 All right, so today we're going to walk through the process or introduce the process of creating a contour map and a contour map is Basically helps show you information by using a set of things called iso lines where iso Basically means the same what we're going to do is we're going to connect points That represent things that have the same value in space For example, if I have a point here that has a value of five units I will connect it to other points that have similar values of five units creating a line that represents a Position in space where everything has the same sort of value This can be useful When you have what we are going to call sparse spatial data In other words, you have information that's associated with some location in space Okay, in this case the location might be on a map. It might be in two-dimensional space It might be in three-dimensional space, but in the case of contour plots, we're going to be working in two-dimensional space and With only a few of the locations that we may be interested in for example that we're able to sample the data from some places in space But we may be interested in having values for more locations than we've actually sampled here's an example Here I have a map and this map is representing a number of Watersheds where water might have might be falling and flowing and We have located somewhere within this map a number of locations where we have sampled the rainfall in the area So these numbers these values the 6.24 the 4.55 Etc. Are all measurements of amounts of rainfall usually depths of rainfall is unclear what units are being used here But there are different depths and notice we might be curious to figure out what depth of rainfall to associate with For example an entire watershed here in watershed D For example, we might go ahead and assume that all the rainfall there was 4.67 units However, if we're looking at something like rain Like watershed C Over here, there is a value measured inside the watershed of 4.39 units But on the other end of the watershed, there's a value here of 5.43 and it calls into question whether or not using this value measured inside is the only value that should be considered When thinking about how much rainfall is actually being measured inside watershed C Similarly this value of 6.01 may not be the only one influencing Values that we might consider when thinking about the rainfall would side of watershed B So if we're interested in finding rainfall values elsewhere in places that we have not measured It will be useful for us to interpolate Where poll the fancy word for the points of our data and enter meaning in between We're going to find out or try to determine values for information. That's between Our poles or our different points Now I've simplified this a little bit. I'm going to go ahead and use I've gone ahead and copied all the values here of my Of my watershed map I'm going to go ahead and create all the locations here So it might be a little bit easier to see what I'm doing on the page without the additional lines But if you do have something you can also go ahead and Follow this process directly on the map if you want to So let's discuss a little bit about what we need to do in a case like this First of all the idea with a contour plot is to create these lines of similar values So first we're going to have to determine what lines of similar values. Do we want to use in? Simple versions of contour plots. You simply could you have a bunch of points that are all the same here None of our values are exactly the same We could make a very simplified contour plot by looking at the values four four four and connecting all the fours Or connecting the fives or connecting the sixes, but that's a little complex It's not exactly clear which ones you should connect and how and that's actually not really the way we're going to build our ISO lines So what we're going to do here the first thing we want to do is establish a range for our data Where the range is the maximum value of the data minus the minimum value of the data So if I peruse this Map I look carefully and I try to locate the maximum value Which appears to be the highest value of six point two four here in the lower right On the right side and our smallest value our minimum value is this value three point eight six So I'm going to record those two values the max minus the min Take my max value and subtract the minimum value to find my range and those values are again the six point two four And again, there are no units supplied if there were I should keep track of what they are six point two four units minus Three point eight six units gives us a value of two point three eight as a range for my data Now my next Part or the next piece is I need to choose how many contours or how many lines Choose number of ISO lines How many lines do I want to use to divide up my information? Usually a good number is somewhere between five or ten a little bit closer to the five Depending on the size of the map and how much data you actually have but in this particular case We're going to go ahead and use what we'll assume five It looks like five divides pretty closely into two point three eight once we choose our number of iso lines If we choose five then we divide our range by iso lines range divided by the number is going to be Approximately point five now I do not have to be exact here in fact I'm going to choose not to be exact because I'm going to choose numbers that are convenient. I'm going to use a Intervals this is actually called an interval size of Point five which means I'm going to count In increments of half a unit once I've chosen the interval size Then I can choose a convenient set of intervals in this case I want to find intervals that range from my 3.86 to my 6.24 and Count up in these values of point five Well, I'm going to choose numbers that work nicely. I'm going to start at four four point five five point five and six and you'll notice by choosing those I have now chosen five iso line values That are going to fit within my range from three point eight six to six point two four so there are my Iso line values and I'm going to end up having six intervals I have five values, but I'm going to have six intervals First of all, there's going to be the interval from three point eight six up to four There's going to be the interval from four to four point five up to five up to five and a half Up to six one two three four five intervals, and then we have one more interval from six up to six point two four So we have six intervals that we're going to have For our data and for our contour plot here So now that I've established that interval size we'll keep that in mind as I return to my map There's my set of intervals as I'm looking at my back And now the next step is I'm going to go ahead and choose a point to start from and I often like to start from one of the points It's near the edge near the sides of the map to start to work from and Or usually one of the lower values That are or the higher values You're the sides of the map in this case. I'm going to go ahead and start with the three point eight six Here's this value three point eight six and Once I've chosen that value. I'm going to go ahead and choose a neighbor figure out what its nearest neighbors are and lightly sketch Direct lines between the value and its nearest neighbor Okay, I'm going to select a couple of neighbors here All right now that I've done so I'm going to start with this closest neighbor here this five point oh six and I'm going to look and see which intervals fit between the 3.86 which happens to be our lowest minimum value and the five point oh six and I recognize that those intervals are four and four and a half and five all three of those intervals fit within this range of three point eight six To five point oh six so there's going to be a range here where I'm going to draw in a line for four a line for 4.5 and a line for five Somewhere along this line now if I want to be particularly careful and particularly exact I want to measure this distance here to figure out exactly how far to go. Let me actually Walk through that process here. I know that my range of values goes from five point oh six To three point eight six And if I subtract those two I get 1.2 units the value Extends over 1.2 units I'm similarly if I want to be careful here. I can measure the distance here And if I measure this distance, I don't know if you can see that but that's a distance of three centimeters Okay, well now I would like a relationship Between how much distance there is for one unit? so I understand here there are our Ratio or our slope is three centimeters per 1.2 units and if I do the math there three divided by 1.2 I Get a value of 2.5 centimeters per unit So if I know there's 2.5 centimeters per unit I can go back to this grid now and I need to know I would like to know where to put the four the five The four the four and a half and the five and we know that there's two and a half centimeters between each one Well, I'm going to work backwards from the five I know the five is very close to this five point oh six So I'm going to go ahead and mark that five right next to the five point oh six And I'm going to move back that two and a half centimeters Two and a half centimeters ends up being right there There is my position for Well, that's one unit away if this is my five This is one unit away that must be four and then halfway between those two at about 1.25 centimeters is the value four point five Notice that the four the four point five and the five are equally spaced here and There's a little bit of space there representing the difference between the three point eight six and the four and a little bit of even less space They're between the five and the five point oh six Now if I want to be very careful I could do these calculations and be carefully do these calculations for every one of these But that's going to probably be a lot more detailed than I actually need To do my math What I'm going to try to do for each of these other lines is do a similar thing But notice the spacing is going to be different here I have a range that goes from three point eight six to four point five five That's going to include four and four and a half, but it's not going to be anywhere near the same Distance as between four and four and a half here. It's going to be much further Spread out so I'm going to have to estimate four and four and a half in a different fashion there Just to get a sense for how much it's that's going to be I look and I see that this is about 10 centimeters and 10 centimeters is going to represent four point five five minus three point eight six, which is about point six nine Units so if point six nine units is equal to ten centimeters Five centimeters is going to be about point three five units. So about halfway there is going to be 3.86 was point three five is four point one or four point Seven or so now I don't need the value four point two seven I'm just trying to get a rough estimate for where things are going to be because what I'd like to put in here is 4.5 and Four well, let's see here four point two seven and four point five five Four point five is going to be very close to the end over here I'm going to estimate it to be right about there and between three point eight six and four point two seven we're going to have four a Little closer to the three point eight six about a third of the way there So I'm going to estimate it as being about a third of the way there again I'm not being extremely careful But the idea here is to sort of get a spacing that's equivalent in length. So if I put this at four and This at four point five notice of roughly half between is a real about where four point two five is I'm not being Extremely careful if I wanted to be extremely careful I could again measure this out and then figure out how much distance would go between 3.86 and four figure out how much that is in centimeters and continue Once I've sort of created a value of four here and a value of four point five there I Can start creating a little bit of an iso line. I can start making some connections. I see there's a four here and a four there I'm going to choose a different color here and with my red I'm going to connect the two values of four and I'm going to connect the two values of 4.5 These are the beginnings of my iso lines Now I can do a similar process With each Part here. I now move from this three point eight six to the six point oh one I Recognize the numbers that have to go in there. Let's see here between the three point eight six and the six point oh one There's going to be one two three four five all five of these Need to fit in which is the little bit On either end so If I need to fit five and change intervals into eight This is approximately a little more than eight centimeters eight divided by five is going to be about 1.6 centimeters per interval So I'm going to start down here at six go about 1.6 centimeters go another 1.6 centimeters go another One point six and that's not one point six centimeters go another one point six centimeters and then my final 1.6 centimeters again sort of a rough estimation there Notice this is a little bit. Let's see here. Does this take into account all of them? There's five if six is right up here at six point oh one There's five point five. There's five. There's four point five and there's four That's a little too far away from the three point eight six. Maybe I need to shift it down to make it a little bit bigger Okay, so I'm going to make these there six again make it slightly wider for each one And now I can go ahead and label those four four point five five five point five And then there's the six down there and perhaps because the six was so close I should divide them Again, we're trying to make even spacing between each of these With a little less space on either side and I'm doing my best to estimate in this case Although again, you can measure them more carefully if you would like and now I can connect the four And the four point five And I can even connect here the five between these two particular points, but I don't know how much further it goes I again repeat that process out here on the side Over here four point six six and three point eight six Well, let me make some estimates there three point eight and four point six Let me go about halfway Which I would recognize as being four point two six is about halfway About halfway between each of those is going to be Four point five Four point oh six and halfway here is four point four six and notice I'm looking for four point five which will place right there and Four which will place over there So again using a couple of different methods to approximate where those pieces are But now I have marks for four and four point five and Again connecting adjacent points that have similar values So now I've sort of established Values between three point eight six and It's various neighbors and notice there's an assumption that I was making in this process And this is kind of a key assumption the assumption is that there is a linear relationship a linear range Along any one of these lines that if I have a value such as my three point eight six and I'm moving through space To a place where the value is six point oh one as I move through space my assumption is That the value rises linearly That is not a perfect assumption for all we know it could dip down or it could rise up or there could be some Strange bumps in there, but because we have no information about what's occurring in the middle our Simplest guess our simplest interpolation is to assume it rises evenly over The segment and those are the assumptions that I'm making when I'm making each of these lines to interpolate Somewhere in between So we've begun our contour plots here What do we do next? Well now we can repeat the process By choosing another point this time. I'll choose an interior point and Choosing a few more Lines connecting it that point with its neighbors. I'll again sketch these lines in lightly In this case it has some neighbors that are significantly further away And once again, we're going to try to make estimates for values between each of these points We'll notice here in the case of four point five five to five point oh six That that range only contains the five point oh I mean that only contains the five and it's going to be relatively close here somewhere along that line So let's see here If we measure this distance here, we'll see that that's about nine and a half centimeters Which is equivalent to about half a unit Well if I want Point oh five Units, which is about the distance between five point oh six Okay, that's going to be a tenth of that nine point five centimeters or roughly Point nine five centimeters are about a centimeter. So my location of The line for five is About a centimeter away and now since I have a five on this line and a five on this line I can connect that contour I'm going to similarly try to make estimations on each of the other lines notice this one is five to six We have the number five in five point oh six to six here So we're going to have basically the values five and a half and six Well that five and a half and six Going up to six point two four Let's see here five and a half if we go halfway between five point oh six and six point two four It's about twelve. So halfway is going to be 5.65 is the point that's halfway there and we're looking for five point five Let me divide that again if I divide this in half Halfway between five point six five and five point oh six is going to be about five point three six and So we can estimate in that particular space there going from three to six that this is roughly three four five We're going to call this position here five point five again. You can be more accurate if you would like I'm trying to get an estimate for where that position is There's five point five And then we had this five point six five let's see if we can figure out where six is Well, if I'm careful I can sort of see that this is a distance of about five centimeters if I go another five centimeters I'm going to get to the point that represents six now notice Oops, this is five point oh six. There's another very close spot right about here representing five and now my Contour curves there, but there isn't a value of five point five here anywhere to connect yet Where is that value going to come from? Well that interpolation is going to come where between this four point five five and the six point two Now I'm going to even be a little less accurate We're going to say it's here four point five five and six means I have a five of five point five and a six to fit in there We're going to see the five is about a full distance a full distance a full distance and maybe half a distance So I'm going to see if I can do that Full distance full distance full distance. That's not quite a half. Maybe I can extend that out a little bit further a Full distance a full distance a full distance. No, that's not quite that So let me go a little bit shorter. We'll call that a full distance a Full distance a full distance and a half a distance So now I'm beginning to eyeball it a little bit more than being accurate if I'm using a computer Or if I care I can measure each of those but now I'm going to label this as being the location of five Which is about half a unit from there. I'm going to label. This is five point five Which is the same half a unit and label this here is six and notice the distance between six and six point two four is less than those other lengths by about half and Now I can make a couple more connections here interestingly We have some relationships. We have to start thinking about in this case Here's our connections between six and six and our connection between five point five and five point five But now we made an interesting sort of connection in turn here And I actually made a mistake Here I said that this was a value five, but if I look carefully this goes from five point oh six up to Six point two four this value here is not a five that value here well, it isn't Five five is less than five point oh six so the five actually doesn't count here because this is the value that's going up so Let me fix that mistake Say oh the curve doesn't go here, which is helpful Because somehow we need to connect this value over here this five. That's over here. Let me go ahead and Interpretate to connect Notice we're beginning to get these segments these intervals across our area I'm going to sketch just a couple more intervals So we get a better sense for what it looks like Notice something very interesting has happened here. I have a range here that goes from five point oh six to five point four three We have none of our intervals actually passing through any of that range So even though I can estimate where five point one two three and four are There is no five point five or even a five for this actually to attach to so where do these two intervals go to? Well, both of those intervals must actually exit out Here in this case six point two to five point four we have Five point eight in the middle. This is about six point oh this would be about five point six so about five point five right about here and so now my contour at this side turns and Turns to connect there Whereas on this side over here The number five that we actually have attached here We do have a value five here, but we don't have anything to connect it to quite yet Neither do we have for the value four point five Here at the moment. I'll continue And now I have some other connections to make 5.5 connecting to five point five connecting to five point five five point oh connecting to five point oh four point five connecting to four point five and You can see how the contours begin to fill out now that I've established some more values for our various contours I can start connecting them. Here's five point five connected to five point five Here's five point five connected to five point five Here's our five connected to our five right here, which can connect to the five over there This five point five needs to find somewhere to sort of escape or get out so it connects to the five over here our six Connects to the six which connects to the six which connects to the six in each of these locations There's a there's a value that's surrounding the six point oh one and you'll notice We actually get sort of a circle here usually when you have something that encapsulates or that goes are all the way around That represents a high point or a peak. We have a high point here around this particular point Similar when I try to connect these five point fives We end up finishing the circle there. You can connect the points to five here and similarly five point four This is roughly five here between four point three. This is four point five All right, and with that we've completed at least the interpolation parts or the inside parts Well, actually I could add one additional piece down here between the five point three seven and the five point four three Notice we actually don't have a piece that can escape there So we must connect these two here So now I've completed the interpolation the inside parts of my contour map notice if We wanted to know information outside the map we would have to do some Extrapolation extra going outside of the points extrapolation Unfortunately since we don't have any information out that about the area outside the points the best we can do is make an estimate by Roughly sketching usually with dotted lines where we think The curves might be taking us however because we have no idea about the values outside of there We don't necessarily have any idea about What might happen it could curve back Down here at the bottom we could have it curving back around Okay, but again this becomes a guess and the further we get from the outside Outside of our points the less valuable this information becomes Let's interpret what this information means Each of these lines Represents a value and if we're smart we'll go ahead and label that value. This is the value for This has the value 4.5. This is our five value. This is our So there's our value five. This is also our value five. This is five point five This is six. This is six and this is five point five and this comes down to four point five So each of those lines has a value that anywhere on that line That's the value that's associated with it. So if this is inches of rainfall I could say reason I I could make a reasonable estimate that the amount of rainfall at this point here Which we didn't measure would actually have a value of five The intervals themselves also have a meaning basically any space that's represented between any two intervals For actualists for example the space I've shaded in Basically states that the rainfall there ranged from four point five to five and if you would like to interpolate any space in there you can do so by Estimating how far you are away from one line or the other if you're halfway in between that might represent a value of about 4.75 if you're a little closer to four point five you could estimate that that value is something around four point six Going this direction four point eight four point nine, etc