Welcome dear students. We are in lecture 12 now of module 3. So, today we shall discussabout the Spatial Interpolation methods which is used in case of disaggregated data, when youhave distributed data and it is not a continuous data. So, in that case, we use different types ofmethods to interpolate the intermediate values. So, we shall see the different methods bywhich we can do this spatial interpolation.
So, the topics or concept that we are going to cover today is we are going to introduce whatis spatial interpolation; we are going to see what is spatial interpolation, we shall see the global
approaches to spatial interpolation. We shall also see what are the local approaches and howthey are different from the global approaches. We shall look into the different spatialinterpolation methods.
These methods includes Trend Surface, Regression, Thiessen Polygons, Triangulated IrregularNetworks that is the vector representation of the 3D surface. We shall look into KernelDensity Estimation and Inverse Distance Weighing. We shall also finally, look into the ThinPlates Spline in this particular lecture.
So, there is a topic which is Kriging which is I mean for constraint of time we may not be ableto cover in this particular lecture. So, we shall continue with this same topic and we shall lookinto the process of Kriging in the subsequent lecture.
So, let us see what is spatial interpolation, it what does it mean. So, if we have a regular gridand in that particular grid, we know only some few values like we had talked about raster datawhich is an array of numbers. So, in that particular array, we may know only few data pointsor we may have a point coverage a GIS point coverage for which we may not we may knowthe data, we may have the data.
For example, you can talk about say weather observatories which might be having saytemperature data values. So, we would want to create a grid, continuous grid of temperaturedata values based on few sets of observation distributed in the space. So, I mean it would be aregular or a irregular distribution of these data points in space. So, there are different methodsby which we can do it.
First, we can use a deterministic process in which the outcomes are determined specificallyand we can establish a relationship among the events and the states and there is no randomvariation in the outcome. So, whether you do the analysis or I do the interpolation, the resultswill always be the same or if you do this interpolation across different iterations, you given thesame inputs your outcome would be the same. So, these processes are the deterministicprocesses.
Now, the second one is the stochastic process which has a random probability distributionassociated with this process or the pattern may be analysed statistically, but it is not predictedprecisely. So, it would be an ensemble of different outputs. So, this is what we call it asstochastic process. So, the outcome of these processes may vary depending on the iteration. Imean if you do a second, I mean iteration based on the same sample points, your output mayslightly vary ok. So, I mean it is not precisely I mean exactly determined as you would behaving, I mean determining these values in the to the deterministic processes.
Now, there are different ways of doing the spatial interpolation. So, first approach is theGlobal approach. So, it uses the entire data at one go simultaneously. So, that is a globalapproach. So, the first approach in the global approach is a deterministic surface which isknown as a trend surface. So, if we have scatter data points, I mean irregularly distributed data
points, we can create a trend circle surface based on deterministic methods using the globalapproaches.
We can also use the stochastic process to I mean project the values of the points in a using aglobal approach. So, these type of approaches would use the regression functions in the forthe stochastic approach. Now, talking about the local approaches we would be using only asubset of the data or it could be referred to as a moving window that I mean we have a smallwindow and in that particular window, whatever data points are selected that is interpolated,those values are interpolated within that particular window and then, we shift the window andtake I mean do it iteratively.
So, it also has a deterministic; there a deterministic method for local approaches as well forspatial interpolation. So, these methods deterministic methods are Thiessen polygons, Densityestimation, Inversed distance weighted methods and Splining. Now, there are also stochasticmethods. Stochastic methods, we have said that it is a probabilistic method which may create asimilar values, but not precise values.
So, this in the stochastic method for local approach, we have one method which is known asKriging. Now, depending on your requirements or the need of your study we would be usingeither of these methods. We had talked about the global methods, we had talked about thelocal methods. In this, we have talked about the deterministic methods and we have alsotalked about the stochastic methods. So, we shall see these processes sequentially.
So, the commonly used approaches that are I mean use for spatial interpolation is trendsurface. We had already talked about it, we can use Thiessen polygon that is used in the localcontext. Triangulated Irregular Networks that is vector representation of 3D surface. We cando a Kernel Density Estimation, we can use the process of Inverse Distance Weighing, we canuse Thin Plate Splining and we can use Ordinary Kriging method.
So, let us see what are the different methods of spatial interpolation one by one. So, first let uslook at the trend surface. Now, this is a surface which has the gravity data for different I meandifferent data points which is known. So, it has been interpolated using trend surface.
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So, when we use a first order trend, you can see you can find out the slope of this particular Imean terrain or I mean the gravity values across this particular terrain. Now, we can also findout the residual, I mean that that is the difference between the initial actual values the and theinterpolated values, the first order interpolated values. So, we can find out the difference andthis gives us the residual value.
We have seen that this trend surface is a deterministic process. I mean calculating the trendsurface is a deterministic process. So, we can use multiple regression; wherein, we can have adependent variable that is the variable of interest. We said as an example we could be workingout or interpolating the values of temperatures which has been recorded by different datapoints like weather stations in a given terrain as a regular network, irregular network. We canalso work out other parameters like humidity precipitation etcetera and we would be havingthe independent variables which are the data coordinates and or sum function of the datacoordinate.
Now, this method is an in exact interpolation method; this is not a exact method ofinterpolation and it would approximate the points using a polynomial equation. I mean thesynthesised points, I mean the data values of unknown points would be the result ofimplementing a polynomial equation. So, this is used, this equation are the polynomial
equation or interpolator, it would be used estimate the values at the other points for which thevalues are not known and when I mean this equation polynomial equation is of first order thenthe equation is linear.
So, this is an example of a linear equation; wherein, Z which is the variable of interest, it couldbe a it could be represented as a first order equation using this equation that is b 0 subscript 0plus b subscript 0 x plus b subscript 2 and into y. So, in this case your the attribute value Z isthe function of x and y coordinates. So, we had earlier seen I mean we had earlier talked aboutthat this the dependent variable is a function of the data coordinates ok. So, and in this casethe b coefficients are the estimates from the unknown data points.
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So, what we do is say suppose we have 5 data points in this particular example and we saysuppose, we have 5 data points and the 6th one is the 0.0. So, in this case what we do is for
the 0.1, we have x and y values that is X 1 and Y 1. So, similarly for 0.2, 0.3, 0.4 and 0.5, wehave these data values of I mean X 1 Y 1, X 2 Y 2. So, these are the different data values foryour X 1 x and y for the corresponding points for which the values are known; these valuesare known.
So, you can get the latitude and longitude coordinates of your weather station, you can plot ina GIS. So, you will know the latitude and the longitude or the projected coordinates as X 1and Y 1 and you would have data points 0 for which you do not know the X and the Y. So,this is a unknown and your X and sorry X and Y is known to you, but the value is unknown.So, in this case the value is unknown. So, we work with the known values of your x and 0.1,0.2, 0.3, 0.4 and 0.5 and make an equation out of it.
So, if we see this, we use a least square method to solve the coefficients b 0, b 1 and b 2 in theequation that we had talked about. So, that is the first order equation. So, in the first step,what we do is we setup 3 normal equation. So, on the left hand side you have sigma z which isthe function of b 0 n plus b1 into sigma y plus b2 into sigma. So, b1 into sigma x plus b 2 intosigma y. Now, the second equation is some is product of x and z and in the third equation, wehave the product of y and the z and we have the this two corresponding equation.
So, we set up this first as a first step, we setup this three normal equations, then we can writeit in the form of a matrix. These three equations can be written as a matrix that you see outhere. So, in this part you can see the values, I mean the I mean known’s that is n that isnumber of points, sigma x, sigma y and then, you have again sigma x, sigma x square comesover here and next element is sigma x, y and similarly these elements come as the third row inthis particular matrix.
So, we can write the coefficients b 0, b,1 and b,2 as the next matrix and then, on the righthand side, we have the I mean sigma x, sigma sigma z, sigma x z and sigma y z as threeelements. So, we can solve these equation and calculate the value of these coefficients b 0, b 1and b 2. So, the deviation or the residual would be there between the observed value and theestimated value.
So, I mean if you have more number of points, you can see that the I mean derived surface orthe I mean the surface that you simulate would be having a deviation or you can work out theresidual. I mean in the last slide, we had seen how we can work out the residual between twodifferent surfaces; one is the known surface and one is the projected surface. So, we can workout the or compute the residual for the two surfaces at each of the known points and we cando a measure of goodness of it and in this model can be tested.
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Now, talking about the trend surface, the distribution of many natural phenomena, it iscomplex, it is more complex than the first order inclined surface that we had seen. So, in thiscase you can see the first one is the actually the natural surface and when we do first orderinterpolation, you get a plain surface. Second order you get some semblance of curvature;third second order polynomial you see a saggy kind of thing and from third order polynomial, Imean it creates semblance of this particular surface.
So, I mean this natural phenomena is generally more complex than the first order or secondorder models. So, what we can do is we can do a higher order surface model fit and it wouldbe able to I mean model the complex surfaces such as if you have hills which has undulatingsurfaces in a given area and it can be this I mean complex terrains, these can be modelled usingthird order model or higher models.
So, or cubic trend surface is based on a equation which looks like this. It is the third orderpolynomial equation. Now, your third order polynomial equation, here you can see that thereare ten coefficients which are unknown. So, in our earlier equation, we had seen we had fewcoefficients; but as you increase the order of polynomial, the number of unknowns would beincreasing.
So, what it means that the number of observations also has to be increased so that I mean thissolution to this system of equation becomes tractable. In GIS package, generally I mean thereare computational limitations, but they would allow up to 12th-order trend fitting for differenttypes of surface model.
Then, there are different types of trend surface analysis the way we do it. So, first one is thelogistic trend; wherein, the known points will be having only having binary values and it
produces a probability surface. The next one it is a local polynomial interpolation which uses asample of known points to estimate your unknown value in the given set and it can thesevalues can be converted into a irregular network, a triangulated irregular network that isknown as tin and which is the vector data model for representing a three-dimensional surface.
And this polynomial equation which I mean can be used to take the vertices of the triangulatedirregular network as points and it can be extrapolated into a digital elevation model. So, weuse the local polynomial interpolation to derive the digital elevation model from a your tinssurface, Triangulated Irregular network surface
Now, we were talking about the triangulated irregular network which is basically a vectorsurface and we can also have a dm which is a raster or a GRID. So, when we are talking aboutthe differences between this two surfaces, when it comes to storing your 3D data points, for
the grids, it is easy to store and operate with raster database and we can integrated with theraster database model.
So, it is more smoother because you have regular array of data points. So, it would be moresmoother and it would have a more natural appearance compared to tin surface. It is notpossible to have varying grid fixes to represent areas; wherein, we have complex relief wherethe relief is very complex where there is a drop in the edge. So, in those cases or there is aprojecting surface wherein, I mean it would be like a cantilever surface a hill which projectsoutwards. So, in those cases it would be difficult to represent those kind of areas using thegrid surfaces.
Now, the triangulated irregular networks represents a surface which is non overlappingtriangles and which is continuous in nature. So, the each of these surfaces are basically plains.So, this I mean each of the surfaces would be triangular plains and we can define or we candescribe the surface at different levels of your spatial resolution and it is also an efficient wayof storing data point, I mean 3D data points.
So, how this tins are created grids we can assume that these are matrix an array of numbers?So, which is regular grid; but this triangulated irregular networks are triangles which are notuniform or which are not I mean have which do not have the same density across a space whenyou create it. So, it is created using a process which is known as Delaunay triangulation.
Now, what is Delaunay triangulation? Delaunay triangulation is either created from contoursor data points. So, the vertices of the contour lines, they are used to mass produce this pointswhich are then used for triangulation. So, you may have two different levels of consecutivecontours. So, say first is the 0 level contours, 0 metre contours; next contours could be a 10meter contours. So, suppose we have 2 contours. So, first one is a level 0 contours, this is a 0meter contours and this is at 10 meters.
So, these are two contours data that we have. So, what we do is this contours will havevertices and these two contours will have vertices. So, these points are joined together tocreate triangulated facets representing the surface. So, this they are used as mass points thevertices of contours for the triangulation. Now, we use proximal method, which this methodbasically uses 3 points, 3 nodes of a triangle and fits a circle through this 3 points which are
almost I mean these triangles are so derived that they are equiangular in nature and will itshould not contain other nodes.
So, any point on the surface is as close as possible to the given node. Now, the triangulation itis a independent process and the points are processed like in this case you can see this 3 datapoints, they are lying on this particular triangle. So, this 3 data points are lying on this on thisparticular triangle. So, likewise we generate the triangles, I mean which are almost equiangularbased on these data points.
So, these triangles triangulated networks are stored in two ways. It can be stored either byTriangle by Triangle method. It is it I mean provides a better solution for storing the attributesthat is you can also include other parameters such as slope or aspect or you can save thistriangle triangulated irregular network as Points and Their Networks. So, when you save thistriangulated points as points and networks, it is useful when you want to generate contoursand it uses less space.
So, but the limitation in this case is it does not it cannot store slope or aspect data along withthis point values. So, it has to be stored or calculated separately. So, which is the advantagewhile we are using a triangle by triangle method.
Now, we make Voronoi polygons based upon this Thiessen polygons I mean Thiessenpolygons or Voronoi polygons based on this triangles that we had talked about. So, what wedo is we create the midpoints of these triangles which we have I mean calculated using yourDelaunay triangulation process, wherein we try to create equiangular triangles using 3 pointsthat I mean are lying or located on a circle.
So, in this case what we do is we take the midpoints of these lines and draw perpendicularbisectors. So, you see that these points will intersect each other and will create a polygon. So,your polygons will be created based on these Voronoi’s I mean Voronoi polygons based onthe Delaunay triangles.
Next, we come down to the density estimation. So, we measure the density using a sample ofpoints and we do the point pattern analysis, it could be the points could be random, they couldbe clustered or they could be dispersed. So, we have different methods to do the densityestimation. The first one is the Simple Density Estimation. So, its basically a counting method.
So, it uses a probability function depending on how dense, I mean what is the density of theestimate and we place a raster over a point distribution and we tabulate the points, I man wecalculate the number of points that fall within itself. We then sum up the point values and wecalculate the density of the cell by dividing the total point value by cell size. So, this is the firstmethod which wherein, we use a simple density estimation process to find out the densityestimation for spatial interpolation.
Now, the next one is the Kernel Density Estimation process, which would associate each pointwith a kernel function. So, this is expressed as a bivariate probability density function. Now, itgenerally would produce a smoother surface in comparison to the simple density estimationand we can have several types of application of this kernel density estimation, like we can do adensity estimation of areas which are prone to traffic accidents or we can I mean estimate orapply it to the I mean working out on urban morphological parameters.
The next method is the Inverse Distance Weighted Interpolation method which is adeterministic method for multivariate interpretation. Now, this are the principle of IDW is toestimate value of point and it is the principal is that this estimated value of a given point isinfluenced by the nearby points, I mean more influenced by the nearby points than those whichare line further away.
So, is it calculates I mean the IDW interpolation is calculated using the this particular equationin which we have a ratio, ratio is the product of the estimated value, the known value at pointI and summation of that into 1 upon kth power of distance between the point I and the pointO, that is the point O is the unknown where we are estimating the value of z.
So, and in this case your s is the number of points which are used in the estimation and k is thespecified power. So, this power k, it would control the degree of influence of the local points.Now, if this value of k is 1, it becomes a linear interpolation that is there is a constant rate ofchange in the values between the points. So, the interpolation is linear interpolation. Now, ifthis k assume the value of 2, then the rate of change in values are higher near a known pointand it levels of when it is these values are away from it, these points are away from it.
So, the predicted values of IDW interpolation are within the range of maximum and minimumvalues of the known points. So, the predicted values won’t go beyond or below the minimumand the maximum values of the points that you are using for interpolation. So, I mean we itcreates a small enclosed isolines, lines of similar values. So, that is typical of Inverse distanceweighted Kriging.
Now, talking about thin plate spline; it is similar to splining. You would have seen yoursoftware’s, CAD software’s wherein, you have data points, you create polylines; wherein, youcreate polylines using points and then, you fit a spline function on that which smoothens thatparticular line. So, in this case, it is a similar concept, but we are going to spline a surface. So,I mean these surface would go through or pass through the control point and has the leastpossible change in slope at all points.
Now, it has a I mean similar analogy to that of a thin sheet metal being bent over different datapoints say suppose you have nails of different heights; nailed into a plywood and then you puta thin sheet metal plate over this and you try to fit a kind of a surface over this differentheights of the nails. So, this metal has rigidity.
So, similarly thin plate spline also I mean fits and it resist bending. So, there is a penaltyfunction which involves smoothness of the fitted surface. Now, the deflection that we see inthe plane is in the direction z that is the I mean third dimension which is orthogonal to theplane. So, I mean it controls the curvature of the surface and it is an approximation of thinplane spline. So, and it assumes equation I mean Q equals to Ai di square sigma of log di plusa plus bx plus cy.
In this case x and y are the coordinates of the point which are to be in interpolated and disquare is x minus xi square plus y minus yi square; where, x and y are the coordinate of thepoint, wherein I mean we are going to find out the value that is find out the value of z in thethird dimension and x and y are the x coordinates of the control points.
There are two components of thin plates spline; first is the local trend function which is yourax plus bx plus cy. This component which is known as local trend function. It is same form aslinear or first-order trend surface that we had seen earlier. Now, there is another term which isknown as the basic basis function which is the log of d. So, it is used to I mean design toobtain the minimum curvature surface.
Now, the coefficients Ai, a, b and c how do we determined it; determine it? If we determine itusing a system of linear equations, so we again can convert it into a matrix form and then, wecan calculate the coefficients of capital Ai, a, b and c; wherein, in this particular case you cansee sigma i is from 1 to n; where, n is the number of control points and fi is the known value atthe control point i. So, we have this I mean this sums up to value fi. So, this is the knownvalue at control point fi. So, we can have based on the observations, we can havesimultaneous equations by which we can convert it into a matrix and we can solve for thecoefficients of Ai, a, b and c like we have done earlier.
So, the estimations of the these coefficients would require n plus 3 simultaneous equation andunlike the IDW method, the predicted values of this TSP that is the thin plates splain, it won’tbe limited within the maximum and minimum range values that we had said for the IDWmethod. The I mean inverse distance weighted method of interpolation, we had seen those
values the outcome values would be ranging between the maximum and minimum valueswhich is not the case for TSP interpolation.
The major problem with TSP is that you may have steep gradience, since the outcome of thevalues are not within the maximum or the minimum values; sometimes you may encountersteep gradience, especially in areas where in the data are not there, your data sample points arenot there. So, it this steep gradients are generally referred to as overshoots and there aredifferent numerical methods and approaches to correct these overshoots.
I mean, we can also have thin plates spline with tension, it controls the tension and it pulls thesurface I mean this TSP to the edges of the surface. Now, there are other methods in the TSPwhich are regularised spline and regularised and splines with tension and they have a I meandiverse group. These methods are belong to a diverse group of functions which are known as
radial basis function. Now, these this method of interpolation that is the TSP, thin plate splineis recommended when you want to create a very smooth or continuous surface like the surfaceof a water table or a elevation or if you want to interpolate the rain.
So, if you have such type of problems, then the TSP is the suggested method. So, you havethe radial basis function which refers to a group of interpolation methods, they are I meanexact interpolation. So, equation functions or equation determinants that governs how thesurface will fit between the control point. So, this is the these are the basis function that wehad talked about in the earlier slide.
So, you are ArcGIS software’s it offers five different methods of your Radial Basis Function;different methods, you can explore them, explore this methods. So, the difference betweenthese points I mean difference method could be very small.
Now, so, we have seen different methods of spatial interpolation, a recap of what we havedone in today’s lecture. We have seen the different methods of spatial interpolation, we hadseen the global approaches, we had gone through the local approaches, we had seen thedifferent spatial interpolation method such as Trend Surface, Regression, Thiessen Polygon,Triangulated Irregular network, Kernel Density Estimation, IDW that is a Inverse DistanceWeighing and Thin Split Plain.
So, thanks and we will continue with the Kriging which is the which is another method ofinterpolation in the next lecture.
Thank you.
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