2012/3/5 1 LSGI 3421: GIS Applications Lecture 1: Introduction to GIS Applications LSGI 3244: Spatial Analysis Lecture 7: Spatial Interpolation & Surface Analysis LSGI 3244: Spatial Analysis Dr. Bo Wu [email protected]Department of Land Surveying & Geo-Informatics The Hong Kong Polytechnic University Lecture 7: Spatial Interpolation and Surface Analysis LSGI 3244: Spatial Analysis Lecture 7: Spatial Interpolation & Surface Analysis 1. Learning Outcomes 2. Spatial Interpolation − Type of Spatial Interpolation − Typical Spatial Interpolation Methods − Kriging 3. Surface Analysis − Slope and Aspect − Viewshed and Hillshed − Contour 4. An Example of Surface Analysis in NASA’s Mars Exploration Rover Mission Contents 2012/3/5 2 LSGI 3244: Spatial Analysis Lecture 7: Spatial Interpolation & Surface Analysis • By the end of this lecture you should be able to: – Know what is spatial interpolation and the main factors affecting interpolation results – Explain the principles of typical spatial interpolation methods – Perform spatial interpolation using Kriging for a given data set – Explain the principles of typical surface analysis techniques – Perform surface analysis for a given data set such as slop analysis Learning Outcomes 2012/3/5 3 LSGI 3244: Spatial Analysis Lecture 7: Spatial Interpolation & Surface Analysis • Spatial Interpolation – Using points with known values to estimate values at other points – A means of creating surface data from sample points Spatial Interpolation • Known Points – Sample points providing the data necessary for development of an interpolator for spatial interpolation – Number and distribution of known points greatly influence the results of interpolation – Assumption – the value to be estimated at a point is more influenced by nearby know points – Control points should be evenly distributed for effective estimation – Poorly distributed areas can cause problems for spatial interpolation 2012/3/5 4 LSGI 3244: Spatial Analysis Lecture 7: Spatial Interpolation & Surface Analysis • Interpolation – Estimating the attribute values of locations that are within the range of available data using known data values • Extrapolation – Estimating the attribute values of locations outside the range of available data using known data values Interpolation & Extrapolation 2012/3/5 5 LSGI 3244: Spatial Analysis Lecture 7: Spatial Interpolation & Surface Analysis Type of Spatial Interpolation • Global vs Local Interpolation – Global interpolation • Uses every known point available to estimate unknown value • Design to capture the global trend • More intensive calculation – Local interpolation • Uses a sample of known points to estimate an unknown value • Design to estimate the local or short range variation • Requires much less computation 2012/3/5 6
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
• Each input point has local influence that diminishes with distance• Estimates are averages of values at s known points within window R• Is an exact method that enforces that the value of a point is
influenced more by nearby known points than those farther away
2012/3/5 11
• z0 is the estimated value at point 0• zi is the z value at known point i• di is the distance between point i and point 0• s is the number of know points used• K is the specified power
− K =1 : constant− K =2 : higher rate of change near a known point
• Kriging is a group of geostatistical techniques to interpolate the value of a random field at an unobserved location from observations of its value at nearby locations.– E.g., the elevation, Z, of the terrain as a function of the geographic
location.
• Kriging is named after the South African engineer, Daniel G. Krige, who first developed the method in 1960
• The kriging estimator is given by a linear combination:
• Assume that there is no structural component (will be handled by Universal Kriging)
• Focuses on the spatially correlated component• Uses the fitted semivariogram directly for interpolation
2012/3/5 20
• Z0 is the estimated value,• Zx is the known value at point x• Wx is the weight associated with point x• S is the number of sample points used in estimation
E.g., for a point (0) to be estimated from three known points (1, 2, 3)
• Assume that the spatial variation in z values has a structural component or a drift in addition to the spatial correlation between the sample points
• Typically incorporates a first-order or a second-order polynomial in the Kriging process
• Higher-order polynomials are not recommended:– Will leave little variation in the residuals for assessing uncertainty– Require to solve a larger set of simultaneous equations
M = b1xi + b2yior
M = b1xi + b2yi + b3x2i + b4xiyi + b5y2
i
• M is the drift• b is the drift coefficients estimated from known points
measured from the horizontal to a plane tangent to the surface at that point
– The value of the slope will depend on the direction in which it is measured. Slope is commonly measured in the direction of the coordinate axes e.g. in the X-direction and Y-directions.
– The slope measured in the direction at which it is a maximum is termed the gradient
– b denotes slope in the x direction – c denotes slope in the y direction – D is the spacing of points (e.g., 30 m)
• tan (slope) = sqrt (b2 + c2)
1 2 3
4 5 6
7 8 9
• Slope is a neighborhood function which creates a grid of maximum rate of change of the cell values of the input grid. The slope is derived based on a 3 x 3–cell neighborhood.