Labs Put your name on your labs! Layouts: Site 1 Photos Title Legend Scale bar North arrow Your name Export as jpeg then paste into your report sheet. Site 1 Photos Lecture 14

– Spatial Interpolation –Part 1 Chapter 12 Lecture 14

INTERPOLATION Procedure to predict values of attributes at unsampled points Why? Can’t measure all locations: Time Money Impossible (physical- legal) Changing cell size Missing/unsuitable data Past date (eg. temperature) Lecture 14

Sampling Sampling Techniques: A shortcut method for investigating a whole population Data is gathered on a small part of the whole parent population or sampling frame, and used to inform what the whole picture is like Techniques: Systematic Random Cluster Adaptive Stratified Lecture 14

Difficult to stay on lines Systematic sampling pattern Easy Samples spaced uniformly at fixed X, Y intervals Parallel lines Advantages Easy to understand Disadvantages All receive same attention Difficult to stay on lines May be biases Lecture 14

Advantages Disadvantages Random Sampling Select point based on random number process Plot on map Visit sample Advantages Less biased (unlikely to match pattern in landscape) Disadvantages Does nothing to distribute samples in areas of high Difficult to explain, location of points may be a problem Lecture 14

Advantages Cluster Sampling Cluster centers are established (random or systematic) Samples arranged around each center Plot on map Visit sample (e.g. US Forest Service, Forest Inventory Analysis (FIA) Clusters located at random then systematic pattern of samples at that location) Advantages Reduced travel time Lecture 14

Advantages Disadvantages Adaptive sampling More sampling where there is more variability. Need prior knowledge of variability, e.g. two stage sampling Advantages More efficient, homogeneous areas have few samples, better representation of variable areas. Disadvantages Need prior information on variability through space Lecture 14

Stratified A wide range of data and fieldwork situations can lend themselves to this approach - wherever there are two study areas being compared, for example two woodlands, river catchments, rock types or a population with sub-sets of known size, for example woodland with distinctly different habitats. Random point, line or area techniques can be used as long as the number of measurements taken is in proportion to the size of the whole. Lecture 14

Example if an area of woodland was the study site, there would likely be different types of habitat (sub-sets) within it. Random sampling may altogether ‘miss' one or more of these. Stratified sampling would take into account the proportional area of each habitat type within the woodland and then each could be sampled accordingly; if 20 samples were to be taken in the woodland as a whole, and it was found that a shrubby clearing accounted for 10% of the total area, two samples would need to be taken within the clearing. The sample points could still be identified randomly (A) or systematically (B) within each separate area of woodland. Lecture 14

INTERPOLATION Many methods - All combine information about the sample coordinates with the magnitude of the measurement variable to estimate the variable of interest at the unmeasured location Methods differ in weighting and number of observations used Different methods produce different results No single method has been shown to be more accurate in every application Accuracy is judged by withheld sample points Lecture 14

INTERPOLATION Raster surface Outputs typically: Raster surface Values are measured at a set of sample points Raster layer boundaries and cell dimensions established Interpolation method estimate the value for the center of each unmeasured grid cell Contour Lines Iterative process From the sample points estimate points of a value Connect these points to form a line Estimate the next value, creating another line with the restriction that lines of different values do not cross. Lecture 14

Sampled locations and values Example Base Lecture 14 Elevation contours Sampled locations and values

INTERPOLATION 1st Method - Thiessen Polygon Assigns interpolated value equal to the value found at the nearest sample location Conceptually simplest method Only one point used (nearest) Often called nearest sample or nearest neighbor Lecture 14

INTERPOLATION Thiessen Polygon Advantage: Ease of application   Accuracy depends largely on sampling density Boundaries often odd shaped as transitions between polygons are often abrupt Continuous variables often not well represented Lecture 14

Thiessen Polygon Draw lines connecting the points to their nearest neighbors. Find the bisectors of each line. Connect the bisectors of the lines and assign the resulting polygon the value of the center point Source: http://www.geog.ubc.ca/courses/klink/g472/class97/eichel/theis.html Lecture 14

Thiessen Polygon Start: 1) 2) 3) Draw lines connecting the points to their nearest neighbors. Find the bisectors of each line. Connect the bisectors of the lines and assign the resulting polygon the value of the center point 1 2 3 5 4 Lecture 14

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Sampled locations and values Thiessen polygons Lecture 14

INTERPOLATION Fixed-Radius – Local Averaging More complex than nearest sample Cell values estimated based on the average of nearby samples Samples used depend on search radius (any sample found inside the circle is used in average, outside ignored) Specify output raster grid Fixed-radius circle is centered over a raster cell Circle radius typically equals several raster cell widths (causes neighboring cell values to be similar) Several sample points used Some circles many contain no points Search radius important; too large may smooth the data too much Lecture 14

Fixed-Radius – Local Averaging INTERPOLATION Fixed-Radius – Local Averaging Lecture 14

Fixed-Radius – Local Averaging INTERPOLATION Fixed-Radius – Local Averaging Lecture 14

Fixed-Radius – Local Averaging INTERPOLATION Fixed-Radius – Local Averaging Lecture 14

INTERPOLATION Inverse Distance Weighted (IDW) Estimates the values at unknown points using the distance and values to nearby know points (IDW reduces the contribution of a known point to the interpolated value) Weight of each sample point is an inverse proportion to the distance. The further away the point, the less the weight in helping define the unsampled location Lecture 14

INTERPOLATION Inverse Distance Weighted (IDW) Zi is value of known point Dij is distance to known point Zj is the unknown point n is a user selected exponent Lecture 14

INTERPOLATION Inverse Distance Weighted (IDW) Lecture 14

INTERPOLATION Inverse Distance Weighted (IDW) Factors affecting interpolated surface: Size of exponent, n affects the shape of the surface larger n means the closer points are more influential A larger number of sample points results in a smoother surface Lecture 14

INTERPOLATION Inverse Distance Weighted (IDW) Lecture 14

INTERPOLATION Inverse Distance Weighted (IDW) Lecture 14

INTERPOLATION Splines Name derived from the drafting tool, a flexible ruler, that helps create smooth curves through several points Spline functions are use to interpolate along a smooth curve. Force a smooth line to pass through a desired set of points Constructed from a set of joined polynomial functions Lecture 14

Spline Surface created with Spline interpolation Passes through each sample point May exceed the value range of the sample point set                                                                                                                                                                                                                               Lecture 14

INTERPOLATION : Splines Lecture 14

Interpolation vs. Prediction Spatial prediction is more general than spatial interpolation. Both are used to estimate values of a variable at unknown locations. Interpolation use only the measured target variable and sample coordinates to estimate the variable at unknown locations. Prediction methods address the presence of spatial autocorrelation. Lecture 14

Tobler’s Law Tobler's First Law of Geography. A formulation of the concept of spatial autocorrelation by the geographer Waldo Tobler (1930-), which states: "Everything is related to everything else, but near things are more related than distant things." Lecture 14

Trend Surface Interpolation Fitting a statistical model, a trend surface, through the measured points. (typically polynomial) Where Z is the value at any point x Where ais are coefficients estimated in a regression model Lecture 14

Trend Surface Interpolation Lecture 14

Global Mathematical Functions Polynomial Trend Surface Lecture 14

Global Mathematical Functions Polynomial Trend Surface Lecture 14

Similar to Inverse Distance Weighting (IDW) INTERPOLATION Kriging Similar to Inverse Distance Weighting (IDW) Kriging uses the minimum variance method to calculate the weights rather than applying an arbitrary or less precise weighting scheme Lecture 14

Cross-correlation – two variables change in concert (positive or negative) Lecture 14

Prediction Kriging Method relies on spatial autocorrelation Higher autocorrelations, points near each other are alike. Lecture 14

Similar to Inverse Distance Weighting (IDW) Prediction Kriging Similar to Inverse Distance Weighting (IDW) A statistically based estimator of spatial variables Kriging uses the minimum variance method to calculate the weights rather than applying an arbitrary or less precise weighting scheme Lecture 14

INTERPOLATION Kriging A statistical based estimator of spatial variables Components: Spatial trend Autocorrelation Random variation Creates a mathematical model which is used to estimate values across the surface Lecture 14

Semivariogram Lecture 14

Lag Distance Kriging uses the concept of lag distance http://www.youtube.com/watch?v=SJLDlasDLEU Lecture 14

Lag Distance and Autocorelataion The semivariance at a given lag distance is a measure of the spatial autocorrelation at that distance. The variogram is used to determine the weights used in the kriging process. Lecture 14

SemiVariogram Range: The distance where the model first flattens out is known as the range. Sample locations separated by distances closer than the range are spatially autocorrelated, whereas locations farther apart than the range are not. Lecture 14

SemiVariogram Nugget: Theoretically, at zero separation distance (lag = 0), the semivariogram value is 0. However, at an infinitesimally small separation distance, the semivariogram often exhibits a nugget effect, which is some value greater than 0. The nugget effect can be attributed to measurement errors or spatial sources of variation at distances smaller than the sampling interval or both. Lecture 14

Semivariogram Nugget Effect: Measurement error occurs because of the error inherent in measuring devices. Natural phenomena can vary spatially over a range of scales. Variation at microscales smaller than the sampling distances will appear as part of the nugget effect. Before collecting data, it is important to gain some understanding of the scales of spatial variation. Lecture 14

Types of Kriging Depending on the stochastic properties of the random field and the various degrees of stationarity assumed, different methods for calculating the weights can be deducted, i.e. different types of kriging apply. Classical methods are: Ordinary Kriging assumes stationarity (flat/no trend)of the first moment of all random variables Simple Kriging assumes a known stationary mean. Universal Kriging assumes a general polynomial trend model, such as linear trend model . Lecture 14

Kriging A surface created with Kriging can exceed the value range of the sample points but will not pass through the points. Lecture 14

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INTERPOLATION (cont.) Exact/Non Exact methods Exact – predicted values equal observed Theissen IDW Spline Non Exact-predicted values might not equal observed Fixed-Radius Trend surface Kriging Lecture 14

INTERPOLATION Kriging A statistical based estimator of spatial variables Components: Spatial trend Autocorrelation Random variation Creates a mathematical model which is used to estimate values across the surface Lecture 14

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INTERPOLATION (cont.) Exact/Non Exact methods Exact – predicted values equal observed Theissen IDW Spline Non Exact-predicted values might not equal observed Fixed-Radius Trend surface Kriging Lecture 14

Which method works best for this example? Thiessen Polygons Fixed-radius – Local Averaging IDW: squared, 12 nearest points Original Surface: Trend Surface Spline Kriging Lecture 14

Interpolation vs. Prediction Spatial prediction is more general than spatial interpolation. Both are used to estimate values of a variable at unknown locations. Interpolation use only the measured target variable and sample coordinates to estimate the variable at unknown locations. Prediction methods address the presence of spatial autocorrelation. Lecture 14

Tobler’s Law Tobler's First Law of Geography. A formulation of the concept of spatial autocorrelation by the geographer Waldo Tobler (1930-), which states: "Everything is related to everything else, but near things are more related than distant things." Lecture 14

Prediction Kriging Method relies on spatial autocorrelation Higher autocorrelations, points near each other are alike. Lecture 14

Cross-correlation – two variables change in concert (positive or negative) Lecture 14

Similar to Inverse Distance Weighting (IDW) Prediction Kriging Similar to Inverse Distance Weighting (IDW) A statistically based estimator of spatial variables Kriging uses the minimum variance method to calculate the weights rather than applying an arbitrary or less precise weighting scheme Lecture 14

INTERPOLATION Kriging A statistical based estimator of spatial variables Components: Spatial trend Autocorrelation Random variation Creates a mathematical model which is used to estimate values across the surface Lecture 14

Lag Distance Kriging uses the concept of lag distance http://www.youtube.com/watch?v=SJLDlasDLEU Lecture 14

Semivariogram Lecture 14

SemiVariogram Range: The distance where the model first flattens out is known as the range. Sample locations separated by distances closer than the range are spatially autocorrelated, whereas locations farther apart than the range are not. Lecture 14

SemiVariogram Nugget: Theoretically, at zero separation distance (lag = 0), the semivariogram value is 0. However, at an infinitesimally small separation distance, the semivariogram often exhibits a nugget effect, which is some value greater than 0. The nugget effect can be attributed to measurement errors or spatial sources of variation at distances smaller than the sampling interval or both. Lecture 14

Semivariogram Nugget Effect: Measurement error occurs because of the error inherent in measuring devices. Natural phenomena can vary spatially over a range of scales. Variation at microscales smaller than the sampling distances will appear as part of the nugget effect. Before collecting data, it is important to gain some understanding of the scales of spatial variation. Lecture 14

Lag Distance and Autocorelataion The semivariance at a given lag distance is a measure of the spatial autocorrelation at that distance. The variogram is used to determine the weights used in the kriging process. Lecture 14

Types of Kriging Depending on the stochastic properties of the random field and the various degrees of stationarity assumed, different methods for calculating the weights can be deducted, i.e. different types of kriging apply. Classical methods are: Ordinary Kriging assumes stationarity (flat/no trend)of the first moment of all random variables Simple Kriging assumes a known stationary mean. Universal Kriging assumes a general polynomial trend model, such as linear trend model . Lecture 14

Kriging A surface created with Kriging can exceed the value range of the sample points but will not pass through the points. Lecture 14

Lecture 14

INTERPOLATION (cont.) Exact/Non Exact methods Exact – predicted values equal observed Theissen IDW Spline Non Exact-predicted values might not equal observed Fixed-Radius Trend surface Kriging Lecture 14