 Hello everyone, welcome to the next lecture in the course remote sensing principles and applications. In this lecture we are going to start a new topic known as passive microwave remote sensing or also known as passive microwave radiometry. So what passive microwave radiometry is, we know earth emits radiation because of its internal energy or its own temperature and that was the major physical reason behind us doing thermal infrared remote sensing. Say here we have this wavelength in x axis with the spectral radiant emittance or what is known as the radiant flux density in the y axis and the curve is plotted for a black body at 300 Kelvin. So this basically approximates earth surface ok. So this we have already seen before. If you look at this curve the radiation will be starting like here it has plotted something around like 1 micrometer. So typically the radiation will start emanating around like say 3 to 5 micrometer range radiation will begin to radiate it will increase and then there will be like a long tail. In thermal infrared remote sensing we use the wavelength range of 8 to 14 micrometers this particular range. So we used or we send satellite sensors to observe in this particular range, observe this radiance and calculate the temperature of earth surface features. But this curve will not stop abruptly earth surface will be keep on emitting radiation and if you have like a long tail which is extending all the way up to the microwave portion of the electromagnetic spectrum. So here even if you observe in millimeter range or centimeter range there is small amount of radiation that is constantly being emitted by earth surface. So this radiation is what we are going to make use of in passive microwave radiometry that is earth starting from short wave infrared or MWR mid wave infrared portion that 3 to 5 microwave range earth will be keep on radiating energy. This radiation will slowly increase and it will reach a peak in the long wave infrared portion around this 9.5 9 to 9 10 micrometer range. But if we since we call it as thermal infrared remote sensing but this radiation will not abruptly stop from earth surface and the radiation will be keep on emitting and earth surface will be emitting energy even in microwave wavelengths that is with wavelengths in the order of centimeters may be from 1 centimeter to 100 centimeters. If we observe this radiation from earth surface in these wavelengths in centimeter wavelength and use it for different applications we call that as passive microwave remote sensing or passive microwave radiometry. So naturally this is an extension of thermal infrared remote sensing. Extension means we are not producing any new form of energy whatever the earth itself is emitting we are observing it but in a different wavelength ok. So we are not observing any more in 8 to 14 micrometer wavelength but we are observing earth emitted energy in microwave wavelength in order of the wavelengths in order of centimeters. In those wavelengths also solar radiation is not going to play any role ok. Because we have already seen once we cross this 5 micrometer wavelength solar radiation reaching the earth surface almost goes to 0. So we can safely neglect any incoming solar radiation. So whether during daytime or night time whatever radiation we observe in microwave wavelengths essentially that energy will be due to emission from earth surface. Just study this in parallel with thermal remote sensing whatever even in the 8 to 14 micrometer wavelength whatever we observe is basically due to emission from earth surface. Same concept is applied here we are applying we are observing at a different wavelength but still whatever we are observing is due to earth surface own emission. So our wavelength only changed but the emission the source is earth surface and the various features present on the earth surface. So what exactly is the microwave portion of electromagnetic spectrum this also we have discussed in the earlier classes in the introductory classes. So in general the microwave portion of electromagnetic spectrum that we use for remote sensing of earth surface spans between the wavelength range of around 1 centimeter to 100 centimeters. So 1 centimeter to 1 meter. So we can classify it from say KA band, K band, KU band, X, C, S, L and P in the range of increasing wavelength. It starts maybe something around 1 centimeter wavelength to 1 meter wavelength. We have lot of divisions in between for our own understanding. This nomenclature this P, L, S, C, X these nomenclature have been developed in old and days as a secret codes during military applications like one of the earliest applications of microwave is in military like for different surveillance needs. So this sort of frequency the random names were given for frequencies in order to keep it secretly and that name prevailed even now. So these are all the common names we give it to the different portion of microwave portions. Like in visible band we call 0.4 to 0.5 as blue, 0.5 to 0.6 as green and all right. Same thing here microwave portion is a pretty long portion of electromagnetic spectrum. We have divided into certain classes and we use this for remote sensing. Whatever the portion we have seen from 1 centimeter to 1 meter from KA band to P band it is not essentially the full microwave portion of spectrum microwave is still really a very huge part of electromagnetic spectrum and it has its applications in various domains like in communications like our cell phones everything depends on microwave based communications. Microwave oven transmits microwaves for cooking of food. So microwave is there everywhere around us and we use it for several applications. Remote sensing is just one of the applications in which microwave wavelength of EMR is being used. Just imagine we have one some transmitter kind of thing emitting energy in one particular microwave frequency. There is a satellite a passive microwave sensor which observes the earth's surface in the same frequency. Just think what will happen whatever this transmitter is radiating is going to reach the satellite sensor and the satellite sensor is going to think something is coming from the earth's surface and as an end user when we take that image we will see what is there on the earth's surface. We will not know that there is some transmitter which is radiating energy maybe a cell phone tower what not. In order to avoid such confusions the microwave spectrum has been divided and for each application each portion of the spectrum has been allotted. So similarly for remote sensing a certain frequencies has been allotted okay only this frequencies has to be used for microwave remote sensing and only this frequency has to be used for communication purposes. If these frequencies clash say if a communication agency if some agency is using a particular frequency allotted for remote sensing if they have a radio if they have a microwave transmitter then that is going to interfere with what the satellite is going to observe. There is some artificial source that is emitting energy that is going to cast interference in satellite observations we call this as RFI radio frequency interference. These sort of things tend to happen but in order to minimize this or avoid this the microwave portion has been split up and several small portions has been created and each application has been allotted certain bands. Say for remote sensing purposes these are all some of the frequencies which we can use for earth observation or observation of atmosphere. You can see here there are two titles active and passive what exactly active and passive means passive is the passive microwave remote sensing that we have just got introduced to or what we are discussing right now that is whatever we are observing from space the energy is primarily due to earth zone emission. There is also an active mode of remote sensing what exactly active mode of remote sensing or active microwave remote sensing is we will have a sensor what is known as the radar this will transmit some electromagnetic radiation with a given frequency. This will interact with the earth surface features and this will be reflected back this will again reach the sensor and the sensor will observe it and use it for imaging purposes. So, this process where the sensor itself will transmit certain energy and receives it back we call it as active mode of remote sensing active remote sensing. So, in microwave it is possible to do active remote sensing we can send a radar to space that will transmit EMR in microwave wavelength get the reflected signals back. So, active remote sensing that is a topic we are going to see next after we finish passive microwave. So, that is possible in microwave. So, even in microwave spectrum for active and passive also the frequencies has been divided for example, let us take L band okay L band frequencies say typically these frequencies correspond to L band 1.2 to 1.3 and 1.4 range frequency gigahertz frequency. If you want to do active remote sensing you are supposed to use this 1.2 to 1.3 gigahertz. If you want to do passive remote sensing in L band you have to choose this 1.4 gigahertz frequency. There should not be any mix why an active remote sensing sensor and a passive remote sensing sensor if they work in same frequency then the active radar itself will became like an interference to the passive sensor. Let us imagine there is a satellite called SMAP soil moisture active passive which is like launched by NASA. So, it had both or it has both a radiometer passive microwave radiometer and also a radar. Both is there both is both will work in L band but the frequency in which the sensor will send in signals is different from the frequency in which the radiometer will observe. So, the radar will send a frequency a different frequency will observe a different frequency like say it will it will transmit in around 1.3 gigahertz receive the signal back in 1.3 gigahertz. Whereas this passive microwave radiometer will not transmit anything but will observe frequency one in 1.4 gigahertz. So, even within a same satellite even if you have active and passive mode of remote sensing mixer together the frequencies will be different because the active radar should not act as an interference to the passive. Imagine the active is also transmitting energy at 1.4 gigahertz what will happen 1.4 gigahertz will go get reflected the surface will come back and passive microwave radiometer will observe it thinking that it is coming only because of earth emission that itself is an interference. So, in order to avoid this interference there has been specific channels created to make sure or to minimize this radio frequency interference. So, for passive microwave remote sensing there are like specified channels or specified bands in which observations will be made. Before moving on to seeing the concepts of passive microwave radiometry we will look at the Planck's law one more time. Planck's law we have seen several times in the introductory classes as well as in thermal infrared remote sensing. Here also in passive microwave radiometry Planck's law plays a major role because essentially the radiation coming out of earth surfaces thermal in nature. The original form of Planck's law that is for the radiant flux density is given by 2 pi hc square lambda power 5 exponential of hc by lambda kt minus 1. So, this is the original form of Planck's law that we have seen earlier. So, this will be like the radiant flux density with units of watt per meter square per micrometer. So, for isotropic radiators of a lambertian surfaces we have seen that radiance is equal to radiant flux density divided by pi. So, by using this particular relationship we have derived this equation for radiance. So, this is the Planck's law for radiance, the only difference is this term pi will not be there. This also we have seen earlier. This particular equation will give us the radiance coming out of a black body. So, essentially these equations are defined for black bodies whose emissivity is equal to 1 always in all the wavelengths. So, this particular equation the radiance equation will give us the radiance of a black body at a given temperature t that will be emitted at a particular wavelength lambda. So, if you look here there are like 2 independent variables t can vary independently and lambda can vary independently. If we assume the black body is at one particular temperature that is L of lambda at a given temperature t then this also will become a constant and this will be like the variable and hence this equation L lambda will tell us how radiance varies with wavelength radiance variation with wavelength. What I mean is this will be like the curve this will be lambda this will be the radiance. So, some sort of curve like this we will draw right say for t is equal to 300 Kelvin like this we will draw. So, if we fix the temperature of an object the equation given here will tell us at which wavelength what will be the radiance. So, essentially the equation given here will tell us what will be the variation in radiance for an with respect to wavelength for a black body at a given temperature t. So, mathematically this can be written as a partial derivative that is this L lambda is nothing but the variation of radiance with respect to wavelength dou L by dou lambda at a temperature t. So, here we are holding temperature as a constant that is why this partial derivative is coming with respect to lambda. In microwave parlance like whenever we enter the microwave domain of remote sensing most of the people like the engineers who develops the systems and everything the microwave antenna the sensor system everything they prefer dealing in terms of frequencies. But as remote sensing people we prefer talking in terms of wavelength. So, essentially that there is always need to convert between expressing something in terms of wavelength and expressing something in terms of frequency. So, it will be easy for us if we can convert this Planck's equation expressed in terms of frequency that is the variation of radiance for an object at a given temperature t with different frequencies that is it rather than having lambda and x axis now we are going to put frequency in the x axis that is what we need is we have to convert this dou L by dou lambda into dou L by dou f where f is the frequency we need to convert this into this. It may seem to be like a very straightforward operation just by replacing lambda with nu you know we know the relationship right c is equal to nu lambda using it we can replace but it is not as straightforward it is not a mere substitution of lambda with frequency it there needs a small mathematical operation in between. So, we will see how to do it. So, once we do it we will be in a position to calculate the radiance with a given frequency. So, what exactly we have to do just examine the equation once more. So, the radiance equation is a function of lambda and using this relationship lambda is a function of nu right if we can like take like c is equal to nu lambda that implies lambda is equal to c by nu. So, lambda varies with frequency where c is a constant like this we can imagine. So, essentially we need dou L by dou f or nu whatever. So, I write it as like f here so dou L by dou f what we have in our hand is dou L by dou lambda. So, essentially if we use the partial on the chain rule of partial differentiation we can write it as dou L by dou f is equal to dou L by dou lambda into dou lambda by dou f. So, if we do this we will be in a position to calculate what will be the change in radiance with respect to frequency. So, that is the thing we are going to do now. So, dou L by dou lambda we already have in our hand that is the Planck's equation given in the previous slide. So, this equation is nothing but the dou L by dou lambda. So, now we have to calculate dou lambda by dou f that is this one the variation of lambda with respect to frequency. So, c is equal to nu lambda or lambda is equal to c by nu just differentiated once dou lambda is equal to minus c by nu square dou nu. So, I am interchangeably using nu and f to represent frequency. So, dou lambda by dou nu is equal to minus c by nu square. So, this is done. This is actually a negative equation where the negative sign represents the direction in which lambda will change with change in frequency that is as frequency increases lambda will decrease and vice versa. So, this negative is an indication of direction. So, derivative is nothing but mathematically it is a slope right. So, normally we attach the concept of derivative with slope. So, in which direction this variable will change. So, as we increase frequency lambda will decrease and vice versa. For us for our particular application the direction is not important, but what we need is the magnitude. So, magnitude means I am just going to take the modulus of this particular function which means c by nu square the minus sign will go off. So, here we are going to calculate the variation of wavelength with respect to frequency. So, when we differentiate lambda like lambda with respect to frequency we get minus c by nu square where minus sign indicates the direction in which lambda will change. So, we are taking modulus in order to avoid direction. We are not interested in seeing in which direction the slope is going to go. So, we are taking modulus of it and we are getting a c by nu square. So, substitute everything there that is l lambda that particular equation you substituted to hc square divided by lambda power 5 exponential of hc by lambda kt minus 1 into this term will come in c by nu square. Now you replace now we have to replace all the lambda with respect to nu because this relationship is newly we found out between lambda and nu. Now we start substituting it now if you start substituting it l rather writing l lambda this is lf. lf is equal to 2 hc square into exponential of hc by c nu kt minus 1 into c by nu square this will come something like this. So, you cancel everything out and finally we will get this particular equation 2 h like c cube this will become c square and then this nu square will cancel this will become nu cube. So, numerator it will become 2 h frequency cube divided by c square exponential of this c will cancel out h nu by kt minus 1. So, this is the way in which we have to derive the Planck's equation to be expressed in terms of frequency. So, it is not a straightforward substitution for lambda and convert it into frequency we have we need we have a chain kind of relationship lambda like the radiance equation the Planck's equation is related to lambda and lambda is related to frequency. So, we have to combine them use the partial chain rule the partial rule chain rule of partial derivative using that we will derive this particular equation. So, this is nothing but the simple Planck's rule expressed in terms of frequency. So, now rather than telling you what is the radiance coming out of an object at a temperature of say 300 Kelvin at 1.4 gigahertz we can directly find it we need not convert frequency to wavelength and then substitute it. So, directly we can use and get the variation of radiance at a given frequency for an object at a temperature t. So, this is like a very simple conversion of one equation in given with respect to one variable into another variable that is all. Now, what we have seen till now is a generic form of Planck's equation and the original form without doing any modification. But in microwave wavelengths if you look at the Planck's curve it will be like the end portion like especially in the longer wavelengths we will have like a linear line. So, here everything is expressed in log scale even if you put everything in like a normal scale we will see it the tail portion can be approximated to be linear it need not be treated perfectly as a exponential curve the tail portion where the microwave emission comes essentially it is it can be approximated as a linear line rather than treating it as a exponential curve. So, we can do one approximation and this approximation is known as Rayleigh gene approximation for black bodies Rayleigh gene approximation of Planck's law to be more specific what this approximation says the Rayleigh gene approximation says that if this condition holds good that is hf by kt is less than much less than 1 or hc by lambda kt is much less than 1 then in this particular equation we can drop this exponential function and simply write it as Lf is equal to 2 hf cube divided by c square hf by kt that is all. So, we are dropping this exponential term this minus 1 this is applicable only when this term is much less than 1. So, if we do this then the frequency the Planck's law variation with respect to frequency will become something like this similarly the Planck's law with respect to wavelength also will become something like this. So, these two simplifications of the Planck's law is what is known as Rayleigh gene approximation. So, what is the advantage of doing Rayleigh gene approximation from Rayleigh gene approximation what we can observe is the relationship between temperature of an object and the radiance coming out becomes linear that is like L lambda becomes 2 ktc by lambda power 4. So, the t is there in a linear relationship with respect to radiance and also all the computations becomes highly simplified highly simplified in the sense we need not take the exponential we do not divide one by other like the many number of like computation steps you need to calculate to perform the certain operations is much simplified the equation is very simple. So, in order to Rayleigh gene approximation first tells us the equation like the Planck's curve or the radiation that is emitted by a black body is kind of like becomes linear towards the end that is one thing and second thing is all the computations that we are going to do becomes much simpler. Till now we have seen the Planck's equation for a black bodies like even when we converted into frequency or even when we did the Rayleigh gene approximation everything was for a black body. But we know most of the earth surface features are non-black bodies they will have emissivity less than 1 and the emissivity will also vary with wavelength. So, for such non-black bodies how it will be? So, normally what we will do for non-black bodies the radiance will be given by emissivity times the Planck's law of the object at a given temperature T. So, this is the usual way we do right even when we did it in thermal infrared remote sensing we calculated this as something like this something like this we would have calculated we would have just multiplied the Planck's equation with respect to spectral emissivity. Same thing we will do here even in this particular wavelength we will multiply the Planck's equation with spectral emissivity. However, here we will use the Rayleigh gene approximation because in microwave wavelengths especially for features of earth surface when the object is at 300 Kelvin with average temperature of 300 Kelvin Rayleigh gene approximation will hold good roughly when the wavelength once crosses 50 micrometers. Even for this sort of micrometer wavelength Rayleigh gene approximation will hold good, but in microwave terms we are talking in terms of wavelength in the order of centimeters say 1 centimeter to 100 centimeters. So, normally at this very long wavelength Rayleigh gene approximation holds good and hence we can write radiation or sorry the radiance using the Rayleigh gene approximation. So, 2 Kc by lambda power 4 and just taking the t and spectral emissivity out together. Let us say a sensor is being sent to space it is going to operate in L band around like 1.4 Giga hertz ok. So, hence as soon as the sensor is designed and sent to space wavelength is also fixed it is going to observe only in this 1.4 Giga hertz frequency or roughly 24 centimeters wavelength. So, this thing is fixed now this entire term within the bracket becomes a constant for that particular sensor. So, the radiance observe or the radiance coming out of any non-black bodies is now a direct is now directly proportional to the product of temperature of the object multiplied by surface spectral emissivity. So, this signifies for a non-black bodies the radiance emitted is directly proportional to the product of temperature and spectral emissivity. So, here both temperature and emissivity has a equal say in defining what will be the radiance that is coming out of an object. So, you just look at the equation in TIR band. So, this is the equation for radiance in TIR band. This is kind of like highly non-linear emissivity is here and the temperature is here in the denominator within the exponential all these things. But if you look at after the Rayleigh gene approximation equation becomes much simpler and the radiance is a is directly proportional to the product of T and emissivity. So, here the weightage of T and emissivity is equal in defining the radiance both have equal say if temperature increases radiance increases to the same extent. If emissivity increases radiance increases to the same extent and vice versa. So, here the influence of emissivity and temperature both are equal in defining the radiance coming out whereas in TIR wavelength in the original form of Planck's law the change in temperature will have a more influence or a larger say in defining what will be the radiance coming out of an object rather than change in emissivity. Maybe like when we do some like numerical problems later we will try to understand how these things work. Here also one thing we have to remember the product of temperature and emissivity for any given object is commonly referred as brightness temperature in microwave parlance. When we defined brightness temperature in TIR remote sensing we defined it in a different way. We defined it as the temperature of black body will have in order to produce the same radiance as observed by the sensor. Here sometimes in most of the literature you can sense like brightness temperature is defined as the product of temperature and emissivity because people assume in microwave wavelength atmospheric fx is negligible we will see it later and hence whatever is observed by the sensor is almost effectively free of atmospheric effects and if you remove this sensor response function and all. So, whatever the radiance received by the sensor is just due to the mere effect of temperature and emissivity. So, this thing you please be clear in microwave parlance brightness temperature the word brightness temperature in several literature will refer to the product of temperature of an object and emissivity of the object. So, as a summary in this particular lecture we have discussed or we have got introduced to the concept of passive microwave radiometry. We have seen what all the spectral bands used to in passive microwave radiometry and also we have seen in detail about the conversion of Planck's law from with respect to wavelength to with respect to frequency. Also we have seen the Rayleigh gene approximation. The Rayleigh gene approximation will help us to simplify our calculations and also to understand the relationship between temperature and radiance on also emissivity and radiance. With this we end this particular lecture. Thank you very much.