 Hello everyone, welcome to the next lecture in the topic of converting radiance to reflectance quantities. In the last class, we discussed about image formation process, how a digital image is formed and what is actually stored in the digital what is actually stored in the digital image, what is the relationship between the energy that came in at the sensor and the digital numbers stored in the image and how to convert it from radiance to DN and DN to radiance we saw in the last class. In this class, what we are going to do is how to further convert the radiance recorded into surface reflectance that is what we are going to see in this particular lecture. So, just as a recap, this was the equation we saw at the end of last lecture that is how to convert the DN stored in the image into the radiance recorded at the sensor. So, for like a remote sensing sensor, essentially it is the radiance recorded at the top of atmosphere. We always denote it with symbol or abbreviation TOA because whatever the energy that came in or went out, it has an influence of the atmosphere and whatever is recorded by the sensor always has effect of atmosphere and that is why we call it as radiance TOA or we always talk it in terms of TOA. So, how to get TOA radiance to reflectance? First of all, why do we need reflectance basically because that is the property we are interested upon. We are in shortwave wavelengths that is less than 3 micrometer wavelengths, we are interested in recording the reflectance property of object. By using the reflectance of the object recorded within this bands, we will be able to understand the object, its behavior, its properties, etc. And hence we are interested in retrieving the reflectance. How to do it? That is what we are going to see now. So, here in this slide, first we are going to calculate what is known as TOA reflectance. What is TOA reflectance? Reflectance, record at the top of atmosphere that is reflectance attained without any atmospheric correction. This is what is meant by reflectance TOA, rho TOA, rho is the symbol used to denote reflectance, TOA is top of atmosphere. So, top of the atmosphere reflectance means reflectance without doing any atmospheric correction. How to let it? So, reflectance we will just go back to the basics. So, reflectance is defined as irradiance or sorry the radiant flux density that got reflected divided by the radiant flux density that irradiated the surface, right. This was the definition of reflectance that we saw earlier. What is the irradiance that illuminated the surface that will be in the denominator? What is irradiance that actually lift the surface after reflection that will be in the numerator. Divide them both, we will get the reflectance of the object within the hemisphere in which we are working. We are going to do the exact same step here. We are going to calculate the two terms in the numerator and denominator, divide them one by the other to get the top of atmosphere reflectance. First we will calculate what is on the denominator that is the energy radiant flux density on the surface. So, first thing in the short wave wavelengths lambda less than 3 micrometers typically sun is the primary source. And as in the previous lectures I told you that for each wavelength band say 0.4 to 0.5, 0.9 to 1.1 like this in different wave band we will be able to calculate what will be the energy from the sun that reaches the top of atmosphere. We will be able to calculate in one of the previous lectures I explained you how to do it. And also we saw there will be some sort of like transmissivity effect of the sun radiation coming in and so on. Lot of atmospheric effects I explained in detail when discussing about the radiance reaching the sensor. Now this is our first step like of getting reflectance we omit all sort of atmospheric effects. We assume atmosphere is not there then what will be the reflectance that is what we are going to calculate now. The coming slides I will explain how to take care of atmospheric effects. So, first we will be able to know what will be the energy that was emitted by sun. We call it as e sun in this particular slide. So, radiant flux density that was emitted by sun which reached the earth surface. We calculated a value of 1368 watt per meter square that is for the entire bandwidth like entire electromagnetic spectrum from 0 to lambda sorry 0 to infinity wavelengths 1368 watt per meter square which we called as solar constant. But this number will vary with the bands with sphere working on. We know that the peak energy from the sun comes in around like 0.55 micrometers around like the green band. So, if you calculate the e sun around the green band it will be much higher in the order of say 1700 or 1800 watts per meter square. So, the e sun values or the irradiance from the sun values will vary with the wavelength you are working on. So, that is normally whenever a sensor is like launched people the sensor people itself will give if this is the bandwidth the sensor is designed to work this will be the value of e sun that you are going to expect at the top of atmosphere that will be clearly given to us. So, e sun actually will be given to us a priori if this is the bandwidth of sensor at that particular band this will be the irradiance from the sun that will be known to us or most of the sensor manifest say NASA will publish it for own satellites ISTO also will have record of what is the e sun for its sensors it is known to us. If we know e sun value that is of the top of atmosphere then if we neglect atmospheric effects as the first step then we can assume whatever the value there comes in and fall on the surface as simple as that. But remember one thing irradiance from the sun is calculated assuming average distance between earth and the sun that is that 1.5 into 10 power 11 meters which we used in all of our numerical problems is essentially the average value overall within an year this is the average value of distance between earth and the sun. But every day this will change because earth and sun like earth is rotating in an elliptical orbit around the sun. So, the distance between earth and the sun is going to change continuously. So, the energy what we calculated for the average distance is not going to be the same every day it is going to vary with the day in which you are with varying months or varying days the position of earth and the distance between earth and sun is going to change continuously. So, we need to correct for this varying distance factors. So, that is why if you look at the equation this d square term is there essentially it is e sun is equal to e naught by d square where e naught is the one that is actually calculated or let us say I say e sun here and e corrected here. So, e corrected is equal to e sun by d square where e sun is the value given for that particular band assuming an average distance and divided by d square where d is the actual distance. So, the average distance we call it as one astronomical unit that 1.5 into 10 to the power 11 meters that we call one astronomical unit the average distance. So, if this number let us say the same 1368 what we will take it if this is the value for one astronomical unit that is for one average distance I have to convert it for say the value may be 1.01 astronomical units 0.98 astronomical units and so on the value will be in terms of like fractions 0.98, 1.02 and so on. If the distance is more than this average value the denominator will be greater than 1 if the actual distance between earth and sun is less than this average value denominator will be less than 1. So, this d square term is actually to correct this essentially this will be in the denominator but instead of writing it in denominator here we have written it here. So, this d square term is to correct for the varying distance between earth and sun. So, then L sat this also we know because from dn we know how to calculate radiance recorded as a sensor we are now ready to calculate it we calculated and kept it ready. For a surface for a Lambertian surface we know E is equal to L pi. So, L is what we have recorded in the sensor in order to convert it into E we are multiplying with pi. So, this is the L sat into pi is the actual numerator term E reflected. So, this E by E sun by d square is the energy that reach the earth surface by sun and cos theta s is to correct for the sun's zenith angle like sun's illumination geometry. If you assume a horizontal surface the normal to it will be the zenith the vertical what angle made by sun with respect to that zenith. Like yesterday in previous classes in some derivations I introduced you to this cos theta term why it is coming it is just to account for the variation in sun's geometry it is not lying exactly overhead it is coming in at an angle in order to correct for it I already told you. So, the energy that came in from sun is actually E sun by d square cos theta where theta s is the solar zenith angle and the irradiance from this surface that is reflected is L sat into pi. So, if you do this we will actually be calculating what is the surface reflectance recorded at the sensor without doing any atmospheric correction or neglecting the effects of atmosphere. So, in order to convert the irradiance recorded at the sensor into top of atmosphere surface reflectance the formula is L sat into pi divided by E sun by d square into cos theta s that is it a very simple equation. However, we know atmosphere is there atmosphere will influence this variation. So, we are essentially have to correct for this effect of atmosphere how to do it? Before going on to this I will just tell you why first of all we need this TOA reflectance can be just anyway TOA reflectance is not corrected for atmospheric effect. So, why do we need to compute it we need to we need to compute it because it corrects for the solar illumination geometry that is the radiance reaching the sensor has effect of solar illumination, atmosphere and topographic effect all these effects that is how what is the variation in suns geometry. What is the variation in surface topography whether the surface is flat surface is like a mountain ridges valleys etc. What is the variation in atmospheric effect and also what is the variation in sensors viewing angle all these things is going to change the radiance recorded at the sensor. So, in the TOA reflectance actually the radiance the radiance when we divided by E sun by d square into cos theta s we are effectively removing the effect of solar radiation based on amount of solar radiation the radiance is going to vary higher incoming radiation higher will be the radiance recorded. So, the radiance based on the day of year the energy from the sun will vary because of seasonal effects and we are actually correcting it. So, one of the effect which affects the radiance is corrected now even without doing TOA sorry the atmospheric correction. So, that is why it is better to do radiance to reflectance correction or reflectance conversion even we do not have any atmospheric information at least one of the effect we are removing from the radiance recorded by the sensor. Now, we proceed on one effect we removed the effect of solar illumination the effect of incoming solar radiation we have removed we have converted it into TOA reflectance actually we need surface reflectance that is after correcting for atmospheric effects how to do this here also we assume the sensor is what to say viewing from nadir when we do our corrections. So, here in this particular slide this is from the previous lectures what I explain. So, the radiance recorded by the sensor has a direct signal component surface reflected diffuse skylight component surface reflected diffuse skylight and the path radiance. So, if you look at this particular figure the else that recorded at the sensor has several components. So, this is actually the surface reflectance term we now want to get this term and remove all these effects. So, our aim is to remove all these effects this equation we derived in one of the previous classes. So, we are going to take this equation work it in a reverse fashion to get reflectance and that is actually given here in this particular slide. So, the surface reflectance is actually is equal to pi times L sat minus L path where this is the path radiance that is what is sent to the sky directly from the atmosphere the diffuse skylight directly reaching the sensor. This term E sun by D square cos theta s this is actually the solar radiation reaching the surface and this tau s is the transmissivity between the sun and the earth surface in order to correct for atmospheric absorption and scattering effects like yesterday we said I told you when I derived that formula in previous classes that transmissivity can be 70 percent 0.7 0.8 and so on what is the transmissivity of the atmosphere between sun and earth that will correct the incoming solar radiation for atmospheric effect because even the incoming solar radiation will be absorbed to some extent by atmosphere so that will be corrected the actual radiance reaching the surface we will corrected if we know tau s E sun by D square cos theta s tau s this will be the radiance at the surface plus E down term is that this E down s is to account for or I write this as direct irradiance E down is the diffuse skylight component that reach the surface so essentially the denominator this term will tell us the total irradiance recorded by any surface on the ground including the effect of atmosphere so the tau s is atmospheric transmissivity how much fraction of sunlight is being allowed inside and this E down is the diffuse skylight component that reach the terrain so this is the total irradiance that is there on the denominator. Now from the radiance recorded from satellite we are subtracting l path by the path radiance so that term is removed and then you multiply with pi in order to assuming a Lambertian surface again E is equal to l pi the same relationship we are using and we are dividing it by tau v tau v is the transmissivity between the earth surface and the sensor because sensor may be somewhere else not exactly in the direction of sun it will be at a different distance based on all these factors atmospheric transmissivity will vary in that particular direction so in order to correct for it we are going to use that. So essentially if we use the equation given in this particular slide we will be able to correct for the effects of atmosphere but we should know what is the path radiance we should know the transmissivities in both the directions from sun to earth and earth to sensor and then we should also know what is the downwelling diffuse skylight that actually irradiate in the surface if we know all these quantities we will be able to correct the radiance recorded at the sensor for atmospheric effects. This equation is derived assuming the surface is horizontal is given here and Lambertian reflecting nature here I am assuming surface is horizontal and Lambertian if the surface is not horizontal we need to do what is known as a topographic correction which I will explain later. So this equation essentially given in the slide will help us to convert the radiance recorded at the sensor to surface reflectance which is of our real interest. We said we need to know l path tau s tau v e down. If we have it it is easy to correct but how to get those values getting those values substituting them in this equation getting surface reflectance is essentially the known as the process of atmospheric correction correcting for the effects of atmosphere how to get these values. Ideally what we should do is at the time of satellite overpass most of the satellite has like a fixed time of overpass over a region at the time of satellite overpass we should send or we should measure atmospheric variables in the entire column above the land surface that is we should measure temperature pressure humidity that is the water vapor content CO2 content etc etc all these parameters we have to measure in the entire atmospheric column above our study area. Feed those values into what is known as a radiated transfer model RTMs. The RTMs will essentially use these measured atmospheric variables and try to model if this is the atmospheric variables how the radiation would have been affected by the atmosphere. RTMs have the capacity to simulate it simulate the radiation provided the atmosphere is known to it. So, the RTM will model it and that will give these variables L path tau S tau V and E down as outputs once we get it is a direct easy way to correct it. This is the ideal way of doing atmospheric correction that is how we should do but normally we will not be having atmospheric data at the time of satellite overpass unless it is like a proper research is going on people are planning everything meticulously for various applications we would not be having this atmospheric information. So, typically what we can do nowadays we have lot of models which simulate atmospheric conditions we have atmospheric models which can simulate at the given place at a given date and time this will be the or this was the atmospheric condition temperature pressure water level content CO2 content etc etc. Take output from those models feed it into RTMs do the correction that is also way which is most of the users are now practicing combine atmospheric models with radiative transfer models. As a normal user say I do not have anything at me I do not have the computation resource like RTMs atmospheric models they are all free nowadays like atmospheric data is now available to us RTMs also nowadays open source RTMs are available but still if I want to use I do not have any computation resource to do it what can I still do I have an image already with me I want to do some sort of atmosphere correction then we can use what is known as an image based atmospheric correction. So, that is the possible most possible and simplest way we can use if we do not have any other information about atmosphere or we do not have access to radiative transfer models. So, these are different ways in which we can correct for the effect of atmosphere. Other than this people also do what is known as a vicarious calibration. Vicarious calibration means like I told you that even before a sensor is launched the calibration or the conversion factor between radiance to DN will be known will be fixed but that will not remain constant as time progresses due to various effects in space the calibration constants will vary. So, most satellite missions the scientists and engineers associated with them will be periodically doing repeated calibration exercises that is whenever a satellite is going to overpass they will fix some certain or they will take some steady area and they will measure the surface reflectance there beforehand. So, the reflectance of the surface is well known at different angles and so on. So, when the satellite overpass over the area they will take the image they will calculate radiance or the reflectance from the image and try to compare them this is the reflectance or radiance recorded in the image this is the actual reflectance and radiance recorded on the ground what is the difference between them they will correct it. So, this will this has two benefits one they correct for the effect of atmosphere itself whenever they do this correction what is recorded in the satellite what is recorded on the ground when they relate them and do a correction they will be able to do atmospheric correction in addition to this they will also be able to understand how much the sensor calibration has changed. So, this is one way of doing atmospheric correction but as I said this is primarily done for operational maintenance of satellites or for experimental studies. But as normal users if we have access to some good computation resources we can combine outputs from atmospheric models feed them into RTMs and use it if we do not have any other resource we can go for image based atmospheric correction models. So, the atmospheric effects can be removed using radiative transfer models with field observed atmospheric observation, a radiative transfer model with modeled atmospheric characteristics fed into an RTM, atmospheric correction using natural or artificial reflectance target within the images this is primarily they using calibration exercises and we can use a image based atmospheric correction. So, these are the four different ways in which we can correct for the effect of atmosphere and what are the different ways we are going to correct for the effect of atmosphere is we are going to correct for an additive path radiance term. So, this is like an added component the L radiance from the surface is added with a path radiance we are going to subtract it we are going to remove it. So, an additive path radiance term we have to correct the reduction in irradiance and refractor radiance by atmospheric transmissivity terms tau v and tau s essentially tau v and tau s will be less than 1 and they are kind of acting as a reducers or they reduce the incoming radiation as well as the outgoing radiation plus there is an additive term e down that is a diffuse skylight. So, atmosphere acts in many different ways it adds radiance to it l path e down it reduces the incoming energy in form of tau s and tau v and so on. So, using the four methods I just mentioned to you we are going to correct for the effect of atmosphere. So, in the next class we will be seeing a simple image based atmospheric correction technique how to use it what are the pros and cons and also we will see one more effect called the topographic correction. So, basically in this lecture we have studied about how to convert the radiance recorded at the satellite into reflectance and basically how to convert this reflectance or how to correct this reflectance for the effect of atmosphere. So, that is what we have seen in this class next class we will continue further in this particular topic. Thank you very