Yes, we have already covered this. In fact, we went a little bit further to understand about sigma naught and gamma images as well, isn't it? Now left is gain of an antenna. So, to understand what is gain of an antenna, let me play a small video. You may have seen it earlier but just to refresh your memory. Remember, one can compare a radar antenna to your ears and remember the example when I mentioned that in a dark room, you hear a sound and your automatic response is going to be, you are going to turn your head and point your ears to the direction from where the sound was heard. That is an automatic response. This is known as directional sensitivity. You feel that by turning your head and hence pointing your ear in a particular direction, you shall be able to listen or hear in a much better manner because it is pitch dark and you are unable to use your vision. Similarly, let me play the video again. Always the total amount of return echo that enters a SAR system that enters a synthetic aperture radar system is dependent on the sensitivity of antenna to direction. I am using the term sensitivity, sensitivity of antenna to direction and we already know that antenna is considered as an aperture. Antenna is considered as an aperture. So, the gain of an antenna refers to the combination of sensitivity to direction and antenna efficiency, gain of an antenna. And we estimate gain of an antenna by using this expression that is antenna efficiency multiplied by physical area of the antenna multiplied by 4 pi by lambda square. So, slowly when we start our modules on active microwave remote sensing, we will try to derive this expression. But for now, I want you to understand what is gain of an antenna because antenna is considered as an aperture. The gain refers to the combination of sensitivity to direction. Remember the example I gave you with the ears and dark room and you pointing your head and ears, use that in your mind. So, sensitivity to direction and antenna efficiency. When we combine both of them, we get something known as gain of an antenna and the expression is antenna efficiency multiplied by physical area of antenna multiplied by 4 pi by lambda square. Now, as we have completed our understanding of key terms required for deriving a radar equation, let us proceed further. So, just to reiterate what is a radar equation? It is the fundamental relationship that explains the measured power of return echo, measured power of return echo. And it tells us what proportion of transmitted energy is received by the receiver, what part of transmitted energy is getting returned from the target. So, to derive a radar equation, we need R that is nothing but range of target from the antenna. We need G that is directional sensitivity of the antenna. We need radar cross section which we denote by the Greek letter sigma. So, let Pt be the power transmitted by the instrument, power transmitted by the instrument. So, what are we going to do? We are going to derive, understand a radar equation. For that, we have already defined R as range, G as antenna gain, sigma as radar cross section. So, now I am going to use Pt to denote the power transmitted by the instrument. And I am going to use P suffix s as the power scattered by the target, power scattered by the target. I am going to use P suffix r to denote the power that is received at the instrument, power received at the instrument. Easy to remember Pt is the power transmitted and Ps is the power that is getting scattered by a target and Pr is nothing but the power that is getting received. Remember Ps is a part of Pt. Okay? Remember also that while the instrument is transmitting Pt that is power transmitted, it is getting intercepted by a target. You know, it is getting intercepted by a target and some amount of energy is going to be redirected towards the antenna, what we call as backscattered. Some amount is going to be redirected, backscattered towards the antenna and we need to write an expression for the power backscattered by target. Okay? So, we want an expression to write what is power backscattered by target. Ps is nothing but it is a small part of the transmitted power, isn't it? The power that is getting scattered, backscattered by a target is a small proportion of Pt which is given by the expression Pt multiplied by g that is gain of an antenna multiplied by, so what is this? This is the proportional area intercepted by target by surface area of a sphere of radius r, r in our case is range. So, you are assuming a sphere, r is nothing but the range. So, sigma by 4 pi r square is nothing but proportional area that is getting intercepted by a target by surface area of a sphere of radius r. So, the power backscattered by a target, the power that a scatterer returns it is given by Ps that is Pt g sigma by 4 pi r square. Now, remember that if we increase the distance that is r, the power returned to antenna is going to reduce, isn't it? So, imagine at what altitude the sensor is available when the platform is a satellite, it is kilometer square more than 500, 600 kilometers. So, when we are increasing the distance that is range distance r, the power that will be returned to the antenna is going to reduce which means the power returned to antenna shall reduce by a factor of 4 pi r square as per this expression, yes? So, here it is worthwhile to include the effective area. We should include the effective area that is area of antenna to the surface area of a sphere with radius r. So, what we will do is we will write the expression for power received. P suffix r, power received is nothing but again it is a proportion of power transmitted multiplied by gain of the antenna multiplied by sigma by 4 pi r square. We now know what it is. I am going to use one more expression that is effective area of antenna by 4 pi r square. So, we have tried to define Ps, power scattered, we have tried to define Pr that is power that is received. We already know that gain of an antenna is nothing but antenna efficiency, antenna efficiency into physical area of antenna 4 pi by lambda square. We already know that. And this expression that is antenna efficiency into physical area of the antenna is nothing but effective area of the antenna. So, let me rewrite g as effective area of antenna into 4 pi by lambda square. We already saw this expression when I was playing the video where years are to be compared with a radar antenna. So, from this we can write effective area of the antenna is nothing but g lambda square by 4 pi, hold this thought in your mind so that we can use this for deriving the radar equation. So, let us rewrite what we have. By now we know that Pr is nothing but Pt multiplied by gain of antenna multiplied by sigma by 4 pi r square into Ae effective area of the antenna by 4 pi r square. I am going to call it as 1, equation 1. And we have already seen that effective area of the antenna is nothing but g lambda square by 4 pi. I am going to call it as equation 2. So, I want to substitute 2 in 1. I want to substitute the value of effective area of antenna in 1 so that now equation 1 is going to become Pr equals Pt into g into sigma by 4 pi r square into already we have 4 pi r square. On top of it we are going to substitute the value of Ae that is effective area of the antenna and this is the expression what we get. Let us make it more refined. Pr is going to be Pt into g square into lambda square sigma by 4 pi cube r raised to 4. Power received is written as a function of power transmitted, gain of the antenna, lambda that is wavelength, sigma radar cross section or that is radar range. Remember as r increases the received power decreases and we can write the radar equation rewritten as signal to noise ratio SNR. Now I hope that you remember the concept of radar cross section RCS. From this expression we can also estimate the target radar cross section that is sigma. From the same expression we can get the value of sigma which will be Pr 4 pi cube r to the power 4 by Pt g square lambda square. We will be covering some numericals on this but you know the same expression I have rearranged to get the value of sigma radar cross section, the target radar cross section. So this is what is known as a radar equation. It gives you the relationship between transmitted power and received power by using gain of an antenna wavelength radar cross section that is to be more specific the target radar cross section and the range. So we have already seen what is target radar cross section by rearranging the terms. We have estimated what is sigma which is nothing but Pr 4 pi cube r to the power 4 by Pt transmitted power g square lambda square. So here remember gain of the antenna wavelength range received power target radar cross section and we have already seen that we can get something known as a normalized radar cross section that is sigma naught which is nothing but sigma by A what you see on the screen. Normalized radar cross section and please note that A is the area over which measurement is being made. So I will let you understand the radar equation and radar cross section let it sink in. This is the radar equation. So what we understood as part of this lecture was we tried to derive the radar equation by first understanding each of the key terminologies that are required such as gain of an antenna, radar cross section RCS and we also understood what is scattering cross section. What is normalized radar cross section? What is sigma naught? What is beta naught? What is gamma? And we began by deriving the radar equation and finally we have also understood that from the radar equation I can get the target area target radar cross section and also I can get sigma naught which is normalized radar cross section. So let me hope that you found this part of the lecture useful and I will see you in the next class. Thank you.