 to the second lecture of the fifth module. So if you remember from the last lecture we were trying to understand more about what is passive microwave remote sensing and we are slowly beginning to understand what are the fundamentals of passive microwave remote sensing. So remember we discussed about the atmospheric windows and what is Planck's law, what is Rayleigh-Jeans approximation and then somewhere in between the lecture I mentioned three terminologies. So what are they, radiometer, radiative transfer model and attenuation. And that is when I said that over the course of upcoming lectures you will get more clarity on what these three terms mean that is radiometer, radiative transfer model and attenuation. So as part of today's lecture we shall try to understand more about these terminologies. Just a quick overview that forward model for passive sensors can be top of atmosphere brightness temperature that is what is recorded by a passive radiometer at the top of atmosphere that can be a sum of transmissivity of entire atmospheric path multiplied by downwelling radiations scattered by the surface plus surface brightness temperature plus upwelling radiation from the atmosphere. We discussed this in the last class but let me refresh your memory. If you have a surface which can be ocean or land there is going to be downwelling atmospheric TB, TB stands for microwave brightness temperature. So from the atmosphere there can be a contribution for downwelling atmospheric TB and this in turn can get scattered by the surface by ocean or land plus there is going to be some emission from the surface themselves and then there is going to be the upwelling atmospheric TB. So you see these four parts and a radiometer that is an instrument on board an aircraft or satellite that is operating in the passive microwave remote sensing a radiometer sees the upwelling radiation in the microwave region that is emanating from the surface. Now a part of this can be downwelling atmospheric TB that is further getting scattered a part of it can be upwelling atmospheric TB and it can also be the TB due to surface emission themselves and that is when in the last class we decided that we will try to develop a generic forward model for passive sensors which can be top of atmosphere brightness temperature that is nothing but transmissivity of entire atmospheric path. So let me write it down transmissivity of entire atmospheric path multiplied by downwelling radiation scattered by the surface so downwelling radiation scattered by the surface plus surface brightness temperature plus upwelling radiation from the atmosphere. So I have repeatedly been mentioning about something known as a radiometer is not it? Just like a radar we have repeatedly tried to understand about a radar in active microwave remote sensing radiometer we will try to understand in passive microwave remote sensing. So a radiometer is nothing but an instrument that produces an output voltage which is proportional to the amount of radiation collected at the antenna an instrument that produces an output voltage proportional to the amount of radiation collected at the antenna and there are different types of radiometers available we can have total power radiometer heterodyne radiometer system we can have decay radiometer and so on. So please note that when we use a total power radiometer it is going to measure the total energy that is incident on the antenna of course we can gain further information by designing a system which measures both the amplitude and phase information and such a system is known as coherent measuring system. Now shown here are the components of a heterodyne radiometer system wherein you can see a local oscillator, a mixer, an IF amplifier, detector and most of the instruments you know whether they are radar or a radiometer they record voltage as the raw output data voltage. But then the quantity which interests us is the backscattered power from a radar and from a radiometer it is the brightness temperature microwave brightness temperature which was abbreviated as TB in the last lecture. So we need to ultimately get or obtain the relationship of the recorded voltage with the physical property of interest. Let me re-hydrate our aim is when a radiometer is used for say atmospheric sounding we need to obtain a relationship between the recorded voltage with the physical property of interest and for a passive radiometer there are many means of calibration. So let us quickly discuss about a total power radiometer I hope you remember the importance of calibration that we covered as part of the last module. When it comes to a total power radiometer it is calibrated by observing black body emitters at two different temperatures. One is going to be a temperature of a hot target, high temperatures, temperature of a hot target and the second is going to be temperature of a cold target. So two extremes need to be observed for calibrating a total power radiometer. So if T hot refers to temperature of a hot target and T cold refers to the temperature of a cold target then the radiometer calibration constant needs to be calculated using this relationship. The antenna temperature further for an unknown target is usually estimated by using the relation calibration constant which was calculated over here multiplied by V minus V naught voltage. Please note that here I am not discussing about the calibration of the antenna because we are just focusing on the calibration of the radiometer and the antenna itself it needs to be calibrated at regular intervals for precise measurements. Now when you look at these relationships you know the calibration process might seem pretty straightforward to here isn't it? But then actually it is not that straightforward. So cold calibration is extremely crucial for us to interpret the physical properties of the object being observed and there may be measurements which are say taken over the same agricultural field that is recorded using an air bond sensor, a satellite bond sensor and a ground based sensor. The same agricultural field can be observed by different sensors on different platforms. It can be airborne, satellite bond or it can be in situ ground based sensors. Now when we try to calibrate it is the calibration that is allowing quantitative limits to be placed on the sensitivity of the instrument and it also allows one to inter compare the different measurements. You know it allows us to compare the measurements from a satellite bond sensor with an airborne sensor with a ground based sensor. So calibration basically it allows the quantitative limits to be placed on the sensitivity of the instrument. So let us move forward. So in this module that is in passive microwave remote sensing as part of this module we shall learn about radiometers which are onboard satellites which are used to measure precipitation and soil moisture in particular. Remember a downward viewing radiometer is going to vertically scan the atmosphere. And as mentioned in the last lecture the medium through which microwaves propagate is of high importance, is not it? And before we begin our focused discussion on the data products from different satellite missions, we should understand the earth's atmosphere as a medium, a medium through which microwaves propagate and we should also understand the effect that various constituents of the atmosphere has on microwaves. So this means that we will be discussing in detail about the earth's atmosphere. I hope you remember this slide which was shown as part of the earlier lecture when we were discussing about earth's atmosphere just to get an idea that how far microwaves have to travel so that it can register a detectable signal at the antenna. So just to refresh your memory the earth's atmosphere it can be described by the lower middle and upper regions of the atmosphere. Let me write that down, the lower region, the middle region as well as the upper regions of the atmosphere. Now when I say lower atmosphere troposphere is the region from the ground to an altitude of say about 15 kilometer troposphere and this is the region in which the maximum weather, most of the weather takes place in the troposphere and in fact the troposphere itself consists of about 90 percentage of the mass of atmosphere and we discussed last time that the top of troposphere is known as tropopause and for middle atmosphere it consists of stratosphere and mesosphere, stratosphere and mesosphere and it is between roughly 15 to 85 kilometers with stratopause the name given to the boundary between stratosphere and mesosphere. So we are talking about the different layers of the atmosphere. Now there is a decline in atmospheric temperature in mesosphere which is due to decreased ozone, increased cooling from carbon dioxide infrared radiation and it is mainly in the middle atmosphere that satellite remote sensing is of particular interest. So now I am going to slightly discuss off track about a related but a very, very important topic known as atmospheric sounding. The reason is because microwaves between 10 and 300 gigahertz are responsive to a number of atmospheric constituents and it is in these frequencies that many atmospheric constituents have rotational as well as vibrational spectral lines going to reiterate rotational and vibrational spectral lines. We will try to understand this further in detail but before that let me ask you a very simple question. Can you break glass with sound waves? There may be many images coming in your memory right now, is not it? With the answer a big yes. So say you are a singer and say you are singing a musical note which is matching with the resonant frequency of the glass giving you one new term resonant frequency. You are a singer assume you are singing a musical note which is matching with the resonant frequency of the glass then what will happen? The sound waves will vibrate the air particles that are around the glass at its resonant frequency and when the pitch is high that is when the singer sings loudly these vibrations cause the glass to vibrate so hard that the glass gets shattered. So here I have used the term resonant frequency, resonant frequency why am I discussing this? You will get to know shortly. So when it comes to the choice of microwave frequencies in satellites it is not chosen randomly. There is a reason why measurement is preferred in particular frequency regions within the microwave region and that is what we are going to understand now. You may have heard about spectroscopy which is an age old science that is explained by quantum mechanics. During the first half of 20th century involving the study of absorption and emission by gases absorption and emission by gases. So according to this there are 5 possible ways in which radiation can interact with atmospheric gases. Remember we are trying to understand about how microwaves propagate in the atmosphere. We had a quick idea about the different layers of the atmosphere and now we are trying to understand why certain frequencies are preferred for atmospheric sounding. And that is when I introduced spectroscopy to you and I am mentioning that there are 5 possible ways in which radiation can interact with atmospheric gases. So let us see what they are. They can be ionization, dissociation interaction, electronic transition, vibrational transition, rotational transition and forbidden transition. Among these say you are a satellite meteorologist then vibrational and rotational transition are going to be more of interest to you. Vibrational transition and rotational transition are going to be of more interest to you because they occur in the infrared and microwave portion of the electromagnetic spectrum. Now resonant frequency as such the term means the natural frequency of an object at which it starts vibrating. Now when it comes to a water vapor and say oxygen, water vapor has a weak absorption line at 22.235 GHz and a very strong line at 183 GHz whereas oxygen has 2 major peaks one is near 60 GHz and another at 118.75 GHz. Why am I telling you these because whenever there is a satellite mission whose aim is to measure the atmospheric water vapor it will certainly have a channel that is sensitive to 22.235 GHz or it will have a channel that is operating in either 183 GHz or 22.235 GHz because these are the frequencies at which water vapor has a weak absorption line resonant frequency. Similarly, say you want to launch a satellite to study more about the oxygen in the atmosphere. So then the channels of 60 GHz and 118.75 GHz is going to be preferred. So my whole point in explaining these is that the choice of frequencies in a satellite it is not random okay, there is a reason why certain frequency channels are more preferred when it comes to atmospheric sounding okay. So now let us try to understand atmospheric attenuation due to cloud liquid water, gaseous atmosphere, ice and finally precipitation. See an extensive study of microwave absorption of atmospheric gases both theoretically as well as experimentally has shown that emission or absorption in gaseous atmosphere is dominated by the presence of oxygen and water vapor. And within the microwave region these molecules of oxygen and water vapor they are subjected to rotational transition, now what is meant by rotational transition wherein a molecule changes the rotational energy states. What happens? Because when a molecule changes the rotational energy states it causes a peak in the microwave to brightness temperature that is measured by a radiometer, Tb measured by a radiometer. So to quantify the magnitude of increase in Tb one needs to estimate the total number of water vapor or oxygen molecules along the propagation path through the atmosphere which means at higher altitudes of the atmosphere there is obviously going to be a decrease in the amount of water vapor or oxygen molecules per unit volume and this in turn is going to reduce the bandwidth of water vapor or oxygen emission or absorption leading to an increase in absorption of the peak of resonance. I am using the term again resonance, resonant frequency and as mentioned earlier the atmosphere itself it can act like a source or a sink of microwave radiation. See in an atmosphere with predominantly cloud water droplets what happens is the prominent sources and sink of microwave energy are going to be local emission and absorption. And whenever the cloud liquid particles are less than 100 mu meter in diameter scattering is negligible. Scattering is negligible and when the microwaves they interact with the rain clouds the phenomenon it is very much similar to an ensemble of drops wherein there is no coherence from drop to drop in the phase of scattered light. Let me repeat I am trying to discuss what happens when there is rain clouds and when microwaves are trying to propagate through the atmosphere which is having rain clouds and I am telling you that this phenomenon is very similar to an ensemble of drops wherein there is no coherence from drop to drop in the phase of scattered light. Now the usual practice is to calculate the scattering and absorption for a single drop assuming it to be spherical dielectric. You are assuming a spherical shape for the drops and then you are calculating the scattering and absorption for a single drop and then you assume that rainfall laden cloud that is rain cloud is nothing but an ensemble of drops which means you can integrate the value obtained for a single drop of spherical dielectric and by integration the scattering and absorption for a rain cloud can be estimated because you are estimating it for a one spherical dielectric and assuming a rain cloud to be a conglomeration of several spherical dielectrics. Here there is a need of course to define something known as a drop size distribution which is abbreviated as DSD which defines the concentration of rain particles per unit volume per unit increment of drop radius. Now whatever I say the important points are written on the screen. So I just want you to listen to me rather than read what is already present in the screen. So the aim of me discussing all these is to understand more about atmospheric sounding in the microwave region of the electromagnetic spectrum. I mentioned that unlike oceans which provide a cold background to a radiometer the land modelling a land it is very very complicated because owing to the spatiotemporal variations of soil features like roughness, like vegetation cover, soil moisture contained etc. So it is very very complicated to model the land surface properties in the microwave region whereas oceans they provide a stable and uniformly cold background for a radiometer and if the surface is no covered the emissivity values tend to depend on the dielectric constant of frozen soil, thickness, water equivalent and liquid water distribution. Now at this juncture let me try to also mention that there are different shapes possible for ice particles and the ice particle shapes they are crucial to understand in the scattering regimes because crystals of ice can exhibit different shapes and different modes and the usual practice is to adopt a Marshall Palmer size distribution. See sensitivity to TB values to the integrated ice or rain it is going to depend on frequency until the optical depth reaches the saturation level. Now there are studies in literature that suggest that even the lowest microwave frequency of say 18 gigahertz even that is significantly affected by deep convective ice mass and generally when the microwave frequency increases say about 60 gigahertz the scattering signatures become more pronounced and at higher frequencies scattering dominates and microwave radiation is seen as relatively transparent to the rain below the freezing level. Okay when it comes to precipitation sized drops or hydrometeors how do they interact with microwave radiation. Now whenever a spherical dielectric interacts with electromagnetic waves it causes scattering or absorption I am going to define scattering as redirection scattering redirection and absorption as conversion to mechanical energy scattering and absorption redirection and conversion to mechanical energy. Whenever a spherical dielectric interacts with electromagnetic waves it causes scattering and absorption depending of course on the size of precipitation particles. Alright so let me re-itrate a few points in surface emission because in the microwave spectrum gray body is emitting surface with emissivity value less than 1 and as we discussed earlier unlike oceans it is very very complicated for us to model land surface properties in the microwave region. And why so? Because owing to the spatio-temporal variations variability of soil features like roughness, like vegetation cover, moisture content etc. So in the screen in front of you you see land region as well as oceans so just to re-itrate for a radiometer that is looking down at the earth oceans they provide a stable and uniformly cold background whereas say the surface is covered with snow then of course the emissivity values are going to depend on the dielectric constant of the frozen soil and the water equivalent and liquid water distribution etc. So while you hold that thought let me also try to discuss about ice particle shapes. So the crystals of ice they can exhibit different shapes and modes and the usual practice is to adopt a Marshall Palmer size distribution because the sensitivity to microwave brightness temperature values their sensitivity to integrated ice or rain depends on frequency until the optical depth reaches the saturation level. Now without going too much into details let me mention that there are numerous studies which suggest that even the lowest microwave frequency of say 18 GHz is significantly affected by deep convective ice mass even the lowest microwave frequency of 18 GHz is significantly affected by deep convective ice mass and generally we say that when microwave frequency increases that is if it is more than 60 GHz the scattering signatures they tend to become more pronounced and at higher frequencies what happens is that scattering tends to dominate and the microwave radiation is seen as relatively transparent to the rain below freezing level. So at microwave frequencies the precipitation sized drops or the hydrometeors they interact strongly with microwave radiation and whenever a spherical dielectric interacts with electromagnetic waves it causes scattering that is redirection or absorption that is conversion to mechanical energy depending of course on the size of precipitation particles. So in this context the expression for my efficiency factors are given here okay here QEX denotes the extinction coefficient and QSC denotes the scattering coefficient and R is nothing but the radius of raindrop, lambda stands for the wavelength, NCE represents the complex index of refraction, symbols that is sigma EX and sigma SC they represent the effective cross sections for extinction and scattering alright. So with this background let us try to understand atmospheric sounding through a forward model. Remember in the last lecture we had discussed briefly about the generic form of forward model and please note that at the end of the day using remote sensing data our aim is to characterize the relationship between a remote physical attribute and a set of numbers. Now the set of numbers can be digital numbers, they can be radar backscatter or they can be microwave brightness temperature anything okay. So in the passive microwave remote sensing what the satellite gives us is the microwave brightness temperature or TB by now we know that is not it. And by a forward model our aim is to characterize accurately the relationship between a physical attribute and the measured TB. So given the physical variables of the atmosphere how do we simulate the measurements and for a forward model please note that a well calibrated measurement quantity like brightness temperature is going to be used and forward model or direct model is fundamental part of the measurement process and what you see in the screen in front of you is how to represent a forward model in atmospheric sounding. So here y refers to the set of measurement values, x refers to the physical attributes of the atmosphere and underlying surface, Pm P suffix m is nothing but the known physical parameters that are important to measurement, P suffix i is the instrument parameters which tend to influence the signal and epsilon is a vector that represents noise or random error. So this is the forward model written in a generic format and just to understand it further assume you have a satellite based radiometer which is measuring in multiple channels by channel I am referring to frequency. So assume that a satellite based radiometer is measuring in multiple channels say 12 channels by channel I am referring to frequency here and assume that this instrument this radiometer is capturing information in multiple view angles, say 20 view angles. So 12 channels, 20 view angles then y that is a set of measurement values is going to be 12 multiplied by 20, 12 channels, 20 view angles. Similarly x is a vector representing the state of atmosphere including the vertical profiles of temperature, pressure etc. And please note that when you look at the forward model like this it may seem straight forward but often it is non-linear with the function f often taking the form of an integral equation. I will not get too much into details of the forward model but this is just to give you an idea that how we can understand atmospheric sounding through a forward model. Now ideally we have different simulated observations and actual observations and what we need is the difference between actual observations and simulated observations that is what we need is not it? And in reality please note that these two values that is PM and PI that is the known physical parameters and the instrument parameters they also will have some amount of uncertainties associated with them, they are not error free ok. They are also going to have some amount of uncertainties associated with them and so this difference between the models and the real observations they are known as residuals ok. So once again for atmospheric sounding we can write a simple forward model as let me write that, write that down, top of atmosphere brightness equal to transmissivity of the entire atmosphere, transmissivity of the entire atmosphere into brightness temperature of the background, brightness temperature of the background plus contribution from atmosphere ok plus contribution from atmosphere. This means there is emission from all layers of atmosphere where each layer is getting attenuated by the layer above ok. So if there is a forward model there should also be an inverse model is not it? So now we are familiar with the forward model and let us try to understand if there is something known as an inverse model. Now through an inverse model we try to estimate the state of atmosphere given a set of satellite brightness temperatures. As we are talking about passive microwave remote sensing here that is why I am specifying passive microwave brightness temperatures. So once again when I talk about inverse model we try to estimate the state of atmosphere given a set of TB satellite brightness temperature and this is also known as retrieval or inversion and as we are speaking specifically to atmospheric sounding let me mention that inverse model for atmospheric sounding is it is not well established ok. And for the inverse model instead of trying to work with the complete radiative transfer equations it is convenient to work with the matrix equivalent wherein the dependences assume to be linear which means we can use something like y equals kx plus epsilon where y is the measurement values, x is the state of atmosphere, epsilon is the noise or error with each measurement of y and k is known as something like a kernel matrix ok kernel matrix and remember that k that is the kernel matrix needs to be known prior and the solution of this problem you know this problem of solving inverse model the solution of this problem can result in many answers which is why it is known as an ill posed problem because there are infinite number of possible solutions that can result in the same set of observations this is very similar to me asking you to solve an equation say I am going to write an equation 4x1 plus 3x2 equal to n ok say I am asking you to solve this equation and there are infinite number of values for x1 and x2 which can be solutions for this equation is not it. Now assume I am going to give you one more equation ok x1 plus x2 equals 3 and now I am asking you solve for x1 and x2 using equation 1 and equation 2. Now there are two equations and two unknowns x1 and x2 which means we have a solution which satisfies both the equations. Remember that in inverse model there is also the factor of noise here that is random error if so the solution will no longer be considered as unique but it will now comprise of a region of solutions and for an inverse model we need to estimate which among a set of solutions are ideal or appropriate to the particular problem and for constraining the solution there are empirical approaches that use regression analysis wherein you have many values of y that are measured by an instrument with simultaneous in situ measurements of x being made and both these values of x and y are being investigated for statistical relationship. Remember we try to get information here through a statistical relationship and not the physical means and the drawback of this method is that of course it cannot be used to simulate extreme cases of atmospheric state. So physics based modeling is ideal and inverse problem is being encountered in many branches of engineering not only in atmospheric sounding using passive microwave remote sensing. So this discussion today's lecture was to share a very small glimpse of what is forward model, what is inverse model when it comes to atmospheric sounding and I hope that you got the answers to the new terminologies that I had introduced as part of the previous lecture. Now we know what is a radiometer. Now we are slowly getting familiar with radiative transfer model. Now we are trying to understand about atmospheric attenuation. We briefly touched about how atmosphere can act as a source or a spink of microwave frequencies. Please remember that this is a vast area of research, active research in fact and there are a lot more to discuss. So through few lectures I shall try to simplify the few points as we try to understand the applications of passive microwave remote sensing through the next lecture. So let me see you in the next class and thank you.