Gov 1016 Spatial Models

Transcription

1 Gov 1016 Spatial Models Lecture 4: Spatial Interpolation Sources: Mitas, L., Mitasova, H., 1999; Spatial Interpolation. In: P.Longley, et al (Eds.), Geographical Information Systems, 2002; Burrough and McDonnell; Bonham-Carter, 1996; Goodchild: Krivoruchko, 2010

2 Outline for Today Why interpolate? Methods (Global/ Local) Global: classification, regression (Trend surface) Local: Deterministic: Thiessen, IDW, Splines Geostatistical: Kriging Ordinary Kriging in depth Discussion

3 when you can measure what you are speaking about and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of science, whatever the matter may be Lord Kelvin

4 What is Spatial Interpolation? Goodchild: A process of determining the characteristics of objects from those of nearby objects From a GIS perspective, spatial interpolation is a spatial transformation

5 Sparse data Why interpolate?

6 Underlying Rationale Points close together in space are more likely to have similar values than points far apart

7 Why is it difficult?

8 Interpolation Methods 1. Point Interpolation/ Areal Interpolation 2. Global/ Local Interpolators 3. Exact/ Approximate Interpolators 4. Stochastic/ Deterministic Interpolators 5. Gradual/ Abrupt Interpolators

9 Exact/ Approximate Interpolators Exact interpolators honor the data points upon which the interpolation is based Approximate interpolators are used when there is some uncertainty about the data points that represent the sampled surface values

10 Point Interpolation Methods Global 1. Classification 2. Regression (Trend surfaces and transfer functions not exact) Local 1. Deterministic (Nearest Neighbors: Thiessen, Pycnophylactic; IDW; Splines can be exact) 2. Probabilistic (Kriging; Simulation can be Exact)

11 Global

12 Fitting surfaces Flat plane to fit data surface is approximated by linear equation Warped plane to fit data surface is approximated by quadratic equation y b xy b x b y b x b b z y b x b b z quadratic: linear:

13 Surface Types

14 Regression Fit the surface to the samples minimizing the sum or squares residual distance Assumes that x and y are independent, that z is normally distributed, and error is independent of location z(x) = b 0 + b 1 x + e

15 Regression: Trend surface analysis Fit polynomial equation to sample points Goal is to minimize deviations between sample points and surface X, and Y coordinates of the area A third quantity (Z) which represents the variation of the surface recorded on the Cartesian plane

16 Trend Surface Analysis Assumptions The surface of Z value is continuous Therefore, the data value at any location can be estimated if sufficient information about the surface is provided The Z value is spatially dependent

17 Example Residuals Trend Isolines for the time of settlement trend surface. Source: Abler, Ronald F., John S. Adams, and Peter Gould. Spatial Organization the Geographer's View of the World. Englewood Cliffs, NJ: Prentice-Hall, 1971, page 136.

18 Local

19 Delauney Triangulation

20 Delauney Triangulation

21 Thiessen Polygons Assign value of nearest sample point

22 Splines Piecewise polynomial fitted on points resulting in a surface that has continuous first and second derivatives

23 Splines Spline functions imitates a thin flexible sheet forced to pass close to the data points The equilibrium shape of the sheet minimizes the bending energy which is closely related to the surface curvature Repeatedly applies a smoothing equation (piecewise polynomial) to the surface Resulting surface passes through all points

24 Splines

25 Inverse distance weighting

26 Inverse distance weighting Estimates value in output surface by weighting the relative influence of input points in the local area by inverse function of distance from location to be estimated

27 IDW

28 Fits function to Kriging Specified number of points All points within specified radius Most appropriate when you know about spatial correlations or directional bias in data

29 Kriging

30 Kriging Kriging is based on the idea that you can make inferences regarding a random function Z(x), given data points Z(x 1 ), Z(x 2 ), Z(x n ) The basis of this technique is the rate at which the variance between points changes over space This is expressed in the semivariogram which shows how the average difference between values at points changes with distance between points

31 Kriging Z(x) = mean(x) + Spatially signal g(h) + error Data = Trend + small scale and microscale variation + measurement error

32 (Semi)variograms The amount and form of spatial autocorrelation can be described by a semi variogram that shows how differences in values increase with geographical separation s ( g h N h i 2 h g j ) 2 b

33 What is a Semi-Variogram The semi-variogram is a function that relates semi-variance (or dissimilarity) of data points to the distance that separates them 1 d or h

34 Example Sample All pairs within 100 ft in EW direction

35 1 N Calculations 2 s ( gi n g j ) 2

36 Calculations

37 Experimental Semi-variogram

38 Parts of a Semivariogram

39 Semi-Variogram components Nugget variance: a non-zero value for g when distance h = 0. Produced by various sources of unexplained error (eg. measurement error) Sill: for large values of h the variogram levels out, indicating that there no longer is any correlation between data points. The sill should be equal to the variance of the data set Range: is the value of h where the sill occurs (or 95% of the value of the sill) In general, 30 or more pairs per bin are needed to generate a reasonable sample variogram

40 Real life semi-variograms Data samples not usually on a regular grid Algebraic definition function of distance with the differences in value

41 Variograms

42 Variogram models Variogram models must be positive definite so that the covariance matrix based on it can be inverted (which occurs in the kriging process). Because of this, only certain models can be used

43 Variogram models

44 Effect of lag size on semivariograms Variogram with a lag size of 5m and a lag tolerance of 2.5m Variogram with a lag size of 10m and a lag tolerance of 5m

45 Anisotropy

46 Geometric Anisotropy

47 Anisotropy?

48 Calculating weights

49 Numeric example of kriging In this example, we want to estimate a value for point 0 (65E, 137N), based on the 7 surrounding sample points. The table indicates the (x,y) coordinates of the 7 sample points, their corresponding values of V (which is the variable we are interested in) and their distance to point 0.

50 Spatial continuity Variogram model Parameters: C 0 = 0, a = 10, C 1 = 10 Covariance function

51 Kriging matrices To solve for the weights, we multiply both sides by C -1, the inverse of the left-hand side semivariance matrix:

52 First, the distance matrix. Kriging matrices variances will be calculated based on the distance between points using our model: Z or g or C(h) = 10 e 0.3 h

53 Kriging matrices

54 Results Kriging weights: Estimated value for point 0:

55 Results

56 x3=1 8 7 x2=4 6 5 x4=5 4 x6? 3 x5=10 2 x1=10 1

57 IDW IDW Location Value Distance of interpolated point Inverse1 Weight1 Distance of interpolated point ^2 Inverse2 Weight2 1 (2,2) (3,7) (9,9) (6,5) (5,3) Total Weight1 Value Weight1*Value Weight2 Weight2*Value

58 Nearest Neighbor nearest 5 6 nearest nearest nearest 2, 1 10

59 Trend Surface bx by constant ,

60 Example Use of inverse distance weighting (IDW) instead of kriging in determining the depth of contamination. This failure clearly results in the creation of significant data gaps, which will prevent the actual measurement of sediments to determine the PCB contamination at locations where they are likely to be found. What methods of data gap sampling were identified by EPA and/or its consultants, and why were each rejected? (From Friends of a Clean Hudson December 14, 2005)

61 NOAA s Comments on the Hudson Intermediate Design Report Aug 22, 2005 (10/18/05) Step 1: Depth of Contamination Step 1 of the dredge prism development process describes a 17-layer approach for interpolating depth of contamination using Inverse Weighted Distance averaging or IDW. EPA, in issuing its July 2004 final determination on the Phase 1 Dredge Area Delineation Report (DAD) dispute, determined that kriging is the appropriate method for vertically delineating depth of contamination. EPA stated that "Vertical delineation of the Phase 1 and Phase 2 dredge areas shall be conducted in accordance with resolution of GE Issue C.2." (EPA 2004, Attachment 1, Page 2) "For the revised Phase 1 DAD, GE shall complete a semivariogram analysis of the vertical extent of contamination in the Phase 1 candidate areas, conservatively estimating the nugget for use in kriging and/or in estimating uncertainty in the depth of contamination." (EPA 2004, Attachment 2, Page 19) "The EPA-approved Remedial Design Work Plan (which is part of the RD AOC) states that "[l]ocally-based geostatistical analysis [i.e., kriging] will be used where weight-of evidence is equivocal." (EPA 2004, Attachment 2, Page 19). There is no technical justification for employing IDW over kriging since IDW does not perform any better than kriging and the 17-layer approach is much more complex than kriging. The IDW approach also requires post-processing of data to smooth out the troughs and peaks and does not have the same capability of quantifying uncertainty in the DOC estimate that kriging has. The general rule for statistically evaluating data is to use the simplest method that provides the best results.

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67 Kriging Types Simple kriging: Known mean value as input Ordinary kriging: Unknown mean estimated as a constant in the prediction neighborhood Universal kriging: Unknown mean which varies locally predicted as a trend

68 Inference Ordinary Kriging produces the best linear unbiased estimate at unmeasured site

69 RMSE Data accuracy is derived by comparing linear interpolation estimates with actual values and computing the statistical standard deviation or root-mean-square error (RMSE)

70 10 Questions to Evaluate 1 What function of distance should we use? How do we handle different continuity in different directions? How many samples should we include in the estimation? How do we compensate for irregularly spaced or highly clustered sampling? How far should we go to include samples in our estimation process? Should we honor the sample values? How reliable is the estimate when we have it? Why is our map too smooth? What happens if our sample data is not Normal? What happens if there is a strong trend in the values? 1 Clark and Harper Practical Geostatistics 2000.

71 Review of 10 Questions to ask 1 1. What function of distance should we use The variogram shows us the spatial structure, and association of the data, and will give us a hint as to what function to possibly use 2. How do we handle different continuity in different directions Here again, the variogram will tell us whether there is any spatial association, and we can determine which direction by evaluating whether anisotropy exists 3. How many samples should we include in the estimation Again, we can look at the variogram 1 Clark and Harper Practical Geostatistics 2000.

72 10 Questions to ask 1 4. How do we compensate for irregularly spaced or highly clustered sampling The variogram defines the relationship between points and their distances from other points. Calculating weights in Kriging takes the distances among all points into account 5. How far should we go to include samples in our estimation process By looking at the variogram we can identify the sill (that area where the spatial correlation has little value). The range tells us the distance where the points are no longer correlated 6. Should we honor the sample values Still lot of debate on this one. IDW says yes, that s why we get the bullseye. The nugget effect in Kriging allows us to say no. But, we can set the nugget to zero with Kriging 1 Clark and Harper Practical Geostatistics 2000.

73 10 Questions to Ask 7. How reliable is the estimate when we have it Kriging allows us to compute the standard error 8. Why is our IDW map too smooth In IDW when you include points far away they become part of the weights. Since the weights have to add up to one, you are basically taking power away from the closer ones 9. What happens if our sample data is not Normal? Make the data normal 10. What happens if there is a strong trend in the values? First, remove the trend, then re-interpolate the points

74 Example 1

75 Spatiotemporal data Spatial-dominated data Example: Temperatures at two neighboring weather stations on the same day may be similar. Temperatures on two different days at the same station may be very different. Temporal-dominated data Example: For the USA presidential election, the outcomes in one voting district may remain the same for many years while the outcomes of two neighboring voting districts may be significantly different.

76 Relationship strength Interpolate based on whether temporal autocorrelation or spatial autocorrelation higher

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78 Spatial

79 Temporal

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81 Discussion: Comparing methods of interpolation

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83 Rules of Thumb: Do they apply? Make sure many sample points at origin to estimate variogram well at origin More than 30 data pairs in each bin Nugget should be small Semivariance (sill) at range should be representative of sample variance