 Hello and welcome to today's lecture. So, today let us begin with a new module and instead of directly showing you the contents of the module, I shall start with a simple question, you know. Speed guns, they use Doppler radars to measure the speed of moving vehicles. We discussed about speed guns and you know Doppler radars in our earlier module and by now we know that radar speed guns, it consists of a transmitter and a receiver. If so, what is the concept of a radar speed gun? As in, can phase of a wave be used as a distance measure? Let me give you another example, okay? So, this is of a sonar, you know sonar standing for sound navigation and ranging. It uses the propagation of sound waves to measure distance, that is for ranging and it detects the distance as well as direction of underwater objects through acoustic mains, is not it? But now, I am slightly confused, you know, because my understanding says that an electromagnetic wave consists of an electric and magnetic field oscillating in mutually perpendicular direction. So, for radar and sonar, can phase be used in some manner to estimate distance? Remember, I am not talking about amplitude or wavelength, I am asking you the question, can radar and sonar be used in some manner? As in, can the phase of waves be used in some manner to estimate distance? So, with this background, let me start with today's module. The tagline of module 6 is, can phase be used as a relative distance measure. So, through this module, we will be learning what is radar interferometry, the fundamentals, the interference pattern and few terminologies I will be introducing you to, like the beam width, what is angular resolution and then what is stereosar radar geometry and also whether we can estimate topography, topography from synthetic aperture radar images. Remember, I am asking you whether a two-dimensional synthetic aperture radar image can be used to represent three-dimensional surface of a terrain, topography using radar images and we will also be learning how to interpret an interferogram and what are fringes. You know, these terminologies, they may sound new to you, but rest assured, they will be explained as part of this module as well as you will be having a hands-on session where using SNAP, you will learn how to create an interferogram and how to interpret about interferogram as well as fringes, alright. So, towards the end, we will also discuss about what are digital elevation models, DEMs, how to access them, how to download them and what are its prominent application in the field of water resources engineering and hydrology, okay. So, with this background, let us start with our first lecture of module 6. We discussed about phase of a wave, okay. In our initial lectures, we discussed about different properties that we can derive from a wave, its amplitude, frequency, wavelength and phase of a wave. So, let me reiterate the small videos that we discussed during module 1 showing wave with phase 0 towards your left side and a wave with phase of 90 degree towards your right side. Now, I started this lecture with a question, isn't it, phase as a relative distance measure. So, let us try to understand whether phase of a wave can be used to estimate distance. So, shown in the screen in front of you, you see three markings X, Y and Z, assume that they signify three points, okay. Now, it is well known that phase of a wave changes with distance, okay. Now, assume that we have a wave 1. So, I am going to call it as wave 1 of a certain amplitude and frequency and the wavelength of the wave is given by lambda wavelength of a wave and a wave is say transmitting from a point X to a point Y, okay. So, the distance between X and Y shall be the number of whole wavelengths plus some fraction of wavelength, isn't it? And if we measure the phase of the wave at a location say Y, it may not convey anything much about the distance, but then it shall convey information about the fraction of a wavelength. Now, assume you have two identical waves, okay, two identical waves and say you are measuring the phase at Y and also at another point Z which is at a small distance from Y. So, now the phase which is being measured at point Y will be different and the phase which is measured at point Z shall also be different. Remember, same wave is being measured, but now instead of measuring phase of one wave, we are measuring the phase difference. Now, let me reiterate the question. Does phase difference convey a crucial piece of information about the distance? By distance I mean distance between the two points Y and Z. Now, let us try to understand that, okay. Assume we slowly increase this distance by a small amount, okay. Now, what happens? Now, does the difference phase difference convey a crucial piece of information to you about the distance D? Now, we will be discussing about phase ambiguity slowly when we start interferometry, but for now the concept of path length and phase is very important and we shall be needing this background in understanding this module when we speak about radar interferometry. So, shown here is a relationship that tells you what is the phase difference, okay. So, remember whenever we have two waves which are propagating say in opposite directions, we also discussed in module 1 that they will be subjected to constructive and destructive interference, okay. Waves can interfere constructively leading to an increased amplitude or waves can interfere destructively leading to an amplitude of 0, okay. Now, just to refresh your memory, I know that we discussed these points in module 1, but because we will need all these fundamentals for this section, I am going to reiterate a few of the basic concepts. That waves can interfere constructively and destructively. Now, let us try to understand things a little bit more in detail. Now, you can visualize what will happen when you throw two stones in water, isn't it? Ripples will be created and when you are throwing two or more pebbles at the same time in still water, there will be waves which will interfere constructively or destructively. Similarly, you can think of the analogy using sound waves, sound waves that also tend to interfere constructively and destructively. Now, what I will do is I will play a small video and then we will try to understand the concepts as the video is being played, okay. Now, before I start assume there are two sources, two sources producing waves in the same direction, okay. And say both the sources are in the same horizontal plane and the waves being emitted are in phase, in the same phase, okay. So, the waves which are being emitted by the two sources, I am going to call it as source 1 and source 2, they are in phase and these waves are having the same frequency and amplitude, they are coherent waves. So, once again, if we consider a point in front of the two sources, you know, in the space in front of the two sources, if we consider one single point, that point is going to be visited by a wave arriving from source 1 as well as another wave arriving from source 2. And there shall be a phase difference between the two waves, isn't it? So, in the video, you can see that I am trying to show you what happens when the properties of a wave are changed, you know, like the amplitude is being changed, what happens to the waves, how do they interfere constructively or destructively. And similarly, you can use the analogy of sound to understand how waves interfere constructively and destructively. So, coming back, now why are we discussing all this? You are coming back to the point. So, if we consider any point in front of these two sources, the phase difference with which both the waves from both sources reach a point can be 0 or it can be some multiple of 2 pi, if they tend to constructively interfere, which means that the resulting amplitude shall be 2a, assuming a is the amplitude of one wave. Now, let me give you the second scenario. At points, now when I say point, it always means point in space which is in front of the two sources emitting radiation. So, at the points wherein the path length is going to differ by half a wavelength, both the waves are going to be out of phase by 180 degrees and they shall be interfering destructively now causing 0 amplitude values. But then, owing to different path length between each point and both the sources, there shall be a phase difference. Now, assume a point that are few wavelengths away from the two sources. Assume a point which is away from both the sources. If you see closely this image, you will be able to notice a regular pattern of peaks and trough, isn't it? Peaks representing high values and trough representing low values. And this pattern, I have shown you a still image, this pattern when you look at it, it will appear to radiate outwards from the two sources in different directions, isn't it? Interference pattern, this is known as interference pattern. Now, let me ask you another question, you know, look at the pattern carefully and then comment on what happens closer to the two sources and what happens farther away from the two sources. I will give it some time, you know. My question is, when you look at this interference pattern in the screen in front of you, what happens closer to the two sources and what happens away from the two sources? You know, closer to the two sources, the pattern may appear slightly complicated, yes? And farther away from the two sources, the pattern appears as a set of radiating beams, okay? So, now is the right time to introduce two terminologies to you known as near field and far field, okay? Near field and far field. Near field is near the two sources that are generating the waves, near field, easy to remember. And far field denotes location that is farther away from the two sources generating the waves, far field. Now, in remote sensing, we are more interested to study about what happens in the far field, isn't it? So, regardless of the distance, any point along the central axis of symmetry shall result in constructive interferences as long as the sources are in phase, okay? And we also refer to this direction as having a look angle of 0 degree. It is also called as, you know, boresight angle or pointing angle. You may have come across this when we were discussing about radar antennas, you know, what are the different types of antennas and what is meant by directional sensitivity. I am just rehydrating a few points. So, remember difference in path length will mean difference in phase, all right? So, I am showing you two interference patterns here, two interference pattern. You can see that both are different, isn't it? If we have the wavelength, even though the path length is same, there shall be twice as many wavelengths which can fit into the same distance. When we have the wavelength, we are doubling the number of beams, okay? Compare between the figures on the left side and the right side. You can see more patterns on the right side, less pattern on the left side, okay? Now, in this context, let me show you a video. Note the frequency and amplitude towards the right side. So, now, I am going to reduce the frequency. Look at how the interference pattern is changing, okay? Look at how the interference pattern is changing. Let me increase the frequency now, okay? Slowly, I am going to increase the frequency and then see for any changes that happens in the interference pattern. I am going to pull the frequency to the maximum possible value, okay? Just to see if there is any change in the interference pattern. Just for you to understand about interference pattern, about boresight angle, about what happens in the near field direction, what happens in the far field direction, alright? So, hold this thought in your mind and let us try to understand whether we can quantify the interference pattern, you know? Visually, I can see the difference in the pattern when I am changing the frequency. Now, how to quantify this pattern? So, by now, we have an understanding that we are more interested to know about the far field solution. And we are interested in considering only about the interference pattern which is far away from the sources, okay? It is not in the distance that is closer to the two sources, but we are more interested to know about what happens to the interference pattern which is far away from the sources. Now, an expression for path length can be written like delta L equal to d sin theta, okay? An expression for path length, d is the distance between the two sources, theta is the angle, delta L is change in path length. Now, the maximum peak occurs only if theta, lambda and d satisfy the following relationship given by theta equals sin inverse n lambda by d. Now, the two sources transmitting waves that were shown as loudspeakers because we were taking the analogy of sound waves. Now, these two sources transmitting waves, they can be considered as two antennas working together in unison. But now, you know, I am a little confused because I can see that from the interference pattern, there are multiple directions in which there is energy radiating. But then, I am more keen to learn about directional sensitivity of an antenna. Our aim is to radiate energy in one narrow direction, okay? So, does it mean that with the two sources or with two antennas, we can steer the interference pattern? Can we change the interference pattern? Isn't it something that is naturally occurring? But then, with the two sources, by some mechanism, can we steer the interference pattern so that the energy gets radiated in a narrow direction? That is what we want to see next. By introducing a phase difference to the signals transmitted from source 1 and source 2, of course, we can make waves to have a phase difference, isn't it? That is what we have been learning so far. But the next question is, can we by some means focus the sensitivity to just one direction? Okay? All right? So, whatever relationships I have written is given in the screen. So, at any point of time, feel free to pause and take a look. The expression for path length is given, D being the separation between two sources, theta being the angle, what angle look angle. And I have also given you the expression wherein maximum in peak occurs if theta, lambda and D satisfy the following relationship, okay? Now, let us try to answer the question. The question was, can we by some means focus the sensitivity to just one direction? So, what we will do is we will try to understand it visually first. The same example of the two sources, emitting waves, have a look at the frequency and amplitude. But this time, I am going to change the separation distance between the two sources, okay? I have stretched it to a maximum value. What happens to the interference pattern? I can see that energy is getting radiated in multiple directions. Now, for a change, let me decrease the distance between the two sources emitting waves. What is happening to the interference pattern? Do you see that energy is radiated more in one particular direction? Directional sensitivity of antennas, you know? The concept of how a radar antenna can be made to be sensitive in certain directions, you know? You get it from here. So, when I change the separation distance between the two sources, there is a change in the interference pattern, which means, yes, I can steer the interference pattern such that the energy gets radiated more in one particular direction instead of it getting scattered in multiple directions, okay? Now, with this background, as part of today's lecture, let me try to introduce two terminologies, okay? Just two terminologies for today. One is angular resolution and the second is beam width, okay? Angular resolution, you may have heard it in parts of module 3 when we were discussing about synthetic aperture radars, okay? Azimuth resolution, range resolution and then somewhere in between I mentioned angular resolution. So, angular resolution of a system shall be defined by how narrow the beam is, okay? How narrow the beam is and the common measure of beam width is given by full width at half maximum, full width at half maximum, okay? Again, look angle is the angle at which first minimum occurs. Now, whenever I mention about these terminologies, recollect the videos I have shown, okay? Look angle, far field, near field, what happens when the distance between the two sources are being changed? What happens when the frequencies are changed? How does the interference pattern look like? What happens when the amplitude is changed? So, remember the videos I showed you, you know, that makes it easy for us to understand and then derive the equations, okay? Now, the reason why we are discussing all these is because these concepts are extremely useful, important to define the pattern formed by an aperture or an antenna, okay? In the far field, a total of n number of waves are there, assume each are having say an amplitude of a and a phase difference of delta phi, assume that in far field, a total of n number of waves exist each of amplitude a and a phase difference of delta phi. The minimum occurs when the waves add in such a way that the result leads to zero amplitude. Minimum occurs when the waves add in such a way that it results in zero amplitude. By definition, you know, this should be when the cumulative angle is 2 phi, is not it? Such that n phi equal to 2 phi, is not it? Think about it. So, let me rephrase the definition a little bit now. The first minimum occurs when the first to last n are exactly in phase again. Then the angle from the center line to the first minimum is given by sin inverse lambda by d. Remember, d is the separation distance between the two sources, lambda is the wavelength, okay? Alright? So, when it comes to an antenna or an aperture as we call it, you may be aware by now that it is a common practice to consider angular resolution instead of beam width, angular resolution and the expression for that is given by angular resolution nearly equal to lambda by d. With this background, we are in a perfect position to understand about what is radar interferometry, which shall be covered in the next part of this lecture. So, in today's lecture, we were pondering on questions as to what happens when two waves which are generated by two sources that are in the same plane separated by a small distance d, what happens to the resulting wave at any point in space in front of the two sources, okay? And then we learnt about interference pattern and what happens to the pattern when I change the distance between the two sources, what happens to the pattern when I change the frequency of a wave, amplitude of a wave and so on. These are fundamental to our understanding of how a three-dimensional topographic information is obtained from two-dimensional synthetic aperture radar images. That was the whole point of, you know, re-iterating the basics once more. So, let me hope that you understood this section of the lecture and I will meet you in the next class. Thank you.