 Hello and welcome to the course. So, right now we are in module 1, lecture 3 and in the last lecture what did we learn? We learn about the different applications of microwave remote sensing and hydrology and in particular we saw about active sensors. So, in the last lecture we learnt about different applications of microwave remote sensing to re-iterate we saw about active sensors. Remember the example of radars that we discussed about and then we moved on to passive sensors and then I mentioned about radiometers and we also understood that there is a distinction difference between imaging sensors and non-imaging sensors. So, as an example of non-imaging sensors we briefly mentioned about radar altimetry and as an example of imaging sensors we mentioned about synthetic aperture radars and we also learnt about the difference between sensors that operate in the visible region as well as those that operate in the microwave region and by now we know that microwaves can penetrate the clouds and they are capable of giving us a vertical profile of the atmosphere. There was a mention in the last lecture about different satellites that use microwave frequencies and towards the end if you remember we discussed about the key applications of data from microwave remote sensing to study about precipitation where satellites and Doppler weather radars are highly useful to study about land subsidence where interferometric synthetic aperture radar finds use to examine soil moisture, to measure water levels from space and for classification of crops, in accurate mapping of flood inundation, in digitally representing the terrain through digital elevation models and finally how each of these can be linked together as inputs in hydrologic and hydrodynamic models that are capable of simulating floods. So that is where we stopped in the last lecture. Now just to clarify a small point you may have wondered by now as to why is microwave able to penetrate through clouds why? So let me try to explain it using something known as atmospheric window. Atmospheric windows strongly influence where we look spectrally with any given remote sensing system which means there are certain wavelength ranges in which the atmosphere is particularly transmissive of energy and there are certain wavelength ranges wherein the atmosphere blocks the energy. So atmospheric window what is it? It is the wavelength range in which the atmosphere is particularly transmissive of energy and microwave region, the wavelengths of microwaves they are able to transmit through the atmosphere because atmospheric window coincides with the wavelength of microwaves. This is also the reason why visible region as well as infrared region they get blocked by the atmosphere because atmospheric window is comparatively non-existent. In microwave region those wavelength ranges that is microwaves they are able to transmit through the atmosphere because it coincides with something known as the atmospheric window. Now by now I am assuming that we all know about electromagnetic waves, microwaves, properties of a wave like its amplitude, wave number, velocity of light, wavelength, direction of travel. So with this understanding let us try to now define a wave mathematically. Again if mathematics is not your forte do not panic because we will try to simplify it. Now many wave properties they can be studied mathematically without even actually defining the exact shape of a wave. And the simplest wave form can be represented using either a sine or a cosine curve. So let us try to mathematically represent a wave and following the convention I am going to use sine curve to mathematically define a wave. Now assume a wave is travelling along the z axis. The profile of a simple wave function can be written as psi of z equals to a sine kz. Here a stands for amplitude, k is nothing but the wave number, z is the axis along which wave is travelling. Now you can assume k as some positive number and kz has units of radiance. And we already discussed what is amplitude that is the maximum disturbance of a wave. Remember a is not from the maximum to minimum, it is the distance from the axis to the maximum height of the wave. And we know that the sine function can have values between plus 1 and minus 1. So the maximum value of psi z is going to be a, isn't it? But then there is a small problem now because you know I have defined the equation as a function of distance along z axis. I have defined the equation as a function of distance. But then I need to describe wave as a function of time as well. We know that waves travel with the speed of light, which means I need to replace this equation with a time varying equivalent because psi of z is a function of z as well as time t. Now how do we do that? Let me think and then give you an example. Say assume you are sitting on the leading edge of a wave. Assume there is a sine wave and just a hypothetical sentence for your imagination. Assume you are sitting on the leading edge of a sine wave. Then as the wave travels that is as it moves in the positive z direction, each time the history of a wave is left behind you, isn't it? Because you are traveling on the leading edge of a wave and as you move forward the history of wave is left behind you. And we know that the best way to describe a wave at any location along the direction of travel which is z in this case at any time t is to write the wave function as, so let me try to rewrite this as psi of z. Now I am going to include t as well because I want to write psi as a function of z and t. So I am going to write it as a sine k. There is a small change. Instead of writing z, I am going to write z minus v of t because v velocity of light t time. Why am I subtracting? Because every time I am moving forward assuming I am sitting on a leading edge of a wave, there is a history of wave that is behind me that is why I am subtracting. So what did we do? We tried to represent a wave profile of a simple wave function. At first we tried to represent it using a sine curve and then we understood that of course wave is a function of distance that is z as well as time t and I need to write an equivalent equation that considers the effect of time as well. So let us move further. We know that lambda is nothing but wavelength of a wave that is the distance over which the function is repeating itself which means any increase or decrease of z by lambda I can denote it as something like this z plus or minus lambda t any increase or decrease in lambda that is wavelength of a wave that is distance over which the function repeats itself. I can denote it like this and for a harmonic wave because now we are considering a sine curve. So for a harmonic wave we can write this as equivalent to sine k z minus v t which is equal to sine k z plus or minus lambda minus v t. What did we do? We are representing it in the form of a harmonic wave and if we do a little bit of rearranging we can get sine k z plus or minus k lambda minus k v t we have taken the k inside for a harmonic wave whatever we have written this is equivalent to changing the argument of the function by plus or minus 2 pi which means I can write it as sine sine k z minus v t plus or minus 2 pi which gives us k z minus k v t plus or minus k lambda equal to k z minus k v t plus or minus 2 pi. So where are we now? We tried to describe the profile of a simple wave using a sine curve and then we added time as well because psi is a function of z as well as t and then we are trying to understand that for a harmonic wave whatever we have tried is an equivalent to changing the argument of the function by plus or minus 2 pi. So we have tried to rewrite the expression in a similar manner. Now at this point it is worthwhile to re-itrate the meaning of phase, phase of a wave. So shown here the diagram shown here tells us that the amplitude of a wave is nothing but its projection onto the axis and phase is denoted by the Greek letter phi. It is the angle that the pointer makes with the horizontal axis. So shown here is a wave which is having a phase difference. Why am I trying to bring it here? Because now I want to complete the derivation. So we already saw that wave number k and wavelength lambda are both positive numbers which means wave number that is the number of waves. I can write it using a simple expression that is 2 pi by lambda. Now again we know that if t capital T is the amount of time it takes for a complete wave to pass a stationary observer we can write it as k Vt equals 2 pi which means 2 pi by lambda into Vt equals 2 pi where t is the number of units per wave cycle and we already know that omega is 2 pi by t which means we can put it all together and try to rewrite the mathematical expression of a wave as psi z t equals a sin kz minus omega t. Omega is nothing but 2 pi by t angular frequency and t is the number of units of time per wave cycle. Which means at t equal to 0 and z equals to 0 the pointer of a wave may start at any angle if there is an initial phase isn't it? Which means we need to add an extra parameter as well which is known as initial phase that I am going to call as phi naught, phi suffix naught. So, let us try to rewrite this expression psi z t equals a sin kz minus omega t plus so what is it? It is the initial phase we have added this extra parameter because we need to consider the initial phase as well. So, I am going to consider this as the complete description of an electromagnetic wave that is travelling at a velocity which is expressed using amplitude a wave number k angular frequency omega time t initial phase phi naught and z is the distance, z is the axis along which wave travels. Complete description of an electromagnetic wave. Remember we have used all the terminologies that were introduced to us as part of lecture 1. So, now let us try to revisit the expressions once more. We started with profile of a simple wave function that psi of z is equal to a sin kz, k is wave number, a is amplitude and then we understood that the profile of a simple wave function has to be a function of z as well as time t which means the complete description of a wave function at any location along its direction of travel z and at any time t can be rewritten as psi of z t a sin kz minus v t v velocity of light and then we tried to describe a wave function again at any location along its direction of travel z and at any time t considering the initial phase as well phi naught which means we arrived at a complete mathematical description of a wave function as psi of z t equals a sin kz minus omega t plus phi naught where phi naught is the initial phase. We have introduced a new parameter as well. So, now all the terms let it be a or k or omega lambda all of these were discussed as part of previous lecture. So, I am hoping that you found this derivation easy, comfortable. Now, so till now what we did is we have tried to understand mathematically how to describe a wave. Remember, we have used a simple sine curve because it is easy for us to mathematically represent. Now, let us try to understand a little bit about something known as a radiation loss. A few fundamental radiation laws that are very much relevant in microwave remote sensing. Now, in remote sensing we typically use the term known as black body that is black body. Now, black body is a hypothetical ideal radiator that totally absorbs and re-emits all the energy that is incident upon it. So, what is it? A black body is a hypothetical hypothetical ideal radiator that totally absorbs all the energy incident upon it and totally re-emits everything hypothetical. So, a few terminologies need to be introduced here for the sake of clarity or for the sake of completeness. We will start with energy. It is the capacity to do work expressed as joules. Now, let us try to look at flux, which is the rate of transfer of energy from one place to the other. Rate of transfer of energy from one place to the other is termed as flux of energy measured in watts. Now, this can be the energy from the sun to the earth. Now, we have seen energy, we have seen flux. Now, just let us add a radiant and then try to understand what is radiant flux. Let us try to understand what is radiant flux, which means the rate of transfer of radiant electromagnetic energy, radiant energy. The rate of transfer of radiant energy with units joules per second is known as radiant flux. So, now we understood flux, we understood radiant flux. Let us try to understand what is radiant flux density now, radiant flux density. So, I am adding the term density as well to see what radiant flux density means. As the name suggests, it is the magnitude of radiant flux that is incident upon or conversely that is emitted by a surface of unit area measured in watts per meter square. So, let me explain again. Imagine this is the earth surface, we have incident energy and we have energy that is getting redirected. So, the magnitude of radiant flux that is incident upon or conversely which is emitted by a surface of unit area is termed as radiant flux density and it is measured in watts per meter square. Now, let me give you one more term that is radiant existence, radiant existence. If the energy flow say it is away from the surface, say the energy flow is away from the surface then we use the term radiant existence. We refer it also as irradiance that is if the radiant energy is incident upon a surface then the term irradiance is used and what is radiant existence? If the energy flow is away from a surface imagine these dashed lines show the surface. So, if the energy flow is away from the surface then we use the term as radiant existence or radiant emittance. It is measured in watts per meter square. Now, radiance is sometimes used to mean the radiant flux density transmitted from unit area of earth surface as viewed through a unit solid angle. Let me repeat, radiance is sometimes used to mean the radiant flux density transmitted from unit area of earth surface as viewed through a unit solid angle and we know that solid angle is measured in steradian. Remember as part of one of the earlier lectures we discussed about degrees, we discussed about radian and now I am introducing steradian. Now, steradian is a three dimensional equivalent of radian. A graphical representation is shown here three dimensional equivalent of radian. All right. So, sphere has four pi steradian. We must be aware of that. Similarly, circle will have a different steradian. Now, whatever I have explained are written here in the slide. So, at any point of time feel free to pause and let it sink in, right? Moving on. So, now as we have covered the basic terminologies, let us try to come back and understand about the radiation laws. I will start with something known as a Stefan Boltzmann law, Stefan Boltzmann law. The amount of energy an object or a body radiates among other things is a function of its surface temperature, okay? And this is expressed as a Stefan Boltzmann law which means the total radiant existence from the surface of a material in watts per meter square according to Stefan Boltzmann law is directly proportional to fourth power of absolute temperature of the emitting material. Let me try to repeat. The total amount of energy an object or a body radiates among other things it is a function of the surface temperature of the object or the body and this is expressed as Stefan Boltzmann law M equals sigma t to the power 4. Here sigma is nothing but Stefan Boltzmann constant. M is nothing but total radiant existence from the surface of a material measured in watts per meter square. So, by now I hope you understand what is radiant existence and capital T is nothing but the absolute temperature in Kelvin of the emitting material Stefan Boltzmann law. So, mathematically Stefan Boltzmann law shows us that higher the temperature of the radiator greater is the total amount of radiation it emits, higher the temperature greater the amount of radiation it emits. Now, imagine you are taking out a very hot metal from a furnace so hot that it is glowing white. By white I mean that the metal is emitting electromagnetic radiation in the middle of the visible region of the electromagnetic spectrum. So, what is color? Color is nothing but reflected light and in that context I want to bring your attention to a very hot metal that you are taking out from the furnace and it is so hot that it is glowing white and by white I mean the metal is emitting electromagnetic radiation in the middle of the visible region of the electromagnetic spectrum. Now, as the metal cools the color shall appear to change slowly from white it is going to become yellow and then maybe orange, red and then eventually this glow shall fade away. Now, let me bring your attention to the Wien's Displacement law here. So, Wien's Displacement law gives us the dominant wavelength at which maximum spectral radiant existence occurs. Dominant wavelength at which maximum spectral radiant existence occurs. The expression shown in front of you here lambda m is nothing but wavelength of maximum spectral radiant existence, capital T is temperature in Kelvin and A is a constant. Wien's Displacement law. Now, there are a few more laws which we shall learn slowly. But for now let me try to give you a trick question that is if the temperature of sun is nearly 6000 Kelvin and T of earth is nearly 300 Kelvin in which wavelength will both sun and earth emit radiation. The answer to that is you get it through the Wien's Displacement law because it gives us the dominant wavelength at which maximum spectral radiant existence occurs. So, think about this answer to this question because I have given you the temperature of sun nearly 6000 Kelvin, temperature of earth nearly 300 Kelvin and then I am asking you in which wavelength will both sun and earth emit radiation. So, while you hold your thought there let me share an interesting information here because you know all of you may be aware of the microwave oven right and as the name suggests obviously it uses microwaves. So, the S band magnetrons are typically used for microwave ovens and typically they are in the range of 2 to 4 gigahertz and the magnetron is a device that uses a magnetic field to force the electrons to rotate and hence accelerate and this in turn results in the generation of electromagnetic radiation in the microwave region of the electromagnetic spectrum. All right. So, just to summarize in this lecture we first learnt about why microwaves are able to penetrate through clouds and that is when I introduced atmospheric windows to you and then we went on to derive mathematically the equation of a wave and next we learnt about two radiation laws, two easy radiation laws namely the Stefan Boltzmann law and the Wien's Displacement law. So, let me hope that you found this section useful and I shall meet you shortly in the next lecture. Thank you.