 to today's class. So, just a quick recap about what we have been learning as part of module 6. So, till now as part of this module, we have been trying to answer the question can phase be used as a relative distance measure, you know that was the tagline of this module. And we also discussed throughout the first and second lectures about the fundamentals of radar interferometry, about stereo SAR radar geometry, about INSAR, the different types. And we also learnt about interferogram and fringes and how to interpret an interferogram. So, I am hoping that now these terminologies are not new to you. So, in today's class, we shall learn about differential SAR interferometry, differential SAR interferometry or it is abbreviated in standard textbooks as D in SAR. We will also try to learn about digital elevation models, how to access them and what are the particular applications of digital elevation models when it comes to hydrology and water resources engineering. So, once again welcome to the third lecture of module 6. So, when we discussed about synthetic aperture radar interferometry, we understood it as a technique to estimate topographic heights. You know just like we use GPS or total station, it is one technique which helps us to estimate the topographic heights. Now, interferometric SAR has a very interesting extension which is known as differential SAR interferometry or D in SAR. Let me write that down, D in SAR or sometimes it is also referred to as D if SAR. So, let us try to understand first the principle behind D in SAR. So, from the last lecture, I hope you remember what is baseline distance. Assume an instrument flies exactly the same flight path such that the baseline distance is 0. Baseline is the distance between two satellites during its subsequent flight paths. So, assume that an instrument is flying the exactly same flight path such that the baseline distance is equal to 0, exactly the same flight path. Now, we will call them as flight path 1 and say flight path 2. Now, in between the flight paths that is in between flight path 1 and flight path 2, assume there has been no change in the surface characteristics. By surface characteristics, I mean the earth surface characteristics. Assume that in between both the flight paths, the earth surface has not changed at all. Then obviously, the resulting synthetic aperture radar images from flight path 1 and flight path 2, they should be exactly identical, is it not? They should be exactly identical and these images will be identical with coherence as 1 and phase difference between both the images will be 0, isn't it? Now, assume that there is a very small movement of the surface, earth surface, very small movement in between both the flight paths. Then what happens? Now, the images that are obtained from flight path 1 and flight path 2 will be completely different, which means after movement, the relative positions of the elements within pixel of course, shall remain same. But now, there is going to be a phase difference between the images acquired from flight path 1 and flight path 2. Now, this is the second case scenario wherein I am asking you to assume that there is very small minute change in the earth surface characteristics in between both flight path 1 and flight path 2. Now, let me try to explain this using a diagram. Assume this is the surface of the ground and assume the green line that you see is the reference surface. Remember, always when we measure vertical heights, it has to be from a reference surface which we call as datum, mean sea level. Assume the green line is the mean sea level and the two colored lines that you see, they assume they represent the earth surface before a movement and after a movement. We have a satellite which is flying at a height at an altitude of say h above the reference surface and what you see here are the same point x which is before the movement, which after movement has shifted to a position which we will call it as x2 dash, same point but it has been shifted due to some phenomenon in between both the flight paths that is flight path 1 and flight path 2. You already know what is a range distance that is the distance of a point from the satellite. Slant range distance is shown here as r. In the diagram x x dash, it represents the relative movement of the point x, x is any point on the surface of the ground. And please note that in the diagram, both the antennas in flight path 1 and flight path 2 are assumed to be in the same location such that baseline distance is 0, which means it is travelling through a path and the next time during the second flight path, it is travelling through the exact same path such that the baseline distance is 0. So, given here are the range h is the altitude and in addition, if you look at the screen closely, I have marked two angles theta 1 and theta 2, okay, two angles. What are they? If I draw a horizontal and vertical line at point x, theta 2 is the angle with respect to horizontal and x x dash that is the relative movement of point x. Similarly, I can define theta 1 that is the angle from vertical to the slant range distance of the point x, okay. The line denoting the slant range distance, the angle that it makes with respect to vertical is marked as theta 1, all right. Let me give you some time to sync this in, okay. I have the reference surface from which all the vertical elevations are being measured that is shown by the green line. I have a satellite which is flying at an altitude of h above the datum, okay. And I have the ground surface which is shown in different colours before movement, after movement. The same point that is x is shifted by a very small amount to x2 dash, there is a relative movement of the point in between two flight paths which we will call it as flight path 1 and flight path 2. In such a way that the baseline distance is 0 which means they are following the exact same path. The range distance is marked and also I have defined two angles which I will call as theta 1 and theta 2 which are self-explanatory as seen in the diagram, all right. So, the resulting range displacement can be written using the above defined diagram, is not it? So, let us write it down. I am going to write down the range displacement. Try to recollect what we just drew to signify what is r h theta 1 theta 2 x x double dash, okay. So, the range displacement I can use this expression to denote the range displacement, is not it? x x double dash sin theta 1 minus theta 2. One of our earlier lectures we already have tried to understand how to define a phase change, change in phase, okay. Because we mentioned that when there is a relative movement of the ground surface between both flight paths, the resulting synthetic aperture radar imagery will be different and there is going to be some change in phase which means we can write an expression for phase change which I am going to denote as phi suffix d phase change as 2 pi multiplied by range displacement, let me write that down, range displacement to whole by lambda, is not it? 2 pi into range displacement by lambda. We have covered this simple relationship to represent phase change as part of our earlier lectures. Now, only difference is instead of range displacement writing it as a distance we are writing it using theta 1, theta 2 and the relative position change of the point x, the distance is given by x x double dash, okay. Let us write that down for the sake of completion, okay x x double 2 dash sin theta 1 minus theta 2 to whole by lambda wavelength, okay. This can be written to estimate the motion vector x x double 2 dash which describes the direction and magnitude of the surface movement. Let us write that down as well. I am just rearranging the equation. Now, it is phi d into lambda the whole by 2 pi into sin theta 1 minus theta 2. Look at this equation very closely, I will name it as 1, okay. This implies that the phase difference can be used to measure the ground displacements of a fraction of wavelength. Let me re-iterate. The phase difference can be used to measure the ground displacements of a fraction of wavelength. So, with INSAR we obtain the component x x 2 dash in the range direction, okay. Now, you may be thinking what is the purpose of estimating x x 2 dash. Let us cover the applications of why we are learning this particular topic. So, few applications are in land subsidence that is the slow sinking of land which can be due to mining activities or due to change in the groundwater, etc. Again, if I talk about an application in the field of glaciology, the slow creep of glaciers can be mapped using INSAR with great precision, okay, creeping of glaciers. Another application can be the crustal deformations. See, you want to study the aftermath of an earthquake using INSAR, okay. The ground displacements after an earthquake can be studied using INSAR. Of course, the measurements need to be made before the earthquake as well as after the event, after the earthquake, alright. So, even though while explaining about DINSAR, we mentioned about zero baseline in practice, you know, it is not that easy to maintain a zero baseline because with satellite systems, there are issues moving to orbital stability. And in such cases, two images, they may not be fully suitable to differentiate the phase difference. And we need to use an accurate digital elevation model. I am giving you a new terminology there, digital elevation model so that the phase caused by topography can be modeled and subtracted from the differential interferogram. Let me re-itrate. Often, it is not possible that we maintain a zero baseline because there is always some inconsistencies when it comes to orbital stability. And in such cases, two images may not be fully suitable to differentiate the phase difference. And we need to use an accurate digital elevation model so that the phase caused by topography can be modeled as well as subtracted from the differential interferogram. Now, one limitation of DINSAR is concerning the extent of ground displacement. That is, the extent we are wanting to estimate, the extent must be large enough such that it can be measured across a set of pixels, okay. Very small displacements may be difficult to estimate using DINSAR but the extent that we want to study must be large enough such that it can be measured across a set of pixels. Now while we are discussing about DINSAR, let me also take the liberty to introduce terminology to you known as permanent or persistent scattering interferometry, PSI. In PSI, it uses a very long time series to detect small differences in relative location across the scene, okay. This requires that the scattering be obtained, observed from permanent scatterers, okay. Just, you know, it is a relatively new area of research and if you are more interested to learn about it, I would urge you to look up PSI that is Persistent Scattering Interferometry. What is the difference between PSI and DINSAR? PSI uses a very long time series to detect small differences in relative location across the scene. Now, if you take an urban area, say Mumbai city, there can be many permanent scatterers, you know. Corner reflectors can also be used but when you look at natural landscapes, you know, it is the rarely permanent. Now, again if you look at rocky terrains with minimal vegetation that of course can act as a natural permanent scatterer, okay. But you know, typically when you say landscape is not permanent but then there are certain landscape, some features like the rocky terrains which doesn't have much of vegetation, they of course can act as natural permanent scatterers, all right. Now, if you remember through the lectures, I did introduce a new terminology to you as part of this lecture, isn't it? So, it is known as digital elevation model abbreviated as DEM, DEM. So, this module would be incomplete if I didn't introduce a digital elevation model. So, let us try to learn more about DEM, you know, what it is and how it is useful in hydrology and is it captured in the microwave region of the electromagnetic spectrum, you know, we will see all that shortly. So, technically a digital elevation model abbreviated as a DEM, I can define it as a digital representation of the continuous variation of relief, okay. Digital representation of continuous variation of relief. When I say relief, I am referring to height of the earth's surface. It is like I am taking a photograph of the earth's surface and I am capturing the undulations of the earth's surface and representing it as a raster. And on a digital environment, information about height can be represented either as a contour lines or in a grid format that is regular square grid or as a tin that stands for triangulated irregular network, okay. Now, when I say DEM in the form of a grid that is grid based DEM, they represent the elevation of the terrain at discrete points in a regular rectangular grid, okay. And the yield of usable datasets, we will see that shortly. The yield usable datasets that represent the elevation of a surface as a function of geographic location at regularly spaced horizontal grids, square grids. And mathematically speaking, grid based DEM is nothing but a matrix of numbers wherein the numbers represent, in some way the numbers represent the elevation above a particular datum, okay, reference surface. Now, let us look at tin. It is created by running an algorithm over a raster to capture the nodes required for tin. Even though several methods exist, typically textbooks, they define Delaunay triangulation method as the most preferred method for generating tin. And contours is something all of you I am sure must be familiar with. Contours are nothing but points representing equal heights. Here, I am going to redefine and say equal elevations with respect to a particular datum. And they can be created by either digitizing the benchmarks from maps or directly from the digital elevation models using the contour options available in any image processing software, RGIS, Erudas, etc. Okay. Now, let us see where to download a digital elevation model data from, where to download a DEM data from. So, shown here is the Earth Explorer USGS.gov website. And you can see that I have already logged in using my user ID. There are options here towards the left side of the screen, which help you to select the area of interest. So, let me select the area of interest as India, okay. There are different ways to select it. I am just going to use the option wherein I specify the coordinates. And if you see towards the left side, I am trying to look at the GTOPO 30 digital elevation model. You know, there are different elevation models with different names and from different sources of which I want to download the GTOPO 30 elevation model. And you can see the size of the file is also being specified, takes a couple of minutes to download. Now, GTOPO 30, it is a global digital elevation model. As the name suggests, it has a horizontal grid spacing of 30 arc seconds. And GTOPO 30 was derived from several raster and vector sources of topographic information. So, once you download the GTOPO 30 elevation model, I can open it in QGIS and this is what you will see on your screen, you know, where the color differences denote the elevation difference represented in a digital format as a raster. Say, I want to have a, you know, better clarity. Let me use the color rendering and there you go. Okay? Elevation represented digitally in the form of a raster is what is known as digital elevation model abbreviated as DEM. Okay? What you see here is the GTOPO 30 digital elevation model, which is freely accessible from USGS website. Right? See, as I mentioned before, there are different sources to download digital elevation models from. I showed you one such source. Here, I also have to mention about shuttle radar topography mission SRTM DEM. So, what you see here is the Krishna River basin wherein SRTM DEM has been downloaded and used to create a tin triangulated irregular network and this represents the Krishna River basin. So, you can see the zoomed out version of how a tin looks like, tin that has been created from a digital elevation model and this particular image has been created from SRTM DEM, one more source from where you get digital elevation model. So, the shuttle radar topographic mission, it was launched in the year 2000 and it consists of two radar antennas on board the space shuttle endeavour. And the topographic or the elevation information was created using a technique known as interferometric synthetic aperture radar, it's R. And the SRTM generates elevation data of resolution 1 arc second released for United States only and with a resolution of 3 arc second, that is 90 meters and this data of 3 arc second is made available globally and it can be accessed from the USGS website. And as I mentioned earlier, these files also they can be imported within a geographical information system that is GIS platform and they can be used for a variety of applications involving three dimensional visualization of the terrain. What you see here are the different operations that are possible in a digital elevation model. For example, the DEM applications in hydrology can be used for drainage pattern extraction. And digital elevation models may be used to extract drainage patterns of a basin that is required for flow routing in most of the distributed models. And the underlying algorithm which is used must be capable to identify the slope variations and possible direction of flow of water. For example, let me show you how hill shade can be created from a digital elevation model. So, what you see here is the same G Topo 30 digital elevation model and hill shade is a grayscale three dimensional representation of the earth's surface after considering the sands position. So, I am directly going to use the hill shade option from analysis raster in QGIS because I want to convert the G Topo 30 digital elevation model into a hill shade image, the one representing a hill shade. Again, hill shade is a grayscale three dimensional representation of the earth's surface after considering the sands position. This is just one operation that is performed on a digital elevation model. There are multiple options that are made available in the QGIS that you should explore. It gives you a different view of the terrain, isn't it? Especially you can view the western heart region as well as the Himalayan region, central India region gives a different aspect of the terrain. All right, moving forward. See, as a hydrologist, I am more interested say to derive the catchment characteristics that influence the generation of runoff. Say, I want to know the catchment area, the catchment width, the shape, the channel or the drainage network or I want to know the slope of the terrain that is when it rains in which direction is the water going to flow. Of course, the elevation is going to dictate that, isn't it? And digital elevation model proves to be an invaluable asset as it is capable of providing information regarding the relief, the slope and the flow direction. Now, shown in front of you is the 8 direction 4 point model. For reference, I would urge you to read this paper by Callaghan and Mark 1984. Assume you have a raster based digital elevation model. Now, each pixel of a digital elevation model is going to have 8 neighbors, isn't it? Towards the left, right, up, down and the four diagonal directions, isn't it? Which means the directions are marked here as 1 for east, 2 for southeast, 4 for south and so on, 8 for southwest, west, northwest, north and 128 denoting northeast. Say water is falling on the central cell. It is raining monsoon season and water is falling on the central cell. Here by cell, I am referring to a certain region on the earth's surface which is represented in the form of a raster in a digital elevation model. So, what happens? Water from each cell shall flow or drain into one of these neighboring cells depending upon the elevation, isn't it? Say you want to estimate the flow network as I mentioned earlier. If towards your right side, you see the direction in which each cell is going to drain or flow towards if water falls on each cell. For example, if I look at the topmost pixel or cell, it is going to drain in this particular direction as seen in the arrow. And if I look at the second cell, it is also going to drain in the direction denoted by 2. Similarly, each cell in a digital elevation model, we can estimate where it is going to flow towards provided water falls on it. Flow network is the data you get once flow direction grid has been obtained, once we estimate the flow direction grid, the flow network can be created by extending the lines of steepest descent within each cell as shown. And remember, I can also estimate the flow accumulation grid. Which means by using a flow direction grid, a flow accumulation function can be defined. So, here I have just replaced the arrow with the direction it represents that is 2 in this case, 2 and so on. And by using a flow direction grid, we can estimate a flow accumulation function. Now, the flow accumulation for a particular cell, it gives us the sum of numbers of all cells that flow to that particular cell. And flow accumulation helps in evaluating the number of upstream cells that are contributing to the lower order downstream cell. And the prerequisite for obtaining the flow accumulation thematic raster is the flow direction thematic layer which is created either with the D8 algorithm model or some other means. Now, stream delineation can be carried out further by specifying a threshold of number of cells contributing. So, I am not going into details of watershed delineation, but I am trying to introduce the concept of how a digital elevation model can be used. So, once again for the Krishna river basin towards the left side, you see a contour image that has been created from SRTM digital elevation model. And towards your right side, you see the hill shade image created from again SRTM digital elevation model. So, with this background, let me try to discuss few case studies which were conducted in house using open source digital elevation model. You see, there are many sources from where one can obtain this data. We already saw one source that is USGS website. There are also many commercial providers of digital elevation models wherein the vertical accuracy of elevation data is very high. And an accurate digital elevation model plays a crucial role in prediction skill of these models by representing the topography, isn't it? So, using a digital elevation model, all we are trying to do is we are trying to realistically represent the earth's surface terrain. To date, global assessment of digital elevation model accuracy was largely evaluated by the validation teams of respective missions. Now, there are studies which indicated that tandem X mission provides global dim with a relative vertical accuracy of 2 meters for a slope of 20 percent. There are many research works wherein digital elevation models were taken from open source and then they were validated using ground survey data. Now, what you see towards your left side is tandem X dem of western heart region and the elevation is shown in meters in different colors. And towards your right side, you see slope map and aspect map for the same region. Now, every digital elevation model, the data product, they will have a particular vertical absolute height error, isn't it? Vertical accuracy will be different for different products. Now, so, what we did is we tried to use different digital elevation models from different sources like Aster, SRTM, Cartosat, ISAT and tandem X. So, the table summarizes the spatial resolution of each data product and the horizontal datum used along with the vertical datum used. Remember, when you have a digital elevation model which has been captured by different missions, when you compare them, it is always important to ensure that the vertical datum is same. So, the study that I am talking about, we tried to explore the accuracy of different open source digital elevation models like SRTM, Aster, Cartosat 1 along with tandem X for performance evaluation over the complex terrains of western Ghats in India. I won't go into the details of the study, but we just wanted to analyze which open source digital elevation model gave us the better representation of earth surface on dilations. So, towards your right side, you see a diagram on the X axis, elevation bins are shown in meters and on the Y axis, the root mean square error is shown in meters and the different colors denote the digital elevation models procured from different sources. So, with the availability of different global digital elevation models, it is imperative to conduct validation studies over rugged terrains like the western Ghats to assess their accuracy and that is what we did as part of this study. This was solely for the western Ghats region. So, similar to this study, there are many studies which are being conducted by trying to validate the existing digital elevation models with a data of a better vertical accuracy such as the LiDAR based digital elevation model. Now, with this background, we have to discuss a little bit as well before I close the lecture, we have to discuss a little bit about the hydrological models. As a hydrologist, I often try to use a model to realistically represent the real world system. What you see here are the different processes that we need to consider when we are talking about a hydrological model. For example, the model should have information about precipitation, transpiration, evapotranspiration. It should be able to simulate runoff and we should be able to route it to say the downstream. So, simply put a model is a simple representation of a real world system and model parameters and complexity are so defined such that the model best represents reality as close as possible and models in hydrology, they are used for predicting the system behavior and in understanding various hydrological processes like the ones which are shown in the screen. For example, a runoff model is a combination of a number of equations which help in quantifying runoff as a function of various parameters and digital elevation models, what you have just heard, it is used as an invaluable input to a hydrological model. Remember, the computer thinks in terms of grids and cells. So, the computer is going to look at the model in the form that you see here. The area of interest divided into a number of cells and one gets to select the size and number of cells as a user. So, that ultimately, if you are trying to understand about water cycle for a computer, it is turned into a series of model cells. We shall be discussing in detail about hydrological models as one of our tutorials. But for now, I want you to understand that digital elevation models are an indispensable part in hydrological modeling. It is required as one of the inputs among others like precipitation forcing. And shown here is a simple schematic on how the various geostationary parameters and the near surface atmospheric forcing and precipitation forcing all are used as inputs to the variable infiltration capacity model to generate runoff. Just a simple schematic. Once again, digital elevation models, they are raster grids of the earth's surface referenced to a datum. Datum is a reference surface from which all the vertical elevations are being measured. And remember, there are open source as well as commercial sources that provide a digital elevation model. I will not get into details of that. But for now, let me leave you with an open question. Towards your left side, what you see is the grace groundwater anomaly trend in centimeter per year where each color gives you the grace that is gray satellite data has been used to create a groundwater anomaly trend which is represented in different colors shown here is the trend in centimeter per year. Now, this was part of a study which was conducted using in situ groundwater data provided by the Central Groundwater Board. And the period of data used for this study was from 1996 to 2018. So, what we did is we tried to use the grace data and the in situ groundwater data provided by the Central Groundwater Board along with SRTM digital elevation data of 30 meter resolution from USGS Explorer. These are the locations of the wells that you see. I am going to keep the question open or I am going to leave the title as an open question for you to explore. Does land subsidence occur due to groundwater pumping? So, please feel free to go through the related articles published by the research group to get an in-depth idea about the same. I am just trying to bring together all the data products that you have been learning as part of this course together to frame a research hypothesis. Does land subsidence occur due to groundwater pumping? I am going to leave it there. So, throughout this lecture we could understand more about differential synthetic aperture radar interferometry. We also learned about what are digital elevation models, what are its applications in hydrology and then I have left you with an open question. So, let me hope that you could understand this part of the lecture and I will meet you in the next class. Thank you.