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GEOG 400, Advanced GIS, Fall 2020; Week 7 Lecture 2

GEOG 400, Advanced GIS, Fall 2020; Week 7 Lecture 2

alan.kasprak

October 07, 2020
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  1. GEOG 400: Advanced GIS - Raster Week 7 – Lecture

    2 In-Situ Measurements and Spatial Interpolation
  2. GEOG 400: Advanced GIS - Raster Remote Sensing is the

    acquisition of information about an object or phenomenon without making physical contact with the object and thus in contrast to on-site observation In-Situ measurements acquire information about an object when the distance between the object and the sensor is comparable to or smaller than any linear dimension of the sensor. In-Situ Measurements and Spatial Interpolation Spatial Interpolation is the process by which we generate continuous raster data from discrete vector (i.e., point-based) measurements using remote and in-situ techniques. LAST LECTURE TODAY This week (and really the next four weeks) will be focused on digital elevation models and terrain data.
  3. DEM source data ▪ A quick note about terminology… ▪

    A digital elevation model (DEM) is a generic term for any rasterized representation of elevation ▪ A digital terrain model (DTM) is a type of DEM where the cells represent height of the terrain (the ground) ▪ A digital surface model (DSM) is a type of DEM where the cells represent height of the surface of the Earth (including natural and artificial features above the ground level)
  4. In-Situ measurements acquire information about an object when the distance

    between the object and the sensor is comparable to or smaller than any linear dimension of the sensor. Common In-Situ Topographic Survey Technique #1: Total Station LASER RANGEFINDER MEASURES ‘SLOPE DISTANCE’
  5. In-Situ measurements acquire information about an object when the distance

    between the object and the sensor is comparable to or smaller than any linear dimension of the sensor. Common In-Situ Topographic Survey Technique #1: Total Station LASER RANGEFINDER MEASURES ‘SLOPE DISTANCE’ THEODOLITE MEASURES ANGLE AND COMPUTES HORIZONAL DISTANCE AND VERTICAL DISTANCE
  6. In-Situ measurements acquire information about an object when the distance

    between the object and the sensor is comparable to or smaller than any linear dimension of the sensor. Common In-Situ Topographic Survey Technique #1: Total Station LASER RANGEFINDER MEASURES ‘SLOPE DISTANCE’ THEODOLITE MEASURES ANGLE AND COMPUTES HORIZONAL DISTANCE AND VERTICAL DISTANCE Total station surveys generally require that we know where (exactly!) our instrument is set up
  7. In-Situ measurements acquire information about an object when the distance

    between the object and the sensor is comparable to or smaller than any linear dimension of the sensor. Common In-Situ Topographic Survey Technique #2: RTK-GPS [or real-time kinematic global positioning system] Most of us have used this… Probably everyone has used this…
  8. In-Situ measurements acquire information about an object when the distance

    between the object and the sensor is comparable to or smaller than any linear dimension of the sensor. Common In-Situ Topographic Survey Technique #2: RTK-GPS [or real-time kinematic global positioning system] Most of us have used this… Probably everyone has used this…
  9. In-Situ measurements acquire information about an object when the distance

    between the object and the sensor is comparable to or smaller than any linear dimension of the sensor. Common In-Situ Topographic Survey Technique #2: RTK-GPS [or real-time kinematic global positioning system] RTK-GPS is a little different: - Base station is set up over a known point - If we know the base point, we know errors in where satellites are saying it is - Base figures out the difference and transmits real-time corrections to the rover
  10. In-Situ measurements acquire information about an object when the distance

    between the object and the sensor is comparable to or smaller than any linear dimension of the sensor. Common In-Situ Topographic Survey Technique #2: RTK-GPS [or real-time kinematic global positioning system] RTK-GPS is a little different: - Base station is set up over a known point - If we know the base point, we know errors in where satellites are saying it is - Base figures out the difference and transmits real-time corrections to the rover
  11. Spatial Interpolation • Introduction • There are a few main

    ways to generate raster data • From remote sensing • Directly • Raw image data collected from digital airborne and satellite remote sensing instruments = raster data • e.g. Landsat 8 OLI satellite image • Indirectly • Products derived from remote sensing data, often based on some image classification or regression analysis • e.g. National Land Cover Database
  12. Spatial Interpolation • Introduction • There are a few main

    ways to generate raster data • From digitizing • The process by which digital data are created in GIS • Usually when we think of digitizing, we think of on-screen (heads-up) digitizing • Manual delineation of vector points, lines, and polygons • But, another common form of digitizing is scanning and georeferencing of analog maps and aerial photographs • Produces raster GIS data!
  13. Spatial Interpolation • Introduction • There are a few main

    ways to generate raster data • Raster data generation through remote sensing and digitizing are examples of converting spatially- continuous data to spatially-continuous data
  14. Spatial Interpolation • Introduction • There are a few main

    ways to generate raster data • From vector to raster conversion • As we know, there are issues with this…
  15. Spatial Interpolation • Introduction • There are a few main

    ways to generate raster data • From vector to raster conversion • As we know, there are issues with this…
  16. Spatial Interpolation • Introduction • There are a few main

    ways to generate raster data • From vector to raster conversion • As we know, there are issues with this…
  17. Spatial Interpolation • Introduction • There are a few main

    ways to generate raster data • From vector to raster conversion • As we know, there are issues with this… • These are all examples of taking a spatially-discrete input (vector point, line, polygon) and converting it into another spatially-discrete output (discrete raster) • Problem is, raster data are not good at representing spatially-discrete geographic features
  18. Spatial Interpolation • Introduction • There are a few main

    ways to generate raster data • But what if we want to take a spatially- discrete input and generate a spatially- continuous output? • For example… • Weather stations • I have temperature measurements at a bunch of points in Colorado, but I need a map that depicts temperature continuously throughout the state have this want this
  19. Spatial Interpolation • Introduction • There are a few main

    ways to generate raster data • But what if we want to take a spatially- discrete input and generate a spatially- continuous output? • For example… • Lidar • I have a billion points that represent the tree height, but I want a rasterized canopy height model Slide #20 have this want this
  20. Spatial Interpolation • Interpolation • Broadly defined, interpolation is the

    process by which unknown values are estimated within a set of known values • Does not have to be spatial! • Simplest example… • Known value 1 • x = 2, y = 2 • Known value 2 • x = 4, y = 4 • Unknown value • x = 3, y = ? x y
  21. Spatial Interpolation • Interpolation • How did you guess that

    y = 3? • Because you interpolated (without even really thinking about it…) • Took an educated guess! • Assumed that the relationship between x and y was linear • More on this in a few… • Drew a mental line between known points 1 and 2 • Used y = ax + b to estimate the value of y at x = 3 x y
  22. Spatial Interpolation • Interpolation • In reality, of course, the

    linear assumption may be totally invalid • Maybe the data would look like this? x y
  23. Spatial Interpolation • Interpolation • In reality, of course, the

    linear assumption may be totally invalid • Maybe the data would look like this? • Or like this? • The only way to truly know would be to actually measure the y value • But you can’t always do that! x y
  24. Spatial Interpolation • Interpolation • That being said, often times

    the simplest explanation for some phenomenon is the best • KISS: “keep it simple, stupid” • The law of parsimony • Ockham’s razor • Thus, the best we can say is that interpolation is an estimation technique • As such, is prone to error… Slide #26 x y
  25. Spatial Interpolation • Interpolation • What is y if x

    = 1? • y = 1, right? • Can we reliably predict that to be the case? • Can we reliably assume that the linear relationship continues outside of the range of known data? • …Maybe? Slide #27 x y
  26. Spatial Interpolation • Extrapolation • Estimating values outside of the

    range of known values is known as extrapolation • Although they are both estimation techniques… • Interpolation is based on known values, and therefore is a valid approach • DO IT! (cautiously…) • Extrapolation is… A wild guess! • DON’T DO IT! Slide #28 x y
  27. Spatial Interpolation • Spatial Interpolation • These have all been

    examples of one-dimensional interpolation • But the same principles apply to two-dimensional (or “spatial”) interpolation! 1-D Interpolation 2-D Interpolation
  28. Spatial Interpolation • Spatial Autocorrelation • Very simple example… Elevation

    • What is your current elevation? 7000 ft • What if you walked 1 foot away? 7000 ft • What if you walked 100 feet away? 6990 – 7010 ft • What if you walked 1 mile away? 5000 – 8000 ft • What if you walked 100 miles away? 2000 – 14000 ft • Clearly, elevation is spatially-autocorrelated • That is, elevations are correlated with other elevations, and that correlation becomes stronger the closer together the measurements are made • Same is true of many spatial variables: temperature, humidity, slope, aspect, population density, socioeconomics, etc. etc. etc.
  29. Spatial Interpolation • Spatial Autocorrelation • Why do we care

    about spatial autocorrelation? • It allows us to perform spatial interpolation! • If your variables aren’t spatially-dependent, then your interpolation will be invalid
  30. Spatial Interpolation • Spatial Autocorrelation • How do we measure

    spatial autocorrelation? • The most common way to measure SA is Moran’s I • Ranges from -1 to 1, where -1 means very low/no SA and 1 means very high SA I = -1 data are perfectly dispersed I ≈ 0 data are perfectly random I = 1 data are perfectly clustered
  31. Spatial Interpolation • Spatial Autocorrelation • How do we visualize

    spatial autocorrelation? • The most common way is using a semivariogram • Displays the semivariance in a variable as a function of distance • What is semivariance? • Looks more complex than it really is… • A measure of how some value varies from itself based on distance • Take each point (xi ), subtract its z value (z(xi )) from every other point and calculate distance between (h), square the differences, sum them all up, and divide by 2 x the number of points (N)
  32. Spatial Interpolation • Spatial Autocorrelation • How do we visualize

    spatial autocorrelation? • The most common way is using a semivariogram • As distance between samples increases, the semivariance (difference between sampled values) increases • At a certain point, the data are no longer spatially-autocorrelated
  33. Spatial Interpolation • Spatial Autocorrelation • How do we visualize

    spatial autocorrelation? • The most common way is using a semivariogram • Nugget provides estimate of measurement error (because at 0 distance, values should be equal to one another) • Range provides distance at which data are no longer SA • Sill provides semivariance at the range (how variable the data are) Slide #36
  34. Spatial Interpolation • Spatial Autocorrelation • How do we visualize

    spatial autocorrelation? • The most common way is using a semivariogram • Let’s take a look at an example… Slide #37 FID x y z 0 244397 4132220 2112.63 1 249363 4134130 2378.68 2 245134 4134600 2391.99 3 245835 4133210 2241.85 4 245744 4132960 2176.04 5 246738 4131240 2024.41 6 244131 4131530 2061.93 7 249346 4133430 2264.28 8 243738 4130550 2059.22 9 246727 4135750 2074.58 10 245824 4134080 2380.01 11 246306 4135550 2153.01 12 246120 4133420 2259.15 … … … … 9999 243961 4134670 2131.38
  35. Spatial Interpolation • Spatial Autocorrelation • How do we visualize

    spatial autocorrelation? • The most common way is using a semivariogram • Let’s take a look at an example… • Results • Elevation data are highly spatially- autocorrelated • Nugget = 0 • Range ≈ 2000 • Sill ≈ 22000 Slide #38
  36. Spatial Interpolation • Spatial Autocorrelation • How do we visualize

    spatial autocorrelation? • The most common way is using a semivariogram • Let’s take a look at an example… Slide #39 FID x y z 0 244397 4132220 2112.63 1 249363 4134130 2378.68 2 245134 4134600 2391.99 3 245835 4133210 2241.85 4 245744 4132960 2176.04 5 246738 4131240 2024.41 6 244131 4131530 2061.93 7 249346 4133430 2264.28 8 243738 4130550 2059.22 9 246727 4135750 2074.58 10 245824 4134080 2380.01 11 246306 4135550 2153.01 12 246120 4133420 2259.15 … … … … 9999 243961 4134670 2131.38
  37. Spatial Interpolation • Spatial Autocorrelation • How do we visualize

    spatial autocorrelation? • The most common way is using a semivariogram • Let’s take a look at an example… • Results • Elevation data are not spatially- autocorrelated at all • Nugget = 30000 • Range = 0 • Sill = 30000 Slide #40
  38. Spatial Interpolation • Spatial Interpolation • If we can safely

    assume (or determine statistically) that our variable of interest is spatially-autocorrelated, then we can proceed with spatial interpolation • Again, most things in nature are spatially-autocorrelated, so interpolation is usually viable • However, the nature and strength of the autocorrelation can help dictate the type of interpolation to be performed
  39. Spatial Interpolation • Spatial Interpolation • (As with most things

    in GIS…) there are several different techniques for spatial interpolation: • Inverse distance weighting • Kriging • Spline • Trend
  40. Spatial Interpolation • IDW • Inverse distance weighted (IDW) interpolation

    • Explicitly assumes Tobler’s 1st law • Everything is related to everything else, but near things are more related than distant things • Calculates unknown cell values based on values of nearby points with known values, weighting the influence of those points based on their proximity to the unknown cell
  41. • IDW • Works like this… 1. Determine neighbors •

    Two options: • Based on fixed radius (e.g. all of the points within 100 m of the unknown cell) • Need to know something about the spacing of your known points • Works better if points are somewhat evenly- distributed Spatial Interpolation
  42. • IDW • Works like this… 1. Determine neighbors •

    Two options: • Based on fixed radius (e.g. all of the points within 100 m of the unknown cell) • Need to know something about the spacing of your known points • Works better if points are somewhat evenly- distributed • Based on a fixed number of neighboring points (e.g. the 5 closest neighbors) • More adaptable to irregular point spacing, but potentially including influence from very distant points Spatial Interpolation
  43. • IDW • Works like this… 1. Determine neighbors •

    Considerations: • Too few neighbors (too small of a statistical sample) means your estimate will be highly subject to outliers • Remember the Law of Large Numbers?! • Too many neighbors might mean that you are “smoothing out” your data too much, missing subtle, local changes • Plus, requires much more computing power • Try to find that sweet spot! Spatial Interpolation
  44. • IDW • Works like this… 2. Determine distance weighting

    • In IDW, the weight is calculated by inverse distance (get it?) • In other words, 1 / d • So as d increases, the relative weight (influence on the unknown cell) decreases • However, we can add a power to the denominator to dictate how quickly that decrease occurs • e.g. 1 / d2, 1 / d3 • The higher the power, the more quickly the decrease occurs Spatial Interpolation
  45. • IDW • Works like this… 3. Calculate the value

    (zj ) of your unknown cell based on known cell values (zi ) and their distances (di ) • Fixed number of neighbors (n) = 5 • Power function (p) = 1.5 25 19 14 12 9 18 15 5 6 7 5 2 9 8 = σ=1 σ =1 1 Spatial Interpolation
  46. • IDW • Works like this… 3. Calculate the value

    (zj ) of your unknown cell based on known cell values (zi ) and their distances (di ) • Fixed number of neighbors (n) = 5 • Power function (p) = 1.5 25 19 14 12 9 18 15 5 6 7 5 2 9 8 = 9 51.5 + 15 61.5 + 18 71.5 + 14 51.5 + 12 21.5 1 51.5 + 1 61.5 + 1 71.5 + 1 51.5 + 1 21.5 = . Spatial Interpolation
  47. • Kriging • The most advanced (and complex!) interpolation technique

    of them all • But, perhaps the best! • Named after Danie Krige, South African mining engineer • Developed this geostatistical technique in order to find mineral-rich locations based on existing mines Spatial Interpolation
  48. • Kriging • Very similar to IDW, except that instead

    of solely using the distance to neighboring points, Kriging weights are determined by distance and spatial autocorrelation of the data! • So, in addition to the steps needed to perform IDW, you need to fit a semivariogram! Spatial Interpolation
  49. • Kriging • There are several different SV models that

    can be fit • Spherical and exponential most common • Selection depends on specific dataset Slide #53 Spatial Interpolation
  50. • Kriging • In ArcGIS, you can use the Geostatistical

    Wizard on the Geostatistical Toolbar to assist in your model selection • Beyond the scope of this class, but I encourage you to inquire into it on your own time/for your own research! Slide #54 Spatial Interpolation
  51. • Kriging • Although this technique can be employed using

    “default” parameters, proper implementation requires an intimate knowledge of the geostatistics underlying the method as well as your data themselves • In some ways, the simpler algorithms (e.g. IDW, Spline) are more defensible, as they are relatively “naïve” methods and thus require a less-than-expert knowledge level Spatial Interpolation
  52. • Spline • This technique is based on the method

    formerly used to draw curves for engineering and design purposes • Long, thin, flexible strips of wood, plastic, or metal bent between nails (or “knots”) causing a nice, smoothly-curving shape • The specific shape would be dictated by number and placement of knots, and tension of the spline Slide #56 Spatial Interpolation
  53. Spatial Interpolation • Spline • In spatial spline interpolation, the

    same principles apply • Surface is generated based on a curved, smooth surface • Unlike other methods that are based on statistical estimations, spline ensures that all z values are maintained exactly from the input in the resulting raster – highly subject to outliers! Slide #57
  54. • Spline • Has several parameters… • Regularized vs. tension

    spline • Regularized produces a generally smoother surface • Tension produces a generally more rigid surface • Weight • For regularized, higher weights mean smoother output • For tension, higher weights mean rougher output • Number of points • For both, the number of nearest neighboring points used to create the localized spline Slide #58 Spatial Interpolation
  55. • Trend • Trend interpolation is based on regression •

    Regression analysis is the optimized fitting of a line or curve to derive a mathematical function that best fits the raw data • In our basic statistics lecture, we talked about the simplest form of regression: ordinary least squares (OLS) regression • Linear regression • y = ax + b • Predicting a dependent variable (y) based on an independent variable (x) Spatial Interpolation
  56. • Trend • However, sometimes the data are not linear

    in nature, and thus require polynomial regression • 1st order: y = ax + b • 2nd order: y = ax + bx2 + c • 3rd order: y = ax + bx2 + cx3 + d • etc. Spatial Interpolation
  57. • Trend • In general, in statistics, we aim for

    the simplest explanations • Higher-order polynomials will always increase the model fit, but at the risk of over-fitting your model • i.e. instead of looking at the broader trend, your function is so specific that it won’t be generalizable to another, similar dataset Spatial Interpolation
  58. • Trend • Trend spatial interpolation is just regression in

    two dimensions raw data 1st order polynomial trend surface
  59. • Trend • Trend spatial interpolation is just regression in

    two dimensions raw data 2nd order polynomial trend surface
  60. • Trend • Trend spatial interpolation is just regression in

    two dimensions raw data 3rd order polynomial trend surface
  61. • Trend • Trend spatial interpolation is just regression in

    two dimensions raw data 4th order polynomial trend surface
  62. • Trend • Trend spatial interpolation is just regression in

    two dimensions raw data 5th order polynomial trend surface
  63. • Trend • Trend spatial interpolation is just regression in

    two dimensions raw data 6th order polynomial trend surface
  64. • Trend • Trend spatial interpolation is just regression in

    two dimensions raw data 7th order polynomial trend surface
  65. • Trend • Trend interpolation is useful when conditions vary

    gradually over relatively broad areas • So, elevation is a bad example • But, atmospheric conditions (temperature, humidity, pollution, etc.) and aquatic conditions (temperature, pH, salinity) are good examples • Can also be used to “remove” broad-scale trends to reveal local phenomena • e.g. compare ambient (background) levels of O3 to local levels Spatial Interpolation