04/18/23 © 2009 Raymond P. Jefferis III Lect 07 - 1

Geographic Information Processing

Data Analysis• Watersheds• Gradient path

- Valleys- Ridges

• Flow analysis• Contours

Chester County, PA Watersheds, Chester County Board of Commissioners

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Computing Watersheds

• Gradient-following– Locating valleys – Locating ridges

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Gradient-Following Methods

• Spatial derivatives computed in both x- and y-directions

• Vector addition of derivatives produces gradient vectors

• Negative gradients point “down”

• Positive gradients point “up”

• Follow gradients to valleys or peaks

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Data Preparation

• Gradient-following methods become “stuck” at local minima

• DTED Level 2 data has had pockets filled [filled DEM data]

• Use digital filtering to smooth contours before computing gradients. Gaussian filtering is best for band-limited random noise.

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Following the Gradient

• Perform Gaussian filtering of data

• Compute gradient at all points, using Savitsky and Golay (or other) filter

• Start at any point

• Move in direction of desired gradient• Stop when gradient is zero

(This can be minimum or maximum point)

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Notes

• Two derivatives computed for each point– x-direction derivative– y-direction derivative

• Gradient is vector sum of x- and y-derivative at each point(Result will have Magnitude and Direction)

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Gradient Computation

• For data field, f(x,y), the gradient vector is a vector of spatial derivatives defined as:

G[ f (x, y)] =∂f / ∂x∂f / ∂y⎡

⎣⎢

⎤

⎦⎥

Where,Gx =∂f / ∂x= lim

Δx→ o[ f(x,y)− f(x+ Δx,y)]

Gy =∂f / ∂y= limΔy→ o

[ f(x,y)− f(x,y+ Δy)]

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Note

Geographic pixels are of finite extent (not infinitesimal), thus the gradients must be computed by finite difference methods that only approximate the derivative definitions above.

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Computational Notes

• Gx can be computed by convolution, using a kernel that gives the x-gradient

• Gy can be computed by convolution, using a kernel that gives the y-gradient

• The actual gradient is vector sum of these(See next slide for polar coordinate vector)

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Derivative Magnitude and Direction

G = Gx2 +Gy

2

ΘG = tan−1[Gy / Gx ]

Can produce gradient vector at each point.

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Computing Gradients

• Nearest neighbor differencing– Follows the finite approximation– Amplifies data noise (no smoothing)

• Convolution– Uses special kernels– Can do smoothing of entire image– Rapid computations (parallel processing)

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Problems with Gradient Methods

• Local minima– Gradient gets “stuck” in a local minimum– Level 2 DTED data should eliminate this– Filter data for smooth contours

• Data resolution limits– Resolution of 1 meter makes valleys flat– Adds Uniform random noise– Results in big “puddles” (See following:)

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Altitude Resolution Problem• Gradient-following

result

• Filtering: 7x7 Gaussian convolute

• Altitude resolution: 1 meter

• Note flat areas where gradient cannot be resolved (black)

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Solutions

• More smoothing (15x15 convolute)[use Gaussian convolution kernel]

• Use 11x11 convolute for derivatives

• More complex descent algorithms

• Using surveyed stream/river data and connecting descending flows to these streams/rivers

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Malvern Gradient Vectors

Heights convolution filteredGaussian filter15 x 15 kernel

Gradients by convolutionSavitsky & Golay11 x 11 kernels

Gradients shown at every 7th point

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Note

Inspect Savitsky & Golay derivative convolution kernels at this point.

(Handout)

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Savitsky & Golay Kernels• Polynomial based• User selects polynomial type

– Quadratic,Cubic, Quartic, Quintic, etc.

• User selects number of terms– Number of data points used in filter calculation– Varies degree of smoothing (more points means

more smoothing - more information used from surrounding points for each point at which derivative is calculated)

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15x15 Gaussian Filter KernelArray[c, 16, 16]; n = 15; sm = 4661.0;

c = {{1, 2, 3, 5, 6, 7, 8, 9, 8, 7, 6, 5, 3, 2, 1}, {2, 3, 5, 8, 10, 13, 14, 15, 14, 13, 10, 8, 5, 3, 2}, {3, 5, 8, 12, 16, 20, 22, 23, 22, 20, 16, 12, 8, 5, 3}, {5, 8, 12, 18, 23, 29, 32, 34, 32, 29, 23, 18, 12, 8, 5}, {6, 10, 16, 23, 31, 38, 43, 45, 43, 38, 31, 23, 16, 10, 6}, {7, 13, 20, 29, 38, 47, 53, 55, 53, 47, 38, 29, 20, 13, 7}, {8, 14, 22, 32, 43, 53, 60, 62, 60, 53, 43, 32, 22, 14, 8}, {9, 15, 23, 34, 45, 55, 62, 65, 62, 55, 45, 34, 23, 15, 9}, {8, 14, 22, 32, 43, 53, 60, 62, 60, 53, 43, 32, 22, 14, 8}, {7, 13, 20, 29, 38, 47, 53, 55, 53, 47, 38, 29, 20, 13, 7}, {6, 10, 16, 23, 31, 38, 43, 45, 43, 38, 31, 23, 16, 10, 6}, {5, 8, 12, 18, 23, 29, 32, 34, 32, 29, 23, 18, 12, 8, 5}, {3, 5, 8, 12, 16, 20, 22, 23, 22, 20, 16, 12, 8, 5, 3}, {2, 3, 5, 8, 10, 13, 14, 15, 14, 13, 10, 8, 5, 3, 2}, {1, 2, 3, 5, 6, 7, 8, 9, 8, 7, 6, 5, 3, 2, 1}}/sm;

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Derivative Kernels (S & G)Array[cx, 12, 12]; n = 11; sm = 110.0;

Array[cy, 12, 12]; n = 11; sm = 110.0;cy = {{0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0}}/sm;cx = {{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}/sm;

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Finding Valleys

• Smooth data, using Gaussian convolution filtering

• Compute gradient at all points, using Savitsky & Golay derivative kernel

• Select starting points

• Follow gradient downward

• Stop when gradient is zero or path reverses

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Malvern Flow TracksHeights convolution filtered

Gaussian filter15 x 15 kernel

Gradients by convolutionSavitsky & Golay11 x 11 kernel

Tracks follow gradients, starting from every 8th pixel

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Gradient-Following CodeFor[jx = 7, jx < nx - 10, jx += 5,{ For[jy = 7, jy < ny - 10, jy += 5, {kx = jx; ky = jy; exit = False;While[zr = gx[[kx - 6, ky - 6]]; zi = gy[[kx - 6, ky - 6]]; zc = 1.0*RandomInteger[{-1, 1}] + ArcTan[zi, zr]/Degree;If[zc < 0.0, zc = 360.0 + zc];rzc = 1 + Mod[(Round[zc/22.5](*+RandomInteger[{-1,1}]*)), 16]; vec = dxy[[rzc]];kx = kx +vec[[1]]; ky = ky + vec[[2]]; If[((kx < 1) || (ky < 1) || y[[ kx, ky]] < 0.0), exit = True]; ]; }] }]

y[[ kx, ky]] = -1.0; sd[[ kx + 7, ky + 7]] = -1.0; kx > 6 && ky > 6 && kx < (nx - 10) && ky < (ny - 10) && ! Exit,

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Possible Applications

• Watershed analysis

• Finding wetlands

• Predicting path of HAZMAT spill

• Back-tracing contaminants to source(s)

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Visualization

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Flow Analysis

• Gradients direct flow to adjacent cells• Flows merge and add (volumetric flow)• Show on plot when flow volume exceeds a given

threshold

Note - Possible improvement:

Add to altitude as flow volume increases

May require re-computation of gradients

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Merging Flows

Set starting point flow to unityMove one pixel along gradientMerge, adding flowsPlot when flow > threshhold

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Finding Ridge Lines

• Smooth data, using Gaussian convolution kernel

• Compute gradients at all points, using Savitsky & Golay derivative kernel

• Select starting points

• Follow against gradient to ridge line

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Malvern Ridge Lines• Ridge lines found

by following gradient backward to maximum

• Starting point every 5 pixels.

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Gradient-Following CodeFor[jx = 7, jx < nx - 10, jx += 5, {If[zc < 0.0, zc = 360.0 + zc]; ]; }] }]

rzc = 1 + Mod[(Round[zc/22.5](*+RandomInteger[{-1,1}]*)), 16]; vec = dxy[[rzc]];kx = kx - vec[[1]]; ky = ky - vec[[2]];If[((kx < 1) || (ky < 1) || y[[ kx, ky]] < 0.0), exit = True];y[[ kx, ky]] = -1.0; sd[[ kx + 7, ky + 7]] = -1.0;For[jy = 7, jy < ny - 10, jy += 5, {kx = jx; ky = jy; exit = False;While[ kx > 6 && ky > 6 && kx < (nx - 10) && ky < (ny - 10) && ! exit,zr = gx[[kx - 6, ky - 6]]; zi = gy[[kx - 6, ky - 6]];zc = 1.0*RandomInteger[{-1, 1}] + ArcTan[zi, zr]/Degree;

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Possible Application

• Locating watershed ridge lines

• Locating source of contamination

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Max., Min., and Curvature

Gradient minimum and maximum points - Filter: S&G 15-point - Point selection; Gradient < 0.4 Curvature > 240

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Contours

Contour map after Gaussian filteringMalvern QuadrangleUSGS DEM Data

• Contours appear as horizontal slices through topography. • Contour spacing can be controlled by software.

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Data Preparation

• Bad data points are removed

• Data is smoothed (Gaussian filter)

• Contour interval is selected(or specific contour set is selected)

• Color scheme is selected

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Plotting Contours

ListContourPlot[y, AspectRatio -> 13.8793/10.6742,

Contours -> {40, 80, 120, 160, 200, 240, 280, 320},

ColorFunction -> "Topographic"]

Notes: Contours produced at indicated heights (meters).Aspect ratio corrects image for latitudeOther color functions possible

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Contour Plot Problems

• Digital filtering required to insure smooth contours

• BEWARE! Contour plots require a LOT of memory in computer.

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Discussion